**4. Statistical mechanics of material transport: Chemical potentials at constant volume and pressure and the Laplace component of pressure in a molecular force field**

The chemical potential at constant volume can be calculated using a modification of an expression derived in (Kirkwood, Boggs, 1942; Fisher, 1964):

$$
\mu\_{iV} = \mu\_{0i} + \int\_0 d\lambda \sum\_{j=1}^N \frac{\phi\_j}{\upsilon\_j} \int\_{V\_{out}}^v g\_{ij}(\vec{r}, \lambda) \Phi\_{ij}(r) \, dv \tag{21}
$$

Here

348 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

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

independent of position in the system but is determined by material transfer across the

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

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

2 0

 

*T*

*T* (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

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

 

2 2 2 1

*T*

21

*P*

21

*P*

*T S* (20)

 2

*N N <sup>N</sup> ij ij k*

1 1 1

*k j k j*

friction in such a system is not considered in the Gibbs-Duhem equation.

in Eq. (17) describes the contribution of that drift to the pressure

*J* is

 *J* 0

system. The term

and 0

coefficient as

 

*k*

*N*

1

*JT*

*kkk*

system boundaries, which may vary over time.

*<sup>t</sup>* , Eq. (18) is transformed into

be observed using dynamic temperature gratings.

*v L*

$$
\mu\_{0i} = -\frac{\Im}{2} kT \ln \frac{2\pi m\_i kT}{h^2} + kT \ln \frac{\phi\_i}{v\_i} - kT \ln Z\_{vt}^i - kT \ln Z\_{vib}^i \tag{22}
$$

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> Z* and *vib <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 (Landau, Lifshitz, 1980)

$$Z\_{rot} = \frac{\sqrt{\pi}}{\gamma h^3} \sqrt{\left(8\pi^2 kT\right)^3 I\_1 I\_2 I\_3} \tag{23}$$

where is the symmetry value, which is the number of possible rotations about the symmetry axes carrying the molecule into itself. For H2O and C2H5OH, 2 ; for NH3, 3 ; for CH4 and C6H6, 12 . 1 2 *I I*, , and <sup>3</sup>*I* are the principal values of the tensor of the moment of inertia.

In Eq. (21), parameter describes the gradual "switching on" of the intermolecular 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 liquid and the center of the considered molecule; , *ij g r* is the pair correlative function, which expresses the probability of finding a molecule of the surrounding liquid at *<sup>r</sup>* ( *r r* ) if the considered molecule is placed at *r* 0 ; and *ij* is the molecular interaction potential, known as the London potential (Ross, Morrison, 1988):

$$\Phi\_{ij} = -\varepsilon\_{ij} \left(\frac{\sigma\_{ij}}{r}\right)^{6} \tag{24}$$

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 351

Using Eqs. (21) and (22), along with the definition of a virtual particle outlined above, we

\*

particle, respectively. In Eq. (27), the total interaction potential *Nik k <sup>j</sup>* of the molecules

*j*

This approximation corresponds to the virtual particle having the size of a molecule of the

In further development of the microscopic calculations it is important that the chemical potential be defined at constant pressure. Chemical potentials at constant pressure are

*iV* by the expression

*i out iP iV i V*

Here *<sup>i</sup>* is the local pressure distribution around the molecule. Eq. (29) expresses the relation between the forces acting on a molecular particle at constant versus changing local pressure. This equation is a simple generalization of a known equation (Haase, 1969) in which the

Next we calculate the local pressure distribution *<sup>i</sup>* , which is widely used in hydrodynamic models of kinetic effects in liquids (Ruckenstein, 1981; Anderson, 1989; Schimpf, Semenov, 2004; Semenov, Schimpf, 2000, 2005). The local pressure distribution is usually obtained from the condition of the local mechanical equilibrium in the liquid around i'th molecular

Semenov, 2010) the local pressure distribution is used in a thermodynamic approach, where it is obtained by formulating the condition for establishing local equilibrium in a thin layer of thickness *l* and area *S* when the layer shifts from position *r* to position *r+dr*. In this case, local equilibrium expresses the local conservation of specific free energy

1

 

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

0

*<sup>v</sup>* . In (Semenov, Schimpf, 2009;

*r*

*<sup>v</sup>* in such a shift when the isothermal system is placed in a force

 

pressure gradient is assumed to be constant along a length about the particle size.

