MODELS OF RECRYSTALLIZATION ACTIVATED BY A DIFFUSION FLOW OF IMPURITIES FROM A THIN-FILM COATING WITH A CONVECTION TERM AT THE CRYSTAL SURFACE: EXACT SOLUTIONS

. Two models of recrystallization are proposed taking into account the convective ﬂux of impurity exchange between the polycrystalline and the thin-ﬁlm coating. The special boundary modes of recrystallization described by the single-phase and two-phase Stefan problems with the boundary condition at coated surface containing the convective term. The exact solutions of the formulated problems corresponding to the grain-boundary concentration of impurities are obtained. The detail theoretical analysis focused on the third type problem shows that the concentration of impurities and the width of the recrystallized layer increase with an increase in the annealing time. An increase in intensity of impurity exchange between the polycrystalline and the coating promotes an increase in the width of the recrystallized layer. The recrystallization front position increases with an increase in the surface concentration of impurities and it decreases with an increase in the intensity of the impurity ﬂux from the surface. The rate of recrystallization kinetics increases with an increase in the intensity of impurity exchange between the polycrystalline and the coating.


Introduction
In order to describe theoretically the phase transitions in solids the heat (diffusion) equation with boundary condition at moving boundary is used often [1]. The moving boundary corresponds to the interface between two phases (phase transition boundary). The problem with unknown time dependence of the motion of the phase transition boundary usually represents the Stefan problem [27]. Various versions of Stefan's problem are well understood mathematically [28,40,43].
Even though such problems have become widely used to simulate processes caused by heat transfer [6,44,46], the formulation of problems for describing diffusion-controlled processes in polycrystalline solids can be considered insufficiently studied [23,26]. Problems with moving boundaries acquire particular importance in the theoretical description of the kinetic regularities of recrystallization of metals and welds under the influence of a flux of impurity atoms from a thin-film coating applied to the surface [33,37]. The kinetics of diffusioncontrolled processes has been experimentally studied in a wide range of metal-coating systems [15,22,25]. It was found experimentally that the motion of the boundary of the recrystallized layer activated by the rest of the impurities obeys the root dependence on the annealing time [3,20,21]. The penetration of impurities from the coating occurs through grain boundary diffusion [13,30,31]. The penetration depth of impurities determines the width of the recrystallized layer in the case of their low solubility in the sample matrix [2,14].
Of interest is a theoretical description of the regularities of recrystallization at various boundary regimes [10,39,45] for controlling the fluxes of impurities from the coating to the sample [8,18,19]. In the present paper, we describe the peculiarities of recrystallization kinetics activated by diffusion taking into account the impurity exchange between the polycrystalline and the coating. We consider two cases. The first one is the case of poorly soluble impurities in the matrix of the polycrystalline sample modeled by single-phase Stefan problem [17]. The second one describes the impurity penetration behind the recrystallization front modeled by two-phase Stefan problem [5,34]. Here we focus on analysis on the third type problem in both cases. Note that the second type condition at fixed boundary corresponding to the sample surface was applied recently to theoretical description of recrystallization kinetics [32]. Recently we also propose the new diffusion models [35,36].
Note that the various transfer processes used in different technical applications can be described theoretically by the diffusion/heat equation with the condition at the moving boundary. In particular, impurity segregation in the crystallization of melt is modeled numerically [38]. The authors of [12] study the segregation during silicon and germanium growth from thin melt layers using the submerged heater method with numeric simulation. The solidification of metals and melt crystallization are investigated for a long time [4,41]. Recent papers [11,42] report the results of modeling thermal conductivity. The authors of [9] investigated the longitudinal lattice temperature distribution of the active region and compared the thermal properties of metal-metal and semiinsulating surface-plasmon THz optical waveguides using a heat diffusion model. Features of crystallization kinetics at laser-powder cladding were actively investigated numerically [7,16,29]. New model of thermal behavior and solidification characteristics during laser welding of dissimilar metals was proposed in [24].

