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ADVANCES IN CHEMISTRY RESEARCH

ADVANCES IN CHEMISTRY RESEARCH VOLUME 21

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ADVANCES IN CHEMISTRY RESEARCH

ADVANCES IN CHEMISTRY RESEARCH VOLUME 21

JAMES C. TAYLOR EDITOR

New York

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Copyright © 2014 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

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CONTENTS Preface

vii

Chapter 1

Antibacterial and Antifouling Properties of Lipophilic Bismuth Compounds Appala Raju Badireddy and Shankararaman Chellam

Chapter 2

Rheology of Carbon Black Suspensions Yuji Aoki

29

Chapter 3

Characterization of Carbon Blacks by High Resolution N2Adsorption Isotherms from P/Po=10-7 to P/Po=0.998: Application of Standard αs Data to Analysis of Microporosity of Activated Carbons Kazuyuki Nakai, Yoko Nakada, Masako Hakuman, Joji Sonoda, Masayuki Yoshida and Hiromitsu Naono

97

Chapter 4

Doubly Bonded Molecules Containing Bismuth and Other Group 15 Elements in the Singlet and Triplet States Ming-Der Su

149

Chapter 5

Growth of Nanocrystals from Amorphous Bi G. N. Kozhemyakin, S. Y. Kovalev, O. N. Ivanov and O. N. Soklakova

185

Chapter 6

Mathematical Theory of Noble Gases V. P. Maslov

197

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vi Chapter 7

Contents Properties of Rarefied Noble Gas Flows Lajos Szalmás

Index

221 237

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PREFACE This book presents original research results on the leading edge of chemistry research. Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum. Topics discussed include the antibacterial and antifouling properties of lipophilic bismuth compounds; rheology of carbon black suspensions; characterization of carbon blacks; doubly bonded molecules containing bismuth and other group 15 elements in the singlet and triplet states; growth of nanocrystals from amorphous bi; mathematical theory of noble gases; and properties of rarefied noble gas flows. The medicinal and biological properties of elemental bismuth are known to be enhanced by chelating it with lipophilic thiols. In Chapter 1, the authors summarize the antibacterial and antifouling potential of selected lipophilic bismuth compounds largely in the context of environmental, medical, and biotechnological applications. They synthesized ionic and nanoparticulate forms of lipophilic bismuth and determined their minimum inhibitory concentrations (MICs) under environmentally relevant conditions for several Gram-negative bacteria. Bacterial production of extracellular polymeric substances can be effectively suppressed by bismuth thiols at sub- or nearMIC concentration with negligible effect on growth. The authors focus on a cationic di-thiol, viz. bismuth-2, 3-dimercapto-1-propanol (BAL) prepared using a 2:1 Bi:BAL molar ratio since it is stable over a wide pH range and does not carry the malodor or toxicity of BAL. This BisBAL formulation is particularly effective in inhibiting bacterial cohesion and adhesion even at subMICs by decreasing EPS secretion, inhibiting acetylation and carboxylation of polysaccharides, and altering protein secondary structures. Results indicate that bismuth thiols could be used to probe and mechanistically understand

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viii

James C. Taylor

specific roles of cell-bound and free-EPS in bioflocculation, cohesion, and adhesion without affecting growth. They also synthesized rhombohedral crystalline BisBAL nanoparticles by reducing the BisBAL aqueous complex using an ice-cold solution of sodium borohydride. These nanoparticles also inhibited bacterial growth, prevented bacterial attachment to surfaces, and impaired preformed/established biofilms. They therefore hypothesize that embedding bismuth nanoparticles will impart antimicrobial and antifouling properties to surfaces given their high surface area-to-volume ratio and their ability to slowly release bioactive bismuth in situ over long timeframes due to their lipophilicity. This may provide the basis for developing bismuthnanoparticle based broad-spectrum antimicrobial agents, which could “seek and destroy” microorganisms in a variety of practical applications. Chapter 2 gives a review of the rheology of carbon black (CB) suspensions. Extensive studies have been made for the CB suspensions. However, a large fraction of these studies was devoted for the suspensions in fairly non-polar media where the CB particles having the polar surface. The rheological properties of CB suspension are affected by the medium affinity to the CB particles. Different types of rheological behavior are observed accordingly, when the affinity changes. In the medium having a low affinity, the CB particles form the continuous network-type agglomerate and exhibit a strong nonlinearity attributable to strain-induced disruption of a fully developed three-dimensional (3D) network structure of the CB particles therein characterized by the yield stress. In contrast, in the medium having a moderate affinity, the suspensions show a sol-gel transition with increasing CB concentration, and the critical gel behavior characterized with a power-law relationship between the modulus and frequency (), G G/tan(n/2) n is observed. This behavior suggests formation of a self-similar, fractal agglomerate of the CB particles. In the medium having a high affinity, the CB aggregates are well dispersed to no agglomerates. These aggregates exhibit a slow relaxation process to their diffusion. Thus, the structure and rheology of the CB particles/aggregates changes with the affinity of the suspending medium. In this chapter, the authors report the rheological properties of CB suspensions in three suspending media, a polystyrene/dibutylphthalate solution (PS/DBP, low affinity), a rosin-modified phenol-type varnish (Varnish-1, moderate affinity), and an alkyd resin-type varnish (Varnish-2, high affinity).The effects of the primary particle size and the structure of CB aggregates on the rheological properties are summarized. For the CB suspensions exhibiting critical gel behavior, heat-induced gelation and the effects of suspending media are also explained.

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Preface

ix

Chapter 3 - The N2 adsorption isotherms at its boiling point (77.4 K) have been extensively utilized in the investigations of surface characters of carbon blacks (CB) and microporous textures of activated carbons (AC) and activated carbon fibers (ACF). When measurements of the N2 adsorption isotherms are extended to an extremely low or high p/po region, information obtained from the N2 isotherms will be remarkably increased. Recent progress of automatic gas adsorption apparatus makes it possible to measure the N2 adsorption isotherms at 77.4 K in a very wide pressure range from p/po=10-8 to p/po=0.999 [6–9]. In addition, liquid argon of high purity (99.995%) (b.p.: 87.3 K) is readily available, which makes it easy to measure the precise N2 isotherms at 87.3 K up top/po=10-8. In the authors’ previous works, the N2 isotherms at these two temperatures have been applied to investigations of surface characters of nonporous carbon blacks or microporosities of activated carbons fibers, where their attention was mainly focused on the isotherms at low relative pressures. Chapter 4 - The lowest singlet and triplet potential energy surfaces for all the group 15 HBiXH (X = N, P, As, Sb, and Bi) systems have been explored through ab initio calculations. The geometries of the various isomers were determined at the QCISD/LANL2DZdp level, and confirmed to be minima by vibrational analysis. In the case of nitrogen, the order of stability is triplet H2NBi > singlet H2NBi > singlet cis-HBi=NH ≈ singlet trans-HBi=NH > triplet HBiNH > triplet H2BiN > singlet H2BiN. For the phosphorus case, the stability decreases in the order triplet H2PBi > singlet trans-HBi=PH > singlet cis-HBi=PH > triplet HBiPH > singlet H2PBi > triplet H2BiP > singlet H2BiP. For arsenic, theoretical investigations demonstrate that the stability of the HBiAsH isomers decreases in the order triplet H2AsBi ≈ singlet transHBi=AsH > singlet cis-HBi=AsH > triplet HBiAsH > triplet H2BiAs > singlet H2AsBi > singlet H2BiAs. For antimony, the theoretical findings suggest that the stability of the HBiSbH system decreases in the order singlet transHBi=SbH > singlet cis-HBi=SbH > triplet H2SbBi > triplet H2BiSb > triplet HBiSbH > singlet H2SbBi > singlet H2BiSb. For bismuth, the theoretical investigations indicate that the stability of the HBiBiH system decreases in the order singlet trans-HBi=BiH > singlet cis-HBi=BiH > triplet H2BiBi > triplet HBiBiH > singlet H2BiBi. Our model calculations indicate that relativistic effects on heavier group 15 elements should play an important role in determining the geometries as well as the stability of HBiXH molecules. The results obtained are in good agreement with the available experimental data and allow a number of predictions to be made.

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Over recent years, there has been an increased interest towards theoretical and experimental investigations of bismuth (Bi) crystal properties. Bi and Birelated materials are of particular interest for thermoelectric applications. Bi is a semimetal with unique electronic structure, and its transport properties have been studied because quantum confinement effects can be observed in lowdimensional systems. The development of Bi as a low-dimensional material has been traditionally done by preparing ordered arrays of 1D quantum wires and studying them in detail. Manipulating nanoparticles sizes in lowdimensional systems is a promising way for fundamental studies and nanotechnological applications of this semimetal. Another method that provides control over nanocrystal formation is low-temperature annealing of amorphous materials. Therefore, in Chapter 5, the authors used this method to study the growth conditions of Bi nanocrystals. The objective of this chapter is further development of Bi nanocrystal formation method by annealing of amorphous Bi. In Chapter 6, the authors single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. The author consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z) diagram by the second sheet. Chapter 7 reviews some of the recent developments of theoretical investigation of rarefied flows of noble gases and their mixtures. These flows can be found in various engineering applications, such as micro- and nanofluidics or vacuum technology. From theoretical viewpoint, rarefied gas flows can be studied on the basis of the Boltzmann or other kinetic equations, which are valid in the whole range of the gaseous rarefaction parameter, i.e. the ratio of the relevant macroscopic size of the flow over the molecular mean free path. These integro-differential equations can be solved by deterministic or probabilistic approaches. Among these techniques, the discrete velocity method and the direct simulation Monte Carlo are most common. The chapter describes the theoretical background of rarefied gases and the applied solution methods of kinetic equations. The slip phenomena are discussed. Results in terms of the so-called slip coefficients, which describe the slip of the macroscopic velocity along solid walls, are presented for noble gas mixtures.

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xi

Flows of rarefied noble gases in long channels are also considered. The effects of the molecular masses, the mass ratios and the concentration for gas mixtures on the results are outlined. The chapter brings the results of recent developments of rarefied gas dynamics to the attention of other researchers, engineers or non-specialized people.

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In: Advances in Chemistry Research. Volume 21 ISBN: 978-1-62948-742-7 Editor: James C. Taylor, pp. 1-28 © 2014 Nova Science Publishers, Inc.

Chapter 1

ANTIBACTERIAL AND ANTIFOULING PROPERTIES OF LIPOPHILIC BISMUTH COMPOUNDS Appala Raju Badireddy1 and Shankararaman Chellam2, 1

Department of Civil and Environmental Engineering, Duke University, Durham, NC, US 2 Departments of Civil and Environmental Engineering and Chemical and Biomolecular Engineering, University of Houston, Houston, TX, US

Abstract The medicinal and biological properties of elemental bismuth are known to be enhanced by chelating it with lipophilic thiols. In this chapter, we summarize the antibacterial and antifouling potential of selected lipophilic bismuth compounds largely in the context of environmental, medical, and biotechnological applications. We synthesized ionic and nanoparticulate forms of lipophilic bismuth and determined their minimum inhibitory concentrations (MICs) under environmentally relevant conditions for several Gram-negative bacteria. Bacterial production of extracellular polymeric substances can be effectively suppressed by bismuth thiols at sub- or near-MIC concentration with negligible effect on growth. We focus on a cationic di-thiol, viz. bismuth-2, 3-dimercapto-1-propanol (BAL) prepared using a 2:1 Bi:BAL molar ratio since it is stable over a wide pH 

E-mail address: [email protected]; Tel: (713) 743-4265. (Corresponding author) Submitted for inclusion in the book “Bismuth: Occurrence, Uses and Health & Environmental Effects,” Nova Science Publishers, Inc. on September 10, 2013.

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Appala Raju Badireddy and Shankararaman Chellam range and does not carry the malodor or toxicity of BAL. This BisBAL formulation is particularly effective in inhibiting bacterial cohesion and adhesion even at sub-MICs by decreasing EPS secretion, inhibiting acetylation and carboxylation of polysaccharides, and altering protein secondary structures. Results indicate that bismuth thiols could be used to probe and mechanistically understand specific roles of cell-bound and free-EPS in bioflocculation, cohesion, and adhesion without affecting growth. We also synthesized rhombohedral crystalline BisBAL nanoparticles by reducing the BisBAL aqueous complex using an ice-cold solution of sodium borohydride. These nanoparticles also inhibited bacterial growth, prevented bacterial attachment to surfaces, and impaired preformed/established biofilms. We therefore hypothesize that embedding bismuth nanoparticles will impart antimicrobial and antifouling properties to surfaces given their high surface area-to-volume ratio and their ability to slowly release bioactive bismuth in situ over long timeframes due to their lipophilicity. This may provide the basis for developing bismuthnanoparticle based broad-spectrum antimicrobial agents, which could “seek and destroy” microorganisms in a variety of practical applications.

Introduction Elemental bismuth, belongs to the Pnictogen group of the periodic table, is a unique inexpensive, non-carcinogenic metal that has been hailed as “green element” [1, 2]. Bismuth naturally exists as bismuthinite (bismuth sulfide) and bismite (bismuth oxide) ores and iridescent crystals in elemental form[2]. Numerous bismuth compounds owing to their low toxicity are widely used in drugs, antifungal andanticancer agents, biomedical devices such as catheters and personal care products[1-6]. For example, LD50 (mg/kg) values of bismuth oxychloride (22,000 (rat, oral)) and bismuth oxide (10,000 (mouse, oral)) are reported to be substantially higher than even that of sodium chloride (3,000 (rat, oral) and 4,000 (mouse, oral)) [7]. Bismuth-based compounds, such as bismuth subsalicylate (BSS, PeptoBismol), colloidal bismuth subcitrate (CBS, De-Nol), and ranitidine bismuth citrate (RBC, TirtecPylorid, GSK) are widely used in combination with antibiotics (amoxicillin, clarithromycin or nitroimidazole) to treat gastrointestinal diseases arising from Helicobacter pylori [3]. 213Biradiolabelled complex is considered to be a promising therapeutic agent for small volume tumors due to its highly localized action and causing negligible damage to the surrounding tissues [3]. However, bismuth and its salts have limited water solubility necessitating very highconcentrations to achieve significant antimicrobial effects. Although, inorganic bismuth compounds can be beneficial to human health, the non-supervised or extended use of drugs containing high bismuth salt concentrations is not recommended due to potential formation of

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Antibacterial and Antifouling Properties …

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toxic bismuth-derivatives (e.g., methylated-bismuth) and ensuing potential adverse effects such as neurotoxicity, renal failure, and genotoxicity[8]. Since water solubility is critical for biocompatibility and antimicrobial effects, bismuth’s solubility can beenhanced by chelating with molecules containing hydroxyl and/or sulfhydryl (thiol) groups to increase its efficacy at lower dosages thereby circumventing problems associated with using high concentrations [9]. Based on this approach, using medically-relevant microorganisms it has been shown that complexation of bismuth with certain thiols would achieve antibacterial effects at concentrations ~1,000-fold lower than inorganic bismuth compounds [10]. Further, a combination of sub-minimum inhibitory concentrations of bismuth thiols and traditional antibiotics not only suppressed capsular secretions by bacteria but also enhanced bacterial susceptibility to antibiotics consequently better controlling pathogenic biofilms [11-14]. We have demonstrated that a certain molar ratio of bismuth-to-thiol (2:1) with bismuth being either in its cationic orzero-valentnanoparticle-form can be used to suppress extracellular polymeric substances (EPS) secretion and biofilm formation by bacteria in suspended cultures and on engineered surfaces (e.g., drinking water filtration membranes)[15, 16]. EPS are chiefly composed of polysaccharides, proteins, nucleic acids, (phospho) lipids, humic substances, and other polymers, which provide a protective and adhesive matrix for microorganisms[17]. The EPS matrix helps to sequester substrates and nutrients from the environment for microbial survival and maintaining a synergistic diverse microbial consortia in a biofilm [17, 18]. Specifically, EPS are classified as either “free” or “bound” depending on whether they are released into surrounding environment or remain in the proximity of the cell surface. Free EPS have been shown to induce microbial aggregation in planktonic cultures and facilitate adhesion through surface conditioning, whereas bound EPS firmly anchors the microorganisms into the biofilm matrix [19-22]. Microbe adherence is thought to be mediated by both free and bound EPS, even though it is still not completely clear whether EPS interacts directly or through an intermediary conditioning film with diverse inanimate surfaces [21-24]. It has been shown that microbial adhesion and colonization on surfaces can be affected by surface chemistry and physicochemical properties of the media [25]. For instance, surface roughness enhances the microbial attachment and facilitates colonization particularly in the regions of reduced shear forces [26]. Hydrophobic and non-polar surfaces are more favorable for bacteria adhesion than hydrophilic surfaces [27]. Microbial surface hydrophobicity, the presence of cell appendages (pilli and flagella), and amount and characteristics of EPS are some additional

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factors that play important roles in microbial attachment (often irreversible) and biofilm formation [19, 26]. Although biofilms play a crucial role in bio-geochemical cycling and can be beneficial in some biotechnological processes (e.g., wastewater treatment) they can also be a menace for purification and separation processes, water distribution systems, ship hulls, cooling towers, clinical and biomedical devices, and processes in the pulp and paper industry [28-33]. In addition to causing odor and taste problems, biofilms are resistant to conventional disinfectants and difficult to remove because of their mechanical stability owing to EPS[34]. Most microorganisms are capable of forming biofilms under the appropriate environmental and developmental conditions and the EPS are responsible for providing favorable milieu for the deposition of microorganisms and biofilm propagation. It has been proposed that development of antifouling strategies should be based on analysis of fouling factors and the biofilm properties[31]. Despite decades of research devoted towards understanding biofilm formation mechanisms[18, 19, 26, 35-41] there is still a considerable knowledge gap in specific molecular level role of EPS components in governing the biofilm formation necessitating more research in this subject [20, 23, 42-44]. For instance, there is a wide consensus that the microbial floc formation is known to depend on total EPS concentration as well as its specific components [27, 45-48]. Therefore, our research has focused on developing detailed information on molecular components of EPS secreted during microbial growth and biofouling processes, which in turn would provide insights on the specific EPS components that promote biofouling and biofilm formation. Recently, we have reported that sub-minimum inhibitory concentrations (sub-MIC) of bismuth thiols could be utilized as the probing agents for understanding specific molecular-level role of cell-bound and free-EPS in bacterial flocculation and adhesion under un-inhibited growth conditions[15, 49, 50]. Other investigations have also shown that bismuth thiols could inhibit biofilm formation on stents, catheters, and water distribution systems[51-53]. We believe that this approach of increasing bismuth solubility by chelating it with appropriate ligands will advance the knowledge of biofilm growth and properties and inspire the development of bismuth-based antimicrobial and antibiofouling strategies. The objective of this chapter is to summarize the use of two-forms of bismuth thiols: (1) the cationic-form for studying the molecular-level role of EPS in governing bioflocculation processes during planktonic growth of bacteria, and (2) the zero-valentnanoparticle-form for creating novel antibacterial and antifouling surfaces. Three formulations of bismuth thiols, namely, bismuth dimercaptopropanol (BisBAL), bismuth ethanedithiol (BisEDT), and bismuth

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pyrithione (BisPYR) were investigated for their antifouling properties in suspended bacterial cultures of single Gram negative bacteria and multi-species microbial cultures (activated sludge microorganisms). Additionally, the free- and cell-bound EPS formed during the treatment were comprehensively characterized using colorimetric, microscopic and spectroscopic techniques. We also report results obtained from the studies with cationic and nanoparticle form of 2:1 molar ratio of Bi-to-BAL (BisBAL) on mitigation of membrane biofouling, bioflocculation of activated sludge cultures, and antibacterial and anti-biofilm effects. A 2:1 BisBAL was chosen due to its stability over a wide range of pH and optimal activity at near-neutral pH without the negative effects of BAL.

Cationic Bismuth Chelates Bismuth can exist predominantly as pentavalent (Bi+5) and trivalent (Bi+3) oxidation states with latter being the most stable and common state whereas the former is a very powerful oxidant (with Bi+5/ Bi+3 reduction potential of 2.03 V) [3]. Trivalent bismuth readily hydrolyzes in aqueous solutions and has a strong affinity to both oxygen and nitrogen containing ligands, however, it prefers to coordinate with ligands containing thiolate groups (e.g. cysteine and glutathione) [54-58]. A wide array of bismuth compounds have been increasingly synthesized and used in clinical and health applications such as antimicrobial and anticancer agents [3]. Until recently, the widespread use of bismuth compounds has been largely limited by its low water solubility. Domenico and co-workers prepared different bismuth complexes using various di-thiols (PDT, BAL, DTT, DMSA), mono-thiols (MBO, βME, MEN, MTG), thiol acids (thiosalicylic acid, thioglycolic acid, and etc.), and non-thiols (D-penicillamine, protocatechuate ethyl ester, etc.) to study their antibacterial activity [10]. They reported that non-thiols and thiol acids had negligible effects on bismuth’s antibacterial activity. However, when bismuth was complexed with di-thiols or mono-thiols at different molar ratios, bismuth’s antibacterial activity was enhanced by 25-300-fold along with its water solubility [10]. Further, they showed that di-thiols were more effective than mono-thiols in the range of bismuth-thiol molar ratios of 3:1 to 1:1. Of particular interest is that a 2:1 molar ratio of bismuth-to-BAL (BisBAL, a di-thiol) prepared in propylene glycol showed strong antibacterial activity, remained stable with excellent solubility over a wide range of pH (3-11.6) for extended time periods, and was relatively stable up to 100 °C (activity lasted for 30 min at 100 °C) [10]. Based on this empirical evidence, 2:1 BisBAL has been investigated to inhibit established and virulent biofilms, pathogenic bacteria, and even drinking water

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biofilms under physiological as well as environmental conditions. For example, sub-MIC of BisBAL increased the susceptibility of Klebsiella pneumonia biofilms to antibiotics by exposing its surface antigens via suppression of capsular exopolysaccharide production [59, 60]. Upon exposure to BisBAL at MIC, Pseudomonas aeruginosa biofilm density and viability was initially unaffected by BisBAL but after prolonged exposure (>100 h) the polysaccharide content of the established biofilms and viability of cells significantly decreased, and consequently inhibited the biofilm proliferation in systems such as colony or dripflow reactors [61, 62]. Unlike chlorine, BisBAL showed persistent residual effects against Escherichia coli and multispecies biofilms and its efficiency increased with temperature although their antimicrobial action was slower compared to chlorine [51]. BisBAL has been shown to prevent bacterial adhesion and biofilm formation by E. coli, K. pneumonia, Enterobacter, and Enterococcus on the surface of biliary stents in an in vitro perfusion system [53]. The mechanism of antibacterial activity of bismuth thiols is primarily attributed to the lipophilicity of thiols and the bioactivity of bismuth. Capsular polysaccharide or exopolysaccharide suppression by bismuth thiols is believed to be due to inactivation of redox enzymes, which could reduce energy levels and inhibit capsule synthesis [63]. It has been shown that when the bacteria were treated with bismuth subsalicylate, ATP levels were substantially decreased (by 90%) due to bypassing of oxidative phosphorylation by salicylate and/or inactivation of thiol containing enzymes [64].

Synthesis of Lipophilic Bismuth Thiols Since Bi(NO3)3 is not readily soluble in either water or non-polar solvents at room temperature, initially a stock suspension of 50 mM Bi+3 was prepared by dissolving bismuth nitrate pentahydrate (Bi(NO3)3•5H2O) in propylene glycol heated to 70°C on a hot magnetic stir plate while stirring for 2 h. Three formulations of cationic Bi:thiol at 2:1 molar ratio (viz., BisBAL, BisEDT, and BisPYR) were prepared by separately mixing 1 mL of 50 mM Bi+3 with 2.5 μL of 10 M 2,3-dimercapto-1-propanol (BAL; British Anti-Lewisite), 2.3 μL of 10.7 M 1,2-ethanedithiol (EDT), and 250 μL of 0.1 M pyrithione (PYR), respectively [10, 50, 60, 65]. BisBAL, BisEDT, and BisPYR are known for exhibiting lipophilicity and stability in the physiological pH range [10]. Additionally, BisBAL has been shown to be stable over a wider pH range (3-11) [10] and therefore, its antimicrobial and anti-biofilm properties under environmentally relevant conditions have been the major focus of our research.

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Bi-thiol Exposure to Microbial Cultures We investigated the growth inhibitory effect over a range of concentrations (typically within 0-40 μM cationic form) of BisBAL, BisEDT, and BisPYR on planktonic cultures of Serratia marcescens and E. coli. BisBAL’s unique stability over several days and optimum activity at near-neutral pH facilitated studies on EPS suppression, microbe susceptibility, biofouling of microfiltration membranes, and bioaggregation of single species (e.g., Brevundimonas diminuta and Pseudomonas aeruginosa) and multispecies (e.g., activated sludge microorganisms) under environmentally relevant conditions. The capability of BisBAL as an antifouling and anti-biofilm agent was studied at its near-minimum inhibitory concentration (MIC) owing to its negligible effect on growth. The culturing procedures for single and multi-species and protocols for exposing microbial cultures to Bi-thiols are described in detail in our earlier publications [15, 16, 49, 50]. In all these experiments, microbial growth was monitored using optical density of single species at 600 nm or volatile suspended solids (Standard Method SM2540E) for multispecies. Membrane biofouling tests were performed using B. diminuta cultures pretreated with and without BisBAL solution and the fouling potential of EPS was determined through the changes in membrane Darcy resistance [15]. Multispecies floc formation (bioaggregation) capability in suspended cultures pretreated with and without BisBAL was evaluated through the reductions in total and specific EPS components [49].

Minimum Inhibitory Concentrations of Bi-thiols The MIC is defined as the concentration at which microbial growth is inhibited for 24 h. MICs of Bi-thiols for suspended microbial cultures were determined from growth inhibition studies in a range of concentrations of Bithiols (0-40 μM). MICs of various bismuth formulations determined by us for suspensions of single species and activated sludge microorganisms are summarized in Table 1. In general, under our experimental conditions the MICs of Bi-thiols are in agreement with the values reported by others for other Gramnegative bacteria [10]. For S. marcescens, MIC was lowest for BisEDT (2-3 μM), highest for BisPYR, and intermediate for BisBAL. A higher BisBAL concentration was needed to inhibit the E. coli growth when compared to BisEDT or BisPYR both of which showed very similar MICs.

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Figure 1. The epifluorescence images of DAPI-CTC stained E. coli treated for 2 days with (b-d) and without (a) sub-MIC Bi-thiols.

EPS Suppression by Bi-thiols EPS from suspended cultures of single and multispecies were extracted using commonly accepted protocols (e.g., centrifugation and cation-exchange resin) and categorized into free and bound EPS [15, 16, 49, 50]. A comprehensive analysis of free and bound EPSusing colorimetry, spectroscopy, and microscopy techniques revealed that generally at near-MICs Bi-thiols could substantially diminish the EPS production from cells while having a negligible effect on their growth. Atomic force microscopy images of E. coli cultures treated with and without Bi-thiols (BisBAL, BisEDT or BisPYR) from day 1 and day 2 samples are shown in Figure 2. These images show that following the treatment with Bithiols the cells were still intact with lower EPS content (Fig.2c-h) when compared with controls (Fig. 2a-b). Interestingly, the MICs for cationic BisBAL were similar for all Gram negative bacteria investigated. As an example, epifluorescence images in Figure 1a-d shows that the viability of E. coli was unaffected even after exposure to subMIC BisBAL, BisEDT, and BisPYR for 2 days (and very similar results were obtained for 5-day exposure) [50].

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Table 1.MIC of 2:1 Bi-thiol and BisBAL NPs for single and multi-species. N/A- refers to “not tested”

S. marcescens

E. coli

B. diminuta

P. aeruginosa

5-7 2-3 7-10

11-12 6-7 5-7

12-15 N/A N/A

10-15 N/A N/A

MIC (μM) for multi-species Activated sludge microorganisms 10-20 N/A N/A

N/A

N/A

N/A

12.5

N/A

2:1 Bi-thiol

BisBAL BisEDT BisPYR BisBAL nanoparticles

MIC (μM) for single species

Colorimetric measurements revealed that significant reductions in free and bound polysaccharides and proteins of stationary-phase E. coli and S. marcescens cultures from 2-day and 5-day samples were measured following exposure to Bithiols with the efficacy decreasing as BisBAL >BisEDT>BisPYR under our experimental conditions. Typical results for total polysaccharides and proteins inbound and freeEPS from E. coli cultures treated with and without Bi-thiols (BisBAL, BisEDT, and BisPYR) after 2 days of growth are shown in Figure 3. BisBAL was highly effective in decreasing free and bound polysaccharides and proteins of S. marcescens and E. coli by >90%. Further analyses using Fourier transform infrared spectrometry (FTIR) and X-ray photoelectron spectroscopy (XPS) further confirmed the EPS reductions trends revealed by colorimetric measurements [50]. Results similar to E. coli and S. marcescens were also observed for B. diminuta cultures wherein reductions in both free and bound polysaccharides and proteins concentrations reached >85% following exposure to near-MIC BisBAL after 5 days [15]. However, beyond the MIC, significant EPS reductions were observed due to growth inhibition including cell death and lysis. More specifically, results further revealed that Bi-thiols not only decreased the free and bound concentrations of polysaccharides, proteins, and other significant components (e.g. humics and uronic acids) but also significantly reduced the O-acetylation of polysaccharides, and altered protein secondary structures (primarily, aggregated strands, β-sheets, random coils, α-helices, 3-turn helices, and antiparallel β-sheets/aggregated strands). FTIR spectrometry was used to monitor total EPS from activated sludge microorganisms treated with near-MIC BisBAL (10 μM) after 3 days of exposure (Fig. 4) [49]. Proteins (protein secondary structures: amide I), O-acetyl ester groups, and carbohydrates were reduced by 67%, 63%, and 47% of control values in the presence of near-MIC BisBAL after 3-days (Fig. 4a). In addition to decreasing protein concentration by 67%, BisBAL treatment also shifted the balance of protein

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secondary structures such as random coils, β-structures, and α-and 3-turn helices (Fig. 4b). For instance, β-structures and random coils content of EPS decreased during exponential phase (1-2 days), whereas α-and 3-turn helices content increased [49]. In contrast, there were negligible changes during the stationary phase (2-3 days) for all the secondary structures.

Figure 2. Atomic force microscopy (AFM) images showing decrease in EPS in day 1 (a, c, e, and g) and day 2 (b, d, f, and h) E. coli samples following exposure to sub-MIC BisBAL, BisEDT, and BisPYR. Control-(a, b), BisBAL-(c, d), BisEDT-(e, f), and BisPYR-(g, h).

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Figure 3. Total polysacccharides and proteins in boound and freee EPS produceed by wth. stationary--phase cultures of E. coli after 2 days of grow

Role of EPS in Mem mbrane Bioofouling To esstablish the roole of EPS in microfiltration m n (MF) membbranes, B. diminuta was emplloyed as a moodel bacterium m. It was chossen because itt is widely used to verify thee sterilizing caapability of MF M membraness. B. diminuta suspensions (~2 ( × 6 10 CFU//mL) treated with w and without near-MIC C BisBAL (12 μM) were filltered onto a traack-etched pollycarbonate MF M membrane in i the dead-ennd mode.

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Figure 4. Reductions R in extracellular e poolymeric substannce proteins, caarbohydrates, annd Oacetyl esteers (a) and prottein secondary structures (b) achieved a by 10 μM BisBAL over o 3 days of gro owth calculatedd from Fourier transform t infrarred spectra.

Figure 5. Improvements I i efficiency off hydrodynamicc backwashing with in w 12 μM BissBAL treatment of o B. diminuta filtered f on a 0.44 μm membranee.

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The MF membranes with filtered bacteria (with and without BisBAL) were monitored daily for 5-days for EPS production and corresponding changes in membrane resistance due to biofouling (Fig. 5). On each day membranes were “backwashed” by rinsing with ultrapure water and measuring the Darcy resistance (Rm) ∆

where, μ is the absolute viscosity of water, J is the flux and ΔP is the applied pressure. The efficacy of backwashing (rinsing) procedure is inversely related to the increase in membrane resistance,    

 

 

 

   

.

Importantly, upon backwashing BisBAL-treated bacteria were easily detached by the simple rinsing procedure, which was attributed to the minimal amounts of EPS that was present to potentially anchor them together and onto membrane surface.

Zero-Valent Bismuth – Bal Nanoparticles Despite excellent anti-EPS and antibacterial effects of the cationic BisBAL at sub- or near-minimum inhibitory concentration (MIC), their use as long-term antifouling agents may be limited. This arises from the ready consumption of bismuth cations upon contact with negatively charged cells. High bismuth demand may induce an impractical (i.e. too frequent) cleaning frequency of surfaces with BisBAL or the need to constantly supply BisBAL along with the feed water for durable biofouling prevention. Therefore, we hypothesized that a nanoparticleform of BisBAL could be a better choice and these nanoparticles could be used for creating engineered surfaces with antibacterial/antifouling properties for longterm use. For example, these nanoparticles could be embedded on surfaces that are relevant for water treatment, energy (semiconducting and thermoelectric materials) [66, 67], biomedical systems [68], synthetic textiles, food processing and packaging, and so forth. The high surface-to-volume ratio, lipophilicity, and antibacterial activity of bismuth could allow BisBAL nanoparticles to be a wide-

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spectrum concoction to combat microbial adhesion and biofilm formation in variety of technological surfaces and biomedical devices.

Synthesis Using BisBAL solution (cationic) as a precursor the nanoparticles were synthesized by reduction reaction with ice-cold sodium borohydride (NaBH4) solution [16]. Typically, upon mixing 9.25 mL of 25 mM Bi+3(the 50 mM Bi+3stock was diluted in ice-cold ultrapure water) and 0.75 mL of 75 mM NaBH4 yieldeda black colored BisBAL nanoparticle suspension. Stock concentrations of 25 mM were prepared just prior to experimentation.

Characteristics of BisBAL Nanoparticles Reduction reaction of cationic BisBAL with NaBH4yielded BisBAL nanoparticles (zerovalent Bi-thiol) having a wide range of diameters from 3 to 400 nm. Since no stabilizing agent was used during the synthesis, BisBAL nanoparticles acquired roughly spherical, cylindrical, and triangular shapes and BAL (being a di-thiol) appears to tether one nanoparticle to another forming short chains (Fig. 6a). SEM images (Fig. 6a) show the larger BisBAL nanoparticles (100-400 nm) with varied geometrical shapes and TEM image (Fig. 6b) show 315 nm predominantly spherical nanoparticles. The X-ray diffraction pattern of BisBAL NPs revealed that they are composed of primarily rhombohedral crystallites of 18.7 nm (Fig. 6c), which suggests that the larger nanoparticles that were observed under the electron microscope (Fig. 6a) may actually be aggregates of these crystalline primary particles. Figure 6d shows that these BisBAL NPs have higher affinity (~70%; measured by UV-Vis spectrophotometry) for the octanol phase rather than for water demonstrating their lipophilic nature [16].

Figure 6. Continued on next page.

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Figure 6. Characterization C n of BisBAL nanoparticles. n Sccanning electroon micrograph shows s BisBAL NPs N having sizees 12 μM completely inhibited the growth of P. aeruginosa for 24 h and it was observed that it remained inhibited for at least a month [16].

Figure 7. The effect of lipophilic BisBAL nanoparticles on bacterial attachment onto a track-etched polycarbonate membrane surface is shown in (a); lipophilic BisBAL nanoparticles shows high affinity for P. aeruginosa cell walls (b); preformed biofilm on the membrane surface after 2 h (control: no nanoparticles) (c) and preformed biofilm on the membrane surface after exposed to lipophilic BisBAL nanoparticles at MIC for 1 h (d). BisBAL nanoparticles not only showed high affinity for bacteria but also lysed them within 1 h of exposure. The initial inoculum consisted of ~107 CFU/mL.

The amount of bacteria attached onto a membrane surface was measured by the optical density at 540 nm of crystal violet-stained bacteria removed from the surface by ethanol treatment [69, 70]. The effect of BisBAL nanoparticles on preformed biofilms on microfiltration membranes was visually assessed by electron microscopy. Figure 7 shows the effect of BisBAL nanoparticles on bacterial attachment and preformed biofilms on microfiltration membranes. A recently published modified-rapid attachment assay was used to evaluate the

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BisBAL nanoparticles activity against P. aeruginosa biofilms [16, 69]. Figure 7a shows that at MIC BisBAL nanoparticles (see Table 1) significantly decreased (by ~99%) the bacterial density on membranes when compared to controls (membranes with no nanoparticles). Electron micrographs of an isolated bacterium after exposure to BisBAL nanoparticles for 1 h, indicates their close association with the bacterium surface as seen in Figure 7b. Closer examination of the bacterium in Figure 7b reveals a likely separation of cell wall from cell membrane potentially due to the antibacterial action of bismuth and attachment of BisBAL nanoparticles onto the cell wall. When preformed biofilms on MF membranes were treated with lipophilic BisBAL NPs for 1 h, nanoparticles were abundantly incorporated into biofilms (Fig. 7d). In the absence of BisBAL nanoparticles, bacteria remained cohesive, viable, and intact (Fig. 7c). In contrast, in their presence, cells were completely and catastrophically damaged, and their intracellular contents appeared to have been released (Fig. 7d). Further, lysed biofilms and lipophilic BisBAL nanoparticles appear to be tightly associated, and the released cellular contents and lysed cells were apparently contained within the preformed biofilm zone on the membrane surface (Fig. 7d).

Discussion Extracellular polymeric substances (EPS) are crucial for microbial adhesion, cohesion, and biofilm formation on animate and inanimate surfaces and for facilitating an ecological niche for thriving microorganisms [71]. It was first observed that by the incorporation of a surface into media containing bacteria, its growth and activity were substantially enhanced and the number of sessile bacteria was dramatically higher than in the surrounding medium [72, 73]. Later it was showed that microbial slimes in industrial water systems were not only cohesive but also very resistant to common disinfectants such as chlorine [74]. These earlier studies clearly demonstrated that microbial slimes or biofilms are chiefly governed by the content and composition of EPS and that the presence of a surface could be a big nuisance to a broad range of systems including those employed for industrial water purification as well as other technical and medical devices. Bismuth thiols have the potential to inhibit biofilm proliferation and eradicating established biofilms thereby counteracting a serious operational challenge in many engineered systems. Our own research has focused on employing cationic bismuth thiols to unravel the role of EPS components in bacterial aggregation, and adhesion, and biofilm formation at the molecular level.

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We have also developed a nanoparticle-form of BisBAL to overcome the potential limitations of cationic BisBAL for biofilm control. Earlier research has shown that bismuth is most bioactive when complexed with di-thiols rather than mono-thiols or any other structurally similar organic chelators. Additionally, significantly lower concentration of di-thiols was needed to enhance bismuth’s antibacterial activity. MICs of 2:1 molar ratio of BisBAL, BisEDT, and BisPYR summarized in Table 1 suggest that beyond MICs these bismuth thiols would exhibit significant antibacterial effects, causing cell death and lysis. Specifically, moderate concentrations (5-7 μM BisBAL, 2-3 μM BisEDT, and 7-10 μM BisPYR for S. marcescens and 11-12 μM BisBAL, 6-7 μM BisEDT, and 5-7 μM BisPYR for E. coli) were found to hamper bacterial growth. Alternately, sub-MICs (5 μM BisBAL, 2 μM BisEDT, and 7 μM BisPYR for S. marcescens and 11 μM BisBAL, 6 μM BisEDT, and 5 μM BisPYR for E. coli) were needed to significantly inhibit EPS production (over 90%) without impacting bacterial growth. Figures 1 and 2 illustrate that the sub-MICs of BisBAL, BisEDT and BisPYR had negligible effects on the viability of E. coli (assessed using epifluorescence microscopy and atomic force microscopy) even after 2 days. The MICs summarized in Table 1 are consistent with values published by other researchers for Gramnegative bacteria including E. coli and S. marcescens[10, 50,52]. Additionally, subMIC of bismuth thiols were very effective at suppressing free and boundEPS of E. coli, S. marcescens, and B. diminuta cultures to the highest level and typical results for E. coli are shown in Figure 3 [50]. Note that at least 250-fold lower concentrations of bismuth thiols are needed to achieve antibacterial effects comparable to bismuth nitrate (MIC ~3,000 μM), while simultaneously enhancing biocompatibility. These sub-MICs are also likely to be better tolerated by mammalian cells [65] while having minimal impact on the environmental biota. EPS produced in BisBAL-treated cultures were generally homologous to those in controls with no BisBAL, yet remarkably different in certain important details. BisBAL-treatment not only decreased proteins (amide I) and carbohydrates concentrations but also altered the extent of O-acetylation and acidification of carbohydrates (Fig 4a) and the expression of various protein secondary structures (aggregated strands, β-sheets, random coils, α-helices, 3-turn helices, and antiparallel β-sheets/aggregated strands) (Fig. 4b). Despite the obvious sensitivity of these EPS components to BisBAL, these data also suggest that specific EPS moieties may be crucial for bioaggregation of microbial flocs. Others have shown that O-acetylation of polysaccharides facilitate the formation of gelation and “hooks”-like structures that keep bacteria cohesive in biofilms [75, 76]. Decrease in total EPS would decrease the gelation and increase electrostatic

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repulsion among negatively charged microorganisms, which prevents bioaggregation and biofilm formation. Similar results were also obtained with single species cultures suggesting that antifouling properties of BisBAL could be due to inhibition of O-acetylation of polysaccharides, specific losses of protein secondary structures, and overall reductions in total EPS [50]. These inferences are further evident since BisBAL-treated B. diminuta were easily removed from membrane surfaces by rinsing with ultrapure water coupled to reductions in free and bound EPS, suggesting the central role of EPS on hydrodynamic backwashing efficiencies during microfiltration of bacterial suspensions (Fig. 5). The long-term activity of bismuth thiols is dependent on their continuous availability, since cationic bismuth thiols are readily consumed during treatment. We developed a nanoparticle-form of BisBAL (zerovalent) to potentially circumvent the limitations of soluble cationic BisBAL while maintaining their ability to prevent bacterial adhesion and disrupt established biofilms. As synthesized, BisBAL nanoparticles had wide range of sizes (from 3-400 nm) due to aggressive nucleation reaction initiated by NaBH4[77]. BAL being a di-thiol likely mediated the tethering of coalesced BisBAL nanoparticles (chains of crystallites) as seen in the electron micrograph (Fig. 6a). In addition to these larger nanoparticles (100-400 nm), there is also a large population of clusters of BisBAL nanoparticles (3-15 nm) held by a lesser electron dense (grey) matter, which is likely the BAL (Fig. 6b). In contrast to the apparent variations in shapes of the nanoparticles, XRD spectrum revealed that these nanoparticles are fundamentally made of rhomohedral crystallites of diameter ~18 nm (Fig. 6c). Further, these nanoparticles are lipophilic in nature, which is evident from their increased affinity for octanol over water (Fig. 6d). MICs of BisBAL nanoparticles and cationic BisBAL were very similar when tested against P. aeruginosa cultures suggesting similar mechanisms of action i.e., lipophlicity and antibacterial activity of bismuth were synergistically acting to inhibit EPS or inactivate bacteria. We hypothesize a multi-step mechanism where first lipophilic BAL increases the association of nanoparticles to bacterial cell walls. Next, following adhesion these nanoparticles dissolve (due to electrochemical activity of the cells) releasing bismuth ions. Finally, bismuth exchanges with the sulfhydryl groups of membrane bound enzymes to arrest the cell’s respiratory activity and eventually cause cell lysis [78]. Minimal P. aeruginosa adhesion onto microfiltration membrane surfaces was observed in the presence of BisBAL nanoparticles (Fig. 7a). This suggests the increased association of nanoparticles with the cells likely diminished cellular activity (inactivation and/or decrease in ATP levels) [10, 63,79] and hindered their diffusion toward the membrane. In addition to growth inhibition by inactivation,

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these nanoparticles appear to rip off the cell wall from the cell membrane, which is consistent with the cell damage mechanism of bismuth [80] (Fig. 7b). Note that bacterial attachment onto microfilters occurred within an hour of exposure to P. aeruginosa in the absence of BisBAL nanoparticles (Fig. 7c). Importantly, no attachment occurred during that period in the presence of nanoparticles (Fig 7a). When attached bacteria on a filter surface in the medium were exposed to BisBAL nanoparticles the cells were calamitously destroyed and cellular contents were contained within the zone of biofilm (Fig. 7d). Lipophilic BisBAL nanoparticles well-control P. aeruginosa biofilms, which are common to both water treatment systems and medical devices [81, 82]. BisBAL nanoparticles (MIC 12.5 μM) could be even better antibacterial agents than silver ions (MIC 18.5-46.3 μM) or silver nanoparticles (MIC 50-1000 μM) due to added attribute of lipophilicity[83]. The efficacy of BisBAL nanoparticles need to be directly compared with other disinfectants such as chlorine, chloramines, copper, and silver ions to determine their relative efficiencies. We speculate that the nanoparticulate form of BisBAL would be highly efficient since they share characteristics of cationic BisBAL such as longer activity and bioavailability, ability to penetrate biofilms, enhanced activity at temperatures relevant to water distribution systems, and no known toxic or mutagenic effects at low doses [51, 84, 85]. Previous research on zerovalent bismuth nanoparticles (in the absence of thiols and stabilizers) showed that bismuth is effective against Streptococcus mutansbiofilms at concentrations greater than 200 μM [86]. It is likely that introducing a di-thiol (like BAL) would substantially lower concentrations necessary to inhibit biofilms. For example, 47 isolates of methicillin-resistant Staphylococcusaureus (MRSA) could be inhibited using ~18 μM of 3:1 BisBAL [52] and P. aeruginosa biofilms could be inhibited at 12.5 μM of 2:1 BisBAL nanoparticles or cationic BisBAL [61]. Additional testing is recommended to better establish the antibacterial and antifouling properties of cationic BisBAL and BisBAL nanoparticles under more realistic conditions. We believe that bismuth’s low toxicity potentially represents only a minimal risk to the environmental microbiome and therefore could be used as broad-spectrum antimicrobial agents with applications in water treatment, pharmaceutical, biomedical and clinical devices, textiles and so forth. In all these applications, the lipophilic nature of BisBAL nanoparticles could potentially ‘seek’ microorganisms from the surrounding medium through its lipophilicity and effectively ‘destroy’ them given its antimicrobial activity. To date, BisBAL nanoparticles (or lipophilic bismuth nanoparticles) remain to be tested under real-world conditions in systems that are challenged by waterborne or airborne pathogens and biofilms arising from multiple species. However, the

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reported results demonstrate significant antibacterial and anti-biofilm effects attributed to the lipophilicity and antibacterial activity of nanoparticulate bismuth. Our preliminary findings may provide basis for the development of variety of bismuth-thiol nanoparticles with the ultimate goal of identifying active thiols that enhance antibacterial activity. At the same time the new formulations should simultaneously allow further derivatization reactions that facilitate incorporation of bismuth nanoparticles onto the active surface of water treatment membranes, medical devices, and others systems that of relevance to environmental, medical, and biotechnological applications.

Acknowledgments This research was made possible through grants from the National Science Foundation (CBET-0966939 and CBET-0134301). A portion of the work described herein was performed in the Environmental Molecular Sciences Laboratory, a national scientific user facility by the Department of Energy’s Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. The contents do not necessarily reflect the views and policies of the sponsors nor does the mention of trade names or recommendations for use.

