Thermodynamic studies on thin liquid films. I. General formulation

Abstract

From the viewpoint that thermodynamic study is essential to elucidate the structure and properties of thin liquid films, thermodynamic equations based on a new convention and employing pressure as a thermodynamic variable are developed for adsorption at film interfaces of a plane-parallel film. The equations together with quasi-thermodynamic ones correlate the dependence of film tension on concentration, temperature, pressure, and disjoining pressure to film density, entropy and volume changes associated with adsorption, and thermodynamic film thickness, respectively. Based on the formulation adopting pressure as a variable, equations are also derived for the differences in thermodynamic quantity between the film and the bulk interfaces coexisting at equilibrium.

Appendices

Appendix I

Let us consider the hypothetical system mechanically equivalent to the actual system, which consists of three phases A, B, and L and two surfaces of tension for the film interfaces at which film interfacial tension is defined as shown in Fig. 1b. We define a rectangular coordinate system (x, y, z) with the z axis normal to the plane-parallel film and directed from the phase A to the phase B and with the (x, y) plane at the surface of tension for the film; positions of the surfaces of tension for the film interfaces are at ${\text{z}} = - {\left( {v^{{\text{L}}}_{{\text{A}}} + v^{{{\text{F}}{\text{,L}}}}_{{\text{A}}} } \right)}\;{\text{and}}\;{\text{z}} = v^{{\text{L}}}_{{\text{B}}} + v^{{{\text{F}}{\text{,L}}}}_{{\text{B}}}$. Then the work δW done on the system by a small strain is given by

$$\begin{array}{*{20}l} {{\delta {\text{W}}} \hfill} & { = \hfill} & {{ - {\int {{\int {{\int_{ - \infty }^\infty {{\left( {p_{{\text{T}}} e_{{xx}} + p_{{\text{T}}} e_{{yy}} + p_{{\text{N}}} e_{{zz}} } \right)}{\text{d}}x{\text{d}}y{\text{d}}z} }} }} }} \hfill} \\ {{} \hfill} & { = \hfill} & {{ - {\int {{\int {{\int {p^{{{\text{ABL}}}} {\left( {e_{{xx}} + e_{{yy}} + e_{{zz}} } \right)}{\text{d}}x{\text{d}}y{\text{d}}z} }} }} }} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + {\int {{\left( {p^{{{\text{ABL}}}} - p_{{\text{T}}} } \right)}{\text{d}}z{\int {{\int {{\left( {e_{{xx}} + e_{{yy}} } \right)}{\text{d}}x{\text{d}}y} }} }} }} \hfill} \\ {{} \hfill} & { = \hfill} & {{ - \left[ {p{\int {{\int {{\int {{\left( {e_{{xx}} + e_{{yy}} + e_{{zz}} } \right)}{\text{d}}x{\text{d}}y{\text{d}}z} }} }} }} \right.} \hfill} \\ {{} \hfill} & {{} \hfill} & {{\left. { + {\left( {p^{{\text{L}}} - p} \right)}{\int_{ - {\left( {v^{{\text{L}}}_{{\text{A}}} + v^{{{\text{F}}{\text{,L}}}}_{{\text{A}}} } \right)}}^{v^{{\text{L}}}_{{\text{B}}} + v^{{{\text{F}}{\text{,L}}}}_{{\text{B}}} } {{\text{d}}z{\int {{\int {{\left( {e_{{xx}} + e_{{yy}} } \right)}{\text{d}}x{\text{d}}y} }} }} }} \right]} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + {\left[ {{\int_{ - \infty }^0 {{\left( {p^{{{\text{ABL}}}} - p_{{\text{T}}} } \right)}{\text{d}}z + {\int_0^\infty {{\left( {p^{{{\text{ABL}}}} - p_{{\text{T}}} } \right)}{\text{d}}z} }} }} \right]}{\int {{\int {{\left( {e_{{xx}} + e_{{yy}} } \right)}{\text{d}}x{\text{d}}y} }} }} \hfill} \\ {{} \hfill} & { = \hfill} & {{ - p\delta {\text{V}} + \Pi \tau \delta \sigma + {\left( {\gamma ^{{\text{F}}}_{{\text{A}}} + \gamma ^{{\text{F}}}_{{\text{B}}} } \right)}\delta \sigma } \hfill} \\ \end{array} $$

