INFLUENCE OF ADATOMS ON THE VACANCY GROWTH OF FACETED PORES IN A CRYSTAL UNDER MECHANICAL LOAD

The process of growth of faceted pores in a crystal under the influence of an applied mechanical load is considered in the framework of the classical Barton-Cabrera-Frank model, taking into account the presence of adatoms on the surface of pore faces. The growth is caused by the flow of excess vacancies from the bulk of the crystal, which arise due to tensile stresses. The recombination of advacancies and adatoms on the surface of pores is taken into account, and it is shown that as a result, a flow of adatoms from steps and fissures to the terrace can occur. This additional flow contributes to the growth of pores under load and, under certain conditions, can be the predominant mechanism of mass transfer, which must be taken into account for a correct assessment of the growth rate and lifetime of the crystal under load before failure. Expressions are obtained for the dependence of the pore growth rate on the applied mechanical load, the diffusion coefficients of vacancies and adatoms, and the rate of their recombination.


INTRODUCTION
Issues of pore formation in crystalline materials occupy an important place in materials science both in terms of strength, plasticity, and durability of products made from such materials, and because of a wide range of practical applications of porous crystals [1][2][3]. Similar materials with a given pore distribution are used in supercapacitors and batteries [4], photonic crystals [5], sensors [6], epitaxy substrates [7,8], and many other applications. Pores determine many properties, ranging from the ability of crystals to absorb various molecules [9] to their brittleness and strength [10]. Thus, it was shown in [11][12][13] that the impact of even small but constant loads can cause a gradual growth of pores according to the vacancy mechanism and lead to the destruction of the material due to pore percolation. In this regard, it is extremely important to understand all the mechanisms of pore formation under the influence of various physical processes in order to assess the durability of the material, as well as to develop methods that allow the controlled formation of crystals with a predetermined pore distribution for various applications. It was shown in [14][15][16] that pores can be nucleated when a mechanical load is applied to a crystal. In this case, the tensile stress causes the formation of excess vacancies in the bulk of the crystal, whose diffusion to the pore leads to its growth. Understanding this growth mechanism, on the one hand, makes it possible to predict the rate of pore growth under load and the lifetime of the material (if the mechanical load is external, uncontrolled), and, on the other hand, to grow pores in the entire volume of the crystal and provide the desired pore size distribution (when applied given, controlled load). As shown experimentally [17], pores can be faceted to reduce the total free energy. In this regard, such pores can be considered as "void" crystals, since they have much in common with real faceted crystals. In previous studies [11][12][13], it was shown that the growth of such faceted pores can be theoretically described in much the same way as the growth of crystals from a gas phase or a solution. In these works, the classical theories of Barton-Cabrera-Frank (BCF) [18] and Chernov [19] were developed to describe the growth of faceted pores from a gas of vacancies in the bulk of a crystal. All processes occurring on the pore faces were analyzed, namely, the diffusion of vacancies, the evaporation of vacancies into the bulk of the crystal, and their deposition on the terrace, and the dependences of the pore growth rate on the applied load were determined.
However, as shown in a series of works by Pimpinelli [20], Latyshev et al. [21] and their colleagues [22][23][24], surface vacancies can play a decisive role in the growth of real crystals by the terrace-step-fracture mechanism. Since the process of growth of an ordinary crystal and a pore are similar, similarly to the effect of advacancies on the growth of a crystal, adatoms on the surface of a pore can affect the rate of its vacancy growth. This work is a continuation of a series of works devoted to the growth and evolution of crystals [11][12][13][25][26][27][28] and pores in single-and multicomponent systems due to various mechanisms, and its main goal is to analyze the effect of adatoms on the pore surface on its vacancy growth of the pore according to the Barton-Cabrera-Frank mechanism, and find an expression for the dependence of the pore growth rate on the applied mechanical load and the properties of the crystal.

