A molecular statistical treatment of pretransitional effects in nematic liquid crystals

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A molecular statistical treatment of pretransitional effects in nematic liquid crystals

J.G.J. Ypma, G. Vertogen

To cite this version:

J.G.J. Ypma, G. Vertogen. A molecular statistical treatment of pretransitional effects in nematic liquid crystals. Journal de Physique, 1976, 37 (5), pp.557-567. �10.1051/jphys:01976003705055700�.

�jpa-00208450�

A MOLECULAR STATISTICAL TREATMENT

OF PRETRANSITIONAL EFFECTS IN NEMATIC LIQUID ^CRYSTALS

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 13 octobre}

1975, accepté

^le

5 janvier 1976)

Résumé. ²⁰¹⁴ Pour discuter les effets de l’ordre à courte distance

dans

les cristaux liquides, ^nous présentons ^{une nouvelle} généralisation ^{de la} méthode de Bethe, qu’on peut ^traiter mathématiquement.

Nous avons appliqué ^{cette méthode au modèle de} Maier-Saupe ^{des cristaux} liquides nématiques ^et

nous avons examiné l’effet des corrélations à courte distance sur la transition nématique-isotrope.

Ensuite nous avons calculé l’effet Cotton-Mouton et la dispersion ^{de la} lumière par les fluctuations de l’orientation dans la phase isotrope.

Abstract. ²⁰¹⁴ In order to discuss short range order effects in liquid crystals ^we present ^a new, mathe-

matically ^tractable generalization of Bethe’s method. We have applied ^this

approximation

^{to the} Maier-Saupe model of nematic liquid crystals and studied the effect of short range correlations ^on the

nematic-isotropic transition. Further we calculate the magnetic birefringence ^{and the} scattering ^of

light

by orientational fluctuations in the isotropic phase.

Classification

Physics ^Abstracts

7.130

1. Introduction. ^- In the

isotropic phase

^{of nematic}

liquid crystals

^some

pretransitional

effects have been observed

[1],

e.g. the

magnetically

induced birefrin- gence

(Cotton-Mouton effect)

^{and the}

scattering

^of

light by

fluctuations of

anisotropy.

^Both

quantities

appear ^to behave

approximately

^as

(T - Tc*)- 1,

where

T,,,*

^lies

slightly (about

¹

K)

below the transition

point Tc.

^The

pretransitional phenomena

^are

usually

described

phenomenologically by

^a Landau model

[1, 2].

^It would be of great ^{interest to treat these} effects in terms of a molecular statistical

theory.

Until _now,

however,

^most statistical theories

neglect

short _range correlations which are essential for the treatment of these

phenomena.

It is the purpose of this paper ^to present ^{a method} which accounts in some way for short _range

order,

and to

apply

^it

subsequently

^{to a} model for nematic

liquid crystals.

^The method is a

generalization

^of

Bethe’s

[3] approximation,

^{which was}

developed

^to

treat the effect of local order on the order-disorder transition in

binary alloys.

^{We shall see} that there are

several _ways for such a

generalization.

In section 2 we discuss the nature of our

approxi-

mation and _compare it with the methods of

Chang [4]

and

Krieger

^{and James}

[5]

^{which can}

hardly

^be

applied

to first-order

phase

transitions. In section 3 the

proposed

^{method is}

applied

^{to a} model of nematic

liquid crystals that,

^{in the mean} field treatment, is

equivalent

^{to Maier and}

Saupe’s [6]

^model. It appears that the behaviour of the order

parameter

^is

nearly

the same as in the

Maier-Saupe theory;

^{the main}

difference, naturally,

^turns up in the results for the

isotropic phase.

^A

preliminary

^{account of this}

part

has been

given

in référence

[7].

