On the possibility of generalized Freedericksz transition in nematics

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On the possibility of generalized Freedericksz transition in nematics

U.D. Kini

To cite this version:

U.D. Kini. On the possibility of generalized Freedericksz transition in nematics. Journal de Physique, 1986, 47 (4), pp.693-700. �10.1051/jphys:01986004704069300�. �jpa-00210250�

On the possibility ^of generalized Freedericksz transition in nematics

U. D. Kini

Raman Research Institute, Bangalore ⁵⁶⁰ 080, ^India (Reçu ^{le 30} septembre 1985, accepté le 28 novembre 1985)

Résumé. ^- Dans un article récent, Lonberg ^et Meyer ^{ont décrit une nouvelle} forme de la transition de Freede- ricksz dans des polymères nématiques ^{soumis à un} champ magnétique ^et l’ont décrite en utilisant la théorie de l’élasticité de Frank et Oseen. Nous utilisons l’approche ^de Lonberg ^et Meyer pour étudier la possibilité d’existence de ce nouvel effet Freedericksz dans les cas d’orientation oblique du directeur ^ou du champ magnétique ^et pour différents rapports ^des constantes élastiques. ^{Les résultats sont} pertinents pour certains polymères nématiques ^et

aussi pour des nématiques qui présentent également ^une phase smectique.

Abstract. ^- In a recent paper, Lonberg ^and Meyer ^have reported ^{a new form of} Freedericksz transition in polymer

nematics under the action of a magnetic ^{field and have described it} using ^the Oseen-Frank elasticity theory. ^In

this communication, using ^the approach ^of Lonberg ^and Meyer ^the possibility ^{of the} occurrence of the new Free- dericksz effect is studied for oblique ^director orientation, oblique magnetic ^{field and for} different ratios of elastic constants. The results _{may be} relevant to the case of certain polymer ^{nematics as also to the case} of nematics

exhibiting ^{a smectic} phase.

Classification Physics ^Abstracts

02.b0 ^- 36.20 ^- 61.30 ^- 62.20

1. Introduction.

The

anisotropic

^elastic behaviour of nematic

liquid crystals

^is well accounted for

by

^the Oseen-Frank

elasticity theory [1-2] (for

^{reviews on the}

elasticity theory

^{and for a more}

complete

^list of references see

[3-6]).

The average local orientation of the nematic molecules is described

by

^{a unit vector called the direc-}

tor. The elastic free energy of ^a nematic is described

by

^{three curvature elastic constants}

Ki, K2

^and

K3

which _are,

respectively,

^the

splay,

twist and bend elastic constants. The orientation of the nematic director can be controlled

by treating

surfaces with which the nematic comes into contact. Uniform orientation of the nematic director, no, between ^two flat

plates

^{is an}

equilibrium

^{state of the} system ^and

corresponds

^to minimum free energy. When ^a dis-

turbing

^{influence tends to create small}

changes

^{in the}

director orientation _away from the

uniformly aligned

state, elastic

restoring

torques ^{come into}

play

^and

tend to restore the

original,

^uniform

configuration.

The

anisotropy

ⁱⁿ

diamagnetic susceptibility,

xa, of nematics makes it

possible

^to disturb the uniform orientation of a nematic

by

^the

application

^{of a} magne- tic field

H ; xa

^{can be}

positive

^or

negative.

^{A nematic}

with Xa > ⁰ tends to

align

with the director

parallel

to H due to the torque ^{exerted on the molecules}

by

the field. This is, ^of course,

opposed by

^{the elastic}

restoring

torques. ^{When H is normal to} no the compe- tition between the elastic and

magnetic

torques ^is such that for field

strengths

^H

Hc,

^{no distortion}

occurs in the director field. When H >

Hc,

^{the director}

field gets ^{deformed as} the distorted state possesses lower total free energy than the uniform orientation.

He

^{is referred to as the} Freedericksz threshold and the transition is known as the Freedericksz transition.

If _Xa is known,

by choosing

^different no and different directions of H normal to no, different elastic constants can be evaluated. In

general, just

above the Freede- ricksz threshold, ^{i.e. in} the limit of small

amplitudes

^of

distortion the deformation is

practically

^{uniform in}

the

plane

^{of the}

sample.