Problem formulation
In this section, we propose the modified model of recrystallization of a polycrystal coated with a thin film of a material, which is poorly soluble in the matrix of the polycrystalline sample (e.g., Ni in Mo or W [14,20,30]). The recrystallization is considered as the phase transition and the recrystallization front defines the phase transition boundary separating the recrystallized region and non-recrystallized one. The saturation of the material with impurities occurs only due to grain-boundary diffusion, since the rate of bulk diffusion is much lower than the rate of movement of the recrystallization front at the temperature at the beginning of recrystallization activation [22].
In the case of an impurity with very low solubility in the matrix of a polycrystalline sample, the width of recrystallized region is determined by the penetration of impurities into the depth of the sample along the grain boundaries. Therefore, the single-phase model of recrystallization can be applied. We consider the system geometry as was described in [32]. The x axis perpendicular to the flat surface of a semi-infinite polycrystalline sample, and a thin-film coating is placed in the yz plane. It is supposed that the diffusion of impurities (diffusant) occurs into polycrystalline sample from the surface (x = 0) in depth uniformly over its, and the position of the moving recrystallization front is the function of time x b (t), which is determined experimentally usually as x b (t) ∼ t 1/2 [22]. Thus, in order to find exact explicit expression of distribution of impurity concentration along the grain boundary, we formulate the model of recrystallization using the problem with moving boundary (the Stefan problem) [27].
In the case considered, the concentration u(x, t) into the grain boundary is determined as a solution of non-stationary diffusion equation with the diffusion coefficient defined by the stepwise function where D 1 is the grain boundary diffusion coefficient at the recrystallized region.
The impurity concentration at the moving boundary of the recrystallization front at x = x b (t) assumed to be negligible. Then, we can write the following condition at the moving boundary (2. 2) The diffusion flux through the recrystallization front is proportional to the velocity of the recrystallization front on this moving boundary. Then, we can use the Stefan condition at the moving boundary where γ is a known positive constant. The diffusion flow of impurities from a thin-film coating at the sample surface x = 0 is given by We take into account the possibility of convective outflow of diffusant at the surface with a thin-film coating. We suppose that the diffusant flow is proportional to the difference between the diffusant concentration at the surface u| x=0 and concentration of the constant diffusant outflow u s . Therefore, the diffusant flow can be written as (2.5) The value of α characterizes intensity of impurity exchange between the polycrystalline and the coating. It is defined as the flux of diffusant from a unit of surface area per unit of time with a difference in the concentration of impurities between the surface and the coating of one unit.
Equating flows (2.4) and (2.5), we obtain the boundary condition at the surface The condition with convective term (2.6) is the boundary condition of the third type. Thus, the grain boundary concentration is the solution of diffusion equation (2.1) satisfying the boundary conditions (2.2) and (2.3) at the moving boundary (phase transition boundary) and the condition (2.6) of the third type at the fixed boundary (surface). The solution of the problem formulated must be a continuous, finite and decreasing function of x.
In order to find exact solution of problem we consider the general case consisting of the conditions at the fixed boundary of the first type the second type and the third type (2.6). For this, we write the general condition at the fixed boundary as follows: where we introduce indicators σ 1,2 . In the case of σ 1 = 1 and σ 2 = 0 condition (2.9) transforms into boundary condition of the first type (2.7); in the case of σ 1 = 0 and σ 2 = 1 condition (2.9) transforms into boundary condition of the second type (2.8); in the case of σ 1 = 1 and σ 2 = 1 condition (2.9) transforms into boundary condition of the third type (2.6).
As well known, the exact solutions of the Stefan problems with boundary condition of the second and the third types exist when the concentration flux of impurities ∼t 1/2 is maintained on the sample surface. The assumption about the possibility of using the root dependence on time is justified by the fact that many processes are described precisely by the root dependence on time, which makes it common in analytical calculations. Then, in order to obtain exact solution of the problem, we put below D 1 /α = h(πD 1 t) 1/2 , where h is a constant determined by the intensity of the impurity flux from the surface.

Exact solutions
As well known, the solution to diffusion equation where constants A and B are found from the boundary conditions. Substituting this solution into the boundary conditions (2.2), (2.3) and (2.9) (we emphasize that it is necessary to replace D 1 /α with h(πD 1 t) 1/2 in equation (2.9) in order to obtain exact solution of the problem), we find that the exact solution to diffusion equation (2.1) satisfying the boundary conditions (2.2), (2.3) and (2.9) in general form can be written as which defines the moving recrystallization front position as follows The solution (2.10) and equation (2.11) of general problem allows to write: i) the solution of the firs type problem, which satisfies the boundary condition of the first type (2.7) at the surface [37], as follows ii) the solution of the second type problem, which satisfies the boundary condition of the second type (2.8) at the surface, as follows iii) the solution of the third type problem, which satisfies the boundary condition of the third type (2.6) at the surface, as follows Typical dependences of the distribution of the grain-boundary concentration of impurities (2.17) in the recrystallized region at the fixed time in the case of third type problem are shown in Figure 1a. The impurity concentration decreases monotonically to zero value at the moving boundary of recrystallized layer x b . The penetration depth decreases with an increasing h. Therefore, an increase in intensity of impurity exchange between the polycrystalline and the coating promotes an increase in the width of the recrystallized layer.
Typical dependence of impurity concentration on annealing time (2.17) at fixed distance from the sample surface in the case of third type problem is shown in Figure 1b. The concentration increases monotonically over time reaching the saturation value of u 0 = u s erf(β)/(erf(β) + h).

