Mutual diffusion of binary liquid mixtures containing methanol, ethanol

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 tetrachloride 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 dynamics 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 equilibrium 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 coefficients 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 prediction

a)

Supplementary Materials available.

b)

Electronic mail: [email protected]

1

Mutual diffusion of binary liquid mixtures 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 properties 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 . Because of these microscopic considerations, force field-based simulation methods can be used for the understanding and interpretation of experimental results, to obtain predictive estimates 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

Mutual diffusion of binary liquid mixtures 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 = −ρ

n−1 X

Dij ∇xj ,

(1)

j=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 X

∇µi xj (ui − uj ) =− , Ðij kB T j6=i=1

(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 , with 3

(3)

Mutual diffusion of binary liquid mixtures

Γ = 1 + x1

∂ ln γ1 ∂x1

!

= 1 + x2 T,p

∂ ln γ2 ∂x2

!

,

(4)

T,p

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 methods, because it is related to the gradient of a measurable quantity, or estimated from molecular simulation, theoretical, semi-empirical or empirical models. Predictive and empirical approaches relate the mutual diffusion coefficients to pure fluid properties or simplify the interaction 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 coefficients 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 complex, time-consuming and usually require very high concentration gradients, which make them impractical to accurately sample the composition dependence of the Fick diffusion coefficient35 . 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 advantages 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

Mutual diffusion of binary liquid mixtures 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 , thermodynamic 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 sensitive to the underlying thermodynamic model. In fact, different GE models may describe experimental VLE data equally well, but yield different values for the thermodynamic factor38,53 . 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 uncertainty 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

Mutual diffusion of binary liquid mixtures 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

ethanol

2

3

a

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

acetone benzene cyclohexane 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, ammonia, 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 conductivity of the pure liquids at 298.15 K and 0.1 MPa. substance

0 Di,sim

0 Di,exp

a

0 ηsim

0 ηexp

mol L−1

mol L−1 10−9 m2 s−1 10−9 m2 s−1 10−4 Pa s 10−4 Pa s Wm−1 K−1 Wm−1 K−1 ref.

exp.

5.44

0.21 (3)

0.201

64–67

10.87 (6) 10.82

0.190 (9)

0.160

68–71

3.07

0.162 (6)

0.156

72–74

(3)

6.14

0.123 (6)

0.141

75–78

7.7

(4)

8.83

0.144 (5)

0.117

74,79,80

2.290

5.3

(3)

5.54

0.109 (7)

0.135

68,76,81,82

1.305

7.6

(4)

8.90

0.114 (3)

0.104

65,75,83,84

24.541 (6) 24.552

2.410 (2)

2.415

ethanol

17.129 (3) 17.046

0.974 (3)

1.075

acetone

13.537 (3) 13.511

4.538 (8)

4.77

3.0

(2)

benzene

10.283 (2) 10.300

2.239 (4)

2.204

6.1

cyclohexane 9.220 (1) 9.200

1.524 (4)

1.424

toluene

9.343 (1) 9.358

2.380 (4)

11.148 (1) 11.183

1.323 (3)

a

λ0exp

ρexp

methanol

CCl4

λ0sim

ρsim

5.3

(4)

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 models 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 experimental 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

Mutual diffusion of binary liquid mixtures 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. Perera 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. Second, 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-bonding, 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

Mutual diffusion of binary liquid mixtures 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 interaction 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 combining 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 necessary 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 E 1 Z ∞ D˙ ˙ dt A(t) · A(0) , γ= G 0

(5)

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

Mutual diffusion of binary liquid mixtures 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 calculated 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

!

(6)

, T,p

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

kB T lnγ1 =

∂GE ∂n1

!

and

T,p

kB T lnγ2 =

∂GE ∂n2

!

,

(7)

T,p

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

Mutual diffusion of binary liquid mixtures 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 Di0 , its shear viscosity η and that of the pure component ηi0 , 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 selfdiffusion coefficients 11


x →1

Dij∞ = Di j x →1

Di j

and

∞ Dji = Djxi →1 ·

(9)

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 = Dij∞

xj

∞ · Dji

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 ηi0 + xj · ln ηj0 + 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 = Di0 · ηi0 /η Di = Di0 · (ηi0 /η) · n0i / 1 + n0i − 1 xi Di =

Pn

j=1 wj

x →1

· Di j

0.5

(a)

xj →1 ) j=1 xj /Di

Pn

Di = 1/(

Carman-Stein

14

Li et al.

15

Krishna-van Baten

89

Liu et al.

99

Mutual Diffusion Coefficients

ref. Darken

11

Vignes

12

/η

Leffler-Cullinan

13

Carman-Stein

14

Li et al.

15

Zhou et al.

16

Bosse-Bart

17

D’Agostino et al.

20

Zhu et al.

23

Ðij = xi · Dj + xj · Di

∞ Ðij = Dij

xj

∞ · Dji

∞ · η0 Ðij = Dij j

xi

xj

∞ · η0 · Dji i

xi

∞ · η 0 + x · D ∞ · η 0 /η Ðij = xj · Dij i i ji j

Ðij = Di · φjj · v/vj + Dj · φii · v/vi

∞ Ðij = Dij

∞ Ðij = Dij

φjj ·v/vj

∞ · Dji

xj

∞ · Dji

xi

φii ·v/vi

(b)

(b)

· exp −gE /(RT )

Dij = (xj · Di + xi · Dj ) · Γ 0.64

Dij = (xjj · Di + xii · Dj ) · Γ 0.64

(c)

(a) n0 = (η0 · D ∞ )/(η0 · D 0 )2 i i ij j i (b) φ = x /(x + x Λ ) and Λ = (v /v ) exp (−∆λ /(RT )), where ∆λ is the Wilson parameter, v is the partial ii i i j ij ij i j ij ij i molar volume of component i and v = xi vi + xj vj (c) x = x /(x + x G ) and G = exp(−α ∆g /(RT )), where α and ∆g are NRTL parameters ii i i j ji ij ij ij ij ij

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 · (λ0j − λ0i ), 13

(12)

Mutual diffusion of binary liquid mixtures 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 categorization 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, benzene + 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


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 ̺

Z

0

r

r 2 gA−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 coefficients Di , Fick diffusion coefficient Dij , shear viscosity η and thermal conductivity λ. system

ρ

Di

Dij

η

λ

no. 1

72,115–117

2

72,117,121

3

64,124

4

129,130,132,133

5

130,131,137,138

6

117,147

7

64,148,149

8

118

118,125

65,139

150

52

117,119

120

122

117,121

123

118,126

127–131

134,135

129–131,136

65,140,141

130,131,137,138,142–144

145,146

117,147

120

16,150–152

148,153–155

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

76,202,212,213

198

76

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


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


FIG. 4. Selected radial distribution functions and the corresponding running coordination numbers (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 methodologies 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


FIG. 5. Selected radial distribution functions and the corresponding running coordination numbers (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 thermodynamic 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 evaluated using the present thermodynamic factor from the Wilson model and its parameters. 20

Mutual diffusion of binary liquid mixtures 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 Dij∞ and Dji 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, covering 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

Mutual diffusion of binary liquid mixtures 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


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 , cyclohexane + 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 toluenetoluene methine sites gCH−CH , CCl4 -CCl4 chlorine sites gCl−Cl and the unlike sites gCH−Cl 23

Mutual diffusion of binary liquid mixtures 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 coefficient 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

Mutual diffusion of binary liquid mixtures 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 statistical 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 composition, 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 curvature, showing a decrease of up to 30% from ideal diffusion behavior, depending on the 25


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

Mutual diffusion of binary liquid mixtures 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 overestimation 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 selfassociation 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


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

Mutual diffusion of binary liquid mixtures 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 coefficients. 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


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

Mutual diffusion of binary liquid mixtures 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 selfdiffusion coefficients of six selected mixtures compared with experimental values and the 31

Mutual diffusion of binary liquid mixtures 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 cyclohexane + 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

Mutual diffusion of binary liquid mixtures 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 discussed in the following. In case of methanol + benzene, five experimental data sets were found at ambient conditions, 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 dependence 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


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 experimental data sets173,178,179 show a fairly linear behavior of the shear viscosity. For benzene + cyclohexane, benzene + toluene, cyclohexane + toluene and toluene + CCl4 , the experimental 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

Mutual diffusion of binary liquid mixtures 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 together with experimental data and the predictions from the Filippov relation114 . The ARD from experimental data for each mixture are shown in Figure 7.

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 conductivity does not exhibit significant deviations from the simple interpolative relation by 35

Mutual diffusion of binary liquid mixtures Filippov114 . This behavior is usually correctly predicted by the present simulations, cf. Figure 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 thermal 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

Mutual diffusion of binary liquid mixtures 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 comparison to the corresponding reference equations of state or experimental data. The Simulation 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

Mutual diffusion of binary liquid mixtures 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 also shown.

ACKNOWLEDGMENTS This work was funded by the Deutsche Forschungsgemeinschaft (DFG) under the grant VR 6/11-1. The simulations were carried out on the national supercomputer hezelhen at the High Performance Computing Center Stuttgart (HLRS) within the project MMHBF2. The authors want to thank Anisha Garg and Alvira Swalin for conducting a large part of the simulation runs.