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[44] Omoike, A.; Chorover, J., Adsorption to goethite of extracellular polymeric substances from Bacillus subtilis. Geochimica et Cosmochimica Acta 2006, 70, 827-838. [45] Flemming, H.-C.; Wingender, J., Relevance of microbial extracellular polymeric substances (EPSs)-Part II: Technical aspects. Water Science and Technology 2001, 43, (6), 9-16. [46] Geesey, G. G.; Van Ommen Kloeke, F., Extracellular enzymes associated with microbial flocs from activated sludge of wastewater treatment systems. In Flocculation in Natural and Engineered Environmental Systems, Droppo, I. G.; Leppard, G. G.; Liss, S. N.; Milligan, T. G., Eds. CRC Press: Boca Raton, FL, 2005; pp 295-315. [47] Liao, B. Q.; Allen, D. G.; Droppo, I. G.; Leppard, G. G.; Liss, S. N., Surface properties of sludge and their role in bioflocculation and settleability. Water Res. 2001, 35, (2), 339-350. [48] Park, C.; Helm, R. F., Application of metaproteomic analysis for studying extracellular polymeric substances (EPS) in activated sludge flocs and their fate in sludge digestion. Water Science and Technology 2008, 57, (12), 2009-2015. [49] Badireddy, A. R.; Chellam, S., Bismuth dimercaptopropanol (BisBAL) inhibits formation of multispecies wastewater flocs. Journal of Applied Microbiology 2011, 110, (6), 1426-1437. [50] Badireddy, A. R.; Korpol, B.; Chellam, S.; Gassman, P. L.; Engelhard, M. H.; Lea, A. S.; Rosso, K. M., Spectroscopic characterization of extracellular polymeric substances from Escherichia coli and Serratia marcescens: suppression using sub-inhibitory concentrations of bismuth thiols. Biomacromolecules 2008, 9, 3079-3089. [51] Codony, F.; Domenico, P.; Mas, J., Assessment of bismuth thiols and conventional disinfectants on drinking water biofilms. Journal of Applied Microbiology 2003, 95, 288-293. [52] Domenico, P.; Baldassarri, L.; Schoch, P. E.; Kaehler, K.; Sasatsu, M.; Cunha, B. A., Activites of bismuth thiols against staphylococci and staphylococcal biofilms. Antimicrobial Agents and Chemotherapy 2001, 45, (5), 1417-1421. [53] Zhang, H.; Tang, J.; Meng, X.; Tsang, J.; Tsang, T.-K., Inhibition of bacterial adherence on the surface of stents and bacterial growth in bile by bismuth dimercaprol. Digestive Diseases and Sciences 2005, 50, (6), 1046-1051. [54] Luckay, R.; Cukrowski, I.; Mashishi, J.; Reibenspies, J. H.; Bond, A. H.; Rogers, R. D.; Hancock, R. D., Synthesis, stability and strucutres of the complex of bismuth(III) with the nitrogen-donor macrocycle 1,4,10-

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[55]

[56] [57]

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Appala Raju Badireddy and Shankararaman Chellam tetraazacyclododecane. The role of the lone pair on bismuth(III) and lead(II) in determining coordination geometry. Journal of the Chemical Society Dalton 1997, (5), 901-908. Stavila, V.; Davidovich, R. L.; Gulea, A.; Whitemire, K. H., Bismuth(III) complexes with aminopolycarboxylate and polyaminopolycarboxylate ligands: Chemistry and structure. Coordination Chemistry Reviews 2006, 250, (21-22 SI), 2782-2810. Sadler, P. J.; Sun, H. Z.; Li, H. Y., Bismuth(III) complexes of the tripeptide glutathione (gamma-L-Glu-L-Cys-Gly). Chemistry-A European Journal 1996, 2, (6), 701-708. Burford, N.; Eelman, M. D.; Mahony, D. E.; Morash, M., Definitive identification of cysteine and glutathione complexes of bismuth by mass spectrometry: assessing the biochemical fate of bismuth pharmaceutical agents. ChemicalCommunications 2003, (1), 146-147. Burford, N.; Eelman, M. D.; Groom, K., Identification of complexes containing glutathione with As(III), Sb(III), Cd(II), Hg(II), Tl(I), Pb(II) or Bi(III) by electrospray ionization mass spectrometry. Journal of Inorganic Biochemistry 2005, 99, (10), 1992-1997. Domenico, P.; Tomas, J. M.; Merino, S.; Rubires, X.; Cunha, B. A., Bismuth-dimercaprol exposes surface components of Klebsiella pneumoniae camouflaged by the polysaccharide capsule. Annals of the New York Academy of Sciences 1996, 797, 269-270. Domenico, P.; Tomas, J. M.; Merino, S.; Rubires, X.; Cunha, B. A., Surface antigen exposure by bismuth dimercaprol supression of Klebsiella pneumoniae capsular polysaccharide. Infection and Immunity 1999, 67, (2), 664-669. Huang, C.-T.; Stewart, P. S., Reduction of polysaccharide production in Pseudomonas aeruginosa biofilms by bismuth dimercaprol (BisBAL) treatment. Journal of Antimicrobial Chemotherapy 1999, 44, 601-605. Folsom, J. P.; Baker, B.; Stewart, P. S., In vitro efficacy of bismuth thiols against biofilms formed by bacteria isolated from human chronic wounds. Journal of Applied Microbiology 2011, 111, (4), 989-996. Domenico, P.; Landolphi, D. R.; Cunha, B. A., Reduction of capsular polysaccharide and potentiation of aminoglycoside inhibition in Gramneagtive bacteria by bismuth subsalicylate. Journal of Antimicrobial Chemotherapy 1991, 28, 801-810. Sox, T. E.; Olson, C. A., Binding and killing of bacteria by bismuth subsalicylate. Antimicrobial Agents and Chemotherapy 1989, 33, (12), 2075-2082.

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[65] Wu, C.-L.; Domenico, P.; Hassett, D. J.; Beveridge, T. J.; Hauser, A. R.; Kazzaz, J. A., Subinhibitory bismuth-thiols reduce virulence of Pseudomonas aeruginosa. American Journal of Respiratory Cell and Molecular Biology 2002, 26, 731-738. [66] Carotenuto, G.; Hison, C. L.; Capezzuto, F.; Palomba, M.; Perlo, P.; Conte, P., Synthesis and thermoelectric characterisation of bismuth nanoparticles. Journal of Nanoparticle Research 2009, 11, 1729-1738. [67] Yarema, M.; Kovalenko, M. V.; Hesser, G.; Talapin, D. V.; Heiss, W., Highly monodisperse bismuth nanoparticles and their three-dimensional superlattices. J. Amer. Chem. Soc. 2010, 132, 15158-15159. [68] Luo, Y.; Hossain, M.; Wang, C.; Qiao, Y.; An, J.; Ma, L.; Su, M., Targeted nanoparticles for enhanced X-ray radiation killing of multi-drug resistant bacteria. Nanoscale 2013, 5, 687-694. [69] Ma, L.; Jackson, K. D.; Landry, R. M.; Parsek, M. R.; Wozniak, D. J., Analysis of Pseudomonas aeruginosa conditional Psl variants reveals roles for the Psl polysaccharide in adhesion and maintaining biofilm structure postattachement. Journal of Bacteriology 2006, 188, (23), 8213-8221. [70] Wozniak, D. J.; Wyckoff, T. J. O.; Starkey, M.; Keyser, R.; Azadi, P.; O'Toole, G. A.; Parsek, M. R., Alginate is not a significant component of the extracellular polysaccharide matrix of PA14 and PAO1 Pseudomonas aeruginosa biofilms. Proceedings of the National Academy of Sciences 2003, 100, (13), 7907-7912. [71] Costerton, J. W.; Geesey, G. G.; Cheng, K.-J., How bacteria stick. Scientific American 1978, 238, 86-95. [72] Heukelekian, H.; Heller, A., Relation between food concentration and surface for bacterial growth. Journal of Bacteriology 1940, 40, 547-558. [73] Zobell, C. E., The effect of solid surfaces upon bacterial activity. Journal of Bacteriology 1943, 46, 39-56. [74] Characklis, W. G., Attached microbial growth-II. Frictional resistance due to microbial slimes. Water Res. 1973, 7, 1249-1258. [75] Nivens, D. E.; Ohman, D. E.; Williams, J.; Franklin, M. J., Role of alginate and its O-acetylation in formation of Pseudomonas aeruginosa microcolonies and biofilms. Journal of Bacteriology 2001, 183, (3), 1047-1057. [76] Tielen, P.; Strathmann, M.; Jaeger, K.-E.; Flemming, H.-C.; Wingender, J., Alginate acetylation influences initial surface colonization by mucoid Pseudomonas aeruginosa. Microbiological Research 2005, 160, 165-176. [77] Richards, V. N.; Shields, S. P.; Buhro, W. E., Nucleation control in the aggregative growth of bismuth nanocrystals. Chemistry of Materials 2011, 23, 137-144.

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[78] Beil, W.; Bierbaum, S.; Sewing, K. F., Studies on the mechanism of action of colloidal bismuth subcitrate I. Interaction with sulfhydryls. Pharmacology 1993, 47, 135-140. [79] Mahony, D. E.; Lim-Morrison, S.; Bryden, L.; Faulkner, G.; Hoffman, P. S.; Agocs, L.; Briand, G. G.; Burford, N.; Maguire, H., Antimicrobial activities of synthetic bismuth compounds against Clostridium difficile. Antimicrobial Agents and Chemotherapy 1999, 43, (3), 582-588. [80] Stratton, C. W.; Warner, R. R.; Coudron, P. E.; Lilly, N. A., Bismuthmediated disruption of the glycocalyx-cell wall of Helicobacter pylori: ultrastructural evidence for a mechanism of action for bismuth salts. Journal of Antimicrobial Chemotherapy 1999, 43, 659-666. [81] Climo, M. W.; Pastor, A.; Wong, E. S., An outbreak of Pseudomonas aeruginosa related to contaminated urodynamic equipment. Infection Control and Hospital Epidemiology 1997, 18, (7), 509-510. [82] Mena, K. D.; Gerba, C. P., Risk Assessment of Pseudomonas aeruginosa in water. Reviews of Environmental Contamination and Toxicology 2009, 201, 71-115. [83] Dankovich, T. A.; Gray, D. G., Bactericidal paper impregnated with silver nanoparticles for point-of-use water treatment. Environmental Science & Technology 2011, 45, 1992-1998. [84] De Beer, D.; Srinivasan, R.; Stewart, P. S., Direct measurement of chlorine penetration into biofilms during disinfection. Appl. Environ. Microbiol. 1994, 60, 4339-4344. [85] Xu, X.; Stewart, P. S.; Chen, X., Transport limitation of chlorine disinfection of Pseudomonas aeruginosa entrapped in alginate beads. Biotechnology and Bioengineering 1996, 49, (93-100). [86] Hernandez-Delgadillo, R.; Velasco-Arias, D.; Diaz, D.; Arevalo-Niño, K.; Garza-Enriquez, M.; De la Garza-Ramos, M. A.; Cabral-Romero, C., Zerovalent bismuth nanoparticles inhibit Streptococcus mutans growth and formation of biofilm. International Journal of Nanomedicine 2012, 7, 2109-2113.

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In: Advances in Chemistry Research. Volume 21 ISBN: 978-1-62948-742-7 Editor: James C. Taylor, pp. 29-96 © 2014 Nova Science Publishers, Inc.

Chapter 2

RHEOLOGY OF CARBON BLACK SUSPENSIONS Yuji Aoki Department of Polymer Science and Engineering, Yamagata University, Yonezawa, Yamagata, Japan

Abstract This article gives a review of the rheology of carbon black (CB) suspensions. Extensive studies have been made for the CB suspensions. However, a large fraction of these studies was devoted for the suspensions in fairly non-polar media where the CB particles having the polar surface. The rheological properties of CB suspension are affected by the medium affinity to the CB particles. Different types of rheological behavior are observed accordingly, when the affinity changes. In the medium having a low affinity, the CB particles form the continuous network-type agglomerate and exhibit a strong nonlinearity attributable to strain-induced disruption of a fully developed three-dimensional (3D) network structure of the CB particles therein characterized by the yield stress. In contrast, in the medium having a moderate affinity, the suspensions show a sol-gel transition with increasing CB concentration, and the critical gel behavior characterized with a power-law relationship between the modulus and frequency (), G G/tan(n/2) n is observed. This behavior suggests formation of a self-similar, fractal agglomerate of the CB particles. In the medium having a high affinity, the CB aggregates are well dispersed to no agglomerates. These aggregates exhibit a slow relaxation process to their diffusion. Thus, the structure and rheology of the CB particles/aggregates changes with the affinity of the suspending medium. In this article, we report the rheological properties of CB suspensions in three suspending media, a polystyrene/dibutylphthalate solution (PS/DBP, low affinity), a rosin-modified phenol-type varnish (Varnish-1, moderate affinity), and an alkyd resin-type

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Yuji Aoki varnish (Varnish-2, high affinity).The effects of the primary particle size and the structure of CB aggregates on the rheological properties are summarized. For the CB suspensions exhibiting critical gel behavior, heat-induced gelation and the effects of suspending media are also explained.

1. Introduction Carbon black (CB) is widely used as a reinforcing agent in rubber products and as an additive for pigmentation, UV protection, electrical conductivity, and rheology control in plastics, ink, and coating. Rheological properties of CB/polymer compounds and CB suspensions are very important from both academic and industrial points of view. In attempting to elucidate these rheological responses, extensive studies have been carried out for the CB/polymer compounds [111] and CB suspensions [1226].The occurrence of both yield stress and thixotropy for these systems has been found. In general, an increase in the CB concentration and a decrease in the particle size enhance the agglomerate formation thereby increase the viscosity. These properties change with the CB particle morphology specified at different levels, “particle”, “aggregate”, and “agglomerate” [27].The “particle” is a primary spherical unit characterized with its surface area. The “aggregate” refers to a body of covalently bonded (fused) “particles” and is a fundamental structural unit in the CB suspension because it generally is considered unbreakable during the normal processing of the materials. This “aggregate”, characterized with an oil absorption value, often has various shapes such as the spheroidal, ellipsoidal, linear, and hyper-branched (network-type) shapes. Finally, the “agglomerate” is a group of aggregates bonded together through the van der Waals force. The magnitude of this binding force may change with chemical properties of CB particles as well as the suspending media, and the shape and size of the agglomerate change accordingly. This structural change is sensitively reflected in the rheological responses of the CB suspensions. In general, an increase in the CB concentration and a decrease in the particle size enhance the agglomerate formation thereby increasing the viscosity. More importantly, the changes in the shape of the agglomerates lead to drastic changes in the nonlinear rheology. For example, the continuous, network-type agglomerates are disrupted even under a slow flow thereby exhibiting significant nonlinearities characterized by the yield stress, while the non-continuous spheroidal/ellipsoidal agglomerates can behave as a flow unit to exhibit Newtonian behavior under slow flow.

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Thus, a wide variety of rheological responses is observed for CB suspensions in various media. In attempt of elucidating these responses, extensive studies have been made for the CB suspensions [12, 1618]. However, a large fraction of these studies was devoted for the suspensions in fairly non-polar media where the CB particles having the polar surface were tightly bound to form the continuous network-type agglomerate. Thus, the effects of the medium affinity to the CB particles on the structure and rheology of the CB suspensions have not been fully elucidated. In attempt to rheologically characterize the CB suspensions, we recently reported a series of studies for CB suspensions [2835].We found that the rheological behavior of CB suspensions changes with the affinity of suspending medium toward the CB particles and exhibits three different types. This chapter deals with our rheological studies of CB suspensions. First, we explain an outline of the three types of rheological behavior. Then, we discuss the properties in details. Lastly, we discuss the effects of the primary particle size and the aggregate structure of CB particles on the rheological properties.

2. Experimental 2.1. Samples Several kinds of CB samples supplied from Mitsubishi Chemical Corp. were used.CB-76 (Furnace Black, mean particle size: 65 nm, average aggregate size (diameter): 230 nm, di-butyl phthalate (DBP) absorption value: 71 ml/100 g) was mainly used, because CB-76 has a relatively large primary particle size and low oil absorption value so that a rather small change in the medium affinity toward the CB particles can results in large changes in the agglomerate structure and rheology. Characteristics of these CB samples are shown in Table 1.Surface area is a measure of particle size, and is measured by absorption of nitrogen gas onto the carbon black.DBP adsorption is a measure of aggregate size and is measured as following; a quality of CB is placed in the mixing chamber of torquemeasuring electric kneader-mixer. The kneader action is started and DBP is added drop-wise from an automatic burette. The transition from a free-flowing powder to a semi-plastic material at the point of maximum absorption (the filling of voids) results in a rapid rise in torque, which activates an automatic cutoff in burette and kneader. The volume of DBP added is registered automatically. Its value per 100g of CB, the DBP number, is a reproducible quantity recognized as a structural index in ASTM method D2415-65-T.Aggregates with a greater number of

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particles and more complex aggregates (higher structure CB) have more voids and thus have a higher oil demand. Table 1. Characteristics of CB samples CB CB-76 CB-24-1 CB-24-2 CB-15-1 CB-15-2 AcB

Particle size (nm) 76 24 24 15 15 40

Surface area (m2/g) 29 130 115 260 294 61

DBP absorption value (mL/100g) 71 69 110 74 147 190

Polystyrene (Mw = 1.13105, Mw/Mn = 2.15; Mitsubishi Chemical Corp.)/DBP (PS/DBP) solutions and two varnishes, Varnish-1 (rosin-modified phenol-type varnish, MS-800, Showa Varnish Co.) and Varnish-2 (alkyd resin-type varnish, F104, Showa Varnish Co.), were utilized as the suspending media. The affinity of the medium toward the CB particle increases in this order.Varnish-1 is a mixture of a rosin-modified phenol resin, linseed oil and petroleum solvent.Varnish-2 is an alkyd resin consisted of linseed oil, iso-phthalic acid, and penta-ellislitol.

2.2. Preparation of CB Suspensions The suspensions were prepared by mechanically mixing. prescribed amount (total 50 g) of dry CB and suspending media was charged into a stainless steel cylinder of diameter 45 mm and stirred with motor-driven two flat disks of diameter 30 mm. The stirring was conducted at 500 rpm for 2 h at 60C. The rheological data of the suspensions thus prepared were reproducible and did not change with time. These reproducible/stable data coincided with the data obtained for samples prepared with a three-roller mill. In the suspensions prepared above, the CB concentration was varied from 0 to 35 wt%.

2.3. Measurements For these suspensions, dynamic viscoelastic and steady-state measurements were carried out with a strain-control rheometer (ARES, Rheometrics) to determine the dynamic storage and loss moduli G and G as a function of angular

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. frequency , and steady-state viscosity  as a function of shear rate .Cone-plate fixtures with a gap angle of 0.0845 rad and a plate diameter of 25 and/or 50 mm were used. Prior to the measurements, the suspensions were sheared at a rate 10 s1 for 2 min and then kept quiescently for 3 min. This pre-shear ensured a good reproducibility of the data. After a quiescent rest for 3 min, the measured G and G data did not change with time. Transmission electron microscope (TEM) observation was performed for solidified CB suspensions (for which the observation could be made in vacuum). Fortunately, the CB/Varnish-1 and CB/Varnish-2 suspensions could be solidified by a removal of a small amount of volatile components. After the solidification, the specimens were microtomed to a ultrathin section of about 70 nm thickness using a ultracryo-microtome with a diamond knife at 70C. The structure in the section was observed under an electron microscope, JEM 100 (100 kV). TEM was used to measure the particle size and the aggregate structure of CB. The average aggregate diameter was measured with Microtrac UPA (HONEYWELL).

3. Three Types of Dynamic Viscoelastic Behavior In this section, the effects of the medium affinity to the CB particles on the structure and rheology of the CB suspensions is elucidated. The CB sample studied here have a relatively larger primary particle size and low oil absorption value so that a rather small change in the affinity toward the CB particles can results in large changes in the agglomerate structure and rheology [28].

3.1. Nonlinearity of Three Kinds of Suspensions In the suspensions prepared as above, the CB concentration was varied from 0 to 20 wt% for CB/(PS/DBP) suspension and to 35 wt% for CB/Varnish-1 and varnish-2 suspensions. The oscillatory strain amplitude 0 was varied over a wide range, 0.1 0 500.For the CB-76/(PS/DBP) suspension with CB-76 concentration cCB = 20 wt%, the CB-76/Varnish-1 suspension with cCB =30 wt%, and the CB-76/Varnish-2 suspension with cCB=30 wt%, Figure 1 shows the dependence of the apparent storage and loss moduli Gapp (a) and Gapp(b) at  = 1 rad s1 on the strain amplitude 0.These moduli were measured from a single- strain sweep test, and the pre-shear (for 2 min at 10 s1) and successive quiescent rest (for 3 min) were made prior to the test.

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4

4 (a)

(b) log (G"app /Pa)

log (G' app /Pa)

3 2 1 0

3 2 1 0

○ △ □

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3

-1 -1

CB-76/(PS/DBP): 20% CB-76/Varnish-1: 30% CB-76/Varnish-2: 30%

0 1 2 log (Strain/%)

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Figure 1. Dependence of the apparent (nonlinear) moduli Gapp(a) and Gapp(b) of the CB76/(PS/DBP) suspension having cCB= 20 wt%, the CB-76/Varnish-1 and CB-76/Varnish-2 suspensions having cCB= 30 wt% at 30C on the oscillatory strain amplitude 0. The measurements were made at  1 rad s1 [28].

For the CB-76/(PS/DBP) suspension, both Gapp (a) and Gapp(b) for small 0 ( 0.3%) are independent of 0 and the linear viscoelastic behavior prevails. However, for larger 0, the apparent moduli (in particular Gapp) continuously decrease with increasing 0; see Figure 1.This strong nonlinearity suggests the elasto-plastic deformation/flow of the network-type agglomerate occurring periodically in each cycle of the oscillatory strain. In fact, elliptic stress-strain patterns (Lissajous patterns) were observed at 0 0.3%, but rectilinear patterns were seen for larger strain. For a similar CB suspension, Onogi and Matsumoto [17] showed a similar distorted stress-strain pattern characteristic of this elastoplastic behavior. The agglomerated network in our CB-76/(PS/DBP) suspension must be disrupted under the large oscillatory strain to exhibit the nonlinearity. The nonlinearity of the CB-76 suspensions in Varnish-1 and Varnish-2 (cCB = 30%) is much weaker than that of the CB-76/(PS/DBP) suspension; see Figure 1. Specifically, the former two suspensions exhibit 0-independent moduli for 0 5%. This difference is naturally related to the affinity of the suspending media. The (PS/DBP) solution has a poor affinity toward the CB particles thereby enhancing the full development of the mechanically nonlinear, network-like agglomerate. In contrast, the Varnish-1 (rosin-modified phenol resin-type varnish) and Varnish-2 (alkyd resin-type varnish) have a moderate/high affinity. Thus,

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only a fragmented network of the CB-76 particles is formed in Varnish-1, as noted from the TEM (Figure 2(a)).Since this network does not spread throughout the system, the applied strain should be mostly absorbed in the soft Varnish-1 matrix phase at low  so that the fragmented network is negligibly disrupted to exhibit the linearity at small 0( 5%).Furthermore, the CB-76 particles are well dispersed in Varnish-2 (having the high affinity) to form no large agglomerates; see Figure 2(b).Those randomly dispersed particles again exhibit the nonlinearity much weaker than that of the strongly agglomerate network in PS/DBP.

Figure 2. Transmission electron micrographs of CB-76/Varnish-1 (a) and CB-76/Varnish-2 (b) suspensions with cCB 10 wt% [28].

3.2. Behavior of CB-76/(PS/DBP) Suspension Figure 3 shows the dependence of the linear viscoelastic modulus G (a) and G (b) of the CB-76/(PS/DBP) suspensions having various cCB. For respective suspensions, 0 was chosen to be sufficiently small so that the linear behavior was observed. The corresponding loss tangent (tan ) data are shown in Figure 4.The G and G increase with increasing cCB, and exhibit -insensitive plateaus at  1 rad s1 for cCB10 wt%. This behavior reflects the solid-like character of the agglomerated network under small strain. Correspondingly, the tan  value decreases with increasing cCB, and the tan vs plot for cCB5 wt% exhibit a peak that shifts to higher  with cCB. In summary, the CB-76/(PS/DBP) suspensions exhibit the rheological behavior characteristic of the agglomerated network structure therein, although we could not observe the network structure directly with the electron microscope

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under high vacuum (because of the presence of the volatile solvent, DBP). This behavior, having been extensively investigated for many similar suspensions [17, 18], naturally reflects the poor affinity of the suspending PS/DBP medium toward the CB particles.

CB-76/(PS/DBP) 5

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Figure 3. Frequency dependence of the linear viscoelastic moduli G (a) and G(b)of the CB-76/(PS/DBP) suspensions having various cCB. The data were obtained at 30C, and the strain amplitude was chosen to be sufficiently small (00.11%) to ensure the linear responses of the respective systems [28].

For some suspensions in polymeric solutions, polymer chains are adsorbed on the particles to enhance the network-like agglomeration of the particles [3638]. This could be the case also for our CB-76/(PS/DBP) suspensions, as judged from a result of an adsorption measurement. This suspension was subjected to centrifugation for 2 h at 18,000 rpm (the rate inducing no significant sedimentation of the free PS chains), and the PS chain concentration in the supernatant was found to be smaller than the nominal concentration (5 wt%). Thus the PS chains would have been partially adsorbed on the CB particles to increase moderately an affinity of the particle surface toward the PS/DBP matrix. Further details of this adsorption behavior will be explained in Section 4.

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CB-76/(PS/DBP) 3

log tan δ

2

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0

-1

-1

0

1

2 -1

log (ω/rad s ) Figure 4. Frequency dependence of the loss tangent, tan , of the CB-76/(PS/DBP) suspensions having various cCB. The data were obtained in the linear viscoelastic regime (00.11%; cf. Figure 3) at 30C. The symbols are the same as in Figure 3 [28].

3.3. Behavior of CB-76/Varnish-1 Suspension For all CB-76/Varnish-1 suspensions having cCB 35 wt%, the linear viscoelastic behavior (0-independence moduli) was observed for 0 5%; see Figure 1.The linear viscoelastic G and G characterizing this behavior are documented in Figure 5.The corresponding loss tangent (tan ) data are shown in Figure 6. As noted in Figure 5, the suspensions with cCB 10 wt% exhibit the flow behavior characterized by the terminal tails, G2 and G at low . These tails vanish and the  dependence of G and Ggradually changes with increasing cCB above 10 wt%. This change reminds us a sol-gel transition on an increase of the concentration [39, 40]. In fact, at the highest cCB examined (35 wt%), the CB76/Varnish-1 suspension exhibits the power-law type  dependence of G and G associated with an -insensitive tan  (see Figure 5 and 6): G() = An and G() = Bn

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Yuji Aoki The loss tangent tan  is given by tan  = G()/G() = tan (n/2)

(2)

CB-76/Varnish-1 5

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2

Figure 5. Frequency dependence of the linear viscoelastic moduli G (a) and G(b) of the CB-76/Varnish-1 suspensions having various cCB. The data were obtained at 30C under the strain amplitude 05% [28].

The value n is called critical relaxation exponent and was obtained to be 0.71 for the CB-76/Varnish-1 suspension. This power-law behavior is characteristic of a critical gel consisting of a self-similarly branched fractal network, as confirmed for many gelling systems [3946].In fact, the critical gelation behavior with a similar power-law index (n = 0.74) was found also for the other CB/varnish suspension [47]. Here it should be emphasized that the critical gelation behavior in the CB76/Varnish-1 suspension is a natural consequence of the moderate affinity of the Varnish-1 toward the CB particles. If the affinity is too low, the densely connected three dimensional network is fully developed at small cCB, as in the case of the CB76/(PS/DBP) suspension; cf. Figures. 1, 3, and 4.In contrast, the CB particles hardly agglomerate if the affinity is too high, as explained in the next section for the CB76/Varnish-2.The affinity of Varnish-1 appears to be just moderate to allow the formation of the fragmented network at low cCB and this fragmented network passively grows to the fractal network (critical gel) at large cCB (35 wt%).

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CB-76/Varnish-1 3

log tan δ

2

1

0 -1

-1

0

1

2

-1

log (ω/rad s ) Figure 6. Frequency dependence of the loss tangent, tan , of the CB-76/Varnish-1 suspensions having various cCB. The data were obtained in the linear viscoelastic regime (05%) at 30C. The symbols are the same as in Figure 5 [28].

3.4. Behavior of CB-76/Varnish-2 Suspension For all CB-76/Varnish-2 suspensions having cCB 35 wt%, the linear viscoelastic behavior characterized by the 0-independent G and G was found for 0 5%. Figure 7 summarizes these G and G data, and Figure 8 documents the corresponding tan  data. For the CB-76/Varnish-2 suspensions with various cCB, the  dependence of G weakens with decreasing . Correspondingly, the tan vs plots exhibit a peak and the tan  value decreases with decreasing  rad s1; see Figure 8. These results indicate that the CB-76/Varnish-2 suspensions exhibit a slow relaxation process not completed in our experimental -window. This slow relaxation behavior of the CB-76/Varnish-2 suspension is qualitatively different from the behavior of the CB-76/Varnish-1 suspensions characterized with the monotonous changes of tan  (without exhibiting the peak; cf. Figure 6), although the magnitude of nonlinearity is not significantly different for these suspensions (cf. Figure 1).

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4 3

(b) 3 2

log (G"/Pa)

2

log (G'/Pa)

(a)

C CB/% 0 5 10 15 20 25 30 35

1

1

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-1

-1

-2

-1

0

1

-1

log (ω/rad s )

2

-2

-1

0

1

-1

2

log (ω/rad s )

Figure 7. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-76/Varnish-2 suspensions having various cCB. The data were obtained at 30C under the strain amplitude 0 5% [28].

In the CB-76/Varnish-1 suspensions, the fragmented agglomerates grow to the fractal network (critical gel network) with increasing cCB to 35 wt%, thereby exhibiting the characteristic power-law type relaxation; see Figure 5 and Eq. (1).In contrast, even at the largest cCB examined (35 wt%), the particles are randomly dispersed and no large agglomerates are formed in Varnish-2 having a high affinity toward the CB particles. Thus, the slow relaxation seen for the CB76/Varnish-2 suspensions are not related to the agglomerate structure. Here, were member that the so-called hard-sphere suspensions such as silica/ethylene glycol suspensions; Shikata and Pearson [48]exhibited a relaxation process similar to that seen for the CB-76/Varnish-2 suspensions. In the hardsphere suspensions, the particles have no long-ranged potential and are randomly/isotropically dispersed at equilibrium. This particle distribution is anisotropically distorted on application of strain, and the placement entropy of the particles decreases accordingly. This decrease gives rise to the viscoelastic stress that relaxes when the particles undergo translational diffusion (Brownian motion) to recover their isotropic distribution [4857].

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CB-76/Varnish-2 3

log tan δ

2 1

0

-1

-1

0

1

2

-1

log (ω/rad s ) Figure 8. Frequency dependence of the loss tangent, tan , of the CB-76/Varnish-2 suspensions having various cCB. The data were obtained in the linear viscoelastic regime (0 5%) at 30C. The symbols are the same as in Figure 7 [28].

The slow relaxation seen for the CB-76/Varnish-2 suspensions appears to be assigned to this diffusion-induced relaxation. However, the CB-76 particles/aggregates utilized are considerably large and their terminal relaxation is too slow to be fully resolved in our experimental window even for the case of the random dispersion. Thus, the Brownian nature of that dispersion is not fully established. The Brownian relaxation becomes faster upon decrease of the aggregate size. Using smaller CB aggregates is possible to detect the terminal relaxation. Details of these results are presented in Section 6.

3.5. Comments of a Conventional Interpretation for the Rheology of Printing Ink The Varnish-1, a rosin-modified phenol resin-type varnish, is a main component in the varnish for printing ink. In Varnish-1, the CB particles form fragmented network-type agglomerates that grow to the self-similar fractal network, and the characteristic power-law type relaxation is observed (cf. Figure 5).

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The Varnish-2, an alkyd resin-type varnish, is a minor component in the varnish for printing ink. In Varnish-2, the CB particles are well dispersed to form no large agglomerates. Thus, these particles exhibit the slow relaxation attributable to their diffusion (cf. Figure 7). In the actually used inks/paints, the Varnish-1 and 2 are mixed and the affinity of this mixture toward CB particles is not necessarily identical to a simple average of the affinities of respective varnishes. However, the above findings pose a caution to a conventional interpretation that the CB particles in the actual inks/paints always form large network-like agglomerates and their rheological properties are dominated by this network structure.

3.6. Summary The rheological behavior of the CB suspensions changes with the affinity of suspending medium toward the CB particles. Specifically, three different types of the behavior were observed, as summarized below. In the PS/DBP solution having a poor affinity, the CB particles form welldeveloped network-like agglomerates. This network structure provides the suspensions with highly nonlinear, elasto-plastic features. In the Varnish-1 (rosin-modified phenol resin-type varnish) having a moderate affinity toward the CB particles, the particles form fragmented networklike agglomerate that grow to a self-similar, fractal network as the CB concentration cCB is increased. Correspondingly, a power-law type relaxation similar to that of critical gels, G Gn with n = 0.71, is observed at a high cCB where this fractal network is formed. In the Varnish-2 (alkyd resin-type varnish) having a high affinity, the particles are randomly dispersed to form no large agglomerates. The slow relaxation seen in this varnish is attributable to diffusion of the covalently-fused aggregates.

4. Steady-State Shear Viscosity of Strongly Flocculated Systems (PS/DBP) solution has a poor affinity toward CB-76, as explained in the previous section. Therefore, to characterize strongly flocculated CB aggregates, steady state measurements are suitable and can provide interesting rheological properties, such as flow curve and yield stress. In this section, steady-shear shear

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viscosities of CB/(PS/DBP) suspensions are reported. The effects of CB aggregates on the rheological properties for these CB systems are elucidated for the design of CB suspensions and that of polymer/CB compounds. The relationship between these rheological quantities and the structure of CB aggregates is discussed in terms of an effective volume fraction eff of CB aggregates. Effects of polymer adsorption on the rheological properties are also discussed [32].

4.1. Flow Curves and CassonPlots Four series of carbon black (CB-76, CB-24-1, CB-15-2, and AcB) suspensions with various CB concentrations cCB in PS/DBP solution were used. AcBis an acetylene black type CB supplied from Denki Kagaku Kogyou Co. Characteristics of these CB samples are shown in Table 1.Aggregates with a greater number of particles and more complex aggregates (higher structure CB) have more voids and thus have a higher oil demand. Accordingly, CB-76 and CB24-1 have almost the same DBP absorption value and are lower structure grades consisting of more compact aggregates. CB-15-2 is higher structure grades containing more branched aggregates. AcB has the highest DBP absorption value among CB samples studied here, although the primary particle size is not the smallest. CB concentration cCB measured depended on CB species because of different aggregate structure, and was ranged from 0 to 20 wt % for CB-76 suspension, to 22.5 wt% for CB-24-1, to 10 wt% for CB-15-2 suspension, and to 3 wt% for AcB suspension. . Figures 9(a), (b), (c), and (d) show the shear rate  dependence of the steady shear viscosity  of the CB-76 (a), CB-24-1 (b), CB-15-2 (c), and AcB (d) suspensions having various CB concentrations cCB. The suspending medium . (PS/DBP solution) shows a constant viscosity independent of  and exhibits a Newtonian flow. On the other hand, all the CB/(PS/DBP) suspensions are characterized by a typical shear thinning behavior. The viscosities seem to approach to a constant viscosity at high shear rate. Both the degree of shear thinning and the viscosity at high shear rate increases with increasing cCB. The shapes of the flow curve are very similar, although the correspondence cCB is different for each suspension.

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(a)

(b) Figure 9. Continued on next page.

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(c)

(d) Figure 9. Flow curves of the CB-76/(PS/DBP) (a), CB-24-1/(PS/DBP) (b), CB-152/(PS/DBP) (c), and AcB/(PS/DBP) (d) suspensions at 25C [32].

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Casson presented the following equation, which is now called the Casson equation [58].This equation has been applied to many kinds of suspensions and found to be useful. . 1/2 = k0 + k11/2

(3)

The Casson equation was derived under an assumption that the dispersed particles are agglomerated in the rod shape and the rods are destroyed in a small size under external shear field. This assumption can be applied to the CB suspensions because the agglomerate structure consisting of CB aggregates corresponds to the rods and the rod structure should be destroyed in smaller agglomerates under high shear. 10

(σ/Pa)1/2

CB-76/(5%PS/DBP)

5

0

0

5

(γ/s-1)1/2

10

Figure 10. Casson plots of the CB-76/(PS/DBP) suspensions at 25C. The symbols are the same as in Figure 9 [32].

At very high shear, the rods must be divided to CB aggregates, as Onogi et al. reported that the Casson equation can be applied to the CB suspensions [1315, 17]. Using the Casson equation, the yield stress y and the viscosity at infinite of shear rate  of the suspensions can be evaluated. Here, the yield stress y (= k02) . is obtained by extrapolation at the shear rate  = 0. The viscosity at infinite of shear rate  is obtained from the slope of Casson plot. As representative examples, Casson plots for the CB-76/(PS/DBP) are shown in Figure 10. . Linearity in all the 1/2 - 1/2 relationships is evident at higher shear rate, although

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the stress decreases at low shear rates for higher cCB samples, as already pointed out by Onogi and Matsumoto[17]. These results indicate that the yield values estimated from Casson plot may lead to an overestimation. However, the values thus evaluated would be a meaningful parameter to characterize CB aggregates. Casson plots for the CB-24-1/(PS/DBP) suspensions are shown in Figure 11. . Linearity in all the 1/2 - 1/2 relationships is evident at shear rates measured.

Figure 11. Casson plots of the CB-24-1/(PS/DBP) suspensions at 25C. The symbols are the same as in Figure 9 [32].

. Figure 12 shows the shear rate dependence of the steady shear viscosity  of the CB-76suspensions with different PS concentrations. The viscosity of the media increases with increasing PS concentration. Casson plots for these . suspensions are shown in Figure 13.Linearity in all the 1/2 - 1/2 relationships is found and the same yield stress values are obtained for the suspensions irrespective of PS concentration. This indicates that CB particles form the same three-dimensional network structure. The y values thus evaluated would be a meaningful parameter to characterize CB aggregates. Similar Casson plots could be obtained for the other three suspensions. From the basic concept of the Casson model, such linearity provides a strong evidence of the presence of connected CB aggregate networks in the CB suspensions.

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Figure 12. Flow curves of the CB-76/(PS/DBP) suspensions with different PS concentrations at 25C.

Figure 13. Casson plots of the CB-76/(PS/DBP) suspensions with different PS concentrations at 25C. The symbols are the same as in Figure 12.

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4.2. Medium Viscosity Change Due to CB Particles/Aggregates It has been reported that, for some suspensions in polymeric solutions, polymer chains are adsorbed on the particles to enhance the network-like agglomeration of the particles [3638]. If the CB aggregates also adsorb PS in the PS/DBP solution, the medium viscosity m must be changed. To analyze relative viscosity r(=/m) versus volume fraction  relationship, the medium viscosity m for the suspensions is necessary. Accordingly, the CB suspensions were centrifuged at 18,000 rpm for 2h.It was found that the supernatant solution after centrifugation was transparent for all the suspensions. Table 2. Viscosities of supernatant solutions for the CB-76/(PS/DBP), CB-24-1/(PS/DBP), and CB-15-2/(PS/DBP) suspensions [32] CB-76 (wt%) 0 5 10 15

 (Pa s) 0.0787 0.0761 0.0736 0.0714

CB-24-1 (wt%) 0 5 10 15

 (Pa s) 0.0787  0.0630 0.0533

CB-15-2 (wt%) 0 4 8 12

 (Pa s) 0.0787 0.0716 0.0589 0.0481

This fact indicates that the supernatant solution does not contain any CB particles. Table 2 shows the viscosity of the three series of supernatant solutions after centrifugation. The viscosity of the supernatant solutions decreases with cCB for the CB-76/(PS/DBP), CB-24-1/(PS/DBP), and CB-15-2/(PS/DBP) suspensions, whereas the viscosity of the AcB/(PS/DBP) suspension is almost the same irrespective of cCB, although the data are not tabulated here. These results indicate that PS molecules are adsorbed on the surface of the neutral furnace blacks (CB-76, CB-24-1, and CB-15-2).The amount of adsorbed PS could be estimated, as it was found that the log viscosity of the PS/DBP solution increases linearly with weight fraction of PS. The amount of adsorbed PS thus estimated is 24 mg/(g of CB) for the CB-76 suspensions, 77 mg/(g of CB) for the CB-24-1 suspensions, and 124 mg/(g of CB) for the CB-15-2 suspensions, but 0 mg/(g of CB) for the AcB suspensions. The difference in PS adsorption due to CB particles/aggregates is considered as follows. Furnace CB aggregates typically contain about 9099% elemental carbon, with oxygen and hydrogen as the other minor constituents [25]. The oxygen and hydrogen of the furnace CB surface may prove the adsorption of the PS onto the CB aggregates. On the other hand, AcB contains almost 100% elemental carbon and is

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constructed of a graphite structure. These adsorption data indicate that the graphite structure never adsorbs the PS chains. However, the flow curves of the AcB suspensions are very similar to those of the furnace black suspensions. The difference of CB adsorption among the furnace CB species is considered to be due to the primary particle size of CB. The surface area becomes larger for CB having small primary particle size. Thus, adsorbed PS chains must be increased.

4.3. Relative Viscosity Figures14 (a) and (b)show the bare volume fraction  dependence of the highshear relative viscosity r=/m for the CB-76 (a) and CB-15-2 (b) suspensions. Here, the bare volume fraction  of the CB is calculated using the densities of CB (= 1.9 g/cc) and PS/DBP solution (= 1.05 g/cc)In these figures, filled circles denote the uncorrected data using the viscosity of 5% PS/DBP and open circles the corrected data using the viscosity of the supernatant solutions as m. As shown in the previous section, the medium viscosity m decreases with increasing CB concentration. Accordingly, the corrected values have higher  dependence, because of low m. It is clear that the correction gives a significant difference for the  dependence of the relative viscosity r.

(a)

(b)

Figure 14. Relative viscosity r versus bare volume fraction  plots of the CB76/(PS/DBP) (a) and CB-15-2 (b) suspensions. Filled small circles denote the uncorrected values and open circles the corrected values by polymer adsorption [32].

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Therefore, we would like to emphasize that the adsorbed PS molecules strongly affect the suspension viscosity by changing the polymer concentration. The medium viscosity change of the CB-15-2 is larger than that of CB-76 as shown in the previous section. Figure 15 shows the  dependence of the relative viscosity r for the CB-76, CB-24-1, CB-15-2, and AcB suspensions. It is clear that the CB suspensions with larger DBP absorption value have larger  dependence. Why do these suspensions have different  dependence? To explain the reason, we utilized effective volume fraction eff instead of bare volume fraction  of the CB aggregates .CB aggregates have non-spherical and non-uniform shape. Therefore, the effective volume fraction eff would be larger than the bare volume fraction .Moreover, the CB aggregates must trap and/or adsorb the PS solution. This “dead” PS solution looses its identity as a solution and behaves as a part of filler in terms of the rheological behavior. These are the reason why the bare volume fraction  is converted to the effective volume fraction eff. The eff must include both effects due to the shape of CB aggregates and immobilized PS solutions on the CB surface.

Figure 15. Relative viscosity r versus volume fraction  plots of the CB-76/(PS/DBP), CB-24-1/(PS/DBP), CB15-2/(PS/DBP), and AcB/(PS/DBP) suspensions [32].

The effective volume corresponds to a volume occupied by sphere having the aggregate radius a. The eff value can be estimated as follows; for mono-disperse hard-core silica suspensions, the high frequency viscosity  normalized by the

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medium viscosity m is universally dependent on  irrespective of the particle radius [48, 52]. The eff of four CB aggregates (eff) was evaluated by comparing the /m ratios of the CB aggregates and the universal /m versus  curve for the hard-core silica particles. The results are shown in Figure 16. From the abscissa coordinates of the plot for the CB aggregates (open symbols), the  value was estimated to be 2.6, 2.6, 5.5, and 16 for the CB-76, CB-24-1, CB-15-2, and AcB suspensions, respectively. This result indicates that the  value or the effective volume is an important parameter to characterize CB aggregates. With increasing shear rate, the density of the three-dimensional network of the CB aggregates is decreased. Under sufficient high shear, the size of flow unit becomes the CB aggregate size. In this shear region, the second Newtonian flow appears. Therefore, the CB aggregates are considered to be a flow unit corresponding to “effective volume”. The  value increases with increasing the DBP absorption value, suggesting that eff depends mainly on aggregate structure morphology irrespective of particle size. This result indicates that the CB aggregates behave as a flow unit at high shear rates and the difference between eff and  reflects mainly the nonspherical shapes of the aggregates.

Figure 16. Relative viscosity r versus effective volume fraction eff plots of the CB-76/(PS/DBP), CB-24-1/(PS/DBP), CB15-2/(PS/DBP), and AcB/(PS/DBP) suspensions. The eff values were determined in a way that the /m versus eff for four suspensions were superposed on the universal /m versus  plots obtained for unimodal hard-core particles [32].

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4.4. Yield Stress Figure 17shows the yield stress y of the CB-76, CB-24-1, CB-15-2 and AcB suspensions as a function of bare volume fraction . The y of these suspensions changes drastically at a certain critical concentration, g, which corresponds to the formation of a space-filling network. At g, dilute suspensions have no yield stress and the discrete clusters settle in the dilute suspensions rather independently. Whereas above g, the suspensions develop a three-dimensional network structure of the CB aggregates therein.

Figure 17. Yield stress y versus volume fraction  plots of the CB-76/(PS/DBP), CB-24-1/(PS/DBP), CB15-2/(PS/DBP), and AcB/(PS/DBP) suspensions [32].

The yield stress y is a force produced by the separation between the particles times the number of inter-particle bonds that cross a unit area of the sample[59]: The latter factor is scaled as 2/a2, where  is particle volume fraction and a particle radius[51].For four CB suspension systems, we used the average aggregate radius as a, and used the effective volume fraction eff instead of , as same as we analyzed the high-shear relative viscosity in the previous section. In Figure 18, the y data of the CB suspensions are normalized by (a/a0)2 and plotted against eff, where a0 100 nm. It is found that these plots also agree with each other for the three furnace black (CB-76, CB-24-1, and CB-15-2) suspensions. We can conclude that the y is attributed to some internal structures formed by CB

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aggregates and the stress to break up into CB aggregates must be almost the same for the furnace black samples. Finally, we mention briefly the reasons why the y values of the AcB suspensions are larger than those of CB-76, CB-24-1, and CB-15-2 suspensions. The surface of AcB develops a graphite structure, although that of the furnace black does not. The surface structure difference may be attributed to different PS adsorption behavior and the eff dependence of the y values, although we cannot explain the mechanism yet.

Figure 18. Yield stress y versus effective volume fraction eff for the CB-76/(PS/DBP), CB-24-1/(PS/DBP), CB15-2/(PS/DBP), and AcB/(PS/DBP) suspensions. a0100 nm [32].

4.5. Summary Steady-shear viscosity of CB/(PS/DB) suspensions are reported as a function of volume fraction () of various kinds of CB aggregates. The effects of the primary particle size and the structure of CB aggregates on the rheological properties are clarified. The suspensions show a typical shear-thinning behavior in the range of a shear rate studied. The Casson model can be applied to evaluate the viscosity at infinite of shear rate  and the yield stress y for the suspensions. Relative viscosity /m, (m: medium viscosity) thus obtained are compared to the high-frequency viscosity for the ideal hard-sphere silica suspensions to

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evaluate the effective volume fraction eff of CB aggregates. The eff values are larger for the higher-structure CB with higher DBP absorption value, irrespective of the primary particle size. This result demonstrates that the effective volume fraction is the most important quantity to characterize the CB aggregates on the rheological properties. It is also reported that the correction of the medium viscosity changes due to polymer adsorption on the CB surface is important since neutral furnace CB adsorbs PS polymers. These flow curves introduced here are the same as those reported by many researchers for CB suspensions and CB/polymer compounds. The yield stress y had almost the same eff dependence for neutral furnace CB/(PS/DBP) suspensions, although it is larger for acetylene black (AcB)/(PS/DBP) suspensions. It is also found that the correction of the medium viscosity changes due to polymer adsorption on the CB surface is important since neutral furnace CB adsorbs PS polymers but AcB hardly adsorbs PS polymers in the solution.

