**2. Thermodynamic theory of material transport in liquid mixtures: Role of the Gibbs-Duhem equation**

The aim of this section is to outline the thermodynamic approach to material transport in mixtures of different components. The approach is based on the principle of local equilibrium, which assumes that thermodynamic principles hold in a small volume within a non-equilibrium system. Consequently, a small volume containing a macroscopic number of particles within a non-equilibrium system can be treated as an equilibrium system. A detailed discussion on this topic and references to earlier work are given by Gyarmati (1970). The conditions required for the validity of such a system are that both the temperature and molecular velocity of the particles change little over the scale of molecular length or mean free path (the latter change being small relative to the speed of sound). For a gas, these conditions are met with a temperature gradient below 104 K cm-1; for a liquid, where the heat conductivity is greater, the speed of sound higher and the mean free path is small, this condition for local equilibrium is more than fulfilled, provided the experimental temperature gradient is below *104 K cm-1*.

Thermodynamic expressions for material transport in liquids have been established based on equilibrium thermodynamics (Gibbs and Gibbs-Duhem equations), as well as on the principles of non-equilibrium thermodynamics (thermodynamic forces and fluxes). For a review of these models, see (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969).

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 345

Gibbs-Duhem equation (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Landau, Lifshitz, 1959; Ghorayeb, Firoozabadi, 2000; Pan S et al., 2007):

> 

*Pn n T*

The Gibbs-Duhem equation defines the macroscopic pressure gradient in a thermodynamic system. In equilibrium thermodynamics the equation defines the potentiality of the thermodynamic functions (Kondepudi, Prigogine, 1999). In equilibrium thermodynamics the change in the thermodynamic function is determined only by the initial and final states of the systems, without consideration of the transition process itself. In non-equilibrium thermodynamics, Eq. (5) plays the role of expressing mechanical equilibrium in the system. According to the Prigogine theorem (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969), pressure gradient cancels the volume forces expressed as the gradients of the chemical potentials and provides mechanical equilibrium in a thermodynamically stable system. However, in a non-isothermal system, the same authors considered a constant pressure and the left- and right-hand side of Eq. (6) were assumed to

*L v <sup>T</sup> <sup>J</sup> <sup>T</sup> T q v T T v <sup>T</sup> <sup>T</sup>* (7)

1

 

1

*<sup>L</sup> <sup>T</sup> <sup>J</sup> T q T v <sup>T</sup> <sup>T</sup>* (10)

*l l ll*

We note that the volume fraction selected for elimination is arbitrary (any other volume fraction can be eliminated in the same manner), and that in subsequent mathematical

<sup>1</sup> is expressed through the other volume fractions using Eq. (8), and the following

 *i ik i k k v*

 

In Eq. (7), the numeric volume concentrations of the components are replaced by their

 1

1 1

 

, ..., ,... <sup>2</sup> *kl l kk k*

*N i i*

*i k i k k*

 

*n T* (6)

 

> 

> >

*v* (11)

 

(8)

(9)

 

*ll l l*

 

*i i*

be zero simultaneously, which is both physically and mathematically invalid. Substituting Eq. (6) into Eq. (5) we obtain the following equation for material flux:

Using Eq. (8) and the standard rule of differentiation of a composite function

*k l i l*

1 2 *N N*

<sup>1</sup> and obtain Eq. (7) in a more compact form:

 

*i i ik ik i k l i i*

*N N N i i i i ik k k i i k l i i*

1 1

 

*i k k ki l k l*

*i ii n v* , which obey the equation

 

 

 