^{In section 4 we use the}

results for the

isotropic phase

^to calculate the _magne-

tically

^induced

birefringence

^{and the}

scattering

^of

light by

orientational fluctuations. The results show that Bethe’s

approximation provides

^a considerable

improvement

^with respect ^to the molecular field

approximation. Finally,

^a summary with conclusions is

given

^{in section 5.}

2. A

generalization

^of Bethe’s method. ^- Bethe’s method was

developed

for the order-disorder tran-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003705055700

sition in

binary alloys,

^{and has also been}

applied

^{to the}

Ising

^{model of}

ferromagnetism.

^{The main}

difficulty

in

extending

this method to the case of nematics is due

to the fact that each molecule now runs

thrôugh

^a

continuous set of

possible

orientational states.

Basically,

the method works as follows :

Consider a small aggregate ^of y + 1

molecules,

one of which is

regarded

^as the central _one,

being

surrounded

by

y nearest

neighbours,

which form the outer shell. It is assumed that these nearest

neighbours

are not nearest

neighbours

of each other. Let the orientation of the symmetry axis of the central mole- cule and of its

neighbours

be described

by spherical

coordinates

(0., ço)

^and

(Oi, ç) respectively, symbo- lically

^{written as}

Qo

⁼

{ 00, ~o}

^and

Qi = { Oi, (pi } (i = 1, 2, ..., y) .

Let

E(Q, Q’)

denote the nearest

neighbour

interaction between two molecules

with

orientations Q and Q’.

(We

confine ourselves to nearest

neighbour

^interac-

tions.)

^The

weight

^{of a}

given configuration

^{of the}

cluster is in the Bethe

approximation given by

where fl

⁼

IlkT

^{and Z is a} normalization constant defined

by

In this

expression z(Qi)

^accounts for the influence of the

remaining

system ^{on molecule i} of the cluster.

This influence is

represented by

^an

orienting

poten- tial

V(Qi),

^{which acts}

only

^{on the outer} shell mole-

cules ;

^therefore

The

potential

is determined in Bethe’s

approximation by

^the

requirement

of translational invariance. At this stage difllerènces _appear between the

approaches

of

Chang [4]

^{and of}

Krieger

^{and James}

[5]

^{and our}

treatment.

a) Chang

^{states that the}

probability

of the central molecule

taking

^a certain direction

Q,

^i.e.

equals

^the

probability

^{of an outer shell molecule}

taking

the direction

Q,

^i.e.

This condition

which can be written as

with

has to be fulfilled for _every orientation Q.

b) Krieger

and James state that the

probability

for the central molecule and one of its

neighbours

^to

assume the directions

Qo and Q, respectively,

^i.e.

must have the same value

regardless

of which of the

two molecules is considered as the central _one, or

which reads in the above mentioned notation

This means

it

being

^a constant,

independent

of the orientation Q.

It is _easy to see that this condition is included in

Chang’s consistency

relation. The reverse,

however,

is not true.

In both treatments the

requirement

of translational invariance leads to a non-linear

integral equation

^for

z(Q ) (for

y >

2).

^{Near a} second-order

phase

^transition

this

equation

^{can be solved}

by expanding z(Q)

^and

exp[- ^03B2E(Q, Q’)]

ⁱⁿ

Legendre polynomials.

^Note

that there is an

isotropic

^solution

z(Q) ==

^{1 at all}

temperatures

^if

E(O, Q’)

^is

rotationally

^invariant.

This

expansion procedure

^{cannot be}

applied

^{to first-}

order

phase transitions,

^{such as the}

nematic-isotropic

transition in

liquid crystals. Consequently

^{one is left}

with an awkward

integral equation

^{which seems}

impossible

^{to solve.}

c)

We propose that the

requirement

of transla- tional invariance should be

interpreted

^{in a different}

way : ^we

only require

^{that the} average

orientation,

as determined

by

^a

suitably

chosen function

S(Q),

will be the same for the central molecule and its

neighbours,

^i.e.

We remark that this condition is still weaker than

Chang’s

condition. In the case of

only

^{two allowed}

orientations,

for instance the

Ising model,

all three mentioned

consistency

^relations

(2.7), (2.11)

^and

(2.13)

^are

completely equivalent.