5. Viscoelastic Behavior of Weakly Flocculated Systems The rheological behavior of weakly attracting CB particles in the medium having a moderately affinity is given in this section. The relationships between the rheology of sol-gel transition of weakly attractive particles and particle morphology is elucidated. The critical relaxation exponent n and the critical gel concentration crit are evaluated and discussed in relation to the CB morphology such as primary particle size and aggregate structure. The gelation at a given particle content occurs at higher temperature, which is the opposite to the gelation behavior of many suspensions explained earlier [30, 31].This peculiar behavior is also discussed.

5.1. Rheology of Sol-Gel Transition Systems Suspensions of weakly attractive particles, exhibiting the sol-gel transition behavior at sufficiently high concentrations, have been attracting industrial as well as academic interest. Most of the rheological studies for such suspensions were devoted to elastic properties of the gels[23, 6064] and to gelation processes[41, 45, 46].Woutersen and de Kruif [65] and Woutersen et al. [66] studied the viscosity of suspensions of silica particles with covalently bound octadecyl chains in benzene at various temperatures (T) and found that the viscosity rises rapidly with decreasing T because of the enhancement of the attractive interactions. Rueb and Zukoski[43]

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investigated the rheology of similar suspensions in decalin and tetradecane and found that the particles interact like hard spheres at high T but the gelation due to the interparticle attraction occurred at low T. Bergstroem and Sjoestroem [67] also found the gelation at low T for sterically stabilized ceramic suspensions. In contrast, Horn and Patterson [68]studied viscoelastic properties of slurries of oxide powders in oxidized polybutene liquids and found a reversible change from a dispersed, fluid state to a coagulated, pasty state on an increase of T. The application of rheology near sol-gel transitions developed by Winter and Mours [69] to these suspensions should result in a much deeper understanding of the rheology of CB suspensions. As explained in Section 3.3, Winter and Chambon [39, 40] found that G() and G() follow power laws with the same exponent n at the gel point; see Eqs. (1) and (2).In other words, the gel point corresponds to the state where G() and G() parallel each other over the experimentally accessible range of frequencies, and is characterized by a single exponent n called the critical relaxation exponent that ranges between 0 and 1. Eq. (2) is of importance, because, Eq. (2) allows the precise value of the gelation point to be determined. The critical relaxation exponent n can be related to the fractal structure of the gelling system [7077]. When the hydrodynamic interaction is completely screened and the excluded volume effect is dominant, the relation of n to mass fractal dimension df can be expressed as [72, 75] n = d/(df + 2)

(4)

where d denotes the space dimension. If the excluded volume effect is screened out near the gel point, the n may decrease as [72] n = d(d + 2  2df)/2(d + 2 df)

(5)

The mass fractal dimension can be described as the structure of an agglomerated one. The distribution of the particle is the straight line for df = 1, plane-like for df = 2, and the three-dimensional space for df = 3.The agglomerated structure having lower fractal dimension can be described as a more open structure. Experimentally determine n in the gelation process was reported to be around 0.75 for zirconiumalkoxide [41], silica gels [45] and SnO2 suspensions [46], and 0.13 and 0.15 for TiO2 suspensions [78].Recently, Uematsu et al.[79, 80]studied rheology of SiO2 suspensions and reported n is 0.45. Despite these studies, critical relaxation exponent n has not been fully elucidated. In this section, the critical relaxation exponent n and the critical gel

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concentration cgel of weakly flocculated CB suspensions are evaluated and discussed [30, 31].

5.2. Viscoelastic Behavior of CB/Varnish-1 Suspensions To elucidate the rheology of sol-gel transitions of weakly attractive particles, the G and G were measured for the CB-24-1/Varnish-1 suspensions. For the suspensions at various CB concentrations, cCB, Figure 19 shows the  dependence of the linear viscoelastic modulus G (a) and G (b) at 30C. The corresponding loss tangent (tan ) data are shown in Figure 20.The behavior of the CB-241/Varnish-1 suspensions is similar to that of CB-76, as shown in Section 3.3. The fluid behavior characterized with the terminal tails (G2 and G at low ) seen for at low cCB gradually changes with increasing cCB to the critical gel behavior characterized with a power-law. Specifically, for cCB=25 wt%, the G and G follow power laws with the same slope over the entire  range, associated with an -insensitive tan  as shown in Eqs. (1) and (2).This power-law behavior of G and G, and -independent tan  are characteristic to a critical gel consisting of a self-similar branched network.

Figure 19. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-24-1/Varnish-1 suspensions having various cCB. The data were obtained at 30C under the strain amplitude 0 5% [31].

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Figure 20. Frequency dependence of the loss tangent, tan , of the CB24-1/Varnish-1 suspensions having various cCB. The data were obtained in the linear viscoelastic regime (0 5%) at 30C. The symbols are the same as in Figure 19 [31].

A method to determine the critical gel concentration crit. and critical relaxation exponent n is known as the -independence of tan .Both values can be determined by a multi-frequency plot of tan  versus concentration .The critical gel point has been well determined using this method when gelation time t or temperature T is a controlling variable for the gelation processes. In this work, this type of plots is the tan  versus  (volume fraction of CB) for different frequencies. An example is shown in Figure 21 for the CB-24-1/Varnish-1 suspensions. All curves at each  pass through the common point at the certain , which is defined as the crit..The critical exponent n was calculated using Eq. (2) from the tan  value at crit..However, this method, the -independence of tan , is a necessary condition but not a satisfied condition, whereas Eq. (1) is a necessary and satisfied condition. We checked whether the power-law type  dependence of G and G with the same n was satisfied and confirmed. For the CB-24-1/Varnish1 suspension, the crit. and n values were 0.16 and 0.74  0.03, respectively. The discussion about the CB morphology dependence of crit. and n will be given in the next section.

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Figure 21. Plots of tan against CB volume fraction  for the CB-24-1 suspensions at various angular frequencies (0.1, 0.398, 1, 3.98, 10, 39.8 and 100 rad s 1). crit. is the critical point [31].

5.3. Effects of CB Morphology Next, we turn our attention to the results of other CB samples with different primary particle sizes and DBP absorption values. CB-24-1 and CB-15-1 have almost the same DBP absorption value, although the primary particle sizes are different; see Table 1. The primary particle size of CB-24-2 is almost identical with that of CB-24-1, and that of CB-15-2 is also identical with that of CB-15-1. But, the DBP absorption values of CB-24-2 and CB-15-2 are larger than those of CB-24-1 and CB-15-1. It means that CB-24-2 and CB-15-2 consist of a more complicated aggregate structure than CB-24-1 and CB-15-1, respectively. Figure 22shows the  dependence of the G (a) and G (b) of the CB-242/Varnish-1 suspensions having various cCB at 30C. At the highest cCB examined (= 20 wt%), the CB-24-2/Varnish-1 suspension exhibits the power-law type  dependence of G and G associated with an -insensitive tan . This cCB corresponds to a critical gel concentration crit. at 30C. The crit. (= 0.13) of the CB-24-2 suspensions is smaller than that of CB-24-1, although the critical exponent n (= 0.71  0.3) is almost the same.

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Figure 22. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-24-2/Varnish-1 suspensions having various cCB. The data were obtained at 30C under the strain amplitude 0 5% [31].

Figures 23 and 24 show the  dependence of the G (a) and G (b) of the CB15-1/Varnish-1 and CB-15-2/Varnish-1 suspensions, respectively, having various cCB at 30C. These two suspensions also exhibit a critical gel behavior. Larger difference between the CB-15-1/Varnish-1 and CB-15-2/Varnish-1 suspensions exists in the critical gel concentration. The crit. (= 0.16) of the CB-15-1/Varnish-1 suspension is about twice larger than that (= 0.074) of the CB-15-2/Varnish-1 suspension. The critical concentrations crit., the critical relaxation exponents n, and mass fractal dimensions df calculated for the dimension d=3 according to eq. (4) are listed in Table 3.The experimental error for the n is 0.03 for all the suspensions. First, we discuss the effect of primary particle size on crit.. As mentioned in “2. EXPERIMENTAL”, CB-24-1 and CB-15-1 have almost identical DBP absorption value, although the primary particle size is different. The comparison of crit. between CB-24-1 and CB-15-1 suspensions indicates that the particle size does not affect the crit. values. As the CB having the same DBP absorption value is considered to have a similar structure each other, then the above results would be a natural consequence of similarity of the aggregate structures of CB-24-1 and

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CB-15-1. In Section 3.3, we reported that crit. of CB-76 (DBP absorption value: 71 ml/100 g) suspensions is 0.25.This crit. value is larger than those of CB-24-1 and CB-15-1, although the DBP absorption values are almost the same. The difference would be due to the bimodal particle distribution of CB-76.On the other hand, the critical gel concentration crit. is strongly affected by the DBP absorption value. The crit. values decrease as the DBP absorption value increases (with developing the aggregate structure). DBP absorption value may be a main factor to determine the crit..

Figure 23. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-15-1/Varnish-1 suspensions having various cCB. The data were obtained at 30C under the strain amplitude 0 5% [31].

Table 3. Experimental values of critical gel concentration crit., critical exponent n, and mass fractal dimension df at 30C [31] CB CB-24-1 CB-15-1 CB-24-2 CB-15-1

crit. 0.16 0.16 0.13 0.074

n 0.74  0.03 0.72  0.03 0.71  0.03 0.70  0.03

df 2.06 2.18 2.22 2.26

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Figure 24. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-15-2/Varnish-1 suspensions having various cCB. The data were obtained at 30C under the strain amplitude 0 5% [31].

The critical relaxation exponent n at 30C does not depend on both particle size and DBP absorption value. This fact implies that there is a strong similarity in the agglomerate structure of CB aggregates that form at different CB species. The same critical exponents reflect the same agglomerate structure at the sol-gel transition point. Here, we use Eq. (3) to estimate the mass fractal dimension df, as the CB agglomerates sense each other only through volume exclusion and hydrodynamic interactions. The connection introduced between the fractal dimension and the critical exponent (in Eq. (3)) show that the fractal dimension does not depend on the agglomerate structure. The mass fractal dimension of df 2 means that the CB agglomerates consist of a plane-like structure. CB-24-1 and CB-15-1 have almost the same DBP absorption value, although the primary particle size is very different. It is important to notice that the effect of primary particle size on the fractal dimension is very small. This fact may result from the nature of the agglomerate structure of CB particles.

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Rheology of Carbon Black Suspensions CB-24-2/Varnish-1. 40 oC

CB-24-2/Varnish-1, 30oC

3

3

(b)

2

log (tan δ)

log (tan δ)

(a)

1

1 0

0 -1

2

-1

0

1

2

-1

-1

0 1 2 log (ω/rad s-1) CB-24-2/Varnish-2, 60oC

-1

log (ω/rad s ) CB-24-2/Varnish-1, 50oC 3

3

(d)

2

log (tan δ)

log (tan δ)

(c)

1

0 -1

2

1

0

-1

0

1

-1

2

log (ω/rad s )

-1

-1

0

1

log (ω/rad s-1)

2

Figure 25. Frequency dependence of the loss tangent, tan , of the CB24-2/Varnish-1 suspensions having various cCB. (a) 30C, (b) 40C, (c) 50C, and (d) 60C. The data were obtained in the linear viscoelastic regime (0 5%) at 30C. The symbols are the same as in Figure 22 [31].

5.4. Temperature Dependence Figure 25shows the  dependence of the loss tangent (tan ) data of the CB-24-2/Varnish-1 suspensions having various cCB at a temperature from 30C to 60C. For the suspensions with low cCB, the  dependence of tan  increases at low .But, for the suspensions with higher cCB, tan  decreases with lowering .The cCB exhibiting an -insensitive tan  value depends on temperature: cCB 20 wt% at 30C to about 15 wt% at 60C. These figures reveal that both the critical gel concentration crit. and critical relaxation exponent n change with

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temperature. To discuss the temperature dependence of crit. and n, it is necessary to estimate the exact values of crit. and n. Figure26 shows the tan  at different  versus  plots for the CB-24-2/Varnish-1 suspensions at different temperatures. From the -independent tan  point, crit. and n values can be determined. The crit.(cgel) value is 0.131 (22.0 wt%), 0.118 (20.1wt%), 0.106 (18.2%), and 0.086 (15.0wt%) at 30C, 40C, 50C, and 60C, respectively. The corresponding critical relaxation exponent n was n=0.71, 0.76, 0.80, and 0.83 at 30C, 40C, 50C, and 60C, respectively. The critical gel concentration shifts to low  and the corresponding tan  values became large with elevating temperature. Figure 27 shows temperature dependence of the critical exponent n for four kinds of CB/Varnish-1 suspensions. The critical exponent n increased with elevating temperature, irrespective of CB species.

Figure 26. Plots of tan  against CB volume fraction  for the CB-24-2 suspensions at various angular frequencies (0.1, 0.398, 1, 3.98, 10, 39.8 and 100 rad s 1). (a) 30C, (b) 40C, (c) 50C, and (d) 60C. crit. is the critical gel concentration [31].

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Figure 27. Temperature dependence of the critical exponent n for the four kinds of CB/Varnish-1 suspensions [31].

On the other hand, the critical gel concentration crit. decreased with elevating temperature. An increase in n and a decrease in crit. indicate that the CB agglomerate changed to a more open structure with increasing temperature. In Figure 28, the crit. is plotted against T. Sol-gel transition occurs at higher temperatures. The heat–induced gelation was found for the suspensions. The reason of this heat-induced gelation will be discussed in the next section. Here, we discuss about the difference of crit. for the CB species. The effective volume fraction eff can be evaluated from the comparison of the relative viscosity /0 for the CB suspension and hard-sphere suspensions, as was described in Section4.3 and will be explained in Section 6 in detail. The eff of CB-24-1 particles is 2.7 times larger than the bare volume fraction of the particles. Accordingly, the other three eff values can be estimated assuming that the effective volume fraction at the critical gel point is the same. Then, the eff/values are evaluated to be 2.7, 3.3, and 5.8 for the CB-151, CB-24-2, and CB-15-2, respectively. In Figure 29, the eff at the critical point is plotted against T. The data points are collapsed, irrespective of CB species. This result suggests that the critical gel concentration is universally determined according the effective volume. For the CB-76/Varnish-1 suspensions with various CB concentrations, cCB, Figure 30shows the  dependence of the loss tangent (tan ) data at temperatures T = 3060C. At low temperatures, T = 30 and 40C, the critical gel behavior are seen. However, the -independent tan  characteristic to the critical gel (Eq (2)) is not observed at 50C and 60C. Furthermore, the tan  versus  plots for various

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 values do not merge at a single point at any  value. Thus, the CB-76/Varnish-1 suspension does not exhibit the critical gelation behavior at 50C and 60C. The suspensions exhibit the G plateau at low  similar to that seen in medium of poor affinity toward the CB particles. Thus, this result suggests that the affinity of the Varnish-1 changes with T.

Figure 28. Change of the bare volume fraction at the gelation point crit. with temperature for the four kinds of CB/Varnish-1 suspensions [31].

Figure 29. Change of the effective volume fraction at the gelation point crit. with temperature for the four kinds of CB/Varnish-1 suspensions [31].

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Figure 30. Frequency dependence of the loss tangent, tan , of the CB-76/Varnish-1 suspensions having various cCB. (a) 30C, (b) 40C, (c) 50C, and (d) 60C [30].

5.5. Mechanism of Heat-Induced Gelation CB/Varnish-1 suspensions exhibit heat-induced gelation. This type of behavior has been reported only for a few systems [68] and is the opposite to the behavior of the majority of gelling systems. The heat-induced gelation should result from an enhancement of the attraction between CB particles with increasing T. A key factor for this enhancement can be found in Figure 31 where the G, G, and tan  data of the neat Varnish-1 in a wide range of T from 5C to 60C are plotted against a normalized frequency aT. (The data at 40C and 50C are not shown for clarity of the figure.) The frequency

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shift factor aT, being common for G, G, and tan , was determined to make the best superposition of the G data. As noted in Figure 31, the excellent superposition of the G plot is associated with systematic deviation of the G and tan  plot at different T. This result indicates a moderate but non-negligible failure of the time-temperature superposition for the neat Varnish-1. A careful examination reveals some details of this failure. Namely, G and tan  at a given normalized frequency aT decreases and increases, respectively, with increasing T. This feature is indicative of a relative decrease of the elasticity of Varnish-1 on heat (in the normalized frequency scale).

Figure 31. G, G, and tan  data of the neat Varnish-1 plotted against a normalized frequency aT. The frequency shift factor aT, being common for G, G, and tan , was determined to make the best superposition of the G data. Note the mild but non-negligible failure of the time-temperature superposition [30].

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The Varnish-1 includes a phenol-resin type polymeric component that sustains the elasticity. Thus, the relative decrease of the elasticity of Varnish-1 suggests a (small) decrease of the solubility of this component in the Varnish-1 with increasing T.(If the solubility is the same, the time-temperature superposition would have been valid for the neat Varnish-1.)The polymeric component should have had a delicate affinity toward the CB-76 particles having a partially oxidized surface. Thus, the decrease of the solubility of the polymeric component (at high T) would enhance adsorption of this component on the CB particles. Then, the adsorbed component would bridge neighboring particles to increase the attraction between the particles and enhance the gelation. The heat-induced gelation of the CB/Varnish-1 suspension appears to be attributable to this adsorption/bridging. It should be also noted that the time-temperature superposition fails for the CB/Varnish-1 suspensions even if the vertical shift of G and G is allowed. This failure appears to be a natural consequence of a change in the particle-particle interaction due to the adsorption of the polymeric component at high T.

5.6. Summary Linear viscoelastic properties of the CB/Varnish-1 suspensions with various CB concentrations cCB is reported at various temperatures T. The suspensions exhibit a sol-gel transition on an increase in cCB and the cgel value at the gelation point decreases with elevating T. This T dependence of cgel is the opposite to the gelation behavior of many suspensions studied earlier. This peculiar behavior of the CB/Varnish-1 suspensions is attributable to a small reduction of the affinity of Varnish-1 to CB particles. The critical relaxation exponent n increases with elevating T. This change shows the CB fractal structure changes from plane-like to straight-lie structure with elevating T. The cgel and n are characteristic values for the suspensions.

6. Viscoelastic Behavior of Well Dispersed Systems The CB-76 aggregates utilized in Section 3.4. are considerably larger and their terminal relaxation is too slow to be fully resolved in our experimental window even for the case of the random dispersion. Thus, the Brownian nature of that dispersion was not fully established. Concerning this problem, we have prepared the random dispersion of smaller CB aggregates to detect the terminal relaxation, as the Brownian relaxation becomes faster upon decrease of the

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aggregate size. For those suspensions, the terminal relaxation is indeed observed and its Brownian nature is confirmed for both relaxation time and mode distribution[29].

6.1. Viscoelastic Behavior of CB/Varnish-2 Suspensions A carbon black sample CB-24-1 (Furnace Black, mean particle size: 24 nm, DBP absorption value: 69 ml/100 g) was used. The average aggregate diameter, measured with Microtrac UPA (HONEYWELL), is a = 120 nm. Varnish-2 was utilized as the suspending medium. The viscosity is slightly different for the Varnish-2 utilized in this and previous studies because of a difference in the supplier’s batch. However, the chemical affinity is indistinguishable for the presently and previously used Varnishi-2.The G and G as a function of angular frequency  were measured at a temperature from 30C to 60C. The oscillatory strain amplitude 0 was kept small (0= 5%), and the linearity of the measured moduli was confirmed for this 0.Time-temperature superposition was valid at the temperature examined. Thus, the G and Gwere reduced at 30C to construct the master curves. For the CB-24-1/Varnish-2 suspensions with various CB concentrations cCB, Figure 32shows the master curves of G (a) and G(b) reduced at 30C. The solid lines present the behavior of the medium (Varnish-2).Figure 33 shows the shift factor aT for the suspensions and medium utilized for constructing the master curves. In the entire range of our experimental window, the medium shows the terminal tails (G2 and G) and behaves as a viscous fluid having G G; see Figure 32.For the CB suspensions, a slow relaxation process is clearly observed through the G that is comparable, in magnitude, with the G at intermediate .At sufficiently low , G becomes much larger than G and the terminal relaxation behavior prevails for the suspensions. Specifically, the terminal tails (from which the relaxation time is evaluated) are observed for the suspensions having cCB = 10 and 20 wt%. These tails are successfully detected because the CB aggregates utilized in this study are considerably smaller than the previously used aggregates. For the suspensions, we also note that G at high  is much larger than G and proportional to . This behavior indicates that the suspensions possess a fast viscous mode (in addition to the slow viscoelastic mode) characterized with a high-frequency viscosity, G/.

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It should be emphasized that the slow viscoelastic mode and fast viscous mode of the suspensions obey the time-temperature superposition with the Arrhenius-type activation energy E identical to that for the medium (E  58 kJ/mole); see Figure 33. This result indicates that the particle/aggregate structure in the suspensions does not change with the temperature T and the motion of the aggregates is governed by the friction from the medium. In other words, the suspensions include no agglomerate structure that changes with T. In fact, the lack of agglomerates in similar CB-24-1/Varnish-2 suspensions was previously confirmed from an electron microscopic observation.

Figure 32. Frequency dependence of the storage (a) and loss (b) moduli of the CB-241/Varnish-2 suspensions reduced at 30C. The solid lines indicate G and G of the medium [29].

For agglomerating CB suspensions having relatively large cCB, Trappe and Weitz[23] reported that the G and G versus  plots for various cCB values can be double-logarithmically shifted to be simultaneously superposed on respective master curves and that the scaling of the dynamic moduli represented by this superposition results from a similarity of the agglomerated networks at various cCB. The master curve of G is characterized with a low- plateau that reflects the rigidity of the agglomerated network. In contrast, for our CB-24-1/Varnish-2 suspensions, Figure 32demonstrates that the G and G data for various cCB values cannot be simultaneously superposed and the G data exhibit no low- plateau.

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The lack of agglomerates in the CB-24-1/Varnish-2 suspensions is also noted from this result.

Figure 33. Temperature dependence of the shift factor a T of the CB-24-1/Varnish-2 suspensions and the medium [29].

6.2. Similarity with Hard Sphere Suspensions The viscoelastic features of CB-24-1/Varnish-2 suspensions are similar to those of the so-called hard-sphere suspensions such as silica/ethylene glycol suspensions by Shikata and Pearson [48].In the hard-sphere suspensions, the particles have no long-range potential and are randomly/isotropically dispersed at equilibrium. This particle distribution is anisotropically distorted on application of strain, and the configurational entropy of the particles decreases accordingly. This decrease gives rise to the viscoelastic stress that relaxes when the particles undergo translational diffusion (Brownian motion) to recover their isotropic distribution [4857]. Thus, from the above similarity, the terminal relaxation of CB-24-1/Varnish-2 suspensions is expected to reflect the diffusion of the CB-24-1 aggregates. We test this expectation for the relaxation time and relaxation mode distribution.

6.3. Terminal Relaxation Time From the G and G data of the CB suspension (Figure 32), terminal relaxation time of the suspensions is evaluated as [48, 56]

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G/20/G/0

(6)

Here, GG is the loss modulus due only to the slow viscoelastic mode (obtained by subtracting a contribution  of fast viscous mode).The  values are summarized in Table 4 together with the high-frequency viscosity evaluated as G/at aT 10  100 s1

(7)

For the hard-sphere particles, a characteristic time (Peclet time) P for diffusion over a distance  particle radius a is expressed as [48] Pa3/kBT

(8)

Here, kB is the Boltzmann constant and T is the absolute temperature. The  gives an effective viscosity for the particle diffusion under the hydrodynamic interaction. For our CB-24-1/Varnish-2 suspensions, we regard the covalently-fused CB24 aggregates as the hard-spheres and utilize their average diameter, 120 nm, as the hard-sphere diameter, 2a. The P values can be evaluated from this a value ( 60 nm) and  data (Table 4). The results are summarized in Table 4.Clearly, the calculated P is close to the measured . Thus, we can attribute the slow relaxation of CB-24-1/Varnish-2 suspensions to the Brownian diffusion of the CB aggregates. A similar assignment has been made for various suspensions containing isotropically/randomly dispersed particles in a quiescent state [48, 56]. In the above evaluation of P, we have utilized the experimentally determined  (Eq. (7)). Here, we utilized a  calculated for ideal hard-sphere suspensions to further test if the terminal relaxation of our CB suspensions is attributable to the diffusion. For the ideal hard-sphere silica suspensions,  is uniquely determined by the particle volume fraction  and medium viscosity m[48]; mf() where the function f() is close to the predictions of the Beenakker theory [81]. Utilizing this  in Eq. (4), we find pma3f()/kBT

(9)

The function f() gradually increase with increasing  and is well approximated by [82]

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(10)

For our CB aggregates having non-spherical and non-uniform shape, the bare volume fraction  included in Eqs. (5) and (6) is to be replaced by an effective volume fraction eff. (This replacement corresponds to the use of the aggregate diameter as the particle diameter 2a in Eq. (8)). The eff is estimated below. For mono-disperse hard-core silica particles, the zero-shear viscosity 0 normalized by the medium viscosity m is universally dependent on  irrespective of the particle radius [48, 52]. The eff of our CB aggregates is estimated by plotting the 0/m data of the CB aggregates on the universal 0/mvs curve obtained for the hard-core silica particles. The result is shown in Figure 34. From the abscissa coordinates of the plot for the CB aggregates (filled circles), the effective volume fraction eff is found to be 2.7 times larger than the bare volume fraction  (cf. Table 4 where eff and  are summarized). This result suggests that the CB aggregates approximately behave as an equivalent hard-sphere having eff=2.7 and the difference between eff and  reflects the non-spherical shape and the shape distribution of the aggregates.

Figure 34. Plots of normalized zero-shear viscosity 0/m of the CB-24-1/Varnish-2 suspensions against the effective volume fraction eff (filled circles). The eff values were determined in a way that the 0/m versus eff plots for CB-24-1 aggregates are superposed on the universal 0/m versus  plots obtained for unimodal hard-core silica particles (unfilled circles)[29].

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Table 4. Composition and rheological data for the CB/Varnish-2 suspensions at 30C [29] eff 0

m/(Pa s) 8.2100

/(Pa s)a

/sb

p/sc

p/sd

0

 0



10

0.058

0.157



1.3101

 1.9

 2.1

 1.8

20

0.122

0.329



2.1101

5.1

3.3

2.9

30 0.192 0.518 Evaluated from Eq. (7) b Evaluated from Eq. (6) c Evaluated from Eq. (8) d Estimated from Eqs. (9) and (10)



1



6.2

7.9

cCB/wt%

3.810

a

From this eff together with the a (= 60 nm) and m (= 8.2 Pa s), the diffusion time P is calculated from Eqs. (9) and (10). The results are summarized in Table 4. This P is close to that calculated from Eq. (8) and also to the  data. This result lends further support to our assignment of the terminal relaxation of the CB suspensions to the aggregate diffusion.

6.4. Relaxation Mode Distribution Shikata and Pearson [48] found that the silica suspensions of various a and  exhibit an almost identical viscoelastic mode distribution. Namely, for the hardsphere silica suspensions, the G normalized by the terminal relaxation intensity Ht is universally dependent on a normalized frequency P with P being the Peclet time given by Eq. (8). We examine if this universality holds for our CB-24/Varnish-2 suspensions. In principle, this universality test is to be made for the modulus GCB due to only CB aggregates; GCB()  Gs() KGm() where Gs and Gm denote the moduli of the suspension and medium, respectively, and K is a factor representing the filler effect ( K  1  2.5eff 14.1eff2;[83, 84]). However, for our suspensions containing relatively small CB aggregates, Gs() is much larger than KGm(). Thus, we make the universality test for the raw Gs() data (without subtracting the medium contribution KGm()). For evaluation of the terminal intensity Ht given by a reciprocal of the steady state compliance Je, we focus on the feature of the hard-sphere silica suspensions. For the silica suspensions, Shikata and Pearson [48] demonstrated that the Je and

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the high-frequency modulus G exhibit a universal relationship irrespective of the  value: GJe 4

(11)

They also showed that the G data are satisfactorily described by an equation: G (0.78kBT/a3) (/m)2gc()

(12)

where gc() ( 1 /213 for  0.5) is a radial distribution function of the spheres in contact[85]. From Eqs. (11) and (12) together with Beenakker function f() ()/m, we evaluate the terminal relaxation intensity Ht = 1/Je as a function of .

Figure 35. Plots of normalized modulus G/Ht(eff) of the CB-24-1/Varnish-2 suspensions having various cCB against the normalized frequency P at 30C. Note that the plots for various cCB almost coincide with each other at low P [29].

Figure 35 shows plots of normalized modulus G/Ht(eff) of the CB-241/Varnish-2 suspensions having various cCB against the normalized frequency

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P. At low P, the plots for different cCB values almost coincide with each other. The normalization factors Ht and P were determined in a way applied for the silica hard-spheres, and this coincidence (similar to that for those hardspheres) was achieved without any adjustable parameters. This result is considered to lend an additional support for the assignment of the terminal relaxation of the CB suspensions to the aggregate diffusion.

6.5. Comparison of Effective Volume Fractions Eff/ for Three Types of CB Suspensions In Section 3, we explained the rheological properties of CB suspensions in three types of media. It was found that the important parameters to characterize the CB aggregates are the effective volume fraction eff of CB aggregates evaluated by plotting the relative viscosity r0/m (m: medium viscosity) on the universal r versus  curves obtained for the hard-core silica particles for CB/Varnish-2 and CB/(PS/DBP) systems, and the critical gel concentration crit. found for CB/Varnish-1 systems. The eff and crit values depend on DBP absorption value irrespective of the primary particle size, and are larger for the higher-structure CB with higher DBP absorption value. We compare three types of effective volume fractions [33]. Table 5summarizes the comparison of eff/and crit/values for three types of suspensions using various kinds of CB samples. The critical gel concentration is the same as the effective volume fraction. This result suggests that the critical gel concentration is universally determined according to the effective volume. The eff/and crit/values are almost the same irrespective of the suspending medium and depend strongly on DBP absorption value which is a measure of CB aggregate structure. Table 5. Comparison of eff/ for three types of CB suspensions [33] CB CB-76 CB-24-1 CB-24-2 CB-15-1 CB-15-2 AcB

DBP abs.(ml/100g) 71 69 110 74 147 190

eff/ (2) 2.7  2.7  

eff/ (2) 2.7 3.3 2.7 5.8 

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6.6. Summary The relaxation time  of the well-dispersed CB/Varnish-2 suspensions was found to be close to the characteristic time (Peclet time) P for the translational diffusion of the CB aggregates, and the terminal mode distribution is scaled in a way similar to that for the hard-sphere suspensions. These results strongly suggest that the behavior of the CB aggregates can be very well described as the behavior of equivalent hard-spheres and the terminal relaxation of the CB suspensions is governed by the aggregate diffusion. The effective volume fractions estimated for suspensions of various CB samples are almost the same irrespective of the suspending media and depend strongly on DBP absorption value which is a measure of CB aggregate structure.

7. Effect of Suspending Media on the Sol-Gel Transition Behavior In Section 7, we are focusing our attention on CB/varnish suspensions with different affinity toward the CB particles. Linear viscoelastic measurements of the CB/(Varnish-1/Varnish-2=60/40 in weight) and CB/(Varnish-1/AF=80/20 in weight) suspensions were conducted as a function of cCB and T. Here, AF is a mixture of saturated hydrocarbons and has a poor affinity toward the CB particles. The former medium has higher affinity and the latter has poorer affinity toward CB particles than Varnish-1, although these two media have almost the same viscoelastic properties. The critical relaxation exponent n and the critical gel concentration cgel of the suspensions are evaluated and discussed in relation to the medium affinity to the CB particles [34].

7.1. Viscoelastic Behavior of Two Suspending Media; Varnish-1/Varnish-2=60/40 and Varnish-1/AF=80/20 For Varnish-1, Varnish-2, and a mixed varnish of Varnish-1/Varnish-2=60/40 in weight, Figure 36 shows the  dependence of the linear viscoelastic modulus G (a) and G (b) at 30C. The component varnishes and the mixed varnish are transparent and exhibit the flow behavior characterized by the terminal tails, G2 and G at low . Such a behavior is similar to the viscoelastic properties of homogeneous polymer solutions. Figure 37 shows the  dependence of the linear

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viscoelastic modulus G (a) and G (b) for the Varnish-1/AF=80/20medium at various temperatures from 30C to 60C. This varnish also exhibits the flow behavior characterized by the terminal tails. The G and G values of the varnish at 30C are almost the same as those of Varnish-1/Varnish-2=60/40. Accordingly, Varnish-1/Varnish-2=60/40 and Varnish-1/AF=80/20 are selected as the media.

Figure 36. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of Varnish-1/Varnish-2 = 100/0, 60/40, and0/100 blends. The data were obtained at 30C under the strain amplitude 0 = 5% [34].

7.2. Behavior of CB-76/(Varnish-1/Varnish-2=60/40) Suspensions Having Various CB Concentrations (cCB) For the CB-76/(Varnish-1/Varnish-2=60/40) suspensions with various cCB, Figures 38 to 41 show the  dependence of G and G at temperatures T = 3060C. The corresponding loss tangent (tan ) data are shown in Figure 42. The behavior of CB-76/(Varnish-1/Varnish-2=60/40) suspensions is similar to that of the previously examined CB-76/Varnish-1 suspensions, namely, the fluid behavior characterized with the terminal tails (G2 and G at low ) seen our suspension at low cCB gradually changes with increasing cCB to the critical gel behavior characterized with a power-law, as shown in Eqs. (1) and (2). For the suspensions with low cCB, the  dependence of tan  increases at low . But, for the suspensions with higher cCB, tan  decreases with lowering . Specifically, Figure

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42suggests that the -independent tan  (characteristic of the critical gel) is observed at all temperatures measured. For more accurate determination of the critical gelation, we make plots of tan  against cCB (not shown here) for a set of  values covering the entire range of our experimental window and evaluate a cCB value where the plots for different  values merge to a point and the critical gelation occurs. This standard method gives a horizontal line in tan  versus  plot. The horizontal lines thus obtained are shown in Figure 42. The cgel is obtained to be 30, 28, 26, and 23 wt% and the corresponding critical exponent is n 0.89, 0.90, 0.91, and 0.92 at 30, 40, 50, and 60C, respectively. The critical gel concentration shifts to low cCB and the corresponding tan  values become large with elevating temperature. The n increases with elevating T. On the other hand, the cgel decreases with elevating T. An increase in n and a decrease in cgel indicate that the CB agglomerate changes to a more open structure with elevating T. In Section 5, we reported cgel 35 wt% and n 0.71 for the CB-76/Varnish-1 suspensions at 30C. The cgel value (30 wt%) of the CB-76/(Varnish-1/Varnish-2=60/40) suspensions is smaller than that of the CB-76/Varnish-1 suspensions. The n value (0.89) of the CB-76/(Varnish-1/Varnish-2=60/40) suspensions is larger than that of the CB76/Varnish-1 suspensions. These facts indicate that the CB/varnish suspensions having higher medium affinity toward CB particles have a coarser fractal structure. Moreover, the increase in n with T indicate that the CB aggregate structure changes from plate-like to more straight with elevating T.

Figure 37. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of Varnish-1/AF blend. The data were obtained at 30C under the strain amplitude 0= 5% [34].

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Figure 38. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-76/(Varnish-1/Varnish-2 = 60/40) suspensions having various cCB. The data were obtained at 30C under the strain amplitude 0= 5% [34].

Figure 39. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-76/(Varnish-1/Varnish-2 = 60/40) suspensions having various cCB. The data were obtained at 40C under the strain amplitude 0= 5% [34].

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Figure 40. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-76/(Varnish-1/Varnish-2 = 60/40) suspensions having various cCB. The data were obtained at 50C under the strain amplitude 0= 5% [34].

Figure 41. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-76/(Varnish-1/Varnish-2 = 60/40) suspensions having various cCB. The data were obtained at 60C under the strain amplitude 0= 5% [34].

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Figure 42. Frequency dependence of the loss tangent, tan , of the CB-76/(Varnish-1/ Varnish-2= 60/40) suspensions having various cCB under the strain amplitude 0= 5%. (a) 30C, (b) 40C, (c) 50C, (d) 60C. The symbols are the same as in Figures 38, 39, 40, and 41 [34].

7.3. Behavior of CB-76/(Varnish-1/AF=80/20) Suspensions Having Various CB Concentrations (cCB) For the CB-76/(Varnish-1/AF=80/20) suspensions with various cCB, the  dependence of G, G, and tan  was measured at T = 3060C. Figure 43 shows the  dependence of G and G of the suspensions at 40C. Figure 44shows the tan  data at T = 3060C. The CB-76/(Varnish-1/AF=80/20) suspensions exhibits the similar behavior as the previously examined CB-76/Varnish-1 suspensions.

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The  dependence of tan  increases with lowering  at low cCB and decreases with lowering  at higher cCB. Specifically, Fig 44 suggests that the independent tan  (characteristic of the critical gel) is observed at all temperatures measured. The cgel and n can be calculated using the standard method explained in Section 5. The cgel is obtained to be 31, 32, 32, and 31 wt% and the corresponding critical exponent n is 0.70, 0.76, 0.80, and 0.84 at 30, 40, 50, and 60C, respectively. The cgel is found to be independent of T. On the other hand, the n values increases with elevating T. The critical gelation behavior with a similar cgel and n is found also for the CB/Varnish-1 suspensions; cgel 37 wt% and n 0.71 for the CB-76/Varnish-1 suspensions at 30C. The cgel (32 wt%) and n (0.70) values of the CB-76/(Varnish-1/AF=80/20) suspensions are similar to those of the CB-76/Varnish-1 suspensions, respectively. These facts indicate that the CB76/Varnish-1 and CB-76/(Varnish-1/AF=80/20) suspensions have almost the same fractal structure, although the media have different viscosity and affinity toward CB particles.

Figure 43. Frequency dependence of the linear viscoelastic moduli G (a) and G (b) of the CB-76/(Varnish-1/AF = 80/20) suspensions having various cCB. The data were obtained at 40C under the strain amplitude 0= 5% [34].

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The n values of the CB suspensions depend strongly on the medium. Using Eq. (4) derived for chemical gels, mass fractal dimension of the CB suspensions was calculated and listed in Table 6.The experimental error for the n is 0.03 for all the suspensions. For the CB-76/(Varnish-1/AF=80/20) suspensions, the mass fractal dimension is about 2.It means that the suspensions consist of a plane-like structure. In contrast, for the CB-76/(Varnishi-1/Varnish-2=60/40) suspensions, the mass fractal dimension is smaller than those of the CB-76/(Varnish1/AF=80/20) suspensions and approaches to df=1, suggesting that the distribution of the particles is the straight line. We would like to emphasize that the critical relaxation exponent n depends on T and medium affinity. This is a big difference between the rheology of polymer gels and flocculated particulate gels.

Figure 44. Frequency dependence of the loss tangent, tan , of the CB-76/(Varnish-1/AF = 80/20) suspensions having various cCB under the strain amplitude 0 = 5%. (a) 30C, (b) 40C, (c) 50C, (d) 60C. The symbols are the same as in Figure 43 [34].

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Table 6. Experimental values of critical gel concentration cgel, critical relaxation exponent n, and mass fractal dimension df for the CB-76/(Varnish1/Varnis-2=60/40) and CB-76/(Varnish-1/AF=80/20) suspensions [34] T/C 30 40 50 60

CB-76/(V-1/V-2=60/40) cgel/wt% n df 30 0.89 1.37 28 0.90 1.33 26 0.91 1.30 23 0.92 1.26

CB-76/(V-1/AF=80/20) cgel/wt% n df 31 0.70 2.29 32 0.76 1.95 32 0.80 1.75 31 0.84 1.57

7.4. Summary The mixtures of Varnish-1/Varnish-2=60/40 and Varnish-1/AF=80/20 with the same viscosity and different affinity toward CB-76 particles were used as a suspending medium. The former medium has higher affinity and the latter has poorer affinity toward CB particles than Varnish-1, although these two media have almost the same viscoelastic properties. Both suspensions exhibit a sol-gel transition on an increase in cCB. For the CB-76/ (Varnish-1/Varnish-2=60/40) suspensions, the cgel value at the gelation point is lower and the critical relaxation exponent n is higher than those of the CB-76/Varnish-1 suspensions. In contrast, for the CB-76/(Varnish-1/AF=80/20) the cgel and n, respectively, are similar to those of the CB-76/Varnish-1 suspensions. These results strongly suggest that the cgel and n are attributable to the medium affinity but not to the medium viscosity, and the CB fractal structure changes from plane-like to straight-line structure with increasing medium affinity. For weakly flocculated gels, the critical relaxation exponent n, in other words, fractal dimension df, depends on temperature and medium affinity.

8. Heat-Induced Gelation and Its Reversibility CB/Varnish suspensions exhibit reversible heat-induced gelation. However, in our previous studies, we reported the viscoelastic properties at the temperature range from 30C to 60C. The reason why the temperature was limited to this range is for the irreversible T response above 60C. In this section, the viscoelastic properties of CB/Varnish-1/Varnish-2=60/40) suspensions are explained in the temperature range from 30C to 80C. The G and Gexhibit an irreversible change

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above 60C. Accordingly, we have been focusing our attention on this temperature range and shearing history of CB/Varnish suspensions. Investigation of these histories of the CB suspensions should result in a much deeper understanding of the rheology of CB suspensions. Three CB concentration samples (cCB=0, 20, and 35 wt%) in the Varnish-1/Varnish-2=60/40are used. The critical gel concentration of the suspension was 30 wt% at 30C. Therefore, the 20 wt% sample is in a sol state and the 35 wt% sample is in a gel state at 30C. Some rheological parameters reported here are non-equilibrium values, intended to demonstrate general trends in the structure of the suspensions [35]. CB-76/(V-1/V-2=60/40)=0/100 4

log (G', G"/Pa)

3

None-Preshear, Strain = 5%  = 1 rad s-1, Parallel : 50mm  o : G' 30 oC to 80 ooC : G" 30 oC to 80 oC : G' 80 oC to 30 oC : G" 80 C to 30 C

2 1 0 -1 -2 20

30

40

50

60

70

80

90

o

T/ C Figure 45. Temperature dependence of the storage (G) and loss (G) shear moduli of the Varnish-1/Varnish-1 = 60/40 solution. (G, filled circle; heating, filled upright triangle; cooling, G, open circle; heating, open upright triangle; cooling) [35].

8.1. Viscoelastic Behavior of Varnish-1/Varnish-2=60/40 Solution Figure 45 shows the T dependence of the G and G for the Varnish1/Varnish-2=60/40 solution at =1 rad s1, when T was heated from 30C to 80C and then cooled down from 80C to 30C. It is clear that both the G and Gof the solution change reversibly with T irrespective of the T history. Figure 46 shows the  dependence of the G and G for the Varnish-1/Varnish-2=60/40 solution at

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30C. The viscoelastic behavior of the solution approached to the flow behavior characterized by the terminal tails, G2 and G at low . Such a behavior is similar to the viscoelastic properties of homogeneous polymer solutions. These facts confirm that there are no significant chemical changes during heating from 30C to 80C.

Figure 46. Frequency dependence of the storage (G) and loss (G) shear moduli of the Varnsih-1/Varnish-2 = 60/40 solution at 30C. (open circle; before temperature sweep, open upright triangle; after temperature sweep) [35].

8.2. Viscoelastic Behavior of CB-76/(Varnish-1/Varnish-2=60/40) Suspensions Figures 47and 48 show the T dependence of the G and G for the CB76/(Varnish-1/Varnish-2=60/40)=20/80 and 35/65 suspensions at =1 rad s1, when the suspensions were heated up from 30C to 80C and cooled down from 80C to 30C. Below 60C, both suspensions exhibit T reversible behavior. However, above 60C, the G and G values increase with T at heating for the 20/80 and 35/65 suspensions. When the suspensions are cooled from 80C to 30C, the higher G and Gare observed. This irreversible behavior is much different from that of the medium solution. Therefore, the data suggest that there

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is a significant amount of association among the CB particles, forming a network structure above 60C.

Figure 47. Temperature dependence of the storage (G) and loss (G) shear moduli of the CB-76/(Varnish-1/Varnish-2=60/40) = 20/80 suspension. (G, filled circle; heating, filled upright triangle; cooling, G, open circle; heating, open upright triangle; cooling) [35].

Figures 49 and 50 show the dependence of the G and G for the CB-76/ (Varnish-1/Varnish-2=60/40)=20/80 and 35/65 suspensions at 30C before and after the T sweep measurements. These figures include the G and G of the suspensions pre-sheared again after the T sweep measurements. It is seen in Figure 49that, before the T sweep measurement, the slope of G is about 1 and that of G is slightly smaller than 1 for the CB-76/(Varnish-1/Varnish-2=60/40)=20/80 suspension. This fact confirms that the suspension is in a sol state at 30C, as the suspension is in a sol state when the slope is larger than the critical relaxation exponent n and is in a gel state when the slope is smaller than the n, and the n of CB-76/(Varnish-1/Varnish-2=60/40) suspensions is found to be 0.89 in the previous section, However, the G and G increase significantly after the T sweep measurement. They exhibit weaker  dependence and the slope of G became

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about 1/2 at low . The increase in Gand G indicates a heat-induced gelation, undergoing a transition from fluid-like to solid-like behavior. As shown in Figure 49, the G and G curves upon shearing after the T sweep, respectively, recover to the same curves before the T sweep measurement. This reversibility of the liquidlike to solid-like transformation is dependent on the history of the sample. It should be emphasized that the heat-induced gelation of the suspension is recovered upon shearing after the T sweep measurement.

Figure 48. Temperature dependence of the storage (G) and loss (G) shear moduli of the CB-64/(Varnish-1/Varnish-2 = 60/40) = 35/65 suspension. (G, filled circle; heating, filled upright triangle; cooling, G, open circle; heating, open upright triangle; cooling) [35].

For the CB-76/(Varnish-1/Varnish-2=60/40)=35/65 suspension, the slope of G in double logarithmic plot is about 0.5.This indicates that the suspension is in a gel state at 30C, as reported previously. The G and G values increase significantly after the T sweep measurement, as same as those of the 20/80 suspension. The dispersion state of the CB particles in the suspension becomes more agglomerated. The agglomerated state is held until the suspension is subsequently sheared after cooled at 30C. The CB/Varnish suspension systems can be characterized by slopes of the G versus  and G versus  curves in the logarithmic plots.

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Figure 49. Frequency dependence of the storage (G) and loss (G) shear moduli of the CB-76/(Varnsih-1/Varnish-2 = 60/40) =20/80 suspension at 30C. (open circle; before temperature sweep, open upright triangle; after temperature sweep, open square; upon shearing after temperature sweep) [35].

8.3. Probable Mechanism of the Observations The changes in the rheological properties of the CB suspensions upon heating are consistent with a change in structure. The agglomerated structure formed in the CB/varnish suspensions should result from polymer-particle interactions, because the rheological properties of the varnish do not indicate significant chemical changes upon heating. We propose a possible mechanism of our new observations according to Section 5.As discussed previously, the heat-induced gelation of the CB/Varnish-1 suspensions can be related to a polymeric component included in the Varnish-1.The solubility of this component in the Varnish-1 would decrease slightly with increasing T, suggesting that this solubility change causes an enhancement of the attraction between CB particles. Below 60C, the rheological behavior of the CB-76/Varnish-1 suspensions is apparently reversible, because the attraction is not so large. However, the solubility of the polymeric component in the varnish continues to decrease with increasing T above 60C. Our new observations indicate that the G and G

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increase irreversibly with increasing T, indicating that the structure evolved to be more strongly agglomerated at elevating T. One possibility to account for the T dependence of the rheology of this suspension system is reduced effectiveness of the polymeric component due to the T-dependent changes in the solubility. This less solvated polymeric component would have been adsorbed more and more on the CB particles, thereby allowing the agglomeration of the particles above 60C. Even the small solubility change must induce the large change in CB structure and the irreversibility of the rheology for the CB/Varnish suspensions. The presence of the adsorbed polymer, however, prevents intimate contact among the CB particles so that the agglomerated structure is easily destroyed upon subsequent cooling and shearing. Our experimental observations suggest that the adsorption of polymer component in the varnish results in the suspensions that are metastable against agglomeration.