1

 

combined chemical potential is introduced:

volume fractions

we can eliminate

Here  *N N*

1 1

Non-equilibrium thermodynamics is based on the entropy production expression

$$
\Sigma = \vec{f}\_e \cdot \nabla \left(\frac{1}{T}\right) - \sum\_{i=1}^{N} \vec{f}\_i \cdot \nabla \left(\frac{\mu\_i}{T}\right) \tag{1}
$$

where *eJ* is the energy flux, *<sup>i</sup> J* are the component material fluxes, *N* is the number of the components, *<sup>i</sup>* are the chemical potentials of components, and *T* is the temperature. The energy flux and the temperature distribution in the liquid are assumed to be known, whereas the material concentrations are determined by the continuity equations

$$\frac{\partial n\_i}{\partial t} = -\nabla \vec{f}\_i \tag{2}$$

Here *ni* is the numerical volume concentration of component *i* and *t* is time. Nonequilibrium thermodynamics defines the material flux as

$$\vec{J}\_i = -n\_i L\_i \nabla \left(\frac{\mu\_i}{T}\right) - n\_i L\_{iQ} \nabla \left(\frac{1}{T}\right) \tag{3}$$

where *Li* and *LiQ* are individual molecular kinetic coefficients. The second term on the righthand side of Eq. (3) represents the cross effect between material flux and heat flux. The chemical potentials are expressed through component concentrations and other physical parameters (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999):

$$
\nabla \mu\_{\kappa} = \sum\_{l=1}^{2} \frac{\partial \mu\_{k}}{\partial \mathbf{n}\_{l}} \nabla \mathbf{n}\_{l} - \overline{\mathbf{v}}\_{k} \nabla P + \frac{\partial \mu\_{k}}{\partial T} \nabla T \tag{4}
$$

Here *P* is the internal macroscopic pressure of the system and *v P k k* is the partial molecular volume, which is nearly equivalent to the specific molecular volume *<sup>k</sup> v* . Substituting Eq. (4) into Eq. (3), and using parameter *qi iQ i L L* , termed the molecular heat of transport, we obtain the equation for component material flux:

$$\vec{J}\_i = -\frac{n\_i L\_i}{T} \left[ \sum\_{k=1}^{N} \frac{\partial \mu\_i}{\partial n\_k} \nabla n\_k - \overline{v}\_i \nabla P + \left( \frac{\partial \mu\_i}{\partial T} - \frac{\mu\_i + q\_i}{T} \right) \nabla T \right] \tag{5}$$

Defining the relation between the heat of transport and thermodynamic parameters is a key problem because the Soret coefficient, which is the parameter that characterizes the distribution of components concentrations in a temperature gradient, is expressed through the heat of transport (De Groot, 1952; De Groot, Mazur, 1962). A number of studies that offer approaches to calculating the heat of transport are cited in (Pan S et al., 2007).

Eq. (5) must be augmented by an equation for the macroscopic pressure gradient in the system. The simplest possible approach is to consider the pressure to be constant (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Landau , Lifshitz, 1959), but pressure cannot be constant in a system with a non-uniform temperature and concentration. This issue is addressed with a well-known expression referred to as the

1 *<sup>N</sup>*

energy flux and the temperature distribution in the liquid are assumed to be known,

 

Here *ni* is the numerical volume concentration of component *i* and *t* is time. Non-

where *Li* and *LiQ* are individual molecular kinetic coefficients. The second term on the right-

The chemical potentials are expressed through component concentrations and other physical parameters (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999):

> 

molecular volume, which is nearly equivalent to the specific molecular volume *<sup>k</sup> v* . Substituting Eq. (4) into Eq. (3), and using parameter *qi iQ i L L* , termed the molecular heat

*n L <sup>q</sup> <sup>J</sup> n vP <sup>T</sup>*

*ii i i i i*

Defining the relation between the heat of transport and thermodynamic parameters is a key problem because the Soret coefficient, which is the parameter that characterizes the distribution of components concentrations in a temperature gradient, is expressed through the heat of transport (De Groot, 1952; De Groot, Mazur, 1962). A number of studies that offer

Eq. (5) must be augmented by an equation for the macroscopic pressure gradient in the system. The simplest possible approach is to consider the pressure to be constant (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Landau , Lifshitz, 1959), but pressure cannot be constant in a system with a non-uniform temperature and concentration. This issue is addressed with a well-known expression referred to as the

*k k l k*

*n vP T*

  *T n T T* (5)

hand side of Eq. (3) represents the cross effect between material flux and heat flux.

1

*l l*

Here *P* is the internal macroscopic pressure of the system and *v P k k*

2

of transport, we obtain the equation for component material flux:

*N*

1

*i k i k k*

approaches to calculating the heat of transport are cited in (Pan S et al., 2007).

