**7. The Soret coefficient in diluted suspensions of charged particles: Contribution of electrostatic and non-electrostatic interactions to thermodiffusion**

In this section we present the results obtained in (Semenov, Schimpf, 2011b). The colloidal particles discussed in the previous section are usually stabilized in suspensions by electrostatic interactions. Salt added to the suspension becomes dissociated into ions of opposite electric charge. These ions are adsorbed onto the particle surface and lead to the establishment of an electrostatic charge, giving the particle an electric potential. A diffuse layer of charge is established around the particle, in which counter-ions are accumulated. This diffuse layer is the electric double layer. The electric double layer, where an additional pressure is present, can contribute to thermodiffusion. It was shown in experiments that particle thermodiffusion is enhanced several times by the addition of salt [see citations in (Dhont, 2004)].

For a system of charged colloidal particles and molecular ions, the thermodynamic equations should be modified to include the respective electrostatic parameters. The basic thermodynamic equations, Eqs. (4) and (6), can be written as

$$\nabla \mu\_i = \sum\_{k=1}^{N} \frac{\partial \mu\_i}{\partial n\_k} \nabla n\_k - \overline{\upsilon}\_i \nabla P + \frac{\partial \mu\_i}{\partial T} \nabla T + e\_i \vec{E} \tag{46}$$

$$\nabla P = \sum\_{i=1}^{N} n\_i \left( \sum\_{k=1}^{N} \frac{\partial \mu\_i}{\partial n\_k} \nabla n\_k + \frac{\partial \mu\_i}{\partial T} \nabla T + e\_i \vec{E} \right) \tag{47}$$

where *i ie* is the electric charge of the respective ion, is the macroscopic electrical potential, and *E* is the electric field strength. Substituting Eq. (47) into Eq. (46) we obtain the following material transport equations for a closed and stationary system:

$$\vec{J}\_i = 0 = -\frac{L\_i}{T} \sum\_k^N \frac{\phi\_i \phi\_k}{\upsilon\_i} \left( \sum\_{l=1}^N \frac{\partial \mu\_{ik}^\*}{\partial \phi\_l} \nabla \phi\_l + \frac{\partial \mu\_{ik}^\*}{\partial T} \nabla T - \frac{\partial \mu\_{ik}^\*}{\partial \Phi} \vec{E} \right) \tag{48}$$

where

356 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

surface. Using Eqs. (36) and (44) we can obtain the following expression for the Soret

22 3 1 21 21 21 11 2 1 21

Here *n* is ratio of particle to solvent thermal conductivity. The Soret coefficient for the

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

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

In this section we present the results obtained in (Semenov, Schimpf, 2011b). The colloidal particles discussed in the previous section are usually stabilized in suspensions by electrostatic interactions. Salt added to the suspension becomes dissociated into ions of opposite electric charge. These ions are adsorbed onto the particle surface and lead to the establishment of an electrostatic charge, giving the particle an electric potential. A diffuse layer of charge is established around the particle, in which counter-ions are accumulated. This diffuse layer is the electric double layer. The electric double layer, where an additional pressure is present, can contribute to thermodiffusion. It was shown in experiments that particle thermodiffusion is enhanced several times by the addition of salt [see citations in

For a system of charged colloidal particles and molecular ions, the thermodynamic equations should be modified to include the respective electrostatic parameters. The basic

> 

*i i i k i i*

 

*n T* (46)

*n v P T eE*

**7. The Soret coefficient in diluted suspensions of charged particles: Contribution of electrostatic and non-electrostatic interactions to** 

*<sup>y</sup> <sup>y</sup>*

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

2

 

*i*

 

5 21 1 2 *R v v*

2 2 *<sup>T</sup> <sup>R</sup> <sup>S</sup>*

<sup>3</sup> \* <sup>1</sup> <sup>21</sup>

1

*i*

Here

21

of 2 21 1 *R v*

 21 *x*

*<sup>R</sup>* since the ratio

hydrodynamic theory.

