Molecular-statistical model for the nematic phase of semiflexible dimers

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Molecular-statistical model for the nematic phase of semiflexible dimers

Eugene Terentjev, Charles Rosenblatt, Rolfe Petschek

To cite this version:

Eugene Terentjev, Charles Rosenblatt, Rolfe Petschek. Molecular-statistical model for the ne- matic phase of semiflexible dimers. Journal de Physique II, EDP Sciences, 1993, 3 (1), pp.41-51.

�10.1051/jp2:1993110�. �jpa-00247812�

J, Pbys. II France 3 (1993) 41-51 JANUARY 1993, PAGE 41

Classification

Physics Abstracts 61.308 36.20

Molecular-statistical model for the nematic phase of semwexible dimers

Eugene

M.

Terentjev,

Charles Rosenblatt and Rolfe G. Petschek

Department ^of Physics, Case Western Reserve, University Cleveland, OH 44106, Great-Britain

(Received

25 March 1992, accepted jn final form 23

October1992)

Abstract. A microscopic model is developed to describe properties of _a nematic phase _con- sisting of semiflexible dimers. The effect of the chain bonding the two mesogens is described by

the local conditional probability of their mutual orientation in terms of

a bare stiffness _param- eter Q _-J

EB/kT,

assuming _{an even} number of carbon atoms in the chain. In the framework of a molecular field approximation _we obtain

a complete statistical description of the nematic with expressions for order parameter, mean-field potential, free _energy and phase transition

parameters, The width of the N-I transition hysteresis is in agreement with observed values.

Comparison with data _on the transition temperatures in three series of nematogenic dimers enables _us to obtain the quantitative dependence of the rigidity Q _on the length of the

(CH2)n

chain connecting the two _monoiners.

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 bound the flexible chain. For rather stiff chains this results in _an

exponential

increase of the effective

persistence lengtlI

of the chain in the direction

along

the nematic direc- tor [1, 2] n.

Recently

_we

developed

consistent theories of linear

elasticity

and

flexoelectricity

in

long-chain polymers

_{[2, 3],}

taking

iiIto account all relevant interactions between _monomers and

expressing corI~esponding

material constants in terms of definite molecular characteristics and the _spacer

I.igidity

_parameter Q.

This latter parameter is

especially important

to

investigate

when

dealing

with chains of nematic mesogens, because it deteI.i~iines the

specific regimes

of the chain behavior. When Q <

I,

one arrives at the _case of

freely joiiIted

_{monomers, or even}

completely free,

at Q

= 0. As

Q _{- cxJ} the spacer

approaclIes

tlIe liiuit of _a

rigid

rod. The intermediate

region

of Q > I defines

the _{most common case} of seiniflexible

bonding,

when the

tendency

of

subsequent

_monomers

to

align along

the _same diI.ection competes ^{with the nematic} mean

field,

which acts _on each

monomer

separately.

This

gives

rise to

complicated dependences

of

macroscopic properties

_on

the parameters of interniononier iiIteractions and _spacer

rigidity.

A _very favorable

object

to

in,,estigate

these

properties

is _a dimer

composed

of two nematc-

genic

_monomers, where the influence of the _spacer is not obscured

by

the entropy ^{effects im-}

posed by

the

long

chain [2]. ^The _purpose ^{of this} _paper ^is to

apply

_a

general

statistical

approach

to the _case of _a

thermotropic

nematic

liquid crystal composed

of these

dimers,

to obtain ob- servable

macroscopic

cliaracteristics of the system

and, by comparison

with

experimental data,

to make certain conclusions about the _nature of the _mesogens

bonding

and estimates of the value of the

rigidity

parameter ^Q.

Before

proceeding,

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

(see,

^for

example

[4,

5]).

The _reason for the observed behavior is

clear,

since _spacers with _an odd iiuniber of

i~ietlIylene (or other)

_groups

impose

_a considerable

equilibrium

bend between

subsequent

_mesogens,

making

the

corresponding

dimer

effectively

biaxial.

Recently

Ileaton and Luckhurst [6] and

Photinos,

Samulski arid Toriumi [7]

developed

a detailed model for seI~iiflexible nematic

dimers,

based _on the isomeric mechanism of _spacer

flexibility.

