Macroscopic dynamics in nematics near the uniaxial-biaxial phase transition

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Macroscopic dynamics in nematics near the

uniaxial-biaxial phase transition

H. Brand, J. Swift

To cite this version:

H. Brand, J. Swift.

Macroscopic dynamics in nematics near the uniaxial-biaxial phase

transition.

Journal

de

Physique

Lettres,

Edp

sciences,

1983,

44

(9),

pp.333-337.

�10.1051/jphyslet:01983004409033300�. �jpa-00232200�

Macroscopic dynamics

in nematics

near

the uniaxial-biaxial

phase

transition

H. Brand

Bell

_Telephone

Laboratories,

Murray

Hill, New _Jersey 07974, U.S.A. and J. Swift

Dept.

Physics, University

of Texas, Austin, Texas 78712, U.S.A.

(Recu le 15 decembre _{1982, accepte} le 17 mars ₁₉₈₃₎

Résumé. 2014 Nous

présentons

une discussion des

_expériences

_possibles

révélant les

_propriétés

macro-scopiques

des

_nématiques

uniaxes

_près

de la transition uniaxe-biaxe. En

_particulier,

nous _proposons une mesure d’un nouveau

_{couplage statique}

entre les

_gradients

des fluctuations du directeur

_(03B4n)

de

la

_phase

uniaxe et du

_paramètre

d’ordre

_(03BEij)

de la

_phase

biaxe. Abstract. 2014 Based

on the derivation of the

_{macroscopic dynamics,}

we _present a discussion of

_possible

experiments

which

_probe

the

_macroscopic

behaviour of uniaxial nematics close to the transition to

the biaxial phase. In

_particular

we _propose an

_experiment

to measure the influence of a novel static cross

_coupling

between

_gradients

of the director fluctuations

_(03B4n)

of the uniaxial

_phase

and the order parameter

(03BEij)

of the biaxial

phase.

Classification

Physics Abstracts 61.30G

Starting

with the identification

_[1]

of

_lyotropic

biaxial nematics in 1980 there has been an

increasing

interest in the

_{experimental [2,}

_3]

and theoretical

_[Ref.

4 and references cited

_therein]

investigation

of the

_{macroscopic properties}

of these

_systems.

_{Very recently}

Jacobsen and one of

the

_present

authors

_{[5] presented}

a

_{hydrodynamic description}

of the uniaxial

_phase

close to the transition to the biaxial

_phase.

In addition to the

_{strictly hydrodynamic}

variables of the uniaxial

phase namely density

p, entropy

density

7,

density

of linear momentum _g and the director fluc-tuations bn

_{(characterizing}

the

_{spontaneously}

broken rotational

_symmetry)

the authors of

reference

_[5]

introduced the order

_parameter

which is slow

_although

with a

_long,

but finite

relaxation time t. The

assumption

which enters this

_description

is,

of course, that no additional

« slow »

_quantities

exist.

In the _present note we

_generalize

the statics

_given

in

_{[5] by}

_including

the effects of external

magnetic

fields. We also

_give

a novel cross

_coupling

term between the

_gradients

of the director

fluctuations and

_gradients

of the

_quantity

_(a

_symmetric,

traceless _tensor; cf

_[5]

for

_details).

It seems

_important

to notice that this cross

_coupling,

_{unlike many of the other features} discussed

below,

has no

_analogue

near the

_{uniaxial-isotropic}

_phase

transition. The latter has been studied in a series of papers

_by

de Gennes

_[6].

To

_probe

the new cross

_coupling

_{experimentally}

we _propose a static

_{configuration}

in an

L-334 JOURNAL DE _{PHYSIQUE -} LETTRES

inhomogeneous magnetic

field. In addition we discuss which

_experiments

can be done to

_study

the static and

_dynamic

behaviour in some

_detail;

as it turns out there are _strong similarities

with the

_{corresponding}

situations near the

_{uniaxial-isotropic phase}

transition. It is

_important

to note,

however,

one

important

difference between the two transitions. The

_{isotropic-uniaxial}

transition is

_weakly

first order

_(and

associated with a latent

_heat)

thus

_giving

rise to finite fluctua-tions of considerable

_magnitude.

The situation near the uniaxial-biaxial transition

is,

from an

experimental point

of

view,

even more favourable because this transition

is,

at least in mean

field

_{approximation [7]}

of second

_order,

and the

_experimental

results available so far seem to confirm these

_predictions.

