Basics of Calculating the Surface Properties of Solid Solutions Taking the Ordering of Components into Account

Abstract

An approach for calculating the surface thermodynamic characteristics of solid solutions (alloys, salts, oxides, ferroelectrics, and nonstoichiometric compounds) with different degrees of ordering at the vapor–solid interface is formulated on the basis of the lattice gas model. This approach is the only one that makes it possible to describe the properties of phases and transition regions between equilibrium coexisting phases with equal accuracy. The model is constructed for the mixture components having comparable (but different) sizes with an arbitrary type of ordered structures in the bulk phase. The type of ordering is determined by some pattern formed by different sublattices periodically repeated in space. The transition region between the solid solution and vapor is a multilayered region with a variable density of components. Interatomic interaction potentials are taken into account in the approximation of pair contributions with preservation of direct correlation effects for several coordination spheres. A change of variables was used to reduce the equilibrium set of equations to component distributions, which allows one to reduce the problem to the dimensionality of the concentration profiles of components in the transition region. The case of nonequilibrium states of a solid solution and description of its evolution by kinetic equations are discussed. Finding the concentration profile of a solid solution allows one to evaluate the state of the interface roughness, the specific area of the rough surface, and the surface segregation of the solution components, and to analyze the effect of the surface segregation on the degree of ordering of the solution components in the transition region.

Appendices

Appendix 1 . Reducing the dimension of the system of nonlinear equations for the distribution of components

An increase in the number of system components increases the dimension of the system of equations for the unary ($\theta _{f}^{i}$) and pairwise ($\theta _{{fg}}^{{ij}}$) functions. The number of equations for $\theta _{f}^{i}$ increases linearly as s increases, and the number of equations of pairwise functions $\theta _{{fg}}^{{ij}}(\rho )$ increases even faster as s(s – 1)/2. An increase in the number of calculated monolayers κ (the system has a nonuniform distribution of the density) gives rise to a further increase in the dimension of the equation set [28, 53].

To reduce the dimension, the system of equations is constructed with respect to the functions of conditional probability $t_{{gf}}^{{si}}$. The change of variables consists in the fact that $u_{{fg}}^{i} = t_{{gf}}^{{si}}$ are selected as independent variables; $\sum\nolimits_{i = 1}^s {u_{{fg}}^{i}} = 1.$ Conditional probabilities $t_{{gf}}^{{si}}$ can be expressed through equations $\theta _{{fg}}^{{ij}}\theta _{{fg}}^{{ss}} = \theta _{{fg}}^{{is}}\theta _{{fg}}^{{sj}}{{e}^{{\beta \varepsilon _{{fg}}^{{ij}}}}}$—which reflect the essence of the QCA—as $t_{{fg}}^{{ij}} = \frac{{\theta _{{fg}}^{{is}}\theta _{{fg}}^{{sj}}}}{{\theta _{f}^{i}\theta _{{fg}}^{{ss}}}}{{e}^{{\beta \varepsilon _{{fg}}^{{ij}}}}}$ = $\frac{{u_{{fg}}^{i}u_{{gf}}^{j}\theta _{f}^{s}}}{{\theta _{f}^{i}u_{{gf}}^{s}}}{{e}^{{\beta \varepsilon _{{fg}}^{{ij}}}}}$. In that case, unary functions $\theta _{f}^{i}$ are calculated through given desired variables $u_{{fg}}^{i}$, instead of set of equations (5), by the following formula:

$$\begin{gathered} \theta _{f}^{i} = \frac{{u_{{fg}}^{i}\xi _{{fg*}}^{i}}}{{{{\psi }_{f}}}},\,\,\,\,\xi _{{qp}}^{i} = 1 + \sum\limits_{j = 1}^{s - 1} {u_{{gf}}^{j}} (exp\{ \beta \varepsilon _{{fg}}^{{ij}}\} - 1), \\ {{\psi }_{f}} = u_{{fg*}}^{s} + \sum\limits_{i = 1}^{s - 1} {u_{{fg*}}^{i}\xi _{{fg*}}^{i}} , \\ \end{gathered} $$

(A1.1)

where $fg{\text{*}}$ is a pair conditionally called the “reference pair,” which extends from site f and possesses the highest interaction energy between the components located at sites of the pair. Using conditional densities in sites $fg{\text{*}}$, unary functions $\theta _{f}^{i}$ and pairwise functions $\theta _{{fg}}^{{ij}}$ are recalculated.

Remaining functions $\theta _{f}^{s}$, $\theta _{{fg}}^{{sj}}$, and $\theta _{{fg}}^{{is}}$ are determined from the following normalizations:

$$\sum\limits_{i = 1}^s {\theta _{f}^{i}} = 1,\,\,\,\,\sum\limits_{j = 1}^s {t_{{fg}}^{{ij}}} = 1{\text{ }} \Rightarrow \sum\limits_{i = 1}^s {\sum\limits_{j = 1}^s {\theta _{{fg}}^{{ij}}} } = 1.$$

(A1.2)

In the QCA, each coordination sphere is analyzed independently; therefore, the above expressions for the nearest neighbors are supplemented by similar expressions for subsequent neighbors.