In the layer forming a closed surface, the change in the free energy is written as:

*ikV N ij kj*

*ik i i ik rot out out*

*kT r dv r dv m vv Z*

*N k j j j V Vj*

\* *ikV* as

*<sup>N</sup>* . We will use the approximation

*<sup>N</sup> ij Nik kj kj <sup>r</sup>* (28)

 

(27)

*dv* (29)

1 1

*<sup>N</sup> Z* are the mass and the rotational statistical sum of the virtual

*i N N i i j j N*

6

*rot kj*

can define the combined chemical potential at constant volume

*m Z*

*ik j*

i'th component and the energetic parameter of the k'th component.

particle, a condition that is written as

<sup>3</sup> ln ln ln

*rot*

included in the virtual particle is written as *ik*

2

where *N k ik ik m mN* and *ik*

related to those at constant volume

 

*i i ij*

*Fr r r*

field of the i'th molecule.

1 *<sup>N</sup> <sup>j</sup>*

*j j*

Here *ij* is the energy of interaction and *ij* is the minimal molecular approach distance. In the integration over *<sup>i</sup> Vout* , the lower limit is *ij r* .

There is no satisfactory simple method for calculating the pair correlation function in liquids, although it should approach unity at infinity. We will approximate it as

$$\log\_{ij}\left(r,\mathcal{A}\right) = 1\tag{25}$$

With this approximation we assume that the local distribution of solvent molecules is not disturbed by the particle under consideration. The approximation is used widely in the theory of liquids and its effectiveness has been shown. For example, in (Bringuier, Bourdon, 2003, 2007), it was used in a kinetic approach to define the thermodiffusion of colloidal particles. In (Schimpf, Semenov, 2004; Semenov, Schimpf, 2000, 2005) the approximation was used in a hydrodynamic theory to define thermodiffusion in polymer solutions. The approximation of constant local density is also used in the theory of regular solutions (Kirkwood, 1939). With this approximation we obtain

$$
\mu\_{iV} = \mu\_{0i} + \sum\_{j=1}^{N} \frac{\oint\_{j} \Phi\_{j}}{\upsilon\_{j}} \int\_{V\_{out}^{i}}^{v} \Phi\_{ij}(r) \, dv \tag{26}
$$

The terms under the summation sign are a simple modification of the expression obtained in (Bringuier, Bourdon, 2003, 2007).

In our calculations, we will use the fact that there is certain symmetry between the chemical potentials contained in Eq. (11). The term *i k v v* can be written as *Nik k* , where *<sup>i</sup> ik <sup>v</sup> <sup>N</sup> v* is

*k k* the number of the molecules of the k'th component that can be placed within the volume *<sup>i</sup> v* but are displaced by a molecule of i'th component. Using the known result that free energy is the sum of the chemical potentials we can say that *Nik k* is the free energy or chemical potential of a virtual molecular particle consisting of molecules of the k'th component displaced by a molecule of the i'th component. For this reason we can extend the results obtained in the calculations of molecular chemical potential *iV* of the second component to calculations of parameter *Nik kV* by a simple change in the respective designations *i k* . Regarding the concentration of these virtual particles, there are at least two approaches allowed:

a. we can assume that the volume fraction of the virtual particles is equal to the volume fraction of the real particles that displace molecules of the k'th component, i.e., their numeric concentration is *i i v* . This approach means that only the actually displaced

molecules are taken into account, and that they are each distinguishable from molecules of the k'th component in the surrounding liquid.

b. we can take into account the indistinguishability of the virtual particles. In this approach any group of the *Nik* molecules of the k'th component can be considered as a virtual particle. In this case, the numeric volume concentration of these virtual molecules is *k i v* .

