Diffusion Coefficients from Molecular Dynamics ... - Springer Link

10 downloads 0 Views 1MB Size Report
Jul 18, 2013 - For any diluted system, one can show that Γ xi →0 ...... G22(R) represents the methanol–methanol O–O correlation and G12(R) represents ... EMD. Margules. MD. (a). (b). Fig. 8 Thermodynamic factor Γ in the binary systems ...
Int J Thermophys (2013) 34:1169–1196 DOI 10.1007/s10765-013-1482-3

Diffusion Coefficients from Molecular Dynamics Simulations in Binary and Ternary Mixtures Xin Liu · Sondre K. Schnell · Jean-Marc Simon · Peter Krüger · Dick Bedeaux · Signe Kjelstrup · André Bardow · Thijs J. H. Vlugt

Received: 12 December 2012 / Accepted: 21 June 2013 / Published online: 18 July 2013 © Springer Science+Business Media New York 2013

Abstract Multicomponent diffusion in liquids is ubiquitous in (bio)chemical processes. It has gained considerable and increasing interest as it is often the rate limiting step in a process. In this paper, we review methods for calculating diffusion coefficients from molecular simulation and predictive engineering models. The main achievements of our research during the past years can be summarized as follows: (1) we introduced a consistent method for computing Fick diffusion coefficients using equilibrium molecular dynamics simulations; (2) we developed a multicomponent Darken equation for the description of the concentration dependence of Maxwell– Stefan diffusivities. In the case of infinite dilution, the multicomponent Darken equaxk →1 which can be used to parametrize the generaltion provides an expression for D ¯ ij ized Vignes equation; and (3) a predictive model for self-diffusivities was proposed for the parametrization of the multicomponent Darken equation. This equation accurately

X. Liu · S. K. Schnell · S. Kjelstrup · A. Bardow · T. J. H. Vlugt (B) Process and Energy Laboratory, Delft University of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands e-mail: [email protected] X. Liu · A. Bardow Lehrstuhl für Technische Thermodynamik, RWTH Aachen University, Schinkelstraße 8, 52062 Aachen, Germany J.-M. Simon · P. Krüger Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Université de Bourgogne, Dijon, France P. Krüger Graduate School of Advanced Integration Science, Chiba University, Chiba 263-8522, Japan D. Bedeaux · S. Kjelstrup Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway

123

1170

Int J Thermophys (2013) 34:1169–1196

describes the concentration dependence of self-diffusivities in weakly associating systems. With these methods, a sound framework for the prediction of mutual diffusion in liquids is achieved. Keywords Matrix of thermodynamic factors · Maxwell–Stefan diffusion · Molecular dynamics · Predictive models · Transport diffusion 1 Introduction Mass transport by diffusion is a phenomenon that occurs due to a gradient in the chemical potential of a component in the system. Diffusion can be a slow process. The slow rate of diffusion is responsible for its importance. In many processes, diffusion occurs simultaneously with other phenomena, such as chemical reactions. When diffusion is the slowest step, it limits the overall rate of the process [1,2]. The necessity of predicting diffusion rates therefore arises. It is important to make the distinction between self-diffusion and transport (or mutual) diffusion. Self-diffusion describes the mean-squared displacements of individual molecules in a medium, while transport/mutual diffusion describes the net transport of a collection of molecules due to a driving force. The generalized Fick’s law describes the mass flux due to transport diffusion as a linear combination of composition gradients. For an n-component system, the molar flux Ji of component i follows from Ji = −ct

n−1 

Di j ∇x j ,

(1)

j=1

in which ct is the total molar concentration, Di j are Fick diffusivities which depend on concentration but not on the magnitude of the driving forces, and x j is the mole fraction of component j [3]. In gases, diffusion coefficients are typically around 10−5 m2 · s−1 . In liquids, diffusion coefficients are about 10−9 m2 · s−1 . In solids, diffusion is even slower [3–5]. In Eq. n 1, the reference frame for the diffusion fluxes is the average molar Ji = 0. Other reference frames, like the barycentric, the mean velocity, so that i=1 volume or the solvent frames of reference, are alternatively used depending on their convenience for experimental conditions. We refer to Refs. [3,6] for the transformation rules from one reference frame to the other. In binary mixtures, the fluxes Ji are related to the mole fraction gradients by the constitutive relations, J1 = −ct D11 ∇x1 , J2 = −ct D22 ∇x2 ,

(2) (3)

which is Fick’s first law of diffusion. As J1 + J2 = 0 and x1 + x2 = 1, we have D11 = D22 ≡ D, so diffusion in a binary system can be described by a single transport coefficient. Fick diffusion coefficients in binary systems are the same for all reference frames provided that one uses the gradient of the appropriate concentration measure [3].

123

Int J Thermophys (2013) 34:1169–1196

1171

In the modeling of multicomponent transport diffusion in liquids, the Maxwell– Stefan (MS) approach is often more convenient [3,6,7]. The key point of this approach is that the driving force for diffusion of component i, i.e., the chemical potential gradient ∇μi at constant temperature and pressure, is balanced by friction forces, resulting in the following equation:



1 ∇μi = RT

n  j=1, j=i

x j (u i − u j ) , D ¯ ij

(4)

in which R and T are the gas constant and absolute temperature, respectively. The friction force between components i and j is proportional to the difference in average velocities of the components, (u i − u j ). The MS diffusivity D ¯ i j is an inverse friction coefficient describing the magnitude of friction between components i and j. We highlight some features of multicomponent Fick and MS diffusivities: (1) MS diffusivities do not depend on a reference frame while Fick diffusivities explicitly depend on a reference frame; (2) MS diffusivities usually depend less strongly on concentration than Fick diffusivities [3]; (3) MS diffusivities are symmetric, i.e., D ¯ ji while Fick ¯ ij = D diffusivities are not, i.e., Di j = D ji , except for binary systems. In a system containing n components, n(n − 1)/2 MS diffusivities are sufficient to describe mass transport while (n − 1)2 Fick diffusivities are needed. This suggests that for n > 2, the Fick diffusivities are not independent; and (4) In multicomponent systems (n > 2), MS diffusivities can be estimated using the MS diffusivities obtained from binary mixtures. Multicomponent Fick diffusivities are not related to their binary counterparts. Therefore, it is very difficult to predict multicomponent Fick diffusivities using only the knowledge of binary systems. Clearly, it is convenient to use MS diffusivities for describing mass transport in multicomponent systems; however, MS diffusion coefficients cannot be directly obtained from experiments as chemical potential gradients cannot be measured directly. As generalized Fick’s law and the MS theory describe the same physical process, it is possible to relate the corresponding transport coefficients [3,7,8]. The corresponding equation to relate the diffusion coefficients in Eqs. 1 and 4 is [D] = [B]−1 [Γ ],

(5)

in which [D] is the (n − 1) × (n − 1) matrix of Fick diffusivities. The elements of the matrix [B] are given by [3,8,9] n  xj xi Bii = + , with i = 1, · · · , (n − 1) D D ¯ in j=1, j=i ¯ i j   1 1 Bi j = −xi − , with i, j = 1, · · · , (n − 1) and i = j. D D ¯ ij ¯ in

(6)

123

1172

Int J Thermophys (2013) 34:1169–1196

The elements of the so-called matrix of thermodynamic factors [Γ ] are defined by [3,7,10]  Γi j = δi j + xi

∂ ln γi ∂x j

  T, p,

,

(7)

in which δi j is the Kronecker delta, and γi is the activity coefficient of component i. The symbol  indicates that the partial differentiation of ln γi with respect to the mole fraction x j is carried n out at constant mole fraction of all other components except the xi = 1 during the differentiation [3,10]. For ideal mixtures, we nth one, so that i=1 have Γi j = δi j . For any diluted system, one can show that Γiixi →0 = 1 and Γixj,i →0 j= i = 0. In binary mixtures, the following relation holds between Fick and MS diffusivities: D = Γ ×D ¯ 12 ,

(8)

in which D is the binary Fick diffusivity, D ¯ 12 is the MS diffusivity, and Γ is the thermodynamic factor given by [3]  Γ = 1 + x1

∂ ln γ1 ∂ x1



  T, p,

= 1 + x2

∂ ln γ2 ∂ x2

  T, p,

.

