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On the emergence of one-dimensional front instabilities in directional solidification and fusion of binary mixtures
B. Caroli, C. Caroli, B. Roulet
To cite this version:
B. Caroli, C. Caroli, B. Roulet. On the emergence of one-dimensional front instabilities in direc- tional solidification and fusion of binary mixtures. Journal de Physique, 1982, 43 (12), pp.1767-1780.
�10.1051/jphys:0198200430120176700�. �jpa-00209560�
On the emergence of one-dimensional front instabilities in directional solidification and fusion of binary mixtures (*)
B. Caroli (**), C. Caroli and B. Roulet
Groupe de Physique des Solides de l’Ecole Normale Supérieure (***),
Université Paris VII, 2, place Jussieu, 75251 Paris Cedex 05, France (Reçu le 21 juin 1982, accepté le 18 août 1982)
Résumé. 2014 Nous étudions, en situation de solidification ou fusion directionnelle des mélanges binaires, la stabilité
des fronts de croissance plans vis-à-vis des déformations unidimensionnelles. Nous étendons le calcul de Wollkind et Segel au cas où la diffusion du soluté dans la phase « fille » est non négligeable. Nous étudions le diagramme
de bifurcation et calculons le coefficient du terme cubique de l’équation d’amplitude. L’analyse qualitative des
résultats montre que la transition liquide/cristal liquide (et, éventuellement, liquide/cristal plastique) semble être
le système le mieux adapté pour une étude expérimentale systématique.
Abstract 2014 We study, for directional solidification or fusion of binary mixtures, the stability of planar fronts against one-dimensional deformations. We extend the perturbative approach of Wollkind and Segel to the case
where solute diffusion in the « daughter » phase in non-negligible. We study the bifurcation diagram and calculate the coefficient of the cubic term in the amplitude equation. From a qualitative analysis of the results, the liquid/
liquid crystal (and, possibly, the liquid/plastic crystal) transition appears to be the most attractive system for systematic experimental investigation.
Classification
Physics Abstracts
64.60 - 61.SOC
1. Introduction. - It has been known from a long
time [1, 2] that various impure materials with atomi-
cally rough solid-liquid interfaces, when submitted
to conditions of directional solidification, may exhibit cellular structures of their solidification front. The
principle of directional solidification is sketched
on figure 1 : a long (quasi-infinite) rod or strip of the
material is pulled at a constant velocity V through
a fixed average temperature gradient G, established via two stationary thermal contacts at temperatures
T1 and T2 (TI > TM > T2, where TM is the melting temperature). Jackson [3], working on thin CBr4
films with a small impurity content, has observed stationary (in the laboratory frame) deformed fronts with a periodic one-dimensional structure.
Such cellular instabilities have been rapidly reco-
(*) We dedicate this work to our colleagues Yuri Orlov
and Vladimir Kislik.
(**) Also : Departement de Physique, UER de Sciences Exactes et Naturelles, Universite de Picardie, 33, rue Saint- Leu, 80000 Amiens, France.
(***) Associe au Centre National de la Recherche Scien-
tifique.
gnized as due to, and regulated by the competition
between surface tension and solute diffusion (which
Fig. 1. - Directional solidification setup. In a directional, fusion experiment, the system is pulled in the opposite direc-
tion.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198200430120176700
is much slower than heat diffusion), interplaying with
the stabilizing effect of the temperature gradient.
The early thermodynamic interpretation in terms of
« constitutional supercooling » [4] has been improved
upon, more recently, by Sekerka [5], Wollkind and
Segel [6] (hereafter referred to as WS), Langer and
Turski [7] and Langer [8]. These authors have studied the appearance of one-dimensional periodic instabi-
lities on the basis of the complete dynamical out-of- equilibrium description of the system. They have
studied the linear stability of the planar stationary solution, thus proving the existence of a bifurcation
threshold, at a finite wavevector, towards a one- dimensional periodic deformation of the front. More- over, WS [6] and Langer [8] have performed pertur- bation expansions up to third order in the amplitude Zk of a deformation of wavevector k, thus obtaining
the Landau-like amplitude equation, valid in the immediate vicinity of the bifurcation :
As usual, at the bifurcation ao vanishes, and four
different situations can be distinguished in its vici-
nity :
(i) ao 0; al > 0 : the planar front is stable.
