Facet destabilization and macrostep dynamics at the smectic-A smectic-B interface

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Facet destabilization and macrostep dynamics at the smectic-A smectic-B interface

F. Melo, P. Oswald

To cite this version:

F. Melo, P. Oswald. Facet destabilization and macrostep dynamics at the smectic-A smectic-B in-

terface. Journal de Physique II, EDP Sciences, 1991, 1 (3), pp.353-373. �10.1051/jp2:1991173�. �jpa-

00247523�

J Phys II1

(1991)

353-373 _MARS 1991, _PAGE 353

Classification

Physics

Abstracts 82 00

Facet destabilization and macrostep dynamics at the smectic-A smectic-B interface

F. Melo and P Oswald

Ecole Normale

Supbrieure

de

Lyon,

Laboratoire de

physique,

46Allbe d'Itahe, 69364

Lyon

Cedex 07, France

(Received

26

September

1990,

accepted

_tn

final form

13 December

1990)

Rksumk. Nous dbcnvons le comportement en croissance directionnelle d'un interface facettd smectique

A-smectique

B lUe matdnau choisi _est le 408

(butyloxybenzihddne octylaniline)

Au- dessus du seuil de ddstabilisation qui ^est bien donna _par le critdre dassique du _« constitutional

supercooling

_», des macromarches _se

ddveloppent.

Plusieurs mdcamsmes sont

possibles

suivant la

vltesse de tlrage ^Ces macromarches sont toujours mstables et _se propagent ^le

long

de l'mterface

macroscopique en

changeant

de forme contlnuellement Nous n'avons jamals observd de solution stationnaire Nous _avons

bgalement

mesurb [es effets cindtlques sur la facette _{que nous} attnbuons I _un mdcamsme de _{croissance par} dislocation _vis Enfin nous ddcnvons le

rdgime

de nuddatlon qui se

ddveloppe quand

_{on augrnente} la vitesse de tirage ainsi que

quelques

effets

vlscodlastiques prdsents

dans la

phase

_smectique A.

Abstract. We describe the behavior _m directional solidification of the faceted smectic-A smectic-B interface of 408

(butyloxybenzihdene octylaniline)

Above the onset of

instability,

which _is given by the classical constitutional-supercooling cntenon, we observe the nucleation of macrosteps ^{Several mechanisms} are

possible, depending

_on the velocity ^These macrosteps are

always

unstable and propagate

along

the macroscopic interface, changing shape continuously. We never~observed any stationary ^{solution We also measured the kinetic} effects _on the facet that _we attribute to a screw dislocation

growth

mechanism Finally, we descnbe the nucleation regime, which appears when one increases the

pulling velocity,

_as well _{as some} vlscoelastic effects that _are present in the smectic A

phase

1. Introduction.

Recently, Bowley

_et al

[I]

Studied the Mullins-Sekerka

instab1llty

Of _a

binary

mixture for directional Solidification _in the _case where the basic

planar solid-liquid

interface _{iS a} facet

They

Showed the ex1Stence Of Solutions consisting ^Of

alternating

cold and hot facets connected

by

curved regions. This crenellated solution _appears above the standard Mullins-Sekerka threshold of

instab1llty Later,

Carob _et al

[2]

showed that this solution _is

always

unstable and

suggested

the existence of another branch of solutions

consisting

^of

partly faceted, large- amplitude

cells. For this _reason,

they

claim that the cellular bifurcation should exhibit _an

anomalously large hysteresis

because _{one must} jump over the unstable branch _to reach this

hypothetical

stationary solution.

In order to test these theoretical

predictions,

_we have

performed

_an

expenment

on the directional

growth

of _a smectic-A smectic-B interface. The matenal studied is 408

(butyloxybenzllidene octylaniline).

Previous

experiments

_on the

equihbnum shape

of _a smectic-B monodomain

[3]

^{showed that this interface} is faceted

parallel

to the smectic

layers.

The facets match

tangentially

with the contiguous

rough regions

whereas faces

perpendicular

to the

layers

_{are missing.} We showed that such faces _are unstable with respect ^"to the formation of _a

hill-and-valley

structure

[4] (Herring's instability [5]).

More

recently,

_we

showed _some

preliminary

results _on the directional

growth

of 408 when the

layers

make _an

angle

with the

temperature gradient [6].

We concentrated _on what

happens

when the macroscopic front

corresponds

to a missing ^onentAtion and showed that cells with _an

angular discontinuity

at the

tip develop

above the

onsit

_of

instability.

In this

article,

_we limit ourselves to the _case where the basic

planar

interface _{is a} facet

(smectic layers perpendicular

to the

temperature gradient)

We describe how _a facet destabilizes _in directional solidification and compare our observations to the theoretical

calculations mentioned above

[1, 2].

In

particular,

_we describe the formation of

macrostep

and their

subsequent propagation along

the interface.