^{In order to} progress further we state

that,

^{in the case that all}

neighbours

are

equivalent,

^the

orienting potential V(Q)

^{is of}

the form -

hS(Q),

^i.e.

This choice for

V(Q)

^is

quite

reasonable from the

physical point

^{of view.}

V(Q),

^which represents ^the interaction of an outer shell molecule with the remain-

ing

system, ^is

simply

considered to be an effective field.

For

simple

interactions like - J cos

Oij

^and

Oij ^being

^the

^angle

between the symmetry ^{axes of two} molecules i

and j,

^the energy of a molecule in such an

effective

(molecular)

field is of the form - h cos 0 and - h

cos’

0

(or

^more

conveniently - h(2 ^cos2 0 20131/2)).

0 denotes the

angle

between the symmetry ^{axis of a} molecule and the

preferred

direction of the aniso-

tropic phase.

As a result the

consistency

^relation

(2.13)

^becomes

an

equation

^{in one}

pàrameter h,

^the

strength

^{of the}

effective field

acting

^{on the outer} shell molecules.

This

equation

^{is of course much easier to} handle than the

original integral equation

^for

z(Q).

^{In the case} that _eq.

(2.13)

allows for more than one solution for

h,

the criterion of the lowest free _energy selects the correct one.

3.

Application

^{to a model for a nematic}

liquid crystal.

- In this section we

apply

^the

method, developed

ⁱⁿ

the

preceding

^{section to a} model for nematic

liquid crystals.

^{In this}

^model,

^first

proposed by

^{Maier and}

Saupe [6],

^{the van} der Waals forces between molecules with

anisotropic polarizabilities

^are considered as the

origin

of the nematic order. The interaction _energy between two

neighbouring

molecules i

and j

is pro-

portional

^to

where _ai and aj ^{are unit vectors}

pointing

ⁱⁿ the direction of the

long

^{axis of the} molecules and rij ^{denotes the}

distance vector between the molecules. In the

following

we

neglect

correlations between the orientations and the center-of-mass

positions

of the molecules : it is assumed that the distribution of the centers of mass

of the molecules is

spherically symmetric.

^{The inter-}

action _energy is

averaged

^{over all}

positions

^{of mole-}

cule j

^{on a}

sphere,

^while

keeping

^the

position

^of

molecule i in the centre of the

sphere

^{and the orien-}

tations _ai and aj fixed. Then we are left with an inter- action of the form

where J is

proportional

^to the inverse sixth _power of the intermolecular distance.

It

easily

follows that this model in the mean field

approximation

^is

equivalent

^to Maier and

Saupe’s

model. In that case the energy of molecule i in the molecular field of its nearest

neighbours equals

where S is the

long

range order parameter.

Expres-

sion

(3.3) is,

except ^{for the}

notation,

^{identical to} Maier and

Saupe’s

^form.

In the Bethe

approximation

^the

weight

^{of a}

given configuration

^{of a cluster of}

molecules,

^{as discussed}

in section

2,

^is

given by

with and

The

strength

h of the effective field

acting

^{on the outer}

shell molecules is determined

by

translational inva- riance

(eq. (2.13)), resulting

^{in :}

with

We note that h ⁼

0, corresponding

^{to the}

isotropic phase with S(a) >

⁼

0,

^satisfies eq.

(3.7)

^{at all}

temperatures. Non-trivial solutions for h have to be obtained with the aid of a computer. ^{Note that} eq.

(3.7) already

involves fourfold

integrations apàrt

from the calculation of

F(ao)7-’.

^{In one of the} appen- dices

(C)

the method to solve this

équation

^is

briefly explained.

^{In the case that eq.}

(3.7)

allows for

more than om solution the free _energy determines

the correct one

corresponding

^to

thermodynamic equilibrium.

^We remark that the classical nature of

our model excludes the appearance of an anti-Curie

point,

^{which is found if} the Bethe-method is

applied

to the

quantum-mechanical Heisenberg

^model

[8].

The entropy ^{of the cluster is}

given by

where the

brackets >

denote the thermal _average.

