On directional solidification of a faceted crystal

HAL Id: jpa-00211003

https://hal.archives-ouvertes.fr/jpa-00211003

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On directional solidification of a faceted crystal

R. Bowley, B. Caroli, C. Caroli, F. Graner, P. Nozières, B. Roulet

To cite this version:

R. Bowley, B. Caroli, C. Caroli, F. Graner, P. Nozières, et al.. On directional solidification of a faceted crystal. Journal de Physique, 1989, 50 (12), pp.1377-1391. �10.1051/jphys:0198900500120137700�.

�jpa-00211003�

On directional solidification of a faceted crystal

R. Bowley (1), ^{B. Caroli} (2, 3), ^{C. Caroli} (2), ^{F. Graner} (1), P. Nozières (1)

and B. Roulet (2)

(1) ^Institut Laue-Langevin, ^BP 156, 38042 Grenoble Cedex, ^France

(2) Groupe ^de Physique des Solides de l’Ecole Normale Supérieure, ^{associé au} Centre National de la Recherche Scientifique, Université Paris VII, ² place Jussieu, 75251 Paris Cedex 05, ^France

(3) Département ^de Physique, U.F.R. des Sciences fondamentales et appliquées, Université de Picardie, ^{33 rue} Saint-Leu, ⁸⁰⁰⁰⁰ Amiens, ^France

(Reçu ^{le 20} janvier 1989, accepté ^{le 8 mars} 1989)

Résumé. 2014 Nous étudions l’instabilité morphologique ^d’un mélange ^{binaire en} solidification directionnelle, ^{dans le cas où} l’interface plan ^{initial est une} facette. Nous montrons que, pour des vitesses de tirage plus grandes que le seuil standard de l’instabilité de Mullins-Sekerka, ^{il existe un} continuum de formes de front stationnaires non planes périodiques ^{et de} petite amplitude,

formées d’une succession de facettes chaudes et froides reliées par des régions courbes. Ces solutions sont instables par rapport ^aux fluctuations d’amplitude ; ^elles peuvent ^{donc être} considérées comme définissant un seuil pour l’amplitude de l’instabilité ordinaire. Celle-ci devrait donc présenter ^une très forte hystérésis.

Abstract.

We study the Mullins-Sekerka instability ^{of a} binary mixture submitted to directional solidification in the case where the basic planar solid-liquid interface is a facet. We show that, ^for pulling velocities larger than the standard MS instability threshold, there exist a continuum of

stationary non-planar periodic ^front profiles ^{of small} amplitude, consisting ^{of an} alternation of hot and cold facets connected by ^curved regions. These solutions are unstable as regards ^their amplitude, ^{so that} they may be viewed ^{as an} amplitude threshold for the usual cellular instability,

which should therefore exhibit an anomalously large hysteresis.

Classification

Physics ^Abstracts

61.50C - 64.60 - 64.70D

Dilute binary mixtures, ^{when submitted to} directional solidification (pulling ^{at an} imposed velocity ^V along ^{the z axis in an} external thermal gradient ^G parallel ^to V) exhibit, ^{at a}

threshold velocity Vc(G), a morphological instability. ^The solid-liquid interface, ^{which is} planar ^{for Vu} Vc, develops, ^{for Vu} Vc, ^a quasi-periodic structure. This instability ^results

from the competition between the destabilizing effect of chemical diffusion and the stabilization due to capillarity ^{and to the} temperature gradient. ^{It was first} analyzed by

Mullins and Sekerka [1] ^who studied the linear stability ^{of the} planar front for materials with

an atomically rough solid-liquid interface and an isotropic surface tension. Much experimental

and theoretical work has been performed since in order to analyze in detail both the vicinity ^of

the bifurcation [2-4] ^{and the} shapes ^of stationary front cells far above threshold [5, 6]. ^{Most of}

these studies deal with the case where the attachment kinetics is instantaneous (local

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500120137700

equilibrium ^{on the} front) and the surface tension isotropic, except ^{for some} extensions ^to systems ^with (isotropic ^or anisotropic [7]) linear kinetics and a smooth capillary anisotropy [8].

That is, ^all existing ^theories of cellular fronts are concerned with solids which do not exhibit facets on their equilibrium shape. ^More generally, ^the question ^of growth shapes ^of faceting

solids when the solidification dynamics is, ^{at least} partly, controlled by ^diffusion [9], ^has

received little attention with the noticeable exception of the work of Chernov and collaborators [10] ^on freely growing crystals. ^A significant opening has been made recently by

Ben Amar and Pomeau. In their work on faceted needle-crystals [11] they ^{were able to}

formulate explicitly ^the equations describing ^the growth ^{of a front} containing ^faceted parts.