(I.1)

where p _T is the tangential component of the pressure tensor at z, directed along the film, p _N is that normal to the film, and e _xx , e _yy , and e _zz are the diagonal components in the matrix of strain tensor. From the condition of mechanical equilibrium, p _N can be expressed by

$$ \begin{array}{*{20}l} {{p_{{\text{N}}} {\left( z \right)}} \hfill} & { = \hfill} & {{p^{{{\text{ABL}}}} {\left( z \right)}} \hfill} \\ {{} \hfill} & { = \hfill} & {{{\left[ {1 - A{\left( {z + v^{{\text{L}}}_{{\text{A}}} + v^{{{\text{F}}{\text{,L}}}}_{{\text{A}}} } \right)}p} \right]}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + {\left[ {A{\left( {z + v^{{\text{L}}}_{{\text{A}}} + v^{{{\text{F}}{\text{,L}}}}_{{\text{A}}} } \right)} - A{\left( {z - v^{{\text{L}}}_{{\text{B}}} - v^{{{\text{F}}{\text{,L}}}}_{{\text{B}}} } \right)}p^{{\text{L}}} } \right]}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + A{\left( {z - v^{{\text{L}}}_{{\text{B}}} - v^{{{\text{F}}{\text{,L}}}}_{{\text{B}}} } \right)}p} \hfill} \\ \end{array} $$

(I.2)

A(z) being the step function

$$ \begin{array}{*{20}l} {{A{\left( z \right)}} \hfill} & { = \hfill} & {{1,} \hfill} & {{{\left( {z \geqslant 0} \right)}} \hfill} \\ {{} \hfill} & { = \hfill} & {{0,} \hfill} & {{{\left( {z < 0} \right)}} \hfill} \\ \end{array} . $$

(I.3)

Here we used the relations

$$\gamma ^{{\text{F}}}_{{\text{A}}} = {\int_{ - \infty }^0 {{\left( {p_{{\text{N}}} - p_{{\text{T}}} } \right)}{\text{d}}z,\;\;\;\;\;\gamma ^{{\text{F}}}_{{\text{B}}} = {\int_0^\infty {{\left( {p_{{\text{N}}} - p_{{\text{T}}} } \right)}{\text{d}}z} }} }$$

(I.4)

and

$$ \begin{array}{*{20}l} {{\delta {\text{V}} = {\int {{\int {{\int {{\left( {e_{{xx}} + e_{{yy}} + e_{{zz}} } \right)}{\text{d}}x{\text{d}}y{\text{d}}z,} }} }} }} \hfill} \\ {{\delta \sigma = {\int {{\int {{\left( {e_{{xx}} + e_{{yy}} } \right)}{\text{d}}x{\text{d}}y,} }} }} \hfill} \\ \end{array} $$

(I.5)

[34].

For the internal energy U of the system, we have from Eq. [I.1]

$$ \begin{array}{*{20}l} {{{\text{d}}U = } \hfill} & {{T{\text{d}}S - p{\text{d}}V + \Pi \tau {\text{d}}\sigma + {\left( {\gamma ^{{\text{F}}}_{{\text{A}}} + \gamma ^{{\text{F}}}_{{\text{B}}} } \right)}{\text{d}}\sigma } \hfill} \\ {{} \hfill} & {{ + \mu _{a} {\text{d}}n_{a} + \mu _{b} {\text{d}}n_{b} + \mu _{l} {\text{d}}n_{l} + {\sum\limits_{i = 1}^c {\mu _{i} {\text{d}}n_{i} } }} \hfill} \\ \end{array} . $$

(I.6)

The enthalpy H and Gibbs free energy G are defined by

$$H = U + pV - \Pi \tau \sigma - {\left( {\gamma ^{{\text{F}}}_{{\text{A}}} + \gamma ^{{\text{F}}}_{{\text{B}}} } \right)}\sigma $$

(I.7)

and

$$ G = H - TS. $$

(I.8)

Hence

$$ \begin{array}{*{20}l} {{{\text{d}}G = } \hfill} & {{ - S{\text{d}}T + V{\text{d}}p - \sigma \tau {\text{d}}\Pi - \sigma \Pi {\text{d}}\tau - \sigma {\text{d}}{\left( {\gamma ^{{\text{F}}}_{{\text{A}}} + \gamma ^{{\text{F}}}_{{\text{B}}} } \right)}} \hfill} \\ {{} \hfill} & {{ + \mu _{a} {\text{d}}n_{a} + \mu _{b} {\text{d}}n_{b} + \mu _{l} {\text{d}}n_{l} + {\sum\limits_{i = 1}^c {\mu _{i} {\text{d}}n_{i} } }} \hfill} \\ \end{array} . $$

(I.9)

It follows by Euler’s theorem that

$$G = n_{a} \mu _{a} + n_{b} \mu _{b} + n_{l} \mu _{l} + {\sum\limits_{i = 1}^c {n_{i} \mu _{i} } }$$

(I.10)

Combination of Eqs. (I.9) and (I.10) yields Eq. (14).

