On the isotropic-nematic transition for polymers in liquid crystals

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On the isotropic-nematic transition for polymers in liquid crystals

Leon Balents, Randall Kamien, Pierre Le Doussal, Eric Zaslow

To cite this version:

Leon Balents, Randall Kamien, Pierre Le Doussal, Eric Zaslow. On the isotropic-nematic transi- tion for polymers in liquid crystals. Journal de Physique I, EDP Sciences, 1992, 2 (3), pp.263-272.

�10.1051/jp1:1992142�. �jpa-00246481�

J. Phys. I France 2

(1992)

263-272 MARCH1992, PAGE 263

Classification Physics Abstracts

64.70M 05.70F

On the isotropic-nematic transition for polymers in liquid crystals

Leon

Balents(*),

^ltandau D.

Kanien(*),

Pierre Le

Doussal(**)

and Eric

Zaslow(***)

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, U-S-A-

(Received

24 June 1991, revised 12 August 1991, accepted 21

November1991)

Abstract A system of dilute polymers dissolved in liquid crystal is studied in the isotropic phase. A _mean field theory is proposed which

undergoes

spontaneous breaking of the rotational

symmetry ^{and reduces to the}

theory

of polymer nematics. In order to eliminate closed loops

of polymers nom entering into the theory, the polymers _are replicated _n times and then

n is taken _{to zero} similarly to the procedures of de Gennes and des Cloizeaux. The transition is first order, and the presence of the polymers shifts the critical point down, in agreement with physical expectations. The effect of polymer-polymer interactions is discussed.

1. Introduction.

Polymer

melts and

gels,

_as well _as

polymer

nematics _are

only

_a few

examples

of the recent and not-sc-recent studies in the

theory

of line

liquids [1-5].

^More

generally, polymers,

considered

as

dynamical lines,

_are related to other systems, ^such as

high

temperature

superconductor

flux lines [6, 7],

electrorheological

fluids [8], ^and ferrofluids [9],

although

these systems ^have an

external field

explicitly breaking

the rotational symmetry. ^{There has also been interest in the} models of random walks in

quenched

and annealed environments

[10].

A field

theory

^{has been}

proposed

which is

mapped

from the

theory

of the individual

polymers interacting

with _a nematic matrix of smau molecules [5]. ^This

theory

^holds

only

in the nematic

phase

of the

liquid crystal,

^{and therefore} in the directed

phase

of the

polymers.

It _can also be

thought

of _{as a}

theory

of the directed

phase

of _a

polymer

melt

by considering

the nematic field to be the

coarse-grained background

of

polymers.

This

approach

would be difserent from the traditional

microscopic approach

of Khokhlov and Semenov

ill, _12].

There has also been

more recent work _on the role that defects

play

in

polymer liquid crystals [13].

In this _{paper, we propose an}

isotropic theory,

which _at low

temperatures spontaneously

breaks to the directed

theory

of [5]. ^The

isotropic theory

is

interesting

to

study

not

only

(*)

Supported in part by National Sdence Foundation Graduate Fellowship.

(**) On leave from Laboratoire de Physique Th60rique de l'Ecole Normale Sup£rieure, Paris.

(***) Supported in part by Fannie and John Hertz Foundation.

because it avows _us to understand the

isotropic-nematic transition,

but also because it _can

serve as a model for _a

polymer

melt. One _{can coarse}

grain

the

background

field of the

polymers

and call this field the

liquid crystal order-parameter. Finally,

_our formulation suggests ^how

one

night

derive _a

theory

of nematic

polymer

melts

alone,

without the aid of the

background

field.

Though

the _zero component field

theory

of de Gennes _was introduced _to

study

the renormalization of

polymers,

it is used here _{near a} first order transition

point.

Since there is _no

universality

at _a first order

transition,

_a renormalization _group calculation would be

uninteresting.

The intuition is borrowed from the

theory

of relativistic bose

particles propagating

in _space-

time, identifying

their world-lines with the

configurations

of the

polymers.