Namely, they ^{show that the kinetic} growth ^{relation on the} front, which is local in the curved

regions, ^{must be} replaced, ^{on the} facet, by ^an integrated ^condition relating the averages of the relevant (temperature ^and/or concentration) ^fields along ^{the facet with its size.} They ^were

then able, ^on the basis of a scaling argument, ^{to make some} predictions ^about velocity

selection for the needle-crystal.

In order to make further _progress on the problem ^{of the} coupling between diffusion and

faceting, and of its incidence _upon dynamical ^front morphologies, ^{it seems useful to address a} question ^a priori simpler than dendritic growth, namely ^{that of the} Mullins-Sekerka instability

of a faceted front under conditions of directional solidification.

As already mentioned, ^{when the} equilibrium shape ^{of the} crystal is smooth i.e. when the Wulff plot of the surface energy is ^not cusped - the existence and position ^{of the} instability

can be established ^on the basis of the MS linear stability analysis. ^{It would of} course seem

natural to try ^and apply ^{the same} perturbative approach ^{to the case} where the basic planar

front is a facet. However, it appears that the presence of ^a cusp in the Wulff plot ^{in the}

direction of the unperturbed interface entails that any deformation of this front, ^however small its amplitude, belongs, ^{from the} point of view of the MS approach, ^{to the} strongly ^non-

linear regime. ^{This can be} easily understood qualitatively.

In all that follows, ^{we will} only consider 1-D front deformations. Consider a situation where :

(i) ^the liquid-solid interface energy a ( 0 ) (where Ois ^the angle between Oz and the normal to the interface pointing ^{into the} liquid ; ^see Fig. 1) ^{is the} equilibrium ^one [12], ^{and its}

temperature ^and concentration dependence ^are negligible in the T and C ranges of interest ; (ii) ^the equilibrium facet matches smoothly (tangentially) ^{with the} contiguous ^curved regions. ^{That is} [13],6 ⁺ d 2o--Id 02 ^is positive everywhere ^{on the Wulff} plot - except, ^of

course, for 0

0 and in the crystallographically equivalent directions, where it is undefined.

We assume for the moment that interface kinetics is everywhere instantaneous. This

assumption, which is unessential for the following qualitative argument, will be relaxed later.

A cusp in a(0 ) ^{in the 0}

0 direction gives ^{rise to a} contribution à o,’ . 8 ( e ) (with

Acr’ = (d a /d 0 )0 + - (d a /d 0 )o _ ) ^{to the}

surface stiffness

(o’ ⁺ 0, " ) which appears in the Gibbs-Thomson equation expressing ^local equilibrium ^on the front. Let us imagine ^{that we}

round off this cusp ^{on a} very small angular ^{width e} (see Fig. 2). (er ⁺ U") ^{becomes a smooth} function, ^{so that the MS linear} expansion ^{can now be} formally performed with the usual results in which the surface stiffness parameter ^{has to} be understood as lQ = (u + u")8 = o.

Note that, ^{due to} the smallness of E, lQ

^{£- 2} a (0 ) ^{is much} larger ^{than for a} rough system.

This entails that the region ^{of the} (G, V ) plane ^{where the} planar ^{front is} linearly ^unstable

becomes correspondingly ^small [3].

Of course, the next question ^{to be asked} is that of the limits of validity of the linear

perturbation treatment. The perturbation expansion in powers of the front deformation

amplitude ^{relies on the} Taylor expansion ^{of the} growth equations. In the presence of the

narrow peak ^{in a + o-"} (Fig. 2), the linear approximation therefore fails as soon as the slope

Fig. ^1.

Schematic sketch of a directional solidification setup.

Fig. 2. - Schematic shape of the surface energy (a) ^and surface stiffness (b) around the facet direction 8

0. The dashed curves correspond ^to the smoothed cusp approximation.

of the front profile 03BE(x ) is such that 0

tan - 1 (d C ldx )

c. In other words, ^{it is} meaningful only ^for profiles containing ^no orientations but those spanned by ^the equilibrium quasi-facet resulting ^{from the} rounded-off _{cusp of} 0152

That is, ^a cusp in a (£ -+ 0) corresponds ^{to a} vanishing range of validity of the linear

approach. This increases considerably the difficulties of an exploration ^{of the} dynamics ^of

deformation of a growing facet. For this reason, ^we limit ourselves in the following ^{to the}

much more restricted problem ^of investigating whether there exists stationary ^front shapes

which are non-planar, periodic ^{and with a small} amplitude ^of deformation with respect ^{to the} original facet. We show that this problem ^{has a} simple analytical solution in the highly ^non-

local limit where the space-period of the front structure is much smaller than the chemical diffusion length ^{- a condition which is} easily satisfied in usual experimental situations.

