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Wavelength selection in rotating solidification of binary mixtures
Chaouqi Misbah
To cite this version:
Chaouqi Misbah. Wavelength selection in rotating solidification of binary mixtures. Journal de
Physique, 1989, 50 (8), pp.971-994. �10.1051/jphys:01989005008097100�. �jpa-00210972�
Wavelength selection in rotating solidification of binary
mixtures
Chaouqi Misbah (*)
Institut für Festkörperforschung, KFA, D-5170 Jülich, F.R.G.
(Reçu le 23 septembre 1988, accepté sous forme définitive le 9 décembre 1988)
Résumé.
2014Nous montrons que l’expérience de solidification par rotation, sur des alliages binaires, comme celle réalisée par de Cheveigné et al. [25], doit conduire, quand la bifurcation est
normale, à une parfaite sélection de la longueur d’onde de la structure périodique de l’interface.
Cette propriété résulte du fait que dans cette géométrie le problème de la dynamique du front de
solidification est formellement équivalent à celui de la geometrie directionnelle en présence d’un
environnement lentement variable ; une telle situation est connue pour conduire à un choix
unique de la longueur d’onde [10-11]. En se basant sur le travail de Kramer et al. [10], nous
dérivons une équation de diffusion de phase, valable à toute distance au-dessus du seuil de Mullins et Sekerka. Nous exploitons cette équation près du seuil de l’instabilité à l’aide d’un
développement usuel en amplitude, et prédisons dans cette région la longueur d’onde
sélectionnée en fonction d’un paramètre de contrôle sans dimension.
Abstract.
2014It is shown that the rotating solidification experiment, as that set up by de Cheveigné
et al. [25], should lead to a perfect wavelength selection of the interface pattern when the bifurcation is normal. This feature stems from the fact that the front dynamics problem in rotating
solidification is formally equivalent to that of directional solidification with a slowly varying environment, a situation which is known to induce a collapse of the finite band to a single wavelength [10-11]. Following the work of Kramer et al. [10], we derive a general phase equation,
valid at arbitrary distances above the Mullins and Sekerka threshold. Evaluating this equation
near the instability point, by means of a standard amplitude expansion, we predict in this region
the value of the selected wavelength as a function of some dimensionless control parameter.
Classification
Physics Abstracts
61.5OC - 68.70
-81.30F
1. Introduction.
There exist a large variety of nonequilibrium systems that spontaneously build up a spatially periodic state from an initially structureless one when some control parameter reaches a certain critical value. Typical examples are observed in hydrodynamic instabilities [1], crystal growth [2], chemical reaction-diffusion systems [3], and so on.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005008097100
A common feature of extended pattern forming systems is the non-uniqueness of the pattern for fixed external parameters : one dimensional patterns, for example, usually exhibit
a continuous family of linearly stable nonequivalent wavelength states. Specifically, linear stability analysis of periodic structures near the instability point shows that if the distance above threshold is of order e, then there is a band of possible wavelengths of order
JlÊ. On the other hand, in most experiments or computer simulations, the pattern wavelength
is greatly restricted or even uniquely determined. Hence, the understanding of whether pattern selection is an intrinsic property of the system, or rather depends on boundary conditions, initial configurations or externally imposed perturbations is a fascinating question
that continues to motivate extensive analytical and numerical investigations.
So far, work directed towards an understanding of the wavelength selection process [1, 4-9]
has dealt for the most part with Rayleigh-Bénard convection and the Taylor system in simple fluids, which provide canonical examples of pattern forming systems. An important step was the treatment of the influence of rigid boundaries on the band of allowed wavenumbers. The result that emerges from different investigations [5, 7, 9] is that the boundary effects do not, in
general, lead to a single wave vector but rather to a drastic restriction of the allowed band.
Here we are interested in another class of boundary conditions, which have proven to lead to a unique wavelength selection. These are the so-called soft boundaries or « ramps », i.e., a region of space where the control parameter is slowly varied from above to below threshold.
Indeed it was demonstrated for a normal bifurcation by Kramer et al. [10] using a reaction-
diffusion model and by Pomeau and Zaleski [11] in a somewhat general framework, that the
presence of an infinitely slow ramp rate leads to a collapse of the band to a single wave
number. Since then, this concept has been applied to the Rayleight-Bénard convection [12] as
well as to Taylor vortex flow [13], and has been studied experimentally by Dominguez-Lerma
et al. [14] and by Rehberg et al. [15].