This

approach

has proven to be useful in the context of flux lines in

high

temperature

superconductors [7,5].

^We

will _use the _n

- 0 limit of _an

O(n)

real scalar field

theory

to describe the

polymers [14].

This limit is taken in order to _remove closed

loops

of

polymers

from the

partition function;

equivalently,

it _removes the functional determinant of the

interacting

scalar

theory.

In order to describe the

liquid crystal,

_we will _use the usual Landau

theory, taking

the order parameter

to be _a

traceless, symmetric matrix,

related to the local dielectric _tensor of the

liquid crystal J51.

We first review the

non-interacting

theories of

polymers

and

liquid crystals.

In section two, we formulate the interaction between the two systems. ^Section three is _a discussion of the transition from the

isotropic

to the nematic

phase,

via the spontaneous

breaking

of the

rotational

symmetry.

The effective theories of

polymer

melts and the

liquid crystal

_are treated in sections four and

five, respectively.

The

non-interacting

free _energy densities _are

~°L § ~ ^(~p~i~~~

^~i~

~~i

~

j ^~i ~j

~

~i~i

_~~'~~

~ ~ ^~

,~

~ ~

~

i#I I,j#I i#I

where the

~i

are scalar fields. All

greek

indices _run from I to 3. The

partition

function of this

theory produces

the

generating

function for

polymers:

lim

/jd~~i

^exp

[- /d~z ^OL]

^"

i§ ^t

^~~~~~

_{Zpm(T) (1.2)}

"~°

m, p=0

(~P)'

'~~'

where

Zpm

^{is the}

partition

function for _p

polymers

with _a total combined

length

of _{m monomers} [3]. The field h is _a

spatially

constant source, and _{we see} from

(1.2)

that h~ _{is the}

fugacity

for

polymer number,

while _~ is the chemical

potential

for _monomer number. The chemical poten- tial controls the

density

of the

polymer

melt.

Additionally,

the

twc-point

function

~(z)#(0) gives

the

partition

function for _a

single polymer

with end

points

0 and _z. In the

single polymer

interpretation,

_{~J ~c} is

conjugate

to the

length

of the

polymer.

For the

liquid crystal

we have:

nc _"

( (fipovp)(fi~Q~~)

+

j

(fi~QpV)(fiPQ~~)

+ ~

T~(Q~)

+ b

T~(Q~)

+ ~

T~(Q~) ("3)

where

Q~v

is _a

traceless,

3 _x

3, symmetric

tensor. Recall that

Q

is related to the

dielectric anisotropy by

E»v

jd»vTr(E)

⁺

^Q»v. (14)

Note,

in

addition,

that

(1.3)

has _a cubic term, ^and so the transition from the

isotropic phase, Q

₌

0,

to the nematic

phase, Q

_cc

diag(-1/3, -1/3, 2/3),

is first order. Other

possible

terms

quadratic

in

Q

^and derivatives _are linear combinations of the _two kinetic terms in

(1.3).

N°3 THE ISOTROPIC~NEMATIC TRANSITION 265

2. Interactions.

In order to formulate the interaction of the two

models,

_we must go back to the

representation

of

polymers

in terms of lines. In this

representation,

each

polymer

is described

by

_a vector

1lq(s), parameterized by

_s.

Additionally,

_we represent the

liquid crystal by

_a unit vector field n, where

Q~v

=

Qo(n~nv (1/3)d~v).

This

parameterization

of

Q

is not

quite

correct in the

isotropic phase,

but is sufficient to understand the

system(~).

The

polymers

tend to

align

with the

nematic,

and the free _energy for _a

non-interacting (phantom) polymer

is

F =

( /

_ds

^~~~~~

ln(R(s))) ^(2.I)

~

S

~

where D is the difsusion coefficient of the

polymers.