We find that, ^{for V} Vcs (G - ^where Vcs (G) is the so-called constitutional supercooling

threshold [14]

^{no small} amplitude stationary non-planar solution exists. Above

Vcs, ^we show that there is a continuum of such stationary

crenellated

profiles, consisting

of an alternation of hot and cold facets connected by ^curved parts, and calculate the relation between the height and size of the facets.

We then show - in the limit where the facets are much longer than the curved regions

that these solutions are unstable against ^small amplitude fluctuations. That is, the branch of crenellated states has to be « jumped ^{over »} in order for the locally stable facet front to

restabilize into a Mullins-Sekerka structure of partly faceted cells. This should result in a very strong hysteresis ^{of the front} morphology, ^{related to the} depth ^{of the} cusp in the Wulff plot,

i.e. to the magnitude ^{of the} step energy for the relevant facet.

1. The front equation.

We describe our two-phase system by the usual one-sided model [2], ^{that is we} neglect

diffusion in the solid. For the sake of simplicity, ^{we assume that the two} phases ^have equal

thermal diffusivities and consider thermal diffusion ^as instantaneous ^on the scale of chemical

diffusion, ^so that the thermal profile is unaffected by the solidification process [3]. ^Let

G

dT/dz ^{be the constant} externally imposed temperature gradient.

We consider here 1-D front deformations. Finally, ^{we assume} that the surface energy

a ( 0 ) ^is locally symmetric about the facet orientation, i.e. that 0

0 is a direction of high symmetry of the solid.

In the laboratory frame, ^the solidifying system, pulled ^at velocity V//Oz, is then described

by ^the following ^{set of} equations [1, 2] : a) ^{in the} liquid (z:>03BE (x, t ) )

b) ^on the front (z = ’(x, t»

-

Solute balance equation :

-

Gibbs-Thomson equation :

In equations (1-3) lengths, ^{times and} concentrations are measured respectively in units of the solute diffusion length f = 2 D /V, ^{the diffusion time} ’T = 4 D /V2 and the solute concentration gap AC

Coo(1 - K)/K ^{for the} planar stationary ^front (i.e. ^for C,

C (0).

Crois ^the imposed concentration of the liquid ^{far ahead of the front, K the} equilibrium partition coefficient of our (dilute) ^{mixture, û the unit vector normal to the front,}

k = - 03BE,l(l ⁺ ep2)3/2 ^{the front} curvature, ^and 8 (x, t )

tan-l(ô(x, t)lax).

u = ÎGI 1 ML AC 1,

^{where mL is the slope of the liquidus line on the} phase diagram.

y ( 0 ) = Tm u ( 0 )/ (Lf mL LlC 1 ) ; ^{L is the} specific ^latent heat, TM ^the melting temperature of the pure solvent. e is measured from the T

Tm ^isotherm.

Finally, ^{the Y term in} equation (3) ^{accounts for the} undercooling effect associated with attachment kinetics. For the sake of simplicity, ^{we assume} this process ^to be instantaneous in all but the facet direction :

where 8 Tkin (v ) is the kinetic undercooling of the facet growing ^at velocity ^v.

The generalized Gibbs-Thomson equation (3) ^{is local} for 0 # 0, where the « surface stiffness » ( y ⁺ y " ) is well defined. In the absence of surface friction (F

o ), it expresses local equilibrium ^{of the} interface. For a facet, the situation is different, ^as equilibrium ^{is now a} global property, corresponding ^{to a} neutral balance of, say, ^terrace nucleation and spreading.

Equation (3) ^{must be} integrated ^{across the} facet, yielding ^the global constraint :

(remember ^{that K}

d 0 Ids). ^{0 y’ -} y’, - yo

0. xo, xi ^are the abscissae of the endpoints

of the facet of height lf. Ef ^{= - 1} (resp. ⁺ 1) ^{for a}

quasi-convex » (resp. concave) ^solid

facet. We define as quasi-convex ^a facet such that 0 (xo- )

0+ ; 9 (xl+ )

^0_ (see Fig. 3).

When F

0, (5) ^is simply ^{a statement} that the energy is stationary ^{when one} full ^{terrace, with} height ^{a, is added to} the facet : the cost of the peripheral steps, 2 | yÓ 1 a ^{= a 0 y’} exactly

balances the bulk energy gained by growing ^{an extra} solid volume a (xl - xo ).

Note that since, ^{as mentioned} above, ^{we assume} that there are no missing orientations on

the equilibrium shape ( y ^{+ y "}

0 for all 0 # 0), mechanical equilibrium of the curved parts, which is instantaneous on the time scale of interest here, imposes that the curved parts ^{of the} profile ^match tangentially with the facets.

Fig. ^3.

