Fluctuations near the convective instability in a cholesteric liquid crystal

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Fluctuations near the convective instability in a cholesteric liquid crystal

J.D. Parsons

To cite this version:

J.D. Parsons. Fluctuations near the convective instability in a cholesteric liquid crystal. Journal de

Physique, 1975, 36 (12), pp.1363-1370. �10.1051/jphys:0197500360120136300�. �jpa-00208384�

FLUCTUATIONS NEAR THE CONVECTIVE INSTABILITY

IN A CHOLESTERIC LIQUID ^CRYSTAL

J. D. PARSONS

Department

^of

Mathematics, University of Strathclyde, Glasgow G1,

^U.K.

(Reçu

^{le 28 avril}

1975,

^{révisé le}

15 juillet 1975, accepté

^le

18 juillet 1975)

Résumé. ²⁰¹⁴ On analyse ^le comportement ^{des modes}

hydrodynamiques

d’une couche cholestérique

horizontale soumise à un gradient thermique qui ^{induit une} instabilité convective pour ^une certaine valeur critique. ^{A cause du}

couplage

de modes induit par la force extérieure, ^la fréquence ^{d’un des}

modes du système (mode ^de conduction-orientation) ^s’annule quand ^le gradient ^de température

atteint la valeur critique. Si le pas de l’hélice cholestérique

dépend

faiblement de la température,

seule la

partie

imaginaire ^{de la} fréquence ^{s’annule au seuil et on trouve des} cellules de convection qui

oscillent.

Abstract. ²⁰¹⁴ An analysis ^is presented of the behaviour of the

hydrodynamic

^{modes in a horizontal}

cholesteric layer

subjected

^{to a} temperature gradient which, ^when reaching ^{a critical} value, ^drives the system into convective instability. It is found that, ^{because of the mode}

coupling

^induced by ^the

external force, ^the

frequency

^{of one} of the modes of the system,

corresponding

^{to a} thermal conduc- tion-orientation mode, goes ^{to zero} when the temperature gradient increases up ^{to a} critical value.

When the cholesteric

pitch depends

weakly ^on temperature, only ^the imaginary part ^{of the}

frequency

vanishes at the instability, ^{and one finds}

oscillating

convection cells.

Classification

Physics ^Abstracts

7.130

1. Introduction. ^- Consider a

simple

^newtonian

fluid confined to a horizontal

layer.

^{If a downward}

directed

temperature gradient

^is maintained across

the

layer,

^{it is well known}

[1]

^that

stationary

^convection

sets in when the

temperature gradient

^{exceeds a}

certain critical value. This is one of the

simplest

^cases

of a

hydrodynamical instability

^{and is}

usually

referred to as the Bénard

Rayleigh problem

^{in classi-}

cal

physics.

^The

instability

^{occurs because the}

liquid,

when heated from

below, develops

^a

buoyancy

^force

due to the volume

expansion

^{of the}

liquid

^{near the}

bottom. For

sufficiently

strong

heating,

^{it is} pos- sible for the

buoyancy

^{force to overcome the vis-}

cous shear forces and the fluid

undergoes

^{a transi-}

tion from a state where heat is

transported by

^thermal

conduction

alone,

^{to a} state of combined heat conduc- tion and convection.

Recently

^the

dynamics

of fluctuations near the

instability

^{threshold has} been studied

[2].

^{In a}

simple

fluid there exists a

propagating

^acoustic

mode,

^two

overdamped

^shear

modes,

^{and an}

overdamped

^ther-

mal conduction mode. The _presence of a

temperature gradient couples

^{these modes}

together. However, provided

that the thermal

conductivity

^{is not too}

large,

^{it is}

possible

^to

approximately decouple

^the

thermal mode from the others. One can then show that as the

instability

^is

approached

from below the thermal modes _goes

soft,

^{i.e. its}

frequency

^vanishes

as

(Re - R )

where R is the

Rayleigh

^number

(the

dimensionless

temperature gradient) and Rc

^{is its}

critical value. This result suggests ^{a formal}

analogy

between the behaviour of the fluid near the

instability

and the situation encountered near the critical

point

in structural

phase

transitions

[3].

The

instability

^{of a} cholesteric

liquid crystal

^under

a

temperature gradient

has been considered

by

^Dubois-

Violette

[4]

^{for the case where the}

temperature gradient

is

applied along

the helical axis and when the

tempe-

rature

dependence

^{of the} cholesteric

pitch

^{can be}

ignored.