FORMULATION OF THE PROBLEM
Let us consider a crystal containing sufficiently large pores with a characteristic size from tens to hundreds of microns or more. As in [11][12][13], we assume that the pores are faceted, which is often observed experimentally at the late stages of their growth [17] and is due to the minimization of the surface energy. The surface of the pore, like that of an "emptiness crystal", is covered with terraces, steps and fractures. We assume that the size of the pore faces is much larger than the characteristic distance between the steps on the terraces, while the distance between the fractures, on the contrary, is very small. In this case, the steps can be considered as continuous sinks of vacancies and adatoms on the surface. Then, considering the progress of an individual step or its group along the terrace, we can consider the face as infinite and neglect the influence of the pore edges on the distribution of vacancies and adatoms near the steps. The distances between the steps on the terraces also significantly exceed the mean free path of vacancies and adatoms over the crystal surface. The problem under consideration is close to the problem of [11,13] with the key difference that in this work we study the effect of adatoms on the pore surface, which can also diffuse along the terrace, recombine with vacancies, and participate in the mass transfer process. Note that the main goal of this work is to identify general dependencies and patterns that describe the growth of a faceted pore according to the terrace-step-fracture mechanism under the application of a mechanical load. In this regard, although we understand that a significant anisotropy is observed in crystalline materials, which determines the elastic stress tensor, the influence of the application direction of the load relative to the orientation of the pore face under consideration, the dependence of the elastic stresses over the face on the distance to the edges of the pore, and so on, we use a simplified model problem with isotropic elastic stress. In it, since the size of the pore faces is considered to be significantly large compared to all other parameters (the distance between the steps and diffusion lengths), edge effects, such as the effect of inhomogeneity in the distribution of elastic stresses near the edges, are not taken into account. Therefore, the magnitude of elastic stresses is assumed to be constant over the entire face of the pore, and stresses are considered only as the cause of the appearance of additional vacancies in the bulk of the crystal, which are born from defects. Note that such an approximation is often used when considering various problems associated with the nucleation and movement of vacancies in a crystal [29]. With this in mind, the problem under consideration can be represented as a one-dimensional problem of the motion of advacancies and adatoms in a half-plane with sinks in the region of steps, the scheme of which is shown in Fig. 1. The influence of the effects mentioned above, in particular, the inhomogeneity of the distribution of elastic stresses over different parts of the face and their dependence on the size of the face, as well as the anisotropy of the applied load, will be studied in subsequent works.
So, in such a model system in equilibrium, all processes and flows are balanced. However, if an isotropic mechanical load is applied to the crystal, then the concentration of vacancies in the bulk of the crystal will change and can be described by the equation [14]: is the equilibrium volumetric concentration of vacancies without an applied load, is the volume of an atom in a crystal cell, a is the lattice parameter, is the Boltzmann constant, is the temperature, and, accordingly, the value of is the supersaturation of vacancies [14]. Note that, if the applied mechanical load is tensile, then an excess of vacancies appears in the volume of the crystal; if it is compressive, then a deficiency. We also note that in this work we assume that the volume of the crystal contains a sufficient amount of dislocations and other defects, which are a strong source of vacancies. This assumption means that the concentration of vacancies in the volume always corresponds to the applied load and there is no depletion of vacancies in the near-surface region. Otherwise, another approach should be used (Chernov mechanism [19]).
Vacancies originating in the bulk diffuse to the pore surface and increase the concentration of advacancies on the terrace. Then the advacancies diffuse towards the steps. They can also "evaporate" from the terrace into the bulk of the crystal and recombine with adatoms on the pore surface. Adatoms behave in a similar way: they can deposit from the gas phase onto the surface, evaporate, diffuse over the surface to or from the step, and recombine with advacancies. Under tensile stresses, the concentration of vacancies in the volume increases, and we should expect the appearance of a surface flow of advacancies to the steps, and, at the same time, an outflow of adatoms from the step to the terrace. Both of these flows lead to the advancement of the "void" step along the terrace and an increase in the pore volume. The main purpose of this work is to analyze these processes and find the rate of advancement of the "void" step and the rate of growth of a faceted pore depending on the applied mechanical load.

