Spinodal decomposition in liquid crystalline materials

(1)

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Spinodal decomposition in liquid crystalline materials

A. ten Bosch

To cite this version:

A. ten Bosch. Spinodal decomposition in liquid crystalline materials. Journal de Physique II, EDP

Sciences, 1991, 1 (8), pp.949-958. �10.1051/jp2:1991119�. �jpa-00247567�

(2)

J

Phys

II France I 11991) 949-958 AOOT 1991, _PAGE 949

Classificatlon

Physics

Abstracts

64.70 64 60 61 30

Spinodal decomposition in liquid crystalline materials

A. Ten BOSCh

Laboratoire de

Physique

de la Matidre Condensie

(*),

Parc Valrose, 06034 Nice Cedex, France

(Received

5 December J990,

accepted

_m

final form

26

April J99J)

Abstract. The

theory

of

phase

separation m a

liquid crystal by spinodal decompoSit1on

is

developed

The time evolution of _two

coupled

parameters, ^the volume concentration (or mole

fraction),

and the orientational order parameter is calculated The conditions for penodic fluctuations _are _gJven and the _structure factor for spinodal

decomposition

_is studied

1. Introduction.

The

dynamJcs

of

phase

_separation in

incompatible

solid _or fluid mixtures has been studied

expenmentally

_as well _as

theoretically

_m a

large

number of different systems from

polymers

to metals

[I]. Cahn[2] proposed long

_ago

that,

besides the usual

phase separation

_m the metastable _region

by

nucleation and

droplet growth,

_a different _process, called

spmodal decomposition,

_may _occur _m the unstable region of the

phase diagram.

The

predicted

structure is descnbed _in terms of _a

superposition

^of

penodic

modulations of _a fixed _wave

length,

random _m

amplitude,

onentation and

phase

On the other

hand,

Binder

[1, 3]

^from

results _on numerical simulation of cluster

growth, suggested

that there _is _no evidence of _a

sharp

distinction between metastable and unstable _states and that the process of

spinodal decomposition

_can be viewed _as _a

generalized

nucleation

phenomen.

Interest _m this

problem

has been revived

by

the advent of _a _new

type

^of ^matenal :

liquid crystal polymers,

and

by

the industnal

applications

of th~s matenal _m composite

design

Penodic _versus random

dispersed

structures could be

advantageous

for mechanical

(or other) properties

of the

composite

and it _is desirable to control and understand the fundamental

processes of

phase

separation in these systems New expenments in the

biphasic

_region of

liquid crystal polymer

_mixtures have been

reported Amsotropic droplets [4]

and even needle type structures

[5]

have been observed and unusual

penodic

structures

[6]

are formed

dunng unmixing. Recently

neutron

scattering

expenments ^have ^shown ^evidence of

spinodal decomposition

_m the structure factor

[7].

In the present paper we

investigate spinodal

decomposition

_in _a

liquid crystal

mixture. In _a

simple

model _we show that concentration fluctuations need not be

accompanied by

order parameter fluctuations and that the _range of

instability,

the critical _wave

length

and the

growth

rate for

spinodal decomposition

_are not afsected

by

the presence of

liquid crystal

order for _a

quench

from the

isotropic phase.

(*)

C N R S-U-A 190

(3)

2.

Theory

^for

early

time behador.

2. I SIMPLE soLvENTs We consider first _a solution of _a

mesomorph~c

_component _in _a _non-

mesomorph~c

matenal

(simple

solvent _or

polymer).

As discussed in the

Appendix,

the free energy is _given

by

F

=

(f(p,

^«

⁾

⁺

^j~ ^(VP ^)~

⁺

^~"

₂

_(V«)~j

^du.

^(2.1)

Here

lip,

_«

)

_is the _mean field free energy

density

of the

initially homogenous

matenal of

composition _p

of

mesomorph~c component

^with nematic order

parameter

S It has been useful _to define the order

parameter

» « as the

product

of

p

^and the usual _nematic order

parameter ^S ^The ^model free _energy _can then be wntten _as _a function of

p

^and

« alone.

lip, «) ~~~

=

gi)- Li«)

j 122)

with the

isotropic

part

@=(in+ii-)inn-)+ii-*)x

L(«)

_is the onentational entropy, which does _not need _to be considered _in _more

detail,

and _u

is the nematic _mean field. The

degree

of

polymenzation

_is _given

by L,

_X is the

isotropic

interaction parameter.

