Cellular instabilities in directional solidification

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Cellular instabilities in directional solidification

S. de Cheveigné, C. Guthmann, M.M. Lebrun

To cite this version:

S. de Cheveigné, C. Guthmann, M.M. Lebrun. Cellular instabilities in directional solidification. Jour-

nal de Physique, 1986, 47 (12), pp.2095-2103. �10.1051/jphys:0198600470120209500�. �jpa-00210403�

Cellular instabilities in directional solidification

S. de Cheveigné, C. Guthmann and M. M. Lebrun

Groupe ^de Physique des Solides de l’Ecole Normale Supérieure, Université Paris VII, ^Tour 23, ^{2 Pl.} Jussieu, 75251 Paris Cedex 05, ^France

(Requ ^{le 3} juillet 1986, accepté le 28 aoat 1986)

Résumé.

Nous avons étudié la morphologie ^de l’interface solide-liquide lors de la solidification directionnelle d’échantillons minces (5-150 03BCm) ^d’un alliage binaire dilué (CBr4/~ ^{0,1 % Br2). Au-dessus}

d’une vitesse de tirage critique, ^à concentration de soluté et gradient ^de températures ^données, l’interface _qui

était plane ^au départ, présente ^une déformation cellulaire périodique. ^La bifurcation d’un front plan ^{à un front}

cellulaire est sous-critique. ^{La vitesse de} tirage critique ^{est fortement} dépendante ^de l’épaisseur ^de

l’échantillon : ^{on ne} peut pas considérer ^le problème ^comme bi-dimensionnel. Nous avons également ^mesuré

les longueurs ^{d’onde au seuil et} au-dessus ainsi que les temps ^de réponse. ^Nous comparons ^{ces résultats aux}

prédictions théoriques existantes.

Abstract.

We have studied the morphology ^{of the} solid-liquid ^interface during the directional solidification of thin (5-150 03BCm) samples ^{of a dilute} binary mixture (CBr4/~ ^{0.1 % Br2) . Above a critical pulling speed,}

for a given temperature gradient and solute concentration, ^the interface, initially planar, breaks down into a

periodic ^cellular pattern. The bifurcation from a planar ^{to a} cellular solidification front is sub-critical. The critical pulling speed ^is strongly dependent ^{on the thickness of the} sample : ^the problem ^{cannot be considered}

two-dimensional. We have also measured wavelengths at, and above the threshold and examined response times. These results are compared ^to existing theoretical predictions.

Classification

Physics ^Abstracts

64.70D

81.30D

1. Introduction.

The existence of morphological instabilities [1] ^defor- ming ^the solid-liquid interface of directionally ^solidi-

fied dilute binary alloys ^{is a} phenomenon ^well

known to metallurgists [2]. Samples ^are pulled ^{at a}

certain velocity ^{V in a fixed} temperature gradient ^G

and are thus progressively solidified. If the equili-

brium concentration of the solute is different in the

liquid and in the solid, ^{an excess} (in ^{the case of the} compounds ^studied here) ^{or a} lack of solute builds up in front of the moving interface and must be evacuated to allow solidification to progress. Above

a threshold pulling speed, ^for given temperature gradient ^and concentration, the interface becomes unstable and breaks down into a regular, periodic pattern (Fig. 1).

Metallic alloys, because of their opacity, ^{are not}

the most practical systems ^{on which to} study ^the

behaviour of such instabilities : samples ^{have to be} quenched ^{or decanted to} allow the observation of a

solid-liquid interface which may be affected by ^this

treatment. The study ^{of thin} samples ^of transparent organic compounds, ^first suggested by ^{Jackson and}

Hunt [3] ⁱⁿ 1966, allows direct observation, ^under

the microscope, ^{of the} dynamics ^{of the} solid-liquid

interface during solidification. Some such

Fig. ^1.

Well-developed ^cells (CBr4’ V = 3V+c, .

A - 50 JLm).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470120209500

2096

compounds, ⁱⁿ general plastic crystals, ^{present a} rough ^interface which does not facet, and in that respect they behave like metals. The most frequently

studied are tetrabromomethane CBr4 ^{and succi-}

nonitrile (CH2CN ) ^2.