Thus,

we can

_expect

_diverging

fluctuations of various

_quantities

when

approaching

the uniaxial-biaxial

_phase

transition from above. For the

_{generalized Ginzburg}

_{Landau free energy} we have

where

_6!!

=

ð¡j

-

ni

nj

and where the first two lines have been

presented

in reference

[5]. Fo

contains the

_purely

_hydrodynamic

contributions which can be found _e.g., in reference

_[8].

The new cross

_coupling

term has the detailed form

The effects of an external

_magnetic

field can be

_easily

included into

_{equation (1) giving}

rise to the additional terms

_(FH)

where

and

_{where x}

contains two

_independent

nontrivial terms

_giving

rise to the contributions

where

_{H 1. symbolizes}

the

_magnetic

field

_{perpendicular}

to n. It should be

kept

in

mind, however,

that x

contains an additional factor

_ç¡j

_compared

to

_X

and is therefore

_presumably

of minor

importance.

The

_dynamics

remains

_unchanged,

_i.e.,

the reversible and the irreversible currents

are a mere

_{superposition}

of the terms

_given

in

_[5]

and

[8].

No additional cross

_couplings

_appear.

A first

_{simple application}

of the basic

_equation

is

_magnetic

_{birefringence}

in the

_plane

perpen-dicular to the uniaxial director. If we assume a

_{homeotropic sample}

with n% ~

(enforced e.g.,

by

a static

_magnetic

field in

_z-direction)

we consider the effects of a static field in the

xy-plane,

say in the x-direction. We then find from

equations (1)

and

(3)

for the

xy-plane

Proceeding

in the same _way as in

_[6]

we find for the

_anisotropy

of the refractive indices in the

plane

where // and -L refer to the directions

_parallel

and

_{perpendicular}

to the

_magnetic

field in the

plane.

Since (X ’"

l/a

the effects

predicted by equation (7)

will become very

large

near the

_phase

transition

_(cf.

also Ref.

_[2]

for a discussion of this

point).

Of course, an

_analogous

_{argument goes}

through

for electric

_{birefringence}

in the

_xy-plane.

Next we consider the case of flow

birefringence.

We assume as above that the uniaxial director is oriented in the

z-direction,

furthermore we

consider a flow

velocity along

x with the

velocity

gradient Dvxl0y.

We then

_get

from the basic

_equations

_(Eq.

₍₁₎

and

_{Eqs. (6~ (9), (12), (16)-(24)}

of ref.

_[5])

where we use the notation of reference

[5]

where

possible

(e.g., Aij

=

2 L(vi V

i +

v

V;)).

For the

_{configuration}

we have

chosen,

we assume a time

_dependence

of the form

eiwt

and obtain

In

_{deriving (9)}

we have

_kept

the viscous

_term’

and the

_bending

_{energy of the}

~-field.

It is well

known for the

_{isotropic-uniaxial}

nematic transition

₍₉₎

that the effects of the

bending

energy

can be

_neglected.

Then we deduce from

_equation

₍₉₎

in the low

_frequency

limit

Proceeding

as in the

_{uniaxial-isotropic}

case

[6]

from

_{equation (10)}

we

_get

for the difference in the

refractive indices

where the characteristic time f

_diverges

at the

_{transition ;}

for the constant S we refer to

_[6].

From the basic

equations

it is

_easily

checked that all other variables

_(e.g.

_bni)

do not

_couple

in the

_specific

configuration

we have chosen. As soon as the

_equilibrium

value of the uniaxial director _no has a

component in the

_plane

of the

shear,

however,

the above

_analysis

has to be

_{changed completely}

and no

_simple

result

_probing

the

_magnitude

_{of ç¡j}

can be obtained.

The influence of fluctuations of

_~1~

can also be

_{investigated by looking}

at the shear wave atte-nuation. We assume, as

_above,

_{no jz}

and the

_velocity

in x-direction with variations in the

y-direc-tion. Then from

_{equations (12),..., (26)}

of

_[5]

we can extract for the stress tensor

L-336 JOURNAL DE _{PHYSIQUE -} LETTRES

Combining

equations

(12)

and

₍₁₃₎

we

_get,

after a Fourier

transform,

the effective

_viscosity

_fleff

Thus we

_get

a fnite correction for

_flerr(0153

=

0)

of the order

7;2/,

which

gives

a

_possibility

to mea-sure this

_quantity

because at

_{higher frequencies}

_{(ro2 ~ (~xo ~ )~ )}

there is no correction. We mention

in

_passing

that our

_expression

for the shear

_viscosity

differs

_{significantly}

from the result derived

by

de Gennes for the

_{uniaxial-isotropic phase}

transition. De Gennes

finds,

for

_example,

no static

correction.

