Accepted Manuscript Title: Microstructure and electrical transport phenomenon of yttria alloyed nanocrystalline ceria solid solution synthesized by mechanical alloying Author: S. Dutta S. Bandyopadhyay A. Dutta S.K. Pradhan PII: DOI: Reference:
S0025-5408(16)31936-5 http://dx.doi.org/doi:10.1016/j.materresbull.2017.05.028 MRB 9343
To appear in:
MRB
Received date: Revised date: Accepted date:
3-11-2016 4-5-2017 10-5-2017
Please cite this article as: S. Dutta, S. Bandyopadhyay, A. Dutta, S.K. Pradhan, Microstructure and electrical transport phenomenon of yttria alloyed nanocrystalline ceria solid solution synthesized by mechanical alloying, Materials Research Bulletin (2017), http://dx.doi.org/10.1016/j.materresbull.2017.05.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Microstructure and electrical transport phenomenon of yttria alloyed nanocrystalline ceria solid solution synthesized by mechanical alloying
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S. Dutta, S. Bandyopadhyay, A. Dutta, S. K. Pradhan*
Corresponding author, e-mail:
[email protected]
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*
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Department of Physics, The University of Burdwan, West Bengal, India-713104
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Abstract:
This article reports the room temperature synthesis of nanocrystalline (4-20 mol%) yttria
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(Y2O3) alloyed ceria (CeO2)-based solid solutions in open air employing a single step mechanical
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alloying technique. Structure and microstructure of these compounds have been investigated and
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the effect of yttria on the electrical transport phenomenon of these compounds is revealed. Structure and microstructure characterizations of all alloyed compounds are carried out in detail
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employing Rietveld refinement method by analyzing respective XRD patterns. Ionic conductivities of all these compounds are measured in open air in the temperature range 633K833K. For unsintered samples, conductivity increases continuously with increase in alloying concentrations up to 12 mol% of Y2O3, which is higher than the alloying concentration synthesized by any other method. However, the total conductivity reaches maximum with 8 mol% of Y2O3 when the samples are sintered and exhibits the highest conductivity of ~ 3.8×103
Ω-1cm-1 at 813K. Electrical relaxation process and transport phenomenon of all compounds are
studied by impedance spectroscopy and complex modulus analysis. Presence of thermally activated relaxation and hopping mechanism in the system are also confirmed and correlated
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with the microstructure. Present findings indicate the possible application of these compounds as oxygen ion conductors in many electrochemical devices at quite low temperature range of 673K-
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873K.
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Keywords: A. Oxides; B. Defects; C. Microstructure; D. X-ray diffraction; E. Electrical
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properties.
1. Introduction:
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The CaF2 structure-based cubic CeO2 shows good oxygen ion conductivity when alloyed with divalent (Ca2+, Mg2+) and trivalent oxides (Gd3+, Sm3+, Y3+, La3+) at intermediate
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temperatures [1-5]. Among these, Gd and Sm doped ceria compounds have higher conductivity values [6]. Steele [7] reported that the ionic conductivity values of Ce.9Gd.1O1.95 and
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Ce.9Sm.1O1.95 are 5.44 × 10-2 Ω-1cm-1and 2 × 10-2 Ω-1cm-1 respectively at 973K. Ionic conductivity of doped ceria compound primarily depends on the nature of dopant, its ionic radius, valence state, concentration and formation of defect pairs. The pair formation between oxygen vacancies and dopant ions depends upon both the size and concentration of the dopant [1,5,8]. Previously, it was reported that the conductivity in doped ceria increase continuously with the increase in dopant concentrations up to the solubility limit. However, in recent study maximum ionic conductivity was obtained before the maximum solubility limit of the dopant in ceria [9]. Eguchi et al. [10] reported that though the solubility limit of CaO in CeO2 is 23mol%, but the ionic conductivity of ceria was found to increase only up to 7-8 mol% of CaO alloying [11]. It was found that nearly 40 mol% of yttria is soluble in ceria [12,4], but the maximum ionic 2 Page 2 of 48
conductivity was obtained with 8-10 mol% of dopant concentrations [11,13]. It was established that the number of defects increases with increasing dopant concentrations. As a result, the interactions between dopant ions and oxygen vacancies increase by producing larger traps for
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mobile oxygen vacancies, which in turn reduces the mobility of vacancies [3,14]. Ionic conductivity in doped ceria also depends significantly on the structure and microstructure of the
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compound [15,5]. In the present study, we have alloyed yttria with ceria by mechanical alloying
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(MA) the stoichiometric mixture of yttria and ceria powders and obtained a single phase ceriabased solid solution. As the ionic radius of Y3+ (r=1.027Ǻ, CN=8) is very close to Ce4+ (r=0.97Ǻ,