We have chosen to use the more general assumption b).

 *ij r* . There is no satisfactory simple method for calculating the pair correlation function in

> *g r ij* , 1

With this approximation we assume that the local distribution of solvent molecules is not disturbed by the particle under consideration. The approximation is used widely in the theory of liquids and its effectiveness has been shown. For example, in (Bringuier, Bourdon, 2003, 2007), it was used in a kinetic approach to define the thermodiffusion of colloidal particles. In (Schimpf, Semenov, 2004; Semenov, Schimpf, 2000, 2005) the approximation was used in a hydrodynamic theory to define thermodiffusion in polymer solutions. The approximation of constant local density is also used in the theory of regular solutions

> 

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

1 *i out*

The terms under the summation sign are a simple modification of the expression obtained in

In our calculations, we will use the fact that there is certain symmetry between the chemical

the number of the molecules of the k'th component that can be placed within the volume *<sup>i</sup> v* but are displaced by a molecule of i'th component. Using the known result that free

chemical potential of a virtual molecular particle consisting of molecules of the k'th component displaced by a molecule of the i'th component. For this reason we can extend the

designations *i k* . Regarding the concentration of these virtual particles, there are at least

a. we can assume that the volume fraction of the virtual particles is equal to the volume fraction of the real particles that displace molecules of the k'th component, i.e., their

molecules are taken into account, and that they are each distinguishable from molecules

b. we can take into account the indistinguishability of the virtual particles. In this approach any group of the *Nik* molecules of the k'th component can be considered as a virtual particle. In this case, the numeric volume concentration of these virtual

*i k k v v*

*r dv*

can be written as *Nik k*

. This approach means that only the actually displaced

*ij* is the minimal molecular approach distance. In

(25)

*v* (26)

by a simple change in the respective

 , where *<sup>i</sup> ik*

is the free energy or

*iV* of the second

*k*

*<sup>v</sup> <sup>N</sup> v* is

liquids, although it should approach unity at infinity. We will approximate it as

Here 

*ij* is the energy of interaction and

the integration over *<sup>i</sup> Vout* , the lower limit is

(Kirkwood, 1939). With this approximation we obtain

(Bringuier, Bourdon, 2003, 2007).

two approaches allowed:

molecules is

numeric concentration is

*k i v* .

potentials contained in Eq. (11). The term

component to calculations of parameter *Nik kV*

energy is the sum of the chemical potentials we can say that *Nik k*

results obtained in the calculations of molecular chemical potential

*i i v*

of the k'th component in the surrounding liquid.

We have chosen to use the more general assumption b).

 0 Using Eqs. (21) and (22), along with the definition of a virtual particle outlined above, we can define the combined chemical potential at constant volume \* *ikV* as

$$\mu\_{\rm ikV}^{\*} = +kT \left( \frac{3}{2} \ln \frac{m\_i}{m\_{\rm N\_k}} + \ln \frac{\phi\_i}{\phi\_k} + \ln \frac{Z\_{\rm nt}^i}{Z\_{\rm nt}^{N\_k}} \right) + \sum\_{j=1}^{N} \frac{\phi\_j}{\upsilon\_j} \int\_{V\_{\rm nt}^i}^{\upsilon\_j} \Phi\_{ij}(r) dr - \sum\_{j=1}^{N} \frac{\phi\_j}{\upsilon\_j} \int\_{V\_{\rm nt}^i}^{\upsilon\_j} \Phi\_{kj}^{N\_{li}}(r) dr \tag{27}$$

