Spinodal decomposition in two-component smectics

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Spinodal decomposition in two-component smectics

G. Sigaud, C. Garland, H. Nguyen, D. Roux, S. Milner

To cite this version:

G. Sigaud, C. Garland, H. Nguyen, D. Roux, S. Milner. Spinodal decomposition in two-component smectics. Journal de Physique II, EDP Sciences, 1993, 3 (9), pp.1343-1355. �10.1051/jp2:1993205�.

�jpa-00247911�

Classification

Physics

Ab.itia<.ts

05.70J 64.70M 64.75

Spinodal decomposition in two-component smectics

G.

Sigaud ('),

C. W. Garland

(', *),

H. T.

Nguyen ('),

D. Roux

(')

and S. T. Milner

(2)

(') Centre de Recherche~ Paul Pascal, CNRS, Universitd de Bordeau~ I, Av. A. Schweitzer, F-33600 Pe%ac. France

(2) ^Exxon Research and

Engineering,

Annandale, NJ 08801, U-S-A-

(Receii.ed lo Mw.cfi 1993, ac<.epted 9.time 1993)

Rdsumd. Les aspects originaux de la

ddcompo~ition

spinodale entre deux phases smectiques A de

compositions

diffdrentes sont

analysds

_par diffusion de la lumibre _et

microscopie

optique. Nous

montrons que cette ddmixtion caractdrisde par un angle entre le~ _vecteurs d'onde [es plus intenses

et la normale _aux couches _est le r6sultat de l'instabilitd du mode

baroclinique.

Abstract. -A two-component ^smectic undergoing _a demixing tran~ition _to two coexisting

smectic pha~es exhibits _a novel

spinodal

pattern as observed by

light

~cattering and

optical

microscopy. We show that this

~pinodal,

characterized by a

prescribed

angle between the _mo~t intense wavevectors and the layer normal, is the result of the instability of the baroclinic mode of

two-cojnponent

^smectic hydrodynamics.

1. Introduction.

When _an

initially homogeneous

_system is driven _{to an} unstable state in such _a way that

nucleation is avoided,

phase separation

_processes

through

the

growth

of fluctuations in

composition

at wavevectors q i q~. this defines _a

spinodal decomposition

and results in _a concentration

profile throughout

the medium.

Scattering techniques

are most suitable to

give

evidence of this

phenomenon

in the

reciprocal

_{space :} the characteristic pattem ^is a

ring

_or halo

in which the maximum

intensity corresponds

to the _wavevector of the fastest

growing

fluctuations

(q~), Experimentally

^the

simplest

_way to achieve

spinodal decomposition

is _a

« critical

quench

_» in which the temperature ^of a

homogeneous sample

of critical

composition

is varied _as

rapidly

_as

possible through

^{the critical}

point.

The

phenomenon

_can also be viewed in the direct _space with an

optical microscope provided

^{that there exists} a

sufficiently large

difference in the refractive indices of the _two

separating phases.

(*) Present address _:

Department

of

Chemistry,

Massachusetts Institute of

Technology, Cambridge,

MA 02139. U-S-A-

A second

important

feature of

spinodal decomposition

is its time

scaling.

At very short times after

quenching

the

intensity,

of the

scattering

is

expected

to grow

exponentially

with its

position

unaffected

(q~

=

Const.,

Cahn's

regime).

At

longer

times the fluctuations _at

longer wavelengths

are

favored,

which results in the

collapse

of the

ring.

For

isotropic

_systems the

radius then decreases with _a power ^law q~ oz

t~

_with

=

for the

early

stage and 3

= -1 later

[I].

The _rate of the evolution

depends

_on the time for diffusion

through

a

correlation

length,

_t~

=

D~ ^'

f~,

which _can be

expressed

in _terms of the

depth

of the

quench (AT), viscosity

_(Y~

),

and characteristic bare correlation

length (fo)

D

=

Do

^{~~ ~ '}

=

~ ~~ ~ ~

and f

=

to

^{~~ ~ A ~~ ~~} (l )

Tc _~1

Tc

Tc

to

Thus,

i~ =

) ) ₁₁ )

=

~

°

) ₍₂₎

c~'~ _~~

^~"

^~~ _c~

^~~~~~

y and _{v are} the critical exponents ^that characterize the

susceptibility

and correlation

length, respectively.

