INTRODUCTION

Heterogeneous systems are widely found in diverse physicochemical processes. They occur due to the formation of different phases and their surfaces during phase transitions and rearrangement in solids, which affects the conditions and mechanisms of adsorption, catalytic, and membrane processes. The best known example of heterogeneous systems are those introduced by Gibbs [1–3] at macro levels in order to consider the true properties of systems in experimental processes. Representations of heterogeneous surfaces [4] and heterogeneous solid solutions that included bulk ordered phases with interstices were then introduced on the macro scale [5, 6]. Other examples of heterogeneous systems are combinations of, e.g., the model of absorption in octa- and tetrahedral interstices with regard to the ordered arrangement of particles, and the ordering of solution components at interfaces.

The diversity of factors resulting in heterogeneous systems is easily illustrated by the example of heterogeneous surfaces [7, 8] that can be attributed to violations of the regular arrangement of surface atoms of a solid (structural heterogeneities) and differences between surface atoms (chemical heterogeneities).

The main factors of the heterogeneous distribution of mixture components in heterogeneous systems are either interfaces or surface force fields in which the mobile vapor and/or liquid components are redistributed. In both cases, the central factor is intermolecular interaction resulting in condensed phases and all effects in them (so-called non-ideal systems). The effect of intermolecular interaction in homogeneous systems is well known because of their behavior deviating from that of ideal systems [9–12]. The problem of describing molecules in space inside heterogeneous systems is complicated, since the effect of external heterogeneous fields of the surface potential of solids or interfaces must be considered along with the mutual influence of molecules.

The simplest model for describing non-ideal systems is the so-called Ising model [13–19]. Total energy Н of this model is the interaction between spin with magnetic moment μ and external field h, plus spin–spin interaction J: Н = \(-\mu h\sum\nolimits_f^{} {{{\sigma }_{f}}} \) – \(J\sum\nolimits_{fg}^{} {{{\sigma }_{f}}{{\sigma }_{g}}} \), where σf is a variable that describes the spin state along or against the external field h; index f is the number of lattice sites; and J is the parameter of spin–spin interaction. In this model, we consider spin–spin interaction J between all nearest neighbors f and g, which determines the cooperative behavior of the system in general. Spins have a tendency to be parallel when J > 0 and antiparallel when J < 0.

The Ising model is analytically solvable for one-dimensional homogeneous and heterogeneous systems. It is partially solvable for two-dimensional homogeneous and ordered systems in a zero external magnetic field, and there are no analytical solutions for any three-dimensional systems [14–19].

The Ising model uses approximate calculations in its adaptation to different problems of physical chemistry. These include low- and high-temperature decompositions [16–18], matrices [14–19], CV [20–22], Monte Carlo modeling [23, 24], and simple algebra. The one-particle (mean field) approximation without correlations, and the more accurate pair quasi-chemical approximation (QCA) with only direct correlations between interacting particles, are the best known algebraic means. Most models in the theory of solutions [25–28], heterogeneous systems [8, 29], and non-ideal reaction systems for calculating kinetics [29, 30] are based on the QCA. CV allows us to consider not only direct QCA correlations but the effects of indirect correlations as well. Considering indirect correlations in CV as clusters grow provides accurate solutions [31, 32]. A general mathematical approach to any size of CV basis clusters was presented in [32] for calculating thermodynamic characteristics of a bulk phase. A universal parametric approximation that included a calculation program was proposed for a planar square lattice, and it was shown that raising the number of sites in the basis cluster to 16 provides solutions with accuracies of up to 2% [32]. Earlier, CV was used mainly for homogeneous systems. So far as we know, the only CV model that can be applied to heterogeneous systems was used to describe the interfaces of ordered systems [20–22].

In this work, we propose a new way of deriving equations for equilibrium particle distributions by means of CV, which generalizes the approach in [32] for heterogeneous spatially distributed lattice structures and allows consideration of clusters of any size. The new approach was reformulated for the planar boundary of the transitional region of coexisting phases in a two-phase system representing a sequence of monomolecular layers with variable fluid density. Explicit expressions for the heterogeneous statistical sum of the transitional region of an interface are given for a 3 × 3 basis cluster and the K1s cluster. For the simplest 2 × 2 cluster, it is shown how explicit equations for the distribution of equilibrium particles in the transitional region of the interface of coexisting phases can be obtained from the heterogeneous statistical sum.

MAIN CONCEPTS OF CV FOR A HOMOGENEOUS LATTICE

A set of basis clusters γ is given in CV. These are clusters of the maximum size that considers all correlations. The basis clusters have a form that reflects the lattice topology as much as possible, provided that the computational resources are sufficient for numerically solving the problem. For simple lattices, it is convenient to select one cluster with several unit cells (or sites) as the basis and denote the number of cluster sites as \({{n}_{\gamma }}\).

After choosing basis cluster γ consisting of f1f2…\({{f}_{{{{n}_{\gamma }}}}}\) sites, it is assumed that its probabilities \(\theta _{{{{f}_{1}}{{f}_{2}}...{{f}_{{{{n}_{\gamma }}}}}}}^{{{{i}_{1}}{{i}_{2}}...{{i}_{{{{n}_{\gamma }}}}}}}\) are distributed independently over the homogeneous lattice. This assumption allows us to estimate the total number of configurations Ωγ of a lattice consisting of N sites, provided that only γ clusters are considered:

$${{\Omega }_{\gamma }} = \prod\limits_{{{\sigma }}_{{{{f}_{1}}}}^{{{{i}_{1}}}}} {...} \,\prod\limits_{{{\sigma }}_{{{{f}_{{{{n}_{\gamma }}}}}}}^{{{{i}_{{{{n}_{\gamma }}}}}}}} {\frac{{N!}}{{[\theta _{{{{f}_{1}}{{f}_{2}}...{{f}_{{{{n}_{\gamma }}}}}}}^{{{{i}_{1}}{{i}_{2}}...{{i}_{{{{n}_{\gamma }}}}}}}N]!}}} .$$
(1)