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47

Supplementary material to:

Mutual diusion of binary liquid mixtures containing methanol, ethanol, acetone, benzene, cyclohexane, toluene and carbon tetrachloride by

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

Thermodynamics and Energy Technology, University of Paderborn, 33098 Paderborn, Germany

A. New molecular models B. Simulation methodology C. Diusion coecient at innite dilution D. Plots for all considered mixtures E. Numerical results

5

5 5% 5 5 ! 5 #"

A. New molecular models The models account for the intermolecular interactions, including hydrogen-bonding, 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 potential energy ukl between two molecules k and l can be written as LJ

ukl (rklab ) =

LJ

Sk S l

4klab

a=1 b=1 Ske Sle

σklab rklab

12

−

σklab rklab

6

Ske Sle 1 qkc qld + 4πε0 rklcd c=1 d=1

1 μkc μld Qld μkc + Qkc μld · f1 (ω k , ω l ) + · f2 (ω k , ω l ) , + 3 4 4πε0 rklcd rklcd c=1 d=1

(1)

where rklab , klab, σklab are the distance, the LJ energy parameter and the LJ size parameter, respectively, for the pair-wise interaction between LJ site a on the molecule k and the LJ site b on molecule l. The vacuum permittivity is ε0, whereas qkc , μkc and Qkc denote the point charge magnitude, the dipole and the quadrupole moments of the electrostatic interaction site c on molecule k. The expression f (ωk , ωl ) stands for the dependence of the electrostatic interactions on the orientations ωk and ωl of the molecules k and l1. The summation limits SkLJ and Ske indicate the number of LJ and electrostatic sites, respectively. Molecular models for benzene and toluene were obtained with the parameterization procedure proposed by Muñoz-Muñoz et al. 2. Benzene was modeled by six LJ sites with superimposed point quadrupole sites. Internal bond angles of 120◦ and dihedral angles of 0◦ were kept constant so that all sites were located in a plane. The quadrupole moment of benzene was equally distributed among all LJ sites to avoid artifacts when mixtures with small molecules are considered. Initially, the quadrupole sites were located at the carbon positions and their value was set to Q = QT /6, where QT is the quadrupole moment magnitude of the benzene model by Bonnaud et al. 3. The site-site distance between the LJ and quadrupole sites were modied in small steps as described in Ref.2 until a suitable combination of parameters was obtained. Finally, the reduced units method by Merker et al.4 was applied to obtain the denitive molecular model parameters, cf. Table I. The site-site distance, quadrupole moment and LJ parameters of the present benzene model agree well with the anisotropic benzene model by Bonnaud et al. 3. Toluene was modeled on the basis of the benzene model with an additional LJ site, representing the methyl group, located a distance δ from the ring. The LJ parameters of the methyl site were taken from Schnabel et al.5. All site-site distances were optimized until accurate results of the thermodynamic properties were obtained, following the procedures described by Muñoz-Muñoz et al.2 and Merker et al.4. The critical point of the present benzene model is located at T = 561 K, ρ = 4.01 mol l−1 and p = 5.0 MPa. The relative deviations with respect to experimental data 6 for critical temperature, critical density and critical pressure of benzene are -0.1%, +2.9% and +2.5%, respectively. The VLE properties predicted from the present benzene model exhibit an average deviation of 7.3% for vapor pressure, 1.0% for the saturated liquid density and 2.4% for the enthalpy of vaporization in the studied temperature range. In the temperature range between 280 and 333 K at ambient pressure, self-diusion coecient and shear viscosity obtained with the present benzene model deviate on average by 5.7% and 6.2% from the correlations of experimental data by Fischer and Weiss7, respectively. The thermal conductivity deviates on average by 11% from a correlation of experimental data8,9. Figure 1 show the calculated VLE and transport properties in comparison with the corresponding reference equations of state or experimental data. The critical temperature of the toluene model is 594 K, the critical density is 3.24 mol l−1 and the critical pressure is 4.4 MPa. The according relative deviations from experiment 10 are +0.5%, +2.1% and +6.4%, respectively. The calculated VLE properties of the present toluene model S2

TABLE I. Lennard-Jones and electrostatic parameters as well as spatial site conguration of the molecular models developed in this work.

site

x

y

z

σ

ε/kB

Q

q

Å

Å

Å

Å

K

DÅ

e

0

0

0

3.446

70.019

-1.0435

0

0

3.446

70.019

-1.0435

0

3.446

70.019

-1.0435

− − − − − −

benzene

CH(1) CH(2) CH(3) CH(4) CH(5) CH(6)

-1.6303 -2.4455

1.4119

-1.6303

2.8238

0

3.446

70.019

-1.0435

0

2.8238

0

3.446

70.019

-1.0435

0.8152

1.4119

0

3.446

70.019

-1.0435

toluene

CH(1) CH(2) CH(3) CH(4) CH(5) C CH3 CCl4 C Cl(1) Cl(2) Cl(3) Cl(4)

0

0

0

3.478

70.662

-1.0730

-1.6456

0

0

3.478

70.662

-1.0730

-2.4684

1.4251

0

3.478

70.662

-1.0730

-1.6456

2.8502

0

3.478

70.662

-1.0730

0

2.8502

0

3.478

70.662

-1.0730

0.8227

1.4251

0

3.823

9.996

-1.0730

2.0377

1.4251

0

3.641

0

0

0

2.81

121.255 12.37

1.18

1.18

1.18

3.25

212.6

-1.18

-1.18

1.18

3.25

212.6

-1.18

1.18

-1.18

3.25

212.6

1.18

-1.18

-1.18

3.25

212.6

− − − − − −

− − − − − − − -0.362 0.0905 0.0905 0.0905 0.0905

exhibit average deviations of 18.3% for the vapor pressure, 1.1% for the saturated liquid density and 3.7% for the enthalpy of vaporization in the studied temperature range. The ability of the present toluene model to predict self-diusion coecient, shear viscosity and thermal conductivity was tested in the temperature range between 273.15 and 350 K at ambient pressure. An ARD of 5% was found for the self-diusion coecient when compared with experimental data 1115 . Shear viscosity and thermal conductivity deviate both on average by approximately 10% from correlations of experimental data 9 , cf. Figure 2. Carbon tetrachloride ( CCl4 ) was modeled by ve LJ sites and ve atom-centered point charges, i.e. one per atom. The magnitude of the point charges, obtained via quantum chemical calculations with the Møller-Plesset 2 method and the 6-31G* basis set, were taken from the NIST database16 . The LJ parameters for the carbon atom, located in the center of the molecule, were taken from Merker et al. 17 . Thus, there were three model parameters left to be determined, two LJ parameters for the four identical chlorine sites and the site-site distance between carbon and chlorine. These three parameters were iteratively optimized by carrying out molecular simulations of the VLE in the range of 300 to 525 K and of liquid CCl4 at 298.15 K and comparing the results to experimental data of vapor pressure, saturated liquid density and self-diusion coecient at 298.15 K. Table I lists the molecular model parameters for benzene, toluene and CCl4 . The critical point of the present CCl4 model is located at T = 551 K, ρ = 3.59 mol l−1 and p = 4.3 MPa yielding relative deviations with respect to experimental data 18 of -1.0%, -5.7% and -0.8%. The predicted VLE properties exhibit average deviations of 24.3% for vapor pressure, 0.6% for saturated liquid density and 13.9% for the enthalpy of vaporization in the studied temperature range. The transport properties of the CCl4 model were calculated in the temperature range between 268 and 343 K at ambient pressure. Compared with correlations of experimental data an ARD of 9% was found for the self-diusion coecient 15,1922 , 27% for the shear viscosity 23,24 and 19% for the thermal conductivity 25,26 , cf. Figure 3. S3

Benzene =

>

?

@

A

B

FIG. 1. Temperature dependence of the vapor-liquid equilibrium and transport properties of benzene. Simulation results (•) for the saturated liquid density (a), vapor pressure (b) and enthalpy of vaporization (c) are compared with experimental data (+) and an equation of state () $ . (d) Simulation results for the self-diusion coecient (•) are compared with experimental data by Rathbun and Babb (+), McCool and Wolf (+), Hiraoka % (+), Graupner and Winter & (+), Falcone et al. ' (+) and the correlation of experimental data by Fischer and Weiss al.% (). (e) Simulation results for the shear viscosity ( •) are compared with a correlation of experimental data% (). (f) Simulation results for the thermal conductivity ( •) are shown together with a correlation of experimental data ' ().

5 "

Toluene =

>

?

@

A

B

FIG. 2. Temperature dependence of the vapor-liquid equilibrium and transport properties of toluene. Simulation results (•) for the saturated liquid density (a), vapor pressure (b) and enthalpy of vaporization (c) are compared with experimental data (+) and an equation of state () 10 . (d) Simulation results for the self-diusion coecient (•) are compared with experimental data by Pickup and Blum 14 (+), Wineld12 (+), Trepadus et al.13 (+), Krüger and Weiss11 (+) and Harris et al.15 (+). The self-diusion coecient from the molecular model by Nieto-Draghi et al.30,31 () is also shown. (e) Simulation results for the shear viscosity ( •) are compared with a correlation of experimental data 8 () and the molecular model by Nieto-Draghi et al. 30,31 (). (f) Simulation results for the thermal conductivity ( •) are shown together with a correlation of experimental data9 ().

5 #

Carbon tetrachloride

=

>

?

@

A

B

FIG. 3. Temperature dependence of the vapor-liquid equilibrium and transport properties of CCl4 . Simulation results (•) for the saturated liquid density (a), vapor pressure (b) and enthalpy of vaporization (c) are compared with experimental data (+) and an equation of state (). (d) Simulation results for the self-diusion coecient (•) are compared with experimental data by Fischer and Weiss al. % (+), McCool and Wolf (+), Collins and Mills , Harris et al.# (+) and Rathbun and Babb (+). (e) Simulation results for the shear viscosity ( •) are compared with experimental data by Luchinskii ! (+) and Ikeuchi et al. " (+). (f) Simulation results for the thermal conductivity (•) are shown together with experimental data by Rowley et al. ! (+), Fischer # (+) and Lei et al. $ (+).

5 $

B. Simulation methodology Green-Kubo equations The Green-Kubo expression for the self-diusion coecient molecule velocity autocorrelation function 1 Di = 3Ni

∞ 0

Di

is related to the individual

dt vik (t) · vik (0) ·

(2)

Here, vik (t) is the center of mass velocity vector of molecule k of component i at some time t and Ni is the number of molecules of component i. The brackets denote the canonical (N V T ) ensemble average. Eq. (2) is an average over all Ni molecules in the ensemble because all contribute to the self-diusion coecient. The self-diusion coecient that describes the mobility of species i in a mixture is also termed intradiusion coecient and in this work it is denoted by Di , whereas the self-diusion coecient of that species in its pure uid state is denoted by Di0 . The MS diusion coecient can be determined from the Onsager coecients Lij with the Green-Kubo expression!! 1 Lij = 3N

∞ 0

dt

Ni

vik (0)

k=1

·

Nj

vjl (t) ,

(3)

l=1

where N is the total number of molecules. In this context, the MS diusion coecient for binary mixtures is given by!!

