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A schematic mechanism for striated growth of solid binary mixtures
B. Caroli, C. Caroli, B. Roulet
To cite this version:
B. Caroli, C. Caroli, B. Roulet. A schematic mechanism for striated growth of solid binary mixtures.
Journal de Physique, 1983, 44 (8), pp.945-951. �10.1051/jphys:01983004408094500�. �jpa-00209677�
A schematic mechanism for striated growth of solid 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 janvier 1983, accepté le 25 avril 1983)
Résumé. 2014 On montre, à l’aide d’un modèle schématique dérivé de celui proposé par Haase et al. pour la croissance de certains feldspaths (plagioclases), que l’effet combiné du transport diffusif et de deux cinétiques d’attachement moléculaire lentes en compétition non linéaire suffit à rendre possible la croissance à striations planes (c’est-à-dire,
la croissance d’un mélange binaire solide présentant des oscillations périodiques de composition dans la direction de croissance).
Abstract.
2014It is shown, on the basis of a schematic model derived from that set up for the growth of plagioclase
rocks by Haase et al., that the interplay between diffusional transport and two slow molecular attachment kinetics in non-linear competition is sufficient to account for the possibility of striated growth (i.e., growth of a solid binary
mixture with planar periodic composition oscillations in the growth direction).
Classification Physics Abstracts
64.60
-61.50C
1. Introduction
Binary mixtures have been observed in various situa- tions to solidify with a planar layered structure parallel
to the solidification front These structures are charac- terized by a periodic behaviour of the concentration,
also called « oscillatory zoning ». Such phenomena
are found, for example, in doped semiconductors grown by directional solidification [1], in the electro-
crystallization of a metal in the presence of an organic
inhibitor [2], and in natural rocks known as plagio-
clase feldspars [3, 4]. The wavelengths of the composi-
tion oscillations, typically in the 10-100 ym range,
are macroscopic. Transient oscillations have also been observed [1].
These striations have often been considered as
resulting from either external fluctuations or modula- tions of the growth conditions. However, several experimental facts seem to rule out such explanations.
In particular, the periodicity observed in the naturally
grown plagioclases gives serious ground to the idea
that oscillatory zoning is, on the contrary, an intrinsic dynamic mechanism [3-5] which may take place under completely stationary growth conditions.
Such an interpretation of plagioclase formation
has been put forward recently by Haase, Chadam,
Feinn and Ortoleva (HCFO) [3]. What we intend to
(*) We dedicate this article to our colleague Yuri Orlov.
(**) Associ6 au C.N.R.S.
do in the present article is the following : reducing
the HCFO model to what we believe to be its essential
physical ingredients, we consider a liquid-solid transi-
tion from a metastable liquid phase made of a mixture
of different (atomic or ionic) species which condense,
at the liquid-solid interface, into two different mole- cular species X and Y. These two competing chemical I
reactions are fed by diffusional transport of the various constituents of the liquid phase, among which one
species, A, involved in the two reactions, is assumed
to have a much smaller diffusion coefficient than all the others (which therefore have quasi-constant con-
centrations in the liquid). The condensation kinetics of X and Y are supposed to be slow, so that the inter- face is very far from local equilibrium, and we neglect
solid - liquid chemical processes. Finally, we assume
that A is present in molecules X and Y in different stoichiometric proportions, so that the formation of,
say, X, consumes more A than that of Y.
We will show that such a system may exhibit self- oscillations of its rate of solidification and, associa- tedly, space modulations of the solid phase composi- tion, due to the following mechanism : assume that,
at a given time, there is a high concentration As of A
on the liquid side of the interface. X formation is favoured and, if it is fast enough, it consumes more A than can be fed by diffusion. AS thus decreases and, when it becomes small enough, Y condensation is enhanced If this reaction is slow enough, diffusion
starts overfeeding the interface, etc..., which may result in the stabilization of self-oscillatory growth.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004408094500
946
In § 2, we define the model and discuss its assump- tions in more detail, and show that it has a steady-
state solution where the (planar) solidification front
moves into the liquid at constant velocity.
In § 3, we perform the linear stability analysis of
this solution with respect to front velocity oscillations,
and show that it exhibits, in the parameter space,
a Hopf bifurcation indicative of the stabilization of an
oscillatory regime.