2. Cridcal

pklling velocity.

2,I THEORETICAL PREDICTIONS. It _is well known that the

morphological instab1llty

of the front results from the

competition

between the

destabihz~ng

effect of solute diffusion and the stabilization due to the temperature

gradient

and the

capillanty

For _an

atomically rough interface,

_{one can}

analyze

front

stability

via linear

stability "theory _[7].

This method _is

nevertheless not valid when there _{is a cusp in} the Wulff

plot y(9),

_say at 9

= 0. This

onentation

corresponds

to a facet _in the

equihbnum shape

The

problem

_is that the _{cusp gives}

nse to a contnbution

[(dy/d9 lo

~

(dy/d9 lo 3(9

_in the surface stiffness _{y +}

y"

and thus in the Gibbs-Thomson

equation

_expressing the local

equilibrium

of the front In this

case, the Gibbs-Thomson

equation

must be

integrated

_across the facet

yielding

_a

global

equihbnum

condition

[8]

^Thls

highly

non-local

problem

has not

yet

^been

completely

solved.

On the other

hand, Bowley

et al.

[ii

showed there exists _a

family

of crenellated solutions when the

pulling velocity

satisfies the condition

(v vc)/vc

»

(do/fc)~'3 (1)

do

is the chemical

capillary length, _i~

=

2D/V

the diffusion

length

and V~ ^{the classical} constitutional

supercooling

threshold

V~ ₌

DGk/mC o(k (2)

where k _is the

partition coefficient,

_m the

slope

of the

hquldus,

G the temperature

gradient

and

Co

the _mean concentration of impurity in the

sample.

In _our system,

do

_m ²⁰

A

_and

i~

_m ⁶⁰ _~Lm so that the

inequality (I)

_can be rewntten _as

(

V

V~)/V~

» 3 _x 10~^~

(3)

Thus V~ ^should give ^to a

good

approximation ^{the cntical}

velocity

above which the faceted front destabilizes.

In order to test this theoretical

prediction,

it _is first necessary ^to determine the

phase diagram

and the

partition

coefficient _as well _as the dilTusion coefficient D

2.2 PHASE

DIAGRAM

DETERMINATION. -Our 408 _was

synthetised by

Germam This

molecule exhibits _a

plastic

smectlc-B

phase

between 30 °C and 49 9 °C

Thelamples

consist of

M 3 FACET DESTABILIZATION AT THE Sm A-Sm. B INTERFACE 355

two

parallel glass plates separated by

two

15-~Lm-thick

_spacers. To obtain _a

planar alignment

of the molecules _on the

glass (smectic layers perpendicular

to the

glass)

_a

300-A-thick _layer

_of

poly1mlde

ZLI-2650

(Merck Corp.)

was

deposited

_on ^the _inner surfaces Thls

layer

_was ^then

rubbed in _a

single

direction _in order to onent the smectic

layers perpendicular

to the scratches To _measure the

phase diagram

_in the presence of _a small amount of

impurities,

_we

left _a

sample

_{in an oven} at 50 °C for several

days.

At this temperature, 40.8

degrades

and it is

possible

to measure at

regular

intervals of time the

liquidus

and the sohdus

temperatures

These temperatures are shown _in

figure

I where _we have assumed that the

hquldus

_{is a}

straight

line. Under this

assumption,

^{the solidus} is a

straight

line too. The

partition

coefficient k _is

approximately

0.56. We also made _a

systematic study

of the

degradation

of _our

samples.

In

figure 2,

_we

reported

the

freezing

_range AT

= T~~~

Ts~j

versus time.

Here,

the

sample

_is

left _{in an oven} at 70 °C and AT _is measured at

regular

intervals of time. Just after

filling

the

sample,

AT

m 0.22 °C. The

degradation

is

rapid dunng

the first

hours,

then it slows down and AT reaches _a

l1mlting

value close to 0 6 °C. The ongln of the 1nltial

degradation

is not clear.

one could think that it _is due to the surface treatment and to the dilTusion of

poly1mlde

into the

sample,

but

equivalent

effects _were observed without surface treatment Another

possibility

is that 408

decomposes

itself _in the presence of adsorbed water, an usual

phenomenon

with schiff's bases. As _we shall _see

below,

the

degradation

of the

samples

_{is a}

severe

expenmental

limitation which must be taken into account in the discussion of the

results

51

50

@

O ^~~

@ 48

~

% ~

₄₇

@

~ 46

45

20 40 60 80

Concentration (arbit(afy units)

Fig I Experimental

phase diagram

2.3 MEASUREMENT oF THE DIFFUSION

COEFFICIENi.

_{The ef§ect of}

temperature gradient

on

non-steady

1nltial transient solute redistribution

during

solidification of the smectic-B

phase

_can be used to determine the diffusion coefficient of the solute _in the

liquid (here,

the

smectic-A

phase).