Ðij = xxj Lii + xxi Ljj − Lij − Lji · i

j

(4)

Because of Onsager's reciprocity theorem, Lij = Lji for i = j . The shear viscosity η is associated with the autocorrelation function of the o-diagonal elements of the stress tensor Jpxy 1 η= V kB T

∞ 0

dt Jpxy (t) · Jpxy (0) ,

(5)

where V stands for the volume. The component Jpxy of the microscopic stress tensor Jp is given by!" Jpxy

=

N

1 x ∂u(rkl ) . − r y 2 k=1 l=k kl ∂rkl N

mk vkx vky

k=1

N

(6)

Here, k and l denote dierent molecules of any species. The upper indices x and y stand for the spatial vector components, e.g. for velocity vkx or site-site distance rklx . Eqs. (5) and (6) may directly be applied to mixtures. Five independent terms of the stress tensor Jpxy , Jpxz , Jpyz , (Jpxx − Jpyy )/2 and (Jpyy − Jpzz )/2 were considered to improve statistics !# . The thermal conductivity λ is given by the autocorrelation function of the elements of the microscopic heat ow Jqx 1 λ= V kB T 2

∞ 0

dt Jqx (t) · Jqx (0) .

(7)

In mixtures, energy and mass transport occur in a coupled manner, thus, the heat ow for a mixture of n components is given by!$ S7

Jq =

⎤ Nj n ij ⎦ · vki ⎣mik vki 2 + wki Iik wki + u rkl ⎡

n Ni 1

2

i=1 k=1

j=1 l=k

Ni n n n Ni ij i ∂u rkl 1 ij i ij − − rkl · vk · + w Γ h vki i k kl ij 2 i=1 j=1 k=1 l=k ∂rkl i=1 k=1 Nj

where

wki

is the angular velocity vector of molecule

k

,

(8)

i and Iik its matrix of angular Γ ij kl is the torque due to i and j denote the components of

of component

ij momentum of inertia. u(r ) is the intermolecular potential energy and kl the interaction between molecules the mixture,

hi

k

and

l.

The lower indices

is the partial molar enthalpy.

Thermodynamic factor In this work, three dierent local composition models were used to determined the thermodynamic factor, i.e.

Wilson

37

, NRTL

38

and UNIQUAC

39

.

These classical

GE

models require

adjustable binary parameters, which were regressed to experimental VLE data. Table II lists 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.

Simulation details Molecular dynamics simulations were performed with the program in two steps:

First, a simulation in the isobaric-isothermal (

N pT )

ms

2

57,58

. These were done

ensemble was carried out

to calculate the density and enthalpy at the desired temperature, pressure and composition. In the second step, a canonic (

NV T )

ensemble simulation was performed at this temperature,

density and composition to determine the transport properties.

Newton's equations of motion

were solved with a fth-order Gear predictor-corrector numerical integrator. was controlled by velocity scaling. The simulations contained

4000

The temperature

In all simulations, the integration time step was

0.877

fs.

molecules and were carried out in a cubic volume with periodic

boundary conditions where the cut-o radius was set to were considered using angle averaging

59

rc = 17.5

Å. LJ long range interactions

. Electrostatic long-range corrections were approximated

by the reaction eld technique with conducting boundary conditions

(RF = ∞).

Analogous

NV T

MD simulations with an extended cut-o radius that reached half of the edge length of the cubic simulation volume were employed to calculate the RDF. Here, starting from well-equilibrated congurations, production runs had a duration between 5 and

10 × 104

time steps.

N pT ensemble were equilibrated over 1.2 × 105 time steps, followed by a 5 production run over 5 × 10 time steps. In the N V T ensemble, the simulations were equilibrated 5 7 over 3 × 10 time steps, followed by production runs of 10 time steps. The self- and MS diusion The simulations in the

coecients, shear viscosity and thermal conductivity were calculated by Eqs. (2) to (7) with up to

4 × 104

independent time origins of the autocorrelation functions. The sampling length of the

autocorrelation functions was

17.5 ps for all mixtures.

That extensive length of the autocorrelation

functions was chosen such that long-time tails corrections were not necessary.

The separation

between the time origins was chosen such that all autocorrelation functions have decayed at least to

1/e

of their normalized value to achieve their time independence

predicted values were estimated with a block averaging method

S 8

61

.

60

.

The uncertainties of the

TABLE II. Wilson, NRTL and UNIQUAC parameters for the studied mixtures and

their average relative deviation (ARD) in terms of the vapor pressure. =

system no.

Wilson

Δλ12 J mol

−1

1

-211.62

2

Δλ21 J mol

−1

NRTL ARD %

Δg12 J mol

−1

488.35

0.03

-459.17

2575.3

-602.30

0.19

781.85

3

8571.3

713.70

0.61

4

8242.2

950.07

1.22

Δg21 J mol

−1

UNIQUAC

α12

ARD %

446.71 0.3

0.05

Δu12 J mol

−1

624.76

Δu21 J mol

−1

exp. ARD ref. %

-500.12 0.03

1107.3

0.3

0.19 -309.87

1743.7

0.19

3577.1

5941.1

0.507 0.14 -339.65

5112.4

1.14

4337.4

5581.2

0.507 0.93 -631.44

6003.8

1.83

5876.8 1139.5 5438.5 5903.4 5222.4 6185.2 1452.6 3743.8 1380.4 3705.3 166.70 1241.2 -812.62 235.47 1388.8 1605.7

0.461 0.3 0.514 0.448 0.546 0.425 0.3 0.506 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

5729.8 1023.6 3746.2 4008.4 4110.1 4745.3 1280.3 2418.9 1540.3 2496.9 699.97 699.43 -779.29 -246.85 677.32 -1190.6

0.79 1.29 0.73 1.26 1.97 0.58 0.12 1.18 0.39 0.40 0.04 0.18 0.09 1.07 0.16 0.05

5 9696.8 675.95 0.21 3230.3 6 1630.3 104.19 1.30 532.02 7 7249.1 598.82 0.26 2297.9 8 8850.1 1867.20 0.20 3627.0 9 6995.8 746.24 0.48 3203.7 10 9340.3 131.19 0.86 1466.1 11 1646.6 -378.92 0.14 -194.79 12 4693.8 1932.66 0.11 3093.0 13 2158.7 -436.10 0.43 285.85 14 3857.1 -719.74 0.39 -674.84 15 862.39 569.70 0.04 1233.7 16 30.846 26.749 0.17 -1066.4 17 -285.15 774.18 0.09 1328.4 18 686.93 607.92 1.17 1026.4 19 775.10 -314.39 0.16 -908.45 20 -878.39 1507.8 0.06 2277.9 ∗ ∗ = ARD = ∗ N |Pcalc − Pexp | /Pexp

5'

0.48 1.29 0.34 1.22 0.50 0.66 0.15 0.10 0.47 0.40 0.04 0.18 0.09 1.06 0.16 0.08

-336.46 -169.20 -625.93 -405.77 -787.93 -876.75 -773.98 -224.32 -790.73 -973.35 -177.97 -590.97 934.41 661.85 -518.62 1558.5

" " " "! 44 45 46 45 47 48 49 49 50 51 52 53 54 55 54 56

C. Diusion coecient at innite dilution TABLE III: Mutual diusion coecients at innite dilution. Present simulation values are listed together with experimental data and predictions with the Wilke-Chang equation. The number in parentheses indicates the statistical uncertainty in the last given digit. ∞ ∞ System D12 D21 Ref. Component 1 Component 2 10−9 m2 s−1 10−9 m2 s−1 Methanol Ethanol Simulation 1.20 (1) 2.04 (9) This work Simulation 1.25 (7) 2.08 (7) $ $! Simulation 1.73 1.83 Wilke-Chang 1.801 2.648 $ Experiment 1.23 2.04 Methanol Acetone Simulation 5.47 (2) 2.84 (2) This work Wilke-Chang 5.815 2.328 $" Experiment 4.86 2.59 Methanol Benzene Simulation 3.40 (2) 2.45 (1) This work Wilke-Chang 3.376 2.048 Experiment 3.700 2.500 $# $$ 3.82 2.66 $% 2.66 $& 2.56 $' 4.05 Methanol Toluene Simulation 3.76 (2) 2.30 (1) This work Wilke-Chang 4.073 1.807 % Experiment 3.69 2.48 $& 2.56 Methanol CCl4 Simulation 2.02 (2) 2.03 (1) This work Wilke-Chang 3.227 2.042 $# Experiment 2.45 2.25 2.635 2.291 % $& 2.25 % 2.30 %! 2.61 Ethanol Acetone Simulation 4.67 (2) 1.58 (1) This work Wilke-Chang 4.576 1.246 Ethanol Benzene Simulation 2.63 (3) 1.40 (1) This work Wilke-Chang 2.656 1.096 $# Experiment 2.90 1.75 $$ 3.30 1.97 $% 1.88 $& 1.81 Continued on next page

5

TABLE III continued from previous page System Substance 1 Ethanol

Substance 2 Cyclohexane

Ethanol

Toluene

Ethanol

CCl4

∞ D12

∞ D21

10−9 m2 s−1

10−9 m2 s−1

Simulation Wilke-Chang Experiment

1.88 (1) 1.917 2.950 1.103

Simulation Wilke-Chang Experiment Simulation Wilke-Chang Experiment

3.18 (3) 3.205 3.12 1.86 (2) 2.540 1.90 2.86 1.95

Acetone

Benzene


2.64 (1) 2.335 2.75

Acetone

Cyclohexane


Acetone

Toluene

Acetone

CCl4


2.28 (2) 1.686 2.745 2.22 2.93 (3) 2.818 2.95 1.88 (1) 2.233 1.690 1.69

5

Ref.

1.24 (1) This work 2.208 1.706 % 1.503 %" %# 1.53 1.30 (1) This work 0.967 %" 1.74 1.11 (1) This work 1.093 %" 1.47 %$ 1.49 %!