In the present study ^{we have tried to} pinpoint ^the

various factors which affect first the onset of cellular

instabilities, ^{then the} periodicity ^{of the} resulting pattern ^{of interface} deformation, ^{in thin} (5-150 pm) samples. Most of the experiments ^were performed

with CBr4, but the results were generally ^checked

with succinonitrile, ^both compounds being ^{used as} supplied. Impurity concentrations were therefore

fixed, pulling speeds, temperature gradient, ^and sample thicknesses were varied. We did not consider the problem ^of dendrite formation during directional

solidification, extensively studied in transparent organic ^materials [4-6]. ^This phenomenon ^takes place well above the threshold at which morphologi-

cal instabilities first appear (above ^{about 10 times}

the threshold pulling speed, ^{for a} given temperature gradient ^and impurity concentration). ^{On the}

contrary, ^most of the work described here was done to gain ^better knowledge ^{of the} behaviour of the system at, ^or near, the threshold.

2. Theory.

A number of authors have attempted ^to give ^a dynamical description ^{of the} phenomena occuring during directional solidification of dilute binary alloys ^{in thin} samples, ^{when the} system ^is definitely

out of equilibrium. The linear stability ^{of the} planar

solution for the solid-liquid ^{interface was} originally

studied by Mullins and Sekerka [7] ^{in 1964.} They

described the balance between the destabilizing

effect of the accumulation of solute in front of the

solid-liquid interface and the stabilizing effects of the temperature gradient and of the surface tension.

Wollkind and Segel [8] reformulated this analysis

and went beyond, giving ^a non-linear stability analy-

sis using perturbation methods valid only ^{close to the}

threshold. Caroli, Caroli and Roulet [9] ^extended

this analysis ^{to the case} where diffusion in the solid

phase ^{is not} negligible. ^{A number of numerical}

calculations have also been performed [10-21]. ^All

these theories tacitly suppose that the sample may be considered infinite in the direction perpendicular ^to

the plane ^{of the} sample, i.e. that the deformations

are one dimensional along ^the interface. We shall

see below that this is not the case.

Let us recall ^some of these results. Diffusion

equations for heat and solute are written in both the solid and the liquid phases. ^{The interface conditions}

are the source of the non-linearity ^{of the} problem,

because the temperature and concentration fields

depend non-linearly ^{on the} shape ^{of the} interface. In the linear stability analysis, ^first performed by ^Mul-

lins and Sekerka [7], ^{a small} sinusoidal perturbation

is superimposed ^{on a} planar interface, ^{and the}

conditions under which it is amplified ^or damped ^are

determined. ^{At low} velocities the equations defining

neutral stability ^{can be} expanded ^{in terms of the}

wavenumber, which is also small, ^{and one in fact} obtains, ^to the lowest order, ^{a modified} (1) ^version

of what is known as the « constitutional supercooling

criterion ^» [22] :

where G, is the critical temperature gradient, Vc ^the

critical pulling speed, Coo ^{the mean solute concentra-} tion, ^{m the} liquidus slope, k ^the partition coefficient and DL the solute diffusion coefficient in the liquid.

The ratio v

V C 00 /G ^{appears as} the control parameter.

Under the conditions of the present experiments,

the second order correction to relation (1) ^{is at most}

of a few percent. ^{It can be} important, particularly ^at higher pulling speeds [23, 24].

To the same order of approximation, ^{one can also}

write the wavelength A MS ^{of the «} critical mode »

(i.e. the first neutrally ^stable mode encountered when the control parameter ^is increased) :

where q is characteristic of the material :

y is the solid liquid ^surface tension, C the latent heat,

p the density. ^The expression ^for A ms is obtained

assuming ^an infinitesimally ^small amplitude perturba-

tion.