On

expanding

the free _energy _to include

spatial

_vanations _m

composition

and order parameter

[9],

again

only

terms _m

VP

and V« _occur

We therefore _wnte _in

(2. I)

terms in

K~

^and

K~

due _to

composition

and _« order

parameter

»

fluctuations

respectively.

^The kinetics of th~s systems are given

by

solutions of the

dynamic

equations

[I]

for the conserved order parameter

M

= r

v2 (- _K~ v2~

₊

^°f~*,

^"

⁾ _j2.3)

at

d~

and for the nonconserved order parameter «

~"

= y

(K~ ^V~« ^~~~~'" ^(2.4)

at am

r and _y _are mobilities for concentration of order parameter diffusion _processes

respectively.

As usual

[2],

we look for _a solution

lllilll ₁₂₅₎

where

(po, «o)

defines the initial average concentration and order parameters which fulfill the stationary state conditions

~° ~o

~ ~"°

~~

~

~~ ~~

~~ Limo)

=

~~

= ~

°~o °~o

(4)

N 8 SPINODAL DECOMPOSITION IN

LIQUID

CRYSTALLINE MATERIALS 951

Here _R _is _a

generalized

chemical

potential [8].

We consider _a

spatially homogenous

initial state and

16 p,

&« denote the fluctuations around these values

The hnearised equations in Founer space are

3p 3p

w = M

(27)

3"~ 3"

Where the _matrix _is given

by

~ dL

rq2(q2 _K~

₊

a

~~ @

(2.8)

~'

dL

y

(q2 K~

+

fl

~

t

In order to obtain _a nontnvial

solution,

the determlnent of the coefficients must vanish.

This condition results in _a relation between _w and _q

lW+rq~(q~K~+a))jw+yjq~K~+p))-yrq~())

^=0.

^j2.9)

The _two control parameters are a =

~~(, ^p

₌

~~(

dpo d«o

We will concentrate here _On the _case Of _a

quench

from the initial disordered state

fro =

0 and ~~

= 0

d«o

We find _a solution _w m0 for a finite range Of _wave vectors from the first

parenthesis

Of

equation (2.9)

identical to the

Cahn-filhard ^theory ^{[1, 2].}

^The ^condition ^on ^the concentration

po

^of ^the ^initial state is a < o _or ^~

(<

o as in _non

mesomorph~c

_systems.

dpo

Schematically,

the mixing free energy ^as ^a function of mole fraction

p

at constant temperature can be given as an isotropic branch

(«

= o at low values of

p

and _an

anisotropic

branch _«

= 0 at

high

values of p. ^Phase separation occurs between

points

^of

equal

chemical

potential

The two

phases

can be both isotropic or one isotropic and _one

anlsotropic phase [10].

The limits for

stab1llty

_or

metastability

of the

quenched

fluid

phase

_are _given

by

the conditions for the

spmodal

of the isotropic ^branch. ^Within ^the

spinodal

_region of the isotropic

branch, phase separation

could take

place though penodic composition

modulations

For _a

quench

from _an initial isotropic state «o #

0,

the

equations (2 7)

are

decoupled.

The order parameter « is stable and _we have _a

positive eigenvalue

associated with the

instab1llty

_in

concentration p ^Similar to the results _on

He3-He~ [1, 20],

the initial behaviour of the system after

quench

_is _one _in which the conserved order parameter

p

evolves towards _a local

equilibnum

state while the nonconserved order

parameter

remains

essentially

constant

land equal

to

zero).

The local

equihbnum

state for

p

w~ll

correspond

to _one of the two

equihbnum

states

p~

or p_ on the

isotropic

^branch ^The ^transition to a stable anisotropic state could then take

place,

for

example, through

nucleation of the anisotropic

minority phase.