Appendix II

The chemical potential of component i can be expressed by

$$ \begin{array}{*{20}c} {{{\text{d}}\mu _{i} }} & { = } & {{s^{{\text{A}}}_{i} {\text{d}}T + v^{{\text{A}}}_{i} {\text{d}}p + \mu ^{{\text{A}}}_{{ib}} {\text{d}}m^{{\text{A}}}_{b} + \mu ^{{\text{A}}}_{{il}} {\text{d}}m^{{\text{A}}}_{l} + {\sum\limits_{j = 1}^c {\mu ^{{\text{A}}}_{{ij}} {\text{d}}m^{{\text{A}}}_{j} } }}} \\ \end{array} $$

(II.1)

$$ \begin{array}{*{20}c} {{}} & { = } & {{ - s^{{\text{B}}}_{i} {\text{d}}T + v^{{\text{B}}}_{i} {\text{d}}p + \mu ^{{\text{B}}}_{{ia}} {\text{d}}m^{{\text{B}}}_{a} + \mu ^{{\text{B}}}_{{il}} {\text{d}}m^{{\text{B}}}_{l} + {\sum\limits_{j = 1}^c {\mu ^{{\text{B}}}_{{ij}} {\text{d}}m^{{\text{B}}}_{j} } }}} \\ \end{array} $$

(II.2)

$$ \begin{array}{*{20}c} {{}} & { = } & {{ - s_{i} {\text{d}}T + v_{i} {\text{d}}p^{{\text{L}}} + \mu _{{ia}} {\text{d}}m_{a} + \mu _{{ib}} {\text{d}}m_{b} + {\sum\limits_{j = 1}^c {\mu _{{ij}} {\text{d}}m_{j} ,} }}} & {{{\left( {i = a,\;b,\;l,\;1, \ldots ,\;c} \right)}}} \\ \end{array} $$

(II.3)

where$m^{{\text{A}}}_{i} ,\;m^{{\text{B}}}_{i} ,\;{\text{and}}\;m_{i}$denote the molalities of component i in the phases A, B, and L, respectively,$y^{{\text{A}}}_{i} ,\;y^{{\text{B}}}_{i} ,\;{\text{and}}\;y_{i}$the corresponding partial molar quantities, and$\mu ^{{\text{A}}}_{{ij}} ,\;\mu ^{{\text{B}}}_{{ij}} ,\;{\text{and}}\;\mu _{{ij}}$the corresponding partial derivatives of the chemical potential of component i with respect to the molality of component j at constant temperature and pressure defined by

$$\mu _{{ij}} = {\left( {\partial \mu _{i} /\partial m_{j} } \right)}_{{m_{j} }} $$

(II.4)

subscript m _j denoting all the molalities except m _j being kept constant. Let us consider the adsorption at film interfaces from the phase L and take T, p, Π, m ₁,...,m _c as experimental variables of the system. Combinations of Eq. (II.3) with Eqs. (II.1) and (II.2) yield

$$\begin{array}{*{20}l} {{\mu _{{ia}} {\text{d}}m_{a} + \mu _{{ib}} {\text{d}}m_{b} - \mu ^{{\text{A}}}_{{ib}} {\text{d}}m^{{\text{A}}}_{b} - \mu ^{{\text{A}}}_{{il}} {\text{d}}m^{{\text{A}}}_{l} - {\sum\limits_{j = 1}^c {\mu ^{{\text{A}}}_{{ij}} {\text{d}}m^{{\text{A}}}_{j} } }} \hfill} \\ {{ = - {\left( {s^{{\text{A}}}_{i} - s_{i} } \right)}{\text{d}}T + {\left( {v^{{\text{A}}}_{i} - v_{i} } \right)}{\text{d}}p + v_{i} {\text{d}}\Pi - {\sum\limits_{j = 1}^c {\mu _{{ij}} {\text{d}}m_{j} } }} \hfill} \\ \end{array} $$

