Interface dynamics and anisotropy effects in directional solidification

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Interface dynamics and anisotropy effects in directional solidification

S. de Cheveigné, C. Guthmann

To cite this version:

S. de Cheveigné, C. Guthmann. Interface dynamics and anisotropy effects in directional solidification.

Journal de Physique I, EDP Sciences, 1992, 2 (2), pp.193-205. �10.1051/jp1:1992133�. �jpa-00246472�

Classification

Physics

Abstracts

81.30 61.50 47.20

Interface dynamics and anisotropy ^{effects in} directional solidification

S, de

CheVeign6

and C. Guthmann

Groupe

^de

Physique

^des Solides

(*),

Tour23, 2 PI. Jussieu, 75251 Paris Cedex 05, France

(Received ⁶

May

1991, revised 4 October I991,

accepted

28 October 1991)

Rksum4. Une 6tude de la

dynarnique

^de l'interface cellulaire _en solidification directionnelle d'un

alliage

binaire dilu6 de CBr4 nous a

perrnis

d'observer _un certain nombre d'instabilit6s secondaires

(propagation

de cellules incl1n6es, modes

optiques,

cellules anorrnales, etc.) dont certaines 6taient bien _connues et traditionnellement attribu£es h des effets de

l'anisotropie

cristalline.

Cependant,

la

plupart

de _ces instabilitds sont observables dons _une

exp6rience qui

est

l'analogue hydrodynamique

de la solidification directionnelle, la

digitation Visqueuse dirig6e

_ou

instabilit6 de

l'imprirneur

(M. RABAUD, S. MICHALLAND and Y. COUDER, Phys. Rev. Lent. 64 (1990) 184). Cette

analogie

_{va nous} perrnettre ^{de discuter (es r61es}

respectifs

de

l'anisotropie

cristalline et de la

dynarnique

_propre aux

systbmes

uni-dimensionnels.

Abstract. A

study

of the

dynamics

of the cellular

solid-liquid

interface in directional solidification of _a dilute

CBr4 alloy

has allowed _us to observe _a number of secondary instabilities

(travelling

states, optical modes, anomalous cells, etc.) _some of which _were well known and

traditionally

attributed to the effect of

crystalline anisotropy.

However, most of these instabilities

are also observed in _a

hydrodynamic analog

of directional solidification, directional viscous

fingering

(M. RABAUD, S. MICHALLAND and Y. COUDER, Phys. Rev. Lent. 64 (1990) 184). This

analogy

enables _us to discuss the

interplay

of

crystalline anisotropy

and of the

dynamics generic

to one-dimensional systems ⁱⁿ directional solidification.

Among

the various

dynamical

_systems that exhibit

regular pattems

^when out of

equilibrium,

those which _can be considered _as one-dimensional _are of

particular

interest

II.

Their

dynamics

have been modelized and it has been shown that

they

can

develop

a number

of

secondary

instabilities _as well _as

spatio-temporal

chaos

[2].

In the present

article,

we

shali

describe _some

aspects

^{of the}

dynamics

of _one such system, directional solidification of thin

samples

and their relations to

wavelength

selection. The

question

which will arise is _: is the behavior of this

system generic,

identical to that of other lD

systems,

or does the

crystalline

substrate

play

_a role ?

(*) Associated with Universities Paris 7 and Pierre-et-Marie-Curie and with the Centre National de la Recherche

Scientifique.

In directional solidification

(DS),

_a

sample

is

pulled

at a

velocity

V in _a constant

temperature gradient

G _{set up} around its

melting temperature

ⁱⁿ such _{a manner as} to

progressively solidify

the

sample [3].

If the material is not pure, solute

rejection (or incorporation,

if the

partition

coefficient is greater ^than

I)

will _{occur at} the

solid-liquid

interface. Above _a critical

pulling speed,

for

given

concentration and temperature

gradient,

diffusion is _no

longer

sufficient to assure the evacuation of solute and the interface

adopts

_a

periodic

cellular pattem ^{in which solute is} accumulated in the cusps of the cells. The present

experiments

focus

particularly

_on

non-stationary phenomena. They

_were carried out on thin

(50

± 5 _~Lm unless otherwise

indicated) samples

of tetrabromomethane

(CBr4),

_a transparent

organic material, containing

a fraction of _a percent

impurities.

