Morphological stability of circular germs in a discotic liquid crystal

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Morphological stability of circular germs in a discotic

liquid crystal

P. Oswald

To cite this version:

Morphological stability

of circular

_{germs in}

a

discotic

_liquid

_crystal

P. Oswald

Université de Paris _Sud, Laboratoire de

_Physique

des Solides, 91405

_{Orsay Cedex,}

France

(Requ

le 30 mai

_{1988, accepté}

le 26 août

₁₉₈₈₎

Résumé. 2014 Nous

étudions dans le cadre de la théorie de Mullins-Sekerka les mécanismes de déstabilisation d’une

_gouttelette

d’un

_{cristal liquide discotique}

en _{train de pousser dans} sa

_phase

_isotrope.

Le matériau choisi

est

_{l’hexaoctyloxytriphénylène.}

On en déduit une mesure de la

_{longueur capillaire do (proportionnelle}

à

03B3/L ;

03B3 est la tension de surface et L la chaleur

_latente)

et des coefficients de diffusion de

_{l’impureté}

dans le

liquide

et dans la

_phase

_hexagonale.

l’évolution non linéaire de la

_{gouttelette après}

destabilisation est

également

décrite. Abstract. 2014

We

study

within the framework of the Mullins-Sekerka

_theory

the destabilization mechanisms of a

droplet

of an

_hexagonal

discotic

_liquid

_{crystal growing}

in its

_isotropic

_liquid.

The chosen material is the

hexaoctyloxytriphenylene.

It

_yields

a measurement of the

_capillary

_{length do (proportional}

to _{03B3/L ;} 03B3 is the surface tension and L the latent

heat)

and of the diffusion coefficients of

_impurity

in the

_liquid

and in the

_{hexagonal phase.}

The non-linear evolution of the

_droplet

after destabilization is also described.

1. Introduction.

In a recent article

_[1]

we

_reported

observations on

the dendritic

_growth

of an

_hexagonal

discotic

_liquid

crystal.

This

_experiment

was

_designed

to test recent

theories on the role of the surface tension

_anisotropy

in dendritic

_growth.

_Satisfactory

_agreement

was

obtained between

_theory

and

_experiment.

In _{this paper} we are interested in the first

_stages

of

the

_growth.

We describe the mechanisms of destabili-zation of a _{circular germ} and we use the results of the classical

_theory

of

_{morphological stability}

_[2]

to

calculate the

_{capillary length}

_{do (proportional}

to the ratio

y/L

where y is the surface tension and

L the latent

_heat)

and the diffusion coefficients of

impurity

in the

_isotropic

_liquid

_{(DL )}

and in the

hexagonal phase

(DB).

In addition we describe the

weakly

non-linear evolution of a

_growing

_germ

_by

expanding

its

_shape

into a Fourier series. 2.

Experiment.

The chosen material is the

_{hexaoctyloxytriphenylene}

(HET).

It is the same as in

[1].

The

_hexagonal

_phase

of HET is formed

_by

a

_{regular packing}

of

_parallel

and

_independent

columns of disc-like molecules. The pure

compound

exhibits a columnar

mesophase

between 67 and 84.4 °C and is an

_isotropic

_liquid

above 84.4 °C

_[3].

The

_{phase diagram}

of HET in the presence of small amount of

_impurities

is

_given

in

_[1].

We shall use it to calculate the

supersatu-ration A. The

_sample

is

_prepared

between two

parallel glass plates

and is

_placed

into a Mettler hot

stage.

The

temperature

is controlled to about

± 0.02 °C. A Leitz

_microspe

is used for

optical

observations. We _{observed that very thin}

_samples

_(a

few J.Lm in our

experiment)

orient

spontaneously

with the columns normal to the

_{glass plates.}

That

way the

hexagonal phase

grows in its basal

plane.

We worked with a

sample

whose

liquidus

and solidus

temperatures

were

respectively

81.2 °C and 75.6 °C.

The

_temperature

was chosen

_equal

to 81 °C. The

undercooling

is then - 0.2 °C.