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CN=8), the solubility limit of yttria in ceria is quite high (~40mol%) [16,4]. MA is a single step, one pot, fast physical method of synthesis and the nanocrystalline ceria-based solid solutions are
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synthesized within a short duration in comparison to the other process like co-precipitation
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[17,18] and other chemical method [4]. Compounds thus prepared do not require sintering at
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high temperature to obtain a single phase, unlike some other methods, such as solid state reaction [19-21], wet chemical [22] and sol-gel [23] methods. Primary objectives of the present work are
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(i) to synthesize nanocrystalline ceria-based solid solutions with different concentrations of yttria alloying by mechanical alloying the stoichiometric powder mixture of ceria and yttria at room temperature, (ii) structure and microstructure characterizations of the synthesized materials, (iii) to measure the ionic conductivity of the alloyed ceria with increasing temperature, alloying concentrations and increasing sintering temperatures, (iv) to find the maximum limit of alloying concentration up to which conductivity increases for the compounds synthesized by mechanical alloying, and (v) to study the electrical relaxation and transport properties of the compounds to correlate their electrical properties and microstructure.
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2. Experimental: Yttria alloyed ceria-based solid solutions were synthesized by mechanical alloying the
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stoichiometric powder mixture of CeO2 (Loba Chemie, purity 99.95%) and Y2O3 (Loba Chemie, purity 99.9%) for six different molar concentrations of yttria (4, 8, 10, 12, 15, 20 mol%) and the
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compounds are abbreviated as CY4, CY8, CY10 CY12, CY15 and CY20 respectively. Powder mixtures were taken in a chrome steel bowl of 80ml volume filled with chrome steel balls of
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10mm diameter and the sealed bowl was then placed in a planetary ball mill (P6, Fritsch,
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Germany) for mechanical alloying. Milling was continued up to 8h with intermediate intervals of 30min. Milled powders were then collected and XRD pattern of each composition was recorded
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with a powder X-ray diffractometer (M/S BRUKER, D8 Advance, DaVinchi) operated at 40 KV and 40mA with step size 0.02º in the 2θ range 20º-80º. Powders were then uniaxially pressed to
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obtain cylindrical pellets of 10mm diameter. Pellets of 4, 8 and 12mol% of yttria alloyed ceria
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compounds were sintered at 973K for 3h (samples are abbreviated as CY4S1, CY8S), and
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CY12S1 respectively). Pellets of 8 and 12mol% yttria alloyed ceria were further sintered at 1373K for 3h (samples are abbreviated as CY8S2, CY12S2 respectively) to achieve more densification. Relative density of the sintered compounds was measured as ~ 92.28% to 96.15% of their theoretical values and the porosity of the sintered compounds was revealed from FESEM images. XRD patterns of the sintered pellets were also recorded as per the above specifications. Both sides of each pellet were then coated with conductive graphite paste (MERCK) and copper wires were connected to the coated pellet. Electrical measurement was then performed using impedance analyzer (HIOKI Model: 3532-50) in open air by two probe method and the impedance data were recorded as a function of temperature in the temperature range 633K-833K as well as in the frequency range 42Hz to 5MHz. 4 Page 4 of 48
3. Method of analysis:
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To obtain different structural and microstructural parameters, XRD patterns of all compounds are analyzed employing Rietveld powder structure refinement method [24-26] using
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Maud software (version 2.26) [27,28]. The least square method is adopted to refine the
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experimental profiles assuming that the shape of the peaks to be a pseudo-Voigt (pV) function, which takes individual care of both particle size and lattice strain broadening in the XRD
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patterns [25,29]. The background intensity of each pattern is fitted with a polynomial of degree 4 and positions of peaks of the experimental profiles are corrected by successive refinements of
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zero shift error and sample displacement error. The quality of fitting is monitored by the ratio of reliability index parameters, Rwp (weighted residual error) and Rexp (expected error), represented
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as GoF (goodness of fitting = Rwp/Rexp). Refinement continues until convergence between
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experimental and refined data is reached and GoF approaches close to 1.0 (here it varies
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between1.07 to 1.67, shown in Table1).
