One of the Known Type of Inhomogeneous Impurity Distribution In
NONLINEAR INSTABILITY ANALYSIS OF UNIDIRECTIONAL
SOLIDIFICATION WITH A MUSHY ZONE
Dmitri V. Alexandrov, Urals State University, Ekaterinburg, Russia
Abstract
This study is based on the exact analytical solutions of the nonlinear model of binary melt solidification with a mushy layer constructed in previous works. The linear analysis of dynamic instability shown possibility of oscillatory instability existence is also taken into account. A set of nonlinear evolutionary equations describing a behavior of perturbations is deduced. The weakly nonlinear approximation under consideration enables us to find two cumbersome equations for frequency and supercriticality of bifurcations. The distance between neighboring layers of impurity distribution in the solid phase is found as well. Theoretical predictions are in a good agreement with experimental data.
1. Introduction
One of the known type of inhomogeneous impurity distribution in industrial production of metals, semiconductor and inorganic materials is the layered impurity distribution (liquation) in the solid phase produced by solidification [1-4]. The phase interface leaves trailes in solids in the form of segments with high impurity concentration, which form as a result of time variations of the impurity concentration in the melt near the solid-liquid interface. The similitude of these trails indicates that the shape of the phase interface is stable. Impurity concentration fluctuations near the interface stem from the fact that the growth rate deviates from its steady-state value of us, which is imposed by external conditions (e.g. by temperature gradients). The regular pattern of the bands points to the existence of some mechanism of self-organization in solidification which is usually the consequence of the manifestation of some type of instability that is typical of the given process. Hence, in order to solve the striation problem, it is necessary to determine the reasons for destabilization and then, analyzing the development stage of the latter, to determine the set of conditions which replaces the unstable steady-state growth. From our point of view this instability, which is subsequently termed dynamic, stems from the fact that the interaction between kinetic factors at the phase interface and heat and mass transfer, causes the amplitude of the small fluctuations in the velocity of the crystallization to start increasing beyond bands. The principal crystallization parameters deviate from values corresponding to the quasisteady mode with constant solidification rate. When the perturbations reach a certain amplitude, a nonlinear interaction sets on between unstable modes which, in the case at hand, causes formation of self-oscillatory solidification at a rate that performs periodic fluctuations about the value of us. These oscillations induce regular fluctuations in the composition of the melt near the solid-liquid interface and growth bands, which repeat the configuration of the interface (striation).
The problem of the planar front stability with respect to infinitesimal perturbations was solved in principle by Mullins and Sekerka, employing the linear theory [5-7]. Then their analysis was extended to the nonlinear case [8,9]. However, often solidification proceeds with a mixed state zone which is placed between pure solid and liquid material. Experimental studies of ref. [10] shows that this solidification scenario leads to formations of periodic structures (bands) in the direction perpendicular to solidification. Therefore, it is very important to be able to predict these formations theoretically.
Recently, the general analytical solutions of nonlinear set of equations describing solidification in the presence of a quasiequilibrium two-phase zone were constructed [11,12]. In spite of approximate solutions obtained earlier (e.g. see refs. [13,14]), these solutions are exact and do not include any a'priori suggestions used in previous studies (e.g. the bulk fraction of the solid phase in the mushy zone, partition coefficient are sufficiently small, concentration at the mush-liquid interface is of the order of concentration far from this boundary in the liquid and so on). On the basis of exact solutions obtained, the quasiequilibrium two-phase zone was changed by the discontinuity surface between pure solid and liquid phases with new boundary conditions [12]. Thus, the mushy region crystallization is reduced to the "frontal" (discontinuity surface) model. Further, the linear instability analysis of this new "frontal" model carried out in ref. [15] shows the existence of self-oscillatory solidification regime at certain thermophysical and operating parameters. In other words, a principal possibility of banded structure formations is demonstrated in ref. [15]. In order to study characteristics of layered impurity liquation in details, it is necessary to carry out the nonlinear instability analysis of the new "frontal" model describing solidification with the two-phase zone. This analysis constructed in spirit of works [8,9] is the main subject of the present theory.