^Non planar periodic ^front profile. ^The planar region - b ^x b is what we define as a quasi-

concave facet, À /2 - a .: X .: À /2 + a corresponds ^{to a} quasi-convex ^one.

The basic stationary ^solution corresponding ^{to an} infinite facet is immediately ^obtained by looking ^{for an} x-independent solution of equations (1), (2) ^and taking ⁱⁿ equation (5) ^the

(xl - xa )

00 limit. It is given by :

The planar ^front position (0) ^{is :}

which simply describes the kinetically induced recoil of the facet from the local equilibrium position.

We now look for small amplitude non-planar stationary periodic solutions of equations (1)- (3). ^The shape ^{of such a} solution is depicted ⁱⁿ figure ^{3. Since} y (0) is ^{even, we} may restrict ourselves to front shapes with reflection symmetry ^{about the centre of a} facet. We call

2 a, e + (resp. 2 b, 03BE - ) ^the length ^and height ^{of a} quasi-convex (resp. concave) ^{facet, c the}

extension along ^{Ox of each curved} region. À = 2 (a + b + c ) is ^then the space period ^{of the} profile.

2. A simple ^{treatment of} stationary ^front profiles.

We first give simple arguments ^{that are} sufficient to provide stationary shapes ^{in the} strongly

non local limit à 1 (i.e., wavelength much smaller than the diffusion length f), ^{but which}

cannot account for the dynamics of these fronts. Detailed calculations will be given ^{in the} following ^section.

A stationary shape ^can be constructed in two steps :

(i) first find a diffusion profile ^that obeys ^the conservation law (2) ^{for a} given ^front profile ).

(ii) Then express surface equilibrium (3) ^and (5) ^{as a} constraint that specifies 03BE(f).

For the one-sided model in the limit À « 1, ^{the first} step ^is indeed trivial : the diffusion

profile that satisfies conservation laws is just the zeroth order _one, (6). ^{Assume for a moment}

that the exponential ⁱⁿ (6) ^is approximated by ^{a linear law}

The diffusion current in the liquid phase is unaffected by interface deformations

03BE(x), ^{and it} guarantees ^solute conservation for any stationary shape of the interface (the

volume swept by the interface is shape independent). ^{In a two-sided} model, ^the concentration fluctuation along the distorted interface would induce ^a short-circuit current in the solid phase

that would spoil conservation ; ^{here this} complication ^{does not occur. Thus C} (0)(z) ^{is the}

appropriate ^diffusion field, ^{but for} corrections due to curvature of the exponential. ^{In order to}

estimate the corresponding ^{error, we note} that the defect of current at the interface is

-(AC)V03BE/l (we ^{restore the} original units in order ^to make the argument ^more

transparent). ^{Thus an extra} correction 8C must appear, yielding ^that missing ^{current on a}

scale - À in the liquid, ^{hence :}

The error is of order (03BEÀ / £2), very small if ^à f. In contrast, ^the short-circuit current in the

solid in a two sided model would be of order (D/A). (oc ) elf, yielding ^{a correction} -- ’1 À, comparable ^{to the main term in the} equilibrium ^condition (3). ^{In this respect, the one}

sided model is much simpler ^{than the two sided one.}

.

From here on, we assume À « l, ^{and we take C} (0)(z) ^{as the} stationary bulk diffusion

profile. We first express the equilibrium ^condition (3) ^on the curved parts, ^{where there is no}

friction. To first order in Ç, the condition is obeyed if the local wave vector, q

’TT lc, ^is

neutral with respect ^to the usual cellular instability, corresponding ^{to an} edge ^{of the} instability

range, which exists only ^{if V exceeds the} appropriate instability ^threshold Vcs (G). ^{Here, it}

must necessarily correspond ^to the upper edge, ^{since we have à} f. Thus, the curved parts

are half arches of ^a sinusoid, ^with wavevector qmax, and with ^an arbitrary amplitude 03BEf (within ^our first order approximation). ^{For a} non-faceted interface, ^these stationary shapes

are unstable as regards ^their wavevector (Eckhaus instability).

We now turn to the integrated equilibrium ^condition for the facet. The facet height, ::t ’f, ^{is fixed} by the curved parts, ^{and it is} symmetric for upper and lower facets : ^we need only

calculate their stationary widths, 2 a and 2 b. Since we ignore ^{for the moment surface} dissipation, equation (5) predicts equal widths for the upper and lower facets :

The smaller the displacement 03BEf, ^the larger ^the equilibrium width. The relationship ^{is more} apparent ^{if we} write it in the form :

where Fo is the surface stiffness near 0

0. Equation (11) may be viewed the opposite way,

as yielding ^the stationary amplitude ef for ^a given global wavelength ^À. Contrary ^{to the non-}

faceted _case, there exists for each À « f a stationary deformed front profile ^{with a} given amplitude, f that varies roughly ^{as A -1 for} long wavelengths.