 *<sup>i</sup> i*

*e i i*

 

1

*<sup>i</sup>* are the chemical potentials of components, and *T* is the temperature. The

*i*

*J J T T* (1)

*<sup>n</sup> <sup>J</sup> t* (2)

*<sup>i</sup> J* are the component material fluxes, *N* is the number of the

<sup>1</sup> *<sup>i</sup> J nL nL i ii i iQ T T* (3)

*n T* (4)

is the partial

 

Non-equilibrium thermodynamics is based on the entropy production expression

whereas the material concentrations are determined by the continuity equations

equilibrium thermodynamics defines the material flux as

where

components,

*eJ* is the energy flux,

Gibbs-Duhem equation (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Landau, Lifshitz, 1959; Ghorayeb, Firoozabadi, 2000; Pan S et al., 2007):

$$\nabla P = \sum\_{i=1}^{N} \boldsymbol{n}\_i \left( \sum\_{k=1}^{N} \frac{\partial \mu\_i}{\partial n\_k} \nabla \boldsymbol{n}\_k + \frac{\partial \mu\_i}{\partial T} \nabla T \right) \tag{6}$$

The Gibbs-Duhem equation defines the macroscopic pressure gradient in a thermodynamic system. In equilibrium thermodynamics the equation defines the potentiality of the thermodynamic functions (Kondepudi, Prigogine, 1999). In equilibrium thermodynamics the change in the thermodynamic function is determined only by the initial and final states of the systems, without consideration of the transition process itself. In non-equilibrium thermodynamics, Eq. (5) plays the role of expressing mechanical equilibrium in the system. According to the Prigogine theorem (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969), pressure gradient cancels the volume forces expressed as the gradients of the chemical potentials and provides mechanical equilibrium in a thermodynamically stable system. However, in a non-isothermal system, the same authors considered a constant pressure and the left- and right-hand side of Eq. (6) were assumed to be zero simultaneously, which is both physically and mathematically invalid.

Substituting Eq. (6) into Eq. (5) we obtain the following equation for material flux:

$$\vec{J}\_{i} = -\frac{\phi\_{i}L\_{i}}{\upsilon\_{i}T} \Biggl( (1-\phi\_{i}) \Bigg( \sum\_{k=1}^{N} \frac{\partial \mu\_{i}}{\partial \phi\_{k}} \nabla \phi\_{k} + \frac{\partial \mu\_{i}}{\partial T} \nabla T \bigg) - \sum\_{k \neq i}^{N} \frac{\upsilon\_{i}\phi\_{k}}{\upsilon\_{k}} \sum\_{l=1}^{N} \frac{\partial \mu\_{k}}{\partial \phi\_{l}} \nabla \phi\_{l} + \frac{\partial \mu\_{k}}{\partial T} \nabla T - \left(\mu\_{i} + q\_{i}\right) \frac{\nabla T}{T} \bigg) \tag{7}$$

In Eq. (7), the numeric volume concentrations of the components are replaced by their volume fractions*i ii n v* , which obey the equation

$$\sum\_{i=1}^{N} \phi\_i = 1 \tag{8}$$

Using Eq. (8) and the standard rule of differentiation of a composite function

*i*

$$\frac{\partial \mu\_k \Big\lfloor \phi\_l, \phi\_l \left( \ldots, \phi\_l \ldots \right) \Big\rfloor}{\partial \phi\_l} \nabla \phi\_l = \frac{\partial \mu\_k}{\partial \phi\_l} \nabla \phi\_l + \frac{\partial \mu\_k}{\partial \phi\_l} \frac{\partial \phi\_l}{\partial \phi\_l} \nabla \phi\_l = 2 \frac{\partial \mu\_k}{\partial \phi\_l} \nabla \phi\_l \tag{9}$$

we can eliminate <sup>1</sup> and obtain Eq. (7) in a more compact form:

$$\vec{J}\_{i} = -\frac{L\_{i}}{T} \sum\_{k}^{N} \frac{\phi\_{i}}{\upsilon\_{i}} \left[ \phi\_{k} \left( 2 \sum\_{l>1}^{N} \frac{\partial \mu\_{ik}^{\*}}{\partial \phi\_{l}} \nabla \phi\_{l} + \frac{\partial \mu\_{ik}^{\*}}{\partial T} \nabla T \right) - \left( \mu\_{i} + q\_{i} \right) \frac{\nabla T}{T} \right] \tag{10}$$