In recent years the pattern selection problem in crystal growth has received an increasing
amount of theoretical and experimental attention. One extensively studied issue concerns
velocity selection in free dendrites. The answer turned out to involve a subtle solvability
mechanism [2] which fixes the tip structure and the velocity propagation of the dendrite as a
function of the undercooling.
A different issue is that of wavelength selection in directional solidification. Indeed, it is
now well established that when a binary mixture is submitted to directional solidification, that is by pulling the sample at a constant velocity in an external thermal gradient, the
solidification front undergoes a transition from the planar configuration to a periodic cellular
one at a critical pulling velocity. The onset of the front instability was first analyzed by Mullins
and Sekerka [16]. The first studies of periodic cellular patterns were performed in the weakly
nonlinear regime [17-21]. The basic picture that emerged [18, 19, 21] is that periodic states can
exist near threshold with a continuous band of wavenumbers, as, for example, in Rayleigh-
Bénard problem. This feature can be understood from translational and rotational symmetry.
It is common to quite a number of different systems and reflected by the universal character of the Landau-Ginzburg equation that describes their slow dynamics in the vicinity of the instability point.
With regards to pattern selection far from threshold, an attempt based on the requirement
that admissible solutions must be smooth at the tip of the cell has been made in the limit of infinite cells. That is when the liquid grooves extend infinitely far. The possibility that this limiting case might exhibit sharp wavelength selection was suggested by Karma [22]. Later
work by Dombre and Hakim [23] and Ben Amar and Moussalam [24], however, showed that
this initial conclusion was incorrect ; a band of physically admissible solutions still survives,
thus leaving the wavelength selection problem open.
This paper is motivated by the rotating solidification experiment of de Cheveigné et al. [25]
in which a sample of impure CBr4 is rotated at a constant rate ú) in a temperature gradient set
up between two ovens (Fig. 1). One oven is above the melting temperature and one below, as
to allow progressive solidification of the sample. If w is large enough the liquid-solid interface
is observed to develop a periodic cellular structure that extends from the outer edge down to a
certain distance from the rotation center, below which the interface remains planar (Fig. 1).
The extent of the planar region decreases with increasing rotation rate ú)" This situation is
analogous to that encountered in Rayleigh-Bénard experiments when the height of the cell and therefore Rayleigh number is progressively varied in a such way that a convective region
is connected to a conductive one.
Fig. 1.
-Schematic plot of the rotating solidification principle. (a) : side view. (b) : top view.
The main novel feature in the rotating solidification experiment is the breaking of the
translational symmetry along the front. Firstly, in the frame attached to the sample, each point of the liquid-solid interface moves with a velocity which increases linearly with the
distance from the rotation center. Secondly, the hot and cold thermal contacts are not parallel
to the mean front position, but make a finite angle with it (Fig. 1). These two facts cause the
concentration and thermal profiles to depend on the coordinates parallel to the solidification front. This feature indicates that the characterization of any planar interface solution, which constitutes the basic starting point in the stability theory in directional solidification, and the investigation of cellular patterns, even in the weakly nonlinear regime is a formidable problem.
However, by inspecting the order of magnitude of the relevant quantities that enter the dynamical equations under the actual experimental conditions, we can show that the breaking
of the translational symmetry can be treated perturbatively. Consequently we can extract
from the whole set of equations, those terms which break this symmetry. They will play the
role of a slowly varying environment. More precisely, thanks to a perturbative extraction, our
equations are essentially formally identical to those of directional solidification with slowly varying pulling velocity and external thermal gradient. In other words our problem is similar to a « ramp » one, which is known [10, 11] to lead to a unique wavelength selection of the
periodic pattern.
Unfortunately, our prediction on the selected wavelength will not apply to the experiment
of de Cheveigné et al. [25]. Because this experiment is now performed with impure BCr4 for which it is known [26] that the bifurcation from the planar structure to the cellular
one is inverted, while our present prediction holds for a normal bifurcation. To have access to the normal bifurcation regime, one should use either a quasi-azeotropic mixture [27], or a liquid crystal [28]. Rotating solidification experiments on one of these two types of material
can be expected to be carried out in the near future.