(9)

The latter equality follows from the Gibbs–Duhem equation. In the limits x1 → 1 and x2 → 1, Γ = 1. In practice, Fick diffusivities are more easily accessible in experiments as they directly relate to the measurable quantities, i.e., concentrations. In fact, in all mutual diffusion experiments Fick diffusion coefficients are measured. Several experimental techniques can be used to study transport diffusion, e.g., Raman spectroscopy [11,12], the diaphragm cell technique [13,14], interferometry [15–18], microfluidics [19], Quasi-elastic neutron scattering (QENS) experiments [20], and the Taylor dispersion method [21,22], etc. However, the investigation of the concentration dependence of diffusivities is tedious. Experimental data on multicomponent diffusion in liquids are therefore limited [23–30]. Due to the difficulties in experiments [5], in this work, we focus on the prediction of diffusion coefficients using molecular simulation and predictive engineering models. This review paper is organized as follows: in Sect. 2, we present molecular simulation approaches for computing mutual diffusivities. The advantages and disadvantages of these approaches are discussed. Compared to simulation approaches, engineering predictive models are often preferred and convenient for the description of the mass transport by diffusion. In Sect. 3, we review and discuss predictive models for diffusion coefficients. We earlier derived a multicomponent Darken equation to predict the concentration dependence of MS diffusivities. The multicomponent Darken equation requires self-diffusivities. We have proposed a model for self-diffusivities to parametrize the multicomponent Darken equation. Using MS diffusivities and the matrix of thermodynamic factors, measurable Fick diffusivities can be calculated. MS diffusivities can be computed in MD simulations while the matrix of thermodynamic factors can be obtained from simulations in the grand-canonical ensemble [31–34].

123

Int J Thermophys (2013) 34:1169–1196

1173

The currently used algorithms for computing the matrix of thermodynamic factors from molecular simulations are inefficient. The thermodynamic factors can also be obtained from experimentally determined equations of state [35–37] or excess Gibbs energy models [3,10]. However, the combination of computed MS diffusivities and experimentally determined thermodynamic factors is inconsistent [38–40]. In Sect. 4, we present a new method for computing thermodynamic factors from equilibrium MD simulations. This method could also be used to study the activity coefficients by integrating the matrix of thermodynamic factors. In Sect. 5, we summarize the main achievements of our work on diffusion in liquids. 2 Molecular Simulation of Self- and Transport Diffusion Molecular dynamics (MD) simulation is a computational technique which uses a (usually classical) force field to compute equilibrium and transport properties of a manybody system. The classical force field uses a functional form to describe the interaction between particles (atoms or molecules), i.e., bonded and non-bonded potentials. The required parameters are usually derived from experiments and/or quantum mechanical calculations. In this paper, bonded interactions typically consist of bond stretching, bond bending, and torsion interactions while non-bonded interactions consist of Weeks–Chandler–Andersen (WCA) [41], Lennard-Jones (LJ), and Coulombic interactions. For studying diffusion, simulations using quantum mechanical interactions are not yet feasible due to the required time scale (i.e., ≥100 ns). The key idea of MD simulations is to compute observable quantities from the time evolution of the system. In MD simulations, the equations of motion (Newton’s second law) are integrated numerically [42–44]. Newton’s second law states that the acceleration of a particle is proportional to the net force on the particle and inversely proportional to its mass, ai =

Fi d2 ri = , mi d t2

(10)

in which ai is the acceleration of particle i, Fi is the net force acting on particle i, m i is the mass of particle i, ri is the position of particle i, and t is the time. To integrate the equations of motion, several algorithms are available. For example, the time-reversible velocity Verlet algorithm is often used [42,43], Fi (t) 2 t , 2m i Fi (t + t) + Fi (t) vi (t + t) = vi (t) + t, 2m i ri (t + t) = ri (t) + vi (t)t +

(11) (12)

in which ri (t) and vi (t) are the position and velocity of particle i at time t, respectively. The velocities of particles are related to the temperature [42]. t is the time step for integration and the typical value for t is 10−15 s [42]. Fi (t) is the net force acting on particle i at time t and can be calculated using a (classical) force field. Typically, we use hundreds to a few thousand molecules in MD simulation. Periodic boundary conditions

123

1174

Int J Thermophys (2013) 34:1169–1196

are usually applied [42–44]. By integrating the equation of motion, typical trajectories are obtained. From these trajectories, thermodynamic and transport properties can be computed. Details are explained in Sect. 2.2. The time scale of MD simulation depends on the properties of interest and typically ranges from a few to hundreds of nanoseconds. For more details on MD simulations, the reader is referred to some excellent standard textbooks [42–45].

2.1 Non-equilibrium Molecular Dynamics Non-equilibrium MD (NEMD) simulation has been treated as the most intuitive way to obtain transport diffusion from MD simulations as some algorithms are similar to physical experiments [46–55]. There are several different NEMD diffusion algorithms that have been used to obtain transport diffusion coefficients [46–52]. The algorithms can be categorized by the method of perturbation used to drive the flux of species. In this section, we briefly review the most often used two methods: inhomogeneous NEMD and homogeneous NEMD, see Fig. 1. Boundary-driven NEMD is an often used inhomogeneous method. Typically, it uses a simulation cell involving reservoirs where the concentration of molecules of a given type is either higher or lower compared to its concentration averaged over the system. Concentration gradients induced by these reservoirs lead to a steady-state

Fig. 1 Schematic overview of computational schemes for obtaining diffusion coefficients. Fick’s law and the MS theory are often used to describe mass transport by diffusion. The two formalisms are related via the matrix of thermodynamic factors [Γ ]. Fick diffusivities can be obtained from inhomogeneous NEMD simulations. MS diffusivities can be obtained from equilibrium MD and homogeneous NEMD simulations. The matrix of thermodynamic factors can be predicted using grand-canonical Monte Carlo (GCMC) simulations. Our recent study shows that it is also possible to obtain the thermodynamic factors from equilibrium MD simulations [38–40,111,112,119,123]

123

Int J Thermophys (2013) 34:1169–1196

1175

flux of molecules. To maintain the concentration difference inside the simulation box, a molecule entering one of the reservoirs has to be deleted from one of the other reservoirs. By calculating the steady-state flux and the concentration gradients in the system, transport coefficients can be evaluated directly. This method has been applied to study diffusion in zeolites [56], membranes [57,58], polymers [59], microporous carbon [60,61], fluid systems [46,47], and across surfaces of different nature [62–64]. Another inhomogeneous NEMD method for computing mutual diffusivities involves the use of gradient relaxation MD techniques [48,65,66]. In these methods, a concentration gradient is set up within a simulation cell. The system is relaxed using MD [48,65,66]. The relaxation rate of the concentration gradient is monitored and fitted to the continuum solution of the time-dependent diffusion equation, i.e., in the binary system, ∂ci = D∇ 2 ci , ∂t

(13)

in which t is the time, D is the Fick diffusivity, and ci is the concentration of component i as a function of position and time. This method is conceptually simpler and has a sound physical basis. However, it suffers from the following issues: (1) a large number of molecules must be tracked to obtain continuum like behavior [48]; (2) many initial conditions should be used to determine whether the simulations were taking place in the linear regime [48]; and (3) the concentration dependence of diffusivities is not easily captured [48,67,68]. In homogeneous NEMD (field-driven NEMD), an external field is applied to the simulation cell. This field is connected to particle properties (e.g., molar mass) to obtain the properties corresponding to a bulk system. Therefore, in principle, concentration gradients are not present in the system. The role of this field is to exert a body force on the molecules. A thermostat is required in order to keep the temperature of the system constant [49]. Diffusivities corresponding to zero field can be obtained by extrapolation [52]. The advantages of this method are that it is easy to implement, computationally efficient, and a range of driving forces may be used. The latter feature enables one to examine both linear and nonlinear responses [49–52]. The disadvantage of this method is that the thermostat interacts with the net motion or flux of each component. The manifestation of this interaction, under large fields, is a free energy advantage to partial phase separation of the system. The system tends to form “traffic lanes” like the opposing lanes of traffic on a road as this reduces the effective friction between molecules of different species (that have different average net velocities). This phase-separation artifact can be suppressed by a minor change to the coupling of the thermostat to the particle momenta [49]. In electrolyte systems, i.e., aqueous KCl and NaCl, homogeneous NEMD is claimed to be more efficient compared to equilibrium MD [52]. 2.2 Equilibrium Molecular Dynamics Equilibrium MD simulations can be used to directly compute the MS diffusivities D ¯ i j from the motion of molecules inside the simulation box. The so-called Onsager

123

1176

Int J Thermophys (2013) 34:1169–1196

coefficients Λi j can be obtained directly from the equilibrium motion of the molecules in MD [7,8,69–79]: ⎞ ⎛ N i  1 1 1 lim × ⎝ (rl,i (t + m · t) − rl,i (t))⎠ Λi j = 6 m→∞ N m · t l=1 ⎛ ⎞ Nj  ⎝ (rk, j (t + m · t) − rk, j (t))⎠ .

