Properties of uniaxial nematic liquid crystal of semiflexible even and odd dimers

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Properties of uniaxial nematic liquid crystal of semiflexible even and odd dimers

Eugene Terentjev, Rolfe Petschek

To cite this version:

Eugene Terentjev, Rolfe Petschek. Properties of uniaxial nematic liquid crystal of semiflexible even and odd dimers. Journal de Physique II, EDP Sciences, 1993, 3 (5), pp.661-680. �10.1051/jp2:1993159�.

�jpa-00247863�

Classification Physics Abstracts

61.308 36.20

Properties of uniaxial nematic liquid crystal of semiflexible

_even

and odd dimers

Eugene Terentjev (~)

^{and Rolfe G. Petschek}

(~)

(~) Physics Department, Case Western Reserve University~ Cleveland~ OH 44106, ^U.S.A.

(~) Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, Great-Britain

(Received

2 November 1992, accepted in final form 8

February1993)

Abstract. A microscopic model is developed to describe statistical properties of _a nematic

phase consisting of semiflexible dimers. The effect of the spacer chain bonding the _{two mesogens} is described by the local conditional probability in terms of _a bare stiffness parameter Q _+J EB

/kT

and _a bare equilibrium angle bo between mesogenic _monomers. In the framework of _a molecular field approximation, taking into account long-range attraction and steric repulsion between

monomers, we obtain _a complete statistical description of the nematic phase with expressions for order parameter, mean-field potential, phase transition parameters and Frank elastic constants

depending

_on details of spe&tic molecular structure. Comparison of results for dimers that _are straight and bent in equilibrium and for corresponding _monomers reveals the effect of molecular biaxiality and biaxial fluctuations.

1. Introduction.

Thermotropic

main-chain

liquid crystalline polymers

and

oligomers

have been the

objects

of extensive

study

in recent years. In the nematic

phase

^there is _a

preferred

axis of orientation of mesogens which _are bound to the flexible _spacer chain. We

develop

here _a consistent molecular-

statistical

theory

of

a

liquid crystal composed

from such molecules. This

approach

is

designed

to

predict macroscopic properties

of the material in terms of

specific microscopic

_parameters

of the

mesogenic

monomers and the characteristics of the

bonding

chain. We treat this chain in _a continuous _way,

assigning

two parameters to it: the bare

rigidity

Q and the bare

angle

00 of the bend between its terminal groups.

If,

for

example,

the

aliphatic

chain

[-CH2-]n

in all-trans

configuration

is

considered,

_one expects 00 ₌ 0° in the

ground

state for all _even

n. For _an odd number of

CH2

_groups the

angle

00 is around 70° for the

ground

state of the free

chain;

^the _presence ^of

bulky

and

strongly interacting

_{mesogens on} ^{both ends}

presumably

reduces this

angle, depending

on the

length

of the chain and its

rigidity

Q.

This latter parameter ^is

especially important

to

investigate

when

dealing

with chains of _ne- matic mesogens, because it determines the

specific regimes

of the chain behaviour of

longer

polymers.

When Q _«

I,

_one arrives at the _case of

freely jointed

_monomers,

becoming

_com-

pletely free,

at Q

= 0. As Q _{- oo} the spacer

approaches

the limit of _a

rigid rod, straight

_or

bent, dependent

_on the value of 00. The intermediate

region

of Q

(

I defines the most _com-

mon case of semiflexible

bonding,

when the

tendency

of

subsequent

_monomers to

align along

their

equilibrium

direction competes with the nematic _mean

field,

which acts _on each _monomer

separately

and tends to

depress

fluctuations of _a molecular

shape.

This

gives

rise to

compli-

cated

dependences

of

macroscopic,

observable

properties

on the parameters of intermonomer interactions and the spacer internal

rigidity

and bend.

A favourable

object

to

investigate

these

properties

is _a dimer

composed

of two

nematogenic

monomers, where the influence of the _spacer is not obscured

by

the entropy ^effects

imposed by

the

long

chain

[Ii.

The _purpose of the present _paper ^is to

apply

_a

general

statistical

approach

to the _case of _a

thermotropic

nematic

liquid crystal composed

of such dimers and to obtain observable

macroscopic

characteristics of the system.

Before

proceeding,

_we note that _over the _years countless observations of so-called "odd- even" effects have been made

(see,

for

example

_[2,

3]).

The _reason for the observed behaviour is clear, since the

ground

state bend between

subsequent

_mesogens makes the

corresponding

dimer

effectively

biaxial. A detailed theoretical

study

of this

question

has been made

recently by

Heaton and Luckhurst [4]. That

work, along

with other related

publications

_[5],

employs

_a

Flory description

of the conformational statistics of _{a spacer} chain based _on the torsion rotations of the carbon-carbon bonds. These authors

calculated,

in the limit of small order parameter, the N-I transition temperature ^{and the} bifurcation

point,

where the _non-zero solution for order

parameter

_appears. ^{Both these}

quantities

show

significant

"even-odd" effect. It has to be noted that this version of the

Flory approach

to the chain

statistics,

while

suiting

the chemical view and often

giving

_a

good

agreement ^with

experiment,

is inconsistent from the

point

of view of statistical mechanics. In this

approach

the

probability

distribution is written

arbitrarily

from symmetry arguments and does not,

generally,

minimize the free _energy. As _a

result, properties

based _on

single particle

_averages, such _as order parameter, bifurcation temperature, etc.,

usually

_are determined

correctly

in this

model,

whereas characteristics based _on the free

energy

(mean-field coupling

constant,

elasticity, etc.)