Figure 50. Frequency dependence of the storage (G) and loss (G) shear moduli of the CB-76/(Varnsih-1/Varnish-2= 60/40) = 35/65 suspension at 30C. (open circle; before temperature sweep, open upright triangle; after temperature sweep, open square; upon shearing after temperature sweep) [35].

Very recently, Uematsu et al. [79, 80] studied the rheological properties of SiO2/(acrylic polymer/epoxy) suspensions. They found that the suspensions also exhibit heat-induced gelation. They suggest that the heat-induced gelation results from an enhancement of the attraction among SiO2 particles with increasing T,

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thereby allowing the gelation due to physical interaction between SiO2 and polymeric components. Therefore, it is considered that such a heat-induced gelation should be a universal feature of weakly attracted particles consisting of particle/polymer solution systems.

8.4. Summary Linear viscoelastic properties of CB suspensions in a mixture of a rosinmodified phenol resin-type varnish (Varnish-1)/an alkyd resin-type varnish (Varnish-2), which exhibit a sol-gel transition on an increase in CB concentration, were investigated from 30 C to 80C. The viscoelastic properties are reversible from 30C to 60C. In contrast, at temperature above 60C, the G and Gare irreversible and increase significantly with increasing temperature. These increases in the moduli are due to a change of the dispersion state to agglomerated state by heating. The agglomerated structure of CB particles is hold with cooling. However, the G and G recover to the original values upon shearing. The heatinduced gelation can be related to a polymeric component in the varnish. At high T, the less solvated polymeric component is adsorbed on the CB particles, thereby allowing the agglomeration of the particles. This heat-induced gelation should be a universal feature for suspensions of weakly attractive particles.

Conclusion In this article, the viscoelastic properties of CB suspensions have been described. The viscoelastic properties of the suspensions are very sensitive to the structural change. Accordingly, different types of rheological behavior are observed, when the medium affinity toward the CB particles is changed. In the low affinity medium, three-dimensional network structure is formed. This suspension exhibits highly nonlinear, elasto-plastic features. On the other hand, in high affinity medium, the viscoelastic features are similar with hard sphere suspensions. The above two types of suspensions have been studied extensively. Our most interesting behavior is the case of a moderate affinity. The suspensions exhibit a sol-gel transition with change of CB concentration and temperature. This critical gel behavior of suspensions has not been fully elucidated. We believe that the viscoelastic properties of CB suspensions give particle design for developing new products. The measurements of viscoelastic properties are a powerful tool in the characterization of CB suspensions.

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Mullins L (1950) J. Phys. Colloid Chem. 54:539. Mullins L, Whorlow RW (1951) Tran. Inst. Rubber Ind. 27:55. Vinogradov GC, Malkin AY, Plotnikova E, Sabsai OY, Nikolayeva NE (1972) Int. J. Polym. Mater. 2:1217. Lobe VM, White JL (1979) Polym. Eng. Sci. 19:617 Montes S, White JL (1982) Polym. Eng. Sci. 55:1359. Montes S, White JL, Nakajima N, Weissert FC (1988) Adv. Polym. Technol. 8:431. Montes S, White JL, Nakajima N J. Non-Newtonian Fluid Mech. 28:183. Gandhi K, Salovey R (1988) Polym. Eng. Sci. 28:877. Shin KC, White JL, Brzoskowski R, Nakajama N (1990) Kautsch. Gummi Kunstst. 43:181. Montes S, White JL (1993) J. Non-Newtonian Fluid Mech. 49:277. Osanaiye GJ, Leonov AI, White JL (1993) J. Non-Newtonian Fluid Mech. 49:87. Seno M (1957) Nihon Kagaku Zasshi, 78:66, 70, 1374. Onogi S, Masuda T, Matsumoto T (1967) Nippon Kagaku Zasshi 88:854. Onogi S, Masuda T, Matsumoto T (1968) Nippon Kagaku Zasshi 89:464. Matsumoto T, Masuda T, Tsutsui K, Onogi S (1969) Nippon Kagaku Zasshi 90:360. Shoukens G, Mewis J (1978) J. Rheol., 22:381. Onogi S, Matsumoto, T (1981) Polym. Eng. Rev., 1:45. Amari, T, Watanabe, K (1983) Polym. Eng. Rev., 3:277. Amari T, Watanabe K (1986) Nihon Reoroji Gakkaishi 14:37. Amari T, Watanabe K (1990) J. Rheol. 34:207. Hayashi T, Morita K, Tateiri M, Amari T (1994) J. Jpn. Soc. Colour Mater. 67:80. Amari T, Uesugi K, Suzuki H (1997) Prog. Org. Coat. 31:171. Trappe V, Weitz DA (2000) Phys. Rev. Lett. 85:449. Kawaguchi M, Okuno M, Kato T (2001) Langmuir 17:6041 Lin Y, Smith TW, Alexandridis P (2002) Langmuir 18:6147. Won Y-Y, Meeker SP, Trappe V, Weitz DA, Digga NZ, Emert JI (2005) Langmuir 21:924. Hagerstrom J (1998) In Polymers, Laminations and Coatings Conference p 553. Aoki Y, Hatano A, Watanabe H (2003) Rheol. Acta, 42:209. Aoki Y, Hatano A, Watanabe H (2003) Rheol. Acta, 42:321.

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Aoki Y, Watanabe H (2004) Rheol. Acta, 43:390. Aoki Y (2007) Colloids Surf. A 308:79. Aoki Y (2008) J. Appl. Polym. Sci. 108:2660. Aoki Y (2008) The XVth Inter. Congress Rheol. p99. Aoki Y (2011) Rheol. Acta 50:779. Aoki Y (2011) Rheol. Acta 50:787. Iller KR (1971) J. Colloid Interface Sci., 37:373. Otsubo Y (1992) J. Colloid Interface Sci., 153:584. Otsubo Y (1993) J. Rheol., 37:799. Winter HH, Chambon F (1986) J. Rheol., 30:367. Chambon F, Winter HH (1987) J. Rheol., 31:683. In M, Prud’homme PK (1993) Rheol. Acta, 32:556. Li L, Aoki Y (1997) Macromolecules 30:7835. Rueb CJ, Zukoski CF (1998) J. Rheol., 42:1451. Borchard W (1998) Ber Bunsenges Phys. Chem., 102:1580. Jokinen M, Gyorvary E, Rosenholm JB (1998) Colloids Surf. A, 141:205. Tokumoto MS, Santilli CV, Pulcinelli SH (2000) J. Non-Cryst. Solids, 273:116. Watanabe H, Matsumiya Y, Kakiuchi M, Aoki Y (2001) Nihon Reoroji Gakkaichi (J. Soc. Rheol. Jpn.), 29:77. Shikata T, Pearson DS (1994) J. Rheol., 38:601. Mellema J, de Kruif CG, Blom c, Vrij A (1987) Rheol. Acta 26:40. Mellema J, van der Werff JC, Blom C, de Kruif CG (1989) Phys. Rev. A 39:3695. Russel WB, Saville DA, Schowalter WR (1989) Colloid dispersion. Cambridge University Press, London. van der Werff JC, de Kruif CG, Blom C, Mellema J (1989) Phys. Rev. A 39:795. Brady JF (1993) J. Chem. Phys. 99:567. Bender JW, Wagner NJ (1995) J. Colloid Interface Sci. 172:171. Bender JW, Wagner NJ (1996) J. Rheol. 40:899. Watanabe H, Yao ML, Yamagishi A, Osaki K, Shikata T, Niwa H, Morishima Y (1996) Rheol. Acta 35:433. Watanabe H, Yao ML, Osaki K, Shikata T, Niwa H, Morishima Y (1999) Rheol. Acta 38:2. Casson N (1959) Rheology of Disperse Systems; Pergamon Press: New York, p 84. Larson RG (1999) The Structure and Rheology of Complex Fluids; Oxford University Press; New York.

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Yuji Aoki Grant MC, Russel WB (1993) Phys. Rev. E 47:2606. Rueb CJ, Zukoski CF (1997) J. Rheol. 41:197. Yanez JA, Laarz E, Bergstrom L (1999) J. Colloid Interface Sci. 209:162. Wu H, Morbidelli M (2001) Langmuir 17:1030. Yorn Y-Y, Meeker SP, Trappe V, Weitz DA (2005) Langmuir 21:924. Woutersen ATIM, de Kruif CG (1991) J. Chem. Phys. 94:5739. Woutersen ATIM, Mellama J, Blom C, de Kruif CG (1994) J. Chem. Phys. 101:542. Bergstroem L, Sjoestroem E (1999) J. Eur. Ceram. Soc. 19:2117. Horn JA, Patterson BR (1997) J. Am. Ceram. Soc. 80:1789. Winter HH, Mours M (1997) Adv. Polym. Sci. 134:165. Cates ME (1985) J. Phys. 46:1059. Muthukumar M (1985) J. Chem. Phys. 83:3161. Muthukumar M (1989) Macromolecules 22:4656. Muthukumar M, Winter HH (1986) Macromolecules 19:1284. Martin JE, Adolf D, Wilcoxon JP (1987) Phys. Rev A 36:1803. Martin JE, Adolf D, Wilcoxon JP (1989) Phys. Rev. A 39:1325. Martin JE, Adolf D (1991) Annu. Rev. Phys. Chem. 42:311. Hess W, Vilgis TA, Winter HH (1988) Macromolecules 21:2536. Ponton A, Barboux-Doeuff S, Sanchez C (2000) Colloids Surf. A 162:177. Uematsu H, Aoki Y, Sugimoto M, Koyama K (2010) Rheol. Acta 49:299. Uematsu H, Aoki Y, Sugimoto M, Koyama K (2010) Rheol. Acta 49:1187. Beenakker CWJ (1984) Physica A 128:48. Lionberger RA, Russel WB (1994) J. Rheol. 38:1885. Guth E, Gold O (1938) Phys. Rev. 53:322. Guth E (1945) J. Appl. Phys. 16:20. Carnahan NF, Starling KE (1969) J. Chem. Phys. 51:635.

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In: Advances in Chemistry Research. Volume 21 ISBN: 978-1-62948-742-7 Editor: James C. Taylor, pp. 97-147 © 2014 Nova Science Publishers, Inc.

Chapter 3

CHARACTERIZATION OF CARBON BLACKS BY HIGH RESOLUTION N2 ADSORPTION ISOTHERMS FROM P/PO=10-7 TO P/PO=0.998: APPLICATION OF STANDARD S DATA TO ANALYSIS OF MICROPOROSITY OF ACTIVATED CARBONS Kazuyuki Nakai, Yoko Nakada, Masako Hakuman, Joji Sonoda, Masayuki Yoshida and Hiromitsu Naono BEL-Japan INC, Haradanaka, Toyonaka-shi, Osaka, Japan

Abstract The N2 adsorption isotherms on nongraphitized and graphitized carbon blacks (CBs) measured in a very wide pressure range of p/po=10-7–0.998 at 77.4 K were separated into three p/po regions; p/po=10-4–0.50, p/po=0.900–0.998, and p/po=107 –10-2. The separated isotherms were analyzed in detail. BET analysis of the N2 isotherms at p/po=10-4–0.50 was performed for nongraphitized CB and graphitized CB. It has been clarified that selection of an appropriate BET plot range is of great importance to obtain a reliable surface area (ABET) and BET constant (CBET). The best-fitted BET plot ranges (cf. Table 2) are different from the usual predetermined BET plot ranges (p/po=0.05–0.30). 

E-mail address: kazu@nippon‐bel.co.jp (Corresponding author)

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Kazuyuki Nakai, Yoko Nakada, Masako Hakuman et al. ABET obtained was used in calculation of the average size of the CB particles CBparticle (primary particle) (e.g. d av =39 nm for sample LGC2102-CB). BJH analysis of the N2 isotherms at p/po=0.900–0.998 at 77.4 K was carried out to obtain the pore size distribution (PSD) in the CB aggregates. The N2 isotherms at 77.4 K by the CB adsorbents have a distinct hysteresis at p/po=0.920–0.998. The  s plots in the p/po range 0.975–0.998were used to estimate the approximate CBaggregate size of the CB aggregates (secondary particle)(e.g., d av =900 nm for sample LGC2102-CB). Analysis of the N2 isotherms at very high p/po gives the important information about macroporosity of CB aggregates. The surface characters of CB particles were investigated using the N2 isotherms at p/po=10-7– 10-2 at two temperatures of 77.4 and 87.3 K. When log(n/n0.4) is plotted against log (p/po), the linear line can be obtained at very low relative pressure ranges below p/po=2  10-5 (cf. Figure 13). Its slope is closely related to the extent of surface homogeneity. The slope by sample Carbotrap-F-CB gives 1.00, and its N2 isotherm strictly obeys Henry‟s law, (n/n0.4)=kH  (p/po). In other words, the surface of Carbotrap-F-CB is highly uniform. The slope decreases from 1.00 with

increase of surface heterogeneity. Isosteric heats of adsorption of N2 ( q st ) on Carbotrap-F-CB were calculated using the N2 isotherms at 77.4 and 87.3 K. From a comparison of the qst–  data for Carbotrap-F-CB with those for nongraphitized (#51-CB) and graphitized carbon black (3845-1-CB) previously reported (cf. Figure 15), it is found that surface homogeneity decreases in the order CarbotrapF-CB3845-1-CB #51-CB. This conclusion is completely consistent with those from the CBET-values and the slope mentioned above. After characterization of CB adsorbents had been finished, the standard  s data of N2 by Carbotrap-FCB (77.4 K and 87.3 K) and by #51-CB (77.4 K) were given in Tables 5–7. These standard  s data can be utilized for investigations of surface area and porosity of unknown carbon adsorbents. In the present work, the standard  s data of N2 by nongraphitized carbon black #51-CB (Table 7) were applied to analysis of microporosity of activated carbons (ACF, SAC). The detailed  s plots made it possible to estimate the volume of ultramicropores (pore width, wd<0.8 nm) and supermicropores (wd>0.8 nm) for ACF and SAC adsorbents. Some comments have given about BET surface area of SAC.

Introduction The N2 adsorption isotherms at its boiling point (77.4 K) have been extensively utilized in the investigations of surface characters of carbon blacks (CB) and microporous textures of activated carbons (AC) and activated carbon fibers (ACF) [1–5]. When measurements of the N2 adsorption isotherms are extended to an extremely low or high p/po region, information obtained from the N2 isotherms will be remarkably increased. Recent progress of automatic gas

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adsorption apparatus makes it possible to measure the N2 adsorption isotherms at 77.4 K in a very wide pressure range from p/po=10-8 to p/po=0.999 [6–9]. In addition, liquid argon of high purity (99.995%) (b.p.: 87.3 K) is readily available, which makes it easy to measure the precise N2 isotherms at 87.3 K up top/po=10-8. In our previous works [8,9], the N2 isotherms at these two temperatures have been applied to investigations of surface characters of nonporous carbon blacks or microporosities of activated carbons fibers, where our attention was mainly focused on the isotherms at low relative pressures. In the present work, characters of several nonporous CB adsorbents will be investigated using the N2 isotherms at the three relative pressure ranges; i.e., (1) p/po=10-7–10-2 (low relative pressure ranges), (2) p/po=10-4–0.50 (medium relative pressure ranges), and (3) p/po =0.900–0.998 (high relative pressure ranges). First, the N2 isotherms (77.4 K) at p/po=10-4–0.50 (medium relative pressure ranges) by nongraphitized and graphitized CB adsorbents will be analyzed by the BET method [10–12]. As pointed out by Sing and coworkers [12], it is very important to select the appropriate relative pressure ranges for BET plots in order to obtain the reliable surface areas and the BET constant values. According to the method recommended by Sing and coworkers [12], we tried the BET plots for the N2 isotherms by nongraphitized and graphitized CB adsorbents. The BET surface areas (ABET) and constants (CBET) of the CB adsorbents will be used in estimation of particle size of primary CB particles, and consideration of surface homogeneity/heterogeneity of CB particles, respectively. Second, the N2 isotherms (77.4 K) at p/po=0.900–0.998 (high relative pressure ranges) will be used in estimation of macroporosity of CB aggregates. The distinct adsorption hysteresis in the N2 isotherms (77.4 K) was detected at very high p/po ranges, usually at p/po=0.950–0.998 (e.g. Figure 7(a)). Because of experimental difficulty, thus far little attention has been paid to the N2 isotherms in such high relative pressure ranges (p/po=0.950–0.998). However, they give the important information about macroporosity of CB aggregates. In this work, the pore size distribution (PSD) of macropore ranges (pore width>50 nm) will be calculated by means of BJH method [13]. On the basis of the PSD and particle size of the CB adsorbents, the macroporous textures will be considered. Furthermore, we tried the evaluation of particles size of the CB aggregates. For this purpose, it is necessary to carry out the  S plots at very high relative pressure ranges (p/po=0.970–0.998). From such  S plots, the approximate size of CB aggregates (secondary particle) may be determined. Third, the N2 isotherms (77.4 and 87.3 K) by graphitized CB adsorbent (Carbotrap-F-CB) with highly homogeneous surface were measured in a very wide pressure range of p/po=2  10-7–10-2. The

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surface characters of Carbotrap-F-CB will be discussed together with those of the other CB adsorbents reported in the previous works [8,9]. Particular attention will be paid to the Henry‟s law at the low coverage of the N2 isotherms (77.4 K) [14]. After characterization of CB adsorbents had been finished, the standard  S data of N2 at 77.4 and 87.3 K by Carbotrap-F-CB (homogeneous surface) and at 77.4 K by #51-CB (heterogeneous surface) will be given in Tables 5-7. The revised standard  S data of N2 at 77.4 K by #51-CB from p/po= 5 108 (  S =0.00219) to p/po=0.90 (  S =2.27) will be compared with the standard  S data by CB adsorbents reported by Kaneko and coworkers [15], Kruk and coworkers [16]. The standard  S data by sample#51-CB will be used in the estimation of microporous textures and surface areas of a series of activated carbon fibers (ACF) and superactive carbons (SAC), because the surface heterogeneity of nongraphitized carbon black#51-CB is similar to that of microporous ACF adsorbents (cf. Figure 21 of ref. 9). Microporosity of a series of ACF and SAC has been analyzed by means of the  S method reported by Sing [17]. On the basis of the high resolution  s plots, Sing and coworkers [18,19] have proposed the two stage micropore fillings; namely, the primary filling into ultramicropores of molecular dimension (less than 0.7 nm in pore width), and the secondary filling into supermicropores (larger than 0.7 nm). Kaneko and his coworkers [15,20,21] have extended the Sing‟ works, and they have reported that there are the two upward swings from the linearity of the  s plots. The first swing called as Filling Swing (FS) which appears at  s <0.3 (p/po<0.001) corresponds to the primary filling into ultramicropores, and the second swing called as Co-operative Swing (CS) which appears at  s =0.7 (p/po=0.13) is due to the secondary filling into supermicropores. We will give the refined data for the primary and secondary fillings. The high resolution  s plots make it possible to differentiate between the primary filling and the secondary filling. From analysis of the high resolution  s plots, the volumes of ultramicropores and supermicropores of a series of ACF samples and three SAC adsorbents will be quantitatively evaluated. In addition, the surface areas of supermicropores and external surfaces can be also determined from slope of  s plots. Some comments will be given on the BET surface areas of SAC.

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Experimental Materials: Nonporous and microporous carbon adsorbents used in this work are listed in Table 1. N2 gas with purity of 99.99995 % and Kr gas with purity of 99.999 % were used as the adsorptives, and He gas with purity of 99.99995 % was used for free volume measurements. The N2 and Ar liquids with purity of 99.995 % were used as the refrigerants. These gases and liquids were supplied from Taiyo Nippon Sanso Corp. Measurement of N2 adsorption isotherms at 77.4 and 87.3 K: High resolution adsorption isotherms of N2 gas by the carbon adsorbents pretreated at 300OC in vacuo were measured at 77.4 K (boiling point of N2 liquid) and87.3 K (boiling point of Ar liquid). Measurements of the N2 isotherms were carried out using two types of automatic volumetric adsorption apparatuses (BELSORP-max and BELSORP-miniII). BELSORP-max [6-9] has three transducers of 0.0133 kPa (0.1 Torr), 1.33 kPa (10 Torr), and 133 kPa (1000 Torr) in full scale. The carbon adsorbents listed in Table 1 were pretreated at 300OC for 3 h under a turbo-molecular pump vacuum at pressures less than 10-4 Pa. Prior to adsorption measurements, it was confirmed that leakage of N2 gas from the adsorption apparatus was less than 0.005 Pa/min, using checking system of BELSORP-max. Using BELSORP-max, the N2 adsorption isotherms were measured from p/po=2 x 10-8 to p/po=0.999. On the other hand, BELSORP-mini II has one transducer of 133 kPa (1000 Torr) in full scale, which was used in the measurements of adsorption isotherms from p/po=10-4 to p/po= 0.999 [22]. The Kr adsorption isotherms at 77.4 K by the nongraphitized and graphitized CB adsorbents were also measured using BELSORP-max. In these two adsorption apparatuses, the continuous dead volume measurement system developed by BEL Japan is adopted (cf. Figure 1 of reference 22), which makes it possible to measure adsorption isotherms with high accuracy, especially at very high relative pressure ranges (p/po=0.900–0.999). In the adsorption measurements in a low pressure range from p/po=2x10-8, thermal transpiration coefficients reported by Takaishi and Sensui were taken into consideration [23]. Total periods of the adsorption measurements were within 20– 60 h, depending on the sample character. Measurements of free volume of the sample tube were carried out by He gas after measurements of the adsorptiondesorption isotherms.

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Table 1. Adsorbents nongraphitized and graphitized carbon blacks (NGCB and GCB), activated carbon fibers (ACF), and superactive carbons (SAC) NGCB: #51-CB by Asahi Carbon Co. Ltd. (ABET=18.5 m2g-1, W$=ca. 0.5 g) GCB: #3845-1-CB by Tokai Carbon Co. Ltd. (ABET=54.4 m2g-1, W$= ca.0.2 g) GCB:#3845-2-CB by Tokai Carbon Co. Ltd.(ABET=49.2 m2g-1, W$=ca. 0.2 g) GCB:LGC2102-CB# by LGC Ltd. (ABET=69.4 m2g-1, W$=ca. 0.2) GCB:Carbotrap-F-CB by Supelco Inc. (ABET=6.79 m2g-1, W$=ca. 1.0) ACF: FT-07-ACF, FT-10-ACF, FT-15-ACF, FT-20-ACF, and FT-25-ACFby Kuraray Chemical Co. Ltd. (W$=ca. 50 mg) SAC: PX21-SAC by AMOCO (Standard Oil Co.)( W$=ca. 50 mg) SAC:MSP-20-SAC and MSP-30-SAC by Kansai Coke and Chemicals Co. Ltd. (W$=ca. 50mg) Note 1: #3845-1-CB and #3845-2-CB are different in lot number. Note 2: W$: the weights of carbon adsorbents used in adsorption measurements at 77.4 K. Note 3: #this sample has been used as the Certified Reference Material of BET Surface Area Measurement.

Results and Discussion A. Characterization of CB Adsorbents by High Resolution Adsorption Isotherms of N2 at 77.4 and 87.3 K In continuation of the previous works [8,9], the N2 adsorption isotherms from p/po =2  10-8 to p/po =0.998 have been utilized in order to investigate the surface characters of CB particles and the macroporous textures of CB aggregates. The N2 isotherms of the whole pressure ranges mentioned above were separated into the three regions of (1) p/po =10-7–10-2, (2) p/po =10-4–0.50, and (3) p/po=0.900– 0.998, and the N2 isotherm of each region will be considered in detail.

1. Intermediate Relative Pressure Ranges (P/Po=10-4–0.50); BET Analysis of the N2 Isotherms The N2 adsorption isotherms in the intermediate relative pressure ranges (p/po=10-4–0.50) by nongraphitized (#51-CB) and graphitized carbon blacks (#3845-1-CB, LGC2102-CB, and Carbotrap-F-CB) are shown in Figures 1 - 5, respectively. As Figure 4 shows, the N2 isotherm by Carbotrap-F-CB has asmall step (6.2% jump) in the pressure range of p/po=0.005-0.01. Therefore, the N2 isotherm of Figure 4 will be discussed separately.

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It is evident from Figures 1–3 that the N2 isotherms have a well-defined Point B, which suggests that the boundary between monolayer and multilayer is unambiguous. The upward deviation above p/po=ca.0.2 detected by samples#3845-1-CB and LGC2102-CB is ascribed to graphitization of carbon surface [24]. The BET plots of N2 at 77.4 K by samples#51-CB, #3845-1-CB, and LGC2102-CB are shown in Figures 1–3, respectively, together with their isotherms.

Figure 1. Adsorption isotherms of N2 at 77.4 K and their BET plots by nongraphitized carbon black (#51-CB) in the region of p/po=10-4– 0.50.

Usually, a pressure range p/po=0.05–0.30 (  =ca. 0.8–ca. 1.2) is selected as the linear line of the BET plot, and the BET monolayer capacity (nm) and the BET C constant (CBET) is calculated from slope and intercept of the BET linear line [10, 11]. However, attention should be paid for selection of the appropriate linear o p/p ranges. As have been pointed out by Sing and coworkers [12], the BET linear regions depend on adsorption systems and temperature, and it is inadvisable to carry out the BET plots at the predetermined p/po ranges (e.g. p/po=0.05–0.30).

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Figure 2. Adsorption isotherms of N2 at 77.4 K and their BET plots by graphitized carbon black (#3845-1-CB) in the region of p/po=10-4– 0.50.

Figure 3. Adsorption isotherms of N2 at 77.4 K and their BET plots by graphitized carbon black (LGC2102-CB) in the region of p/po=10-4– 0.50.

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Figure 4. Adsorption isotherms of N2 at 77.4 K and their BET plots by graphitized carbon black (Carbotrap-F-CB) in the region of p/po=10-4– 0.50.



adsorption data for Carbopack-F-CB cited from the paper of Kruk et al. [16].

Figure 5. Adsorption isotherms of N2 at 77.4 K and their BET plots by graphitized carbon black (Carbotrap-F-CB) in the regions of p/po=0.0005-0.005 (before step) and p/po=0.010.04 (after step).

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Their recommended procedure [12] is that, after selecting the appropriate p/po region including Point B, the „best‟ fit linear line is determined by the statistical analysis of the repeatedly measured isotherms. We employed the recommended procedure for the five or six N2 isotherms of N2 at 77.4 K by samples#51-CB, #3845-1-CB, and LGC2102-CB. The „best‟ fit lines thus determined are given in Figures 1-3, respectively, and the results of BET plots are summarized in Table 2. It is very important to mention that the linear p/po ranges given in Figures 1–3 and Table 2 are significantly different from the generally recommended range ofp/po=0.05–0.30. For example, if p/po=0.05–0.30 is employed as the BET plot range, nm and CBET of #3845-1-CB are estimated to be 0.592 mmol g-1 and 108, respectively. These figures are significantly different from nm=0.558 mmol g-1 and CBET =1900 given in Table 2; especially remarkable difference is observed in CBETvalue. Accordingly, it is understood that selection of the appropriate p/po ranges is very important. As was pointed out above, the N2 isotherm by Carbotrap-F-CB has a small step (6.2 % jump), whose phenomenon is known as the two dimensional phase transition of N2 adsorbed layer from a liquid to a solid state on very uniform carbon surface [25]. In the case of the N2 isotherm by Carbotrap-F-CB, therefore, it is necessary to carry out the BET plots before and after step. The two linear lines of the BET plots at p/po=0.0005-0.005 and p/po=0.01-0.04 are shown in Figure 5, from which monolayer capacities before and after step are estimated to be nm(before)=0.0676 mmol g-1 and nm(after)=0.0718 mmol g-1, respectively. Table 2. Summary of BET and s analysis

Sample name #51-CB #3845-1-CB #3845-2-CB LGC2102-CB Carbotrap-F-CB (before step) (after step)

0.001–0.16 0.001–0.18 0.001–0.18 0.001–0.18

BET monolayer capacity (nm) mmol g-1 0.190 0.558 0.504 0.712

0.0005–0.005 0.01–0.04

0.0676 0.0718

linear p/po range of BET plot

adsorbed amount of Point B(nPointB) mmol g-1 0.187 0.562 0.509 0.705 ―――― 0.0709

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Characterization of Carbon Blacks by High Resolution N2 … Table 2. (Continued)

Sample name

#51-CB #3845-1-CB #3845-2-CB LGC2102-CB Carbotrap-F-B (before step) (after step)

Average particle

BET constant (CBET)

BET surface area (ABET) m2 g-1

 s surface area

260 1600 1900 1300

18.5 54.4 49.2 69.4

―― ―― ―― 9±1

nm 160 50 55 39

5200 5900

―――― 6.79

4±2

408

( ACB aggregate) S

m2 g-1

CB size ( d av . )

This difference between nm(before) and nm(after) is calculated to be 6.2 %, which suggests that the packing density of adsorbed N2 molecules increases 6.2 % by a transition of liquid state to solid state. Table 2 clearly shows that all nm values are in excellent agreement with nPointB-values, and that CBET of graphitized CB samples (#3845-CB, LGC2102CB, Carbotrap-F-CB) is notably larger than that of nongraphitized CB sample (#51-CB). Furthermore, we tried to re-calculate CBET-values using the other N2 isotherms (standard  s data) reported by Kaneko et al. [15] and Kruk et al. [16]. All CBET values including ours indicate that the CBET–values of nongraphitized CB adsorbents having heterogeneous carbon surface are in the vicinity of 260, while those of graphitized CB samples varies from values of1300-1900 forsamples#3845CB and LGC2102-CB having partially heterogeneous surface to a very high value of 6000 for Carbotrap-F-CB of highly uniform surface. Magnitude of CBET is, therefore, closely related to the extent of surface homogeneity of CB particle; in other words, increase of surface homogeneity brings about remarkable increase in CBET. On the homogeneous surface of a highly graphitized carbon black, the first and the second layers of N2 are formed at distinctly different p/po regions, respectively. This results in sharpening of knee in the monolayer region of the isotherm. The BET surface areas (ABET) of the present carbon black adsorbents were calculated from nm by assuming the adsorbed area of a N2 molecule (  N 2 ) to be 0.162 nm2 for the CB adsorbents without step. When there is a small step, the

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 N 2 after step is reported to be 0.157 nm2 [25].Therefore, we employed 0.157 nm2 as  N 2 for Carbotrap-F-CB. The values of ABET given in Table 2 will be used to estimate the average CB particle size in the CB aggregate. Scanning electron micrograph shows that shape of the CB particle is spheroidal (nongraphitized) or polyhedron (graphitized) [2]. In the present paper, the shape of the CB particle is assumed to be spherical; the relation CB between its average diameter ( d av ) and ABET is given by the following equation.

CB d av 

CB Vsolid can be calculated as

cm and  -3

CB particle

1



CB particle

CB 6  Vsolid ABET

(1.1)

, where  CB particle (graphitized CB) = 2.2 g

(nongraphitized CB) = 2.1 g cm-3 have been obtained from

CB X-ray diffraction measurements of carbon blacks [2]. When Vsolid are inserted into equation (1.1), the equations (1.2) and (1.3) will be obtained;

GCB d av 

NGCB d av 

6.0  103 nm( ABET  2.2

ABET : m2g-1) for graphitized CB

(1.2)

2 -1 6.0  103 nm ( A BET : m g ) for nongraphitized CB ABET  2.1

(1.3)

Table 3. Important parameters for evaluation of macroporous textures of CB aggregates Sample name #51-CB #3845-2-CB LGC2102-CB CarbotrapF-CB

CB d av

CBaggregate d av

CB Vsolid 3 -1

isoth. V pore 3 -1

nm

nm

cm g

160 55 39

―― ―― ca. 900

0.47 0.45 0.45

cm g 0.40 1.53 0.84

408

ca. 1100

0.45

0.30

porosity/packing (po-%)$/(pa-%)$

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Table 3. (Continued) Sample name #51-CB #3845-2-CB LGC2102-CB Carbotrap-F-CB $

definition of po-%;

$

definition of pa-%;

V pore (V pore  Vsolid )

ads.branch d peak

des.branch d peak

PSD V pore .

nm 300 250 70 240

nm 150 110 50 190

cm3g-1 0.39 1.51 0.85 0.29

 100 .

Vsolid 100. (V pore  Vsolid )

cf. body centered cubic packing; (po-%=32)/(pa-%=68). cubic packing; (po-%=48 %)/(pa-%=52). NGCB GCB From the equations (1.2) and (1.3), the values of d av and d av were calculated and listed in Tables 2 and 3, which will be used in discussion of macroporosity of CB aggregates.

2. High Relative Pressure Ranges (P/Po=0.900–0.998); Macroporous Textures and Particle Size of CB Aggregates Prior to discussion of the N2 isotherms at high relative pressures, terminologies of carbon black adsorbents used in the present paper will be given at first. According to ASTM [26], a CB aggregate is defined as a discrete, rigid colloidal entity that is smallest dispersible unit, and it is composed of extensively coalesced particles, and a CB particle is defined as a small spheroidally shaped, paracrystalline, non-discrete component of an CB aggregate, and it is separable from the aggregate only by fracturing. Hess and Herd have pointed out that sometimes the term “agglomerate” is confused with aggregate [26]. Definition of CB aggregates by ASTM [26] differs apparently from that by IUPAC [27]. Accordingly, it is very important to make clear which definition is used. We adopted the terminologies defined by ASTM in this paper, whose definition is generally used in the field of CB technology. The purpose of this section is to investigate the macroporous texture formed in the CB aggregates. As is seen from Table 3, the average size of the CB particles (components of the CB aggregates) is in the ranges of 40 nm – 400 nm. Accordingly, it is expected that the inter-particle voids formed by network of the

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CB particles will be also in the ranges of macropores (d>50 nm). In order to examine the macroporous textures of the CB aggregates, it is absolutely necessary to measure in detail the N2 adsorption-desorption isotherms (77.4 K) up to the very high relative pressures, e.g., p/po=0.995–0.998. Because of experimental difficulty, few works have been reported with respect to measurements of the N2 isotherms at such high relative pressures. In Figure 6(a), the two N2 adsorption-desorption isotherms (77.4 K) by nongraphitized CB (#51-CB) are shown, where the isotherm A is measured in the range of p/po=0.950–0.992 and the isotherm B in the range of p/po=0.950–0.998. This result clearly shows that the isotherm A until p/po=0.992 is insufficient in calculation of the pore size distribution (PSD) of sample#51-CB, because all macropores of the sample are not necessarily be filled by condensed N2 liquid at p/po=0.992. In Figures 7(a)–9(a), respectively, the N2 isotherms by three kinds of graphitized CBs (#3845-2-CB, LGC2102-CB, and Carbotrap-F-CB) are shown. As is seen from Figures 6–9, all N2 isotherms by the present CB adsorbents give the distinct adsorption hysteresis at very high relative pressures. Such hysteresis can be explained by condensation/evaporation of N2 into/from macropores in the CB aggregates.

Figure 6(a). Adsorption-desorption isotherms of N2 at 77.4 K by nongraphitized carbon black (#51-CB) in the region of p/po=0.950–0.998.

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Figure 6(b). Pore size distribution curves for CB aggregates of #51-CB.

Figure 7(a). Adsorption-desorption isotherms of N2 at 77.4 K by graphitized carbon black (#3845-2-CB) in the region of p/po=0.950–0.998.

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Figure 7(b). Pore size distribution curves for CB aggregates of #3845-2-CB.

Figure 8(a). Adsorption-desorption isotherms of N2 at 77.4 K by graphitized carbon black (LGC2102-CB) in the region of p/po=0.900–0.998.

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Figure 8(b). Pore size distribution curves for CB aggregates of LGC2102-CB.

Figure 9(a). Adsorption-desorption isotherms of N2 at 77.4 K by graphitized carbon black (Carbotrap-F-CB) in the region of p/po=0.950–0.998.

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Figure 9(b). Pore size distribution curves for CB aggregates of Carbotrap-F-CB.

The pore size distribution (PSD) of macropores in the CB aggregates will be calculated by means of the BJH method [13]. The fundamental data used in PSD calculation of macropore ranges (p/po=0.900–0.999) are given in Table 4.  S of the second column were calculated on the basis of the FHH equation (2.1) [28]. ln (po/p)=0.818 /  S 2.5

(2.1)

Coefficients of 0.818 and 2.5 in equation (2.1) were determined from the FHH plots shown in Figure 10. The standard  S data of N2 at 77.4 K by nongraphitized(#51-CB) and graphitized CB(#3845-2-CB)in the relative pressure ranges of p/po=0.800–0.950 give the common linear line in the FHH plots. Here, it should be mentioned that the  S curves at p/po>0.800 (  >2.7) are represented as the common reduced isotherm, and that effect of surface character on the adsorption isotherms can be neglected at p/po>0.800. Coverage (  ) in the third column was evaluated by equation (2.2) (cf. Table 5).

n     S   0.4  =   1.63 S  n 1 

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Characterization of Carbon Blacks by High Resolution N2 …

The relation between  and statistical adsorbed thickness (t) has been given by Lippens, Linsen, and de Boer [29]. t =0.354  nm

(2.3)

The core radius ( rcore ) of macropores in the fifth column was calculated by Kelvin equation of semi-spherical meniscus (Equation 2.4) [30].

 p  2VL 1 ln o     RT rk p 

(2.4)

In the case of nitrogen liquid at 77.4 K, the following data were used [31]. γ (surface tension of liquid N2 at 77.4 K) = 8.85 x 10-3 N m-1, 3 -1 VL (molar volume of liquid N2 at 77.4 K)= 34.71 cm mol ,

R (gas constant)= 8.3145 J mol-1 K-1, T (temperature at boiling point)=77.4 K. These numerical data are inserted into Eq. 2-4, and the equation (2.5) was obtained; rcore  

0.416 nm log( p / p o )

(2.5)

And pore radius ( rpore ) is given by equation (2.6). rpore = rcore  t

(2.6)

By using the equations (2.5) and (2.6), and the fundamental data given in Table 4, the PSD curves from both of adsorption and desorption branches for nongraphitized (51-CB) and graphitized carbon blacks (3845-2-CB, LGC2102CB, and Carbotrap-F-CB) were calculated by the BJH method [13], whose results are shown in Figure 6(b) – Figure 9(b), respectively. des.branch

ads.branch The peak values ( d peak , d peak

) of PSD curves for adsorption and des.branch

CB ads.branch desorption branches are listed in Table 3. Digital data of d av , d peak , d peak

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porosity (Table 3), and PSD curves (Figures 6(b) – 9(b)) are the important parameters for evaluation of the macroporous textures of CB aggregates.

Figure 10. FHH plots of N2 adsorption isotherms at 77.4 K by nongraphitized CB (#51CB), graphitized CB (#3845-2-CB).

From these data, it is possible to construct three-dimensional macroporous textures of CB aggregates. In the present paper, we will propose the simplified model of macroporous textures of Carbotrap-F-CB in Figure 11.If the diameter of a sphericalCB particle (dI) is assumed to be 400 nm, the window diameter (dII) and the inner sphere diameter (dIII) can be calculated to be 160 nm and 290 nm, respectively. The CB aggregate of Carbotrap-F-CB may be explained by the model des.branch

CB of Figure 11; that is, d av =408, d peak

ads.branch =190 nm and d peak =240 nm may

correspond approximately to dI, dII, and dIII of the simplified model, respectively. The next problem is to evaluate the external surface areas of CB aggregates by analyzing the N2 isotherms at very high relative pressures. It is well known that in the case of microporous or mesoporous adsorbents, their external surface areas ( Aex , Aex ) of pore walls can be precisely evaluated from the slope of  S plot (e.g. Figure 19) [17,32]. In the case of macroporous CB aggregates, however, micro

meso

CB aggregate

evaluation of Aex is not an easy task. In the case of LGC2102-CB, as is shown in Figure 8(a), capillary condensation into macropores of CB aggregates (process I) finishes in the vicinity of p/po=0.985, and capillary evaporation from macropores (process II) starts from the vicinity of p/po=0.970.

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Characterization of Carbon Blacks by High Resolution N2 … Table 4. Fundamental data used in PSD calculation of macropores (adsorptive: N2, temperature: 77.4 K) Relative pressure ranges: p/po=0.900 – 0.999 p/po 0.900 0.905 0.910 0.915 0.920 0.925 0.930 0.935 0.940 0.945 0.950 0.952 0.954 0.956 0.958 0.960 0.962 0.964 0.966 0.968 0.970 0.972 0.974 0.976 0.978 0.980 0.982 0.984 0.986 0.988 0.990 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999

αs 2.270 2.320 2.373 2.430 2.493 2.561 2.635 2.717 2.809 2.911 3.028 3.079 3.133 3.190 3.252 3.317 3.387 3.463 3.544 3.633 3.729 3.835 3.952 4.083 4.229 4.395 4.586 4.809 5.075 5.400 5.811 6.062 6.356 6.706 7.134 7.675 8.394 9.419 11.08 14.62

θ 3.707 3.788 3.875 3.969 4.071 4.182 4.303 4.438 4.587 4.754 4.944 5.028 5.116 5.210 5.310 5.417 5.531 5.655 5.788 5.932 6.090 6.263 6.454 6.667 6.906 7.177 7.489 7.853 8.288 8.818 9.489 9.900 10.38 10.95 11.65 12.53 13.71 15.38 18.09 23.88

t / nm 1.312 1.341 1.372 1.405 1.441 1.480 1.523 1.571 1.624 1.683 1.750 1.780 1.811 1.844 1.880 1.918 1.958 2.002 2.049 2.100 2.156 2.217 2.285 2.360 2.445 2.541 2.651 2.780 2.934 3.122 3.359 3.505 3.674 3.877 4.124 4.437 4.852 5.445 6.405 8.453

rcorespr /nm 9.10 9.61 10.17 10.80 11.50 12.30 13.22 14.27 15.50 16.95 18.70 19.50 20.37 21.31 22.35 23.49 24.75 26.16 27.72 29.49 31.49 33.77 36.40 39.48 43.11 47.47 52.80 59.46 68.02 79.44 95.42 106.1 119.4 136.5 159.4 191.3 239.3 319.2 479.0 958.5

ln(po/p) = 0.818/αs2.5. n0.4 = 0.310 mmol g-1. nθ=1 = 0.190 mmol g-1.

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rporespr /nm 10.41 10.95 11.54 12.20 12.94 13.78 14.74 15.84 17.12 18.64 20.45 21.28 22.18 23.16 24.23 25.41 26.71 28.16 29.77 31.59 33.64 35.99 38.69 41.84 45.56 50.01 55.45 62.24 70.96 82.56 98.78 109.6 123.1 140.4 163.5 195.8 244.1 324.6 485.4 967.0

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On the other hand, in the cases of #51-CB, #3845-2-CB, and Carbotrap-F-CB, as is seen from Figure 6a, Figure 7a, and Figure 9a, the process I finishes near p/po=0.996, and the process II starts from p/po=0.994. These experimental facts clearly indicate that in order to evaluate

AexCBaggregate, it is necessary to measure in detail the desorption isotherms at extremely high relative pressures from p/po=0.998 to p/po=0.970.

Figure 11. Simplified model of macroporous texture of CB-aggregate of Carbotrap-F-CB.

In the present work, we tried the  S plots by LGC2102-CB and Carbotrap-FCB (Figure 12), and the slopes of  S plots by LGC2102-CB and Carbotrap-F-CB were estimated to be 0.15 mmol g-1 and 0.063 mmol g-1, respectively. When the slope in the  S plot is expressed as mmol g-1, the surface area from the  S plot ( A S ) is given by equation (2.7) (see section 4 and equation 5.3).

A S =60.8  slope

(2.7)

A S corresponds to the external area of CB aggregates ( AexCB aggregate). From CB aggregate

Figure 12, the Aex values of LGC2102-CB and Carbotrap-F-CB are 2 -1 estimated to be 9±1 m g and 4±2 m2g-1, respectively.

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CB aggregate Next, the average diameter of CB aggregate ( d av ) were calculated by using equation (2.8), where the shape of CB aggregate is assumed to be spherical.

CBaggregate d av 



CB isoth. 6  VexCBaggregate 6  Vsolid  V pore  CBaggregate CBaggregate Aex Aex



(2.8)

CB

isoth. The values of Vsoilid , V pore , and AexCBaggregate listed in Table 3 are put into CB aggregate values. The aggregate sizes of equation (2.8) to estimate the dav LGC2102-CB and Carbotrap-F-CB are ca. 900 and 1100 nm, respectively.

Figure 12.  S plots of N2 at 77.4 K by CB aggregates (LGC2102-CB and Carbotrap-F-CB).

It can be estimated that the CB aggregate size of LGC2102-CB (ca. 900 nm) is twenty-three times of the CB particle size (39 nm), while the aggregate size of Carbotrap-F-CB (ca. 1100 nm) is ca. 2.7 times of its particle size (408 nm). The CB aggregate aggregate sizes of d av =ca. 1100 nm for Carbotrap-F-CB approximately corresponds to the cluster size of the simplified model illustrated in Figure 11.

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Kazuyuki Nakai, Yoko Nakada, Masako Hakuman et al. The present work has showed that analysis of  S plots at very high relative

pressures (p/po=0.970–0.999) makes it possible to evaluate the CB aggregate size. However, as is seen from Figure 12, there is large fluctuation in the  S plots. CBaggregate The present work is the first step of evaluating d av from N2 adsorption data. In future, we will try to improve the automatic adsorption apparatus so as to decease fluctuation in the  S plots at the extremely high relative pressures of p/po=0.970 – 0.999.