(14)

k=1

In this equation, N is the total number of molecules in the simulation, and i, j are the molecule types. rl,i (t) is the position of lth molecule of component i at time t. The mean-squared displacement (MSD) can be efficiently updated with different frequencies according to the order-n algorithm described in Refs. [42,80]. By plotting the MSD as a function of time on a log–log scale [81], one can determine the timescale for the diffusive regime and extract diffusivities. The diffusive regime starts when the slope in this plot equals 1 provided that the MSD is sufficiently large (MSD > (box size)2 ). An alternative but equivalent expression for obtaining Λi j is 1 Λi j = 3N

∞ dt 0



N i 

vl,i (t) ·

l=1

Nj 



vk, j (t + t ) ,

(15)

k=1

in which vl,i (t) is the velocity of the lth molecule of type i at time t. Note that the matrix [Λ]  is symmetric, i.e., Λi j = Λ ji and that the Onsager coefficients are constrained by i Mi Λi j = 0 in which Mi is the molar mass of component i [8]. The MS diffusivities directly follow from the Onsager coefficients Λi j . In binary systems, the MS diffusivity D ¯ 12 is related to the Onsager coefficients by [8] x2 x1 D ¯ 12 = x Λ11 + x Λ22 − 2Λ12 . 1 2

(16)

For ternary and quaternary systems, the expressions are more complex and we refer the reader to Refs. [8,82]. As MS diffusivities relate to a collective property of moving molecules, usually long simulations are required to obtain accurate data, i.e. >100 ns [38,39,83]. In principle, MS diffusivities obtained from equilibrium MD and NEMD are identical. This agreement has been observed for the NaCl–water system [52] and methane diffusing in silicalite [48]. Unlike MS diffusivities that describe collective mass transport, self-diffusivities describe the motion of individual molecules. The self-diffusivity Di,self of component i is related to the average molecular displacements described by the Einstein equation [42]

Di,self

N i  1 1 2 = lim (rl,i (t + m · t) − rl,i (t)) 6Ni m→∞ m · t l=1

123

(17)

Int J Thermophys (2013) 34:1169–1196

1 = 3Ni

∞

1177

N i  dt  (vl,i (t) · vl,i (t + t  )) ,

(18)

i=1

0

in which Ni is the number of molecules of component i. It is much easier to obtain accurate self-diffusivities than to obtain MS diffusivities. In a system containing Ni molecules of component i, the statistics for obtaining self-diffusivities is improved as one can take the average over all molecules of component i. However, at each point in time, there is only one data point for the Onsager coefficients Λi j . Due to the much better statistics in self-diffusivities, simulations of a few nanoseconds are usually sufficient to obtain accurate self-diffusion data [38,39,83]. In the limit of infinite dilution (x j → 1), MS, Fick, and self-diffusivities become identical for a x →1

j binary system, i.e., D ¯ ij

x →1

x →1

j = D x j →1 = Di,self . There are several predictive models

j available for Di,self of which the Wilke–Chang equation is still the most popular [4,84].

3 Predictive Models for Maxwell–Stefan and Self-Diffusion Coefficients In Fig. 2, we present several models for obtaining diffusion coefficients that have been categorized as Darken-type and Vignes-type equations. These models are discussed in the following. 3.1 Darken-Type Models The Darken relation postulates that the composition dependence of the binary MS diffusivity is given by [8,85,86] D ¯ i j = xi D j,self + x j Di,self ,

(19)

where Di,self denotes the self-diffusion coefficient of species i in the mixture which depends on the composition. The appeal of the Darken equation originates from the fact that self-diffusivities are more easily accessible than mutual diffusivities, both experimentally [9] and computationally [8]. While often regarded as empirical, the Darken equation can be rigorously derived based on the linear response theory and Onsager’s reciprocal relations [83,87,88]. Considering the Green-Kubo form of Onsager coefficients (Eq. 15), for the terms Λii , we can write [82] Λii

1 = 3N

∞ dt



l=1

0

1 = 3N

∞ 0

N i 

dt 

N i 

vl,i (t) ·

Ni 



vg,i (t + t )

g=1

vl,i (t) · vl,i (t + t  )

l=1

123

1178

Int J Thermophys (2013) 34:1169–1196

Fig. 2 Schematic overview of predictive models for diffusion coefficients. The Darken-type and the Vignestype equations are often used to describe the concentration dependence of MS diffusivities D ¯ i j . The Darken equation requires self-diffusivities, i.e., Di,self while the Vignes-type equation requires MS diffusivities at x j →1

infinite dilution, i.e., D ¯ ij

x →1

k and D ¯ ij

. As in the limit of infinite dilution, self-diffusivities are identical

x j →1 = Di,self , the Darken-type equation can be used to parametrize the to the MS diffusivities, i.e., D ¯ ij

Vignes equation

1 + 3N

∞ dt 0



N Ni i  



vl,i (t) · vg,i (t + t )

l=1 g=1,g=l

≈ xi Cii + xi2 N Cii ,

(20)

in which Cii and Cii account for self- and cross-correlations of the velocities of molecules of component i, respectively. One can assume here that Ni2 − Ni ≈ Ni2 . From Eqs. 18 and 20, it follows that Cii = Di,self .

(21)

For Λi j with i = j, i.e., the correlations between unlike molecules, we can write [82] 1 Λi j = 3N

∞ 0

123

dt 

N i  l=1

vl,i (t) ·

Nj  k=1

vk, j (t + t  )

Int J Thermophys (2013) 34:1169–1196

Ni N j ≈ 3N

∞

1179

  dt  v1,i (t) · v1, j (t + t  ) = N xi x j Ci j .

(22)

0

For ideal-diffusing mixtures, it can be assumed that the velocity cross-correlations, i.e., Ci j and Cii are small compared to self-correlations Cii at finite concentrations. Using this assumption, one can obtain a multicomponent Darken equation for an n-component system: D ¯ i j = Di,self D j,self

n  i=1

xi , Di,self

(23)

For binary systems (n = 2), the multicomponent Darken equation reduces to the Darken equation (Eq. 19) [8,86,89–91]. In WCA fluids in which the molar mass differences between components are small (i.e., M j /Mi ≤ 10), the velocity crosscorrelations are small compared to the self-correlations, Cii , while in the n-hexane– cyclohexane–toluene system, velocity cross-correlations are comparable to the selfcorrelations [82,83]. In the highly associating methanol–ethanol–water system and ionic liquid systems, these cross-correlations are particularly strong due to the collective motion of molecules, i.e., formation of hydrogen bonds and electrostatic interactions [82,83,92]. The coefficients Cii , Ci j , and Cii , can be used to quantitatively study the association of molecules and form the basis for developing predictive models for diffusion. 3.2 Vignes-Type Models The Vignes-type equation is another way to describe the concentration dependence of MS diffusivities [93]. However, the Vignes equation is purely empirical. In binary systems, the Vignes equation is   x →1 x j  xi →1 xi j D . D ¯ ij ¯ ij ¯ ij = D

(24)

x →1

xi →1 j The terms D and D describe the interactions between components i and j if ¯ ij ¯ ij one component is infinitely diluted in the other one. These binary diffusion coefficients are easily obtained from both simulations and (semi-)empirical equations [86,94–98]. For systems containing three and more components, a generalization of the Vignes equation was proposed by Wesselingh and Krishna [99]:

  x →1 x j  xi →1 xi j D D = D ¯ ij ¯ ij ¯ ij

n 

  xk →1 xk D . ¯ ij

(25)

k=1,k=i, j xk →1 The term D describes the interactions between components i and j while both i ¯ ij

xk →1 and j are infinitely diluted in a third component k. This diffusion coefficient D ¯ ij

123

1180

Int J Thermophys (2013) 34:1169–1196

is not directly accessible in experiments. The past 20 years, several predictive models xk →1 xk →1 have been proposed for D [8,10,99,100]. These models suggest to predict D ¯ ij ¯ ij x →1

xi →1 j using either (1) the friction between solutes, i.e., D and D ¯ ij ¯ ij

[99]. For example,

xk →1 xk →1 (2) the friction between solute and solvent, i.e., D and D [8,10], or (3) a ¯ ik ¯ jk combination of the first two categories [100]. Since there is no physical basis for any of these models, it is unclear which one to use for a specific system. A physically based model can be obtained from the multicomponent Darken equation (Eq. 23). In the limit of infinite dilution, one can easily show that the multicomponent Darken equation is reduced to [82]

xk →1 D ¯ ij

=

xk →1 k →1 · D xj,self Di,self xk →1 Dk,self

.