_are often

non-adequate.

An

alternative, Onsager's approach

based _on the virial

expansion,

is free from this

disadvantage

and has been

preferably

used _over the years in molecular theories of

polymers

and

liquid crystals.

This

approach

offers _a different set of

approximations

from that in the model of _a torsion chain and is

specifically

suitable for _mean field calculations of the free energy.

In this _{paper we} present an alternative

(to

the Ref. [4]) way of

calculating properties

of

semiflexible dimers that is based _on the virial

expansion

of the free _energy. This

approach

for dimers is

essentially

the

opposite limiting

_case of the model that _we used to calculate _curvature elastic _constants of _a nematic of

long polymer

chains

iii.

The difference between these _two

limiting

_cases is that for the

long

semiflexible chain _one is allowed to _use the

(approximate)

differential

equation

for the chain propagator,

neglecting

end effects.

Very

short

chains,

dimers in

particular, require

the discrete treatment of

subsequent

_monomers. Our model

employs

the concept ^of a local conditional

probability

for the relative orientation of bonded _monomers that describes the _spacer

flexibility by

_means of _an

integrated

parameter ^Q. In this _way both conformational

(trans-gauche) changes

and C C bonds fluctuations of the _{spacer are} accounted for _as thermal excitations around _some

ground-state configuration parallel

for _even dimers and bent to the

angle

00 for odd dimers. We do not take into account biaxial

ordering and, therefore,

_our results _are

quantitatively applicable only

to the _case of _a

regular

uniaxial

nematic.

Nevertheless,

since the effective

biaxiality

of the odd dimer is included in this model in the

form, averaged

over the molecular rotation about its

long axis,

_many of _our results will

show _a considerable effect of the

shape

_on the observable

macroscopic properties.

The purpose of this work is _to describe in _a framework of the

relatively simple theory broad, qualitative effects,

caused

by

the

flexibility

of the spacer and the

equilibrium

bend between the bounded _mesogens. We shall _use the mean-field

approximation

^{in order} to obtain _some of the results in

analytical

^{form. We}

emphasize,

that this is _a first _necessary step ⁱⁿ

describing

such

a

complex

system; some

phenomena (such

_as biaxial

ordering,

mentioned

above)

will

require

_a

more detailed treatment.

2.

Equifibriurn

statistics.

We consider _a dimer _{as a sequence} of two rod-like _mesogens each of

length

and width

d, figure

I. It is necessary ^to

give

_a consistent

sign

to the orientation _u of each of these

rods,

because the

probable position

of the

adjacent

_monomer

depends

_on this

sign.

We define

P~(r r',

_u,

u')

to be the

probability

that _{on an} isolated dimer the mesogen with its center of symmetry

position

_r and orientation _u is followed

by

another _mesogen, which has the

position

r' and orientation u'. Relative coordinates of the two _{monomers are}

always

bounded

by

the constraint

&(r-r'- )I[u+u'])

if _{we assume a} constant

length

for the _spacer. We do not consider here the limit of

freely jointed

_mesogens for their

properties

_are not very much different from those for free _monomers. In the

opposite limit,

when the _mesogen orientation is

strongly

determined

by

the

adjacent

_{mesogen, a} reasonable

approximation

for the

bond-probability ^P~

is the Gaussian. For the _{even spacer}

chain,

_I-e-, for the dimer that is

straight

in its

equilibrium

all-trans

configuration,

_we write:

P~(u, u')

_m

(

~ exp

[Q(u u')] _&(r

r'

)I[u

⁺

^{u']), (I)}

sin

where the

coupling

Q is determined

by

_an effective _energy of

bending

of the _spacer of constant

length,

Q

r-

EB/kT

> I and

essentially incorporates

all mechanisms of the chain

flexibility.

In the _case when the _spacer chain

imposes

_a finite

equilibrium

bend 00 on the dimer the bond-

probability

has its

ground

state maximum around

(u. u')

_{= cos} 00, ^{the width of this} distribution is still determined

by

_a bare

rigidity

parameter Q. We will _{use a} similar Gaussian

P~ (u, u')

m

~

exp

~~

(u

u')

₊

2V5sin( fij

&(r

^r'

jl[u

⁺

^{u']), (2)}

+ _C°S

o

~~

i

~~ (~

So

Fig-I.

Mesogenic _monomer and its corresponding dimer with inherited bend.

where

Z~

_{is the} normalization.

Equation (I) implies

that if _a

long polymer

chain of these

monomers was

considered,

its

isotropic persistence length (the

so-called Kuhn

segment)

would

be

equal

to al.

