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We present a detailed discussion of the boundary conditions of the directed crystallization problem, a formulation of the model considering temperature fields of external sources, the mechanism of attachment of particles to the growing solid surface, the influence of interphase component absorption on the phase distribution ratio of the components as well as the calculation of the period of the morphological interface instability which is made with due regard of all the aforementioned conditions.

So far the problem of interface behavior upon phase transition has not yet acquired a satisfactory mathematical formulation due to a variety of the physical phenomena involved. Analytical solutions exist only for elementary problems describing the free interface behavior in directed crystallization conditions, for instance, for those implying a clearly shaped isothermal interface (ellipsoid, paraboloid, hyperboloid) [

Crystallization of two-component liquid solution is described by the system of heat conduction and diffusion equations for solid and liquid phases as well as the corresponding boundary conditions. The boundary crystallization problem is described in detail in a number of studies. For instance in [

Here the values related to the solid phase are denoted by the prime, index r indicates the dimensional quantities, the overline indicates that the functions are written in the lab coordinates, T is the temperature, C concentration, thermal diffusivity, t time, D diffusion coefficient. The diffusion coefficient of the solid phase is assumed to be equal to zero. The problem does not take into account the liquid convection and crystal anisotropy. The heat transfer is written as a heat conduction equation. The interface thickness is assumed to be zero and on the interface have to be fulfilled the temperature continuity condition

and the heat flow continuity condition

Here, is the interface velocity along the normal to the interface toward the liquid phase. The mass flow condition should also be fulfilled

Here is the direction of the normal to the interface surface. These boundary conditions express the general conservation laws between the contacting phases. In analysis of interface stability in the one-dimensional case the interface temperature and infinitely remote point temperatures are specified [

The quasi-equilibrium problem setting is the most popular approximation for directed crystallization problems. The quasi-equilibrium setting implies that the interface temperature is equal to the equilibrium temperature of the phase transition with due allowance made for the effect of the interface curvature, i.e. the Gibbs—Thomson effect. The distribution ratio is assumed to be equilibrium. The temperature distribution is found from homogeneous heat conduction Equation (1) and specified are the values of the temperature phase gradients on the interface rather than the temperature in infinitely remote points. Such problem setting enabled to obtain a number of simple and practically useful solutions of directed crystallization problems [

We introduce environmental heat transfer into the heat conduction equation for two reasons. First, it enables us to state a problem with specified temperature values in infinitely remote points, second, it allows for eliminating physically unrealizable solutions of the stationary problem. Such solutions may occur due to the fact that in interface stability problems material heating is frequently simulated by specified temperature gradients. Such boundary conditions can be accepted provided the aim is to show the feasibility of unstable stationary modes of interface movement. However, in the experiments a temperature field is usually formed in the material by way of environmental heat exchange or induction heating [

1) Temperature continuity in point.

2) Heat flow continuity in point.

3, 4) Phase temperatures T'_{inf} and T_{inf} are set in infinitely remote points in the vicinity of the liquid and solid phases.

5) Temperature gradient is set in point.

6) Temperature is set in point.

The function satisfying the first five conditions has the form

Here is an unknown parameter to be found on solving the stationary problem.

Let us consider the value of interface temperature. In the quas-equilibrium problem setting the interface temperature is assumed to be equal to equilibrium temperature of phase transition. However, from statistical physics it is known that for phase transition particles have to overcome a potential barrier, so transition occurs at a temperature which is different from equilibrium phase transition temperature. This temperature difference is known as kinetic overcooling. The interface velocity is a function of kinetic overcooling. The function form depends on the growth mechanism. If the temperature of the interface is equal to that of the phase transition, the kinetic overcooling is zero. In this case, in accordance with the thermodynamics law, any growth mechanism yields a moveless interface. If the problem involves equal interface and phase transition temperatures, the velocity is independent of the phase transition non-equilibrium grain. Then the goal is to find the movement of the geometric surface where the temperature is equal to that of the phase transition, rather than the interface. If kinetic overcooling is neglected, then linearization of the problem involving a moving interface disregards the linear approximation of the interface velocity related to the growth kinetics. And it should be noted that the interface temperature in the stationary mode practically coincides with the equilibrium temperature of the phase transition. This follows from the limiting transition when the kinetic coefficient tends to zero as it is shown in [

