Statics and dynamics near the nematic-columnar phase transition in liquid crystals

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Statics and dynamics near the nematic-columnar phase

transition in liquid crystals

J. Swift, B.S. Andereck

To cite this version:

J. Swift, B.S. Andereck.

Statics and dynamics near the nematic-columnar phase transition

in liquid crystals.

Journal de Physique Lettres, Edp sciences, 1982, 43 (12), pp.437-440.

�10.1051/jphyslet:019820043012043700�. �jpa-00232073�

Statics

and

_dynamics

near

the

nematic-columnar

_phase

transition

in

_liquid

_crystals

J. Swift and B. S. Andereck

Department of

_Physics,

The

_University

of Texas at _{Austin, Austin,} Texas 78712, U.S.A.

(Re~u le ll mars 1982, accepte le 22 avril ₁₉₈₂₎

Résumé. 2014 Un Hamiltonien de

Landau-Ginzburg

décrivant un

_{cristal liquide nématique près}

de la transition

_{nématique-phase}

colonne est _{utilisé pour} calculer la renormalisation des constantes

élas-tiques

et des viscosités dans la _{phase nématique.} Nous trouvons que les constantes

_élastiques,

sauf la

constante

_élastique

de flexion, sont

_augmentées

_{par les fluctuations du}

_paramètre

d’ordre et _que cette

amplification

est

_{proportionnelle}

au carré de la _longueur de corrélation. Une des viscosités seulement n’est pas

amplifiée près

de la transition.

Abstract. 2014 A

Landau-Ginzburg Hamiltonian

_describing

a

_{liquid crystal}

in the nematic

_phase

near

the nematic-columnar _phase transition is used to find the renormalized nematic elastic constants

and viscosities. The elastic constants, with the

_exception

of the bend elastic constant, are found to be enhanced

_by

order _parameter fluctuations, the enhancement

_{being proportional}

to the second power of a correlation

_{length. Only}

one of the viscosities is not enhanced near the transition. Classification

Physics Abstracts 64. 70E

The

_{long-range stability}

of the two-dimensional

_crystal

in a three-dimensional _space was

demonstrated

_by

Landau and Lifshitz

_{[1] long}

before a

_physical

realization was discovered

_by

Chandrasekhar, Sadashiva,

and Suresh

_[2]

in a

_{thermotropic liquid crystal}

_system. This

_phase,

with

_long-range

order in two directions

_(a

_{triangular lattice)}

and fluid

_properties

in the third direction is known

_by

a

variety

of labels

(tubular,

disk-like, canonic, calamitic, discotic,

hexa-gonal),

but we shall

adopt

Prost’s

[3]

use of the name columnar

_phase.

The

_{hydrodynamics}

and

elastic

_properties

of columnar

_{liquid crystals}

were discussed

_by

Prost and Clark

_[4].

Kats

_[5]

described the free energy and behaviour of the elastic constants near the

_{columnar-crystalline}

transition and also near the nematic-columnar

_(N-Col)

transition. As we describe our

_theory

we

will comment on some differences between our work and that of Kats. In this letter we consider

the N-Col

_phase

transition and the enhancement of the elastic constants and viscosities in the nematic

_phase

due to order

_parameter

fluctuations near the transition.

For

_{liquid crystals}

of the

_type

which can form a columnar

_phase

the director is

_{perpendicular}

to the disk-like molecules which

_comprise

the

_{liquid crystal.}

In the nematic

_phase

the _average orientation of the director is non-zero, i.e. the disks

generally

face in a common direction. At the N-Col transition we have the onset of a

_{two-dimensional}

_triangular

lattice

_{perpendicular}

to

the mean director orientation. The

_long-range

_density

correlations of the columnar

_phase

can be

L-438 JOURNAL DE PHYSIQUE - LETTRES

represented by choosing

an order _parameter

_analogous

to the NAC-model of Chen and

Luben-sky (CL) [6].