4. Results and Discussion:
4.1 Structural characterization
The powder XRD patterns of all six unsintered compounds are shown in Fig.1. All reflections in these XRD patterns belong to cubic ceria phase and there is no other reflection either from yttria or from the milling media. It confirms that contamination free single-phased yttria alloyed (up to 20mol%) ceria-based complete solid solutions are formed within 8h of
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mechanical alloying of stoichiometric powder mixture of ceria and yttria. A careful observation reveals that all reflections become broadened continuously as well as shifted slowly towards higher scattering angle with increase in yttria concentrations from 4 to 20 mol%. It indicates that
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particle size and lattice parameter of cubic ceria solid solutions decrease continuously with
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increasing yttria concentrations in ceria lattice.
For Rietveld analysis, XRD pattern has been simulated using cif file of ceria (ICSD code
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# 167160, cubic, space group: Fm3m, a=0.54169nm) by substituting equivalent amount of Ce4+
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ions by Y3+ ions in the ceria lattice for each composition. The Rietveld refined fitted patterns (IC) with respective experimental XRD patterns (IO) of the unsintered compounds are shown in Fig.
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2(a) and those of the sintered compounds in Fig.2b. Residue of fittings (Io-IC) of the experimental (IO) and fitted (IC) intensities of all compounds are shown by green lines below
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each respective XRD pattern. Almost linear variation of residues over the entire angular range of
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2θ indicates the good quality of fitting between the refined and experimental XRD patterns. The
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change in peak broadening due to high temperature sintering of 8mol% yttria alloyed compound is clearly revealed in Fig. 2c. It is evident that the peak broadening reduces significantly after sintering the compound at 1373K for 3h. It indicates that there is a noticeable change in the microstructure of the compound after the sintering treatment. Lattice parameters of all alloyed ceria-based solid solutions are obtained from the Rietveld analysis of the XRD patterns. The variation of lattice parameters of all compounds with increase in alloying concentrations and sintering temperatures are shown in Fig. 3(a). It is seen that the lattice parameters of both unsintered and sintered compounds decrease with the increase in alloying concentrations, which may be due to the increase in oxygen vacancies with the increase in alloying concentrations [30]. This nature of the variation, for the unsintered 6 Page 6 of 48
compounds, is almost similar to that reported by Longo and Podda [31]. It may be noted that as the ionic radius of Y3+ is greater than that of Ce4+, thus the lattice parameter of ceria-based solid solution should increase continuously with increase in yttria concentrations. However, the
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decrease in lattice parameter with yttria concentrations signifies the presence of substantial amount of oxygen vacancies in ceria lattice, which supersedes the expanding nature of the ceria
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lattice [30]. It is also seen from Fig. 3(a) that the lattice parameter decreases appreciably and
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continuously when compounds are sintered at different elevated temperatures. It implies that the thermal energy of sintering is primarily absorbed by the ceria lattice and has been utilized to
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release the lattice distortion, generated by high energy milling, by rearranging ions and vacancies in the lattice of the compound towards achieve a minimum energy configuration. This thermally
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activated redistribution of ions and vacancies inside the lattice helps to reduce the stored
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mechanical energy in ceria lattice, which in turn results in contraction of the lattice. Estimations
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of crystallite (particle) size and microstrain of all six unsintered and sintered compounds are obtained from Rietveld analysis and their nature of variations with increasing yttria
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concentrations are plotted in Fig. 3(b). The error bars of respective results are shown by vertical lines over the data points in the figure. Particles of the unsintered compounds are found to be isotropic in shape and their sizes reduce slowly from17.6nm up to 12mol% yttria and then rapidly to 11.8nm with increasing yttria concentrations. Microstrains show diminishing nature up to 12 mol% of ytria doping concentrations and then increases to relative higher values. As the particle size does not reduce appreciably up to 12 mol% of alloying concentrations, the milling energy is utilized in relaxing the lattice by releasing the lattice strain [15]. On the other hand, energy accumulated by milling is used to reduce in particle size with increasing alloying concentrations from 12 mol% to 20 mol% of yttria. With the high energy milling, particle size is
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reduced by generation of severe plastic deformation through repeated cold welding and fracturing mechanism of mechanical alloying. This produces shear stress inside the grains and results in increase of lattice strain [29]. After sintering the compounds at 973K, particle size
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increases nominally from 17.9nm to 19.8nm and microstrain reduces appreciably due to the release of lattice strain and reaches a least value for 8 mol% yttria alloyed compound. However,
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after sintering the compounds (CY8S2, CY12S2) at 1373K, particle size increases noticeably up
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to ~ 61 nm and microstrain is almost removed from compounds. This significant growth in particle size and reduction in microstrain are the consequences of coarsening of grains of the
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pellets due to sintering at a higher temperature as well as release of lattice strain by the lattice.