This iuodel is based _on the

assumption

of fixed tetrahedral

configuration

of bonds of each carbon atom, ^overall

flexibility

is due to torsional rotations of C C bonds. As _a

result the statistical

properties

of the neiuatic

phase

_are determined

by population

of

gauche

bonds _on the _spaceI. chaiiI. TlIis

model, seemingly

_very

realistic,

has certain

disadvantages.

First,

the cost of such _a detailed

description

of _every bond

configuration

and orientation is that authors have to make

severe

approximations

in order to obtain _any

practical

result. It

does not look

plausible

tlIat this model _caiI be

successfully applied

for the

description

of

longer oligomers

and

polymers.

Second, tlIe isomeric mechanism of

flexibility completely

overlooks the

possibility

of _a continuous

bending

of the

chain,

which is distributed between each C C bond in

equal portions.

A

straightforward

calculation with the

help

of the molecular

modelling

software

package "Sybyl"

shows that the _energy of _one

gauche

bond (~- ^0.02

eV)

is sufficient for continuous

bending

of the all-trails

(ClI2)Io

chain for the

angle

~- 15°. Creation of the

pair

of

gancfie

^bonds

requires

the _{same energy as} the continuous

bending

of this chain for 24°.

Therefore _{we argue}

that,

in

spite

of the

quite

detailed and

specific description,

the isomeric model of the spacer chain

flexibility

cannot be considered _{as a} final realistic

approach, obviously

a combination of discrete torsional rotations and continuous bond deformations takes

place

in

a real system.

For these _{reasons we} addressed the

problem

of the nematic

liquid crystalline phase

of semi- flexible dimers

applying

the

coIuputationally siiuple

continuous model of the _spacer

flexibility,

which does not _assume any

specific

atomic iuechanism and _{uses an}

integrated phenomenologi-

cal

rigidity

paI~aiueter Q. In the present work _we do not take into account the dimer

biaxiality and, therefore,

_our results _are

quantitatively applicable only

to the _case of _an even-numbered

spacer when the two

Iuesogenic

iuoiioiuers are

parallel

in

equilibrium.

Let _us

emphasize

that the

configuration

of the spacer chain

connecting

the two

mesogenic

units is

strongly

influenced

by

these

interacting

multi-atomic aggregates, and differs from the

configuration

of the free

chain,

for

example

_a teriuinal group. It _seems

highly improbable

to find _an odd number of

gancfie

conforinat,ions in spacers of

niesogeiiic

diniers in _a nematic

phase.

In the

isotropic phase gauche

excitations _{are niore easy} to appear, still their _{energy must} be much

higher

than that for the free

(CII2)n

chain. We present ^{below the}

expression

^{that shows} how the distribution of the local conditional

probability

of the two _monomers orientation _narrows due _to the effect of the nematic _mean field. The central result of this _paper is the

dependence

of the bare spacer

rigidity

Q _on the

length

of the

(CII2),1 chain,

obtained

by

the

analysis

of

experimental

data for dimers with even-numbered _spacers. This

dependence,

Q

~-

I/n,

indicates that the continuous meclianisiu of

flexibility

_niay

prevail

_even in the

isotropic phase.

We will return to the discussion of this

question

below: in the

remaining

of this work _{we assume} that the value of Q is the _saute for all diiners and expect ^{that this}

simplification

does not

change qualitative

predictions.

N°i NEMATIC OF SEAIIFLEXIBLE DIAIERS 43

n ~,

u u'

u~ ,

.. 8

~ U

~~ / fi'

Fig-I-

Mutual orientation of rod-like _mesogens, bounded by _a semiflexible chain, in _a nematic director field.

2. Mean field

theory.

We consider _a dimer _{as a sequence} of _t,vo rod-like _mesogens of

length

I and width

d, figure

16.

In order to be able to compare our results with observations for

corresponding

_{monomers we}

assume that the

length

of these is

exactly

I, ^which includes the hard _core and terminal _groups;

the

length

of the dimer in

equilibriuiu

is 21, so that the _spacer

actually

consists of _two linked terminal

(CiI2)n/2

_groups. The effective

flexibility

of such _spacer

depends

_on the total number

n of

CH2

units between two hard _cores, but _we do not consider this spacer as a separate steric

object, assuming

that its two parts _are

already

included in the

shapes

of the two _monomers.