Considering equations (1,2),

it is

important

to

emphasize

that

P~

_is

a condi- tional

probability

of mutual orientation of the two bound _{mesogens. It} does not include the effect of external fields

or of other

particles,

which will _come into the

theory later,

when

writing

down the

one-particle probability

of the

given

_monomer orientation. For this _{reason we} refer to the parameters ^Q and 00 as "bare" characteristics of the _spacer; in the

interacting

system ^their

effective

(observable)

values will be

different,

_{as we} shall _see below. Note that

P~ depends

on the mutual

position

of the centers of _mass of bound _monomers

only through

the exact delta-functional constraint. However, ^the

symmetric

Gaussian form of equations

(I)

and

(2)

is _an

approximation.

This

approximation

is

certainly

valid in the _case of the weak

long-range

interaction between mesogens. If the _two

mesogenic

_monomers

strongly

attract each

other,

in order for the

P~

distribution to

keep

the

symmetric

form of

equation (2)

^it is necessary ^to restrict ourselves from consideration of

large

bend

angles and/or

_very flexible _spacer chains.

As _a

practical

matter this

requires

₍₀₀ +

/fl)

_< 90°. One _{can see} that the

approximation

is not very

demanding and, therefore, equation (2)

is

general.

In the nematic

phase

the orientational

probability

distribution function

P(u)

for each

given

mesogen may be determined

by integration

of the dimer

pair probability P(u,u')

_over the variables of the

adjacent

_monomer ^u'. We _use

P~(u,u')

as a correlator and _account for the

mean orientational

field, acting separately

_on each

mesogenic

_monomer:

P(u,u')

=

je~P~l(~'~)lP~(u,u')e~P~l(~"~ll

P(U)

=

/ P(U> U')dU' (3)

Here Z is the normalization

factor, fl

₌

(kT)~~, U((u n))

is the mean-field

potential acting

on a

given

_monomer, and _n is the nematic director.

Generally,

this

potential U((u n))

has

an

axially symmetric

form _on the unit

sphere

with two

deep

wells at the

poles (n, -n)

and _a barrier of

height

J at the equator

(uIn).

To make the molecular-statistical

description

of _our system self-consistent _{we now} need the free _energy functional in the _mean field

approximation,

which

depends

_on the

single particle probability P(u), equation (3),

and for which the _mean field

potential

is _a

minimizing

function.

This free energy takes the form

(compare

with

[Ii):

F = p~

/ _{P(ui)P(u~ )U~~'(1~ 2) dridr2dui}

_du2

jNkT

^{In Z N}

/ U(u, n)P(u)du (4)

where the

partition

function of _a semiflexible dimer is

Z =

e~P~(~'~lP~(u,u')e~P~(~"~ldu'du (5)

and N

=

pV

is the total number of _mesogens in the system. The last two terms of

the free energy

equation (4)

reduce to the usual orientational entropy contribution Fjoc ₌

pkT J

PlnPdudr in both

limiting

_cases

(Q

_-

0)

^and

(Q

_{- oo;} 00 _-

0).

These

correspond

respectively

to

freely jointed

_or ^unbounded _mesogens, ^and to

rigidly

bounded _mesogens which form _a

straight rigid

^rod.

Direct

varying

of

equation (4)

shows that in order for the

probability P(u)

in the form

equation (3)

to be the minimum of the free _energy

functional,

the mean-field

potential

must be

exactly

the

one-particle

_average of _an effective

pair potential:

U(ui, n)

_{m p}

/ P(u2)U~~'(1, 2)dr12du2

_{" p}

/ _U~~'(1, 2)P(u2 u[)dr12du2du[. (6)

Here p is the _monomer number

density

and the effective

pair potential

_[6] is _{a sum}

(see Eq.(7)

below)

of _a

long-range

attractive part, which is modulated

by

steric

repulsion,

and the

packing

entropy term, which is _non-zero

only

when the two momoners are in contact. Note that these

interacting

_monomers

belong

to different

dimers,

because all

pair

interactions between the bounded mesogens are assumed to be accounted for in the

bond-probability ^P~

The effective

interaction

potential

is

given by

U~~.

(1, 2)

re U~~~.

(1, 2) 8(1, 2)

+

~~

[l 8(1, 2)] (7)

VOP where

U~~~'(1,

2)

G3

-( _{((Ul 'U2)~ ((Ul 'U2)(Ul} 'r12)(U2 'r12)

+..

(8)

12 12 12

The factor

(I-vop)

in

equation (7),

where _vo is the _monomer

volume,

accounts for the correction to the

Onsager approximation

for the dense

packing,

which _was discussed

by

Gelbart and Baron

iii.

The steric

repulsion

is accounted for

by

_a step ^function

8(1, 2)

₌

8((12

_r12)> which is

equal

to

unity

when the distance between the two _monomers centers of _{mass r12} is greater than the

corresponding

contact form-factor

(12 (closest approach distance).

In the _case of unbound

rod-like _monomers this form-factor in the absence of steric

dipoles

is

~12 * d +

~j~

[(Ul r12)~

+ (U2

'r12)~j (~)

With _{an accuracy}

r-

d/I (cf.