Therefore the main objective of the present work is to demonstrate how the mathematical method works and illustrate the wavelength selection in the vicinity of threshold. We leave the systematic investigation of the phase equation in the highly nonlinear regime to the future.
The scheme of this paper is as follows. In section II we write down the basic equations and
compare them briefly to those of directional solidification. We then show how our problem
can be put into a « ramp » picture. In section III the general phase equation that governs the slow temporal and spatial behaviour of the local wave number, valid at arbitrary distances
above the Mullins and Sekerka threshold, is derived. In section 4 we exploit this equation analytically in the weakly nonlinear regime by using a standard amplitude expansion. In this regime, we present the prediction of the phase equation for static situations. Section 5 is devoted to concluding remarks.
2. Basic equations.
We consider the following situation : a binary dilute mixture with solute concentration
C , is rotated at a constant frequency to in a thermal gradient generated by two thermal
contacts (Fig. 1). We consider one dimensional interfacial deformations only and assume that
the system is quasi-infinite (on the scale of all wavelengths of interest) in the directions
parallel to the mean front position.
As usual, we neglect mass diffusion in the solid phase, since the corresponding diffusion
coefficient D, = 10-1°-10-11 cm2/s is small compared to that in the liquid D ’" 10-5 cm2/s. We
also neglect hydrodynamic flows. This implies, in particular, that we neglect the density
difference between the two phases [29]. The temperature fields in both phases ; tf and
Ts, are assumed to be in a quasi-steady state and satisfy Laplace’s equation. This assumption is justified by the fact that thermal diffusivities are several orders of magnitude larger than
chemical diffusivities, so that the temperature fields relax very quickly to their steâdy-state
value. For simplicity we will also assume that heat diffusion takes place symmetrically in the liquid and solid phases. For reasons that will become clear, we will exploit this assumption only later in this section.
Let us first introduce dimensionless variables. As unit of length we use Dlù)f, where
f is a characteristic distance of a point on the solidification front from the rotation center. As unit of time we take D / lJJ 2 e2. Finally temperature and concentration are measured in unit of
Tm and C.Ik respectively, where T. denotes the melting temperature of the pure material, Coo the concentration far in front of the interface (1), and k the equilibrium partition
(1) Later, we will specify what we mean by far in front of the interface.
coefficient. In what follows all the quantitites are dimensionless, unless otherwise indicated.
In the laboratory frame, the equations governing mass and heat transport read :
where v = DI W f2.
On the interface (for z
=z,(x, t», energy and mass conservation take the form
where n
=Ksi Ke, K,, Ke being the heat diffusion coefficients in the solid and liquid phases respectively, and C, is the interface concentration on the solid side. We must mention that, as usual, the source term in equation (2a), which is proportional to the latent heat that evolves
during solidification, has been neglected.
These interface conservation conditions must be supplemented with kinetic equations relating the mass and energy currents to the chemical potential and temperature differences
across the interface. However, for most materials of interest, and in the range of growth rates
encountered in standard experiments, the interface dissipation is negligibly small compared to
the bulk dissipation, so that deviation from chemical and thermal equilibrium at the solid- liquid interface can safely be ignored. The thermodynamic conditions to be satisfied on the solidification front then read
where M
=C. me/kT m’ with me the liquidus slope, and do
=ywflLD is the dimensionless
capillary length, where y is the interfacial tension, and L the latent heat of fusion per unit volume. Finally K is the surface curvature defined as positive for a convex solid.
The most important difference between our equations and those of directional solidification is the breaking of the translational symmetry along the x-direction, which is reflected on
equations (la, 2b). Therefore the concentration and thermal fields should in general depend
on the x coordinate even for a planar steady-solution (if any).
If we inspect our dimensionless equations we discover that the parameter v is small compared to unity in the experiment of de Cheveigné et al. [25]. Indeed, D is typically of
order 10- 5 cm2/s, the rotation rate to used in this experiment is of order 10- 3 S-l. With this value of w the cellular instability is observed to develop beyond a distance of order 1 cm from the rotation center (2). Then by taking the length f, wich fixes the characteristic velocity of the
solidification front in the cellular region of order 1 cm one finds that v - 10- 2. This constitutes the small parameter that enters our theory. It allows us to extract, in a
(2) Below this distance the front remains planar. In de Cheveigné et al. [25] experiment this distance is of the order of that obtained from the Mullins and Sekerka threshold by assuming that each point of
the front is « pulled » at the velocity wf.
perturbative way, from the complete set of our equations, those terms that break the translational invariance.