The temperature

gradient

_was

G

w loo °/cm which

gives

_a threshold

V~

₌ ⁸ ~Lm/s for the

planar-cellular

front transition. The

experiments

_were observed under _a

microscope

and

video-taped. (The experimental

_set-up

and

previous stationary-state

results

conceming

this material _can be found elsewhere

[4, 5]).

In the present ^work we have made extensive _use of

spatio-temporal diagrams

made

by extracting

a

given

line from the

digitalized

video

image

_at

given

time intervals

(typically

0. I to I

s)

and

recreating

an

image (with

_one

spatial

dimension and _one

temporal one) by

juxtaposing

these lines

[6].

Coullet and Iooss

[7]

have

shown, by

symmetry arguments

only,

that one-dimensional

interfaces _can be

expected

to present a series of

instabilities,

_a number of which have

already

been observed in directional solidification of

liquid crystals [8]

^and eutectic

alloys [9],

in _one- dimensional

Rayleigh-Bdnard

convection

[10]

and in directional viscous

fingering (DVF),

also known _as the

printer's instability

it

].

^This last system is of

particular

interest for _us because it is _a

hydrodynamic equivalent

to DS the

physical

mechanisms

driving

the instabilities _are

equivalent

but the medium in which

they

_appear has _no

crystalline anisotropy.

In the

experiment,

oil is

trapped

between two

nearly tangential rotating cylinders,

_one inside the other. The interface observed is that between the oil and air forced into the oil _as the

cylinders

come apart.

(DVF

can be in fact _more

complex

than DS because the

cylinders

can be _co- rotated or

counter-rotated.)

A close

comparison

^of our results with those obtained in DVF will lead _{us to} reconsider the role of

crystalline anisotropy

in DS.

Travelling

waves.

Tilted cells _were observed in the very first

experiments

_on transparent ^{materials carried} out

by

Jackson and Hunt

[12]

in 1965. The effect is

traditionally

attributed _to

crystalline anisotropy (essentially

of the attachment

kinetics)

: for cubic

crystals

such _as

CBr4, growth

in the

(100)

direction is faster than in others. If the direction of fastest

growth

^is not

perpendicular

to the

interface,

the cells grow

asymmetrically.

The

importance

of attachment kinetics increases with

pulling speed

until the

growth

is oriented

along

the

preferred direction,

_as is believed to be the _case for dendrites.

Anisotropic

interface kinetics _were indeed shown to

produce

such

effects,

both

analytically by

Coriell and Sekerka

[13]

and

numerically (in

the

case of _a low

partition coefficient) by Young

et al.

[141.

However, _a number of observations

bring

us to

question

^this

interpretation.

Inclined cells,

by definition,

_{grow at an}

angle

to the normal to the solid

liquid

interface

(Fig, la)

and appear

as

travelling

_{waves on} a

spatio-temporal diagram (Fig. lb).

A similar

diagram

has been

presented by

Gleeson et al.

[5]

for succinonitrile-ethanol. The lateral

velocity V~

of the cells is

simply

related _to their

angle

of inclination _a:

V/V~

= cotan _a.

Figure

2 shows how the lateral

velocity V~

of the

travelling

_wave increases with

pulling speed

V in a

given

_«

grain

_».

(This

_term

normally

refers _{to a zone} of

given crystalline

orientation. We shall

only

mean here

a zone of cells of _a

given inclination.)

Note the presence of _a well defined

threshold,

at about 25

~Lm/s,

below which _no lateral _movement is observed, to within _our

precision

of

b)

a)

Fig.

I. a) A cellular front

presenting

inclined cells and both _{a source} and _a sink (V

= 66 ~Lm/s). b) The

corresponding spatio-temporal diagram.

Note the

following

features _: A) an

asymmetric

source

(cells only ^divide on the upper side) _; B) the _{same source} has become symmetric (cells divide _on both sides) C) _an

asymmetric

sink where altemate cells _are

pinched

off D) is _a gas bubble.

