Mutual diﬀusion of binary liquid mixtures · NRTL and UNIQUAC. It is found that the Wilson model...

Mutual diffusion of binary liquid mixtures

Mutual diffusion of binary liquid mixtures containing methanol, ethanol, acetone,

benzene, cyclohexane, toluene and carbon tetrachloridea)

Gabriela Guevara-Carrion,1 Tatjana Janzen,1 Y. Mauricio Muñoz-Muñoz,1 and Jadran

Vrabec1, b)

Thermodynamics and Energy Technology, University of Paderborn,

33098 Paderborn, Germany

(Dated: 6 March 2016)

Mutual diffusion coefficients of all 20 binary liquid mixtures that can be formed out

of methanol, ethanol, acetone, benzene, cyclohexane, toluene and carbon tetrachlo-

ride without a miscibility gap are studied at ambient conditions of temperature and

pressure in the entire composition range. The considered mixtures show a varying

mixing behavior from almost ideal to strongly non-ideal. Predictive molecular dy-

namics simulations employing the Green-Kubo formalism are carried out. Radial

distribution functions are analyzed to gain an understanding of the liquid structure

influencing the diffusion processes. It is shown that cluster formation in mixtures

containing one alcoholic component has a significant impact on the diffusion process.

The estimation of the thermodynamic factor from experimental vapor-liquid equilib-

rium data is investigated, considering three excess Gibbs energy models, i.e. Wilson,

NRTL and UNIQUAC. It is found that the Wilson model yields the thermodynamic

factor that best suits the simulation results for the prediction of the Fick diffusion

coefficient. Four semi-empirical methods for the prediction of the self-diffusion co-

efficients and nine predictive equations for the Fick diffusion coefficient are assessed

and it is found that methods based on local composition models are more reliable.

Finally, the shear viscosity and thermal conductivity are predicted and in most cases

favorably compared with experimental literature values.

Keywords: Fick diffusion coefficient, Maxwell-Stefan, Green-Kubo, thermodynamic

factor, radial distribution function, shear viscosity, thermal conductivity, model pre-

diction

a)Supplementary Materials available.b)Electronic mail: [email protected]

1


I. INTRODUCTION

Almost all separation processes in chemical engineering, such as distillation, absorption

or extraction, are affected by diffusion in liquids. Traditionally, equilibrium stage methods

predominate for modeling, design and control of these unit operations because they are

comparably easy to implement and solve. The weakness of equilibrium approaches is that

they often yield a solution that is far from the physical process, which is usually corrected for

with an empirical efficiency factor1. In the last decades, advances in this field have been made

with the continuous development of rate-based or non-equilibrium methods, which are much

nearer to physical reality. Nowadays, rate-based methods are employed to solve complex

modelling and simulation issues in steady and unsteady state operations including start-up

and shut-down2,3. These non-equilibrium methods involve mass and energy transfer models,

which require not only diffusion data, but also other transport coefficients like shear viscosity

and thermal conductivity for pure components as well as mixtures4. Thus, there is a growing

need for accurate transport properties, which experimental measurements alone are not able

to satisfy5. Traditionally, transport data have played a lesser role than time independent

properties, like vapor-liquid equilibria (VLE), so that the availability of experimental data on

transport coefficients is still low5. Hence, there is an increasing interest in better methods for

their prediction. Owing to the rapid development of computing power, molecular modelling

and simulation has emerged as an alternative for such predictions, especially when dealing

with hazardous substances or challenging thermodynamic conditions6,7.

Molecular modeling and simulation comprises computational techniques derived from

quantum chemistry and statistical mechanics to study macroscopic thermodynamic proper-

ties by means of particle ensembles based on interaction potentials. These force fields provide

a fundamental molecular-level physical description of the nature of matter. Thus, a detailed

insight into the physics of equilibrium and non-equilibrium processes can be gained8. Be-

cause of these microscopic considerations, force field-based simulation methods can be used

for the understanding and interpretation of experimental results, to obtain predictive esti-

mates and to inter- or extrapolate experimental data into regions that are difficult to access

in the laboratory9.

To describe diffusive mass transport in liquid mixtures, two approaches are commonly

used: Fick’s law and Maxwell-Stefan (MS) theory1. Both relate a mass flux to a driving

2


force1. In the case of Fick’s law, the driving force is expressed in terms of the mole fraction

gradient ∇xj , which is a quantity that can be measured in the laboratory. The diffusive

molar flux of component i is then

Ji = −ρn−1∑

j=1

Dij∇xj , (1)

where n is the number of components in the mixture, ρ is the molar density and Dij denotes

the Fick diffusion coefficient coupling the flux of component i with the gradient of the mole

fraction of component j.

On the other hand, MS theory1 expresses the driving force in terms of the gradient of

the chemical potential ∇µi, which is assumed to be balanced by a friction force that is

proportional to the mutual velocity between the components ui − uj

n∑

j 6=i=1

xj(ui − uj)Ðij

= −∇µi

kBT, (2)

where kB is the Boltzmann constant and T the temperature. The MS diffusion coefficient

Ðij thus plays the role of an inverse friction coefficient between components i and j.

The MS approach accounts for thermodynamics and mass transfer separately such that

only the latter contribution is characterized by the MS diffusion coefficient. Because Ðij is

related to the chemical potential gradient, it cannot directly be measured in the laboratory.

However, the MS diffusion coefficient can well be sampled by molecular dynamics (MD)

simulation. It is usually calculated via equilibrium molecular dynamics (EMD) simulation

from velocity correlation functions with the Green-Kubo formalism or, alternatively, from the

mean square displacement with the Einstein formalism1,10. The thermodynamic contribution

is considered by the so-called thermodynamic factor Γ.

Eqs. (1) and (2) describe the same phenomenon so that a relation between both sets of

diffusion coefficients exists1. For binary mixtures, because there is only a single independent

MS and Fick diffusion coefficient, it is simply

Dij = Ðij · Γ , (3)

with

3


Γ = 1 + x1

(

∂ ln γ1∂x1

)

T,p

= 1 + x2

(

∂ ln γ2∂x2

)

T,p

, (4)

where γi stands for the activity coefficient of component i. The MS diffusion coefficient can

thus be transformed to the Fick diffusion coefficient and vice versa, if the thermodynamic

factor is known.

The Fick diffusion coefficient can either be determined directly with experimental meth-

ods, because it is related to the gradient of a measurable quantity, or estimated from molec-

ular simulation, theoretical, semi-empirical or empirical models. Predictive and empirical

approaches relate the mutual diffusion coefficients to pure fluid properties or simplify the in-

teraction between unlike molecules, which may lead to inaccurate approximations for many

liquid mixtures1. Most methods for the prediction of the composition dependence of the

Fick diffusion coefficient have been related theoretically and empirically to the diffusion co-

efficients at infinite dilution or to the self-diffusion coefficients in the form of composition

weighted averages. The classical interpolation methods by Darken11 and Vignes12 have been

extended to better consider non-idealities. Leffler and Cullinan13 as well as Carman and

Stein14 introduced the shear viscosity of the mixture and that of its neat components. The

concept of local composition according to the Wilson model was employed by Li et al.15

and Zhou et al.16 to improve Darken’s and Vignes’ models, respectively. Bosse and Bart17

proposed an excess Gibbs energy (GE) correction term for Vignes’ equation. Several authors

have introduced modifications of the thermodynamic factor18–23. Other predictive methods

rely on the knowledge of a variety of properties of the involved pure fluids and/or binary

adjustable parameters which are often not available24–32.

The Fick diffusion coefficient can be also calculated directly with molecular simulation

employing non-equilibrium MD methodologies7,33,34. However, these approaches are com-

plex, time-consuming and usually require very high concentration gradients, which make

them impractical to accurately sample the composition dependence of the Fick diffusion co-

efficient35. Recently, Nichols and Wheeler36 reported a novel Fourier correlation method to

calculate the Fick diffusion coefficient directly from systems in equilibrium. In the present

work, EMD simulation and the Green-Kubo formalism were preferred. One of the advan-

tages of this route is that all transport coefficients can be sampled simultaneously. On the

other hand, the thermodynamic factor is needed to obtain the Fick diffusion coefficient.