Using Stirling’s formula and probability normalization \(\theta _{{{{f}_{1}}{{f}_{2}}...{{f}_{{{{n}_{\gamma }}}}}}}^{{{{i}_{1}}{{i}_{2}}...{{i}_{{{{n}_{\gamma }}}}}}}\), contribution Sγ from the entropy of the γ cluster is written as

$$\begin{gathered} \frac{{{{S}_{\gamma }}}}{{{{k}_{{\text{B}}}}}} = \ln {{\Omega }_{\gamma }} = - N\sum\limits_{{{\sigma }}_{{{{f}_{1}}}}^{{{{i}_{1}}}}} {\sum\limits_{{{\sigma }}_{{{{f}_{1}}}}^{{{{i}_{1}}}}} {...} \,\sum\limits_{{{\sigma }}_{{{{f}_{{{{n}_{\gamma }}}}}}}^{{{{i}_{{{{n}_{\gamma }}}}}}}} {\theta _{{{{f}_{1}}{{f}_{2}}...{{f}_{{{{n}_{\gamma }}}}}}}^{{{{i}_{1}}{{i}_{2}}...{{i}_{{{{n}_{\gamma }}}}}}}} } \ln \theta _{{{{f}_{1}}{{f}_{2}}...{{f}_{{{{n}_{\gamma }}}}}}}^{{{{i}_{1}}{{i}_{2}}...{{i}_{{{{n}_{\gamma }}}}}}} \\ = - N\sum\limits_{{{\sigma }_{\gamma }}} {{{\theta }_{\gamma }}({{\sigma }_{\gamma }})} \ln {{\theta }_{\gamma }}({{\sigma }_{\gamma }}). \\ \end{gathered} $$
(2)

The second equality reflects the reduced form of recording the configuration array of the γ cluster.

It is easy to see that Ωγ for only one γ cluster overestimates the true number of configurations Ω in the system. Basis γ clusters are considered only once in formula (2). On the other hand, γ clusters can overlap one another are because they are distributed independently, forming subclusters β contained more than once in (2). To consider all subclusters only once, geometric coefficients aβ, β ≤ γ are introduced for all subclusters β contained in γ, aγ = 1. Using geometric coefficient aβ, entropy S and the number of configurations Ω can be written as

$$\Omega = \prod\limits_{\beta \leqslant \gamma } {{{{[{{\Omega }_{\beta }}]}}^{{{{a}_{\beta }}}}}} ,\quad S = \sum\limits_{\beta \leqslant \gamma } {{{a}_{\beta }}{{S}_{\beta }}} ,$$
(3)

where aβ is determined by the topology of the lattice, and the shape and size of clusters. They can be negative if β subclusters are considered when large γ clusters overlap more than once, and positive if they are considered less than once. They are zero if β subclusters are considered only once.

The total number of subclusters inside a γ basis cluster is generally \({{2}^{{{{n}_{\gamma }}}}}\), where nγ is the number of sites inside it. Nevertheless, sequence {aβ} quickly converges to zero when β ≤ γ. All non-overlapping subclusters are already contained in the basis cluster and therefore considered once already, so for them aβ = 0 and β ≠ γ ∩ γ. Coefficients aβ for overlapping clusters when β = γ ∩ γ are determined in order, starting from the largest overlapping subcluster until zero is reached. All coefficients aβ are found by solving a system of linear equations written for each α subcluster:

$$\sum\limits_{{{\beta }} \geqslant \alpha } {q_{{{\beta }}}^{{{\alpha }}}{{a}_{{{\beta }}}}} = 1,$$
(4)

where β = γ ∩ γ, ∀α = γ ∩ γ, and \(q_{{{\beta }}}^{{{\alpha }}}\) is the number of α subclusters in a β cluster.

The system

$$\begin{array}{*{20}{c}} \circ & \circ &{...}& \circ \\ \circ & \circ &{...}& \circ \\ \vdots & \vdots &{}& \vdots \\ \circ & \circ &{...}& \circ \end{array}$$
(5)

is solved by selecting an unknown α subcluster obtained by superimposing two γ basis clusters. Summation is then done over all overlapping β ≥ α subclusters. We finally obtain a linear equation for unknown coefficient aα, which is then determined. Equation (4) is written for the overlapping α subcluster next in size, and the process continues until α clusters with zero coefficients (aα = 0) are obtained.

FORMULATING CV FOR A HETEROGENEOUS LATTICE

Let us consider a planar square lattice for which we can easily draw figures. All sites are equivalent in a homogeneous lattice (denoted by the symbol \( \circ \)), so CV equations for (5) have form (3).

Sites of a heterogeneous lattice are non-equivalent:

$$\begin{array}{*{20}{c}} {{{ \circ }_{1}}}&{{{ \circ }_{2}}}&{...}&{{{ \circ }_{q}}} \\ {{{ \circ }_{p}}}&{{{ \circ }_{r}}}&{...}&{{{ \circ }_{s}}} \\ \vdots & \vdots &{}& \vdots \\ {{{ \circ }_{u}}}&{{{ \circ }_{w}}}&{...}&{{{ \circ }_{t}}} \end{array},$$
(6)

where the index of the site indicates its type. A repeated fragment of a spatially distributed lattice is shown. Periodic boundary conditions are used to describe the entire lattice. For heterogeneous lattice (6) we use CV with β clusters, as we did for homogeneous lattice (5). The clusters are now characterized by both the β form and the set of q-type sites contained in this cluster, β(q). CV for a heterogeneous lattice thus takes the form

$$\Omega = \prod\limits_{\beta \leqslant \gamma } {{{{\prod\limits_{q \in \beta } {[{{\Omega }_{{\beta (q)}}}]} }}^{{{{\alpha }_{\beta }}}}}} ,\quad S = \sum\limits_{\beta \leqslant \gamma } {{{{{\alpha }}}_{{{\beta }}}}} \sum\limits_{q \in \beta } {{{S}_{{{{\beta }}(q)}}}} ,$$
(7)

in which additional index q considers all possible ways of accommodating a β cluster on the heterogeneous lattice that reflect translating this cluster over the whole system. When a heterogeneous fragment is translated over the lattice periodically, geometric coefficients aβ remain the same as for a homogeneous lattice.