Therefore the effect of the _spacer is

purely

orientational in _our model. It is _necessary to

give

a consistent

sign

to the orientation _u of each of these

rods,

because the

probable positions

of

subsequent

_monomers

depend

_on this

sign.

We define

P~(r r',u,u')

to be the

probability

that for _an isolated diiuer 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'. _{We assume} that this bare bond

probability

^P~ does not

depend

on parameters of other mesogens, other than the

linearly adjacent

_ones. ~fe do not consider lIere the limit of

freely jointed

_mesogens; it has

been shown

(see ill,

for

example)

that its

properties

_are not very much different from those for

free _monomers. III the

opposite liiuit,

wlIen the _mesogen orientation is

strongly

determined

by

the

adjacent

_{mesogen on} the

chain,

_a reasonable

approximation

for the

bond-probability

^P~ is

~~

~~~_~~'~ ~

8il~

_Q ^~~~

^~~~~

^{~'~~~ ~~~}

where the

coupling

Q is determined

by

_an effective energy of

bending

of the spaceri Q

~-

EB/kT

> I.

Equation (I)

_iueans tlIat if _a

long polymer

^chaili of these _{monomers was} consid-

ered,

its

isotI.opic peI.sisteiIce length (tlIe

sc-called I(uhn

segment)

would be

equal

to al.

In the nematic

phase

the oI.ientational

probability

distribution function

P(u)

for each _mesc- gen is determined

by

tlIe statistical

weiglIt

of the

remaining

part ^{of the dimer and}

by

the

interaction with the _iuean field in the media.

Ileducing

the

general expression

_[2] to the _case of

dimers,

_one obtains

P(u,ii)

₌

je~~l~"

^~~~l/~~

/ P~(u, u')e~~l~"'~~l/~~du' (2)

where Z is the normalization

factor, U((u .n))

is the mean-field

potential,

and _n b the nematic director.

Generally,

this

potential U(u, n)

has _an

axially syuinietric

form _on the unit

sphere

with the two

deep

wells at the

poles

_(~i,

-u)

and _a barrier of

height

J at the equator

(uLn).

The

simplest possible

form of U is U ₌ Uo

J(u n)~,

_an

approximation

which works

fairly

well if the

coupling

constant J is

sufficiently large.

The

analytical expression

for

J,

^which

is

proportional

to the nematic order parameter, ^will be derived below. We note here that since the N-I transition is first

order,

there is _no

region

of nematic

corresponding

to small

J/kT (actually,

in _{many cases}

J/kT

_may be considered _{as a}

large parameter). Therefore,

the

expansion

of

expressions

like that in

equation (2)

in powers of

Legendre polynomials (like

it is done in [6], ^for

example) is, strictly speaking, incorrect,

and _one must

always

retain the full

exponential

form.

Three

pairs

of unit _{vectoI.s are} involved in the

equation (2):

_u,

u';

_{u, n} and

u',

_n

(see Fig. I).

Substituting

the

expI.essions

for P~ and U and

using

the

relationship

_cos7

=

cosbi

cosb2 + sin

bI

sin b2 cos ~, _{one can}

diI.ectly integrate equation (2)

_over the azimuthal

angle

~ to obtain

p~

_~^{& cos2} @1

j

~

g~cos2 @2~n ^{cos @i cos} @2z_o

~~

~i~ ~₁~i~ ~₂ ~i~ ~₂d~₂ ~3j Z

where lo is the Infeld function of _zeI.o oI.der.

Since the mean-field

coupling

constant

Ji together

with

Q, plays

the most

important

role in the present

iuicroscopic theory,

it is desirable to derive _an

analytic expression

for J

using

interaction parameteI.s in the system. ^One can do this

by

minimization of the free _{energy over} the _mean field

potential which, strictly speaking,

_appears as an unknown function in

equation (2).

In the molecular field

approximation

for

probability

distributions the free _energy of _a

system of flexible diiuers contains two contributions: from the

single

molecule internal free energy and from the interaction between _iuesogens which _are located _on the different dimers.