3. Low Relative Pressure Ranges (P/Po=10-7–10-2); Surface Homogeneity or Heterogeneity of CB Particles 3.1. Henry’s Law and Deviation from Its Law at Very Low Relative Pressures In the section (1), the effect of graphitization on the surface homogeneity of CB particles has been considered. The CBET values, which are evaluated from the BET plot (p/po=0.001–0.20), depend remarkably on surface homogeneity. It has been pointed out that increase in surface homogeneity brings about the remarkable increase in the CBET value. In the present section, effect of the surface homogeneity on the N2 isotherms at 77.4 K will be examined in detail at the very low relative pressures. The reduced adsorption isotherms of N2 at 77.4 K by nongraphitized CB (#51-CB) and graphitized CB (#3845-2-CB, LGC2102-CB, and Carbotrap-F-CB) are shown in Figure 13, where n0.4 is an adsorbed amount at p/po=0.400 and n an adsorbed amount at arbitrary relative pressure. Here, the reduced adsorption isotherms are expressed as log10 (n / n0.4 ) vs.

log10 ( p / p o ) . From Figure 13, it is found that the reduced isotherms by the present CB adsorbents are classified into three groups below p/po=2  10-5. From this experimental fact, it has been suggested that the surface characters (homogeneity/heterogeneity) of CB particles are characterized by the N2 adsorption isotherms in very low relative pressure ranges. The three groups of reduced isotherms can be expressed as the following experimental formulas. For Carbotrap-F-CB of our work and Carbopack-F-CB by Kruk et al. [16].  p   n  o -6 -5   1.00  log o   3.00 (p/p =10 –2  10 ) log p  n  0.4   

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Figure 13. Reduced adsorption isotherms of N 2 at 77.4 K by nongraphitized carbon black(#51-CB) and graphitized carbon blacks (#3845-2-CB, LGC2102-CB, and CarbotrapF-CB) in the relative pressure regions of p/po=10-7-10-2.

for graphitized samples#3842-2-CB and LGC2102-CB,

 p   n    0.62 log o   1.34 (p/po=10-7–10-5) log  p   n0.4   

(3.2)

and for nongraphitized sample#51-CB,

 n   p    0.49  log o   1.18 (p/po=10-7–4  10-5) log  p   n0.4   

(3.3)

The equations (3.1), (3.2), and (3.3) can be re-arranged as follows; 1.00

 n  3  p       n   1.0010   p o   0.4   

(0.002<  <0.02)

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Kazuyuki Nakai, Yoko Nakada, Masako Hakuman et al.  p   n       n   22  p o   0.4     p   n     15  o   n0.4  p 

0.62

0.49

(0.002<  <0.04)

(0.01<  <0.16)

(3.5)

(3.6)

Measurement of the N2 isotherms at 77.4 K until p/po=2  10-8 made it possible to determine the three experimental formulas mentioned above. Equation (3.4) clearly shows that the N2 adsorption isotherms at 77.4 K by highly graphitized sample Carbotrap-F-CB obey Henry‟ law quite closely at very low surface coverages. It is well known that the simplest behavior of adsorbed phase is expressed as Henry‟s law of equation (3.7) [33], n=kH  p

(3.7)

where n is the specific surface excess amount and kH the Henry‟s law constant. Equation (3.7) is derived by assuming that the dilute adsorbed phase behaves as a two-dimensional ideal gas on the uniform surface. The experimental fact given by equation (3.4) and Figure 13 clearly indicates that the N2 adsorbed phase at 77.4 K on Carbotrap-F-CB behaves as the ideal two-dimensional gas at the very low surface coverages of 0.002<  <0.02 and that the surface of Carbotrap-F-CB is highly homogeneous or uniform until very low coverage,  =0.002. Rouquerol and co-workers [33] have pointed out that deviation from linearity (convex curvature with respect to the adsorption axis) may be due to surface heterogeneity [33]. In our previous paper [8], we have reported that the surface of #3845-1-CB is partially heterogeneous (ca. 20%), and the surface of #51-CB is heterogeneous. From the present results of equations (3.4), (3.5), (3.6) and the previous qst results (cf. Figure 9 of reference 8 and Figure 15), the following conclusion may be obtained. If the relation between n n0.4  and (p/po) is expressed by equation (3.8), k

 2 (3.8) n n0.4   k1   po  p  the constant k2of equation (3.8) is the important parameter of evaluating surface heterogeneity of CB adsorbents; in other words, the high adsorption energy sites of heterogeneous CB surfaces (#51, #3845-1-CB) brings about k<1, and as the

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result, the N2 isotherms at 77.4 K have convex curvature against the adsorption axis below p/po=2  10-5. In the present adsorption experiments, we were able to show that the surface heterogeneity of CB adsorbents is closely related to the k2value in equation (3-8). In the next section, 3.2, the relation between equations (3.4), (3.5), (3.6) and the qst curves will be considered. 3.2. Isosteric Heats of Adsorption of N2by Highly Graphitized CB of Carbotrap-F-CB For samples#51-CB and #3845-1-CB, the relationship between qst and  has been already given in the previous work (see Figure 9 of reference 8). In the present work, the relationship between qst and  for Carbotrap-F-CB was investigated on the basis of the two N2 isotherms at 77.4 K and 87.3 K shown in Figure 14. The two N2 isotherms at 77.4 and 87.3 K run precisely parallel until very low coverage, suggesting that the carbon surface of Carbotrap-F-CB may be highly homogeneous. Calculation of qst was carried out by the integral form of Clausius-Clapyron equation (3.9) [34].

q  st



  adsH diff 

RT1T2  p2  ln  T1  T2  p1 

(3.9)

where T1 =77.4 K (boiling point of N2 liquid) and T2 =87.3 K (boiling point of Ar liquid). When the equation (3.9) is applied to the N2 adsorption isotherms at 77.4 K and 87.3 K by Carbotrap-F-CB, particular attention should be paid. As has been pointed out previously in Figure 5, the small step is detected in the N2 isotherm at 77.4 K, whose step is understood as the two dimensional phase transition near  =1 [25]. On the other hand, such a small step cannot be detected in the N2 isotherm at 87.3 K. Clausius-Clapyron equation is derived under the assumption that no phase transition occurs in the range of measuring temperatures. Accordingly, we have judged that equation (3.9) cannot be used in the vicinity of monolayer coverage. The qst-values by Carbotrap-F-CB is shown as a function of  in Figure 15, where the previous qst results by #51-CB and #3845-1-CB are also given. The qst vs.  data in the coverage range below =0.2 are represented as the three separated lines with quite different slopes, I, II, and III.

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Figure 14. Adsorption isotherms of N2 at 77.4 K and 87.3 K by graphitized carbon blacks (Carbotrap-F-CB) in the region of p/po=10-6-0.1.

Figure 15. Isosteric heat of adsorption of N2 by nongraphitized carbon black (I: #51-CB) and graphitized carbon blacks (II: #3845-1-CB and III: Carbotrap-F-CB) in the region of  =0.03–1.5.

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These lines demonstrate the different surface characters of the present CB adsorbents. In this work, we will focus our attention to the qst-values below  =0.20 in order to examine the surface homogeneity/heterogeneity of the CB adsorbents. Figure 15 shows that the qst-values by Carbotrap-F-CB are constant (10.7±0.1 kJ mo-1) in the range of  =0.03–0.20, while those by #3845-1-CB decrease from 11.8±0.2 kJ mo-1(  =0.05), to 10.9±0.2 kJ mo-1 (  =0.20), and those by #51-CB decrease from 13.8±0.8 kJ mo-1 (  =0.05) to11.8±0.5 kJ mo-1 (  =0.20). From the qst-behavior shown in Figure 15, it is evident that Carbotrap-F-CB has a very uniform (homogeneous) surface, while the surface of #3845-1-CB has partially heterogeneous (ca. 20 %), and the surface of #51-CB is heterogeneous. The conclusion obtained from the qst-data is completely consistent with that obtained from equations (3.4), (3.5), (3.6). 3.3. Adsorption Isotherms of Kr at 77.4 K by Nongraphitized (#51-CB) and Graphitized CB (#3845-2-CB, Carbotrap-F-CB) In this section, the surface homogeneity/heterogeneity of the CB adsorbents will be investigated through the Kr adsorption isotherms at 77.4 K. In Figure 16, it is seen that the Kr isotherm (77.4 K) by Carbotrap-F-CB gives the stepped isotherm up to three molecular layers. Similar stepped adsorption isotherms at 77.4 K in the adsorption system of Kr gas and exfoliated graphite have been reported [35]. Stepped isotherms, which are classified as Type VI [36], have been detected for several other adsorbents of highly uniform surface (e.g. MgO) [35]. The purpose of the Kr adsorption measurements is to investigate the relationship between sharpness of the steps and surface homogeneity. In the case of the Kr isotherm (77.4 K) by Carbotrap-F-CB, the step is very sharp, indicating that boundary between monolayer and double layer appears in the very narrow relative pressure ranges of p/po=0.40–0.42. Furthermore, the Kr isotherm in the 1st and 2ndlayer regions runs almost parallel to the p/po axis. These experimental facts strongly suggest that the graphitized surface of Carbotrap-F-CB is highly homogeneous. In the case of #3845-2-CB, the sharpness of the step decreases, and boundary between 1st and 2nd layers broadens into p/po=0.38–0.50. In addition, the Kr isotherm in the 1st and 2nd layer gradually increases with the relative pressures. These experimental facts lead to conclusion that the surface of #3845-2-CB is partially heterogeneous. In the case of nongraphitized #51-CB, the step disappears completely from the Kr isotherm (77.4 K). The Kr isotherm becomes Type II.

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Figure 16. Adsorption isotherm of Kr at 77.4 K by nongraphitized CB (#51-CB) and graphitized CB (#3845-2-CB and Carbotrap-F-CB).

These data indicate that the surface of #51-CB is heterogeneous. In conclusion, the results obtained from the Kr adsorption isotherm (77.4 K) are in good agreement with the results obtained from the N2 adsorption isotherms (77.4 K and 87.3) (cf. sections 3.1 and 3.2).

B. Application of s Method to Analysis of Activated Carbon Adsorbents 4. Standard s Data of N2at 77.4 K and 87.3 K by Carbotrap-F-CB In the previous papers [8,9], we have reported the standard  S -data of N2 at 77.4 K and 87.3 K by nongraphitized (#51-CB) and graphitized CB (#3845-1-CB) as low as  S =0.002 (77.4 K). In the present paper, we will report the standard

 S -data of N2 at 77.4 K and 87.3 K by highlygraphitized CB (Carbotrap-F-CB) having highly uniform surface. The high resolution N2 adsorption isotherms at 77.4 and 87.3 K by Carbotrap-F-CB had been repeatedly measured until p/po=10-7 using the same adsorbent, and the standard  S -data of N2 at 77.4 K and 87.3 K were calculated and listed in Tables 5 and 6.

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Characterization of Carbon Blacks by High Resolution N2 … Table 5. Standard s data of N2 at 77.4 K by Carbotrap-F-CB Relative pressure ranges: p/po=2  10-7– 0.900 p/po 0.0000002 0.0000003 0.0000004 0.0000005 0.0000007 0.0000010 0.0000013 0.0000020 0.0000030 0.0000040 0.0000050 0.0000060 0.0000070 0.0000080 0.000010 0.000013 0.000020 0.000030 0.000035 0.000040 0.000050 0.000060 0.000070 0.000080 0.00010 0.00013 0.00020 0.00030 0.00040 0.00050 0.00060 0.00080 0.0010 0.0013 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 0.0080 0.0090 0.010 0.011

αs 0.000190 0.000277 0.000366 0.000466 0.000640 0.000923 0.00121 0.00190 0.00283 0.00376 0.00470 0.00560 0.00653 0.00733 0.00916 0.0118 0.0183 0.0267 0.0313 0.0360 0.0453 0.0540 0.0666 0.0766 0.0980 0.135 0.217 0.300 0.341 0.375 0.390 0.416 0.430 0.445 0.464 0.479 0.488 0.496 0.502 0.509 0.516 0.525 0.535 0.540

θ 0.000350 0.000509 0.000675 0.000859 0.00118 0.00170 0.00222 0.00350 0.00521 0.00693 0.00865 0.0103 0.0120 0.0135 0.0169 0.0218 0.0337 0.0491 0.0577 0.0663 0.0834 0.0994 0.123 0.141 0.180 0.248 0.399 0.552 0.629 0.690 0.719 0.766 0.792 0.819 0.855 0.883 0.899 0.914 0.925 0.937 0.950 0.967 0.985 0.994

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t / nm 0.000124 0.000180 0.000239 0.000304 0.000417 0.000602 0.000786 0.00124 0.00185 0.00245 0.00306 0.00365 0.00426 0.00478 0.00597 0.00771 0.0119 0.0174 0.0204 0.0235 0.0295 0.0352 0.0434 0.0500 0.0639 0.0880 0.141 0.195 0.223 0.244 0.255 0.271 0.280 0.290 0.303 0.313 0.318 0.324 0.328 0.332 0.336 0.342 0.349 0.352

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Kazuyuki Nakai, Yoko Nakada, Masako Hakuman et al. Table 5. (Continued) p/po 0.013 0.016 0.020 0.025 0.030 0.040 0.050 0.060 0.080 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80

αs 0.544 0.548 0.552 0.555 0.558 0.563 0.568 0.572 0.582 0.591 0.601 0.615 0.629 0.646 0.666 0.695 0.724 0.757 0.796 0.835 0.870 0.907 0.941 0.971 1.000 1.023 1.045 1.068 1.089 1.111 1.134 1.156 1.180 1.208 1.235 1.268 1.301 1.338 1.381 1.424 1.474 1.525 1.578 1.638 1.697

θ 1.001 1.009 1.016 1.022 1.027 1.037 1.045 1.054 1.071 1.088 1.107 1.132 1.158 1.190 1.226 1.279 1.333 1.394 1.465 1.537 1.601 1.670 1.733 1.788 1.841 1.883 1.925 1.966 2.006 2.047 2.088 2.129 2.173 2.224 2.275 2.334 2.396 2.464 2.544 2.621 2.713 2.807 2.905 3.015 3.125

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t / nm 0.354 0.357 0.360 0.362 0.364 0.367 0.370 0.373 0.379 0.385 0.392 0.401 0.410 0.421 0.434 0.453 0.472 0.493 0.519 0.544 0.567 0.591 0.614 0.633 0.652 0.667 0.681 0.696 0.710 0.725 0.739 0.754 0.769 0.787 0.805 0.826 0.848 0.872 0.900 0.928 0.961 0.994 1.028 1.067 1.106

Characterization of Carbon Blacks by High Resolution N2 … Table 5. (Continued) αs 1.771 1.854 1.894 1.947 2.005 2.071 2.150 2.231

p/po 0.82 0.84 0.85 0.86 0.87 0.88 0.89 0.90

θ 3.262 3.414 3.487 3.586 3.691 3.813 3.960 4.107

t / nm 1.155 1.209 1.234 1.269 1.307 1.350 1.402 1.454

n0.4 = 0.131 mmol g-1. nθ=1 = 0.0719 mmol g-1. t= 0.354×θ nm. ABET =6.79 m2 g-1.

Table 6. Standard s data of N2 at 87.3 K by Carbotrap-F-CB Relative pressure ranges: p/po=5  10-7– 0.400 p/po 0.0000005 0.0000007 0.0000010 0.0000013 0.0000020 0.0000030 0.0000040 0.0000050 0.0000060 0.0000070 0.0000080 0.000010 0.000013 0.000020 0.000030 0.000035 0.000040 0.000050 0.000060 0.000070 0.000080 0.00010 0.00013 0.00020 0.00030 0.00040 0.00050

αs 0.000234 0.000333 0.000472 0.000606 0.000922 0.00135 0.00177 0.00222 0.00262 0.00301 0.00340 0.00422 0.00535 0.00787 0.0116 0.0135 0.0154 0.0188 0.0223 0.0262 0.0300 0.0379 0.0514 0.0833 0.133 0.183 0.227

θ 0.000414 0.000590 0.000835 0.00107 0.00163 0.00238 0.00314 0.00392 0.00464 0.00533 0.00602 0.00747 0.00948 0.0139 0.0205 0.0238 0.0272 0.0333 0.0395 0.0464 0.0530 0.0671 0.0910 0.147 0.236 0.324 0.402

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Kazuyuki Nakai, Yoko Nakada, Masako Hakuman et al. Table 6. (Continued) p/po 0.00060 0.00080 0.0010 0.0013 0.0020 0.0030 0.0050 0.0060 0.0080 0.010 0.013 0.020 0.030 0.040 0.050 0.060 0.080 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.35 0.37 0.40

αs 0.264 0.316 0.348 0.383 0.424 0.453 0.484 0.494 0.508 0.516 0.526 0.542 0.557 0.568 0.578 0.586 0.604 0.621 0.638 0.653 0.670 0.688 0.709 0.730 0.755 0.780 0.810 0.837 0.867 0.898 0.914 0.950 1.000

θ 0.468 0.559 0.615 0.678 0.750 0.802 0.856 0.874 0.899 0.914 0.931 0.959 0.985 1.00 1.02 1.04 1.07 1.10 1.13 1.16 1.19 1.22 1.26 1.29 1.34 1.38 1.43 1.48 1.53 1.59 1.62 1.68 1.77

n0.4 = 0.120 mmol g-1. nθ=1 = 0.0682 mmol g-1.

Our standard  S -data at 77.4 K by Carbotrap-F-CB (  S =0.00019 (p/po=2  10-7) to  S =2.231 (p/po=0.90)) are in good agreement with those by the same adsorbent reported by Kruk and coworkers [16] until  S =0.004. As has been seen in section (3.1.), these two standard  S data of N2 at 77.4 K by Carbotrap-F-CB play an important role for confirmation of the Henry‟ law.

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5. Relation between the s-Surface Area ( A ) and the Slope of s Plots S

According to the  S method proposed by Sing [17], the slope of the  S plot is proportional to the surface area of test adsorbent; that is, A s =constant  (slope of  S -plot)

(5.1)

Constant of equation (5-1) is defined as the ratio of specific surface area of reference adsorbent (Aref) to the adsorbed amount of adsorptive at p/po=0.4 (n0.4). A s =(Aref/n0.4)  (slope of  S -plot)

(5.2)

When the equation (5.2) is applied to the adsorption isotherms of N2 (77.4 K) by the CB adsorbents (reference adsorbents), we used ABET as Aref. Furthermore, the adsorbed amount at p/po=0.4 (n0.4) and the ordinate of the  S -plot is expressed as mmol g-1. For nongraphitized CB (#51-CB), 2 -1 A s =60.8  (slope of  S -plot)m g

(5.3)

for graphitized CB (#3845-2-CB), 2 -1 A s =53.5  (slope of  S -plot)m g

(5.4)

and for highlygraphitized CB (Carbotrap-F-CB), 2 -1 A s =52.5  (slope of  S -plot)m g

(5.5)

The constant 60.8 in equation (5-3) for nongraphitized CB is in good agreement with those reported by Kaneko and coworkers [15] and by Kruk and coworkers [16]. And the constant 52.5 in equation (5-5) for Carbotrap-F-CB coincides with that for the same adsorbent reported by Kruk and coworkers [16]. In conclusion, equation (5.3) can be applied to determination of surface areas of carbon adsorbents of heterogeneous surface. Surface areas of many commercial nongraphitized CB adsorbents can be evaluated through equation (5.3). On the

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other hand, equation (5.5) can be used in determination of carbon adsorbents of homogeneous surface such as graphite et al. In the present work, the external surface areas of the CB aggregates as well as the surface areas of supermicropores were determined by means of equation (5.3).

6. Analysis of Microporosity of ACF and SAC In our previous work [9], the standard  S -data of N2 at 77.4 K by nongraphitized sample#51-CB was used in the research of microporous textures of the activated carbon fibers (FT-07-ACF and FT-25-ACF). In the present work, the revised standard  S -data N2 at 77.4 K by #51-CB will be used for analysis of microporosity of a series of ACF and SAC adsorbents. In the revised  S -data, the number of digital  S -data are 1.5 times compared with the previous  S -data given in Table 2 of reference 8. As a result, it is possible to analyze microporosity by the  S plots in detail. In addition, in Table 7, ABET=18.5 m2g-1 for #51-CB is used instead of ABET=18.9 m2g-1 which has been used in the previous paper [8]. Table 7. Standard s data of N2 at 77.4 K by #51-CB Relative pressure ranges: p/po=5  10-8– 0.900 p/po 0.00000005 0.00000006 0.00000007 0.00000008 0.00000009 0.00000010 0.00000012 0.00000015 0.00000020 0.00000025 0.00000030 0.00000035 0.00000040 0.00000045 0.0000005 0.0000007 0.0000010 0.0000013 0.0000017 0.0000020 0.0000025

αs 0.00219 0.00280 0.00342 0.00403 0.00451 0.00495 0.00571 0.00659 0.00763 0.00851 0.00933 0.0102 0.0111 0.0118 0.0126 0.0152 0.0184 0.0210 0.0237 0.0256 0.0283

θ 0.00357 0.00458 0.00559 0.00657 0.00737 0.00808 0.00932 0.0108 0.0125 0.0139 0.0152 0.0166 0.0181 0.0193 0.0205 0.0249 0.0301 0.0343 0.0386 0.0418 0.0463

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t / nm 0.00126 0.00162 0.00198 0.00233 0.00261 0.00286 0.00330 0.00381 0.00441 0.00492 0.00539 0.00588 0.00640 0.00682 0.00727 0.00881 0.0106 0.0121 0.0137 0.0148 0.0164

Characterization of Carbon Blacks by High Resolution N2 … Table 7. (Continued) p/po 0.0000030 0.0000035 0.0000040 0.0000045 0.0000050 0.0000055 0.0000060 0.0000070 0.0000080 0.0000090 0.000010 0.000013 0.000016 0.000020 0.000023 0.000026 0.000030 0.000035 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009 0.00010 0.00012 0.00013 0.00016 0.00020 0.00025 0.00030 0.00035 0.00040 0.00045 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0013 0.0016 0.0020 0.0025 0.003

αs 0.0309 0.0335 0.0359 0.0380 0.0402 0.0424 0.0445 0.0473 0.0503 0.0531 0.0560 0.0618 0.0673 0.0733 0.0774 0.0814 0.0862 0.0919 0.0977 0.106 0.115 0.122 0.129 0.137 0.144 0.155 0.161 0.175 0.190 0.208 0.223 0.237 0.250 0.260 0.270 0.283 0.298 0.309 0.319 0.331 0.352 0.367 0.382 0.398 0.410

θ 0.0505 0.0547 0.0587 0.0620 0.0656 0.0693 0.0727 0.0772 0.0822 0.0866 0.0914 0.101 0.110 0.120 0.126 0.133 0.141 0.150 0.160 0.174 0.188 0.199 0.211 0.223 0.235 0.253 0.263 0.285 0.310 0.339 0.364 0.387 0.409 0.425 0.441 0.463 0.487 0.505 0.521 0.540 0.575 0.599 0.625 0.649 0.669

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t / nm 0.0179 0.0194 0.0208 0.0219 0.0232 0.0245 0.0257 0.0273 0.0291 0.0307 0.0323 0.0357 0.0389 0.0424 0.0447 0.0470 0.0498 0.0531 0.0565 0.0615 0.0665 0.0705 0.0746 0.0790 0.0833 0.0894 0.0931 0.101 0.110 0.120 0.129 0.137 0.145 0.150 0.156 0.164 0.172 0.179 0.185 0.191 0.204 0.212 0.221 0.230 0.237

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αs 0.427 0.441 0.453 0.462 0.472 0.477 0.485 0.499 0.509 0.522 0.534 0.546 0.557 0.568 0.575 0.582 0.589 0.597 0.608 0.621 0.633 0.644 0.668 0.687 0.711 0.733 0.754 0.780 0.802 0.825 0.848 0.874 0.899 0.922 0.948 0.974 1.000 1.023 1.049 1.075 1.099 1.124 1.147 1.178 1.207

θ 0.697 0.721 0.740 0.755 0.770 0.779 0.791 0.815 0.832 0.852 0.872 0.892 0.910 0.927 0.939 0.951 0.961 0.974 0.993 1.014 1.033 1.052 1.091 1.122 1.161 1.197 1.232 1.274 1.309 1.347 1.384 1.427 1.469 1.506 1.549 1.591 1.633 1.671 1.713 1.755 1.795 1.835 1.872 1.924 1.971

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t / nm 0.247 0.255 0.262 0.267 0.273 0.276 0.280 0.288 0.294 0.302 0.309 0.316 0.322 0.328 0.332 0.337 0.340 0.345 0.352 0.359 0.366 0.372 0.386 0.397 0.411 0.424 0.436 0.451 0.463 0.477 0.490 0.505 0.520 0.533 0.548 0.563 0.578 0.591 0.606 0.621 0.635 0.650 0.663 0.681 0.698

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Table 7. (Continued) αs 1.239 1.272 1.300 1.330 1.365 1.402 1.451 1.494 1.537 1.595 1.652 1.710 1.782 1.868 1.911 1.968 2.040 2.112 2.180 2.270

p/po 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.85 0.86 0.87 0.88 0.89 0.90

θ 2.022 2.076 2.123 2.172 2.229 2.290 2.370 2.440 2.511 2.604 2.698 2.792 2.909 3.050 3.121 3.214 3.332 3.449 3.560 3.707

t / nm 0.716 0.735 0.752 0.769 0.789 0.811 0.839 0.864 0.889 0.922 0.955 0.988 1.030 1.080 1.105 1.138 1.179 1.221 1.260 1.312

n0.4 = 0.310 mmol g-1. nθ=1 = 0.190 mmol g-1. t= 0.354×θ nm. ABET = 18.5 m2 g-1. CBET = 260.

6.1. Microporosities of a Series of ACF by High Resolution s Plots of N2 at 77.4 K Figure 17 shows the adsorption-desorption isotherms of N2 at 77.4 K by a series of activated carbon fibers (a: FT-07-ACF, b: FT-10-ACF, c: FT-15-ACF, d: FT-20-ACF, and e: FT-25-ACF). It is evident from Figure 17 that, according to increase in the adsorbed amounts of N2, the pattern of the N2 isotherms change from Type Ia (FT-07-ACF) to Type Ib (FT-20-ACF), which suggests that supermicropores successively increase from FT-07-ACF to FT-25-ACF. ACF adsorbents (a), (b), (c), and (d) give no adsorption hysteresis at the relative pressure ranges of p/po=0.1–0.95, but ACF adsorbent (e) has a small adsorption hysteresis suggesting formation of a small amount of mesopores. To see the adsorption behavior of N2 molecules at low relative pressures of p/po=10-8–1.0 in detail, the reduced amounts of N2 (n/n0.4) are plotted against log(p/po) for a series

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of ACF adsorbents (Figure 18). In Figure 18, the reduced N2 isotherms by #51-CB and Carbotrap-F-CB are also listed. Prior to analysis of the N2 isotherms by the  S method, it is of considerable importance to discuss which  S data are appropriate as the standard  S data. We selected the standard  S data by #51-CB (cf. Table 7) as the reference data. The reason of selection is due to the following experimental facts. One is that the reduced N2 adsorption isotherms by a series of ACF adsorbents, as is seen from Figure 18, run almost parallel to that by #51-CB (heterogeneous carbon adsorbent), but not parallel by Carbotrap-F-CB (highly homogeneous carbon adsorbent).

Figure 17. Adsorption-desorption isotherms of N2 at 77.4 K by a series of activated carbon fibers (a: FT-07-ACF, b: FT-10-ACF, c: FT-15-ACF, d: FT-20-ACF, e: FT-25-ACF).

The second is that, as has been reported in our previous paper [9], the qst behavior of N2 by FT-25-ACF is similar to that by #51-CB (cf. Figure 21 of reference 9). Furthermore, k2 of equation (3-8) is smaller than 1; e.g., k2 =0.28 by FT-25-ACF, whose value is smaller than k2 =0.49 by #51-CB. Such small value of k2 by FT-25-ACF may be due to microporosity effect in addition to surface heterogeneity. Discussion between k2 values and microporosity will be reported elsewhere. The high resolution  S plots by FT-10-ACF, FT-15-ACF, and FT-20-ACF are shown in Figure 19. The  S plots by FT-07-ACF and FT-25-ACF have been

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already reported in the previous paper [9] (cf. Figures 11, 14, 18 of reference 9). The three  S plots in Figure 19 clearly indicate that the initial steep rise appears in the ranges of  S = 0 – 0.15, which is due to the primary filling into ultramicropores, and the linear line is followed in the ranges of  S =0.15–0.50, which is due to the secondary filling into supermicropores. The upper deviation of  S plots from the linear line (co-operative swing named by Kaneko and coworkers [20, 21]) can be detected above  S =0.50 by FT-15-ACF and FT-20-ACF.

Figure 18. Reduced adsorption isotherms of N 2 at 77.4 K by a series of ACF in the region of p/po=10-8-1.0 together with the isotherms by nonporous reference carbon black (#51-CB and Carbotrap-F-CB). ACF adsorbents; a: FT-07-ACF, b: FT-10-ACF, c: FT-15-ACF, d: FT-20-ACF, e: FT-25-ACF.

The volumes of ultramicropores and supermicropores ( Vultramicropore ,

Vsup ermicropore ) were estimated from the adsorbed amounts ( n0I , n0II ) at  S =0, which were obtained by back extrapolation of lines I and II to  S =0 (see Figure 19).

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Figure 19.  S plots of N2 at 77.4 K by a series of activated carbon fibers (b: FT-10-ACF, c: FT-15-ACF, d: FT-20-ACF).

In calculation of Vultramicropore and

Vsup ermicropore , the density of the N2

adsorbate was assumed to be equal to the density of liquid N2 at 77.4 K (0.808 cm3g-1). The surface areas of supermicropore ( Asup ermicropore ) as well as the external surface area ( Aexternal ) were evaluated from the slopes of the lines I and II, where equation (5.3) was used. Average pore width of the supermicropores av ( d supermicro pore ) can be calculated by equation (6.1), where pore shape is assumed

to be slit-shaped. av d supermicro pore 

2  Vsupermicropore Asupermicropore

(6.1)

The digital data of Vultramicropore , Vsupermicropore , Asupermicropore , Aexternal , and av d supermicro pore for FT-10-ACF, FT-15-ACF, and FT-20-ACF are listed in Table 8

together with FT-07-ACF and FT-25-ACF. Some comments will be given about the surface area of ultramicropores ( Aultramicropore ). As is seen from Figure 19, it is

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 S =0–0.15, because the slope of

the  S plots remarkably changes. As a result, it is difficult to estimate Aultramicropore .

Table 8. Results obtained from analysis of s plots shown in Figures 19, 22, 23, 24 surface area (Aαs) /m2g-1 3 -1 FT-07 pore volume (Vp) /cm g -ACF Vp/Vp(total) / % pore width /nm surface area (Aαs) /m2g-1 3 -1 FT-10 pore volume (Vp) /cm g -ACF Vp/Vp(total) / % pore width /nm surface area (Aαs) /m2g-1 3 -1 FT-15 pore volume (Vp) /cm g -ACF Vp/Vp(total) / % pore width /nm surface area (Aαs) /m2g-1 3 -1 FT-20 pore volume (Vp) /cm g -ACF Vp/Vp(total) / % pore width /nm surface area (Aαs) /m2g-1 3 -1 FT-25 pore volume (Vp) /cm g -ACF Vp/Vp(total) / % pore width /nm surface area (Aαs) /m2g-1 3 -1 AX21 pore volume (Vp) /cm g -SAC Vp/Vp(total) / % pore width /nm surface area (Aαs) /m2g-1 3 -1 MSP-20 pore volume (Vp) /cm g -SAC Vp/Vp(total) / % pore width /nm surface area (Aαs) /m2g-1 3 -1 MSP-30 pore volume (Vp) /cm g -SAC Vp/Vp(total) / % pore width /nm

ultramicropore supermicropore mesopore external Uncertain 203 0 14 0.18 0.09 0 67 33 0 0.90 0.4 ~ 0.7 Uncertain 392 0 6 0.21 0.15 0 58 42 0 0.79 0.4 ~ 0.7 Uncertain 694 0 9 0.23 0.29 0 44 56 0 0.83 0.4 ~ 0.7 Uncertain 1022 0 20 0.22 0.48 0 31 69 0 0.94 0.4 ~ 0.7 Uncertain 1538 72 40 0.14 0.95 0.14 10 80 10 1.20 3.9 0.4 ~ 0.7 Uncertain 1775 0 45 0.16 1.05 0 13 87 0 1.18 0.4 ~ 0.7 Uncertain 1603 0 36 0.27 0.79 0 25 75 0 0.99 0.4 ~ 0.7 Uncertain 2343 0 73 0.14 1.56 0 8 92 0 1.33 0.4 ~ 0.7

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The digital data in Table 8 are represented as the histograms of micropore volumes in Figure 20. It is evident from Figure 20 that the ultramicropore volumes for a series of ACF adsorbents are almost constant, while the supermicropore volumes gradually increase from FT-07-CB to FT-20-CB. In the case of FT-25-CB, a small amount of mesopores appears together with a slight decrease in Vultramicropore . 6.2. Microporosities of a Series of Superactive Carbons by High Resolution s Plots of N2 at 77.4 K Figure 21 compares the reduced N2 adsorption isotherms at 77.4 K by the SAC adsorbents with those by #51-CB and Carbotrap-F-CB. The N2 isotherm by the SAC adsorbents run almost parallel to that by #51-CB, but not parallelto that by Carbotrap-F-CB. In the  S plot analysis for SAC adsorbents, we selected the N2 isotherm by #51-CB as the reference isotherm, because the surface heterogeneity of #51-CB may be similar to that of SAC adsorbents. The N2 adsorption isotherms at 77.4 K and their  S plots by three kinds of the SAC adsorbents are shown in Figures 22 – 24, respectively. All isotherms by SAC are of Type Ib. The  S plots in these figures clearly show the initial steep rise at  S = 0–0.15, followed by the linear line at  S =0.15–0.50, and finally the upper deviation at  S =0.5–1.0. The behavior of the  S plots by SAC adsorbents is very similar to that by ACF-adsorbents. The values of Vultramicropore , av Vsupermicropore , Asupermicropore , Aexternal , and d supermicro pore were estimated according

to the procedures described in section 6.1. These values are listed in Table 8 and Figure 20 together with those of ACF adsorbents. From the numerical data in Table 8, the ratio of Vsupermicropore by Vtotal is found to be 87%, 75% and 92% for AX21-SAC, MSP-20-SAC, and MSP-30SAC, respectively, which indicates that supermicropores mainly occupy the pore volume of SAC-adsorbents. And the ratio of Aexternal by Asupermicropore is estimated to be ca. 2%, meaning that the external surface is negligibly small. From the BET analysis, the BET monolayer capacities ( nm ), the BET SAC constant ( CBET ), and the BET surface areas ( ABET ) of AX21-SAC, MSP-20SAC, and MSP-30-SAC were determined, and the results are listed in Table 9. It

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is evident that these SAC adsorbents have large BET surface areas; e.g. SAC =3416 m2g-1 for MSP-30-SAC. ABET

Figure 20. Pore volume histograms of ultramicropores, supermicropores, and mesopores for a series of activated carbon fibers (FT-07-ACF, FT-10-ACF, FT-15-ACF, FT-20-ACF, and FT-25-ACF)、and for superactive carbons (AX-21-SAC, MSP-20-SAC, MSP-30-SAC).

Figure 21. Reduced adsorption isotherms of N 2 at 77.4 K by a series of SAC in the region of p/po=10-8-1.0 together with the isotherms by nonporous reference carbon black (#51-CB and Carbotrap-F-CB). I: AX-21-SAC, II: MSP-20-SAC, III: MSP-30-SAC.

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Figure 22. (a) Adsorption-desorption isotherm of N2 at 77.4 K and (b) its

 S plot by

superactive carbon (AX-21-SAC).

Figure 23. (a) Adsorption-desorption isotherm of N2 at 77.4 K and (b) its superactive carbon (MSP-20-SAC).

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Here, some comments will be given for the BET surface areas of SAC. It has been well known that the superactive carbon adsorbents have extremely large BET surface areas from 2000–3700 m2g-1[38]. The BET areas of SAC adsorbents serve one of the important parameters of adsorption capacity of SAC adsorbents. However, it should be mentioned that the BET areas of SAC adsorbent do not indicate the true surface area. As is seen from the  S plots in Figures 22–24, the SAC adsorbents have always ultramicropores. It has been pointed out that it is difficult to determine the surface area of ultramicropores by the BET method [12].

Figure 24. (a) Adsorption-desorption isotherm of N2 at 77.4 K and (b) its  S plot by superactive carbon (MSP-30-SAC).

Furthermore, as is seen from  S plots in Figures 22-24, nm corresponds to

 S =0.60-0.65, which means that co-operative swing into the supermicropores is superimposed in addition to the adsorption of N2 molecules on the supermicropore surface. These two factors make it difficult to evaluate the true surface areas of SAC adsorbents.

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Kazuyuki Nakai, Yoko Nakada, Masako Hakuman et al. Table 9. BET analysis of N2 isotherms at 77.4 K by SAC adsorbents Monolayer Capacity -1 nm /mmol g

BET surface area SAC / m2 g-1 ABET

BET constant CBET

type of isotherm

AX21-SAC

27.1

2645

101

Ib

MSP-20-SAC MSP-30-SAC

25.0 35.0

2440 3416

887 60

Ib Ib

adsorbent name

6.3. Microporous Textures of ACF- and SAC-Adsorbents On the basis of the  S plot analysis (Table 8), we propose the simplified micropore model of ACF-adsorbents and SAC-adsorbents (Figure 25).

Figure 25. Simplified model of microporous texture obtained from detailed analysis of  S plots of N2 isotherms by a series of ACF and SAC adsorbents.

The primary micropore filling into the ultramicropores is observed in the ranges of  S =0–0.15, where the pore width may be in the vicinity of single layer of N2 molecules. The secondary micropore filling into the supermicropores is detected in the ranges of  S =015–050, where the pore width may be ca. double layers of N2

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molecules. And the upper deviation of the  S plots (co-operative swing) occurs in the  S =0.5–1.0, where the pore width is more than three layers of N2 molecules. It seems that the co-operative swing of the  S plots [15,20,21] appears as the filling of N2 molecules into the free space between the walls whose surface is already occupied by the adsorbed monolayer of N2 molecules. In our present work, it was able to differentiate between the primary filling and secondary filling. The boundary between the two fillings lies at  S =0.15 for both of ACF adsorbents and SAC adsorbents, and  S =0.15 corresponds to p/po=10-4 of the N2 isotherm by the reference adsorbents (#51-CB). And the co-operative swing begins at  S =0.5, which corresponds to p/po=0.013 of the N2 isotherm by #51-CB.

Acknowledgments We wish to express our sincere thanks to Professor K. Morishige for giving us the valuable comments about our manuscript.

References [1] [2]

[3] [4] [5] [6] [7]

Bansal, R. C., Donnet, J.-B. and Stoeckli, F., Active Carbon, Marcel Dekker, Inc., NY, 1988, p.119-162. Hess, W. M. and Herd, C. R., Carbon Black (2nd Ed., Revised and Expanded), Donnet, J.-B., Bansal, R. C. and Wang, M-J.; Ed.: Marcel Dekker, Inc., NY, 1993, p. 89-173. Marsh, H. and Rodríguez-Reinoso, F., Activated Carbon, Elsevier, Amsterdam, 2006, p. 143-242. Sing, K. S. W., Adsorption by Carbons, Bottani, E. J. and Tascón, J. M. D.; Ed.: Elsevier, Amsterdam, 2008. p. 3-14. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 219-285. Nakai, K., Sonoda, J., Yoshida, M., Hakuman, M., and Naono, H., Stud. Surf. Sci. Catal., 2007, 170, 831. Nakai, K., Sonoda, J., Yoshida, M., Hakuman, M., and Naono, H., Adsorption, 2007, 13, 351.

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[17] [18] [19] [20] [21] [22] [23]

[24]

[25] [26]

Kazuyuki Nakai, Yoko Nakada, Masako Hakuman et al. Nakai, K., Yoshida, M., Sonoda, J., Nakada, Y., Hakuman, M., and Naono, H. J. Colloid Interface Sci.2010, 351, 507. Nakai, K., Nakada, Y., Hakuman, M., Yoshida, M., Senda, Y., Tateishi, Y., Sonoda, J., and Naono, H., J. Colloid Interface Sci. 2012, 367, 383. Brunauer, S., Emmett, P. H. and Teller, E., J. Am. Chem. Soc., 1938, 60, 309. Gregg, S. J. and Sing, K. S. W., Adsorption, Surface Area and Porosity, 2nd ed., Academic Press, London, 1982, p. 41-61. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 166-176. Barrett, E.P., Joyner, L. G. and Halenda, P. P., J. Am. Chem. Soc., 1951, 73, 373. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 94-96. Kaneko, K., Ishii, C., Kanoh, H., Hanzawa, Y., Setoyama, N., and Suzuki, T., Adv. Colloid Interface Sci., 1998, 76-77, 295. Kruk, M., Jaroniec, M. and Gadkaree, K. P., J. Colloid Interface Sci., 1997,192, 250; Kruk, M., Li, Z., Jaroniec, M., and Betz, W. R., Langmuir, 1999, 15, 1435. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 176-179. Sing, K. S. W., Colloids and Surfaces, 1989, 38, 113. Carrott, P. J. M., Roberts, R. A. and Sing, K.S. W., Stud. Surf. Sci. Catal., 1988, 39, 89. Kaneko, K., Ishii, C., Ruike, M., and Kuwabara, H., Carbon, 1992, 30, 1075. Setoyama, N., Suzuki, T. and Kaneko, K., Carbon, 1998, 36, 1459. Nakai, K., Sonoda, J., Iegami, H., and Naono, H., Adsorption, 2005, 11, 227. Takaishi, T. and Sensui, Y., Trans. Faraday Soc.,1963, 59, 2503; Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 87-89. Isirikyan A. A. and Kiselev, A.V., J. Phys. Chem., 1961, 65, 601; Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 242-247. Rouquerol, J., Partyka, S. and Rouquerol, F., J. Chem. Soc., Faraday Trans. 1, 1977, 73, 306. ASTM D 3849, Annual Book of ASTM Standards, 1990, Vol.09.01, p. 630; W. M. Hess, W. M. and Herd, C. R. Carbon Black (2nd Ed., Revised and

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[27] [28] [29] [30] [31] [32] [33] [34] [35]

[36] [37] [38]

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Expanded), Donnet, J.-B., Bansal, R. C. and Wang, M.-J.; Ed.: Marcel Dekker, Inc., NY, 1993, p. 106-108. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 6-10. Gregg, S. J. and Sing, K. S. W., Adsorption, Surface Area and Porosity, 2nded., Academic Press, London, 1982, p. 89-90. Lippens, B. C., Linsen, B. G. and de Boer, J. H., J. Catalysis, 1964, 3, 32. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 192-197. Kagaku-Binran (Fundamental Data) (4th rev. ed.), Chemical Society of Japan; Ed.: Maruzen Co. Ltd., Tokyo, 1993, II-207, II-245. Hakuman, M. and Naono, H. J. Colloid Interface Sci.2001, 241, 127. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p.439-443. Gregg, S. J. and Sing, K. S. W., Adsorption, Surface Area and Porosity, 2nd ed., Academic Press, London, 1982, p. 13-18. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 103-107, p. 250252, p. 333-336. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 18-20. Gregg, S. J. and Sing, K. S. W., Adsorption, Surface Area and Porosity, 2nd ed., Academic Press, London, 1982, p. 94-100. Rouquerol, F., Rouquerol, J. and Sing, K. Adsorption by Powders and Porous Solids, Academic Press, San Diego, CA, 1999, p. 404-412.

Reviewed by Professor Kunimitsu Morishige, Okayama University of Science.

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In: Advances in Chemistry Research. Volume 21 ISBN: 978-1-62948-742-7 © 2014 Nova Science Publishers, Inc. Editor: James C. Taylor, pp. 149-183

Chapter 4

DOUBLY BONDED MOLECULES CONTAINING BISMUTH AND OTHER GROUP 15 ELEMENTS IN THE SINGLET AND TRIPLET STATES Ming-Der Su* Department of Applied Chemistry, National Chiayi University, Chiayi, Taiwan

Abstract The lowest singlet and triplet potential energy surfaces for all the group 15 HBiXH (X = N, P, As, Sb, and Bi) systems have been explored through ab initio calculations. The geometries of the various isomers were determined at the QCISD/LANL2DZdp level, and confirmed to be minima by vibrational analysis. In the case of nitrogen, the order of stability is triplet H2NBi > singlet H2NBi > singlet cis-HBi=NH ≈ singlet trans-HBi=NH > triplet HBiNH > triplet H2BiN > singlet H2BiN. For the phosphorus case, the stability decreases in the order triplet H2PBi > singlet trans-HBi=PH > singlet cis-HBi=PH > triplet HBiPH > singlet H2PBi > triplet H2BiP > singlet H2BiP. For arsenic, theoretical investigations demonstrate that the stability of the HBiAsH isomers decreases in the order triplet H2AsBi ≈ singlet trans-HBi=AsH > singlet cis-HBi=AsH > triplet HBiAsH > triplet H2BiAs > singlet H2AsBi > singlet H2BiAs. For antimony, the theoretical findings suggest that the stability of the HBiSbH system decreases in the order singlet trans-HBi=SbH > singlet cis-HBi=SbH > triplet H2SbBi > triplet H2BiSb > triplet HBiSbH > singlet H2SbBi > singlet H2BiSb. For bismuth, the theoretical investigations indicate that the stability of the *

E-mail address: [email protected]

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Ming-Der Su HBiBiH system decreases in the order singlet trans-HBi=BiH > singlet cisHBi=BiH > triplet H2BiBi > triplet HBiBiH > singlet H2BiBi. Our model calculations indicate that relativistic effects on heavier group 15 elements should play an important role in determining the geometries as well as the stability of HBiXH molecules. The results obtained are in good agreement with the available experimental data and allow a number of predictions to be made.

1. Introduction During the last two decades, the synthesis of monomeric group 15 compounds of the type RM=MR (M = pnicogen elements) has received wide interest because of their unusual structure and properties when compared with compounds containing second-row elements, such as olefins (R2C=CR2) and ketones (R2C=O). Most have been isolated as stable compounds by taking advantage of kinetic stabilization by using appropriately bulky substituents as steric protection groups [1]. Of these, the syntheses of stable double-bond compounds containing bismuth have been a particular challenge to synthetic chemists [1(f)]. Through the elegant studies performed by Tokitoh, Power, and many coworkers, kinetically stabilized molecules bearing a Bi=Bi double bond, dibismuthenes, have been synthesized and fully characterized [2]. They are TbtBi=BiTbt (Tbt = 2,4,6tris[bis(trimethylsilyl)methyl]phenyl) [2(a)] and 2,6-Ar2C6H3Bi=BiAr2C6H3 (Ar = mesityl or 2,4,6-triisopropylphenyl) [2(b)]. To the best of our knowledge, two examples of heteronuclear doubly bonded systems between bismuth and other group 15 elements have been reported. One is phosphabismuthene, BbtBi=PMes* (Bbt = 2,6-bis[bis(trimethylsilyl)methyl]-4-[tris(trimethylsilyl)methyl]phenyl and Mes* = 2,4,6-tri-tert-butylphenyl) [3], a novel doubly bonded system involving bismuth and phosphorus. The other is the first stable stibabismuthene (BbtBi=SbBbt), which has been synthesized again by taking advantage of the Bbt group [4]. As a result, doubly bonded systems between heavier group 15 elements are no longer imaginary species even in the case of bismuth. Nevertheless, attempts to isolate other analogues, iminobismuthene (RBi=NR) and arsanbismuthene (RBi=AsR), have all been unsuccessful up to now. There have been only a few theoretical studies on the compounds containing bismuth double bonds reported so far. The theoretical investigation of HM=MH (M = P, As, Sb, and Bi) was first explored by Nagase et al [5]. These authors predicted that all the doubly bonded compounds between the heavier group 15 elements could be isolated as stable species with the use of appropriate steric protection groups. Schoeller et al reported MCSCF studies on HBi=BiH and HBi=YH2 (Y = C, Si, Ge, and Sn) systems [6]. These authors investigated the -

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bond strengths and singlet-triplet splittings for the various structures of HBi=BiH and HBi=YH2. Density functional theory (DFT) calculations for the PhBi=BiPh model have been first reported by Cotton, Cowley and Feng [7]. Further, Tokitoh et al reported DFT results for some substituted RBi=PR species [3]. In a later study by Tokitoh and coworkers [1(i)], the experimental investigations of dibismuthenes were supplemented by theoretical studies on both parent and phenyl substituted Bi=Bi doubly bonded systems (i.e., HBi=BiH, PhBi=BiH, PhBi=BiPh, HBi=BiTbt, and TbtBi=BiTbt). It should be stressed that these authors were mainly concerned with the evaluation of the groundstate geometry. However, none of the studies have dealt with either the magnitude of the energy barriers to isomerizations or the singlettriplet energy separations in the bismuth double bond systems. Moreover, to our knowledge, no theoretical study concerning either HBi=NH or HBi=AsH has been reported to date. In this paper, we present a theoretical study of HBi=XH potential energy surfaces, with X varying from nitrogen to bismuth. Our goal is to obtain more insight into the bonding character of their various isomers, and to provide structural and energetic data. In addition, the purpose of the work reported in this paper is to provide theoretical information on the thermodynamic and kinetic stability of the various HBiXH isomers. To achieve this, it is necessary to determine the transitionstate (TS) geometries of the molecules as well as the groundstate energies of the products of isomerization. Unimolecular reactions pertinent to the stability of HBi=XH considered here are as follows: (A) trans-HBi=XH  TS1  cis-HBi=XH, (B) trans-HBi=XH  TS2  H2X=Bi, and (C) trans-HBi=XH  TS3  H2Bi=X. For comparison, the singlet and triplet BiXH2 potential energy surfaces were investigated using the QCISD method. It is believed that this study will aid further developments in group 15 chemistry. In particular, the predicted molecular parameters and harmonic vibrational frequencies will assist experimental study of the presently unknown HBiXH (X  N, P, As, Sb, and Bi) isomers.