(26)

This equation is valid if the cross-correlations are small compared to the selfxk →1 correlations. Equation 26 allows us to predict the ternary diffusivity D based ¯ ij

xk →1 xk →1 on binary diffusion coefficients D and D and the pure component self¯ ik ¯ jk diffusivity Dk,self . The quality of this equation has been tested for several systems. For all cases, Eq. 26 is as least as good as the other empirical equations. For WCA fluids, Eq. 26 is very accurate resulting in a maximal deviation of 7 % compared to xk →1 D [82]. In the system n-hexane–cyclohexane–toluene in which molecules are ¯ ij weakly associated, the error of the prediction using Eq. 26 is less than 23 % [82]. In the system ethanol–methanol–water in which molecules are highly associating, Eq. 26 leads to an error of 78 % [82]. As hydrogen bonding becomes even more important in systems with ionic liquids, water and DMSO, Eq. 26 cannot accurately predict the magnitude of MS diffusivities at infinite dilution for these systems, suggesting the velocity cross-correlations should be taken into account [92]. The rigorous derivation of Eq. 26 allows the identification of the physical cause of its failure which was not possible for the previous empirical models. It is important to note that errors introduced by modeling the concentration dependence of MS diffusivities using the generalized Vignes equation (Eq. 25) may be either xk →1 [9,92,101]. In the reduced or enhanced by a particular choice for a model for D ¯ ij xk →1 limit of xk → 1, the quality of the model for predicting D determines whether the ¯ ij generalized Vignes equation (Eq. 25) is accurate. For finite concentrations, the quality of the generalized Vignes equation (Eq. 25) and Eq. 26 play a role in the prediction of the concentration dependence of D ¯ ij. In Fig. 3, we compare predicted MS diffusivities in a ternary WCA system using Eqs. 23 and 25 to the computed MS diffusivities from MD. Good agreement between the multicomponent Darken equation (Eq. 23) and simulations was observed. The generalized Vignes equation does not accurately describe the concentration dependence of MS diffusivities, especially not for D ¯ 23 . In quaternary WCA systems ¯ 13 and D and in the n-hexane–cyclohexane–toluene system, Eq. 23 also accurately describes the MS diffusivities as a function of concentration. Data for more systems can be found in Ref. [83]. The multicomponent Darken equation should therefore be viewed

123

Int J Thermophys (2013) 34:1169–1196

1181

1.1

1.1

1.0

1.0

0.9

0.9

13

(b) 1.2

12

(a) 1.2

0.8

0.8 Generalized Vignes

0.7

Generalized Vignes

0.7

Multicomponent Darken

Multicomponent Darken

0.6

0.6 MD

MD

0.5

0.5 0

0.2

0.4

0.6

0.8

1

x1

0

0.2

0.4

0.6

0.8

1

x1

(c) 1.2 Generalized Vignes

1.1 Multicomponent Darken

1.0 23

MD

0.9 0.8 0.7 0.6 0.5 0

0.2

0.4

0.6

0.8

1

x1 Fig. 3 Computed and predicted MS diffusivities in ternary systems as a function of composition in which particles interact with a WCA potential. All quantities are reported in dimensionless units. x1 is varied while keeping x2 = x3 . Open symbols represent computed MS diffusivities using MD. Red lines represent predictions using the multicomponent Darken equation (Eq. 23) with the self-diffusivities obtained from k→1 MD simulation. Blue lines represent predictions using the generalized Vignes equation (Eq. 25) with D ¯ ij obtained from Eq. 26. Simulation details: ρ = 0.2; M1 = 1; M2 = 5; M3 = 10; T = 2; N = 400 (Color figure online)

as a physically-based model for the prediction of MS diffusivities in ideal diffusing mixtures. Due to the fact that velocity cross-correlations are excluded in the multicomponent Darken equation, in the systems where molecules are highly associating, the multicomponent Darken equation will fail in predicting MS diffusivities [40]. In these systems containing associating molecules, the Darken-type equation may lead to accurate results in some cases but this is due to cancellation of errors. For such mixtures, a sound description of velocity cross-correlations would be required. 3.3 Prediction of the Composition Dependence of Self-Diffusivities Many engineering predictive models, e.g., the Darken equation, need information on the concentration dependence of self-diffusivities. The following models have been proposed in the literature [8] for relating the self-diffusivity Di,self at any mixture

123

1182

Int J Thermophys (2013) 34:1169–1196 x →1

j composition to the self-diffusivity in a infinitely diluted solution Di,self :

Di,self = Di,self = Di,self =

n  j=1 n  j=1 n 

x →1

j x j Di,self ,

x →1

j w j Di,self ,

x →1

j v j Di,self .

(27)

(28)

(29)

j=1 x →1

j In these equations, w j is the mass fraction, v j is the volume fraction, and Di,self is the self-diffusivity of component i in an i- j mixture when the mole fraction of component j is approaching one. In liquid mixtures of linear alkanes [8], the mass weighted equation (Eq. 28) is the best for describing MD simulation data. In Ref. [83], we proposed a model for the self-diffusivity of species i in the mixture as the inverse mole-fraction weighted sum of its pure component value and its values at infinite dilution [83],

1 Di,self

=

n 

xj . x j →1 j=1 Di,self

(30)

One can show that in the limit xi → 0, Eq. 30 is exact [82,101]. The validation of Eq. 30 in ternary WCA fluids is shown in Fig. 4. The predictions using Eq. 30 perfectly agree with the computed self-diffusivities. This excellent agreement was also observed for the binary systems acetone–methanol and acetone–tetrachloromethane; see Fig. 5. In Fig. 6, we compared the computed self-diffusivities to the predictions from Eq. 30 in the ternary system chloroform–acetone–methanol. We observed that over the whole concentration range, Eq. 30 quantitatively agrees with the computed self-diffusivities. The averaged absolute differences are approximately 10 %. The good agreement between predictions and computed self-diffusivities suggests that Eq. 30 can be used to describe the composition dependence of self-diffusivities, even for systems that show moderately strong interactions. Recently, Vrabec and co-workers have shown that Eq. 30 works less well for systems with strong hydrogen bonding [102]. 4 Thermodynamic Factors The matrix of thermodynamic factors (Eq. 7) describes how ideal a solution is. Knowledge of the thermodynamic factors allows one to relate Fick to MS diffusivities (Eq. 5). By integrating the thermodynamic factors, activity coefficients can be obtained. In this section, we briefly review the methods for obtaining thermodynamic factors from experiments and simulations.

123

Int J Thermophys (2013) 34:1169–1196

1183

(b) 1.6

1.4

1.4

1.2

1.2

1.0

1.0

Di,self

Di,self

(a) 1.6

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

x1

0.0 0.0

0.2

0.6

0.8

1.0

x2

(c) 1.6 1.4 1.2

Di,self

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

x3 Fig. 4 Computed and predicted self-diffusivities in ternary systems in which particles interact with a WCA potential. All quantities are reported in dimensionless units. Open symbols represent computed self-diffusivities using MD. Solid lines represent predictions using Eq. 30. Blue triangles represent the self-diffusivities of component 1. Red squares represent the self-diffusivities of component 2. Green circles represent the self-diffusivities of component 3. (a) x1 varies, x2 /x3 = 1; (b) x2 varies, x1 /x3 = 1; and (c) x3 varies, x1 /x2 = 1. Simulation details: ρ = 0.2; M1 = 1; M2 = 5; M3 = 10; T = 2; N = 400 (Color figure online)

4.1 Experimental Determination Typically, the matrix of thermodynamic factors [Γ ] can be calculated if activity coefficients are known (Eq. 7). By differentiation of the activity coefficients with respect to composition, the matrix of thermodynamic factors can be calculated [3,10]. Experimental vapor–liquid equilibrium data [103] and experimentally determined equations of state [35–37] are often used to obtain activity coefficients. The most important advantages of using equations of state are: (1) they lead to simple expressions and enable fast calculations; (2) they are applicable over a wide range of pressures and temperatures; and (3) many existing databases and correlations are available. However, there are still limitations: (1) predictions for binary and multicomponent systems are difficult and often rely on mixing rules [104]: and (2) predictions are sometimes inaccurate, especially in the presence of associating molecules [105].

123

Int J Thermophys (2013) 34:1169–1196

Di,self , 10 -9 m2.s-1

(a) 5

4

3

2

(b)

5

Di,self , 10 -9 m2 .s-1

1184

4

3

2

1

1 0

0,2

0,4

0,6

0,8

1

0

0,2

0,4

x1

x1

0,6

0,8

1

Fig. 5 Self-diffusivities in binary liquid systems (a) acetone (1)–methanol (2) and (b) acetone (1)–CCl4 (2) mixtures at 298 K, 1 atm. Filled symbols are the computed self-diffusivities using MD simulations. Squares are the self-diffusivities of acetone. Triangles are the self-diffusivities of either methanol or CCl4 . Solid lines are the predictions using Eq. 30. Details of the used force field can be found in Refs. [38,39]

(b) 6 D1,self D1,self

D2,self D2,self

D3,self D3,self

1D1,self

2D2,self

3D3,self

4

2

0 0,0

0,2

0,4

0,6

0,8

D1,self D1,self

D2,self D

D3,self D

1D

2D

3D

1,self

Di,self , 10 -9 m2.s-1

Di,self , 10 -9 m2.s-1

(a) 6

1,0

x1

2,self

2,self

3,self

3,self

4

2

0 0,0

0,2

0,4

0,6

0,8

1,0

x2

Di,self , 10 -9 m2.s-1

(c) 6 D1,self D1,self

D2,self D2,self

D3,self D3,self

D1,self Series4

Series5 D2,self

Series6 D3,self

4

2

0 0,0

0,2

0,4

0,6

0,8

1,0

x3 Fig. 6 Self-diffusivities in ternary liquid systems chloroform (1)–acetone (2)–methanol (3) at 298 K, 1 atm. Open symbols are the computed self-diffusivities using MD simulations. Solid lines are the predictions using Eq. 30. Details of the used force field can be found in Ref. [40]