4


The thermodynamic factor is usually extracted from experimental VLE data or excess

enthalpy measurements37,38. For this purpose equations of state34,39, excess Gibbs energy

GE models or direct numerical integration20–22 can be employed. However, it can also be

obtained from molecular simulation. E.g., it can be estimated from the integration of the

radial distribution function (RDF) based on Kirkwood-Buff theory40–47. Another molecular

simulation approach is to determine the composition dependence of the chemical potential

using free energy perturbation methods, like Widom’s test particle insertion48,49, thermody-

namic integration50,51 or gradual insertion52. However, these approaches are challenging in

case of dense liquids consisting of strongly interacting molecules.

The advantage of the classical approach to determine the thermodynamic factor is the

good availability of experimental VLE data. However, this method should be employed

carefully, taking two important issues into account. First, the thermodynamic factor is sen-

sitive to the underlying thermodynamic model. In fact, different GE models may describe

experimental VLE data equally well, but yield different values for the thermodynamic fac-

tor38,53. Second, the thermodynamic factor determined with this approach corresponds to

thermodynamic conditions under which the underlying VLE data were measured, i.e. for a

given temperature the resulting thermodynamic factor may not correspond to the desired

pressure, but to the varying phase equilibrium pressure. In order to investigate the uncer-

tainty of the thermodynamic factor introduced by the GE model, the thermodynamic factor

was calculated here for all studied mixtures with three different GE models, i.e. Wilson54,

NRTL55 and UNIQUAC56.

This work relies on the capability of molecular modeling and simulation to predict the

diffusion coefficients of liquid mixtures. All binary systems that can be formed out of the

seven components methanol, ethanol, acetone, benzene, cyclohexane, toluene and carbon

tetrachloride (CCl4) were studied in a fully combinatorial manner, cf. Figure 1. The selection

of these mixtures was driven by the unusually good availability of experimental transport

data and by the presence, in many cases, of interesting thermodynamic and structural

characteristics. Table I provides an overview of the 20 systems that were investigated at

ambient conditions. The mixture methanol + cyclohexane was not considered because of its

miscibility gap. Previous simulation results for the Fick diffusion coefficient of methanol +

ethanol of our group52 were complemented here with more thorough simulations for the sake

of consistency. Present simulation results were compared, wherever possible, to experimental

5


data and to a set of predictive equations. Further, self-diffusion coefficients, shear viscosity

and thermal conductivity were also predicted.

FIG. 1. Graphical representation of the binary liquid mixtures studied in this work

TABLE I. Overview of the binary liquid mixtures studied in this work.

ethanol acetone benzene cyclo- toluene CCl4

hexane

methanol 1 2 3 a 4 5

ethanol 6 7 8 9 10

acetone 11 12 13 14

benzene 15 16 17

cyclohexane 18 19

toluene 20

Group I, Group II, Group III

a Presence of a miscibility gap at ambient conditions.

In the present work, rigid and non-polarizable molecular models were used, i.e. a simple

model class that is suitable to predict thermodynamic and structural properties of fluids,

including hydrogen-bonding, with a good accuracy, e.g. for water, methanol, ethanol, am-

monia, dimethylamine and some of their binary mixtures57–59. Further, both alcohol models

6


TABLE II. Present results for density, self-diffusion coefficient, shear viscosity and thermal con-

ductivity of the pure liquids at 298.15 K and 0.1 MPa. a

substance ρsim ρexp D0i,sim D

0i,exp η

0sim η

0exp λ

0sim λ

0exp exp.

mol L−1 mol L−1 10−9m2s−1 10−9m2s−1 10−4 Pa s 10−4 Pa s Wm−1K−1 Wm−1K−1 ref.

methanol 24.541 (6) 24.552 2.410 (2) 2.415 5.3 (4) 5.44 0.21 (3) 0.201 64–67

ethanol 17.129 (3) 17.046 0.974 (3) 1.075 10.87 (6) 10.82 0.190 (9) 0.160 68–71

acetone 13.537 (3) 13.511 4.538 (8) 4.77 3.0 (2) 3.07 0.162 (6) 0.156 72–74

benzene 10.283 (2) 10.300 2.239 (4) 2.204 6.1 (3) 6.14 0.123 (6) 0.141 75–78

cyclohexane 9.220 (1) 9.200 1.524 (4) 1.424 7.7 (4) 8.83 0.144 (5) 0.117 74,79,80

toluene 9.343 (1) 9.358 2.380 (4) 2.290 5.3 (3) 5.54 0.109 (7) 0.135 68,76,81,82

CCl4 11.148 (1) 11.183 1.323 (3) 1.305 7.6 (4) 8.90 0.114 (3) 0.10465,75,83,84

a The number in parentheses indicates the statistical uncertainty in the last given digit.

have successfully been tested in preceding work for the prediction of transport properties

in their ternary mixture with water and the according binary subsystems52. The mod-

els for methanol, ethanol, acetone and cyclohexane were taken from previous work of our

group60–63, whereas the molecular models for benzene, toluene and CCl4 were devised here.

These three models were developed starting from quantum mechanical calculations and a

subsequent optimization of the site-site distances and model parameters to experimental

VLE and self-diffusion coefficient data following a recently published methodology63. The

employed molecular models well reproduce the transport properties considered here under

the relevant thermodynamic conditions. Table II lists the simulation results for self-diffusion

coefficients, shear viscosity and thermal conductivity of the seven pure fluids. The exper-

imental values were reproduced with an average relative deviation (ARD) of 3%, 6% and

13%, respectively.

Mutual diffusion of binary liquid mixtures by molecular simulation has been the subject of

several publications. Not only simple mixtures of Lennard-Jones (LJ) spheres33,85–88, but also

more complex systems of hydrocarbons89 or hydrogen-bonding components90,91 have been

regarded in this sense using equilibrium and non-equilibrium MD simulations. Among the

binary mixtures considered in this work, benzene + cyclohexane has been the one that was

most widely studied by molecular simulation with respect to transport properties. Schaink

7


et al.92 and Hoheisel and Würflinger93 calculated Fick and self-diffusion coefficients, shear

viscosity and thermal conductivity using EMD and rigid molecular models. Zhang and

Müller-Plathe94 calculated the thermal and transport diffusion coefficients of this mixture

using reverse-NEMD and EMD methods with flexible all-atom molecular models95. Liu et

al.35 investigated the MS diffusion coefficient of the binary mixtures methanol + acetone

and acetone + CCl4 using EMD and rigid molecular models96. The thermodynamic factor

reported by Liu et al. was determined with the molecular simulation approach proposed

by Schnell et al.41,42. Wheeler and Rowley90 predicted the shear viscosity of methanol +

acetone, employing rigid molecular models and non-equilibrium simulation methods. Per-

era et al.97 also studied this mixture with molecular simulation techniques, however, only

time independent thermodynamic and structural properties were considered. The diffusion

coefficient at infinite dilution of the binary mixture cyclohexane + toluene as well as their

self-diffusion coefficients in their ternary mixture with n-hexane were predicted by Liu et

al.98,99. We are not aware of any other molecular simulation studies on transport properties

covering the remaining 16 binary mixtures studied here.

This paper is organized as follows: First, the simulation methodology is described. Sec-

ond, the results for the thermodynamic factor, the MS and the Fick diffusion coefficients

are presented for the 20 studied binary mixtures. The calculated Fick diffusion coefficient is

compared with experimental data and different predictive methods for mutual diffusion. The

observed behavior of the diffusion coefficients is analyzed based on the microscopic physical

structure of the mixtures as provided by radial distribution functions (RDF). Subsequently,

the predictions for self-diffusion coefficients, shear viscosity and thermal conductivity of the

mixtures are compared with the available experimental data. Finally, conclusions are drawn.

A detailed description of the new molecular models for benzene, toluene and CCl4 and the

technical simulation details are given in the Supporting Information.

