On Dipole Association and Short-Range Order in Nematic Discotic Liquid Crystals

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On Dipole Association and Short-Range Order in Nematic Discotic Liquid Crystals

T. Phillips

To cite this version:

T. Phillips. On Dipole Association and Short-Range Order in Nematic Discotic Liquid Crystals. Jour-

nal de Physique I, EDP Sciences, 1995, 5 (12), pp.1667-1680. �10.1051/jp1:1995222�. �jpa-00247165�

Classification

Physics

Abstracts

61.30Cz 61.30Gd 64.70Md

On Dipole Association and Short-Range Order in Nematic Discotic Liquid Crystals

T.J.

Phillips

Defence Research

Agency,

St. Andrews

Road,

Malvern, Worcestershire, WR14

3PS,

UK

(Received

30 March 1995, revised and

accepted

8

August 1995)

Abstract. The Kirkwood

dipole

correlation factor _gi of _a nematic discotic

liquid crystal

is measured _{as a} function of temperature. A

large degree

of

anti-parallel dipole

association

is

found,

which persists into the isotropic

phase.

This _{is in} conflict with previous mean-field mortels for

dipole

correlation, which

predict parallel

association _m

discogens.

A cluster mortel of

ordering, including short-range dipolar

orientational order,

quantified by

_a parameter

T,

and

induding

the

expenmeutal

fact of anti-

parallel dipole association,

is used _to attempt a more

reasonable treatment of

dipole

correlation _m these systems. The theory _is found to

predict

the

dipole

association rather well, with _a great

improvement

_over mean-field results, and shows other improvements _over mean-field mortels,

including

_a reduction _in transition

enthalpy

and

drop

_in the mean

permittivity,

I, when the system _is cooled into the nematic

phase.

1. Introduction

The first

theory

of the formation of the nematic

phase,

based

purely

_on

dipolar interactions,

was

proposed by

Born in 1916

[ii.

This

theory

_was discredited

by

measurements that showed _no surface free

charges [2],

^{and the}

discovery

of

quinquephenyl _[3],

which has _no permanent

dipole

moment but still shows _a nematic

phase. Nevertheless,

the vast

majority

^of mesogens contain

polar

_groups, and

yet

the

dipolar

contribution to the formation

of,

and

ordering

_m,

liquid crystalline phases

is

usually

assumed to be small

[e.g. 4,5].

^The _area m which permanent

dipoles

are most

routinely

considered is for the dielectric

properties

of

mesogens-dipolar

interactions then have _a

significant

effect [6]. Various models of

dipole

association in

anisotropic

media have been

proposed [7-9], mcluding

_a mean-field

theory

^for nematics

by

Dunmur and

Palffy- Muhoray

_[10]

Recently _[11],

_some nematic

discogens

have been studied. These _were found to have _a

large

permanent molecular

dipole

moment

parallel

to the short

(1.e. unique)