The outline of this paper is as follows: Sec. 2 is a theoretical description of solidification scenario; nonlinear instability analysis is given in Sec. 3; results are reported and discussed in Sec. 4.
2. The Model
Let us consider a unidirectional solidification of a binary melt or solution with the quasiequilibrium mushy region replaced by the discontinuity surface between pure solid and liquid phases. The melt and solid occupy regions xS(t) and xS(t), respectively whereas the discontinuity surface x=S(t) replaces the actual two-phase zone (here t is the time). This new "frontal" model includes temperature and diffusion conductivity equations in both the phases (diffusion in the solid is traditionally droped out):
, , , (1)
, , (2)
where q and qs are the temperatures in the liquid and solid phases, s and D are the impurity concentration and diffusion coefficient in the liquid. The aforementioned equations do not include temperature derivatives with respect to the time t because a relaxation time ta=2/a of temperature fields is essentially less than a relaxation time tD=2/D of the diffusion field, that is ta/tD is of the order of 10-3-10-4 ( is a characteristic length scale, and a is a temperature diffusivity coefficient). The impurity concentration, s∞, far from the "front" will be regarded as known, that is,
, . (3)
At the discontinuity surface, x=S(t) the following boundary conditions hold true [12]:
, (4)
, (5)
, (6)
. (7)
Here and m are the phase transition temperature for a pure melt and the liquidus slope, ls and l are the thermal conductivity coefficients in the solid and liquid, LV is the latent heat parameter, and k is the equilibrium partition coefficient.
As is seen from the model above, conditions (4) and (6) are the same as in the classical Stefan thermodiffusion model with a planar front. The boundary condition (5) expresses the constitutional supercooling in the mushy zone changed by the discontinuity surface S(t) while condition (7) characterizes a jump in concentration fields at both boundaries of the mushy layer.
Let us also assume that the temperature asymptotics in the solid is given, that is,
, , , (8)
where gs stands for the temperature gradient in the solid and the value of ssf is determined from the quasi-steady-state solutions and will be given below.
We consider the steady-state solidification along x-axis with a constant rate us. In the frame of reference x=us(x-ust)/D, the process is stationary.
Introducing new dimensionless variables and parameters
, , , , ,
, , , , ,
we rewrite the set (1), (2) supplemented by the boundary conditions (3)-(8) in the form
, , , (9)
, , (10)
, , (11)
, , (12)
, , (13)
, , (14)
, , (15)
, . (16)
In the case of steady-state solidification scenario nothing depends on the time t. Therefore, the set of equations (9), (10) supplemented by the boundary conditions has the steady-state solutions. For the sake of simplicity we will not dwell on this point and reproduce these solutions from ref. [15]. They are
, , (17)
, . (18)
Here subscript "s" denotes steady-state distributions.
Moreover, the expression connecting the steady-state rate us and temperature gradients gs in the solid and g in the liquid was deduced in our previous work [12] and has the form
. (19)
It is easy to see that expression (19) completely coincides with the same one for the planar front solidification scenario.
The subsequent procedure of nonlinear instability analysis is implemented as follows. The principal solidification mode consists of the quasi-steady mode with planar discontinuity surface and constant interface velocity. Section 3 is devoted to nonlinear analysis of infinitesimal perturbations on the basis of the linear approximation of the principal equations studied in ref. [15]; it is demonstrated that the quasi-steady mode is unstable under certain conditions, with the instability consisting of fluctuations about some quasi-steady value. This type of destabilization is analyzed in Sec. 4, where we validate the transition from the quasi-steady case with constant solidification rate us to the mode with solidification rate performing periodic fluctuations about some average value. The characteristics of this self-oscillatory mode are determined and parameters of layered distribution of impurity in the solid phase are calculated.