Here <sup>1</sup> is expressed through the other volume fractions using Eq. (8), and the following combined chemical potential is introduced:

$$
\mu\_{ik}^\* = \mu\_i - \frac{\upsilon\_i}{\upsilon\_k} \mu\_k \tag{11}
$$

We note that the volume fraction selected for elimination is arbitrary (any other volume fraction can be eliminated in the same manner), and that in subsequent mathematical

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 347

even for large (micron size) particles, the energy difference is no more than a few percent of *kT.* But the local equilibrium is determined by processes at molecular level, as will be

**Thermodynamic equations of material transport with the Soret coefficient as** 

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

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

zero. In both these cases, the Gibbs-Duhem equation can no longer be used to determine the

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

> 

Summing Eq. (16) for each component and utilizing Eq. (8) we obtain the following equation

 

Comparing Eq. (18) with Eq. (15) for a stationary mixture shows that former contains an

2 *N N <sup>N</sup> ij ij i ii*

 1 2 *N i ii i i*

*i k k i P JT L T Lv*

*N N N i i i i k iii*

1 1 1

 

*t T <sup>T</sup>* (18)

 

proportional to the total material flux through the open

*jjj k kkk*

*j k k k <sup>L</sup> JT v L T vL*

1 1 1

*k i*

*<sup>L</sup> vP T*

 

*t T <sup>T</sup>* (16)

 

*T* (17)

 

 

 1

*i J vJ* , is

*i i*

*N*

*<sup>i</sup> J* can be nonzero even though the total material flux in the system,

**3. Dynamic pressure gradient in open and non-stationary systems:** 

discussed below, and this argumentation cannot be accepted.

pressure in the system, and an alternate approach is necessary.

transport equations obtained by non-equilibrium thermodynamics.

for the dynamic pressure gradient in an open non-stationary system:

additional drift term

 

*k*

1

*iii N*

*vLJ*

*kkk*

*v L*

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

2

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

*k k*

**a thermodynamic parameter** 

and open system.

fluxes

expressions, we express the volume fraction of the first component through that of the others using Eq. (8).

Equations for the material fluxes are usually augmented by the following equation, which relates the material fluxes of components (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Ghorayeb, Firoozabadi, 2000; Pan S et al., 2007):

$$\sum\_{i=1}^{N} v\_i \vec{J}\_i = 0 \tag{12}$$

Eq. (12) expresses the conservation mass in the considered system and the absence of any hydrodynamic mass transfer. Also, Eq. (12) is used to eliminate one of the components from the series of component fluxes expressed by Eq. (10). That material flux that is replaced in this way is arbitrary, and the resulting concentration distribution will depend on which flux is selected. The result is not significant in a dilute system, but in non-dilute systems this practice renders an ambiguous description of the material transport processes.

In addition to being mathematically inconsistent with Eq. (12) because there are *N+1* equations [i.e., *N* Eq. (10) plus Eq. (12)] for *N-1* independent component concentrations, Eq. (10) predicts a drift in a pure liquid subjected to a temperature gradient. Thus, at 1 *<sup>i</sup>* Eq. (10) predicts

$$\vec{J}\_{i} = -\frac{\mathcal{L}\_{i}}{T} \frac{\left(\mu\_{i} + q\_{i}\right)}{\upsilon\_{i}} \frac{\nabla T}{T} \tag{13}$$

This result contradicts the basic principle of local equilibrium, and the notion of thermodiffusion as an effect that takes place in mixtures only. Moreover, Eq. (13) indicates that the achievement of a stationary state in a closed system is impossible, since material transport will occur even in a pure liquid.