The physical meaning of our parameter v is easy to understand. Indeed, assume that we
watch locally a point on the liquid-solid interface located at a distance i from the rotation center. This point moves with a mean velocity wf in the frame attached to the sample. This
means that the rate by which the velocity changes when one moves along the liquid-solid
interface is given by w. This quantity is to be compared to the inverse of the relevant characteristic relaxation time of concentration fluctuations, T - 1 = ú) 2 £2 / D. The product
ú) T is nothing but the parameter v. It measures how rapidly the concentration profile adapts locally to the change of the front velocity in the x-direction. The smaller the quantity
co,r, the more the concentration profile is slaved to the local velocity, and the more efficient is the adiabatic adaptation of the front structure. In other words the smallness of v implies that
the change of the environment felt by the front varies smoothly.
Besides the effect of rotation, there is in fact another source of the breaking of translational invariance that is imposed by the « hot » and « cold » thermal contacts which are not parallel
but make a finite angle with each other (Fig. 1). This means that the external thermal gradient depends on the x-coordinate. This variation with x can however be rendered smooth by choosing the angle between the two thermal contacts sufficiently small. In the experiment of
de Cheveigné et al. [25], this angle is in general of order few percents rd, that is, of order
v.
We are now in a position to formulate our problem in terms of « ramps » in the spirit of
Kramer et al. [10]. The first step is to notice that x occurs in equations (la, 2b) in front of
v only. This dictates us to introduce the slow (or « ramp ») variable X
=vx.
First we can notice that if we set V=X in equations (1-3) and put formally v
=0, we obtain
the directional solidification equations where the usual constant pulling velocity is replaced by
the slow function (3) V
=X.
Since in the usual directional solidification problem the characteristic diffusion length is
fixed by the ratio of the diffusion coefficient to the pulling velocity, we want to scale here the z-variable by the local diffusion length, given by the ratio of D to the local physical velocity wi, x being the physical coordinate. Since lengths are already scaled by D / ú)£, the new
dimensionless coordinate is written as (4)
On the other hand we will use the V (X) p X notation instead of X in order to be able to
distinguish in our phase equation, we will derive later, the contribution that comes from he variation of the velocity along the front, which will appear as a derivative of V, from that
originating from the « centrifugation » effect. The two effects will indeed be distinguished
from each other by the fact that the drift term due the velocity variation will vanish by setting
V
=constant, which is the pure directional solidification case. Also this notation allows our
results to apply to a general slowly varying velocity without adding any algebraic complexity.
(3) For the moment we have not yet defined the boundary conditions on the two thermal contacts.
We will see later in this section that to leading order, that is when v is formally set equal to zero, the
directional solidification limit is recovered provided that, in addition to the fact that the constant
velocity is substituted by X, the distance between the front and the two contacts is replaced by the local
one, which also varies slowly with the x-coordinate.
(4) The scaling of the x variable by the same length will not be performed directly on
x but rather on the phase we will introduce later in this section.
We should however mention that since it is of course hard to imagine an experiment other
than rigid rotation, the result for a general V (X), other than X, is not regarded as having
interest for real experiments, except, may be, for numerical simulations.
In terms of the new variable e equations (1-2) read, to order v :
and the interface boundary conditions (2-3) become
where so is defined by so =- 1 + V - 2 2, v - log (V) and primes will always signify
derivation with respect to the slow variable X. Note that for the rotating case
V’ = V’ /V =1/X.
_We should point out that the derivative ax in equations (4a-f) acts on x as well as on
X. We could in fact split ax into ax + v ax and treat the two variables as being independent in a spirit of a multiscale analysis. However, it turns out in this kind of problems that the appropriate fast variable is not x, as is the case in the usual amplitude theory for example, but
rather the total phase of the pattern (a nonlinear multiscale analysis ; see below). Therefore,
in order to avoid confusions we do not treat the couple (X, x ) in a multiscale picture. The only purpose of introducing X at this stage is to indicate explicitly the quantities that vary
slowly in space.