CV FOR A TRANSITIONAL REGION

Let us consider wide transitional region κ∗ between coexisting vapor and liquid phases on a planar square lattice:

$$\begin{array}{*{20}{c}} {{{ \circ }_{1}}}&{{{ \circ }_{2}}}&{...}&{{{ \circ }_{k}}} \\ {{{ \circ }_{1}}}&{{{ \circ }_{2}}}&{...}&{{{ \circ }_{k}}} \\ \vdots & \vdots &{}& \vdots \\ {{{ \circ }_{1}}}&{{{ \circ }_{2}}}&{...}&{{{ \circ }_{k}}} \end{array}\begin{array}{*{20}{c}} {{{ \circ }_{{k + 1}}}} \\ {{{ \circ }_{{k + 1}}}} \\ \vdots \\ {{{ \circ }_{{k + 1}}}} \end{array}\begin{array}{*{20}{c}} {...} \\ {...} \\ {} \\ {...} \end{array}\begin{array}{*{20}{c}} {{{ \circ }_{{k + n}}}} \\ {{{ \circ }_{{k + n}}}} \\ \vdots \\ {{{ \circ }_{{k + n}}}} \end{array}\begin{array}{*{20}{c}} {...} \\ {...} \\ {} \\ {...} \end{array}\begin{array}{*{20}{c}} {{{ \circ }_{t}}} \\ {{{ \circ }_{t}}} \\ \vdots \\ {{{ \circ }_{t}}} \end{array},$$
(8)

where the index of the site denotes the number of the layer (inside a monolayer, all sites are of the same type). We take cluster m × n as our basis, and the layers between k and k + n are distinguished explicitly in scheme (8). For a homogeneous lattice, \({{a}_{{m \times n}}}\) = \({{a}_{{(m-1) \times (n-1)}}}\) = 1 and \({{a}_{{(m-1) \times n}}}\) = \({{a}_{{m \times (n-1)}}}\) = –1 [32]. From expression (7) in the explicit form of \({{a}_{{m \times n}}}\), we then obtain the formula

$$\begin{gathered} {{\Psi }^{{(k)}}} = {{\left\{ {\frac{{\Omega _{{(m - 1) \times (n - 1)}}^{{(k - 1).}}}}{{\Omega _{{m \times (n - 1)}}^{{(k - 1).}}}}} \right\}}^{{1/2}}}{{\left\{ {\frac{{\Omega _{{(m - 1) \times (n - 1)}}^{{.(k)}}}}{{\Omega _{{m \times (n - 1)}}^{{.k}}}}} \right\}}^{{1{\text{/}}2}}} \\ \times \;\left\{ {\frac{{\Omega _{{m \times n}}^{{(k)}}}}{{\Omega _{{(m - 1) \times n}}^{{(k)}}}}} \right\}{{\left\{ {\frac{{\Omega _{{(m - 1) \times (n - 1)}}^{{(k).}}}}{{\Omega _{{m \times (n - 1)}}^{{(k).}}}}} \right\}}^{{1{\text{/}}2}}}{{\left\{ {\frac{{\Omega _{{(m - 1) \times (n - 1)}}^{{.(k + 1)}}}}{{\Omega _{{m \times (n - 1)}}^{{.(k + 1)}}}}} \right\}}^{{1{\text{/}}2}}}, \\ \end{gathered} $$
(9)

where index k is associated with, e.g., the first layer of the cluster and must cover all layers of the transitional region, thereby moving the cluster over all its layers. Limits for k depend on the type of the cluster, so the sign of the product is given separately with its index for each cluster.

In the transitional region, if there is one phase (fluid) on the left and another (gas) on the right. A cluster (n – 1) wide can be placed on both the left (.k) and right (k.) sides in an n wide cluster. The dot on the left in (.k) in upper index Ω means that averaging is done over one left layer in an m × n cluster. The dot on the right in (k.) means it is done over one right layer to obtain an m × (n − 1) cluster. Formula (9) has the same structure as the QCA equation for a transitional region.

For an m × n = (3 × 3) cluster, function Ψ(k) is presented in explicit graphic form as

$$\begin{gathered} {{\Psi }^{k}} = {{\left\{ {\frac{{{{{\left[ {\begin{array}{*{20}{c}} \circ & \circ & \cdot \\ \circ & \circ & \cdot \\ \circ & \circ & \cdot \end{array}} \right]}}_{{k - 1}}}{{{\left[ {\begin{array}{*{20}{c}} \cdot & \circ & \circ \\ \cdot & \circ & \circ \\ \cdot & \circ & \circ \end{array}} \right]}}_{k}}}}{{{{{\left[ {\begin{array}{*{20}{c}} \cdot & \cdot & \cdot \\ \circ & \circ & \cdot \\ \circ & \circ & \cdot \end{array}} \right]}}_{{k - 1}}}{{{\left[ {\begin{array}{*{20}{c}} \cdot & \cdot & \cdot \\ \cdot & \circ & \circ \\ \cdot & \circ & \circ \end{array}} \right]}}_{k}}}}} \right\}}^{{1{\text{/}}2}}} \\ \times \;\left\{ {\frac{{{{{\left[ {\begin{array}{*{20}{c}} \cdot & \cdot & \cdot \\ \circ & \circ & \circ \\ \circ & \circ & \circ \end{array}} \right]}}_{k}}}}{{{{{\left[ {\begin{array}{*{20}{c}} \circ & \circ & \circ \\ \circ & \circ & \circ \\ \circ & \circ & \circ \end{array}} \right]}}_{k}}}}} \right\}{{\left\{ {\frac{{{{{\left[ {\begin{array}{*{20}{c}} \circ & \circ & \cdot \\ \circ & \circ & \cdot \\ \circ & \circ & \cdot \end{array}} \right]}}_{k}}{{{\left[ {\begin{array}{*{20}{c}} \cdot & \circ & \circ \\ \cdot & \circ & \circ \\ \cdot & \circ & \circ \end{array}} \right]}}_{{k + 1}}}}}{{{{{\left[ {\begin{array}{*{20}{c}} \cdot & \cdot & \cdot \\ \circ & \circ & \cdot \\ \circ & \circ & \cdot \end{array}} \right]}}_{k}}{{{\left[ {\begin{array}{*{20}{c}} \cdot & \cdot & \cdot \\ \cdot & \circ & \circ \\ \cdot & \circ & \circ \end{array}} \right]}}_{{k + 1}}}}}} \right\}}^{{1{\text{/}}2}}}, \\ \end{gathered} $$
(10)

where (⋅) denotes averaging over the cluster site, (\( \circ \)) means that all states of the site are considered, and lower index k is the number of the layer (in the transitional region) to which the cluster belongs. The braces around the cluster in its reduced form denotes the product over all states of the non-averaged sites of the cluster. It is worth noting the conditions for the consistency of subclusters inside one cluster when averaging is shifted inside the same layer and the adjacent clusters between the layers, e.g.,

$${{\left[ {\begin{array}{*{20}{c}} \cdot & \cdot & \cdot \\ \circ & \circ & \circ \\ \circ & \circ & \circ \end{array}} \right]}_{k}} = {{\left[ {\begin{array}{*{20}{c}} \circ & \circ & \circ \\ \circ & \circ & \circ \\ \cdot & \cdot & \cdot \end{array}} \right]}_{k}},\quad {{\left[ {\begin{array}{*{20}{c}} \cdot & \circ & \circ \\ \cdot & \circ & \circ \\ \cdot & \circ & \circ \end{array}} \right]}_{k}} = {{\left[ {\begin{array}{*{20}{c}} \circ & \circ & \cdot \\ \circ & \circ & \cdot \\ \circ & \circ & \cdot \end{array}} \right]}_{{k + 1}}}.$$
(11)

Consistency equations for subclusters can be considered automatically by means of correlation factors (CFs). Equations for the distribution of equilibrium clusters in the transitional region are obtained by automatically differentiating Ω (11) with respect to all types of CFs in the program.