When no is

parallel

^{to the}

plates

^{and H normal to}

the

plates,

^the transition which occurs with

increasing

field

strength

^{is known as the}

splay

Freedericksz transition. This is because

just

^above the threshold the deformation

approximates

^{to a}

splay

and is uniform in the

plane

^{of the}

sample. Recently, Lonberg

^and

Meyer [7]

studied the

splay

Freedericksz transition in

a

polymer

^nematic

(a

^racemic mixture of levo- and dextroversions of

PBG). They

discovered above threshold a

spatially periodic

structure, the

periodicity being

^{in the}

plane

^{of the}

sample.

^{Such a structure}

would involve a mixture of

splay

^{and twist} distortions.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004704069300

694

Using

the Oseen-Frank

theory Lonberg

^and Meyer showed that the threshold for the

periodic

^distortion

(PD)

^{is less than the}

splay

Freedericksz threshold because

K1 >

KZ

[8]

^{for the} system studied. ^The elastic free _energy of a pure

splay

deformation can be further reduced if the director _escapes out of the

(H, no) plane

^{via a twist} distortion. Indeed, ^it appears that the

periodic

^{structure is more} favourable than pure

splay

^for

K1/K2 >

^3.3

[7].

^The

importance

^of

this result is that when

K1

> 3.3

K2,

^{the usual} way of

measuring K 1

may ^no

longer

^{be feasible.}

In this communication, ^the

analysis of [7]

is extended to other cases. As

explained

ⁱⁿ

[7],

^a proper

approach

would be to consider non-linear

perturbations

^and

to prove that the free energy gets ^minimized

by

^the

distortion of the director field.

Unfortunately,

^this

results in a formidable set of

coupled

non-linear

partial

differential

equations.

^Hence,

following [7],

^the

possi- bility

^of occurrence of the PD is studied

by calculating

the linearized threshold

(HLM say)

^{for a}

given

^case

and

comparing HLM

^with

Hc, the

^usual Freedericksz threshold for that case.

Oblique

orientation of the

magnetic

^{field in a}

plane

^{normal to} no

(such

^{as that}

studied

by Deuling et

^al.

[9])

^{as well as}

oblique

^director

orientation at the boundaries are considered for different ratios of elastic constants.

2. Mathematical model, differential

equations

^and

boundary

conditions.

Consider a nematic confined between flat

plates

Z. ⁼ ± h and

uniformly

oriented in the _xy

plane

^such

that the director makes

angle

^{0 with x axis}

(Fig. 1).

^The

uniform director orientation is

A

magnetic

^field

is

applied

^{in a}

plane

^{normal to} no and

making angle 03C8

^{with the xz}

plane.

^Under the action of linear pertur- bations

0’(x,

y,

z)

^and

cp’(x, y, z)

the director field becomes

To second order in director

gradients

^{the total} free energy

density

^is

where

0’.,

⁼

^00’/Dx

^etc. The free energy

density

^{is minimized with} respect ^{to 0’ and}

(p’

^{and the}

Euler-Lagrange

equations

that result ^are

In a realistic situation it is known that the

anchoring

energy of the director at the

boundary

^{is finite and this}

has to be

explicitly

^taken into account,

especially

for

oblique

^director orientation at the boundaries.

However, ^{for a}

preliminary

calculation it seems suffi- cient to assume

[7]

that the director is

firmly

^anchored

at the boundaries. Thus, ^the

boundary

^conditions

on the

perturbations

^{are taken as}

3. Results for

homogeneous

^director orientation;

symmetric

field orientation.

Let 9 ⁼ 0

and ýJ

^{= 0 so that H =}

(0,

0,

Hz) (Fig. la).

Let 0’ and

T’ depend

on y and 03BE. ^Then

equations (1)

reduce to

which are the differential

equations

derived in

[7].

These support ^two

independent

^{modes :}

With

[0’, lIJ1

⁼

[f(ç)

^cos py,

g(ç)

^sin

py], ⁽⁴⁾

^{reduce to}

a

pair

^of

ordinary

differential

equations.