_According

to the

phase diagram

of [1]

the

_{supersaturation}

is

A = 0.1 ± 0.01. At this

_temperature

the rate of nucleation is very small so it is

_possible

to follow the evolution in time of a

_single

_{germ. The sequence} of

pictures

of

_figure

2 shows the

_growth

of an

_initially

smooth and _{circular germ. Below} a certain radius

R6

of the order of 15 J.Lm the germ is stable and

circular. Above this radius it becomes unstable and

an

_hexagonal

modulation

_develops.

At

_long

times

this deformation leads to six dendrites

_growing

at 60

2120

Fig.

1. -

a),

Hexaoctyloxytriphenylene

b)

Hexagonal

columnar

_phase.

The disc-like molecules

pack

in

_{long parallel}

columns

_forming

an

_hexagonal

array.

deg.

each from the other

_along

_{the (1120)}

directions

[1, 4].

Let us see now in more detail how these

observations can be

_compared

with the

_theory.

3. Time evolution of the area.

We measured with a

_planimeter

the surface area A

of the germ as a function of time. The accuracy is

about 1 %. As shown in

_figure

3 the curve

_{A (t)}

is a

straight

line whose

_slope

is

_equal

to 5.15 x

10 - 8

cm2/s.

This behaviour shows that the

size of the germ increases at

t1/2

which means the

growth

is controlled

_by

diffusion. Because our

sample

is very thin and sandwiched between two

glass plates,

thermal effects are

_negligible.

As a

matter of

fact,

heat can _escape

_through

the

_confining

walls whereas matter cannot. So we

_only

have to

consider the two-dimensional

_problem

of

_impurity

Fig.

2. -

Sequence

of

_{micrographs showing}

the evolution of a _germ

_growing

at fixed

_{supersaturation A}

= 0.1. Below

a certain radius

_{R6 -}

15 J.Lm the germ is circular. Above

this radius an

_hexagonal

modulation

_develops.

1 :

t = 100 s

2 t = 150 s 3 t = 250 s ; 4 : t = 315 s ; 5 :

t = 376 s 6 t = 526 s

7 : t = 1 065 s ; 8 : t = 1 785 s ;

9: t = 3 120 s; 10: t = 4 695 s;

11: t = 5 955 s; 12 :

t = 7 950 s.

diffusion. Let

_{C (r)}

be the

_impurity

concentration

and

_{u (r)}

the dimensionless concentration

_{[5] :}

Co

is the concentration in the

_liquid

on the interface

(assumed

to be

_flat)

and k the

_partition

coefficient of

impurity.

The

_equations

to solve are

_{[5] :}

They

are

_subject

to the

_boundary

conditions :

u (R, t )

= -

do/5t (Gibbs-Thomson equation) (4)

Fig.

3. -

Surface area A of the germ as a function of time.

The

_slope

is

_equal

to 5.15 x

10- 8 cm2/s.

supersaturation

defined to be

_{( C 0 - Coo) /}

(Co(l - k ) ) and do

the

_capillary

_length

which is

given

in the chemical model

_{by :}

m is the

slope

of the

liquidus

and T * the

melting

temperature

of the pure

compound.

The resolution of

_{equations (2)-(5)}

can be found in

[6].

The radius is

given by :

where A satisfies the

_equation

_{[7] :}

v is the Euler’s constant

(Ln v2

_{= 0.5772).}

Equation (7)

shows there is a critical radius

Rc

above which the germ will grow rather than melt for the

_given

_{supersaturation A :}

In our

_experiment

R >

_Rc

and the solution of

(Eq.

(7))

is

_{given by :}

dA / dt

does not

_depend

_upon the

_shape

_{of the germ}

[9]. Consequently

the linear law Aat we

_{have just}

demonstrated for a _{circular germ still}

_applied

when the germ becomes unstable.

Experimentally,

A = 0.1 ± 0.01 whence

by using

(Eq. (8))

A 2

= 0.0366 ± 0.070. From the

slope

of

A (t )

and

_{(Eq. (11))}

we calculate

_DL

= 1.1 ± 0.3 x

10-7

em2/s.

This value is in accord with the one

we obtained in directional solidification

_[10].

4.

Comparison

of

theory

and

_experiment

in the linear

_regime.