where *N k ik ik m mN* and *ik rot <sup>N</sup> Z* are the mass and the rotational statistical sum of the virtual particle, respectively. In Eq. (27), the total interaction potential *Nik k <sup>j</sup>* of the molecules included in the virtual particle is written as *ik j <sup>N</sup>* . We will use the approximation

$$\boldsymbol{\Phi}\_{\boldsymbol{j}}^{N\_{\vec{k}}} = \boldsymbol{N}\_{i\boldsymbol{k}} \boldsymbol{\Phi}\_{\boldsymbol{k}\boldsymbol{j}} = -\boldsymbol{\varepsilon}\_{\boldsymbol{k}\boldsymbol{j}} \left(\frac{\sigma\_{\boldsymbol{i}\boldsymbol{j}}}{r}\right)^{\boldsymbol{\Phi}} \tag{28}$$

This approximation corresponds to the virtual particle having the size of a molecule of the i'th component and the energetic parameter of the k'th component.

In further development of the microscopic calculations it is important that the chemical potential be defined at constant pressure. Chemical potentials at constant pressure are related to those at constant volume *iV* by the expression

$$
\nabla \mu\_{i\mathcal{P}} = \nabla \mu\_{i\mathcal{V}} + \int\_{V\_{out}^i} \nabla \Pi\_i d\upsilon \tag{29}
$$

Here *<sup>i</sup>* is the local pressure distribution around the molecule. Eq. (29) expresses the relation between the forces acting on a molecular particle at constant versus changing local pressure. This equation is a simple generalization of a known equation (Haase, 1969) in which the pressure gradient is assumed to be constant along a length about the particle size.

Next we calculate the local pressure distribution *<sup>i</sup>* , which is widely used in hydrodynamic models of kinetic effects in liquids (Ruckenstein, 1981; Anderson, 1989; Schimpf, Semenov, 2004; Semenov, Schimpf, 2000, 2005). The local pressure distribution is usually obtained from the condition of the local mechanical equilibrium in the liquid around i'th molecular

$$\text{Particle, a condition that is written as}\\
\nabla \left[ \Pi\_i + \sum\_{j=1}^{N} \frac{\phi\_j}{\upsilon\_j} \Phi\_{ij}(r) \right] = 0 \quad \text{In (Semenerov, Schimplf, 2009;)}$$

Semenov, 2010) the local pressure distribution is used in a thermodynamic approach, where it is obtained by formulating the condition for establishing local equilibrium in a thin layer of thickness *l* and area *S* when the layer shifts from position *r* to position *r+dr*. In this case, local equilibrium expresses the local conservation of specific free energy *<sup>N</sup> <sup>j</sup> i i ij Fr r r <sup>v</sup>* in such a shift when the isothermal system is placed in a force

field of the i'th molecule.

1

*j j*

In the layer forming a closed surface, the change in the free energy is written as:

Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 353

where *N N* 1 21 is the number of solvent molecules displaced by molecule of the solute,

we expect the use of assumption a) in Section 3 for the concentration of virtual particles will

 1 1 1 1 1

Using Eqs. (29), (34), we obtain the following expression for the temperature-induced

component of the bulk temperature gradient. After substituting the expressions for the interaction potentials defined by Eqs. (23), (24), and (28) into Eq. (36), we obtain the

> 

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

In Eq. (37), the subscripts *2* and *N*1 are used again to denote the real and virtual particle,

The Soret coefficient expressed by Eq. (37) contains two main terms. The first term corresponds to the temperature derivative of the part of the chemical potential related to the solute kinetic energy. In turn, this kinetic term contains the contributions related to the translational and rotational movements of the solute in the solvent. The second term is related to the potential interaction of solute with solvent molecules. This potential term has the same structure as those obtained by the hydrodynamic approach in (Schimpf, Semenov,