%% 1.50 %& 1.45 4.14 (2) This work 3.581 %' 4.12 & 4.28 $% 4.25 & 4.16 3.64 (2) This work 3.181 3.564 % & 4.06 3.85 (2) This work 3.123 &! 3.70 3.51 (1) This work 3.529 3.480 $# &" 3.57 Continued on next page

TABLE III continued from previous page System Substance 1 Benzene

Substance 2 Cyclohexane

Benzene

Toluene

Benzene

CCl4



Cyclohexane

Toluene

Simulation Simulation Wilke-Chang Experiment

Cyclohexane

CCl4


Toluene

CCl4


5

∞ D12

∞ D21

10−9 m2 s−1

10−9 m2 s−1

1.88 (1) 1.483 1.88 1.876 1.880 1.896 1.975 1.885 1.92 1.95 1.90 1.874 2.55 (2) 2.480 2.545 1.59 (1) 1.965 1.42 1.45 1.41 2.24 (1) 2.73 2.202 2.420 2.228 1.31 (1) 1.745 1.265 1.275 1.283 1.47 (4) 1.734 1.479 1.404

Ref.

2.01 (2) This work 1.825 %' 2.09 2.101 &# 2.104 &$ 2.090 &% 2.052 && &' 2.07 $% & ' '

2.10 (2) This work 1.813 1.847 &% 1.92 (1) This work 2.049 &" 1.91 $% '

1.79 (9) This work '! 2.10 1.309 1.569 &% 1.323 '" 1.54 (9) This work 1.479 1.486 %% 1.486 &% 1.481 '# 2.11 (9) This work 2.472 2.143 '$ %!

D. Plots for all considered mixtures

5 !

Methanol + Ethanol

=

>

?

@

A

B

C

FIG. 4. Results for methanol (1) + ethanol at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of methanol ( •) and ethanol (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−). 5 "

=

>

?

@

B

A

FIG. 5.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of methanol (1) + ethanol at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of methanol gO−H (a,b), the hydroxyl hydrogen sites of methanol and ethanol gH−H (c,d) and the oxygen and hydroxyl hydrogen sites of ethanol gO−H (e,f). Data for pure methanol and ethanol ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 #

Methanol + Acetone

>

=

@

A

?

B

C

FIG. 6. Results for methanol (1) + acetone at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of methanol ( •) and acetone (•) are compared with the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity (•) are shown together with the viscosity of the ideal mixture ( −−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and with experimental data (+). Simulation results by Liu et al. are also given. 5 $

=

>

?

@

B

A

FIG. 7.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of methanol (1) + acetone at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of methanol gO−H (a,b), the oxygen sites of methanol and acetone gO−O (c,d) and the oxygen sites of acetone gO−O (e,f). Data for pure methanol and acetone ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 %

Methanol + Benzene

>

=

@

A

?

B

C

FIG. 8. Results for methanol (1) + benzene at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. The thermodynamic factor obtained via direct numerical integration! () is also shown. (c) Simulation results for the Maxwell-Stefan diusion coecient ( •) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of methanol ( •) and benzene (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−). 5 &

=

>

?

@

A

B

FIG. 9.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of methanol (1) + benzene at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of methanol gO−H (a,b), the methylene and methine sites of methanol and benzene gCH3−CH (c,d) and the methine sites of benzene gCH−CH (e,f). Data for pure methanol and benzene ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.3 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 '

Methanol + Toluene

>

=

?

@

A

B

C

FIG. 10. Results for methanol (1) + toluene at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of methanol ( •) and toluene (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5

=

>

?

@

A

B

FIG. 11.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of methanol (1) + toluene at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of methanol gO−H (a,b), the methylene and methine sites of methanol and toluene gCH3−CH (c,d) and the methine sites of toluene gCH−CH (e,f). Data for pure methanol and toluene ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.3 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5

Methanol + CCl

>

=

4

?

@

A

B

C

FIG. 12. Results for methanol (1) + CCl4 at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. The thermodynamic factor obtained via direct numerical integration! () is also shown. (c) Simulation results for the Maxwell-Stefan diusion coecient ( •) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of methanol ( •) and CCl4 (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5

=

>

?

@

A

B

FIG. 13.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of methanol (1) + CCl4 at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of methanol gO−H (a,b), the methylene and chlorine sites of methanol and CCl4 gCH3−Cl (c,d) and the chlorine sites of CCl4 gCl−Cl (e,f). Data for pure methanol and CCl4 (· · · ) as well as for the mixtures with x1 = 0.1 (), 0.3 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5

!

Ethanol + Acetone

=

>

?

B

@

A

C

FIG. 14. Results for ethanol (1) + acetone at 298.15 K and 0.1 MPa. (a) Simulation results for the density ( ◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental the models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of ethanol ( •) and acetone (•) are compared with the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity (•) are shown together with the viscosity of the ideal mixture ( −−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5

"

=

>

?

@

A

B

FIG. 15.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of ethanol (1) + acetone at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of ethanol gO−H (a,b), the oxygen sites of ethanol and acetone gO−O (c,d) and the oxygen sites of acetone gO−O (e,f). Data for pure ethanol and acetone ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 #

Ethanol + Benzene

=

>

?

@

A

B

C

FIG. 16. Results for ethanol (1) + benzene at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. The thermodynamic factor obtained via direct numerical integration! () is also shown. (c) Simulation results for the Maxwell-Stefan diusion coecient ( •) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of ethanol ( •) and benzene (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−). 5

$

>

=

?

@

A

B

FIG. 17.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of ethanol (1) + benzene at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of ethanol gO−H (a,b), the methylene and methine sites of ethanol and benzene gCH3−CH (c,d) and the methine sites of benzene gCH−CH (e,f). Data for pure ethanol and benzene ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.3 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5

%

Ethanol + Cyclohexane

=

>

?

@

A

B

C

FIG. 18. Results for ethanol (1) + cyclohexane at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of ethanol (•) and cyclohexane (•) are compared with the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture ( −−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5

&

=

>

?

@

B

A

FIG. 19.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of ethanol (1) + cyclohexane at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of ethanol gO−H (a,b), the methylene and methyl sites of ethanol and cyclohexane gCH3−CH2 (c,d) and the methyl sites of cyclohexane gCH2−CH2 (e,f). Data for pure ethanol and cyclohexane ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.3 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5

'

Ethanol + Toluene

=

>

?

@

B

A

C

FIG. 20. Results for ethanol (1) + toluene at 298.15 K and 0.1 MPa. (a) Simulation results for the density ( ◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of ethanol ( •) and toluene (•) are compared with the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity (•) are shown together with the viscosity of the ideal mixture ( −−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5 !

=

>

?

@

B

A

FIG. 21.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of ethanol (1) + toluene at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of ethanol gO−H (a,b), the methylene and methine sites of ethanol and toluene gCH3−CH (c,d) and the methine sites of toluene gCH−CH (e,f). Data for pure ethanol and toluene ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.3 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 !

Ethanol + CCl4

=

>

?

@

A

B

C

FIG. 22. Results for ethanol (1) + CCl4 at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. The thermodynamic factor obtained via direct numerical integration! () is also shown. (c) Simulation results for the Maxwell-Stefan diusion coecient ( •) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of ethanol ( •) and CCl4 (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5 !

=

>

?

@

A

B

FIG. 23.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of ethanol (1) + CCl4 at 298.15 K and 0.1 MPa between the oxygen and hydroxyl hydrogen sites of ethanol gO−H (a,b), the methylene and chlorine sites of ethanol and CCl4 gCH3−Cl (c,d) and the chlorine sites of CCl4 gCl−Cl (e,f). Data for pure ethanol and CCl4 (· · · ) as well as for the mixtures with x1 = 0.1 (), 0.3 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 !!

Acetone + Benzene

=

>

B

@

A

?

C

FIG. 24. Results for acetone (1) + benzene at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of acetone ( •) and benzene (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−). 5 !"

=

>

?

@

A

B

FIG. 25.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of acetone (1) + benzene at 298.15 K and 0.1 MPa between the oxygen and sites of acetone gO−O (a,b), the oxygen and methine sites of acetone and benzene gO−CH (c,d) and the methine sites of benzene gCH−CH (e,f). Data for pure acetone and benzene ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 !#

Acetone + Cyclohexane

=

>

B

A

@

?

C

FIG. 26. Results for acetone (1) + cyclohexane at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of acetone (•) and cyclohexane (•) are compared with the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture ( −−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−). 5 !$

=

>

?

@

A

B

FIG. 27.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of acetone (1) + cyclohexane at 298.15 K and 0.1 MPa between the oxygen and sites of acetone gO−O (a,b), the oxygen and methylene sites of acetone and cyclohexane gO−CH2 (c,d) and the methylene sites of cyclohexane gCH2−CH2 (e,f). Data for pure acetone and cyclohexane ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 !%

Acetone + Toluene

=

>

B

@

A

?

C

FIG. 28. Results for acetone (1) + toluene at 298.15 K and 0.1 MPa. (a) Simulation results for the density ( ◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of acetone ( •) and toluene (•) are compared with the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity (•) are shown together with the viscosity of the ideal mixture ( −−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5 !&

=

>

?

@

A

B

FIG. 29.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of acetone (1) + toluene at 298.15 K and 0.1 MPa between the oxygen and sites of acetone gO−O (a,b), the oxygen and methine sites of acetone and toluene gO−CH (c,d) and the methine sites of toluene gCH−CH (e,f). Data for pure acetone and cyclohexane ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 !'

Acetone + CCl4

=

>

@

B

A

?

C

FIG. 30. Results for acetone (1) + CCl4 at 298.15 K and 0.1 MPa. (a) Simulation results for the density ( ◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of acetone ( •) and CCl4 (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). Simulation results by Liu et al. are also given. 5 "

=

>

?

@

A

B

FIG. 31.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of acetone (1) + CCl4 at 298.15 K and 0.1 MPa between the oxygen sites of acetone gO−O (a,b), the oxygen and chlorine sites of acetone and CCl4 gO−Cl (c,d) and the chlorine sites of CCl4 gCl−Cl (e,f). Data for pure acetone and CCl4 (· · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 "

Benzene + Cyclohexane

=

>

@

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A

B

C

FIG. 32. Results for benzene (1) + cyclohexane at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of benzene (•) and cyclohexane (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5 "

=

>

?