A

large

_t~

corresponds

to a slow

decay

in the diameter. For

example,

the

study

of the

dynamic~

^of

phase separation

_over reasonable time scales

Ii-e-,

100

s)

in

simple liquid

mixtures

requires

_very shallow

quenches

(~ ⁵ mK ), which _means

equipments

with

extremely good

thermal stabilization

(10~~

K

[I ]. Experiments

on systems ^{in which} t~ ^is

large

are thus

highly

attractive. Some such

experiments

have been conducted in viscous media such _as

polymer

solutions _or blends

[2, 3]

or in systems ^with

large

correlation

lengths

such as

microemulsions

[4].

Also

investigated

more

recently

have been the characteristics of

phase separation involving mesophases

in _a nematic solution of _a

liquid crystalline polymer [5]

and between _a

lyotropic

lamellar

mesophase

and _an

isotropic

_sponge

phase [6].

The exponent ^of the power law which fits the decrease of the wavevector _{as a} function of time looks

quite

similar to the classical value obtained in

simple

^fluid systems

although

unusual

anisotropic

features could be noticed in the domain

growth.

Recently

_a critical consolute

point

has been observed in _a two-component ^{smectic A}

phase [7].

We present a

study

of the

spinodal decomposition

in this system. In the

experimental

section of this paper,

light scattering

and

optical microscopy pictures

recorded

during phase separation

_are described and

analyzed.

These

experimental

observations _are consistent with the

theory

for

spinodal decomposition

in such ordered fluids which is

presented

in the theoretical section.

2.

Experimental

section.

2, SAMPLE PREPARATION. The system ^{in which} a

phase separation

between two smectic A

phases

is obtained is _a

binary

mixture

involving

the

6-cyanonaphtyl-2 4-decyloxybenzoate (compound A)

^and the

4-hexyloxyphenyl 4-perfluoroheptylbenzoate (compound

B).

CioH2i°-)O_ O ~

(A)

~

C7Fiifi-~°2-filC6H13

(B) The

phase diagram

has been

published previously [7]

and is shown in

figure1.

Isotropic liquid

N cJ

~

F- Smectic A

phase

0,0 0,2 0,4 0,6 0,8 1,0

Volume fraction of mesogen B

Fig.

I.-Composition-temperature

phase diagram for the

liquid crystalline

binary ~ystem 6-

cyanonaphtyl-2

4-decyloxybenzoate 5 (left) ₊

4-hexyloxyphenyl 4-perfluoroheptylbenzoate

(right) (from Ref. [7]). N denotes _a nematic

region.

The

experiments

have been

performed

on a mixture with _a

composition

_as close _as

possible

to the critical

composition (volume

fraction of B _~~

~ ⁼

0.44

corresponding

to mole fraction

~i~

~ ⁼

0,40 and

weight

fraction

w>~ _~ =

0.51).

The basic set-up ^for

light scattering experiments

is described in

figure

2. A

typical quench

was a few tenths of

degrees deep starting

^from about

? °C above the critical temperature

(final

temperatures were 0.2 °C and 0.4 °C below the

critical temperature ^{in the} two

experiments

for which

quantitative

data _are

given below).

Cooling

was obtained _I>ia two thermostats.

Using

_a three way ^valve the fluid from _a water bath maintained at 10 °C _was allowed _to circulate in the heat

exchanger

of the oil bath

regulating

the temperature ^{of the}

sample just

the time necessary to reach the final temperature of the

quench.

It took

typically

one minute to

bring

the cell at the set temperature. ^For

optical

observations the

thermostated cell _was

simply

moved _{on to a}

microscope

_stage. The _same

sample

and

sample

holder have been used for both

light scattering

and

microscope

observations. The

sample

semi transparent

screen cold bath

z

video recorder

He-Ne )aser

Fig.

2.

Light

scattering set-up.

holder _{was a} 100 _~m thick

glass

cell. Suitable

coatings

were

deposited

to obtain two different orientations of the smectic

planes

either

homeotropic (planes perpendicular

to the

light

beam,

Fig. 3a)

or

homogeneous planar (planes parallel

to the

light

beam,

Fig. 3b).

The orientation of the

sample

with

regard

to the

polarization

of

light

is

specified

in

figure

4.