II. MOLECULAR MODELS

Throughout this work, rigid and non-polarizable molecular models of united-atom type

were used. The models account for the intermolecular interactions, including hydrogen-bon-

ding, by a set of LJ sites and superimposed point charges, point dipoles or point quadrupoles

which may or may not coincide with the LJ site positions. The molecular models for

8


methanol, ethanol, acetone and cyclohexane were taken from prior work60–63, whereas the

models for benzene, toluene and CCl4 were developed here. For detailed information on

the molecular models the interested reader is referred to the Supplementary Material of this

work and to the original publications60–63.

To define a molecular model for a binary mixture on the basis of pairwise additive pure

substance models, only the unlike interactions have to be specified. In case of polar in-

teraction sites, this can straightforwardly be done by following the laws of electrostatics.

However, for the unlike LJ parameters there is no physically sound approach100 and combin-

ing rules have to be employed for predictions. Vrabec et al.101 have shown in a systematic

study on 267 binary mixtures that in many cases an adjustable binary parameter is nec-

essary to describe the VLE with a high accuracy. Thus, the use of such parameters may

be important to accurately cover the phase behavior of binary mixtures near liquid-liquid

phase separation, i.e. when Γ → 0. However, in this work, a strictly predictive route was

followed that exclusively relies on the pure fluid models. Thus, the interactions between LJ

sites of unlike molecules were specified by the Lorentz-Berthelot combining rules.

III. METHODOLOGY

A. Transport properties

Transport data were sampled by EMD simulation and the Green-Kubo formalism102,103.

This formalism establishes a direct relationship between a transport coefficient and the time

integral of the correlation function of the corresponding microscopic flux in a system in

equilibrium. The general Green-Kubo expression for an arbitrary transport coefficient γ is

given by

γ =1G

∫ ∞

0dt〈

Ȧ(t) · Ȧ(0)〉

, (5)

Therein, G is a transport property specific factor, A the related perturbation and Ȧ its time

derivative. The brackets denote the ensemble average. In case of the self-diffusion

coefficient, A(t) is the position vector of a given molecule at some time t and Ȧ(t) is its center

of mass velocity vector. In this way, the self-diffusion coefficient is related to the velocity

autocorrelation function. On the other hand, the shear viscosity is associated with the time

9


autocorrelation function of the off-diagonal elements of the stress tensor and the thermal

conductivity to the autocorrelation functions for the energy. The detailed expressions can

be found in the Supporting Information and in previous publications58,104.

The partial molar enthalpy, which is necessary to calculate the heat flow in a mixture105,

was determined in two steps. First, the molar enthalpy of the binary mixture h was calcu-

lated in the isobaric-isothermal (NpT ) ensemble over the entire composition range. Second,

a third order polynomial h = h(x1) was fitted by a least squares optimization to these data.

Finally, the partial molar enthalpy was calculated analytically by

hi = h + xj

(

∂h

∂xi

)

T,p

, (6)

where h represents the molar enthalpy of the mixture at the desired composition.

B. Thermodynamic factor

For the calculation of the thermodynamic factor, the composition dependence of the

activity coefficients is required. Following the Gibbs-Duhem equation, the excess Gibbs

energy GE of the binary mixture is related to the individual activity coefficients by106

kBT lnγ1 =

(

∂GE

∂n1

)

T,p

and kBT lnγ2 =

(

∂GE

∂n2

)

T,p

, (7)

where ni is the number of moles of component i in the mixture. Therefore, a mathematical

expression for GE as a function of composition is required to calculate the thermodynamic

factor from Eq. (3). In this work, three different well-established local composition models

were considered to estimate the uncertainty of the thermodynamic factor, i.e. Wilson54,

NRTL55 and UNIQUAC56. These classical GE models require adjustable binary parameters,

which were regressed to experimental VLE data assuming that the vapor phase is an ideal gas

and that the Poynting correction factor is negligible. The according regressions were carried

out with the RecPar tool from the Dortmund Data Bank107. The composition derivatives of

the activity coefficient models were evaluated analytically to determine the thermodynamic

factor as described by Taylor and Kooijman38.

The quality of the underlying experimental VLE data is crucial for the calculation of

the thermodynamic factor, therefore, only thermodynamically consistent VLE data should

10


be employed17. For this purpose both the point to point test108,109 and the integral or

area consistency test110,111 were applied. The VLE data sets with the largest number of

measured data points were chosen at or close to the target temperature of 298.15 K. For the

regarded mixtures no significant changes of the thermodynamic factor were observed for a

temperature interval of ±10 K around the target temperature for a given GE model. Further,

the thermodynamic factor was calculated for other VLE data sets, if available, in order to

confirm the consistency of the data. The parameters of the Wilson, NRTL and UNIQUAC

models determined for all studied mixtures and their ARD in terms of the vapor pressure

together with the respective experimental VLE data sources are given in the Supporting

Information.

C. Predictive equations

There are numerous correlations in the literature to determine the composition depen-

dence of the self-diffusion coefficients in mixtures. Four of these were assessed in this work:

The correlation by Carman and Stein14, which relates the self-diffusion coefficient in the

mixture Di with that of the pure liquid D0i , its shear viscosity η and that of the pure com-

ponent η0i , its correction proposed by Li et al.15, the correlations by Krishna and van Baten89

and by Liu et al.99, which relate the self-diffusion coefficients with their values at infinite

dilution and its mass fraction wi or molar fraction xi. These predictive equations are listed

in Table III.

Because the present simulations provide MS and self-diffusion coefficients simultaneously,

a comparison with the classical interpolation approach for the determination of the mutual

diffusion coefficients suggested by Darken11 is straightforward. Darken’s model takes only

self-correlations into account, resulting for binary mixtures to11

Ðij = xi · Dj + xj · Di· (8)

Hence, Darken’s model is applicable for ideally diffusing mixtures, where the contribution

of the velocity cross-correlations to the net velocity correlation function is negligible112.

Further, the limiting values of the mutual diffusion coefficient are given by the self-

diffusion coefficients

11


D∞ij = Dxj→1i and D

∞ji = D

xi→1j · (9)

Dxj→1i is the self-diffusion coefficient of component i when it is infinitely diluted in compo-

nent j. It is thus possible to obtain the mutual diffusion coefficient at infinite dilution by

extrapolation of the self-diffusion coefficients. The logarithmic average proposed by Vignes12

to predict the composition dependence of the MS diffusion coefficient

Ðij =(

D∞ij

)xj·

(

D∞ji

)xi, (10)

requires the knowledge of the diffusivities in the infinite dilution limit. Eq. (10) has a simple

form and it is easy to use so that it is widely applied. However, it may lead to large deviations

for mixtures containing associating components1.

There is a variety of methods attempting to improve Darken’s and Vignes’ interpolation

methods. Among the most widely used are those that relate mutual diffusion with the shear

viscosity, e.g. by Leffler and Cullinan13 and by Carman and Stein14. Li et al.15 modified

Darken’s equation using local volume fractions according to the Wilson model, whereas Zhou

et al.16 applied the same concept to Vignes’ equation. Bosse and Bart17 expanded Vignes’

equation by an excess Gibbs energy term. On the other hand, D’Agostino et al.20 and Zhu

et al.23 proposed to rise the thermodynamic factor by an empirical factor α = 0.64. Table III

gives an overview on the nine predictive equations that were assessed here.

The shear viscosity of liquid mixtures is in many cases very sensitive to association

effects among its components. Thus, the shape of the mole fraction dependence of the shear

viscosity can be linear for ideal mixtures or exhibit a maximum, a minimum or both for

highly non-ideal mixtures106. Predictive equations are usually interpolative and relate the

shear viscosity of the mixture with that of its neat components. Because a comprehensive

discussion of the different predictive shear viscosity equations is out of scope here, present

simulation results were only compared with experimental data and the widely used relation

by Grunberg and Nissan113

ln η = xi · ln η0i + xj · ln η0j + xi · xj · Gij , (11)

where the binary interaction parameter was set to zero, i.e. Gij = 0, which is termed as

behavior of the ideal mixture.

12


TABLE III. Predictive equations for the mole fraction dependence of self-diffusion, Maxwell-Stefan

and Fick diffusion coefficients of binary mixtures.