We will check in section 4 that such a stationary ^state is unstable as regards ^its amplitude

a quite unsurprising ^{result, since} ultimately ^{one must recover} the usual Mullins-Sekerka cellular profiles ^at large amplitudes ^when faceting effects become minor. The stationary ^state

described here must consequently ^{be viewed as an} amplitude ^threshold for the cellular

instability, ^{which must be overcome} before the deformation grows ^to its usual, ^non-linear

value. Since that state is unstable, the issue of phase stability ^{is of} no concern - we should remember only ^{that the} amplitude threshold is lower at long wavelength.

The above discussion is easily generalized ^{to the case of a} finite surface dissipation, F 96 0. As will be shown in the next section, ^{it is} found that the upper facets broaden, ^while

the lower facets shrink :

in which £ is a characteristic length ^{which measures friction :}

As the wavelength grows, (f decreases and the upper facets become progressively ^dominant.

When ef f-->. L, the upper facets invade the front altogether and the distorted stationary ^state

disappears.

3. Detailed description ^{of the} stationary ^front.

Since we only investigate deformations of small amplitude (03BE +- ^03BE- ), ^we linearize the diffusion problem ^described by equations (1), (2) [15]. ^{We set :}

where k

2 7r/À, ^{with À the} period ^{of the} profile.

After linearization of equation (2), ^one gets [1, 2] :

Inserting ^this expression of the concentration field into the generalized Gibbs-Thomson

equation (3) ^and linearizing C (x, (0) + ((1)(.X)) in ((1), ^one obtains the front equation ^for

our small amplitude problem ^{as :}

Integrating equation (14) along ^{each facet, one} obtains the linearized version of equation (5),

so that the front equation explicitly ^{reads :} (i) ^{for a} quasi-concave ^facet (- b - x - b ) :

where we have made use of the reflection symmetry of C (x) ; (ii) ^{for a} quasi-convex ^facet (À /2 - a : X : À /2 ⁺ a ) :

(iii) in each curved part

where :

In equation (18), ^{since the} singular part ^of y ( 0 ) ^{does not appear, we} have linearized the

capillary ^term.

We must now solve the three coupled equations (16), (17), (18) together ^{with the} matching

conditions :

Note that (16), (17), (18) ^{are a} system of linear but inhomogeneous equations. It is the

inhomogeneous capillary ^term, characteristic of the presence of ^a facet, ^{which accounts for}

the fact that our problem ^{is not} fully linearizable in the sense of the Mullins-Sekerka analysis.

The fact that the kinetic term also gives ^{rise to a} non-homogeneous contribution results from

our assumption (Eq. (4)) ^{that the} 8-dependence ^of 37(v, 6 ) ^is discontinuous ; ^{this model-} dependent feature is unessential to our results, since it does not modify ^{the basic structure of} equations (16), (17) ^{which is} determined by ^the capillary singularity.

The strongly non-local limit is, ^a priori, characterized by A « 1, ^{i.e. the} physical wavelength

À must be much smaller than the diffusion length.

In this limit, ^{for n _} 1, mn - nk > ^{1 and the sum on} the r.h.s. of equation (14), ^which

describes the distortion of the diffusion field by the deformation of the front profile, ^is easily

checked to be, ^at most, of order À . In k . Max ( 1 C(l) 1

l ,03BE(1)1). ^{So, it can be neglected as}

compared ^{with terms on} the I.h.s. of equation (18) ^{as soon as :}

which provides ^{a more accurate} definition of what we call the strongly non-local limit. That is,

in this limit we approximate ^the diffusion field in the presence of the actual distorted front by

that associated with its planar average.

The same argument applies ^to equations (16), (17) ^{so that, in the} non-local limit, ^{we are left}

with solving ^the following simplified ^front equations :

together ^with equations (20), (21) and with the consistence condition :

The solution of equation (23c) ^{is :}

with

It is immediately ^{seen that the} continuity conditions (20), (21) together with the constraint

that £’ # ⁰ along the curved part except ^{at its ends} impose that q ^be real, 0

^{0 and :}

That is, ^{we find that the} length ^c of each curved part is fixed. Note that, ^as expected, q ^{is the}

wavevector of the neutral mode (more precisely, ^{of that} with q larger than the critical value

qMS) of the classical MS problem with surface tension Fo ⁱⁿ the q > qMS limit where space modulations of the diffusion field can be neglected.