In mean,field

theory,

the

polymers

will

align exactly

^with the

nematic,

~~P(~)

~~

(J~( ~)) (~ ~)

~~ ^P

This cannot be

easily

translated into the

language

of the scalar fields and the dielectric

anisotropy.

We note,

though,

that if _we define

T(?~(z)

e

fds (R~(s)Rv(s)b~(R(s) x)j,

and

T~v

%

T(y~ (1/3)d~vTl T~°~,

then

(2.2)

_can also _come from the minimization of

F ₌

/ _{d~z lF(f 12Q)2. (2.3)}

2

This has also been discussed in [16].

Additionally,

T is the

equivalent

of

Q

for the

polymers.

That

is,

T is the dielectric

anisotropy generated by

the

polymers.

This

suggests

that _a

theory

of

polymer

melts _can be attained

by considering

_a

Landau-Ginzburg theory involving only

T

Indeed,

_on symmetry

grounds, (2.3)

is the lowest order allowed _term in _a Landau

theory

for

polymer nematics,

and is therefore the correct

coupling

term in the

path-integral

formalism.

Any

field theoretic formulation of

(he _problem

_is

_ultimately

_true

_only

_{in that it}

_reproduces (albeit

in _a convenient

way)

the results order

by

order of the basic

path-integral

model. Nev-

ertheless,

it is

possible

to guess the

approximate form

of the interaction term

by

_an intuitive

argument.

The

important

observation is that

T(?~

^is the contribution _to the euclideanized

energy-momentum

tensor for _a collection of

non-interacting particles ii?].

In

fact,

the expecta- tion value of T~°~ will be

proportional

to the _outer

product

of the unit tangent vectors of the free _mean field

polymer configurations.

This identification allows _us to write the interaction

(2.3)

in terms of the scalar fields! Recall from the

theory

of classical

fields,

if F is the free energy

density

for the system, ^{the euclidean} energy-momentum tensor is

T4

"

&Jp~~

~~

i

fi(fi»~;) ^~v~i

⁺

^d»vfpo~ (2A)

and thus

f f

fi

~

fi

~

~

(fi ^~

)2

(~ $)

PV P I _v I

j

Pv I

I=1

We have

neglected

^the

higher

momentum ternls in

(I.I)

because

they

_are ^irrelevant to the

coupling.

(~ ^If the liquid crystal elements _are uniaxial, this parameterization is adequate. However, if they

are biaxial, this simplification cannot be made in the isotropic phase.

Though

the scalar fields in

(I.I)

_are

interacting, f

_is

independent

of the interactions because the trace has been removed. This is _a convenient

simplification

when it _comes to

including

_any

variety

of

polymer

interaitions the

coupling

to the

liquid crystal

will remained

unchanged

since

TI(Q)

₌ 0.

Expanding (2.3)

^{will lead} to terras

quadratic

ⁱⁿ T. These _are

higher

order in the

interaction,

and will be

neglected. Thus, (2.3)

suggests an interaction of the form

f~

=

l~Tr(QT)

=

l~o~vfi~~fi~~. (2.6)

Another

possible interaction, equivalent

_up to operator

ordering

of

(2.1) [10],

is

J~~~ =

-l~Q"~~fi~fiv~

_e

-l~Tr(QS) _(2.7)

where S _%

~fi~fiv~

^{is the}

polymer alignment

tensor. We choose between the _{two terms}

by requiring

that the

theory

of [14] ^{is borne out in the}

path-integral language.

If _we consider the

arc-length

to be

time,

and

(2.3)

to be _a

Lagrangian,

_{we can} derive _an

imaginary-time Schr6dinger (Fokker-Planck) equation

for the

partition

function. We have

lj

^S

Z(s, ri)

=

fl[dllq]

_exp ^ds

lij(s) _{(d"~ l~Q~~)} R$(s)

+ interactions

(2.8)

4D

o

Thus the canonical momentum is

p[

₌

(dj >2Qi) k$(s) (2.9)

This leads _us to fiZ

=

£

_D

_{(dPv 12~JPv)}

^{~~ i_}

fi,

_{Z +}

interactions, (2.10)

3~

firj firj

where the cobrdinates r'

are the

endpoints

of the _n

polymers. Twc-point

functions of _our

theory

will

reproduce

the difsusion kernel in

(2.10)

to lowest order in l~. A scalar field

theory

with this

propagator

^will

reproduce (1.2).