123

Int J Thermophys (2013) 34:1169–1196

1185

Obtaining the matrix of thermodynamic factors from physical models (e.g., excess Gibbs energy (G E ) models, equations of state) suffers from large uncertainties. Uncertainties in Γi j of more than 20 % are expected [3,106]. That is, even if a G E model shows an excellent fit to activity coefficients determined from experimental vapor liquid equilibria data, the calculated elements of the matrix of thermodynamic factors may deviate significantly. In Ref. [106], Taylor and Kooijman further illustrate the dangers of estimating thermodynamic factors using G E models: i.e., in the binary system 2-butanone–water, the Margules, van Laar, and NRTL models provide a negative value of Γ around an equimolar composition implying demixing, while the Wilson model provides a positive Γ (no demixing) in the same concentration range. Different G E models may also lead to different behavior of [Γ ]. 4.2 Widom’s Test Particle Insertion Method The conventional Widom test particle insertion method can be used to obtain the matrix of thermodynamic factors from molecular simulations [31–33]. Essentially, calculating the thermodynamic factor involves second derivatives of the Gibbs energy with respect to the number of particles. In the frame of Widom’s test particle insertion method, this corresponds to the simultaneous insertion of two test particles. The disadvantages of Widom’s test particle insertion method applied to two test particles is that it is very inefficient to insert test particles into high density systems. Recently, a permuted Widom test particle insertion method has been developed by Balaji et al. [107,108]. The permuted Widom’s test particle insertion method involves the use of combinatorics to make the Widom’s method more efficient for the simulations insertion of two test particles [107,108]. So far, the method has been applied only to WCA systems although, in principle, it is applicable to any system. 4.3 Kirkwood–Buff Integrals The thermodynamic factors [Γ ] can be calculated using the so-called Kirkwood–Buff (KB) coefficients. The KB coefficients can be obtained from density fluctuations in the grand-canonical ensemble [88]:    Ni N j − Ni N j V δi j   =V , − Ni Ni N j 

Gi j

(31)

in which V is the volume and δi j is the Kronecker delta. The brackets · · · denote an ensemble average in the grand-canonical ensemble. The thermodynamic factors are related to the KB coefficients G i j [38–40,109]. In binary systems, the thermodynamic factor Γ is related to the KB coefficients G i j by [38,39,109] Γ = 1 − xi

c j (G ii + G j j − 2G i j ) . 1 + c j xi (G ii + G j j − 2G i j )

(32)

in which c j = N j /V . The expression relating the thermodynamic factors [Γ ] to the KB coefficients G i j in multicomponent systems can be found in Refs. [82,109].

123

1186

Int J Thermophys (2013) 34:1169–1196

Kirkwood and Buff showed that in the thermodynamic limit (V → ∞), the KB coefficients can be related to the integrals of radial distribution functions over volume [109,110], resulting in the following expression [111,112]: G i∞j

∞ = 4π



 gi j (r ) − 1 r 2 dr.

(33)

0

In this equation, gi j (r ) is the radial distribution function for molecules of type i and j. As one usually does not have data for gi j (r ) for an infinite range, it is common practice to truncate the integration at a distance R: R G i j (R) = 4π



 gi j (r ) − 1 r 2 dr.

(34)

0

It is important to note several issues associated with Eq. 34: (1) Eq. 34 only has a physical meaning when the upper bound of the integration is infinity, and therefore it can only be applied to infinitely large systems (this issue is further discussed in Sect. 4.5); (2) gi j (r ) is the radial distribution function in an open system while MD simulations usually consider closed systems; (3) for infinitely large systems, gi j (r ) → 1 for r → ∞. For finite systems with periodic boundary conditions, gi j (r ) does not converge to 1 for large r , and therefore the convergence of the integral is often slow and poor [109]. It is very difficult to find a plateau in a plot of G i j (R). In Fig. 7, we show that the computed function G i j (R) obtained using Eq. 34 for systems of (200, 500, and 5800) molecules. It is clear that even with a system of 5800 molecules, one cannot obtain an accurate estimate for the KB coefficients. Some effort has been made to solve the problem regarding to the poor convergence of Eq. 34 [113–115]. The simplest and in the past frequently used approach is to simply use a switching function to force the radial distribution function g(r ) to converge to 1 for large distance r . However, it turns out that the final result depends on the choice of the switching function [115,116]. Other approaches use the extension of g(r ) due to the method by Verlet [43,113,117]. 4.4 Scaling of Small System Fluctuations Recently, Schnell et al. developed an alternative approach to compute G i j from the local density fluctuations in small subsystems of volume V embedded in the simulation box (Eq. 31) [111,112]. As these small subsystems can exchange energy and particles with the rest of the system, it can be considered as grand-canonical like. In this way, KB coefficients follow directly from Eq. 31 in which concentrations, particle numbers, and volume refer to those inside the subvolume V . One can show that Eq. 31 scales as 1/L, where L is the linear length of subvolume V in one dimension [112,118,119]. To obtain KB coefficients in the thermodynamic limit (G i∞j ), one can fit the simulation data to [112,118,119]

123

Int J Thermophys (2013) 34:1169–1196

1187

(a) 100

(b) 300 200

50

500

100

5800

5800

G22(R) , Å3

G11(R) , Å3

0 -50 -100 -150

0 -100 -200

-200

-300

-250

-400

-300

200

200

500

0

10

20

30

40

50

-500

0

R,Å

10

20

30

40

50

R,Å

(c) 150 200

100

500

G12(R) , Å3

50

5800

0 -50 -100 -150 -200 -250 -300

0

10

20

30

40

50

R,Å

Fig. 7 Running integrals of KB coefficients calculated using Eq. 34 for an equimolar acetone (1)–methanol (2) mixture at 298 K, 1 atm. G 11 (R) represents the acetone–acetone C–C correlation (carbon atom connected to the oxygen atom). G 22 (R) represents the methanol–methanol O–O correlation and G 12 (R) represents the acetone–methanol C–O correlation. Black lines represent G i j (R) using gi j (r ) obtained from a system containing 200 molecules. Red lines represent G i j (R) using gi j (r ) obtained from a system containing 500 molecules. Green lines represent G i j (R) using gi j (r ) obtained from a system containing 5800 molecules. Details of the used force field can be found in Ref. [40] (Color figure online)

G i j (L) = G i∞j +

A , L

(35)

where A is a constant which does not depend on L , G i j (L) is the KB coefficient obtained from subsystems of size L and G i∞j is the KB coefficient in the thermodynamic limit and can be obtained by extrapolation of Eq. 35 to L → ∞. This method has been validated for WCA and LJ systems [112], and recently also for molecular systems [38,39,120]. The extrapolated KB coefficients G i∞j using the so-called small subsystem method agree very well with those obtained from the grand-canonical ensemble [112]. Here, we validated our method in the binary systems acetone–methanol and acetone–tetrachloromethane, see Fig. 8 [38,39]. We compared the computed thermodynamic factors to those predicted using G E models fitted to the experimental vapor–liquid equilibrium data. Very good agreement was found suggesting that it is possible to access the grand-canonical ensemble from MD simulations. The resulting

123

1188

Int J Thermophys (2013) 34:1169–1196

(a) 1.2

(b) 1.1

EMD MD

1.0

1.0

Margules

0.9 0.8 0.8 0.6 0.7 0.4

0.6

MD EMD

0.2 0.0

0.5

Margules

0

0.2

0.4

0.6

0.8

0.4 0.0

1

0.2

0.4

x1

0.6

0.8

1.0

x1

Fig. 8 Thermodynamic factor Γ in the binary systems (a) acetone (1)–methanol (2) and (b) acetone (1)– tetrachloromethane (2) at 298 K, 1 atm. Open symbols are the computed Γ using MD simulations. The solid lines represent Γ calculated from the Margules model fitted to experimental vapor–liquid equilibrium data [125]. Details of the used force field can be found in Refs. [38,39]

(a) 8

(b) 5 MD

MD EMD

D , 10 -9 m2 .s-1

D , 10 -9 m2.s-1

4

Experiment experiment

6

Experiment

4

3

2

2 1

0

0 0

0,2

0,4

0,6

x1

0,8

1

0

0,2

0,4

0,6

0,8

1

x1

Fig. 9 Fick diffusivities in binary systems (a) acetone (1)–methanol (2) and (b) acetone (1)– tetrachloromethane (2) at 298 K, 1 atm. Open symbols are Fick diffusivities D calculated using D ¯ 12 and Γ , both computed from equilibrium MD simulations. The solid lines represent Fick diffusivities D obtained from experiments [23,126]. Details of the used force field can be found in Refs. [38,39]