From equation (26), and since A > 2 TT / q, ^{our non-local} approximation (22) ^{is valid} only

when

where Qcs = 2 D/Vcs ; V cs == V cs (G) = DG 1 1 mL dC 1 is the constitutional supercooling instability threshold, ^to which the MS one reduces when qms » 1, i.e. when the system ^is

highly ^{non-local at} the bifurcation. do

T M( U + U ">0+ IL 1 ML AC | 1 ^{is the} capillary length ^of

our problem. ^Note that condition (28) precisely guarantees that q > qMS, i.e. that the

approximate expression (26) ^{of the} wavevector of the neutral MS mode is justified.

So, ^{we cannot} predict exactly ^the position of the threshold which is only defined, ^{in the} present approximation, ^{to within an} uncertainty ^measured by (do/fcs)1I3. ^{However, typically,}

for chemical diffusion, do ^{cannot exceed a} few hundred angstrôms, ^while Vcs lies in the 1-

10 u/sec range, ^so that (do/fcs)1I3 is ^{of order 10-1 1 at most. So, although our non-local}

assumption ^{forbids us to} analyze the mathematical details of the « bifurcation », the restrictions that it implies ^{should be} of very little consequence for the analysis ^of experiments.

From equations (24), (25), ^{one obtains :}

We are thus left with the five equations (20), (23a, b) (27) relating ^{the six} unknowns À, a, b,

e (1), ^{e (1), A -}

That is, ^{one of these} quantities, ^{which we choose to be} À, ^{remains a free} parameter. ^The lengths ^and heights ^of the faceted regions ^{of a} profile ^of wavelength ^{À are then} given by :

with, ^of course,

Note that, ^from equations (30), (31), a :::. b, ^{i.e. the hot} (quasi-convex) ^{facet is} longer ^than

the cold (quasi-concave) ^one, and the average displacement of the deformed front from the infinite facet position (JI)

² 7T’F(V)/qÀjL ^is positive. These features of course result from the fact that, ^{due to kinetic} effects, the curved parts of the front grow ^more

easily

^{than the} facets, which favors the hot against the cold facet parts of the crenellated profile.

The deformation amplitude, ^{for a} given ^{À, is :}

It is seen on equation (32) ^that (e, - e- ) ^{is a} decreasing function of A. In the range of

validity ^{of our} calculation (À 1 ), it remains larger than the limit value 2 F / (2 - g ). ^Since Dry’ - do LBu’ / f ( u 0 + o- 0 ") - 10- 4 LBu’ / u 0 this limit can be approached ^rather closely ^for

materials with slow facet kinetics. The amplitude ^defined by equation (32) diverges ^{when À}

tends towards its minimum value 2 TT /q, i.e. when the facets become vanishingly small. This

can be understood in the following way : when a, b -->+ 0, ^the profile reduces almost

everywhere ^{to a cosine} deformation, ^{i.e. to} the standard MS linear mode. However, ^the difference between our problem and the standard MS one lies in the fact that each zero-slope region ^{of the} profile, ^{even if its} length vanishes, ^{costs a finite} capillary energy ^oc 0 y’. ^When À --+ À min’ ^this plays the role of a finite driving ^term for the relevant MS mode, ^whose amplitude, since this mode is neutral, blows up. This divergence ^{is of course an} artefact of our

linearization of both the curvature and the diffusion terms in equations ((1), (3)). ^{So, the}

above results are valid only when non-linear corrections are negligible, i.e. when :

where al is the coefficient of the third order term of the amplitude expansion of the standard MS problem [2, 3]. ^{Note that, due to} the above mentioned smallness of 0 y’, ^{this is not a} very

stringent ^condition.

4. Linear stability of the crenellated front.

We now want to analyze ^the stability of the continuum of solutions determined above with respect ^{to small} perturbations. Modulations of such periodic structures can be classified into

amplitude ^and phase ^modes. Usually (i. e. , for the solidification problem, ^{in the case of} rough interfaces), phase ^modes correspond ^to space modulations of the wavelength ^{À, while} amplitude ^{ones describe} perturbations of the basic structure at constant À. Here, ^{due to the} presence of facets, ^{a new class of}

optical » phase modes should appear, corresponding roughly ^to modulations of the relative lengths of hot and cold facets at constant À. We will

only study ^{here the} amplitude ^modes, since it will appear that the crenellated structures are

unstable against ^such perturbations.

That is, ^we look for the time evolution of an infinitesimal deviation 8( (x, t ) ^{from the} stationary solution (x) = (0) + (l)(x), ^{where be has the same} wavelength ^{and the same}

facet lengths as 03BE (x). That linear eigenmodes satisfying ^these constraints exist is of course not obvious and, indeed, only ^{true, as} will appear in the following, ⁱⁿ the non-local limit and when the facets are much longer than the curved front regions.