@

B

A

FIG. 33.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of benzene (1) + cyclohexane at 298.15 K and 0.1 MPa between the methine sites of benzene gCH−CH (a,b), the methine and methylene sites of benzene and cyclohexane gCH−CH2 (c,d) and the methylene sites of cyclohexane gCH2−CH2 (e,f). Data for pure benzene and cyclohexane ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 "!

Benzene + Toluene

=

>

@

?

A

B

C

FIG. 34. Results for benzene (1) + toluene at 298.15 K and 0.1 MPa. (a) Simulation results for the density ( ◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of benzene ( •) and toluene (•) are compared with the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity (•) are shown together with the viscosity of the ideal mixture ( −−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5 ""

=

>

B

A

@

?

FIG. 35.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of benzene (1) + toluene at 298.15 K and 0.1 MPa between the methine sites of benzene gCH−CH (a,b), the methine sites of benzene and toluene gCH−CH (c,d) and the methine sites of toluene gCH−CH (e,f). Data for pure benzene and toluene ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 "#

Benzene + CCl4

=

>

?

@

A

B

C

FIG. 36. Results for benzene (1) + CCl4 at 298.15 K and 0.1 MPa. (a) Simulation results for the density ( ◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of benzene ( •) and CCl4 (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−). 5 "$

=

>

?

@

A

B

FIG. 37.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of benzene (1) + CCl4 at 298.15 K and 0.1 MPa between the methine sites of benzene gCH−CH (a,b), the methine and chlorine sites of benzene and CCl4 gCH−Cl (c,d) and the chlorine sites of CCl4 gCl−Cl (e,f). Data for pure benzene and CCl4 (· · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 "%

Cyclohexane + Toluene

=

>

@

?

A

B

C

FIG. 38. Results for cyclohexane (1) + toluene at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion

coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (−−) based on present simulation data are shown. Innite dilution simulation results by Liu et al. '! are also given. (e) Simulation results for the self-diusion coecients of cyclohexane ( •) and toluene (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity (•) are shown together with the viscosity of the ideal mixture ( −−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5 "&

=

>

?

@

B

A

FIG. 39.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of cyclohexane (1) + toluene at 298.15 K and 0.1 MPa between the methylene sites of cyclohexane gCH2−CH2 (a,b), the methylene and methine sites of cyclohexane and toluene gCH2−CH (c,d) and the methine sites of toluene gCH−CH (e,f). Data for pure cyclohexane and toluene ( · · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 "'

Cyclohexane + CCl4

=

>

@

?

A

B

C

FIG. 40. Results for cyclohexane (1) + CCl4 at 298.15 K and 0.1 MPa. (a) Simulation results for the density (◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of cyclohexane (•) and CCl4 (•) are compared with the models by Li et al. $# (−−) and Liu et al. () and . (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5 #

=

>

?

@

B

A

FIG. 41.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of cyclohexane (1) + CCl4 at 298.15 K and 0.1 MPa between the methylene sites of cyclohexane gCH2−CH2 (a,b), the methylene and chlorine sites of cyclohexane and CCl4 gCH2−Cl (c,d) and the chlorine sites of CCl4 gCl−Cl (e,f). Data for pure cyclohexane and CCl4 (· · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 #

Toluene + CCl4

=

>

?

@

A

B

C

FIG. 42. Results for toluene (1) + CCl4 at 298.15 K and 0.1 MPa. (a) Simulation results for the density ( ◦) are compared with experimental data. (b) Thermodynamic factor; the shaded area represents the range of the results of the three considered GE models. (c) Simulation results for the Maxwell-Stefan diusion coecient (•) are compared with the models by Darken '% (◦), Vignes'& (−−), Li et al.$# (− −) and Zhou et al.'' () based on present simulation data. (d) Simulation results for the Fick diusion coecient ( •) are compared with experimental data (+). The models by Li et al. $# (− −), Zhou et al.'' () and Zhu et al. (− −) based on present simulation data are also shown. (e) Simulation results for the self-diusion coecients of toluene ( •) and CCl4 (•) are compared with experimental data (+) and the models by Li et al. $# (−−) and Liu et al. (). (f) Simulation results for the shear viscosity ( •) are shown together with the viscosity of the ideal mixture (−−) and experimental data (+). (g) Simulation results for the thermal conductivity ( •) are compared with the predictions from the Filippov relation (−−) and experimental data (+). 5 #

=

>

?

@

A

B

FIG. 43.

Selected radial distribution functions (left) and the corresponding running coordination numbers (right) of toluene (1) + CCl4 at 298.15 K and 0.1 MPa between the methine sites of toluene gCH−CH (a,b), the methine and chlorine sites of toluene and CCl4 gCH−Cl (c,d) and the chlorine sites of CCl4 gCl−Cl (e,f). Data for pure toluene and CCl4 (· · · ) as well as for the mixtures with x1 = 0.1 (), 0.5 () and 0.9 mol mol−1 () are depicted. The inset shows the function r2 h(r), where h(r) = g(r) − 1.

5 #!

E. Numerical results TABLE IV: Density ρ, self-diusion coecients Di , shear viscosity η and thermal conductivity λ of 20 binary liquid mixtures at 298.15 K, 0.1 MPa and specied composition from present simulations. The number in parentheses indicates the statistical uncertainty given in the last digits. x1 mol mol−1

ρ mol L−1

methanol (1) + ethanol (2) 0.05 17.386 (5) 0.10 17.655 (5) 0.20 18.221 (5) 0.30 18.816 (4) 0.40 19.459 (4) 0.50 20.155 (4) 0.60 20.891 (4) 0.70 21.699 (4) 0.80 22.564 (4) 0.90 23.512 (3) 0.95 24.020 (3) methanol (1) + acetone (2) 0.05 13.851 (3) 0.10 14.178 (3) 0.20 14.882 (3) 0.30 15.658 (4) 0.40 16.514 (4) 0.50 17.466 (4) 0.60 18.537 (4) 0.70 19.746 (4) 0.80 21.116 (5) 0.90 22.704 (5) 0.95 23.582 (5) methanol (1) + benzene (2) 0.05 11.450 (2) 0.10 11.778 (2) 0.20 12.498 (2) 0.30 13.317 (2) 0.40 14.243 (3) 0.50 15.313 (3) 0.60 16.567 (3) 0.70 18.031 (4) 0.80 19.792 (4) 0.90 21.920 (5) 0.95 23.162 (5)

D1 −9 2 10 m s−1

D2 −9 2 10 m s−1

10−4 Pa

1.227 (8) 1.272 (6) 1.367 (5) 1.472 (5) 1.586 (4) 1.692 (4) 1.833 (4) 1.964 (3) 2.119 (3) 2.271 (4) 2.362 (4)

1.014 (8) 1.050 (5) 1.126 (5) 1.220 (4) 1.315 (4) 1.409 (4) 1.530 (4) 1.649 (4) 1.775 (3) 1.903 (3) 2.034 (3)

11.6 11.0 10.6 9.2 9.3 8.3 7.3 6.9 6.7 5.8 5.4

(4) (4) (4) (3) (3) (3) (3) (3) (3) (2) (2)

0.17 0.17 0.18 0.18 0.21 0.19 0.20 0.23 0.25 0.27 0.26

5.16 (2) 4.79 (2) 4.32 (1) 4.01 (1) 3.74 (1) 3.524 (9) 3.332 (8) 3.134 (8) 2.921 (7) 2.688 (6) 2.575 (7)

4.512 (8) 4.481 (7) 4.394 (8) 4.281 (8) 4.140 (8) 4.004 (8) 3.843 (8) 3.632 (8) 3.399 (9) 3.14 (1) 3.00 (2)

3.1 2.9 3.0 3.2 3.6 3.7 4.1 4.3 4.6 5.0 4.4

(2) (2) (2) (2) (2) (3) (3) (3) (3) (3) (4)

0.172 (5) 0.171 (6) 0.167 (7) 0.189 (9) 0.18 (1) 0.19 (1) 0.21 (1) 0.22 (2) 0.23 (2) 0.25 (2) 0.27 (3)

2.53 (2) 2.32 (1) 2.176 (9) 2.104 (7) 2.115 (7) 2.121 (6) 2.145 (6) 2.172 (6) 2.226 (6) 2.308 (6) 2.368 (6)

2.261 (4) 2.275 (4) 2.290 (4) 2.294 (5) 2.314 (5) 2.331 (5) 2.338 (5) 2.346 (6) 2.349 (7) 2.371 (9) 2.40 (1)

5.9 6.6 6.9 7.5 7.6 8.5 8.6 8.4 7.9 6.4 6.1

(3) 0.128 (5) (3) 0.132 (6) (3) 0.139 (7) (4) 0.15 (1) (4) 0.15 (1) (3) 0.18 (2) (3) 0.16 (1) (5) 0.17 (2) (3) 0.20 (2) (3) 0.21 (2) (3) 0.29 (3) Continued on next page

5 #"

η s

W

λ m−1 K−1

(1) (1) (1) (2) (2) (2) (2) (2) (2) (3) (3)

x1 mol mol−1

TABLE IV continued from previous page

ρ mol L−1

methanol (1) + toluene (2) 0.05 9.635 (2) 0.10 9.950 (2) 0.20 10.650 (2) 0.30 11.459 (2) 0.40 12.404 (2) 0.50 13.515 (2) 0.60 14.852 (3) 0.70 16.482 (3) 0.80 18.515 (3) 0.90 21.114 (4) 0.95 22.706 (5) methanol (1) + CCl4 (2) 0.05 10.567 (2) 0.10 10.887 (2) 0.20 11.586 (2) 0.30 12.388 (2) 0.40 13.308 (3) 0.50 14.388 (3) 0.60 15.659 (4) 0.70 17.188 (3) 0.80 19.070 (5) 0.90 21.439 (4) 0.95 22.885 (3) ethanol (1) + acetone (2) 0.05 13.675 (3) 0.10 13.819 (3) 0.20 14.119 (3) 0.30 14.436 (3) 0.40 14.763 (3) 0.50 15.106 (3) 0.60 15.465 (3) 0.70 15.845 (3) 0.80 16.246 (3) 0.90 16.670 (3) 0.95 16.894 (3) ethanol (1) + benzene (2) 0.05 11.332 (2) 0.10 11.524 (2) 0.20 11.948 (2) 0.30 12.408 (2) 0.40 12.905 (2) 0.50 13.459 (2) 0.60 14.058 (2) 0.70 14.713 (3) 0.80 15.442 (2) 0.90 16.243 (3) 0.95 16.669 (3)