In this section we are interested in what

_{happens just}

after destabilization

_{(R >}

_R6).

In this linear

_regime

the

_shape

_{of the germ is}

_given

to a

_good

approxi-mation

_(see

the

_{following section)}

_{by :}

0 is the

_{polar angle}

in the

_plane

of the

_sample.

We

measured

_{ð 6}

and we

_plotted

Ln

_{ð 6}

versus time in

figure

4. Its

_{slope gives}

the

_growth

rate _W6 =

86/ ð6.

This curve seems to admit a

_point

of inflexion. At

this

_point

the

_growth

rate is maximum. We shall discuss this

_point

in the

_following.

Let us now

remember the

_predictions

of the linear

_theory.

The Mullins-Sekerka

_theory

of a

_{slightly perturbed}

germ has been

presented

in detail in reference

_[6].

For the time

_being

we leave out

_anisotropy.

This

Fig.

4. - Ln

S6

versus time.

_S6

is the

_amplitude

of the

hexagonal

modulation which grows

just

after destabili-zation. This curve _passes

_through

a

_point

of inflexion at

time t = 370 _{s :} the radius is then R =

R6* ’"

23

>m. The

slope

of the curve at this

_{point gives}

the maximum

growth

2122

theory

shows there is a radius

_Rj

above which the .

amplitude

_6j

of

_{the j-th}

Fourier

_component

of the surface grows :

This

_{theory gives}

also the

_growth

rate

_wj

=

Sj/ðj

of the

_{j-th perturbation.}

Calculations are made

_by

looking

for a diffusion field of the form :

where

Substitution into

_(Eqs.

₍₄₎

and

₍₅₎₎

leads after

some

_algebra

to :

where

f3

=

DH/DL.

The

j-th

mode

develops

when

wj >

0 i. e. when R >

Rj

with :

By using

(Eqs. (16)

and

₍₁₇₎₎

the

_growth

rate of the

j-th

mode can be rewritten in the form :

or

_according

to

_{(Eq. (7))}

and on the

_assumption

that

R0 > Rc :

wj

passes

through

a maximum for

_Ro

=

Ri* = (3/2) Ri

with:

This theoretical model shows that all the modes

j >

2 can

_develop.

The first one to _{appear should be}

the j

= 2 mode.

Experimentally,

_we observe that the

, first

mode is

_always

_{the j}

= 6 mode. This

discrepancy

is due to the fact we have

_neglected

the

_crystalline

anisotropy.

As

_pointed

out be Cahn

_[11],

the main effect of a

_slight

_anisotropy

of the surface _energy

_(of

the order of 2.5 x

10- 3

in our

_system

_[1])

is to

provide

an initial

perturbation

which is then

en-hanced

_by

the diffusion field. In other

_words,

the

anisotropy

selects the modes which will grow. In our

experiment they

are the

_multiples

of

_{the j}

= 6 mode. Some recent numerical simulations

_fully

confirm this

interpretation [12].

The

_question

is now to know whether formulas

₍₁₇₎

and

₍₁₉₎

are still valid. Coriell and

_Hardy

_[13]

have taken into account the

ani-sotropy

86 of the surface energy in the

stability

analysis. They

have shown that the

_only

modification is to introduce a

_{multiplicative}

factor which is

_given

for

_{the j}

= 6 mode

by :

This factor is

_negligible

when

_{56 > Rb}

’-6 - 0.04 )JLm.

Experimentally,

we are not able to

detect so small deviations so we shall

_disregard

this factor in the

_following.

Experimentally

we have found

_{R6 -}

15 J.Lm.

Using

(Eq. (17)),

we can now calculate the critical radius of nucleation

_Rc

=

do/..1.

If we assume

₁₃

= 0 _we calculate

_{Rc ~}

2 600

A

and

_{do -}

260

A.

This value of the

_{capillary length}

is a little

_larger

than the one we

directly

calculate from

_{(Eq. (6)) : do -}

190 A

_[14].

An

_explanation

of this difference may be that

13

is different from zero. This is

_{quite possible}

because

the columns are

_liquid

and not correlated

_[3].