2. The model and its constant growth rate solution.
According to the assumptions of § 1, in the liquid phase, the only limiting transport is that of species {A}, which concentration A obeys the diffusion
equation :
with the boundary condition, at infinity in the liquid
We assume that diffusion in the solid phase is completely negligible.
We take, for the rates of formation Gx,y of the mole-
cular X and Y species at the solid surface, the follow-
ing laws :
where A. is the value of A on the liquid side of the interface. The two constants N and B depend on the
concentrations of the fast diffusing liquid species, the
temperature and the microscopic mechanisms of the two reactions. Equations (3) imply that the formation reactions of X and Y, which we assume for simplicity
to be direct, consume respectively two and one A
units. So, the condition of conservation of A at the
moving interface reads :
where v is the velocity of the interface, and n is the unit vector normal to the surface pointing into the liquid phase.
Assuming, for simplicity, that the molecular den-
sity p of f A } in the solid phase is composition inde- pendent, the interface velocity and the rates of forma-
tion are related by :
Note that the model implies the following simplifica-
tions :
(a) convection effects are negligible, i.e. the density
differences between the solid and liquid phases are neglected;
(b) the heats of reaction are not too large so that,
since the diffusion of heat is much faster than that of
concentration, the temperature can be assumed cons-
tant ;
(c) the rate constants N and B are assumed inde-
pendent of the local solid composition close to the
surface. This represents an important simplifying
difference with the existing models of plagioclase growth. Indeed, the HCFO theory [3] implies the
idea that the dependence of N and B on the solid
surface composition is crucial to explain oscillatory growth, while the model of All6gre et al. [4] assumes
that the important phenomenon is a retardation effect which can be understood as resulting from the dependence of N, B on the composition profile of a
finite depth of the solid We will show that these effects
are inessential to describe the oscillatory zoning of a binary mixture with condensation kinetics of different orders such as those chosen in equation (3).
,Finally, it must be pointed out that the rate cons-
tants N, B do in fact depend on the local orientation of the interface
-which reflects the anisotropy
of growth velocities and surface energies. However,
we will make the important assumption that the growth front is planar, and that the possibility of
cellular instabilities [6] is excluded This restriction,
motivated by the planar character of the striations observed in many experimental situations [1, 2, 4], is coherent, from a theoretical point of view, with the assumed slowness of the condensation reactions
(i.e. with the strong departure from local equilibrium
at the interface which we invoked to neglect the
« melting » processes inverse of the condensation
ones). Such a situation is usually characteristic of facetted growth, in which surface distortions are
prevented by the high energy cost of surface step creation.
We now look for a solution of the growth equa- tions (1)-(5) with a constant growth velocity V in the
x-direction. In the reference frame moving at velocity
V (where the interface is stationary at position x = 0)
A must be time-independent, and equations (1)-(5)
become :
From equations (6 . a, b) :
Plugging equation (7) into (6. c, d), and introducing
the parameter
we find
rw W di
Since Ao must be positive, it is seen from equa- tion (9) that growth at constant velocity can occur only if 0 f 1, i.e.
This condition expresses the fact that a steady state (in the moving frame) imposes that the (constant) A
concentration in the solid must be equal to that in the liquid reservoir, Aoo. If condition (11) is not satisfied,
no steady solution exists, and the front moves at non- constant velocity [7]. We will assume from now on that
condition (11) holds.
3. Linear stability analysis.
We now assume that the front velocity undergoes a
small time modulation, so that the position of the interface, in the frame moving at the velocity V of
the steady solution, is given by :
where s is a dimensionless smallness parameter and ç
an amplitude. In the same frame, the front velocity is :
This modulation induces a modulation 6A(x, t)
of the A concentration field. Expanding equations (1)-
(5) to first order in e about the steady solution { A (’)(x), V } calculated in 2, one gets :
From the diffusion equation and interface conditions, one obtains :
i.e.
where
and the square root in equation (16. b) is chosen to
have a positive real part. Then Re (q+ - q-) > 0.
When applying the boundary condition lim A1(x) _
x-+ 00
0, one immediately checks that, except in a layer of
the order of the diffusion length 1D = D/V along the
external boundary of the liquid, the contribution to
A1(x) of the exp( - q- x) term is exponentially small,
and thus negligible in the interface conditions since the liquid reservoir is assumed to be large with respect
to 1L, Then, for all purposes :
and, with the help of
the first order expansion of equations (3-5), together
with (17), gives :
where Ao z A (0)(0) is given by equation (9).