A non-linear analysis ^{can describe} analytically ^the

way in which ^an initially ^instable perturbation ^evol-

ves close to the threshold, using ^{an «} amplitude equation » :

The various possible evolutions ^are discussed ⁱⁿ references [8] ^and [9]. ^{Under the} present experimen-

tal conditions (low velocity, ^a partition coefficient k 0.45 and a temperature distribution decoupled

from the instability because it is imposed ^{from the}

outside by ^the glass ^cell [25]) ^a subcritical bifurcation is predicted - ^{and as we} shall discuss below, ^{it is} indeed observed. This means that the’ amplitude ^of

the deformation is never infinitesimal ^so the ampli-

tude equation ^{cannot be used to make further} predictions:

One would nevertheless like to have predictions concerning ^{the cell} shape ^{and the} wavelength ^at

(1) ^{Mullins and Sekerka’s result} replaces ^the tempera-

ture gradient by the average of its values in ^the solid and in the liquid. ^{In the} present experiments ^the gradient ^is imposed externally ^{and has the same} value in the solid and in the liquid. ^{In that case, as in the case of} equal ^thermal

conductivities in both phases, Mullins and Sekerka’s result

reduces to the ^« constitutional supercooling ^{criterion ».}

velocities above the threshold. To answer this ques-

tion, ^a certain number of numerical calculations have been performed. The first models relied on the

amplitude equation ^{and the} hypothesis ^of infinitesi-

mally ^small deformations. Langer [10,11] developed

a « symmetrical » ^{model -} equal ^{solute diffusion}

coefficients in the solid and in the liquid

which, although ^{it was not} very realistic, gave ^a qualitative picture ^{of the} phenomena. ^For example, ^{with this} model, ^Dee [12], assuming ^that pattern ^selection

takes place by propagation along ^the front of the

« wall » between two regions of different cellular

wavelengths, ^{finds a} well-defined wavelength ^that

decreases with increasing velocity. On the other hand he does not predict ^cell shapes.

Using ^{a more realistic «} one-sided » model

-

which neglects solute diffusion in the liquid

^but

still using ^the amplitude equation, Kerszberg [13-15]

derived the equation ^{of movement of a deformed}

front. He obtained quite ^a good representation ^of

the shape ^{of a} cellular interface. The wavelength ^was

not uniquely determined but is restricted to an

interval.

To approach ^the problem of wavenumber selec-

tion, Kerszberg ^{studied the effect of Gaussian white}

noise on the dynamics ^{of the} interface, again by ^a

numerical simulation. A unique wavevector is selec- ted in this ^manner. The noise induces the disappea-

rance or the division of cells in a manner very similar to what is observed experimentally (compare Fig. ⁴

of Ref. [15] ^to Fig. ^{4 of the} present article). ^The

value of k ^« selected ^» by the noise is given ^{as a}

function of the constraint parameter v ^a V/DL ^G.

Unfortunately his calculation does not go far beyond

the threshold ( A v / v == 5 X 10-3) . ^{The selected}

wavelength ^{increases with} increasing pulling speed,

at a given temperature gradient.

McFadden and Coriell [16] apply ^{a one-sided}

model to an Al-Ag alloy, ^without using ^the ampli-

tude equation. They ^{use the} solute concentration

Coo ^as the control parameter. (It should be noted that for the experimentalist, ^{V is a more « natural »}

parameter ^{since it can be varied} rapidly ^and easily).

Their predictions concerning ^the wavelength ^are particularly pertinent ^{to the case of a} subcritical bifurcation : a relatively large amplitude perturba-

tion of wavelength smaller than that of the most unstable linear mode ^can develop. It appears explici- tly, ^as expected, that the critical wavelength ^calcula-

ted by the linear analysis ^{is not} valid in the case of a

subcritical bifurcation. Above the threshold ^concen- tration C * , ^the wavelength ^evolves towards smaller values. When values larger ^{than the most unstable}

linear wavelength A ^{* are tested,} they ^{are shown to} decay ^{and to be} replaced by their first harmonic at half wavelength which is smaller than A *.

Ungar ^{and Brown} [17-20] also avoid the amplitude equation. They ^{use -a one-sided} mode and suppose

equal ^thermal conductivities in the solid and in the

liquid. They work with the temperature gradient ^{as a}

control variable at fixed pulling speed (once again,

these are not the most practical experimental ^condi-

tions). Using finite element methods, they predict

the presence of ^two successive bifurcations, ^{the first}

to a deformed interface with a periodicity ^{À, the}

second to a periodicity A/2, but the second bifurca- tion is predicted ^{to be so close to the first} (AV/V - 191,,) ^{as not to} be visible experimentally.

