**8. Material transport equation in binary molecular mixtures: Concentration dependence of the Soret coefficient**

In this section we present the results obtained in (Semenov, 2011). In a binary system in which the component concentrations are comparable, the material transport equations defined by Eq. (18) have the form

$$\frac{\partial \phi}{\partial t} = \nabla \left[ L\_2 \phi (1 - \phi) \left( 2 \frac{\partial \mu^\*}{\partial \phi} \nabla \phi + \frac{\partial \mu^\*}{\partial T} \nabla T \right) \right] \left( T \left( 1 - \phi + \frac{L\_2 v\_2}{L\_1 v\_1} \phi \right) \right) \tag{68}$$

Eq. (68) can be used in the thermodynamical definition of the Soret coefficient [Eq. (59)]. The mass and thermodiffusion coefficients can be calculated in the same way as the Soret coefficient. The microscopic models used to calculate the Soret Coefficient in (Ghorayeb, Firoozabadi, 2000; Pan S et al., 2007) ignore the requirement expressed by Eq. (10) and cannot yield a description of thermodiffusion that is unambiguous. Although the material transport equations based on non-equilibrium thermodynamics were used, the fact that the chemical potential at constant pressure must be used was not taken into account. In these articles there is also the problem that in the transition to a dilute system the entropy of mixing does not become zero, yielding unacceptably large Soret coefficients even for pure components. An expression for the Soret coefficient was obtained in (Dhont et al, 2007; Dhont, 2004) by a quasi-thermodynamic method. However, the expressions for the thermodiffusion coefficient in those works become zero at high dilution, where the standard expression for osmotic pressure is used. These results contradict empirical observation.

Using Eq. (27) with the notion of a virtual particle outlined above, and substituting the expression for interaction potential [Eqs. (24, 28)], we can write the combined chemical potential at constant volume \* *V* as

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 363

related to intermolecular interactions in dilute systems. These parameters can be both positive

intermolecular interaction is stronger between identical solutes, thermodiffusion is positive, and vice versa. This corresponds to the experimental data of Ning and Wiegand (2006). When simplifications are taken into account, the equations expressed by the nonequilibrium thermodynamic approach are equivalent to expressions obtained in our hydrodynamic approach (Schimpf, Semenov, 2004; Semenov, Schimpf, 2005). Parameter *kin ST* in Eq. (71) is the kinetic contribution to the Soret coefficient, and has the same form as the term in square brackets in Eq. (37). In deriving this "kinetic" Soret coefficient, we have made different assumptions regarding the properties and concentration of the virtual

In deriving the temperature derivative of the combined chemical potential at constant pressure in Eq. (70) we used assumption a) in Section 4, which corresponds to zero entropy of mixing. Without such an assumption a pure liquid would be predicted to drift when subjected to a temperature gradient. Furthermore, the term that corresponds to the entropy

unacceptably high negative values of the Soret coefficient. However, in deriving the concentration derivative we must accept assumption b) because without this assumption the term related to entropy of mixing in Eq. (70) is lost. Consequently, the concentration derivative becomes zero in dilute mixtures and the Soret coefficient approaches infinity. Thus, we are required to use different assumptions regarding the properties of the virtual particles in the two expressions for diffusion and thermodiffusion flux. This situation reflects a general problem with statistical mechanics, which does not allow for the entropy of mixing for approaching the proper limit of zero at infinite dilution or as the difference in

particle properties approaches zero. This situation is known as the Gibbs paradox.