We _see that the interaction

(2.7)

_comes from the lowest order _term in the

expansion

of

(I l~Q)~~.

Thus _our identification of the directed

phase

of _our model with that of [5] ^is

only

valid in the weak

coupling

limit.

We _note that the interaction

(2.6), which,

based _on motivational

grounds,

_{seems more}

intuitive,

adds to the field

theory

_an additional

coupling

^{of the form}

$DD _~

~p~/~~~~V~. (2.ii)

Translating

back to the

physical path-integral language,

this becomes _a linear

coupling

between the

polymer

tangents ^{and the}

divergence

of Q~

~n ~

/~

~~ ~

~pv~~~ _~~~dRv(S)

_~~

j~~

ADD p ~

0 S

In the

path-integral language,

it is clear

why

this term should not be present ^{in the}

theory.

Since it is linear in

R,

it does not

obey

the symmetry

k

- -R. Thus

(2.7)

is the correct lowest order

coupling

in the field

theory.

The free energy we propose for the full

interacting theory

of

polymers

in _a

liquid crystal

solvent is thus

~ ~POL + ~iC + ~NT.

~2.13)

The individual terms _are

given respectively by equations (I.I), (1.3),

and

(2.7).

N°3 THE ISOTROPIC~NEMATIC TRANSITION 267

3.

Spontaneous

symmetry

breaking

and directed

polymers.

The free _energy of the

liquid crystal

is minimized

by

_a

spatially

constant dielectric

anisotropy, Q. Thus, only

^the bulk

part

of the free _energy need be minimized:

V(Q)

=

aTr(Q~)

+

bTr(Q~)

+

CTr(Q~) (3.')

If there _{are non-zero}

minima,

the dielectric

anisotropy

will be of the form

Q

_"

Qodiag(-1/3, -1/3, 2/3),

_up to

global

rotations

[18].

^{In terras of}

Qo,

this

potential

has the form

v(Qo)

₌

(Qi(cot

⁺

^boo

⁺

^{3a) (3.2)}

If this has _{a nonzero}

miminum,

then the transition to the directed

phase

will _occur. This

will

happen

if the second

quadratic

factor in

V(Qo)

_assumes

negative

values. The condition for

being

below the critical temperature ^{is thus b~} 12ac >

0,

where

equality

is achieved at

transition.

Assuming

_a is linear in the reduced temperature, we have the relation

(a b~/12c)

_oc

(T Tc), (3.3)

with

positive

constant of

proportionality.

Note that

(3.I)

is

symmetric

under

Q

_-

OQO~,

where O is _an

orthogonal

matrix.

Since the

potential

terms _are invariant under such

rotations,

the

theory

will have Goldstone fluctuations

corresponding

to _a slow

spatial

variation of the matrix O. We examine the efsect of these

goldstone

modes and

neglect

^the

remaining

massive

degrees

of freedom

by making

the

decomposition

Q

₌

o(x)Q~~poT(x), (3.4)

where QREp _"

Qodiag(-1/3, -1/3,2/3)

is _a

spatially

constant reference matrix.

Writing

O

=

exp(I£)~~ n;(x)7j),

where the

2j

are generators ^of

SO(3),

_we find _to lowest order in n;, F~~~ ₌

Q(

^~~

)

^~~

_(T7.dn)~

+

Q] j

(T7x

dn)~

+

Q(

^{~~ ~ ~~}_(fiz

_dn)~

~ ~~ ~

~

(3.5)

+

)Q(

₊

_~Q(

₊

)Ql

where dn

= -n2i+

niY

is the transverse

projection

of _n and T7 denotes the

gradient

transverse to the _z direction. At

sufficiently long length scales,

_{we can}

neglect

the massive fluctuations in

Q.

