Behaviour of mesomorphic systems of conformationally rigid and flexible molecules in external orienting fields

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Behaviour of mesomorphic systems of conformationally rigid and flexible molecules in external orienting fields

Vl. Pershin, V. Konoplev

To cite this version:

Vl. Pershin, V. Konoplev. Behaviour of mesomorphic systems of conformationally rigid and flexible molecules in external orienting fields. Journal de Physique II, EDP Sciences, 1991, 1 (2), pp.91-96.

�10.1051/jp2:1991149�. �jpa-00247512�

1

Phys.

II 1

(1991)

91-96

FtvwER1991,

PAGE 91

Clhssification

PhysicsAbsvuctg

M.70E

Short Communication

Behaviour of mesomorphic _{systems ~of} conformationally rigid and flexible molecules in external orienting fields

Vl.K Pershin and VA

Konoplev

Ural

Polytechnical Insjitute,

621XXQ Sverdlovsk-2, U.S.S.R.

(Received

6

August

199fl revtsed 10 December199fl

accepted

12

December1990)

Abstrnct. A comparative

analysis

of the influence of external orienting

iuadrupolar

_symmetry

fields on

mesophases

formed

by conformationally rigid

and flexible

particles

with positive suscepti-

bility

anisotropy ^for which weak and strong ^{first order} transitions to the isotropic

liquid

state are,

respectively,

^realized

(m

the absence of

fields)

is fulfilled

theoretically.

Peculiarities

allowing

to dis-

tinguish experimentally mesomorphic

systems described

adequately by

the approximations ^of

rigid

and flexible molecules are noted

Investigation

of

liquid crystals (LC)

ⁱⁿ electric and

magnetic

fields is of fundamental and prac- tical

importance.

lb date _a

large

number of papers

(see,

^for

example,

^{review articles}

[1,2])

^have

been devoted to thin

question.

The _main

part

^of

expenmental

and theoretical data relates _to

mesophases

formed

by conformationally rigid particles.

Great efforts have been done for an'ac- count of molecular

flexibility

in

orientationally

ordered

systems [3-13].

The first

attempt

^for a

theoretical

description

of ~the

orienting

fields action on the nematic

phase

of

thermotropic

LC

polymers

in terms of the Worm-like chain model based _on the

Maier-Saupe theory [14]

has been carried out in

[15]. However,

essential distinctions from the conclusions of earlier papers

[16-20]

in which LC

mesophases

formed

by rigid particles

had been studied and _a

description

had also been fulfilled in terms of the

Maier-Saupe

model _were not found in

[15]. Apparently,

the results of

[15]

^{reflect well}

enough

a behavlour of

systems

^{of flexible chain molecules in}

magnetic

^and electric fields

which,

in the absence of the

latters,

behave like

monomerij LC,

in

particular, they undergo

^first order

phase

transitions dose to the second order ones

exhibiting insignificant

orien-

tational order

parameter ((P2) jumps (AS

_~ 0 3 _÷ 0

4)

in

points

of nematic

(N) isotropic liquid (IL)

transformations

(compare

curves I and 2 in

Fig. la).

Such structures

include,

for instance,

comb-like LC

polymers

of the

polysiloxane type

^{in which}

rigid

mesogeneous groups ^connected with the main chains

by

flexible

servings

are

responsible

for orientational

ordering [21].'However,

there is _a

large

number of

mesomorphic systems

^{of flexible molecules}

(soaps [22],

lecithins

[23],

aromatic

polyesters [24,25j, cynnamates [26], cyanobiphenyls [27j,

etc. with _an

essentially

differ- ent behaviour: _a weak

(P2) change

^with the

temperature

^{rise and}

large jumps (AS

~ 0 7 _÷ 0

9)

in

points

of transitions into IL _are characteristic for them

(see,

^for

instance, Fig. lb). Obviously,

92 JOtJRNALDEPHYSIQUEII N°2

peculiarities

^of the behaviour in external fields _are to be observed in such structures which distin-

guish

them

fundamentally

from low molecular LC described

adequately by

the

approximation

of

rigid particles.

^The

present paper

^{b devoted to the}

comparative analysis

of field effects _on

systems

of

conformationally rigid

and flexible molecules.