Fick diffusivities for the systems acetone–methanol and acetone–tetrachloromethane obtained via the computed MS diffusivities and the matrix of thermodynamic factors are in excellent agreement with the experimental data as shown in Fig. 9. We applied this method also to the ternary system chloroform–acetone–methanol [40]. The computed thermodynamic factors using the small subsystem method are compared to predictions obtained from the COSMO-SAC theory [121,122]. In Fig. 10, the concentration of chloroform increases while the concentration of acetone and methanol are identical. For the whole concentration range, the data obtained from MD simulation are in excellent agreement with experiments. The COSMO-SAC theory qualitatively describes the concentration dependence of thermodynamic factors; however, errors of more than 100 % were observed. The resulting Fick diffusivities for the ternary system

123

Int J Thermophys (2013) 34:1169–1196

(a)

1189

2.0

2.0

(b) MD

1.5

COSMO-SAC

NRTL

1.0

COSMO-SAC

NRTL

0.4

0.8

1.0

0.5

Γ12

Γ11

MD

1.5

0.5

0.0

0.0

-0.5

-0.5

-1.0 0.0

0.2

0.4

0.6

0.8

-1.0

1.0

0.0

0.2

x1

(c)

COSMO-SAC

NRTL

1.5

1.5

1.0

1.0

Γ 22

Γ21

1.0

2.0

(d)

2.0 MD

0.6

x1

0.5

0.5

0.0

0.0

-0.5

-0.5

MD

COSMO-SAC

NRTL

-1.0

-1.0 0.0

0.2

0.4

0.6

0.8

1.0

x1

0.0

0.2

0.4

0.6

0.8

1.0

x1

Fig. 10 Thermodynamic factors Γi j in the ternary system chloroform (1)–acetone (2)–methanol (3) at 1 atm. Open circles are the computed values of Γi j using MD simulations at 298 K. Filled circles are the computed values of Γi j using the COSMO-SAC approach at 298 K. Solid lines represent Γi j calculated from the NRTL model fitted to experimental vapor–liquid equilibrium data at 303 K of Ref. [127]. x1 is varied while keeping x2 = x3 . Details of the used force field can be found in Ref. [40]

acetone–methanol–chloroform can be found in Fig. 11 [40]. It can be observed that: (1) the diagonal Fick diffusivities are always positive and the off-diagonal Fick diffusivities may be negative; (2) the diagonal Fick diffusivities are about one order of magnitude larger than the off-diagonal Fick diffusivities meaning the diffusion flux of component i mainly depends on its own concentration gradient while the concentration gradients of other components play a minor role. This behavior is in accordance with the bound placed on off-diagonal coefficients by the entropy production for ternary diffusion [2]. 4.5 Kirkwood–Buff Integrals in Finite Systems As discussed in Sect. 4.3, density fluctuations in the grand-canonical ensemble can be related to integrals of the radial distribution function over volume. Kirkwood and Buff have shown this for systems in the thermodynamic limit (Eq. 33). However, in molecular simulation one deals with systems of finite-size, so very often Eq. 34 is used

123

1190

(a)

Int J Thermophys (2013) 34:1169–1196

(b) 6

6 11

12

21

22

4

Dij , 10 -9 m2.s-1

Dij , 10 -9 m2.s-1

4

2

2

0

0

-2 0,0

-2 0,0

11

0,2

0,4

0,6

0,8

1,0

0,2

0,4

21

22

0,6

0,8

1,0

x2

x1

(c)

12

6 11

12

21

22

Dij , 10 -9 m2.s-1

4

2

0

-2 0,0

0,2

0,4

0,6

0,8

1,0

x3

Fig. 11 Fick diffusivities in the ternary system chloroform (1)–acetone (2)–methanol (3) at 298 K, 1 atm. Fick diffusivities Di j are calculated using the computed D ¯ i j and Γi j . (a) x1 varies while keeping x2 = x3 , (b) x2 varies while keeping x1 = x3 , (c) x3 varies while keeping x1 = x2 . Details of the used force field can be found in Ref. [40]

as an approximation. Recently, Krüger et al. [123] have shown that the approximation of Eq. 34 is in fact incorrect for finite-size systems, and significantly deviates from G i∞j . This can be understood as follows. Consider a finite and open system of volume V . We assume that this volume is spherical with a radius R. The KB coefficients G iVj for this finite system are defined as:    Ni N j − Ni N j V δi j   ≡V − Ni Ni N j   1 = (gi j (r12 ) − 1)dr 1 dr 2 V 

G iVj

(36) (37)

V V

in which r12 = |r 1 − r 2 | and gi j (r12 ) is the radial distribution function for i, j pairs. For an infinitely large system, the double integral of Eq. 37 can be reduced to a single integral by the transformation r 2 → r = r 1 − r 2 leading directly to Eq. 33. For a

123

Int J Thermophys (2013) 34:1169–1196

1191

finite-size system, one cannot apply this transformation because the integration domain of r depends on r 1 . In this case, the correct integral becomes   1 G iVj = (gi j (r12 ) − 1)dr 1 dr 2 (38) V V V

= 4π

2R 

   r3 3r r 2 dr. + gi j (r ) − 1 1 − 4R 16R 3

(39)

0

which is only identical to Eq. 33 when R → ∞. One can show that [G iVj (R) − G i∞j ] scales as 1/R [123]. Note that the KB coefficients in Eqs. 31 and 36 correspond to the grand-canonical ensemble, and so gi j (r ) must, in principle, be calculated in an infinite system. The radial distribution functions computed in a finite, closed system, do not converge to one for r → ∞, which leads to a divergence of the KB coefficients, as discussed in Sect. 4.3. However, the radial distribution functions of the infinite system, gi∞j (r ) can be accurately estimated from those of two finite, closed systems with particle numbers N1 and N2 . This is based on the fact that in the expansion, giNj (r ) ≈ gi∞j (r ) +

c(r ) , N

(40)

the function c(r ) does not depend on N [118]. Therefore, c(r ) and thus gi∞j (r ) can be readily estimated from giNj (r ) computed for two different system sizes N1 and N2 . In Fig. 12, we compare the KB coefficients calculated using Eq. 39, using exactly the same radial distribution functions as in Fig. 7. We computed G iVj (R) using gi j (r ) obtained from a finite system, and also using gi∞j (r ) defined in Eq. 40. Using gi j (r ) obtained from a finite system in Eq. 39 has the disadvantage that it is difficult to determine the linear regime for which [G iVj (R) − G i∞j ] scales as 1/R. This problem becomes less severe by using a very large system. Using gi∞j (r ) in Eq. 39, Fig. 12 clearly shows that it is much easier to find the linear regime and extrapolate the KB coefficients to the thermodynamic limit. To obtain gi∞j (r ), two small systems are sufficient and simulations of large systems are not needed. 5 Conclusions In this work, we briefly reviewed methods for predicting mutual diffusion coefficients [38–40,82,83,92,101,124]. The main achievements of our research team can be summarized as follows: – We introduced a consistent method for computing Fick diffusion coefficients using MD simulations. In experiments, Fick diffusivities are measured while molecular simulation usually provides MS diffusivities. These diffusivities are related via the matrix of thermodynamic factors which is usually known only with large uncertainties. This leaves a gap between experiment and molecular simulation. To overcome this problem, we calculated thermodynamic factors from MD simulations using

123

1192

Int J Thermophys (2013) 34:1169–1196 0

(b) 250

-20

200

(a)

200 5800

GV22 (R) , Å3

GV11 (R) , Å3

-40

200

-60 -80 -100

50 0

500-200

0

0.1

0.3

0.2

1/R , Å-1

(c)

500-200

100

5800

-120 -140

200

150

-50

0

0.2

0.1

0.3

1/R , Å-1

0

GV12 (R) , Å3

-50

-100 200

-150

5800 500-200

-200

-250

0

0.1

0.2

0.3

1/R , Å-1 Fig. 12 KB coefficients calculated using Eq. 39 for an equimolar acetone (1)–methanol (2) mixture at 298 K, V (R) represents the acetone–acetone C–C correlation (carbon atom connected to the oxygen atom). 1 atm. G 11 V V (R) represents the acetone–methanol G 22 (R) represents the methanol–methanol O–O correlation and G 12 C–O correlation. Black lines represent G iVj (R) using gi j (r ) in Eq. 39 (a system containing 200 molecules). Green lines represent G iVj (R) using gi j (r ) in Eq. 39 (a system contains 5800 molecules). Red solid lines

∞ represent G iVj (R) using gi∞ j (r ) in Eq. 39 with gi j (r ) calculated from two systems consisting of 500 and 200 molecules. The red dashed lines are a guide to the eye and show the extrapolation of KB coefficients to the thermodynamic limit. Details of the used force field can be found in Ref. [40] (Color figure online)

the small subsystem method. Our approach allows for an efficient and consistent prediction of multicomponent Fick diffusion coefficients from molecular models. – We developed a multicomponent Darken equation for the description of the concentration dependence of MS diffusivities. A multicomponent Darken equation was developed based on the linear response theory and the Onsager theory. The multicomponent Darken equation provides consistent and accurate predictions for ideal diffusing systems. Its sound theoretical background also enables the identification of the physical cause in case it fails. This was not possible for the previous empirical models. In the case of infinite dilution, the multicomponent Darken xk →1 which can be used to parametrize the equation provides an expression for D ¯ ij generalized Vignes equation.