_axis, but to have

only

_a small

positive permittivity anisotropy,

due to

anti-parallel dipole

association. This is _a counter-

example

to the mean-field

theory _[10],

which

predicts parallel

association for such

materials,

due to the

assumption

that the

dipoles

_are ~buried' within the molecule

[11]. Although

the

uniaxial

long-range quadrupolar

orientational order

parameter

was found [12] ^to have _a similar

©

Les Editions de Physique 1995

1668 JOURNAL DE PHYSIQUE I N°12

~M

~

_~) _{il °CioH~i}

DB126

~

~~~7~5

HET7

Fig. 1. Molecular structures of the materials.

value to calamitic

(rod- like)

materials

(and

the

Maier-Saupe

mean-field

prediction

_[13]

),

short- range columnar order _was also found to be present. The

dipole

association and columnar order both influenced the

physical properties

and _were found to be

linked,

with the columnar

ordering favouring

correlation between

dipoles _[11].

In the past,

discrepancies

have been found between the

equations

^for the

permittivities

of

a nematic derived

by

àllaier and Meier [14]

(based

_on the mean-field

theory

of order

[13])

and

experimental

results. An

example

is the

drop

_m the _mean

permittivity,

z, seen as some

materials _are cooled into their nematic

phase [e.g. 15].

This has been

explained by

Madhusu- dana and Chandrasekhar [16] as

being

due to _a

change

_m the

anti-parallel dipole association, using

_a

theory

where the usual mean-field is

perturbed by

_a

short-range

interaction between

a small nttmber of

partiales. However,

for

simple nematics,

such _as the

cyano-biphenyls,

the

discrepancies

have

usually

been

small,

and the

theory largely ignored.

Nematics that have

a

strongly negative permittivity anisotropy might

well be

expected

to be _a

counter-example

to the

theory

of Dunmur and

Palffy- Muhoray,

but here molecular

biaxiality comphcates

the situation

[17, 18].

The nematic

discogens

studied here have the benefit of

being

_an obvious

counter-example, having

a correlation

opposite

_m

sigq

to that

predicted,

while

retaining

the

simplifying

feature of little _{or no} molecular

biaxiality.

In this paper, the Kirkwood

dipolar

correlation factor _gi~

of _a discotic _is determined _{using some} of the

properties

of these materials that _were measured

previously [il,12].

A cluster model of

short-range order,

based _on those used

previously-,

_is

developed

to

produce

_a

prediction

for the

temperature dependence

of the correlation factor.

The model _is shown to be _a

great improvement

over the mean-field

result,

and to give more accurate

predictions

of other

physical properties

of discotics.

2.

Experimental

2.1. REFRACTIVE INDEX _AND PERMITTIVITY l/IEASUREMENT. The material used is the

mixture Tla

[12], containing

DB126

[19] (triphenylene hexa-(2-methyl- 4-n-decyloxy)benzoate)

and

triphenylene hexa-n-heptanoate.

HET7 [20]

(Fig. 1).

The mixture has

phase

transitions

(+2°C)

K

66[40] ND

90 1

(°C).

The refractive indices of Tla _were

measured,

to a tolerance of

0.lsl,

_{using an} Abbé _re- fractometer with

light

of

wavelength

_ÀNaD

" 589.6 _{nm, as} described

previously _[12].

The

permittivity

of Tla

parallel

to the

director, _ell,

_tuas

measured,

to _a tolerance of

0.25%,

_us-

mg a HP4192A

impedance analyser,

1v-ith the

sample

_{m a} 9 /tm thick cell ~v-ith

homeotropic

ahgnment-inducing

rubbed

polyimide layers _[11].

2.2. KIRKWOOD CORRELATION FACTOR DETERMINATION. The I(irkwood

dipolar

_corre-

lation factor gi~ ^is a factor

showing

the

degree

and

sign

of

dipolar association,

defined

by [î]

~~~'~~

J#~

where _~j

=

II,1

is _a

macroscopic axis,

with

respect

to _a director _n, and

~1~,~ ^is the

component

of _a

unique

axis _m the i~'~ molecule

along

the

j>-axis.

Trie factor _can be defined in terms of _any

unique direction _m the molecule _a direction

coinciding

v~ith the electric

dipolar

direction

is used here _as electrical

properties

ai-e to be discussed. The summation is _over the molecules

in a

macroscopic spherical

volume V centred _on

dipole

1. The effective mean-square

dipole

iuoment of _a

phase (per molecule)

_is

gi,-en by (Pi _)Phase

"

9~(Pi)mol> ^(~)

lvhere

(p()moi

is the mean-square ^moment of _a

single

molecule. Hence the factor is less than

unity

for

anti-parallel

correlation and

greater

than unity for

parallel

correlation. Previous

study

_[11] has shown gill ^{< 1 for DB126. Trie factor} gi~ ^{as a} function of temperature was

calculated

using

the uniaxial Kirk,vood-Frôhlich

equation

_[7]