The predictions ^of Ungar ^{and Brown} [20] ^concer- ning deep ^cell morphology well above the threshold

are particularly interesting. They ^{are based on the} hypothesis ^{that the} wavelength ^{does not} change ^with increasing ^constant

^a hypothesis contrary ^to expe- rimental results. Nevertheless, the evolution of cell

shape is very well described (compare ^for example Fig. 10 ^{of Ref.} [20] ^to Fig. 1 ^{of the} present ^article- both are at V - 3-5 Vc.)

Very recently, ^Karma [21] ^has applied ^a boundary

model to simultaneously obtain the shape ^{and wave-} length ^{of well} developed ^cells. This work is still in progress, but the preliminary ^{results seem} promising.

3. Experimental.

Most of the work presented ^{here was done with}

tetrabromomethane CBr4 . ^Some experiments

were also carried out on succinonitrile (CH2CN) ^2’

for comparison. Succinonitrile has been extensively

characterized both when pure and mixed with ^ace- tone, by ^Glicksman [4]. ^{Table I} gives ^data concerning CBr4. ^{We use the} compound ^as provided by Fluka,

Table 1.

Physico-chemical ^constants for CBr4.

without further purification. ^{The main} impurity ^is given ^as Br2. ^We performed Differential Scanning Calorimetry measurements on the material. This

provided ^{us with a curve of the} energy absorbed

versus temperature during melting. By fitting ^succes-

sive melted fractions (from 1/8 ^to 1/3) ^versus

temperature ^{to the van t’Hoff} formula, ^{we obtained}

an impurity concentration _Coo

0.12 ± 0.02 mol. %,

a liquidus slope ^m

^{2.9 ± 0.2 K/mol. % and a} parti-

tion coefficient k

0.16 ± 0.01. A melting tempera-

ture of 93.3 ± ^{0.2 °C} for the pure compound ^was calculated. ^The phase diagram ^for Br2 ⁱⁿ CBr4 reported ^{in reference 26} gives ^m

^2.73

± 0.2 K/mol. % with TM

^{92.5 °C. We also} perfor-

2098

med the same measurements and calculations on

succinonitrile (Koch-light ^and Eastman) ; they gave Coo

0.45 ± 0.05 mol. %, ^m

^{2.20 ±} 0.05 K/mol.

% and k

0.12 ± 0.01. The melting temperature ^for the pure compound ^was calculated as 58.1 ± 0.3 °C.

The diffusion coefficient for Br2 ^is missing ^{but we}

shall estimate below a value of 1.2 ± 0.4 x

10- 5 cm2/s.

The pulling stage ^is essentially ^{similar to that} proposed by Jackson and Hunt [3]. ^Cell preparation

is described in reference [25] ; sample thicknesses

are measured to within ± 2-3 _pm and are uniform to within ± 5 _pm. The temperature ^{of the} sample ^is imposed by ^the glass cell which is a far better conductor of heat than the sample itself. Particular

care was taken to avoid residual vertical temperature gradients ^which strongly affect the experimental

results (see below).

The experiments ^are video-taped which allows the

pulling speed ^{to be measured} precisely (± 0.1 Rm/s).

Wavelengths ^are averages ^{over at} least 10 cells.

4. Results and discussion.

Before presenting ^our experimental results, ^we

believe that some preliminary ^{remarks are} necessary.

First, ^when studying dynamical phenomena, ^where

fluctuations play ^{such an} important role, ^{a certain}

amount of dispersion in the results is inevitable.

Another point is that all the important parameters ^of the problem ^{have not} yet been isolated and can

easily ^{be missed}

^{this was the case of the} sample thickness, the effect of which is decribed below. Just

as in the case of hydrodynamic instabilities, ^nume-

rous and repeated experiments ^{must be} performed

before definite conclusions can be drawn.