Smith _et al

[9]

_give the

following

formula C

=

(Co/2 k)

I ₊ erf

(1/2) ( ^V~

t

/D)'/~]

₊

(2

k I exp

[-

k

(

I k

V~

_t

/D]

x

x erfc

[(1/2) (2

k I

(V~.t/D)~/~]) (4)

o

- A

~ _, A

~06

~

~ _A

@ a

Cl

(04

^'

~

Cl _A

fi

#

A

@ 0 2

~

o

0 5 10 5 2 0

Time t

(hour)

Fig

2

Freezing

_range AT versus time. The

degradation

is rapid

dunng

the first hours that follow the

filling

of the

sample

and next slows down

for the solute concentration at the interface _in the smectic-A

phase

at time t. This equation

gives, according

to the

phase diagram,

the front temperature

T=m(C-Co)+To (5)

where

To

is the front

temperature

at time t ₌ 0.

Putting

_x

=

Vt,

the distance which _is covered

by

the

sample

at time _{t, we}

get

T

To

₌

(mco/2 k) (1

^{2 k} + erf

[(1/2) ( Vx/D)~'~]

₊

(2

k I

)

_exp

[- k(

I k

Vx/D]

_x

x erfc

[(k 1/2) (Vx/D)~/~]) (6)

In order _{to measure} the front temperature T

accurately,

_we have

supenmposed

_a

planar sample

of 40.8 and _a

homeotropic sample

of the mixture 9CB-10CB

(75

fb and 25 fb _in

weight respectively) (Fig. 3).

This mixture of classical

cyanobiphenyl liquid crystals

has _a nematic

smectic-A

phase

transition at

TN~~_s~Am5o°C

_very close to the smectic-B smectic-A transition

temperature

of 40 8

Furthermore,

this nematic smectic-A

phase

transition is very

weakly

first

order,

_so that the

corresponding freezing

range is very small

AT~~~_s~A

_m ^{5 mK.}

Since

cyanobiphenyls

are very stable

chemically

too, the nematic smectic-A transition

temperature

can be considered _{as a} reference

temperature

^which does not vary with the

pulling velocity (within

_a few

mK) Thus,

the front

temperature

is measured

by locating through

^the

microscope

the

position

^{of the} two fronts. Because the temperature

gradient

does not

change

at the very small velocities that _are used _in this expenment, we can calculate from the shift

betwien

_{the two}

fronts,

the real temperature of the SrnA-SmB interface. The _main

advantage

of this method _is to avoid the _{errors coming} from _an accidental shift of the temperature

gradient.

Moreover, we have taken into account the _errors due to the vertical

gradient

of

temperature (m

o.5

°C/cm )

that we were able to calibrate

by supenmposing

two

M 3 FACET DESTABILIZATION AT THE Sm A-Sm B INTERFACE 357

licroscope

hot _oven cold oven

Nemotic Smectic A

~ V

Smectic A Smectic B

Fig

3. Double

sample

used for the measurement of the front temperature

identical

samples

of the mixture

9CB-JOCB Finally,

this method allows to _measure

temperature

vanations as small _as o,ol

°C,

much _more

easily

than with _a

thermocouple.

In

figure

4 _we

reported

T

To

versus x, the distance that _is covered

by

the

sample

at time t The

expenment

was

performed

with _a

sample

saturated _in

impurity

_in order to minimize the

parasitic

effects due to the

degradation.

For this

sample

AT

m 0 56

°C,

G

= 56.2

°C/cm

and

Vm02~Lm/s

of the order of

0.2V~.

The fit with equation

(6)

_is

good

and

gives

o

-oos

-o i o

o

(

~ _o

-O 25

-O 30

.

200 400 600 800 1000

x

(~m)

Fig

4 Front temperature _versus the distance which _is covered

by

the sample (transient regime)

Ji

is the initial front temperature The dots are

expenmental

and the solid line is the best fit with

equation

(6)

D m 5 _x 10~^?

cm~/s.

Note that D is the

only adjustable parameter

since k and

mco

_are known from the

phase diagram.

Note also that D _is the diffusion coefficient

along

the director that _we called Djj m reference

[6].

Despite

the

good

fit shown _in

figure 4,

this method is not very sensitive and

only

_{gives a}

rough

estimation of the diffusion coefficient. We have indeed found _a

fairly large dispersion

in

the values of D obtained _in this way,

ranging

from 3 _x 10~ ^?

cm~/s

_up to 7 _x 10~ ^?

cm~/s.

One of the difficulties lies _in the control of the initial

conditions,

which differ from _one

sample

to

another

Indeed,

the calculation _assumes C=

Co

_m the smectlc A

phase

at time

t ₌

0,

_a condition that is

certainly

not fulfilled

expenmentally.