~

e~ ⁺

(cf _e~)Q(

_iii

~~~ ~~ ~

e~~(?°°

+

2)~ 9eokBT~~~

^{~~~~~~°~' ~' ~~"}

^~'

^~~~

where e~ ^are the

required components

^{of the}

permitti,;ity

tensor of the material and

e~i

are

the _zero

frequency extrapolation

of the _atomic and electronic

pei~mittivity contributions,

_ap-

proximated by e~i

= 1.05 + 0.0.5

n( [21].

^{N is} the molecular number

density,

calculated

using

trie

approximation

for trie

density

_p

= 970 + 30

kg ^m~~ Q(

is a

shape

factor

depending

on

the

anisotropy

of _c~j, defined in

[7],

^{and the} mean-square molecular

dipole _components parallel

and

perpendicular

to the director _are taken _as

(p)

_/)>moi =

[t1Î(1

+

25)

+

PI (i S)]

,

(t1[),moi

₌

t1Î(1 _S)

+

PI (i

₊

jS)j ⁽⁴⁾

respectively

~KThere the in _vacuo molecular

dipole

_{components are} mdexed _s and t for

longitu-

dinal and transverse to the molecular short axis

[22].

^This

neglects

_any biaxial nature of trie molecules concerned

(1.e.

an average ^transverse

dipole

moment is

assumed,

and the order _pa- rameter D taken to be

zero).

In

addition,

the _use of these

equations implies

trie

neglect

of trie

anisotropy

^{of induced}

polarisation _[21],

_1-e- that lia _« ii. This is _a rather

good approximation

for this

example,

_as shown

pre,>iously _[12].

The values for the molecular

dipole components (taken

to be the

time-average

^of the molecular resultant

here)

_are measured using the method

adopted

for DB126

previously _[11].

These _{ai-e ps}

= 20.7 +1.0 _x

10~~°

cm, /tt " 0 due to

the rotational

symmetry

^of the molecule. The measured value of _e

II ^ruas

used,

and I and _ei calculated

by extrapolating

e;sot<opte into the nematic

phase, giving

_a

systematic uncertainty

^of

~-

2Sl.

This

approximation

is

justified by

the

large

_{errors m p} and _ps that dommate the final

uncertainty,

calculated _as

~-

6%.

with _a further

possible systematic

errer of

~-

5%

due _to

e~i.

The results for the

dipole

correlation factor, _{gi~, ai-e} shown _m

Figure

2 versus reduced temperature. The factor _is

isotropic

with _a value of about 0.32 at temperatures

greater

than TNT. This shows that there _{is a} considerable

degree

of

anti-parallel

coi-relation _in the

isotropic

1670 JOURNAL DE

PHYSIQUE

I N°12

~Î

O ^a

lÎ

°

~i g °O~a

C

i

°°Oa

£

°°~

ÎÎ

~~O

1

°O~

~

°Oa

o

°O

U

°q

© O O

~

é ^g

f~~

g aaoooooooooooooooooooooooq

0.80 0.85 0 90 1.00 1.05

Reduced temperature,

T/T~

Fig.

2. Kirkwood correlation factor gi~ of T1a _versus reduced temperature.

phase

_as

expected

for these materials. When the

sample

is cooled into the nematic

phase,

the factor becomes

anisotropic.

The

parallel component

^has a value of about _0.24

just

below the

transition;

this value continues to

drap

_as the

temperature

_is lowered.

Comparison

calamitic values [15] are for

SPCH,

which has _giII

drap

from 0.70 to 0.61 at the

transition,

and

SCB,

which sholv-s _a

di~op

from 0.49 to 0.43. We _see that T1a has _a

significantly higher degree

of

association, parallel

to the

director,

than these calamitic materials.

The

perpendicular

correlation

factor,

_gii, has _a rather

larger

value than _{gi / /,}

showing

that there _is less correlation in this direction. There

is, however,

a

significant degree

of

perpendicular association,

^{which is found} to decrease _as the

temperature

is lo~KTered; this is

expected

when the association

parallel

to the director is

increasing.

3. Theoretical

3.1. CLUSTER MODEL OF SHORT-RANGE ORDER. In this

section,

_a

simple

model of short- range

order,

derived from _a

picture

of the

phase

_as small clusters of

molecules,

is introduced.

Each cluster consists of _a central molecule surrounded

by

_~i nearest

neighbours,

_none of which

are nearest

neighbours

of each other

(this

is the Bethe

approximation [23],

^which

obviously

becomes less reahstic _{as ~i}

mcreases).

A

representation

of this _is shown in

Figure

3. Each of the pairs is taken to mteract with _a

potential

E(cos _ô~j)

= e

(ÀPI(cos ô~j) P2(cosd~j))

,

(5)

where

Pi

_are the

Legendre polynomials, d~j

is the

angle

between the two molecular

unique

_axes,

e is the

strength

of the

potential,

and is _a parameter ^{of the model}

giving

the

proportion

of

dipolar

and

quadrupolar

interaction

(usually dipolar

interaction _is

neglected

in models of this

type,

e.g. that of

Ypma

and

Vertogen _[24],

and _{so =}

0).

Notice that

anti-parallel dipole

correlation _is introduced

by

_{using a}

positive

value of and is not fundamental to the model.

Parallel coi-relation could be included

by

_usmg

negative

values of À. A considerable

simplifica-

tion has been made to _arrive at this

potential, namely