Appendix 2 . Basics of the kinetic approach

In a nonequilibrium state, total distribution function Р({$\gamma _{f}^{i}$},τ) varies because of a certain process. Let the total process consist of many steps, and let the step number of the elementary process be denoted by α [28, 29]. The main kinetic equation for the evolution of the total distribution function of the system in state {I} (for brevity, notation {I} ≡ {$\gamma _{f}^{i}$} is used) can be written, owing to implementation of α elementary processes in condensed phases, in the following fashion:

$$\begin{gathered} \frac{d}{{d\tau }}P(\{ {\text{I}}\} ),\tau ) \\ = \sum\limits_{\alpha ,\{ {\text{II}}\} } {[{{W}_{a}}(\left\{ {{\text{II}}} \right\} \to \left\{ {\text{I}} \right\})P(\left\{ {{\text{II}}} \right\},\tau ),~} \\ - \,\,{{W}_{a}}(\left\{ {\text{I}} \right\} \to \left\{ {{\text{II}}} \right\})P(\left\{ {\text{I}} \right\},\tau )], \\ \end{gathered} $$

(A2.1)

where W_α({I} → {II}) is the probability of implementation of elementary process α (the probability of transition through channel α), as a result of which the system passes from state {I} to state {II} at time point τ. The sum with respect to index α is taken over all possible transitions for all realizable states of the system.

If the elementary process proceeds at one site, then the lists of occupancy states {I} and {II} of the system sites differ only for this site. One-site processes are the processes associated with a change in the internal degrees of freedom of a particle, with the adsorption and desorption of nondissociating molecules, and with a reaction proceeding by a shock mechanism. If the elementary process proceeds at two neighboring sites of the lattice, then the lists of states {I} and {II} differ in the occupancy states of these two sites. Two-site processes are the exchange reactions, the adsorption and desorption of dissociating molecules, the migration processes by vacancy and exchange mechanisms, etc. The partition function of state {II} corresponds to the change in the occupancy states of all lattice sites. The relationship between states {I} and {II} depends on the process mechanism, which determines the set of elementary steps α.

Equation (A2.1) is written in the Markov approximation, for which it is believed that the relaxation processes of the internal degrees of freedom of all particles proceed faster than the processes of changing the occupancy states.

Transition probabilities α obey the following condition of detailed equilibrium:

$$\begin{gathered} {{W}_{a}}(\left\{ {\text{I}} \right\} \to \left\{ {{\text{II}}} \right\}){\text{exp}}(-\beta {\text{H}}\left( {{\text{\{ I\} }}} \right)) \\ = {{W}_{a}}(\left\{ {{\text{II}}} \right\} \to \left\{ {\text{I}} \right\}){\text{exp}}(-\beta {\text{H}}\left( {{\text{\{ II\} }}} \right)), \\ \end{gathered}$$

(A2.2)

where H({I}) is the total energy of the lattice system in state {I}.

Within the QCA, all probabilities of the multiparticle configurations describing the influence of surrounding particles on the rates of elementary processes are expressed through unary ($\theta _{f}^{i}$) and pairwise correlators. A closed system of equations for the unary and pairwise correlators can be written in general form as follows [28]:

$$\begin{gathered} \frac{d}{{dt}}\theta _{f}^{i} = \sum\limits_\alpha {\left[ {U_{f}^{b}(\alpha ) - U_{f}^{i}(\alpha )} \right]} \\ + \,\,\sum\limits_h {\sum\limits_j {\sum\limits_\alpha {\left[ {U_{{fh}}^{{bd}}(\alpha ) - U_{{fh}}^{{ij}}(\alpha )} \right]} } } , \\ \end{gathered} $$

(A2.3)

$$\frac{d}{{dt}}\theta _{{fg}}^{{ij}} = \sum\limits_\alpha {\left[ {U_{{fg}}^{{bd}}(\alpha ) - U_{{fg}}^{{ij}}(\alpha )} \right]} + P_{{fg}}^{{ij}} + P_{{gf}}^{{ji}},$$

(A2.4)

$$\begin{gathered} P_{{fg}}^{{ij}} = \sum\limits_\alpha {\left[ {U_{{fg}}^{{(b)j}}(\alpha ) - U_{{fg}}^{{(i)j}}(\alpha )} \right]} \\ + \,\,\sum\limits_h {\sum\limits_m {\sum\limits_\alpha ^{} {\left[ {U_{{hfg}}^{{(cb)j}}(\alpha ) - U_{{hfg}}^{{(mi)j}}(\alpha )} \right]} } } , \\ \end{gathered} $$

(A2.5)

where unknown unary and pairwise functions obey normalization conditions (A1.2), which are fulfilled for any point in time. The right-hand sides of Eqs. (A2.3)–(A2.5) contain the rates of elementary steps. The presence of equations for pairwise correlators makes it possible to reflect any nonequilibrium state of the components and provides a description of the effect of the prehistory of the system state on the process dynamics.