~" ^-',,

'lfl(#t9+~-,".-

i^°"

~%lfi#I

(

E

°ating

~

@JaSi jl&SS

al ' ' ' '

_b

' ' ' ' ' '

' ' ' ' ' '

be%m

belm1

' ' ' ' ' '

' ' ' ' ' '

Fig.

3. The _two different orientation~ of the ~mectic A layer~ in the cell a)

homeotropic

b)

planar

homogeneou~.

Z beam

i

' n

" Y

X

Ey

Fig.

4. Orientation of the sample with respect to the polarized ^{laser beam in} the planar

configuration.

2.2 RESULTS.

Light scattering

_patterns recorded in the course of

spinodal decomposition

from

planar

oriented

samples

are

anisotropic they

consist of four blobs

symmetrically

located, as shown in

figure

5 in which the orientation of both the director and the

polarization

is vertical. This four-leaved clover geometry ^is reminiscent of the pattern

previously reported

for _a shear induced transition in

lyotropic

systems

[6] involving

a lamellar

phase.

Moreover the sequence of

pictures

in

figure

5

gives

evidence for the

collapse

of the

scattering

pattern. Two series of

images corresponding

to two different

quenches

have been

analyzed

in order to

study

the time evolution of the

position

of the maximum scattered

intensity.

The

quench depth

AT

= T~ T _was 0.2 K for

quench

and 0.4 K for

quench

2.

Figure

6 shows _a

logjo-logjo

plot

^of the modulus of the _wavevector in

(angstr0ms)~'

_{as a} function of time in seconds for both

quenches.

These results _can be

compared

to the classical power law

(represented by

the dashed line

corresponding

to a 0.33

slope)

over some time range. One _can note the saturation of _{q at} the very

early

stage ^{of the}

decomposition

for the shallower

quench although

the

Fig.

5.

Sequence

of

light scattering

patterns ^{selected from} the video-recording _on

quench

I for _a planar homogeneou~ sample. Time interval between _two pictures i~ 5 _s. axis (orientation of both the

director and the

polarization,

_see Fig. 4) is vertical.

-3

+ AT=0.2K

+ X + x AT=OAK

x x.

x +

x +

+

~ ^{x *}

w ^X +

o +

-4

0 2

log

t

Fig. 6.-logj~-logj~ plot

of the wavevector q a~ a function of time t for quenches (+:

T T 0.2 K) and 2 (x T T 0.4 K). ^The dashed line

corresponds

to 4

~ ' 3

faintness of the

scattering

at these

early

times does _not allow _a

highly

reliable observation.

Also noticeable from these _curves is the trend toward

larger slopes

at

longer

times in both

cases, which suggests a cross-over as observed with other systems

[1,

4,

6].

In addition,

image

analysis

indicates that the

angle

between the direction of maximum

scattering

and the director of the smectic A

phase

decreases

regularly

with time from _a finite value

ranging

between 60°

and 70° _{at t}

= 0, _as shown in

figure

7.

~~

fi

x AT ₌ 0.4 K

65 ^~ +

~

x +

~

*

+

~

~

~

+

z X

~

x

~

x .

55

x

50

~ ~~ ~~

t/ _s ^{~~ ~~ ~~}

Fig. 7. Plot of the angle & between the direction of maximum scattering and the director (Y axis) _{as a} function of time. _{+ :}

quench

I, _x quench 2.

It is also

interesting

to discuss the duration of the _two

experiments.

As

expected,

the faster

collapse

_occurs for the

deeper quench. Comparatively

the

decomposition proceeds

at a slower rate in _our system ^{than in}

simple liquids owing

to the

higher viscosity.

Thus

spinodal decomposition

_can be observed _{over a} reasonable interval of time for

fairly deep quenches.

Indeed the kinematic

viscosity

of _a smectic A

liquid crystal

is

roughly

two hundred times

larger

than the

viscosity

of the usual

isotropic liquid

_(Y~s~

w 200 cP and the time scale is

expanded by

the _same factor. Then,

provided

that the bare correlation

lengths

_are

comparable

in both

cases, one obtains from

equation (2)

~~S~ 'iS,

~Y ' '

~~simple_liquid ~l~impleliquid

Using

mean-field values for the exponents _y + v =

1.5 _{one can} estimate that _a 0? K

quench

on the smectic A mixture would result in the _{same rate} of

decomposition

_{as a} 5-10 mK

quench

on a

simple liquid mixture,

which is consistent with the results in reference

[I].