8

6

~

E

~ 2

e

0

measurement

(w0.5 ~Lm/s).

If the video _camera is at all

misaligned

an apparent ^lateral movement will appear in the

spatio-temporal diagrams.

Whenever necessary, it has been subtracted out for

quantitative

measurements. In _our

experiments,

the _{curve was} not

unique

_:

in different

grains

and

samples,

the lateral

velocity

increased with

pulling velocity

from the

same threshold but its value

beyond

^{this threshold}

varied, depending probably

on the

crystalline

orientation of the

grain.

A _curve similar _to that of

figure

2 _was

reported

in succinonitrile-ethanol

[13].

On the other

hand,

Trivedi

[16]

has studied the effects of

anisotropy

in

pivalic acid-ethanol,

_a material for

which

they

_are believed to be stronger ^{than in}

CBr4

_or succinonitrile. In that _{case, no}

threshold in lateral

velocity

is

reported

for misorientations of the

crystal

_greater than 20° and the threshold observed

(at

15°

misorientation)

is

only

of 0.5

pm/s. Indeed,

_no threshold in lateral

velocity

is

predicted by

the modelizations of the effect of cristalline

anisotropy

mentioned above

[13, 14]

_: the lateral

velocity

was

predicted

to be

roughly

constant. So the threshold in lateral

velocity

does not seem to be related _to cristalline

anisotropy.

In this context, ^{it is}

particularly interesting

to consider that inclined

travelling

states with

asymmetric

cells _are observed in DVF

[10], although

_no

crystalline anisotropy

is present.

Figure 3, kindly provided by Couder,

Michalland and

Rabaud,

shows _a

spatio-temporal diagram

of such _{waves, on} either side of _{a source}

(time

_runs from top to

bottom).

It is

possible

that in this system, translational symmetry

along

the air-oil interface be broken for instance

by

lateral flow of the oil. But there is _no

equivalent

to the

microscopic anisotropy

of attachment kinetics.

These two observations lead _us to believe that

microscopic

cristalline

anisotropy

_may not be the

only

factor

influencing

the

growth morphology

of cells in directional solidification

although

cristalline orientation

apparently

^{affects their}

precise angle

of inclination and that

more

generic dynamic

instabilities also

play

_a role. We shall _see further evidence of this in what follows.

Fig.

3. A

spatio-temporal diagram

with _a

symmetric

_source observed in directional viscous

fingering (kindly provided

by COUDER, MICHALLAND and RABAUD. The

spatial

coordinate is horizontal and time

increases downward). Note that _a single central cell splits altemately to one side then _to the other.

Let _us first examine in _more detail the

spatio-temporal diagram

of

figure16.

Boundaries

between

grains

_{appear on} the

spatio-temporal diagrams

as sources where cells widen then

split

and sinks where cells _narrow and

pinch

off. Sources _can be _{more or} less

symmetric

with cells

appearing

_on both sides of the

boundary,

or

they

_can be

asymmetric

with

only

cells

on one side

splitting.

In the upper part ^of

figure

16 _{one can see an}

asymmetric

_source become

symmetric.

We have

only

observed

asymmetric

sinks _{: one}

grain

_appears to be blocked

by

the other.

Figure

¹⁶

^(feature C)

also shows _a

particular

oscillation of the sink where the last cell and the second last cell _are

altemately pinched

off.

In

DVF,

sinks and

asymmetric

_{sources appear} to be identical to those

reported

here.

Symmetric

_sources,

however,

_are different

(see Fig. 3)

_{: a}

single

central cell

splits altemately

to the left and _to the

right

whereas in

figure

16 there _are two central cells

clearly separated by

_a continuous line

probably

a

grain boundary

and each cell

splits

to its _own side. This is _an illustration of _one

possible

interaction between

dynamic

instabilities and the

crystal

_:

sinks and _{sources seem} to lock _{onto a} defect of the

crystal,

I-e- _a

grain boundary.

Asymmetric

cells and

wavelength

decrease.

A number of

secondary

instabilities _can be described _{as cases} where _a

cell,

too wide

compared

to the normal

wavelength

at that

velocity, unsuccessfully attempts

to

tip-split (Fig. 4).

