Dynamics of the director alignment in cholesterics under applied magnetic field

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Dynamics of the director alignment in cholesterics under applied magnetic field

J. Marignan, G. Malet, O. Parodi

To cite this version:

J. Marignan, G. Malet, O. Parodi. Dynamics of the director alignment in cholesterics under applied magnetic field. Journal de Physique, 1976, 37 (4), pp.365-368. �10.1051/jphys:01976003704036500�.

�jpa-00208431�

DYNAMICS OF THE DIRECTOR ALIGNMENT

IN CHOLESTERICS UNDER APPLIED MAGNETIC FIELD

J.

MARIGNAN,

G. MALET and O. PARODI

Groupe

^de

dynamique

^des

phases

condensées

(*),

Laboratoire de

cristallographie

Université des sciences et

techniques

^du

Languedoc, place Eugène-Bataillon,

³⁴⁰⁶⁰

Montpellier,

^France

(Rep

^{le 26}

septembre

^1975,

accepté

^{le 25 novembre}

1975)

Résumé. ²⁰¹⁴ Nous étudions, ^au moyen des théories élastiques ^et

hydrodynamiques,

^{le mouvement}

de rotation des molécules, provoqué par l’action d’un champ

magnétique

relativement faible sur un

échantillon

cholestérique

d’épaisseur déterminée avec ancrage fort aux parois. La solution est une

fonction sinusoidale simple, ^dont l’amplitude

dépendant

^du temps ^est caractérisée par ^un temps de relaxation de l’ordre de 10-1 s.

Abstract. ²⁰¹⁴ By ^means of static and hydrodynamic theories, ^we study ^the rotary ^{motion of the} molecules in a cholesteric slab (of given thickness and strong ^anchored surfaces),

subject

^{to a weak} magnetic field. The tilt angle ~(z, t) of the molecules is a simple ^{sine wave with a time}

dependent

amplitude characterized by ^a relaxation time of the order of 10-1 s.

Classification

Physics ^Abstracts

7.130

1. Introduction. ^- In a recent letter

[ 1 ],

^{we have}

briefly presented

^{the main}

experimental

and theoretical results

concerning

^the

dynamics

^{of the first}

Grandjean-

Cano line under weak

magnetic

fields. The theoretical

approach

^{to this}

problem

^{was based on}

dividing

^the

dynamical study

^{in two} parts :

alignment

of director without _any motion of the

line,

^and

then,

translation of the line.

The _purpose of this _{paper is} to present ^{in a more}

complete

form the theoretical

analysis

of the first

point (director adjustment).

In absence of _any

perturbing forces,

the cholesteric

mesophase

possesses ^a

typical

^helical structure. The

possibility

^of

untwisting

the cholesteric

spiral by

^a

magnetic

^{field H was} first noted

by

^Frank

[2].

^The

coupling

^between

susceptibility

xa

(xa

>

0)

^{and a static}

magnetic

field normal to the helical

axis,

^{tends to} unwind the

spiral

structure,

and,

^{as a} consequence, the

spatial period

increases. When the

magnetic

^field

reaches a critical value

Hc,

the cholesteric

liquid crystal

^{takes a nematic}

configuration.

This cholesteric- nematic transition has been first observed

by

^Sack-

mann et al.

[3].

^The variation of the

pitch

^with magne- tic field has been measured in a few cases

[4, 5].

^A

static

theory

of this effect has been

given by

De Gennes

[6]

^and

Meyer [7].

The continuum

theory

was

completed by hydrodynamical

^{studies on fluctua-}

tions of cholesterics in a static

magnetic

^field

[8, 9].

An

experimental

and theoretical

study

^has

already

been

performed by

^Prost

[10]

^on cholesteric

drops,

(*) Laboratoire ^{associe au C.N.R.S.}

i.e. in the case where one of the

limiting

surfaces is a

free-surface.

In this _paper we

study

^{the case of a} cholesteric film

placed

^{between two treated}

surfaces,

^with strong

anchoring.

^{Hence our}

analysis

and results are

quite

different from those obtained

by

^Prost.

In section 2 of the present paper, ^we rework the static

theory by introducing

^a cholesteric slab of finit thickness and strong

anchoring

^{at walls. In section 3}

we present, ^{for twist}

mode,

^a very

simple

solution of motion

equation

of the director under weak

magnetic

fields

(H Hc).

2. Static

theory.

^{- Let us start with a} cholesteric

sample

^of thickness d ⁼

Po/2 (Po being

the unper- turbed

pitch).

^{We assume}

that,

^at the walls there is one

easy direction for the

molecules, parallel

^{to the x-axis}

and that strong

anchoring prevails (Fig. 1).