DISTRIBUTION OF ADATOMS AND VACANCIES ON THE PORE SURFACE
To find the pore growth rate, we will use a combination of approaches proposed in the classical work of Barton, Cabrera and Frank [18], Pimpinelli [20], and recent works [11,13]. Let us consider the onedimensional problem of the motion of a group of equidistant steps with a distance l between them. To find a solution in the general case, we introduce the following variables: and are the surface concentrations of adatoms and advacancies; and are their equilibrium values; and are their diffusion coefficients; and are the times of adatom evaporation into the pore volume and advacancy into the crystal volume, respectively. Let us also introduce an additional parameter . Using approaches from [18,20,28], we write an equation for the flow of vacancies from the bulk of the crystal to the surface and a similar one for adatoms arriving at the surface from the gas phase: The first term on the right side of each equation describes the deposition of adatoms/advacancies on the terrace, the second describes the "evaporation" process, the third describes the recombination of adatoms and advacancies, and the last one describes the formation of a new adatom-advacancy pair directly on the terrace. K is the coefficient of proportionality between the recombination rate and surface concentrations. Note that, in contrast to the work by Pimpinelli [20], we consider the possibility of evaporation of vacancies into the bulk of the crystal; that is, the second equation also contains a term describing evaporation. In addition, in the model [20] there is no flow to the surface from the gas phase, while in our studies it is taken into account. Compared to [24], system (3.1) takes into account the recombination of vacan- Since we are considering a quasi-stationary process in which the step motion is very slow compared to the characteristic times of other processes, the surface flows must obey the continuity equation . This allows us to write the final system of equations: When solving this problem, we use the following boundary conditions: near the step ( ), the concentrations of adatoms and advacancies correspond to their equilibrium values ( , ), from which it follows that and . In the center of the terrace between the steps, the flow over the surface should be equal to zero, since the distribution is symmetrical: , . The solution of the system of equations is easy to find in the following form: (3.4) where and λ are the characteristic length, which is determined by the diffusion lengths of vacancies and adatoms in all considered processes. , where superscripts "e" and "r" correspond to evaporation and recombination. Let us note that λ is determined by the smallest diffusion length present in the system. The term containing the factor I describes the co-dependence of the stationary concentrations of adatoms and vacancies due to recombination if the surface concentration of advacancies deviates from the equilibrium one. Solution (3.4) allows one to find the distribution functions of advacancies and adatoms over the surface: Let us consider some special cases: 1) If the system is in equilibrium, i.e. , , Eqs. (3.5) give equilibrium concentration values, and as expected (Fig. 2a). 2) If the supersaturation of vacancies ( ) is increased, but the recombination coefficient is small (it can be neglected), then advacancies do not affect the distribution of adatoms on the surface of the pore (Fig. 2b) and the vacancy growth of the pore can be described in the same way as in [13]. If, on the contrary, the recombination rate is sufficiently high, then the concentration of both adatoms and vacancies In l decreases away from the step relative to the unperturbed values (without recombination). In this case, the influx of vacancies to the step decreases, but an additional diffusion flow of adatoms from the step to the terrace appears (Fig. 2c), accelerating the pore growth.
Knowing the distribution functions of adatoms and advacancies, it is easy to calculate their diffusional fluxes to the step: The final equation for the advancement speed of the echelon of equidistant steps has the form [18]: (3.6) where is the area of the atom. The velocity is considered positive if the pore grows. Note that it follows from Eq. (3.6) that the main contribution to the step motion of the mass transfer can be due to both diffusion of vacancies and adatoms, if recombination is sufficiently effective. Indeed, at a higher diffusion coefficient of adatoms over the surface compared to advacancies ), which is observed in many crystals, the contribution of adatoms to mass transfer becomes significant. In addition to the direct contribution of adatoms, when recombination is taken into account, the effective diffusion length included in the denominator (3.6) also changes compared to the mechanism [11], and this factor also strongly affects the pore growth rate.