123

Int J Thermophys (2013) 34:1169–1196

1193

– Eq. 30 for self-diffusivities was proposed for parametrization of the multicomponent Darken equation. This equation accurately describes the concentration dependence of self-diffusivities in weakly associating systems. – The Kirkwood–Buff (KB) theory is correct and applicable for any open system. The original KB integrals, which are valid only for infinite systems, have often incorrectly been applied to finite systems. We have shown that with Eqs. 39 and 40, KB integrals can be calculated accurately for open systems of any finite volume, requiring only radial distribution functions computed in closed systems of moderate size. The presented methods thus provide a hierarchy of tools to predict multicomponent diffusion. For ideal diffusing mixtures, the multicomponent Darken equation allows for a physically based comprehensive engineering model. In combination with Eq. 30 for self-diffusivities, multicomponent diffusion coefficients can be predicted using binary data of infinite dilution and pure component data. For non-ideal diffusion mixtures, MD allows one to quantify the complex diffusion behavior. The small subvolume method opens the route for efficient determination of Fick diffusion coefficients from MD. Acknowledgments This work was performed as part of the Cluster of Excellence “Tailor-Made Fuels from Biomass”, which is funded by the Excellence Initiative by the German federal and state governments to promote science and research at German universities. TJHV, SKS, and SK acknowledge financial support from NWO-CW through an ECHO grant. This work was also sponsored by the Stichting Nationale Computerfaciliteiten (National Computing Facilities Foundation, NCF) for the use of supercomputing facilities, with financial support from the Nederlandse Organisatie voor Wetenschappelijk onderzoek (Netherlands Organization for Scientific Research, NWO).

References 1. S. Kjelstrup, D. Bedeaux, E. Johannessen, J. Gross, Non-equilibrium Thermodynamics for Engineers, 1st edn. (World Science Publishing Co. Pte. Ltd., Singapore, 2010) 2. S. Kjelstrup, D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems, 1st edn. (World Science Publishing Co. Pte. Ltd., Singapore, 2008) 3. R. Taylor, R. Krishna, Multicomponent Mass Transfer, 1st edn. (Wiley, New York, 1993) 4. B. Poling, J. Prausnitz, J.P.O. O’Connell, The Properties of Gases and Liquids, 5th edn. (McGraw-Hill, New York, 2004) 5. E.L. Cussler, Diffusion, Mass Transfer in Fluid Systems, 3rd edn. (Cambridge University Press, New York, 2005) 6. G.D.C. Kuiken, Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology, 1st edn. (Wiley, New York, 1994) 7. R. Krishna, J.A. Wesselingh, Chem. Eng. Sci. 52, 861 (1997) 8. R. Krishna, J.M. van Baten, Ind. Eng. Chem. Res. 44, 6939 (2005) 9. A. Bardow, E. Kriesten, M.A. Voda, F. Casanova, B. Blümich, W. Marquardt, Fluid Phase Equilib. 278, 27 (2009) 10. H.A. Kooijman, R. Taylor, Ind. Eng. Chem. Res. 30, 1217 (1991) 11. A. Bardow, V. Göke, H.J. Koß, W. Marquardt, AIChE J. 52, 4004 (2006) 12. A. Bardow, W. Marquardt, V. Göke, H.J. Koß, K. Lucas, AIChE J. 49, 323 (2003) 13. R.L. Robinson Jr, W.C. Edmister, F.A.L. Dullien, J. Phys. Chem. 69, 258 (1965) 14. R.H. Stokes, J. Am. Chem. Soc. 72, 763 (1950) 15. M.L. Wagner, H. Scheraga, J. Phys. Chem. 60, 1066 (1956) 16. W.J. Thomas, I.A. Furzer, Chem. Eng. Sci. 17, 115 (1962) 17. H.Z. Cummins, N. Knable, Y. Yeh, Phys. Rev. Lett. 12, 150 (1964)

123

1194

Int J Thermophys (2013) 34:1169–1196

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

D. Ehlich, M. Takenaka, T. Hashimoto, Macromolecules 26, 492 (1993) E. Häusler, P. Domagalski, M. Ottens, A. Bardow, Chem. Eng. Sci. 72, 45 (2012) H. Jobic, N. Laloué, C. Laroche, J.M. van Baten, R. Krishna, J. Phys. Chem. B 110, 2195 (2006) I.M.J.J. van de Ven-Lucassen, M.F. Kemmere, P.J.A.M. Kerkhof, J. Solut. Chem. 26, 1145 (1997) I.M.J.J. van de Ven-Lucassen, F.G. Kieviet, P.J.A.M. Kerkhof, J. Chem. Eng. Data 40, 407 (1995) A. Alimadadian, C.P. Colver, Can. J. Chem. Eng. 54, 208 (1976) H.T. Cullinan, H.L. Toor, J. Phys. Chem. 69, 3941 (1965) T.K. Kett, D.K. Anderson, J. Phys. Chem. 73, 1268 (1969) F. Shuck, H.L. Toor, J. Phys. Chem. 67, 540 (1963) R.A. Graff, T.B. Drew, Ind. Eng. Chem. Fundam. 7, 490 (1968) A. Sethy, H.T. Cullinan, AIChE J. 21, 571 (1975) S. Käshammer, H. Weingärtner, H. Hertz, Phys. Chem. Chem. Phys. 187, 233 (1994) J. Butchard, H. Toor, J. Phys. Chem. 66, 2015 (1980) B. Widom, J. Chem. Phys. 39, 2808 (1963) A. Lotfi, J. Fischer, Mol. Phys. 66, 199 (1989) A. Lotfi, J. Fischer, Mol. Phys. 71, 1171 (1990) J. Vrabec, H. Hasse, Mol. Phys. 100, 3375 (2002) M.K. Kozlowska, B. Jurgens, C.S. Schacht, J. Gross, T.W. de Loos, J. Phys. Chem. B 113, 1022 (2009) C.S. Schacht, C. Schuell, H. Frey, T.W. de Loos, J. Gross, J. Chem. Eng. Data 56, 2927 (2011) J. Gross, G. Sadowski, Ind. Eng. Chem. Res. 40, 1244 (2001) X. Liu, S.K. Schnell, J.M. Simon, D. Bedeaux, S. Kjelstrup, A. Bardow, T.J.H. Vlugt, J. Phys. Chem. B 115, 12921 (2011) X. Liu, S.K. Schnell, J.M. Simon, D. Bedeaux, S. Kjelstrup, A. Bardow, T.J.H. Vlugt, J. Phys. Chem. B 116, 6070 (2012) X. Liu, A. Martín-Calvo, E. McGarrity, S.K. Schnell, S. Calero, J.M. Simon, D. Bedeaux, S. Kjelstrup, A. Bardow, T.J.H. Vlugt, Ind. Eng. Chem. Res. 51, 10247 (2012) J.D. Weeks, D. Chandler, H.C. Andersen, J. Chem. Phys. 54, 5237 (1971) D. Frenkel, B. Smit, Understanding Molecular Simulation: from Algorithms to Applications, 2nd edn. (Academic Press, San Diego, 2002) M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, 1st edn. (Oxford University Press, New York, 1987) M.E. Tuckerman, Statistical Mechanics: Theory and Molecular Simulation, 2nd edn. (Oxford University Press, Oxford, 2010) D. Rapaport, The Art of Molecular Dynamics Simulation, 2nd edn. (Cambridge University Press, Cambridge, 2004) A.P. Thompson, D.M. Ford, G.S. Heffelfinger, J. Chem. Phys. 109, 6406 (1998) A.P. Thompson, G.S. Heffelfinger, J. Chem. Phys. 110, 10693 (1999) E.J. Maginn, A.T. Bell, D.N. Theodorou, J. Phys. Chem. 97, 4173 (1993) D.J. Evans, G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, 2nd edn. (Academic Press, London, 1990) D. MacGowan, D.J. Evans, Phys. Rev. A 34, 2133 (1984) S. Sarman, D.J. Evans, Phys. Rev. A 45, 2370 (1992) D.R. Wheeler, J. Newman, J. Phys. Chem. B 108, 18362 (2004) T. Ikeshoji, B. Hafskjold, Mol. Phys. 81, 251 (1994) B. Hafskjold, T. Ikeshoji, S.K. Ratkje, Mol. Phys. 80, 1389 (1993) B. Hafskjold, S.K. Ratkje, J. Stat. Phys. 78, 463 (1995) J.R. Hill, A.R. Minihan, E. Wimmer, C.J. Adams, Phys. Chem. Chem. Phys. 2, 4255 (2000) J.M.D. MacElroy, J. Chem. Phys. 101, 5274 (1994) J.M.D. MacElroy, M.J. Boyle, Chem. Eng. J. 74, 85 (1999) D.M. Ford, G.S. Heffelfinger, Mol. Phys. 94, 673 (1998) R.F. Cracknell, D. Nicholson, N. Quirke, Phys. Rev. Lett. 74, 2463 (1995) D. Nicholson, R.F. Cracknell, Langmuir 12, 4050 (1996) I. Inzoli, S. Kjelstrup, D. Bedeaux, J.M. Simon, Chem. Eng. Sci. 66, 4533 (2011) I. Inzoli, S. Kjelstrup, D. Bedeaux, J.M. Simon, Microporous Mesoporous Mater. 125, 112 (2009) J.M. Simon, D. Bedeaux, S. Kjelstrup, J. Xu, E. Johannessen, J. Phys. Chem. B 110, 18528 (2006) I. Inzoli, J.M. Simon, S. Kjelstrup, D. Bedeaux, J. Colloid Interface Sci. 313, 563 (2007)