The condition of compatibility of the two linear homogeneous equations (19) gives the relaxation rate c) of the velocity modulation. Introducing the
dimensionless variable :
one finds
JOURNAL DE PHYSIQUE.
-T. 44, N° 8, AOUT 1983
where
948
Equation (21) shows that one always finds the
trivial solution z = 0, which expresses that the
steady solution is degenerate with respect to the position of the front in the moving frame. The non
trivial solution is given by :
The steady solution becomes unstable at the
point(s) of the parameter space
-if it exists
-,where Re z = 0 (if Re z 0 (resp. > 0) the steady
solution is linearly stable (resp. unstable) with respect
to velocity fluctuations
Let us first show that such a bifurcation cannot occur with Im z = 0. z = 0 is a,solution of equa- tion (23) if Q = 1. Using equations (8-10) and intro- ducing the « competition parameter »
Q can be rewritten
so that Q = 1 only for f = 0 or f = 1, i.e. in the two limits where the steady solution (7-10) disappears.
So, if the system exhibits a bifurcation, this has to
be of the Hopf type, i.e., at the bifurcation, z = ± ia, and, from equation (23), one must have :
which have a unique solution
if and only if
K (Eq. (22. a)) can be rewritten as :
K and Q only depend on the two parameters f and À..
The first one is directly related to the concentration of
species {A } in the reservoir while A measures the
competition between the kinetics of the two chemical reactions. Condition (28.b) results in a polynomial equation, of degree 2 in À., which has two real roots for 0 f 1. One checks that only the larger root
satisfies condition (28. a). It is given by :
The bifurcation curve (30), which separates in the (A, f) space the linearly stable (Re co 0) and unstable
(Re (o > 0) regions, is plotted in figure 1.
Using equation (27), one finds that, on the bifur-
cation curve :
roc D/B2 is plotted on figure 2.
If the Hopf bifurcation defined by equation (30)
is of the normal (that is, not subcritical) type, close to the bifurcation in the unstable region the growth velocity is no longer constant, but oscillates around the steady value (Eq. (10)) with frequency roc and a
small amplitude. This small limit cycle then manifest
itself by oscillations of the solid phase composition
with a wavelength Lc given by :
The dimensionless quantity BLc/2 nD is plotted on figure 3. Note that, in this case, if the system lies, in parameter space, close below the bifurcation (in the
stable region), its transient regime will exhibit a
slowly damped oscillatory behaviour.
If the bifurcation is subcritical, no prediction can
be made about the nature of the restabilized growth regime, which should be looked for by numerical computation.
The nature of the bifurcation can be determined by
a non-linear perturbation expansion of the system ( 1-5).
Such an expansion, although systematic, is very
Fig. 1.
-The bifurcation curve in parameter space (A = BIpN, f = (Aoo/p) - 1).
heavy and difficult to exploit, due to the fact that
the amplitude of the limit cycle is a complicated
function of the parameters B, N, p, D, f. So, at the present stage of the model
-which remains too schematic to describe quantitatively the existing experiments
-it seems quite pointless to perform
such a calculation in detail. We simply outline in the
appendix the principle of the appropriate expansion.
Finally, in order to get an order of magnitude for
the wavelength of the composition oscillations, we
have calculated L, (Eq. (32)) with the values of f, D, B resulting from the numerical data for plagioclase growth as estimated by HCFO [3]. We find
Lc = 100 J1m ,
that is, the observed order of magnitude. Let us insist, however, that our model is oversimplified as compared with that of HCFO, that the geochemistry
of plagioclase formation does not seem completely
clear yet, and that, finally, Lr is calculated at the bifurcation. So the agreement between the estimated and observed wavelengths must not by any means be taken literally, but only as an indication of the plau- sibility of the physical mechanism. The HCFO model
-
although its assumed f-dependence of B and N finally appears inessential
-thus provides an attrac-
tive qualitative explanation of oscillatory zoning in plagioclases. It is difhcult, at the moment, to compare in more detail its plausibility with that of the mecha- nism put forward by Allegre et al. [4], as long as the
Fig. 2.