Now the boundary conditions can be specified :
(i) the temperature is fixed at the two thermal contacts, namely :
The dimensions of « the thermal box » are assumed to be much smaller than the thermal diffusion length. This is justified by the fact that for most material of interest, mainly metallic alloys, that form the large part of materials whose solidification front do not facet, the thermal diffusion length is very large compared to the dimension of samples used in standard
directional solidification (see for example Ref. [20]) ;
(ii) the physical concentration is fixed at a value Coo far ahead of the front in the liquid phase
-i.e. at a distance much larger (5) than Dlwf. For all practical purposes, this amounts, in dimensionless form, to
The set of equations (4-6) completely describes the motion of the solidification front. To
leading order v ° equations (4-5) reduce formally to those of directional solidification problem
(5) This possible as long as we consider regions not too close to the rotation center.
provided one substitutes the constant pulling velocity by the slowly varying function
V (X) and the distances from the mean front position to the hot and cold thermal contacts by L, (X) and L2 (X) respectively. The perturbative terms which are proportional to v have two
sources : one is the variation of the velocity along the x-direction (term proportional to v’) and the other appears as « centrifugation » effect produced by the rotation (terms proportional to C ax in Eqs. (4b, d)).
To this order one can specify formally, in the adiabatic sense, a cellular steady-solution for
the local values V (X), L1(X) and LZ(X) that play the role of the slowly varying environment of our problem. The wavenumber q (X) of the adiabatic solution is arbitrary at this order. The next order of iteration produces an inhomogeneous linear problem. Due to the translational invariance the linearized operator has a nontrivial kernel, which is noting but the translational mode. Therefore for the solution to remain uniform a solvability condition must be imposed,
which results in an equation that relates the wavenumber to the control parameters.
First, the basic steady-state solution has to be determined to order v. We thus seek solutions of the form
To order v 0, one easily discovers that equations (4-6) support a planar wave solution ( So =0) characterized by :
where is the reduced thermal gradient in the liquid phase.
To order v, one obtains a trivial solution
This means that, to order v, the basic solution is formally identical to that of directional solidification in which we substitute the (constant) pulling velocity and the imposed thermal gradient by the slow functions V (X ) and G (X) respectively (6).
In order to treat the pattern selection problem we follow closely Kramer et al.’s strategy
[10]. That is we seek solutions which are 2 7r-periodic in a certain phase q, to be determined below. In the absence of the ramp q
=qo x, qo being a constant wavenumber characterizing
the periodic structure. In the presence of a ramp, however, the wavenumber is no longer
constant but varies in space and eventually in time.
The demand that the wave number is a slow function of space (q’ - 0(1)) is satisfied by requiring that the total phase q scales as q - 0 (X, t )/ v. On the other hand because we are
(6) One should mention, for example, that in Rayleigh-Bénard convection with ramped cell [12], the
basic solution contains to first order in the small parameter, a non vanishing contribution, which is characterized by a large scale flow. This is a consequence of the presence of a slow transverse
component of the imposed thermal gradient which automatically induces convection.
interested in a diffusion process, we may expect that the characteristic time scale of motion of interest is of order v - 2. We therefore introduce the slow time variable T
=v 2 t and write the
phase 11 as
with p a slow phase (the factor V is introduced for convenience).
We then write formally
as if they depended separately on the rapid variable 11 and the slow variables X and
T.
Here we are to understand that we must make the substitutions
whenever we differentiate u, 0, and e,. q denotes the local wavenumber defined by
Now our scheme is to insert equations (lla-llc) together with equation (12) into the basic
equations (4-6) to deduce successively higher terms in an expansion of u, 0, e, in powers of
v. Because we expect the solution to coincide with that of the perfectely periodic state in the
absence of a ramp, i.e. for v
=0, we start this expansion at order v° :
Before going further, we exploit our assumption of identical thermal properties for both phases, mentioned at the begining of this section. This assumption has the advantage to simplify greatly the algebra without altering the generality of the method developed below,
nor the main conclusion which will emerge from our analysis. Under this assumption the temperature and the concentration fields decouple completely ; in this limit the interface is thermaly inactive. In other words the temperature profile is simply that fixed by the external thermal contacts. It is given to order v, by expression (8a). The reason for exploiting this assumption only now is that we prefered to introduce first the ramp notion before eliminating
the thermal profile (Eq. (8)), which is written in a ramp approximation, from the whole
equations.