Another

difficulty

_is,

despite

our

precautions,

^the

inhomogeneous degradation

of 408

dunng

the

expenment (which

_can be

more important near the hot

side).

This leads to _a further drift of the front _m the

temperature gradient

On the other

hand,

there _is

always

in

practice

a thermal _transient which _{is not} taken

into account

by

the

theory

and which will be all the _more

important

^{that the}

velocity

_is

large.

Furthermore,

it _is well known that the solution _given

by

Smith et al. _is not exact

(and

_even

wrong near t ₌

0)

because of the

assumption

that the

velocity

of the interface _is

equal

to the

pulling velocity Recently, Huang

et al.

[10]

have discussed this effect and have

improved

the

~calculation of Smith _et al.

They

show that this

hypothesis

leads _one to overestimate the diffusion coefficient and _is

only

valid when M

=

mco(I k) V/GD

=

V/V~

_« I. In _our

expenment, M

m 0.2 _so that the _error which _is made

(overestimation) by

_using equation

(6)

_is

of the order of 20 fb

(from Fig.

3 of Ref.

[10]).

This calculation also _assumes that there _{is no} dilTusion _in the solid. This _is not

ngorously

true _in our

system

since the ratio of the diffusion

coefficients _in the two

phases p

=

Ds~~/DS~A

_m 0.2. This

phenomenon

leads _us again ^to overestimate the diffusion coefficient. It _is

possible

to show

that,

within the

approximation

of Smith _et

al.,

the characteristic relaxation time _is modified

by

_a factor I ₊

k~ _p

m 1,05 _so that

the diffusion coefficient it overestimated of about 5 fb.

Taking

into account these two

effects,

_we shall take in the

following

D ₌ 4 _± 2 _x 10 ^?

cm~ Is.

Another _way to estimate the diffusion constants

Di

^and Djj is to measure the cntical

velocity

and the dnft

velocity

of the cells when the

layers

make _an

angle

with the

temperature gradient.

In this _case, the interface _is

rough

and the classical

constitutional-undercoollng

cntenon

(which

_we have

generalized

to the

anisotropic

case, in Ref.

[2])

holds. This method gives

Di

₌ 6 _x

10~? cm~/s

and Djj = 3 5 _x

10~? cm~/s.

2.4 CRITICAL _VELOCITY. In Order to measure the Onset Of

instability,

_{We increase Very}

slowly

the

pulling Velocity by

increments Of 0. I ~Lm/S. ^{At each}

increment,

We Wait for _a time Of the order of

D/kV~

_{which is the}

minimum time that is _necessary to reach _a

quasi-stationary

regime In this way, we obtain _a first cntical

Velocity

that _we call

VI Then,

_we decrease

slowly

the

Velocity

till the front restabilizes. This _occurs below _a cntical

Velocity Vi.

^The

hysteresis VI Vi

is charactenstic of _a subcntical bifurcation.

For _a temperature

gradient

G

= 50

K/cm

and _a

sample

thickness of 15 _~Lm, the threshold

on increasing

speed

is

VI

= 0.55

~Lm/s

whereas it is _on

decreasing speed Vj

=

0.4

~Lm/s.

The shift

VI Vi

is

small,

which _means that the bifurcation _is

only weakly subcntical,

_in contrast to what _was

expected

from the

theory.

The

large

Value of the

partition

coefficient _is

perhaps

responsible

for this small

hysteresis

It _is indeed well known that for _a

rough

interface the bifurcation becomes

supercritical

_{as soon as} the

partition

coefficient is

large enough, typically greater

than 0.45

[11].

Nevertheless it is not at all clear that this cntenon _remains valid for _a facet.

In order to test _more

quantitatively

the theoretical

predictions concerning

the onset of

instability,

_we measured

Vf

for several

samples

of different

composition

and for various

M 3 FACET DESTABILIZATION AT THE Sm A-Sm. B INTERFACE 359

temperature gradients.

For each

sample,

_we measured

carefully

the

freezing

_range

AT

=mco(I/k-I).

In

figure 5,

_we

plotted VI/G

_versus

I/AT

and found _a linear

dependence.

The

slope

is

proportional

to _a diffusion coefficient and _is close to 4, I _x 10~ ^?

cm~/s.

This value _is

equal,

within _our

experimental

_error

(±

30

fb),

to the diffusion

coefficient Djj given

previously

Thls _means that the classical constitutional

undercoohng

threshold _{gives a}

good

estimation of the

stability limit,

_{even in} ^{the faceted} case

~_

a

~U~ _a

o

~u~

~

~

~

₅₀

~

>

0 2 3 4

1/AT(°K'

Fig

5

VI /G

against

lib

T.

VI

is the critical velocity obtained

by

_{increasing very} slowly the pulling velocity ^from zero, G

(he

temperature

gradient

and AT the

freezing

_range The solid line _is the best fit with _a linear law Its

slope equals

41 ~Lm~/s.