The fabric-like textures observed in the direct space are also

highly

characteristic with _a mesh size

increasing

with time _{; see}

figure

⁸ (the orientation of the director is

approximately

S-W--N-E- in both

Figs.

⁸ and

9).

These observations

corre~pond

to a late stage ^{of the}

decomposition compared

to the

light-scattering experiments

^since

deeper quenches (AT

=

0.9-1 K

typically)

_are

performed

in transmitted

light microscopy

to

produce enough

contrast.

The ?D-Fourier transforms of these

pictures nicely

confirm the geometry ^{of the}

light scattering

patterns

(Figs. 9a-d).

The

corresponding

wavevectors have been estimated from

figure

9.

Fig. 8.

Sequence

of

microscopic

textures observed

during

the _course of

decomposition

(quench ^is different from quenches and 2). Time interval between two pictures is 5 _s, with A

being

the earliest picture. The orientation of the director is

approximately

S-W--N-E- The actual size of the field in each picture is (175 ~m x 175 ~m).

These

microscope

data _can be combined with the data from the

light-scattering experiments

to

produce

the

scaling plot

^{shown in}

figure

10. This

scaling plot

has been made for mean-field values for the exponents. ^{It confirms} an increase in the

slope

_~fi at

longer

times. In the _case of smectic A-smectic A transitions dominated

by

difference in

layer

thickness

(where coexisting phases

do _not differ

appreciably

in

composition),

a

special non-Ising

critical

point

is

predicted [8],

but the exponent ^{values for these} systems are not yet ^well established and such

non-Ising scaling

cannot be tested.

Finally

the

quenches performed

_{on a}

homeotropically aligned sample provided

_no noticeable

scattering.

One _can

explain

this behavior if the difference in the refractive indices of the components is

quite

small in this orientation. One does not expect an

anisotropic

pattern in this

case and the

scattering

of

light

over a

ring

instead of

being

focused within four spots ^would result in _an insufficient

intensity.

A very

fuzzy picture

^is observed with the

optical microscope

and _no useful data could then be obtained from this

configuration.

3. Theoretical section.

We may summarize _our

description

of this novel

spinodal

as follows. The

spinodal

_pattern

results from the

growth

^of the most unstable modes of the two-component ^smectic

hydrodynamics

below the critical

point.

A

simple description

of two-component ^smectic

Fig. ^{9. 2D-Fourier} transforms of the

pictures

in

figure

8.

'

"

+ +

~ 'x+

*+~

*

$

_-2

o

o o

o

3

~ ~

log

~

~ ~

Fig. 10. Log-log plot of the scaled _{wavevector q} *

as a function of scaled time _r for the light

scattering experiments

(+: AT =0.2K, _x: AT=0.4K) and _an

optical

observation lo AT=0.9K).

T T~ v T T~ ^(Y +v

~~ _-

T~ ~ and _T

=

T~

t. MeanfIeid exP°rents Y ₌ ~ =

0.5 have been

used. The dashed lines

correspond

to the classical values 4

= and ~b -1.

3

Fig,

il. Sketch of the

layer displacement

in the baroclinic mode. Note the

oblique

wavevector.

hydrodynamics [9, 10]

above T~ contains

coupled equations

for the smectic

layer displacement

u, the concentration _c, and the _{transverse momentum} g~. The

layer displacement

and

concentration variables _are

coupled

in the free energy because the

layer spacing

is different for

the two mesogens

[7]

and thus a function of concentration. The three first-order time-

dependent hydrodynamic equations give

rise _to three

hydrodynamic

modes _{: a}

pair

of

propagating

second-sound modes, and _an

overdamped

«baroclinic

» mode, in which the

eigenvector

is _a mixture of

layer displacement

and concentration variation (see

Fig.

II ).

Near the critical

point

of such _a system, ^{the slowest}

[I II hydrodynamic

mode is made slower

by

the

vanishing susceptibility

_(« critical

slowing

down

») [12]

in the present case this is the

overdamped

baroclinic mode.

Upon

_a critical

quench,

this

critically

slowed mode becomes unstable and grows

[13].

Near the

spinodal,

where the

growth

or

decay

rate of the baroclinic mode is small

compared

to the

frequencies

_or relaxation _rates of all other modes, the

dynamics

is well described

by

an adiabatic

approximation,

in which the time derivatives of all _non-

critical variables have

ordinary

relaxation _rates, and _so relax to values such that the

hydrodynamic

_« forces _{» on} these variables

vanish,

and _we obtain _a

single

first-order time-

dependent equation

for the

growth

of the baroclinic mode.