When

tip-splitting

succeeds, two _new cells appear, but when it

fails,

the result is _an

asymmetric

cell with _a shoulder that is advected back towards the solid. The

instability

_can

_propagate

_or not

along

the interface. It has been shown

[1,

7,

17]

that the presence of such modes is related _to the

development

of the second harmonic of the interface deformation. The cell

shape

_goes

from sinusoidal to _« square » with _a flattened _center that _can allow

tip-splitting.

The _term

« mitten cells _»,

suggested by

Cladis _et al.

[18],

describes

large amplitude

cells with _a thumb-like shoulder. A

given

cell

periodically begins

to

tip split,

but the

dip slips

off-

i

x ~

$ _e

~

x ~ °

_

x

m

i lo loo

vl~L1n/s)

Fig.

4. Average

wavelength

^I _versus

pulling velocity

V for cells

presenting

optical modes (crosses) and mitten _or

pre-splitting

instabilities (diamonds). The line represents ^the average

wavelength (dispersion

is about 10fb) for normal cells [4].

JOURNAL DE PHYSIQUE I T 2, N'2, FEBRUARY 1992 8

center _so the deformation is advected

back, giving

the characteristic shoulder. The cell then

regains

its initial

shape

and the whole process starts

again.

The _same

thing

can

happen

to its

neighbours, slightly dephased

in time, hence the

impression

of

propagation.

This is best seen

in

figure

5b. The

phenomenon,

first

reported by Venugopalan

and

Kirkaldy [19]

and observed in succinonitrile

by

Heslot and Libchaber

[20],

_can also be described _{as a} failed

tip- splitting.

The cells _are

abnormally

^wide

(here

68 _~Lm on average instead of _a normal value of 50 ± 5 ~Lm)

(Fig. 4).

Mitten cells _can also be observed in DVF

[I il.

G ""

,. "z

~' -~ _'T '$ ~~'

~ w'~

' r-L'

~~ Z '.S~_- _S,

i n©' _j~b ~G'

~

', ~$

"~ f .'

.I

~~ 'l '[

z ''

"

»,' M')_. t~)~.-~ ~l't

~Q~- j

~

~"~' -"-'

, .~~

&'~ ~~" _-~

3 /- /.

_' ^{~- -~}

_' ,' l-?i~

i -,,'

~

II- /,

t

b)

a)

Fig.

5. a) An interface

presenting

_« mitten cells _». A

tip split

off center

gives

_a shoulder that is advected back (V

= 10 ~Lm/s). b) The

corresponding spatio-temporal diagram.

Figure

6a shows two

pairs

of _« anomalous cells

».

Again,

these cells are also observed in DVF

by

Rabaud et al.

[I II

and in DS of

liquid crystals [2 II.

The cells _are

abnormally

wide and each _one is flattened _on the side

facing

the other. Different

aspects

^{of their}

dynamics

_are

illustrated in

figure

6b. The

pair

can remain stable for

quite long periods,

in which _case it has

approximately

the width of three normal cells. It can be

progressively squeezed

out

(feature C).

If it is _too wide it _can

disappear through

the

splitting

of both cells

(feature D).

In these two cases, the

pair disappears

without

leaving

_any trace. But another type ^of

instability

can also be observed the

pair

widens and the wall between the cells appears ^{to oscillate until} one of the two

splits (feature E,

_or

Fig. 6c).

In fact the two cells

altemately undergo

_a series of unsuccessful

tip-splittings

the oscillation of the middle wall is due to the advection back of

the shoulder _on altemate sides until _one of them

finally

succeeds. This _can

happen

repeatedly.

In this _case the

pair

seems to be locked _{onto a}

grain boundary (we

have checked this

by examining

the

sample

at rest :

grain

boundaries form cusps where

they

^{reach the} solid-

liquid interface).

Rabaud et al.

[22]

observe _a similar

locking phenomenon

in DVF _{on a} scratch made _{on one} of their

rotating cylinders.

A-

C)

a)

w-

b)

Fig.