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4.2. Impedance spectroscopy analysis
The Cole-Cole or Nyquist plots of 8 mol% yttria alloyed ceria (CY8) solid solution nanoparticles
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recorded at different temperatures are shown in Fig. 4(a). For comparative study, Cole-Cole plots
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of all the six compounds (unsintered) recorded at 793K are shown in Fig. 4(b).
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All these plots comprise of semicircular arcs whose centers are below the real axis, indicating the non-Debye type behavior of the compounds. The intersections of these semicircular arcs on the Z′ (real) axis on lower frequency sides produce the total resistance of the samples. Electrical conductivities (σ) of all compounds are calculated using the relation σ = l⁄
, where R and A
are width and area of the pellets respectively. Variations of conductivity with inverse of temperature, for the all six unsintered compounds, are plotted in Fig. 5(a), which follow the Arrhenius equation: σ = σ ⁄ exp (−
⁄
)
(1)
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Where σ is pre-exponential factor,
is
activation energy, k is Boltzman′s constant, T is the
absolute temperature.
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The probable value of activation energies obtained from the slopes of the Arrhenius plots are tabulated in Table 2. The variation of the total conductivity (σtotal) value of these compounds
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with alloying concentration at 793K is also shown in the inset of Fig. 5(a). It is seen that the total
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conductivity value increases up to 12mol% of yttria and then decreases with further increase of alloying concentrations. In earlier studies, Fu [32] reported that the conductivity of yttria alloyed
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ceria reached a maximum at 9 mol% of Y2O3 concentration and exhibited a conductivity value ~ 10-4 Ω-1cm-1 at 833K. According to Shing et al. [12] best conductivity (σ873K ~ 2.32 × 10-3 Ω-1cm) was obtained for 10 mol% yttria alloyed ceria. Glass et al. [13] found that the conductivity
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1
(σ833K ~ 1.5 × 10-3 Ω-1cm-1) reached a maximum at doping level of 8 mol% of yttria. In the
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present study, in case of unsintered compounds, the maximum conductivity of ceria is achieved
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with 12 mol% of yttria alloying, which is slightly higher than the previously reported values of
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yttria concentration levels [12,13,32]. This significant difference may be attributed to the method of preparation of compounds by mechanical alloying, which plays a significant role on the structure and microstructure of the system and its properties [11]. In present case, unsintered compounds exhibit a maximum conductivity of ~ 6.08 × 10-5 Ω-1cm-1 at 833K which is lower than the conductivity values reported in the earlier studies [12,13,32]. It may be noted that we have used less conducting graphite paste as electrode instead of high conducting Pt or Au paste which were used in the previous studies and compounds were also synthesized by different methods and sintered at higher temperatures [11,13]. However, it is interesting to note that the highest conductivity of ceria is obtained for 8 mol% yttria alloyed ceria after sintering the compounds at higher temperatures and the conductivity decreases with further increase in yttria 9 Page 9 of 48
concentrations as shown in Fig. 5(b). This variation is also in good agreement with the previous reports [11,13]. The measured conductivity of CY8S2 compound (sintered at 1373K), ~ 3 × 10-3 Ω-1cm-1 at 793K, is quite comparable to the previously reported values [12,13] and significantly
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higher than conductivity values, ~ 2.67 × 10-5 Ω-1cm-1 and ~ 2.58 × 10-5 Ω-1cm-1 of CY8S1 the same
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(sintered at 973K) and CY12 (unsintered) compounds respectively, measured at temperature.