In

pnncipal,

_a

quench

from _an initial

homogenous anisotropic phase

with mom o _into _a

biphasic

_zone is also feasible but has not yet ^been studied

expenmentally

The _situation _is then _more

complex

and has _not been

analyzed

here. A _new type of

spinodal decomposition

with

coupled

composition ^and ^order parameter fluctuations may then be

possible

from

equation (2.9).

(5)

2.2 BINARY MESOMORPHiC MiXTURES. We _now go on to the

expenmental

situation of interest of _a

binary

_mixture of

mesomorph~c components

^The ^model ^free energy in this _case

is given as

F

=

dr

I/i~,

^«,

^«~)

⁺

l~ ^iv~)~

₊

l' _iv«1)~

K~2 ~

+

K,~(V«i

V«

~)

+

(V«~) ^(2.10)

2

Agaln lip, «1,«~)

is the free energy

density

of the initial

homogeneous phase.

The

compos1tlon

of

component

^I ^is given

by p,

that of

component

²

by ii p)

The respective

orientational _« order parameters » are then «j and a~, again

products

of the usual nematic order parameters of the

components Si

and S~ ^and ^the ^relevant concentrations : «j =

pS,,

«~ =

(l p)

_S~

(see Appendix)

The terms in

K~, ^Kjj,

K~~ ^and

Kj~

take

compositional (K~)

and order parameter

(K,~)

fluctuations in to account. The model free energy can be

wntten _as before

[8, lo].

lip,

_«1,

«~)

= g

i*) Lip,

_«1

«~) 12.i1)

where

~~~~

=

~

ln

p

+

~~

ln

ii p)

+

#(I #)

_X

KT

Li L2

L(~l,

_"1,

"2)

"

~l~l("1, "2)

₊

Ii

~

L2("1, "2) (Ull "~

₊ _U22

"~

~ U12 "1

"2)

Lj(aj, a~)

and

L~(aj, a~)

_are the onentational entropies of the two

liquid crystalline components

^and ^the ^terms ⁱⁿ _u,~ are the contnbutlons from the nematic _mean fields.

Lj

and

L~

are the

degrees

of

polyrnenzation.

Three

dynamic equations

_now ^control the kinetics of

phase separation

_; one for the conserved vanable

#

~4'

= r

v2 (- _K~ v2~

+

°f j~,

_«1,

«~) j2.12)

St al

and two for the non-conserved variables «j, «~

~~~

= yi

(Kij ^V~«j

⁺

^{Kj~ V~«~}

~~ ii,

_«

j,

«2)

~~

~"~

(2.13)

~~~

= y~

K~~ V~«~

+

Kj~ V~«j ^~~ ii,

_«

j,

«~)

at

d«~

As before _we investigate ^the ^existence of

penodic

fluctuations

16 #, &«j, &«~)

^around an

initial isotropic

phase.

After hneanzation of

equations (10)

and

ill),

the condition for existence of nontnvial solutions leads to the

dispersion

relation. For _an initial isotropic state with ml = «2 =

0,

_we find for the linearized

equations

in Founer space :

&# #1

w

&"2~ &«j

^=M

3"2 Sal (2.14)

(6)

LIQUID

Where the hneanzed matrix _is now

rq ~(K~

_q^~ + a 0 0

M

= 0 -1

°

(2_15)

z

=

Y>

iK»

q[

⁺

^{P >i)}

^Yi

^iK12 q[

⁺

^P12)

Y2(3°12

_~ +

~12) Y2(~°22

_~ +

~

₂₂₎

The control parameters are

a _#

~j~

,

fl

~j ^#

~~~

d

~

^d« ^da

j

The condition for _a non-tnvlal solution is then det M

= 0 _or

iw rq 21a

+

K~ q2))

det z

= o

Again

the

composition

and order

parameter

instab1llties are

decoupled

We find _an

instability

for #

equivalent

to the

spinodal

of

isotropic

mixtures for wl~ich «j and

«~ remain constant

land equal

to

zero)

A second

instability

for «j and «~ ^at ^constant

#

also exists for _certain values of material constants and has been discussed

previously Ill].