The internal _energy of the cluster

equals

the last term

representing

^the contribution of the effective field which has to be distributed

equally

among the outer shell molecules and the

surroundings

of the cluster. The free _energy of the cluster is

given by

We find that the system ^{exhibits a} first order

phase

transition between the nematic

phase (h

>

0)

^{and the}

isotropic phase (h

⁼

0), provided

^that the number of nearest

neighbours

^is greater ^{than two. As will be} shown in

appendix

^{A the Bethe}

approximation yields

in the case y ⁼ 2 the exact answer in the

thermody-

namic limit. For various numbers of

neighbours

^the

transition temperatures ^are

given

^{in table I.}

They

^{are of}

course lower than in the mean field

approximation

^of

Maier and

Saupe.

^The

long

range order is

only slightly

less than the mean field value

(see

^table

I).

TABLE 1

Transition temperature

Pc

^{J and}

corresponding long

range order

Sc for

the Bethe and the mean

field (MF) approximation for

^{various numbers}

of

^nearest

neigh-

bours _y

From the internal energy per

particle

^which

equals

it follows that the

specific

^heat at constant volume

Cv

^can be written as

h

+ 03B2 dfi

is obtained

by differentiating

^the

consistency

^relation

(3.7)

^with

respect to 03B2.

^{After some} calculations

dp

we find :

In this

expression

ao

specifies

the orientation of the central

molecule,

^and a, and _a2 denote the orientations of two of its nearest

neighbours.

^{We remark that in the}

approximation

^of

Krieger

and James mentioned in the

preceding section,

the correlation functions

will be identical. In our

approximation

^these

quanti- ties, although

^not

being identical,

^{differ at most a few}

tenths of a

percent

in the whole relevant

temperature

range. In

figure

^{1 we} compare the

specific

^{heat as a}

function of temperature for the Bethe

approximation

and the mean field

approximation.

^{The most remark-}

able feature of our

approximation

appears from the

behaviour above the critical temperature, ^where

long

range order has

disappeared

^{but short} range order is still

present.

^{This short} range

order,

^{which causes the}

tail in the

specific

^heat curve, ^can be described

by

Figure

^{2 shows J as a} function of temperature ^for various numbers of nearest

neighbours.

FIG. 1. ^- The specific ^{heat versus} the scaled temperature ^for y ⁼ 3, 6, 12 and in the mean field approximation (MF).

FIG. 2. ^- The short range order u versus the scaled temperature for _y ⁼ 3, 6, 12.

The short _range

order,

^{which is} absent in mean

field

theories,

^{enables us to calculate}

pretransitional

effects in the

isotropic phase.

^This

program

^{is carried}

out in the next section.

4. Pretransitional effects. ^- 4.1 MAGNETIC ^BIRE-

FRINGENCE. ^- When an external

magnetic

^{field is}

applied

^{to the}

isotropic phase

^{of a}

nematic,

^{the aniso-}

tropic

molecules will be

slightly aligned, producing

^a

birefringence

^{An which can be} calculated from the dielectric tensor eij. ^{It holds}

and

where _ni, nj ^are the Cartesian components ^{of a unit}

vector

parallel

^{to the}

optic axis,

ap

and 03B5t

^{are the} dielectric constants

parallel

^and

transverse to the

optic

^{axis for a}

completely aligned material,

and S is the induced

long

range order :

with a

denoting

the orientation of the symmetry ^axis of a molecule. For S « 1 we can write :

The induced

long

range order S was

already

^calcu-

lated

by

de Gennes

[2]

and Stinson and Litster

[1]

from a Landau model. We will follow the molecular statistical

approach

derived in section 2 and 3 for the calculation of S. An

attempt

^to compute ^{the Cotton-} Mouton effect

by

^means of Bethe’s method as

proposed by Krieger

^{and James}

[5],

has been made

by

^Madhu-

sudana and Chandrasekhar

[9, 10]. However,

^their evaluation of

Tr

^is

incorrect,

because their assumed form for

z(Q) (eq. (3.5))

^{does not}

satisfy

^the

integral

eq.