^{For Mode 1}

the solution ^can be written as

Fig. ^{1. - The} sample geometry. (a) Homogeneous ^initial orientation of the director, no ⁼ (1,0, ). OA ^{is the} projection

of the perturbed ^{director n =} (1, cp’, 9’) ^{on the xz} plane. ^OB

is the projection ^{of n on the xy} plane. ^The oblique magnetic

field H ⁼ (0,

H 1. S.", -

Hl

C,)

^{lies in the yz} plane making

an angle 0 with the - z axis. When 4/ ⁼ 0, ^{H acts} along ^z.

When # = x/2, ^{H is} along ^y. (b) Tilted initial alignment ^of

the director, no ⁼ (Co, 0, Se) ; the director lies in the xz plane making ^an angle ^{0 with x} axis. OA is the projection ^{of the} perturbed ^{director n =} (Co - So 0’, T’, So ⁺ Co 9’) ^{on the}

xz plane. ^{OB is the} projection ^{of n on the} xy plane. ^The oblique magnetic ^{field H =} (H, So

C,y, H1. S.", - H1.

^Co

Cl&)

lies in a plane ^{normal to} no and makes an angle 41 ^{with the}

xz plane. ^{OC is the} projection ^{of H on the} yz plane ^with

OD = H1. S."

^{and OE} = - Hl Co

CO.

^{When 41 = 0, H =} (H, SiP 0, - H, Co) ^{lies in the xz} plane ^{and is normal to} no.

When 0 n/2, ^{H is} along y. In the above diagrams ^the position ^{of the} perturbed ^{director n has not been shown}

to avoid confusion.

with the wave vectors

being

calculated from (4). ^The

boundary

conditions

(3)

result in the threshold condi- tion

which has been derived in

[7].

^For

K1

>

K2, (5) yields

a field HZ ⁼

HzT

^{for a}

given

^{value of} p, the wave

vector

along

y. Variation

of p

results in ^a neutral

stability

^{curve of}

HZT( p)

^vs. p. If

HzT

^{takes on a mini-}

mum value

HLM

^for p ⁼ Pc,

HLM

^is

regarded

^{as the}

threshold for the PD and _pr the domain wave vector at threshold. For the

given

^{value of}

K1,

^the

splay

Freedericksz threshold

Hc1 = (7c/2 h) (K1/xJ1/2

^is

calculated. If

HLM Hc1’

^{the PD} may be taken to occur instead of the

splay

deformation.

Figure

² summarizes the results of

[7].

^{The ratio}

HRc = HLMiHc,

^{has been}

plotted

^{as a function of}

KR

⁼

K1/K2 (Fig. 2a). Figure

^2b contains the

plot of Pc

^vs.

KR.

^The main results of

[7]

^{can be summarized}

as follows :

(i)

^For

KR

> 3.3,

HRc

^{1. Hence, as}

KR

^increases

from 3.3, the occurrence of PD is favourable. For

KR

^{3.3, the}

splay

distortion can occur. Large

KR

are known to occur in certain

polymer

^nematics

[8],

such as the system ^{studied in}

[7].

(ii)

^When

KR

decreases from a

high

^value, p,

decreases; ^{i.e. the threshold wave}

length

^{of the PD}

increases. As

K R -+

3.3, Pc -+ 0. In the same limit, ^for

the wave vectors connected with f ^one finds that qi ^-+ 0 and _{q2 -+}

n/2.

As Mode 2 has a much

higher

threshold than Mode 1, ^{it is}

only

^{of academic interest.}

The results of

[7]

^{have been stated} above with a

purpose which becomes clear when the effect of ^a field H ⁼

(0, Hy, ⁰⁾ corresponding to g/ = n/2

^is

studied. For

dependence

of 0’ and

T’

on y and f,

equations (1)

^become

Equations (4)

^and

(6)

^are

isomorphic

^{under the trans-}

formation

Thus for instance, if the Mode 1 threshold for

K1/K2

^*

e1 is

HZ --_

e2 ^at Pc ^--_ e3, the Mode 2 threshold for

KZ/K1 - ei

will be H _{= e2} at Pc == e3. Hence, ^if

Hc2

⁼

(n/2 h) (K21x.)’72

^is the twist Freedericksz threshold and if we calculate the Mode 2 threshold

HLM using (6)

^and

plot

^{the ratio}

HRr

⁼

HLMlHc2

^vs.