Consider the expression for interface temperature and the terms of the linear expansion of the boundary-value problem which are lost if no account is taken of the dependence of the interface velocity on kinetic overcooling. Let us write the interface velocity as a function of kinetic overcooling

Here

is kinetic overcooling, is the equilibrium temperature of the phase transition. Expand the velocity by kinetic overcooling into a Maclaurin series

Here the kinetic coefficient is introduced

From (8,9) we find

Crystallization temperature is a function of the equilibrium temperature of the phase transition for the given component concentration and interface curvature [

where is the interface curvature, is the surface tension coefficient. At large values of the kinetic coefficient the last term becomes small and the equation takes the form

i.e. the interface temperature is equal to the temperature of phase transition. This is the limiting case considered in the works on directed crystallization. There is a formal reason to take the interface temperature as equal to the temperature of phase transition at the nonzero velocity. Here the smallest term rather than kinetic overcooling is neglected. A different situation occurs when the dependence of interface velocity on kinetic overcooling is neglected, and interface perturbation is used instead of temperature and concentration perturbation expansion of the velocity If the kinetic overcooling in (9) is assumed to be equal to zero, then. This result conceals an error which occurs when solving nonstationary problems, in particular, in analysis of the stability of the stationary regime of interface movement. Consider expansion of velocity by kinetic overcooling in the vicinity of the stationary crystallization regime for small perturbations of stationary temperature and concentration values. Then kinetic overcooling is written as a sum of perturbation and the stationary part

Then the linear approximation of Taylor series expansion of rate in the vicinity of the stationary value of kinetic overcooling takes the form

Here the following notations are used

The expansion yields

For the stationary mode expansion (10) can be written as

Here expansion (9) is used for function (11). For sufficiently large values of the kinetic coefficient the interface temperature is equilibrium which does not imply that kinetic overcooling is zero. In any case, formal substitution of zero kinetic overcooling leads to a zero interface velocity. When solving the stationary problem, kinetic overcooling is not introduced not due to its smallness, rather due to its constant value which is unambiguously related to the stationary interface velocity. Yet, decomposition of velocity into its stationary part and small perturbation, , without considering kinetic overcooling has no physical meaning in the problem involving perturbation of stationary concentration and temperature distribution. It does not allow for the fact that the velocity is a function of kinetic overcooling which, in its turn, is a function of phase transition and interface temperatures. At zero kinetic overcooling the equation means that the interface is shifted with respect to the liquid phase without regard for the kinetics of particle attachment to the growing solid phase surface and the effect of the thermodynamic conditions on the liquid phase in front of the interface. As such an approach is applied to problems of linear analysis of interface stability, the initial perturbation is commonly that of the interface. Consider the physical values for the corresponding problem of small perturbations [

The use of the equilibrium distribution coefficient requires a more precise definition of the diffusion problem since it results in the fact that phase redistribution of components is independent of interface velocity. This is inconsistent with the experimental data. Phase transition is a nonequilibrium process and the value of the equilibrium distribution coefficient is taken from the equilibrium phase diagram. The latter is calculated at equal chemical phase potentials. The values of the chemical potentials correspond to an infinite volume of each phase. It is known, however, that component adsorption occurs on the interface [

Here is the effective distribution coefficient, k_{0} the equilibrium distribution coefficient, the equilibrium adsorption distribution coefficient, the adsorption rate constant. The formula has the following physical meaning. To maintain the composition of the adsorption layer, the rate of component atom diffusion from the melt to the crystal must be higher, the higher the growth velocity. This may give rise to a dissolved component concentration gradient in the melt in the direction opposite to the previous one and, as a result, a component depleted zone may form instead of an accumulation region corresponding to the equilibrium phase diagram without regard for interphase adsorption. In this case the effective distribution coefficients pass from region with the equilibrium distribution coefficient into the region where. In the high interface velocity limit the distribution coefficient will approach the adsorption distribution coefficient,. In the quasi-equilibrium boundary conditions interphase adsorption is commonly neglected. We present linear analysis of interface stability with due regard for the distribution coefficient as a function of interface velocity (12) and derive the period of morphological instability structure.