Again, ^{we linearize} the, ^now time-dependent, ^diffusion problem, ^{and then} repeat ^the procedure (Eqs. (12), (13)) leading ^{to the} quasi-linearized ^front equation. ^{We moreover}

assume, and will check later, ^{that the} quasi-static approximation [16], ^{which amounts to}

neglecting ^the eC/et ^{term in} equation (1), ^{is valid.} Setting Z(x, t) = e (1)(x) ⁺ 8( (x, t), ^one easily finds that the dynamical ^front equation ^which generalizes equation (14) ^{now reads :}

where :

and Zn (t )

azn/at.

We are looking ^{for a} pure amplitude mode, ^{that is a} deviation from the stationary ^solution

which leaves the average position of the front invariant, i.e. such that :

As appears in equations (33), (35), ^the dynamics of front deviations is controlled by ^two

mechanisms :

-

facet kinetics, ^which gives ^{rise to the} 2VY’ /2 ^{term in} equation (33),

-

diffusion, ^which gives ^{rise to the} non-stationary ^terms proportional ^to Zn

stn ⁱⁿ

jé (Eq. (35)). ^{It is of course} necessary ^to compare the importance of this effect with that of the kinetic one, the strength of which varies widely ^{with V and with the} microscopic ^nature (e.g.

dislocation spirals, ^{or 2-D} nucleation) ^{of the} growth ^mechanism.

We have not been able to build a complete solution of the set of dynamical ^front equations generated by equations ((33), (35)), ^{i.e. to} get ^a general consistent solution for the infinite set of d03BEn (t ). ^{We therefore limit ourselves to the} simple ^{case where the facet} half-lengths ^{a, b are}

much larger than the x-extension 7r /q of the curved regions of the front. In this _case, the contribution of these curved regions ^to the diffusion term can safely ^be neglected. ^That is, ⁱⁿ

jé, ^{Z can be} approximated by :

where, in order for (36) ^{to be} satisfied :

That is :

Plugging expression (39) ^into equation (33), neglecting, ^{for the} same reason as in the

stationary ^{case, the} Zn ^{terms in} fC, integrating equation (33) along ^{the hot} facet, ^and

subtracting ^{out the} stationary ^facet equation (23b), ^{we obtain :}

with :

The same procedure, ^when applied ^{to the cold} facet, generates ^an equation ^for d03BE_ (t ). ^{It is} easily ^checked that, ^{when use} is made of equation (38), ^this degenerates ^with equation (40).

That is, ^to lowest order in the ratios 7T /qa, 7T /qb, ^we find that the amplitude ^{mode is}

described by :

where :

F ( V ) ^is proportional ^to the kinetic undercooling (Eq. (4)), that is increases with V. So, w (Eq. 43)) ^is always positive, ^{and we} find that the stationary crenellated states are unstable

against amplitude fluctuations.

Of course, the above calculation of the amplitude growth ^{rate is} only ^a lower order

estimate, ^{since it} ignores completely ^the problem ^{of the} dynamical shape of the curved parts, which would come into play ^{at the next} step ^{of an} expansion ^{in 7T} /qa, 7T /qb, ^{and would} certainly lead, ^{in a more elaborate} calculation, ^to small variations of the facet lengths.

5. Conclusion.

Inspite of its limitations (small amplitude ^and strong non-locality approximations) ^{the above} analysis permits ^{to draw a first} picture ^{of the} destabilization of a faceted directional solidification front.

Two main results emerge from ^our calculation :

(i) ^small amplitude crenellated stationary solutions, characteristic of faceted fronts, ^exist beyond ^a threshold V

V cs (G ) . (1 ^{+ o} «do/fcs)1I3» ^{which is, in practice,} the usual MS

one.

There is, for each value of V beyond ^this threshold, ^a continuum of such ^states. Our

approximations, although they ^{do not} permit ^{to locate} precisely the limits Àmin, Àmax ^{of this} continuum, ^{enable us to} determine the front shapes for space periods À ^{such that} (in physical variables)

(ii) these crenellated solutions are unstable against ^small amplitude fluctuations. This entails that, ^{for V}

Vcs, ^the planar front is unstable against ^finite amplitude deformations.

These, ^of course, ^cannot be treated within our approximations, ^{but it can} reasonably ^be guessed ^{that the} corresponding stable branch of stationary ^states corresponds ^to globally

cellular fronts with faceted regions, analogous ^{to what is observed on} faceted dendrites [17].

The crenellated solutions must be

jumped ^over

by ^{the front} configuration in order for cellular shapes ^to be reached. So, ^we predict that the facet to cell transition should exhibit a

very large velocity hysteresis.