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In yttria doped ceria compound, Y3+ ions substitute Ce4+ ions in the host lattice and generates
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oxygen vacancies (VO) to maintain the charge neutrality of the system, which can be represented
Y2O3+2Ce× +O× →2Y +V .. +2CeO2
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by Kröger-Vink notation [33]:
(2)
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A polyhedral arrangement of 10 mol% Y2O3 alloyed CeO2 is shown by 3D modeling in Fig.6. An
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oxygen vacancy (VO), created due to Y3+ doping in ceria lattice is shown in yellow color. These
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generated oxygen vacancies are responsible for the oxygen ion conduction through the lattice. As the alloying concentration increases from 4 mol% to 12 mol%, the conductivity increases due to the generation of more oxygen vacancies with alloying concentrations. However, further increase in alloying concentrations produces more defects and generates more oxygen vacancies in the system. This, in turn, increases defect interactions between alloying ions and oxygen vacancies (VÖ) by forming deep traps for mobile oxygen vacancies and results in depression of conductivity [4]. Another probable cause may be due to variation of bulk conductivity of the unsintered compounds for the unusual behavior of lattice strain, which reduces up to 12mol% of alloying concentrations and then increases to relatively higher values. The reduction in microstrain value, shown in Fig. 3(b), is manifested by the relaxation of the ceria lattice, which
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facilitates to improve the conductivity [15]. In sintered compounds the thermal energy is used to rearrange the ions and vacancies in the lattice towards release of strain generated in the compound by high energy milling as well as to increase the density of the sintered pellets. The
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conductivity increases up to 8mol% of yttria and shows the highest value of ~ 3.8 × 10-3 Ω-1cm-1
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at 813K after sintering at 1373K. The enhancement of conductivity with increase in sintering temperature, as shown in Fig. 5(b), is due to the increase in the compact density of the pellets.
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The relative density of the sintered pellets reaches to ~ 92.28% - 96.15% of their theoretical values after sintering at 1373K. However, decrease in conductivity with further increase in yttria
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concentrations may be attributed to the formation of trimer (Y -V .. -Y ) at higher alloying
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concentrations. These trimer type defects formed due to densification during sintering reduces the oxygen ion conductivity for their immobile nature [34, 35]. The densification nature of the
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sintered pellets is observed by the FESEM images from the surfaces of these pellets. Figs. 7(a)
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and (b) show the SEM microstructures of 8 and 12mol% of yttria alloyed ceria pellets sintered at 1373K respectively, in which the presence of insignificant porosity in both compounds is
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evidenced.
The variations of real part of the impedance for 20mol% yttria alloyed ceria (CY20) nanoparticles in the frequency range (42Hz – 5MHz) recorded at different temperatures are shown in Fig.8(a). It is seen that the impedance values decrease with increase in temperature showing the semiconducting behavior of the compounds and also indicates the improvement in the conductivity value with increase in temperature. The Z′ values at different temperatures follows a sigmoidal type variation in the lower frequency side followed by plateau region after particular frequencies and merges at the higher frequency (ω>1MHz) side and almost reduces to zero. This is may be due to possible release of space charge [36] as well as the reduction of 11 Page 11 of 48
barrier properties of the materials [37] which may be responsible for the enhancement of the ionic conductivity at higher frequency. From Fig.8(a), it may also be observed that there is a particular frequency onset of plateau region for the curves of all temperatures which shift
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towards the higher frequency side with the increase in temperature indicating the presence of electrical relaxation in the system [38]. This nature of variation is also observed in all alloyed
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ceria solid solutions. Fig.8(b) represents the loss spectrum i.e. the variation of imaginary part of
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the impedance with frequency. It shows that each curve of a particular temperature has a peak at a particular frequency which is known as relaxation frequency (ωmax) characterized by the
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structure and microstructure of the system [39]. It is also observed that these peak-shifts towards the higher frequency value with increase in temperature indicating the presence of thermally
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activated relaxation in the materials [38]. A significant amount of asymmetric peak broadening is
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observed with the increase in temperature. It suggests the presence of thermally activated
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relaxation phenomenon with a spread of relaxation time which may be due to the simultaneous contribution of following relaxation processes: (a) at lower temperatures due to the presence of
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immobile species and (b) at higher temperature due to presence of defect vacancies [38]. For all plots at different temperatures, Z′′ decreases and merges at higher frequency side which is again possibly due to the release of space charge in the materials [40]. The relaxation time (τ) for the electrical relaxation in the material is obtained from impedance data using the relation ωmaxτ =1, where ωmax is the relaxation frequency and it is calculated from Z′′ plots. The variations of relaxation time with inverse of temperature for CY10 and CY20 compounds are plotted in Fig.9. This variation also follows the Arrhenius relation τ =τ0 exp (−
⁄
)
(3)
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Where τ0, is the pre exponential factor,
is activation energy, k is Boltzman′s constant, T is the
absolute temperature. The figure shows that the relaxation time decreases with the increase in temperature indicating the presence of temperature dependent relaxation process in the system
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which may be due to the faster movement of the vacancies in the system with the increase in
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temperature [38]. The activation energies (Eτ) estimated from the plot (shown in Fig.9) is 1.15 ev and 1.13ev, which agree well with the value of activation energies (Ea) ≈1.17ev and 1.15ev
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estimated from Arrhenius plot (shown in Fig.5(a)) for the compounds CY10 and CY20 respectively. The close resemblance of these values indicates that both the relaxation and
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conductivity process are thermally activated phenomena and due to the mobility of same type of
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4.3. Modulus spectroscopy analysis
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charge carriers [38].