^The critical value of the

temperature gradient

is found to be much lower than in a

simple

^{fluid and}

depends

^{on the}

anisotropy

of the thermal conducti-

vity

Ka. The mechanism is

briefly

^as follows : the

temperature gradient couples

^{to the} orientation

through Ka

^{and leads to a} distortion in the orientation.

The

instability

^{occurs when the}

buoyancy

^{force of}

the

liquid

^near thé bottom overcomes the elastic forces associated with the orientation. Because these elastic forces are much smaller than the viscous shear

forces,

^{a much smaller}

buoyancy

^{force can}

balance them and this leads to a smaller value for the critical

temperature gradient.

^{A rather}

striking aspect

^{of the}

problem

^{is that the} direction of the

temperature gradient

necessary for

instability depends

upon the

sign

^of ka. Thus it is

possible

^{to reach an}

instability by heating

^the

sample

^{from the}

top !

^This

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197500360120136300

1364

situation also occurs in the case of a nematic

liquid crystal

^{oriented in a} horizontal direction

[5].

In this _paper we discuss the

hydrodynamic

^fluctua-

tions for temperature

gradients slightly

^{below the}

critical value. We consider the case where a

magnetic

field is

applied parallel

^{to the} helical axis and the

temperature gradient,

^and where the

anisotropy

^of

the

magnetic susceptibility,

X. is

negative.

^A

simple

set of

hydrodynamical equations

^{has been}

developed by Lubensky

^{for this case}

[6].

^These

equations

^are

equivalent

^{to the director} formulation for

slowly varying hydrodynamic ^motions,

^{but are more conve-}

nient to use because of the

spatial dependence

^{of the}

unperturbed

^{director in} cholesterics. The

equations

also

apply

^{to the case of a}

parallel

electric field with

a

negative

dielectric

anisotropy, provided

^{the effect}

of conduction

impurities

^{can be}

neglected.

The critical

temperature gradient

^and normal modes

are evaluated from the

hydrodynamical equations

first

assuming

^{that the} cholesteric

pitch

^is

indepen-

dent of

temperature,

^{as in}

[4].

^{In this case one finds}

steady

convection

cells,

and both the critical

tempe-

rature

gradient

and size of the convection cells increase with the field H. Like the case of a

simple fluid,

^one of the modes _goes soft

(its frequency vanishes)

^at

the critical

point.

The soft mode is a

coupled

^orienta-

tion-thermal conduction mode. The

frequencies

^of

the other modes are well behaved at the critical

point.

Next,

the effect of the

temperature dependence

^of

the

pitch

is considered. The critical temperature

gradient

is decreased

slightly

and in this case one

obtains

oscillating

convection cells. The

frequency

^of

the slow mode

acquires

^{a real}

part

^which stays ^finite

at the

instability,

^{while the}

imaginary

part ^vanishes

as before. Thus a

propagating

^thermal conduction- orientation mode can exist

sufficiently

^{close to the}

instability.

A distorted

equilibrium

^{helical structure}

is crucial for this effect. It could not occur in a nematic

or a cholesteric whose

pitch

^is

independent

^of

tempe-

rature.

2.

Governing équations.

^{- 2. 1 BASIC HYDRODYNA-}

MICS. ^- The cholesteric is

imagined

^to be confined to a horizontal

layer

of thickness 1 and of infinite extent in the transverse directions. The

magnetic

field is

applied parallel

^{to the} helical axis which is in the vertical direction. At the lower

boundary

^the

temperature

^{is fixed at}

To

^{and at the} upper

boundary

it is fixed ^at

To -

AT. AT" _may be

positive

^or

negative

but its

magnitude

^{must be small}

enough

^to

keep

^the

entire

sample

^{in the} cholesteric state.

As usual we introduce a unit vector n

(the director)

which lies in the direction of _average molecular orientation at each

point

ⁱⁿ space. In the _presence of the

temperature gradient

but in the absence of fluctuations we have

where the helical axis is chosen as the ^z axis.

qJs(z)

will be determined below and

depends

upon the temperature

gradient ATI1.

^When

ATIL =

^{0 we have}

qJs(z)

⁼ qo ^z where qo ⁼

n/P

^{with P}

being

^the

pitch.

The

subscript

^{s refers to the}

steady

^state.