With allowance made for distribution of ambient temperature, heat conduction Equation (1) take the form

Here is the heat-transfer coefficient. The second term of Equation (13) is the density of heat sources. It takes into account the heat exchange with the environment. In this approximation it is not take into account the temperature distribution over the cross section. The thickness of the rod is assumed to be small. Mass flow balance condition (5) with due account of the effective distribution coefficient and the one-dimensionality of the problem takes the form

Introduce dimensionless parameters into Equations (2)-(4), (13), (14). To this end, multiply heat conduction Equations (13) by factor, where, are auxiliary parameters,

is the equilibrium temperature of phase transition at original liquid solution component concentration C_{0}. Multiply diffusion Equations (2) by factor. Multiply boundary conditions (3,4), (14) by factors,

and, respectively. Interface coordinate in the lab coordinate system us written as

Interface velocity is connected with vn by the expression

Now let us write the boundary-value problem in the moving coordinate system rigidly bound to the interface. Note that the introduced coordinate system is curved with respect to the lab coordinate system. It is connected to the interface, whose velocity, in the general case, is a function of the temperature and concentration of the component, rather than to the interface moving in the stationary regime, i.e. in the lab coordinate system with a constant velocity. New variables are introduced in accordance with the expressions

In Equations (1)-(5), we neglect the solid phase diffusion coefficient, and in the moving coordinate system which is rigidly bound to the interface the equations have the form

These conditions should be supplemented with specified temperature in infinitely remote points

The interface velocity as a function of kinetic overcooling (7) is also specified. Let the melt overcooling conditions be such that in the stationary regime the planar crystallization front moves in the lab coordinate system with constant rate. We study the stability of the stationary crystallization mode to small temperature and concentration perturbations in the linear approximation. To obtain the linear approximation of boundary problem (16)-(22) we assume that the solutions take the form

where

are the solutions of the stationary problem. Equation (15) for the constant rate yields Boundary problem (16)-(22) for stationary concentration and temperature distribution takes the form

Hence, the solution of the stationary problem is

Here, is the constant of integration determined from boundary conditions (24), (25), , are the characteristic numbers of homogeneous equations. In the stationary problem the interface velocity is specified which, thus, determines the interface temperature whose deviation from equilibrium temperature of phase transition is defined by the growth mechanism. Parameter x is found by equating the solution of the problem to this temperature. We do not present the detailed solution of the stationary problem. The numerical calculations are given in the brief communication [

This solution is different from the known one by the distribution coefficient which is here equal to the effective distribution coefficient (12). The linear approximation for small perturbations is

Let interface velocity as a function of kinetic overcooling (7) be written as

Consider linearization of kinetic Equation (33). It describes the dependence of the interface velocity on the kinetics of molecule attachment to the growing surface. The form of dependence is set by the model of mechanism growth. In accordance with the above change of variables, kinetic overcooling (8) is given by the expression

which is general for any growth model. Expansion of rate (33) into a Maclaurin series by small temperature and concentration perturbations takes the form

For the model of normal growth [

For the model of screw dislocation growth

For growth involving two-dimensional nucleation

Write linear approximation (34) by small temperature and concentration perturbations as

Here denotes the small perturbation of the interface velocity, the small perturbations of temperature and concentration take the form

We will seek as the linear combination of the temperature and concentration perturbations:

where and are as yet unknown expressions. Substituting (39) into (34) yields

By definition, the curvature is given by

To explain the curvature in temperature and concentration perturbations, we first find an expansion for the function. To this end, we pass to the curvilinear coordinates in (15):

where

Differentiating (42), we derive from (41)

