**5. The Soret coefficient in diluted binary molecular mixtures: The kinetic term in thermodiffusion is related to the difference in the mass and symmetry of molecules**

In this section we present the results obtained in (Semenov, 2010, Semenov, Schimpf, 2011a). In diluted systems, the concentration dependence of the chemical potentials for the solute and solvent is well-known [e.g., see (Landau, Lifshitz, 1980)]: 2 *kT* ln , and <sup>1</sup> is practically independent of solute concentration <sup>2</sup> . Thus, Eq. (20) for the Soret coefficient takes the form:

$$S\_T = -\frac{\frac{\tilde{\mathcal{C}}\mu\_p^\*}{\tilde{\mathcal{C}}T}}{2kT} \tag{33}$$

where \* *<sup>P</sup>* is \* <sup>21</sup>*<sup>P</sup>* .

The equation for combined chemical potential at constant volume [Eq. (28)] using assumption b) in Section 3 takes the form

$$\mu\_V^\* = -kT \left( \frac{3}{2} \ln \frac{m\_2}{m\_{N\_1}} - \ln \frac{\phi}{1 - \phi} + \ln \frac{Z\_{\text{rot}}^1}{Z\_{\text{rot}}^{N\_1}} \right) + 4\pi \left[ \frac{\Phi\_{21}(r) - \Phi\_{11}^{N\_1}(r)}{v\_1} r^2 dr \right] \tag{34}$$

 

*r r*

<sup>0</sup>*r* is the unit radial vector. The pressure gradient related to the change in surface area has the same nature as the Laplace pressure gradient discussed in (Landau, Lifshitz, 1980).

 

2 0

2 ' '

*r r dr*

'

*r*

 

0

*v r* (32)

2 

<sup>2</sup> . Thus, Eq. (20) for the Soret coefficient

*kT* (33)

<sup>1</sup>

*N*

(34)

1

 *kT* ln , and <sup>1</sup> is

0

(30)

(31)

 

*N N j j i ij i ij j j j j dF r r lSdr r ldS v v*

where we consider changes in free energy due to both a change in the parameters of the layer volume ( *dV Sdr* ) and a change *dS* in the area of the closed layer. For a spherical layer, the changes in volume and surface area are related as *dV rdS* 2 , and we obtain the

1 1

where

**molecules** 

takes the form:

where \* *<sup>P</sup>* is \* <sup>21</sup>*<sup>P</sup>* .

Solving Eq. (31), we obtain

following modified equation of equilibrium for a closed spherical surface:

 

1

expression here, rather we will derive the expression for binary systems.

and solvent is well-known [e.g., see (Landau, Lifshitz, 1980)]:

1

*V N*

2 1

practically independent of solute concentration

assumption b) in Section 3 takes the form

*i ij j j*

1 1

 

*N N <sup>j</sup> j ij ij i j j j j*

Substituting the pressure gradient from Eq. (32) into Eq. (29), and using Eqs. (24), (27), and (28), we obtain a general expression for the gradient in chemical potential at constant pressure in a non-isothermal and non-homogeneous system. We will not write the general

**5. The Soret coefficient in diluted binary molecular mixtures: The kinetic term in thermodiffusion is related to the difference in the mass and symmetry of** 

In this section we present the results obtained in (Semenov, 2010, Semenov, Schimpf, 2011a). In diluted systems, the concentration dependence of the chemical potentials for the solute

> 

 \*

*P*

2

The equation for combined chemical potential at constant volume [Eq. (28)] using

1

 

1 \* 2 21 11 2

*rot rot*

*<sup>m</sup> <sup>Z</sup> r r kT r dr m v Z*

*T T S*

<sup>3</sup> ln ln ln 4

*N R*

*<sup>N</sup> <sup>r</sup> <sup>j</sup> ij*

*v v r*

where *N N* 1 21 is the number of solvent molecules displaced by molecule of the solute, <sup>1</sup> 11 *<sup>N</sup>* is the potential of interaction between the virtual particle and a molecule of the solvent. The relation <sup>1</sup> 1 is also used in deriving Eq. (34). Because ln 1 at 0 , we expect the use of assumption a) in Section 3 for the concentration of virtual particles will yield a reasonable physical result.

In a dilute binary mixture, the equation for local pressure [Eq. (32)] takes the form

$$\Pi\_i = -\sum\_{j=1}^{N} \frac{\Phi\_{i1}\left(r\right)}{\upsilon\_1} + \int\_{\sigma} \frac{2\Phi\_{i1}\left(r\right)}{\upsilon\_1 r'} dr' \tag{35}$$

where index *i* is related to the virtual particle or solute.