^We

consider the case where the static

magnetic

^{field is}

along

^{the ox}

direction;

the helical axis oz is then normal to the

applied

field. This is a situation of _pure twist with the

following

components ^{of the director n :}

FIG. 1. ^- Cholesteric slab of thickness d, under static magnetic field parallel ^{to the} easy axis and normal to the helical axis oz.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003704036500

366

Q(z)

^{is the}

angle

^{between n} and the x-axis. The boun-

dary

conditions are Q ⁼ 0 for z ⁼ 0 and _T = 1t

for z ⁼ d.

The Frank free _energy

(per

^{unit area in the} xoy

plane),

^can be written in the _presence of the field

[6] :

K2

^{is the} twist elastic constant, qo the wave vector

corresponding

^{to H =}

0, (qo

^{= 2}

nIPo).

2.1 EQUILIBRIUM CONFORMATION OF CHOIESTERIC.

-

Minimizing

^{the free} energy, eq.

(2 .1 ),

ⁱⁿ

Q(z),

^one

finds the

equilibrium

condition :

where ç = (K2IXa)1/2 jH

^{is the}

magnetic

^coherence

length.

Multiplying by

²

dcpjdz,

^and

integrating,

^{one finds}

the

following

^first

integral :

where C is a constant

integration.

Introducing ql =

Q ^-

(n/2)

^{and o}

we

finally

write the first

integral

^of eq.

Recalling

^that

and

integrating

eq.

(2.3),

^one

gets

^the

following implicit equation

for the modulus k :

where

K(k)

^{is the}

complete elliptic integral

of the first kind

[11].

The

integral

^of eq.

(2.3)

^{can be}

expressed

^{as :}

where

F( t/J, k)

is the normal

elliptic integral

of the first kind. The constant zo is fixed

by

^the

boundary

^condi-

tions. We

finally find, making

^use

of eq. (2.4) :

where am

(u, k)

^{is a Jacobian}

elliptic

function of argument ^u and modulus k.

2.2 EQUILIBRIUM FREE ENERGY. ^-

Taking again

§

= qJ

-, n/2

^and

using

eq.

(2.3)

^one

finds,

^after

integration of eq. (2 .1 )

Or, introducing

^the

complete elliptic integral

^{of the}

second kind

E(k) :

2.3 WEAK MAGNETIC FIELD APPROXIMATION. ^-

Under weak

magnetic fields, ç

^{- oo and k - 0. We}

can then use the

expansion

^of

K(k), neglecting

^{all terms}

of order

higher

^than

^{k3 :}

and _eq.

(2.4)

^becomes

which leads to :

and

finally :

Define now u as the argument ^{of the am function} in the r.h.s. of _eq.

(2.5) :

Then,

^from eq.

(2.4),

^{one has}

Introduce the Jacobi’s _{nome q :}

One can now

expand

^am

(u, k)

in Fourier series :

Finally making

^use of the four _eq.

(2. 10)

^and

(2. .11 ),

eq.

(2. 5)

^{becomes :}

-

In the

expression

^{for the}

free-energy (2.6),

^{one can}

expand E(k),

Using

^{now the}

developments of k - 2 and k -1

from _eq.

(2. 8),

^{one finds :}

Eq. (2. 6)

then reduces to :

It must be noted that the same results could have been found

by inserting directly

into eq.

(2.1)

^the

expression

^for Q

given by

eq.

(2.12).

3.

Hydrodynamic theory

under weak

magnetic

^field.

- Let us now consider the case where a

magnetic

field H is

applied

^{at time t = 0. At t = 0} the helical structure is still

undistorted, corresponding

^{to the}

wave vector qo ⁼

n/d.

^The

equilibrium

^distortion

under field H is

given by

eq.

(2.12).

Introducing

^{a time}

dependent amplitude a(t)

^{of the}

simple

^sine wave, ^we then can choose :

with the

help

^of eq.

(3.1),

eq.

(2.1)

^is

readily integrated

and one gets : ^.

Jl(2 a(t))

^{is the Bessel} function of first kind of order 1.

Let us assume that the cholesteric

liquid crystal

^is

incompressible

and that all friction processes in the fluid are isothermal. The

dissipation ^Tg resulting

from these _processes is

given by

the fundamental

equation of nematodynamics [12] :

A is the shear ^rate

tensor, a’

^the

symmetric

part ^{of the} viscous stress tensor, ^h the molecular

field,

^{N the}

velocity

of the director relative to the fluid.

It can be shown from the Leslie

equations

^{that for}

pure twist there is no

hydrodynamic

backflow. Then the tensor A

drops

^out and the evolution

equation

of the director

gives :

where yi

^{is a} Leslie friction coefficient and N ⁼

dn/dt.

The entropy ^source

TS

is therefore

Using

^now eq.

(3.1),

^one gets

The

dissipation TS

is

equal

^to the decrease in stored

free-energy

The time derivative of

F(t)

^{is from} eq.