Let p be a unit vector

parallel

^{to the helical axis.}

Then n can rotate in a

plane perpendicular

^to p and also in a

plane containing

p. When

H //

p, the mole- cules will tend to lie

perpendicular

^{to the field when}

the

anisotropy

^{of the}

magnetic susceptibility, xa

^is

negative.

^The

principal

effect of the field in this case

is to suppress rotations in the

plane containing

p.

The form of the director in the presence of the fluc- tuations is :

where _ç ⁼

(p,,(z)

⁺

qJ’(r, t),

^with

ç’

^{the small fluctua-}

tion. The

only

fluctuations in _{nz that} are

slowly varying

and

hydrodynamic

^are those due to a

splay

^distor-

tion of the

pitch

^vector p. Bend and twist distortions of _p lead to finite

energies

in the limit of

vanishing spatial gradients,

and therefore fluctuations asso-

ciated with these distortions can be

neglected

^{in a}

hydrodynamic

treatment. This

point

^has been discuss- ed in detail elsewhere

[6, 7, 8].

Since the fluctuations in _nz are determined

by

^{those in} ç,

only

^the

single

variable

ç’

is needed to describe

hydrodynamic

fluctuations in orientation. In the _presence of both

a

temperature gradient

^{and the field}

H,

the contant À in

(2)

^{can be shown to be}

given by

where

kil

^and

k33

^{are the Frank}

[9]

^{elastic constants}

for

splay

^and

bend, respectively.

The

hydrodynamical equations

in the standard

Boussinesq approximation [1]

^are,

following [6] :

where v is the fluid

velocity, ps

^{is the}

density

^{in the}

presence of the

temperature gradient, g

^{is the acce-}

leration due to

gravity,

^{and G is the} energy

density.

The fluxes and forces are

given by

In these

equations

p is the pressure and h is the vector

variationally conjugate

^to

Vç.

^{It can be shown that}

where

k22

^{is the Frank constant for}

twist,

^and

with _X. o..

The

dissipative

parts ^of

(8), (9), (10)

^are

given by

kl

(Kjj)

^{is the thermal}

conductivity perpendicular (parallel)

^{to the} helical axis. We will use the nota-

tion xa = KII

^{- xl.}

Finally

y has units of a

viscosity

^{and is}

usually

^{called the}

director friction coefficient. In these

equations

^we

have

ignored

^a

possible coupling

^between orientation and the temperature

gradient

^discussed

by

^Leslie

[10] ;

this term is also

ignored

ⁱⁿ

[4]

^{where it is}

argued

that it is small.

2.2 STATIC SOLUTION. ^- A

possible

^{solution to}

these

equations

^{when the}

temperature gradient

^is

small

enough

^{is the}

equilibrium

^{situation where}

convection is absent :

We will assume the

following equation

^{of state}

dp = px dp - pp dT (18)

where _x is the isothermal

compressibility

^and

fl

^is

the volume

expansion

coefficient :

For thin

layers

^{we will have}

where po, po ^are constants.

Finally,

ⁱⁿ

equilibrium

we must have

Now

where we

ignore

the pressure

dependence

^{of the}

pitch.

^The

following thermodynamic identity applies :

iôh. ^{Ô T a,Z}

= ^- 1 .

(24)

(Ghz) ( OT) (ÔlfJ’J

^{oT ({J,z}

^{a lfJ,z}

^hz

^{ah z}

^{T = - 1 .}

⁽²⁴⁾

Using (12)

^and

noting

^that

(olfJ,z/oT)hz

⁼

^oqoloT,

(23)

^{leads to the}

equation

,

and

using (17)

^this

integrates

^to

, 2.3 LINEARIZED EQUATIONS. ^- Next we consider small fluctuations from the

equilibrium

^state

(17), (20), (21)

^and

(26)

and write _p ⁼ _Ps

+ p’,

p ⁼ Ps ⁺

p’,

T ⁼

Ts

⁺

^T’,

9 = ({Js ⁺

q’

and linearize the

hydro- dynamical equations

^{in the}

primed

variables and v.

For thin

layers

^we may

replace

p. ^s

by po in

^{the first}

term in

(5).

The second term is

approximately

po

gflT ’

and afl viscosities and thermal conductivities are

assumed

independent

^of

temperature.

These assump- tions form the basis of the

Boussinesq approxi-

mation

[1].