KR

⁼

KZ/K1

^we get

identically

^{the curve in}

figure

^2a;

needless to say, the curve

depicting

^{variation of}

Pc with

KR

⁼

K2/K1

^{will be identical to the curve}

in

figure

^2b.

Following [7],

the results for Mode 2

Hy

^threshold

can be summarized :

Hence, ^as

KR

^increases from 3.3 the incidence of PD is favourable. This _may be

possible

^{near a nematic-}

smectic A transition

[10,11]

^where

K2

^and

K3

^become

much

larger

^than

K 1.

696

Fig. ^{2. -} Homogeneous initial orientation of the director;

0=0. (a) ^{Plot of} HR, ⁼ HLm/H,,,, the ratio of the PD threshold and the splay Freedericksz field, ^vs. KR ⁼ K1/K2

for Mode 1 ; ^field along ^z. (b) ^Plot of Pc, ^the dimensionless domain wave vector at Mode 1 PD threshold, ^vs. KR. ^These plots ^{summarize the} results of [7]. ^As explained ^{in section} 3,

these curves are identically ^valid for Mode 2 when the field is along ^{y. We redefine} HR, ⁼ HLMiHc2 ^where H,2 ^{is the}

twist Freedericksz threshold. In either _case, PD is possible for KR > 3.3. For KR 3.3, ^the corresponding Freedericksz deformation is favourable.

(ii)

^When

KR

decreases from a

high

value, per decreases. When KR -+ 3.3, p, -+ 0. Thus, the twist Freedericksz transition is favourable

only

^for KR ⁼

K2/K1

^{3.3. Mode 1 has a}

higher

^threshold

than Mode 2 and is therefore not considered. The appearance of PD in the twist Freedericksz geometry for

large enough K2

^{may pose}

problems

in the determi- nation of

K2.

^{So far, such}

periodic

distortions do not appear ^to have been observed in the twist Freedericksz geometry.

4. Results for

homogeneous

^director

orientation ;

^obli-

que field.

With 0 ⁼ 0, ^{H =}

(0, H 1. S"" - ^{H 1.} c*) ^{(Fig. la)}

^and

dependence

^{of 0’ and}

rp’

on y and f

equations (1)

^reduce

to

The

oblique

orientation of H in the

y( plane

^{has the}

following

consequences :

(i)

^new

coupling

^{terms arise}

between 0’ and 9’ (ii) ^{the modal} symmetry ^{of the}

problem

^{is lost.}

Solution of (7)

by

the conventional method is no

longer

convenient. A series solution method can be used

(Appendix)

^{to determine the PD} threshold. It is also

possible

^{to calculate} the f ^{wave vectors at}

threshold. The

problem again

^{falls into two}

categories : K1 > K2

^and

K1 K2. Noting that 03C8

^{= 0 and}

03C8 n/2 correspond

^{to fields}

Hz

^and

Hy, respectively,

these two cases can be discussed

separately.

For

K1

>

K2,

^{three values of}

KR

⁼

K /K2

^are

chosen such that

KR

> 3.3. For

given

KR ^and

03C8,

^the

PD threshold

Hl

⁼

HLM(ql)

^is determined at p ⁼ pr

as

explained

^{in section 3. For that}

0,

the Freedericksz threshold is

given by [9]

The ratio

HRc{t/J)

⁼

HLM{t/J)/Hc{t/J)

^is

plotted against 0

for each value of

KR (Fig. 3a); figure

^3b illustrates the

plots of Pc

^vs.

0.

^The

following

conclusions can be drawn.:

(i)

^{For a}

given KR = K1/K2, HRc(t/I)

^increases

and p, decreases

when

^{increases from zero. When}

03C8 - 0.(KR), HR. - 1.

^Thus,

for # 0,,

^{the occur-}

rence of PD is favourable and

for ql

>

qlr ,,

^{the Free-}

dericksz transition can occur.