With this assumption we are left with only one field variable u. To order vO one finds that
u (0)
obeys
On the interface «( = (8(0» the conservation equation and the Gibbs-Thompson condition are
with the boundary condition
are, for a given material, the only dimensionless parameters that enter our equations.
Equations (14a-14d) represent the static and rescaled version of the dynamical interface equations in directional solidification. The slow variable X occurs only parametrically.
Therefore, according to Mullins and Sekerka [16] (see also Wollkind and Segel [17]), for
T> Tc (,B ), with Tc (0 ) given by the parametric equations
the planar interface is unstable ; it develops a cellular structure which is 2 7r-periodic in
,q with a wave number undetermined at this order.
At order v, a straightforward calculation yields
On the interface
with the boundary condition
Do is defined by Do= 1 + (q a n s (0»2 , and the functions f, g, h are listed in Appendix A.
3. The phase diffusion equation.
Equations (17a)-(17b) constitute a set of linear and inhomogeneous equations. One can show
that the homogeneous problem admits the translational mode as a solution. This implies that
the particular solution of the whole inhomogeneous set would exhibit a nonuniform behaviour with q, unless a certain solvability condition is fulfilled. This condition is nothing but the phase equation we want to determine.
To derive this equation, we proceed as follows. We first multiply equation (17a) by the adjoint function t/1 (ry, C, X, T ), chosen to be 2 7T-periodic in q, and integrate over one period
in ’Y}, and over (, s(O), oo ) in e. Integrations by part lead to
where we have made use of equation (17d) and imposed to li to vanish at =00.
L + denotes the adjoint of the operator acting on u (1) in equation (17a) :
The next step is to use equation (17b) to extract a{u(l) as a function of u(1), ’s(l) and their
derivatives with respect to 17. Upon integration by parts, the r.h.s. ; of equation (18) becomes
In expression (20), the two integrals whose integrands are proportional to e s (1) and
u (1) are not independent but related to each other by the Gibbs-Thompson condition (17b).
Expressing, from this equation u (1) in terms of e s (1) and its derivatives, integrating by parts, and inserting the result in equation (18), we obtain
where A is an operator acting on V/ ; it is listed in Appendix A.
To define the adjoint homogeneous problem, we consider the homogeneous problem (i.e.
f = g
=h
=0). The adjoint bulk equation is obtained by setting (7) :
It then follows from équation (21) written for the homogeneous problem ([ = g = h = 0) that , evaluated on the interface, must satisfy 0 d’Tl , P) .ae ’"
=0. This condition is fulfilled
by choosing the adjoint boundary condition on the real surface’ = ’s(O) as
Equations (22-23) with the boundary condition & ( -+ ao ) = 0 completely define the adjoint problem.
Having defined the adjoint problem in this way, we immediately obtain from equation (21)
the solvability condition, which results in the sought after phase diffusion equation
Substituting in equation (24) f, g, h by their expressions given in Appendix A we can write (24) as
(7) We may mention, if need be, that we are at liberty to define the adjoint problem in this way.
Other choices are of course possible. The present definition is, however, very natural.
(recall that q
=ax(OV» where
The ax in equation (26e) is understood to act on all the X-dependent functions except on q ; this is because in writing equation (25) we have extracted from A4 the terms that are proportional to q’ (= a2 OV) to put them together in the diffusion term Al.
Equation (25) constitutes the phase equation that describes the slow evolution of the wavenumber. In the absence of the ramp, equation (25) reduces to
(note that V
=1 in this limit) which describes the phase dynamics at arbitrary distances above threshold for the free ramp situation. In this case the expansion parameter is the gradient of
the phase.
Besides the usual diffusion term (Al), equation (25) contains the drift introduced by the slowly varying environment. This is expressed by the v’ A2 and A4 terms which involve terms proportional to v’ and G’. Indeed one can write formally the derivative aux that appears in
expression (26e) as (see Eq. (15))
so that v’ A2 + A4 can be written as a sum of two factors proportional v’ and G’ respectively.