This is the standard form for the

free-energy

of _a nematic

configuration jig],

where

the identification with the usual Frank constants is

Ks

₌

Q((ki

₊

_k2),

_Kt

"

Q(ki,

and

Kb

_"

Q~(k1

+

k2).

If the

liquid crystal

is in the nematic

phase,

there will be _a chosen direction.

Moreover,

due to the

coupling (2.7),

the

original

~~ scalar field

theory

will

develop

_an

anisotropic

propagator.

Thus in the nematic

phase,

J~~~ ₌

l~oo

1-

^{(T7~)~ +}

~

(fiz~)~

+

2fiz~

dn.T7~ + ~T7

.dnfiz~

+ ~fiz ^dn.

7~j

(3.6)

3 3

The

quadratic

part of the free energy becomes

F~uAo

₌

~~ ^(fiz~)~

⁺

) _{(V~)~ ~~, (3.7)}

where _we have

dropped replica

^indices aJ~d the diffusion constants _are

Dz = +

(4/3)l~Q(

Di ₌ 1-

(2/3)l~Q(.

^~~'~~

At this

point

_we now have _a

theory

of

polymers

with

anisotropic

^diffusion

coefficients,

_so that

(z~)

c3 Dzs ^and

(z~)

c3 Dis.

In order _to write _our

theory

_as a directed

theory,

_we introduce _a

dummy

variable of

integra-

tion for each field

~;.

These fields

correspond

to the canonical _momentum of the scalar fields.

We have

z ₌

/ldKil ld~;lldo»vle~ '~~~'~

x exP

/d~z ^17iM,

^x,

^Ql

⁺

^{Kfizfl) 13.9)}

~h~~~

yi = ruin

j Jpo~ ii

+

TNT], Ql

~~fi~

~l

_~~'~~~

a~ j

is the

Legendre

transform of the free _energy. The

theory

is then put into _a standard form

by

the transformation

~

₌

(~ +1~*)/V§,

_x

=

I(~ -1~*)/v§.

These _new fields _are creation and annihilation operators, and the functional

integral

is _{now over} coherent _states. The

resulting theory

is still

isotropic, although

written in _an

apparently anisotropic

_way. Below the

isotropic-

nematic transition temperature, ^{the nematic} may

pull

the

polymers

into

alignment.

This

situation,

in which the

polymers

become directed

along

the

spontaneously

chosen nematic

axis,

is

expected

to be described

by

the

theory

of [5]. An

important

check of any

theory

of the transition is that it reduces in this limit to the directed

theory.

In

general, however,

it is not

necessarily

the _case that nematic

alignment

forces the

polymer

melt _to become directed

it

depends

_upon the

coupling strength.

In order for the

polymers

to

align

at the

liquid crystal transition,

the

persistence length

must be

sufficiently large.

^{This condition will be}

demonstrated below. It should be noted that _xi has

changed

due to the interaction with the nematic field. In

(3.9)

_we must set

iKi ₌

aj~)_~

⁼

^fiz~i

⁺

21~Q»zfi~~;

+

i~fi~o»z~;

_13.

ii)

z i

and then eliminate

fiz#

from F.