36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

123

Int J Thermophys (2013) 34:1169–1196 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118.

1195

J.M. Simon, A.A. Decrette, J.B. Bellat, J.M. Salazar, Mol. Simul. 30, 621 (2004) M. Tsige, G.S. Grest, J. Chem. Phys. 120, 2989 (2004) M. Tsige, G.S. Grest, J. Chem. Phys. 121, 7513 (2004) G. Guevara-Carrion, J. Vrabec, H. Hasse, Int. J. Thermophys. 33, 449 (2012) G.A. Fernandez, J. Vrabec, H. Hasse, Int. J. Thermophys. 26, 1389 (2005) G. Guevara-Carrion, C. Nieto-Draghi, J. Vrabec, H. Hasse, J. Phys. Chem. B 112, 16664 (2008) G.A. Fernandez, J. Vrabec, H. Hasse, Int. J. Thermophys. 25, 175 (2004) G. Guevara-Carrion, J. Vrabec, H. Hasse, Fluid Phase Equilib. 316, 46 (2012) G. Guevara-Carrion, J. Vrabec, H. Hasse, J. Chem. Phys. 134, 074508 (2011) R. Krishna, J.M. van Baten, Chem. Eng. Technol. 29, 516 (2006) R. Krishna, J.M. van Baten, Chem. Eng. Sci. 64, 3159 (2009) R. Krishna, J. Phys. Chem. C 113, 19756 (2009) R. Krishna, J.M. van Baten, J. Membr. Sci. 360, 476 (2010) R. Krishna, J.M. van Baten, Langmuir 26, 10854 (2010) D. Dubbeldam, D.C. Ford, D.E. Ellis, R.Q. Snurr, Mol. Simul. 35, 1084 (2009) D. Dubbeldam, E. Beerdsen, T.J.H. Vlugt, B. Smit, J. Chem. Phys. 122, 224712 (2005) X. Liu, A. Bardow, T.J.H. Vlugt, Ind. Eng. Chem. Res. 50, 4776 (2011) X. Liu, T.J.H. Vlugt, A. Bardow, Ind. Eng. Chem. Res. 50, 10350 (2011) C.R. Wilke, P. Chang, AIChE J. 1, 264 (1955) L.S. Darken, Transactions of the American Institute of Mining and Metallurgical Engineers 175, 184 (1948) R. Krishna, J.M. van Baten, Chem. Eng. Technol. 29, 761 (2006) H.J.V. Tyrell, K.R. Harris, Diffusion in Liquids, 2nd edn. (Butterworths, London, 1984) M. Schoen, C. Hoheisel, Mol. Phys. 53, 1367 (1984) Y.H. Zhou, G.H. Miller, J. Phys. Chem. 100, 5516 (1996) Z.A. Makrodimitri, D.J.M. Unruh, I.G. Economou, J. Phys. Chem. B 115, 1429 (2011) M.A. Granato, M. Jorge, T.J.H. Vlugt, A.E. Rodrigues, Chem. Eng. Sci. 65, 2656 (2010) X. Liu, T.J.H. Vlugt, A. Bardow, J. Phys. Chem. B 115, 8506 (2011) A. Vignes, Ind. Eng. Chem. Fundam. 5, 189 (1966) D.J. Keffer, A. Adhangale, Chem. Eng. J. 100, 51 (2004) D.J. Keffer, B.J. Edwards, P. Adhangale, J. Non-Newton. Fluid Mech. 120, 41 (2004) A. Leahy-Dios, A. Firoozabadi, AIChE J. 53, 2932 (2007) D. Bosse, H. Bart, Ind. Eng. Chem. Res. 45, 1822 (2006) J.A. Wesselingh, A.M. Bollen, Chem. Eng. Res. Des. 75, 590 (1997) J.A. Wesselingh, R. Krishna, Elements of Mass Transfer, 1st edn. (Ellis Hoewood, Chichester, 1990) S. Rehfeldt, J. Stichlmair, Fluid Phase Equilib. 256, 99 (2007) X. Liu, T.J.H. Vlugt, A. Bardow, Fluid Phase Equilib. 301, 110 (2011) S. Parez, G. Guevara-Carrion, H. Hasse, J. Vrabec, Phys. Chem. Chem. Phys. 15, 3985 (2013) J.M. Smith, H.C. van Ness, M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, 6th edn. (McGraw-Hill, New York, 2001) G.M. Kontogeorgis, G.K. Folas, Thermodynamic Models for Industrial Applications, 1st edn. (Wiley, New York, 2010) J. Gmehling, R. Wittig, J. Lohmann, Ind. Eng. Chem. Res. 41, 1678 (2002) R. Taylor, H.A. Kooijman, Chem. Eng. Commun. 102, 87 (1991) S.P. Balaji, S.K. Schnell, E.S. McGarrity, T.J.H. Vlugt, Mol. Phys. 111, 285 (2013) S.P. Balaji, S.K. Schnell, T.J.H. Vlugt, Theor. Chem. Acc. 132, 1333 (2013) A. Ben-Naim, Molecular Theory of Solutions, 2nd edn. (Oxford University Press, Oxford, 2006) J.G. Kirkwood, F.P. Buff, J. Chem. Phys. 19, 774 (1951) S.K. Schnell, T.J.H. Vlugt, J.M. Simon, D. Bedeaux, S. Kjelstrup, Chem. Phys. Lett. 504, 199 (2011) S.K. Schnell, X. Liu, J.M. Simon, A. Bardow, D. Bedeaux, T.J.H. Vlugt, S. Kjelstrup, J. Phys. Chem. B 115, 10911 (2011) R. Wedberg, J.P.O. Connell, G.H. Peters, J. Abildskov, J. Chem. Phys. 135, 084113 (2011) J.W. Nichols, S.G. Moore, D.R. Wheeler, Phys. Rev. E 80, 051203 (2009) D. Mukherji, N.F.A. van der Vegt, K. Kremer, L. Delle Site, J. Chem. Theory Comput. 8, 375 (2012) A. Perera, L. Zoranic, F. Sokolic, R. Mazighi, J. Mol. Liq. 159, 52 (2011) R. Wedberg, J.P. O’Connell, G.H. Peters, J. Abildskov, Mol. Simul. 36, 1243 (2010) J.J. Salacuse, A.R. Denton, P.A. Egelstaff, Phys. Rev. E 53, 2382 (1996)

123

1196

Int J Thermophys (2013) 34:1169–1196

119. 120. 121. 122. 123.

S.K. Schnell, T.J.H. Vlugt, J.M. Simon, D. Bedeaux, S. Kjelstrup, Mol. Phys. 110, 1069 (2012) P. Ganguly, N.F.A. van der Vegt, J. Chem. Theory Comput. 9, 1347 (2013) S.T. Lin, S.I. Sandler, Ind. Eng. Chem. Res. 41(5), 899 (2002) C.M. Hsieh, S.I. Sandler, S.T. Lin, Fluid Phase Equilib. 297, 90 (2010) P. Krüger, S.K. Schnell, D. Bedeaux, S. Kjelstrup, T.J.H. Vlugt, J.M. Simon, J. Phys. Chem. Lett. 4, 235 (2013) X. Liu, A. Bardow, T.J.H. Vlugt, Diffusion-fundamentals.org 16, 81 (2011) J. Gmehling, U. Onken, Vapor–Liquid Equilibrium Data Collection, 1st edn. (DECHEMA, Frankfurt, 2005) D.K. Anderson, J.R. Hall, A.L. Babb, J. Phys. Chem. 62, 404 (1958) P. Oracz, S. Warycha, Fluid Phase Equilib. 137, 149 (1997)

124. 125. 126. 127.

123