Finally the A3 term originates from the « centrifugation » terms which appears in the mass
conservation equations (la, 2b).
Note that our phase equation applies to the rotating solidification problem as well as to the
directional solidification one with a general ramp. The later limit is obtained by simply setting A3
=0, and replacing G and V by their expressions relative to the ramp under consideration.
For a static situation, equation (25) is a first order differential equation for the
wavenumber. If this one is prescribed at one point, e.g. the critical point, the phase equation
fixes q (X ) in the whole system.
4. Prédiction of the phase équation near threshold.
At arbitrary distances from threshold, the coefficients 1B that enter equation (25) can only be
determined numerically. However, close to the Mullins and Sekerka threshold, analytic
evaluation is possible by means of an amplitude expansion. The solution u (0) and
e s «» are then obtained as powers of some amplitudes expressed in terms of the local values of the control parameters, by assuming that the amplitudes adiabatically follow the control parameters. In other words we assume that the scale implied by the amplitude theory, which
is of order £-1/2, £ being the distance from threshold, is much smaller than that implied by the
« ramp », v
If we define the dimensionless distance from threshold E by
where
(a is defined in such a way that e coincides with the dimensionless growth rate of the critical mode) and expand e s (0) and u ° in powers of lie-
one finds to order E 1/2 that (1’ f/11 and ul can be written as [17, 20]
with al = (1 - k) (T - i3 q2) and Ao is an amplitude (’), undetermined at this level of
approximation.
To order e we find
(g) Recall that q accounts for the total phase.
with
A is the amplitude of the homogeneous solution ; it is also undetermined at this order. The coefficients 8 ij and (ij are listed in Appendix B. Note that the absence of the
eo iq mode in C2 simply expresses the fact that since the interface is thermally inactive in the limit n
=1, its mean position over one period is fixed (at e
=0) by the external thermal
gradient.
At order e 3/2@ removing the « secular » terms from equations (14a-14c), we obtain the
amplitude equation, which reads to leading order in the deviation from the bifurcation point
0
where p- (q - q,, ,) /30, 130 = - 2 aq.a, qc is the critical wavenumber given by equation (16b) as a function of the control parameter 8, and Q is the Mullins-Sekerka growth rate ; it
is solution of the equation [17]
The quantity p is defined in equation (30) in such a way that p 2 = e is, to lowest order, the
neutral curve. Finally the Landau coefficient à) , first derived by Wollkind and Segel [17], is given by
Using equations (30) and (28a-28b) we can write e s (0) and u (0) to leading order as follows
with the demand that úJ1 :> 0, which means that the bifurcation must be super-critical (we will specify later under which conditions úJ1:> 0).
Using the same procedure for the adjoint problem (Eqs. (22-23) with the boundary
condition at oo) we find that the adjoint problem is solved by
(Note that a Ç$° represents the translational mode.)
i p -y
(9) In terms of the amplitude Ào
=Ao e F"8’ , (Y = x Je), equation (30) reads Âo +
f30 a}yÃo - úJ llÃo 12 Âo
=0 which is the Newell-Whitehead [30] and Segel [31] equation.
Inserting (33-34) into the 7l/s coefficients given in Appendix A, performing integrations,
we find to leading order in the deviation about the bifurcation point
where the ai’s coefficients are given by :
Due to the square root singularity of the deformation amplitude Ao (see Eq. (33a)) the
determination of A4 to the same order requires higher order contributions to u , (s(O) and gk given in equations (33-34). This means, in particular, that we have to determine the second order amplitude A, that appears on the r.h.s. of equation (29). For that purpose we must push our expansion scheme to the next order. We first write the expressions for e3 and U3 that entered the previous order. They are given by
with
where A2 is the amplitude originating from the homogeneous problem (undetermined at this order). As a consequence of the solvability condition (30), the coefficients (31 and ,831 that multiply the fundamental harmonic in equations (36a, b) are not independent. The
relation between these two coefficients, that we need below for the evaluation of
A4 is given in Appendix B. Finally the (33’ f333 terms, as well as the amplitude A2 turn out to give no contribution to A4 and therefore do not need to be specified here.