**thermodiffusion** 

(Dhont, 2004)].

coefficient of a colloidal particle:

colloidal particle is proportional to

. Thus, the colloidal Soret coefficient is

thermodynamic equations, Eqs. (4) and (6), can be written as

*N*

1

*k k*

21 1

is practically independent of molecular size. This proportionality

*n v kT v* (45)

. In practice, this means that *ST* is proportional to

*<sup>R</sup>* times larger than its molecular

(44)

1 1 ln 6 22

*v yy y*

$$-\frac{\partial \mu\_{ik}^{\bullet}}{\partial \Phi} = e\_i - N\_{ik} e\_k \tag{49}$$

We will consider a quaternary diluted system that contains a background neutral solvent with concentration<sup>1</sup> , an electrolyte salt dissociated into ions with concentrations *n v* , and charged particles with concentration2 that is so small that it makes no contribution to 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

$$
\upsilon \upsilon\_- \phi\_+ = \upsilon\_+ \phi\_- \tag{50}
$$

In this case we can introduce the volume concentration of salt as 1 1 *<sup>s</sup> v v v v* and formulate an approximate relationship in place of the exact form expressed by Eq. (8):

$$
\phi\_s + \phi\_1 = 1 \tag{51}
$$

Here the volume contribution of charged particles is ignored since their concentration is very low, i.e. 2 1 *<sup>s</sup>* . Due to electric neutrality, the ion concentrations will be equal at any salt concentration and temperature, that is, the chemical potentials of the ions should be equal: (Landau, Lifshitz, 1980).

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

$$\vec{J}\_2 = 0 = -\frac{\phi\_2 L\_2}{v\_2 T} \left[ \frac{\partial \mu\_{21}^\*}{\partial \phi\_2} \nabla \phi\_2 + 3 \frac{\partial \mu\_{21}^\*}{\partial \phi\_s} \nabla \phi\_s + \frac{\partial \mu\_{21}^\*}{\partial T} \nabla T + e\_2 \vec{E} \right] \tag{52}$$

$$\vec{f}\_{-}=0=-\frac{\phi\_{-}L\_{-}}{\upsilon\_{-}T}\left(\Im\frac{\partial\mu\_{-1}^{\*}}{\partial\phi\_{s}}\nabla\phi\_{s}+\frac{\partial\mu\_{-1}^{\*}}{\partial T}\nabla T-e\vec{E}\right)\tag{53}$$

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 359

For an isolated particle placed in a liquid, the chemical potential at constant volume can be calculated using a modified procedure mentioned in the preceding section. In these calculations, we use both the Hamaker potential and the electrostatic potential of the electric double layer to account for the two types of the interactions in these systems. The chemical potential of the non-interacting molecules plays no role for colloid particles, as was shown

In a salt solution, the suspended particle interacts with both solvent molecules and dissolved ions. The two interactions can be described separately, as the salt concentration is usually very low and does not significantly change the solvent density. The first type of interaction uses Eqs. (25) and the Hamaker potential [Eq. (44)]. For the electrostatic interactions, the properties of diluted systems may be used, in which the pair correlative function has a Boltzmann form (Fisher, 1964; Hunter, 1992). Since there are two kinds of ions, Eq. (21) for the "electrostatic" part of the chemical potential at

*e e e e*

<sup>0</sup> 2 0 *e e*

<sup>0</sup>*r* is the unit radial vector. In Eq. (61) it is assumed that the particle radius is much larger than the characteristic thickness of the electric double layer. Solving Eq. (62) assuming a Boltzmann distribution for the ion concentration, as in (Ruckenstein, 1981; Anderson,

*e e r r e e*

 

Substituting the pressure gradient calculated from Eq. (62) into Eq. (29), utilizing Eq. (60), and considering the temperature-induced gradients related to the temperature dependence of the Boltzmann exponents, we obtain the temperature derivative in the gradient of the chemical potential for a charged colloidal particle, which is related to the electrostatic

4 ' '

*n kR <sup>r</sup> dr e e dr*

 

Here *n* is again the ratio of particle to solvent thermal conductivity. For low potentials

*e r e e P s e kT kT*

*R*

( *<sup>e</sup> kT* ), where the Debye-Hueckel theory should work, Eq. (63) takes the form