(ii) ao > 0 ; a 1 > 0 : the stationary cellular defor- mation is stable, with amplitude (ao/al)1/2.
(iii) ao 0; al 0 : the planar front is locally stable, but unstable to finite amplitude deformations.
This is a case of subcritical instability.
(iv) ao > 0 ; ai 1 : the planar front is completely unstable, and equation (1) is insufficient to predict
which (or whether any) stationary solution is stable.
These calculations have been performed in two
different regimes for the physical parameters of the liquid-solid system : while WS assume that solute diffusion in the solid phase is negligible (q = DS/DL
= 0, with Ds and DL the solute diffusion coefficients in the solid and the liquid), Langer has studied the symmetric limit: n = 1; n = Xs/J{,L = 1 (J{,S,L are
the thermal conductivities).
The bifurcation diagrams G = Gc(V) they obtain
are qualitatively similar, but their predictions about
the sign of ai (i.e. about which structure may stabilize
beyond the bifurcation) are different : for instance,
in the low velocity limit, Langer finds a stable cellular
structure (al > 0), while WS predict al 0.
This naturally raises the question of the influence of the value of il (and n) on the nature of the bifurcat- ing solution. This problem is not, as could appear at first sight, only of academic interest : indeed, finite ,,’s can be found not only, as pointed out by Langer,
in solid-solid transitions, but also, as will be discussed in more detail in § 3, in the liquid-liquid crystal case.
On the other hand, front instabilities can, as we shall see, occur as well in directional fusion, the expe- rimental study of which may lead to complementary
information about the non-linear behaviour of such systems. It is clear that, in the fusion situation, solute diffusion in the low temperature phase, even if very
small, cannot be neglected : indeed, the planar sta- tionary solution, which stability we study, can only
be sustained by a space variation of solute concentra- tion in the « mother >> phase [2].
In § 2, we therefore extend to any value of j7 and n,
with the method of reference [6], the analysis of linear stability of a stationary planar solidification front, obtain the corresponding bifurcation diagram, and
calculate the third order coefficient al. We show that the corresponding results for fusion can be deduced from the solidification ones by a straightforward
transformation of the parameters. It will appear, in the course of the calculation, that :
(i) Although heat diffusion is much faster than solute diffusion, in the" =1= 0 (and in the n # 1 ) case, one must be more careful about the thermal boundary
conditions than for" = 0.
(ii) The WS calculation of the cubic coefficient a 1
contains an error, which can easily be cured, and, though it changes the value of ai, has no bearing on
its sign for" = 0, i.e. on the qualitative predictions
of reference [6].
An exhaustive analysis of the results thus obtained is made difficult by the large number of physical parameters involved, and by the present lack of systematic experiments. So, in § 3, we try to clear the ground to some extent by classifying the main realistic experimental situations, on the basis of plausible
orders of magnitude for the most relevant parameters.
2. Bifurcation from the planar front and third order
amplitude expansion. - In this section, we use a systematic perturbation expansion method which has been developed at length and used by various
authors on problems of hydrodynamic stability. We
follow especially closely the work of Wollkind and
Segel [6], using, as far as possible, their notations.
Most of the steps in the calculation are only briefly
sketched here (the reader being referred to their
article for algebraic details), and we only develop fully
those points on which we differ from them.
Z .1 BASIC DYNAMICAL EQUATIONS FOR SOLIDIFICA- TION. - The system is pulled with velocity V in the ( - z) direction (Fig. 1). We define reduced solute
concentrations C (resp. C’) and temperatures T (resp. T’) in the liquid (resp. solid - or, more gene-
rally, low temperature) phase, and length and time variables, related to the (tilded) corresponding phy-
sical quantities by :
TM is the melting temperature of the pure solvent. The
length scale l = DL/ V is the concentration diffusion
length in the liquid (typically 10- 3 cm), DL (resp.