that the many

dipoles

and

quadrupoles

that _{can occur on}

real,

flexible molecules _are taken _{as one,}

fixed, resultant,

_so that the

potential

is

only

considered to be

dependent

_on the

angle

between the molecules. This

assumption

has been made

by

most other

authors,

and is necessary in this _case to

preclude

the

large

number of

a)

b)

~

',

l

,Î

1"

Fig.

3.

a)

Schematic

representation

of the _{way m} which the actual molecules _are

arrangea

in the mixture;

b)

the way in which

they

_are

represented

in the cluster mortel, with _a coordination number of three. The dotted fines mdicate the

pair-wise

interactions.

unknown parameters ^{that would} otherwise be introduced. Various

simple

molecules _are rather well described

by

this

approximation,

such _as the

cyanobiphenyls. Unfortunately,

the discotic molecule whose

experimental

results _were

presented

earlier is

considerably

_more

complicated.

However,

_we shall _see later that the results _may well be

applicable despite

this.

The outer molecules _are taken to interact with the rest of the medium via the

Maier-Saupe

pseudo-potential _[13]

V(cosdj)

=

-uP2(cosdj), (6)

where

dj

is the

angle

between the normal to the

j~~

molecule and the

director,

and _u is the

strength

of the effective

potential acting

_on the outer molecules _m trie cluster. This

approximation neglects

the effect of _a

dipolar

term m the

pseudo-potential,

which is believed to be

justified

_as the

dipolar

correlation between nearest

neighbours

_is

anti-parallel,

and _so

tends to cancel _over

larger

distances.

Retaining

a

dipolar

term _m the

pseudo-potential

also tends to

give

a non-zero value for the

dipolar

order parameter,

(Pi (cos dj )),

which is undesirable for _a nematic

phase.

The

angle d~j

^{is found from the}

spherical

harmonic addition theorem [25]

m=+1

n

_(COSiY~j

"

~j (~i)'~'Cl,m(Ùl> i~i)Cl -m(iYj

_i~J)>

(7)

m=-1

where çJ~,j are the azimuthal

angles

of the molecular short-axes about the

director,

and

Ci m(d,

_{çJ~) are} modified

spherical harmonics,

_giiren

by (for

1=

1)

Cl,Ù(Ùi>§~1)

~ COSÙz,

Ci,+i(Ùi, ç7i)

₌

+2~i

sm

d~e~J~' (8)

In this _{case, we} have

Pi (cas _ô~j

= cas d~ _cas

dj

+ sind~ sm

dj

_{cas çJ~j}

,

(9)

16î2 JOURNAL DE

PHYSIQUE

I N°12

where çJ~j = ~~ pj. ~ie _can then construct

(un-normalised)

distribution functions

f(Qo, Qj)

₌

exp(-E(cosdoj)fl), g(Qj)

=

exp(-17(cosçJj)fl) (10)

where

fl

=

III.BT

and

Qj

₌

([, çJj),

fi.om ~vhich _we obtain the statistical

~veight

of _a

given

cluster of molecules

~

P(Qo; Qi, Q~,

,

Q~ =

j _{~j f(Qo. Qj)g(Qj), (ii)}

j=i

~vhere molecule '0' is the central _one and Z is the

partition

coefficient.

The

potential,

_{u, is} then found from the

Ypma

and

Vertogen

condition

[24], namely

that the condition of translational invariance

(the

model net

depending

_on which molecule _is taken _as the central

one)

_is

interpreted by requiring

that the average orientation of the central molecule and its

neighbours

be the _same. This is _a weaker condition than that of

Krieger

and James

[26],

who

required

that the

~veight

^{P of} _a

pair

^{of molecules be the} same

regardless

of which molecule

is chosen _as the central _{one, i e.}

P(Qo, Qj)

₌

P(Qj, no)- (12)

The

Ypma

and

ifertogen

condition _can be written

/dQoP2(Qo)F(Qo)~

^"

/ dQoF(Qo)~~~ / dQiP2(Qi)f(Qo, Ri )g(Qi), (13)

where

F(fia)

"

/ dQJf(Qo, Qj)§(QJ). (14)

The external field

strength

is therefore found

by solving

/d(cas do)P2(cas do)F(cas ôo)~

₌

d(cas ôo)F(cas do)~

^~~

d(cas ôi)P2(cas di)

x

Î Î

xG(cas ôo,casai)g(cas ôi), (là)

for

flv,

_{as a} fonction of

temperature (used

in the reduced inverse form

fie here),

with

G(cas ôo,casai)

=

dçJoif(cosôo,

_cas

ôi,ç7oi), (16)

and

F(cosdo)

=

/ d(cas ôi)G(cosdo, casai)g(casai). (lî)

This _is found to be the method that reduces

computation

time to _a minimum.

The entropy ^{of the cluster} is given

by

-kB / _doo / _dei /

dQ~ P(Qo Qi,

_Q~ In

P(Qo; Qi,

,

Q~

=

-kB

In Z

~ikBfle(a ÀT) j,kBflvs. (18)

(19)

where

~ç =

p~jcos doj))>

is _a

short-range dipolar

order

parameter,

a =

(P2(cas ôoj)), (20)

is _a

short-range quadrupolar

order

parameter (the angle

brackets

indicating

the ensemble

average),

and S is the usual

long-range quadrupolar

order

parameter

S =

jP2(cosÙo)). j21)

The internal energy of.the cluster _is

U =

-~ie(a ÀT) ~ivS, (22)

where the last term

represents

the contribution of the effective

field, giving

^for the Helmholtz free _energy

~ ~~ ~ _~

Î~'~~'

^~~~~

This function is found to exhibit _a first-order transition for all values of _~i

greater

^than two.

This _{is as}

expected,

since _a system ^{with two} nearest

neighbours

_is