3.

Description

^of _some macrostep ^{nucleation mechanisms and} measurement of the front

temperature.

Above the cntical threshold

VI,

the front _is

composed

from _{a succession} of hot and cold facets

separated by

macrosteps. These macrosteps

always

drift

along

the interface _{: we} have

never observed

penodic

crenellated solutions. This observation _is in fact

compatible

with the

stability analysis

of Carob _et al.

[2]

who found that crenellated fronts _are

always

unstable.

On the other

hand,

the model _is unable to

predict

how

macrosteps

occur. Several mechanisms have been

observed,

which _{we now} descnbe.

The first _one, which _was mentioned _in reference

[6], corresponds

to the

heterogeneous

nucleation of _a small

bump

on the interface

(Fig 6).

This localized

perturbation quickly facets,

_{giving nse} to a pair of

wide, rough

macrosteps

separated by

a new facet which is hotter than the

preceding

_one. This mechanism _{is common} at low

velocity

_near the onset of

instability.

It _can appear below the onset of

instability but,

_in this _case, the

bump rapidly

shrinks and

disappears.

There _are other mechanisms which _are

frequently

encountered.

One of them _is associated with the

dynamics

of the macrosteps themselves. As _we have

already emphasized,

the macrosteps

propagate along

the interface.

Furthermore,

their

shape changes

_m time. As _a

rule,

_an

initially

wide and smooth macrostep ^becomes first

step-like

and then

develops

_a

sharp edgi

and _{a cusp}

_T(e

^drift

velocity

then increases

sharply

and _one often

sees on the hot

facet,

behind

thq tip,

_{a new,}

wide,

smooth macrostep

(Fig. 7).

It then evolves

as the former and the

mec%hiiin I@ _begin igaii.

_{This mechanism}

seems to be related to the

change

of

velocity

which _occurs when _a

sharp edge develops.

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small

_bumps

can onthe _interface ^hey acet, eading to

two

At

large velocity (V

m2

VI ),

another mechanism _very

frequently

_{seen is} the sudden nucleation of _a small pit _on a facet

(Fig. 8)

This pit can either

heal,

_{giving a} small smectic~A inclusion which

rapidly disappears,

_or it _can grow. In the latter _case the _two

portions

^{of the} facet evolve

separately

at different velocities. This leads

rapidly

to a

sharp~edged macrostep

that

propagates

towards the colder facet.

We _sometimes observed that _a

pit

can destabilize

by

_giving birth to _an

adjacent

hill

(Fig. 9) This'process

_is

always asymmetncal,

two hills _never appeanng

simultaneously

_on the two sides of the pit This hill then

quickly

facets

leading

to _an

abrupt

_{macrostep on} the side of the

pit

and to a smooth macrostep on the other Then these two macrosteps

begin

to propagate the first _one

(on

the side of the

pit) develops rapidly

_a

sharp edge

and _a cusp while the other _is at first smooth.

The nucleation _rate of the

pits

fixes the _{mean size} of facets at

large velocity (Fig. 10).

We found that _varies

roughly

with the

pulling velocity

_as

V~°~ (Fig 11)

We also noted that the interface often _« oscillates

locally,

between stable state _» with few macrosteps ^and one that _is unstable

(Fig. 12).

These

cycles

_are

irregular

which hinders

their

quantitative analysis. They

_are

only

visible _near the onset of

instability

and have _a

penod roughly equal

to the diffusion time

D/V~.

Finally,

we measured the front temperature versus the

pulling velocity,

_using the _same method _as in subsection 2.3. Our data _are

reported

in

figure

13. Measurements _were made _on

a «clean»

sample (AT

=

0,22 °C

), immediately

after

filling

with 40.8 and _as

rapidly

_as

possible

in order to _minimize the

degradation

effects

(we

measured _again AT after the expenment, about half _an hour

later,

and found the _same value to about _± 0 01

°C).

At very small

velocity,

the front _is

planar

and its temperature T is

equal

to the sohdus

temperature

(that

_we

previously

measured _{in a}

separate oven).

As the

velocity

_increases the front

N 3 FACET DESTABILIZATION AT THE Sm A-Sm B INTERFACE 361

ioopm

Fig

7.

Shape

evolution of _an

initially

rounded macrostep and nucleation of _{a new} macrostep ^when

the first _one becomes

sharp-edged

and

changes velocity

G=31K/cm, AT=12°C and

V

= 012 ~Lm/s ^The time interval between two

photographs

_is I _mn.

temperature begins

to decrease

slightly,

then the front becomes unstable _{: we next measure its}

mean

temperature

^It does make _sense because the

macrostep height always

_remains small in

comparison with the _retreat of the front it

corresponds,

for

example,

to _a

temperature

span

~

fs~j;

,~~j _~~+

~

< & W~

~l'w fl

~~

~~

-- &~

1"2

^{'i %}

,~#

'~

b&§1

~~ , .~~

~~j.q~i:?ji

^W~/! /

J

ioo

pm

Fig.