Using Eqs. (29), (34), we obtain the following expression for the temperature-induced gradient of the combined chemical potential of the diluted molecular mixture:

$$k\nabla\mu\_P^\star = -k\nabla T\left(\frac{3}{2}\ln\frac{m\_2}{m\_{N\_1}} + \ln\frac{Z\_{\rm net}^1}{Z\_{\rm net}^{N\_1}}\right) + \int\_{V\_{\rm net}} \frac{a\_1 dv}{v\_1} \nabla\_\beta T\left[\frac{\Phi\_{21}\left(r'\right) - \Phi\_{11}^{N\_1}\left(r\right)}{r'} dr'\right] \tag{36}$$

Here<sup>1</sup> is the thermal expansion coefficient for the solvent and *T* is the tangential component of the bulk temperature gradient. After substituting the expressions for the interaction potentials defined by Eqs. (23), (24), and (28) into Eq. (36), we obtain the following expression for the Soret coefficient in the diluted binary system:

$$S\_T = \frac{1}{2T} \left[ \frac{3}{2} \ln \left( \frac{m\_2}{m\_{N\_1}} \right) + \ln \left( \frac{\gamma\_{N\_1} \sqrt{\left(I\_1 I\_2 I\_3\right)\_2}}{\gamma\_2 \sqrt{\left(I\_1 I\_2 I\_3\right)\_{N\_1}}} \right) \right] + \frac{\pi^2 \alpha\_1 \sigma\_{\gamma\_2}^3 \varepsilon\_{12}}{18 v\_1 kT} \left( \frac{\varepsilon\_{11}}{\varepsilon\_{12}} - 1 \right) \tag{37}$$

In Eq. (37), the subscripts *2* and *N*1 are used again to denote the real and virtual particle, respectively.

The Soret coefficient expressed by Eq. (37) contains two main terms. The first term corresponds to the temperature derivative of the part of the chemical potential related to the solute kinetic energy. In turn, this kinetic term contains the contributions related to the translational and rotational movements of the solute in the solvent. The second term is related to the potential interaction of solute with solvent molecules. This potential term has the same structure as those obtained by the hydrodynamic approach in (Schimpf, Semenov, 2004; Semenov, Schimpf, 2005).

According to Eq. (37), both positive (from hot to cold wall) and negative (from cold to hot wall) thermodiffusion is possible. The molecules with larger mass ( <sup>1</sup> *m m* <sup>2</sup> *<sup>N</sup>* ) and with a stronger interactions between solvent molecules ( 11 12 ) should demonstrate positive thermodiffusion. Thus, dilute aqueous solutions are expected to demonstrate positive thermophoresis. In (Ning, Wiegand, 2006), dilute aqueous solutions of acetone and dimethyl sulfoxide were shown to undergo positive thermophoresis. In that paper, a very high value of the Hildebrand parameter is given as an indication of the strong intermolecular interaction for water. More specifically, the value of the Hildebrand parameter exceeds by two-fold the respective parameters for other components.

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 355

Using the above parameters and Eq. (42), we obtain 3 1 *a K <sup>m</sup>* 5.7 10 , which is still about six-times greater than the empirical value from (Wittko, Köhler, 2005). The remaining discrepancy could be due to our overestimation of the degree of symmetry violation upon

isotopic substitutions occur at random positions. Thus, it may be more relevant to use Eq. (42) to evaluate the characteristic degree of symmetry from an experimental measurement of

**6. The Soret coefficient in diluted colloidal suspensions: Size dependence of** 

While thermodynamic approaches yield simple and clear expressions for the Soret coefficient, such approaches are the subject of rigorous debate. The thermodynamic or "energetic" approach has been criticized in the literature. Parola and Piazza (2004) note that the Soret coefficient obtained by thermodynamics should be proportional to a linear combination of the surface area and the volume of the particle, since it contains the

coefficient is directly proportional to particle size for colloidal particles [see numerous references in (Parola, Piazza, 2004)], and is practically independent of particle size for molecular species. By contrast, Duhr and Braun (2006) show the proportionality between the Soret coefficient and particle surface area, and use thermodynamics to explain their empirical data. Dhont et al (2007) also reports a Soret coefficient proportional to the square

Let us consider the situation in which a thermodynamic calculation for a large particle as said contradicts the empirical data. For large particles, the total interaction potential is assumed to be the sum of the individual potentials for the atoms or molecules which are