Self-Diffusion Coefficients ref.

Di = D0i · η

0i /η Carman-Stein

14

Di = D0i · (η

0i /η) ·

(

n0i /(

1 +(

n0i − 1)

xi))0.5 (a) Li et al. 15

Di =∑n

j=1 wj · Dxj→1i Krishna-van Baten

89

Di = 1/(∑n

j=1 xj/Dxj→1i ) Liu et al.

99

Mutual Diffusion Coefficients ref.

Ðij = xi · Dj + xj · Di Darken11

Ðij =(

D∞ij

)xj·

(

D∞ji

)xiVignes 12

Ðij =(

D∞ij · η0j

)xj·

(

D∞ji · η0i

)xi/η Leffler-Cullinan 13

Ðij =(

xj · D∞ij · η

0j + xi · D

∞ji · η

0i

)

/η Carman-Stein 14

Ðij = Di · φjj · v/vj + Dj · φii · v/vi(b) Li et al. 15

Ðij =(

D∞ij

)φjj ·v/vj·

(

D∞ji

)φii·v/vi (b) Zhou et al. 16

Ðij =(

D∞ij

)xj·

(

D∞ji

)xi· exp

(

−gE/(RT ))

Bosse-Bart 17

Dij = (xj · Di + xi · Dj) · Γ0.64 D’Agostino et al. 20

Dij = (xjj · Di + xii · Dj) · Γ0.64 (c) Zhu et al. 23

(a) n0i

= (η0j

· D∞ij

)/(η0i

· D0i)2

(b) φii = xi/(xi + xjΛij) and Λij = (vi/vj) exp (−∆λij/(RT )), where ∆λij is the Wilson parameter, vi is the partial

molar volume of component i and v = xivi + xjvj

(c) xii = xi/(xi + xjGji) and Gij = exp(−αij ∆gij/(RT )), where αij and ∆gij are NRTL parameters

Predictive equations for the thermal conductivity of mixtures are usually also based on

the interpolation of pure component data. In this work, the simple expression by Filippov114

was chosen for comparison

λ = wi · λ0i + wj · λ0j + 0.72 wi · wj · (λ

0j − λ

0i ), (12)

13


where λ and λ0i stand for the thermal conductivity of the mixture and of the pure component

i, respectively. This equation is expected to predict the thermal conductivity of binary

mixtures within 5%106.

IV. RESULTS AND DISCUSSION

In order to facilitate the discussion of the results obtained in this work, the 20 studied

mixtures were divided according to their mixing behavior into three groups. This catego-

rization was based on the maximum deviation of the thermodynamic factor from that of

the ideal mixture, i.e. less than 10% (group I), up to 45% (group II), and greater than 60%

(group III). Five mixtures constitute group I: methanol + ethanol, benzene + toluene, ben-

zene + CCl4, cyclohexane + CCl4 and toluene + CCl4. Group II contains seven mixtures,

i.e. methanol + acetone, ethanol + acetone, acetone + benzene, acetone + toluene, acetone

+ CCl4, benzene + cyclohexane and cyclohexane + toluene. The remaining eight mixtures,

methanol + benzene, methanol + toluene, methanol + CCl4, ethanol + benzene, ethanol

+ cyclohexane, acetone + cyclohexane, ethanol + toluene and ethanol + CCl4, form group

III, cf. Table I.

A. Density

The density specified in the simulations to determine the transport properties was ob-

tained from MD simulations in the NpT ensemble under ambient conditions and compared

with experimental data. For all studied mixtures, a very good agreement was found between

simulation and experiment with an ARD of 0.35%. The largest relative deviation of 1.5%

was found for the strongly non-ideal mixtures containing CCl4. The simulation results of

six selected mixtures are exemplarily shown in Figure 2 together with experimental values.

Among these mixtures a diverse density behavior is present, i.e. from “ideal” mixing to large

positive or negative excess volume. Tabulated numerical simulation data and a graphical

representation for all 20 mixtures are given in the Supporting Information.

14


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FIG. 2. Mole fraction dependence of the density of (a) methanol (1) + ethanol, (b) benzene (1) +

CCl4, (c) methanol (1) + acetone, (d) benzene (1) + cyclohexane, (e) ethanol (1) + toluene, (f)

acetone (1) + cyclohexane at 298.15 K and 0.1 MPa. Present simulation results (◦) are compared

with experimental data (+) as referred to in Table IV. The simulation results by Liu et al.35 (△)

are also shown.

B. Structure

To gain an insight into the underlying microscopic structure RDF were sampled for all

studied mixtures. The RDF gA−B(r) between like and unlike sites were calculated for the

pure liquids and mixtures at different compositions. Further, the running coordination

number NA−B(r) between the sites A and B, was determined from the integral of the RDF

NA−B(r) = 4π ̺∫ r

0r2gA−B(r) dr, (13)

where r is the distance from the reference site and ̺ is the bulk number density of site B.

The RDF of all studied mixtures are given in the Supporting Information. Exemplarily,

the RDF of six selected mixtures are shown in Figures 3 to 5. Relevant structural aspects

15


TABLE IV. Sources of experimental binary liquid mixture data on density ρ, self-diffusion coeffi-

cients Di, Fick diffusion coefficient Dij , shear viscosity η and thermal conductivity λ.

system ρ Di Dij η λ

no.

1 72,115–117 118 52 117,119 120

2 72,117,121 122 117,121 123

3 64,124 118,125 118,126 127–131

4 129,130,132,133 134,135 129–131,136

5 130,131,137,138 65,139 65,140,141 130,131,137,138,142–144 145,146

6 117,147 117,147 120

7 64,148,149 150 16,150–152 148,153–155

8 117,156 157,158 117,143,155,159 120

9 68 158 160,161 162

10 163 164 140,141,158,165,166 144,165 120

11 167,168 73,169 73,151,170 171–173

12 117,174,175 176 117,175

13 133 177 133,173,178,179 145

14 117 164 16,151,170,180 117,181 120

15 79,138,182–185 125,186,187 79,183,188–190 155,182,184,185,189,191–197 198

16 79,183,199 79,183 199–202 198

17 75,203 150,164 118,170,204 144,191,201,204,205

18 79,183,206 207 79,135,183 197,202,206 198

19 79,117,183,208,209 79,165,183,210 117,138,144,165,191,194,196,197,211 120

20 76,212 76 76 76,202,212,213 198

provided by the RDF analysis are discussed together with the results for mutual diffusion.

C. Thermodynamic factor

The thermodynamic factor was calculated with the Wilson, NRTL and UNIQUAC models

for all mixtures. The model parameters, determined from adjustments to carefully selected

VLE data, yield relative deviations in terms of the vapor pressure of below 2% in all cases. It

is well known that GE models that fit the same VLE data set equally well may lead to quite

different values for the thermodynamic factor38. In the present work, the thermodynamic

factor calculated with the different models differs by less than 2% for the mixtures in group

16


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FIG. 3. Selected radial distribution functions and the corresponding running coordination numbers

(inset) of methanol (1) + ethanol (left) and toluene (1) + CCl4 (right) at 298.15 K and 0.1 MPa

between (a) the oxygen and hydroxyl hydrogen sites of methanol gO−H, (b) the hydroxyl hydrogen

sites of methanol and ethanol gH−H, (c) the oxygen and hydroxyl hydrogen sites of ethanol gO−H,

(d) the methine sites of toluene gCH−CH, (e) the methine and chlorine sites of toluene and CCl4

gCH−Cl, (f) the chlorine sites of CCl4 gCl−Cl. Data for pure methanol, ethanol, toluene and CCl4

(· · · ) as well as the mixtures with x1 = 0.1 (–), 0.5 (–) and 0.9 mol mol−1 (–) are depicted.