Which of the _two instabilities will be chosen

depends

_on the initial concentration, ^the ^final

temperature

and the parameters of the mixture, _in

particular

the

phase diagram.

The

instability

_m composition occurs for

quenches

witl~in the

spinodal

_curves of the isotropic mixture. The

instability

in order

parameter

occurs for T<

T*,

the

limiting temperature

^for

metastab1llty

of _a stable

isotropic

state m the mixture. At _a given

temperature,

th~s

requires

that the initial concentration

correspond

to _a value for which the

isotropic phase

is unstable after the

quench.

The order

parameter

^then ^evolves ^towards ^the ^value

corresponding

to lower energy anisotropic

phase (Figs

1,

2)

T

Tc

-O

b ^~

Spinodal

+

Fig

I

Typical phase diagram

for _a mixture of two mesogenlc components. ^The cntical temperatures 7~ (limite ^of

stab1llty

of

isotropic phase)

and T* (limit of

instability

of isotropic

phase)

are

given

A

typical quench

from the

isotropic phase

to within the

spinodal ix

or to within the anisotropic zone

IO)

is marked.

(7)

Free Energy F

ANisoTRopic 5'o

Concentration ^?

Fig 2

Typical

free energy plot corresponding to figure I If the

quench

results _in point x spinodal

decomposition

of compos1tlon could occur

leading

to _a separation in isotropic states ~~, #_ If the

quench

results in point O, penodic fluctuations of order parameter could _occur

leading

to an anisotropic

state #o

3. Calculations of the structure factor

[12, 131.

31 SOLUTIONS IN A NON-MESOMORPHIC SOLVENT We

Study

first SOlut1OnS _m _a _non-

mesomorphic

solvent. In these systems three correlation functions _can be defined

(&#~&#_~),

the fluctuation of concentration, given

by

the Founer transfortn

&#~, (&a~ 8a~),

the fluctuation of «order

parameter

given

by &a~,

^and ^the cross

coupling

correlation function

(

a

~

3

#

~

These functions _can be calculated from the set of

dynamic

_equations

(2.6). Following

the method of references

[12]

and

[13],

_we must

supplement

these equations

by

a random force

~j~)