Ds) being the solute diffusion coefficient in the liquid (resp. solid) at temperature TM. The unit of concen-
tration, Cg(0), is chosen to be the value of C, for the planar stationary solution, on the liquid side of the interface.
Following WS and Langer, we only study stability against the simplest type of interfacial deformations, namely 1-dimensional ones. Let z = sC(x, t) be the (reduced) position of the front in the laboratory frame, s being a smallness parameter. The system is then completely described [2, 6] by the following
set of equations :
(i) Volume diffusion of concentration and heat in each phase :
(ii) Interface conditions, at z = e(x, t)
- continuity of temperature :
- heat balance :
- local phase equilibrium :
- concentration balance :
- curvature induced local interface temperature shift :
where
K is the solute distribution coefficient: K = (dT/dC)L/
(dT/dC)s IT=TM (K > 0), where (dTjdC)L,s are the
slopes of the liquidus (solidus) curves on the binary phase diagram (Fig. 2) at T = TM.
y UV ’Y = where C is the specific latent heat and 6
fD-L ’ p
the solid-liquid surface tension.
Fig. 2. - Low concentration part of the phase diagram
of a binary mixture.
Equations (3) and (4) imply the following assump-
tions :
- convection in the liquid is neglected;
- instantaneous local thermodynamic equilibrium
on the front is assumed, which implies that the inter- face is rough on the microscopic scale;
- the mixture is dilute (the solidus and liquidus
lines are approximated by their tangents at T = TM) ;
- heat diffusion is quasi-instantaneous compared
with solute diffusion, so that all terms of order DL/Dth
are neglected in equations (3b, d) and (4b) (Dtb is the
heat diffusion coefficient).
(iii) Boundary conditions : The temperature is fixed at the two thermal contacts, namely :
Note, however, that the dynamical terms (DL/Dth) (OTIOZ - aT/at) in equations (3b, d can legitimately
be neglected only if the dimensions L,,2 of the « ther-
mal box » are small compared with the thermal diffusion length Th = Dth/V. This limitation, which is
not clearly stated by WS, must be kept in mind, as
will appear below, in the ?I :A 0 case. From now on,
we therefore assume that the thermal box is small
(L 1, 2 Th). This assumption will be discussed in § 3.
The solute concentration is fixed at the value C.
far ahead of the front in the liquid phase - i.e., at a
distance much larger than T. So, for all practical
purposes, this amounts to
We assume that the system is infinite along x, and thus forget about transverse boundary conditions.
This implies that the scale of front structures is small
compared with the transverse size of the sample, as
is the case in Jackson’s experiments [3].
2.2 BASIC EQUATIONS FOR FUSION, AND THE SOLIDI- FICATION-FUSION TRANSFORMATION. - Fusion is des- cribed by the same physical equations as solidification,
the only difference being the reversal of the drag velocity (V --> - V). Using the same reduced variables (Eqs. (2)) as for solidification, we find that equa- tions (3b, d), (4a, b, c, e) and (5) remain unchanged,
while equations (3a, c), (4d ) and (6) become :
One easily checks that the solidification equations
can be recast into the fusion ones by applying to equations (3)-(6) the transformation :
which expresses the fact the « mother >> and « daugh-
ter » phases are interchanged, while the temperature boundary conditions remain the same.
So, we will perform all calculations for solidification,
and simply apply (8) to the final results.
2.3 THE PLBNAR STATIONARY SOLUTION. - Choos-
ing the position of the front as the origin of the z-coor- dinate, one finds the following planar stationary
solution of equations (3)-(6) :
G, the (positive) reduced temperature gradient in the liquid phase, is given by :
and choosing z = 0 on the front imposes that :
(the only length fixed in the experiment is L1 + L2).