[8], ^for

example)

the

corresponding

steric

integral

for dimers will

give

Ill _8(f12 r12)jdr12

_*

2dl~

ui x u2 + ui x

u[

₊

u[

_{x u2 +}

u[

_x

u[ (10)

where the

prime

denotes the _monomer

adjacent

to the

given

_{one on} its dimer. In the

limiting

case 00 " 0 and Q _{= oo}

(rigid straight dimer)

this

integral

will be four times

bigger

than that

for unbound _monomers. It is

important

to

emphasize

that the attractive

potential

contribution to the _mean field is not so sensitive to the difference between dimers and unbound _monomers.

Indeed,

since the attraction

potential equation (8)

is _a

rapidly decreasing

function of _r12> the main contribution to the

integral J

_U~~~(1,

_{2)8(1, 2)dr12}

is determined

by

the distances _r12

'~

d

(side

to side

packing

of

monomers). Therefore,

with _{an accuracy}

r-

d/I,

effects of attraction to

adjacent

_{monomers are}

insignificant

in

equation (6).

One has to carry out the

angular integration

in the

equation (6)

with the

two-particle prob-

ability

distribution

P(u, u')

in order to account for the

flexibility

of the _spacer. The

simplest possible

form of _mean field allowed

by

the symmetry is U

= Uo

J(u n)~,

_an

approximation

that works

fairly

well if the

coupling

constant J is

sufficiently large.

An

approximate analytical expression

for

J,

which is

proportional

to the nematic order parameter, takes the form [9]:

J re

)p ($G

^~ _{+ (12 +}

_)lo)

₊ ^~~~

^Tj

S _e

)HS ^(II)

VOP

where

G,

lo and 12 are the coefficients

appearing

in

equation (8),

the nematic order parameter is S

= -0.5

+1.5( (u n)~)

and where

equation (II)

defines the material parameter H with

dimensionality

of _energy.

3. N-I

phase

transition.

It is

interesting

to

investigate

how the

properties

of the N-I

phase

transition in _a system ^of semiflexible dimers

depend

_on the spacer

rigidity

Q and bend

angle

00. Since _{we are}

using

the

simple

_mean field

approach

in this paper, the limit Q ₌ 0

exactly corresponds

to _a Maier-

Saupe

_case, ^with the

only important

difference that _we estimate steric

(entropic)

and

energetic

contributions to the

coupling

constant J

independently. Thus,

the

properties

of _monomer nematic will _{serve us as a} reference basis with the transition _at

kTjjl

_m 0.15H

(I.e. J/kT

m

4.55, _see Eq.

(II)),

and order parameter

jump

at the transition AS _m 0.44. We define the

point

of the first order N-I transition _as the temperature at which the free energy of the

non-symmetric phase equation (4)

becomes lower than that of the

isotropic phase.

^There are two other

important (observable)

characteristic temperatures,

describing

this transition the ultimate

superheating point

Ti where the bifurcation of the non-trivial

(S # 0)

^local minimum of the free _{energy occurs,} and the

supercooling point

T* at which the

isotropic phase

becomes

unstable.

So,

_we determine

Ti

_{as a} solution of

equation

S

= -0.5

+1.5((u n)~)

₌

l.5J/H,

and T*

by deriving

the first term of the free _energy

equation (4) expansion

in _powers of small S: F Fo _'~

ao(T T*)S~

+ The

equilibrium

transition temperature TNT is deternfined

by

_a direct

integration

of the free energy

equation (4).

For small Q « I _{we can}

expand

the

complicated bond-probability exponential

and calculate

corresponding integrals analytically. Keeping

in nfind that the

approximation

for P~

equation (2)

will not be

good

at very small Q when there is _a strong

long-range

attraction in the system,

we write down the

following

estimate for the transition temperature

(determined by

the Landau free _energy

expansion

_[9]):

~

~(oj ~

~ ~

~Q sin(00/2)

~~

Q~

~~ ~

~~ ~

l + cos 00 ^~

(l

+ cos 00)~

~

Q~[0.44sin(00/2)

+ 1.76

sin~(00/2)]

+

(12) (1+

_C°S

°o)

One _can

clearly

observe the

competing

action of the spacer

rigidity

Q and the bend

angle

00.

Straight (even)

dimers have 00 _" 0. _TNT increases

together

with Q until it reaches saturation at very

rigid,

Q »

I,

dimers. Nonzero

equilibrium

bend between _mesogens

composing

the flexible dimer reduces its

mesogenic

_power

and, consequently,

decreases the N-I transition temperature when the _spacer

rigidity

increases.

When Q is not small

(and

_we expect

typical

values of Q to be between 3 and 20 for

[CH2]n

spacers with _n

varying

from 16 down to

2,

[9]) one has to

perform

calculations

numerically.

Figure

2 shows the evolution of the transition temperature ^with

increasing

_spacer

rigidity

for several fixed values of 00. We _{can see} that in

spite

of considerable

bend,

if the spacer is

rigid

2

a

~~

_i.

~~

~_

0.

O lo 20 30 40 50

Q

Fig.2.