D1 −9 10 m2 s−1

D2 −9 10 m2 s−1

η

10−4 Pa

s

W

λ m−1 K−1

2.64 (2) 2.41 (1) 2.265 (9) 2.208 (7) 2.183 (7) 2.146 (6) 2.164 (6) 2.189 (6) 2.226 (6) 2.303 (6) 2.364 (6)

2.399 (4) 2.387 (5) 2.396 (5) 2.384 (5) 2.366 (5) 2.340 (5) 2.343 (5) 2.295 (6) 2.267 (7) 2.238 (8) 2.25 (1)

5.2 5.0 5.4 5.2 5.0 5.1 5.4 5.4 5.8 5.5 5.5

(3) (3) (2) (2) (3) (3) (3) (2) (2) (2) (3)

0.119 (5) 0.126 (6) 0.135 (9) 0.135 (8) 0.11 (1) 0.14 (1) 0.15 (1) 0.16 (2) 0.15 (2) 0.19 (2) 0.24 (2)

1.61 (2) 1.57 (1) 1.598 (7) 1.644 (7) 1.695 (6) 1.766 (6) 1.808 (6) 1.899 (5) 1.990 (5) 2.170 (6) 2.284 (7)

1.354 (4) 1.369 (3) 1.394 (3) 1.428 (3) 1.455 (4) 1.496 (4) 1.543 (4) 1.617 (5) 1.692 (6) 1.835 (8) 1.92 (1)

7.1 7.4 7.1 7.1 6.6 6.6 6.2 6.5 5.7 5.7 5.9

(3) (4) (3) (3) (3) (2) (3) (4) (2) (4) (2)

0.108 (4) 0.123 (5) 0.107 (7) 0.125 (8) 0.115 (9) 0.14 (1) 0.14 (1) 0.16 (1) 0.18 (2) 0.19 (2) 0.22 (2)

4.09 (1) 3.75 (1) 3.26 (1) 2.877 (9) 2.545 (5) 2.268 (5) 1.990 (5) 1.724 (4) 1.470 (4) 1.216 (4) 1.100 (3)

4.453 (6) 4.344 (8) 4.115 (7) 3.851 (7) 3.566 (5) 3.284 (6) 2.968 (5) 2.644 (5) 2.285 (7) 1.925 (8) 1.745 (8)

2.9 2.7 3.3 3.6 3.9 4.3 5.1 6.3 7.1 8.3 9.9

(1) (2) (2) (2) (2) (2) (2) (3) (4) (4) (4)

0.158 (6) 0.164 (7) 0.165 (6) 0.161 (7) 0.174 (7) 0.184 (7) 0.174 (8) 0.178 (7) 0.179 (7) 0.179 (9) 0.175 (8)

2.02 (1) 1.849 (9) 1.662 (7) 1.557 (6) 1.479 (5) 1.388 (4) 1.320 (4) 1.250 (4) 1.162 (4) 1.068 (3) 1.027 (4)

2.245 (5) 2.256 (4) 2.228 (4) 2.185 (4) 2.122 (4) 2.042 (5) 1.948 (5) 1.829 (5) 1.694 (6) 1.513 (6) 1.436 (9)

5.7 5.5 5.9 6.4 6.8 7.0 7.5 8.1 8.7 10.0 10.4

5 ##


x1 mol mol−1


ρ mol L−1

D1 −9 10 m2 s−1

ethanol (1) + cyclohexane (2) 0.05 9.410 (2) 1.374 (9) 0.10 9.626 (2) 1.309 (8) 0.20 10.095 (1) 1.270 (6) 0.30 10.618 (2) 1.243 (5) 0.40 11.203 (2) 1.223 (5) 0.50 11.860 (2) 1.204 (4) 0.60 12.607 (2) 1.166 (4) 0.70 13.473 (2) 1.131 (4) 0.80 14.472 (3) 1.067 (4) 0.90 15.673 (2) 1.032 (3) 0.95 16.358 (4) 1.034 (5) ethanol (1) + toluene (2) 0.05 9.548 (1) 2.12 (2) 0.10 9.768 (1) 1.94 (1) 0.20 10.250 (2) 1.758 (7) 0.30 10.784 (2) 1.657 (6) 0.40 11.382 (2) 1.538 (5) 0.50 12.052 (2) 1.458 (5) 0.60 12.808 (2) 1.381 (5) 0.70 13.664 (2) 1.288 (4) 0.80 14.653 (3) 1.207 (4) 0.90 15.787 (4) 1.099 (3) 0.95 16.432 (4) 1.038 (4) ethanol (1) + CCl4 (2) 0.05 10.460 (2) 1.30 (2) 0.10 10.667 (2) 1.238 (7) 0.20 11.104 (2) 1.172 (6) 0.30 11.590 (2) 1.157 (5) 0.40 12.128 (2) 1.131 (4) 0.50 12.719 (2) 1.109 (4) 0.60 13.385 (3) 1.096 (4) 0.70 14.136 (3) 1.067 (3) 0.80 14.977 (3) 1.051 (4) 0.90 15.970 (3) 1.023 (3) 0.95 16.522 (3) 0.996 (4) acetone (1) + benzene (2) 0.05 11.249 (2) 2.71 (1) 0.10 11.350 (2) 2.780 (9) 0.20 11.557 (2) 2.939 (7) 0.30 11.781 (2) 3.096 (7) 0.40 12.012 (2) 3.255 (6) 0.50 12.243 (2) 3.442 (6) 0.60 12.487 (2) 3.629 (6) 0.70 12.739 (2) 3.842 (6) 0.80 12.995 (3) 4.060 (7) 0.90 13.260 (3) 4.300 (7) 0.95 13.402 (3) 4.419 (7)

D2 −9 10 m2 s−1

η

10−4 Pa

s

W

λ m−1 K−1

1.603 (3) 1.614 (3) 1.626 (2) 1.630 (3) 1.628 (4) 1.634 (4) 1.609 (4) 1.576 (5) 1.518 (4) 1.389 (5) 1.310 (7)

7.7 7.2 7.0 7.2 7.5 7.7 7.6 8.3 9.2 9.6 10.5

(3) (3) (2) (2) (4) (4) (4) (5) (3) (4) (4)

0.143 (4) 0.143 (4) 0.141 (5) 0.143 (5) 0.147 (5) 0.150 (5) 0.162 (6) 0.161 (7) 0.157 (7) 0.172 (8) 0.160 (8)

2.380 (8) 2.375 (5) 2.328 (5) 2.268 (5) 2.187 (5) 2.099 (5) 1.991 (5) 1.845 (5) 1.688 (5) 1.486 (7) 1.38 (1)

5.5 5.5 5.6 5.7 6.2 6.3 7.2 7.6 8.1 9.9 10.7

(2) (3) (2) (3) (2) (3) (4) (4) (5) (4) (4)

0.120 (4) 0.115 (4) 0.128 (5) 0.121 (5) 0.128 (5) 0.141 (7) 0.135 (7) 0.143 (7) 0.144 (8) 0.165 (8) 0.189 (9)

1.359 (5) 1.359 (3) 1.370 (3) 1.372 (3) 1.371 (3) 1.352 (3) 1.344 (4) 1.314 (4) 1.269 (4) 1.202 (5) 1.158 (8)

7.6 7.1 7.5 7.1 8.1 7.5 8.6 8.3 9.1 10.4 10.2

(3) (3) (2) (4) (2) (3) (3) (3) (4) (4) (4)

0.115 (4) 0.112 (4) 0.115 (4) 0.113 (5) 0.122 (5) 0.118 (5) 0.122 (6) 0.133 (6) 0.147 (6) 0.172 (7) 0.167 (8)

2.306 (4) 2.375 (5) 2.520 (5) 2.653 (5) 2.809 (6) 2.999 (6) 3.185 (7) 3.391 (8) 3.607 (9) 3.87 (1) 3.99 (2)

5.8 5.7 4.5 4.3 4.3 4.2 3.7 3.6 3.3 3.3 2.7

5 #$


x1 mol mol−1


ρ mol L−1

D1 −9 10 m2 s−1

acetone (1) + cyclohexane (2) 0.05 9.343 (2) 2.39 (2) 0.10 9.481 (2) 2.494 (9) 0.20 9.774 (2) 2.706 (7) 0.30 10.099 (2) 2.913 (7) 0.40 10.458 (2) 3.127 (7) 0.50 10.850 (2) 3.343 (6) 0.60 11.274 (2) 3.548 (7) 0.70 11.749 (2) 3.777 (6) 0.80 12.278 (3) 4.003 (7) 0.90 12.866 (3) 4.263 (7) 0.95 13.193 (3) 4.399 (9) acetone (1) + toluene (2) 0.05 9.489 (2) 2.97 (2) 0.10 9.641 (2) 3.00 (1) 0.20 9.962 (2) 3.096 (8) 0.30 10.305 (2) 3.217 (8) 0.40 10.673 (2) 3.355 (7) 0.50 11.068 (2) 3.485 (7) 0.60 11.488 (2) 3.659 (7) 0.70 11.943 (2) 3.840 (7) 0.80 12.434 (2) 4.036 (7) 0.90 12.963 (3) 4.273 (7) 0.95 13.244 (3) 4.395 (9) acetone (1) + CCl4 (2) 0.05 10.379 (2) 2.01 (1) 0.10 10.490 (2) 2.105 (8) 0.20 10.725 (2) 2.350 (7) 0.30 10.985 (2) 2.593 (6) 0.40 11.259 (2) 2.838 (6) 0.50 11.563 (2) 3.051 (6) 0.60 11.895 (3) 3.296 (6) 0.70 12.248 (3) 3.565 (6) 0.80 12.636 (3) 3.865 (7) 0.90 13.067 (3) 4.177 (7) 0.95 13.299 (3) 4.347 (7) benzene(1) + cyclohexane (2) 0.05 9.290 (2) 1.94 (1) 0.10 9.366 (1) 1.967 (7) 0.20 9.522 (1) 2.048 (5) 0.30 9.688 (1) 2.111 (5) 0.40 9.859 (2) 2.169 (5) 0.50 10.041 (2) 2.216 (4) 0.60 10.236 (2) 2.251 (4) 0.70 10.440 (2) 2.279 (4) 0.80 10.661 (2) 2.273 (4) 0.90 10.897 (2) 2.260 (4) 0.95 11.019 (2) 2.247 (6)