(2. 12)

in the ordered

phase,

^{as we checked}

by

^°

numerical calculation.

The orientational _energy of a molecule with orien- tation a in a

magnetic

^{field H}

along

the z-axis reads :

where

S(a) = 2 ai - 2

^and

AX

^{is the}

anisotropy

^{of the}

diamagnetic susceptibility

of the molecule. The distri- bution function

(3.4)

^{for a} cluster of molecules is now

given by :

where h is determined

by

^the

consistency

^relation

(3. 7) :

with

In the

isotropic phase

^{where h « 1}

(and also J1

^«

1)

^{we can}

expand

part ^{of the}

exponentials. Taking only

^terms

linear

in J1

^and

^h,

^the

consistency

^relation

(4.7)

^{reduces to}

Note that the correlation functions have to be evaluated with the aid of the

unperturbed

distribution function

(u

⁼

0) given

ⁱⁿ

(3.4). Eq. (4.9) implies immediately, using

As a result the induced

long

range order reads

We can relate this

quantity

^{to the short} range order

(see (3.15)) by realizing

^{that in the}

isotropic phase

^the

orientations of the outer shell molecules are correlated

only

^{via the} central molecule of the cluster. This

gives

rise to the

following relations,

^{to be}

proved

ⁱⁿ appen-

dix B,

This leads to the

following expression

^{for the}

long

range order :

The Cotton-Mouton coefficient can with the aid of

(4 . 4), (4.5)

^and

(4.14)

^be

expressed

^{as : ·}

It appears from our calculations that the

reciprocal

of the Cotton-Mouton coefficient behaves ^to a very

good approximation

^like

a fact which is also

experimentally

^observed

[1].

Table II shows the ratio

(Tc - Tc*)/I;;

^{and an estimate}

of the coefficient C for M.B.B.A. Also the calculted

mean field results and the

experimental

^values

[1]

are

given.

^{The mean} field results can

easily

be obtained from the

consistency

^relation

which can be solved

by expanding

^the

exponentials

(u, S « 1).

^It results in :

with 1 y/3c*

^{J =} 5. For the estimate of C we used the values of M.B.B.A.

reported by

^{Stinson and}

Litster

[1]

^and

by Gasparoux et

^al.

[11] : é =

^2.605

[l],

AE = 1.09

[1], (AX)macroscopic ==

^{1.25 x l0 ’}

erg/G2

^g

at 18 °C

[11],

^{which is related to} the molecular ani- sotropy

Ax

^via

where n is the number of molecules _{per gram} and S is the

long

range

order, roughly

^{estimated at}

1.

^We

then obtain

Ax =

^{83 x}

^{10- 3°} erg/G2. Using

^the

value

Tc

^{= 318 K for M.B.B.A. we find}

in the molecular field

approximation

^{and as best}

value

(y

⁼

3) Tc - T*c

^{= 3.5 K in the Bethe}

approxi- mation,

whereas Stinson and Litster observed _expe-

rimentally

^{1 K.}

TABLE Il

The ratio

(Tc - T:)ITc

^{and the}

coefficient

^C

for M.B.B.A.,

^as

defined

ⁱⁿ

(4.16), for

^{various numbers}

of

^nearest

neighbours

ⁱⁿ the Bethe

approximation,

and

compared

^{with the}

experimental

^{and the mean}

field ( M F )

^results.

(u) ^{See reference} [1].

From the

results,

^{shown in table}

II,

^{it can be} concluded that the

approximation

^of

Bethe,

^which includes the effect of local

order, improves

^the

descrip-

tion of the

magnetic birefringence considerably

^with respect ^to the molecular field

theory.

4.2 LIGHT SCATTERING. ^- The relevant formulas for a molecular

description

^of

light scattering

^have

been

reported by Lubensky [12].

