**6. The Soret coefficient in diluted colloidal suspensions: Size dependence of the Soret coefficient and the applicability of thermodynamics**

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 parameter *ik* given by Eq. (11). They argue that empirical evidence indicates the Soret 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 of the particle radius, as calculated by a quasi-thermodynamic method.

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 contained in the particle

$$\Phi\_{i1}^{\*}\left(r\right) = \int\_{V\_{iw}^{i}} \frac{dV\_{in}}{\upsilon\_{i}} \Phi\_{i1}\left(\left|\vec{r}\_{i} - \vec{r}\right|\right) \tag{43}$$

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 liquid placed at *<sup>r</sup>* ( *r r* ) and an internal molecule or atom placed at *ir* . Such potentials are 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. For a colloidal particle with radius *R* >> *ij* , the temperature distribution at the particle 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 corresponds to the assumption that *r R* ' and <sup>2</sup> *dv R dr* 4 in Eq. (36). To calculate the Hamaker potential, the expression calculated in (Ross, Morrison, 1988), which is based on the London potential, can be used:

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 357

 

*P n n T eE*

 

*E* is the electric field strength. Substituting Eq. (47) into Eq. (46) we

 

*i i k ik ik ik*

 

*n T* (47)

 

*i ik k e Ne* (49)

2 that is so small that it makes no contribution to

<sup>1</sup> 1 *<sup>s</sup>* (51)

(50)

*n v* ,

*T v <sup>T</sup>* (48)

and formulate an approximate relationship in place of the exact

*<sup>s</sup>* . Due to electric neutrality, the ion concentrations will be equal at

 

*v T <sup>T</sup>* (52)

*v T <sup>T</sup>* (53)

 

> 

*i i ik i*

*ie* is the electric charge of the respective ion, is the macroscopic electrical

*i l k l i l*

*N N*

0

where

where

very low, i.e.

equal: 

with concentration

 

1 1 *<sup>s</sup>*

form expressed by Eq. (8):

2 1 

(Landau, Lifshitz, 1980).

2 2

 

*v v v v*

 

and charged particles with concentration

*i*

potential, and

*N N*

1 1

*i k k*

obtain the following material transport equations for a closed and stationary system:

1

 *ik*

We will consider a quaternary diluted system that contains a background neutral solvent

the physicochemical parameters of the system. In other words, we consider the thermophoresis of an isolated charged colloidal particle stabilized by an ionic surfactant. With a symmetric electrolyte, the ion concentrations are equal to maintain electric neutrality

> *v v*

 

In this case we can introduce the volume concentration of salt as

Here the volume contribution of charged particles is ignored since their concentration is

any salt concentration and temperature, that is, the chemical potentials of the ions should be

 1 1 0 3 *<sup>s</sup> s <sup>L</sup> <sup>J</sup> T eE*

 2 2 21 <sup>21</sup> <sup>21</sup> 22 2

*<sup>L</sup> <sup>J</sup> T eE*

 

> 

*s*

 

Using Eqs. (48) – (51) we obtain equations for the material fluxes, which are set to zero:

0 3 *<sup>s</sup>*

<sup>1</sup> , an electrolyte salt dissociated into ions with concentrations

 

*<sup>L</sup> <sup>J</sup> T E*

$$\boldsymbol{\Phi}\_{i1}^{\*}\left(\boldsymbol{y}\right) = -\frac{\varepsilon\_{i1}}{6} \frac{\sigma\_{21}^{3}}{v\_{2}} \left(\frac{1}{y} + \frac{1}{2+y} + \ln\frac{y}{2+y}\right) \tag{44}$$

Here 21 *x y* , and *x* is the distance from the particle surface to the closest solvent molecule

surface. Using Eqs. (36) and (44) we can obtain the following expression for the Soret coefficient of a colloidal particle:

$$S\_T = \frac{\pi^2 \alpha\_1 R \sigma\_{21}^2 \varepsilon\_{21}}{2 \left(n + 2\right) v\_2 kT} \frac{\sigma\_{21}^3}{v\_1} \left(\frac{\varepsilon\_{11}}{\varepsilon\_{21}} - 1\right) \tag{45}$$

Here *n* is ratio of particle to solvent thermal conductivity. The Soret coefficient for the colloidal particle is proportional to 5 21 1 2 *R v v* . In practice, this means that *ST* is proportional to

 21 *<sup>R</sup>* since the ratio 6 21 1 2 *v v* is practically independent of molecular size. This proportionality

is consistent with hydrodynamic theory [e.g., see (Anderson, 1989)], as well as with empirical data. The present theory explains also why the contribution of the kinetic term and the isotope effect has been observed only in molecular systems. In colloidal systems the potential related to intermolecular interactions is the prevailing factor due to the large value

of 2 21 1 *R v* . Thus, the colloidal Soret coefficient is 21 *<sup>R</sup>* times larger than its molecular

counterpart. This result is also consistent with numerous experimental data and with hydrodynamic theory.
