Extending the Equation of State for Three-Aggregate Systems to Their Interfaces

Abstract

An analysis is performed of the state of a molecular theory oriented toward describing three-aggregate systems from a single viewpoint that includes their three bulk phases and three types of interfaces (vapor–liquid, solid–vapor, and solid–liquid). It is established that the theory, which is based on the lattice gas model (LGM), provides a common structure to express the pressure of not only three different bulk aggregate states but in their transitional regions of interface boundaries. The existence of vacancies in the LGM allows it to be used for all concentrations of the components of a multicomponent mixture from zero (which is characteristic of rarefied gas) to unity (in mole fractions) (which is characteristic of solids). Mechanical characteristics are calculated for heterogeneous systems using equations of local partial isotherms with the same structure in different regions of a common heterogeneous system. General expressions are derived for the pressure in different aggregate phases and the local pressure in the transitional regions of the three interfaces, through which local pressures the surface tensions of diverse interfaces are determined.

APPENDIX

The free energy of a fully distributed heterogeneous lattice system is expressed as $F(s) = \sum\nolimits_{f = 1}^M {\sum\nolimits_{i = 1}^s {\theta _{f}^{i}} } M_{f}^{i}$, where 1 ≤ f ≤ М, М is the number of system sites, function $M_{f}^{i}$ determines the contribution made by i particles to the free energy of a system for a site with number f (for simplicity, the interactions between nearest neighbors are written) [20, 22, 24]

$$\begin{gathered} \mathop M\nolimits_f^i = \nu _{f}^{i} + kT\ln \theta _{f}^{i} \\ + \;\frac{{kT}}{2}\sum\limits_{g \in {{z}_{f}}} {\ln \left[ {{{\hat {\theta }_{{fg}}^{{ii}}\hat {\theta }_{{fg}}^{{ik}}} \mathord{\left/ {\vphantom {{\hat {\theta }_{{fg}}^{{ii}}\hat {\theta }_{{fg}}^{{ik}}} {{{{(\theta _{f}^{i})}}^{2}}\hat {\theta }_{{fg}}^{{ki}}}}} \right. \kern-0em} {{{{(\theta _{f}^{i})}}^{2}}\hat {\theta }_{{fg}}^{{ki}}}}} \right],} \\ \end{gathered} $$

(A.1)

where $\theta _{f}^{i}$ is the probability of an i component being at an f site; $\theta _{{fg}}^{{ik}}$ is the probability of two components i and j being at neighboring sites f and g; $\nu _{f}^{i}$ is the one-particle contribution to the free energy of the i component, $\hat {\theta }_{{fg}}^{{ik}} = \theta _{{fg}}^{{ik}}\exp ( - \beta \varepsilon _{{fg}}^{{ik}})$, β = (kT)^–1; $\varepsilon _{{fg}}^{{ik}}$ is the energy of i and j components interacting at neighboring f and g sites; and k corresponds to one component of the mixture (the main sort of particles being 1 ≤ k ≤ s, k ≠ i). The particular number of k is unimportant, due to the ease of solving the problem. The above functions are defined as the average over the ensemble of identical heterogeneous lattices, for which the following normalization relations are satisfied: $\sum\nolimits_{i = 1}^s {\theta _{{fg}}^{{ij}}} = \theta _{g}^{j}$, $\sum\nolimits_{j = 1}^s {\theta _{{fg}}^{{ij}}} = \theta _{f}^{i}$, $\sum\nolimits_{i = 1}^s {\theta _{f}^{i}} = 1$.

The molecular distribution in the quasi-chemical approximation ($\hat {\theta }_{{fg}}^{{ij}}\hat {\theta }_{{fg}}^{{kl}} = \hat {\theta }_{{fg}}^{{il}}\hat {\theta }_{{fg}}^{{kj}}$) is found from the solution to the equations for local partial isotherms [19, 22]:

$$\begin{gathered} \beta ({{\mu }_{i}} - {{\mu }_{s}}) = \beta (\nu _{f}^{i} - \nu _{f}^{s}) + \ln (\theta _{f}^{i}{\text{/}}\theta _{f}^{s}) \\ + \;\sum\limits_{g \in {{z}_{f}}} {\ln \left[ {\sum\limits_{j = 1}^s {\hat {\theta }_{{fg}}^{{ij}}} \exp (\beta \varepsilon _{{fg}}^{{sj}}){\text{/}}\theta _{f}^{i}} \right]} {\kern 1pt} . \\ \end{gathered} $$

(A.2)

The relation g ∈ z_f is the sum over all nearest sites of the central f site. For the layered structure of the interface, this sum becomes z = $\sum\nolimits_{p = q-1}^{q + 1} {{{z}_{{qp}}}}$, where z is the number of the neighboring sites in the bulk phase, and z_qp is the number of sites in the р layer relative to the central site in the q layer. For the bulk phase,

$$\begin{gathered} F(s) = \sum\limits_{i = 1}^s {{{N}_{i}}{{M}_{i}}} {\text{,}} \\ {{M}_{i}} = {{\nu }_{i}} + kT\ln {{\theta }_{i}} + \frac{{kT}}{2}z\ln [{{{\hat {\theta }}}_{{ii}}}{\text{/}}{{({{\theta }_{i}})}^{2}}]. \\ \end{gathered} $$

(A.3)

The Gibbs potential of the fully distributed heterogeneous lattice system is defined through the above functions as $G(s) = \sum\nolimits_{f = 1}^M {\sum\nolimits_{i = 1}^s {\theta _{f}^{i}} } \mu _{f}^{i}$, where

$$\mu _{f}^{i} = \nu _{f}^{i} + kT\ln \theta _{f}^{i} + \frac{{kT}}{2}\sum\limits_{g \in {{z}_{f}}} {\ln [\hat {\theta }_{{fg}}^{{ii}}{\text{/}}\theta _{f}^{i}\theta _{g}^{i}]\,} .$$

(A.4)

In the bulk, expression (A.4) for μ_i becomes the expression for M_i (A.3).