In this

case a

_good

_agreement

is found between static and

dynamical

measurements

_by

_taking

_{13 -}

0.35 or

equivalently DH ~

0.35

_DL.

Let us note that

_DH

is the diffusion coefficient of

_impurity

normal to the

columns. It must be much

larger

than the one

_along

the columns.

As

_regards

to the

_growth

rate, the linear

_theory

predicts

it must _pass

_through

a maximum as the

radius is

_equal

to

R6

= 3/2

R6 -

23 _)JLm

namely

at time t - 370 s

_(see

_Fig.

_5a).

This is

_compatible

with

the curve of

_figure

4

which,

in

_spite

of the

_large

error

bars in this

_region,

seems to have a

_point

of inflexion

at time t = 370 s. At this

point

_{we measure}

w6Max. -

4 x

10- 3

s-1.

From

_{(Eq. (20))}

we calculate

DL -

2 x

10- 7

cm2/s.

This value is of the same order of

_magnitude

as the one we have deduced from the area measurement and we obtained in directional

solidification.

In conclusion our

_experiment

is in

_agreement

with the

_predictions

of the Mullins-Sekerka

_theory.

In the

following

section we shall look at what

_happens

in the non-linear

_regime

i.e. at

_large

deformations. 5. Non-linear evolution.

A

_good

_way to

_study

the

_shape

_{of the germ is} to

expand

R (0, t )

into a Fourier series. Because of the

six-fold

_symmetry

we have :

with

Fig.

5. -

a)

Mean radius of the germ versus time.

b)

Fourier _component

_5,

versus time :

0 j

=

6 ;

o j = 12; V j = 18 ; ð j = 24.

We have

_plotted

_R0(t)

in

_figure

5a and

_{86 k}

in

figure

5b for various values of

_j

=

6k,

k

_{= 1, 2, 3, 4.}

As

_{expected Ro (t )}

follows a

t 1/2 law.

The first mode which

_develops

is

_{the j = 6}

mode. The

_following

one is

_{the j}

= 12 mode. It appears for

Ro > R12 -

60 J.Lm. At this

point

56/Rl2

-4 x

10- 2

« 1 so that the linear

_theory

is still

appli-cable.

_According

to it

_(see

_Eq.

₍₁₇₎₎

_{RI2IR6(Th.) =}

12 x 13/6 x 7 = 3.71 whereas the

experimental

value is

_{R121 R6 (Exp. )}

= 60/15 = 4. Here too there is a

satisfactory

agreement

between

_theory

_and

exper-iment. The fact that

₈₆

is

_positive

whereas

₈₁₂

is

negative

is

_certainly

due to the

_anisotropy

of the

surface tension y. It means,

according

to

_Cahn,

that the initial

_perturbation

which is enhanced

_by

the

diffusion field is of the form :

The

_shape

of a _germ is

_given

at

_equilibrium

_by

,y/,R

= Cst. We then calculate

[15] :

Hence

_according

to the

_previous

_{inequalities :}

It would be

_interesting

to test

_numerically

these

predictions

and to see for what values _{of y6 and}

y12 one can

_reproduce

the curves of

_figure

5.

6.

_Concluding

remark.

We have

_just

seen that

_freezing

an

_initially

_very

small germ

always

leads to the formation of six branches

_growing

_along

_{the 1120>}

directions. In fact it is

_possible

to select modes of

_higher

order

_by

cooling

initially large

and circular germs

(that

we

make grow at _{very small}

_{undercooling).}

An

_example

is

_given

in

_figure

6. This

_experiment

was made with a

Fig.

6. -

By

quickly

cooling

(10 °Clmin)

an

_{initially large}

and circular germ

(that

we make grow at _{very small}

undercooling)

we observe the

_preferential

development

of

the j

= 18 mode.

sample

different from the

_previous

one. Its

_liquidus

and solidus

_temperatures

were

_respectively

83 and

80.5 °C. The initial _{radius of the germ} was

R¿ ,....,

15 J.Lm.

By

quickly

decreasing

the

temperature

2124

grows. At this

temperature

the

supersaturation

is

A - 0.4. This observation can be

_{easily explained by}

looking

at

_{equations (19)}

and

_(20).