<P~>

o,a o

I

~~~ ~"~ 2

~"~~

, '

,J

°"2

o.9O ow'J5 ~woo °"°

o,a iwoo t

,p ~ <p~>

n,R

~ ,/

~~o

~

~

l'

(f>J

~-

_~"'~

_l'

/ / / / owe

ow8 ow9 two ows two t

Fig.

I.

Experimental dependences

of the onentational order parameter

(P2)

on the temperature t'

T/Tor(Tor

is the temperature ^{of the} nematic-isotropic

liquid phase transition).

of _monomeric

azoccmpounds

=

(curve

I _in

Fig. la),

their comb-like

polymer analogs

on the basis of siloxanes

(curve

2 in

Fig. la)

[21], ^lin-

ear

liquid crystalline polymers

of the _aromatic

pJlyester type (Fig. lb)

[25], ^and

corresponding

theoretical

dependences

calculated _by means of formulae

(3)

at _{e =} 6 _in the approximation of

conformationally rigid (Fig. lc,

_{7 "}

§.9)

^and

ccnformationally

flexible

(Fig. ld,

_{7 =} 0

2)

molecules _in the absence of external fields.

We consider _a model of

mesophase

with inner

degrees

^of freedom in the

quadrupolar symmetry magnetic

^field 7i

allowing

to

reproduce

both

types

of

dependences

of

(P2)

on

temperature

ⁱⁿ

figure la,b

with weak and

strong

^{first order N-IL} transitions at 7i ₌ 0

(compare Fig. la,b

and

Fig.

lc,d).

This model is described

by

the Hamiltonian

N 2 N

H =

j £ £ _{V«p ii j)P2 (cos 0,j}

na

(I,)

_np

(lj

+

£

E

(1,)

'~~ ~~~~

~q

~

(l)

)Ax ^7i~ £

_j~2

_{(cos 0,}

>=i

in which intra- and intermolecular interactions _are taken in the

pseudo-spin approximation [4-6, 11]

^{and in the}

Maier-Saupe

one

[14], respectively.

We _note

that,

from the mathematical

point

of

view,

additional contributions to the

system's

_energy are

qualitatively

identical at the effect of both the

magnetic

and electric fields

[28-30].

In

(I):

N h _a number of

particles; P2

is the second

order

Legendre polynomial; 0,j

is an

angle

between the

long

axes of the

i and _j

panicles;

values

N°2 BEHA3/IOUR OF MESOMORPIIIC SYSTEMS 93

<P~> ^<P~>

~~

""'~(

^{o.~ ',}i

(o) ^~°~ ₃

I'°~

^{~ ~} _A ^{5 ~~~}

O.2 ~."

2

°°°~

i~o i~i i~ °°°

~

i~oo i~os £'

O.8 O.B

O.6

~~

(~~ ^..

~

o.7 "'.. ~ (b)

O-I I 2 3 ~

i,~?3

_{ij 5}

,1'

fi

O~O O.6

i~O ~.~ ~ i~OO i~05 ~

Fig.

I

(a,b) lbmperature dependences

of the onentational order

(P2) (a)

and ccnformational

disord/r z(b)

_parameters of the

mesomorphic

system ^of

conformationally ngid

molecules

(7

= 0

9)

in the magnetic field _p

= 0

(1),

0.004

(2),

0.008

(3),

0.012

(4),

0.016

(5).

(c,d) lbmperature dependences

of the onentational order

(P2) (c)

and ccnformational disorder

z(d)

_pa-

rameters of the

mesomorphic

system ^of

ccnformationally

flexible molecules

(7

= 0

2)

in the magnetic field p = 0

(1),

^1.2

(2),

2A

(3),

3.6

(4),

4.8

(5).

_{r =}

t/to,

where to is the onentational

phase

transition tempera-

ture at p = 0 Points "ill> and "B" in the qurve ^{4 indicate states of} systems of

conformationally rigid (a,b)

and flexible

(c,d)

molecules in magnetic ^fields at whose

reaching

differences between neurotic and paranematic

phases disappear.

np

(I, equal

to I _{or zero} describe the

presence

or absence of the _i

particle

ⁱⁿ the conformational state

pi Ax

is _a

positive anisotropy

of the

diamagnetic susceptibility;

0, is the

angle

between the

long

axis of the

particle

and the

magnetic

field direction at coinddence with the

system's

director.

Formula

(I)

is written in the

approximation

of two conformational states

[4-6]

^{in which all the}

posslle configurations (I,)

of

panicles

with

configurational energies

E

(I,)

_are

conventionally

divided into two subsets

corresponding

to "unfold"

(n2

_"

1)

^and "fold"

(ni

₌

I)

conformations with

keeping

the molecular

geometric anhotropy.

lb calculate the

system's thermodynamic potential

tY, we use a variational

prindple [31]

tY < tYv ₌ -kT In

Sp [exp (-Ho/kT)]

+

(H Ho)

^{where k is the} Boltzman constant, ^{T is} the

temperature,

^{the brackets} <...> indicate _a

thermodynamic

_average with the Hamiltonian

N N N

Ho

= -p

£ _{P2 (cos 0,)}

_{+ h}

£

ni

(I,)

+

£

_E

_(I,

, p~h _are variational

parameters. lbking

into

>=I >=I i=I

account the formulae introduced above _we write _a variational

thermodynamic potential

in the

sizeless form

'

+

tvv/ _(NICV22)

= -t

lnJo(a)

+ la

(2 lP2)

_{+ 11}

/3

₊ t

'n(I z)+

j~j

+

tzRlz,

_Cl

lP2) Q~(z, 7)/2

_~

lP2)

and

equations Vi

₌ 0 of _state

~ =

2at/3 lP2) Q~lzi ₇₎₁ tRlz, 6)

₌

Ii 7) lP2)~ Q(~i 7) (3)

94 JOtJRNALDEPHYSIQUE.II N°2

of the

mesophase

n~ the

magnetic

field. In

~(2,

3): (P2)

₌

3Ji(a) [2Jo(a)]~~ l/2

where _a

=

3p/2kT, Jm (a)

=

/ _u~'~

exp

(au~) du, (m

₌

0, ₁₎

_are the

Maier-Saupe integrales,

_{~ =}

(ni (I, ii

₌

o

[Zi

+

22 exp (h/kT)]~~ Zi

ha conformational disorder

parameter reflecting

a

portion

of molecules in the "fold"

conformationj Zm

₌

£ _{exp [-E (I, kTj (m}

= l~2)~ c = In

(Zi /22)

are

I,(ni(I,)=m-I)

energetic parameters; R(z~ c)

₌

In[z/(I xii

_c,

Q(z, 7)

₌

(1 7)z I;

t

=

kT/ (nV22),

~ =

Ax 7i~ / (3nV22)