Each molecule in a

nematic has an

anisotropic

^electric

polarizability

of the form

from which we can define a

polarizability density

where a ⁼

1/3(ap

^{+ 2}

^o:J,

^{Cla =} ap - at ^and ap

and 03B1t

are the molecular

polarizabilities parallel

^{and trans-}

verse to the electric

field ; R’(t)

denotes the

position

of molecule i at time t and

Q,,’,o

^{can be}

^expressed

^{as :}

with

ai denoting

^{a unit vector} in the direction of the symmetry ^axis of molecule i. The

intensity

^of

light

scattered

by

fluctuations of the

polarizability

^tensor

is

proportional

^to

where A is the

wavelength

of the incident

light,

q is the

scattering

^{vector and e’ and eF are the}

polarization

vectors of the incident and the

outgoing

^beam respec-

tively.

^We have assumed that the

positions

^{Ri of the}

molecules are

independent

of time and do not corre-

late with the orientations _ai. The

polarization

^{of the}

incident beam e’ is taken

perpendicular

^{to the scatter-}

ing plane,

ⁱⁿ agreement ^{with the}

experimental

^confi-

guration [13].

We further assume that

q . (Ri - Ri) «

¹

which

implies

^{that the}

exponential

^{will be}

equal

^to

one. This

assumption

^is

justified by

^the

experimental

fact

[14, 15]

^{that the}

light intensity

^I

depends only

very

slightly

^{on the}

scattering

^{wave vector.}

Experi- mentally,

^two intensities have been measured with

polarizations

^e’

parallel

^and

perpendicular

^{to the}

polarization ^e’

of the incident

light.

^We

find,

^because

eF . Qi . el > = 0

^{in the}

isotropic phase,

^that

and

where

Q003B103B2 ^belongs

^{to a}

particular

molecule and the index i runs over all the other molecules of the

sample.

The correlations of the central molecule with its _y nearest

neighbours give

^{a term}

y Q’ Q;p ^),

^where

Qlo ^belongs

^{to a nearest}

neighbour.

Correlations with next nearest

neighbours

^with

corresponding Q203B103B2,

^run

via nearest

neighbours giving

^{a term}

which is put

equal

^to

Generally

^we put

Because of translational invariance this means

Using

^the definition

(4.22)

^for

Q’o

^and

^summing

^over

all correlations between the

molecules,

^the

following expressions

^{result :}

In

appendix

^{B it is}

proved

^{that these}

quantities

^{can be}

related to the short _range order parameter

We find

The ratio of the

polarized

^{and the}

depolarized

compo- nent is seen to be

4/3,

^{which is confirmed}

by

^the

experi-

ments of Stinson and Litster

[1].

^{The total}

intensity

of the scattered

light

^is

given by

Our calculations show that the

intensity

^{behaves as}

(T - Tc*)-l.

^For the ratio

(Tc - Tc*)/Tc

^{we can}

refer to the values found for the

magnetic

^birefrin-

gence

(Table II)

^{which are the same.}

5:

Summary

and conclusions. ^- In order to discuss short _range order effects we

proposed

^{in section 2 a}

generalization

of the Bethe

approximation

^{to the case}

of rotational transitions as observed in molecular and

liquid crystals.

^{From the}

computational point

^of

view it is obvious that our method is easier to handle than the methods

proposed by Chang

^and

Krieger

and

James, especially

^{in the case of} first-order

phase

transitions.

We

applied

in section

3

Bethe’s

approximation

^to

the

rotationally

invariant form of the

Maier-Saupe

interaction. It has to be stressed that the use of a

rotationally

invariant interaction is

imperative

^as

soon as the molecular field

approximation

^{is aban-}

doned. The

original symmetry breaking potential

^of

Maier and

Saupe,

^{which is} of the form

does not

produce

^an

isotropic phase

^{in Bethe’s}

approxi-

mation and in fact in _any

approximation

^that goes

beyond

^{the mean field}

approximation.

This result was

also obtained

by

Raich et al.

[16] using

the two-site

cluster

approximation

^and

by

^Kimura

[17] using

^an

inverse temperature

expansion.