Reduced transition temperature

TNT/7f~,

_vs. the _spacer stiffness Q: _even dimers

(a),

odd dimers vith equilibrium bend 20°

(b),

^40°

(c),

60°

(d)

respectively.

Table I. Estimated _spacer

rigidity Q,

and the

equilibrium

bent

angle,

00, for

aliphatic

_spacer chains with

increasing

number of

carbons,

_n.

n 2 3 4 5 6 7 8 9 10 II 12

Q 33.3 24.4 17.2 13.9 11.8 10.2 8.7 7.9 7.0 6A 5.8

bo(°)

0 47 0 40 0 35 0 32 0 30 0

enough,

the dimer remains

mesogenic, though

^with

significantly

decreased transition tempera-

ture. For the _even,

straight

in

equilibrium,

dimers _we have obtained [9] ^from comparison with

experiment

that Q has _a universal

dependence

_{on n,} the number of carbons in _an

aliphatic

spacer chain: Q

r-

70/n.

If _{we assume on} this basis that the whole chain

rigidity

is determined

by

the

independent bending

of each C- C

bond,

it is

possible

to estimate Q for odd-numbered

spacers

by interpolation.

Then the

only

unknown parameter ^{for the}

given

odd chain is the

bare

angle

00. For each

given

odd-numbered

dimer,

_{we can} estimate the actual bend _an-

gle

of _{a spacer} 00

by fitting

the

appropriate

_curve in

figure

2 to the

experimental

transition temperature for its

particular

value of Q. For the series of dimers of mesogen

4-n-alkyl-N-[4- n-alkyloxy-benzyliden]-aniline

_{[10] we} obtain in this _way the

expected

values for the dimer's

spacer

rigidity

and its _average

equilibrium

bend

(see

the Tab.

I).

^The numbers

presented

in this table _cannot be considered _{as very} accurate, since

they

involve _a considerable

experimental

error, ^our

simplifying approximations

and the

averaging

_over all

possible

trans and

gauche

_con-

formations in the real system. However, the effect of

straightening

_a bent dimer with

decreasing

spacer stiffness under the influence of interaction between _monomers (00 '- 70° _x n~°.~~ _{at the} transition

point)

is

clearly

described

by

this model.

Note,

that this is _a different effect from the

(also present)

additional extension of _a dimer affected

by

the nematic _mean field.

There _are several characteristic parameters of the first order N-I

phase transition,

which _can be measured

experimentally.

We consider two of

them,

the width of the transition

hysteresis

AT =

(Ti

T* and the

jump

of the order parameter AS at the transition

point. Again, using

the

expansion

at small Q _we obtain the

following expressions,

which

help

to understand the

- 7

~y

~

b

O

fl

ig.3.

60°

equflibrium bend.

trends of AT and AS behaviour:

~~

~~(~j ~

~

~~Q sin(00/2)

~ ~~

Q~

~ l + cos 00 ^~

(l

_{+ cos 00)~}

(l

+

~s

00)~~~~~ ~~ ~~~~~°~~~ ~'~~ ~~~~~~°~~~~ ^~ ~~~~

~ ~ ~~~~j

~

~ ~~ ^Q

sin(00/2)

~ ~~

Q~

~ l + cos 00

(1

+ cos 00)~

Q~

[2.60 sin(00/2)

1.96

sin~(00/2)]

+

(14) (1+

_C°S

°o)~

For _an

arbitrary

Q these parameters have to be calculated

numerically: they

_are determined

by isotropic (at

J

=

0)

averages of

corresponding

mean-field

expressions

if _{we assume a}

weakly

first-order N-I transition. In

figure

3 _we

plot

the reduced

hysteresis

width _TNT T*. This result then _can be

compared

with

experimental

data for dimers of

4,4' dialkoxyphenylbenzoate

"5005"

monomer

ill,

_12], where the

supercooling

temperature T* _was determined

by extrapolation

of the inverse Cotton-Mouton coefficient

C~~

r-

(T T*).

Since _{we are} unable to estimate

microscopic

attraction

potential

constants

G,

lo and 12 ⁱⁿ

equation (8)

with _any reasonable accuracy, we take J

=

)HS

from the data for

corresponding

_monomers

ill]. Although

_our

data for _even (00 "

0)

^and

small-angle

odd dimers exhibit _some increase in the N-I transition

hysteresis

at

relatively

small

Q,

this increase is _{not so} dramatic _as has been observed

by

Rosenblatt and Griffin

[I Ii

(TNT T* _was observed to increase for the dimer

by

_a factor of 7 with respect to its

monomer).

The similar weak

dependence

_on Q

(which

describes the

gradual change

from _monomers to even

dimers)

is

presented

in

figure

4

(curve a)

the

plot

of the order parameter

jump

AS at the transition. When the

rigidity

of the _even dimer

increases,

the characteristic behaviour of the

phase

transition is

exactly

that of the _monomers

(except

for the transition temperature

itself, Fig.

2) as is

expected

in _a self-consistent mean-field

theory.

O.

0.

c w

-n

lo 20 30 40 50

Q

FigA.