*kT kT <sup>s</sup> kT kT e s <sup>e</sup> <sup>n</sup> n kT e e e e r dr R* 

2

*n d e e r r dr n kT e e r dr* (60)

is the numeric volume concentration of salt, and *<sup>e</sup> e* is the

<sup>2</sup> <sup>2</sup> ' '

(62)

 

2

2

*T n kT* (63)

2 2

*r*

*R* (61)

 

*<sup>e</sup> kT kT kT kT s es R R*

Eq. (32) expressing the equilibrium condition for electrostatic interactions is written as

*nn r nn r*

4 4 2

above.

where

where

1989), we obtain

constant volume can be written as

 

 *<sup>s</sup> ns v v*

interactions in its electric double layer:

2

electrostatic interaction energy.

1

0

2

 

$$\vec{J}\_{+} = 0 = -\frac{\phi\_{+} L\_{+}}{\upsilon\_{+} T} \left( \Im \frac{\partial \mu\_{+1}^{\*}}{\partial \phi\_{s}} \nabla \phi\_{s} + \frac{\partial \mu\_{+1}^{\*}}{\partial T} \nabla T + e \vec{E} \right) \tag{54}$$

where *e ee* (symmetric electrolyte). We will not write the equation for the flux of background solvent <sup>1</sup>*J* because it yields no new information in comparison with Eqs. (52) - 54), as shown above. Solving Eqs. (52) – (54), we obtain

$$\nabla \phi\_s = -\nabla T \frac{\partial \left(\mu\_{+1}^\* + \mu\_{-1}^\*\right)}{\partial \Gamma} \bigg/ 3 \frac{\partial \left(\mu\_{+1}^\* + \mu\_{-1}^\*\right)}{\partial \phi\_s} \tag{55}$$

$$2e\vec{E} = 3\frac{\partial \left(\mu\_{-1}^{\star} - \mu\_{+1}^{\star}\right)}{\partial \phi\_{\rm s}}\nabla\phi\_{\rm s} + \frac{\partial \left(\mu\_{-1}^{\star} - \mu\_{+1}^{\star}\right)}{\partial T}\nabla T \tag{56}$$

Eq. (55) allows us to numerically evaluate the concentration gradient as

$$
\nabla \phi\_s \approx \phi\_s \mathbf{S}\_T^\* \nabla T \tag{57}
$$

where <sup>3</sup> <sup>10</sup> *<sup>s</sup> ST* is the characteristic Soret coefficient for the salts. Salt concentrations are typically around *10-2-10-1 mol/L,* that is <sup>4</sup> 10 *<sup>s</sup>* or lower. A typical maximum temperature gradient is <sup>4</sup> *T K cm* 10 / . These values substituted into Eq. (57) yield 4 31 10 10 *<sup>s</sup> cm* . The same evaluation applied to parameters in Eq. (56) shows that the first term on the right side of this equation is negligible, and the equation for thermoelectric power can be written as

$$\vec{E} \approx \frac{\partial \left(\mu\_{-1}^{\star} - \mu\_{+1}^{\star}\right)}{\partial T} \frac{\nabla T}{2e} = \frac{v\_{+} - v\_{-}}{2ev\_{1}} \frac{\partial \mu\_{1}}{\partial T} \nabla T \tag{58}$$

For a non-electrolyte background solvent, parameter <sup>1</sup> *T* can be evaluated as 1 1 *T kT* , where <sup>1</sup> is the thermal expansion coefficient of the solvent (Semenov, Schimpf, 2009; Semenov, 2010). Usually, in liquids the thermal expansion coefficient is low enough ( 3 1 <sup>1</sup> 10 *K* ) that the thermoelectric field strength does not exceed *1 V/cm*. This electric field strength corresponds to the maximum temperature gradient discussed above. The electrophoretic velocity in such a field will be about *10-5-10-4 cm/s*. The thermophoretic velocities in such temperature gradients are usually at least one or two orders of magnitude higher.