(ii)

^{For a}

given KR, as 0 -+

0,

HRc(t/I) --+

^{the Mode}

1

HR,

value for the

HZ

^field

(Sect.

3,

Fig. 2a).

(iii) When KR increases,

^{so does}

03C8c ;

i.e.,

the 0

range

over which the PD is favourable increases.

(iv)

^{At a}

given 03C8,

Pc increases

(or, equivalently,

^the

domain wave

length decreases)

^when

KR

^increases.

(v)

^For

sufficiently high KRI P,,, generally

^{does not}

go ^{to zero as}

HRc --+

^1.

Fig. ^{3. -} Homogeneous initial director orientation; ^{0 = 0.}

The oblique ^{field makes} angle 0 (radian) ^{with z.} HLM(O) ^is

the PD threshold and H c( t/J) ^{the usual} Freedericksz

threshold. KR = K,IK2- (a) Plot of HR,(ql) = HLM(O)IHR ,(ql) vs. ql ^for KR ⁼ (1) 4 (2) ⁷ (3) ^10. (b) ^{Plot of the} dimensionless domain wave vector at PD threshold, p,,(ql) vs. t/J ^{for the}

above values of KR. As shown in section 4, ^{these curves are} identically ^{valid if} the 0 ^{axis is} renumbered to decrease from

7T/2 (instead ^of increasing ^from 0) ^{and if} KR ^{is redefined to be} K2/K1. ^{The dashed} parts ^{of the curves} correspond ^{to the} regions ^{of existence} of the Freedericksz distortion. For a

given KR there exists a # range ^over which the PD is more

favourable than the Freedericksz threshold.

The results for K2

K1

^and

for ql varying

^from

n/2

^{can be deduced from the} above results

by

^remark-

ing

^{that the} transformation

leaves

(7)

invariant. This is a

generalized

version of the symmetry transformation encountered in section 3.

It is sufficient to renumber the abscissae of

figures

^3a,

3b such

that g/

decreases from

7r/2

instead of

increasing

from _zero;

KR

^{is redefined as}

KZ/K1.

^The

qualitative

differences between the two cases are :

For a

given KR

⁼

K2/K1,

(i) HRc(03C8)

^increases

as V/

^{decreases from} n/2.

(ii) As 0 - n/2, HR,(41)

^{- the}

HRc

value obtained for the

Hy

^{field in section 3.}

(iii)

^When

KR

increases,

ý¡ c

^decreases.

The ç ^{wave vectors are four in number. Two of} them, qi and _q2, are real. The

remaining

^{two are}

complex conjugates

^{of one another.}

5.

Oblique

^director orientation ;

symmetric

^{field orien-}

tation.

Let #

^{= 0 so that H =}

(Hl So,

0,

- H 1. CB)

lies in the

xz

plane

^such that no is normal to it

(Fig. lb).

For y,

03BE

dependence

^{of 0’ and}

T’ equations (1)

take the form

The formal

similarity

^of

(10)

^and

(4)

^{shows that}

(10)

can support ^two

uncoupled

modes and that the threshold condition will be similar to

(5).

^The

only

difference from section 3 is that

K3

^{enters the}

picture

as an additional parameter ^{due to} the initial

oblique

orientation; ^{the bend}

couples

^with

splay

^{and twist.}

Four sets K ⁼

(K1, K2, K3)

have been chosen to

study

the effect of

changing

the ratios of the elastic

constants.

For a

given

^{set K with}

K,IK2

> 1 and a

given oblique

^{director tilt 0, the} Mode 1 PD threshold

HLM(8)

^{is found at} p ⁼

p,(O).

^The Freedericksz threshold

H1(9) _ (n/2 h) (d 2/Xa) 1/2 .

^Now,

HR,, (0) H LM(O)jHet (0)

^and

Pe(O)

^are

plotted against

⁰

(Fig. 4)

for different

KR.

^The

following points

^{can be men-}

tioned :

(i)

^{For a}

given

^{K with}

K 1 /K2

⁼

KR

> 3.3, ^{as 0} increases from _zero,

HRe

^{increases and} Pc decreases;

when 0 -+

Oe(KJ, HRe -+

^1. Thus, ^{for 0}

Or,

^the

PD is favourable and for 0 >

0c,

^the Freedericksz transition should occur.