-Plot of the dimensionless frequency D.wc(f)/ B2
of the neutral mode (along the bifurcation curve).
Fig. 3.
-Plot of the dimensionless critical wavelength BL,/2 nD (along the bifurcation curve).
delay between concentration changes at the interface and the crystal growth rate response introduced in the latter model remains purely phenomenological.
4. Conclusion:
The above simple analysis, although performed on an extremely schematic model, shows that the interplay
between diffusional transport and two attachment kinetics in non-linear competition is sufficient to open the possibility of growth with oscillatory zoning
of purely dynamic origin. This type of mechanism,
950
based on the non-linear competition of two chemical kinetics, appears to us to be essential for the under-
standing of such phenomena - although, of course,
more sophisticated models should be built up for each specific situation (including, for instance, the
diffusion of several rather than one species).
This qualitative conclusion meets with the point
of view underlying a model recently proposed by
Favier [5], who also attributes striations to the dyna-
mic competition between diffusion and chemical kinetics. However, his model implies the existence of a transient surface state, the nature of the exchanges
of which with the liquid and solid phases appears
somewhat arbitrary.
Let us mention here that a more realistic model should also include convection effects, which are currently held responsible for some striation effects in semiconductors. This is still an open problem in the description of growth morphologies.
The above discussion points to the fact that it would be highly desirable to develop systematic
controlled experimental studies of the oscillatory zoning phenomenon, in the absence of which more
detailed theoretical treatments will remain specula-
tive. The development of such experiments will also
be helpful to clarify the problem of the influence of the relative rates of attachment kinetics and diffusion
on growth instabilities.
Acknowledgements.
It is a pleasure to thank J. Chadam for enlightening
discussions on the mathematical aspects of this and related problems.
Appendix
The non-linear perturbation expansion, close to the bifurcation, follows the general principles of such
calculations [8]. We will simply sketch out here its
main lines.
Fig. 4.
-Schematic representation of the critical surface in the parameter space. P is the point corresponding to the
actual parameters (Â., f, B). P, is the corresponding zeroth-
order point (to be determined) around which the perturba-
tion expansion is performed
The main complication, in the present problem,
lies in the fact that the bifurcation is defined, not by a
critical value of a single parameter, but by a critical
surface in the (B, N, A.) or, equivalently, ( f, A, B)
space, defined by the equation H( f, A, B) = 0. So (see Fig. 4), given a point P in this space close to the critical surface, no obvious criterion can tell a priori
which of the critical points Po must be chosen to
define the zeroth-order of the perturbation expansion.
In fact, Po is determined, at each order, by the expan- sion itself.
One sets :
and
with the boundary condition
s is a smallness parameter, and we have chosen to work in the reference frame moving at the constant (zeroth-order) velocity V0, related to the zeroth- order values Â.o, Bo, fo by equation (10). One then
selects the various orders in 8 in the diffusion equation
and the boundary conditions (Eqs. (3-5)).
The zeroth-order solution is the steady one (§ 2).
Then :
(i) To first order, from equations (A. 4) and (3-5),
one shows that At (x, s) and ç t (s) are of the form :
The solution of the dynamic equations then gives :
-
for the terms in exp( ± is), the linear dispersion
relation (21). The corresponding A 1 (x) is that calculat-
ed at the bifurcation by the linear stability analysis
of § 3.
-
for the s-independent terms, a solvability condi- tion, which reads :
This does not provide a unique determination of
Å,1’ Bl, fi (and E to first order), which can be completed only by going to order B2 in the expansion.
(ii) From the second order equation, it is found that :
-
due to the linearity of the diffusion equa- tion (A. 4), and from the standard condition of elimi- nation of diverging resonant terms, the first order correction OJ 1 to the oscillation frequency is zero.
This is true, for the same reason, to all orders, so that :
(It is the same linearity argument which makes an
expansion of D superfluous.)
-
the equations for the {21} terms yield one complex, i.e. two real solvability conditions, linear and homogeneous in Ât, Bl, fl, which, together
with (A. 6), give
-
the { 20 } equations lead to a solvability condi- tion, linear in (À,2’ B2, f2) and non homogeneous, with
coefficients expressible in terms of the zeroth-order solution,
-