The influence of tlIe link between bound _mesogens lIas been accounted for with the

help

of the bond

probability equation (I).

In

addition, anisotropic

_mesogens interact nia _an effective

pair potential

_{[8] which} contains attractive and hard-core

repulsive

parts. The

repulsive

part contributes both to _an

oI.ientatioiI-dependent

cut-off for the effective attraction and to the

packing (translational)

entI.opy ^{of the} system

(see Eq. (6) below).

In this work _{we assume,}

following StI.aley

_[9] the simplest model for the

repulsive potential

which reflects the symmetry

requiremeiIts:

the contact

step-function B(ri, uilr2,u2),

which is

equal

to

unity

outside the

restricted

regions

where the hard _cores of the t,vo iuesogens would otherwise penetrate into each other. These

regions

are deteI.ruined

by

the condition that the distance between two

centers of symiuetI.y _rI r? is less tlIaiI the

anisotropic

form-factor

i~~

~s d +

j

^~

_{i(u~ i~~)2}

₊

_(u~

_r~~121

₍₄₎

where the unit _vector I.12 _"

ri?/r12.

A reasonable

approximation

for the attractive part ^{of the}

pair potential

wlIiclI accounts for botlI

isotropic

and

anisotropic

contributions is

U~~~'(1,

2)

=

( [Io(uI u2)~ +12(ui u2)(ui FI2)(u2 FI2)] (5)

l'I2 l'I2

Note that in iuost known _cases the

isotropic

Van der lvaals attraction is much

stronger

^than

anisotropic

_one, ^which is determined

by anisotropies

of molecular

polarizabilities, viz.,

G

~-

(5 7)

_x

lo,?

_'-

(1- 5)

x 10~~~

eI.gcni~.

Following

_{[8] ,ve} define the effective

paiI. potential

for interaction of _monomers "I" and "2"

as the _sum of attractive part and the

packing

_entropy term, which both have the

twc-particle

N°i NEMATIC OF SEAIIFLEXIBLE DIR,IERS 45

microscopic

form

(see

also

[9]):

U~~'(1, 2)

₌ U~~~'(1,

2)E(r12i

_ui,

u2)

+

~~

(l E(r12i

_ui,

u2)] (6)

nap

Here the second _term differs from _zero

only

in

regions

^which are restricted for the _mesogens

"I" and "2" due to their steric

repulsion.

The factor

(I nap),

where _no is the _mesogen

volume,

accounts for the correction to the

Onsager approximation

for the dense

packing,

which _was discussed

by

Gelbart and Baron

[10].

^The

twc-particle

interaction part ^{of the internal} _energy takes the form

F~~t

=)p( ^£ / l§(ui,u(ri))Pk(u2,n(r2))U~~'(1,2)dridr2duidu2 (7)

j,k=1,2

where pd "

p/2

is the

density

of

dimers,

and the

probability

distributions Pk _are determined

by equation (2).

The summation in

equation (7)

is carried _{out over mesogens} located _on different

molecules. The interaction of

adjacent

_{iuesogens on} the _same chain is accounted for

by

the bond

probability

^P~ and contributes to the

local, single-chain

free energy. This local free energy of _a clIain in _{a iuean} field is the _excess of the

logarithm

of the dimer

partition

function

(-kT

In

Z)

_over the

backgI.ound

_average of the _mean field U in the system; ^it consists of the orientational eiItropy ^of a semiflexible dimer and the internal _energy and entropy ^associated with its

bending:

Fjo~ ₌ NAT In Z N

£ f _{U(u, ~i)Pk (u, n)du (8)}

~ ~

k=1,2

The partition function of _a semiflexible diiner is

Z ₌

f e~~~~>~~)/"~e~(~ "'~e~~(""")/~~du'du ₍₉₎

and N

=

pV

is the total number of iuesogens in the system. ^This

expression

reduces _to the usual orientational entropy contribution Fjo~ ₌

pkT J

P In Pdudr in both

limiting

cases Q

- 0

and Q

- cxJ. These

correspond respectively

to

freely jointed

or unbounded _mesogens, and to

rigidly

^bounded _mesogens ^wlIiclI foriu _a

rigid

rod. Note that _an increase of the nematic _mean field _{causes an} effective incI.ease of tlIe bare

rigidity

_so ^{that the actual}

(observed)

^{width of the}