one-dimensional,

and it is well known that one-dimensional

systems

^exhibit no transitions at finite

temperatures [27j.

The

equilibrium

conditions _at the

temperature

^of interest _can then be found from the minimum of F with

respect

to the order

parameters,

^which are calculated from trie

consistency

relations

d(cosdo)P2(cosdo)F(cosdo)~

S =

,

(24)

Z

d(cas do) d(cas di) dçJoi P2(cas ôoi)~b(cas do,cas dl, _ç7oi)

~ =

(~~)

Z

and

/ _{d(cas do)} / _{d(cas di)} / dçJoiPi(cas dei)ù(cas do,cas di, ç7oi)

~f

=

(~6)

Z where

Z =

/d(cas ôo)F(cas ôo)~, (27)

and

#(cas ôo,cas di, _ç7oi)

"

f(cas do,

_cas

di, ç7oi)g(cas ôi)F(cas ôo)~~~ (28)

Given the

parameters

~i and À, ^{the above model is}

universal,

if the volume

dependence

of

e is

neglected (it

~vill tend to

depend

_on the inter-molecular distance to the inverse power of six

[24j).

The calculated order parameters ~s a function of reduced

temperature

_are shown _in

Figures

4 to

7,

with _~i

=

3,

^for various values of À. Transition data _are shown _in Table I. The first-order

transition

temperature, (fie)NI,

is found from the condition

that,

at the

transition,

^{the free} energies of the nematic and

isotropic phases

_ai-e

equal.

The reduced

temperature

_is then

equal

to

(fie)NI Ifle.

The transition

enthalpies

_in Table I show _a considerable reduction _{in comparison} with the _mean field

theory,

however the calculated

IiHNI

for Tla of 850 J

mol~~

is still

larger

than the measured value of around 200 J

mol~~

_{Notice that the}

parameter

^{T is}

negative

at

16î4 JOURNAL DE

PHYSIQUE

I N°12

à=o,y=3

~

~ ""...-...

Q~

~

É E

~

~

⁰

(

~

O

07 08 09 10 1-1

Reduced temperature,

T/T~

Fig.

4. Calculated order parameters ^for ~ = 3, à

= 0. The shaded fine shows the

34aier-Saupe

mean-field S order parameter ^for _companson.

= 0.25, y= 3

$

""""1,,_

Î$

É E

~

~

_,_~,-.~.,-

(

~ T _~___~_,,,.-~.-"j~j_~__,,,~---.

O ~-."jzziZIlCI1°Z~~~~~'~~~~~~"'1'

0.7 10

Reduced

temperature, T/T~

Fig.

5. Calculated order parameters ^for _~ = 3, à

= 0.25. The shaded fine shows the

Maier-Saupe

mean-field S order parameter for

comparison.

all

temperatures,

as

expected

for _a parameter

describing anti-parallel dipole

correlation. It

is also _{non-zero in} the isotropic

phase,

as is _a,

although

_a

(small)

discontinuous

drop

in its

magnitude

is _seen,

showing that,

_as

expected, short-range

order

persists

even when

long-range

order has vanished. While the calculated form for T is

qualitatively

similar to that for _a similar order

parameter predicted by

~fadhusudana and Chandrasekhar

[16j,

^the

magnitude

of T is

rather

larger.

Madhusudana and Chandrasekhar used _an

approximate compliance

with the

Krieger-James condition,

which has been found to

give

incorrect results

[24j). They

also do not state the coordination number _in their calculation.

One important

point

about this model _is that it makes no transparent distinction between discotic and calamitic nematics.

However,

differences between the two types ^of molecules _can be covered in several _ways, not least because the

magnitude

and volume

dependence

of the

potential

e

(and

therefore the transition

temperature)

_may well be different for the different

aspect

ratios. The main distinction between discotics and calamitics _is that _a

larger

number of

À=1.0, y=3

i

oo io i-i

Reduced temperature,

T/T~

Fig.

6. Calculated order parameters for _~

= 3, à

= 1.0. The shaded line shows the

Maier-Saupe

mean-field S order parameter for comparison.

À=2.5, _{y= 3}

0.7 0.8 0 9 10 1 1

Reduced temperature,

T/T~

Fig.

7. Calculated order parameters ^for ~ = 3, à

= 2.5. The shaded fine shows the

Maier-Saupe

mean-field S order parameter for

comparison.

nearest

neighbours

_may be tolerated in calamitic

examples

without the Bethe

approximation

breaking

down

severely.

One would

expect

_~i in _a calamitic version of the model to have values of around 6 to

8,

whereas the _minimum

integral

value of 3 is used here for discotic calculations.

This bas the desired effect of

making

the deviation from mean-field less in

calamitics,

which does _seem

supported by experimental

observations. For

example

the

enthalpy

of transition is

an

increasing

function of _j, and transition

enthalpies

_are found to be lower in discotics than in calamitics. Results with _an

example parameterisation

^for a

polar calamitic,

_{~i =}

8,

₌

o-à,

are shown in

Figure 8,

and in Table I.

The _ttse of _a low coordination number in the discotic calculations is

supported by

_computer

simulations of _a

polar discogen [28j,

which find the molecules associated in short

columns,

and

by

NMR measurements of

benzene-hexa-n-heptanoate [29j,