Rewriting

_our model in terms

of1~,

_1~* and

bn, keeping only

the lowest order _terms in

bn, yields

v

'~i(l~~ ^l~l~l~l ~~llll~llllll

~~~

~

~/°fi~dn.(1~T7l#

+

l#*V'~*)

+

~

~~'~" ~'~~~~

~

~~~~~

_{~~ ~~~}

where the

grouping

in the

quartic

terms indicates the _{sum over}

replica

indices. The difsusion constant Di was

given

in

(3.8),

and _we have identified

i 1

~ ~ ^~

~

~

~~~~~~~~°

(3,13)

l +

/~~12Qo'

N°3 THE ISOTROPIC-NEMATIC TRANSITION 269

One _can also _go from the directed limit to the

isotropic

limit _as discussed in [5]. ^The

equality

of the chemical

potential

with the coefficient of (1~)~ ⁺(1~*)~ ^is a result of the hidden rotational symmetry. ^{This would not be enforced if} one were to start from _a directed

theory,

with

no

underlying

rotational invariance. The final term of

(3.12)

will become irrelevant at

long wavelengths,

_{as can} be _seen

through

the renormalization _group treatment in [5].

Some of the ternls in

(3.12)

_are not present ^{in the}

theory

of [5]. ^These are the operators which

produce hairpin configurations

of the

polymers

and _are not invariant under the

phase

symmetry _{1~ -}

e'~l~. However, they

_may be

neglected

when the

directing

field is

sufficiently

strong. We estimate the

importance

of the

hairpin

terms

by considering

the

overlap

of _terms such _as S ₌

if1~~).

If the fields _are

strongly peaked

in _{qz away} from

0,

then the

overlap

will be small. When the system is directed the interaction between the

directing

field and the

polymer

field will _suppress fluctuations with

large

_qi Thus the

dispersion

for _1~ will be centered around

it.

Because Di decreases _as the system ^becomes more

directed,

without _a

compensating spread

in _qi, the

spread

in _qz will become smaller and smaller. Thus the

hairpin

terms will be

suppressed. Equivalently,

_{we can} estimate the

importance

of the

hairpin

terms

by

_{a mean} field

theory calculation,

which

yields

the condition

DIA~

~w

Dill(

_«

_(~(,

_~~'~~~

where A is the cutofs and should be set to the inverse of the

polymer persistence length, lp.

^We expect

(3.14)

to hold for

polymers

made _up of the

liquid crystal molecules,

as the transitions

should _occur

simultaneously.

Thus if the

coupling

is not

strong enough

to drive Di

down,

we may not

neglect

the additional _terms in

(3.12).

^Thus we would

only

have _a

theory

of

anisotropic polymers

where

(Ri(L) Ri(0))~

oc

iRi(L) Ri(0))~

+~ L.

(3.15)

In _a directed

theory

these would not scale the _{same way} with L.

Aligned polymers

_are directed

polymers.

4. Effective

theory

of

polymer

melts.

In addition to

reducing

to the correct directed

theory

in the nematic

phase,

the full

theory

allows _us to

study

the system ^{both in the}

isotropic regime

and in the

vicinity

of the transition.

In order to understand the nature of the

physics

in this

regime,

it is natural to

attempt

to

integrate

out

alternately

the

polymers

and the nematic and examine the effective

theory governing

^the

remaining

component.

First _we win examine the form of the effective field

theory

for the

polymer

melt in the

isotropic phase.

Just _as in the _case of the directed

theory

[5], one expects the

liquid crystal

to mediate interactions between the

polymers. Intuitively,

^the directional

properties

of the

liquid crystal

dictate that the induced

polymer-polymer

interaction will

depend

_on the

tangent

vectors _as well _as the

polymer density.

By perturbatively integrating

out the

Q field,

we can arrive at an efsective interaction for the

polymers