8 At

large velocity,

pits

suddenly

appear on the facets and break them This mechanism leads to new macrosteps ^when the two parts of the facets evolve later at different velocities G

= 38 K/crn, V

= 2 5~Lm/s The time interval between _{two successive}

photographs

is 1s

of the order of 0.015 °C whereas the retreat of the front _is rather 0.06 °C _at

i'

= 10

~m/s.

On the other

hand,

it _is

possible

to show

theoretically

that the _mean position of the unstable front must coincide with that of the

planar front,

if there _{is no} diffusion _in the solid

[12].

For this reason, we believe that the

cooling

down of the _mean front _is the _{same as} the _one that would have the

planar

front and

consequently

_is due to kinetic effects _on the

facets,

which _we shall discuss _in section 5.

In the

impure samples,

it _was not

possible

to measure the kinetics. This _is due to the small range of usable

pulling velocities, mainly

because of the

parasitic

nucleation of smectic~B germs ahead of the interface. Since the _mean

temperature

of the unstable front _is close to that of the

planar stationary

front

(the _macrostep height

_is much smaller than the diffusion

length too),

it _is

possible

to make _some theoretical

predictions

on the nucleation rate ahead of the

interface and to test them

expenmentally (see Appendix I).

Let _{us now} describe _{in more} detail the

dynamics

of the

macrosteps

^{and their} interactions,

N 3 FACET DESTABILIZATION AT THE Sm A~Sm B INTERFACE 363

fi

Fig 9

Asymmetncal

destabilization of _a small pit appeanng on a facet G ₌ 32

K/s,

V

= 0 8 ~Lm/s and AT

= 0 2 °C The time interval between two successive

photographs

_is 7_s

'g,-q

)m)<~

'"j~

,

~~~~~~~~~'~~'~~

^'

~

,i;vi~i=iii:.ill.

~

~" ~ ~@Wlfliiilflt

~.

~

»

.

ioopm

Fig 10 Front aspect at increasing

velocity (indicated

_{in ~Lm/s on} each

photograph)

G

=

38 K/cm and AT

= 0 2 °C

30

ji

~i

~ 20

~j

w

i~

~

o

2 4 6 8

Velocity (~m/s)

Fig 11 Facet _{size versus} the

pulling velocity

N 3 FACET DESTABILIZATION AT THE Sm A~Sm. B INTERFACE 365

sopm

Fig.

12 Sequence ^of pictures

showing

the cyclic behavior of the interface _near the critical velocity.

G

= 31 K/cm, AT

= 0.2 °C and V

=

0 8 ~Lm/s 0) t ₌ 0, ₁₎ t ₌ 85 _s, 2) t ₌ 230 _s

on

80

11

©~ 60

~o

~i 4

E40

20

o

° ~

~veloci(y

V

(~n~/s)

^{~° ~~}

Fig

13 Front

undercoohng

8T~~ = Ts~j ^T versus

pulling velocity

V The solid line _is the best fit

wtth _a

@

_law

4.

Macrostep dynamics.

We measured the

velocity

of the macrosteps at _a fixed temperature

gradient

and found that it

strongly depends

_upon their

shape,

the

sharp-edged

_macrosteps

propagating

^{faster than} the rounded _ones. We found that the

velocity

of _a rounded macrostep

is,

within _our

expenmental

error, close to 0 3 v

irrespective

of their width On the contrary, the

velocity

of _a

sharp-edged

macrostep ^{lies between V and 2 V}

depending

_on the tip

sharpness.

The

sharper

the

tip,

^the

larger

_is its

velocity

In

figure14

_we show the dnft

velocity

_Vd~i~ of the

sharp~edged macrosteps

as a function of the

pulling velocity

V We _see that Vd~i~ is

proportional

to V _in spite ^of a

large dispersion.

^The same result holds for rounded

macrosteps.

Another

interesting question

_is how the macrosteps interact. We have thus observed the

dynamics

of _an isolated _pair of

macrosteps

^of

opposite

_signs,

approaching

each other.

« Isolated _means that the other macrosteps are far away, ^at a distance which is greater than

twice the diffusion

length

Such situation _occurs

only

_near the onset of

instability.

Expenmentally,

either the two

macrosteps

are rounded and

they

_remain

unchanged

till

they

join

together,

or

they

are

pointed

with _{a cusp} where the

impurity

^{molecules collect.}

5

(Sharp~edge macrosteps(

4

o

3

>

~ l

~

_* ~

~ ^{~ . ~ #}

i

~ *

$ ~

$

* *

o

~

o _o

o

f O

I

o o

o

0 2 4 6 8 10

V(~m/S)

Fig

14 Dnft

velocity

of

sharp~edged

_{macrosteps versus} the puffing

velocity.

The second _case is

by

far the _more

frequent. Figure

15 shows _two

sharp~edged

_macrosteps

approaching

each other.

Here,

the two

macrosteps

are

slightly asymmetrical.

For macrosteps

having exactly

the _same

height,

the distance d separating them

(Fig. 16a)

decreases

linearly

versus time wuich _means that the two

macrosteps

move at constant

velocity. Figure

16b shows