Analysis of the recrystallization front kinetics
The kinetics of recrystallization front is determined by equation (2.12). The position of recrystallization front versus t 1/2 is the straight line with an angle of inclination determined by parameter β. It determines the rate of recrystallization kinetics. The transcendental equation (2.11) can be solved in general case numerically only. We focus below on the case of the third type problem.
The results of numerical analysis of equation (2.18) are presented in Figure 2. Note that the numerical solutions to equation (2.18) can be easy obtained from the explicit function h = k/βe −β 2 − erf(β), which we inverse and then plot the dependence β = β(h).   with an increase in the intensity of impurity exchange between the polycrystalline and the coating. An increase in k promotes an increase in β. Therefore, the rate of crystallization kinetics increases with an increase in the surface concentration u s and with a decrease in γ. The solution of transcendent equation (2.11) can be solved analytically in the case of small values of the parameter β. In this case of β 1 we can put in equation (2.11) exp(β 2 ) ∼ 1 and erf(β) ≈ 2β/ √ π, then we get the equation the positive root of which is given by In the case of the boundary condition of the first type (2.7) from equation (2.19) we find β = (u s /2γ) 1/2 , which can be obtained directly from equation (2.14). Then, the dependence of the position of the recrystallization front (2.12) takes the form The recrystallization front position (2.21) increases with an increase in the surface concentration of impurities, and it decreases with increasing parameter γ of Stefan condition.
In order to demonstrate the comparison of exact analytical and approximate solutions, we plot in Figure 3a kinetic dependencies x b (t) calculated from equation (2.12) with exact root of equation (2.14) and x b (t) calculated from approximate equation (2.21). Figure 3b demonstrates the absolute difference ∆x b between these exact and approximate solutions. We find that the absolute difference ∆x b is very small (about 0.001-0.016 at 0 < t < 1) and the relatively difference ∆x b /x b is about 2.8% at 0 < t < 1. This indicates a good accuracy of the approximate equation (2.21) for small values of k.
Approximate expression (2.21) allows simple estimating the depth of diffusant penetration into the polycrystalline during annealing with known surface concentration of impurities and the coefficient of grain-boundary diffusion [37]. From experiments of nickel grain-boundary diffusion in molybdenum an approximate estimate of the recrystallized layer depth in pre-deformed molybdenum gives a value about x b ≈ 158, µm,according with equation (2.21) after 10 h of annealing with D 1 = 10 −12 m 2 /s [20][21][22], and γ = 1.44 and the nickel concentration on the surface u s = 50%. Note that the calculation of position of recrystallization front (2.12) with exact root β of equation (2.14) with the same values of diffusion parameters gives the value about 150 µm. The obtained estimations of penetration depth of nickel in molybdenum are in satisfactory agreement with the experimental data [14,[20][21][22].
In the case of the boundary condition of the second type (2.8) from equation (2.19) we find β = k/h, which can be obtained directly from equation (2.16). Then, the dependence of the position of the recrystallization front (2.12) takes the form The recrystallization front position (2.22) decreases with an increase in the intensity of the impurity flux from the surface.
In the case of the boundary condition of the third type (2.6) from equation (2.19) we find where β 1 = h √ π/4 and β 2 2 = k √ π/2. The recrystallization front position defined by root (2.23) increases with an increase in intensity of impurity exchange between the polycrystal and the coating. The smallness of the β defined by equation (2.23) is ensured by the fulfillment of the condition h 1 or k h.

Problem formulation
Here we consider the modified two-phase model of activated recrystallization [33]. This model is used in the case when the diffusant can penetrate behind the recrystallization front. In this case, the penetration depth of the diffusant will be greater than the width of the recrystallized layer. Then the concentration of diffusant after the recrystallization front at x > x b (t) cannot be considered negligible although the concentration at the region 0 < x < x b (t) before the phase transition boundary x b (t) is greater than the concentration after the phase transition boundary x b (t) at the region x > x b (t). It is necessary to take into account the distribution of the diffusant after the recrystallization front at x > x b (t).
The system geometry is the same as in Section 2. The diffusion coefficient in equation (2.1) is given now by where D 1,2 are the known positive constants. The initial condition is specified in the two-phase Stefan problem: where u 0 is known impurity concentration of at initial moment of time t = 0. We suppose that the impurity concentrations of both phases are the same at the moving boundary: where u b is known impurity concentration of at the moving boundary (the phase transition concentration).
The diffusion flow through the recrystallization front suffers a jump proportional to the speed of its movement that it can be written with using the standard Stefan condition at the moving boundary: The condition at the fixed boundary is defined by the same equation (2.9), which can be rewritten as follows Thus, the two-phase model of recrystallization is an initial-boundary value problem (2.1), (3.1)-(3.5). The detail analysis we hold in [32] in the case of the problem of the second type. We focus below on analysis of the problem of the third type.