4.1 RESPONSE TIMES.

In the present experiments

we have tried to observe the solid-liquid interface in

a stationary ^{state and this can take a} long ^{time. To}

illustrate this we have plotted the time it takes the interface to begin ^{to deform} visibly ^{after the} pulling speed ^is suddenly changed ^{from a zero value} (the

time it takes to reach a steady ^{state is even} longer).

Figure 2 shows the curves of T versus the final speed

V for CBr4 ; ^the points ^we obtain for succinonitrile fit the same curve, ^as do most of the response times

reported by Somboonsuk ^et al. [27] ^{for a} succinonitri- le-acetone solution. The final speed ^{is the} pertinent parameter since it is at that speed ^{that the concentra-}

tion profile adjusts.

The response time in fact contains two terms. The

first, ^a solute diffusion term, ^is roughly the time it takes to establish the steady ^state concentration

profile ^after begining ^to solidify ^a sample. ^{This can}

be estimated as [22] :

The second term is due to the thermal response of

Fig. ^2.

Response ^{time versus} pulling speed (see text).

the glass ^{cell and is} expected ^{to be} preponderant ^at higher pulling speeds. ^To explain this, ^{the cell can be}

considered as a heat conducting ^bar moving ^at velocity ^{V across the two heat} reservoirs (Fig. 3a). ^In

the laboratory ^{frame, the} temperature 7y ^{at z} satisfies, ^{in the} stationary ^{state, the} equation :

where DG is the thermal diffusivity ^of glass. ^With Ty(O) = T, ^and Ty(L) = T2, the solution is :

Fig. ^3.

a) ^Schematic representation ^{of the cell} moving ^at velocity ^{V across} the heat reservoirs. Tl > T M > T2;

b) Temperature profiles ^{for V}

^{0 and V} > 0.

i.e. the temperature profile ^{is no} longer ^{linear as it is}

when V

0 (Fig. 3b) and the average temperature of the cell has increased. The term in the response time which corresponds ^to the time it takes the cell to fit the new thermal conditions is preponderent ^at high pulling speeds, It is the value at which ^T levels off: about 10 s in the present ^case. The thermal component ^is negligible ^at speeds lower than

~

10 f.L/s. ^{When that term} is subtracted from the

experimental ^{values of T} (Fig. 2) the result varies as

V-2 . ^The proportionality factor is

2-3 x 10-5

cm2/s which is less than the value of D/k ^estimated

as 1.2 x 10-5/0.16

^{7.5 x 10-5} cm2/s. ^{One must}

however remember that T is not sufficiently precisely defined, experimentally ^or theoretically, ^{to allow}

any quantitative conclusions to be drawn.

The important result is that at pulling speeds ^of

10 R/s, ^{one must} wait about 60 s for the system ^to react, ^{at 1} R/s about 30 min and extrapolating ^to

0.1 ii/s, about 16 hours ! Such times make extensive and repeated ^studies extremely difficult. That is why

we found it preferable ^to work under conditions (in particular ^{at low} concentrations) ^{where critical} speeds ^{are above 1 R/s.}

4.2 A SUBCRITICAL BIFURCATION.

We have

previously ^shown [25] ^{that the} bifurcation from a

planar ^{to a cellular} morphology of the interface is

subcritical, ^{i.e. the critical} pulling speed V+c ^for increasing speeds ^is greater than the critical value

Vy ^for decreasing speeds. Figure 4 shows the ampli-

tude of the deformations of the interface (measured

from the cell tip ^{to the} point where the cell sides

meet) ^{versus the} pulling speed. ^Above V’ ^a finite amplitude deformation sets in. As mentioned above,

the non-linear analysis ^{of the} problem [8, 9] predicts

this result which is of practical consequence ^for calculations. It means that the amplitude ^{of the}

deformations is never infinitesimally small, ^{so the} amplitude equation ^{cannot be} used above the thres- hold. Note that the threshold values are somewhat lower than those given in reference [25] : this is due to the presence of ^a residual vertical gradient ^{in the}

Fig. ^4.

Amplitude ^{of the} periodic deformation versus

pulling speed (e

^{30 J..Lm, G} =120°/cm). ^Full segments : increasing speed. ^Dashed segments: decreasing speed.