(II.5)

and

$$\begin{array}{*{20}l} {{\mu _{{ia}} {\text{d}}m_{a} + \mu _{{ib}} {\text{d}}m_{b} - \mu ^{{\text{B}}}_{{ia}} {\text{d}}m^{{\text{B}}}_{a} - \mu ^{{\text{B}}}_{{il}} {\text{d}}m^{{\text{B}}}_{l} - {\sum\limits_{j = 1}^c {\mu ^{{\text{B}}}_{{ij}} {\text{d}}m^{{\text{B}}}_{j} } }} \hfill} \\ {{ = - {\left( {s^{{\text{B}}}_{i} - s_{i} } \right)}{\text{d}}T + {\left( {v^{{\text{B}}}_{i} - v_{i} } \right)}{\text{d}}p + v_{i} {\text{d}}\Pi - {\sum\limits_{j = 1}^c {\mu _{{ij}} {\text{d}}m_{j} } }} \hfill} \\ \end{array} $$

(II.6)

Adding the sum of Eq. (II.5) multiplied by $m^{{\text{A}}}_{i}$ and the one of Eq. (II.6) by $m^{{\text{B}}}_{i}$ yields

$$ \begin{array}{*{20}l} {{{\left( {{\sum\limits_{i = a}^c {m^{{\text{K}}}_{{\text{i}}} \mu _{{ia}} } }} \right)}{\text{d}}m_{a} + {\left( {{\sum\limits_{i = a}^c {m^{{\text{K}}}_{{\text{i}}} \mu _{{ib}} } }} \right)}{\text{d}}m_{b} } \hfill} \\ {{ = {\sum\limits_{i = a}^c {m^{{\text{K}}}_{{\text{i}}} } }{\left[ { - {\left( {s^{{\text{K}}}_{i} - s_{i} } \right)}{\text{d}}T + {\left( {\nu ^{{\text{K}}}_{i} - \nu _{i} } \right)}{\text{d}}p + \nu _{i} {\text{d}}\Pi - {\sum\limits_{j = 1}^c {\mu _{{ij}} {\text{d}}m_{j} } }} \right]},{\left( {{\text{K}} = {\text{A}}{\text{, B}}} \right)}} \hfill} \\ \end{array} $$

(II.7)

where we used the Gibbs-Duhem equations for the phases A and B at constant T and p,

$${\sum\limits_{i = a}^c {m^{{\text{K}}}_{i} \mu ^{{\text{K}}}_{{ij}} = 0,\;\;\;\;\;{\left( {j = a,\;b,\;l,\;1, \ldots ,\;c;\;{\text{K}} = {\text{A}},\;{\text{B}}} \right)}} }$$

(II.8)

Equation (II.7) can be rewritten as

$$ D^{{\text{K}}}_{a} {\text{d}}m_{a} + D^{{\text{K}}}_{b} {\text{d}}m_{b} = - D^{{\text{K}}}_{s} {\text{d}}T + D^{{\text{K}}}_{v} {\text{d}}p + D^{{\text{K}}}_{{\text{L}}} {\text{d}}\Pi - {\sum\limits_{j = 1}^c {D^{{\text{K}}}_{j} {\text{d}}m_{j} } }.{\left( {{\text{K}} = {\text{A}},\;{\text{B}}} \right)} $$

(II.9)

where $D^{{\text{K}}}_{j} ,\;D^{{\text{K}}}_{y} ,\;{\text{and}}\;D^{{\text{K}}}_{{\text{L}}}$ denote

$$ \begin{array}{*{20}l} {{D^{{\text{K}}}_{j} = m^{{\text{K}}}_{a} \mu _{{aj}} + m^{{\text{K}}}_{b} \mu _{{bj}} + m^{{\text{K}}}_{l} \mu _{{lj}} + {\sum\limits_{i = 1}^c {m^{{\text{K}}}_{i} \mu _{{ij}} {\left( {j = a,\;b,\;1, \ldots ,c;\;{\text{K}} = {\text{A}},\;{\text{B}}} \right)}} }} \hfill} \\ \end{array} $$