2. Methodology The geometries of all the stationary points were fully optimized at the QCISD(FC) level of theory [8]. QCISD calculations were carried out using the Gaussian 03 program [9] with relativistic effective core potentials on group 15 elements modeled using the double-zeta (DZ) basis sets [10] augmented by a set of d-type polarization functions [11]. The DZ basis set for the hydrogen element

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[10(a)] was augmented by a set of p-type polarization functions (p exponents 0.356). The d exponents used for N, P, As, Sb and Bi are 0.736, 0.364, 0.286, 0.207, and 0.192, respectively. Accordingly, all the QCISD(FC) calculations are denoted by QCISD/LANL2DZdp. All the structures obtained were confirmed to be real minima or transition states via frequency analysis, which was also used to calculate zero-point energies (ZPEs) without scaling. For all the transition states, motion corresponding to the imaginary frequency was checked visually, and most structures were visually optimized to the minima they connected after perturbing the TS geometry.

3. Results and Discussion As mentioned in the Introduction, there are only three kinds of molecular RBi=XR species that have been identified experimentally, i.e., RBi=PR [3], RBi=SbR [4], and RBi=BiR [2]. In order to evaluate the theoretical method, QCISD, we will first make comparisons between our computational results and the experimental observations. We will begin by discussing the HBiBiH molecule. Then, we will turn our attention to the other species as follows: HBiPH, HBiSbH, HBiNH, and finally HBiAsH.

3.1. HBiBiH Dibismuthene (TbtBi=BiTbt) [2(a)] was the first compound synthesized containing a bismuthbismuth double bond, the heaviest double bond containing stable elements. Later on, Power and co-workers also synthesized another type of stable dibismuthene substituted by bulky 2,6-Ar2C6H3 groups [2(b)]. Although only two crystallographic investigations on substituted dibismuthenes have been carried out during the last five years, no experimental geometries are so far available in the literature for the parent HBiBiH isomers. Due to this, the reliability of our predicted geometries can only be estimated by comparison between different levels of theory. Indeed, the HBiBiH species has attracted the interest of several theoretical groups, as mentioned in the Introduction [1(i),5,6]. For comparison, the optimized geometries of the transHBi=BiH species calculated at the QCISD/LANL2DZdp level of theory are listed in Table 1 along with previous theoretical work and some available experimental data [2].

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Considering that QCISD is the highest level ab initio computational method, these computed structures should be the most accurate. Apparently, the effects of electron correlation are relatively large. For instance, it is seen from Table 1 that electron correlation typically increases the RHF bond length by 0.078Å and changes the bond angle by 2.2°. Moreover, our QCISD calculations predict that the Bi=Bi bond length and the HBiBi angle are 2.80Å and 89.6°, which are in good agreement with the experimental values (2.822.83Å and 100.2°92.5°) as shown in Table 1, bearing in mind that the experimental structures contain bulkier groups. In any event, the good agreement between our computational results and available experimental data is quite encouraging. We therefore believe that the QCISD/LANL2DZdp level employed in this work can provide accurate molecular geometries for those unimolecular isomerization reactions, for which experimental data are not available. It has to be emphasized here that all the computational results presented in Table 1 predict the HBiBi angle about 90°, while the experimentally corresponding angles are 100.5° [2(a)] and 92.5° [2(b)]. These values deviate greatly from the ideal sp2 hybridized bond angle (120°). The reason for is because the size-difference and energy-gap between the valence s and p orbitals are known to increase upon going from N to Bi atoms due to the relativistic effect [1(i),2,5,12,13]. As a result, bismuth prefers to maintain the (6s)2(6p)3 valence electron configuration. Consequently, the use of almost pure 6p orbitals instead of more hybridized orbitals leads to a bond angle of ~ 90° at each bismuth [13]. In other words, the reason for RBi=BiR adopting the trans-bent form can be attributed to the phenomenon of orbital nonhybridization (or the so called “inert spair effect”) [12,13]. In this work, three stable closed-shell structures on the HBiBiH potential energy surface (i.e., trans-HBi=BiH, cis-HBi=BiH, and H2BiBi) have been studied by using the QCISD level of theory. In addition, the open-shell triplet states of these isomers have also been investigated. The optimized geometries and relative energies of the HBiBiH species on the energetically lowest singlet and triplet potential energy surfaces from QCISD calculations are shown in Figure 1. The vibrational frequencies, as well as the rotational constants, dipole moments, and net atomic charges calculated for both singlet and triplet HBiBiH isomers are collected in Table 1.

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Table 1. Calculated Harmonic Vibrational Frequencies, Rotational Constants, Dipole Moments, and Net Atomic Charges of the Species in HBiXH Isomerization Reactions at the QCISD/LANL2DZdp Level of Theory

Species

Frequencies (cm-1)

trans-HBi=NH

3384, 1772, 947, 763, 655, 605

TS2

3445, 767, 651, 445, 266, 1203i

H2NBi

3635, 3529, 1561, 716, 584, 363

TS3

1751, 1710, 667, 576, 477, 1179i

H2BiN

1714, 1697, 780, 631, 385, 329

TS1

3281, 1744, 803, 548, 450, 2019i

cis-HBi=NH

3430, 1722, 897, 738, 644, 565

Rotational Constants (MHz) A 121416.12 B 8719.43 C 8135.20

Dipole Moments (Debye)

q(Bi)

q(X)

q(H)

3.363

0.6663

-0.5419

-0.2492(Bi) 0.1248(N)

A 349834.37 B 7361.49 C 7209.77

0.6085

0.2539

-0.7163

0.2312

A 108884.23 B 9138.96 C 8712.20

3.469

0.8797

-0.4817

-0.1990

A 122081.04 B 8738.27 C 8154.59

2.571

-0.6062

0.7132

-0.2877(Bi) 0.1807(N)

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Table 1. (Continued)

Species

Frequencies (cm-1)

trans-HBi=PH

2296, 1786, 775, 591, 498, 361

TS2

2274, 1443, 645, 541, 217, 1848i

H2PBi

2377, 2376, 1080, 489, 368, 101

TS3

1746, 1707, 711, 535, 305, 1065i

H2BiP

1791, 1790, 790, 458, 407, 227

TS1

2107, 1731, 514, 459, 232, 1895i

cis-HBi=PH

2335, 1778, 681, 543, 543, 354

Rotational Constants (MHz) A 95369.14 B 3050.04 C 2955.52

Dipole Moments (Debye)

q(Bi)

q(X)

q(H)

2.028

0.3821

-0.0508

-0.2267 (Bi) -0.1045 (P)

A 203274.23 B 2945.06 C 2903.00

0.1989

0.0814

0.0187

-0.0501

A 80761.12 B 2852.57 C 2844.06

1.757

0.5684

-0.1569

-0.2057

A 96225.20 B 3.013.80 C 2922.27

1.630

0.3860

-0.0925

-0.2298(Bi) -0.0638(P)

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Table 1. (Continued)

Species

Frequencies (cm-1)

trans-HBi=AsH

2114, 1784, 707, 550, 462, 241

TS2

2055, 1318, 599, 450, 159, 1167i

H2AsBi

2165, 2158, 980, 533, 406, 169

TS3

1751, 1691, 713, 519, 208, 1008i

H2BiAs

1794, 1793, 789, 417, 398, 174

TS1

1920, 1740, 510, 454, 150, 1679i

cis-HBi=AsH

2147, 1784, 617, 515, 503, 236

Rotational Constants (MHz) A 89358.57 B 1397.79 C 1376.26

Dipole Moments (Debye)

q(Bi)

q(X)

q(H)

1.625

0.3687

-0.0348

-0.1099(As) -0.2241(Bi)

A 110704.10 B 1253.99 C 1252.13

0.8006

0.0941

0.0482

-0.0711

A 79666.25 B 1272.21 C 1271.08

1.484

0.5462

-0.1322

-0.2070

A 90380.05 B 1380.90 C 1360.12

1.354

0.3652

-0.0671

-0.0770(As) -0.2211(Bi)

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Table 1. (Continued)

Species

Frequencies (cm-1)

trans-HBi=SbH

1878, 1783, 626, 491, 406, 186

TS2

1758, 1674, 709, 502, 167, 604i

H2SbBi

1796, 1794, 787, 405, 354, 142

TS3

1816, 1251, 807, 513, 161, 912i

H2BiSb

1913, 1909, 831, 447, 364, 134

TS1

1756, 1657, 467, 400, 107, 1534i

cis-HBi=SbH

1901, 1791, 543, 462, 445, 182

Rotational Constants (MHz) A 80400.69 B 871.99 C 862.64

Dipole Moments (Debye)

q(Bi)

q(X)

q(H)

0.8069

0.2720

0.1409

-0.1920(Sb) -0.2210(Bi)

A 78778.40 B 779.49 C 779.26

1.112

0.4697

-0.0534

-0.2082

A 87047.22 B 794.88 C 794.21

0.5083

0.0323

0.3204

-0.1764

A 81290.78 B 860.68 C 851.66

1.014

0.2551

0.1187

-0.1679(Sb) -0.2059(Bi)

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Table 1. (Continued)

Species

Frequencies (cm-1)

trans-HBi=BiH

1781, 1770, 598, 473, 378, 154

TS2

1811, 1478, 1008, 517, 155, 639i

H2BiBi

1775, 1768, 746, 576, 443, 118

TS1

1779, 1750, 863, 476, 114, 1679i

cis-HBi=BiH

1804, 1784, 517, 443, 427, 150

3

HBiNH

3

TS2

3357, 1770, 768, 514, 476, 431

Rotational Constants (MHz) A 77287.77 B 615.11 C 610.26

A 77507.02 B 524.55 C 524.54

A 78226.14 B 606.27 C 601.61 A 121091.20 B 7382.89 C 7137.63

Dipole Moments (Debye)

q(Bi)

q(X)

q(H)

0.0005

0.2257

0.2257

-0.2257

0.9889

0.4292

-0.0015

-0.2139

1.311

0.2021

0.2021

-0.2021

2.259

0.5225

-0.4678

-0.2265(Bi) 0.1718(N)

3349, 1680, 863, 723, 475, 1598i

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Table 1. (Continued)

Species

3

H2NBi

3

TS3

3

H2BiN

3

HBiPH

3

TS2

3

H2PBi

3

TS3

3

H2BiP

Frequencies (cm-1) 3541, 3447, 1574, 777, 655, 477

Rotational Constants (MHz) A 281315.29 B 6892.76 C 6809.41

Dipole Moments (Debye)

q(Bi)

q(X)

q(H)

1.530

0.2967

-0.6564

0.1799

2.695

0.7450

-0.3128

-0.2161

1.526

0.3244

-0.0394

-0.2113(Bi) -0.0738(P)

A 124076.92 B 2481.17 C 2477.45

1.325

0.1147

0.0512

-0.0829

A 78050.03 B 2608.56 C 2607.94

1.684

0.5454

-0.1163

-0.2145

1784, 1641, 713, 567, 490, 1699i 1801, 1796, 769, 501, 488, 439 2291, 1772, 593, 479, 286, 271

A 78158.88 B 7391.62 C 7383.41 A 95128.22 B 2584.45 C 2568.52

2305, 1547, 650, 536, 285, 1165i 2337, 2332, 1096, 596, 577, 263 1782, 1465, 704, 459, 276, 1337i 1795, 1790, 766, 472, 471, 260

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Table 1. (Continued)

Species

3

HBiAsH

3

TS2

3

H2AsBi

3

TS3

3

H2BiAs

3

HBiSbH

3

TS2

Frequencies (cm-1) 2108, 1773, 535, 466, 259, 182

Rotational Constants (MHz) A 89243.54 B 1193.65 C 1191.22

Dipole Moments (Debye)

q(Bi)

q(X)

q(H)

1.285

0.3116

-0.0182

-0.0850(As) -0.2084(Bi)

1.074

0.1076

0.0609

-0.0843

1.458

0.5260

-0.0967

-0.2146

0.9572

0.2456

0.1335

-0.1733(Sb) -0.2058(Bi)

2117, 1480, 599, 494, 193, 1097i 2144, 2134, 979, 542, 521, 178

A 106251.79 B 1168.29 C 1167.86

1782, 1371, 686, 445, 186, 1201i 1794, 1790, 768, 458, 458, 175 1874, 1773, 460, 446, 221, 143

A 78003.47 B 1185.99 C 1185.88 A 80375.87 B 750.61 C 750.33

1890, 1371, 707, 473, 212, 1180i

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Table 1. (Continued)

Species

3

H2SbBi

3

TS3

3

H2BiSb

3

HBiBiH

3

TS2

3

H2BiBi

Frequencies (cm-1) 1903, 1903, 816, 491, 487, 200

Rotational Constants (MHz) A 84433.65 B 1484.72 C 1483.97

Dipole Moments (Debye)

q(Bi)

q(X)

q(H)

0.6093

-0.0619

0.4251

-0.1816

0.7996

0.0484

0.3189

-0.1837

1.015

0.2073

0.2073

1.071

0.4218

0.0106

1884, 1334, 548, 429, 152, 932i 1898, 1897, 818, 457, 449, 140 1771, 1769, 432, 430, 204, 118

A 84.196.48 B 741.08 C 740.93 A 77311.59 B 529.17 C 529.08

-0.2073

1779, 1235, 545, 405, 125, 849i 1790, 1785, 768, 429, 425, 116

A 77832.37 B 523.63 C 523.62

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As one can see in Figure 1, the QCISD results show that the singlet HBi=BiH species have a planar structure in both trans and cis forms, with the former being lower in energy than the latter by 2.3 kcal/mol. The inversion barrier for such a cis-trans isomerization is predicted to be 32 kcal/mol. It is therefore reasonable to expect that such a small energy difference between cis- and trans-HBi=BiH would be enlarged when the hydrogen atoms are replaced by the bulkier substituents. Indeed, all the experimentally substituted RBi=BiR species reported so far adopt a trans-bent structure [2]. Additionally, our QCISD calculations on the relative stability of the three closed-shell HBiBiH isomers show an order trans-HBi=BiH > cis-HBi=BiH > H2BiBi. In particular, the QCISD results also indicate that the isomerization barrier for singlet trans-HBi=BiH  singlet H2BiBi is quite large (i.e., 44 kcal/mol), while its reverse barrier is relatively small (i.e., 7.6 kcal/mol). All these suggest that singlet H2BiBi is kinetically and thermodynamically unstable, and thus should not be detected experimentally. Let us now consider the first excited state of HBiBiH, a triplet, which, to our knowledge, has not been studied by any theoretical treatments. Likewise, no experimental study of triplet HBiBiH isomers has appeared to date. As expected, all of the triplet structures are nonplanar. For instance, as shown in Figure 1, the lowest triplet of HBiBiH is twisted (torsion angle 91.4°), and the lowest triplet H2BiBi structure is pyramidal (out-of-plane angle 89.4°). So it is not surprising that the transition state between these structures (i.e., triplet HBiBiH  triplet H2BiBi) is nonplanar (torsion angle 86.1°) as well. As one can see in Figure 1, our QCISD calculations show that the energies of the triplet structures H2BiBi and HBiBiH both lie significantly above the corresponding singlets. H2BiBi is the more stable of the two triplets. Moreover, the singlettriplet separation for H2BiBi is estimated to be 30 kcal/mol (at QCISD). However, the energy of triplet HBiBiH is predicted to lie 12 kcal/mol above singlet trans-HBi=BiH. To our knowledge, no theoretical data are available for the singlettriplet splitting in dibismuthene. Additionally, the activation energy for the 1,2-hydrogen shift (i.e., triplet HBiBiH  TS2  triplet H2BiBi) is calculated to be 11 kcal/mol, while its reverse barrier is anticipated to be 17 kcal/mol at the QCISD level. It is therefore predicted that the triplet H2BiBi species should have a lifetime sufficient for spectroscopic observation. In short, our theoretical findings based on the present QCISD level of theory indicate that the order of stability calculated for the HBiBiH species is singlet trans-HBi=BiH > singlet cis-HBi=BiH > triplet H2BiBi > triplet HBiBiH > singlet H2BiBi.

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Figure 1. The optimized geometries (in Å and deg) of the HBiBiH isomers and relative energies (in kcal/mol) of the pathways for isomerization of the singlet (plain line) and triplet (dashed line) HBiBiH species at the QCISD/LANL2DZdp level.

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Figure 2. The optimized geometries (in Å and deg) of the HBiPH isomers and relative energies (in kcal/mol) of the pathways for isomerization of the singlet (plain line) and triplet (dashed line) HBiPH species at the QCISD/LANL2DZdp level.

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Figure 3. The optimized geometries (in Å and deg) of the HBiSbH isomers and relative energies (in kcal/mol) of the pathways for isomerization of the singlet (plain line) and triplet (dashed line) HBiSbH species at the QCISD/LANL2DZdp level.

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3.2. HBiPH To the best of our knowledge, only one substituted phosphabismuthene (RBi=PR) has been isolated and characterized unequivocally [3]. In order to parallel the study of dibismuthene, HBiBiH, four stable closed-shell structures were studied on the HBiPH potential energy surface using the QCISD method. They are trans-HBi=PH, cis-HBi=PH, H2BiP, and H2PBi. In addition, the openshell triplet states of these isomers were also investigated. The optimized geometries and relative energies of the HBiPH species on the energetically lowest singlet and triplet potential energy surfaces from QCISD calculations presented in this work are shown in Figure 2. The vibrational frequencies, as well as the rotational constants, dipole moments, and net atomic charges calculated for both singlet and triplet HBiPH isomers are collected in Table 1. Our QCISD results show that the singlet HBi=PH species have a planar structure in both trans and cis forms. Both BiPH and PBiH angles are larger in the cis form (94° and 96°, respectively) than in the trans form (91° and 92°, respectively). It has to be pointed out here that the QCISD bond angles at both bismuth and phosphorus for trans-HBi=PH are predicted to be closed to 90°, indicating a concentration of s-character in the lone pair orbitals due to the relativistic effect (vide infra) [12,13]. Similarly, the Bi=P bond length in cis conformation (2.453Å) is somewhat larger than that in the trans structure (2.442Å). Although little is known experimentally about the HBiPH species, we may compare some of our results with those experimentally obtained for the substituted phosphabismuthene. The Bi=P bond length determined by X-ray diffraction for substituted trans-phosphabismuthene is 2.454Å [3]. In addition, this also displays wider bond angle of 102.2° at bismuth and narrow angle of 96.40° at phosphorus. The wider angle at bismuth is somewhat surprising and is presumably due to the larger size of the aryl substituent at bismuth [3]. Our predicted QCISD bond length for trans-HBi=PH (2.442Å) is in excellent agreement with the experimental value (2.454Å), keeping in mind that the synthesized molecule contains bulkier substituents. On the other hand, our QCISD calculations predict that singlet H2BiP has a nonplanar Cs structure with a pyramidalization angle (or out-of-plane angle, defined as the angle between the BiP bond and the H2Bi plane) of 90°, whereas the singlet H2PBi adopts a planar geometry. As can be seen in Figure 2, our computational results indicate that the singlet trans-HBi=PH isomer is lower in energy than the singlet cis-HBi=PH isomer by 1.8 kcal/mol at the QCISD level of theory. This small energy difference is presumably to be enlarged when the hydrogen atoms are replaced by the bulkier

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substituents, owing to steric effects. Accordingly, our model calculations predict that the bulkily substituted trans-RBi=PR should be more stable than its corresponding cis isomer. This prediction is confirmed by one available experimental structure [3], as mentioned earlier. In addition, our QCISD calculations on the relative stability of the four closed-shell HBiPH isomers show an order trans-HBi=PH > cis-HBi=PH > H2PBi > H2BiP. In particular, in the singlet states the QCISD barriers for the unimolecular isomerization reactions decrease in the order trans-HBi=PH  H2BiP (57 kcal/mol) > trans-HBi=PH  H2PBi (32 kcal/mol)  trans-HBi=PH  cis-HBi=PH (24 kcal/mol). As a result, our computational results indicate that the isomerization barriers are large enough for both trans-HBi=PH and cis-HBi=PH to have lifetime sufficient for spectroscopic observation. Moreover, it is also predicted that 1,2-H shifts in the singlet HBiPH species are unlikely to proceed at room temperature. Let us now consider the first excited state of HBiPH, an open-shell triplet. In this work, we considered five stationary points: HBiPH, H2PBi, H2BiP, and the transition states for the isomerization of triplet HBiPH. As expected, triplet phosphabismuthene is twisted, with a BiP bond length 0.21 Å (QCISD) longer than the trans singlet ground state. In addition, triplet H2BiP is pyramidal with a large pyramidalization angle on bismuth (89° at QCISD), while the BiP bond length is considerably longer (by 0.21 Å) than that in the corresponding pyramidal singlet state. Likewise, the same phenomenon can also be found in the triplet H2PBi structure, with a pyramidalization angle on phosphorus (92°) and a long PBi single bond (2.656 Å). The most noteworthy feature about the triplet HBiPH species is that triplet H2PBi is predicted to be the most stable isomer of phosphabismuthene from both kinetic and thermodynamic viewpoints. For instance, the QCISD results indicate that H2PBi has a triplet ground state with a singlet-triplet separation of 24 kcal/mol. From Figure 2, it is apparent that triplet H2PBi is estimated to be only 5.8 kcal/mol below the most stable singlet isomer, trans-HBi=PH. Moreover, it is found that the barrier height for the isomerization triplet H2PBi  triplet HBiPH is about 31 kcal/mol at QCISD level. Besides these, triplet H2PBi is calculated to possess a dipole moment of 1.33 D. Consequently, all of the above results reveal that the triplet H2PBi species should exist and it should be possible to isolate for experimental observations. Furthermore, as shown in Figure 2, the energy difference between singlet HBi=PH and triplet HBiPH is as large as 17 kcal/mol, favoring the former over the latter. Additionally, the QCISD results suggest that the isomerization barrier for triplet HBiPH  triplet H2BiP is 25 kcal/mol, while the inversion barrier for triplet H2BiP  triplet HBiPH is 16 kcal/mol. These

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results clearly indicate that any experimental detection of the triplet H2BiP species formed during the reaction is highly unlikely. In short, our theoretical investigations based on the present QCISD level of theory indicate that the order of stability for the HBiPH species is triplet H2PBi > singlet trans-HBi=PH > singlet cis-HBi=PH > triplet HBiPH > singlet H2PBi > triplet H2BiP > singlet H2BiP.

3.3. HBiSbH As mentioned in the Introduction, only one stable compound with a Bi=Sb double bond has been reported recently, i.e., BbtBi=SbBbt by Tokitoh and coworkers [3]. Nevertheless, definite structural parameters for this substituted Bi=Sb doubly bonded species have not obtained yet owing to the inevitable disorder of the antimony and bismuth atoms [3]. Likewise, neither experimental nor theoretical results for the parent HBiSbH isomers are available so far for a definitive comparison. Again, four minima corresponding to trans-HBi=SbH, cis-HBi=SbH, H2SbBi, and H2BiSb were located on the singlet potential energy surface. As usual, the open-shell triplet states of these isomers have also been investigated. The optimized geometries and relative energies for the HBiSbH species on the energetically lowest singlet and triplet QCISD potential energy surfaces are shown in Figure 3. The calculated vibrational frequencies for singlet and triplet HBiSbH isomers as well as the calculated rotational constants, dipole moments, and net atomic charges are collected in Table 1. It should be noted that the order of stability of the HBiSbH isomers is somewhat different from that calculated for the previous HBiXH (X = Bi and P) systems, in particular for the triplet species. For instance, the order of stability of the singlet HBiSbH species increases in the order (in kcal/mol): trans-HBi=SbH (0.0) < cis-HBi=SbH (2.1) < H2SbBi (29) < H2BiSb (38) at the QCISD level. A similar trend is also found for the analogue HBiPH system as discussed previously. On the other hand, our QCISD calculations show that the trans-HBi=SbH  H2BiSb isomerization occurs via a non-planar transition state, and is predicted to possess a large energy barrier of 38 kcal/mol with respect to trans-HBi=SbH. This obviously suggests that such a 1,2-H shift reaction is energetically unfavorable and would be highly endothermic (~ +38 kcal/mol) if it occurs. We thus conclude that the H2BiSb species cannot exist. Similarly, there exists a high activation barrier to separating the singlet H2SbBi and trans-HBi=SbH isomers. Our QCISD results predict that the forward barrier for the trans-HBi=SbH  H2SbBi isomerization is about 32 kcal/mol, while its reverse barrier is about 10 kcal/mol.

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As a consequence, if H2SbBi were directly formed by the appropriate gas-phase experiments, it should easily interconvert into the trans-HBi=SbH species. Moreover, according to our QCISD calculations, trans-HBi=SbH and cisHBi=SbH isomers are found to be close to each other in energy, with the transHBi=SbH species being lower by about 2.1 kcal/mol. It should be noted that the barrier height for such an isomerization reaction is estimated to be 23 kcal/mol. In any event, all of the above results confirm that substituted trans-RBi=SbR should be stable in both a kinetic and a thermodynamic sense. With the triplet states, we considered five stationary points: HBiSbH, H2SbBi, H2BiSb, and the transition states for the isomerization of triplet HBiSbH. It is apparent from Figure 3 that the energy difference between singlet HBi=SbH and triplet HBiSbH is as large as 13 kcal/mol, favoring the former over the latter. In addition, the QCISD calculations indicate that the isomerization barrier for triplet HBiSbH  triplet H2BiSb is 14 kcal/mol, while that for unimolecular rearrangement (i.e., triplet HBiSbH  triplet H2SbBi) is estimated to be about 11 kcal/mol. On the other hand, their reverse barriers are anticipated to be 16 and 21 kcal/mol, respectively. Accordingly, all of these results suggest that the triplet HBiSbH species should not be easily detected experimentally. In short, our theoretical investigations based on the present QCISD level of theory suggest that the stability of the HBiSbH species is in the order singlet trans-HBi=SbH > singlet cis-HBi=SbH > triplet H2SbBi > triplet H2BiSb > triplet HBiSbH > singlet H2SbBi > singlet H2BiSb.

3.4. HBiNH To date, neither experimental nor theoretical results for the bismuazene species are available for a definitive comparison. Again, four minima corresponding to trans-HBi=NH, cis-HBi=NH, H2NBi, and H2BiN were located on the singlet potential energy surface. As usual, the open-shell triplet states of these isomers have also been investigated. The optimized geometries and relative energies for the HBiNH species on the energetically lowest singlet and triplet QCISD potential energy surfaces are shown in Figure 4. The calculated vibrational frequencies for singlet and triplet HBiNH isomers as well as the calculated rotational constants, dipole moments, and net atomic charges are collected in Table 1.

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Figure 4. The optimized geometries (in Å and deg) of the HBiNH isomers and relative energies (in kcal/mol) of the pathways for isomerization of the singlet (plain line) and triplet (dashed line) HBiNH species at the QCISD/LANL2DZdp level.

It should be noted that the order of stability of the HBiNH isomers is similar to that calculated for the HBiPH species. For instance, for the singlet HBiNH species, H2NBi is shown to be the lowest lying isomer, with trans-HBi=NH, cis-

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HBi=NH, and H2BiN lying about 11, 9.9, and 88 kcal/mol higher in energy, respectively, at the QCISD level. This is in contrast to its six row analogue HBiBiH, where trans-HBi=BiH is the most stable singlet compound. Moreover, according to our QCISD calculations, trans-HBi=NH and cis-HBi=NH isomers are found to be nearly degenerate in energy, with an energy difference estimated to be only 0.80 kcal/mol. In addition, the trans-HBi=NH  H2BiN isomerization occurs via a non-planar transition state, and is predicted to possess a considerable energy barrier of 85 kcal/mol with respect to trans-HBi=NH. This obviously suggests that such a 1,2-H shift reaction is energetically unfavorable and would be highly endothermic (+77 kcal/mol) if it occurs. We thus conclude that the H2BiN species cannot exist. On the other hand, there exists a high activation barrier to separating the H2NBi and trans-HBi=NH isomers. Our QCISD calculations predict that the forward barrier for the trans-HBi=NH  H2NBi isomerization is about 64 kcal/mol, while its reverse barrier is about 75 kcal/mol. As a consequence, if H2NBi and/or trans-HBi=NH were directly formed by the appropriate gas-phase experiments, they should not easily interconvert. In any event, all of the above results confirm that H2NBi should be stable in both a kinetic and a thermodynamic sense. With the triplet states, we considered five stationary points: HBiNH, H2NBi, H2BiN, and the transition states for the isomerization of triplet HBiNH. It is apparent from Figure 4 that the energy difference between singlet HBi=NH and triplet HBiNH is as large as 19 kcal/mol, favoring the former over the latter. In addition, the QCISD calculations indicate that the isomerization barrier for triplet HBiNH  triplet H2BiN is sizable (i.e., 46 kcal/mol). This suggests that, like the case of singlet H2BiN discussed above, any experimental detection of the triplet H2BiN species formed during the reaction is highly unlikely. Additionally, the energy barrier for the unimolecular rearrangement (i.e., triplet HBiNH  triplet H2NBi) is estimated to be about 16 kcal/mol at QCISD. Besides, our theoretical investigations pinpoint triplet H2NBi as the global minimum on the HNBiH potential energy surfaces. In particular, it is of interest to note that triplet H2NBi is lower in energy than singlet H2NBi by 15 kcal/mol, a value that is the smallest of any of the singlet-triplet energy separations found for the isoelectronic system triplet H2XBi  singlet H2XBi. Namely, 24, 24, 26, and 30 kcal/mol for X = P, As, Sb, and Bi, respectively, at the QCISD level.

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Figure 5. The optimized geometries (in Å and deg) of the HBiAsH isomers and relative energies (in kcal/mol) of the pathways for isomerization of the singlet (plain line) and triplet (dashed line) HBiAsH species at the QCISD/LANL2DZdp level.

In short, our theoretical investigations based on the present QCISD level of theory suggest that the stability of the HBiNH species is in the order triplet H2NBi

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> singlet H2NBi > singlet cis-HBi=NH ≈ singlet trans-HBi=NH > triplet HBiNH > triplet H2BiN > singlet H2BiN.

3.5. HBiAsH For consistency with the above studies, four stable closed-shell structures on the HBiAsH potential energy surface have been considered in the present work, i.e., trans-HBi=AsH, cis-HBi=AsH, H2AsBi, and H2BiAs. Likewise, the openshell triplet states of these isomers have also been investigated. The optimized geometries and relative energies for the HBiAsH species on the energetically lowest singlet and triplet QCISD potential energy surfaces are shown in Figure 5. The calculated vibrational frequencies for singlet and triplet HBiAsH isomers as well as the calculated rotational constants, dipole moments, and net atomic charges are collected in Table 1. It is our conviction that for the HBiAsH species under study, for which the rotational constants and vibrational IR spectra are still unknown, the presented QCISD results (Table 1) should be of the same accuracy as we have already discussed earlier. It should be noted that the order of stability of the HBiAsH isomers is essentially similar to that calculated for the HBiPH species. For instance, for the singlet HBiAsH species, trans-HBi=AsH is shown to be the lowest lying isomer, with cis-HBi=AsH, H2AsBi, and H2BiAs lying about 2.0, 24, and 49 kcal/mol higher in energy, respectively, at the QCISD level. Moreover, according to our QCISD calculations, trans-HBi=AsH and cis-HBi=AsH isomers are found to be nearly degenerate in energy, with an energy difference estimated to be only 2.0 kcal/mol. In addition, the trans-HBi=AsH  H2BiAs isomerization occurs via a non-planar transition state, and is predicted to possess a considerable energy barrier of 49 kcal/mol with respect to trans-HBi=AsH. This obviously suggests that such a 1,2-H shift reaction is energetically unfavorable and would be highly endothermic (+48 kcal/mol) if it occurs. We thus conclude that the H2BiAs species cannot exist. On the other hand, there exists a high activation barrier to separating the H2AsBi and trans-HBi=AsH isomers. Our QCISD calculations predict that the forward barrier for the trans-HBi=AsH  H2AsBi isomerization is about 34 kcal/mol, while its reverse barrier is about 10 kcal/mol. As a consequence, all of the above results confirm that trans-HBi=AsH should be stable in both a kinetic and a thermodynamic sense. With the triplet states, we considered five stationary points: HBiAsH, H2AsBi, H2BiAs, and the transition states for the isomerization of triplet HBiAsH. It is apparent from Figure 5 that the energy difference between singlet HBi=AsH

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and triplet HBiAsH is as large as 15 kcal/mol, favoring the former over the latter. In addition, the QCISD calculations indicate that the isomerization barrier for triplet HBiAsH  triplet H2BiAs is large (i.e., ~ 20 kcal/mol). This suggests that, like the case of singlet H2BiAs discussed above, any experimental detection of the triplet H2BiAs species formed during the reaction is highly unlikely. Additionally, the energy barrier for the unimolecular rearrangement (i.e., triplet HBiAsH  triplet H2AsBi) is estimated to be about 11 kcal/mol at QCISD. Besides these, our theoretical investigations show that triplet H2AsBi is the global minimum on the HBiAsH potential energy surfaces. In particular, it is of interest to note that triplet H2AsBi is found to be nearly degenerate to singlet transHBi=AsH by 0.13 kcal/mol. In short, our theoretical investigations based on the present QCISD level of theory suggest that the stability of the HBiAsH species is in the order triplet H2AsBi ≈ singlet trans-HBi=AsH > singlet cis-HBi=AsH > triplet HBiAsH > triplet H2BiAs > singlet H2AsBi > singlet H2BiAs.

3.6. Overview of HBiXH (X = N, P, As, Sb, and Bi) Systems Since the relative QCISD energies of the various states are already shown in Figures 1-5, Figures 6 and 7 present the trend in relative stabilities for the closedshell and open-shell triplet minima on the HBiXH (X = N, P, As, Sb, and Bi) potential energy surfaces. For each figure, the energy reference is the transHBi=XH isomer. The major conclusions that can be drawn from Figures 6 and 7 are as follows: 1) Both trans- and cis-HBi=XH molecules are the minima on the singlet potential energy surfaces. Moreover, the sizable barriers calculated show that these compounds should be kinetically stable with respect to isomerization. As such, one may then foresee that the trans- and cisHBi=XH species would serve as candidates for experimental observation. 2) When X is phosphorus or arsenic or antimony or bismuth, the transHBi=XH isomer is more stable than its corresponding cis isomer. It is found, however, that the cis-HBi=NH isomer is slightly more stable than the trans-HBi=NH species. Conventionally, the trans isomer is more stable than the corresponding cis isomer due to reduced steric repulsion. In the case of HBi=NH, however, this situation may change. It is well established that the charge distribution is determined by the electronegativities. For most cases the

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charge distribution leads to X(+)H(), except in the case of N where it leads to N()H(+). In particular, the atomic charges of these two hydrogens would be enhanced as X is a much heavier element (such as bismuth; see Table 1). In this regard, the electrostatic attraction between the NH and XH moieties tends to favor the cis isomer over the trans isomer, as demonstrated in 1.

attraction 

H

H Bi



 (1)

N



3) In the singlet trans-HBi=XH species, both HBiX and BiXH angles decrease and the Bi=X bond length increases as X changes from N to Bi. For instance, as seen in Figures 1-5, the bond angle HBiX in the singlet trans-HBi=XH species decreases in the order N (93.1°) > P (92.3°) > As (91.6°) > Sb (90.4°) > Bi (89.6°). Likewise, their BiXH angles decrease in the order: N (105°) > P (90.6°) > As (89.9°) > Sb (89.8°) > Bi (89.6°). In addition, the Bi=X bond length increases in the order: N (2.011 Å) < P (2.442 Å) < As (2.546 Å) < Sb (2.741 Å) < Bi (2.797 Å). The same phenomena can also be found in the singlet cis-HBi=XH species as given in Figures 1-5. In singlet trans-HBi=XH, it thus appears that, as the X atom becomes heavier, both HBiX and BiXH angles approaching 90° are preferred. The reason for this may be due to the relativistic effect [12, 13]. As X changes from nitrogen to bismuth, the valence s orbital is more strongly contracted than the corresponding p orbitals [13]. Namely, the size difference between the valence s and p orbitals increases upon going from N to Bi (the significant 6s orbital contraction originates mostly from the relativistic effect). Consequently, the valence s and p orbitals differ in spacial extension and overlap less to form strong hybrid orbitals [12,13]. As a result, the heavier atoms have a lower tendency to form hybrid orbitals and prefer to maintain the (ns)2(np)3 valence electron configuration even in compounds. In other words, the so-called “nonhybridization effect” (or “inert s-pair effect”) [12,13] occurs on moving from nitrogen to bismuth, where sp hybridization is preferred to sp2, i.e., bond formation of type 2 is preferred over type 3. Consequently,

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Ming-Der Su the bond angles HBiX and BiXH (X = N, P, As, Sb, and Bi) in both singlet trans- and cis-HBi=XH species approach 90° as the X atom varies from N to Bi.

.. . . . ... H

H Bi H

H

..

..

.. . . ..

H H

(2)

..

H

..

H

X

(3) 4) Singlet HBi=XH, H2Bi=X, and H2X=Bi isomers possess a pyramidal conformation, except for the cases of singlet H2NBi and H2PBi with planar geometries. On the other hand, all HBiXH species with open-shell triplet states adopt a nonplanar geometry. Namely, the triplet HBiXH molecules are twisted, while the triplet H2BiX and H2XBi compounds are pyramidal. The reason for these phenomena can be easily understood with reference to their electronic structures. In the singlets, two kinds of interactions exist that determine the geometries of the singlet H2XBi species [14]. One is the delocalization of the p- lone pair of the central atom into the  p-(Bi)), as empty p valence orbital of the terminal atom (i.e., p- shown in 4.

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(4) The other is a “back-bonding” effect, in which delocalization of the p- lone pair of the terminal atom into the unoccupied d orbital (of the appropriate  p-(Bi)), as shown in 5.

(5) If the above two affects are dominant, then the singlet H2XBi molecule definitely prefers a planar structure, since these effects lead to a maximum overlap between bismuth and X atoms. In contrast, if the delocalizations are weak, then the central atom will tend to adopt a nonplanar geometry at the expense of  bonds. In a phenomenological sense, this can be considered to be a higher-row manifestation of the “nonhybridization effect” [12, 13]. For example, when X is N

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or P, the tendency to planar prevails over these weak delocalizations, allowing the H2NBi and H2PBi isomers to adopt the planar geometry. All these predictions have been confirmed by our calculations as given in Figures 1-5. On the other hand, as can be seen in 6, due to the repulsion between the electron pair on the central atom and the single electron on the terminal atom, triplet H2BiX and H2XBi species should favor the formation of pyramidal structures over planar structures.

(6) 5) For X = N, the stability of the singlet isomers decreases in the order H2NBi > cis-HBi=NH > trans-HBi=NH > H2BiN. Similarily, for X = P, As, Sb and Bi, the stability is in the order trans-HBi=XH > cis-HBi=XH > H2XBi > H2BiX. This finding strongly suggests that Bi is more reluctant to form doubly bonded compounds than the other pnicogen elements. Again, the answer lies in the dramatic contraction of the Bi 6s orbital due to the relativistic effect [12,13], which reduces their tendency to form hybrid orbitals. This results in Bi preferring to maintain the (6s)2(6p)3 valence electron configuration. The use of these three orthogonal 6p orbitals without significant hybridization leads to the presence of two lone pairs at Bi. It is therefore apparent that the structures with the most lone pairs are more stable. In short, the reason for the reluctance in forming Bi=X multiple bonds is the significant contraction of the Bi valance s orbital. 6) The H2XBi isomer is always more stable than the corresponding H2BiX isomer for X = N, P, As, and Sb, irrespective of the singlet or triplet states they adopt. In fact, the theoretical investigations strongly indicate that both singlet and triplet H2BiX (for X = N, P, As, and Sb) molecules

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are thermodynamically and kinetically unstable, and thus should not be observable. The reason for this trend can readily be understood in terms of the BiH vs XH bond energies. In the H2BiX isomers, there are two BiH bonds, whereas there are two XH bonds in the H2XBi isomers. Experimental values of the XH bond energy are in the order: NH (93 kcal/mol) > PH (78 kcal/mol) > AsH (71 kcal/mol) > SbH (61 kcal/mol) > BiH (56 kcal/mol) [15,16]. Therefore the H2XBi species should always be more stable than the corresponding H2BiX isomer. 7) The other feature of interest concerns the pyramidalization angle at the central atom X of the triplet H2XBi species. In particular, the out-of-plane angle (defined as the angle between the BiX bond and the H2X plane) increases in the order N (69.8°) < P (86.8°) < As (88.0°) < Sb (88.3°) < Bi (89.4°). Moreover, the pyramidalization angles in triplet H2BiX are all close to 90°. Again, these data provide evidence for the nonhybridization nature of the Bi 6s and 6p orbitals, namely, the so-called “nonhybridization effect” [12,13], discussed earlier. 8) For X = N, P, and As, the stability of the triplet isomers decreases in the order H2XBi > HBiXH > H2BiX. In contrast, for X = Sb, and Bi, the stability is in the order H2SbBi > H2BiSb > HBiSbH and H2BiBi > HBiBiH, respectively. Moreover, ab initio calculations indicate that the ground state of the HBiNH, HBiPH and HBiAsH species is the openshell triplet H2XBi, whereas for the HBiSbH and HBiBiH species it is the singlet trans-HBiXH, as X goes from antimony to bismuth. Figure 7 clearly illustrates that when going down group 15, the triplet H2XBi structure appears to be less and less favored, while the triplet pyramidal trans-HbiXH structure becomes more favored. In other words, triplet H2XBi (X = N, P, and As) is the globally most stable structure and lies 26, 5.8, and 0.14 kcal/mol, respectively, below the most stable singlet HBi=XH species. However, for triplet state H2SbBi and H2BiBi, the pyramidal structure was found to be the local minimum isomer lying 3.3 and 6.7 kcal/mol in energy above singlet trans-HBi=SbH and transHBi=BiH, respectively. 9) As expected, the triplet state is characterized by a large twist angle or a high degree of pyramidalization on the central atom, and a BiX distance which is considerably longer than that in the singlet state. All these changes are not surprising since the triplet states result from the excitation from a bonding  or nonbonding n to an antibonding * orbital. Our calculations

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Ming-Der Su have confirmed these predictions as already given in Figures 1-5. It is interesting to note that the energy difference favoring triplet H2XBi over singlet H2XBi is calculated to be smaller than that favoring triplet H2BiX over singlet H2BiX, when X goes from N to Sb.

Figure 6. Evolution of QCISD relative energies (in kcal/mol) of the closed-shell singlet states (minima on the potential energy surface) with regard to the trans-HBi=XH (X = N, P, As, Sb, and Bi) isomer, taken as energy reference.

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Finally, we hope that the present study will convince experimental chemists that a molecule containing a Bi=N or Bi=As bond is not an unrealistic proposition. Successful schemes for the synthesis and isolation of such doubly bonded species are thus expected to be devised soon. We eagerly await experimental results to confirm our predictions.

Figure 7. Evolution of QCISD relative energies (in kcal/mol) of the open-shell triplet states (minima on the potential energy surface). Zero energy corresponds to the closed-shell singlet trans-HBi=XH (X = N, P, As, Sb, and Bi) isomer.

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Acknowledgments The author is grateful to the National Center for High-Performance Computing of Taiwan for generous amounts of computing time. He also thanks the National Science Council of Taiwan for their financial support.

References [1]

[2]

[3] [4] [5] [6] [7] [8] [9]

For recent reviews, see: (a) Cowley, A. H. Polyhedron, 1984, 3, 389. (b) Cowley, A. H. Acc. Chem. Res. 1984, 17, 386. (c) Cowley, A. H.; Norman, N. C. Prog. Inorg. Chem. 1986, 34, 1 (d) Scherer, O. J. Angew. Chem. Int. Ed. Engl. 1990, 29, 1104. (e) Weber, L. Chem. Rev. 1992, 92, 1839. (f) Tokitoh, N. Pure Appl. Chem. 1999, 71, 495. (g) Power, P. P. Chem. Rev. 1999, 99, 3463. (h) Jones, C. Coord. Chem. Rev. 2001, 215, 151. (i) Sasamori, T.; Arai, Y.; Takeda, N.; Okazki, R.; Furukawa, Y.; Kimura, M.; Nagase, S.; Tokitoh. N. Bull. Chem. Soc. Jpn., 2002, 75, 661. (a) Tokitoh, N.; Arai, Y.; Okazaki, R.; Nagase, S. Science, 1997, 277, 78. (b) Twamley, B.; Sofield, C. D.; Olmstead, M. M.; Power, P. P. J. Am. Chem. Soc. 1999, 121, 3357. Sasamori, T.; Takeda, N.; Fujio, M.; Kimura, M.; Nagase, S.; Tokitoh, N. Angew. Chem. Int. Ed. Engl. 2002, 41, 139. Sasamori, T.; Takeda, N.; Tokitoh. N. J. Chem. Soc., Chem. Commun. 2000, 1353. Nagase, S.; Suzuki, S.; Kurakake, T.; J. Chem. Soc., Chem. Commun. 1990, 1724. Schoeller, W. W.; Begemann, C.; Tubbesing, U.; Strutwolf, J. J. Chem. Soc., Faraday Trans. 1997, 93, 2957. Cotton, F. A.; Cowley, A. H.; Feng, X. J. Am. Chem. Soc. 1998, 120, 1795. Pople, J. A.; Head-Gordon, M.; Raghavachaic, K. J. Chem. Phys. 1987, 87, 5968. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, Jr., J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.;

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[10]

[11] [12] [13]

[14] [15] [16]

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Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian, Inc., Wallingford CT, 2003. (a) Dunming, T. H., Jr.; Hay, P. J. In Modern Theoretical Chemistry, Schaefer, H. F., III, Ed.; Plenum: New York, 1976; pp1-28. (b) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 270. (c) ibid., 1985, 82, 284. (d) ibid., 1985, 82, 299. Check, C. E.; Faust, T. O.; Bailey, J. M.; Wright, B. J.; Gilbert, T. M.; Sunderlin, L. S. J. Phys. Chem. A, 2001, 105, 8111. Nagase, S. in The Chemistry of Organic Arsenic, Antimony, and Bismuth Compounds; Ed. Patai, S., Wiley: New York, 1994, pp.1-24. (a) Pyykko, P.; Desclaux, J.-P. Acc. Chem. Res., 1979, 12, 276. (b) Pykko, P. Chem. Rev. 1988, 88, 563. (c) Kutzelnigg, W. Angew. Chem. Int. Ed. Engl. 1984, 23, 272. (d) Schwerdtfeger, P.; Health, G. A.; Dolg, M.; Bennett, M. A. J. Am. Chem. Soc. 1992, 114, 7518. (e) Schwerdtfeger, P.; Laakkonen, L. J.; Pyykko, P. J. Chem. Phys. 1992, 96, 6807 and references cited therein. Trinquier, G. J. Am. Chem. Soc. 1982, 104, 6969. Purcell, K. F.; Kotz, J. C. Inorganic Chemistry; Holt-Saunders International Editions: Toronto, 1977. The BiH bond energy was previously estimated to be ≤ 67.7 kcal/mol. See: Lindgren, B.; Nilsson, Ch. J. Mol. Spectrosc. 1975, 55, 407. On the other hand, our QCISD/LANL2DZdp calculations estimate the BiH bond energy to be 56.2 kcal/mol.

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In: Advances in Chemistry Research. Volume 21 ISBN: 978-1-62948-742-7 © 2014 Nova Science Publishers, Inc. Editor: James C. Taylor, pp. 185-195

Chapter 5

GROWTH OF NANOCRYSTALS FROM AMORPHOUS Bi G. N. Kozhemyakin1,, S. Y. Kovalev1, O. N. Ivanov2 and O. N. Soklakova2 1

Department of Micro and Nanoelectronics, Volodumur Dahl East Ukrainian National University, Lugansk, Ukraine 2 Belgorod State University, Belgorod, Russia

Abstract Over recent years, there has been an increased interest towards theoretical and experimental investigations of bismuth (Bi) crystal properties. Bi and Bi-related materials are of particular interest for thermoelectric applications. Bi is a semimetal with unique electronic structure, and its transport properties have been studied because quantum confinement effects can be observed in low-dimensional systems. The development of Bi as a low-dimensional material has been traditionally done by preparing ordered arrays of 1D quantum wires and studying them in detail. Manipulating nanoparticles sizes in low-dimensional systems is a promising way for fundamental studies and nanotechnological applications of this semimetal. Another method that provides control over nanocrystal formation is lowtemperature annealing of amorphous materials. Therefore, we used this method to study the growth conditions of Bi nanocrystals. The objective of this paper is further development of Bi nanocrystal formation method by annealing of amorphous Bi. 