^{which find the molecules} associ-

ated in pairs. It is also _consistent with the distribution function results from the computer

1676 JOURNAL DE

PHYSIQUE

I N°12

Table I. Transition

temperatures (fie)Ni,

transition order

parameters SNI

and

enthaipies liHNi/TNT for

_a

variety of

values

of

the model

parameters

_~i and À. Note that

liHNi

_"

~IUNI,

as the volume

dependence of

the model has been

neglected.

~i

(fie)Ni SNI liHNi/TNT

J

I<~l mol~~

3

3 0.25 2.202 0.404 2.387

3 0.50 2.011 0.404 2.414

3 1.00 1.660 0.404 2.444

3 1.50 1.406 0.404 2.422

3 1.80 1.288 0.403 2.402

3 2.50 1.078 0.401 2.349

8 0.50 0.5954 0.424 3.344

oJ 0.00 4.452 0.429 3.473

S ~~~'~'~~~

0.7 0.8 0 9 1-o 1 1

Reduced temperature,

T/T~

Fig.

8. Calculated order parameters for _~

= 8, à

= o-à- The shaded fine shows the

Maier-Saupe

mean-field S order parameter for comparison.

simulations of Emerson et ai.

[30]

^{and de Luca} et ai.

[31].

^The paper of Biscarini et ai.

[28],

which _uses two-site cluster

theory

_as well _as Monte-Carlo

simulation, provides

values for the short- and

long-range

order

parameters

^that are

comparable

with those

presented

in this _pa- per. The results for the second-rank

long-range

order parameter

presented

here _are well _m

agreement

^with

previous experiment [12].

^In contrast, the results from Emerson et ai.

[30]

^do

not

correspond

with

experiment, being

much too

large

at the neiuatic to

isotropic

transition.

Of _course the

parameter

allows additional iTariation to be

introduced,

and _a

complete

theory

of intermolecular interactions should allow this parameter to be calculated for

particular

molecules. For _{now, we can} estimate

by attempting

to calculate the ratio of

dipolar

and

quadrupolar

interactions for

typical

molecules. There _{is no} reason, either mathematical _or

physical, why

^the parameter i need be

integral,

_as

long

_as it is

thought

of _as the _average coordination nttmber in the

liquid~

This will

provide

further freedom for

description

of realistic

liquid crystal phases,

and allow transition

enthalpies

to be reduced still further.

~Î

Iso~opic

(

E~

é

~

Î

Nematic

~a~

O

T/T~,=0.85

00 0 2.0 3.0

Model parameter, 1

Fig.

9. Values of the parameter T' for _vanous values of the mortel parameter ^à at diiferent reduced temperatures, I.o

(in

the

isotropic phase),

I.o

(in

the nematic

phase)

and 0.85.

3.2. USE OF SHORT-RANGE ORDER PARAMETER IN DIPOLE CORRELATION CALCULATION.

Using

the definition of

gill

^{given in}

equation (2) above,

_{we can express} the correlation factor in terms of the cluster model _as

gill

" 1 ₊

(cosdo cosdj)

₌ 1 ₊

T', (29)

with

d(cos do) d(cos di) dçJoi

cos

do

_cos

diifi(cos do,

_cos

ôi, ç7oi)

T'

=

(30)

Z

In

principle

_{we can now use} this

equation

to fit trie

experimental data,

with _one free parameter, À. The _range of

possible

values for T' is shown in

Figure 9,

and the calculated correlation factor

shown in

Figure 10,

for

= o-à and

2.5,

wîth the

experimental

values for

comparison.

The resultant calculated value for the

dipole

correlation factor _is found to be rather too

large,

1-e- the

magnitude

^{of T'} is still too small to

give

_a

good

fit _to the data.

Figure

9

suggests

that

larger

values cannot be obtained without _iismg

unrealistically large

values of À. Indeed the value of 2.5 is still rather

larger

than is

likely

in

reality.

However the

theory

is

clearly

far

better than

previous

_mean- field

results,

_as these would

give

_a correlation factor much

greater

than

unity.

3.3. PERMITTIVITIES _{IN THE} SHORT-RANGE ORDER À~ODEL. À~adhusudana and Chan-

drasekhar

[16]

give expressions for the _average

dipole

moment

parallel

and

perpendicular

to the

director,

with _/tt

"

0,

_as

(in

this

notation)

l~ll/)ajuster

= ~IÎ

(~~

+ (COSlYo COS

YJ)ajuster) ₁₃₁₎

and

l~llcluster

"

~Î l~ j

^~ ₊

_jslll do

_Slll

iJj

COSÇ7o ^COS

7j)cj~~t~~.) (32)

1678 JOURNAL DE

PHYSIQUE

I N°12

bù

fl

<

E

(

^_____,,.

Q À~_§;j~__-,,----~-""~"'

~'~""""' À=25

~ jf

ù

^ooo

oooooooooooooooooooooooooo

o.ss o 9o o

Reduced temperatare,

T/T~

Fig.

10. Calculated values for gill versus reduced temperature, ^for two values of

à,

with the measured values

(open circles)

for

comparison.

=0.5, _y=3

42 ~,

)

~ Em

'@

i _,__....:...

à

.'"'"'

Ei

0.8 0 9 1-Ù 11

Reduced temperature,

T/T~

Fig.

Il. Calculated

permittivities

for _~

= 3, à

= o-à, and other parameters calculated _using the measured

properties

of T1a.

Substitution of these results into the Maier-Meier

equations [14]

8~ ^-1 =

@ ^(a~)

+

)1J1j)j

, ~t =

II,1, (33)

gives results for the

permittivities

in the

short-range

order

model,

with the _mean

permittivity given by

1 1

=