The

fastest-growing

wavevector

points

in _an

oblique

direction for two basic _reasons. First, the

diffusivity

of the mesogen concentration is

strongly anisotropic,

with molecules

diffusing mostly along

the

layers

_; this suppresses the

growth

of fluctuations with wavevectors

parallel

to

n

(z-axis).

Second, the

spinodal corresponds

to the

vanishing

of B, the smectic

layer compressibility

at constant chemical

potential,

_so that the

growth

of the unstable baroclinic mode is driven

by

_{a «}

negative layer compressibility

_». This

requires

the wave,,ector to have

some

projection along

n. Our detailed calculations below show that the

angle

^{between the} most unstable _wavevector and the =-axis is related _to the fourth-order elastic coefficients.

Our

description

of the free energy

density

and

hydrodynamic equations

for _a two-component smectic is taken from Nallet _et al.

[9]

this system was

originally

described

by

Brochard and de Gennes

[10].

The free energy

density

may be written in terms of the

layer displacement

_u

and the deviation _c. from the _mean mesogen concentration _as

f

=

B(J,u)~

₊

K(V(

a)~ + x ^' c~ ₊

C,

_c d-u

(3)

2 2 ?

The smectic

compressional

modulus at constant concentration is denoted

by

B, and the

bending elasticity by

K. The

coupling

term

C,

_c d-ii expresses the fact that the

layer spacing

depends

on

composition.

If pure

phases

have _a relative difference

(~

ⁱⁿ

^{layer spacing,}

we

~

Sh°U'd ~~P~~t

C<

_~ ^B

(~

0

It is useful to define _{a new} concentration variable fi

= c +

XC,

_c

d=u,

and to include _{a source}

term hc. In terms of F

land

_upon Fourier

transforming)

we have

f(q)=)(hq)+Kq[)ju~j~+)x-'jBjq)j~+xc,q=hj-q)ujq)-h(-q)Bjq), j4)

where B

=

B

XC ).

^The fluctuations of

u(q)

and

F(q)

_are

independent.

We _see that _{as x} becomes

large,

B is driven _{to zero,} for _a finite value of _x. When

B

vanishes,

the fluctuations in

u(q)

with

q~10

become

large. Moreover,

fluctuations in

c(q)

become

large

we have

in iq) Hi-

_q

)i

=

ihq]

₊

Kq[ )-

^'

(F(q) Ii- q))

= x

(5)

(c (q

_c (- _q

))

= x

(Bq)

₊

Kq[ )(Bq)

₊

Kq[ )-

^' We say that

u(q=) [and c(q=)] undergo

critical fluctuations _as B

- 0, but B is non-critical.

Reference

[9]

describes the

hydrodynamics

of two-component ^{smectics in} a

simplified

and useful

form,

under the

assumptions

of

I) incompressibility

_;

2)

_constant temperature ;

3)

_no

«flexodiffusion»,

I-e-, no

off-diagonal Onsager couplings

between _c and _u

4) highly anisotropic

diffusion of the

concentration,

_so that diffusion is confined _to the directions

along

the smectic

layers;

and

5) isotropic viscosity.

With these

assumptions,

the linearized

hydrodynamic equations

_are

d~u =

~~

g~

q

~

~tgt

_~

~)~

^{gi +}

~j $ ₍₆₎

"q~

_dF

~tC " fi

p

P ^~

(Here

F

=

f

dV is the free _energy. )

These three

equations

have three

eigenmodes

_one

pair

_or

propagating

second-sound

modes, with

frequency

_w

=

~ ^~ ~°~~

,

and _one

overdamped

_« baroclinic _» mode. The

P q

baroclinic mode has _a relaxation

proportional

_to B _as the

spinodal

is

approached

and

B

vanishes,

the baroclinic mode is

critically

slowed. Because B

land

_g~) do _not have _a

diverging susceptibility,

_we expect an adiabatic

approximation,

_d,g~

= d~B = 0, to be

appropriate.