CV FOR THE K 1-SQUARE (K 1s) APPROXIMATION

The nearest associated sites that form convex polygons with short bonds are traditionally selected as basis clusters in CV. This is explained by the tendency of sequentially considering correlations in local regions of two- and three- dimensional lattices [20–22, 31]. Clusters not closed by short bonds are found among others, one of which is cluster K1 consisting of one central site and its z nearest neighbors. This cluster emerges in the cluster approach when deriving equations for equilibrium distributions in the CF for discrete systems, and the rates of elementary stages of processes on one (central) site in the theory of absolute reaction rates [29, 30].

There are two basis clusters in the CV approximation of K1s: K1 and a 2 × 2 square. Triplets, pairs, and isolated sites are overlapping clusters that we present as sequences for the number of sites in the cluster to diminish from left to right. The power of the cluster is its geometric coefficient (weight):

$$\begin{gathered} {{\Omega }_{{{{K}_{1}} + sq}}} = {{\left[ {\begin{array}{*{20}{c}} {}& \circ &{} \\ \circ & \circ & \circ \\ {}& \circ &{} \end{array}} \right]}^{{{{a}_{5}}}}} \times {{\left[ {\begin{array}{*{20}{c}} \circ & \circ \\ \circ & \circ \end{array}} \right]}^{{{{a}_{4}}}}} \\ \times \;{{\left\{ {\left[ {\begin{array}{*{20}{c}} \circ &{} \\ \circ & \circ \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} {}& \circ \\ \circ & \circ \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ & \circ \\ \circ &{} \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ & \circ \\ {}& \circ \end{array}} \right]} \right\}}^{{{{a}_{3}}}}} \\ \times \;{{\left\{ {\left[ {\begin{array}{*{20}{c}} \circ \\ \circ \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ & \circ \end{array}} \right]} \right\}}^{{{{a}_{2}}}}} \times {{\left[ \circ \right]}^{{{{a}_{1}}}}}. \\ \end{gathered} $$
(12)

There are triplets in four orientations. Since all orientations are equal, the weight of each triplet must be the same, so all triplets are combined in one group with braces. As with vertical and horizontal pairs, K1 and the square are the largest clusters, so we immediately conclude that a5 = a4 = 1. The intersection of K1 and the square is a triplet. In this orientation, we have one triplet in K1 and one triplet in the square, so the equation for the weight of the triplet is 1 × a5 + 1 × a4 + a3 = 1, and we obtain a3 = −1.

Let us consider the vertical pair. It occurs twice in K1 and twice in the square, \(q_{5}^{2}\) = \(q_{4}^{2}\) = 2, once in each triplet, \(q_{3}^{2} = 1\). The equation for the pair (cofactor 4 of \(q_{3}^{2}\) considers the total number of triplets) is

$$q_{5}^{2}{{a}_{5}} + q_{4}^{2}{{a}_{4}} + 4q_{3}^{2}{{a}_{3}} + {{a}_{2}} = 1,$$
(13)

from which we obtain a2 = 1.

We write the equation for isolated sites in a similar way:

$$5{{a}_{5}} + 4{{a}_{4}} + 4 \times 3{{a}_{3}} + 2 \times 2{{a}_{2}} + {{a}_{1}} = 1.$$
(14)

By substituting coefficients 5 + 4–12 + 4 + a1 = 1, we obtain a1 = 0.

Finally, for the homogeneous lattice we can write

$$\begin{gathered} {{\Omega }_{q}}_{{({{K}_{1}} + s)}} = \frac{{\left[ {\begin{array}{*{20}{c}} \circ &{} \\ \circ & \circ \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} {}& \circ \\ \circ & \circ \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ & \circ \\ \circ &{} \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ & \circ \\ {}& \circ \end{array}} \right]}}{{\left[ {\begin{array}{*{20}{c}} {}& \circ &{} \\ \circ & \circ & \circ \\ {}& \circ &{} \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ & \circ \\ \circ & \circ \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ \\ \circ \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ & \circ \end{array}} \right]}} \\ \, = \frac{{{{{\left[ {\begin{array}{*{20}{c}} \circ &{} \\ \circ & \circ \end{array}} \right]}}^{4}}}}{{\left[ {\begin{array}{*{20}{c}} {}& \circ &{} \\ \circ & \circ & \circ \\ {}& \circ &{} \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} \circ & \circ \\ \circ & \circ \end{array}} \right] \times {{{\left[ {\begin{array}{*{20}{c}} \circ & \circ \end{array}} \right]}}^{2}}}}. \\ \end{gathered} $$
(15)

The second expression explicitly considers the equivalence of all triangular and pair clusters (not done below for the transitional region).

APPROXIMATION K 1 + A SQUARE FOR THE TRANSITIONAL REGION

To obtain the statistical sum for the transitional region, we must move each subcluster from the first expression for Ω (15) through all layers of the noted transitional region sequentially. We then obtain the formula