3. Nonlinear Instability Analysis
The set (9)-(16) of governing equations and boundary conditions can be significantly simplified. Since the second derivative of the solid phase temperature with respect to x turns to zero (eq. (10)), we conclude that the function ps(x) is linear. Then the boundary condition (16) gives the solid phase temperature at the "front"
. (20)
Eliminating from the boundary conditions (11)-(14) temperatures ps, p and their first derivatives with respect to x, and using expression (20), we come to the boundary conditions dependent only on the solute concentration C and its first derivative with respect to x. Namely, at we have
, (21)
. (22)
Thus, the problem for the nonlinear instability analysis is reduced to the second equation in (9), and the boundary conditions (21) and (22) at x=X supplemented by the boundary condition (15).
In the following presentation we investigate the behavior of perturbations of the steady state (17). For this, we derive the applicable equations for the perturbations. We shall analyze small perturbations and relative to Cs and Xs=0, corresponding to quasi-steady conditions. The boundary conditions (21) and (22) will be moved from plane x=X(t) to plane x=0 by means of an expansion in a Taylor series. For example, for the impurity concentration we have
Using expansions like this in equations (21) and (22), we obtain the following equations for perturbations and at x=0
,(23)
, (24)
where , .
Here we also used the steady-state distribution Cs(x) determined in expression (17). Conditions (23) and (24) contain only terms which will be needed in investigating the Landau-Hopf bifurcation [16,17]. In other words, only terms to the third order with respect to perturbations have been written out.
Substitution of perturbations in the second equation (9) and the boundary condition (15) gives
, ; , . (25)
Generally speaking, the appearance of dynamic instability can bring about both of regular and irregular unsteady-state solidification, characterized respectively by periodically or randomly varying velocity of the front and corresponding to the soft or hard scenario of evolution of perturbations [18]. In the case of the "soft" scenario the amplitude of fluctuations of any of the major parameters of this mode builds up from zero, upon passing through the point corresponding to parameters of the quasi-steady mode, through the neutral stability curve to the domain of instability on the phase plane of parameters. If the supercriticality, proportional to the shortest distance between this point and the neutral stability curve is small, then the secondary oscillatory mode does not differ from the harmonic mode with frequency of unstable perturbations on the neutral stability curve and can be investigated using equations (23)-(25).
In the case of "hard" instability, the amplitude of fluctuations changes jumpwise from zero to some final value when the characteristic point passes through the neutral stability curve. This induces oscillations with a wide frequency spectrum and with an amplitude which is not small even if the supercriticality is. As a result, the secondary mode is not periodic and equations (23)-(25) are inapplicable.
The possibility of either the "soft" or "hard" type of instability evolution is shown in ref. [15]. However, departure from dynamic stability occurs only in the soft fashion [15] and, as a result, only a regular self-oscillatory mode with periodically varying crystal growth rate may set on. Therefore, taking into consideration aforementioned arguments, we subsequently will pay our attention only on the "soft" type instability evolution scenario.
Further, we will assume that the unstable quasisteady mode is replaced by a weakly nonlinear self-oscillatory mode. The solution of equation (25) with the boundary condition at infinity is presented in the form of the Fourier expansions (i is the imaginary unity)
, , (26)
, , , (27)
Here b2 stands for the frequency of perturbations and asterisk denotes the complex conjugation. Perturbation amplitudes Cn and An as well as the frequency b2 will be found in the instability analysis carried out below.
Substituting expansions (26) and (27) into the boundary conditions (23) and (24) at x=0 and separating terms with different harmonics, we obtain a set of equations for amplitudes An and Cn. It is known that, when instability develops in the soft fashion, A1 and C1 are proportional to , whereas for the remaining amplitudes we have: , , and , [17-20] (here is the supercriticality). When is sufficiently small it is natural to retain in expansions (26) and (27) only the fundamental harmonic with n=1, and also "secondary" harmonics, for which |n|=0 and 2. Such an approximation corresponds to the accuracy of the evolution equations (23) and (24), in which terms of order higher than have been discarded.