Curve : interpretation ^{in terms of a} subcritical bifurcation.

earlier experiments. The vertical temperature gra- dient produces ^an inclined interface which favorizes the macroscopic evacuation of solute along ^the

bottom of the cell, since the concentration gradient

has a component parallel ^{to the} interface. This process, also observed with succinonitrile, ^{is suffi-}

cient to retard the ’destabilization of the interface.

According ^{to relation} (1), the critical speed predic-

ted by the linear theory (V ’ in the notation of the

previous paragraph) ^{should be} proportional ^{to the} temperature gradient G, ^{with a} slope ^{that is a}

function of the diffusion coefficient DL’ ^{We checked} (for relatively ^thick samples - ^see explanation below) that the linear relation holds and used it to estimate DL ^{= 1.2 ± 0.4 x 10- 5} cm2/s. This value is

quite reasonable and ^we shall use it for further calculations.

4.3 WAVELENGTH SELECTION.

Our experimental

values for the wavelength ^{versus the} pulling speed

are shown in figure ^{5, for two different} temperature gradients. (The samples considered were thicker than 50 _Rm to allow comparison with the thresholds

predicted by ^relation (1) - ^see explanation below).

We have drawn on the ^same figure ^{the neutral} stability ^curve deduced from the linear analysis

described above [7, 9]. Periodic deformations of

wavelengths within the band delimited by ^{the curve}

will not a priori relax, whereas those outside the band will, ^{but no} clear selection mechanism within the band has been described theoretically. Experi- mentally ^a wavelength ^is selected, ^{to within a few}

tens of percent (the ^{shaded area on the curve shows}

the maximum experimental dispersion). ^{The value}

Fig. ^5.

Wavelength ^versus pulling speed ^{for two}

different temperature gradients. The full line ^is the calculated neutral stability curve ; the points ^are experi-

mental. The shaded strip gives ^the experimental disper-

sion. (e > 50 03BCm; (a) ^{and full circles :} G= 120 7cm;

(b) ^and empty circles : G

70*/cm).

2100

of the critical wavelength A ms predicted by ^{the linear} analysis is shown for both values of the temperature gradient. ^{It is 2 to 3 times} greater than the threshold value A c ^observed experimentally (180 ^{pm versus 75-}

80 pm for G

70*/cm, 125 R ^{versus 50-55} pm for G = 120 7cm). This tends to confirm the existence of a subcritical bifurcation since the prediction ^made

under the hypothesis ^of infinitesimal deformations does not hold. The fact that we observe a critical

wavelength A c - A ms/2 ^{may mean that we are}

indeed observing the second bifurcation predicted by Ungar ^{and Brown} [16, 20].

The wavelength easily adjusts by ^{birth or death of}

cells (Fig. 6) ^{in a manner} very similar ^to that found

numerically by Kerszberg [15].

Above the threshold ^A varies roughly ^as V-’/2 (we ^{find a} functional dependence ^of V - o-’ ± 0. 1). ^This

dependence is consistent with predictions by

Karma [21]. ^{For a} symmetric ^{model the} system should evolve at constant

where 1

2 DL/V is the diffusion length ^and do

’Y T M/ Lco m ^{( 1- k ) is the} capillary length. ^{In the}

present case, ^{one can} write

u

6.6 x 10- 4/A 2 ^{V (A in J.Lm, V in Rm/s) .}

We find u - 5 - 6 X 10-3; ^Karma _predicts

at

3.1 x 10-2 (symetric model, k = 1).

Experimentally, ^{a ,} monotonous decrease in À ( V ) ^{as V-1/2 or} V-1/4 has often been reported

in literature, generally for three-dimensional metal

alloy samples (see ^reviews by ^Hunt [29] ^{and Klaren}

et al. [30]). ^{It is not} always ^{clear at} what distance above the threshold the experiments ^are performed.

For speeds ^{above = 3} Vc’ ^{Jin and} Purdy [31] report ^a

Fig. ^6.