~~~

a +

~~)

l +

(cos do

_cos d

j +

2(sin do

_sm

dj

cos çJo cos çJj

(34)

eo

3kB

~

We _can then calculate these

permittivities

_usmg values for

dipole

moments and

polarisabilities

calculated

earlier,

and

taking

both h and F _as

approximately 1.3,

_as calculated from the

mean

permittitivity

and

polarisability _[14].

The calculated

permittivities

for the _{case ~i}

=

3,

À= 2.5, y=³

3

""1'

E~ ^~

~

'@ _...""'

)@ _,,:""'

Î

__:..."' ^E

~ ,,:."'

0.8 0 9 1-Ù 1 1

Reduced temperature,

T/T~

Fig.

12. Calculated

permittivities

for _~

= 3, à

= 2.5, and other parameters ^calculated

using

the measured

properties

of Tia.

= o-à _are shown in

Figure

11. The

large drop

in the _mean

permittivity

is

clearly

_seen, due to the

change

in

dipole

correlation at the transition. The results _{are m} fact rather similar to those

presented

for DB126

[11],

and

provide

further evidence for the

efficacy

of this model.

For

comparison,

the mean-field result for the _same values

predicts

_a

permittivity anisotropy

of around +7, which is much too

large. Very large

values of the

parameter give

^different results

(Fig. 12),

_as the

large

amount of

anti-parallel

correlation _removes the effectiveness of the

parallel dipole

_moment,

leaving

the

polarisability _amsotropy,

which is

negative,

_as the

only

contributor to the

permittivity anisotropy. Density

variations _are

ignored

in the

permittivity calculations,

_as ~v-ell _as the

temperature dependencies

of h and

F,

which would make the

discontinuities at the transition

greater,

and trie

temperature dependencies

somewhat different.

4. Conclusions

The Kirkwood

dipole

association factor _gi~ bas been measured _{as a} function of temperature for _a nematic discotic mixture. The

system

showed _a considerable

degree

of

anti-parallel dipole

correlation,

_as

suggested by previous study [11],

^which

persisted

into the

isotropic phase.

This is in conflict with

previous

models of

dipole correlation,

which

predict parallel

association

m

discogens.

