The effect of permanent dipoles on the nematic-isotropic phase transition

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The effect of permanent dipoles on the nematic-isotropic phase transition

J.G.J. Ypma, G. Vertogen

To cite this version:

J.G.J. Ypma, G. Vertogen. The effect of permanent dipoles on the nematic-isotropic phase tran- sition. Journal de Physique, 1976, 37 (11), pp.1331-1333. �10.1051/jphys:0197600370110133100�.

�jpa-00208530�

1331

THE EFFECT OF PERMANENT DIPOLES ON THE NEMATIC-ISOTROPIC

PHASE TRANSITION

J. G. J. YPMA Solid State

Physics Laboratory University

^of

Groningen,

The Netherlands

and G. VERTOGEN Institute for Theoretical

Physics University

^of

Groningen,

The Netherlands

(Reçu

^{le 5 mai 1976,}

accepté

^le

17 juin 1976)

Résumé. ²⁰¹⁴ On étudie l’influence de

dipôles

moléculaires permanents ^{sur la} transition nématique- isotrope ^en

ajoutant

^une interaction

polaire simple

^{au modèle de}

Maier-Saupe.

^{Ce nouveau modele}

est traité suivant

l’approximation

^de Bethe-Peierls pour tenir compte de l’ordre à courte distance.

La

possibilité

d’un ordre (anti) ferroélectrique ^à longue ^{distance a été} considérée, ^il peut ^{avoir une} forte influence sur la stabilité de la

phase.

Abstract. ²⁰¹⁴ The influence of permanent

dipoles

^on the nematic isotropic transition is studied by adding ^a

simple polar

interaction to the

Maier-Saupe

model of nematic liquid crystals. ^{The model}

is treated in the Bethe-Peierls approximation ^{in order} to account for short range order. We have also included the

possibility

^of (anti) ferroelectric

long

range order. This ferroelectric order strongly

affects the

stability

of the nematic

phase.

LE JOURNAL DE PHYSIQUE ^TOME 37, ^NOVEMBRE 1976, :

Classification Physics ^Abstracts

7.130

One of the characteristic

properties

of nematics is the

indistinguishability

^{of states} with director n and

- n. When individual molecules _carry a permanent electric

dipole moment, just

^as many

dipoles point

up

as down. Otherwise the system would be ferroelectric.

Ferroelectric nematics

have,

^as yet, ^{not been found}

[1].

There are a number of

nematogenic

^materials

[2]

which do not carry ^a

permanent dipole

moment, i.e.

the influence of

permanent dipoles

ⁱⁿ

establishing

nematic order cannot be dominant. In the model of Maier and

Saupe [3]

^{the Van} der Waals interaction between

mutually

^induced

dipole

^moments

gives

^rise

to a

phase

transition from the

isotropic

^to the nematic

phase.

The interaction _energy can be written as

where ; specifies

^the orientation of the

long

^{axis of a}

molecule i,

^and

P2

denotes the second

Legendre polynomial.

^{In the mean field}

approximation,

^as

used

by

^{Maier and}

Saupe,

^the energy of a molecule is

given by

where y

^{is the number of nearest}

neighbours,

^and

S ⁼

( p 2 (aiz) >

^{is the}

long

range order parameter,

to be determined self

consistently.

In two recent papers

[4]

^we studied the interaction

(1)

in the

approximation

of Bethe and Peierls in order to account for short _range order. The

long

range pro-

perties appeared

^{to be} very similar to Maier and

Saupe’s

^mean field results. The

description

^of pre- transitional effects in the

isotropic phase, namely

^the

magnetically

^induced

birefringence

^{and the}

scattering

of

light by

orientational

fluctuations,

^was

improved considerably.

Although

permanent

dipoles

^should

play a

^minor

role in

establishing

^nematic

order,

^the effects of a

polar

interaction on the transition temperature, ^the

jump

in the order parameter, etc., ^{is still} open ^to

question.