6. a) A front

presenting optical

modes (A) and 4

pairs

of anomalous cells (B). The lower cell of the lowest

pair

is splitting. b) The

corresponding

spatio-temporal diagram, with _: A)

optical

modes B)

a

pair

of anomalous cells C) _a

pair

becomes

progressively

normal D) _a

pair disappears

after both cells

split

_; E) a

pair

that

regularly splits

after

presenting

_«

pre-splitting

_» oscillations. c) A

blow-up

(x 2.5) of _{a «}

pre-splitting

_» oscillation

(T~~

w 0.6 s) and _an

optical

mode

(T~~~ w 1.8 s).

Figure

7 shows the

propagation along

the front of _a

single

cell deformation _{on a} low-

amplitude

cellular front. These deformations _are

analogous

to the mittens that _are observed

on

higher amplitude cells,

in that

they

_appear to be unsuccessful

tip,splits. They

_are also observed in DS of

liquid crystals [81,

but above about 3 times the threshold

velocity

for the

transition from _a

planar

to a cellular front. Here

they

can be observed much closer _to

threshold,

due _no doubt to the fact that the bifurcation is subcritical: the interface deformation is _never

purely sinusoidal, higher

harmonics are present ^{from the} start and the

cells look square. This _can also be _{seen on} the

marginal stability

_curve for

impure CBr4

(critical velocity V~

vs. wavenumber

q),

that is

particularly

flat-bottomed

[4]

_: modes at 2 q

(and beyond)

_are accessible

practically

at threshold.

Fig.

7. A sequence of

pictures

of the interface taken every 5.7 _s

showing

the

propagation

of _a solitary mode (V

= 14 ~Lm/s).

The

periods

of these _«

pre-splitting

_»

oscillations,

I.e. those of the central cell wall of anomalous cells and of the mitten

instability,

decrease _as the

pulling speed increases, approximately

_as

T~~~ ^oc

V~~ (Fig. 8).

This is also the time

scaling

of the diffusion time

(D/V~w

_lo

s for V

= lo

~Lm/s),

_as remarked

by

Gleeson et al.

[15].

The

periodicity

^of

dendrite

side,branching

in DS of

CBr4

falls _on the _{same curve}

[23] (in

succinonitrile-acetone

[24]

the order of

magnitude

of the

periodicity

is the _same but it decrease _as V

~i.

_One

can

speculate

that the _same type ^of fluctuation of the

tip

_can

(a) split

_a

flat-tipped (rich

in second

harmonic) cell, (b)

be _a little off center, fail to

split

it and be advected back _or

(c)

in the _case of

sufficiently pointed cells,

where

tip-splitting

is blocked

[5],

form side-branches I.e.

high frequency

shoulders that have _room to

develop laterally.

Optical

modes and

wavelength

increase.

In

regions

^of the

sample

where the cellular

wavelength ^I

is _too small

(Fig. 4),

either because of _{a recent} local

adjustment

^{of cell} widths _a cell has

split

in the

vicinity

or because the

loo

lo

]

w

~~4

o

~

I

T

i io

8. -

of

oscillation _ceases and the

adjustment

then

spreads

out over the

neighboring

cells. Even in

cases where the cell walls do not touch _at the

interface,

_a rearrangement can take

place

behind

the interface

causing

the cells _to

pinch

off

alternately, producing

the checkered pattem seen in

the middle of

figure

6a

[26].

The

dependence

of the

period

on

pulling velocity

is about

T~~~ ^oc

V~~'~ (Fig. 9),

I-e- it decreases _more

slowly

than

T~~~. The _reason for this different

dependence

is _not clear.

Combined instabilities.

In another type ^of

secondary instability,

known _{as a} tilt _{wave, a zone} of inclined cells propagates

through

normal _{ones, as} has been

clearly

observed in eutectic solidification

[9].

Tilt _{waves are not}

frequent

in the

experiments

described

here,

but _we have observed three

cases.

Figure

lo shows _a tilt _wave associated with _an

optical

mode. This is _once

again

_very

similar to _a

phenomenon reported

in DVF

[27].

The _{cases we} observed _were all at low

velocities

(6

_to lo

~Lm/s)

and the inclined cells _were at 7 _± 1° off the normal _to the interface.