$$\begin{gathered} {{\Omega }_{{{{K}_{1}} + sq}}} = {{\Psi }_{{liq}}}\left\{ {\prod\limits_k {{{\Psi }_{k}}} } \right\}{{\Psi }_{{{\text{gas}}}}}, \\ {\text{where}}\quad {{\Psi }_{k}} = Y_{{k - 1,k}}^{{1{\text{/}}2}}{{X}_{{k,k}}}Y_{{k,k + 1}}^{{1{\text{/}}2}}; \\ \end{gathered} $$
(16)
$${{X}_{{k,k}}} = {{\left\{ {{{{\left[ {\begin{array}{*{20}{c}} \cdot & \circ & \cdot \\ \circ & \circ & \circ \\ \cdot & \circ & \cdot \end{array}} \right]}}_{k}} \times \left[ {\begin{array}{*{20}{c}} \circ \\ \circ \\ \cdot \end{array}} \right]_{k}^{{1{\text{/}}2}} \times \left[ {\begin{array}{*{20}{c}} \cdot \\ \circ \\ \circ \end{array}} \right]_{k}^{{1{\text{/}}2}}} \right\}}^{{ - 1}}},$$
(17)
$${{Y}_{{k - 1,k}}}\, = \,\frac{{\left[ {\begin{array}{*{20}{c}} \cdot & \circ & \cdot \\ \cdot & \circ & \circ \\ \cdot & \cdot & \cdot \end{array}} \right]_{{k - 1}}^{2}{\kern 1pt} \times \left[ {\begin{array}{*{20}{c}} \circ & \cdot & \cdot \\ \circ & \circ & \cdot \\ \cdot & \cdot & \cdot \end{array}} \right]_{k}^{2}}}{{\left[ {\begin{array}{*{20}{c}} \cdot & \circ & \circ \end{array}} \right]_{{k - 1}}^{{1{\text{/}}2}} \times \left[ {\begin{array}{*{20}{c}} \cdot & \circ & \circ \\ \cdot & \circ & \circ \\ \cdot & \cdot & \cdot \end{array}} \right]_{{k - 1}}^{{1{\text{/}}2}}{\kern 1pt} \times \left[ {\begin{array}{*{20}{c}} \circ & \circ & \cdot \\ \circ & \circ & \cdot \\ \cdot & \cdot & \cdot \end{array}} \right]_{k}^{{1{\text{/}}2}}{\kern 1pt} \times \left[ {\begin{array}{*{20}{c}} \circ & \circ & \cdot \end{array}} \right]_{k}^{{1{\text{/}}2}}}},$$
(18)

where lower index k is the number of the layer (in the transitional region) to which the mentioned cluster belongs. Expressions for \({{\Psi }_{{{\text{liq}}}}}\) and \({{\Psi }_{{{\text{gas}}}}}\) are described by formulas (15).

EQUILIBRIUM DISTRIBUTION OF CLUSTERS OF THE TRANSITIONAL REGION IN THE 2 × 2 APPROXIMATION

To illustrate the derivation of equations for the equilibrium distribution of particles (in which we use particle А for spin +1 and particle В for spin −1), we write explicit equations relating them in the transitional region for a 2 × 2 cluster. The list of probabilities of specific particle configurations for 2 × 2 squares and their ratios of symmetry are given below:

\(\theta {{\left[ {_{{VV}}^{{VV}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{VV}}^{{VA}}} \right]}_{{k,k + 1}}} = \theta {{\left[ {_{{VA}}^{{VV}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{VV}}^{{AV}}} \right]}_{{k,k + 1}}} = \theta {{\left[ {_{{AV}}^{{VV}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{VV}}^{{AA}}} \right]}_{{k,k + 1}}} = \theta {{\left[ {_{{AA}}^{{VV}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{VA}}^{{VA}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{AV}}^{{AV}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{AV}}^{{VA}}} \right]}_{{k,k + 1}}} = \theta {{\left[ {_{{VA}}^{{AV}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{AA}}^{{AV}}} \right]}_{{k,k + 1}}} = \theta {{\left[ {_{{AV}}^{{AA}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{AA}}^{{VA}}} \right]}_{{k,k + 1}}} = \theta {{\left[ {_{{VA}}^{{AA}}} \right]}_{{k,k + 1}}},\) \(\theta {{\left[ {_{{AA}}^{{AA}}} \right]}_{{k,k + 1}}}.\)

The condition for the consistency of probabilities between layers is written as

$$\begin{gathered} \theta {{\left[ {_{{{{{{\sigma }}}_{3}}}}^{{{{{{\sigma }}}_{{\text{1}}}}}}} \right]}_{k}} = \sum\limits_{{{{{\sigma }}}_{2}}} {\sum\limits_{{{{{\sigma }}}_{4}}} {\theta {{{\left[ {_{{{{\sigma }_{3}}{{\sigma }_{4}}}}^{{{{\sigma }_{1}}{{\sigma }_{2}}}}} \right]}}_{{k,k + 1}}}} } \\ = \sum\limits_{{{{{\sigma }}}_{2}}} {\sum\limits_{{{{{\sigma }}}_{4}}} {\theta {{{\left[ {_{{{{{{\sigma }}}_{{\text{4}}}}{{{{\sigma }}}_{{\text{3}}}}}}^{{{{{{\sigma }}}_{{\text{2}}}}{{{{\sigma }}}_{{\text{1}}}}}}} \right]}}_{{k - 1,k}}}} } . \\ \end{gathered} $$
(19)

As in [32], we introduce ξ CF to automatically satisfy condition (19):

$$\begin{gathered} \theta {{\left[ {_{{{{{{\sigma }}}_{3}}{{{{\sigma }}}_{4}}}}^{{{{{{\sigma }}}_{1}}{{{{\sigma }}}_{2}}}}} \right]}_{{k,k + 1}}} = \frac{1}{{16}}\sum\limits_{{{{{{\hat {\sigma }}}}}_{{\text{1}}}}} {\sum\limits_{{{{{{\hat {\sigma }}}}}_{2}}} {\sum\limits_{{{{{{\hat {\sigma }}}}}_{3}}} {\sum\limits_{{{{{{\hat {\sigma }}}}}_{4}}} {\left\{ {\prod\limits_{i = 1\text{`}}^4 {{{\sigma }_{i}}{{{\hat {\sigma }}}_{i}}} } \right\}} } } } \\ \times \;\xi {{\left[ {_{{{{{{{\hat {\sigma }}}}}_{3}}{{{{{\hat {\sigma }}}}}_{4}}}}^{{{{{{{\hat {\sigma }}}}}_{1}}{{{{{\hat {\sigma }}}}}_{2}}}}} \right]}_{{k,k + 1}}}, \\ \end{gathered} $$
(20)
$$\xi {{\left[ {_{{{{{{{\hat {\sigma }}}}}_{3}}{{{{{\hat {\sigma }}}}}_{4}}}}^{{{{{{{\hat {\sigma }}}}}_{1}}{{{{{\hat {\sigma }}}}}_{2}}}}} \right]}_{{k,k + 1}}}\, = \,\sum\limits_{{{\sigma }_{1}}} {\sum\limits_{{{\sigma }_{2}}} {\sum\limits_{{{\sigma }_{3}}} {\sum\limits_{{{\sigma }_{4}}} {\left\{ {\prod\limits_{i = 1\text{`}}^4 {{{{{\sigma }}}_{i}}{{{{{\hat {\sigma }}}}}_{i}}} } \right\}} } } } \theta {{\left[ {_{{{{{{\sigma }}}_{3}}{{{{\sigma }}}_{4}}}}^{{{{{{\sigma }}}_{1}}{{{{\sigma }}}_{2}}}}} \right]}_{{k,k + 1}}}{\kern 1pt} .$$
(21)