D2 −9 10 m2 s−1

η

10−4 Pa

s

W

λ m−1 K−1

1.663 (4) 1.749 (3) 1.902 (4) 2.062 (4) 2.208 (4) 2.382 (5) 2.585 (6) 2.802 (7) 3.052 (8) 3.320 (9) 3.48 (1)

7.1 6.5 5.8 5.1 4.9 4.1 4.0 3.4 3.3 3.1 2.8

(3) (2) (2) (2) (2) (2) (2) (1) (1) (1) (1)

0.137 (4) 0.144 (4) 0.138 (4) 0.136 (4) 0.143 (8) 0.143 (5) 0.140 (4) 0.149 (7) 0.139 (5) 0.150 (5) 0.156 (6)

2.421 (9) 2.469 (5) 2.550 (5) 2.654 (5) 2.759 (5) 2.882 (6) 3.032 (6) 3.188 (7) 3.363 (8) 3.59 (1) 3.71 (2)

4.9 4.9 4.4 4.5 4.3 4.2 3.7 3.6 3.1 3.0 3.0

(2) (3) (3) (2) (2) (2) (2) (2) (2) (2) (1)

0.106 (5) 0.123 (5) 0.124 (5) 0.113 (4) 0.137 (5) 0.134 (5) 0.139 (5) 0.156 (5) 0.164 (6) 0.159 (5) 0.144 (6)

1.420 (3) 1.482 (3) 1.637 (3) 1.787 (3) 1.953 (4) 2.129 (4) 2.330 (4) 2.568 (6) 2.862 (7) 3.146 (9) 3.32 (1)

6.8 7.1 5.9 5.1 4.9 4.5 4.1 3.6 3.3 3.0 3.1

(2) (3) (2) (2) (2) (2) (2) (2) (2) (2) (2)

0.112 (4) 0.111 (4) 0.115 (4) 0.123 (4) 0.115 (4) 0.120 (4) 0.125 (5) 0.124 (5) 0.148 (5) 0.161 (5) 0.162 (5)

1.605 (4) 1.639 (3) 1.708 (3) 1.778 (4) 1.839 (4) 1.891 (4) 1.934 (4) 1.982 (5) 1.991 (5) 2.008 (7) 2.00 (1)

7.2 7.3 6.6 6.4 6.1 6.3 5.8 5.7 5.4 5.6 5.8


5 #%

x1 mol mol−1


ρ mol L−1

D1 −9 10 m2 s−1

benzene (1) + toluene (2) 0.05 9.420 (1) 2.51 (2) 0.10 9.500 (1) 2.476 (8) 0.20 9.652 (2) 2.454 (7) 0.30 9.818 (1) 2.433 (5) 0.40 9.990 (2) 2.423 (5) 0.50 10.163 (1) 2.387 (5) 0.60 10.343 (2) 2.372 (5) 0.70 10.532 (2) 2.320 (4) 0.80 10.730 (1) 2.312 (4) 0.90 10.940 (2) 2.265 (4) 0.95 11.039 (2) 2.252 (9) benzene (1) + CCl4 (2) 0.05 10.312 (2) 1.625 (7) 0.10 10.348 (2) 1.662 (6) 0.20 10.418 (2) 1.737 (5) 0.30 10.488 (2) 1.819 (5) 0.40 10.569 (2) 1.881 (4) 0.50 10.651 (2) 1.953 (4) 0.60 10.738 (2) 2.028 (4) 0.70 10.832 (2) 2.090 (4) 0.80 10.931 (2) 2.141 (4) 0.90 11.035 (2) 2.198 (4) 0.95 11.089 (2) 2.219 (6) cyclohexane (1) + toluene (2) 0.05 9.333 (1) 2.19 (1) 0.10 9.322 (2) 2.175 (8) 0.20 9.303 (1) 2.122 (5) 0.30 9.287 (1) 2.068 (5) 0.40 9.272 (1) 2.002 (5) 0.50 9.259 (1) 1.946 (4) 0.60 9.247 (1) 1.867 (4) 0.70 9.240 (2) 1.801 (4) 0.80 9.230 (1) 1.725 (3) 0.90 9.223 (2) 1.647 (3) 0.95 9.220 (1) 1.60 (1) cyclohexane (1) + CCl4 (2) 0.05 10.224 (1) 1.365 (9) 0.10 10.163 (1) 1.386 (5) 0.20 10.045 (2) 1.406 (4) 0.30 9.930 (2) 1.424 (4) 0.40 9.820 (1) 1.451 (4) 0.50 9.710 (1) 1.474 (3) 0.60 9.608 (1) 1.491 (3) 0.70 9.507 (1) 1.511 (3) 0.80 9.409 (1) 1.531 (3) 0.90 9.311 (1) 1.550 (3) 0.95 9.266 (2) 1.558 (4)

D2 −9 10 m2 s−1

η

10−4 Pa

s

W

λ m−1 K−1

2.368 (8) 2.368 (4) 2.340 (5) 2.303 (4) 2.286 (5) 2.258 (5) 2.242 (5) 2.201 (5) 2.192 (6) 2.145 (7) 2.13 (2)

5.7 5.1 5.1 5.4 5.5 5.2 5.5 5.7 5.5 5.6 6.4

(3) (4) (4) (2) (2) (2) (2) (3) (2) (2) (3)

0.120 (4) 0.123 (5) 0.117 (5) 0.133 (5) 0.123 (4) 0.118 (4) 0.120 (4) 0.127 (4) 0.122 (5) 0.126 (4) 0.133 (5)

1.354 (3) 1.393 (3) 1.460 (3) 1.531 (4) 1.584 (3) 1.655 (4) 1.716 (4) 1.772 (4) 1.828 (5) 1.885 (6) 1.91 (1)

6.8 7.7 6.9 6.6 6.2 6.5 5.9 6.0 6.1 5.9 5.7

(4) (4) (4) (3) (3) (3) (3) (2) (2) (2) (4)

0.114 (3) 0.110 (4) 0.114 (3) 0.114 (3) 0.114 (4) 0.121 (5) 0.115 (4) 0.117 (4) 0.124 (4) 0.120 (4) 0.125 (4)

2.358 (6) 2.396 (4) 2.299 (4) 2.259 (4) 2.189 (4) 2.143 (4) 2.070 (4) 2.013 (5) 1.934 (5) 1.864 (7) 1.82 (2)

4.8 4.9 5.4 5.9 5.5 5.4 5.9 6.5 7.2 6.9 8.1

(3) (3) (3) (2) (3) (2) (3) (3) (3) (4) (5)

0.119 (4) 0.118 (8) 0.120 (8) 0.133 (7) 0.137 (7) 0.124 (4) 0.129 (6) 0.128 (4) 0.136 (5) 0.145 (4) 0.140 (5)

1.332 (4) 1.347 (3) 1.371 (3) 1.395 (3) 1.421 (3) 1.443 (3) 1.461 (4) 1.483 (4) 1.511 (4) 1.522 (5) 1.54 (3)

7.3 6.8 6.6 7.3 6.6 7.1 7.1 7.2 7.1 7.1 7.4


5 #&

x1 mol mol−1

toluene (1) + 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95


ρ mol L−1 CCl4 (2)

10.225 (2) 10.169 (1) 10.059 (1) 9.956 (1) 9.855 (2) 9.763 (1) 9.670 (1) 9.586 (1) 9.500 (1) 9.423 (1) 9.380 (1)

D1 −9 10 m2 s−1

1.54 (1) 1.591 (6) 1.696 (5) 1.769 (4) 1.871 (4) 1.956 (4) 2.045 (4) 2.117 (4) 2.204 (4) 2.286 (4) 2.338 (4)

D2 −9 10 m2 s−1

1.364 (4) 1.407 (3) 1.499 (3) 1.569 (3) 1.662 (3) 1.737 (4) 1.826 (4) 1.886 (5) 1.983 (5) 2.059 (6) 2.16 (2)

5 #'

η

10−4 Pa

7.2 6.9 6.7 6.2 6.5 6.0 6.0 5.6 5.1 5.0 5.5

(3) (4) (3) (3) (3) (3) (3) (2) (2) (2) (2)

s

W

λ m−1 K−1

0.103 (8) 0.120 (3) 0.113 (3) 0.112 (3) 0.110 (4) 0.113 (3) 0.112 (3) 0.120 (4) 0.120 (4) 0.114 (4) 0.124 (8)