In this

approximation

_we have _a

single equation

of motion

0

=

Ii

+

~~~' _~~~

_~~

d~u +

~~

(7)

qj «q au _r

[hql

₊

_Kq~ f(4)1 ql

W

lq

₌

~ ~ ~

'~~~ ~~~~~~

(8) Kq~ f (4

m

Kq[

_{+ y}

q]

_{+ Y2}

ql ql

where _we have defined

q(w ^~~

^{'~ ~'} In

equation (8)

_we have included additional terms

a

~

(yj qi

_{+ y~}

qi q(

₎

u

(q)

^~ in the free energy

given by equation (4),

since _{we are} interested in

cases where B is small _or

negative.

^We expect ^from

scaling

considerations that these

higher-

order elastic coefficients _y and y~ will be of the order

Bf~,

_where

f

is the mean

layer spacing.

We _note that in

typical thermotropic

smectics, the

penetration length

A defined

by

~

WA ^~ is close to the

layer spacing f [14]

_; hence the function

f(()

_may be

only weakly

B

dependent

_on the direction of _cl.

We expect ^{the wavenumber} _qo to

correspond

to a

microscopic length

as well _{; note} that

qo satisfies ~

q(

=

'~

q() ^"~~ _q(~,

^{where the left-hand side is a}

^typical

second-sound

P P p

frequency squared,

and the

right-hand

side is the

product

of relaxation rates for viscous

diffujion

^and concentration diffusion.

Using

_a

viscosity

Y~ ⁱ

I,

a molecular

diffusivity

"~~

i 10~ ^~ smectic modulus B1

10~

_and

density

_{p i} I for

typical

^smectics (cgs units), P~

[15, 16]

_we find ~ _~

i

201.

qo

Now consider the _case of a

quench

below the

spinodal, I-e-,

such that B _~ 0.

Thy

^relaxation

rate for the baroclinic mode in the limit of small q

(except

^for _{q- -}

0)

i~ _w

(q )

i

~~j

,

which is

~lqo

negative

in fact, _a small baroclinic distortion with such _a wavenumber would grow with this

growth

rate. The fastest

growing

modes, which dominate the initial

spinodal

_{pattern, are} those wavevectors that minimize the

decay

rate,

equation (8).

Consider _a

relatively

shallow

quench,

such that ~

« IA

qo)2 (the

latter

quantity,

_as

argued

B

above,

being

of order of

unity).

Then the denominator of

equation (8)

_may be

approximated

_as

q( q)

_;

minimizing

with respect to q and am tan~ ^'

~~ leads _to

qi

sin² 9)

~ A B

~ 2

f

a

)

~

(9)

~ ^d sin

(2

9

do

f

⁹

If

f(

9

)

m

f ((

9 ) is not too

anisotropic,

the

optimum angle

will be _near to 9

= ± ^~ ; we have

4

then q~

~

For _a

sufficiently deep quench

such that ~

w

(Aqo)~,

the adiabatic

approximation

is B

questionable,

because the

growth

rate of the unstable mode is _no

longer

small

compared

to the other characteristic time scales.

Making

the

approximation

_{anyway, one} ^obtains

~2_

_~

-i~ _)~) ~

~

~~~

(10) o

= +

~

4

Then the

fastest-growing

wavenumber is of the order of the inverse

layer spacing,

and the

preferred angle

^between _q ^{and the} z-axis is

again

45°.

To summarize our results the

fastest-growing

modes of a two-component ^smectic

liquid

crystal quenched

below the critical

point

for the

demixing

^of the two

species

of mesogens are

growing

baroclinic modes. The

fastest-growing

wavenumber satisfies q~

~~

A ^~ which is B

the usual behavior for

spinodal decomposition

of _a conserved

density (that is,

_q~ is

proportional

to the reduced temperature, ^{since B} _w ^{B far from the critical}

point).

The

preferred angle

between the

fastest-growing

wavevectors and the

layer

normal _n should be in

general

^different from _zero and _more

specifically

of the order of 45° for systems ^with fourth-order elastic

coefficients that _{are not too}

anisotropic.

This

preferred angle

results from the

dispersion

relation for the baroclinic mode, and reflects the fact that II the diffusion of molecules is much faster within the smectic

layers

^than across

the

layers,

_so that modes with _q

parallel

to n are very slow, and

2)

the

driving

force for the

growing

baroclinic mode is the

negative layer compressibility ^fi

which _acts

along

the _z-

direction,

_so that the wavevector must have _some

projection along

n to be unstable.