probability

distribution of tlIe I.elative _monomers orientation is: 11 _cS Q +

J/kT.

It has been shown [2] tlIat the variation of the total free energy of the system Tint, + Fioc

over the _mean field

potential gives

the

expression

for U,

exactly corresponding

to that for the

free _monoIuer systeni:

U(iI,

ii cS pd

=1,2~ ~ _l§ (u')U~~'(u, u'; r12)dr12du' ₍₁₀₎

This

justifies

the _use of tlIe

equation (2)

for the _mesogen orientational

probability. Again using

the initial

assumption

U m Uo-

J(u.u)~,

_{one can}

perform

^both the radial

(r12)

and

angular (u) integrations

in

equation (10) by

_iueans of successive saddle

point approximations

at

J/kT

> I

on each

non-analytical

step. ^This transforiiis the

integral equation (10)

into _an

algebraic

_one.

Then, in order to deterniine the mean-field

coupling

constant

J,

it is convenient to consider small

angles

91 ^of deviation of _u from _ii

(such

_a liiuitation

corresponds

to the

approximation

J/kT

> I

above).

TlIen _{u,e are} able to consider the _mean field

coupling

constant to

obey

the relation J

=

)3~U/3b][~~=o.

^{~Ve thus obtain}

J _cS

)p(~)/~G

⁺

~(12

⁺

)Io)

^{+ ~~~}

Tj

S ₊

)HS ⁽¹¹⁾

Yap

where the nematic order paraiueter ^S ₌

-0.5+1.5(cos~ bi)

contains the additional

dependence

on Q and T. We estimate tlIe "baI.e" Iuean-field

coupling

constant H

using typical

values for the molecular parameters.

Taking

~- 30

I,

_d

~- 5

I,

_p

~- 10~~

cm~~,

temperature kT ~- 4 _x 10~~~ _erg

and, approxiIuately,

lo _" 12 ₌

G/5;

G

~- 3 _x

10~~~ergcm~ (clarify

with the Ref. [10], ^for

exaniple).

The three contributions to

equation (11) give, respectively,

17 _x

10~~~,

5 _x 10~~~ and 10~~~ _erg,

so that the total

coupling

constant H

~- 25 _x kT.

If the

anisotropic

attractioiI betiveeii

mesogenic

molecules is much weaker

(Io,2

<

G),

^ther-

motropic

nematic

ordeI.ing

is deterniined

priniarily by

_an

isotropic

attraction modulated

by

an asymmetry of molecular

shape

(12. ^{The first} term then dominates

equation (11)

and _one

can expect muclI lower values for H

~-

(8 10)

x

kT, corresponding

to a lower transition temperature, _{as ,ve} will _see belo,v.

The molecular-statistical

description

of the nematic

phase composed

of semiflexible dimers with the

only

^model parameter

being

^Q _{iuay now} ^{be considered}

complete.

There _are several

simple

_ways to obtain the

niacroscopic,

observable parameters ^of the system. ^One

method, analogous

to that of

lklaier-Saupe,

is to describe the

branching

of the _nonzero solution for the order parameter

S,

I-e-

by looking

for tlIe nontrivial solution of the

equation

S

= -0.5 ₊

1.5(cos~ bi)

#

1.5J/H.

This

gives

the ultimate

superheating

temperature ^T* of the transition.

It is also

helpful

to _use the fact that the N-I transition is

weakly

first

order,

and to derive the

microscopic expressions

for the coefficients of the Landau free _energy

expansion

F _m

aS~+bS~+cS~+..

as functions of the iuolecular parameters. ^One can then obtain the

complete description

of the

phase

transition and the

low-temperature

nematic

phase.

In

particular,

the

instability

threshold of the

isotropic phase (ultimate supercooling

temperature

TI)

and the width of the first-order transition

hysteresis

T*

TI,

which both _are

easily

observable

quantities,

_can be deteriniiIed iiI this model.

3. Phase transition

properties.

Using equations (2)

and

((7)-(9)),

_we find the

following

relations:

j

_~ ²~

f

_~J(1111)~/kT~fl(11 U')~J(11'.1l)~^/kT

j~ ~j2

~~i ~~

fi_{~Il ~}

°~ ~ Z

32J/kT j~~j

The Landau coefficients _are determined

by isotropic

_averages at J

= 0:

a(Q,

H,

ET)

₌

pH

I

$

[((cos~

bI + cos~

b2)~)o (cos~ bi

+ cos~

b2)(])(13)