The N-Col order _parameter

_is,

then

where

_p(k)

is the Fourier-transformed

_density

and the wave vector

_integration

is over a domain D

which is a torus of minor radius A and

_major

radius _qlo, where _qlo is the

_magnitude

of the

smal-lest

_reciprocal

lattice vector of the

_triangular

lattice

_(A’

_qlo).

The torus is centred at the

_origin

and its

_major

axis is

_parallel

to the mean director. The choice for order

_parameter

establishes no

preferred

direction in the

_{plane perpendicular}

to the director.

Our

_{Landau-Ginzburg}

Hamiltonian

_{(H )}

is _very similar to that of CL but with obvious

_changes

so that

density

waves

_{perpendicular}

to the director are

_preferred.

Our H is written in the form

where the first term in

_/!N

is

where n is the

_director,

~

=

8~~ - nt n~, a

=

a’(T -

TJ/T~, (a’ being

_a

positive

constant and

_Te

the N-Col transition

_{temperature),}

and

_C-L

0 and

_CII

0.

Equation (2)

can be written in terms of the

_spatial

Fourier transform

of m,

If we

_neglect

fluctuations in the director in

₍₂₎

then we have

where q is

the

_component

_{of q}

_parallel

to the

_director,

_qjL is the

_magnitude

of the

_component

of _q

perpendicular

to the

_director,

_{and no is the}

_equilibrium

director.

_Obviously,

in this model

_density

waves in the

_{plane perpendicular}

to _{the average director and with}

_{magnitude ql}

=

qlo are

favoured.

Note that our Hamiltonian

(3)

reflects the

_symmetry

of the nematic state and that order

para-meter fluctuations associated with _any wave vector _q

_pointing

to the

_ring

defined

_{by q ~~}

=

0,

qj_ =

~10 are

_{equally likely [6,}

_7,

_8].

This means that the

_strong

_{X-ray scattering}

from order

para-meter fluctuations would be associated with this

_{single ring}

in wave vector _space.

On the

_contrary,

Kats has

_mpdified

_{the free energy of}

_Jahnig

and Brochard

_[9]

and even in the

nematic

_{phase apparently}

assumes a definite orientation for the

_reciprocal

lattice vectors in the

plane perpendicular

to the _average director. Since no such

preferred

direction exists in the

sym-metry

of the nematic

_phase

we assume that he has

_{implicitly imposed}

an external field or some

In

_{equation (1) PHel}

_{is the Frank free energy and is}

_{exactly given by equation (2.4)}

in

_CL,

_pH4

is

_{given by equation (2. 5)}

in their _paper,

and,

since a cubic invariant exists for our _case, we have

where v is a constant. Our calculation is a direct

_parallel

of that of CL and assumes that the

tran-sition is not

_strongly

first order so that the fluctuations are observable.

In

_{equation (2), PHm}

_{may be}

_expanded

out in terms of the deviation of the director from its

average value. This leads to

_couplings

between the order

_parameter

and director of the form

_given

by equations (3.4)

and

_(3.5)

in CL. The vertex

_{ri(k, q)}

is

_{given by}

their

_equation

_(3.5)

but with

D

II = 0 in this

expression.

Their formalism for the self energy of the director

propagator

may

then be used to calculate the enhancements of the elastic constants near the N-Col transition.

We find for the enhancements

and

Our enhancement for the

_splay

elastic constant is

_proportional

to the square of a correlation

length

whereas Kats’ result is linear in a correlation

_length.

The difference stems from the

integra-tion

_regions

used in

_calculating

the free energy. Kats’

preferred

direction for the

reciprocal

lat-tice vector _po

_gives

him a small three-dimensional

_{spherical integration region}

around _po,

_making

his

_problem

most similar to the nematic-smectic-A

_problem.

Our

_expression

for _{the free energy} involves no

_unique

direction in the

_plane

of the

_triangular

lattice and hence

_requires

a toroidal

integration region,

similar to the nematic-smectic-C case.