^In

addition,

^the

problem only

^seems

tractable for the case where the cholesteric

pitch depends weakly

^on

temperature.

^Provided

the

quantity dCPsldz

^{may be}

replaced by

qo in

(3), (8), (10)

^and

(13),

^{and the Frank} elastic constants, qo, and

oqolor

^{may all be}

regarded

^{as constants.}

Finally,

^{we will}

simplify

the calculation

by assuming

that all of the

fluctuating quantities depend

upon ^z and one of the transverse coordinates

only,

say ^x.

Thus our convection cells will be one dimensional rolls. It can be shown that vy ^{= 0} and then the _equa- tions reduce ^{to :}

1366

In

(30),

_cp ^{is the}

specific

^heat at constant pressure.

3. Normal modes near the

instability.

^{- 3.1 DIS-}

PERSION RELATION. ^- We look for solutions of the

equations

^{of the}

following special

^{form :}

q

with similar

expansions

^{for v,}

p’

^and

({J’.

ql is

quasi-

continuous in

(32)

^{but we} will limit _q3 to the values q3 ⁼

nnll

where n is a nonzero

integer.

These values of _q3 can be shown to lead to solutions

satisfying q’

^{- T’ =} Vz ⁼

èvxlèz

^{= 0 at} both boundaries. This is also the

assumption

^{used in}

[4]. Rigid boundary

conditions

where vx

^{= 0 at} both boundaries

requires

numerical solution even for

simple

^fluids

[11]

^since

the

expansion (32)

^{is not}

appropriate

^{for this case.}

Inserting (32)

^into

(27) through (31)

^{leads to five}

linear

homogeneous equations

^{for the}

amplitudes

vq,

Tq,

^({Jq, pq. A nontrivial solution can exist

only

if the determinant of the coefficients vanishes. This leads to a

dispersion

relation between the

frequencies

of the modes and their wavevectors :

where the

following

^{notation has} been introduced :

and ql + q23

⁼

q2.

^{The rest} of this paper will be concemed with

approximate

solutions of the

disper-

sion relation

(33).

First,

suppose that the

temperature gradient

^vani-

shes. An

approximate

solution in the case where

dissipative

^{terms are} small is :

where

1- -1

with _{ql = q} sin

0,

q3 = q ^cos 0. This is the propa-

gating

shear-orientation mode discussed in

[6].

^Notice

that the

phase velocity ct(O)

vanishes for

propagation along

^the helical axis. In this case the shear and orientation modes are

uncoupled

^{and both are}

overdamped [12].

^{The solution}

(37)

^is

only

^{valid for}

small _q. Corrections involve

dissipative

^{terms which}

occur at

higher

^{order in} q. If these terms are small the solution to next order is

where

For most cholesterics it is true that

Then the

damping

^factor

(40)

is dominated

by

^the

viscous term. The other solution to

(33)

^when

A T/1

^{= 0}

is the

overdamped

thermal mode

3.2 STEADY CONVECTION.

ôqolôt

^{= 0. - Now}

let us look for solutions to

(33)

^{when the}

temperature gradient

^{is near} the critical

value, ignoring

^the

ôqo/ôT

^{term as in}

[4].

^{We know that near the insta-}

bility

^one of the solutions for ^co is _very small. At the critical value of

AT/1

^it

vanishes,

^{hence we can find}

the critical value

by setting

^{co = 0 and}

solvifig (33)

for

A7Y/.

^{We make the}

assumption

^that

in addition to

(41).

Under these conditions one finds for the critical value

The true threshold is found

by minimizing (44)

^with respect ^{to q.}

AT/1

^{will be} smallest when _q3 assumes its smallest

possible

^value

nll

⁼

Q. Minimizing (44)

with respect ^{to ql, it is found that}

AT11

^{attains its}

smallest value when ql = qc where

The minimum value is

where

The critical

temperature gradient

^{can be}

positive

^or

negative depending

upon the

sign

^of xa. We get ^an

instability by heating

from below when

and an

instability by heating

from above when xa > 0

(K

^jj >

KJ.).

This unusual behaviour also occurs

in a nematic but there the critical

temperature gradient

is

proportional

^to

Q4

rather than to

Q 2 q2.

^{Thus the}

instability

threshold is smaller in a nematic and it has been observed in

samples

^{submitted to a}

tempe-

rature difference between boundaries of

only

^{a few}

degrees [13]

^{for 1} c>i 1 mm. For

cholesterics,

^{in order}

to

keep

^AT within the few

degrees required by

^the

Boussinesq approximation,

^we will need thick

samples

with

relatively large

cholesteric

pitches.