_Furthermore,

there is a

_{high frequency}

correction in the latter

_phase

transition

_[6].

Finally,

we discuss the

possibilities

of

measuring

the static cross

_{coupling presented}

here for the first time.

_Light

_scattering

in

_simple

_geometries

has been discussed in reference

_[5].

As is

easily

checked,

the

_{coel~icient ~}

does not enter for these

_special

cases,. thus

leaving

the

presentation

of [5] unchanged.

It is

_{straightforward}

to convince

oneself however,

by

looking

into more

compli-cated

_{configurations,}

_{that ~}

enters in

_general

_situations,

_although

the

_dependence

_{on 0}

becomes rather involved. In the

_present

_paper we _propose a static

_experiment

to

_gain

some

_qualitative

insight

into the effects of

_~.

To achieve this one _sees,

_by

_inspection

of

_equation

(1),

that one needs

inhomogeneous

external

_fields,

and a

_magnetic

field

_might

be a

_particularly

_good

candidate.

We examine the situation sketched in

_figure

1. We assume that one

_pole

of the

_magnet

is

_wedge

shaped

in the

_xy-plane

and extends to ± oo in

_y-direction

which renders the situation

_essentially

two-dimensional. In the uniaxial

_phase

the n field is then

distorted,

but in a

_symmetric

_way in z and - z-direction because the twist term

_K2

does not enter. If the uniaxial

_phase

is cooled towards the biaxial

_phase

transition the

_~-term,

which breaks the z -+ - z

_symmetry

in the

configuration

chosen

here,

becomes more and more

important

and thus the

_pattern

(observed

e.g., in a

_{microscope) picks}

_up an

asymmetric

distortion

(in

z-direction),

which

should be

detec-table

_{experimentally.}

This can be inferred from the variational

_equations

obtained from

_{(1), (3)}

In conclusion we have discussed

_{possible experimental}

_{configurations}

to

_study

the influence of the order

_parameter

near the uniaxial-biaxial nematic

_phase

transition. In

_particular

we have

proposed

an

_experimental

set _up to

_study

the novel

_coupling

term between the director field

Fig.

1. -

Proposed experimental

set _up to detect the

_{cross-coupling coe~cient ~ by application}

of an

inhomogeneous

magnetic

field. Without the

inhomogeneous

field one can assume a

_homogeneous

texture, if _necessary this can be enforced

by

a homogeneous field in z-direction.

and the

_{slowly varying quantity}

_~;~

which should allow for a

_{simple qualitative}

determination of

the

_importance

of this term.

Acknowledgments.

It is a

_pleasure

to thank Pat

Cladis,

W. D. McCormick and Ron Pindak for useful discussions.

The research of one of us

_(J.S.)

was

_supported

in

_part

_by

the Robert A. Welch Foundation under

grant

number F-767.

References

[1]

Yu, L. J. and _SAUPE,

_{A., Phys.}

Rev. Lett. 45

₍₁₉₈₀₎

1000.

[2]

SAUPE, A., BOONBRAHM, P. and Yu, L. J.,

preprint.

[3]

BARTOLINO, R., CHIARANZA, T., MEUTI, M. and COMPAGNONI, R.,

Phys.

Rev. A 26

₍₁₉₈₂₎

1116.

[4]

BRAND, H. and PLEINER, H.,

Phys.

Rev. A 24

₍₁₉₈₁₎

2777,

Phys.

Rev. A 26

₍₁₉₈₂₎

1783.

[5]

JACOBSEN, E. A. and SWIFT, J., Mol. _{Cryst. Liq. Cryst.} 87

₍₁₉₈₂₎

29.

[6]

DE GENNES, P. G.,

Phys.

Lett. A 30

₍₁₉₆₉₎

454, Mol. _Cryst.

_Liq.

_Cryst. 12

₍₁₉₇₁₎

_193,

_{Physics of Liquid}

Crystals

(Clarendon Press,

Oxford)

1974; for a review of the experiments near the

uniaxial-isotropic

phase transition we refer to de

Gennes’

book p. 211-215 and to CHANDRASEKHAR, S.,

Liquid Crystals,

p. 64-74.

[7]

FREISER, M. _J.,

_Phys.

Rev. Lett. 24

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_{1081 ;} ALBEN, R.,

Phys.

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₍₁₉₇₃₎

778.

[8]

FORSTER, D.,

Hydrodynamic

Fluctuations, Broken

_Symmetry

and Correlation Functions

_(Benjamin,