~~~' ~' ~~~ II ^~~ ~~

^~~~~°~~~~ ~ ~°~~_{~~~~~° ~~~~}

-3((cos~

₉₁ + cos~

b?)~)o(cos~

bi + cos~

b2)o

+

2(cos~

bi + cos~

b2)(]

N°i NEMATIC OF SEAIIFLEXIBLE DIAIERS ⁴⁷

~~~'~'~~~ 2~3~~ ~~~

~~~~°~~^{~~ ~} ~°~~_{~~~~~° ~~~~}

-4((cos~

bi +

cos~ b2)~)o(cos~

i

+

cos~ b2)o 3((cos~

bi +

cos~ b2)~)(

+

+12((cos~

bI + cos~

b2)~)o(cos~ bi

+

cos~ b2)( 6(cos~ bi

+

cos~ b2)(]

where

((cos~ bi

+ cos~

b2)~)o

+

) _{f e~}

~~ ~i _~"

~?Zo(Q

^sin

bi

sin

b2) (16)

o

(cos~

bi + cos~

b2)~'sin bi

sin

b2d81db2 and, clearly, (cos~ bi

+ cos~ b~)o +

2/3.

ExamiIIing equations ((12)-(16))

one can

easily

notice that in the limit Q <

I,

the coefficients

approach

the _monomer form with _a

=

ao(T T~'°~°.)

cS

~p$[kT ^0.0445H];

^b cS -8.3 _x

10~~pH(H/kT)~

and _{c cS} 5.6 _x

10~~pH(H/kT)~.

These 9values

correspond quite

well to the

predictions

of _a

Maier-Saupe-like theory: recalling

the value of H in

equation (I I)

and

assuming

that the

isotropic-nematic

transition temperature is of the _same order of

magnitude

as the ultimate

supercooling

temperature Ti in _{a, we} find

kTf°~°.

_~-

kTflJ~°.

~- 5 _x

10~~~

ergi which is of order 90 °C. In the

opposite limiti

Q _{- cxJ,} the

integrands

in

equation (17)

take the form

exp(Q cos(bi b2))

^and the Landau coefficients become _{a cS}

(p$[kT 0.088H];

m

-3.2 x

10~~pH(H/kT)~

and _{c cS} 4.3 _x

10~~pH(H/kT)~.

These values for the _monomers with doubled

length

exhibit

a shift of the transition temperature ^and _a more

strongly

^{first order}

transition with

corresponding

increases of the order parameter

discontinuity

and

enthalpy.

In order to describe the

crossover behavior of the system in the _most

interesting region (Q

>

I),

one must calculate the

integI.als

in

equations ((12)-(16)) numerically.

From the theoretical

point

of

view,

there _are three characteristic temperatures

describing

the N-I first order

plIase

traiIsition the ultimate

superheating

T* and ultimate

supercooling Ti points,

and the

equilibrium

transition

point

T~ ^at which the free energy of the

non-symmetric phase

becomes lower tlIan that of the

isotropic phase.

The ultimate

superheating

temperature T* at which the bifurcation of tlIe _nonzero order parameter S takes

place,

_can be determined very

precisely

in _our statistical model. All other characteristic

points

on the temperature scale

can be referenced to T*. The difference between T* and the ultimate

supercooling temperature

TI

(the

ultimate widtlI of the weak first order transition

hysteresis)

is T* -TI ₌

9b~/32uoc.

The

actual transition temperature Tc ^lies between TI and T* T* T~ cS 0.022b~

lace

_« T* TI for Q ₌ 0 "short monomer"

case) kTm°~°

_cS 0.04995H. In the

figure

^{2 the} _{upper curve} represents

the behavior of

Tc(Q),

^which is normalized

by Tm°~°.

The lower _{curve on} this

plot

shows the relative variation of the ultimate

supercooling

temperature TI.

One _can

clearly

_see the

sharp change

in both characteristic temperatures ^{in the}

region

of

Q between 2 and 5, as well _as their saturation when the bond between the _{two mesogens}

becomes _very

rigid.

^{Since both} Tc and Ti _are

easily

measurable

quantities,

_we show in

figure

3 the

dependence

of their ratio _on the bond

rigidity

Q.

Again,

it is clear that the

region

of intermediate Q has the most dramatic influence _on the

thermodynamic properties

of the

system

of semiflexible dimers.