_According

to reference

[7]

the

_high

degree

of

_degeneracy

in this case

_effectively

lowers the

_{dimensionality}

of the wave vector

integra-tion from three to two and leads to an additional factor

of ~,

as was seen in CL’s results. We find

similar enhancement for the twist elastic constant

_(as

seen in

_Eq.

_(4b)).

Kats states that his for-mulation

_{only gives}

an increase in

_{K 1.}

The enhancements of the viscosities near the N-Col transition are found

_{by techniques}

parallel-ing

Hossain, Swift,

Chen and

_{Lubensky [ 11]. Again}

corrections to the various

_propagators

(asso-ciated with the director and

velocity)

are calculated. The inverse

_propagators

are not

_readily

separable,

but form a matrix in which the

_simple

mode

coupling

fluctuation corrections can be

included and from which the

_perturbed

values of the viscosities can be found. For a sketch of the

procedures

involved,

see the

_Appendix

of reference

_[10].

Enhancements are

_again

found

_[11].

In the notation of reference

_[10]

the enhancement of the «

viscosity »

associated with the director is

L-440 JOURNAL DE _{PHYSIQUE -} LETTRES

In

addition,

there is a correction to the reactive coefficient A

_{given by ~,}

=

(y 1 ) -1 (~o

y?

+ ~~i)

and near the N-Col

transition X

= 1.

We have found enhancements due to order

_parameter

fluctuations for all but

_{K33 and vs}

as the N-Col transition is

_approached

in the nematic

_phase.

The critical behaviour of elastic

constants and viscosities

_{can, be}

studied

_{experimentally by measuring}

the

_speed

and attenuation of sound. Such measurements would

provide

a test of the

_assumption

that the N-Col transition

can be treated as

_nearly

second order and could also be a check on the

_accepted

_symmetry

of the columnar

_phase.

Acknowledgments.

- This work was

_{supported by}

the Robert A. Welch Foundation

_through

Grant No. F-767.

References

[1]

LANDAU, L. and LIFSHITZ, E. M., Statistical

_{Physics (Addison-Wesley, Reading}

Mass.) 1969, p. 402.

[2]

CHANDRASEKHAR, S., SADASHIVA, B. K. and _SURESH, K. A., Pramana 9 ₍₁₉₇₇₎ 471.

[3] PROST, J.,

preprint.

[4]

PROST, J. and CLARK, N. A.,

Bangalore Int.

Liq. Cryst.

Conf.

(Addison-Wesley)

1980.

[5]

KATS, E. I., Sov.

_Phys.

JETP 48

₍₁₉₇₈₎

916.

[6]

CHEN, J. and LUBENSKY, T. _C.,

_Phys.

Rev. A 14

₍₁₉₇₆₎

1202.

[7]

SWIFT, J.,

Phys.

Rev. A 14 ₍₁₉₇₆₎ 2774.

[8]

ALEXANDER, S. and MCTAGUE, J.,

Phys.

Rev. Lett. 41 ₍₁₉₇₈₎ 702.

[9]

JÄHNIG, F. and BROCHARD, F., J.

_Physique

35

₍₁₉₇₄₎

301.

[10]

HOSSAIN, K. A., SWIFT, J., CHEN, J., LUBENSKY, T. C.,

Phys.

Rev. B 19 ₍₁₉₇₉₎ 432.

[11]

We find, for the N-Col transition, in the notation of reference

_{[9] :}

03B1 =

03B303(- C~)1/2 D-1/2~q2~0/64 03C0a

03B2 =

03B303(-

C~)3/2

D1/2~

q2~0

In

(039B^a-1/2)/32 03C0

0394 = 03B303(- C~)-1/2D~1/2q4~0/6403C0a.

We find that 03A6 and 03C3 have a

_{logarithmic dependence}

on _a, but their contributions are dominated