If we add a

polar

interaction in its

simplest form,

the interaction _energy reads

The first term could result from the interaction between permanent

dipoles ^which,

^for

cylindrically symmetric molecules, point effectively along

^the

long

^molecular

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370110133100

1332

axis.

Krieger

^{and James}

[5]

studied this model in the

mean field

approximation. They

showed that the

system

can exist in three

phases :

^an

isotropic phase (Pl

⁼

P2

⁼

0),

^an even

phase,

^{which we would call}

the nematic

phase (Pl

⁼

^0, j52 :0 0)

^{and a mixed}

phase with (anti)- ferroelectric ^long

^{range order}

(P1 = 0, P2 =1= 0). P1

^and

P2

^{are the}

long

range order

parameters Pl(aiz) > and P2(aiz) >,

^{which are}

determined self

consistently.

^In

figure

^{1 we have}

FIG. 1. ^- Transition temperatures between the isotropic, ^nematic

and (anti) ferroelectric phase ^{as a} function of J1/J2, in the Bethe

approximation (solid line) ^{and the mean field} approximation (MF, ^dashed line).

drawn the transition

temperatures

between the three

phases

^{as a} function of

J1/J2,

^as found in the ^mean field

approximation (dashed line).

^It shows that a

nematic

phase

^cannot exist for

J1/J2 >

^{0.35. All}

properties

of the nematic

phase, except

^{for its}

stability,

are

independent

^of

Jl,

^the

strength

^{of the}

polar

interaction. This is a consequence of the mean field

approximation,

^which

neglects

short range order.

In any

approximation

^{that accounts for local}

order,

there

might

^{be an} effect of the

Pi type

^{of short} range order on the characteristics of the

nematic-isotropic

transition. We studied this

question

^{in the}

approxi-

mation of Bethe and Peierls. We remark that Madhu- sudana and Chandrasekhar

[6]

treated the

problem

in a

mathematically

^{much more}

complicated

^version

of the Bethe

approximation. They obtained,

^inaccurate

results

however,

e.g. the order parameter ^{at the}

nematic-isotropic

transition is too low

by nearly

^a

factor of two. Whether their version of Bethe’s method

can

give

^accurate

results,

^is open ^to

question.

In Bethe’s

approximation [4, 7]

^a cluster of _y + 1 molecules is

considered,

^one of which is

regarded

^as

the central one with orientation _ao,

being

^surrounded

by

y ^nearest

neighbours

with orientations

The

weight

^{of a}

given configuration

of the cluster is

given by

where Z is a normalization constant

and fl

⁼

1 /kT.

In this

expression z(ai)

^accounts for the influence of the

surroundings

^on the cluster and is

given by

where we included the

possibility

^of

P1

type ^of

long

range order

(h1 = 0).

^The

strengths h1

^and

h2

^{of the}

effective

fields,

^{which act}

only

^{on the nearest}

neigh-

bours of the central

molecule,

^are determined

by

^the

condition that the _average orientations of the central molecule and its

neighbours should

^be

equal :

and

The thermal _averages, denoted

by ( >,

^{have to be}

evaluated with the distribution function

(4).

^The

relations

(6)

^and

(7)

^{form a set of two}

coupled

^transcen-

dental

equations

ⁱⁿ

h1

^and

h2, which,

^{with some}

effort,

can be solved

numerically.

We find a similar behaviour of the transition

temperatures

^{as a function}

of JIIJ2

^{as in the mean field}

approach (see figure ^1,

^solid

line).

^{The whole}

diagram

is

only

^{shifted to lower} temperatures ^{and the}

triple point

^{lies at a}

slightly higher J,IJ2 value,

^{which also}

depends

^{on y.}

Since

(anti)

ferroelectric nematics have not been

found, Ji/J2

^{has to} be chosen such that a reaso-

nable width of the nematic

temperature region ^results, something

like 0.1 times the

nematic-isotropic

^tran-

sition temperature. ^This

implies J, IJ2

^{0.3. One}

could

imagine

^{that the}

hypothetical

transition from the nematic to the mixed

(ferroelectric) phase

^{is then}

hidden

by

^the transition to the

crystalline

^{or a smectic}

phase,

which would take

place

^earlier.