~

fq ~q

~

~~~~)

⁼

M~~~~)

⁺

^~~ ⁽³¹⁾

at

&«~ &«~ f~

Where M _is the linear matrix glvcn earlier in

(2.8).

In vector form _we wnte

(

*

It)

=

Malt)

₊ G

it)

with _a

it )

=

~

~~

^~~^~~

&"q

We will

investigate

^here

only

correlations _over

sufficiently large distances,

_so that the structure of the

particules

will not

play

_a role The linear

equation (3 2)

_is solved _as usual

i

a = no

e~~

₊

e o e~'Gil')

_dt'

(3 3)

leading

to the correlation functions

given by

the

dyad inn _*)~~

= a,

at

Inn *I

=

e~~~l I«o ail

₊

(2 M)-i _Q(q)] (2 M~-~ Q(q)

^~~'~~

(8)

LIQUID

We have used the

following

relations which hold

by

definition

[14]

:

iG it)>

= o

iG it)

G

it')>

=

Q iq) Sit t')

¹³

5)

and the coefficient

Q(q)

_is related to the

Onsager

coefficients

~~

_~

(3.6) Q(q)

= 2 KT

/

Y

It _is _now

simple

to denve the static correlation functions

a a ^*

)

~ ⁼

(2 M)-

^'

Q (q ) (3.7)

For _a

quench

from the

isotropic

state, _we find for the static structure factor

j

a

w~ 2j

=

~°~

~

13.8) K~ q~+

^~

ajo

The concentration fluctuations _give identical behaviour to

spinodal decomposition

_in

isotropic liquid

mixtures. As _in

tjese

^systems, ^the ^static ^structure ^factor ^shows ^a

^divergency

^at ^a ^cntical

wave

length q)=

^~

~~.

^If ^the concentration

#o

is within the

isotropic spinodal,

°~o ~°~

exponential growth

of the

intensity

with time _is

predicted [14].

The _cross

coupling

correlation functions _are _zero and for the order parameter fluctuations _we find

ii&«qi~i

=

~

~~

~flj ^~~

«

~«j

Th~s is _same form _as for

homophase

fluctuations of the order parameter ⁱⁿ ^the

isotropic phase [15].

The

typical divergency

at q = 0 of the

amplitude

at the temperature ^T* ^{of the} mixture is

found T* _is thus the

temperature

^below ^which ^the isotropic

phase

is

absolutely unstable,

and

given

here

by

of d~f

o _~

~ o

am

d«2

^'

3.2 BINARY MIXTURE oF MESOGENS. An identical treatment _can be made _in the _case of _a

nuxture of two

liquid crystal

components. ^For

example,

two

liquid crystal polymers

of

different

degree

of

polymenzation,

_or _a solution of _a

liquid crystal

and _a

polymer liquid crystal

have been studied in

optical

and _in neutron scattenng expenments. ^In ^tins case two order

parameters

on each of the

components

exist. The correlation functions of interest _are

the fluctuation of concentration, and the fluctuation of total «order

parameter»

« ₌ «, ⁺ «~. The linear matnx for _a

quench

from the isotropic state is given

by (2 15)

and the static structure factor _is _again of the form

(3 8)

with behaviour _as discussed _m 3 On the

other hand the order parameter fluctuations _are

given by Sag _(~)

=

~ (- _q~(Kll

+ 3°22 ^~

~°12)

+ ~

~

>2

~

>l

~

₂₂₎

(3 1°)

det

~j

As discussed in 2.2 and in reference

[I I],

a second type ^of«

spmodal

_» _can exist, with _a cntical

(9)

wave vector for

divergency

of the scattered

amplitude,

which follows from the condition det

jj

= 0

Decay

of the isotropic state

through periodic

vanations of the order

parameter

can occur

for _a

quench

from the

isotropic phase

if the initial concentration

corresponds

to a value

greater

^than ^the ^limit ^of

metastabihty

of the

isotropic phase.

The concentration _remains

constant

during

this _process

(Fig 1, 2).

Discussion.

In

conclusion,

_we have

investigated

the existence of

periodic

mstab1llties m the concentration and in the nematic order parameter ^for

liquid crystal

mixtures. We have concentrated _on the

early

_stage, _coarse

pawed

limit of the

phase

separation. ^We have shown that for _a solution of

a

liquid crystal

m a

simple solvent,

the usual isotropic

spinodal composition

_occurs for

incompatible components

^when

quenched

from the

homogenous isotropic phase.

ThJs results

m a local metastable

isotropic phase

and _can then be followed

by

a

phase

trans1tlon

by

nucleation to a stable

amsotropic phase

if such _a

phase

exists at the final concentration after

phase

separation. For _a mixture of _two

liquid crystal components,

^the same scenano can take

place.

In addition _m certain

materials,

if the

quench

leads to an unstable

(relative

to the

anisotropic phase) isotropic phase,

_a

periodic instability

^of the order parameter can occur for which the concentration remains constant and wh~ch leads to _a stable

anisotropic phase.

In th~s paper, we have assumed the director fluctuations to be

damped

and the director to be fixed

parallel

to _an extemal

magnetic

field. Director fluctuations have been studied _m reference

[16]

and may be

important

m non-onented

samples. Strong

correlations _m

position

may lead to concentration

dependence

of the constants of the

gradient

terms

K~,

K~

etc. Th~s would lead to

h~gher

order terms and a different concentration

dependence

of the correlation

lengths

of the fluctuations Such effects have been studied in isotropic

polymer

mixtures

[12]

We have concentrated _on the

quench

from the

isotropic phase

wh~ch

corresponds

to the

present expenmental procedure

and the present work

applies

to side chain

[17j

as well _as

main chain

[7] liquid crystal polymers

Concentration fluctuations with

spinodal

type ^behavior should be visible _m small

angle

neutron

scattering

for which order

parameter

fluctuations would be

negligable.

The _case of

light

_scattenng is _more

complex,

both concentration

[18, 19]

and order parameter

[15]

fluctuations

playing

_a role Late stage

dynamics [20]

can also be observed and would be of interest to

study

_as well _as critical behavior

[21]

_as has been done _m

the related _case of

phase

_separation with

coupled

conserved and nonconserved order

parameters

in

He~-He~

_mixtures

[22].

Appendix.

The free _energy

density

of _a solution of _a

liquid crystal polymer

and a

simple

solvent has been derived in _a _mean field

approximation [8]

f(4r,«)= [(in~+ ji-~)inji-j)+~ji-~)xjKT-~Ljs,~)-~2us2j2jAi)

# is the volume concentration of the

liquid crystal polymer

with nematlc

polyTner

S. The first terms

give

the

isotropic part

; the entropy ^of

mixing

and the

isotropic

interactions.

The fourth term L

IS,

#

)

is calculated from the chain

partition

function and

depends

_on the model of the

liquid crystal

chain. The final term is the _mean field contribution from the nematic interaction _u corrected for the effect of dilution

by

_a factor

#.

The expression denved

m

(8)

for _u _is _a

spatial

_average of the

pairwise

orientation

dependent

interaction.

(10)

LIQUID

The free _energy of defornlation

f~ jr)

_m onentation

dependent

_systems _is _given _in reference [91.

(x~

₌

^ix,

_y, z

)