According to Eq. (37), both positive (from hot to cold wall) and negative (from cold to hot wall) thermodiffusion is possible. The molecules with larger mass ( <sup>1</sup> *m m* <sup>2</sup> *<sup>N</sup>* ) and with a

thermodiffusion. Thus, dilute aqueous solutions are expected to demonstrate positive thermophoresis. In (Ning, Wiegand, 2006), dilute aqueous solutions of acetone and dimethyl sulfoxide were shown to undergo positive thermophoresis. In that paper, a very high value of the Hildebrand parameter is given as an indication of the strong intermolecular interaction for water. More specifically, the value of the Hildebrand parameter exceeds by

 

*rot out*

1

1

*rot*

*N V*

<sup>1</sup> is the thermal expansion coefficient for the solvent and

following expression for the Soret coefficient in the diluted binary system:

*m III*

2 2 18 *N*

*<sup>N</sup> <sup>N</sup>*

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

*<sup>m</sup> <sup>Z</sup> dv r r k T <sup>T</sup> dr*

In a dilute binary mixture, the equation for local pressure [Eq. (32)] takes the form

*i*

 

1

where index *i* is related to the virtual particle or solute.

*P N*

*j*

gradient of the combined chemical potential of the diluted molecular mixture:

*N r i i*

*<sup>N</sup>* is the potential of interaction between the virtual particle and a molecule of the solvent.

is also used in deriving Eq. (34). Because ln 1

*r r dr v vr*

> 

2 1 21 11 1 <sup>3</sup> ' ' ln ln ' 2 '

2 ' '

'

 

2 123 1 12

*T m III v kT* (37)

2 3

1 12

123 2 1 12 2 11

*m vr <sup>Z</sup>* (36)

 

1

11 12 ) should demonstrate positive

*r N*

 at0 ,

*T* is the tangential

(35)

 <sup>1</sup> 11

Here

respectively.

The relation

<sup>1</sup> 1 

yield a reasonable physical result.

*T*

2004; Semenov, Schimpf, 2005).

stronger interactions between solvent molecules (

two-fold the respective parameters for other components.

*S*

$$dF\_i(r) = \nabla \left[ \sum\_{j=1}^{N} \frac{\phi\_j}{\upsilon\_j} \Phi\_{ij} \left( r \right) + \nabla \Pi\_i \left. \left| lS dr + \sum\_{j=1}^{N} \right| \frac{\phi\_j}{\upsilon\_j} \Phi\_{ij} \left( r \right) \right| dS = 0 \tag{30} \right]$$

where we consider changes in free energy due to both a change in the parameters of the layer volume ( *dV Sdr* ) and a change *dS* in the area of the closed layer. For a spherical layer, the changes in volume and surface area are related as *dV rdS* 2 , and we obtain the following modified equation of equilibrium for a closed spherical surface:

$$\nabla \left[ \sum\_{j=1}^{N} \frac{\phi\_j}{\upsilon\_j} \Phi\_{ij} \left( r \right) + \Pi\_i \right] + 2 \sum\_{j=1}^{N} \frac{\phi\_j}{\upsilon\_j} \frac{\Phi\_{ij} \left( r \right)}{r} \vec{r}\_0 = 0 \tag{31}$$

where <sup>0</sup>*r* is the unit radial vector. The pressure gradient related to the change in surface area has the same nature as the Laplace pressure gradient discussed in (Landau, Lifshitz, 1980). Solving Eq. (31), we obtain

$$\Pi\_i = -\sum\_{j=1}^{N} \frac{\phi\_j}{\upsilon\_j} \left[ \Phi\_{ij} \left( r \right) + \int\_{\alpha} \frac{2 \Phi\_{ij} \left( r' \right)}{r'} dr' \right] \tag{32}$$

Substituting the pressure gradient from Eq. (32) into Eq. (29), and using Eqs. (24), (27), and (28), we obtain a general expression for the gradient in chemical potential at constant pressure in a non-isothermal and non-homogeneous system. We will not write the general expression here, rather we will derive the expression for binary systems.