These evaluations show that temperature-induced diffusion and electrophoresis of charged colloidal particle in a temperature gradient can be ignored, so that the expression for the Soret coefficient of a diluted suspension of such particles can be written as

$$S\_{2T} = -\frac{\nabla \phi\_2}{\phi\_2 \nabla T} = -\frac{\frac{\partial \mu\_{21P}^\*}{\partial T}}{\phi\_2 \frac{\partial \mu\_{21P}^\*}{\partial \phi\_2}} = -\frac{1}{kT} \frac{\partial \mu\_{21P}^\*}{\partial T} \tag{59}$$

Eq. (59) can also be used for microscopic calculations.

For an isolated particle placed in a liquid, the chemical potential at constant volume can be calculated using a modified procedure mentioned in the preceding section. In these calculations, we use both the Hamaker potential and the electrostatic potential of the electric double layer to account for the two types of the interactions in these systems. The chemical potential of the non-interacting molecules plays no role for colloid particles, as was shown above.

In a salt solution, the suspended particle interacts with both solvent molecules and dissolved ions. The two interactions can be described separately, as the salt concentration is usually very low and does not significantly change the solvent density. The first type of

interaction uses Eqs. (25) and the Hamaker potential [Eq. (44)]. For the electrostatic interactions, the properties of diluted systems may be used, in which the pair correlative function has a Boltzmann form (Fisher, 1964; Hunter, 1992). Since there are two kinds of ions, Eq. (21) for the "electrostatic" part of the chemical potential at constant volume can be written as

$$\mu\_2^{\varepsilon} = -4\pi n\_s \left| \int\_0^{\varepsilon} d\lambda \right| \left| \left( e^{\frac{\lambda \Phi\_\varepsilon}{kT}} - e^{-\frac{\lambda \Phi\_\varepsilon}{kT}} \right) \Phi\_\varepsilon(r) r^2 dr = -4\pi n\_s kT \right|\_R^{\varepsilon} \left| e^{\frac{\Phi\_\varepsilon}{kT}} + e^{-\frac{\Phi\_\varepsilon}{kT}} - 2 \right| r^2 dr \tag{60}$$

where *<sup>s</sup> ns v v* is the numeric volume concentration of salt, and *<sup>e</sup> e* is the

electrostatic interaction energy.

358 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

where *e ee* (symmetric electrolyte). We will not write the equation for the flux of

11 11 <sup>3</sup> *<sup>s</sup>*

11 11 2 3 *<sup>s</sup>*

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

*T*

Eq. (55) allows us to numerically evaluate the concentration gradient as

*s*

 *<sup>s</sup>*

Soret coefficient of a diluted suspension of such particles can be written as

*T S*

2

Eq. (59) can also be used for microscopic calculations.

*T*

 

For a non-electrolyte background solvent, parameter

1 1

54), as shown above. Solving Eqs. (52) – (54), we obtain

typically around *10-2-10-1 mol/L,* that is

background solvent

power can be written as

 1 1 *T kT* , where

3 1

enough ( 

as 

higher.

 

> 

 

*eE T*

where <sup>3</sup> <sup>10</sup> *<sup>s</sup> ST* is the characteristic Soret coefficient for the salts. Salt concentrations are

gradient is <sup>4</sup> *T K cm* 10 / . These values substituted into Eq. (57) yield

 

Schimpf, 2009; Semenov, 2010). Usually, in liquids the thermal expansion coefficient is low

electric field strength corresponds to the maximum temperature gradient discussed above. The electrophoretic velocity in such a field will be about *10-5-10-4 cm/s*. The thermophoretic velocities in such temperature gradients are usually at least one or two orders of magnitude

These evaluations show that temperature-induced diffusion and electrophoresis of charged colloidal particle in a temperature gradient can be ignored, so that the expression for the

21 2 21

*P*

 

2 21 2

2

*P*

 

*Tv v E T*

<sup>1</sup> 2 2

<sup>1</sup> 10 *K* ) that the thermoelectric field strength does not exceed *1 V/cm*. This

 4 31 10 10 *<sup>s</sup> cm* . The same evaluation applied to parameters in Eq. (56) shows that the first term on the right side of this equation is negligible, and the equation for thermoelectric