Jump of the order parameter at the N-I transition _vs. Q for _even

(a)

and odd dimers lvith 20°

(b),

40°

(c),

60°

(d)

equilibrium bend.

When odd dimers _are

considered,

the

equilibrium biaxiality

of _a molecule _causes noticeable

changes

in the transition behaviour. Both AT and AS _are related to the

enthalpy

of the first order

phase

transition which is _{a measure} of

discontinuity

of this transition.

Therefore,

_a

significant

decrease of AT and AS for

considerably

bent dimers indicates the

narrowing

of the

region

of two

phase

coexistence when the molecule becomes _more biaxial. This

phenomenon

has been observed

by

several authors

(see

_[14], for

example)

and it is caused

by

_an

approach

to the isolated Lifshits

point

for biaxial nematic _on the

corresponding phase diagram.

4. Uniaxial nematic

phase.

In

spite

of _a certain _success of _our

theory

to describe the N-I

phase

transition in the system ^of semiflexible dimers _we note that this

simple

mean-field model is not

designed

for this _purpose.

Its weak

point

is its omission of fluctuation effects. It is obvious that biaxial fluctuations _are

important

at the

transition, especially

for the

significantly

bent and

rigid

dimers. Far below the transition

point, however,

_we expect ^{this model} to work

quite

well.

One of the

questions

which has to be clarified at this

point,

is the

configuration

of bent semiflexible dimers in the nematic

phase.

One _may expect ^{that such}

dimers, being

affected

by

a strong ^uniaxial mean

field,

will be

effectively extended,

_or

straightened

with respect to their

equilibrium shape

in the

isotropic phase,

_PO- It

depends, apparently,

_on the

magnitude

of Q:

for _{a very}

rigid

_spacer the dimer will remain in its

V-shape (Fig.I)

and _{even as} T _- 0 the uniaxial order parameter ^will never reach

unity. Figure

5 shows the order parameter, calculated at the _same scaled temperature (TNT

T)/TNT

for dimers with different bare bend

angle

00 and

several fixed values of

Q,

which represent ^different characteristic

regimes.

One _can observe two

phenomena in the

figure

5 _an effective

straightening

of the dimers and _an increase of effective

stiffness,

_{as a}

secondary

effect.

First,

consider the

points

where all

plots

intersect the ordinate axis these

correspond

to the

even (00 "

0)

^{dimer. When the} _spacer is rather

flexible,

the order parameter

slightly decreases,

then it returns back _to the hard-rod value _as Q becomes _very

large (compare

with

Fig.4).

In

spite

of the

increasing

bare bend

angle

_Ro> dimers with _an

odd,

but flexible _spacer do not

O.

monomer

w O.

lo 20 30 40 50 60 70

6

Fig-S-

Order parameter at _r

(=

I T/TNT _" 0.05 _vs. equilibrium bend angle bo for odd dimers lvith different _spacer stiffness: Q

= 2

(a);

Q

= 4,

(b);

Q

= 10

(c);

Q

= 18

(d);

Q

= 45

(e).

show _any further decrease in the order parameter; we can even observe _a

slight

raise of S. This shows that in the nematic

phase

bent dimers with flexible

(Q

<

5)

spacers are almost

completely

extended. In this

straightened configuration,

relative fluctuations of the bounded _monomers

are

depressed

and the

degree

^of

ordering

in the system becomes closer _to the hard-rod limit.

Bent dimers with _a considerable _spacer stiffness

(Q

r-

10)

also tend to extend their

isotropic equilibrium configuration,

when affected

by

the nematic _mean field. Its

strength, however,

is not sufficient _to make the dimer

completely straight,

when the initial

angle

_PO is

large enough.

Flatter

regions

_on each of

corresponding

curves indicates the limit of initial bend _00> after which the total

straightening

cannot _occur.

Finally,

when the _spacer is _very stiff

(Q

>

20),

the shape of the dimer is not

significantly

affected

by

_an external

ordering

field

acting

_on its

monomers

and, therefore,

the uniaxial nematic

ordering

decreases

monotonically,

_as cos~ 00,

when the molecular asymmetry ^increases.

Our model allows

investigation

of the

properties

of the _spacer and the molecular

configura-

tion in _more detail since

equation (3)

^{is the}

joint probability

for the two

adjacent

_monomers

orientations. After _some

geometric simplifications,

which do not affect the

qualitative

be- haviour of the

dimer,

_{we can} estimate _an

increasing

effective stiffness of the spacer and _a

decreasing

effective

bending angle

in _a nematic

phase (which

_we define _{as a} location of the

joint probability

maximum with respect to the

angle

between _u and

u'):

Q _m Q ₊

flJ;