Expansion into series gives the expression

In the linear approximation

Substituting (43) in (40) yields

From Equations (44) and (45), we find expressions for f and that are linear in temperature and concentration perturbations:

It follows from (46) and (43) that

Grouping the coefficients by the temperature and concentration perturbations, we derive a set of equations for the unknown coefficients. From this set, we find

A similar calculation was made in [

(49) is the solution of the heat conduction equation, (50) is the solution of the diffusion problem. Here, , are the roots of the characteristic equations for Equations (26)-(28), and are dependent on the parameters of the system with at. Substituting (38) in (49), (50), we obtain a dispersion equation in the form

where and are the parameters depending on time frequency and the wave number. The following notation is also introduced

To obtain an analytical solution for the instability period, consider dispersion Equation (51) at zero frequency. In this case all the parameters of the dispersion equation are real numbers. We consider a stationary mode with, but with fulfilled inequalities and. Let the dispersion equation be written as

The fraction in the brackets is expanded into series by the small parameter

For further simplifications the results of numerical calculations are required. To this end, we specify the values of the segregation coefficient. The parameter value is difficult to explain only by the influence of adsorption. In [

As shown by the numerical calculations made in [

Characteristic numbers and take the form

where

Substitution of the numerical values leads to the relationships

which bring the characteristic numbers (54) to the form

Substituting the expressions for the characteristic numbers in dispersion Equation (53)

Whence we find

At the minus before the root gives the trivial value. Therefore, we consider the solution with a positive root. Linearize the expression by.

Expression (39) is written as

And substitute (56) and (40) in (55). Following elementary transformations, the equation obtained is written as a fraction. Equating the numerator to zero, we arrive at an equation as related to

Instead of, we introduce the relationship, where. According to the numerical calculation [

Thus, and from (57) we find

Whence we find the period of spatial perturbations

The expression obtained is distinguished from the time expression in [

Condition (58) is essential for obtaining the desired solution. The validity of this condition follows from the numerical calculations of the solutions of the system dispersion equation [

It should also be noted that in contrast to the case when interface diffusion is neglected, in normal growth the time depends on the rate of the stationary mode. For the screw dislocation growth model

The growth model for two-dimensional nucleation yields

The expressions obtained for the morphological instability period are distinguished from the similar dependences in [_{V} are equal expressions obtained in [_{1}, A_{2} and B are constant. In the

The accomplished analysis of the boundary conditions reveals the reasons for interface instability and enables their simple physical explanation. Interface instability indicates that at small concentration or temperature perturbations the interface velocity increases and (in the linear approximation) tends to infinity. Let us outline the reasons for interface instability. Kinetic overcooling is driving force of the crystallization. The interface is moveless if the kinetic overcooling is zero and the interface velocity increases monotonically with increasing kinetic overcooling. The latter changes for two reasons: on changing the equilibrium temperature of phase transition or interface temperature. One of the reasons of instability is the change of the temperature of phase transition due to the changing concentration on the interface caused by interface adsorption. According to Hall [

Let the stationary phase transition regime proceed in the system. Assume that with constant the concentration on the interface increases by. Since in this case and, the equilibrium temperature of the liquid phase transition on the interface becomes somewhat smaller along with the

kinetic overcooling. This also involves a decrease in the interface velocity and, hence, an increased concentration of liquid on the interface. Therefore, the initial increase of concentration leads to its further increase. The system is unstable. In the interface velocity region, where, the component concentration on the interface increases with increasing interface rate, the equilibrium temperature of phase transition decreases which results in decreasing kinetic overcooling and interface velocity. The system is stable.