Exact solutions
As well known, the solution to diffusion equation (2.1) with diffusion coefficient defined by equation (3.1) can be presented as where constants A 1,2 and B 1,2 are found from the boundary conditions. Substituting this solution into the boundary conditions (3.2)-(3.5), we find that the exact solution of the problem (2.1), (3.1)-(3.5) in general form is given by where θ = D 2 /D 1 , erfc(z ) = 1 − erf(z ), and the phase transition boundary is given by the same equation (2.12) with β defined by the positive root of equation The solution (3.6) and equation (3.7) of general problem allows to write: i) the solution of the firs type problem, which satisfies the boundary condition of the first type (2.7) at the surface, as follows ii) the solution of the second type problem, which satisfies the boundary condition of the second type (2.8) at the surface, as follows iii) the solution of the third type problem, which satisfies the boundary condition of the third type (2.6) at the surface, as follows Typical dependences of the distribution of the impurity grain-boundary concentration (3.12) in the recrystallized layer and ahead of the recrystallization front at the fixed time in the case of third type problem are shown in Figure 4. A flow jump at the boundary of the recrystallization front causes a non-smooth profile of the impurity concentration distribution. A decrease in h leads to the increase in impurity concentration at fixed distance from the sample surface (Fig. 4a). The impurity concentration in recrystallized layer (x < x b ) in the case of D 1 < D 2 (θ > 1) is lower than in the case of D 1 > D 2 (θ < 1) at the fixed time, and the concentration in non-recrystallized region (x > x b ) in the case of D 1 < D 2 is higher than in the case of D 1 > D 2 (Fig. 4b). The concentration of impurities and the width of the recrystallized layer increase with an increase in the annealing time (Fig. 4c).

Analysis of the recrystallization front kinetics
We mention above that the value of β completely determines the kinetics of the recrystallization front. The analysis showed that the values of the root of equation ( where Note that in the case of the second and the third type problems, the equation (3.14) becomes the linear one. The quadratic term in equation (3.14) is necessary only for the first type problem. In this case, the solution of equation (3.14) can be written as It is interesting to note that the solution of equation (3.14) exists in the case of γ = 0. This case is corresponding to the smooth concentration distribution [36].
It is sufficient to restrict oneself to a linear term in equation (3.14) in the case of the second boundary value problem and the solution is given by In the case of the third boundary value problem, we restrict to a linear term in equation (3.14) too, and find the solution as follows . (3.17) In the case of small value of θ (this case corresponds to the condition of D 2 D 1 ) from equation (3.17) we find the simplest estimation of the root as which allows to obtain the estimation of the recrystallization front position as In the case of small value of h from equation (3.17) we find that the parameter β is linearly decreases with an increase in h as follows: where We focus below on numerical analysis of kinetics of recrystallization front in the case of the third type problem. The results of numerical analysis of equation (3.13) are presented in Figures 5 and 6. Note that the numerical solutions to equation (3.16) can be easy obtained from the explicit function which we inverse and then plot the dependence β = β(h), and from explicit function γ = γ(β), which we inverse and then plot the dependence β = β(γ). Figure 5 demonstrates the typical dependencies of equation (3.13) root β (Fig. 5a) and dependencies of recrystallization front position calculated with root of equation (3.13) (Fig. 5b) on characteristic parameter of the third type problem h. It is found that β monotonically decreases with an increase in h (Fig. 5a). The width of recrystallized layer decreases with an increase in h. Therefore, the rate of recrystallization front motion decreases with a decrease in the intensity of impurity exchange between the polycrystalline and the coating. The root of equation (3.13) β monotonically decreases with an increase in γ and it increases with a decrease in θ (Fig. 6a). The kinetics of recrystallization front calculated with root of equation (3.13) is shown in Figure 6b. The rate of recrystallization in the case of D 1 < D 2 (θ > 1) is lower than in the case of D 1 > D 2 (θ < 1) at the fixed time.

Conclusion
We proposed two models of recrystallization taking into account the convective flux of impurity exchange between the polycrystalline and the thin-film coating. We have shown that the single-phase and two-phase Stefan problems with the boundary condition at coated surface containing the convective term can be applied to description of the special boundary modes of recrystallization.
We obtained the exact solutions of the formulated problems and analyzed in details the case of the third type problem. In was found that an increase in intensity of impurity exchange between the polycrystalline and the coating promotes an increase in the width of the recrystallized layer, the rate of recrystallization, and the impurity concentration at fixed distance from the sample surface.