(II.10)

$$ \begin{array}{*{20}r} {\hfill {D^{{\text{K}}}_{y} = m^{{\text{K}}}_{a} {\left( {y^{{\text{K}}}_{a} - y_{a} } \right)} + m^{{\text{K}}}_{b} {\left( {y^{{\text{K}}}_{b} - y_{b} } \right)} + m^{{\text{K}}}_{l} {\left( {y^{{\text{K}}}_{l} - y_{l} } \right)} + {\sum\limits_{i = 1}^c {m^{{\text{K}}}_{i} {\left( {y^{{\text{K}}}_{i} - y_{i} } \right)}} }}} \\ {\hfill {{\left( {y = s,\;v;\;{\text{K}} = {\text{A}},\;{\text{B}}} \right)}}} \\ \end{array} $$

(II.11)

$$D^{{\text{K}}}_{{\text{L}}} = m^{{\text{K}}}_{a} v_{a} + m^{{\text{K}}}_{b} v_{b} + m^{{\text{K}}}_{l} v_{l} + {\sum\limits_{i = 1}^c {m^{{\text{K}}}_{i} v_{i} ,\;\;\;\;\;{\left( {{\text{K}} = {\text{A}},\;{\text{B}}} \right)}} }$$

(II.12)

respectively, or in the matrix form

$$ {\left( {\begin{array}{*{20}c} {{D^{{\text{A}}}_{a} }} & {{D^{{\text{A}}}_{b} }} \\ {{D^{{\text{B}}}_{a} }} & {{D^{{\text{B}}}_{b} }} \\ \end{array} } \right)}{\left( {\begin{array}{*{20}c} {{{\text{d}}m_{a} }} \\ {{{\text{d}}m_{b} }} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {{D^{{\text{A}}}_{s} }} & {{D^{{\text{A}}}_{v} }} & {{D^{{\text{A}}}_{{\text{L}}} }} & {{D^{{\text{A}}}_{1} }} & { \cdots } & {{D^{{\text{A}}}_{c} }} \\ {{D^{{\text{B}}}_{s} }} & {{D^{{\text{B}}}_{v} }} & {{D^{{\text{B}}}_{{\text{L}}} }} & {{D^{{\text{B}}}_{1} }} & { \cdots } & {{D^{{\text{B}}}_{c} }} \\ \end{array} } \right)}{\left( {\begin{array}{*{20}c} { - } & {{{\text{d}}T}} \\ {{}} & {{{\text{d}}p}} \\ {{}} & {{{\text{d}}\Pi }} \\ { - } & {{{\text{d}}m_{1} }} \\ {{}} & { \vdots } \\ { - } & {{{\text{d}}m_{c} }} \\ \end{array} } \right)} $$

(II.13)

Hence we have the dependence of m _a and m _b on T, p, Π, m ₁,...,m _c

$${\text{d}}m_{a} = D^{{ - 1}} {\left( { - D^{a}_{s} {\text{d}}T + D^{a}_{v} {\text{d}}p + D^{a}_{{\text{L}}} {\text{d}}\Pi - {\sum\limits_{j = 1}^c {D^{a}_{j} {\text{d}}m_{j} } }} \right)}$$

(II.14)

$${\text{d}}m_{b} = D^{{ - 1}} {\left( { - D^{b}_{s} {\text{d}}T + D^{b}_{v} {\text{d}}p + D^{b}_{{\text{L}}} {\text{d}}\Pi - {\sum\limits_{j = 1}^c {D^{b}_{j} {\text{d}}m_{j} } }} \right)}$$

(II.15)

where $D,\;D^{k}_{s} ,\;D^{k}_{v} ,\;D^{k}_{{\text{L}}} ,\;{\text{and}}\;D^{k}_{j} ,\;{\left( {k = a,\;b} \right)}$ are the determinants:

$$ \begin{array}{*{20}l} {D \hfill} & { = \hfill} & {{{\left| {\begin{array}{*{20}c} {{D^{{\text{A}}}_{a} }} & {{D^{{\text{A}}}_{b} }} \\ {{D^{{\text{B}}}_{a} }} & {{D^{{\text{B}}}_{b} }} \\ \end{array} } \right|},} \hfill} & {{D^{a}_{x} } \hfill} & { = \hfill} & {{{\left| {\begin{array}{*{20}c} {{D^{{\text{A}}}_{x} }} & {{D^{{\text{A}}}_{b} }} \\ {{D^{{\text{B}}}_{x} }} & {{D^{{\text{B}}}_{b} }} \\ \end{array} } \right|},} \hfill} & {{D^{b}_{x} } \hfill} & { = \hfill} & {{{\left| {\begin{array}{*{20}c} {{D^{{\text{A}}}_{a} }} & {{D^{{\text{A}}}_{x} }} \\ {{D^{{\text{B}}}_{a} }} & {{D^{{\text{B}}}_{x} }} \\ \end{array} } \right|}} \hfill} \\ \end{array} ,{\left( {x = s,\;v,\;{\text{L}},\;1, \ldots ,c} \right)} $$