^{We note that}

although

^a

negative

^critical

temperature gradient

^is

unusual,

^{it is not confined to}

liquid crystals.

^In

binary

fluids one can get ^an

instability by heating

^{from the}

top

when the thermal diffusion coefficient is

negative [14].

The above results _may be

compared

^{to those in}

[4]

with H ⁼ 0

by setting kl(H)

^{= 0 in}

(34).

^However

then we must include the next order term in _q1. Accord-

ing

^to

[4]

^{this term is}

(3 k33/8) q4lq’

^{and is}

negligible compared

^{to the term}

k_L(H) q2 1

^for fields which

satisfy

when H = 0 the critical value

of ql

^becomes

and since

Àqo

^{= 1 when H =}

0,

this leads to

which is the result in

[4].

^{The critical}

wavevector qc

(the

wavevector of the first mode to become

unstable)

is now much greater than before

(see

eq.

(45)).

^Thus

the size of the convection cells increases with the field H. There is a

simple physical explanation

^for

this. The field ^H increases the energy necessary ^to distort the

layers.

^{Since the available} energy ^comes from the

bouyancy

forces which are

independent

of

H,

^{we must have less} distortion as H increases so

that the cell size

qc-1

increases with H. From

(46),

it is seen that

(AT11).i.

also increases with

H,

^as would be

expected

^{in a}

parallel

field that stabilizes the helical structure.

For

temperature gradients

^close to, ^but

slightly

below the critical

value,

^{one of the} roots of

(33)

will be small. To find that root, ^linearize

(33)

^{in w.}

With

oqoloT

⁼

^0,

^{and the}

approximations (40), (43)

and

(48),

^{it is}

straightforward

^to show that the solu- tion is :

where

when

AT/1 = 0, R

^{= 0 and}

(51)

^{reduces to}

(42),

an

overdamped

thermal mode. However at the cri- tical value we have

R/Rc(q)

^{= 1 and the}

frequency

of the mode vanishes. We _say the mode _goes

soft

^{at the}

critical value of R. The form

(51)

is similar to what

happens

^{in an}

ordinary

fluid but the critical

tempe-

rature

gradient

^is

différent ;

^{note the}

striking

^absence

of the

viscosity

^{v from}

(52).

^{Also near the critical}

point (51)

^{is not a} pure thermal mode because the fluctuations in

temperature

^are

strongly coupled

^to

those in orientation. From

(30)

^and

(31)

^{it is}

possible

to

show,

^{when co - 0 :}

where

Tq

^{is the}

^amplitude

^{of the}

^divergent

^thermal

fluctuations. It is also

possible

^{to show that the other}

solutions to

(33)

^{are well behaved near the}

instability ;

the

frequency

of the shear-orientation mode is

changed only slightly

^from

(39).

3 . 3 OSCILLATING CONVECTION.

ôqolê T :0

^{0. - Next}

we consider

(33)

^when

oqoloT ¥=

^{0. Since the term}

in

AT/1

^is

complex

it follows that the

frequency

^ce

must be

complex

also. Write ro ⁼ (j),

+ io)i

^{and use}

the fact that near the critical

point

coi will be small.

This suggests ^{that a} linearization of

(33)

ⁱⁿ roi would be

appropriate. Separating

^the

equation

^{into real}

and

imaginary parts :

1368

At the

instability

^threshold Wi ⁼ 0 and

(54)

^and

(55)

^{become two}

coupled equations

^for Wc and

i1TII.

^If

K a

Vq 2 «

Po c

kq6

^{one obtains the}

following

^{solution to} lowest order in

(ôqo/ôT) :

The

principal

effect of the

temperature dependence

of the

pitch

^{is to}

give

^{a real}

part

^{to the}

frequency

^of

the

soft

thermal-orientation mode. To first order in

(ôqo/ôT),

^the

imaginary

part

of o)i

^{is still}

given by (51).

^{It vanishes at R =}

Rc(q),

^whereas

(56) clearly

remains finite. The convection cells are

oscillating

rather than

steady.

^{The mechanism}

leading

^{to the}

oscillation is shown in

figure

^{1 for the case where}

the

pitch

decreases with

temperature

^and

Ka 0 (this

^{is the usual case in}

cholesterics).