in terms of S. In order to compare with [16], we can find the efsective Frank

constants for _a

polymer melt, making

contact with the results in

[12].

This

requires

_us to

reproduce

terms with the _same

symmetries

_as the terms of

(1.3).

^The

quadratic

terms _come from the

propagator

^of the

Q field,

while the coefficients of the nonlinear ternls

require

the

evaluation of

loop diagraras.

We find:

This

generalizes

the result of [16]. ^{Since the transition is first}

order,

_a is _non-zero. Thus the interactions _are short

ranged

in the

isotropic phase.

We _can

expand (4.I)

in _powers of momentum to find the effective free energy of the

polymer

melt in tend of S. We have to lowest order:

TNT

_" ^>~

d~Z

l~)TI(S)~

⁺

$(fiPS°fl)(fi~S~~)

⁺

(fiOS~~)(fi'S7fl)j

(4.2)

From this

we can read off the "FraJ~k" constants for the

polymer

melt. We note that the ratio of the

polymer

Frank constants is the _{same as} that for the

liquid crystal.

This suggests ^that

using

a

liquid crystal background

_as a

coarse-grained

model of _a

polymer

melt should be

good

at

long-wavelengths-

In the

isotropic phase

the terms in

(4.2)

_are irrelevant. However, in the directed

phase,

these terms become relevant.

5. Effective

theory

of the

liquid crystal

and the transition temperature.

More

interesting

is the effect of the

polymers

_on the

liquid crystal.

Because of their entropy, the

polymers

resist

alignment by

the

liquid crystal.

This win afsect the Frank _{constants as} well _as the parameters ^{of the bulk free} energy. We expect the _presence of the

polymers

to lower the temperature at which the

liquid crystal

_goes from the

isotropic

to nematic

phase.

In

principle,

it should be

possible

to calculate this shift in ternls of the parameters in the

polymer

free energy. This calculation is

possible

in _a

simple

_{way away} from the

vicinity

of the critical

point

of the

polymer theory

I-e- for dense

polymer

systems. ^{In this} case, the

polymer

free

energy can be

integrated

out

simply using

fluctuation corrected mean-field

theory.

By integrating

out the

polymers,

corrections to the

potential

terms in the

liquid crystal

free energy are

generated,

and it is

possible

to calculate these order

by

order in the dielectric tensor

Qpv.

It is

straightforward

to see that the transition temperature will be shifted down.

We examine the corrections to the

liquid crystal

_mean field

theory by generating

terms in the liquid crystal free energy

density

_upon

integration

of the

polymer

fields.

First,

_we must

linearize the

polymer

interactions in

(I.I), including

_{a source} term h~

= hb~~, ^which

couples

to

just

_one of the fields. To

generate TI(Q~)

terms, _we note that _constant

Q implies

_a

spatially

constaJ~t minimum for the

polymer

fields. We linearize

by expanding

about the

mean field result

~°

hi

(v/2)~(

_~ ^~~'~~

where

~(

is determined

by squaring

both sides and

solving.

We define the deviation

by

~

₌

~o

_{+ n.}

Rewriting

the free _energy to second order in _q

yields

F~

₌

I(fi»n,)(fi»n~) (~ v~l)n;n,I

+

@nl >~n,Q»vfi~fivn~. (5.2)

From this _{we see} that _n I fields

decouple

with chemical

potential

~J~ ⁼ ~J

v~(, (5.3)

while _qi

acquires

chemical

potential

_~J~

v~(/2.

From

(3.3)

, we see that if

(a

b~

/12c)

is shifted

positively,

then the transition temperature decreases. In

general,

the connected

diagram contributing

to the shift in

Landau-Ginzburg

parameters _a, b, ^and _c, ^is a

polygon

with two,

three,

_or four vertices.

Noting

that each vertex

N°3 THE ISOTROPIC~NEMATIC TRANSITION 271