Wavelength adjustment by birth and death of cells. Time increases from left to right.

h oc V - 1,,14 behaviour in a Fe-8 wt % Ni alloy. ^An exception ^{to this} monotonous variation ^was found in Pb alloys [30, 32] ^{where a} very slight ^{increase was}

first observed, ^followed by ^{a decrease at} higher pulling speeds. ^Trivedi [5] ^has reported ^{a pronoun-}

ced maximum in À ( V ) ^{for a} succinonitrile-acetone system, ^{but the} region studied is far above the threshold predicted by the constitutional supercoo-

ling criterion. The question of how the ^A ( v ) ^curves

near the cellular threshold and in the dendritic

region join each other is quite open, both theoreti-

cally ^and experimentally.

Relation (4) predicts ^{no direct} dependence ^{of A on}

the temperature gradient because the latter is not yet introduced into the model. We find A slightly ^lower

for the lower temperature gradient ^{at a} given speed (Fig. 5) but the shift is barely greater ^{than our} experimental dispersion. ^This tendency ^is reported

in the literature but at speeds well above the threshold. Various dependences ^{on the} temperature gradient G have been reported ^{for metal} alloys,

from G- 1/3 for Pb alloys [30, 32] ^{to G-1 in Al} alloys [33] (see ^Ref. [29] ^and [30] ^{for a review of} literature).

According ^{to Mason et al.} [32] ^the variables V and T

are not separable, ^{and the} exponent ^{of G} varies with V.

More work, ^{nearer the} threshold, ^and concerning

various systems, ^is clearly necessary ^to confirm the

experimental picture. ^{We insist once} again ^{on the} importance ^of doing experiments sufficiently slowly : ^{if a} pulling speed above the critical value is

applied suddenly ^{a transient} pattern ^of wavelength

smaller than the steady ^state wavelength is observed.

This effect, which has been mentioned by ^various

authors [27, 33-35], ^{can be a source of error in}

measurements of À, particularly ^{in the case of} high

response times, ^{i.e. low} pulling speeds.

4.4 EFFECT OF SAMPLE THICKNESS.

Our most

unexpected result is the effect of the thickness of the

sample. Samples thicknesses are of the order of tens of microns, compared ^to lengths ^{and widths of}

several centimeters, ^{and the} problem ^is usually

treated theoretically ^{as two} dimensional. In fact, sample thickness is an essential parameter : ^{the solid} liquid interface becomes less and less stable when the sample thickness decreases. Figure ⁷ gives ^the

dependence ^we observe for the critical pulling speed V; ^versus sample thickness, ^{for two} temperature gradients (70 ^and 120 °Icm). We have observed a

similar effect with succinonitrile.

Convection in the liquid could have been a

possible explanation ^for this effect. We have checked that none takes place, ^{at least on a time}

scale comparable ^{to that of the} phenomena ^under study, by introducing ^small (5 pm) particles ^{of silica} (of roughly ^{the same} density ^as CBr4). ^The particles

are definitely ^{mobile since in} one case we were able to observe their movement induced by Marangoni

effect on the liquid-gas interface of a bubble.

We believe that the solid-liquid meniscus is the

cause of this strong dependence ^{of the critical} pulling

speed ^{on the} sample thickness. As drawn in figure ^8,

Fig. ^7.

^Critical pulling speed V+c ^{versus sample}

thickness for two different temperature gradients : a) ^G =120°/cm ; b) ^G

^707cm.

Fig. ^8.

^{A schematic} representation ^{of the} solid-liquid

interface.

the interface is curved to assure mechanical equili-

brium at the point ^{of contact with the} glass. ^The

contact angle depends ^{on the} pulling speed ^because

the solute distribution along the interface varies.

The shape of the interface adjusts ^{to assure local} thermodynamic equilibrium (Gibbs-Thomson ^rela-

tion ; ^{see Ref.} [7, ^{8 or} 9]). ^The meniscus is not easily

visible in a true sample, ^but figure 9 shows the one

formed by ^the solid-liquid interface of succinonitrile

on the side of an optical ^cell (on ^a polished glass surface). The meniscus affects a distance of about

50 tL which is of the order of the sample ^thickness

range ^over which the threshold pulling speed ^{is seen}

to vary.