 

 Eq. (72) yields the main features for thermodiffusion of molecules in a one-phase system. It describes the situation where the Soret coefficient changes its sign at some volume fraction. Thus a change in sign with concentration is possible when the interaction between molecules of one component is strong enough, the interaction between molecules of the second component is weak, and the interaction between the different components has an intermediate value. Ignoring again the kinetic contribution, the condition for changing the

22 11 12 11

 2 or

example of such a system is the binary mixture of water with certain alcohols, where a

Upon refinement, a model for thermodiffusion in liquids based on non-equilibrium thermodynamics yields a system of consistent equations for providing an unambiguous

 

<sup>12</sup> 1 2 *S a iT i ii kT* is the "potential" Soret coefficient

will approach infinity at low volume fractions, yielding

1 , Eq. (71) is transformed into Eq. (37) at any temperature,

1 2 1 *kin S S SS T T TT* (72)

22 11 12 11

 2 . A good

1 , when the system is miscible at all concentrations, *ST* is a linear

*ii* and

<sup>12</sup> . When the

 

and negative depending on the relationship between parameters

(Kondepudi, Prigogine, 1999).

particles for different terms in Eq. (70).

 change of sign was observed (Ning, Wiegand, 2006).

of mixing *k* ln 1

In a diluted system, at

function of the concentration

sign change can be written as

 \* <sup>1</sup> . At 

**9. Conclusion** 

provided

$$\begin{split} \boldsymbol{\mu}\_{\boldsymbol{\nu}}^{\*} &= -kT \Big[ \frac{3}{2} \ln \frac{m\_{2}}{m\_{N\_{1}}} - \ln \frac{\phi}{1-\phi} + \ln \frac{Z\_{\text{act}}^{2}}{Z\_{\text{act}}^{N\_{1}}} \Big] + \\ &+ \frac{\phi}{\upsilon\_{2}} \left[ \int\_{\upsilon\_{\text{act}}^{2}}^{\upsilon} \Phi\_{22} \left( r \right) dv - \int\_{\upsilon\_{\text{act}}^{1}}^{\upsilon} \Phi\_{12}^{N\_{1}} \left( r \right) dv \right] + \frac{1-\phi}{\upsilon\_{1}} \left[ \int\_{\upsilon\_{\text{act}}^{2}}^{\upsilon} \Phi\_{21} \left( r \right) dv - \int\_{\upsilon\_{\text{act}}^{1}}^{\upsilon} \Phi\_{11}^{N\_{1}} \left( r \right) dv \right] \end{split} \tag{69}$$

In order to proceed to the calculation of chemical potentials at constant pressure using Eq. (29), we must calculate the local pressure distribution *<sup>i</sup>* using Eq. (32). We can subsequently use Eqs. (29) and (33) to obtain an expression for the gradient of the combined chemical potential at constant pressure in a non-isothermal and non-homogeneous system:

$$\begin{split} \nabla \mu\_{\ \rho}^{\*} &= \left[ \frac{kT}{\phi (1-\phi)} - a \left( \frac{\varepsilon\_{11} + \beta \varepsilon\_{22}}{\varepsilon\_{12}} - 1 - \beta \right) \right] \nabla \phi + \\ &+ a \left[ \alpha\_{2} \beta \phi \left( 1 - \frac{\varepsilon\_{22}}{\varepsilon\_{12}} \right) - \alpha\_{1} \left( 1 - \phi \right) \left( 1 - \frac{\varepsilon\_{11}}{\varepsilon\_{12}} \right) \right] \nabla T - \\ &- k \left( \frac{3}{2} \ln \frac{m\_{2}}{m\_{N\_{1}}} - \ln \frac{\phi}{1-\phi} + \ln \frac{Z\_{\text{ref}}^{2}}{Z\_{\text{ref}}^{N\_{1}}} \right) \nabla T \end{split} \tag{70}$$