 

 

<sup>1</sup>*J* because it yields no new information in comparison with Eqs. (52) -

 

1

<sup>1</sup> is the thermal expansion coefficient of the solvent (Semenov,

*P*

*<sup>T</sup> kT T* (59)

 

1

*s*

*v T <sup>T</sup>* (54)

*T* (55)

*s sTS T* (57)

<sup>4</sup> 10 *<sup>s</sup>* or lower. A typical maximum temperature

*T e ev T* (58)

<sup>1</sup> *T* can be evaluated

*T* (56)

Eq. (32) expressing the equilibrium condition for electrostatic interactions is written as

$$
\nabla \left[ \left( n\_+ - n\_- \right) \Phi\_\varepsilon \left( r \right) + \Pi \right] + \mathbf{2} \left( n\_+ - n\_- \right) \Phi\_\varepsilon \left( r \right) \frac{\vec{r}\_0}{R} = 0 \tag{61}
$$

where <sup>0</sup>*r* is the unit radial vector. In Eq. (61) it is assumed that the particle radius is much larger than the characteristic thickness of the electric double layer. Solving Eq. (62) assuming a Boltzmann distribution for the ion concentration, as in (Ruckenstein, 1981; Anderson, 1989), we obtain

$$
\Pi\_e = n\_s kT \left( e^{\frac{\Phi\_e}{kT}} + e^{-\frac{\Phi\_e}{kT}} - 2 \right) - \frac{2n\_s}{R} \int\_{\sigma}^{r} \left( e^{\frac{\Phi\_e}{kT}} - e^{-\frac{\Phi\_e}{kT}} \right) \Big|\_{\sigma}^{r} \Phi\_e(r') dr' \tag{62}
$$

Substituting the pressure gradient calculated from Eq. (62) into Eq. (29), utilizing Eq. (60), and considering the temperature-induced gradients related to the temperature dependence of the Boltzmann exponents, we obtain the temperature derivative in the gradient of the chemical potential for a charged colloidal particle, which is related to the electrostatic interactions in its electric double layer:

$$\frac{\partial \mu\_{2P}^{\varepsilon}}{\partial T} = \frac{4\pi n\_s k R}{\left(n+2\right)} \Big|\_{R}^{\infty} dr \Big|\_{\ast}^{r} \Big( e^{\frac{\Phi\_{\varepsilon}}{kT}} + e^{-\frac{\Phi\_{\varepsilon}}{kT}} \Big) \frac{\Phi\_{\varepsilon}^{2}\left(r'\right)}{\left(kT\right)^{2}} dr' \tag{63}$$

Here *n* is again the ratio of particle to solvent thermal conductivity. For low potentials ( *<sup>e</sup> kT* ), where the Debye-Hueckel theory should work, Eq. (63) takes the form

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 361

first two of these four terms are always significantly distinct from unity. The characteristic length of the interaction is much higher for electrostatic interactions. Also, the characteristic density of ions or molecules in a liquid, which are involved in their electrostatic interaction with the colloidal particle, is much lower than the density of the solvent molecules. The

concentrations in water at room temperature. The energetic parameter may be small, (~*0.1*) when the colloidal particles are compatible with the solvent. Characteristic values of the energetic coefficient range from *0.1-10*. Combining these numeric values, one can see that the ratio given by Eq. (67) lies in a range of *0.1*-*10* and is governed primarily by the value of

electrostatic or the Hamaker contribution to particle thermophoresis may prevail, depending on the value of the particle's energetic parameters. In the region of high Soret coefficients, particle thermophoresis is determined by electrostatic interactions and is positive. In the region of low Soret coefficients, thermophoresis is related to Hamaker

**8. Material transport equation in binary molecular mixtures: Concentration** 

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

> 

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

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

   

*<sup>t</sup> <sup>T</sup> L v* (68)

1 2 <sup>1</sup> *L v <sup>L</sup> T T*

values of these respective coefficients are

**dependence of the Soret coefficient** 

defined by Eq. (18) have the form

potential at constant volume

2

interactions and can have different directions in different solvents.

pressure is used. These results contradict empirical observation.