~" ~° ~ ~

"~~~~°~~~

ii

+

(flJ/2Q)(1

+ cos 00)]~

~~~~

(compare

with the Tab. I in the

previous Sect.).

When the order parameter ^{is close} to

unity (which

means that

J/kT

>

I),

dimers may be considered

effectively straightened

since the

average

cos00

- 1. There is _a

competition

between

ordering

and _spacer stiffness in

equation (IS)

_very

rigid

dimers

require

_very low temperatures to achieve the _same

degree

of

ordering.

In the

isotropic phase,

when J ₌

0,

the effective and the bare

bending angle

coincide: 00 ₌ 00.

5. Elastic constants.

The Frank elastic constants _are

important

parameters of _a nematic

liquid crystal

because

they

influence almost _every observable property,

including

those used in

applications.

Therefore _we derive

microscopic expressions

for

Ki,

K2 and K3> ^which

correspond

to

splay (i7 n)~,

twist

(n

i7 _x

n)~

and bend

in

_x i7 _x

n]~

deformations. The molecular-statistical

theory

of curvature

elasticity

for _an

ordinary

rod-like nematic has been

developed by

Gelbart [6] ^for a

thermotropic

system

and, earlier, by Straley

_[8] and Priest [15] ^for a system with

only

steric interactions.

The basic assertion of this

approach

is that

(I)

^the response of the

liquid crystal

to _an

ordering field,

which varies

slowly

in _space, is

just

that the

preferred

orientation

aligns everywhere along

that field and

(it)

the relative

probability

of _a molecule at the

point

_r

having

the orientation _u is

P(u, n(r)),

where P is the _same distribution function _as in the uniform _{nematic. Then the} free energy

equation (4),

^{which describes the}

coupling

of the _two molecules

(u,r)

and

(u', r')

must be

expanded

in _powers of the small director

gradients: n(r')

=

n(r)

+

(r

r' i7

n(r)

+...

up to second-order _terms.

By averaging

of the

corresponding

correction _to F _one obtains the

phenomenological expression

for the Frank free _energy with _curvature constants

expressed

in

terms of molecular parameters. ^The orientational entropy term _r- PIn P does not contribute

to the elastic free _energy because of its local _nature.

In _{our case} the

single-particle

distribution

P(u,n(r))

includes characteristics of the other mesogen at the

point

r'

(where

_r r'

=

) flu

+

u']

is the distance between centers of _mass of the _two bound

monomers). So,

in contrast to the established theories of nematics for immo- bile

molecules,

there is _an additional functional correction to the

single-particle

distribution

equation (3),

^which is due _to the different director orientation at the

points

r and r'. The variables of the

P(u, n(r))

must include

only positional

coordinates _r of the

monomer under

consideration,

thus _we have to transform

exp[-flU(u')]

in

equation (3)

to refer to this

point.

Expanding n(r')

in

equation (3)

as indicated above and

keeping only

terms up to the second

order,

_we obtain:

~~~~ _" ~~~~jiiiil,~l'l lll'lllllll~ fl m i _i~~U'~~~' +

~~

?~~~~' ~~~~

here

we

_{the derivative} of

the ean

field

/~" _{dU/d(u .} n)

and Z =

J P(u)du.

Since the

mean

_field

potential

U(u, n)

is

determined by _{a free} energy minimum

no

steric _dipoles or

chirality

in _monomers which would generate ^on-zero averages with linear

i7n),

there will be no additional change in P(u) due

to the

U(u,n), which would effect theecond-order elastic free energy. The

microscopic expression

for this

F ₌ p~

/ P(ui)U~~'(1, 2) I

+

(r12 i7)

+

)(r12 7)~j

P(u2)drdr12dui

du2

jNkT

^{In Z N}

/ _{U(u, n)P(u)du (Ii)}

where

equation (16)

must be used for the

single-particle probability density P(u).

After sub-

stituting

the _mean field

potential

_we obtain the elastic free _energy of

a dimer nematic:

Y-Yom

/(Fi+F2]dr,

where the

leading

contribution to the free energy

density

_can be written _as follows:

~l

~2P~(fl~)~~i

_~k Vu_llj

VP

ill

~ / _{II ~~~} )~(2~~2~~12j (18)

I I k

~-i-I

k ~ ^{i I-k-I}

-I-I-k-1)

~2~2~l~l ~2~2~l~l ~2~2~l~l + ~2~2~l~l

e~P~(~'l P~ (ui

fir

)e~P~(°il e~P~(~21P~(u2, fi2)e~P~(~?l duidu2dfii dfi2

where the

subscript

indexes

(I

and

2)

characterize the

given

monomer of the

interacting pair

under

consideration,

and the

corresponding

fi

gives

the orientation of the

adjacent

_monomer

on a

given

dimer. Other terms, that did not contribute to the elastic free _energy in the casd of unbound _{monomers, now} take the form:

F2 ₌

~Pl~ninki7anji7pni (U(u, n)[2(flJ)~fi;fik flJ&;k](fijfiiuoup

+

fijfiifinfip))

8

((U(u, n))

+

kT) ([2(flJ)~uiuk flJ&;k](ujuiunup

+

ujuifiofip )) (19)

Integration

_{over r12} in the

equation (18)

_can be carried out

analytically,

if _{one uses}

approximate

methods with small parameter

(d/I).

The

result, however,

is very

complicated.

In this _paper

we are

dealing

with

qualitative

^{effects which describe} the difference between monomeric and dimeric

liquid crystals.