Returning

to the definition of the bare bond

rigidity

Q in the

equation (I),

_we note _{as an} aside that the value of Q _cS 3

corresponds

to the characteristic

angle

₇

~- 45°

of thermal fluctuations of the dimer iii _an

isotropic

inert solvent

(see Fig. I).

Recently

several series of dimers with two identical

nematogenic

monomers and _spacers of

varying lengths

_[12]

(CH2),1

have been

synthesized.

^These

molecules,

with _{an even} number of

2

1.8

u1

§

l.6

/

#

~

,

u1 lA

~

~

~ l.2

~1 O

fj

~

~

~~0

5 lo 15 20 25

Q

Fig.2.

Reduced transition (n /T~~°~°' solid curve) and supercooling

(Ti

/T~~°~° dashed

curve)

temperatures _ns. ^Q

0.965

0.964

0.963

~

0.962

fl

b~ _0.961

0.96

o.959

~'~~~0

_{5 lo 15 20 25 30}

Q

Fig.3.

The ratio Ti/T~ _vs. Q

methylene units,

are

likely

to be

good

candidates which _can be

reasonably

well described

by

our model. Consider, ^for

example,

the molecule

7l-~-N=HC-#-O-(CII?)n-O-~-CH=N-~-7l (17)

where

~

denotes _a benzene

ring

and terniinal _groups 7l _are

CH3, C2H5,

and

C3H7. Figure

4 shows the N-I transition temperatures ^for even

homologues

of these series

[13].

^{One of the}

predictions

of _our model is the univeI.sal _curve for the

dependence

_on Q of the temperatures

Tc(Q),

cf.

figure

2.

Using

tlIis _curve and the

experimental

transition temperatures it is

possible

to obtain values for Q _{as a} funct.ion of _{a spacer}

length.

In order to refer these temperatures

to the theoretical _curve

Tc(Q),

it is convenient to normalize them

by

the

extrapolated

value of Tc for _n

=

0, which, presumably, corresponds

to very

large

Q and therefore _to the saturation

N°i NEAIATIC OF SEAIIFLEXIBLE DIAIERS 49

540

520 ^*

A

soo ^"

m

A

480

~

~46 +

m

440

m

A .

420

400

O ²

n

region

in

figure

2. Linear

extrapolation gives

the

following

values for

Tc(n

₌

0):

for

CH3

terminal _group 537 + 8

Ii;

for

C2H5

520 + 8 K and for

C3H7

529 + 8 K.

Referring

to these temperatures as the saturation values

on the _curve

Tc(Q) (Fig. 2)

^and

normalizing

the transition temperatures for _{n =}

214, 6, 8,

10 and 12 to

Tc(n

₌

0),

_we obtain the

corresponding

values for

Q(n)

for all three materials under consideration. It is

important

to note that there is _{an error} of

~- 1.5 $l in the calculation of reduced temperatures. This _error propagates to

Q(n)

and

rapidly

increases when

approaching

the flatter

(close

to

saturation) region

of the

Tc(Q)

_curve.

Figure

5 represents ^{the variation of Q with inverse} _spacer

length

I

In,

and the solid line is _a linear least _squares fit to the function I

IQ

= An+B. For these three series at

least,

which differ

only by

their terminal _groups

7l,

it is clear that this

dependence

is

universal: Q _cS

(70 +10) In;

within _{our accuracy} the coefficient B is not

distinguishable

from

zero.

4. Discussion.

The

preceeding result,

Q

~-

70/n,

_can be

explained by

_a

simple

argument ^which assumes

that each

[CH2 CH~]

unit in the _even- numbered bond has _an

equilibrium angle

0° and

an effective

bending

_energy Ea. The

probability

of its

bending

to _some

angle

₇ is then p ~-

exp[(Eo/kTcos7].

If _we

neglect

^both correlations between

subsequent

units and the influence of terminal _monomers, the total

probability

of

bending

of the all-trans

configuration

of the

spacer chain

[CH2 CII2]n/2 by

the overall

angle

₇ must be P~

~-

exp[(2Eo/nkT)cos7]

=

exp[Q(u u')].