~~~~~ il~°~~~~"'~iPir,W~iPir,w~~w~w, ^~~~~

The distribution function of

position

_r and onentation _w _is

p(r,w)

and the term

c~~(w,w')

is related to the direct correlation function _m the

homogenous

material. We consider monodomaln

samples

of uniform

alignment

_n.

For

systems

charactenzed

by

a second order

expansion

_m

Legendre Polynomials Pi(w)

for _a

Maier-Saupe type

of amsotropic

potential

c~_~

(w,

w'

=

£

_c~

~, i P_i

(w ) Pi (w' ) (A3)

I o,2

We set as usual

PIT,

_w

)

=

~ jr) fir,

_w

),

where

fir, w)

_is the orientational distribution function

[23].

Then

fir,

_w

)dw

= I With the usual definition of the local _nematic order

parameter

fir,

w

P~(w

dw

=

Sir ),

_we find

P

(r,

_W

P2(W>

dW

= #r

jr) S(r)

= «

(r) (A4)

From

(A2)

to

(A4),

_we obtain

Fdir)

=

i ^ca~.o k

_~

_(r) ) _~ _ir)

₊

_C«~.

2

k

«

ir) ) _«ir)1 _iA5)

Consistent with the

symmetry

of the nematic for _a radial distnbution of

liquid crystal entities,

equation

(A5) simplifies

to

F~jr)

=

( iV#r jr)

)~ +

( _iV«ir))~.

iA6)

If the

angular dependence

of distance

dependent

correlations _are considered in

(A3),

_a

ternl m

in.

V« )~ will appear m

(A6)

and lead _to direction

dependent

order parameter

fluctuations

[15].

In

nonaligned samples

fluctuations in the director _n could also _occur

[15].

The

K~

term _is well known from interfacial and

dynamic

studies _on

incompatible

mixtures

[12]

The term in the elastic constant

K~

relative to the nematic order parameter

(VS)~

has been used in studies of the

biphase

of pure

liquid crystals [15, 23]

In the present case,

coupling

between order parameter and concentration

gradients

also _occurs

To denve the model free _energy _m the mixture of

mesomorph~c

components, the

proceedure

_is identical. The nematic order parameters on each individual

component

are

given

by Si

and 6~, the volume concentrations _are

#

and I

#

As

before,

_we find that the free energy

density

can be written _as _a function of #, «, =

pS,

and «~ =

ii p ) 6~

and their

gradients.

(11)

References

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chapt

3, Phase transitions and cntical

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1983)

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METIU H., KITAHARA K, ROSS J,

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Chapt

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liquid

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Wojtowicz,

P

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ii?]

NOIREz L, thesis, Unlv

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Sci

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Phys

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Phys

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Phys

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