> \* 1 1 *i in*

Here *<sup>i</sup> Vin* is the internal volume of the real or virtual particle and *i i* <sup>1</sup> *r r* is the respective intermolecular potential given by Eq. (24) or (28) for the interaction between a molecule of a

referred to as Hamaker potential, and are used in studies of interactions between colloidal particles (Hunter, 1992; Ross, Morrison, 1988). In this and the following sections, *<sup>i</sup> v* is the specific molecular volume of the atom or molecule in a real or virtual particle, respectively.

surface can be used instead of the bulk temperature gradient (Giddings et al, 1995), and the curvature of the particle surface can be ignored in calculating the respective integrals. This

Hamaker potential, the expression calculated in (Ross, Morrison, 1988), which is based on

*<sup>r</sup>* ( *r r* ) and an internal molecule or atom placed at

*in i i i V i dV r r r*

*ik* given by Eq. (11). They argue that empirical evidence indicates the Soret

<sup>2</sup> is to some extent conditional because the

*<sup>v</sup>* (43)

*ij* , the temperature distribution at the particle

<sup>2</sup> *dv R dr* 4 in Eq. (36). To calculate the

*ir* . Such potentials are

<sup>2</sup> 2 3 . One

isotopic substitution. The true value of this parameter can be obtained with

*am* rather than trying to use theoretical values to predict thermodiffusion.

**the Soret coefficient and the applicability of thermodynamics** 

of the particle radius, as calculated by a quasi-thermodynamic method.

should understand that the value of parameter

parameter

contained in the particle

liquid placed at

For a colloidal particle with radius *R* >>

the London potential, can be used:

corresponds to the assumption that *r R* ' and

Since the kinetic term in the Soret coefficient contains solute and solvent symmetry numbers, Eq. (37) predicts thermodiffusion in mixtures where the components are distinct only in symmetry, while being identical in respect to all other parameters. In (Wittko, Köhler, 2005) it was shown that the Soret coefficient in the binary mixtures containing the isotopically substituted cyclohexane can be in general approximated as the linear function

$$S\_T = S\_{iT} + a\_m \Delta M + b\_i \Delta I \tag{38}$$

where *SiT* is the contribution of the intermolecular interactions, *am* and *<sup>i</sup> b* are coefficients, while *M* and *I* are differences in the mass and moment of inertia, respectively, for the molecules constituting the binary mixture. According to Eq. (37), the coefficients are defined by

$$a\_m = \frac{3}{4T m\_{N\_1}}\tag{39}$$

$$b\_i = \frac{\left(\gamma\_{N\_1}\right)^2}{4T\left(\gamma\_2\right)^2 \sqrt{\left(I\_1 I\_2 I\_3\right)\_{N\_1}}}\tag{40}$$

In (Wittko, Köhler, 2005) the first coefficient was empirically determined for cyclohexane isomers to be 3 1 *a K <sup>m</sup>* 0.99 10 at room temperature (*T=300 K*), while Eq. (39) yields 3 1 *a K <sup>m</sup>* 0.03 10 ( *<sup>M</sup>*<sup>1</sup> <sup>84</sup> ). There are several possible reasons for this discrepancy. The first term on the right side of Eq. (38) is not the only term with a mass dependence, as the second term also depends on mass. The empirical parameter *am* also has an implicit dependence on mass that is not in the theoretical expression given by Eq. (39). The mass dependence of the second term in Eq. (37) will be much stronger when a change in mass occurs at the periphery of the molecule.

A sharp change in molecular symmetry upon isotopic substitution could also lead to a discrepancy between theory and experiment. Cyclohexane studied in (Wittko, Köhler, 2005) has high symmetry, as it can be carried into itself by six rotations about the axis perpendicular to the plane of the carbon ring and by two rotations around the axes placed in the plane of the ring and perpendicular to each other. Thus, cyclohexane has <sup>1</sup> *<sup>N</sup>* 24 . The partial isotopic substitution breaks this symmetry. We can start from the assumption that for the substituted molecules, <sup>2</sup> 1 . When the molecular geometry is not changed in the substitution and only the momentum of inertia related to the axis perpendicular to the ring plane is changed, the relative change in parameter *bi* can be written as