^{The bottom}

plate

is hotter than the

top

^{so that the}

spacing

^{of the cho-}

lesteric

layers (the

^{surfaces of constant}

phase)

^increases

from bottom to

top.

^{Consider a} distortion like that in

figure

^{1. It}

requires

^{more elastic} energy ^to distort the

layers

^near the bottom because the local value of _qo ⁼

(dPs/dz)

^is

larger

^there

(see

^eq.

(26)).

^For

xa 0 the heat flux will be deviated

along

^the

layers

because the thermal

conductivity

^is

largest

^{in this}

FIG. 1. ^- Horizontal cholesteric subjected ^{to a} temperature gra- dient. When xa ⁰ the heat flux is deviated along ^the layers ^and there _appear warmer (w) and cooler (c) regions. ^The gravity ^forces

induce opposite velocities in regions (w) ^and (c). ^For ôqo/ôT > 0 the layers ^{near the} top ^{distort more} than those near the bottom.

direction. This causes lateral warmer

(w)

^{and cooler}

(c) regions.

^The

bouyancy

forces induce

opposite hydrodynamic

velocities in the

regions (w)

^and

(c)

and one can see from

figure

¹ that this will tend to distort the

layers

^{more. This is the}

origin

^{of the insta-}

bility,

^as discussed in reference

[4].

^The

instability

first appears when the

bouyancy

^{forces are}

large enough

^{the overcome the} elastic forces of the

layers

near the

top. However,

^{near the}

bottom,

^more elastic energy is

required

^than is available from the

bouyancy

forces. There will be an

exchange

^of energy between the

bouyancy

^forces

giving

^{rise to} convection

(kinetic energy)

and the elastic forces of the cholesteric

(potential energy).

This leads to an oscillation with the

period being

determined

by

^the

restoring

^force

of the bottom

layers,

^and

by

^the time it takes for

an elastic distortion to

develop

^{under an}

applied temperature gradient.

^{It is}

possible

^to

develop

^an

analogy

^{with a}

simple

^mass

spring

system. ^From

(26)

the effective value of _qo at each

point

in space is :

The différence in elastic

energies

^{between that at}

the bottom

plate

^{and that at a}

point (z)

^{in thé fluid}

is then

This has the same form as the

potential

energy of a

linear

spring.

^The

restoring

^force

(and

^{hence the}

frequency

^{of the}

oscillation)

will increase with

ATIL

and

(8qo/8T).

Let us estimate the order of

magnitude

^{of the}

oscillating frequency (56). Suppose

^that

this

corresponding

^{to a 1}

% change

ⁱⁿ

pitch

per

degree.

With

fipo g -- 1, (KI Po Cp) ^{10-3 (these}

^{values are}

typical for

^most

^{organic liquids), k22 10-6,}

qo

103

we get

where all

quantities

^are in cgs units and

N(H)

^is

given by

For 1 ^-

10-1

_cm,

(60) gives

co, --

10-1 N(H) (rad/s).

For Ka

⁰

(the

^{usual case in}

cholesterics), N(H)

will be of order

unity except

^{when H is close to a}

cri tic al field He :

For H

Hc, N(H)

^becomes

large

^{and the} oscillation

frequency

will reach a maximum. In

principal,

^the

oscillations could be observed

by placing

^{a thermo-}

couple

^{in the} system,

provided

^that this does not

appreciably

distort the orientation.

Oscillating

^tem- perature fluctuations would then be observed at the

frequency

úJr ^near the critical

point.

The effect of the

temperature dependence

^{of the}

pitch

^{is to lower the threshold}

(57) slightly. ,Using

the same estimates as before we find that the

ôq,IôT

term in

(57) represents

^{about a 1}

%

correction. For H -

Hc

this correction can be

considerably larger,

however.

It is not difficult to show that the shear-orientation mode

(39)

^is

only slightly

^affected

by

^the

instability.

To first order in

ôqo/ôT we

^find

that ct

^{and T are}

modified to :

where

co)

^is

given by (38).

^The

damping

^{factor is}

decreased when

ATIL

^and

ôqolôt

^{have the same}

sign,

and increased when

they

^have

opposite signs.

4. Discussion. ^- We have

presented

^{a treatment}

of the

problem

^{of a} horizontal cholesteric fluid

layer subject

^{to an external}

force resulting

from the combi- nation of a vertical temperature

gradient

^and

gravity.