I and by less than 4.5% for the mixtures in group II. Therefore, the related error is not

expected to exceed 5% for these groups. The consistency of the thermodynamic factor

calculated with the three different models breaks down for the mixtures in group III, which

can be considered as thermodynamically more challenging. For these mixtures, the difference

among the calculated thermodynamic factor data can even achieve one order of magnitude if

the thermodynamic factor approaches zero, being near liquid-liquid phase separation, which

may translate to significant errors when the Fick diffusion coefficient is determined. Figure 6

exemplarly shows the uncertainty of the thermodynamic factor introduced by the GE model,

17


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(inset) of methanol (1) + acetone (left) and acetone (1) + benzene (right) at 298.15 K and 0.1

MPa between (a) the oxygen and hydroxyl hydrogen sites of methanol gO−H, (b) the oxygen sites

of methanol and acetone gO−O, (c) and (d) the oxygen sites of acetone gO−O, (e) the oxygen and

methine sites of acetone and benzene gO−CH, (f) the methine sites of benzene gCH−CH. Data for

pure methanol, acetone and benzene (· · · ) as well as for the mixtures with x1 = 0.1 (–), 0.5 (–)

and 0.9 mol mol−1 (–) are depicted.

indicated as a shaded area, for six selected mixtures belonging to the three introduced groups.

As can be seen, the thermodynamic factor may change significantly, not only in magnitude

but also in shape when different GE models are used.

Further uncertainties of the thermodynamic factor may arise when different methodolo-

gies are used to obtain it. To illustrate this point, the thermodynamic factor sampled with

molecular simulation by Liu et al.35 is compared in Figure 6 with the present values for

methanol + acetone. Accordingly, a larger uncertainty of the thermodynamic factor could

18


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(inset) of methanol (1) + benzene (left) and ethanol (1) + cyclohexane (right) at 298.15 K and

0.1 MPa between (a) the oxygen and hydroxyl hydrogen sites of methanol gO−H, (b) the methyl

and methine sites of methanol and benzene gCH3−CH, (c) methine sites of benzene gCH−CH, (d)

the oxygen and hydroxyl hydrogen sites of ethanol gO−H, (e) the methyl and methylene sites of

ethanol and cyclohexane gCH3−CH2, (f) the methylene sites of cyclohexane gCH2−CH2. Data for

pure methanol, ethanol, benzene and cyclohexane (· · · ) as well as for the mixtures with x1 = 0.1

(–), 0.3 (–), 0.5 (–) and 0.9 mol mol−1 (–) are depicted.

be inferred, however, the calculations by Liu et al.35 are based on molecular models and

not on experimental data. On the other hand, Moggridge22 determined the thermodynamic

factor via a piecewise fit of experimental vapor pressure data in three regions followed by

numerical integration. The difference between the thermodynamic factor calculated here

and that by Moggridge22 suggests an increase of approximately 5% in terms of the thermo-

dynamic factor uncertainty for mixtures in group III, cf. Figure 6.

To select a model for the thermodynamic factor that is most suitable for the present

19


FIG. 6. Mole fraction dependence of the thermodynamic factor of (a) benzene (1) + CCl4, (b)

cyclohexane (1) + CCl4, (c) methanol (1) + acetone, (d) acetone (1) + CCl4, (e) methanol (1)

+ CCl4, (f) ethanol (1) + benzene. The shaded area represents the range of the results of the

three considered GE models. The thermodynamic factor obtained via Kirkwood-Buff integrals35

(△) and via direct numerical integration of experimental VLE data22 (�) is also shown.

simulation results, the Fick diffusion coefficient was determined with the data from the

three GE models and compared with the available experimental data. It was found that the

Fick diffusion coefficient calculated on the basis of the Wilson model yields, on average, the

smallest deviation from the experiment data for all regarded mixtures, i.e. 16%. Therefore,

the Wilson thermodynamic factor was chosen to be applied in the following.

D. Mutual diffusion coefficients

A set of nine predictive equations for the mutual diffusion coefficients based on Vignes’

and Darken’s models was assessed. For a fair comparison, all predictive equations were eval-

uated using the present thermodynamic factor from the Wilson model and its parameters.

20


In case of the Darken based equations, the values of the self-diffusion coefficients in the

mixture were taken from the best polynomial fit of the experimental values, if available, or

from present molecular simulation values otherwise. For the assessment of the predictive

diffusivity equations that include a shear viscosity correction13,14, a polynomial function of

third order was fitted to the available experimental data sets in order to minimize data

scatter inaccuracies. In case of Vignes’ based equations, which require the mutual diffu-

sion coefficients at infinite dilution D∞ij and D∞ji as an input, the average of the available

experimental data was employed.

The diffusion coefficients at infinite dilution were estimated from molecular simulation

results by extrapolation of the corresponding self-diffusion coefficients in the dilute region to

the limiting value at vanishing concentration. Note that infinite dilution diffusion coefficients

may also be obtained from the extrapolation of the MS diffusion coefficient, but self-diffusion

coefficient data were preferred because of their inherently lower statistical uncertainty. On

the other hand, semi-empirical relationships like the Wilke-Chang equation214 can also be

applied for such predictions. The diffusion coefficients at infinite dilution predicted by

simulation yield in general a better agreement with experimental data than those predicted

by the Wilke-Chang equation. The overall ARD from experimental data is 8% for the

simulation results compared with 19% for the Wilke-Chang equation, which is recommended

by Poling and Prausnitz106 for non-aqueous mixtures. Present numerical values together with

experimental data from the literature and predictions from the Wilke-Chang equation214 are

listed in the Supporting Information.

Among the nine tested predictive models, those by Zhu et al.23, Zhou et al.16 and Li et

al.15 were found to be the best three with overall ARD from polynomial fits to experimental

data for 19 binary mixtures of 12%, 13% and 15%, respectively. The model by D’Agostino et

al.20 follows with an overall ARD of slightly above 15%, whereas the remaining five models

yield an overall ARD between 19 and 25%.

For all 20 mixtures, the Fick diffusion coefficient was determined by molecular simulation

in combination with the Wilson thermodynamic factor for 11 different mole fractions, cov-

ering the entire composition range, and was compared with the available experimental data.

In general, a good agreement was found between simulative predictions and experimental

data, having an overall ARD of 16%. Figure 7 depicts the ARD of the present simulation

results for the Fick diffusion coefficient. All data are plotted and the numerical values are

21


listed in the Supporting Information. In the following, the results for the MS and Fick

diffusion coefficients are analyzed in detail.

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FIG. 7. Average relative deviation (ARD) of present simulation results for Fick diffusion coefficient,

shear viscosity and thermal conductivity from the best polynomial fit of the available experimental

data.

1. Group I

The mixtures methanol + ethanol and benzene + toluene exhibit an almost ideal behavior

because their components have very similar molecular structures. Thus, the MS diffusion

coefficient is almost a linear function of the mole fraction and corresponds to the Fick

diffusion coefficient because Γ ≃ 1, cf. Figure 8. In these simple cases, the interpolation

equations by Darken11 and Vignes12 are able to accurately predict the mutual diffusion

coefficient with deviations below 1%. The RDF of nearly ideal mixtures are expected to

be similar215, and indeed, the RDF sampled here for both mixtures show solvation shells

located at practically the same distances with similar magnitudes that undergo only small

changes when the composition is varied, explaining the linear behavior of the MS, Fick and

self-diffusion coefficients, cf. Figure 3.

22


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FIG. 8. Mole fraction dependence of the Maxwell-Stefan (left) and Fick (right) diffusion coefficients

of (a) benzene (1) + toluene, (b) benzene (1) + CCl4, (c) cyclohexane (1) + CCl4, (d) toluene (1) +

CCl4 at 298.15 K and 0.1 MPa. The simulation results for the Maxwell-Stefan diffusion coefficient

(•) are compared with the models by Darken11 (◦), Vignes12 (−−), Li et al.15 (− ⋄ −) and Zhou et

al.16 (–) based on present simulation data. The simulation results for the Fick diffusion coefficient

(•) are compared with experimental data (+) as referred to in Table IV. The models by Li et al.15

(−⋄−), Zhou et al.16 (–) and Zhu et al.23 (−⋄−) based on present simulation data are also shown.