\* *V* as

*kT* is the ratio of energetic parameters for the respective interactions. Only the

2

2 21

11 21 . Thus, calculation of the ratio given by Eq. (67) shows that either the

3

10 *<sup>D</sup>* and <sup>2</sup> <sup>3</sup> 1 <sup>10</sup> *n vs*

and the difference in the energetic parameters of the Hamaker

 

2 2

1 1

*<sup>T</sup>* for typical ion

 

11 21 *e*

interaction

2

the electrokinetic potential

 

 

$$\frac{\partial \mu\_{2P}^{\varepsilon}}{\partial T} = \frac{8\pi n\_s k R}{\left(n+2\right)} \Big|\_{R}^{\prime} dr \Big| \frac{\Phi\_{\varepsilon}^{2}\left(r^{\prime}\right)}{\left(kT\right)^{2}} dr\,\tag{64}$$

Using an exponential distribution for the electric double layer potential, which is characteristic for low electrokinetic potentials , we obtain from Eq. (64)

$$\frac{\partial \mu\_{2P}^{\varepsilon}}{\partial T} = \frac{8\pi m\_s k R \lambda\_D^2}{\left(n+2\right)} \left(\frac{e\zeta}{kT}\right)^2\tag{65}$$

where *<sup>D</sup>* is the Debye length [for a definition of Debye length, see (Landau, Lifshitz, 1980; Hunter, 1992)].

Calculation of the non-electrostatic (Hamaker) term in the thermodynamic expression for the Soret coefficient is carried out in the preceding section [Eq. (45)]. Combining this expression with Eq. (65), we obtain the Soret coefficient of an isolated charged colloidal particle in an electrolyte solution:

$$S\_T = \frac{8\pi n\_s R \lambda\_D^2}{T\left(n+2\right)} \left(\frac{e\zeta}{kT}\right)^2 + \frac{\pi^2 a\_1 R \sigma\_{21}^2 \varepsilon\_{21}}{2\left(n+2\right) v\_2 kT} \frac{\sigma\_{21}^3}{v\_1} \left(\frac{\varepsilon\_{11}}{\varepsilon\_{21}} - 1\right) \tag{66}$$

This thermodynamic expression for the Soret coefficient contains terms related to the electrostatic and Hamaker interactions of the suspended colloidal particle. The electrostatic term has the same structure as the respective expressions for the Soret coefficient obtained by other methods (Ruckenstein, 1981; Anderson, 1989; Parola, Piazza, 2004; Dhont, 2004). In the Hamaker term, the last term in the brackets reflects the effects related to displacing the solvent by particle. It is this effect that can cause a change in the direction of thermophoresis when the solvent is changed. However, such a reverse in the direction of thermophoresis can only occur when the electrostatic interactions are relatively weak. When electrostatic interactions prevail, only positive thermophoresis can be observed, as the displaced solvent molecules are not charged, therefore, the respective electrostatic term is zero. The numerous theoretical results on electrostatic contributions leading to a change in the direction of thermophoresis are wrong due to an incorrect use of the principle of local equilibrium in the hydrodynamic approach [see discussion in (Semenov, Schimpf, 2005)].

The relative role of the electrostatic mechanism can be evaluated by the following ratio:

$$\frac{8}{\pi} \frac{n\_s v\_2}{a\_1 T} \frac{\lambda\_D^2}{\sigma\_{21}^2} \frac{v\_1}{\sigma\_{21}^3} \frac{\left(e\zeta\right)^2}{\left(\varepsilon\_{11} - \varepsilon\_{21}\right) kT} \tag{67}$$

The physicochemical parameters contained in Eq. (67) are separated into several groups and are collected in the respective coefficients. Coefficient 2 1 *n vs <sup>T</sup>* contains the parameters related to concentration and its change with temperature, 2 2 21 *<sup>D</sup>* is the coefficient reflecting the respective lengths of the interaction, 1 3 21 *<sup>v</sup>* reflects the geometry of the solvent molecules, and

Using an exponential distribution for the electric double layer potential, which is