In expression (21), the sum is taken over the sigma + cap variables [32] describing all CFs (coefficient 16 = 24 represents normalization to the number of repeated configurations.) Here, \({{\hat {\sigma }}_{i}} = 1\) if this site belongs to the considered subcluster; otherwise \({{\hat {\sigma }}_{i}} = {{\sigma }_{i}}\). In expression (21), the sum is taken over spins.

Below we list CFs. Zero means the site is not in the subcluster; 1 means that it is. With regard to their symmetries, \(\xi {{\left[ {_{{00}}^{{00}}} \right]}_{{k,k + 1}}} = 1\) is the normalization of all CFs; \(\xi {{\left[ {_{{00}}^{{10}}} \right]}_{{k,k + 1}}} = \xi {{\left[ {_{{10}}^{{00}}} \right]}_{{k,k + 1}}}\) = \(\xi {{\left[ {_{{00}}^{{01}}} \right]}_{{k - 1,k}}} = \xi {{\left[ {_{{01}}^{{00}}} \right]}_{{k - 1,k}}}\) is the CF of a site in layer k; \(\xi {{\left[ {_{{10}}^{{10}}} \right]}_{{k,k + 1}}} = \xi {{\left[ {_{{01}}^{{01}}} \right]}_{{k - 1,k}}}\) is CF of a vertical or internal pair of sites in layer k; \(\xi {{\left[ {_{{00}}^{{11}}} \right]}_{{k,k + 1}}} = \xi {{\left[ {_{{11}}^{{00}}} \right]}_{{k,k + 1}}}\) is the CF of a horizontal or interlayer pair of sites between layers k and k + 1; \(\xi {{\left[ {_{{01}}^{{10}}} \right]}_{{k,k + 1}}} = \xi {{\left[ {_{{10}}^{{01}}} \right]}_{{k,k + 1}}}\) is the CF of a diagonal pair of sites between k and k + 1; \(\xi {{\left[ {_{{01}}^{{11}}} \right]}_{{k,k + 1}}} = \xi {{\left[ {_{{11}}^{{01}}} \right]}_{{k,k + 1}}}\) and \(\xi {{\left[ {_{{11}}^{{10}}} \right]}_{{k,k + 1}}} = \xi \left[ {_{{10}}^{{11}}} \right]\) are the CFs of the left and right triplets of a site; and \(\xi {{\left[ {_{{11}}^{{11}}} \right]}_{{k,k + 1}}}\)is the CF of the four sites of a square.

Entropy of the heterogeneous system is written as

$$\begin{gathered} S = \sum\limits_k {\left\{ {\sum\limits_{{{\sigma }_{1}}} {\sum\limits_{{{\sigma }_{2}}} {\sum\limits_{{{\sigma }_{3}}} {\sum\limits_{{{\sigma }_{4}}} {\theta {{{\left[ {_{{{{\sigma }_{3}}{{\sigma }_{4}}}}^{{{{\sigma }_{1}}{{\sigma }_{2}}}}} \right]}}_{{k,k + 1}}}} } } {\kern 1pt} \ln \theta } {{{\left[ {_{{{{\sigma }_{3}}{{\sigma }_{4}}}}^{{{{\sigma }_{1}}{{\sigma }_{2}}}}} \right]}}_{{k,k + 1}}}} \right.} \\ - \;\sum\limits_{{{\sigma }_{1}}} {\sum\limits_{{{\sigma }_{2}}} {\theta {{{\left[ {{{\sigma }_{1}}{{\sigma }_{2}}} \right]}}_{{k,k + 1}}}} \ln \theta {{{\left[ {{{\sigma }_{1}}{{\sigma }_{2}}} \right]}}_{{k,k + 1}}}} \\ \left. { - \;\sum\limits_{{{\sigma }_{1}}} {\sum\limits_{{{\sigma }_{2}}} {\theta {{{\left[ {_{{{{\sigma }_{2}}}}^{{{{\sigma }_{1}}}}} \right]}}_{k}}} \ln \theta } {{{\left[ {_{{{{\sigma }_{2}}}}^{{{{\sigma }_{1}}}}} \right]}}_{k}} + \sum\limits_{{{\sigma }_{1}}} {\theta {{{\left[ {{{\sigma }_{1}}} \right]}}_{k}}\ln \theta {{{\left[ {{{\sigma }_{1}}} \right]}}_{k}}} } \right\}, \\ \end{gathered} $$
(22)

where the first sum in (22) is the sum over all squares; the two second sums are over pairs of sites; and the last sum is taken over single sites.

Equilibrium distributions are obtained by taking derivatives of the entropy of the CF and equating them to zero: \(\partial S{\text{/}}\partial \xi {{\left[ {_{{{{{\hat {\sigma }}}_{3}}{{{\hat {\sigma }}}_{4}}}}^{{{{{\hat {\sigma }}}_{1}}{{{\hat {\sigma }}}_{2}}}}} \right]}_{{k,k + 1}}} = 0\) at constant energy Е and number of particles N

$$\begin{gathered} E = \sum\limits_k {\sum\limits_{i = A,B} {\sum\limits_{j = A,B} {\left( {\varepsilon _{{kk}}^{{ij}}\theta _{{kk}}^{{ij}} + \varepsilon _{{kk + 1}}^{{ij}}\theta _{{kk + 1}}^{{ij}}} \right),} } } \\ N = \sum\limits_k {\sum\limits_{i = A,B} {\theta _{k}^{i}} } . \\ \end{gathered} $$
(23)