If

_{R’ 0}

is close to

R*

it is

_clear,

_according

to these

formulas,

that the

j = 18

mode is the one which has the

_{largest growth}

rate. So it must dominate and

_{firstly develop}

as

observed

_{experimentally.}

From

_equation

₍₆₎

we cal-culate

_{do -}

330

A

whence

_{Rc ~}

820

A.

We then calcu-late

_{R18 (Th.) ~}

8.5 J.Lm and

R*18 (Th.) ~

13 J.Lm

[16].

This value is well of the same order of

_magnitude

as

the initial radius of _{the germ.}

Acknowledgments.

I would like to thank Dr.

_Nguyen

Huu Tinh for

providing

me with the

_{hexaoctyloxytriphenylene}

and M. Mihailovic who lent me the

_planimeter.

I am

grateful

to Dr. C. Caroli and Pr. R. F. Sekerka for

fruitful discussions.

References

[1]

OSWALD, P., J.

_Phys.

France, to be

_{published (July}

1988).

[2]

MULLINS, W. W., SEKERKA, R.

F.,

J.

_Appl.

_Phys.

34

(1963)

323.

[3]

DESTRADE, C., MONCTON, M.

C.,

MALTHÊTE, J., J.

Phys.

France 40

₍₁₉₇₉₎

C3-17.

[4]

BOULIGAND, Y., J.

_Phys.

France 41

₍₁₉₈₀₎

1307.

[5]

LANGER, J. S., Rev. Mod.

_Phys.

52

₍₁₉₈₀₎

1.

[6]

CORIELL, S. R., PARKER, R. L., J.

_{Appl. Phys.}

36

(1965)

632.

[7]

This formula is

_only

valid if 0394 ~ 1. In this

case

03BB2

~ 1. This condition is verified

experimen-tally.

[8]

SEKERKA, R. _F., Private communication.

[9]

This result is

_only

valid when 0394 ~ 1. In this limit 0394C = 0

(quasi-stationary

approximation)

and

dA/dt =

-DL/2 C0(1 - k)

~c

grad C n ds.

C is an

_arbitrary

circuit

_surrounding

the germ. It

can be chosen as far from the germ as we want.

Consequently

dA/dt

is

_independant

of the

shape

of the germ.

[10]

The critical

_velocity

above which the

hexagonal-isotropic

front destabilizes is

_given

_by

Vc

=

DL

Gk/mC~(k - 1)

where G is the

tem-perature gradient

(see

Ref.

_[5]).

In our

exper-iment,

everything

is known _except for the dif-fusion coefficient

_DL.

We measured

_Vc

for various

_samples

and _temperature

_gradients.

We found

_{DL ~}

1.4 ± 0.2 x

10-7

cm2/s.

Our results on directional solidification will be

published

in a

forthcoming

publication.

[11]

CAHN, J. W., in

_Crystal

Growth, Ed. H. S. Peiser,

Pergamon

Press,

Oxford,

1967, p. 681.

[12]

LANGER, J. S., in Les Houches XLVI

_1986,

Chance and Matter, Ed.

_by

J.

Souletie,

J. Vanninemus and R.

_Stora,

_Elsevier, 1987.

[13]

HARDY, S.

C., CORIELL,

S. R., J.

_{Cryst. Growth,}

5

(1969)

329.

[14]

The values of the constants we used to calculate

do

are the

_following

ones

_(from

Ref.

_{[1]) :}

mC0

= 3.5

K, k

= 0.35,

03B3/L

=1.2

Å.

This last

value has been obtained from static

measure-ments.

[15] Experimentally

we have

_{03B36/03B30 ~}

0.09

_(see

Ref.

_[1]).

To derive the second relation we

assumed

_{03B312/03B30 ~}

_(143/352)

_(03B36/

_{03B30)2 ~}

10-3.

[16]

In this

_experiment

0394 = 0.4 and

_(Eq.

_[8])

is

not valid any more. In this case 03BB is

_{given by}

the

most

_general

_{equation (see}

Ref.

_{[6]) :}

03BB2 exp (03BB2) Ei (- 03BB2) + 0394 = 0

where Ei is the

_exponential

_integral

function.