 

Calculation of the non-electrostatic (Hamaker) term in the thermodynamic expression for the Soret coefficient is carried out in the preceding section [Eq. (45)]. Combining this expression with Eq. (65), we obtain the Soret coefficient of an isolated charged colloidal

> 

This thermodynamic expression for the Soret coefficient contains terms related to the electrostatic and Hamaker interactions of the suspended colloidal particle. The electrostatic term has the same structure as the respective expressions for the Soret coefficient obtained by other methods (Ruckenstein, 1981; Anderson, 1989; Parola, Piazza, 2004; Dhont, 2004). In the Hamaker term, the last term in the brackets reflects the effects related to displacing the solvent by particle. It is this effect that can cause a change in the direction of thermophoresis when the solvent is changed. However, such a reverse in the direction of thermophoresis can only occur when the electrostatic interactions are relatively weak. When electrostatic interactions prevail, only positive thermophoresis can be observed, as the displaced solvent molecules are not charged, therefore, the respective electrostatic term is zero. The numerous theoretical results on electrostatic contributions leading to a change in the direction of thermophoresis are wrong due to an incorrect use of the principle of local equilibrium in the

<sup>8</sup> <sup>1</sup>

2 2 22 3

 

*<sup>P</sup> n kRs D e*

 2 2

*<sup>D</sup>* is the Debye length [for a definition of Debye length, see (Landau, Lifshitz, 1980;

2

*n kR <sup>r</sup> dr dr*

8 ' '

 

2

, we obtain from Eq. (64)

*T n kT* (64)

*T n kT* (65)

*<sup>T</sup> kT* (67)

*<sup>v</sup>* reflects the geometry of the solvent molecules, and

*<sup>T</sup>* contains the parameters related

*<sup>D</sup>* is the coefficient reflecting the

1 21 21 21 11 2 1 21

*T n kT n v kT v* (66)

2

 

2 *e r P s e R*

characteristic for low electrokinetic potentials

where

Hunter, 1992)].

particle in an electrolyte solution:

2

*e*

<sup>2</sup> 8

2 22

*n R e R <sup>S</sup>*

hydrodynamic approach [see discussion in (Semenov, Schimpf, 2005)].

are collected in the respective coefficients. Coefficient

to concentration and its change with temperature,

respective lengths of the interaction,

The relative role of the electrostatic mechanism can be evaluated by the following ratio:

2 1 2 3 1 21 21 11 21

8 *<sup>s</sup> <sup>D</sup> n v v e*

 

1 3 21

 

 

The physicochemical parameters contained in Eq. (67) are separated into several groups and

2 2

2 2 21

2 1 *n vs*

 

*T*

*s D*

 2 11 21 *e kT* is the ratio of energetic parameters for the respective interactions. Only the

first two of these four terms are always significantly distinct from unity. The characteristic length of the interaction is much higher for electrostatic interactions. Also, the characteristic density of ions or molecules in a liquid, which are involved in their electrostatic interaction with the colloidal particle, is much lower than the density of the solvent molecules. The

$$\text{values of these respective coefficients are } \frac{\lambda\_0^2}{\sigma\_{21}^2} \ge 10^3 \text{ and } \frac{n\_s v\_2}{a\_1 T} = 10^{-3} \text{ for typical ion.}$$

concentrations in water at room temperature. The energetic parameter may be small, (~*0.1*) when the colloidal particles are compatible with the solvent. Characteristic values of the energetic coefficient range from *0.1-10*. Combining these numeric values, one can see that the ratio given by Eq. (67) lies in a range of *0.1*-*10* and is governed primarily by the value of the electrokinetic potential and the difference in the energetic parameters of the Hamaker interaction 11 21 . Thus, calculation of the ratio given by Eq. (67) shows that either the electrostatic or the Hamaker contribution to particle thermophoresis may prevail, depending on the value of the particle's energetic parameters. In the region of high Soret coefficients, particle thermophoresis is determined by electrostatic interactions and is positive. In the region of low Soret coefficients, thermophoresis is related to Hamaker interactions and can have different directions in different solvents.