the position h+

(resp h~)

of the hot

(resp. cold)

facet _versus time h+ _is constant, which

means that the hot facet _{is quasi} immobue _in the

temperature gradient,

while h~ decreases _so

that the other facet recedes and

consequently

cools down This _is the consequence of _a

global

increase of the

impurity

concentration at the cold facet, As _a

general rule,

the colder _a

facet,

the faster it recedes in the temperature

gradient

and cools

The

situation

_is s1mllar when _two rounded macrosteps meet before

deve16ping

_a

sharp edge, except

^for one point, in tills _case, the cold facet retreats faster than

before, mainly

because there is _{no cusps into} wuich

impurities

_{can escape}

Finally,

we

emphasize that,

_in both _cases, the dnft

velocity

starts to decrease when the _two

macrosteps

are very close each

other, namely

_a few ~m, i e at _a distance much smaller than

N 3 FACET DESTABILIZATION AT THE Sm A-Sm. B INTERFACE 367

loopm

Fig

15. Two

sharp~edged

_macrosteps

approaching

each other

They

leave _a rounded

macrjstep

_after

partial

annihilation

the diffusion

length.

In

general,

the two macrosteps ^anneal in the

end,

either

completely

if

they

_are of the _same

height

_or,

partially, by leaving

_{a new} rounded macrostep, if

they

_are not

exactly

of the _same

height (see Fig. 15)

0 loo 200 300

t(s)

~~

^{° h+}

. h~

~ a o o ~ o o ^{o a} o o Da _{a ~ ~ ~ o} a

E

/5

~

~~

jf

_{o o.}

°

o ° o o

o ~

~~

^o o.

o o

o o

5

0 loo 200 300

t(s)

Fig ¹⁶ a) Distance between _two

sharp~edged

_macrosteps propagating m opposite directions _versus time b) Position h+ (resp. h~ of the hot (resp

cold)

facet _m the temperature

gradient

_versus time (the ongln has been chosen

arbitranly)

5. Discussion.

One of the _main results of tuis

study

is that there _{is no}

stationary

_{regime, even near} the onset of

instability.

The facet advances

by

_generating macrosteps which propagate

along

the

macroscopic interface. The

leading edge

of these macrosteps is

rough just

after formation. It

is

always unstable,

the bottom _{reentrant corner}

advancing

_more

slowly

than the

top

corner

and results in an

overhang

with a

sharp tip

and _an

impurity~nch

_cusp.

Let _us mention that similar

behavior, namely

loss of

stability

of

macrosteps

^and formation of _a flat inclusion under the

overhang,

_is sometimes observed

during growth

from solution of

faceted materials

(sucrose

for

example [13]).

N 3 FACET DESTABILIZATION AT THE Sm A-Sm B INTERFACE 369

So

far,

there is _no unified

theory describing

the behavior of the

macrosteps

in directional solidification To understand

qualitatively

their

dynamics,

we focus _{on opposing}

physical

mechanisms

[14]

Let _us first _assume that _an isolated macrostep has been formed _on the

growing

interface and that G

= 0. We

neglect capillarity

and attachment kinetics. The

equiconcentration

lines of the solute may be

represented

_as in

figure17a. Here,

the surface _{is an}

equlconcentration

^line.

Over the top comer of the macrostep, the

equlconcentration

^lines

shrink, enhancing

the

growth

rate

there,

^while in the vicinity ^of the bottom _comer, the lines _move aside _so that the

growth

rate decreases This _is the consequence of the conservation law of the solute.

V~

AC

=

D aC

Ian (7)

Thls diffusion

mechanism,

which is

responsible

for the dnft of the macrostep, ^its

amplitude

increase, and its

shape instability

is just ^the well-known Mullins~sekerka

instability adapted

to the geometry of _a

macrostep.

On the other

hand,

the

temperature gradient

and the surface free energy both _oppose the

growth

of the

macrostep.

In this _case, the

equiconcentration

^lines cut the solid _as shown

schematically

_in

figure

17b. The _main effect is to reduce the

growth

rate of the top corner and

conversely

_{to increase} that of the bottom _corner

Nevertheless,

the

expenment

shows that these two effects cannot suppress the

destabilizing

effect of the diffusion field

Concernlng

the kinetic

effects,

we first

emphasize

that

they

are much smaller than _in usual

plastic crystals

such _as salol

[15].

On the other

hand,

the

macrostep

^behavior is

qualitatively

the same in all the

samples

_near the onset of

instability

and does not

change significantly

^with

the

velocity.

As the kinetics _are

naturally strongly dependent

_on the

velocity,

_we conclude that