By differentiating the entropy of the CF corresponding to all four sites of the square, \(\xi {{\left[ {_{{11}}^{{11}}} \right]}_{{k,k + 1}}}\), we obtain

$$\frac{{\theta {{{\left[ {_{{VV}}^{{VV}}} \right]}}_{{k,k + 1}}}\theta {{{\left[ {_{{AV}}^{{AV}}} \right]}}_{{k,k + 1}}}{{{\left( {\theta {{{\left[ {_{{VV}}^{{AA}}} \right]}}_{{k,k + 1}}}\theta {{{\left[ {_{{AV}}^{{VA}}} \right]}}_{{k,k + 1}}}} \right)}}^{2}}\theta {{{\left[ {_{{VA}}^{{VA}}} \right]}}_{{k,k + 1}}}\theta {{{\left[ {_{{AA}}^{{AA}}} \right]}}_{{k,k + 1}}}}}{{{{{\left( {\theta {{{\left[ {_{{VV}}^{{AV}}} \right]}}_{{k,k + 1}}}\theta {{{\left[ {_{{VV}}^{{AV}}} \right]}}_{{k,k + 1}}}\theta {{{\left[ {_{{AV}}^{{AA}}} \right]}}_{{k,k + 1}}}\theta {{{\left[ {_{{VA}}^{{AA}}} \right]}}_{{k,k + 1}}}} \right)}}^{2}}}} = 1.$$
(24)

Via the sequential differentiation of \(\xi {{\left[ {_{{11}}^{{01}}} \right]}_{{k,k + 1}}}\), we have

$$\frac{{\theta {{{\left[ {_{{AV}}^{{AV}}} \right]}}_{{k,k + 1}}}\theta \left[ {_{{VV}}^{{VA}}} \right]_{{k,k + 1}}^{2}\theta {{{\left[ {_{{AA}}^{{AA}}} \right]}}_{{k,k + 1}}}}}{{\theta {{{\left[ {_{{VV}}^{{VV}}} \right]}}_{{k,k + 1}}}\theta \left[ {_{{AV}}^{{AA}}} \right]_{{k,k + 1}}^{2}\theta {{{\left[ {_{{VA}}^{{VA}}} \right]}}_{{k,k + 1}}}}} = 1,$$
(25)

so the CF for \(\xi {{\left[ {_{{11}}^{{10}}} \right]}_{{k,k + 1}}}\)is

$$\frac{{\theta \left[ {_{{VV}}^{{AV}}} \right]_{{k,k + 1}}^{2}\theta {{{\left[ {_{{VA}}^{{VA}}} \right]}}_{{k,k + 1}}}\theta {{{\left[ {_{{AA}}^{{AA}}} \right]}}_{{k,k + 1}}}}}{{\theta {{{\left[ {_{{VV}}^{{VV}}} \right]}}_{{k,k + 1}}}\theta {{{\left[ {_{{AV}}^{{AV}}} \right]}}_{{k,k + 1}}}\theta \left[ {_{{VA}}^{{AA}}} \right]_{{k,k + 1}}^{2}}} = 1.$$
(26)

The CF for \(\xi {{\left[ {_{{10}}^{{01}}} \right]}_{{k,k + 1}}}\) is

$$\frac{{\theta {{{\left[ {_{{VV}}^{{VV}}} \right]}}_{{k,k + 1}}}\theta \left[ {_{{AV}}^{{VA}}} \right]_{{k,k + 1}}^{2}\theta {{{\left[ {_{{AA}}^{{AA}}} \right]}}_{{k,k + 1}}}}}{{\theta {{{\left[ {_{{AV}}^{{AV}}} \right]}}_{{k,k + 1}}}\theta \left[ {_{{VV}}^{{AA}}} \right]_{{k,k + 1}}^{2}\theta {{{\left[ {_{{VA}}^{{VA}}} \right]}}_{{k,k + 1}}}}} = 1.$$
(27)

Expressions (24)–(27) are identities with rules of factorization

$$\begin{gathered} \theta {{\left[ {_{{cd}}^{{ab}}} \right]}_{{k,k + 1}}} = \frac{1}{{{{Z}_{{k,k + 1}}}}}U_{{k,k + 1}}^{{ab}}U_{{k,k + 1}}^{{cd}}U_{{k,k}}^{{ac}}U_{{k,k}}^{{bd}}, \\ a,\;b,\;c,\;d = A,\;B. \\ \end{gathered} $$
(28)

Minimizing the entropy of the CF for horizontal pairs \(\xi {{\left[ {_{{00}}^{{11}}} \right]}_{{k,k + 1}}}\) we obtain

$$\begin{gathered} {{\left( {\frac{{\theta {{{\left[ {_{{VV}}^{{VV}}} \right]}}_{{k,k + 1}}}\theta \left[ {_{{VV}}^{{AA}}} \right]_{{k,k + 1}}^{2}\theta {{{\left[ {_{{AA}}^{{AA}}} \right]}}_{{k,k + 1}}}}}{{\theta {{{\left[ {_{{AV}}^{{AV}}} \right]}}_{{k,k + 1}}}\theta \left[ {_{{AV}}^{{VA}}} \right]_{{k,k + 1}}^{2}\theta {{{\left[ {_{{VA}}^{{VA}}} \right]}}_{{k,k + 1}}}}}} \right)}^{{1{\text{/}}2}}} \\ \times \;{{\left( {\frac{{\theta {{{\left[ {VV} \right]}}_{{k,k + 1}}}\theta {{{\left[ {VV} \right]}}_{{k,k + 1}}}}}{{\theta {{{\left[ {AV} \right]}}_{{k,k + 1}}}\theta {{{\left[ {VA} \right]}}_{{k,k + 1}}}}}} \right)}^{{ - 1}}} \\ = \exp \left( {\beta \left[ {\varepsilon _{{k,k + 1}}^{{AA}} + \varepsilon _{{k,k + 1}}^{{VV}} - \varepsilon _{{k,k + 1}}^{{AV}} - \varepsilon _{{k,k + 1}}^{{VA}}} \right]} \right). \\ \end{gathered} $$
(29)

This expression becomes the identity after substituting

$$\theta _{{k,k + 1}}^{{ab}} = V_{{k,k + 1}}^{a}V_{{k,k + 1}}^{b}{{\left[ {U_{{k,k + 1}}^{{ab}}} \right]}^{2}}\exp \left( { - \beta \varepsilon _{{k,k + 1}}^{{ab}}} \right).$$
(30)

