**9. Conclusion**

362 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

2

*rot*

*rot*

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:

 

1 11

 

 

*a T*

 

12 12 2

12 22 11

 

1

*rot rot N*

*N*

1

22 12 21 11

1

2 1 2 1

*VV VV*

*out out out out*

1

*N*

2 1

*v v*

<sup>3</sup> ln ln ln 2 1

*m Z*

 

*m Z*

*kT*

1

*P*

 

2 1

<sup>3</sup> ln ln ln 2 1

\* 11 22

*a*

*m Z k T m Z*

 

*<sup>i</sup>* is the thermal expansion coefficient for the respective component,

parameter characterizing the geometrical relationship between the different component

the van der Waals equation (Landau, Lifshitz, 1980) but characterizing the interaction

 

*<sup>c</sup> T T* is the ratio of the temperature at the point of measurement to the critical

 

*<sup>k</sup>* , where phase layering in the system begins.

11 2 , which is equivalent to the equation that defines the

 11 22 12 

 

*T TT*

 

*S SS*

1 2 2

4 12 1

between the different kinds of molecules. Then, using Eqs. (20), (70), we can write:

1

*T*

 

stronger than those between different molecules. When

At temperatures lower than some positive*Tc* , when

 

11 22 12 1 *<sup>c</sup>*

 

predicts absolute miscibility in the system.

*<sup>a</sup> <sup>T</sup>*

exist, where \* 1,2 

1 , the condition for parameter *Tc* to be positive is as

concentration range can exist. It this temperature range, only mixtures with \*

means that phase layering is possible when interactions between the identical molecules are

boundary for phase layering in phase diagrams for regular solutions, as discussed in

1

2

*N*

\* 2

*kT*

*V*

Here 

where 

temperature

Assuming that

molecules, and

 2 3 12 12 <sup>1</sup> 9

*v*

*a*

 

*r dv r dv r dv r dv*

1 1

(69)

1

 

  is the energetic parameter similar to the respective parameter in

*kin*

*S* (71)

 

(70)

is the

*v v*

1 only solutions in a limited

 1 , \* <sup>2</sup> can

 11 22 12 

2 , the present theory

 2 . This

*N N*

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

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 365

*Nik* Number of the molecules of the k'th component that are displaced by a

*N N* 1 21 Number of solvent molecules displaced by the solute in binary systems

*r* Coordinate of the correlated molecule when the considered particle is

*SiT* Contribution of the intermolecular interactions in Eq. (38)and in the Soret

*Tc* Critical temperature, where phase layering in binary systems begins

*x* Distance from the colloid particle surface to the closest solvent molecule

*y* Dimensionless distance from the colloid particle surface to the closest

*<sup>N</sup> Z* Rotational statistical sum for the virtual particle of the molecules k'th

Parameter characterizing the geometrical relationship between the

*ni* Numeric volume concentration of the respective component

*ir* Coordinate of internal molecule or atom in the particle

*S* Surface area of a spherical layer around the particle

*kin ST* Contribution of kinetic energy to the Soret coefficient

*<sup>i</sup> Vout* Volume external to a molecule of the i'th component

*<sup>k</sup> v* Partial molecular volume of respective component

*Zrot* Rotational statistical sum for polyatomic molecules

*<sup>i</sup> Z* Rotational statistical sum for the respective component

*<sup>i</sup> Zvib* Vibrational statistical sum for the respective component

component displaced by the molecule of i'th component

*<sup>i</sup>* Thermal expansion coefficient for the respective component

*<sup>i</sup> Vin* Internal volume of a molecule or atom of the i'th component

molecule of i'th component

*<sup>i</sup> q* Molecular heat of transport

*R* Radius of a colloidal particle

*ST* Soret coefficient in binary systems

<sup>3</sup> <sup>10</sup> *<sup>s</sup> ST* Characteristic Soret coefficient for the salts

coefficient for diluted systems.

*<sup>k</sup> v* Its specific molecular volume

different component molecules

solvent molecule surface

 placed at *<sup>r</sup>* <sup>0</sup> <sup>0</sup>*r* Unit radial vector

*T* Temperature

*t* Time

surface

*rot*

*ik rot*

*n* Ratio of particle to solvent thermal conductivity

*P* Internal macroscopic pressure of the system

*ns* Numeric volume concentration of salt

description of material transport in closed stationary systems. The macroscopic pressure gradient in such systems is determined by the Gibbs-Duhem equation. The only assumption used is that the heat of transport is equivalent to the negative of the chemical potential. In open and non-stationary systems, the macroscopic pressure gradient is calculated using modified material transport equations obtained by non-equilibrium thermodynamics, where the macroscopic pressure gradient is the unknown parameter. In that case, the Soret coefficient is expressed through combined chemical potentials at constant pressure. The resulting thermodynamic expressions allow for the use of statistical mechanics to relate the gradient in chemical potential to macroscopic parameters of the system.

This refined thermodynamic theory can be supplemented by microscopic calculations to explain the characteristic features of thermodiffusion in binary molecular solutions and suspensions. The approach yields the correct size dependence in the Soret coefficient and the correct relationship between the roles of electrostatic and Hamaker interactions in the thermodiffusion of colloidal particles. The theory illuminates the role of translational and rotational kinetic energy and the consequent dependence of thermodiffusion on molecular symmetry, as well as the isotopic effect. For non-dilute molecular mixtures, the refined thermodynamic theory explains the change in the direction of thermophoresis with concentration in certain mixtures, and the possibility of phase layering in the system. The concept of a Laplace-like pressure established in the force field of the particle under consideration plays an important role in microscopic calculations. Finally, the refinements make the thermodynamic theory consistent with hydrodynamic theories and with empirical data.