The components of the remaining mixtures of this group, i.e. benzene + CCl4, cyclohex-

ane + CCl4 and toluene + CCl4, do not have similar molecular structures. However, they

behave nearly ideal mainly because their components have similar sizes and interactions,

which is reflected by their RDF. Exemplarily, Figure 3 shows the RDF between toluene-

toluene methine sites gCH−CH, CCl4-CCl4 chlorine sites gCl−Cl and the unlike sites gCH−Cl

23


at three different toluene mole fractions of 0.1, 0.5 and 0.9 mol mol−1. Here, the peaks of

the first and second solvation shells are found at similar distances, i.e. around 4 and 6 Å.

The running coordination numbers indicate a rather homogeneous structure and the small

changes of the RDF for different compositions suggest an insensitivity of the structure upon

mixing because of the similarity of the interaction sites. The homogeneous distribution of

the molecules in the mixture can clearly be seen in the snapshots shown in Figure 9. For

these mixtures, the mole fraction dependence of the MS diffusion coefficient is only slightly

above a linear course. The equations by Darken and Vignes are thus still adequate with

deviations of 3 and 4%, respectively. However, other predictive equations, e.g. by Li et al.15

or Zhou et al.16, achieve an even better accuracy. Consequently, the Fick diffusion coeffi-

cient shows only slight deviations from a linear behavior, cf. Figure 8. The ARD between

simulation and experiment is only 6% for this group of mixtures.

FIG. 9. Snapshots of (a) toluene (1) + CCl4, (b) acetone (1) + benzene, (c) ethanol (1) +

cyclohexane at 298.15 K, 0.1 MPa and three mole fractions x1 = 0.1 (left), 0.5 (center) and 0.9 mol

mol−1 (right). At mole fractions of 0.1 and 0.9 mol mol−1 the solvent molecules are not depicted

to improve visibility. The methyl and methylene groups are shown in orange, the methine groups

in brown, the oxygen atoms in red and the chlorine atoms in green.

24


2. Group II

Mixtures in this group show a moderate deviation from ideality in terms of Γ. In general,

the mole fraction dependence of the MS diffusion coefficient is not linear, but exhibits

a convex curvature which is centered near the equimolar composition in most cases, cf.

Figure 10. This curvature can be explained with the presence of some grade of association

between the molecules, like solvation. In case of the mixtures of acetone with benzene and

toluene, the RDF between the methyl and the oxygen sites of acetone gCH3−O shows a sharp

main peak, which indicates the presence of an ordered nearest-neighbor structure related

to the tendency of acetone molecules to associate into dimers216. Further, the RDF remain

alike when the composition is changed. Thus, the packing structure of the solvents changes

little with composition, indicating that acetone tends to stay in segregated pockets216, cf.

Figure 4. This fact can also clearly be observed in the simulation snapshots for acetone +

benzene shown in Figure 9 and explains the moderate decrease of the self-, Fick and MS

diffusion coefficients.

For mixtures of acetone with methanol and ethanol, the composition dependence of the

MS diffusion coefficient has a maximum at an alcohol mole fraction of around 0.2 mol

mol−1. This is related to the pronounced presence of alcohol association at this composition

as revealed by the corresponding RDF. The main peak of the RDF between the oxygen

and hydroxyl hydrogen sites of the alcohol gO−H, which is related to the hydrogen-bonding

structure, becomes sharper and higher when acetone is added, cf. Figure 4. The relatively

low main peak observed for the mixtures with a high alcohol content is a result from sta-

tistical standardization, i.e. more methanol molecules can be found in the far range of the

simulation volume217. However, the increase of the first neighbors peak with decreasing

methanol concentration is important, suggesting the enhancement of the self-associating

alcohol structure97. Further, shape and amplitude of the peaks do not change with compo-

sition, indicating that at low concentrations the alcohol molecules form clusters which are

surrounded by acetone218. These findings lead to the observed maximum of the MS diffusion

coefficient. Therefore, deviations of the MS diffusion coefficient from Vignes’ and Darken’s

interpolation methods become important.

The mole fraction dependence of the Fick diffusion coefficient exhibits a concave cur-

vature, showing a decrease of up to 30% from ideal diffusion behavior, depending on the

25


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FIG. 10. Mole fraction dependence of the Maxwell-Stefan (left) and Fick (right) diffusion coeffi-

cients of (a) methanol (1) + acetone, (b) ethanol (1) + acetone, (c) acetone (1) + benzene, (d)

benzene (1) + cyclohexane at 298.15 K and 0.1 MPa . The simulation results for the Maxwell-

Stefan diffusion coefficient (•) are compared with the models by Darken11 (◦), Vignes12 (−−), Li

et al.15 (− ⋄ −) and Zhou et al.16 (–) based on present simulation data. The simulation results

for the Fick diffusion coefficient (•) are compared with experimental data (+) as referred to in

Table IV and to the simulation results by Liu et al.35 (△). The models by Li et al.15 (− ⋄−), Zhou

et al.16 (–) and Zhu et al.23 (− ⋄ −) based on present simulation data are also shown.

thermodynamic factor. Here, the Darken based methods by Li et al.15 and D’Agostino et

al.20 achieve the best agreement with experimental values, the ARD is approximately 5%.

Present simulation results are in good agreement with experimental data for the mixtures

acetone + benzene, acetone + toluene, benzene + cyclohexane and cyclohexane + toluene

with ARD between 4 and 10%. Unfortunately, the present simulation results poorly predict

26


the Fick diffusion coefficient of the mixture acetone + CCl4 in the acetone-rich composition

range, resulting in an ARD of approximately 45% for this mixture. Here, the displacement

of the peaks of the unlike RDF gCH3−Cl towards larger distances suggests that intermolecular

interactions debilitate when CCl4 is added. This observation may explain the present over-

estimation of the Fick diffusion coefficient that increases the overall ARD of the simulation

data for this group to 15%.

To the best of our knowledge, there are no experimental Fick diffusion coefficient data

of the mixture ethanol + acetone. As mentioned above, the shape of the mole fraction

dependence of the MS diffusion coefficient indicates the presence of self-association at low

ethanol concentrations, similar to the one observed from experimental methanol + acetone

data, cf. Figure 10.

3. Group III

Most systems containing one alcoholic component exhibit a well pronounced peak in the

mole fraction dependence of the MS diffusion coefficient with a maximum located between

0.2 and 0.3 mol mol−1 of alcohol content, cf. Figure 11. This sharp increase of the MS

diffusion coefficient at low alcohol concentration is typical for this group of mixtures and is

related to cluster formation due to solute self-association219. The MS diffusion coefficient of

acetone + cyclohexane exhibits a pronounced peak centered at an acetone mole fraction of

0.4 mol mol−1, indicating significant self-association. The presence of clusters due to self-

association can also be inferred from the RDF. The sharpness and magnitude of the main

double peak, corresponding to the hydrogen-bonding structure of the alcohol gO−H, increases

significantly as the alcohol is depleted, cf. Figure 5. This fact suggests that the nearest

neighbor hydrogen-bonding structure at low alcohol concentration is more stable than that

of the pure alcohol. Moreover, the insensitivity of the location of the RDF peaks to a

change in composition indicates that the alcohol molecules conserve their local environment

of nearest neighbors of the neat liquid, supporting the thesis of the presence of strong alcohol

self-association in clusters, causing the maximum of the MS diffusion coefficient. The running

number of nearest oxygen neighbors around the hydrogen site has a well defined step-like

form with a plateau at approximately unity. It is noteworthy that upon mixing, the value at

which the plateau is reached remains almost constant at least up to an alcohol mole fraction

27


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FIG. 11. Mole fraction dependence of the Maxwell-Stefan (left) and Fick (right) diffusion coeffi-

cients of (a) methanol (1) + benzene, (b) methanol (1) + toluene, (c) ethanol (1) + benzene, (d)

acetone (1) + cyclohexane at 298.15 K and 0.1 MPa. The simulation results for the Maxwell-Stefan

diffusion coefficient (•) are compared with the models by Darken11 (◦), Vignes12 (−−), Li et al.15

(− ⋄−) and Zhou et al.16 (–) based on present simulation data. The simulation results for the Fick

diffusion coefficient (•) are compared with experimental data (+) as referred to in Table IV. The

models by Li et al.15 (−⋄−), Zhou et al.16 (–) and Zhu et al.23 (−⋄−) based on present simulation

data are also shown.

of 0.3 mol mol−1. For the lowest alcohol concentration, the coordination number is usually

lower, suggesting a smaller size of the alcohol clusters. This conclusion is supported by the

analysis of the RDF between the like sites of the solvents. Again, it can be observed that the

structure of the pure liquid remains almost unchanged for alcohol mole fractions of 0.1, 0.3

28


and 0.5 mol mol−1, but looses its long range behavior for 0.9 mol mol−1. Here, the structure

beyond the first solvation shell exhibits less stable long range structures. These findings are

supported by the analysis of the simulation snapshots of this type of mixtures, which back

up the theory of Pozar et al.216 that species are microsegregated and explain the low values

of the Fick diffusion coefficient, cf. Figure 9.