We

have

investigated

^the

dynamics

^{of the} system which evolves towards

instability

when the extemal force increases _up to a critical value.

An

important

feature of the

temperature-indepen-

dent

pitch problem

^{is that the}

frequency

^{of one of}

the normal modes of the system ^{vanishes at the critical}

temperature gradient (a

^soft

mode).

^{This means the}

relaxation time associated with random fluctuations

of the mode tends towards

infinity

^{as the}

instability

is

approached

from below. Near

enough

^{to the insta-}

bility

this slow mode will dominate the

hydrodyna-

mical _response of the fluid because all of the other modes of the system

decay

^{much faster. In other}

words,

^{if an}

arbitrary

^extemal

perturbation

^is

applied

to the

fluid,

^the

dynamical

response function will contain several terms for each of the different

possible

modes of the system. ^{As the critical}

point

^is

approached (R - Rc)’

^{one of the terms}

(that corresponding

^to

the soft

mode)

^will

diverge

^as

(Rc - R ) -1

^{while the}

other terms will remain finite at R ⁼

Rc.

An infinite response function indicates ^{that the} system is unstable

against

fluctuations of the soft

mode,

^{and will}

undergo

a transition to a new stable state ^- in this case a state

representing steady

convection.

This behaviour is somewhat similar to what occurs near the critical

point

^{in an}

equilibrium phase

^tran-

sition. It is well known

[15]

that there is a considerable

slowing

down of certain

thermodynamic

fluctuations

near the critical

temperature Tc.

^{One or more of the}

modes _goes soft at a critical

wavevector qc;

^i.e.

w(qc)

^{= 0 at T =}

Tc.

^The system becomes unstable

against

^the excitation of these modes and the static response function

(which

^is

usually

^{the w = 0 limit}

of the

dynamical

response

function) diverges.

^The

soft mode is the one that is

predominantly

^excited

and we say that the system condenses into this soft mode. The

divergent

response function indicates an

instability

^{and the} system evolves into a new stable state with a

qualitatively

different symmetry

(a phase transition).

Of course the nature of the

soft

^mode near a convec-

tive

instability

will differ from system ^to system, ^as in the case of

phase

transitions. In a

simple

^{fluid it is}

essentially

^the thermal conduction mode that is affected. Both of the

overdamped

^{shear waves are}

approximately uncoupled

and well behaved at R ⁼

Rc.

In the case of a nematic lined _up

parallel

^{to the boun-}

daries it can be shown

[16]

^that the director

mode nz

is the one to go

soft ;

^the thermal mode is finite at R ⁼

Rc. Finally,

^{we have seen in the case} of a choleste- ric that a

coupled

thermal-orientation mode is affected while the other modes are finite at R ⁼

R .

In the

appendix

^a

general

argument ^is

given

^which

shows that the nature of the soft mode is determined

by

the slowest mode of the

unperturbed

system.

When the

temperature dependence

^{of the}

pitch

^is

taken into account

only

^the

imaginary part

^{of the} mode

frequency vanishes;

^{the real} part ^remains finite at the critical

point R

⁼

Rc.

^This

implies

that the convection cells are

oscillating

rather than

steady,

^{and that a}

propagating

thermal conduction mode can exist in a cholesteric under an extemal

temperature gradient. Oscillating

convection cells also occur in certain

binary

^fluids

[14],

^{and in}

general

viscoelastic fluids

[17]

^{under certain} conditions.

Finally,

^our calculation is

only

valid for a choles- teric whose

pitch

^is

weakly dependent

upon tempe-

1370

rature

(oqoloT small).

^{The case of the}

pitch strongly dependent

upon

temperature

^will

require

^{a numerical}

solution and will

probably

^{lead to} very

complex stability diagrams

which will

depend

^{on the form of}

the function

qo(T).

APPENDIX

In the _presence of the

temperature gradient

^the

mode structure of the system ^can be written as

The number of modes N

equals

the number of inde-

pendent hydrodynamical

variables of the system.

F is the external force which is

proportional to AT/1.

If F ⁼

0,

^{the N} solutions of

(A. .1)

^{are w}

= iii,

^and

these are the normal modes of the system.

They

may be

propagating

^or

overdamped depending

^{on whether}

the

Qn

^are

real, complex,

^or

imaginary.