PORE GROWTH RATE
Let us note that the supersaturation of atoms in the gas phase (in the pore) in the quasi-stationary case is assumed to be identically equal to zero for the following reasons. Although the application of a mechanical load also changes the equilibrium concentration of adatoms on the very surface of the pore, we con-   sider the pore as a closed system of limited volume containing no additional gas sources. At the first moment after a load is applied to the crystal, excess adatoms evaporate from the surface into a pore or, conversely, precipitate from a pore onto the surface, changing the pressure in the pore. The process ends when the pressure in the pore exactly corresponds to the equilibrium pressure under the given conditions, and therefore the equilibrium concentration will be maintained on the terraces far from the step. Substituting the value instead of , we can rewrite (3.6) in the form: where . Knowing this expression, we can find the pore growth rate using different mechanisms. The first case is the growth of a pore due to the displacement of a group of equidistant parallel steps of the "void" [17], which is described by the equation: Second mechanism is the helical growth. In [11,13], the authors suggested that, by analogy with the growth of a real crystal by embedding atoms in growth spirals, the same process can also take place during the growth of a pore. If a growing pore contains a screw dislocation on its face, then such a dislocation can become a source of "void" steps and lead to further helical growth of the pore due to the influx of vacancies. This causes the formation of etch pits [30] (Fig. 3) and other phenomena inherent in the helical growth of ordinary crystals.
To estimate the rate of helical pore growth, let us use the approach proposed in [11]. During spiral growth, the distance between the steps is equal to [20] where ρ c is the critical radius of two-dimensional vacancy nuclei [14] , γ is the surface energy density of the crystal). The final expression for the helical pore growth rate can thus be written as: Note that, as mentioned in [11], the elastic stresses caused by the presence of the screw dislocation itself can significantly affect the distance between the steps and the concentration of vacancies and can even lead to the formation of a cavity along the core of the screw dislocation. In this regard, Eq. (4.2) should be used only for a qualitative assessment of the rate, and further analysis of the effect of the dislocation's own elastic stresses on growth is required. It should also be noted that in the absence of recombi- In D n In n kT sp a a a a c B a l R D n I n D nI n k T Fig. 3. Schematic representation of the growth of pores due to the incorporation of advacancies into steps and the reverse flow of adatoms according to the considered mechanism on the "negative" helical mound.
R sp U c ℓ nation, Eq. (4.2) coincides with the equation obtained for the rate of helical pore growth in [13]. Finally, we note that Eq. (4.2) is valid if the characteristic dimensions of the pore and the distance between the exit points of screw dislocations is greater than the distance between adjacent steps and the critical radius . Otherwise, the approaches used to describe the growth of small faceted crystals proposed in [31,32] should be used.

CONCLUSIONS
Within the framework of the classical Barton-Cabrera-Frank model, the growth of faceted pores in a crystal under the influence of a mechanical load is studied, taking into account the influence of adatoms present on the pore surface. The distribution functions of adatoms and advacancies are found and it is shown that, in a linear approximation, the recombination of vacancies and adatoms reduces their surface concentrations relative to the values that are due to the applied mechanical load. This affects the rate of pore growth. In the case of an excess of vacancies, recombination reduces the surface flow of vacancies towards the step, simultaneously causing the appearance of a counter flow of adatoms from the step to the terrace. An analytical expression is obtained that describes the dependence of the speed of steps moving along the pore surface on the applied mechanical load, as well as their kinetic properties. It is shown that, in the case of sufficiently fast recombination, mass transfer in some cases can be carried out mainly due to the diffusion of adatoms, even if the process of pore growth itself is caused by supersaturation of vacancies. In addition, the characteristic lifetimes of adatoms and advacancies on the surface and the characteristic diffusion length can be determined by the recombination rate rather than evaporation, as in the classical BCF theory, which also strongly affects the pore growth rate. The results obtained open up new possibilities for the analysis of a wide range of experimental data on the growth of faceted pores. The results can also be used to predict the failure of crystalline materials under low but constant mechanical loads, where slowly growing and percolating pores determine the durability of the material. It is shown that the growth of pores according to the considered mechanism and the destruction process can be slowed down by increasing the pressure in the pore system, which reduces the surface concentration of vacancies and their flow to the steps. Part of the work (analysis of spiral growth) was supported by the state task of IPME RAS (no. FFNF-2021-0001).

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