contributes _a factor of

l~

_{to the}

shift,

we may, in the limit of small

coupling,

look at the correction _to lowest order in l~. The twc-vertex

diagram

is

proportional

to

TI(Q~),

and thus afsects the shift in _a. Recall that _we have _n

polymer

fields in _our

problem,

and _we wish to take the _n-+ 0

limit,

as discussed

previously.

We have

-bJ~c

=

(->~)~Q~~Q~~ IA

P»PUP«PP

(p~

_~~

l

v~i/~)~

⁺

(p'-i~~~1 ^(5.4)

Taking

n-

0,

we note that the

right

hand side of

(5A)

is

manifestly negative,

and

multiplies TI(Q~). Therefore,

_a is shifted

upward,

with

leading dependence

on the parameters as follows:

~~ ~

(4x)d/2(d2

+

2d)r(d/21'

^~~'~~

Recall that

~]

is the

polymer density.

Now since db and dc _are

higher

order in

l~,

_we conclude that in the weak

coupling ^limit, (a

b~

/12c)

is shifted

positively.

In

particular, (3.3) gives

the shift in the transition temperature:

dTc

_cc -da.

Note that if there is _no

polymer

excluded volume interaction

(v

₌

0),

_a constant

Q implies

no shift in the transition temperature ^within _mean ^field

theory

for the

liquid crystal and,

in

fact,

_no shift in _any of the

potential

terms of the

liquid crystal.

To

study

the effects of

(v

₌

0)

interactions _on the transition temperature, we must consider nonconstant

Q.

In this _case, the minimal

polymer

field

configuration

^is _no

longer

constant.

However,

_we

expand

about

(5.I) again.

This results in _anew term in

(5.2), namely -A~~[QP~fi~

au

q~.

This interaction generates

a new kinetic _term in the effective

liquid crystal

free _energy of the form

-I((fi~ au Q~~)~,

where K ₌

l~(~[)~ /(-

_~J)is

positive.

This term thus makes the system ^{less still}

against

fluctuations in

Q, resulting

in _a lower transition temperature.

Thus,

we see onci

again that,

in agreement ^with

physical expectations,

the

polymer

interactions shift the

liquid crystal

transition temperature

downward.

Acknowledgements.

It is _a

pleasure

to

acknowledge stimulating

discussions with

Sidney Coleman,

Howard

Georgi,

Mehran

Kardar,

Onuttom

Narayan,

and

especially

David Nelson. This work _was

supported

in part

by

the National Science

Foundation, through

Grant No.

DMR-91-15491,

and

through

the Harvard Materials Research

Laboratory.

References

ill

Kamien R-D-, Le Doussal P. and D.R. Nelson, submitted _to Phys. Rev. A

(1991).

[2] ^Nelson D.R., Physica A 177

(1991)

220.

[3] ^{des Cloizeaux} J., J. Phys. ^France 36

(1975)

281.

[4] ^{A. Ciferri and W-R-} Kringbaum Eds., Polymer Liquid Crystals

(Academic

Press, New York,

1982).

[5] ^{Le Doussal P. and Nelson} D-R-, Europhys. Lent. 15

(1991)

161.

[6] ^Nelson D-R-, Phys. Rev. Lent. 60

(1988)

1973.

[7] Nelson D-R- and

Seung

H.S.,Phys. ^{Rev. B} 39

(1989)

9153.

[8] Halsey T.C. and Toor W., Phys. Rev. Lent. 65

(1990)

2820.

[9] Rosensweig R. E., Ferrohydrodynamics

(Cambridge

University Press, New York,

1985).

[10] ^Le Doussal P. and Kamien R.D., in preparation

(1991).

[11] ^{Khokhlov A. and} Simeuov A.N., Physica108A

(1981)

546.

[12]

Vroege

G.J. and Odijk T., Macromolecules 21

(1988)

2848.

[13]

Selinger

J. and Bruinsma R., Phys. Rev. A,

(1991).

[14] ^{de Genues} P-G-, Phys. Lent. A 38

(1972)

339.

[15] Wright D.C. and Mermiu N-D-, Rev. Mod. Phys. 61

(1989)

385.

[16] ^{de Geuues} P.G., Mol. Cryst. Liq. Cryst. 102

(1984)

95.

[17] ^{Landau L.D. and Lifshitz E.} M., The Classical Theory of Fields

(Pergamon

Press, Oxford, 1975)

pp. 77-87.

[18] ^M.J. Freiser, Phys. Rev. Lent. 24

(1970)

1041.

[19] P-G- de Geuues, Physics of Liquid Crystals

(Oxford

University, London,

1974).