$$\frac{\left(\left(\boldsymbol{\gamma}\_{N\_{1}}\right)^{2}\sqrt{\left(I\_{1}I\_{2}I\_{3}\right)\_{2}}-\left(\boldsymbol{\gamma}\_{2}\right)^{2}\left(I\_{1}I\_{2}I\_{3}\right)\_{1}}}{4T\left(\boldsymbol{\gamma}\_{2}\right)^{2}\sqrt{\left(I\_{1}I\_{2}I\_{3}\right)\_{N\_{1}}}}=\frac{\left(\boldsymbol{\gamma}\_{N\_{1}}\right)^{2}\left(m\_{2}-m\_{N\_{1}}\right)}{4T\left(\boldsymbol{\gamma}\_{2}\right)^{2}m\_{N\_{1}}}+\frac{\left(\boldsymbol{\gamma}\_{N\_{1}}\right)^{2}-\left(\boldsymbol{\gamma}\_{2}\right)^{2}}{4T\left(\boldsymbol{\gamma}\_{2}\right)^{2}}\tag{41}$$

Eq. (41) yields

$$a\_m = \frac{\mathbf{1}}{4Tm\_{N\_1}} \left[ \mathbf{3} + \left( \frac{\boldsymbol{\gamma}\_{N\_1}}{\boldsymbol{\gamma}\_2} \right)^2 \right] \tag{42}$$

Since the kinetic term in the Soret coefficient contains solute and solvent symmetry numbers, Eq. (37) predicts thermodiffusion in mixtures where the components are distinct only in symmetry, while being identical in respect to all other parameters. In (Wittko, Köhler, 2005) it was shown that the Soret coefficient in the binary mixtures containing the isotopically substituted cyclohexane can be in general approximated as the linear function

where *SiT* is the contribution of the intermolecular interactions, *am* and *<sup>i</sup> b* are coefficients, while *M* and *I* are differences in the mass and moment of inertia, respectively, for the molecules constituting the binary mixture. According to Eq. (37), the coefficients are defined

*a*

*i*

*b*

occurs at the periphery of the molecule.

the substituted molecules,

Eq. (41) yields

4 *<sup>m</sup>*

 

In (Wittko, Köhler, 2005) the first coefficient was empirically determined for cyclohexane isomers to be 3 1 *a K <sup>m</sup>* 0.99 10 at room temperature (*T=300 K*), while Eq. (39) yields 3 1 *a K <sup>m</sup>* 0.03 10 ( *<sup>M</sup>*<sup>1</sup> <sup>84</sup> ). There are several possible reasons for this discrepancy. The first term on the right side of Eq. (38) is not the only term with a mass dependence, as the second term also depends on mass. The empirical parameter *am* also has an implicit dependence on mass that is not in the theoretical expression given by Eq. (39). The mass dependence of the second term in Eq. (37) will be much stronger when a change in mass

A sharp change in molecular symmetry upon isotopic substitution could also lead to a discrepancy between theory and experiment. Cyclohexane studied in (Wittko, Köhler, 2005) has high symmetry, as it can be carried into itself by six rotations about the axis perpendicular to the plane of the carbon ring and by two rotations around the axes placed in

partial isotopic substitution breaks this symmetry. We can start from the assumption that for

substitution and only the momentum of inertia related to the axis perpendicular to the ring

 

2 2 2 2 2 123 2 123 2 1 2 2 2 2 2 2 123 <sup>2</sup> <sup>2</sup> 4 44

1 1

*N N*

1

*N*

<sup>1</sup> <sup>3</sup>

*T III Tm T*

1 1 11

*N N NN*

*III III m m*

4

*m*

*a*

the plane of the ring and perpendicular to each other. Thus, cyclohexane has

plane is changed, the relative change in parameter *bi* can be written as

 

  *T III*

2 2 123 4 *N*

1

2

1

<sup>2</sup> 1 . When the molecular geometry is not changed in the

 

2

1

*N*

2

*N*

*N*

1

3

by

*S S a M bI T iT m i* (38)

*Tm* (39)

(40)

 

 

*Tm* (42)

<sup>1</sup> *<sup>N</sup>* 24 . The

(41)

Using the above parameters and Eq. (42), we obtain 3 1 *a K <sup>m</sup>* 5.7 10 , which is still about six-times greater than the empirical value from (Wittko, Köhler, 2005). The remaining discrepancy could be due to our overestimation of the degree of symmetry violation upon isotopic substitution. The true value of this parameter can be obtained with <sup>2</sup> 2 3 . One should understand that the value of parameter <sup>2</sup> is to some extent conditional because the isotopic substitutions occur at random positions. Thus, it may be more relevant to use Eq. (42) to evaluate the characteristic degree of symmetry from an experimental measurement of *am* rather than trying to use theoretical values to predict thermodiffusion.