Rates of elementary steps. In Eq. (A2.3), $U_{f}^{i}(\alpha )$ are the rates of elementary two-site processes i ↔ b (here, h ∈ z_f), $U_{{fg}}^{{ij}}(a)$ are the rates of elementary two-site processes i + j_α ↔ b + d_α (h ∈ $z_{f}^{*}$) at neighboring sites, and the second term in (A2.5) describes step i + m ↔ b + c at neighboring sites f and h.

All rates $U_{f}^{i}(\alpha )$ and $U_{{fg}}^{{ij}}(\left. r \right|\alpha )$ of elementary steps are calculated within the theory of absolute reaction rates for nonideal reaction systems, which are written in the QCA considering the interparticle interaction [28, 29, 49]. The properties of the activated complex (AC) in the theory of absolute reaction rates for nonideal reaction systems depend on the interaction between particles in the transition and ground states. This requires, in addition to knowledge of the ${{\varepsilon }_{{ij}}}$ values in the ground state, knowledge of interparticle interactions $\varepsilon _{{ij}}^{*}$ in the transition state. Therefore, the formulas for rates $U_{f}^{i}(\alpha )$ and $U_{{fg}}^{{ij}}(\left. r \right|\alpha )$ depend on both ${{\varepsilon }_{{ij}}}$ and energy $\varepsilon _{{ij}}^{*}$.

As an example, we give the following expressions for diffusion shift rates $U_{{fg}}^{{iV}}(\alpha )$ taking into account the interaction of nearest neighbors (the general case is given in [49]):

$$U_{{fg}}^{{iV}}(\alpha ) = K_{{fg}}^{{ij}}(\alpha )\theta _{{fh}}^{{iV}}\prod\limits_{\eta \in z_{f}^{*}} {S_{{f\eta }}^{i}} \prod\limits_{\chi \in z_{g}^{*}} {S_{{g\chi }}^{V}} ,$$

(A2.6)

where $K_{{fh}}^{{ij}}(\alpha )$ is the rate constant of the elementary step of migration, $S_{{f\eta }}^{i} = \sum\nolimits_{m = 1}^s {\theta _{{f\eta }}^{{im}}} \times$$\exp {{[\beta (\varepsilon _{{f\eta }}^{{im*}} - \varepsilon _{{f\eta }}^{{im}})]} \mathord{\left/ {\vphantom {{[\beta (\varepsilon _{{f\eta }}^{{im*}} - \varepsilon _{{f\eta }}^{{im}})]} {\theta _{f}^{i}}}} \right. \kern-0em} {\theta _{f}^{i}}}$, and $S_{{h\chi }}^{V} = \sum\nolimits_{m = 1}^s {\theta _{{h\chi }}^{{Vm}}} \times$$\exp {{(\beta \varepsilon _{{h\chi }}^{{im*}})} \mathord{\left/ {\vphantom {{(\beta \varepsilon _{{h\chi }}^{{im*}})} {\theta _{f}^{V}}}} \right. \kern-0em} {\theta _{f}^{V}}}$ is the factor of the imperfection function for the hop rate. The product in (A2.6) is taken with respect to neighboring sites η (for central site f) and χ (for central site g) by excluding bond fg, which is indicated by asterisks in z_f* and z_g*.

Equations (A2.4) contain terms $U_{{fg}}^{{\left( i \right)j}}(\alpha )$ and $U_{{hfg}}^{{\left( {mi} \right)j}}(\alpha )$ related to one- (i"b) and two-site (i + m"b + c) reactions of particle i in the presence of neighbor particle j. In this case, particle j itself does not take part in elementary process α, but changes the value of the activation barrier for reacting particle i in the case of a one-site process and for reacting particles i and m in the case of a two-site process. In the QCA, terms $U_{{fg}}^{{\left( i \right)j}}(\alpha )$ and $U_{{hfg}}^{{\left( {mi} \right)j}}(\alpha )$ are closed through functions $\theta _{f}^{i}$ and $\theta _{{fg}}^{{in}}\left( r \right)$, and no new unknown correlators appear. For example (A2.6), the $U_{{hfg}}^{{(VA)A}}$ functions in Eqs. (A2.4) have the following form: $U_{{hfg}}^{{(ij)A}}$= $U_{{hf}}^{{ij}}Y_{{fg}}^{{jA}},$$\Psi _{{fg}}^{{jA}} = t_{{hg}}^{{jA}}{\text{exp}}{{(\beta \delta \varepsilon _{{fg}}^{{jA}})} \mathord{\left/ {\vphantom {{(\beta \delta \varepsilon _{{fg}}^{{jA}})} {S_{{fg}}^{j}}}} \right. \kern-\nulldelimiterspace} {S_{{fg}}^{j}}},$ where $S_{{fg}}^{j} = {{\sum }_{k}}t_{{hg}}^{{jk}} \times$${\text{exp}}(\beta \delta \varepsilon _{{fg}}^{{jk}}),$ and $t_{{hg}}^{{jk}} = {{q_{{hg}}^{{jk}}} \mathord{\left/ {\vphantom {{q_{{hg}}^{{jk}}} {q_{h}^{j}}}} \right. \kern-0em} {q_{h}^{j}}}.$

In the limit of large times, system of kinetic equations (A2.4) transforms into Eqs. (6) for the equilibrium distribution of components.