are a sizeless

temperature

^and

field;respectively,

_n is the number of nearest

neighbours; _V«p

₊

_Vap (it

+

1)

7 is the effective molecular

rigidity parameter

set

by

the

equality

V22

%2 Vii

₌ 1 7 :

7~, (0

_{< 7} <

1) [11,12].

Figure lc,d

shows

temperature dependences

of the

parameter (P2)

calculated

by

means of formulae

(3)

at _{c =}

6,

_~

= 0 for

rigid (7

₌

0.9)

and flexible

(7

₌ ⁰

2) partides~

It is seen from

figure

2 how these

dependences_(Fig. 2a,c)

and those of the conformational disorder

parameters

~fig. 2b,d) change

in the

magnetic

field. Calculations fulfilled

by

means of formulae

(2, 3)

_{at ~}

#

0 indicate that isostructural

N-paranematic (pN)

transitions take

place

at 0 < ~ ^< ~A cs 0.012 in

sysjems

^of

conformationally rigid particles

and at 0 _< ~ < ~B m 3 6 in

systems

^{of flexible}

ones[

Diflerences between low and

high temperature phases disappear

while

reaching magnetic

fields _~

= ~A and ~ = ~B,

respectively, just _a§

it

hjppens

ⁱⁿ

"pressure-temjerature"

variables

1n the classical critical

liquid-gas point. By

order of

magnitude

_a value of the sizeless critical field _~A at the fixed molecular

parameters

c =

6,

₇

= 09 coincides with the _one calculated in [17~. ^Thb indicates that in absolute units the

magnetic

field

7iA

_~ 0.8 _x

10~

_koe

corresponds, accqrding

to

[32],

to the value _~A and the value

7iB exceeding

it

approxim£tely _by

two orders

corresponds

to the value ~B We note that to date the critical 'W'

point

is revealed within the limit of

experimental opportunities

for creation of

orienting

fields

[19,32] and, hence,

^{the critical} "B"

point

is most

likely

unachievable in the

experiment.

A conclusion of

practical importance

follows from here:

investigation

of

properties

of

mesophases

in

magnetic

fields in the critical

point vicinity

is

coilnected with

lowest efforts in

thq

class

of,systems

of

rigid

molecules.

Figure 3a,b

shows the

dependences

of the orientational order

(P2)~

,

(P2)~~

^{and conforma-}

tional disorder _zN, zpN

parameters

ⁱⁿ

points

of

N-pN phase

transitions _on the value _~ in sys- tems of

conformationally rigid

^molecules. It is _seen that the curves ~A ~ = pi

((P2)~)

and

~A ~ = w2

((,P2)~g)

^form

together

a

parabola symmetrical

with

respect

to the _axis

(~A ~)

which coincides

precisely

with the

corresponding

result of the

papers [2,20,32]. However,

_a simi-

lar _curve in coordinates _zN

(~A ~)

and zpN

(~A ~)

is

essentially asymmetrical ("skewed

parabola"

in

Fig. 3b).

A situation

contrary

^{to the described} one h observed in

systems

^{of conforma-}

tionally

flexible molecules: the combined _curve in coordinates _zN

(~B ~)

and ~pN

(~B ~)

is

parabolic (Fig. 3d)

and the _one in coordinates

(P2)~ (~B ~)

and

(P2)~~ ^(~B ~)

is _an

"asymmetrical parabola" (Fig. 3c).

This conclusion _can be considered _{as a} characteristic _one for

distinguishing experimentally mesomorphic systems

described

adequately by approximations

of

rigid

and flexible molecules.

Figure

4 shows

phase diagrams

of the model in the

"field-temperature"

coordinates calculated

by

means of formulae

(2, 3)

at c,= 6 and different values of the molecular

rigidity parameter

₇

It is _seen from

figure 4a,

b that _a standard situation described in theoretical

papers [2,16,17]

and

supported experimentally [20]

^takes

place

ⁱⁿ the

approximation

of

rigid particles (7

_-

1).

^And

besides,

the results of the

paper [17j

^are

quantitatively reproducible.

It follows from

figure 4c,d

that in the

approximation

^{of flexible}

particles

the _curve

dividing

N and

pN phases corresponds

to a

square dependence T(7i)

in

relatively

weak

magnetic

fields

only

and passes ^into a

dependence

close to the linear _one in

strong

^{fields. These results} are characteristic for the class of

systems

N°2

BEHA3/IOUROFA~SOMORPIIICSYSTEMS

₉₅

(a) (o)

+

<p~>~y <p~>~ <p~>y <p~>~y <p~>~ <p~>y

(b) (d)

~R

~A ~ _~pR ~B

~

Fig.

3.

(a,b)

Functional

relationship bet~veen magnitudes

of the §nagnetic ^field and ^{values of} onentational order parameters m the nematic

(P2)N

and paranematic

(P2)pN Phases (a)

^and,

respectively,

of ccnforma- tional disorder parameters _zN ^and _zpN m these

phases (b)

in points of drientational transitions in the system of

ccnformationally rigid particles (7

= 0

8).

(c,d)

Functional

relationship

between

magnitudes

of the magnetic ^field and values of onentational order parameters in the _nematic

(P2)~

and paranematic

(P2)~~ ^phases (c) and, respectively,

of ccnformational disorder parameters _zN and zpN ^{in these}

phases (d)

in prints ^of onentational transitions in the system ^of

ccnformationally

flexJble

particles (7

₌

0.3).

_{pA> PB} are critical magnetic field values at which the first order

neurotic- paranematic

phase

transition

changes

for the second order _{one in} systems ^of

ccnformationally rigid

and flexible molecules,

respectively.

~"~

p