We sometimes observe

spatio-temporal diagrams (Fig. ii)

with _an

apparently

chaotic

mixture,

in time and space, of anomalous

cells,

mitten instabilities and

optical

modes. Similar

diagrams

can be observed in DVF

[I ii

when the

cylinders

_are co-rotated. _« Chaotic _»

grains

can coexist with

quite

stable _ones at the _same velocities _so the orientation of the

grain

_may

play

a role _as

suggested by

Heslot and Libchaber

[24].

A proper

spectral analysis

of the time evolution of the

interface,

such _as has been carried _{out on} convection

experiments [28],

is be needed _to be able to say if it is _true

« chaos _{» we}

observe, and,

if _so, of what

type.

Fig.

10. -A

spatio-temporal diagram presenting

_a «tilt _wave». The tilted cells _are affected by oscillations of the

optical

mode type (V

=

20 ~Lm/s).

Tip-splitting

in dendrites.

To

complete

^{this discussion} of the

respective

roles of cristalline

anisotropy

and

dynamic

instabilities in

DS,

_we wish to mention _some

experiments

carried out _on dendritic

growth

in very thin

samples (e

^it ~Lm). ^The

stability

of dendrites is _{a case} in which cristalline

b)

a)

Fig.

ll.-a) A «chaotic» interface, b) The

corresponding spatio-temporal diagram showing

_a

mixture of anomalous cells,

optical

modes, etc. (V

= 80 ~Lm/s).

anisotropy,

either of attachment kinetics _or of interfacial

tension,

is essential. This has been established

by

numerical simulations

[29-34], analytical

calculations

[35, 36], experiments

in viscous

fingering [37, 211,

where _« cristalline _»

anisotropy

is introduced

locally by scratching

to

plates

^{of the Hele-Shaw cell} and in solidification

[38, 391

and

phase

transitions in

liquid crystals [40, 411.

More

indirectly,

dendrite

instability

has been observed when the effect of

cristalline

anisotropy

is

thought

to

weaken,

I.e. when

growth

^{slows down}

[42, 43]

_or in

homeotropic liquid crystal samples [44].

In the absence of

anisotropy,

_or under conditions where

crystalline

and surface tension

anisotropy, aligned along

different

directions,

cancel each

other,

dendrites are unstable _to

tip-splitting

and the result is _a dense

branching morphology (DBM).

In

experiments [181

_on

samples

of the order of I _~Lm

thick,

_one

easily

obtains dendritic _{structures at} velocities of the order of _a few micrometers per second

(Fig. 12)

because the critical

velocity

for the transition from _a

planar

to a cellular interface decreases

sharply

_as

sample

thickness

decreases,

due _to meniscus effects.

Therefore, simply by decreasing

the thickness

along

the

sample,

one increases the constraint. These dendrites

are unstable to

tip-splitting

and

produce

a structure similar to DBM. We suppose ^that a such

velocities, anisotropy

effects in

particular

kinetic

anisotropy

_are too weak _to stabilize

dendritic

growth.

Conclusion.

We have observed _a number of

secondary

instabilities of cellular fronts in DS. These

instabilities

(optical modes,

anomalous

cells, solitary modes,

mitten

cells, travelling waves)

a) b)

Fig.

12.

Tip-splitting instability

^of dendrites in very thin

samples

(= I ~Lm) : a) V

= 3 ~Lm/s b) V

=

6 ~Lm/s.

appear

quite

similar to those observed in

DVF,

the

hydrodynamic analogue

of DS.

Crystalline anisotropy

is _not therefore essential _to their

apparition although

the

crystalline

structure of the material affects inclination

angles

and locks

particular

structures

(anomalous cells,

_sources

and sinks of

travelling waves).

When

growth

becomes

dendritic, crystalline anisotropy

is

necessary ^to stabilize the dendrite

tips against tip-splitting

and _we have shown that in _our

experiments

it is not

strong enough

at low velocities.

Acknowledgments.

It is _a

pleasure

to thank Y. Couder, C.

Misbah,

S. Michalland and M. Rabaud for very fruitful discussions and for

allowing

us use their results

prior

to

publication.

This work _was

supported by

the Centre National d'Etudes

Spatiales.

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