The factor 2 is needed to account for the fact that each unit contains two

methylene

_groups. If _we consider the

i-O-j bonds, connecting

the spacer with

mesogenic

cores as flexible units _as well witlI the

corresponding bending

_{energy w (w} >

Eo),

the

resulting

expression

for Q _must be corI.ected in the

following

_way:

~

nw +

4Eo'

Q 2Eo ^~ _w ^~

~~~~~°~

~~~~

If there _{are in}

double-gauche (g-t-g) conformations,

located somewhere _on the _spacer

chain,

similar arguments about _non- correlated _sequence of elastic units will lead _{to an}

equation Q(n),

similar to

equation (18)

,vith two types ^{of units with}

bending energies Eo

and E21

1/Q

_~-

in (6 2E2/Eo)m].

This is not the kind of

dependence

that _one observes in the

fig-

ure 5.

The

proportionality represented by equation (18)

has been tested

by

direct quantum ^chem-

istry

calculations [14] ^of

bending energies

of various conformations of _a free

[CH2]n

^{chain and}

averaging

_over the rotation around the cliain axis. The

dependence Din),

that _one observes in the

figure 5, corresponds

to the

equation (18)

with _very little deviation from

(I In) dependence.

This suggests, ^that in the materials [12] ^the concentration of double-

gauche

conformations _was very low _even at the transition

point.

There is _a reasonable

quantitative agreement

^between the

figure 5, equation (18)

and the value of the

bending

_energy

Eo,

calculated with the

help

of molecular

modeling

software

Sybyl. Indeed,

if _we take the molecule

OH-CH2-CH2-OH

_as

representing

the unit

[-CH2 CH2-]

in the _spacer and find the coefficient for the

potential

energy for

bending

U cS const Eo _{cos 7,} this calculation

gives

Eo _cS 2.2 + 0.I

eV, [14, 15].

We have obtained _a similar value for the

bending

constant Eo

by

numerical calculation of energy of different

[CH2]n

chains _{as a} function of the

angle

between the terminal C H bonds. For

temperatures ^{of order} 520 II tlIis

corresponds

to _a coefficient I

IA

₌

2Eo/kT

~-

100,

_very ^close

to the

expected

_mean value

I/A

₌ 70 from the

plot I/Q(n), figure

5; as it _was

expected, 2/w

< I.

This remaI.kable

agreement

^is not

IIecessarily

universal.

llepresentation

of the _{spacer as}

a

non-correlated _sequence of

symmetrical

elastic units is

obviously

too gross an

approximation

to

N°1 NEAIATIC OF SEAIIFLEXIBLE DIAIERS 51

be considered

appropriate

for all niaterials. For

example,

_{one can} expect

significant

deviations from the linear

dependence

Q

~-

I/n

for dimers which _possess

dipole

moments in the terminal groups close to the _spacer. However the theoretical

predictions

of this

work,

such _as

figures 2,3,

_are

universali

with the parameter ^Q

effectively describing

the _{spacer as a} whole. The

three series which have been used for

comparison

with _our

predictions

_possess an

relatively

flexible connection between the

bonding

chain and hard

mesogenic

_cores

through

the ether group, therefore it

might

be

relatively

_easy to rotate these _cores around the

long

molecular axis. Moreover, there is _an absence in the terminal _mesogens of

large

transverse electric _or steric

dipoles.

It is for these _reasons that such _a remarkable

"universality"

in

Q(n)

is found for these series of dimers.

At this

juncture

we have

found,

for three series of

dimers,

_an ^excellent

agreement experiment

and theoretical

predictions

for _spacer

energies. Nevertheless,

the

theory

makes other

predic-

tions _as

well,

such _as

Ti/Tc.

Tests of these results

require

further

experimentation,

which _are

currently underway. Finally, although

the niodel does not

exhaustively

account for all

possible

intramolecular

interactioiIs,

it does consider tlIe

bending

of _a spacer group in _a self-consistent way. It thus makes _an

important

contI.ibution to _our

understanding

of the role of the

spacer(s)

in

polymeric

and

oligomeI.ic liquid crystals.

Ack~iowledgiiieiits.

We _are

grateful

to A.C.GI.iffin for

providing

^his

experimental

data for N-I transition temper- atures and to I<.NatlI _foI~ useful discussions. This work _was

supported through

the Advanced

Liquid Crystal Optical

MateI.ials Science and

Technology

Center

by

the State of Ohio

Depart-

ment of

Development

and board of regents ^{and the National Science Foundation}

through

_grant

No.: DMR-8920147.

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