~~~

~ A

~~

~,o

~

~,o

o.9 , ~*~

o o,o~ o,02

~ o

~.~

(b)

~

~.6 iA

o ~

* y

O.9 O.~

Fig.

4.

"magnetic

field

(p

temperature

(r)" phase diagrams

of

mesomorphic

systems ^of conformation- ally rigid

(a,b)

and flexible

(c,d)

molecules with different values of the

rigidity

parameter 7 = 0 9

(a),

0.8

(b),

0.3

(c),

0.2

(d).

% JOURNAL DE PHYSIQUE II N°2

of flexible molecules and can

be,

as a matter of

principle,

^verified

experimentally,

in

spite

of the fact that critical

points,

as has been shown

above,

cannot be achievable in such

systems

^{in the}

experhnenL Figure

4 shows tendencies of alteration of the critical fields _~A and ~B values de-

creasing

with the molecular

rigidity parameter

^increase both in

systems

^of

conformationally rigid

molecules

~fig. 4a,b)

and in those of flexible

particles (Fig. 4c,d).

In conclusion _we note

that,

since the

diamagnetic susceptibility

of dielectrics has small

values,

then for verification of the conclusions obtained

investigations

of

mescphases

in electric fields _can prove ^{to be} more

preferential

than in

magnetic

_ones.

Designations.

N

nematic, pN paranematic; A~B

end critical

points

in lines of isostructural transformations N-

pN16systems

of

conformationally rigid (Fig, 4a,b)

and flexible

~fig. 4c,d) molecules, respectively.

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