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Modulus spectroscopy analysis is one of the effective and necessary tools for the detailed study of the electrical transport phenomenon, relaxation mechanism etc. The real (M′) and imaginary (M′′) part of the complex modulus of all the six compounds (unsintered) are obtained and described as M′=C0ωZ′′ and M′′= C0ωZ′, where C0= ε0A/t (ε0 is permittivity of free space, A is the area of the surface of the pellets, t is the thickness of the pellets) [30]. Variation of real part (M′) and imaginary part (M′′) of the complex modulus with frequency for CY08 compound is shown in Fig. 10(a-b). From the real part variation (Fig.10(a)), it is observed that M′ value decreases with the increase in temperature indicating enhancement in conductivity and almost approaches to zero at lower frequency sides. But, with increase in frequency M′ value increases and shows continuous dispersion which tends to reach at a saturation value for all temperatures.
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This may be due to the lack of restoring force responsible for the motion of vacancies for an induced electric field. As the frequency increases the vacancies move through shorter path and get confined in a potential energy well due to the rapid change in the electric field [41]. It may
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also be observed that the dispersion shifts towards the high frequencies as the temperature
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increases.
The imaginary part of electric modulus (M′′), shown in Fig. 10(b) approaches to zero value at the
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lower frequency region indicating the absence of significant amount of polarization in the
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modulus data. These spectra also shows that each curve has a single peak where M′′ is maximum and the peak shifts towards the higher frequency region with the increase in temperature
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indicating the presence of thermally activated hopping mechanism in the electric conduction. The frequency corresponding to the maximum value of M′′ is described as ωmax = 2Пfmax. Long
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range and short range conduction take place in the frequency region of ω< ωmax and ω>ωmax
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respectively [39]. The presence of peak at maximum of M′′ may be possibly due to the change in
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reorientation of relaxation of defects due to the jump of oxygen vacancies from one site to other site in the lattice [42]. The migration energy (Em) of oxygen vacancies associated with the reorientation process can be calculated from the Arrhenius plot of log(fmax) vs 1000/T. The migration energies (Em) for CY08 and CY10 are 1.15 ev and 1.16 ev respectively as obtained from Fig.11.
The spectrum can also be used to evaluate the capacitance (C) of the sample using the relation given as M′′ = ε0/2C or C= ε0/2 M′′ [38]. Values of the capacitance thus obtained for CY10 compound is represented by Fig.12. The diminishing behavior of capacitance with increase in temperature implies the loss of dielectric properties with the increase in temperature which in turn enhances the ac conductivity of the sample with increase in temperature. 14 Page 14 of 48
4.4. AC conductivity
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Fig. 13(a) shows the variation of AC conductivity for all the six unsintered compounds at 793K. It is observed that each curve consists of a plateau region in the lower frequency side and a
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dispersion region in the higher frequency side. This type of conductivity variation obeys
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Jonscher’s Universal power law [43] described as: σ(ω )= σdc(0,T)+ Αωn
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(4)
Where A, is a pre-exponential factor depends on temperature and 0< n