they

do not

play

_a dominant role _in the drift and the destabilization of the macrosteps

By

d)

b)

Fig 17

-

a) Isoconcentratlon lines

near a rounded _macrostep

dashed lines) hen G = 0 _and

capillanty is egligible, the interface

itself is

contrast,

they

_are

perhaps

_more

important

in the

high~velocity

_regime which _is observed _in the

purest samples, favonng

hot _over cold facets _{as was} shown _in reference

II

Finally,

_we discuss the microscopic

growth

mechanism that _is

responsible

for the observed kinetics. Two mechanisms _{are a priori}

possible.

the first _{one is} set

by

the

nucleaiion

_{and the}

growth

of two~dimensional nuclei _on the facet. Th1s _{is a} slow process which controls the facet

growth only

if the

crystal

_is

nearly perfect.

This _is not,

by far,

the _{case in our}

expenment

_;

the second _{one is a screw} dislocation

growth

mechanism It is much _more

likely

for

many reasons

first,

_we know that _screw dislocations _{are numerous in our}

samples

because

there

always

exists _a small

angular

^mismatch 0 between the scratches of the surface treatment on the two

glass plates

Thls mismatch induces _a twist deformation of the

layers

which _can be

relaxed

by introducing elementary

screw dislocations with _a

density ^A~'

=

o/b

where b is the

layer

thickness and A the _mean distance between two dislocations

Expenmentally,

0

=10~~

_{rd and b}

=

3

x10~~

cm so that A~ ^'

= 3

x10~~ cm~'.

_{In other}

words,

there _{is a}

screw dislocation _every 0.3 _~m wuich _means that _even the smallest facet _is

pierced by

several

dislocations.

Second,

there

ale

theoretical

predictions according

to which the

resulting

kinetic

undercooling 8T~,~

^is

proportional

to

vi _[16].

_{This behavior}

is observed

expenmentally,

the best fit with the data of

figure

13 giving 8

T~,~(°C)

= 0.023

@ _{(~m ). Finally,}

_it

is

possible

to evaluate the minimum

undercooling 8Tz~

that _{is necessary in} order that _a

step joining

two

screw dislocations of

opposite

_signs works _{as a} Frank Read _source. A

simple

calculation

gives

8Tz~=2flT~/Lbd

where d _is the distance between the two

opposite dislocations,

fl

the free energy of the step

(fl =3x10~~erg/cm[3]),

L the latent heat

(L

= 5 _x

10~ erg/cm~)

and T~ the transition temperature

(T~

₌ 323

K). Expenmentally,

the shortest distance d _is of the order of 0.3 _~m from which _we calculate

8Tz~

₌ 0 05 °C. Of course, this _{is an} overestimation of the

undercoolmg

that _{is necessary} to make _{grow a} facet because there _{is a}

large

distribution of distances d and in

particular

_some values of d much

larger

than 0.3 _~m. In

conclusion,

a very small

undercoolmg

is sufficient to make

grow a facet _{via a screw} dislocation

mechanism,

_{m agreement} with _our measurements.

6.

Concluding

remarks.

So

far,

_we have treated the smectic~A

liquid crystal

_{as an}

ordinary isotropic liquid

In

fact,

this

is incorrect for at least two reasons :

first,

diffusion _m the sniectic~A _is amsotropic ^We showed in reference

[2]

that this effect _is

responsible

for the dnft of the cellular

pattern

^{when the}

layers

make _an

angle

with the

temperature gradient

We tuink that this

amsotropy (D~ _/Djj

=

7

)

must accelerate the dnft of

the"macrosteps

_because

_impurity

diffuses faster inside the

layers

than

perpendicular

to

them

second,

the smectic~A _{is an}

organized phase

Its lamellar structure inhibits the solute~

dnven convection

present

in many expenments Such convection

leads

_to

an increase of the

critical

velocity

with the thickness of the

sample (see,

for

instance,

the _case of the nematic-

isotropic

interface

II 7])

We checked that in _our system the cntical

velocity

is

independent

of the thickness

(which-we

varied from 10 to 50

~m).

On the other

hand,

there _are elastic effects

which _are due both to the

change

of

density

^and to the difference of

layer

_spacing between the two smectic

phases [18].

In

particular,

the smectic A

phase

can sustain _a stress normal to the

layers. However,

_{one can}

easily

show

(see Appendix II)

^that in the situations encounted in this

article,

elastic effects _may be

neglected

In

conclusion,

we have shown that the faceted Sm.A~Sm.B interface is unstable above _a critical

velocity,

^which is given

by

the classical constitutional

undercoohng

criterion We have