(II.16)

Eliminating dm _a and dm _b in Eq. (II.3) by use of Eqs. (II.14) and (II.15), we have

$$ \begin{array}{*{20}l} {{{\text{d}}\mu _{i} = } \hfill} & {{ - {\left[ {s_{i} + D^{{ - 1}} {\left( {D^{a}_{s} \mu _{{ia}} + D^{b}_{s} \mu _{{ib}} } \right)}} \right]}{\text{d}}T} \hfill} \\ {{} \hfill} & {{ + {\left[ {v_{i} + D^{{ - 1}} {\left( {D^{a}_{v} \mu _{{ia}} + D^{b}_{v} \mu _{{ib}} } \right)}} \right]}{\text{d}}p} \hfill} \\ {{} \hfill} & {{ - {\left[ {v_{i} - D^{{ - 1}} {\left( {D^{a}_{{\text{L}}} \mu _{{ia}} + D^{b}_{{\text{L}}} \mu _{{ib}} } \right)}} \right]}{\text{d}}\Pi } \hfill} \\ {{} \hfill} & {{ + {\sum\limits_{j = 1}^c {{\left[ {\mu _{{ij}} - D^{{ - 1}} {\left( {D^{a}_{j} \mu _{{ia}} + D^{b}_{j} \mu _{{ib}} } \right)}} \right]}{\text{d}}m_{j} } }.} \hfill} \\ \end{array} $$

(II.17)

Substitution of Eq. (II.17) into Eqs. (6) and (15) leads to

$$ \begin{array}{*{20}l} {{{\text{d}}\gamma ^{{\text{f}}} = } \hfill} & {{ - {\left\{ {s^{{\text{f}}} - {\sum\limits_{j = 1}^c {\Gamma ^{{\text{f}}}_{j} {\left[ {s_{j} + D^{{ - 1}} {\left( {D^{a}_{s} \mu _{{ja}} + D^{b}_{s} \mu _{{jb}} } \right)}} \right]}} }} \right\}}{\text{d}}T} \hfill} \\ {{} \hfill} & {{ + {\left\{ {v^{{\text{f}}} - {\sum\limits_{j = 1}^c {\Gamma ^{{\text{f}}}_{j} {\left[ {v_{j} + D^{{ - 1}} {\left( {D^{a}_{v} \mu _{{ja}} + D^{b}_{v} \mu _{{jb}} } \right)}} \right]}} }} \right\}}{\text{d}}p} \hfill} \\ {{} \hfill} & {{ + {\left\{ {v^{{\text{L}}} + {\sum\limits_{j = 1}^c {\Gamma ^{{\text{f}}}_{j} {\left[ {v_{j} - D^{{ - 1}} {\left( {D^{a}_{{\text{L}}} \mu _{{ja}} + D^{b}_{{\text{L}}} \mu _{{jb}} } \right)}} \right]}} }} \right\}}{\text{d}}\Pi } \hfill} \\ {{} \hfill} & {{ - {\sum\limits_{i = 1}^c {{\left\{ {{\sum\limits_{j = 1}^c {\Gamma ^{{\text{f}}}_{j} {\left[ {\mu _{{ji}} - D^{{ - 1}} {\left( {D^{a}_{i} \mu _{{ja}} + D^{b}_{i} \mu _{{jb}} } \right)}} \right]}} }} \right\}}{\text{d}}m_{i} } }} \hfill} \\ \end{array} $$

(II.18)