Let _us make _a

rough

estimate of this

integral

which will _preserve all effects of

bonding,

but which will make

equations

much _more

simple

and clear. This is

possible

if _we consider

separately

the

integration

of

U~~.(r12)

and

r(~r(~

in equation

(18).

This will

cause an error of the order

(d/1)

r- 0.2 in

typical

materials. This _error,

however,

does not exceed the _accuracy of other

approximations

_we have made, for

example,

in

equations (8)

and

(9)

for the effective

pair potential

^U~~. Denote

li

_the

average

pair potential

and _v12 the excluded volume for the two monomers

li(1, ₂₎

=

/ U~~'(1, 2)dr12

G3 A ₊

B(ui u2)~

+ V12

v12 =

Ill 8(1, 2)]dr12

G3

2dl~

ui X u2

The

averaged pair

interaction _energy

depends only

_on the

product (ui

_.u2 and _can be

expanded

in

Legendre polynomials

of _even index. This

expansion

has _a small parameter

(d/I) (in

addition to

independent anisotropic

contributions _see

equation (8))

and is well

justified. Comparing

with the mean-field

potential gives readily:

J ₌

-pBS.

Now _we rewrite the second virial part of the elastic free energy

density equation (18)

note that the term with A vanishes

exactly

for symmetry reasons:

~

p2j3

Fi m

jpl~~nknpi7nn1i7pnq (u[u[u(u[)(u[u(u(u(u[u~)+

~

₍₂₀₎

+(~~~~~~~~)(~~~~~~~~~~~~)+(~~~~~i~~~~~~)(~~~~~~~~)+

(~~~~~~~~)(~~~~~(~~~~~~))

where the

averaging

must be carried out with the

two-particle probability

distribution

P(u, fi) given by equation (3).

In the _case Q _- 0

only

the first average survives in the

equation (20)

and it

gives

_us Frank elastic constants:

ICI ₌

3K2

=

~~

Pl~

fI~H~S~ I

+

~S ^~~

4j

(1

^~~P4 +

~~

P6)

,

(21)

42525 7 7 II II

1(~ ₌ ^~~

pl~fl~H~s~ 1

+

~

S ^~~

4j

II

+

~

S ~~

P4 ^~

P6) (22)

42525 7 7 3 385 231

Here P4 and P6 _are

equilibrium

_averages of the fourth and sixth order

Legendre polynomials (which

_are

normally

much smaller than the main order parameter

S,

I-e-, than the _average of

the second

Legendre polynomial)

and the energy parameter ^{H is} determined

by equation II):

J =

2HS/3.

The _exact ratio

Iti/1(2

" 3 is a consequence of

accounting only

for the first term

B(u .u')~

in

the average

pair potential li(1, _2);

_we have

already acknowledged

that corrections of order

(d/I)

and

(I/G) (see equation (8))

are

expected

in _{a more} exact evaluation of the free _energy

density

Fi

Keeping

in mind this level of

approximation

of the Frank elastic constants

estimates,

_we

still _can observe several

important

features of their behaviour.

First,

at small order parameter

all constants increase _as

S~,

which is consistent with observations and other models [6]. ^The

ratio

K3/1(1

_'~ 1+ 0.71S ₊

O(P4)

is

increasing

when the temperature decreases from the N-I transition.

And, finally,

the order of

magnitude

of Frank elastic constants is close to the

observed values

It _m I-S _X

l0~~pl~S~(flH)~kT

r-

~kTS~

r- 3 _X

10~~S~ dyn

Other terms in the elastic free energy,

given by equation (19),

are caused

by

the non-local correction to the

probability

distribution

equation (Ii).

It is not not _so easy to estimate these

terms

analytically. However,

there _are two

limiting

_cases: Q _- 0 and Q _{- oo,} that

correspond

(at

bo _"

0)

to the

ordinary

rod-like nematic with molecules of

single (I)

and double (21)

lengths,

in which such estimation _can be carried out. It is _very

important

to

emphasize,

that these two _cases do not

simply correspond

to _{a -} 21substitution in results of other models [6,

8],

^{because such}

change

must be

accornpanied by

_a

corresponding change

in parameters ^of

pair

interaction: the

strength

and

anisotropy

of

dispersion

forces for the

effectively

doubled

particles.

This effect is

naturally

accounted for in _our

approach,

which considers each _monomer

separately. So,

both for Q _- 0 and Q _{- oo} the

single-particle

corrections in the free energy transform into

J

P In Pdu. For Q

= 0 the correction to

P(u)

vanishes

exactly,

while for Q >

flJ

we obtain the additional contribution to the free energy

density:

F2 m

pl~J 2flJ[(uiujukuiupuq

In

P(u)) (uiujukuiupuq)(In P(u))]np

_nq +

+(u;ujukui

In

P(u)) (uiujukui)(In P(u)) i7;nji7kni (23)

where P

r-

exp[-2flU].

This contribution is

small, compared

to the

pair

interaction free energy

Fi which becomes for Q

- oo

Fl *

)P~~~~~ l'~~'~~'~~'~~)l'~~'~~'~~'~~'~~'~~)~kllpi7alll~flllq (~~)