Although

^this

procedure

is somewhat

artificial,

^{we ihink it more}

legitimate

^{than to}

ignore completely

^the

possibility

of

p 1 type

^of

long

range order. The last

approach

^is

followed

by

Madhusudana and Chandrasekhar

[6].

They

consider the

nematic-isotropic

transition also in the cases

J1/J2

^{= 0.5 and}

J1/J2

⁼

3.2,

^{where the} nematic

phase

^is

always

^less stable than the ferroelectric

phase.

^We remark that

changing

^the

sign

^of

J1’

pro- duces an

antiferroelectric

instead of ferroelectric mixed

phase (Pi - - Pi).

We studied the effect of ferroelectric short _range order on the

nematic-isotropic

transition in the Bethe

approximation

for values

of J1/J2

between 0 and 0.3.

Our results can be summarized as follows :

1333

1)

^The transition temperature

T,

^for

J1 =I

^{0 is}

somewhat raised as

compared

^{to the case}

J 1

^{= 0.}

The

change

^{is at most a few} percent ^for y ⁼ 3 and less for

higher

^{values of} y. The latent heat of the tran- sition

changes

^{in the same} way

(see

^Table

I).

TABLE I

Change of

^the transition temperature ^and

of

^the

latent heat

of

^the

nematic-isotropic

transition in the

case

J1

^{= 0.3}

J2

^as

compared

^with

J1

⁼

0, for

^various

numbers

of

^nearest

neighbours

y.

2)

The nematic

long

range order parameter S

= P2(ao,- ,) >

^{as a} function of

TITc,

^is

independent

of

Jl. (The variations

are less than 0.01

%.)

3)

The ferroelectric short _range order parameter 6p

= P, (ao. a,) >

^increases

steadily

^{as a function}

of

J1/J2.

This short _range order is

responsible for

^the

small variations in the transition temperature, ^the latent heat and the

specific

^heat.

4)

The nematic short _range order parameter

US

= P2(aO.a1) >

^is

independent

^of

Jl. (The

^varia-

tions are less than 0.01

%.)

This short _range order determines the value

of (Tc - Tc*)Tc

^{for the}

magnetic birefringence

^{and the}

scattering

^of

light

^{in the}

isotropic phase [4].

^{As a}

result,

^the

description

^{of these} pre- transitional

phenomena

^{is not}

changed by adding

^a

polar

interaction of the form -

J1 P1(a1.aj)

^{to the}

original Maier-Saupe

interaction

(1).

Acknowledgments.

^{- We wish to} thank Prof. A.

J. Dekker for _many

helpful

discussions. This work is part of the research _program of the Foundation for Fundamental Research on Matter

(F.O.M.).

References

[1] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Oxford, University Press) ^1974.

[2] ^{VAN DER} VEEN, J., ^DE JEU, W. H., GROBBEN, ^{A. H. and} BOVEN, J., Mol. Cryst. Liq. Cryst. ¹⁷ (1972) ^291.

[3] MAIER, ^{W. and} SAUPE, A., ^Z. Naturforsch. ^14a (1959) ^882.

[4] YPMA, J. G. J. and VERTOGEN, G., Solid State Commun. 18

(1976) 475 and J. Physique ³⁷ (1976) ^557.

[5] KRIEGER, T. J. and JAMES, ^H. M., J. Chem. Phys. ²² (1954) ^796.

[6] MADHUSUDANA, ^{N. V. and} CHANDRASEKHAR, S., ^Pramãna Suppl. ¹ (1975) ^57.

[7] BETHE, H. A., Proc. R. Soc. 149 (1935) ^1.