The contradiction that a system cannot reach a stationary state, as expressed in Eq. (13), can be eliminated if we assume

$$
\eta\_i = -\mu\_i \tag{14}
$$

With such an assumption Eq. (10) can be cast in the following form:

$$\vec{J}\_i = -\frac{L\_i}{T} \sum\_k^N \frac{\phi\_l \phi\_k}{\upsilon\_i} \left( 2 \sum\_{l>1}^N \frac{\partial \mu\_{lk}^\*}{\partial \phi\_l} \nabla \phi\_l + \frac{\partial \mu\_{lk}^\*}{\partial T} \nabla T \right) \tag{15}$$

Because the kinetic coefficients are usually calculated independently from thermodynamics, the material fluxes expressed by Eq. (15) cannot satisfy Eq. (12) for the general case. But in a closed and stationary system, where <sup>0</sup> *<sup>i</sup> <sup>J</sup>* , Eqs. (12) and (15) become consistent. In this case, any component flux can be expressed by Eq. (15) through summation of the other equations. The condition of mechanical equilibrium for an isothermal homogeneous system, as well as the use of Eqs. (l) – (6) for non-isothermal systems, are closely related to the principle of local equilibrium (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969). As argued in (Duhr, Braun, 2006; Weinert, Braun; 2008), thermodiffusion violates local equilibrium because the change in free energy across a particle is typically comparable to the thermal energy of the particle. However, their calculations predict that

expressions, we express the volume fraction of the first component through that of the

Equations for the material fluxes are usually augmented by the following equation, which relates the material fluxes of components (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Ghorayeb, Firoozabadi, 2000; Pan S et al., 2007):

Eq. (12) expresses the conservation mass in the considered system and the absence of any hydrodynamic mass transfer. Also, Eq. (12) is used to eliminate one of the components from the series of component fluxes expressed by Eq. (10). That material flux that is replaced in this way is arbitrary, and the resulting concentration distribution will depend on which flux is selected. The result is not significant in a dilute system, but in non-dilute systems this

In addition to being mathematically inconsistent with Eq. (12) because there are *N+1* equations [i.e., *N* Eq. (10) plus Eq. (12)] for *N-1* independent component concentrations, Eq.

> *<sup>i</sup> i i*

This result contradicts the basic principle of local equilibrium, and the notion of thermodiffusion as an effect that takes place in mixtures only. Moreover, Eq. (13) indicates that the achievement of a stationary state in a closed system is impossible, since material

The contradiction that a system cannot reach a stationary state, as expressed in Eq. (13), can

*qi i* 

 

1 2 *N N*

Because the kinetic coefficients are usually calculated independently from thermodynamics, the material fluxes expressed by Eq. (15) cannot satisfy Eq. (12) for the general case. But in a closed and stationary system, where <sup>0</sup> *<sup>i</sup> <sup>J</sup>* , Eqs. (12) and (15) become consistent. In this case, any component flux can be expressed by Eq. (15) through summation of the other equations. The condition of mechanical equilibrium for an isothermal homogeneous system, as well as the use of Eqs. (l) – (6) for non-isothermal systems, are closely related to the principle of local equilibrium (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969). As argued in (Duhr, Braun, 2006; Weinert, Braun; 2008), thermodiffusion violates local equilibrium because the change in free energy across a particle is typically comparable to the thermal energy of the particle. However, their calculations predict that

 

*i ik ik ik*

 

 

*<sup>L</sup> J T T v <sup>T</sup>* (15)

*i*

0

*v J* (12)

*<sup>L</sup> <sup>q</sup> <sup>T</sup> <sup>J</sup> Tv T* (13)

(14)

1 *<sup>i</sup>* Eq.

 1

*i*

practice renders an ambiguous description of the material transport processes.

*i*

With such an assumption Eq. (10) can be cast in the following form:

*i l k l i l*

(10) predicts a drift in a pure liquid subjected to a temperature gradient. Thus, at

*N i i*

others using Eq. (8).

(10) predicts

transport will occur even in a pure liquid.

be eliminated if we assume

even for large (micron size) particles, the energy difference is no more than a few percent of *kT.* But the local equilibrium is determined by processes at molecular level, as will be discussed below, and this argumentation cannot be accepted.