Whether or not this transition can be termed a bifurcation is not clear to us, since no

linearization of the basic facet state is legitimate. ^Our calculation does not predict any upper

velocity threshold for crenellated states. However, ^{we cannot} describe the high velocity regime since, ^{when f}

² D/V decreases, ^{our non-local} approximation fails. In particular, ^we

cannot exclude that, ^{in our} model, ^there might ^{be no} such upper threshold. However, ^we have assumed here that the facet is perfect, ^i.e. that the Wulff plot ^{exhibits a} perfect cusp whatever the velocity. ^This approximation becomes incorrect at high growth ^{rates, where the}

cusp is rounded-off by ^kinetic roughening [12]. ^{In real} systems, this effect will always ^come

into play ^to produce a cross-over between the regime analyzed ^{here and a} high-V ^{one where}

the standard MS analysis ^is, again, valid, giving ^{rise to an} ordinary inverted bifurcation.

Clearly, ^the deeper ^the equilibrium cusp, i. e. the larger ^the step energy relative ^to the facet, the higher ^the velocity ^at which this cross-over occurs.

It is of course essential at this stage ^that systematic quantitative experimental ^{studies of}

directional solidification morphologies ^of transparent faceting materials be developed. ^{On the}

basis of the present study, ^we believe that they ^{should aim} primarily ^at the observation of the

anomalously large hysteresis predicted ^above.

Let us also remark that, ^as appears ^on equation (43), large ^{values of} VF’ (V ) ^{lead to small} amplification ^{rates for} amplitude fluctuations. That is, ^{as is} reasonable, strongly

^{locked »}

facets correspond ^{to slow} amplitude dynamics. So, ^it might ^{not be excluded} that, ⁱⁿ strongly faceting materials, crenellated states - possibly triggered by ^{an external} perturbation ^{- be}

observable as transients.

References

[1] ^{MULLINS W. W., SEKERKA R. F., J.} Appl. Phys. ³⁵ (1964) ^444.

[2] ^{WOLLKIND D.} J., ^{SEGEL L.} A., Philos. Trans. R. Soc. London 51 (1970) ^268.

[3] ^{CAROLI B., CAROLI C., ROULET B., J.} Phys. ^{France 43} (1982) ^1767.

[4] de CHEVEIGNE S., GUTHMANN C., LEBRUN M. M., ^J. Phys. ^{France 47} (1986) ^2095.

[5] See for instance : KESSLER D. A., ^LEVINE H., Preprint INLS 1007 and references therein.

[6] ^{SAITO Y., MISBAH C.,} MÜLLER-KRUMBHAAR H., Proc. of the 3rd UC Conf. on Statistical Mechanics Univ. Calif. Davis (1988) ^{Ed. C.} Garrod, ^{to be} published.

[7] CORIELL S. R., SEKERKA R. F., ^J. Cryst. ^{Growth 34} (1976) ^157.

[8] MCFADDEN G. B., CORIELL S. R., ^{SEKERKA R.} F., ^J. Cryst. ^{Growth 91} (1988) ^180.

[9] This excludes the case where facetted growth is controlled by ^kinetics only (see UWAHA, M., ^J.

Cryst. ^{Growth 80} (1987) 84).

[10] CHERNOV A. A., NISHINAGA T., Morphology ^of Crystals, ^{Part A, Ed. I.} Sunagawa (Terra ^Scient.

Publ. Co.) 1987, p. 207.

[11] ^{BEN AMAR} M., ^POMEAU Y., Europhys. ^{Lett. 6} (1988) ^609.

[12] ^{We assume} here that the melting temperature ^{is far} enough ^{below the} roughening ^{one for us to} neglect any dynamical roughening ^effect (see ^{NOZIERES P., GALLET F., J.} Phys. ^{France 48} (1987) 353).

[13] ^NOZIERES P., Cours au Collège ^{de France} (1984), unpublished ;

See also CAHN J. W., ^{HOFFMAN D.} W., ^{Acta Metall. 22} (1974) ^1205.

[14] ^{TILLER W.} A., The Art and Science of Growing Crystals, Ed. J.J. Gilman (J. Wiley & Sons, ^New

York) ^1966.

[15] ^{Note that, in the} highly non-local limit we will restrict to here (03BB ~ 1), ^{this is a much} more severe

limitation for our one-sided model than for the symmetric ^one (see ^{LANGER J. S., TURSKI}

L. A., Acta Metall. 25 (1977) 1115). Indeed, ^{it can} be checked that the ratio of the coefficients a1 of the cubic ^terms in the amplitude expansion ^{are such} that, in this limit (see ^Ref. [3]) :

aOS1/aSym1 ~ 03BB.

[16] ^{LANGER J. S.,} Rev. Mod. Phys. ⁵² (1980) ^1.

[17] ^{MAURER J., BOUISSOU P., PERRIN B., TABELING P., Europhys. Lett. 8} (1989) ^67.