A cluster model of

ordering, mcluding short-range dipolar

orientational

order, quantified by

_a

parameter T,

_was used _to

attempt

a more reasonable

prediction

of

dipole

correlation in these

systems.

The

theory

_was found to

predict

the

dipole

association rather

well,

with _a

great improvement

_{over mean~} field

results,

and also had other desirable

features, including

_a reduction _m transition

enthalpy

and

drop

_m the _mean

permittivity,

?, when the

system

was cooled into the nematic

phase.

Acknowledgments

Workers at the School of

Chemistry, University

of

Hull,

_are thanked for

synthesis

of the _ma- terials described in this _paper. The invaluable contribution of J.C. Jones _to this work is

acknowledged.

This work _was

supported by

the

Strategic

Research Committee of the U-Ii-

Ministry

of Defence. British Crown

Copyright, 1995/DRA.

^Published

by

_permission of Her Brittanic

Majesty's Stationery

^Office.

1680 JOURNAL DE

PHYSIQUE

I N°12

References

Ill

Bom M., Sitz.

Phys.-Math.

25

(1916)

614.

[2] Szivessy G., Z.

Phys.

31

(1926)

159.

[3] ^Vorlander D., ^Z.

Phys.

Chem. A126

(1927)

449.

[4] Cotter

àI.A.,

Phu. Trans. R. Soc. Lond. A309

(1983)

127.

[Si ^Gelbart iii-M-, J.

Phys.

Chem. 86

(1982)

4298.

[6]

Toriyama

K. and Dunmur

D.A.,

Mol.

Cryst. Liq. Cryst.

139

(1986)

123.

[î]

Bordewijk

P.,

Physica

75

(1974)

146.

[8] Dunmur D.A. and Miller

iii-H-,

Mot.

Cryst. Liq. Cryst.

60

(1980)

281.

[9]

Toriyama

K. and Dunmur D.A.. Molec.

Phys.

56

(1985)

479.

[10] ^{Dunmur D.A. and}

Palify~muhoray

P., Motec. Phys. 76

(1992)

1015.

[Il] Philhps

T.J., Jones J.C. and McDonnell D.G., Liq.

Crystatsls (1993)

.203.

[12]

Philhps

T.J. and Jones J.C.. Liq. Crystals 16

(1994)

805.

[13] ^{Maier W. and}

Saupe A.,

Z.

Naturforsch.

15a

(1960)

287.

[14] ^{à>laier W. and Meier} G., ^Z.

Naturforsch.

16a

(1961)

262; à'laier W. and Meier G., Z.

Naturforsch.

16a

(1961)

470.

[15] ^Bradshaw M.J., Ph.D.

Thesis, University

of Exeter

(1985).

[16] Madhusudana N.V. and Chandrasekhar S., Proc. Int.

Liq. Cryst.

Conf.

Bangalore,

Pramana.

Suppl. (1973)

57.

[iii Toriyama

K., Dunmur D.A. and Hunt S-E-,

Liq. Crystals

5

(1989)

1001.

[18] ^Jones

J.C.,

Proc. FLC '93

(1993).

[19] ^Beattie D.R., Ph.D. Thesis,

University

of Hull

(1993).

[20] ^Destrade C., Mondon M.C. and Malthete J., J.

Phys.

France 40

(1979)

C3-1î.

[21]

Bordewijk

P. and De Jeu iV.H., J. Chem.

Phys.

68

(1978)

116.

[22] ^{De Jeu} ~V.H.,

Physical Pioperties

of

Liquid Crystalhne

Materials

(Gordon

and Breach. London

1980).

[23] ^Bethe H.A., Froc. R. Soc.149

(1935)

1.

[24] Ypma J.G.J. and

Vertogen

G., Sohd State Commun. 18

(1976)

475.

[25] Bleaney B.I. and

Bleaney

B.,

Electricity

and Magnetism, 3rd. edition

(Oxford

University Press

1976)

p.40.

[26]

Krieger

T.G. and -James

H-M-,

J. Chem.

Phys.

22

(1954)

796.

[27]

Sheng

^P. and

Wojtowicz P.J., Phvs.

Rev. A 14

(1976)

1883.

[28]

Zarragoicoechea

G-J-.

Levesque

D. and uleis J.J., Mot.

Phys.

78

(1993)

1475.

[29] ^Mabmak A., J. Chem.

Phys.

96

(1992)

2306.

[30] ^Emerson A.P.J., Luckhurst G-R- and

Whatling S.G.,

Mot. Phys. 82

(1994)

113.

[31] ^{De Luca}

hl.D.,

Neal M.P. and Care

C.M., Liq. Cryst.

16

(1994)

257.