Interesting

theoretical

que~tions

still _to be addressed include the nature of the

coarsening

process

beyond

^the

early-time regime

^of the

exponential growth

^of the unstable mode. The two-component ^{smectic with}

demixing

nematogens ^bears some resemblance to the

problem

of

phase separation

in

gels,

where the elastic

degrees

of freedom of the

gel

_are

coupled

_to the

concentration variable in _{a manner} similar to

equation

(3)

[17].

There, the

long-range

elastically

mediated forces between

separated

domains affects the domain

morphology

these effects have been the

subject

of _a recent

study [18].

4. Conclusion.

Light scattering

as well _as

microscopic

observations have revealed the

anisotropic

nature of

spinodal decomposition

in two-component ^{smectics. This behavior is}

theoretically interpreted

as

being

the result of _a destabilization of the baroclinic mode in two-component ^smectics

where the

layer spacing

is different in the _two pure

compounds

thus

depending

_on

concentration in the mixtures. This

coupling

is crucial _to

S~-S~ phase separation

_as

experimentally

evidenced

by miscibility

studies conducted _{on numerous}

binary

_systems in which the

layer

^{thickness of} one component ^{is modified with}

regard

to the other

[19, 20].

The kinetics of _a

spinodal decomposition

ⁱⁿ a smectic A solution presents a classical

behavior. The decrease of the characteristic wavevector with time is in agreement with the usual behavior observed in

liquids

or other

liquid crystal

systems. ^More

specifically

it is observed that the

angle

between the direction of

scattering

and the director decreases

regularly

as

decomposition proceeds.

viscosity

of the

mesophase

and the

experiments require

_a less

sophisticated

temperature control. From this

point

of view it _seems worth

extending

the

study

of

spinodal decomposition

to smectic A solutions of

liquid crystalline polymers

in which

phase separation

have also been observed

[20].

References

[ii Chou Y. C.,

Goldburg

W. I., Phys. Rev. A 20 (1979) 2105.

[2] Snyder ^H. L., Meakin P., J. Chem. Phys. 79 (1983) 5588.

[3]

Krishnamurthy

S., Bansil R.,

Phys.

Ret>. Lent. 50 (1983) 2010.

[4] Roux D., J. Phys. Fiance 47 (1986) 733.

[5] Casagrande C., Veys~ie M., Knobler C. M., Phy.I. Ret>. Lent. 58 (1987) 2079.

[6] Roux D., Knobler C. M., Pfiy.I. Rev. Lent. 60 (1988) 373.

[7]

Sigaud

G., Nguyen H. T., Achard M. F.,

Twieg

R. J., Phj,s. Rev. Lent. 65 (1990) 2796.

[8] Park Y., Lubensky T. C., Barois P., Prost J., Phys. Ret> A 37 (1988) 2197.

[9] Nallet F., Roux D., Prost J., J. Phys. Fiance 50

(1989)

3147.

[10] Brochard F., de Gennes P.-G., Pianiana

suppl.

1 (1975) 1.

[I I] A brief argument ^{that the slowest} hydrodynamic mode in _a system ^{with several modes is made} slower by the approach to the critical point is given in MiIner S. T., Martin P. C., Ph»s. Ret,.

A 33 (1986) 1996.

[12] Ma S.-K., Modern Theory of Critical Phenomena, chap. XI (Addison-Wesley, Reading, MA.

1975).

[13] For _a review _see Gunton J. D., San Miguel M., Sahni P. S., Phase Transitions and Critical Phenomena, vol. 8, C. Domb, J. Liebowitz Eds. (Academic Pres~, New York) _p. 319.

j14] ^de Gennes P.-G., The

Physics

of

Liquid Crystals

(Clarendon Press, Oxford, 1975) _p. 291.

[15]

Bhattacharya

S., Sarrna B. K., Ketterson J. B., Pfiys. Ret, B 23 (1981) 2397.

[16] Cheng B. Y., Sarma B. K., Calder J. D., Bhattacharya S., Ketterson J. B., Pfiys. Ret,. Lent. 46 (1981) 878.

[17] Onuki A., J. Ph_vs. Sac. Jpn 58 (1989) 3065.

[18] Onuki A., J. Phys. Sac. Jpn 58 (1989) 3069.

[19] Sigaud G., Nguyen H. T.,

Twieg

R. J., to be

published.

[20] Laffitte J. D., Sigaud G., Nguyen H. T., to be

published.