The concentration scale C*(O) (Eq. (2)) is now
related, via equation (9a), to the boundary value Coo by
2.4 LINEAR STABILITY OF THE PLANAR FRONT. -
One now assumes that the planar front undergoes a
small harmonic deformation :
which induces responses bv(z, x, t) (with v == (C, C’,
T, T’)) of the concentration and temperature fields.
Expanding equations (3)-(4) about the planar statio-
nary solution, and since this is translationally inva-
riant in x and t, one gets, to first order in E :
Equations (3) then give :
with :
From which :
where
In expressions (I 6c, d) for T 11 1 and T 11, we have
made use of the fact that wL 1, 2 >> 1- that is, we
only consider deformations with wavelengths much
smaller than the dimensions of the sample.
The interface conditions (4) become, to the same
order :
The condition of compatibility for system (18) gives
the dispersion relation :
where
The values of the four constants A1 A;, B1, B’
which then result are given in Appendix B.
The linear stability of the planar front against the
deformation (13) is given by the sign of (Re ao). We
show in Appendix A that equation (19) entails that, if (Re ao) = 0, then (Im ao) = 0 (principle of exchange
of stabilities). So, when looking for the threshold of
instability, we can simply set ao = 0 in equation (19), and the condition of neutral stability then reads :
where
The minimum of the bc(W2) curve in the physical region 3 1, if it exists, defines a bifurcation from
the stationary planar solution to a one-dimensional front structure. We show in Appendix A that either
bc(W2) > 1 (for p > 1/K), or bc(W2) has a single
minimum in the physical region (for 0 p 1/K).
So, the bifurcation curve G = Gc(fJ) in the (G, b) plane is defined by the parametric equations (21 ) and :
v
The linear stability diagram is sketched on figure 3.
The corresponding diagram for fusion is obtained
by applying to equations (21) and (23) the transfor-
Fig. 3. - Schematic plot of the bifurcation curve G = G c(P).
For G Gc(fl) the planar front is unstable; it is linearly stable for G > G c(P). In the solidification case Pmax = 1 /K
and
For fusion Pmax = r¡ and
mation (8), which must be supplemented with :
The bifurcation curve has the same qualitative shape as that for solidification, but the [3 and G scales
are different (see Fig. 3).
2 . S THE NON-LINEAR EXPANSION. SECOND ORDER TERMS. - We now look for a reduction of the full
dynamics of the problem valid in the close vicinity
of the bifurcation. As explained in detail in WS, let
us consider solutions of the basic equations of the
form :
_.l_ _. ,
with
and we assume that
with
where v 11 (z) is the solution of the linearized system obtained in § 2.4, and, if one keeps to 1 st order, A(t) = exp(ao t).
As soon as one goes beyond the linear expansion,
linear superposition does not apply. Assumption (26) only enables us to study periodic solutions with the
single basic wavevector (9, its harmonics being gene- rated by higher orders in E. This is not too drastic
a restriction : on the one hand, observed experimental
structures are periodic, on the other hand, close to the bifurcation, only a very narrow band of wavevectors are linearly unstable, while their harmonics are stable.
One then straightforwardly checks that (25) and (26), when inserted into the basic equations, imply :
etc..., , and an analogous expression for C2, C3, ...
In order to ensure the coherence of the perturbation expansion, one must set (see [6])
Selecting in the expansion of equations (3) up to
order c’ the x-independent terms, one immediately finds, using the boundary conditions (5-6) :
Let us point out that equations (29c, d) differ from the WS expressions for T2o, T’o (which would vanish
everywhere) : they neglect the finite thermal size of the
box, and impose T20 = 0 at z = + oo, and T 2 0 = 0
at z = - oo. This, if assumed here (with r¡ =F 0) would
lead to an incompatibility between the interface equa- tions for the { 20 } terms. This expresses a simple physical fact : the { 20 } terms are associated with a
uniform displacement of the planar front. This induces
a rearrangement of the temperature profile with a range in the z-direction min (l,h, Ll,2), i.e., here,
L1,2, since we deal with a small thermal box. This effect only appears (to all even orders in c) for x- independent terms. Temperature corrections in
cos (rmx) (r =A 0) have a range along z of order
(Tlr(o) Ll,21 so, for them, the thermal box is effec-
tively infinite in the z-direction.
