**3. Dynamic pressure gradient in open and non-stationary systems: Thermodynamic equations of material transport with the Soret coefficient as a thermodynamic parameter**

Expressing the heats of transport by Eq. (14), we derived a set of consistent equations for material transport in a stationary closed system. However, expression for the heat of transport itself cannot yield consistent equations for material transport in a non-stationary and open system.

In an open system, the flux of a component may be nonzero because of transport across the system boundaries. Also, in a closed system that is non-stationary, the component material

fluxes *<sup>i</sup> J* can be nonzero even though the total material flux in the system, 1 *N i i i J vJ* , is

zero. In both these cases, the Gibbs-Duhem equation can no longer be used to determine the pressure in the system, and an alternate approach is necessary.

In previous works (Schimpf, Semenov, 2004; Semenov, Schimpf, 2005), we combined hydrodynamic calculations of the kinetic coefficients with the Fokker-Planck equations to obtain material transfer equations that contain dynamic parameters such as the crossdiffusion and thermal diffusion coefficients. In that approach, the macroscopic gradient of pressure in a binary system was calculated from equations of continuity of the same type as expressed by Eqs. (2) and (8). This same approach may be used for solving the material transport equations obtained by non-equilibrium thermodynamics.

In this approach, the continuity equations [Eq. (2)] are first expressed in the form

$$\frac{\partial \phi\_i}{\partial t} = \nabla \frac{\phi\_i L\_i}{T} \left( 2 \sum\_{k>1}^N \frac{\partial \mu\_i}{\partial \phi\_k} \nabla \phi\_k - \overline{\upsilon}\_i \nabla P + \frac{\partial \mu\_i}{\partial T} \nabla T \right) \tag{16}$$

Summing Eq. (16) for each component and utilizing Eq. (8) we obtain the following equation for the dynamic pressure gradient in an open non-stationary system:

$$\nabla P = \left[ \vec{f}T + \sum\_{i=1}^{N} \phi\_i L\_i \left( 2 \sum\_{k>1}^{N} \frac{\partial \mu\_i}{\partial \phi\_k} \nabla \phi\_k + \frac{\partial \mu\_i}{\partial T} \nabla T \right) \right] \Bigg/ \sum\_{i=1}^{N} \phi\_i L\_i \overline{v}\_i \tag{17}$$

Substituting Eq. (17) into Eq. (16) we obtain the material transport equations:

$$\frac{\partial \phi\_i}{\partial t} = \nabla \frac{\phi\_i L\_i}{T} \left[ \left[ \vec{J} \Gamma + \sum\_{j=1}^N \phi\_j v\_j L\_j \left( 2 \sum\_{k>1}^N \frac{\partial \mu\_{ij}^\*}{\partial \phi\_k} \nabla \phi\_k + \frac{\partial \mu\_{ij}^\*}{\partial \Gamma} \nabla T \right) \right] \left\langle \sum\_{k=1}^N \phi\_k v\_k L\_k \right\rangle \tag{18}$$

Comparing Eq. (18) with Eq. (15) for a stationary mixture shows that former contains an additional drift term 1 *iii N kkk k vLJ v L* proportional to the total material flux through the open

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 349

where subscript *P* is used to indicate that the derivatives are taken at constant pressure, as is the case in Eqs. (4) and (6). We can solve Eqs. (19) to express the "partial" Soret coefficient

*<sup>v</sup>* (21)

*<sup>h</sup>* (23)

12 . 1 2 *I I*, , and <sup>3</sup>*I* are the principal values of the tensor of the

describes the gradual "switching on" of the intermolecular

 , *ij g r* is the pair correlative function,

(24)

2 ; for NH3,

*<sup>r</sup>* ( *r r* )

*<sup>k</sup>* and *T* .

*<sup>i</sup> Z* and

*<sup>k</sup> ST* for the k'th component through a factor of proportionality between

expression derived in (Kirkwood, Boggs, 1942; Fisher, 1964):

0 2

0

**molecular force field** 

Here

*vib*

where

moment of inertia. In Eq. (21), parameter

(Landau, Lifshitz, 1980)

3 ; for CH4 and C6H6,

**4. Statistical mechanics of material transport: Chemical potentials at** 

 

*h v*

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

 

0 1

**constant volume and pressure and the Laplace component of pressure in a** 

The chemical potential at constant volume can be calculated using a modification of an

<sup>1</sup>

*i out*

<sup>3</sup> <sup>2</sup> ln ln ln ln 2 *rot*

 