(3.2) :

Replacing J1’(2 a) by Jo(2 a) - ( 1 /2 a) J1(2 a),

^and

expanding

the Bessel functions

Jo

^and

J1

^and

neglect- ing

^{terms in}

a2 (a2 1),

^{one finds :}

We now substitute _eq.

(3.3)

^and

(3.5)

^into eq.

(3.4),

and obtain :

Taking

^{now into account the}

boundary

^condition

a(O)

⁼

0,

^{one finds :}

where the relaxation time ⁱ is

given by

368

The solution

a(t) gives

^the

expected equilibrium

value for the tilt

angle Q(z, t).

Note that this relaxation time is

quite typical

^of

such processes. A similar

expression

^{had been}

already

found

by Fan,

^{Kramer and}

Stephen

^and

by

^Prost.

4. Conclusion. ^- In the

experiments reported

^{in a}

previous

paper

[1]

^{we have used} cholesterol cinna- mate diluted in an

equimolar

mixture of

4-methoxy 4’-pentyl

tolane and

4-propoxy-4’-heptyl

^{tolane. For}

such a

sample

^{one has the}

following

^{values :} y 1 ~ 2

poises (1), d ~

^{10 pm and}

^{K2 - 5 x}

^10-7

^{dyne [13] ;}

then the relaxation time i would be of order 10 -1 s.

In this

previous

paper ^on the motion of

Grandjean-

Cano lines under weak

magnetic ^fields,

^{our basic}

assumption

^{was that} the relaxation time for the director

alignment

^was much shorter than the relaxa- tion times for the lines motion.

This last relaxation was found to be of order

102

s.

We therefore find that our

assumption

^was

completely justified.

An

experimental

confirmation of the present ^theo- retical results is not easy ^to find. As far as the

pitch

^is

much greater ^{than the}

wavelength,

^{and the}

magnetic

(1) Achard, ^M. F., Gasparoux, H., private communication.

distortion

small,

^the so-called adiabatic

approximation

for

light propagation along

^the helical axis holds.

This means that a small

magnetic

distortion has no

effect on the

light polarisation

^{at the} exit-face. On the other

hand, light

refraction

by

^a cholesteric

prism

should be very sensitive ^{to a} distortion of the helical structure. An

experimental study relying

^{on this idea}

is to be done in our

laboratory.

It is worth

finally

^to

point

^{out that our results are}

quite

^different from those found

by

^{Prost. This is}

essentially

^{due to} the difference in

boundary

^condi-

tions. In Prost case, the _presence of a free surface allowed for the _escape of free walls. Hence a critical field

appeared.

^{In our} case, the presence of ^a strong

anchoring

^{on both}

plates

forbids such an escape, and this is the reason

why

the nematic-cholesteric transi- tion cannot take

place

^{in a}

perfect sample.

^{Of course,}

in real

samples,

^{for very}

high magnetic fields,

^this

constraint should be relaxed

by

the appearance of disclination lines which would thus allow the transi- tion to occur.

Acknowledgments.

^{- It is a}

pleasure

^{for us to thank}

here Drs M. F. Achard and H.

Gasparoux

^{who have}

kindly

measured for us the twist viscous coefficient _{y 1.}

References

[1] MALET, G., MARIGNAN, ^{J. and} PARODI, O., J. Physique ^Lett.

36 (1975) L-317.

[2] FRANK, ^{F. C., Discuss.} Faraday ^{Soc. 25} (1958) ^1.

[3] SACKMANN, E., MEIBOOM, ^{S. and} SNYDER, ^L. C., ^{J. Am. Chem.}

Soc. 89 (1967) ^5982.

[4] DURAND, G., LÉGER, L., RONDELEZ, ^{F. and} VEYSSIÉ, M., Phys. Rev. Lett. 22 (1969) ^227.

[5] MEYER, ^R. B., Appl. Phys. ^{Lett. 14} (1969) ^208.

[6] ^DE GENNES, ^P. G., Solid State Commun. 6 (1968) ^163.

[7] MEYER, ^R. B., Appl. Phys. ^{Lett. 12} (1968) ^281.

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

2482.

[9] PARSON, ^{J. D. and} HAYES, C. F., Phys. ^{Rev. A 9} (1974) ^2652.

[10] PROST, J., Thèse, ^Bordeaux (1973).

[11] Our notations about elliptic integrals ^are from : Handbook of elliptic integrals ^for engineers ^and physicists. By BYRD, ^{P. F. and} FRIEDMANN, ^{M. D.} (Ed. Springer-Berlin)

1971.

[12] ^DE GENNES, ^P. G., ^The physics of Liquid Crystals (Clarendon

Press. Oxford) ^1974.

[13] MARIGNAN, J., MALET, ^{G. and} PARODI, O., ^J. Phys. (to ^be published).