(ii)

^{For a}

given KR,

^{as 0 -+} 0,

HRe -+

^the

HR,

^value

of

figure

^2a.

(iii)

^At

given

^{0 and}

K3,

^when

KR

increases,

HR,,

decreases and _p, increases.

(iv)

^{At a}

given KR

⁼

KlIK2

^when

K3

increases,

0, (the

0 range for the occurrence of

PD)

^decreases.

This is

presumably

^{a result of} increased elastic _energy associated with

K3.

(v)

^At

given KR

⁼

KlIK2

and 0 when

K3

increases,

HRe

increases and _p, decreases.

For

K2

>

Kl,

^the transformation

(9)

^{enables a}

study

of the Mode 2

Hy

^threshold

(corresponding

^to

698

Fig. ^{4. -} Oblique director orientation 0 (radian) ^{with x axis.}

Field Hl ^{normal to} no in the xz plane. H LM( 8) is the Mode 1 PD threshold and He 1 (8) the Freedericksz field. K ⁼

(K1, K2, K3); ^case K1 ¹ > K2. (a) ^{Plot of} HRc(8) = H,m(O)IH,, (0) ^{as a} function of 0 for K ⁼ (1) 21, 3, 7 (2) 15, 3,

7 (3) 6, 1.5, ³ (4) 6, 1.5, ^7. (b) ^{Plot of} Pc(8), the dimensionless domain ^wave vector at PD threshold, vs. 0 for the above values of K. As indicated in section 5, ^for K2 > K1, ^the

curves hold for Mode 2 with the field applied along y ^if K1 ¹

and K2 ^are interchanged ^{in K and if} HRc(8) ^{is redefined to be} HLM(O)/H.2(0) ^where Hc2 ^is the Freedericksz field along ^y.

03C8 = 7r/2). Figures

4a, ^{4b are}

identically

valid for this

case if we

interchange K1

^and

K2 in

^the

figure legends.

This also _means, in

particular,

^{that in the above}

discussion we redefine

KR

⁼

K2/Kl.

The Freede- ricksz threshold is now

given by Hr,2(o) = (n/2 h)

(dglX.)’I’

and the ratio

HR,(O)

⁼

HLM(O)/Hc2(O). (The validity

^of

(8)

^{has been}

explicitly

^checked

by

^com-

putation).

^This

particular

^{case, where}

K2

^and

K3

are

larger

^than

K 1,

may be of interest in the

vicinity of a N - SA

transition

[10,11].

6.

Oblique

^director orientation ;

oblique

field orien- tation.

With 0’,

cp’ depending

on y and (

(Fig. lb), equations (1)

become

As in section 4, ^{the modal}

purity

^{of the} system ^{is lost.}

The solution is described in the

Appendix. Noting again

^the

limits # -

⁰

and # - n/2,

^{the cases}

K1

>

K2

and

K1 K2

^{are discussed}

separately.

For a

given

^{set K =}

(K1, K2, K3)

^with

K1

>

KZ,

^a

given

⁰

0c (Sect. 5) and #

^{close to zero, the PD}

threshold

Hl

⁼

H LM(4 /)

^{is found} at p ⁼ PC. The Freedericksz threshold is

given by

The ratio

HRc(t/J)

⁼

HLM(t/J)/Hc(t/J)

^{and the wave}

vector

Pc(t/J)

^are determined as functions

of #

^and

plotted

^{for different K and 0}

(Fig. 5).

Most of the conclusions follow

along

^{the lines of those}

given

ⁱⁿ

section 4.

(i)

^For

given

^{K and 0,}

HRr

^increases

and Pc

^decreases

when 03C8

increases.

When 03C8 - t/Jc«(J), HRc

^{1. The PD}

is favourable

for 03C8 03C8c(03B8)

and the Freedericksz transition will occur

for #

>

4/,r

(ii)

^{For a}

given

^K,

qlr ,(0)

^{decreases as 0 increases.}

Thus, ^{as 0} increases,

the 4/

range of occurrence of the PD decreases.