^{Remarks in favour of}

a

rotationally

^invariant interaction have been

given by

^Priest

[18],

^{while some} fallacies of the mean field

approximation

^{in the case} of the interaction

(5.1)

have been

pointed

^out

by

^Schultz

[19].

We found that the

long

range

properties

^{of the}

rotationally

invariant model in the Bethe

approxi-

mation are

nearly

^{the same as} those obtained

by

^the

mean field

approximation

of Maier and

Saupe.

Differences with respect ^{to the mean field}

approxi-

mation _{appear in} the discussion of the

magnetic

^bire-

fringence

^{and the}

light scattering

^{in the}

isotropic phase,

^where short range order is of the utmost

importance.

^These

effects,

^{which cannot}

adequately

be described in the molecular field

approximation,

are

usually

discussed in terms of a

phenomenological

Landau model. Bethe’s

approximation provides

^a

molecular statistical basis for the treatment of these

pretransitional phenomena.

^It

appeared

^{that the}

essential features of these

phenomena

^{can be}

quali- tatively

well described

using

Bethe’s method.

Quanti-

tative agreement ^with

experimental

results is reason-

able

only

^{for a} small number of nearest

neighbours,

y ⁼ 3 or

4,

^and gets ^{worse with}

increasing

^{number of}

neighbours.

This is due to the

topological

^structure

of the

lattice,

which underlies the Bethe method.

This structure, which is of the so-called

Cayley

^tree

type, results from our

assumption

^{that the nearest}

neighbours

^{of a molecule are} not nearest

neighbours

of each other. In a

Cayley

^{tree lattice two molecules}

are

only

correlated via those intermediate molecules that lie on the shortest and

only path connecting

^the

two molecules. AU other correlations

running

^via

roundabout _ways are

neglected.

This results in a

reduction

of

the actual

correlations,

which becomes

more serious with

increasing

^{number of}

neighbours.

In order to

get

^a reasonable estimate of the actual correlations ^one should therefore take a y that is lower than the actual number of nearest

neighbours

in the

physical

system. ^{In this} way y is

interpreted

as a measure for the amount of short _range corre-

lation,

rather than as the actual number of nearest

neighbours.

Concluding,

^Bethe’s

approximation provides

^a

considerable

improvement of

^{the value}

of (Tc - Tc*)ITc

upon the molecular field

approximation. Consequent- ly,

the remark of de Gennes

[20]

^{that the}

experi- mentally

observed small value cannot be understood in terms

of

^the

Maier-Saupe theory,

^{or in}

fact of

any

theory

^{where the} interactions are described in terms

of a single coupling

constant, is open ^to

question,

^because

the effect of short _range order is still rather under- estimated in the Bethe

approximation.

Acknowledgments.

^- The authors wish to thank Professor A. J. Dekker for _many

interesting

^and

stimulating

discussions.

They

^are indebted also to Dr. W. J. A. Goossens and Dr. W. H. de Jeu from

the

Philips

Research Laboratories Eindhoven for a

critical

reading

^{of the}

manuscript

^{and a number of}

useful remarks.

This work was

performed

^{as a} part ^{of the research} program of the

Stichting

^voor Fundamenteel Onder- zoek der Materie

(F.O.M.)

with financial support ^from the Nederlandse

Organisatie

^voor

Zuiver-Wetenschap- pelijk

^Onderzoek

(Z.W.O.).

Appendix

^{A. The exact solution of} the one-dimen- sional

Maier-Saupe

^{model. - Consider a one-dimen-}

sional chain of N molecules with nearest

neighbour interaction,

^i.e. y ⁼

2,

in the presence of a

magnetic

field H directed

along

^the z-axis. The interaction

Ei,i + ^between

^two

neighbouring

molecules is

given by

while the _energy of molecule i in the

magnetic

^{field is}

described

by (see (4.5)) :

where ai

^{is a unit vector}

denoting

the orientation of molecule i. We define

ZN(aN)

^{to be the}

partition

function of the chain with the orientation of end molecule N fixed. Then we add an extra molecule next to molecule N. The

partition

^function

ZN+ 1(aN + 1)

of the chain with N + 1 molecules

having

the orienta- tion of the end molecule N + 1

fixed,

^{is now}

simply

related to

ZN(aN) through

^the

^equation

Because this relation should hold for all N and because of the

positivity

^{of the}

partition ^function,

^{it follows}

that the free energy per molecule is

given by

where is determined

by

^the

homogeneous

^Fredholm

integral equation

in such a way that À is the

positive eigenvalue belong- ing

^{to an}

eigenfunction g(a),

which does not

change sign.