and

$$ \begin{array}{*{20}l} {{{\text{d}}{\left( {\gamma ^{{\text{F}}}_{{\text{A}}} + \gamma ^{{\text{F}}}_{{\text{B}}} } \right)} = } \hfill} & {{ - \left\{ {{\left( {s^{{\text{F}}}_{{\text{A}}} + s^{{\text{F}}}_{{\text{B}}} } \right)} - {\sum\limits_{j = 1}^c {{\left( {\Gamma ^{{{\text{F}}{\text{,A}}}}_{j} + \Gamma ^{{{\text{F}}{\text{,B}}}}_{j} } \right)}} }} \right.{\left[ {s_{j} + D^{{ - 1}} {\left( {D^{a}_{s} \mu _{{ja}} + D^{b}_{s} \mu _{{jb}} } \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ + \left. {\Pi {\left( {\partial \tau /\partial T} \right)}_{{p,\Pi ,m}} } \right\}{\text{d}}T} \hfill} \\ {{} \hfill} & {{ + \left\{ {{\left( {v^{{\text{F}}}_{{\text{A}}} + v^{{\text{F}}}_{{\text{B}}} } \right)} - {\sum\limits_{j = 1}^c {{\left( {\Gamma ^{{{\text{F}}{\text{,A}}}}_{j} + \Gamma ^{{{\text{F}}{\text{,B}}}}_{j} } \right)}} }} \right.{\left[ {v_{j} + D^{{ - 1}} {\left( {D^{a}_{v} \mu _{{ja}} + D^{b}_{v} \mu _{{jb}} } \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ - \left. {\Pi {\left( {\partial \tau /\partial p} \right)}_{{T,\Pi ,m}} } \right\}{\text{d}}p} \hfill} \\ {{} \hfill} & {{ + \left\{ {v^{{\text{L}}} + {\sum\limits_{j = 1}^c {{\left( {\Gamma ^{{{\text{F}}{\text{,A}}}}_{j} + \Gamma ^{{{\text{F}}{\text{,B}}}}_{j} } \right)}} }} \right.{\left[ {v_{j} - D^{{ - 1}} {\left( {D^{a}_{{\text{L}}} \mu _{{ja}} + D^{b}_{{\text{L}}} \mu _{{jb}} } \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ - \left. {{\left[ {\partial {\left( {\Pi \tau } \right)}/\partial \Pi } \right]}_{{T,p,m}} } \right\}{\text{d}}\Pi } \hfill} \\ {{} \hfill} & {{ - {\sum\limits_{i = 1}^c {\left\{ {{\sum\limits_{j = 1}^c {{\left( {\Gamma ^{{{\text{F}}{\text{,A}}}}_{j} + \Gamma ^{{{\text{F}}{\text{,B}}}}_{j} } \right)}} }} \right.{\left[ {\mu _{{ji}} - D^{{ - 1}} {\left( {D^{a}_{i} \mu _{{ja}} + D^{b}_{i} \mu _{{jb}} } \right)}} \right]}} }} \hfill} \\ {{} \hfill} & {{ + \left. {\Pi {\left( {\partial \tau /\partial m_{i} } \right)}_{{T,p,\Pi ,m_{i} }} } \right\}{\text{d}}m_{i} .} \hfill} \\ \end{array} $$

(II.19)

If the phase L is assumed to be an ideal dilute solution, then we have

$$\mu _{i} = \mu ^{{\text{o}}}_{i} {\left( {T,p^{{\text{L}}} } \right)} + RT\ln m_{{i,}} \;\;\;\;\;{\left( {i = 1, \ldots ,c} \right)}$$

(II.20)

and

$$\begin{array}{*{20}l} {{\mu _{{ij}} } \hfill} & {{ = RT/m_{i} ,} \hfill} & {{{\left( {i = j} \right)}} \hfill} \\ {{} \hfill} & {{ = 0,} \hfill} & {{{\left( {i \ne j} \right)}} \hfill} \\ \end{array} $$

(II.21)

where $\mu ^{{\text{o}}}_{i}$ is the standard chemical potential of component i in the phase L.Substituting Eq. (II.21) into Eqs. (II.18) and (II.19) yields Eqs. (21) and (22).

Appendix III: Nomenclature

$c^{{\text{K}}}_{i}$ :

number of moles of component i per unit volume in phase K (K = A, B, L)

c _i (z):

number of moles of component i per unit volume at point z

G :

Gibbs free energy

Δg ^f :

Gibbs free energy change associated with adsorption at film interfaces

H :

enthalpy

Δh ^f, Δh ^F :

enthalpy change associated with adsorption at film interfaces and the one at film interface of symmetric film

Δh ^fH, Δh ^FH :

difference in enthalpy change between film interfaces and bulk ones and the one between film interface and bulk one for symmetric film

m _i :

molality of component i

n _i :

number of moles of component i

$n^{{{\text{K*}}}}_{i}$ :

number of moles of component i in part K* divided by surface of tension for film

$n^{{\text{f}}}_{i}$ :

excess quantity of component i

$n^{{{\text{F}}{\text{,K}}}}_{i}$ :

excess quantity of component i ascribed to film interface against phase K (K = A, B)

p :

pressure

p ^L :

pressure in phase L

S :

entropy

s _i :

partial molar entropy of component i

s ^K :

entropy per unit volume in phase K (K = A, B, L)

s ^f :

excess entropy per unit film area

$s^{{\text{F}}}_{{\text{K}}} ,\;s^{{\text{H}}}_{{\text{K}}}$ :

excess entropies per unit area ascribed to film interface and bulk one against phase K (K = A, B)