E-mail address: [email protected] (Corresponding author)

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G. N. Kozhemyakin, S. Y. Kovalev, O. N. Ivanov et al. The spinning method for obtaining of amorphous Bi was used. Bi of 99.9999% purity was used as the source material. Before using this Bi charge, it was purified by drop cleaning in vacuum, followed by directional crystallization. After crystallization, a Bi polycrystal was obtained in pieces smaller than 7 g. These pieces were placed into a quartz ampoule with 8–20 mm inner diameter and 115 mm length, which had an external resistance heater and an outer thermal ceramic insulator. High purity Ar at a gauge pressure of 0.1 atm was passed through the ampoule to prevent Bi melt oxidization. The melt temperature during spinning was controlled from 750 K to 800 K. After the Bi melted, and a steady state established, the melt was poured out onto a cold copper plate with 300 mm diameter, rotating at 1100 rpm. Amorphous Bi formed as a film with 5–30 μm thickness, 2–3 mm width, and 15–35 mm length. The Bi samples were annealed at temperatures 150–220 °C for 1 h in a special cylindrical furnace with less than 1 K/cm temperature gradient in the axial and radial directions. A crossection of the films was prepared by breaking the samples in liquid nitrogen. The film microstructure was studied using a scanning electron microscope “Quanta 200 3D”. The Bi films had a few microcrystals with sizes 10–30 μm after spinning process, with thickness 20-30 μm. The amorphous Bi films had thickness near 5 μm. Nanocrystals were detected in amorphous Bi films only after annealing at temperature above 150 °C. Individual nanocrystals with dimensions from 10 to 50 nm were sparsely located in a central part of the films after annealing at 150°C. Increasing the annealing temperature above 200 °C led to an increased amount of nanocrystals with 50 – 100 nm dimensions and growth of crystallites with sizes up to 500 nm. For these growth conditions, Bi crystallites had an orientation of the (111) plane perpendicularly to the film surface.

Introduction 1.1. Crystal Structures Bismuth (Bi) is a semimetal, and has a rhombohedral primitive unit cell containing two atoms (Figure 1) [1]. Principal crystallographic axes of Bi are the following: the trigonal axis of three-fold symmetry, the binary axes of two-fold symmetry, and the bisectrix axis, which is perpendicular to the former two axes [2]. The bismuth structure may be regarded as a slightly distorted form of a simple cubic lattice having superposition of two interpenetrating face-centered sublattices. Bi crystals have double layer structure, which is shown in Figure 2 [3]. Within a layer, the atoms are covalently bonded with some metallic bonding [4]. Between the layers, there act metallic and van der Waals binding forces, so that the first coordination atoms with smaller distance are placed within the layers. On other hand, the next coordination sphere consists of atoms of the adjoining double layer.

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Figure 1. The T rhomboheggral unit cell off bismuth-type crystals containning two atomss. The axes X, Y and Z are the binary, b bisectrixx and trigonal axxes, respectivelyy.

Figure F 2. Doublee layer structuree of bismuth wiith the atoms inn two layers.

1.2. Cry ystals and Nanostructu N res Invesstigation of physical propeerties of Bi sttarted after thhe developmeent of growth method m of these crystals by the horizontal Bridgm man techniquee [5]. Several methods m have been applied in order to im mprove the quality of Bi single s crystals and a alloys: vertical Bridgm man, horizontall zone meltingg, and Czochrralski [2,6-13]. Advances in optimization of the crystaals growth proocesses allowed to

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grow high quality Bi single crystals, which made it possible to study their fundamental properties and electronic structures [8,14-16]. Over the recent years, there has been an increased interest towards theoretical and experimental investigations of Bi crystal properties [17-19]. The unique electronic structure and electron transport properties of Bi have been studied because quantum confinement effects can be observed in low-dimensional systems [20-22]. Such low-dimensional systems of Bi and Bi-related materials are of particular interest for thermoelectric applications. The development of Bi as a lowdimensional material has been traditionally done by preparing ordered arrays of 1D quantum wires and studying them in detail [23–25]. It was found that in quantum regime, as the wire diameter decreases to less than 50 nm, the material makes a transition from a semimetal to a semiconductor with a bandgap between the valence and conduction bands [26]. This phenomenon can be used for control and optimization of nanomaterials to enhance their thermoelectric properties. Manipulating nanoparticles sizes in low-dimensional systems is a promising way for fundamental studies and nanotechnological applications of this semimetal. Another method that provides control over nanocrystal formation is lowtemperature annealing of amorphous materials [27, 28]. Therefore, we used this method to study the growth conditions of Bi nanocrystals.

Experimental Procedures 2.1. Bismuth Purification In our experiments, bismuth of 99.9999% purity was used as the source material. It is known, that the nanostructure properties are extremely sensitive to impurities, but bismuth may be oxidized in air at elevated temperatures. Therefore, before using this Bi charge, it was purified by drop cleaning in vacuum (development by A.A. Baukov Institute Metallurgy RAN, Moscow), followed by directional crystallization. Drop cleaning and crystallization were performed in quartz glass ampoules, evacuated up to 10-6 mbar. A schematic drawing of the apparatus for Bi purification is shown in Figure 3. The cleaning apparatus consisted of a container and two crucibles. The container was a quartz glass ampoule with 75 mm inside diameter, 83 mm outer diameter, and 600 mm length. Two crucibles were bottom quartz glass ampoule and top quartz glass ampoule. Bismuth was placed in the top quartz glass ampule with 40 mm inside diameter, 45 mm outer diameter and 250 mm length. This ampule had a capillary with 2 mm inner diameter and 10 mm height at the bottom. Molten bismuth flowed through the capillary in

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bottom qu uartz glass amppule with 40 mm m inside diam meter, 45 mm outer diameteer and 200 mm length. l Bi mellt was heated to 500 °C andd held at tempperature for 1 hour. Then, the heater movedd up at 10 mm m/h, and Bi direectionally solidified. After thhat, a ystal was obtaained in pieces smaller than 7 g. Bi polycry

Figure 3. Schematic drawinng of the apparaatus for Bi puriffication.

2.2. Bism muth Amorrphization Amorphous alloyys create possibilities p for fundameental studiess of a I would seem It m that nanostrucctured materiaals and nanoteechnological applications. the low melting m point and a the large kinematic k viscosity are advaantages for bissmuth amorphizzation. Howevver, amorphouus alloys are prepared p by adding a components into the melt, m which sloow down the crystallization c process[27]. But B we have to t use high puritty Bi melt for the growth off Bi nanocrystals.

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G. N. Kozzhemyakin, S. Y. Kovalev, O. N. Ivanov et al. The setup s for prepaaration of amoorphous Bi film ms is shown inn Figure 4.

Figure 4. Schematic S draw wing of the setuup for Bi amorpphous film prepparation by spiinning method: I – Bi melting; III – Bi amorphizzation.

Polyccrystalline pieeces were placed into a quarrtz ampoule with w 8–20 mm inner diameter and 115 mm length, l which had an externnal resistance heater h and an outer t insulaator. High puurity Ar at a gauge pressurre of 0.1 atm m was ceramic thermal passed th hrough the amppoule to preveent Bi melt oxxidization. Thee melt temperrature during sp pinning was controlled c from m 750 K to 800 K. Afterr Bi melted, and a a steady staate establishedd, the melt was w poured ouut onto a cold copper plate with 300 mm diameter, rotating at 11000 rpm. Amorpphous Bi form med as a film with m thickness, 2––3 mm widthh, and 15–35 mm length. The T Bi amorpphous 5–30 μm samples were w annealedd at temperaturres 150–220 °C for 1 h in a special cylinddrical furnace with w less than 1 K/cm tempeerature gradiennt in radial dirrections (Figurre 5). A crosseection of the films was prepared p by breaking b the samples in liquid l nitrogen. The film microstructure m e was studieed using a scanning eleectron pe “Quanta 2000 3D”. microscop

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Figure 5. Schhematic drawinng of the heater for annealing Bi B films.

Experiimental Reesults and d Discussioon The Bi films had a few microcrystals with sizes 10–30 μm after spinnning w thicknesss 20-30 μm. The T amorphouus Bi films hadd thickness upp to 5 process, with – 10μm. The film surfface, which was w not in conntact with the copper disc in i the m had the influx chhanging its thhickness (Figuure 6). The crracks cooling moment, perpendiccular to the fiilm length weere observed inn crossection after breakinng the samples in i liquid nitroggen. Previous studies showeed that the nannocrystals beggin to grow in Bi0.9Sb0.1 amoorphous alloyys with the melting m point ~ 290 °C onnly at ures above 150 °C [29]. Therefore, we annnealed Bi film ms at temperaatures temperatu from 150 °C to 220 °C,, considering that t the meltinng point of Bi is 271 °C. ocrystals weree detected inn amorphouss Bi films after a annealinng at Nano temperatu ure 150 °C. Inndividual nanoocrystals withh dimensions from f 10 to 100 nm were spaarsely located in a central part and neaar the film surface (Figurre 7). Increasing g the annealinng temperaturee above 200 °C led to an inncreased amouunt of nanocrysttals with 50 – 100 nm dimennsions and groowth of crystaallites with sizzes up to 500 nm m. We determiined the preferred orientatioon of growingg nanocrystals from amorphou us Bi. Most off the crystallittes had a pyraamid shape, likke a rhombohhedral primitive unit cell (Figuure 8). The pyyramid base was w an equilateeral trianglepaarallel p This orrientation indicates that the crystallites grrew with the (111) ( to (111) plane. plane perrpendicular too the film suurface. Probaably, the orieentation of grown g crystallitees is the resuult of the tem mperature graadient in Bi films, which was observed with the increease of annealing temperatuure.

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Figure 6. The crossectioon of amorphoous Bi film aftter breaking thhe samples in liquid nitrogen.

Figuree 7. Nanocrystalls in Bi amorphous film after annealing a at tem mperature 150 °C C.

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Figure 8. Pyramidal crysttallites in Bi am morphous film affter annealing att temperature 2000°C.

It sho ould be noted that the therm mal conductivitty of bismuth is anisotropicc with the differrence up to 70% 7 dependinng on directioon, and has a minimal valuue in diirection [2]. Therefore, T bism muth crystallittes grown durring annealingg had the orienttation of (111)) plane paralleel to the lager temperature gradient, g or paarallel to the heaater axis (Figuure 4). Indeedd, air convectioon above the films can deccrease the tempeerature on the film surface. In this case, the t vertical tem mperature graadient will exceeed the horizonntal one in the film.

Conclu usion We used u the spinnning method and a annealingg for obtainingg of amorphouus Bi nanocrysttals. Amorphoous Bi formedd as a film wiith 5–30 μm thickness, t 2–33 mm width, an nd 15–35 mm length. The Bi B film samplees were annealled at temperaatures 150–220 °C for 1 h. A crossectionn of the filmss was prepareed by breakingg the i liquid nitroogen, and thee microstructuure of this suurface was stuudied samples in using a sccanning electron microscopee.

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After annealing at 150 °C temperature, it was found that the amorphous Bi films had individual nanocrystals with dimensions from 10 to 100 nm, which sparsely located in a central part and near the surface. Increasing the annealing temperature above 200 °C led to an increased amount of nanocrystals with 50 – 100 nm dimensions and growth of crystallites with sizes up to 500 nm. For these growth conditions, Bi crystallites had an orientation of the (111) plane perpendicularly to the film surface. This preferred orientation of grown nanocrystals and crystallites is the result of anisotropy of thermal conductivity in amorphous Bi with its minimal value in direction.

Acknowledgment The author thanks Dr. A. Churilov for helpful discussions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Saikawa, K. J. Phys. Soc. Jpn. 1970, 29, 562 – 569. Jim, W.M.; Amith, A. Sol. Stat. Electr. 1972, 15, 1141 – 1165. Vigdorovich, V.N.; Uhlinov, G.A.; Maruchev, V.V. Sborniknauchnuhtrudov MIET (Moscow) 1972, XVIII, 24 – 38. Wagner, N.; Brummer, O. Phys. Stat. Sol. B 1976, 75, 157 – 161. Kapitza, P. Proc. Roy. Soc. (A) 1928, 119, 358. Jain, A.L. Phys. rev. 1959, 114, 1518 – 1528. Coucka, P.; Barrett, C.S. ActaCryst. 1962, 15, 865 – 872. Gallo, C.F.; Chandrasekhar, B.S.; Sutter, P.H. J. Appl. Phys. 1963, 34, 144, 152. Brown, D.M.; Heumaun, F.K. J. Appl. Phys. 1964, 35, 1947 – 1951. Noothoven Van Goor, J.M.; Trum, H.M.G.J. J. Phys. Chem. Solids 1968, 29, 341 – 345. Bhatt, V.P.; Pandya, G.R.; Rao, R.D. J. Cryst. Growth 1972, 16, 283 – 286. Zemskov, V.S.; Belaya, A.D.;Beluy, U.S.; Kozhemyakin, G.N. J. Cryst. Growth 2000, 212, 161 – 166. Mase, S. J. Phys. Soc. Jpn. 1959, 14, 584 – 589. Balla, D.; Brandt, N.B. J. Exper. Theor. Phys. 1965, 20, 1111 – 1117. Brandt, N. B.; Lyubutina, L. G. J. Exper. Theor. Phys. 1965, 20, 1150 – 1153. Liu, Y.; Allen, E. Phys. Rev. 1995, B 52, 1566–1577. Ast, C. R.; Hochst, H. Phys. Rev. 2003, 113, 102.

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Hengsberger, M. et al., Eur. Phys. J. 2000, B 17, 603–608. Ast, C. R.; Hochst, H. Phys. Rev. Lett. 2001, 87, 177602. Jezequel, G.; Thomas, J; Pollini, I. Phys. Rev. B 1997, 56, 6620–6626. Behnia, K.; Balicas, L.; Kopelevich, Y. Science 2007, 317, 1729–1731. Hirahara, T.; et al., Phys. Rev. 2007, B 76, 153305. Heremans, J.; Thrush, C. M.; Lin, Y.-M.; Cronin,S.; Zhang, Z.; Dresselhaus, M.S.; Mansfield, J. F.Phys. Rev. B: Condens. Matter Mater.Phys. 2000, 61, 2921. Takaoka, S.; Murase, K. J. Phys. Soc. Jpn.1985, 54, 2250. Lin, Y.-M.; Dresselhaus, M. S.; Ying, J. Y. Advances in Chemical Engineering, Academic, York, PA 2001, Ch. 5, 167–203. Dresselhaus, M. S. et al., Adv. Mater.2007, 19, 1043–1053. Lu, K.; Wang, J.T.; Wei, W.D. Scrip. Metallurgica Mater., 1991, 25, 619–623. Cheng, T. Nanostruct. Mater., 1992, 1, 19–27. Kozhemyakin, G.N.; Kovalev, S.Y.; Shevchuk, V.A.; Ivanov, O.N.; Maradudina, O.N. Nauchnuevestidalevskogouniversiteta (Lugansk) 2012, 6E, http://www.nbuv.gov.ua/e-journals/ Vsunud/

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In: Advances in Chemistry Research. Vol. 21 ISBN: 978-1-62948-742-7 c 2014 Nova Science Publishers, Inc. Editor: James C. Taylor, pp. 197-219

Chapter 6

M ATHEMATICAL T HEORY OF N OBLE G ASES V. P. Maslov∗ National Research University “Higher School of Economics,” Bolshoi Tryokhsvyatitelsky Pereulok, Moscow, Russia

Abstract We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P, Z) diagram by the second sheet.

Keywords: Noble gas, equilibrium thermodynamics, Bose condensate, number theory, fractal dimension, Maxwell rule, tunnel quantization, ultrasecond quantization ∗

E-mail addresses: [email protected], [email protected]

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V. P. Maslov

Heuristic Considerations. Significance of Low Viscosity

Experts in molecular physics usually argue proceeding from the symmetry of the molecule motion averaged in all six directions. In the scattering problem, we use the principle of symmetry in all directions, which is standard in molecular physics, but apply it to define not the mean free path but other molecular physics quantities. Therefore, the fraction of all particles that moves head-on is 1/12. There are three such directions; hence, one quarter of all molecules collide. For the interaction potential, we consider the Lennard-Jones potential u(r) = 4ε

a12 a6  − 6 , r 12 r

(1)

where ε is the energy of the depth of the well and a is the effective radius. The mean energy of particles is kT = 4εE, where k is the Boltzmann constant, ε is a constant in the Lennard-Jones potential, and E is the energy of the pair (p1 − p2 )2 /m, where p1 , p2 are the momenta of two particles scattering on each other. In the absence of an external potential, the two-particle problem reduces to the one-dimensional radial-symmetric one. As is well known [1], two quantities: the energy E and the momentum M are conserved in this problem. In the scattering problem, it is convenient to consider, instead of the momentum M , another preserved constant, namely, the impact parameter b, so that √ M = Eb. (2) By solving with respect to energy E the well-known relation E=

p2 M 2 + 2 + u(r), m r

(3)

we obtain the attractive Hamiltonian H: H=

p2 /(2m) + u(r) , 1 − b2 /r 2

a < r ≤ b.

(4)

The repulsive Hamiltonian is separated from H by the barrier. Repulsive particles put obstacles in the way of particles of the Hamiltonian H, by creating “viscosity”.

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As the temperature decreases, the barrier height increases up to the value Ec = 0.286ε, and then starts decreasing (see 1). According to rough energy estimates [2], an additional barrier must form when the clusters form. This barrier can be given for a carbon dioxide gas by micelles and for neutral gases and methane by germs of droplets, i.e., three-dimensional clusters that contain at least one molecule surrounded by other molecules (a prototype of a droplet)1 .

Figure 1. Values of E(b, r) for different values of the impact parameter (left-toright): 1.81378 1.89344, 2.00178, 2.1559, 2.39252, 2.80839, 3.79391. However, to study the penetration through the barrier of the incident particle, we must plot E along the y axis and turn the wells upside down. Then the minimum becomes the barrier and the maximum becomes the depth of the well. A dimer can form in a classical domain if the scattering pair has an energy equal to the barrier height, slipping into the dip in an ”infinite” time and getting stuck in it as a result viscosity (and hence some little energy loss) because this pair of particles having lost energy hits the barrier on the return path. If the pair of particles has passed above this point, then the viscosity may be insufficient for the pair to become stuck: such a pair would return above the barrier after a reflection. Therefore, only the existence of a point E = Emax plus an infinitesimal quantity, where Emax is the upper barrier point, is a necessary condition for c the pair to be stuck inside the dip; Emax is the height of the maximum barrier. c We can compare the values Tc with the values Emax in the table below. 1

In mathematics, a domain is an open region containing at least one point.

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V. P. Maslov Table 1. Substance Ne Ar Kr

ε, K 36.3 119.3 171

Tcr /4 11 37 52

Ecr · ε/k 10.5 35 50

Above the value EB = 0.8ε, the trap disappears. At the value 0.286ε, the depth of the trap is maximum and corresponds to Tc = 1.16ε k . For neon and krypton, as can be seen from the table, the concurrence is sufficiently good. Because TB = 3.2ε/k, it follows that TB /Tc = 2.7, which corresponds to the known relation of “ the law of corresponding states” [3]. The temperature corresponding to 4EB /k, is the temperature above which dimers do not appear. Exactly this is what we call the Boyle temperature (in contrast to [4]). The dressed or “thermal” potential ϕ(r) is attractive [5]. In addition, because the volume V is a large parameter, it follows that if N εb r ϕ(r) = √ U( √ ), 3 3 V V r ) is the smooth function, N is the number of particles, is expanded where U ( √ 3 V √ in terms of 1/ 3 V , then   r C2 r C3 r 2 1 √ √ √ √ U ( 3 ) = C1 + 3 + 3 +O . (5) 3 V V ( V )2 ( V )3

Expanding C1 + r 2 =

(r − r0 )2 (r + r0 )2 + , 2 2

(6)

where C1 = r02 , we can, just as in [5], separate the variables in the two-particle problem and obtain the scattering problem for pairs of particles and the problem √ of their cooperative motion for r1 + r2 . The term C2 r/ 3 V does not depend on this problem and the correction aε 1 √ NO 3 V V

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is small. Then, in the scattering problem, an attractive quadratic potential (inverted parabola) is added to to the Lennard-Jones interaction potential. For this problem, we can find just as in (3)–(4), for all ρ = N/V , a point corresponding to the temperature at which the well capturing the dimers vanishes, and thus determine the so-called Zeno-line. It is really a straight line (up to 2%), on which Z = Emin /Emax = 1 (i.e. an ideal curve).

2.

Main Axiom and Theorem Consider the Maxwell distribution ω(p) =

1 p2 exp{− }. 2mT (2πmT )3/2

(7)

As is well known, it is associated with the potential Ω=

V π 3/2 T 5/2 m5/2 Γ(2 + 1/2)

Z



µ

t3/2 e−(t− T ) dt.

(8)

0

Now consider an ultrarelativistic gas in which the kinetic energy is proportional to |p|. In this case, ω(p) =

c3 c|p| exp{− }, 3 (πT ) T

(9)

where c is the velocity of light. For this gas, the potential Ω is of the form π2 Ω = VT Γ(3)c3 3

Since N =−

∂Ω , ∂µ

Z



µ

t3 e−(t− T ) dt.

(10)

0

P =−

∂Ω , ∂V

it follows that, in both the first and the second case, we obtain the equation of an ideal gas PV = NT (11) .

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In the thermodynamics, there is an important notion, namely, l, the number of the degrees of freedom of the molecule. The energy of the molecule is E∼l

p2 . 2m

Assuming that the value of l depends on the momentum, we replace this relation by the following one [6]: Eσ = cσ |p|2+σ ,

cσ = const.

(12)

We can say that both the average momentum and the number of the degrees of freedom depend on the temperature, and hence the number of the degrees of freedom depends on the momentum. We assume that this dependence is the simplest one, namely, it is a power dependence (the parameter σ of in the exponent characterizes the molecule). Here the potential Ω takes the form Z ∞ µ m2+γ Ωγ = π 1+γ V T 2+γ t1+γ e−(t− T ) dt, (13) Γ(2 + γ) 0 where

3 1−σ −1= . 2+σ 2+σ It is readily verified that, in this case, relation (11) will remain valid for an ideal gas. Thus, we replace the parameter of the integer degrees of freedom by some continuous parameter that characterizes the given molecule. In principle, this parameter γ is of the same physical origin as the number of the degrees of freedom. But since it is continuous, it takes into account more details of the spectrum of the molecule. Now we can rigorously formulate the axiom of thermodynamics corresponding to the approximate conservation of the gas density (this corresponds to the physicists’ statement in equilibrium thermodynamics: “the density is homogeneous in a vessel”; physicists consider equilibrium thermodynamics as a separate discipline, and then, separately, fluctuation theory). γ=

Main mathematical axiom of thermodynamics. Consider a vessel of volume V containing N > 1019 identical molecules corresponding to the parameter γ = γ0 . Consider a small convex volume of

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6 size Vδ containing Nδ particles, where √ Nδ is not less than 10 . Let P be the probability of the deviation Nδ ± Nδ for any volume of size Vδ inside the vessel. The claim of the axiom is that the probability P is sufficiently small. Namely, √   N Nδ Nδ 0.01 P − > ≤√ . (14) V Vδ Vδ ln Nδ

This implies that the numbering of particles in the volume Vδ is arbitrary and does not affect the density. We can rearrange the numbers and this also does not affect the density: the rearrangement of summands does not change the sum2 . Thus, we have already proved that the approximate conservation of density inside a gas volume implies that the particles are independent of the numbering and hence are Bose particles. Therefore, the potential Ωγ (13) must be replaced by the Bose gas potential 3: Z ∞ 1 t1+γ 1+γ 2+γ Ωγ = π VT dt = π 1+γ V T 2+γ Liγ+2 (eµ/T ), µ Γ(2 + γ) 0 e(t− T ) − 1 (15) where Liγ (·) is the polylogarithm: Liγ+2 (1) = ζ(γ+2), where ζ is the Riemann zeta function. Thus, in our case the Bose gas also depends on another parameter, γ. We call it the fractional (“fractal”) dimension in momentum space (cf. [4], Sec. Degenerate Bose gas, footnote). More precisely, it should be called a parameter of the characteristic of the spectrum of the molecule. Theorem 1. The point of degeneracy of the Bose distribution for µ = 0 for molecules of characteristic γ coincides with the critical points of these molecules up to normalization. Therefore, the dimensionless quantity Zc =

Vc Pc , RTc

2

The set of initial data for an N -particle dynamical system can be called chaotic if a relation of the form (14) arises in a sufficiently large time interval. 3 The Bose gas is usually regarded as a quantum gas, but it is related to number theory (see [5]) and number theory is related to the main axiom. Therefore, the potential Ωγ (15) can be applied to classical gas.

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where R is the gas constant, coincides with the dimensionless point of degeneracy of an ideal Bose gas of fractal characteristic γ. Hence for µ = 0   V ∂Ωγ ∂Ωγ ζ(γ + 2) Zc =  = , γ = γc , (16) T ∂V ∂µ µ=0 ζ(γ + 1) which is a consequence of the number-theoretic interpretation of the Bose distribution. This is a property of the new ideal gas without interaction. Apfelbaum and Vorob’ev [7] computed the critical isotherms in the {P, V }-plane for different gases, where the value of γ is determined by this formula, and the critical isotherm of the Bose gas with the calculated characteristic γ was computed by the relevant formulas for an ideal Bose gas 4 (see Figs. 2–4).

Figure 2. (a) Isotherms of pressure for the Van-der-Waals equation are shown by continuous lines. The small circles show the corresponding lines computed with γ = 0.312 for ϕ(V ) = V (i.e., ideal Bose gas), Zc = 3/8. p = P/Pc , n = N/Nc. (b) Isobars of density for the Van-der-Waals equation are shown by continuous lines. Line 1 is the binodal curve. The small circles correspond to Bose-Einstein isobars for γ = 0.312. At first glance, it looks as if the notion of new ideal gas leads to an alteration of the famous relation PV = NT (17) 4

More general than in [8] and corresponding to the fractal dimension γ in momentum space. In [8], γ = 1/2. There is no classical gas of this dimension.

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Figure 3. Critical isotherms for the Lennard–Jones system. The symbols 1 and 2 are related to results of numerical simulation. Line 3 corresponds to the isotherm of ideal Bose gas with γ = 0.24, Zc = 0.29. (which, besides, is similar to the main economics law, the Irving Fisher formula; it is used to calculate the “turnover rate” of capital [9]). This would be surprising indeed. But this is not so. The relation P V = N T or, equivalently P V = RT (because the number of particles in the vessel remains the same), defines an imperfect gas and, in contemporary experimental thermodynamical diagrams, it is called the Zeno line or, sometimes, the ideal curve, the Bachinskii parabola, etc. On the diagram (ρ, T ) for pure gases, this is the straight line Z = 1. It is a most important characteristic particularly for an imperfect gas. Since, for imperfect gases, it has been calculated experimentally and is an “almost-straight” line on the (ρ, T ) diagram, it follows that the Zeno line is determined by the two points TB and ρB called the “Boyle temperature” and the “Boyle density.” In contrast to Zc , these points are related to the interaction and scattering of a pair of gas particles with the interaction potential peculiar to this gas, as was shown in Section 1 and in other papers of the author (see, e.g. [10]). Therefore, the Zeno line on which the following relations hold: P V = N T,

ρ T + = 1, ρB TB

N = ρ, V

(18)

where ρ is the density (concentration), is a consequence of pairwise interaction and thus is a relation for an imperfect gas.

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Figure 4. (a) Isotherms for argon. The continuous lines correspond to the experimental data, and the line formed by small circles are computed according ζ(γ+2) the isotherm of ideal Bose gas. Zc = ζ(γ+1) = 0.29. Zc = 3/8. p = P/Pc , n = N/Nc . (b) Isotherms for water. Zc = 0.23. (c) Critical isotherms for mercury. Zc = 0.39.

3.

Coincidence of the Solution of Distributions of Bose–Einstein Type for Fractional Dimension with the Solutions Obtained in the Van-der-Waals Model

Below we present the plots (see Fig. 5) obtained by Vorobiev comparing the results of the VdW model and the distribution of Bose–Einstein type for new

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ideal gas. On the standard (P, V )-diagram, which is presented in all thermodynamics textbooks as illustrating the VdW equations, let us compare the gas isotherms for temperatures less than or equal to the critical temperature. We see that the theoretical distribution and the empirical model coincide with precision better than 3%.

Figure 5. Isotherms on the (P, V )-diagram. The solid lines show the isotherms for the VdW gas. The dotted lines show the isotherms for the distribution of Bose–Einstein type with the number of degrees of freedom D, where D = T 2.624: (Zc = 83 = ζ(γ+2) ζ(γ+1) , γ = 0.312, D = 2γ + 2). Here T = Tr = Tc . It is not hard to see that he gas spinodal for the VdW equation coincides very precisely with the quarter of an ellipse. In the general case, we know two points through which the ellipse must pass. One point is the critical point and the other one is the point at γ = 0.312 and Z = 3/8 in the case of isotherms corresponding to the VdW equation which are close to the isotherm of mercury. Now that we know this, we can say that when the spinodal for P = 0 reaches a point lower than 0.5, then the number of degrees of freedom near this point is slightly less than 3. But as was indicated above, the experimental isotherms cannot reach this point, since the spinodal is bounded by the value γtr

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corresponding to the temperature Ttr of the triple point. Let us explain the reasons for which we approximate the spinodal by an ellipse. We know the two endpoints of the spinodal and the two tangents to it at these points are parallel to the coordinate axes of the (Z, P ) plane. The ellipse can be transformed to a circle by changing the scale of the axes. The circle has a constant radius of curvature , and from this point of view the approximation by a circle is as natural as the simplest approximation of a curve joining two points by the line segment with those endpoints. This reminds us of the geometry of Poincar´e’s famous thermodynamical two-dimensional model of the universe, called Poincaria by Gindikin [11]. In that model, there is a circle (the absolute) as unaccessible to the isotherms as, in the situation described above, the spinodal is unaccessible to isotherms.

4.

The Law of Preference of Cluster Formation over Passage to Liquid State

“The law of economic profitability” asserts that at the same temperature Tr < 1, smaller than Tc , gas particles tend to increase the number of degrees of freedom from the initial value γ = γ0 up to γ(Tr ), which is critical for the given temperature Tr . In economics, this preference should be understood as the preference of uniting (associating) over incurring debts. In thermodynamics it should be understood as preferring to form clusters over passing to the liquid state. It is easy to show that the correction to the Bose–Einstein distribution related to this law improves the coincidence of our theoretical isotherms with the isotherms coming from the VdW equation to such an extent that they are practically undistinguishable. The geometric locus of points where the phase transition gas to liquid occurs is known as the binodal. The geometric locus of points where the tangent to the isotherm is perpendicular to the P axis in the (Z, P ) diagram is known as the spinodal. Suppose γc corresponds to the critical isotherm of the given gas, i.e., Zc =

Pc Vc ζ(γc + 2) = . Nc Tc ζ(γc + 1)

The point Ttr which lies on the binodal, and at which three phases coexist:

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gas, liquid, and solid, is called the triple point. So this point lies on the binodal, and isotherms for temperatures lower than that point cannot be observed experimentally. The plot in Fig. 6 shows part of a picture appearing in [12].

Figure 6. The upper thick line is the binodal, the lower one, the spinodal. The thin lines are isotherms Tr = TTc . A good agreement with the latest experimental values for nitrogen takes place for the isotherms from Z = 1.0 up to the point (Z = 0.29, P = 1). The spinodal is the thick lower line; it comes from the empirical model devised by the authors of the paper [12]. Here we see a significant difference from the Van-der-Waals model. In the latter, the spinodal is a quarter of an ellipse on the (Z, P ) diagram which joins the points Z = 3/8 and Z = 0.5. The tangency point of the isotherm with the spinodal is at the end point of the spinodal and is an accumulation point. Let us define the Poisson adiabat for fractional dimension. To do this, con-

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sider the limit as a → 0, i.e., the case of the old ideal gas. We have a2

+ O(a3 ), 22+γ a2 Liγ+1 (a) = a + 1+γ + O(a3 ). 2

Liγ+2 (a) = a +

Z(a) =

(19)

1 + a/2γ+2 a = 1 − γ+2 . γ+1 1 + a/2 2

(20)

On the (Z, Pr ) diagram with Pr = P/Pc as a → 0, we have

Therefore,

dP T γ0 +2 = . da ζ(γ0 + 2)

(21)

dP (2T )γ0+2 =− . dZ ζ(γ0 + 2)

(22)

We have calculated the angle at which the the gas isotherm issues from the point P = 0, Z = 1 whose crirical value of the compressibility factor is Zc =

ζ(2 + γ0 ) . ζ(1 + γ0 )

(23)

Thus, we have obtained a generalization of the Poisson adiabat to the case of a fractional number of degrees of freedom. The equation of the isotherm for a fixed value of γ = γ(T ) is of the form Z=

Liγ+2 (a) , Liγ+1 (a)

P = C(γ)

Liγ+2 (a) , ζ(γ0 + 2)

γ = γ(T ),

(24)

where a is the activity, ∞ > a ≥ 0, while C(γ) is determined by using the coincidence of the relations  ζ(2 + γ) ζ(2 + γ)   , P = C(γ) ,  Z= ζ(1 + γ) ζ(2 + γ0 ) (25) γc = 0.222,    2 > γ ≥ 0.222

with the spinodal constructed above. In this situation C(γ) is determined uniquely.

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Indeed, let the equation of the the quarter of the ellipse for 0 ≤ P ≤ 1 be P = f (Z). Then   ζ(γ + 2) ζ(γ0 + 2) C(γ) = f . ζ(γ + 1) ζ(γ + 2) If γ = γc , Tr,γc = 1, we can write −

dP C(γ(T ))2(γc+2) (2T )(γ0+2) = = . dZ ζ(γc + 2) ζ(γc + 2)

Finally, we obtain (2+γ )

C(γ(Tr ))2γ(Tr ) 2γ0 Tr 0 = , ζ(γ(Tr) + 2) ζ(γ0 + 2)

Tr =

T . Tc

Now the relation for determining γ(T ) acquires the form   2γ(T ) ζ(γ(T ) + 2) 2γ0 T 2+γ0 f = . ζ 2 (γ(T ) + 2) ζ(γ(T ) + 1) ζ 2 (γ0 + 2)

(26)

(27)

The final main equations of the “critical” isotherms of a pure gas corresponding to the value Zc (23) will be  Liγ(T )+2 (a)   ,  Z= Liγ(T )+1 (a) (28)  T (2+γ0 ) · 2(γ0 −γ(T )) ζ(γ(T ) + 2)   P = Liγ(T )+2 (a) . ζ 2 (γ0 + 2)

For the VdW equation to which the ellipse passing through the points (Z = 3/8, P = 1) and (Z = 1/2, P = 0) corresponds, the coincidence of the isotherms is so precise that on the plot they are completely undistinguishable (unlike those in Fig. 5). As V. S. Vorobiev has shown, if the law of preference stated above is not taken into account, the discrepancy with the experimental data for nitrogen are greater. 0 0 Since, as Tr → Ttr , where Ttr is the triple point corresponding to a gas with the given value of Zc (23), the isotherms become denser, it follows that the derivative of the spinodal with respect to P on the (P, Z) diagram is zero. Therefore, instead of ending on the line P = 0, the quarter of the ellipse terminates at the point γ = γtr with the corresponding value of Ptr (µ → 0). This uniquely determines the elliptic spinodal of the gas.

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0 Consider a virtual gas whose critical temperature, denoted by T˜c = Ttr , coincides with the temperature of the triple point of the given gas. Let us consider two isotherms: the critical isotherm of the virtual gas and the isotherm of the given gas for T = Ttr . From relation (25), for a fixed C(γ) with Tp = Ttr , we can find γtr (the value of γ at the triple point). For known values of Ztr and Ptr , using formula (13), we obtain the value of C(γtr ). Thereby we find on the (P, Z) diagram the terminal point of the spinodal (24) at which the derivative of the quarter of the ellipse vanishes. We have found this by using the experimental values of the two endpoints of the quarter of the ellipse, which is our gas spinodal. Thus, the positions of the critical point and the triple point (well known from standard tables) entirely determine the isotherms of the given gas. It should be noted that the theory constructed above did not take into account the interaction of particles anywhere. As was already pointed out in the author’s papers, it is more natural to pass from momenta p to energies ε = p2 /2m and to generalize the problem of the transition of the energy p2 /2m to the energy H = p2 /2m + c/r k , i.e., to the Hamiltonian containing repulsion. In this case, the following correspondence is used:

p2 p2 c → + k, 2m 2m r

1 ep2 /2m−µ/T

−1



1 2 k e(p /2m+c/r −µ)β

−1

,

(29)

where β = 1/T . Here integration of the distribution corresponding to the number of particles must be performed with respect to the measure pγ dp r 2 dr (integration over angles is omitted). Integration of the pressure distribution must be performed with respect to the measure Hpγ dp r 2 dr over the three-dimensional volume V .

5.

Supercritical State as the Parastatistics of Clusters

When Tr > 1, in order to determine C(γ), we use parastatistics in which µ varies from minus infinity to plus infinity. In this case the isotherm issued on the (Z.P ) plane from the point Z = 1, P = 0 comes to the line Z = 1 for certain positive µ. The value of K(Tr , γ) satisfies the condition   γ  1 (γ) 1+γ Nc = C(γ)ζ(1 + γ) = Tr C(γ) 1 − ζ(1 + γ). (30) 1+K

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Hence Tr = (1 − (K + 1)−γ )−1/(γ+1) ,

γ = γ(Tr ).

(31)

Thus on the isochor N − Nc we can find common points of this isochor and the isotherm T = Tr , by writing  Nc = ζ(γ + 1) 1 −

1 (K + 1)γ



,

(32)

where K = K(Tr ). In the discrete case, for γ < 0 we have the law of reflection Nc =

∞ X j=1





j=1

j=1

X 1 X jγ γ = j − j γ F (bj). ebj − 1 bj

(33)

Since the function f (x) = xγ F (bx) is monotone decreasing, it follows that ∞ X j=1

j γ F (bj) =

∞ X j=1

f (j) ≤

Z

0



f (x) dx =

Z



xγ F (bx) dx = b−γ−1

0

Z



xγ F (x) dx.

0

(34)

Hence for µ = 0 the isochor N = Nc coincides with the isochor of the second sheet, corresponding to negative values of γ (i.e. second phase) for µ ˜ = 0 and γ˜ < 0, so, that −˜ γ = γ and µ = 0. Thus we shall pass from the sheet µ ≤ 0 to the sheet µ ˜ ≥ 0 and γ < 0.

6.

Asymptotic Continuation of a Perfect Liquid to the Second Sheet

The gas spinodal, which is defined in a new way as the locus of isotherms of a new ideal gas, is formed at the maximum entropy at the points at which the chemical potential µ vanishes. Therefore, on the diagram (Z, Pr ), the spinodal is a segment Pr ≤ 1, Z = Zc (in the case of the Van-der-Waals normalization Tr = T /Tc and Pr = P/Pc ). For Tr ≤ 1, the Bose condensate occurs and, consequently, for the liquid phase on the spinodal, the quantity N = C(γ)Trγc +1 ζ(γc + 1)

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remains constant on the liquid isotherm. This means that the isotherm of the liquid phase that corresponds to a temperature T is given by Z=

Pr Pr = γc +2 . Tr N Tr ζ(γc + 1)

(35)

All isotherms of the liquid phase (including the critical isotherm at Tr = 1) pass through the origin Z = 0, Pr = 0 and then fall into the negative region (or to the second sheet). The point Z = 0 corresponds to the parameter γ = 0, and hence to the continuation to γ < 0, since, for µ = 0, the pressure Pr = C(γ)Tr2+γ

ζ(2 + γ) ζ(2 + γc )

(36)

can be extended to 0 > γ > −1. We shall see below that the value of Z as µ/T → o(1/ ln N ) is also positive, and therefore the spinodal for 0 > γ > −1 gives another sheet on the diagram (Z, P ) (the second one); it is more convenient to take this sheet onto the negative quadrant. Under the assumption that the transition to the liquid phase is not carried out for T = 1, we equate the chemical potentials µ and µ ˜ for the “liquid” and “gaseous” phase on the isotherm T = 1 (this fact is proved below). After this, we find the point µ which is the point of transition to the “liquid” phase for T < 1 by equating the chemical potentials of the “liquid” and “gaseous” phase. In this section, we find the point of the isotherm-isochore of the liquid as the quantity κ = −µ/T slowly tends to zero. First of all, we take into account the fact that Nc is finite, although it is large, and hence we are to use a correction. Recall the relation Ω = −Λγc −γ T

X

ln

k

 1 − exp µ−εk N  T , k 1 − exp µ−ε T

Λγc −γ = C(γ)

(37)

for the Ω-potential. Applying the Euler–Maclaurin with regard to the parameter γ, we obtain n  X j=1

jγ ebj+κ − 1



kj γ 

ebkj+κ

=

1 α

Z

0

∞

1 ebx+κ − 1



k ebkx+κ − 1

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where α = γ + 1, k = N , and b = 1/T . Here the remainder R satisfies the bound 1 |R| ≤ α

Z



0

|f 0 (x)| dxα,

where

f(x) =

1 ebx+κ

−1



k ek(bx+κ)

−1

.

Evaluating the derivative, we see that bk2 ek(bx+κ)

0

f (x) =

(ek(bx+κ) − 1)2



bebx+κ (ebx+κ − 1)2

,

|R| ≤

1 αbα

Z ∞˛ ˛ ˛ 0

k2 ek(y+κ) (ek(y+κ) − 1)2

ey+κ



(ey+κ − 1)2

˛ ˛ α ˛ dy . (38)

We also have [13] ey 1 = 2 + ψ(y), (ey − 1)2 y

where ψ(y) is a smooth function and |ψ(y)| ≤ C(1 + |y|)−2 .

Substituting this formula into (38), we obtain |R| ≤

1 αbα

Z

∞ 0

2−α ˛ ˛ 2 ` ´ ˛k ψ k(y+κ) −ψ(y+κ)˛ dyα ≤ k bα

Z





|ψ(y)| dyα +

1 bα

Z



κ

|ψ(y)| dy ≤

C bα

with some constant C. Consequently, k = Nc and T = Tc, and we obtain the following formula for the integral with µ = 0: Z Z Λγc−γ ξ dξ α Λγc−γ ∞ ηdη α M= = 1+α , αΓ(γ + 2) ebξ − 1 b eη − 1 0 where α = γ + 1. Hence, b=

1 M 1/(1+α)



Λγc −γ αΓ(γ + 2)

Z

∞ 0

ξ dξ α eξ − 1

1/(1+α)

.

We obtain (see [14]) Z

=



ff « Z ∞„ 1 k 1 1 1 − kbξ dξ α = α − dξ α ξ −1 e −1 b e −1 ξ 0 0 «  ff Z ∞„ Z 1 1 k1−α ∞ kα 1 kα + α − dξ α − α − dξ α kξ b ξ ξ(1 + (k/2)ξ) b e −1 kξ(1 + (k/2)ξ) 0 0 

ebξ

z(γ) 1−α (k − 1) + O(b−α). bα

Write k = N |µ˜/T =0 , N |µ˜/T =0 ∼ = (Λγc −γ z(γ))1/(1+γ)T,

where z(γ) =

Z

0

∞

1 1  γ − ξ ξ dξ. ξ e −1 (39)

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The relation N = C(γ)Trγ+1 ζ(γ + 1), where γ = γ(Tr), according to (27) meets the linear relation N = A(˜ γ )Tr , where A(˜ γ ) = (Λγc−˜γ z(˜ γ ))1/(1+˜γ). We can make the normalization of activity a at the point Tc , and we can find a0 by matching the liquid and gaseous branches at Tc for the pressure, in order to prevent the phase transition on the critical isotherm at Tr = 1. In what follows, we normalize the activity for Tr < 1 with respect to the value of a0 computed below. Then the chemical potentials (in thermodynamics, the thermodynamic Gibbs potentials for the liquid and gaseous branches) coincide, and therefore there can be no phase transition “gas-liquid” at Tr = 1. Now, for the isochore-isotherm of the “incompressible liquid” to take place, we must construct it with regard to the relation Nc = ζ(γc + 1), namely, Nγ = C(γ)Trγ+1 ζ(γ + 1),

(40)

where γ = γ(Tr) according to (27), remains constant on the liquid isotherm. We obtain the value γ˜ = γ(Tr) from the implicit equation A(˜ γ ) = C(γ(Tr))Trγ(Tr ) ζ(γ(Tr) + 1). Thus, for each Tr < 1, we find the spinodal curve in the domain of negative γ [14], Λ(˜γ −γc )/(1+˜γ ) z(˜ γ )1/(1+˜γ) = C(γ(Tr))Trγ(Tr ) ζ(γ(Tr) + 1),

(41)

We choose the least value of γ˜ (it has the largest absolute value) which is one of the two solutions of equation (41) and denote it by γ(Tr). In particular, for Tr = 1, we write γ0 = γ˜(1). Let ag = e−µ/T be the activity of the gas and al = e−˜µ/T the activity of the liquid. We present the condition for the coincidence of M and of the activities at the point of the phase transition,   al γ(Tr ) |˜ γ (Tr )|+γc −|˜ γ (Tr )| C(γ(Tr))Tr Li2+γ(Tr )(ag ) = Λ Tr Li2−|˜γ (Tr )| , a0 (42) Λγc−γ0 al Li2+γ0 (a0 ) = 1, ag = . (43) ζ(2 + γc ) a0 These two equations determine the value of the chemical potential µ = µ ˜= T ln ag at which the phase transition of the “ gas” into the “liquid” occurs.

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217

The Positive Chemical Potential for the Number of Degrees of Freedom Less than 2 (γ < 0)

Similarly, for µ ˜ > 0 we obtain the following relations. Let us find the constant Tr and κ according (31) from the following relations by taking K = K(Tr ) into account, A

  1 K+1 ξ − ξ γ dξ = P, b(ξ−κ) − 1 b(K+1)(ξ−κ) − 1 e e 0  Z A 1 K+1 − b(K+1)(ξ−κ) ξ γ dξ = N, b(ξ−κ) − 1 e e − 1) 0

Z

where κ = −˜ µ, b = 1/Tr , and A  1. Write  Z A 1 1 (A) z (γ, µ ˜) = − ξ γ dξ, ξ 1−κ eξ−κ − 1 0

γ < 0.

(44) (45)

(46)

Hence after the change (K + 1)ξ = η we obtain N (A) = −

„ « ff  Z 1 (A) 1 K + 1 1−α (K + 1)1−α ∞ 1 1 z + α z(A) − − dη α α α η b b 2 b e −1 η(1 − η/2) 0 Z (K + 1)1−α 1 (K + 1)1−α (A) 1 ∞ dη α · = α z(A) + z , γ = 1. − α 2 0 1 + η/2 b b bα (47)

P (A) = −

1 (A) z bα+1 γ+1

+

(K + 1)−α (A) zγ+1 . bα+1

The relation

(48)

(A)

Z = lim

A→∞

((K + 1)−α−1 − 1)zγ+1 (A)

((K + 1)−α − 1)zγ

=1

(49)

gives the relationship between K, γ and κ on the Zeno line. We denote µr = µ/Tc , ρr = ρ/ρc, where ρc can be found from the condition ρc Tc + = 1. ρB TB Now from (49) and the relation on the Zeno line we can find isotherms of the second sheet.