The solvability condition which emerges at order 62, results in an equation for
i p Y
1/ p 71 @-. y
Ai . In terms of the amplitudes Ãp= Al e " , and  - A e " , (Y= 17x) this
equation reads :
where the (real) coefficients a i , {3i, yi are listed in Appendix B. Equation (36) is the analog of
that derived by Hohenberg et al. [32] for the reaction-diffusion model (Eq. (2.35) of this reference) .
Using the fact that one can write the solution for A 1 as
For the adjoint problem we find that tP2 and 0/3 are given by
Using equations (28-30), (33-38), inserting in equation (26e), we obtain, to leading order in
p and e
where
and
All the coefficients .1 and àij that multiply E and p in the .’s terms are understood to be evaluated at the critical point. They appear as functions of Tc, /3, q,. However, these quantities are related to each other by the parametric equations (16a, b). This means that all ài and àij coefficients are in fact functions of a single variable, for instance /3 (this represents
the rescaled critical velocity at which the front instability takes place). In other words equation (25) is a (nonlinear) partial differential equation for the phase l/J parametrized by
the control parameter /3 (and the partition coefficient k).
Inserting expressions (35, 40) in equation (25), one obtains
which represents the general phase equation in the weakly nonlinear regime. The directional solidification limit is obtained with v and e constant, and by setting formally Li3
=0. Using the
definition of 4o and ài together with those of /3 and /30, equation (41) reads in that limit
which reproduces exactly the known result close to onset for the Eckhaus instability at
p2= 8 that can directly be derived from the lowest-order amplitude equation of Newell,
3
Whitehead [30] and Segel [31].
Equation (41) has the same form as that derived by Kramer and Riecke [12] in the context
of Rayleigh-Bénard convection. This is of course comforting because the dynamics is
universal in the vicinity of threshold in the sense that it does not depend on the model details but rather on the symmetry properties.
Following Kramer and Riecke [12] one can show that, if the velocity and thermal gradient
vary along the front in a such way that (1°) e’
=0, non-moving pattern solutions on the Eckhaus-stable band can exist only if the variation of the velocity is confined to
In the case of Kramer and Riecke [12] the coefficients are numbers. This is to be contrasted to the present case where the coefficients Ai are complicated functions of the critical wavenumber and the segregation ratio k. We have checked that for typical values of qc’s, namely from qc = 1 to 10 and from (11 ) k
=0.5 to k
=2 by a step of 0.1, condition (43) is violated, when W 10- 3 s as in the experiment of de Cheveigne et al. [25] and
e
=10- 2 to 10-1, only if the physical distance from the rotation center is much larger than a centimeter ; this limit is too large compared to the sample size, which is of the order of a
centimeter. Of course for an azeotrope or for a liquid crystal, which are two candidates we
have in mind for experiments in the normal bifurcation regime, the 4!s that enter equation (43) are different and we cannot a priori exclude the possibility that equation (43) may be violated for a reasonable value of the distance from the rotation center. In that case,
according to Kramer and Riecke [12] one expects a moving pattern to take place.
We now focus attention on the static situations where the left-hand side of equation (25)
must vanish. This constitutes a first-order differential equation for q (X ) and thus determines the local wave number if it is prescribed at one end. Specifically, since the variation of the control parameters in rotating solidification is such that a planar structure is connected to a
cellular one (Fig. 1), the wave number is determined everywhere because it is unique at
threshold (q
=qc).
Kramer and Riecke [12] noted that when the terms proportional to p2 except that appearing
as a factor of p’ are neglected in their equation (3.6), which is the analog of the static version of our equation (41) after X is expressed as a function of e, the differential equation is exact.
This allowed them to write explicitly the selected wavenumber as a function of the control
eo) One can show from the definition of e that this correspond to a situation where the distance from the hot and cold thermal contacts, say L is related to the velocity by L’ /L
= -V ’ /V . For the rotating
case L decreases with X, which means that the stabilizing thermal gradient increases when the
destabilizing effect, which is proportional to the velocity, increases.
(11 ) We have started from k
=0.5 because as will be mentioned below the bifurcation is subcritical at
k $ 0.4.
parameter. In fact this was possible also because the p-coefficient that appears on the r.h.s. of their equation (3.6) is equal to - 1. This means in our case that A4,/AO must be equal to
-
1 for our equation to reduce to a total differential. This ratio is, as mentioned above, a complicated function of qc and k and has been checked to be noticeably different from
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