Δs ^f, Δs ^F :

entropy change associated with adsorption at film interfaces and the one at film interface of symmetric film

Δs _K :

entropy change associated with adsorption at bulk interface against phase K (K = A, B)

Δs ^fH, Δs ^FH :

difference in entropy change between film interfaces and bulk ones and the one between film interface and bulk one for symmetric film

T :

temperature

U :

energy

Δu ^f, Δu ^F :

energy change associated with adsorption at film interfaces and the one at film interface of symmetric film

Δu ^fH, Δu ^FH :

difference in energy change between film interfaces and bulk ones and the one between film interface and bulk one for symmetric film

V :

volume

V ^K :

volume of phase K (K = A, B, L)

v _i :

partial molar volume of component i

v ^f :

excess volume per unit film area

$v^{{\text{F}}}_{{\text{K}}} ,\;v^{{\text{H}}}_{{\text{K}}}$ :

excess volumes per unit area ascribed to film interface and bulk one against phase K (K = A, B)

$v^{{{\text{F}}{\text{,K}}}}_{{\text{K}}} ,\;v^{{{\text{F}}{\text{,L}}}}_{{\text{K}}}$ :

parts of$v^{{\text{F}}}_{{\text{K}}}$divided by surface of tension

$v^{{{\text{H}}{\text{,K}}}}_{{\text{K}}} ,\;v^{{{\text{H}}{\text{,L}}}}_{{\text{K}}}$ :

parts of $v^{{\text{H}}}_{{\text{K}}}$ divided by surface of tension

v ^L :

volume of phase L per unit film area

$v^{{\text{L}}}_{{\text{K}}}$ :

part of v ^Ldivided by surface of tension for film

Δv ^f, Δv ^F :

volume changes associated with adsorption at film interfaces and the one at film interface of symmetric film

Δv _K :

volume change associated with adsorption at bulk interface against phase K (K = A, B)

Δv ^fH, Δv ^FH :

difference in volume change between film interfaces and bulk ones and the one between film interface and bulk one for symmetric film

v ^fH,L, v ^FH,L :

difference in part of excess volume per unit area between film interfaces and bulk ones and the one between film interface and bulk one for symmetric film

$\bar{y}^{{\text{f}}}_{i} ,\;\bar{y}^{{\text{I}}}_{i}$ :

mean partial molar quantity of component i in film and the one inherent in film

γ ^f :

film tension

$\gamma ^{{\text{f}}}_{{\text{K}}} ,\;\gamma ^{{\text{F}}}$ :

film interfacial tension against phase K and the one of symmetric film (K = A, B)

γ _K, γ :

interfacial tension against phase K and the one in symmetric film system (K = A, B)

$\Gamma ^{{\text{f}}}_{i}$ :

film density of component i

$\Gamma ^{{{\text{F}}{\text{,K}}}}_{i} ,\;\Gamma ^{{{\text{H}}{\text{,K}}}}_{i}$ :

interfacial densities of component i ascribed to film interface and bulk one against phase K (K = A, B)

$\Gamma ^{{\text{F}}}_{i}$ :

interfacial density of component i in film interface of symmetric film

$\Gamma ^{{\text{I}}}_{i} ,\;\Gamma ^{{{\text{I}}{\text{,K}}}}_{i}$ :

film densities of component i inherent in film and portions of film region (K = A, B, L)

$\Gamma ^{{{\text{fH}}}}_{i}$ :

difference between film density and interfacial densities of component i

$\Gamma ^{{{\text{FH}}{\text{,K}}}}_{i} ,\;\Gamma ^{{{\text{FH}}}}_{i}$ :

difference in interfacial density of component i between film interface and bulk one against phase K and the one in symmetric film system (K = A, B)

μ _i :

chemical potential of component i

μ _ij :

partial derivative of chemical potential of component i with respect to molality of component j

Π:

disjoining pressure

σ :

film area

τ :

distance between surfaces of tension for film interfaces

τ ^f :

thermodynamic film thickness