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Thus values µ ˜r , TB,r and ρB,r are dimensionless, and Bachinskii’s relation on the Zeno line has the form N (A) Tr + = 1, ρB,r TB,r

(50)

for A  1. This relation restricts value µ ˜r , which turns out to be small. Therefore, (A) (A) “velocity” at which Nγ reaches the line Z = 1 for Nγ > Nc is high, and an interval of isochor between P = 1 and Z = 1 on the (P, Z) diagram form an almost straight line. Remark 1. Let us note the important equality which determines the constant Λ in the distribution for Ω potential. According to (33)–(34), we have γ˜liquid = −γc . Hence by substituting in (39) the value Λ is determined uniquely for γ < 0.

References [1] Yu. G. Pavlenko Lectures on Theoretical Mechanics (Fizmatlit, Moscow, 2002) [in Russian]. [2] B. D. Summ, Foundations of Colloidal Chemistry, Akademiya, Moscow (2007) [in Russian]. [3] E. A. Guggenheim, “The Principle of Corresponding States,” Journal of Chemical Physics, 13, 253–261, (1945). [4] L. D. Landau and E. M. Lifshits, Statistical Physics (Nauka, Moscow, 1964) [in Russian]. [5] V. P. Maslov, “Thermodynamics of fluids: The law of redestribution of energy, two-dimensional condensate, and T-mapping,” Teoret. Mat. Fiz. 161 (3), 422–456 (2009). [6] W.-S. Dai, M.Xie, “Gentile statistics with a large maximum occupation number,” Annals of Physics 309, 295–305 (2004).

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[7] E. M.Apfelbaum, V. S. Vorob’ev, “Correspindence between of the ideal Bose gas in a space of fractional dimension and a dense nonideal gas according to Maslov scheme”, Russian J. Math. Phys. 18 (1), 19–25 (2011). [8] L. D. Landau and E. M. Lifshits, Quantum Mechanics (Nauka, Moscow, 1976) [in Russian]. [9] I. Fisher, The Purchasing Power of Money: Its Determination and Relation to Credit, Interest and Crises (Izd. Delo, Moscow, 2001) [in Russian]. [10] V. P. Maslov, Zeno-line, Binodal, T − ρ Diagram and Clusters as a new Bose-Condensate Bases on New Global Distributions in Number Theory, arXiv 1007.4182v1 [math-ph], 23 July 2010. [11] S. G. Gindikin, Tales about Physicists and Mathematicians (MTsNMO, Moscow, 2001). [12] W.-G. Dong and J. H. Lienhard, “Corresponding states of saturated and metastable properties,” Canad. J. Chem. Eng. 64 158–161 (1986). [13] V. E. Nazaikinskii, “On the Asymptotics of the Number of States for the BoseMaslov Gas,” Math. Notes, 91 (6) 816–823, (2012). [14] V. P. Maslov, Threshold Levels in Economics, arXiv:0903.4783v2 [qfin.ST], 3 Apr 2009. Reviewed by prof. V. S. Vorob’ev, Institute for High Energy Densities RAS, Joint Institute of High Temperatures of the Russian Academy of Sciences, 125412, Moscow, Izhorskaya ul., 13/19, E-mail address: [email protected].

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In: Advances in Chemistry Research. Vol. 21 ISBN: 978-1-62948-742-7 c 2014 Nova Science Publishers, Inc. Editor: James C. Taylor, pp. 221-235

Chapter 7

P ROPERTIES OF R AREFIED N OBLE G AS F LOWS Lajos Szalm´as∗ Hatvan, Hungary

Abstract This chapter reviews some of the recent developments of theoretical investigation of rarefied flows of noble gases and their mixtures. These flows can be found in various engineering applications, such as microand nano-fluidics or vacuum technology. From theoretical viewpoint, rarefied gas flows can be studied on the basis of the Boltzmann or other kinetic equations, which are valid in the whole range of the gaseous rarefaction parameter, i.e. the ratio of the relevant macroscopic size of the flow over the molecular mean free path. These integro-differential equations can be solved by deterministic or probabilistic approaches. Among these techniques, the discrete velocity method and the direct simulation Monte Carlo are most common. The chapter describes the theoretical background of rarefied gases and the applied solution methods of kinetic equations. The slip phenomena are discussed. Results in terms of the so-called slip coefficients, which describe the slip of the macroscopic velocity along solid walls, are presented for noble gas mixtures. Flows of rarefied noble gases in long channels are also considered. The effects of the molecular masses, the mass ratios and the concentration for gas mixtures on the results are outlined. The chapter brings the results of ∗

E-mail address: [email protected]

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Lajos Szalm´as recent developments of rarefied gas dynamics to the attention of other researchers, engineers or non-specialized people.

PACS: 05.20.Dd, 47.45.-n, 51.10.+y Keywords: Rarefied gas dynamics, noble gases, Boltzmann equation, slip phenomena, flows in channels AMS Subject Classification: 76P05, 82B40

1.

Introduction

The flows of rarefied gases have attracted considerable attention over the last years in fluid dynamic communities. This interest can be justified by the appearance of micro- and nano-fluidics [1, 2] and the relevance of more traditional scientific areas, such as vacuum science [3]. When the flow is under rarefied condition, the details of the molecular interactions are non-negligible, and its proper description should be based on the molecular distribution function and the Boltzmann or other kinetic equations [4]. The theoretical analysis of rarefied gas flows requires the solution of these equations, which is a non-trivial task, since kinetic equations are integrodifferential and typically can be solved only by numerical methods. Rarefaction phenomena affect the flow, and the dynamics of rarefied gases significantly differs from ordinary hydrodynamics. While the properties of single gas flows are almost known, flows of gaseous mixtures are less studied. The description of mixtures is more complicated, and their characterization requires more parameter, like the gaseous concentration and the molecular masses. The scope of this chapter is to discuss some important aspects of flows of rarefied gaseous mixtures. The theoretical description of rarefied flows is presented. Recent development of the solution of the associated kinetic equations is reviewed. The slip phenomena and the main characteristics of flows through long channels are considered. Results for the slip coefficients and flows of selected noble gas mixtures through long channels are presented.

2.

Main Concept

A gaseous mixture consisting of K components is considered. The components have molar masses and molar densities mα and nα . The total molar

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PK density and the mass density of the mixture are given by n = α=1 nα and PK ρ = α=1 nα mα . The average mass of the mixture is defined as m = ρ/n, P and the bulk velocity of the mixture can be obtained by u = K α=1 nα mα uα /ρ, where uα is the velocity of species α. The proper description of rarefied gases requires the consideration of the distribution function of the molecular velocity fα (v, r, t) for component α, where v is the velocity of the molecules, r is the coordinate vector and t is the time variable. In the following, steady flows will be considered; hence, the time variable is omitted. The macroscopic molar density and velocity of the components are derived by using the moments of the distribution function Z nα = dvfα, (1) Z 1 uα = dvvfα. (2) nα The flow can be characterized by the local concentration of the first species n1 n

(3)

P Lc , µv0

(4)

C(r) = and the local rarefaction parameter δ(r) =

where P and µ are the total pressure and the viscosity of the mixture, v0 = p 2Rg T (r)/m is the characteristic molecular speed and Lc is the characteristic length of the flow. Here, Rg = 8.314J/K is the global gas constant and T (r) denotes the temperature. The distribution function obeys the steady Boltzmann equation v

∂fα X = Qαβ (fα , fβ ), ∂r

(5)

β

where Qαβ (fα , fβ ) denotes the collision operator, which encodes the interactions among the molecules. The collision operator can be based on various interaction potentials, like hard-sphere, variable soft-sphere [5] or the LennardJones potentials. However, it can be replaced by simplified kinetic models. For

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single gases, the Bhatnagar-Groos-Krook operator is the most known collision term [6]. For gaseous mixtures, numerous operators, like the Sirovich model [7], the Hamel-Morse model [8, 9], the model of Andries et al. [10] and the model of Kosuge [11] have been developed. The most used linearized kinetic model for mixtures is the McCormack operator [12]. If the gas is slightly perturbed, the distribution function can be linearized such that fα (v, r) = fα0 (v, r)[1 + hα (v, r)],

(6)

where hα (v, r) is the perturbation function and fα0 (v, r) = n0α (r)π −3/2 vα−3 exp(−v 2 /vα2 )

(7)

is the equilibrium distribution function ofp component α. Here, n0α (r) is the equilibrium value of the density, and vα = 2Rg T /mα is the mean velocity of the molecules for species α. The variation of the relative density and the macroscopic velocity around the equilibrium values can be obtained as the moments of the perturbation function Z nα − n0α −3/2 −3 =π vα dvhα exp(−v 2 /vα2 ), (8) n0α Z uα =π −3/2 vα−3 dvvhα exp(−v 2 /vα2 ). (9) The perturbation function is governed by the linearized Boltzmann equation v

∂hα X ∂f 0 1 = Lαβ (hα, hβ ) − α 0 , ∂r ∂r fα

(10)

β

where Lαβ (hα, hβ ) is the linearized collision term, and the last term on the right hand side represents a driving force arising from the spatial variation of the equilibrium properties. The kinetic equations, Eqs. (5) and (10), are supplemented by boundary conditions at the solid-gas interfaces. There are numerous models to describe the gas-surface interaction. In typical applications under isothermal conditions, however, the diffuse or diffuse-specular type boundary conditions can provide physically accurate results. In the diffuse boundary condition, the molecules hitting the solid surface are reflected in accordance with the local equilibrium

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of the wall. In the diffuse-specular condition, some part of the molecules is reflected in a specular fashion. The relative weight of the diffusely reflected molecules is the so-called accommodation coefficient, σ, which can be in the range of 0 < σ ≤ 1.

3.

Numerical Methods

Kinetic equations are integro-differential mathematically. Their analytical solution exists only in very special cases, and they need to be solved numerically in typical applications. The numerical methods for their solution can be either deterministic or probabilistic. The deterministic approaches are often referred as the discrete velocity methods, when the spatial, temporal and velocity spaces are discretized [13]. The coordinate derivatives are approximated by finite differences, and the integrals of the collision term are evaluated by quadratures. The resulting discrete equations are solved computationally. Time-dependent problems can be solved directly by a real-time integration, while steady flow problems are typically solved iteratively. The kinetic iterators can be slow in the hydrodynamic limit. In order to overcome this difficulty, accelerated iteration schemes can be used for kinetic models [14, 15, 16]. The most known probabilistic method is the direct simulation Monte Carlo (DSMC) [17], in which the gas dynamics is modeled by the simulation of a large number of test particle in the real geometry. The volume is divided into a collection of cells, where the macroscopic quantities are obtained as cell averages of the dynamic quantities of the particles. The Boltzmann equation is solved by the splitting approach. The algorithm consists of the steps of free motion of molecules and of collision, when the interactions between the particles are simulated in a probabilistic sense. The results of the DSMC are always subject to statistical fluctuations, which make the simulation unpractical for low-speed flows. In that case, the low-variance or variance-reduced DSMC can be used to speed up the scheme by modeling the departure from equilibrium [18, 19, 20].

4.

Slip Phenomena

The problem of flows of rarefied gases may be modeled by an extended version of the fluid dynamic equations near the hydrodynamic limit. In this

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Table 1. Viscous slip coefficient versus concentration for mixtures of noble gases on the basis of the McCormack kinetic model. Results in the first and second columns for the same mixture stand for hard-sphere and Lenard-Jones potentials, respectively.

C 0.00 0.01 0.10 0.25 0.50 0.75 0.90 0.99 1.00

N e/Ar 1.018 1.018 1.019 1.019 1.024 1.025 1.032 1.033 1.040 1.040 1.037 1.036 1.028 1.027 1.019 1.019 1.018 1.018

σP He/Ar 1.018 1.018 1.021 1.021 1.044 1.044 1.084 1.084 1.151 1.150 1.193 1.191 1.154 1.153 1.041 1.041 1.018 1.018

He/Xe 1.018 1.018 1.022 1.022 1.057 1.059 1.122 1.127 1.253 1.259 1.402 1.401 1.413 1.412 1.116 1.119 1.018 1.018

case, the Navier-Stokes equations can be used by considering slip (and jump) boundary conditions for the macrodynamic quantities. The slip phenomenon is a rarefaction effect and can be characterized by solving the kinetic equations.

4.1.

Viscous Slip Coefficient

Shear flow of gaseous mixture along a plane solid wall causes an apparent velocity slip on the wall. Suppose that the normal and tangential vectors of the wall are along the x and y coordinate directions. For a shear flow, the extrapolated tangential velocity slip at the wall can be expressed as uy = σ P

∂uy l0 , ∂x

(11)

where σP is the so-called viscous slip coefficient, and l0 =

µv0 P

(12)

is the viscosity based mean free path of the molecules. The spatial derivatives of the velocity in Eq. (11) is taken far from the wall, where the so-called Knudsen

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layer does not affect the velocity profile. The above definition of the mean free path, Eq. (12), provides a unique identification, since the viscosity is always available for every interaction potentials or kinetic models, while otherwise the physical interpretation of the mean free path for kinetic models would be problematic. The above mentioned so-called Kramers problem can be described at the kinetic level. The resulting equations are typically solved by the discrete velocity method to deduce the viscous slip coefficient. Additionally, the Knudsen layer characteristics is also determined by the kinetic solution. By using this direct numerical solution, the slip coefficient has been calculated for several kinetic models or the Boltzmann equations. For gaseous mixtures, comprehensive results are available on the basis of the McCormack linearized kinetic model [21]. Table 1 shows the slip coefficient calculated by the McCormack model for selected noble gas mixtures as a function of the concentration. The applied boundary condition is diffuse reflection. The results of the hard-sphere and Lenard-Jones potentials are close to each other. The intermolecular potential has small effect on the slip coefficient. As it can be seen the coefficient has a maximum at an intermediate value of the concentration, and it is always larger for mixtures than single gases. The coefficient is larger for mixtures having a larger mass ratio.

4.2.

Diffusion Slip Coefficient

Concentration gradient along the tangential direction of a plane wall causes a uniform non-zero velocity in the bulk phase for gas mixtures. This phenomenon is know as the diffusion slip. The uniform tangential velocity far from the wall can be expressed as uy = σ C

v0 ∂C l0 , 2 ∂y C

(13)

where σC is the diffusion slip coefficient. The kinetic problem of the diffusion slip can also be solved directly. Such a calculation provides σC and the corresponding velocity profile in the boundary layer. For gaseous mixtures, the diffusion slip coefficient has been calculated by using the McCormack kinetic model in a comprehensive manner for diffuse reflection boundary condition [22]. Table 2 shows the coefficient as a function of the concentration for selected noble gas mixtures for hard-sphere and realistic

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Table 2. Diffusion slip coefficient versus concentration for mixtures of noble gases on the basis of the McCormack kinetic model. Results in the first and second columns for the same mixture stand for hard-sphere and realistic potentials, respectively.

C 0.00 0.01 0.10 0.25 0.50 0.75 0.90 0.99 1.00

N e/Ar 0.000 0.000 0.005 0.006 0.047 0.062 0.118 0.152 0.231 0.291 0.335 0.416 0.391 0.486 0.421 0.526 0.424 0.530

σC He/Ar 0.000 0.000 0.010 0.013 0.107 0.134 0.297 0.363 0.731 0.862 1.468 1.681 2.288 2.628 3.139 3.708 3.264 3.879

He/Xe 0.000 0.000 0.016 0.024 0.172 0.253 0.483 0.690 1.228 1.646 2.654 3.281 4.699 5.612 8.018 10.210 8.700 11.360

potentials [24]. This latter potential is derived from the experimental values of the transport parameters of gaseous mixtures and expected to provide reliable results. As it can be seen in the table, the diffusion coefficient is always positive, increases with increasing concentration and molecular mass ratio of the mixture. If the mixture has a larger mass ratio, the diffusion effects are stronger. The values at C = 1 are the limiting results for C → 1, since C = 1 indicates a single gas.

4.3.

Thermal Slip Coefficient

A constant temperature gradient along the tangential direction of a plane wall generates a uniform gaseous velocity in the bulk phase far from the wall. This phenomenon is the so-called thermal slip. The resulting tangential velocity far from the wall can be expressed as uy = σ T

v0 ∂T l0 , 2 ∂y T

(14)

where σT is the thermal slip coefficient. The thermal slip phenomenon can be described at the kinetic level, and by solving the kinetic problem the thermal

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Table 3. Thermal slip coefficient as a function of the concentration for mixtures of noble gases on the basis of the McCormack kinetic model. Results in the first and second columns for the same mixture stand for hard-sphere and realistic potentials, respectively.

C 0.00 0.01 0.10 0.25 0.50 0.75 0.90 0.99 1.00

N e/Ar 1.175 1.175 1.173 1.175 1.164 1.179 1.153 1.184 1.144 1.186 1.151 1.183 1.163 1.178 1.173 1.175 1.175 1.175

σC He/Ar 1.175 1.175 1.170 1.174 1.127 1.172 1.066 1.171 1.004 1.183 1.025 1.209 1.101 1.217 1.167 1.184 1.175 1.175

He/Xe 1.175 1.175 1.173 1.181 1.162 1.243 1.146 1.345 1.133 1.503 1.168 1.592 1.231 1.523 1.202 1.252 1.175 1.175

slip coefficient and the velocity profile in the boundary layer can be determined. Comprehensive results for the thermal slip coefficient for noble gas mixtures are available on the basis of the direct solution of the McCormack linearized model kinetic equation for diffuse reflection boundary condition [23]. In Table 3, σT is shown for selected noble gas mixtures with hard-sphere interaction and realistic potential. It can be seen that the thermal slip coefficient is positive and quit sensitive to the interaction potential. For the realistic potential, which is expected to provide physically reliable result, the coefficient is larger at intermediate values of the concentration than the limiting values for the single gas case.

5.

Flows through Long Channels

Isothermal flows of gas mixtures in long channels are considered. The axis of the channel is in the x coordinate direction, while its cross section is located in the y, z coordinate sheet. The flow is driven by the pressure and/or concentration difference between the two ends of the channel. Due to the pressure and/or concentration difference, the flow is driven by the local pressure and concentra-

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tion gradients XP =

∂P Lc , ∂x P

XC =

∂C Lc . ∂x C

(15)

Since the channel is long, the driving forces are small |XP |  1, |XC |  1 and the flow speed is much smaller than the characteristic molecular velocity v0 . Under these conditions, the linearized description can be used. The component densities vary only along the axial direction, and the component flow velocities are also along this direction.

5.1.

Kinetic Coefficients

The flow problem can be described at a particular cross section of the channel. Within this framework, the gas mixture with concentration C and rarefaction parameter δ is locally driven by XP and XC causing a non-zero molar flux of the components. It is convenient to introduce the following thermodynamic fluxes [25] Z JP = − dA(n1 u1x + n2 u2x ), (16) AZ JC = −n1 dA(u1x − u2x ). (17) A

These fluxes are the linear functions of the driving forces nAv0 [ΛP P XP + ΛCP XC ], 2 nAv0 JC = [ΛP C XP + ΛCC XC ], 2 JP =

(18) (19)

where Λij are the so-called kinetic coefficients. The usefulness of choosing JP and JC is that the kinetic coefficients are not independent. The cross coefficients are equal with each other ΛCP = ΛP C due to the Onsager relation [26]. The local flow problem can be solved at the kinetic level. Comprehensive results in terms of the kinetic coefficients are available on the basis of the McCormack model at various values of the concentration and the rarefaction parameter. Flows of gas mixtures between two parallel plates [27] and through long rectangular [28], circular [29], triangular and trapezoidal channels [30] have been calculated by using this model. Figure 1 shows the kinetic coefficient ΛP P for He/Xe and N e/Ar noble gas mixtures for a special triangular

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10

231

C=0.0 C=0.2 C=0.5 C=0.8

ΛPP

ΛPP

C=0.0 C=0.2 C=0.5 C=0.8

1

1

0.1

1

10 δ

100

0.1

1

10

100

δ

Figure 1. Kinetic coefficient ΛP P for He/Xe (left) and N e/Ar (right) mixtures for flows through a triangular channel.

channel at C = [0, 0.2, 0.5, 0.8] [30]. The diffuse reflection boundary condition is used at the wall. Here, C = 0 corresponds to the single gas case. As it can be seen LP P is strongly sensitive to the gaseous concentration in case of He/Xe, while it does not depended on the concentration too much in case of N e/Ar. This situation can be explained by the larger molecular mass ratio of He/Xe mixture. Figure 2 presents the kinetic coefficients ΛCP and ΛCC for He/Xe. These coefficients decrease with increasing δ and tend to zero in the hydrodynamic limit. Both coefficients describe the gaseous diffusion, which is intensified by increasing rarefaction. The corresponding coefficients for N e/Ar can be found in Ref. [30] and are much smaller than those of He/Xe.

5.2.

Global Behavior

After the local flow problem is solved and the kinetic coefficients are deduced in a wide range of concentration and rarefaction, the global flow behavior, the distributions of pressure and concentration and the molar flow rates of the components can be determined. To achieve this goal, the conservation of mass along the axis of the channel is to be solved. Figure 3 shows typical pressure and concentration profiles of a global pressure driven flow through a long rectangular channel with height 1.88µm, width 21.2µm and length L = 5000µm. The outlet pressure is PB ≈ 2kP a, the inlet and outlet concentrations are CA = CB = 0.501 and the temperature is T = 298.5K [31]. As it can be seen the pressure profile is nearly linear, which corresponds to the high rarefaction.

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1

ΛCC

ΛCP

1

0.1

0.1 C=0.2 C=0.5 C=0.8

0.01

C=0.2 C=0.5 C=0.8

0.01 0.1

1

10 δ

100

0.1

1

10

100

δ

Figure 2. Kinetic coefficients ΛCP (left) and ΛCC (right) for flows of He/Xe mixture through a triangular channel.

The concentration is non-uniform in the channel. Such behavior is the consequence of the gaseous separation, when the lighter species travels faster than the heavier one in the channel. The gaseous separation and the non-uniformity of the concentration are intensified by the increasing pressure drop. For the pressure driven flow, the molar flow rates of the components are also calculated on the basis of the mass conservation. With increasing rarefaction, due to the diffusion effects, the ratio of the flow rates of the lighter and heavier components is increasing. Its minimal value occurs in the hydrodynamic limit, when the ratio is solely determined by the mole fraction of the components and equals to CA /(1 − CA ). The separation phenomena can be important in various applications. In the rarefied domain, flows through meshes can be used to separate gaseous components by their weights. In this situation, the gas separation is useful and its role should be maximized in order to achieve maximal efficiency of the separator. However, there are applications, when the gaseous separation should be avoided. For example, when rarefied gases are transported through long channels, the separation should be minimized in order to reach an efficient transportation of all components of the gas.

6.

Conclusion

In this chapter, some features of flows of rarefied noble gases have been reviewed. The kinetic description of these flows, involving the molecular distri-

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0.52

6

0.5

4

0.48

C

P/PB

Properties of Rarefied Noble Gas Flows

2

233

0.46

0

0.44

0

0.2

0.4

0.6 x/L

0.8

1

0

0.2

0.4

0.6

0.8

1

x/L

Figure 3. Distribution of pressure (left) and concentration (right) for He/Ar flow driven by global pressure difference through a rectangular duct. Symbols M, N, , , ◦ represents the results for PA /PB = [3.03, 4.06, 5.03, 5.91, 6.42], respectively.

bution function and the Boltzmann equation, is presented. The typical numerical methods used for the solution of kinetic equations, the discrete velocity method and the direct simulation Monte Carlo, are discussed. Results for the viscous, diffusion and thermal slip coefficients are presented for noble gas mixtures. The determination of the flow in long channels is presented. Results in terms of the kinetic coefficients and the distribution of pressure and concentration for pressure driven flows of selected noble gas mixtures are shown. The chapter may be useful for scientists, engineers or non-specialized people interested in the dynamics of flows of rarefied gases.

References [1] Kandlikar, S.G.; Garimella, S.; et al. Heat transfer and fluid flow in minichannels and microchannels; Elsevier: Oxford, 2006. [2] Li, D. Encyclopedia of Microfluidics and Nanofluidics; Springer-Verlag, 2008. [3] Jousten, K. Handbook of Vacuum Technology; Wiley-VCN, 2008. [4] Cercignani, C. The Boltzmann Equation and its Application; SpringerVerlag: New York, 1988.

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[5] Koura, K.; Matsumoto, H. Phys. Fluids 1991, 3, 2459-2465. [6] Bhatnagar, P.L.; Gross, E.P.; Krook, M. Phys. Rev. 1954, 94, 511-525. [7] Sirovich, L. Phys. Fluids 1962, 5, 908-918. [8] Morse, T.F. Phys. Fluids 1964, 7, 2012-2013. [9] Hamel, B.B. Phys. Fluids 1965, 8, 418-425. [10] Andries, P.; Aoki, K.; Perthame, B. J. Stat. Phys. 2002, 106, 993-1018. [11] Kosuge, S. Eur. J. Mech. B/Fluids 2009, 28, 170-184. [12] McCormack, F.J. Phys. Fluids 1973, 16, 2095-2105. [13] Aristov, V.V. Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2001. [14] Lihnaropoulos, J.; Naris, S.; Valougeorgis, D. Transp. Theory Statist. Phys. 2007, 36, 513-528. [15] Szalmas, L.; Valougeorgis, D. J. Comput. Phys. 2010, 229, 4315-4326. [16] Szalmas, L. Comput. Phys. Commun. 2013, 184, 1432-1438. [17] Bird, G.A. Molecular Gas Dynamics and the Direct Simulation of Gas Flows; Oxford University Press, 1994. [18] Radtke, G.A.; Hadjiconstantinou, N.G. Phys. Rev. E 2009, 79, 056711. [19] Szalmas, L. J. Comput. Phys. 2012, 231, 3723-3738. [20] Szalmas, L. Comput. Fluids 2013, 74, 58-65. [21] Sharipov, F.; Kalempa, D. Phys. Fluids 2003, 15, 1800-1806. [22] Sharipov, F.; Kalempa, D. Phys. Fluids 2004, 16, 3779-3785. [23] Sharipov, F.; Kalempa, D. Phys. Fluids 2004, 16, 759-764. [24] Kestin, J.; Knierim, K.; Mason, E.A.; Najafi, B.; Ro, S.T.; Waldman, M. J. Phys. Chem. Ref. Data 1984, 13, 229-303.

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[25] De Groot, S.R.; Mazut, P. Non-Equilibrium Thermodynamics; Dover, New York, 1984. [26] Sharipov, F. Physica A 1994, 209, 457-476. [27] Naris, S.; Valougeorgis, D.; Kalempa, D.; Sharipov, F. Physica A 2004, 336, 294-318. [28] Naris, S.; Valougeorgis, D.; Kalempa, D.; Sharipov, F. Phys. Fluids 2005, 17, 100607. [29] Sharipov, F.; Kalempa, D. J. Vac. Sci. Technol. A 2002, 20, 814-822. [30] Szalmas, L.; Valougeorgis, D. Microfluid. Nanofluid. 2010, 9, 471-487. [31] Szalmas, L.; Pitakarnnop, J.; Geoffroy, S.; Colin, S.; Valougeorgis, D. Microfluid. Nanofluid. 2010, 9, 1103-1114.

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INDEX A accommodation, 225 acetylation, vii, 2, 9, 18, 19, 27 ACF, x, 98, 100, 102, 132, 135, 136, 137, 138, 139, 140, 141, 144, 145 acid, 5, 32 activated carbon, x, 98, 99, 100, 102, 132, 135, 136, 138, 141 activation energy, 71, 162 adhesion, vii, viii, 2, 3, 4, 6, 14, 15, 17, 19, 23, 24, 27 adsorption, ix, 23, 31, 36, 43, 49, 50, 54, 55, 69, 92, 97, 98, 99, 101, 102, 103, 105, 110, 114, 115, 116, 120, 121, 122, 123, 124, 125, 126, 131, 135, 136, 137, 140, 141, 143 adsorption isotherms, ix, 97, 98, 99, 101, 102, 114, 116, 120, 121, 122, 123, 125, 126, 131, 136, 137, 140, 141 adverse effects, 3 AFM, 10, 23 aggregation, 3, 17, 23 algorithm, 225 amplitude, 33, 34, 36, 38, 40, 57, 60, 61, 62, 70, 79, 80, 83, 84, 85 anisotropy, 194 annealing, xi, 185, 186, 191, 193, 194 annihilation, xii, 197 antifouling potential, vii, 1 antigen, 26 antimony, x, 149, 168, 174, 179 aqueous solutions, 5 argon, 99, 206 arrest, 19

arsenic, x, 149, 174 atomic force, 18 atoms, 153, 168, 175, 177, 186, 187 ATP, 6, 19 attachment, viii, 2, 3, 4, 15, 16, 17, 20, 22

B Bacillus subtilis, 23, 25 bacteria, vii, 1, 3, 4, 5, 6, 7, 8, 13, 15, 16, 17, 18, 19, 20, 26, 27 bacterium, 11, 17 bandgap, 188 barriers, 151, 167, 169, 174 base, 191 benzene, 55 bile, 25 bioavailability, 20 biocompatibility, 3, 18 biokinetics, 22 biotechnological applications, vii, 21 bismuth, vii, viii, x, 1, 2, 3, 4, 5, 6, 7, 13, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 149, 150, 151, 152, 153, 166, 167, 168, 174, 175, 177, 179, 185, 186, 187, 188, 189, 193 Blacks, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147 blends, 79 Boltzmann constant, 73 bonding, 151, 177, 179, 186 bonds, 53, 177, 178, 179 Brownian motion, 40, 72

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Index

C candidates, 174 capillary, 116, 188 capsule, 6, 26 carbohydrates, 9, 12, 18 carbon, vii, viii, ix, x, 24, 29, 31, 43, 49, 70, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 121, 123, 124, 131, 132, 136, 137, 141, 142, 143, 199 carbon dioxide, 199 carboxylation, vii, 2 cation, 8 CB adsorbents, ix, 98, 99, 100, 107, 110, 120, 122, 123, 125, 131 CBS, 2 cell death, 9, 18 cell surface, 3 ceramic, xi, 56, 186, 190 chemical, 30, 70, 85, 88, 91, 213, 214, 216 chemical properties, 30 chlorine, 6, 17, 20, 28 clarity, 67 cleaning, xi, 13, 186, 188 clusters, 19, 53, 199, 208 colonization, 3, 22, 27 commercial, 131 communities, 23, 222 compliance, 75 composition, 17, 24 compounds, vii, 1, 2, 3, 5, 21, 28, 30, 43, 55, 150, 174, 175, 176, 178 compressibility, 210 computing, 182 condensation, 110, 116 conditioning, 3 conduction, 188 conductivity, 193, 194 configuration, 153, 175, 178 confinement, 188 Congress, 95 consensus, 4 conservation, 202, 203, 232 constituents, 49 consumption, 13 conviction, 173 cooling, 4, 87, 89, 90, 92, 93, 191 coordination, 26, 186

copper, xi, 20, 186, 190, 191 correlation, 153 corrosion, 24 covering, xii, 80, 197 cracks, 191 crystal structure, 15 crystalline, viii, 2, 14 crystallites, xi, 14, 19, 186, 191, 193, 194 crystallization, xi, 186, 188, 189 crystals, 2, 186, 187 cycling, 4 cysteine, 5, 26

D DBP, viii, 29, 31, 32, 33, 34, 35, 36, 37, 38, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 59, 60, 61, 62, 70, 77, 78 debts, 208 degenerate, 171, 173, 174 Department of Energy, 21 deposition, 4 depth, 198, 199, 200 derivatives, 3, 225, 226 desorption, 101, 110, 111, 112, 113, 115, 118, 135, 136, 142, 143 detection, 168, 171, 174 deviation, 68, 103, 122, 137, 140, 145, 203 DFT, 151 differential equations, xii, 221 diffusion, viii, 19, 29, 40, 41, 42, 72, 73, 75, 77, 78, 227, 228, 231, 232, 233 digestion, 25 dipole moments, 153, 166, 169, 173 diseases, 2 disinfection, 28 disorder, 168 dispersion, 41, 69, 90, 93, 95 distribution, ix, 4, 20, 40, 56, 61, 70, 72, 74, 75, 78, 85, 98, 99, 110, 111, 112, 113, 114, 174, 175, 201, 203, 204, 206, 207, 208, 212, 218, 222, 223, 224, 233 distribution function, 222, 223, 224, 233 double bonds, 150 drawing, 188, 189, 190, 191 drinking water, 3, 5, 25 drugs, 2

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Index

E economics, 205, 208 electrical conductivity, 30 electron, xi, 14, 15, 16, 19, 33, 35, 71, 108, 153, 175, 178, 186, 188, 190, 193 electron microscopy, 16 electronic structure, x, 176, 185, 188 endothermic, 168, 171, 173 energy, x, 6, 13, 122, 149, 151, 153, 162, 166, 167, 168, 169, 171, 173, 174, 179, 180, 181, 183, 198, 199, 201, 202, 212, 218 engineering, xii, 221 entropy, 40, 72, 213 environment, 3 environmental conditions, 6 enzymes, 6, 19, 25 EPS, vii, viii, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 17, 18, 19, 25 equality, 218 equilibrium, 40, 72, 87, 197, 202, 224, 225 equipment, 28 ester, 5, 9 ethanol, 16 ethylene, 40, 72 ethylene glycol, 40, 72 evaporation, 110, 116 evidence, 5, 28, 47, 179 excitation, 179 exclusion, 62 exopolysaccharides, 23 experimental condition, 7, 9 exposure, 6, 8, 9, 10, 16, 17, 20, 26

F fibers, 98, 99, 100, 102, 132, 135, 136, 138, 141 films, xi, 186, 190, 191, 193, 194 filtration, 3 financial, 182 financial support, 182 flocculation, 4 flow curves, 50, 55 fluctuations, 225 fluid, 56, 57, 70, 79, 90, 222, 225, 233 food, 13, 27

force, 8, 10, 30, 53, 224 formation, viii, xi, 2, 3, 4, 6, 7, 14, 17, 18, 19, 21, 22, 25, 27, 28, 29, 30, 38, 53, 135, 175, 178, 185, 188 formula, 204, 205, 212, 215 fouling, 4, 7 fractal dimension, xi, 56, 60, 61, 62, 85, 86, 197, 204 fractal structure, 69, 80, 84, 86 free volume, 101 freedom, 202, 207, 208, 210 friction, 71 FTIR, 9

G gel, viii, 29, 30, 38, 40, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 77, 78, 79, 80, 84, 86, 87, 89, 90, 93 gelation, viii, 18, 30, 38, 55, 56, 58, 65, 66, 67, 69, 80, 84, 86, 90, 91, 92, 93 geometry, 26, 151, 152, 166, 176, 177, 178, 208, 225 glucose, 23 glutathione, 5, 26 glycol, 5, 6 grades, 43 grants, 21 graphite, 50, 54, 125, 132 growth, vii, viii, xi, 1, 2, 4, 7, 8, 9, 11, 12, 15, 16, 17, 18, 19, 22, 25, 27, 28, 185, 186, 187, 188, 189, 191, 194

H Hamiltonian, 198, 212 health, 5 height, 167, 169, 188, 199, 232 Helicobacter pylori, 2, 28 heterogeneity, ix, 98, 99, 100, 120, 122, 123, 125, 136, 140 histogram, 140, 141 history, 87, 90 homogeneity, ix, 98, 99, 107, 120, 125 human, 2, 26 human health, 2 Hungary, 221 Hunter, 24

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hybrid, 175, 178 hybridization, 175, 178 hydrogen, 24, 49, 151, 162, 166 hydrogen atoms, 162, 166 hydrogen peroxide, 24 hydrophobicity, 3 hydroxyl, 3 hysteresis, ix, 98, 99, 110, 135

I ideal, 54, 73, 122, 153, 201, 202, 204, 205, 206, 207, 210, 213, 219 identification, 26, 227 identity, 51 image, 14 images, 8, 10, 14 impurities, 188 in vitro, 6 independence, 37, 58 industry, 4, 24 inferences, 19 inhibition, 7, 9, 19, 26 inoculum, 16 integration, 212, 225 inversion, 162, 167 ionization, 26 ions, 19, 20 IR spectra, 173 isolation, 181 isomerization, 151, 153, 162, 163, 164, 165, 167, 168, 169, 170, 171, 172, 173, 174 isomers, x, 149, 151, 152, 153, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 176, 178, 179 isotherms, ix, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109, 110, 111, 112, 113, 116, 118, 120, 122, 123, 124, 125, 135, 136, 137, 140, 141, 144, 204, 205, 206, 207, 208, 209, 211, 212, 213, 214, 217 issues, 210 iteration, 225

J Japan, 29, 97, 101, 147

K ketones, 150 kinetic equations, xii, 221, 222, 224, 226, 233 kinetic model, 223, 225, 226, 227, 228, 229 krypton, 200

L laws, 56, 57 lead, 26, 30, 47, 125, 177 leakage, 101 lifetime, 162, 167 light, 201 linear function, 230 lipids, 3 lipophilic bismuth compounds, 1 liposomes, 22 liquid phase, 213, 214 liquids, 56, 101 locus, 208, 213 low temperatures, 65 low-dimensional material, x, 185 Luo, 27 lying, 170, 171, 173, 179 lysis, 9, 18, 19

M macropores, 110, 114, 115, 116, 117 magnitude, 30, 39, 70, 151 majority, 67 mammalian cells, 18 mapping, 218 mass, xii, 26, 56, 60, 61, 62, 85, 86, 221, 223, 227, 228, 232 mass spectrometry, 26 materials, x, xi, 13, 24, 30, 185, 188 mathematics, 199 matrix, 3, 27, 35, 36 matter, 19 measurement, 28, 36, 89, 90, 101 measurements, 9, 32, 33, 34, 42, 78, 89, 93, 98, 101, 102, 108, 110, 125 media, viii, 3, 15, 17, 29, 30, 31, 32, 34, 47, 77, 78, 79, 84, 86 medical, vii, 1, 17, 20, 21

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Index melt, xi, 186, 189, 190 melting, 187, 189, 190, 191 membranes, 3, 7, 11, 13, 16, 17, 21, 24 mercury, 206, 207 microorganisms, viii, 2, 3, 4, 5, 7, 9, 17, 19, 20 microscope, xi, 14, 33, 35, 186, 190, 193 microscopy, 8, 10, 18 microstructure, xi, 186, 193 microtome, 33 minimum inhibitory concentrations (MICs), 1 mixing, 6, 14, 31, 32 models, 224 modulus, viii, 29, 35, 57, 73, 75, 76, 78, 79 molar ratios, 5 molar volume, 115 mole, 71, 232 molecular mass, xii, 221, 222, 228, 231 molecules, vii, x, 3, 49, 51, 107, 135, 143, 144, 145, 150, 151, 174, 176, 178, 198, 199, 202, 203, 223, 224, 225, 226 momentum, xi, 197, 198, 202, 203, 204 monolayer, 103, 106, 107, 123, 125, 140, 145 morphology, 52, 55, 58 Moscow, 188, 194, 197, 218, 219 mucoid, 22, 27

N nanocrystals, vii, xi, 27, 185, 186, 188, 189, 191, 193, 194 nanomaterials, 188 nanoparticles, viii, xi, 2, 9, 13, 14, 15, 16, 17, 19, 20, 21, 27, 28, 185, 188 nanostructured materials, 189 Nanostructures, 187 negative effects, 5 neon, 200 Netherlands, 22, 24, 234 network-type agglomerate, viii, 29, 30, 31, 34, 41 neurotoxicity, 3 neutral, 5, 7, 49, 55, 199 nitrogen, x, xi, 5, 25, 31, 115, 149, 151, 175, 186, 190, 191, 192, 193, 209, 211 nitrogen gas, 31

noble gases, vii, xi, xii, 197, 221, 222, 228, 229, 233 non-polar, viii, 3, 6, 29, 31 nucleation, 19 nucleic acid, 3 nuisance, 17 nutrients, 3

O obstacles, 198 oil, 30, 31, 32, 33, 43 olefins, 150 optical density, 7, 16 optimization, 187, 188 ores, 2 overlap, 175, 177 oxidation, 5 oxygen, 5, 49

P Pacific, 21 paints, 42 parallel, 56, 123, 125, 136, 140, 166, 193, 208, 230 particle morphology, 30 pathogens, 20 pathways, 163, 164, 165, 170, 172 perfusion, 6 petroleum, 32 pharmaceutical, 20, 26 phenol, viii, 29, 32, 34, 41, 42, 69, 93 phosphorus, x, 149, 150, 166, 167, 174 phosphorylation, 6 photoelectron spectroscopy, 9 physical interaction, 93 physical properties, 187 physicochemical properties, 3 physics, 198 pigmentation, 30 plastic deformation, 34 plastics, 30 pneumonia, 6 polar, viii, 29, 31 polarization, 151, 152 polycarbonate, 11, 16

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polymer, 30, 36, 43, 49, 50, 51, 55, 78, 85, 88, 91, 92, 93 polymer chain, 36, 49 polymer chains, 36, 49 polymer solutions, 78, 88 polymers, 3, 55 polysaccharide, 6, 26, 27 polysaccharides, vii, 2, 3, 9, 11, 18, 19, 22 polystyrene, viii, 15, 29 population, 19 porosity, x, 98, 108, 116 preparation, 24, 190 prevention, 13 probability, 203 probe, vii, 2 profitability, 208 proliferation, 6, 17 propagation, 4 proposition, 181 propylene, 5, 6 protection, 30, 150 proteins, 3, 9, 11, 12, 18, 22 prototype, 199 Pseudomonas aeruginosa, 6, 7, 22, 24, 26, 27, 28 pulp, 4 purification, 4, 17, 188, 189 purity, xi, 99, 101, 186, 188, 189, 190

Q quantization, xii, 197 quantum confinement, x, 185 quartz, xi, 186, 188, 189, 190

R radial distribution, 76 radiation, 27 radius, 51, 52, 53, 73, 74, 115, 198, 208 reactions, 21, 151, 153, 167 recommendations, 21 relaxation, viii, 29, 38, 39, 40, 41, 42, 55, 56, 58, 60, 62, 63, 64, 69, 70, 72, 73, 75, 76, 77, 78, 85, 86, 89 relaxation process, viii, 29, 39, 70 relevance, 21, 222 reliability, 152

renal failure, 3 repulsion, 19, 174, 178, 212 researchers, xii, 18, 55, 222 resistance, xi, 7, 13, 27, 186, 190 resolution, 100, 101, 126, 136 response, 86 rheology, vii, viii, 29, 30, 31, 33, 55, 56, 57, 85, 87, 92 risk, 20 rods, 46 room temperature, 6, 167 roughness, 3 rubber, 30 rubber products, 30 Russia, 185, 197

S salt concentration, 2 salts, 2, 28 saturated hydrocarbons, 78 scaling, 71, 152 scattering, 198, 199, 200, 201, 205 science, 222 scope, 222 secretion, vii, 2, 3 sedimentation, 36 semiconductor, 188 sensitivity, 18 shape, 30, 46, 51, 74, 108, 119, 138, 191 shear, 3, 33, 42, 43, 46, 47, 50, 52, 53, 54, 74, 87, 88, 89, 90, 91, 92, 226 shear rates, 47, 52 showing, 10 silica, 40, 51, 52, 54, 55, 56, 72, 73, 74, 75, 77 silver, 20, 28 simulation, xii, 205, 221, 225, 233 single crystals, 188 SiO2, 56, 92, 93 sludge, 5, 7, 9, 25 sodium, viii, 2, 14 sol-gel, viii, 29, 37, 55, 56, 57, 62, 69, 86, 93 solid state, 106, 107 solid surfaces, 27 solidification, 33 solubility, 2, 3, 4, 5, 69, 91, 92

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Index solution, viii, xii, 2, 7, 14, 29, 34, 42, 43, 49, 50, 51, 55, 87, 88, 93, 221, 222, 225, 227, 229, 233 solvents, 6 species, 5, 7, 9, 19, 20, 43, 50, 62, 64, 65, 150, 151, 152, 153, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 178, 179, 181, 223, 224, 232 specific surface, 122, 131 spectrophotometry, 14 spectroscopic techniques, 5 spectroscopy, 8 stability, x, 4, 5, 6, 7, 25, 149, 150, 151, 162, 167, 168, 169, 170, 172, 173, 174, 178, 179 stabilization, 150 stabilizers, 20 staphylococci, 22, 25 state, xi, 5, 24, 32, 33, 42, 56, 73, 75, 87, 89, 90, 93, 107, 151, 162, 167, 168, 171, 173, 179, 186, 190, 208 states, vii, 5, 152, 153, 166, 167, 168, 169, 171, 173, 174, 176, 178, 179, 180, 181, 200, 219 statistics, 218 steel, 24, 32 storage, 32, 33, 71, 87, 88, 89, 90, 91, 92 stress, viii, 29, 30, 34, 40, 42, 46, 47, 53, 54, 55, 72 structure, viii, 26, 27, 29, 30, 31, 32, 33, 35, 40, 42, 43, 46, 47, 50, 52, 53, 54, 55, 56, 59, 60, 61, 62, 65, 69, 71, 77, 78, 80, 85, 86, 87, 89, 91, 92, 93, 150, 162, 166, 167, 177, 179, 186, 187 substrates, 3 Sun, 21, 26 supplier, 70 suppression, 6, 7, 25 surface area, viii, ix, x, 2, 30, 50, 97, 98, 99, 100, 107, 116, 118, 131, 132, 138, 139, 140, 141, 143, 144 surface chemistry, 3 surface component, 26 surface structure, 54 surface tension, 115 survival, 3 susceptibility, 3, 6, 7, 24 suspending media, viii, 29, 30, 32, 78

suspensions, vii, viii, 7, 11, 19, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93 symmetry, 177, 186, 198 synthesis, 6, 14, 150, 181

T Taiwan, 149, 182 techniques, xii, 8, 221 technology, xii, 24, 109, 221 TEM, 14, 33, 35 temperature, xi, 6, 55, 58, 63, 64, 65, 66, 68, 69, 70, 71, 73, 80, 86, 87, 88, 91, 92, 93, 103, 115, 117, 185, 186, 188, 189, 190, 191, 192, 193, 194, 199, 200, 201, 202, 205, 207, 208, 212, 214, 223, 228, 232 temperature annealing, xi, 185, 188 temperature dependence, 64 testing, 20 textbooks, 207 textiles, 13, 20 texture, 109, 118, 144 therapy, 22 thermodynamics, xii, 197, 202, 207, 208, 216 thinning, 43, 54 three-dimensional space, 56 time periods, 5 torsion, 162 toxicity, vii, 2, 20, 22 Toyota, 182 trade, 21 transducer, 101 transformation, 90 transmission, 15 transpiration, 101 transport, x, 185, 188, 228 transportation, 233 treatment, 4, 5, 8, 9, 12, 13, 16, 18, 19, 20, 21, 25, 26, 28 trial, 22 tumors, 2 turnover, 205 twist, 179

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U Ukraine, 185 uniform, ix, 51, 74, 98, 106, 107, 122, 125, 126, 227, 228, 232 universality, 75 universe, 208

W wastewater, 4, 25 water, 2, 3, 4, 5, 6, 13, 14, 15, 17, 19, 20, 21, 23, 24, 28, 206 wells, 199 wires, xi, 185, 188 workers, 5, 122, 152

V X vacuum, xi, xii, 33, 36, 101, 186, 188, 221, 222 valence, 153, 175, 176, 178, 188 variables, 200 variations, 19 vector, 223 velocity, xii, 201, 218, 221, 223, 224, 225, 226, 227, 228, 229, 230, 233 viscoelastic properties, 56, 69, 78, 86, 88, 93 viscosity, 13, 30, 33, 43, 46, 47, 49, 50, 51, 52, 53, 54, 55, 65, 70, 73, 74, 77, 84, 86, 189, 198, 199, 223, 226, 227

XPS, 9 X-ray diffraction, 14, 15, 19, 108, 166 xylem, 23

Y yield, viii, 29, 30, 42, 46, 47, 53, 54, 55

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