The presence of the meniscus, which deforms the interface in a manner that depends ^{on the} pulling speed, should have ^an effect similar to that of grain

boundaries described by Voorhees et al. [28] ^and by Ungar ^{and Brown} [18], i.e. the threshold is shifted to lower values. The description of the interface is in itself a non-linear problem ; ^{one has then to treat}

Fig. ^9.

Solid-liquid meniscus in succinonitrile in an

optical ^cell (full picture height ²⁶⁰ J.Lm).

the coupling of deformations in perpendicular pla-

nes. Caroli, Caroli and Roulet [37] ^{have calculated}

the meniscus analytically ^to the third order and then

performed ^{a linear} stability analysis ^{of the solid-} liquid ^interface (in ^a symmetric model). This calcula- tion is valid only. ^{for a} very weakly ^curved meniscus,

but it shows qualitatively that the effect of the curvature is to decrease the critical pulling speed ^for

thicknesses smaller than the cellular wavelength ;

the shift increases as thickness decreases.

A similar effect may well be responsible ^{for the}

unusual tip velocity

tip ^curvature relation obser- ved by Sawada and co-workers [38] ^{for dentritic} growth in very thin (5 J.Lm) samples.

Because of this dependence of the critical pulling speed ^{on the} sample thickness, ^{one must} be cautious in comparing with the value given by the « constitu- tional supercooling ^{criterion »} (Eq. (1)). ^{The rela-}

tion is expected ^to apply ^{to thick} samples ^{for which}

the effect of the meniscus is weak. It is for this

reason that we used the value of V+c ^{at which} V+c (e) levels off to calculate DL ^{and that we drew} figure ^{5 for} samples with thicknesses in the plateau region.

Note that the wavelength of the cellular deforma- tion of the solidification front gives ^the length ^scale

to which the thickness of the sample ^{should be} compared. Since the wavelength varies with the

pulling speed, ^a sample ^{is « thick » or « thin »} depending ^{on the} speed ^at which it is being pulled.

In the calculation performed by ^{Caroli et al.} [37] ^of

the effect of the sample ^{thickness on the critical} pulling speed, ÀMs is the scale which‘ appears. In fact the experimentally ^observed value Àc ^is probably ^the appropriate scale. For example ^{the curve of} V’

versus e given ⁱⁿ figure 7a intersects the curve of A versus V (given ^{in a} log-log diagram ⁱⁿ Fig. 5) ^at

about e

A

50 _pm. It is in that same region ^that

it saturates.

We have found that the samples ^become definitely

« thick _», and present ^several layers ^of cells, ^once

the wavelength becomes smaller than 1/2 to 1/3 of

2102

Fig. ^10.

Multiple layers ^{of cells in a thick} sample (e ^{= 170} ttm, k

³⁵ fJ.m, V

45 Rm/s). a) Focusing ^{on the} top ^of the sample ; b) Focusing ^on the bottom of the sample. ^{A vertical} temperature gradient staggers ^the layers. ^{Note the}

effect of cristal orientation observed at high pulling speeds.

the sample thickness. At that point, ^several layers ^of

cells are observed (Fig. 10) ^{and A is} larger ^than expected according ^to the relation A 2 V = constant.

Apparently, ^the wavelength ^{is not thickness} dependent. This is somewhat surprising ^{since one}

would expet the meniscus to increase in some way the « effective » curvature of the interface and

perhaps ^{cause A to decrease. If such an effect exist, it}

is weak. To a first approximation, Caroli et al.

predict theoretically ^that A MS ^{will not} be affected.

5. Conclusion.

In conclusion, ^we have studied the nature of the bifurcation from a planar ^{to a} cellularly ^deformed

interface and have found it to be subcritical, ^as

predicted by the non-linear analysis ^{of the} problem.

This result is confirmed by the values found for the critical wavelength. Response ^{times are shown to}

become very long ^{at low} pulling speeds ^{and to be a} possible ^{source of} experimental ^{error. We have}

discovered that the critical pulling speed ^is strongly dependent ^{on the} sample thickness. This last result should be stressed : the sample ^cannot be considered two-dimensional.

Acknowledgments.

It is a pleasure ^{to thank B. Caroli, C. Caroli, B.}

Roulet and C. Misbah for frequent ^and stimulating

discussions.

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