*i vib i m kT kT kT kT Z kT Z*

is the chemical potential of an ideal gas of the respective non-interacting molecules (related to their kinetic energy), *h* is Planck's constant, *mi* is the mass of the molecule, *rot*

*<sup>i</sup> <sup>Z</sup>* are its rotational and vibrational statistical sums, respectively, and *<sup>i</sup> Vout* is the volume external to a molecule of the i'th component. The molecular vibrations make no significant contribution to the thermodynamic parameters except in special situations, which will not be discussed here. The rotational statistical sum for polyatomic molecules is written as

> 

symmetry axes carrying the molecule into itself. For H2O and C2H5OH,

liquid and the center of the considered molecule;

known as the London potential (Ross, Morrison, 1988):

<sup>3</sup> <sup>123</sup> 8 *Zrot kT I I I*

interaction. A detailed description of this representation can be found in (Kirkwood, Boggs, 1942; Fisher, 1964). Parameter *r* is the distance between the molecule of the surrounding

if the considered molecule is placed at *r* 0 ; and *ij* is the molecular interaction potential,

 

*ij ij r*

6 *ij*

which expresses the probability of finding a molecule of the surrounding liquid at

<sup>3</sup> <sup>2</sup>

is the symmetry value, which is the number of possible rotations about the

 

,

*i i i i*

(22)

*d g r r dv*

system. The term *N kkk JT v L* in Eq. (17) describes the contribution of that drift to the pressure

gradient. This additional component of the total material flux is attributed to barodiffusion, which is driven by the dynamic pressure gradient defined by Eq. (17). This dynamic pressure gradient is associated with viscous dissipation in the system. Parameter *J* is independent of position in the system but is determined by material transfer across the system boundaries, which may vary over time.

If the system is open but stationary, molecules entering it through one of its boundary surfaces can leave it through another, thus creating a molecular drift that is independent of the existence of a temperature or pressure gradient. This drift is determined by conditions at the boundaries and is independent of any force applied to the system. For example, the system may have a component source at one boundary and a sink of the same component at opposite boundary. As molecules of a given species move between the two boundaries, they experience viscous friction, which creates a dynamic pressure gradient that induces barodiffusion in all molecular species. The pressure gradient that is induced by viscous friction in such a system is not considered in the Gibbs-Duhem equation.

Equations (6), (7), and (15) describe a system in hydrostatic equilibrium, without viscous friction caused by material flux due to material exchange through the system boundaries. Unlike the Gibbs-Duhem equation, Eq. (17) accounts for viscous friction forces and the resulting dynamic pressure gradient. For a closed stationary system, in which *J* 0 and 0 *<sup>t</sup>* , Eq. (18) is transformed into

$$\sum\_{k>1}^{N} \left( 2 \sum\_{j=1}^{N} \frac{\partial \mu\_{ij}^{\star}}{\partial \phi\_{k}} \right) \nabla \phi\_{k} + \left( \sum\_{j=1}^{N} \frac{\partial \mu\_{ij}^{\star}}{\partial T} \right) \nabla T = 0 \tag{19}$$

There are thermal diffusion experiments in which the system experiences periodic temperature changes. An example is the method used described by (Wiegand, Kohler, 2002), where thermodiffusion in liquids is observed within a dynamic temperature grating produced using a pulsed infrared laser. Because this technique involves changing the wall temperature, which changes the equilibrium adsorption constant, material fluxes vary with time, creating a periodicity in the inflow and outflow of material. A preliminary analysis shows that material fluxes to and from the walls have relaxation times on the order of a few microseconds until equilibrium is attained, and that such non-stationary material fluxes can be observed using dynamic temperature gratings.

The Soret coefficient is a common parameter used to characterize material transport in temperature gradients. For binary systems, Eq. (19) can be used to define the Soret coefficient as

$$S\_T = -\frac{\frac{\partial \mu\_{21P}^\*}{\partial T}}{2\phi\_2 \left(1 - \phi\_2\right) \frac{\partial \mu\_{21P}^\*}{\partial \phi\_2}}\tag{20}$$

*k*

1

where subscript *P* is used to indicate that the derivatives are taken at constant pressure, as is the case in Eqs. (4) and (6). We can solve Eqs. (19) to express the "partial" Soret coefficient *<sup>k</sup> ST* for the k'th component through a factor of proportionality between *<sup>k</sup>* and *T* .