(iii)

^For

given

^{K and}

g/, HR,

^{decreases and} p, increases when 0 increases.

(iv)

^For

given K3

^{and 0,}

t/Jc«(J)

increases with

KR - KI/K2*

(v)

^For

given KR

⁼

KlIK2

^{and 0, an increase in}

K3

^decreases

0,,(0).

Using

^the transformation

(9)

^{results for the case}

K2

>

K1

^can be written down when the field

angle 03C8

^{is varied from}

n/2.

7. Conclusions ; limitations of the mathematical model used.

In conclusion, ^it is noted that

periodic

^distortion patterns ^{similar to those}

reported

ⁱⁿ

[7]

may be obser- vable in different situations, ^{viz. over certain} ranges of

oblique

^director orientation and over certain _ranges of the

oblique

orientation of the

magnetic

^field

applied

in a

plane

^{normal to the} initial director orientation.

The results have been stated without much discussion

Fig. ^{5. -} Oblique director orientation 0 with x. Oblique magnetic ^{field in a} plane ^{normal to} no making angle 1/1 ^with

z. 0 and 0 ^{are measured in radian.} K1 > K2 ^K =(K1, K2, K3).

H LM( 1/1,.0) ^and H c( 1/1, 0) ^{are the PD threshold and the}

Freedericksz threshold, respectively, ^{for the} given 1/1 ^{and 0.}

Pc( 1/1, 0) ^is the dimensionless PD wave vector at threshold.

HRc( 1/1, 0) ⁼ HLM( 1/1, 9)/Hc( 1/1, 0). ^{K =} (21, 3, 7). (a) HR, ^vs.

03C8 ^{for 0 =} (1) ^0.05 (2) ^0.55 (3) ^0.9. (b) p. vs.1/1 ^for the same 0 as

as

they

^{can be}

easily

understood on the basis of the Oseen-Frank

theory.

^{The case}

K1 > K2

may be relevant to

polymer

^nematics of the kind studied in

[7,

8]

^{while the}

opposite

^{case of}

K2

>

K, may’

^be

of interest in the

vicinity

^{of a N -}

SA

transition.

In all the above ^cases

only

^the

y, f dependence

^of

the

perturbations

has been considered. It can be

shown,

using (1)

^and

(3),

^that

periodic

patterns ^with

x, 03BE dependence

may ^not be

possible,

^{at least in the}

linear

approximation.

^{The existence of such a threshold}

in _every case seems to be associated with

unphysical

conditions and unrealistic restraints on the material parameters.

(This

^{is also true of the}

special

^case

of

homeotropic

orientation which can be dealtwith

by taking

^{0 = 0, x = ± h as the}

plates

^and y, ^{z as} the free directions, ^Here

again,

^the occurrence of PD is found to be not

possible.)

Before

concluding,

^{the main} limitations of the model must be stated. As

explained

ⁱⁿ

[7],

^{a non-}

linear calculation would be more authoritative than the linear

analysis. Secondly,

the fmiteness of the

above. K ⁼ (15, 3, 7). (c) HRc vs. 03C8^{for 0 =} (1) ^0.05 (2) ^0.4 (3)

0.7. (d) Pc vs. # for the same 0 values. K ⁼ (6, 1.5, 7). (e) HR, vs. 03C8 ^{for 9 =} (1) ^0.05 (2) ^0.2 (3) ^0.3 (f) Pc vs. # ^{for the}

above 0. K ⁼ (6, 1.5, 15). (g) HRc vs. # ^{for 0 =} (1) ^0.05 (2)

0.1 (3) ^0.2 (h) p, vs. # ^{for the same 0 values as above. As}

mentioned in section 6, ^when K2 > K1, ^{all the above}

curves are valid if 03C8 ^{decreases from} n/2 ^and if K1 ^and K2 ^are interchanged ^{in K.}

anchoring

energy may

greatly

^influence the results of this communication, ^{at least}

quantitatively

^{if not}

qualitatively, especially

^{in the case of}

oblique

^director

orientation.