In the case H ⁼ 0 we find

g(a)

^{to be a constant and}

Clearly f

^{is an}

analytic

function of the

temperature,

i.e. the one-dimensional

Maier-Saupe

model does not exhibit a

phase transition,

^{which contrasts the result} of the molecular field

approximation. Application

of the Bethe

approximation

^also

yields

expres- sion

(A. 4)

^with

given by (A. 6) (compare (3 .11)

^with

h ⁼ 0 and Z ⁼

À(03B2J»).

^We remark that the exact result has also been obtained

by

Vuillermot and Romerio

[21], using

^a

slightly

^different

argument.

Finally,

^{we note that the Bethe}

approximation

for H # 0 does not

yield

^{the exact answer for} y ⁼ 2 in the

isotropic phase

because of the chosen form of

z(Q).

Appendix

^{B. Some} useful relations for T > T . - In this

appendix

^we prove the

following

^relations

which were used in the calculations of section 4 :

In these relations ao denotes the orientation of the central molecule of the

cluster,

al and _a2 refer to the orientations of two outer shell molecules.

1)

^{It holds}

with.

i.e.

of ao

because

of the rotational invariance of

(ao.al)2.

Application

^{of the}

following orthogonal

^transfor-

mation R to the

integration

^variable ai in

(B. 4) :

(i.e.

^{the new} z-axis will lie

along ao) yields

because of

Next we

apply

the inverse

transformation R -1

to the

integration

^{variable in}

(B. 6)

^{and we} get :

which had to be

proved.

2)

^Because

F(ao)

^{is a constant in the}

isotropic phase

^{it holds that :}

Applying

the transformation R to the

integral

^between square brackets it results in

Transforming

^back

gives

Using

the relation

(B. 1)

^{we rewrite}

(B. 9)

^as

3)

^{In the}

isotropic phase

^{we have}

From the definition of o-

(3.15)

it follows that

while it results from

which also

only

^{holds in the}

isotropic phase,

^that

Substituting (B .11)

^and

(B.13)

^into

(B. 10)

^and

using (B. 1)

^{one verifies}

easily

^that

It can also

easily

be verified that

which

completes

^the

proof

^of

(B. 3).

Appendix

^C. Numerical solution of the

consistency

relation. ^In this

appendix

^{we will}

give

^{a brief}

outline of the method to solve the

consistency

^rela-

tion

(3.7).

The numerical evaluation of this relation involves a lot of

multiple integrations,

^{which without}

taking special

care, will consume much

computing

time.

Eq. (3.7)

^can be rewritten in

spherical

^coordi-

nates :

with

and

Given an inverse temperature

f3J

^{a value of}

f3h

^should

be found that satisfies _eq.

(C .1). Using

^Gaussian

quadrature

^{it suffices to evalùate}

G(O, 0’) only

^once

at each

f3J

^{in N x M Gauss}

points (cos ai,

^cos

Oi)

The values of

G(0;, Oj)

^{can be used each time when}

varying ph

^{in the}

computation

^of

F(Oi)

^{and both sides}

of eq.

(C .1).

^The

integrals

ⁱⁿ

(C .1)

^{were evaluated}

for various numbers of Gauss

points (N, M)

^and

several values of

pl, ph

^and y. It

appeared

^sufficient

to use at most 32 Gauss

points

for each dimension

(lV

^{= M =}

32).

^{In that} way the

required

computer time was reduced

considerably

^without

introducing

numerical errors in the evaluation of the

integrals

greater ^{than 1} part ⁱⁿ

10g.

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