Here *<sup>i</sup>* is the thermal expansion coefficient for the respective component, 3 1 22 3 2 12 *v v* is the parameter characterizing the geometrical relationship between the different component molecules, and 2 3 12 12 <sup>1</sup> 9 *a v* is the energetic parameter similar to the respective parameter in the van der Waals equation (Landau, Lifshitz, 1980) but characterizing the interaction between the different kinds of molecules. Then, using Eqs. (20), (70), we can write:

$$S\_T = \tau \frac{\left(1 - \phi\right) S\_{1T} - \phi S\_{2T} + S\_T^{kin}}{4\left(\phi - 1/2\right)^2 + \tau - 1} \tag{71}$$

where *<sup>c</sup> T T* is the ratio of the temperature at the point of measurement to the critical temperature 11 22 12 1 *<sup>c</sup> <sup>a</sup> <sup>T</sup> <sup>k</sup>* , where phase layering in the system begins. Assuming that 1 , the condition for parameter *Tc* to be positive is as 11 22 12 2 . This means that phase layering is possible when interactions between the identical molecules are stronger than those between different molecules. When 11 22 12 2 , the present theory predicts absolute miscibility in the system.

At temperatures lower than some positive*Tc* , when 1 only solutions in a limited concentration range can exist. It this temperature range, only mixtures with \* 1 , \* <sup>2</sup> can exist, where \* 1,2 11 2 , which is equivalent to the equation that defines the boundary for phase layering in phase diagrams for regular solutions, as discussed in (Kondepudi, Prigogine, 1999). <sup>12</sup> 1 2 *S a iT i ii kT* is the "potential" Soret coefficient related to intermolecular interactions in dilute systems. These parameters can be both positive and negative depending on the relationship between parameters *ii* and <sup>12</sup> . When the intermolecular interaction is stronger between identical solutes, thermodiffusion is positive, and vice versa. This corresponds to the experimental data of Ning and Wiegand (2006).

When simplifications are taken into account, the equations expressed by the nonequilibrium thermodynamic approach are equivalent to expressions obtained in our hydrodynamic approach (Schimpf, Semenov, 2004; Semenov, Schimpf, 2005). Parameter *kin ST* in Eq. (71) is the kinetic contribution to the Soret coefficient, and has the same form as the term in square brackets in Eq. (37). In deriving this "kinetic" Soret coefficient, we have made different assumptions regarding the properties and concentration of the virtual particles for different terms in Eq. (70).

In deriving the temperature derivative of the combined chemical potential at constant pressure in Eq. (70) we used assumption a) in Section 4, which corresponds to zero entropy of mixing. Without such an assumption a pure liquid would be predicted to drift when subjected to a temperature gradient. Furthermore, the term that corresponds to the entropy of mixing *k* ln 1 will approach infinity at low volume fractions, yielding unacceptably high negative values of the Soret coefficient. However, in deriving the concentration derivative we must accept assumption b) because without this assumption the term related to entropy of mixing in Eq. (70) is lost. Consequently, the concentration derivative becomes zero in dilute mixtures and the Soret coefficient approaches infinity.

Thus, we are required to use different assumptions regarding the properties of the virtual particles in the two expressions for diffusion and thermodiffusion flux. This situation reflects a general problem with statistical mechanics, which does not allow for the entropy of mixing for approaching the proper limit of zero at infinite dilution or as the difference in particle properties approaches zero. This situation is known as the Gibbs paradox.

In a diluted system, at 1 , Eq. (71) is transformed into Eq. (37) at any temperature, provided \* <sup>1</sup> . At 1 , when the system is miscible at all concentrations, *ST* is a linear function of the concentration

$$S\_T = \left(1 - \phi\right) S\_{T1} - \phi S\_{T2} + S\_T^{kin} \tag{72}$$

 Eq. (72) yields the main features for thermodiffusion of molecules in a one-phase system. It describes the situation where the Soret coefficient changes its sign at some volume fraction. Thus a change in sign with concentration is possible when the interaction between molecules of one component is strong enough, the interaction between molecules of the second component is weak, and the interaction between the different components has an intermediate value. Ignoring again the kinetic contribution, the condition for changing the sign change can be written as 22 11 12 11 2 or 22 11 12 11 2 . A good example of such a system is the binary mixture of water with certain alcohols, where a change of sign was observed (Ning, Wiegand, 2006).