Usually, the maximum of the mole fraction dependence of the MS diffusion coefficient

coincides with the presence of a dominating minimum in the mole fraction dependence of

the Fick diffusion coefficient, which implies slower diffusion due to molecular association.

The differences in the location of the minimum can be traced back to the contribution

of the thermodynamic factor to the Fick diffusion coefficient. In general, the presence of

strong non-idealities in these mixtures is a challenge for predictive equations and molecular

simulation techniques because of the proximity of liquid-liquid phase separation at certain

compositions. The predictive equations by Zhu et al.23 and Zhou et al.16 yield the best results

for this group, with ARD of 19 and 22% from the experimental values, respectively. The

other predictive equations yield ARD above 30%. Present molecular simulation results for

the individual mixtures in this group deviate between 15 and 20% from experimental data,

except for the mixtures containing CCl4. The Fick diffusion coefficient in the CCl4-poor

region was strongly overpredicted, leading to an overall ARD of 29% for this group.

E. Self-diffusion coefficients

Self-diffusion coefficients of the individual species in their binary mixture were predicted

in this work with an estimated statistical uncertainty between 1 and 2%. Figure 12 shows

present simulation values for six selected mixtures together with experimental data and

the two predictive equations that were found to be the best in the present assessment.

According figures for all mixtures can be found in the Supporting Information together with

the numerical simulation data.

Four semi-empirical equations were assessed for the prediction of the self-diffusion coeffi-

cients. However, none of the them was found to be satisfactory for all considered mixtures.

The model by Liu et al.99, based on a development of the linear response theory, yields the

best overall agreement with the available experimental data for eleven mixtures. Its overall

ARD is 14% for Di and 5% for Dj, where i is the component with the lower molecular

29


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FIG. 12. Mole fraction dependence of the self-diffusion coefficients of (a) benzene (1) + CCl4, (b)

toluene (1) + CCl4, (c) acetone (1) + benzene, (d) benzene (1) + cyclohexane, (e) methanol (1)

+ benzene, (f) ethanol (1) + benzene at 298.15 K and 0.1 MPa. Present simulation results for

components (1) (•) and (2) (•) are compared with experimental data (+) as referred to in Table IV.

mass. Comparatively, present predictions by molecular simulation yield deviations from

experiment of 9% and 7%, respectively.

1. Group I

The self-diffusion coefficients of these mixtures vary almost linearly with the mole fraction

and the lighter component always shows a higher mobility, cf. Figure 12. As expected, the

simple interpolative equations by Krishna and van Baten89 and Liu et al.35 yield on average

predictions within 3% of the experimental values. The ARD of present simulation results

from experimental data are 6 and 3% for Di and Dj, respectively.

30


2. Group II

For mixtures belonging to this group, a departure from linearity for the self-diffusion

coefficients can be observed. Therefore, the ARD increases up to 6% for the interpolative

models by Krishna and van Baten89 and Liu et al.99, similarly to present molecular simulation

values. In case of acetone + benzene, acetone + toluene, acetone + CCl4 and benzene +

cyclohexane, the self-diffusion coefficient of the lighter component is higher in the whole

composition range, cf. Figure 12. Here, the influence of molecular size on molecular mobility

prevails over any association effect. In case of methanol + acetone, ethanol + acetone and

cyclohexane + toluene, molecular association overcomes the size effect such that the lighter

component propagates slower than the heavier one at least in one part of the composition

range.

3. Group III

As expected, the largest deviations between simulation and experiment were found for

mixtures in this group. For the alcoholic mixtures, the self-diffusion coefficient of the lower

mass component, i.e. the alcohol, is in a large composition range significantly smaller than

that of the heavier component. The slower movement of the lighter molecules in the mixture

can be explained by the presence of association among the alcohol molecules, which increases

their effective diameter and hinders their mobility. The strong self-association of the alcohol

molecules into clusters explains the sharp decrease on the alcohol self-diffusion coefficients

and may be inferred from the RDF, cf. Figure 5. Only molecular simulation is able to predict

this complex molecular behavior, while all four tested predictive equations fail. Among

the predictive equations, the relation by Li et al.15 yields the best agreement with the

experimental data. This model, which is highly dependent on shear viscosity data, predicts

deviations from linearity of self-diffusion in non-ideal mixtures, however, in many cases the

self-diffusion coefficients exhibit a strong overcorrection, yielding an ARD of 25% for Di.

The inadequacy of the model by Liu et al.99 for this kind of mixtures exposes its major

weakness, i.e., the assumption of negligible velocity cross-correlations.

Figure 12 shows the simulation results for the mole fraction dependence of the self-

diffusion coefficients of six selected mixtures compared with experimental values and the

31


results from the relations proposed by Li et al.15 and Liu et al.99. In case of the self-diffusion

coefficients of methanol + benzene and ethanol + benzene, only molecular simulation is able

to qualitatively correctly predict the composition dependence of the alcohol self-diffusion

coefficient.

F. Shear viscosity

Although the simulation results for the Fick diffusion coefficient may deviate strongly

from experimental data in particular cases, the predicted shear viscosity does usually agree

qualitatively and quantitatively with the experimental data, having an overall ARD of only

8% from the best polynomial fit of the experimental data given in Table IV. The ARD

for each mixture are represented graphically in Figure 7. Figure 13 exemplarily shows the

predicted shear viscosity for six selected mixtures together with experimental data and the

ideal mixture model by Grunberg and Nissan113, cf. Eq. (11). Analogous Figures for all

mixtures are given in the Supporting Information together with the numerical data.

1. Group I

As expected, the shear viscosity of these mixtures shows a nearly linear mole fraction

dependence so that it can be accurately predicted by the ideal mixture model by Grunberg

and Nissan191, cf. Figure 13. Present molecular simulation results deviate with an overall

ARD of 10% for this group, mainly because of the underestimation of the shear viscosity

of pure CCl4. If this offset would be corrected, a significant improvement of the agreement

between simulation and experiment is expected.

2. Group II

The shear viscosity of these mixtures shows some negative deviation from ideal behavior.

Especially for methanol + acetone, ethanol + acetone, benzene + cyclohexane and cyclo-

hexane + toluene, a significant negative deviation from the ideal behavior is observed. Here,

the shear viscosity of the mixture is lower than that implied by their individual components,

which is related to their differences in size and shape and to weak unlike intermolecular

32


interactions220. Present simulation work was able to predict the shear viscosity composition

dependence with an overall ARD of 7% for this group.

3. Group III

In this group of mixtures, positive and negative deviations from the ideal shear viscosity

behavior were found. The binary mixtures of methanol with benzene, toluene and CCl4

exhibit a positive deviation from ideality, cf. Figure 13. This is most pronounced for

methanol + benzene, where the shear viscosity is 50% higher than its ideal value around

equimolar composition, indicating strong intermolecular interactions, which is related to

the presence of alcohol self-association leading to clusters. The negative deviation from

ideal behavior found for the mixtures ethanol + benzene, ethanol + cyclohexane, ethanol +

toluene and acetone + cyclohexane is a consequence of the combination of the interacting and

non-interacting forces221 between unlike molecules. With the exception of methanol + CCl4,

present simulations are able to predict the composition dependence of the shear viscosity

well also for mixtures belonging to this group, where the overall ARD from experimental

values is 7%.