The well-known concentration instability related to the so-called concentration overcooling is caused by a simultaneous change of equilibrium temperature of phase transition and interface temperature. If the temperature gradient of the liquid solution on the interface is less than the gradient of the equilibrium temperature of phase transition, the interface stability may fail. This can be schematically described as follows. Let interface temperature decrease by virtue of fluctuations. Decreasing temperature involves an increase in kinetic overcooling and, hence, an increase of the interface velocity. This manifests itself as a ridge occurring in the region of concentration overcooling, i.e. the equilibrium temperature of phase transition on the interface gets higher. This change involves a further increase of the kinetic overcooling and the interface velocity. Thus, the region of concentration overcooling brings about interface instability. On the other hand, at, in the situation in question, the increase in the interface velocity leads to increasing component concentration on the interface and, as a consequence, to decreasing kinetic overcooling and interface velocity. These two opposite processes can be illustrated by a diagram.

We have obtained opposite changes of equilibrium temperature of phase transition. On the one hand, it increases due to a local movement of the interface upon temperature fluctuation, on the other hand, it decreases due to edging of the component by the interface and the change of kinetic overcooling. The two opposite processes can lead to or fail to lead to interface instability which depends on the external conditions as well as the physical parameters of the system.

In conclusion, an additional comment should be made on the setting of liquid solution crystallization problems. In [

The assumption as to the existence of an unstable solution layer complicates significantly the setting of the problem. Hence, two situations are feasible. The case of metastable solution is considered in the directed crystallization theory and has been analyzed in the present paper. The unstable solution in the non-equilibrium layer exhibits outward diffusion and tends to decay into equilibrium zones. This creates a region where the diffusion of the components differs from their diffusion in solid and liquid quasi-equilibrium solutions. In such a problem there occurs an additional interface separating the non-equilibrium and quasi-equilibrium liquid phases. Then the component distribution is described by three rather than two diffusion equations. As a result, the equilibrium phase diagram cannot be used to determine the values of concentration in the overcooled layer, or, more precisely, the condition of equality of chemical potentials is not valid on the solution interface. Therefore, one has to apply the conditions connecting the parameters of the system on the basis of the dynamics of the physical process. All the phases differ in component concentrations and diffusion coefficients. The analysis of the problem is outlined in [

1) Specified concentration in infinitely remote point of quasi-equilibrium solution.

2) Zero diffusion coefficient in solid phase.

3) Equality of constant concentration plane velocity on both sides of solid phase-non-equilibrium liquid interface.

4) Equality of constant concentration plane velocity on both sides of non-equilibrium liquid - quasi-equilibrium liquid interface.

The other two constants are found from the condition of mass component flow conservation. Besides these conditions, the concentration in the solid phase and the quasi-equilibrium phase concentration on the nonequilibrium solution interface are bound by an equilibrium phase diagram which determines the coordinate of the interface between the non-equilibrium and quasi-equilibrium phases. The problem has an analytical solution at a small difference between the component diffusion coefficients [

Purpose of the article is to build a model of crystallization, which takes into account the processes that significantly affect on the phase transition from liquid to solid phase. For this we use the boundary value problem heat and mass transfer, which takes into account the following phenomena.

1) Heat exchange with the environment, which forms the temperature field in the crystallizing solution.

2) Nonequilibrium redistribution of components between the liquid and solid phases due to interface adsorption and various solubility of components in the phases.

3) Kinetics of attachment of particles to the surface of the solid phase, the growth rate of which depends on the overcooling of the solution in front of the interface.

4) In this paper, we also take into account the state of the overcooled solution. Overcooled solution can be found in the metastable or nonequilibrium state [

The resulting model can describe many cases of the component distribution in the solid phase. For example, the periodic eutectic structure is formed due to the interaction of the instability of the phase boundary, spinodal decay of an unstable solution, due to the kinetics of attachment of particles to the growing solid phase and due to the interphase adsorption. Environmental heat transfer does not affect the dependence of the period eutectic structure on the growth rate because of the smallness of the corresponding term in the dispersion equation of the system. This explains the unsuccessful attempts to control the structure of the eutectic composites by environmental heat transfer in the crystallization zone.

This work was supported by the Russian Foundation for Basic Research, Grant N 11-03-01259.