Expanding the interface equations and selecting
the { 20 } terms, one can solve for the five constants
A20, A 2 o B20, B’20 C20. The results are given in Appen-
dix B, in the limit ao = 0 (i.e. at the bifurcation),
which will be sufficient to calculate the third order term in the amplitude equation.
Only for il = 0 does one recover the WS result
B2O = Bio = 0, in which limit the thermal boundary
conditions are irrelevant (1).
The calculation of the other second order correc-
tions v22, C22 proceeds exactly as in WS, and one
finds :
with (in the ao = 0 limit)
where m is to be understood as the critical wavevector at the bifurcation. The values of A22, ..., ’22 are
listed in Appendix B.
2.6 THIRD ORDER TERMS. CALCULATION OF THE CUBIC COEFFICIENT a 1. - In order to calculate the coefficient at of the cubic term in the amplitude equation, we only need to consider, among the 0(83) contributions, the term v31 (Eq. (27b)). Equations (3) give :
Taking into account the boundary conditions, one gets :
( 1) This is also trivially true in the n = 1 limit, where temperature and concentration decouple completely, since,
as the latent heat production can be neglected on the time
scale of solute diffusion, the phase boundary is thermally
inactive.
The interface conditions yield (following the nota-
tions of WS) :
The constants do, ..., d5 can be easily [9] expressed
in terms of the amplitude coefficients of vo, vi i, V20, v22. They are listed (for ao = 0) in Appendix B.
Due to the simplicity of the differential operators appearing in equations (32), it is possible to by-pass
the details of the adjoint operator method used by WS
with the help of the following remark : insertion of
expressions (33) into the system (34) transforms it into a system of five linear equations for the five unknowns A 31, A 31, B31 , B3 > > (31. The associated
determinant is easily checked to be A(3 ao, m) -
where A(ao, w), given by equation (19), is the corres- ponding quantity for the linear problem. So, at the bifurcation, the determinant vanishes exactly, and the system (34) degenerates. One must therefore impose a
condition of compatibility between equations (34)
for ao = 0, which determines the coefficient a,.
This condition is obtained by multiplying equa- tions (34a) to (34e) respectively by - dm n/( 1 + n) ;
- Dm/( 1 + n) ; m’ ; 1 and Dm, with
(m and m’ are defined in Eq. (22)), adding up and tak-
ing the ao = 0 limit. Note that the singular terms proportional to (ao) - 1 (see Eq. (33)) vanish identically,
which ensures that the perturbation expansion is
indeed regular at the bifurcation.
One thus immediately obtains :
Using the expressions for the d’s given in Appen-
dix B, this can be rewritten, in terms of the ampli-
tudes of VII v2 :
It can be checked easily that expression (36) for a, differs from the WS result (Eq. (6.46) of reference [6]) by an overall factor of
where -6, 3, w are connected by the bifurcation con- ditions (21), (23). We believe that this difference stems from the following error in the WS calculation : the Fredholm condition, when they apply it to the pro- blem, leads to their equation (6.45), which reads :
where Vil is the solution of the adjoint linear problem.
They then take the bifurcation limit l3 -> 13c, w >- We, where ao - 0, and from this they conclude that the second term on the l.h.s. of equation (39) vanishes at
the limit. However, one must notice that V31(Z) is not regular at the bifurcation (1) since, as can be seen on equations (33), it contains terms proportional to (ao)-1. So, the limit of this term is in fact finite. One
can compute it straightforwardly with the help of equations (33), and it is found that this precisely leads
to our result (Eq. (36), (37)).
(2) Although, when the cellular structure stabilizes,
e oc (ao)1/2, so that the correction C3 C31 to the concentra-
tion profile is indeed regular and contains a contribution of order (ao)1/2. In the other cases, the perturbation expansion
is coherent for E (ao)1/2.