4. Data discrimination

One of the powerful applications of molecular simulation is data discrimination. There-

fore, inconsistencies found for several mixtures among experimental literature data are dis-

cussed in the following.

In case of methanol + benzene, five experimental data sets were found at ambient condi-

tions, three sets127–129 disagree with the other two130,131. While the former three suggest a

nearly ideal behavior, present simulation results predict a strongly non-linear shear viscosity

mole fraction dependence, which is consistent with the data sets by Rathore at al.130 and

Goyal et al.131, thus backing up their results. A similar observation was made for methanol

+ toluene, where four different experimental data sets are available. The composition de-

pendence of the shear viscosity of three of them has a convex shape129,131,136, whereas the

fourth one130 exhibits a concave shape, cf. Figure 13. In this case, the simulation results do

not show a strong deviation from ideality and agree well with the data sets by Han et al.129,

33


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FIG. 13. Mole fraction dependence of the shear viscosity of (a) methanol (1) + ethanol, (b)

benzene (1) + toluene, (c) acetone (1) + toluene, (d) benzene (1) + cyclohexane, (e) methanol

(1) + benzene, (f) methanol (1) + toluene at 298.15 K and 0.1 MPa. Simulation results for the

shear viscosity (•) are shown together with the viscosity of the ideal mixture (−−). Experimental

data: (+)117,119 for methanol + ethanol, (+)200,201, (+)202 for benzene + toluene; (+)178, (+)179,

(+)133 for acetone + toluene; (+)138,155,185,194,197, (+)202 for benzene + cyclohexane; (+)130, (+)131,

(+)129 for methanol + benzene; (+)130, (+)131, (+)129, (+)136 for methanol + toluene.

Goyal et al.131 and Wanchoo et al.136, reducing the credibility of the data set by Rathore

et al.130. For acetone + toluene, the experimental data set by Rajagopal et al.133 indicates

a quite significant deviation from ideality. Molecular simulation results and other experi-

mental data sets173,178,179 show a fairly linear behavior of the shear viscosity. For benzene

+ cyclohexane, benzene + toluene, cyclohexane + toluene and toluene + CCl4, the experi-

mental data by Pandey et al.202 are always higher than other data sets from the literature

and than the present simulation results. In case of benzene + cyclohexane and cyclohexane

+ toluene, again the data by Pandey et al.202 suggest a rather ideal behavior, where present

results and other experimental data indicate a negative deviation from ideality. In case of

34


toluene + CCl4, further inconsistencies among experimental data sets were found, present

results show a good agreement with the experimental data by Reddy et al.212.

G. Thermal conductivity

All present simulation results for the thermal conductivity are listed and plotted in the

Supporting Information. Figure 14 shows simulation results for six selected mixtures to-

gether with experimental data and the predictions from the Filippov relation114. The ARD

from experimental data for each mixture are shown in Figure 7.

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FIG. 14. Mole fraction dependence of the thermal conductivity of (a) benzene (1) + toluene, (b)

toluene (1) + CCl4, (c) methanol (1) + acetone, (d) acetone (1) + CCl4, (e) methanol (1) + CCl4,

(f) ethanol (1) + toluene at 298.15 K and 0.1 MPa. Present simulation results (•) are compared

with the predictions from the Filippov relation114 (−−) and experimental data (+) as referred to

in Table IV.

As expected, for all studied mixtures, the mole fraction dependence of the thermal con-

ductivity does not exhibit significant deviations from the simple interpolative relation by

35


Filippov114. This behavior is usually correctly predicted by the present simulations, cf. Fig-

ure 14. Nonetheless, in the case of methanolic mixtures with ethanol, acetone and benzene,

the predicted thermal conductivity may show a strong scatter and overestimation in the

methanol-rich composition range that is related to the high noise to signal ratio of the ther-

mal conductivity autocorrelation function of both alcohols and to the offset for the pure

fluids. Nonetheless, the overall ARD is only 11% for 14 mixtures for which experimental

data are available.

V. CONCLUSIONS

This work is aimed at the understanding of transport property behavior in liquid mixtures,

considering diffusion coefficients, shear viscosity and thermal conductivity. 20 binary liquid

mixtures were studied on the basis of classical rigid force fields by molecular dynamics in

a strictly predictive way. It was shown that these properties can be predicted from little

sophisticated molecular models with a good accuracy.

Values for the MS diffusion coefficient were sampled directly from equilibrium simulations

with the Green-Kubo formalism, whereas the thermodynamic factor was obtained from

selected experimental VLE data. The uncertainty of the thermodynamic factor was assessed

employing three different GE models: Wilson, NRTL and UNIQUAC. In this way, it was

estimated that the corresponding uncertainty does not to exceed 5% for mixtures with up to

moderate deviations from the ideal behavior. For strongly non-ideal mixtures, it was found

that the thermodynamic factor determined with different GE models may vary even by an

order of magnitude, if the value of the thermodynamic factor approaches zero, i.e. near

phase separation. The best results for the Fick diffusion coefficient were obtained with the

thermodynamic factor based on the Wilson model with an overall ARD from experimental

data of 16%. For the mixtures not deviating significantly from the ideal mixture behavior,

the achieved overall ARD was usually below 10%, whereas for the most challenging strongly

non-ideal mixtures, ARD between 15 and 20% were found. Further, an excellent qualitative

agreement for the composition dependence of the Fick diffusion coefficient was obtained for

the majority of the studied mixtures. Exceptions are mixtures of an alcohol and CCl4, where

the Fick diffusion coefficient was strongly overestimated in the alcohol-rich region.

In addition, nine different predictive relations for the mutual diffusion coefficients were

36


tested against the available experimental data. The predictive equations based on local

composition models, i.e. those by Li et al.15 and Zhou et al.16, as well as those including a

correction of the thermodynamic factor, i.e. by D’Agostino et al.20 and Zhu et al.23, yield the

best results with overall ARD from 12 to 15%. Further, four different predictive expressions

for the self-diffusion coefficients in the mixture were assessed. The method by Liu et al.99

was found to be the one with the lowest overall ARD from the available experimental values.

For the highly non-ideal mixtures considered here, it was found that the equation by Liu

et al.99 is not able to predict the strong decrease of the self-diffusion coefficient of the most

polar substance in the mixture. The only predictive method, based on physical arguments,

that was able to predict this extreme behavior of the self-diffusion coefficients is molecular

simulation.

The EMD method employed here yields the MS and self-diffusion coefficients as well as

the shear viscosity and thermal conductivity directly from one simulation run. Therefore,

the simulation results for shear viscosity and thermal conductivity were also compared with

the available experimental values. An overall ARD of only 8% was found for the shear

viscosity, whereas the ARD for the thermal conductivity is 11%.

The microscopic structure of the studied mixtures was analyzed thoroughly. The RDF

of mixtures that exhibit a nearly ideal behavior, albeit their components may be quite

different, have similar distances and characteristics of the nearest-neighbors solvation shells.

The similarity of their intramolecular interactions explains their almost ideal behavior. The

RDF of the mixtures containing an alcohol and a less polar component suggest strong alcohol

self-association, the presence of clusters and, in many cases, microheterogeneity. These are

related to a sharp decrease of the self-diffusion coefficient of the alcohol in the mixture and

to low values of the Fick diffusion coefficient.

SUPPLEMENTARY MATERIALS

A detailed description and parameters of the new molecular models for benzene, toluene

and CCl4 are presented together with the calculated VLE and transport properties in com-

parison to the corresponding reference equations of state or experimental data. The Simu-

lation methodology is explained and details of the carried out simulations are reported. GE

model parameters for the Wilson, NRTL and UNIQUAC models for all mixtures are given.

37


The obtained diffusion coefficients at infinite dilution are listed for all mixtures compared

to the Wilke-Chang equation an experimental data. Tabulated numerical simulation data

and graphical representations for the density, Fick and MS diffusion coefficients, self-diffusion

coefficients, shear viscosity and thermal conductivity for all mixtures are given. The RDF

of all studied mixtures are

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