This is, however, ^{not an} easy task

(see,

^{for instance,}

[12, 13]).

^One other aspect ^which

might

^{be worth}

studying

^is the relaxation of the domains when the

applied

^field is removed. The viscoelastic data of

[8]

may be of

help

ⁱⁿ

determining

^the relaxation time,

at least in the linear limit. The

possibilities

^of

including

surface effects and of

studying

relaxation are

being

considered.

Acknowledgments.

The author thanks Professor V.

Rajaraman,

^Chair-

man,

Computer

Centre, ^{I. I.} Sc.,

Bangalore,

for pro-

viding

the author facilities at the

Computer

^Centre

during

^{a crucial}

period

^{in the course of this work.}

The author also thanks the Referees for useful com- ments.

Appendix.

We present ^{the series} solution method which is found to be convenient for

solving equations (7)

^and

(11). Seeking

solutions of the form

equations (11)

^and

(3)

take the form

JOURNAL DE PHYSIQUE. ^- T. 47, ^No 4, ^AVRIL 1986

700

Substituting

N ⁼ a

positive integer , equations (A. .1)

^{lead to} recurrence relations between Tr ^and Pr ; these show that

Using (A. 3)

^and

(A. 4)

the various coefficients in the series can be calculated _up to r ⁼ N + 2, say. The

boundary

condition

(A. 2)

results in four

equations

The

compatibility

^of

(A. 5) implies

^the

vanishing

^{of the 4 x 4} determinant of the coefficient matrix. At

given

K

= (K1, K2, K3),

⁰

and 4/

^the

compatibility

^{condition makes it}

possible

^{to determine} H1- ⁼

HT

^{as a function}

of p.

The minimum value

of HT

^{which occurs} at p = Pc

is

^{taken as the PD threshold}

HLM. Convergence

^is

generally good

^{for N = 10.}

Higher

^{values of}

N(20-40)

have been used. The above

technique

^is

equally

^{valid for}

solving (7). By seeking

solutions of the form

exp(qc)

ⁱⁿ

(A. 1)

^a

biquadratic equation in q

results. After the PD threshold is determined, ^this

biquadratic equation

^{is solved}

by

conventional methods and the four roots q ^{are obtained.}

For the two sets of

equations (7)

^and

(11)

^{one finds two real roots and two}

complex

^{roots which are}

conjugate

^to

one another.

The series solution method can be

conveniently

^{used to solve the} systems

(4)

^and

(10)

^also. Here, the series for the

perturbations

^{have to be chosen}

keeping

ⁱⁿ view the modal

symmetries.

References

[1] OSEEN, ^C. W., ^Trans. Faraday ^{Soc. 29} (1933) ^883.

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

[3] ERICKSEN, ^J. L., ^{Advances in} Liquid Crystals, ^{G. H.}

Brown, ^editor (Academic Press) ^1976.

[4] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Clarendon Press, Oxford) ^1974.

[5] DEULING, ^H. J., Solid State Phys. Suppl. 14 (1978) ^77.

[6] CHANDRASEKHAR, S., Liquid Crystals (Cambridge ^Uni- versity Press, Cambridge) ^1977.

[7] LONBERG, ^{F. and} MEYER, ^R. B., Phys. Rev. Lett. 55

(1985) 718.

[8] TARATUTA, ^V. G., HURD, ^{A. J. and} MEYER, ^R. B., Phys. Rev. Lett. 55 (1985) ^246.

[9] DEULING, ^H. J., GABAY, M., GUYON, ^{E. and} PIERA0143SKI, P., ^J. Physique ³⁶ (1975) ^689.

[10] CHEUNG, L., MEYER, R. B. and GRULER, H., Phys.

Rev. Lett. 31 (1973) ^349.

[11] DELAYE, M., RIBOTTA, ^{R. and} DURAND, G., Phys.

Rev. Lett. 31 (1973) ^443.

[12] BARBERO, G., BARTOLINO, ^{R. and} MEUTI, M., ^J. Phy- sique ^{Lett. 45} (1984) ^L-449.

[13] MADA, H., ^Mol. Cryst. Liquid Cryst. ⁵¹ (1979) ^43.