TABLE V: Maxwell-Stefan and Fick diusion coecients of 20 binary liquid mixtures at 298.15 K, 0.1 MPa and the specied composition from present simulations. The number in parentheses indicates the statistical uncertainty given in the last digits. The relative deviation (RD) of the predicted Fick diusion coecient from experimental values is also given, RD = |Dijcalc − Dijexp|/Dijexp. x1 Ð12 D12 RD Ð12 D12 RD mol mol−1 10−9 m2 s−1 10−9 m2 s−1 % 10−9 m2 s−1 10−9 m2 s−1 % methanol (1) + ethanol (2) methanol (1) + acetone (2) 0.05 1.38 (11) 1.38 10.2 6.51 (33) 6.06 34.5 0.10 1.24 (10) 1.24 3.4 6.89 (34) 6.02 41.9 0.20 1.41 (11) 1.42 5.8 7.55 (36) 5.85 51.1 0.30 1.44 (10) 1.45 2.5 7.15 (32) 5.05 37.5 0.40 1.60 (11) 1.61 8.6 6.74 (30) 4.47 24.2 0.50 1.56 (10) 1.58 0.9 6.30 (29) 4.07 13.5 0.60 1.72 (11) 1.73 5.1 5.68 (26) 3.73 4.1 0.70 1.68 (12) 1.68 3.1 4.89 (23) 3.39 4.1 0.80 1.79 (11) 1.79 2.3 3.95 (20) 3.00 11.4 0.90 1.76 (12) 1.76 9.2 3.51 (18) 3.02 1.9 0.95 1.95 (12) 1.94 2.3 3.17 (16) 2.93 2.8 methanol (1) + benzene (2) methanol (1) + toluene (2) 0.05 6.10 (29) 2.11 9.3 6.25 (36) 2.73 2.0 0.10 6.54 (37) 1.16 20.6 7.74 (42) 1.87 6.3 0.20 7.51 (42) 0.68 29.9 9.55 (49) 1.07 18.9 0.30 7.38 (41) 0.61 36.9 9.20 (48) 0.74 27.8 0.40 6.35 (34) 0.64 30.8 8.54 (43) 0.68 20.9 0.50 5.64 (29) 0.78 11.3 7.57 (43) 0.76 9.3 0.60 5.21 (25) 1.02 5.4 6.13 (35) 0.86 10.0 0.70 4.25 (26) 1.21 5.2 4.76 (27) 0.99 17.4 0.80 3.66 (17) 1.53 10.8 3.87 (23) 1.27 16.8 0.90 2.97 (17) 1.88 13.3 3.24 (19) 1.77 6.9 0.95 2.66 (16) 2.10 12.4 2.96 (17) 2.15 0.5 methanol (1) + CCl4 (2) ethanol (1) + acetone (2) 0.05 5.29 (28) 1.29 22.5 5.17 (21) 4.84 0.10 6.51 (37) 0.73 2.6 5.36 (29) 4.68 0.20 7.97 (44) 0.47 17.8 5.16 (29) 4.03 0.30 9.56 (51) 0.60 23.0 4.89 (29) 3.53 0.40 9.39 (47) 0.79 56.0 4.54 (17) 3.14 0.50 10.30 (49) 1.25 105 4.09 (19) 2.80 0.60 7.60 (43) 1.35 73.2 3.44 (17) 2.42 0.70 6.39 (36) 1.68 66.3 2.97 (15) 2.16 0.80 4.39 (23) 1.73 32.4 2.66 (16) 2.14 0.90 2.65 (16) 1.62 4.9 2.17 (14) 1.94 0.95 2.19 (12) 1.70 13.7 1.88 (11) 1.77 Continued on next page

5 $

x1 mol mol−1

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95

TABLE V continued from previous page Ð12 D12 RD Ð12 D12 RD 10−9 m2 s−1 10−9 m2 s−1 % 10−9 m2 s−1 10−9 m2 s−1 % ethanol (1) + benzene (2) ethanol (1) + cyclohexane (2) 3.98 (29) 1.61 20.5 4.32 (24) 1.14 23.9 4.88 (28) 1.12 23.0 6.46 (25) 0.78 1.4 5.24 (29) 0.74 25.8 7.96 (32) 0.44 20.7 4.88 (30) 0.71 23.0 7.83 (37) 0.38 11.4 4.57 (26) 0.84 12.4 6.69 (40) 0.41 1.4 4.01 (26) 0.97 5.2 6.32 (39) 0.55 22.1 3.22 (19) 1.04 9.9 5.42 (33) 0.71 24.2 2.79 (17) 1.20 9.9 4.38 (27) 0.88 18.3 2.46 (15) 1.40 6.6 2.96 (18) 0.94 2.2 1.72 (11) 1.29 19.7 1.99 (11) 1.07 12.4 1.53 (10) 1.37 22.4 1.48 ( 9) 1.07 21.0 ethanol (1) + toluene (2) ethanol (1) + CCl4 (2) 4.79 (34) 2.32 15.2 4.54 (24) 0.92 3.9 6.05 (36) 1.75 25.6 5.25 (24) 0.57 27.4 6.32 (32) 0.99 32.3 6.90 (31) 0.70 1.9 6.19 (35) 0.82 26.1 8.02 (43) 1.11 66.8 5.36 (29) 0.78 23.5 7.42 (33) 1.45 114 4.51 (27) 0.84 10.3 6.07 (27) 1.63 115 4.05 (23) 1.01 2.2 4.67 (21) 1.67 97.1 2.90 (18) 1.00 13.1 3.46 (17) 1.62 60.3 2.36 (15) 1.14 16.6 2.29 (13) 1.39 20.8 1.82 (11) 1.25 20.7 1.67 ( 9) 1.30 1.2 1.42 ( 9) 1.18 26.0 1.20 ( 9) 1.06 24.5 acetone (1) + benzene (2) acetone (1) + cyclohexane (2) 2.87 (19) 2.72 2.2 2.66 (17) 1.84 1.3 3.17 (19) 2.83 8.8 3.19 (19) 1.60 1.6 3.30 (19) 2.72 6.6 4.43 (22) 1.34 3.7 3.32 (20) 2.61 1.3 5.20 (25) 1.10 18.3 3.43 (19) 2.64 0.3 6.16 (27) 1.10 22.7 3.56 (20) 2.76 0.8 6.18 (28) 1.11 28.4 3.90 (22) 3.11 4.9 5.85 (25) 1.23 28.3 4.18 (21) 3.48 9.0 5.27 (24) 1.47 24.3 4.10 (22) 3.60 4.4 4.40 (21) 1.76 23.2 3.99 (21) 3.73 0.4 4.10 (19) 2.50 16.4 3.93 (23) 3.80 2.5 3.76 (19) 2.90 14.9 acetone (1) + toluene (2) acetone (1) + CCl4 (2) 2.96 (18) 2.74 3.1 2.75 (14) 2.19 37.4 3.36 (19) 2.90 5.9 3.13 (15) 2.10 36.4 3.57 (20) 2.76 6.5 4.18 (20) 2.31 55.5 3.61 (20) 2.59 3.2 4.93 (21) 2.60 73.2 3.76 (21) 2.60 4.2 5.50 (27) 3.04 90.7 3.88 (22) 2.69 5.8 5.41 (22) 3.27 86.0 4.01 (22) 2.88 8.6 5.43 (21) 3.64 73.0 3.94 (17) 3.00 6.3 4.96 (24) 3.70 62.0 3.78 (21) 3.11 1.8 3.99 (19) 3.30 24.1 3.52 (19) 3.18 5.1 3.64 (18) 3.32 8.0 3.64 (16) 3.45 2.0 3.57 (18) 3.41 1.4 Continued on next page 5 $

x1 mol mol−1

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95

TABLE V continued from previous page Ð12 D12 RD Ð12 D12 RD 10−9 m2 s−1 10−9 m2 s−1 % 10−9 m2 s−1 10−9 m2 s−1 % benzene (1) + cyclohexane (2) benzene (1) + toluene (2) 1.89 ( 9) 1.81 3.2 2.44 (15) 2.44 2.5 1.87 (10) 1.71 7.8 2.41 (16) 2.41 2.2 2.00 ( 9) 1.70 7.3 2.46 (17) 2.45 3.0 2.22 ( 9) 1.73 4.3 2.34 (14) 2.33 1.3 2.04 ( 9) 1.55 13.7 2.28 (13) 2.27 2.4 2.17 (10) 1.60 10.9 2.29 (13) 2.28 6.4 2.32 (11) 1.72 5.2 2.32 (13) 2.31 11.5 2.20 (12) 1.67 9.3 2.18 (11) 2.17 8.0 2.20 (11) 1.79 6.1 2.22 (12) 2.22 13.9 2.14 (12) 1.90 4.7 2.08 (11) 2.09 9.8 2.01 (11) 1.89 7.6 2.07 (12) 2.06 10.3 benzene (1) + CCl4 (2) cyclohexane (1) + toluene (2) 1.61 (10) 1.59 10.2 2.19 (13) 2.08 12.0 1.74 (11) 1.71 15.4 2.22 (14) 2.01 12.8 1.83 (10) 1.77 15.2 2.20 (12) 1.85 15.5 1.81 (11) 1.72 9.6 2.18 (13) 1.72 16.0 1.99 (11) 1.87 16.6 2.16 (14) 1.65 15.4 1.95 (11) 1.81 11.2 2.12 (13) 1.61 13.1 2.01 (11) 1.86 12.4 2.12 (12) 1.63 7.1 1.98 (10) 1.83 8.4 2.13 (12) 1.70 1.8 1.98 (11) 1.85 6.5 1.84 (10) 1.56 3.1 1.97 (10) 1.90 4.7 1.89 (12) 1.73 10.3 1.82 (11) 1.78 4.3 1.92 (13) 1.83 16.4 cyclohexane (1) + CCl4 (2) toluene (1) + CCl4 (2) 1.26 ( 8) 1.23 4.2 2.11 (14) 2.10 0.4 1.31 ( 9) 1.27 2.5 1.89 (12) 1.88 9.6 1.43 ( 8) 1.29 2.3 2.04 (11) 2.02 0.9 1.50 ( 8) 1.34 0.6 2.09 (12) 2.05 6.4 1.46 ( 7) 1.40 2.4 1.90 (12) 1.85 0.2 1.47 ( 7) 1.37 1.2 1.88 (11) 1.82 2.8 1.47 ( 7) 1.39 1.3 1.99 (11) 1.90 12.4 1.45 ( 7) 1.39 2.9 1.84 (10) 1.75 7.9 1.53 ( 8) 1.49 2.7 1.70 ( 9) 1.61 3.1 1.49 ( 8) 1.46 0.4 1.58 (10) 1.52 0.5 1.52 (10) 1.51 2.4 1.51 (10) 1.47 1.4

5$

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Transporteigenschaften reiner Flüssigkeiten und binärer Mischungen mit unterschiedlichen Wechselwirkungsparametern,

Experimentelle und theoretische Untersuchung des Einusses der thermischen Strahlung auf die eektive Wärmeleitfähigkeit von Flüssigkeiten

Introduction to Molecular Simulation and Industrial Applications: Methods, Examples and Prospects Statistical Mechanics of Nonequilibrium Liquids S 63

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