Minimizing the entropy of CF for vertical pairs \(\xi {{\left[ {_{{01}}^{{01}}} \right]}_{{k - 1,k}}}\) and \(\xi {{\left[ {_{{10}}^{{10}}} \right]}_{{k,k + 1}}}\) we obtain

$$\begin{gathered} \frac{{{{{\left( {U_{{kk}}^{{VV}}U_{{kk}}^{{AA}}} \right)}}^{2}}}}{{{{{\left( {U_{{kk}}^{{AV}}} \right)}}^{4}}}}{{\left( {\frac{{\theta {{{\left[ {_{V}^{V}} \right]}}_{k}}\theta {{{\left[ {_{A}^{A}} \right]}}_{k}}}}{{{{{\left( {\theta {{{\left[ {_{V}^{A}} \right]}}_{k}}} \right)}}^{2}}}}} \right)}^{{ - 1}}} \\ = \exp \left( {\beta \left[ {\varepsilon _{{k,k}}^{{AA}} + \varepsilon _{{k,k}}^{{VV}} - 2\varepsilon _{{k,k}}^{{AV}}} \right]} \right), \\ \end{gathered} $$
(31)

which after substitution becomes the identity

$$\theta _{{k,k}}^{{ab}} = V_{{k,k}}^{a}V_{{k,k}}^{b}{{\left[ {U_{{k,k}}^{{ab}}} \right]}^{2}}\exp \left( { - \beta \varepsilon _{{k,k}}^{{ab}}} \right).$$
(32)

Finally, we obtain the equation for unary probabilities by differentiating entropy S with respect to the CFs of single \(\xi {{\left[ {_{{01}}^{{00}}} \right]}_{{k - 1,k}}}\) and \(\xi {{\left[ {_{{00}}^{{10}}} \right]}_{{k,k + 1}}}\). With factorization (28), we obtain

$$\begin{gathered} {{\left\{ {\frac{{U_{{k,k - 1}}^{{VA}}U_{{k,k - 1}}^{{AA}}U_{{k,k}}^{{AA}}}}{{U_{{k,k - 1}}^{{AV}}U_{{k,k - 1}}^{{VV}}U_{{k,k}}^{{VV}}}} \times \frac{{U_{{k,k + 1}}^{{AV}}U_{{k,k + 1}}^{{AA}}U_{{k,k}}^{{AA}}}}{{U_{{k,k + 1}}^{{VA}}U_{{k,k + 1}}^{{VV}}U_{{k,k}}^{{VV}}}}} \right\}}^{{1{\text{/}}2}}} \\ \times \;{{\left\{ {\frac{{\theta {{{\left[ {VA} \right]}}_{{k - 1,k}}}\theta {{{\left[ {AA} \right]}}_{{k - 1,k}}}}}{{\theta {{{\left[ {VV} \right]}}_{{k - 1,k}}}\theta {{{\left[ {AV} \right]}}_{{k - 1,k}}}}}\frac{{\theta {{{\left[ {AV} \right]}}_{{k,k + 1}}}\theta {{{\left[ {AA} \right]}}_{{k,k + 1}}}}}{{\theta {{{\left[ {VV} \right]}}_{{k,k + 1}}}\theta {{{\left[ {VA} \right]}}_{{k,k + 1}}}}}} \right\}}^{{ - 1{\text{/}}4}}} \\ \times \;{{\left( {\frac{{\theta {{{\left[ {_{A}^{A}} \right]}}_{k}}}}{{\theta {{{\left[ {_{V}^{V}} \right]}}_{k}}}}} \right)}^{{ - 1{\text{/}}2}}}{{\left( {\frac{{\theta {{{\left[ A \right]}}_{k}}}}{{\theta {{{\left[ V \right]}}_{k}}}}} \right)}^{{1{\text{/}}2}}} = \exp \left( {\beta \left[ {\mu _{k}^{A} - \mu _{k}^{V}} \right]{\text{/}}2} \right). \\ \end{gathered} $$

If substitutions (30) and (32) are used here for the second and third cofactors in separate parentheses, the first brace disappears and the last expression is rewritten as

$$\begin{gathered} {{\left( {\frac{{V_{{kk - 1}}^{V}{{{\left( {V_{{kk}}^{V}} \right)}}^{2}}V_{{kk + 1}}^{V}}}{{V_{{kk - 1}}^{A}{{{\left( {V_{{kk}}^{A}} \right)}}^{2}}V_{{kk + 1}}^{A}}}} \right)}^{{1{\text{/}}2}}}\exp \left\{ {\beta [{{\varepsilon }_{{AA}}} - {{\varepsilon }_{{VV}}}]} \right\}{{\left( {\frac{{\theta {{{\left[ A \right]}}_{k}}}}{{\theta {{{\left[ V \right]}}_{k}}}}} \right)}^{{1{\text{/}}2}}} \\ \, = \exp \left( {\beta \left[ {\mu _{k}^{A} - \mu _{k}^{V}} \right]{\text{/}}2} \right), \\ \end{gathered} $$
(33)

which with the natural relationships between energy parameters \(\varepsilon _{{k,k + 1}}^{{AV}} = \varepsilon _{{k,k + 1}}^{{VA}} = \varepsilon _{{k + 1,k}}^{{AV}} = \varepsilon _{{k + 1,k}}^{{VA}}\) and \(\varepsilon _{{k - 1,k}}^{{AA}} = \varepsilon _{{k,k + 1}}^{{AA}} = \varepsilon _{{k,k}}^{{AA}} = {{\varepsilon }_{{AA}}},\) \(\varepsilon _{{k - 1,k}}^{{VV}} = \varepsilon _{{k,k + 1}}^{{VV}} = \varepsilon _{{k,k}}^{{VV}} = {{\varepsilon }_{{VV}}},\) gives the final expression

$$\frac{{\theta _{k}^{i}}}{{V_{{k,k - 1}}^{i}{{{\left[ {V_{{k,k}}^{i}} \right]}}^{2}}V_{{k,k + 1}}^{i}}} = \exp \left( {\beta \left[ {\mu _{k}^{i} - 2\varepsilon _{{kk}}^{{ii}}} \right]} \right).$$
(34)

CONCLUSIONS

We generalized the new approach in [32] to spatially distributed heterogeneous systems. As a special case, we considered a model of the transitional region of the interface between coexisting phases in magnets and vapor–liquid systems. The capabilities of [32] were generalized for clusters of any sizes in heterogeneous lattices. As in a homogeneous system, increasing the size of clusters yielded precise results. We generalized ordered systems and interfaces considerably in [20–22]. Expressions for density concentration profiles in transitional regions were written for two-dimensional 3 × 3, K1s, and 2 × 2 basis clusters.

Our CV principles for locally heterogeneous spatially distributed systems form the mathematical apparatus for calculating the surface (interface) tension that we subsequently discussed in [33].