The threshold is

given by

Suppose

^{that one of the}

Qm

^say

Qg

is much smaller than any of the others for the

appropriate q

^which

is determined

by

^the

geometry

^{of the} system. ^Since

we know that OJ --+ 0 at the threshold for

steady convection,

^{we can write}

(A. 1)

^as

The soft mode is

given by

For Q.

^real

(the

^usual

case), (A. 4)

^{describes an over-}

damped

^{mode which goes soft at} the threshold when

(A. 2)

is satisfied. If it so

happens

^that

Qs

^is

imaginary,

the soft mode will be a

propagating

^{one and since}

propagating

^{modes occur in}

pairs (A. 4)

^{will be}

replaced by

so that the critical exponent ^is

now 1/2

^{rather than 1 for}

the

overdamped

^case.

In a

simple

^{fluid there are} five modes : two sound waves, two

overdamped

^{shear waves, and one over-}

damped

thermal conduction mode.

Except

^for

liquid

metals it is true that

(x/po cp) « v/po

^{so we have}

Qr,

⁼

kq2/ po _cp,

^{and the} soft mode is a thermal conduction mode. Next consider a nematic

initially

lined _up

parallel

^{to the}

plates.

^{There are in addition}

two

overdamped

director modes of the form

Qd kq2/y,

^but

only

^{one of these is}

coupled

^into

thé

problem

^for

parallel alignment.

^Since

kly *« x/po

cp

we have

Qs

⁼

Qd

and the director mode is the one to go

soft;

^{the thermal} conduction mode remains finite at the threshold.

Finally,

consider the case of a

cholesteric. The director mode is

strongly coupled

to the shear mode

corresponding

^{to a reactive}

exchange

of _energy between mass motion and the twist elastic forces of the helix

(see

^eq.

(37)).

^{In this case the thermal}

mode has the lowest

frequency

^{so that the soft mode}

is

again

the thermal conduction mode.

References [1] See e.g. CHANDRASEKHAR, S., Hydrodynamic ^and Hydroma-

gnetic Stability (Clarendon, Oxford) 1961, Chap. ^2.

[2] LEKKERKERKER, ^{N. and} BOON, ^J. P., Phys. ^{Rev. A 10} (1974)

1355.

[3] THOMAS, H., ^{IEEE Trans.} Magn. ⁵ (1969) ^874.

[4] DUBOIS-VIOLETTE, E., ^J. Physique ³⁴ (1973) ^107.

[5] DUBOIS-VIOLETTE, E., ^{C. R. Hebd.} Séan. Acad. Sci. 273B

(1971) 923.

[6] LUBENSKY, T. C., ^Mol. Cryst. Liq. Cryst. ²³ (1973) ^99.

[7] LUBENSKY, ^T. C., Phys. ^{Rev. A} 6 (1972) ^452.

[8] MARTIN, ^P. C., PARODI, ^{O. and} PERSHAN, ^P. S., Phys. ^Rev.

A 6 (1972) ^2401.

[9] FRANK, ^F. C., ^Discuss. Faraday ^{Soc. 25} (1958) ^19.

[10] LESLIE, ^F. M., Proc. R. Soc. A 307 (1968) 359; ^{Arch. Ration.}

Mech. Anal. 28 (1968) ^265.

[11] ^See e.g. MONIN, A. S. and YAGLOM, ^A. M., Statistical Fluid Mechanics (MIT, Cambridge, Mass.,1971) Vol.1, p.179 ^ff.

[12] FAN, C., KRAMER, ^{L. and} STEPHEN, M. J., Phys. ^{Rev. A 2} (1970) ^2482.

[13] GUYON, E., PIERANSKI, P., ^{C. R. Hebd. Séan. Acad. Sci. 274B} (1972) 656 ;

DUBOIS-VIOLETTE, E., GUYON, E., PIERANSKI, P., ^Mol. Cryst.

Liq. Cryst. ²⁶ (1974) ^193.

[14] SCHECHTER, ^R. S., PRIGOGINE, ^{I. and} HAMM, ^J. R., Phys. ^Fluids

15 (1972) ^379.

[15] SCHNEIDER, T., SRINIVASAN, ^{C. and} ENZ, C. P., Phys. ^{Rev. A 5} (1972) 1528.

[16] PARSONS, ^J. D., to be published.

[17] SOKOLOV, ^{M. and} TANNER, ^R. I., Phys. ^{Fluids 15} (1972) ^534.