Effect of weak anchoring on the generalized Freedericksz transition in nematics

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Effect of weak anchoring on the generalized Freedericksz transition in nematics

U.D. Kini

To cite this version:

U.D. Kini. Effect of weak anchoring on the generalized Freedericksz transition in nematics. Journal

de Physique, 1986, 47 (10), pp.1829-1842. �10.1051/jphys:0198600470100182900�. �jpa-00210379�

Effect of weak anchoring ^{on the} generalized Freedericksz transition in nematics

U. D. Kini

Raman Research Institute, Bangalore, 560080, ^India (Reçu ^{le 28 mai} 1986, accepté ^{le 23} juin 1986 )

Résumé.

On utilise la théorie de courbure des cristaux liquides nématiques, ^{due à Oseen et Frank,} pour étudier la formation de domaines statiques périodiques ^induits par ^un champ magnétique ^{dans une situation où} l’ancrage du directeur aux surfaces de l’échantillon est faible. Les orientations planaire ^et obliques du directeur sont étudiées pour différentes inclinaisons d’un champ magnétique appliqué ^{dans un} plan ^{normal à}

l’orientation initiale du directeur. En comparant ^avec les résultats obtenus dans une situation d’ancrage fort,

on montre qu’un ancrage faible peut influencer profondément la formation de la distortion périodique. ^Ces

résultats peuvent ^être importants pour les nématiques possédant ^{une forte} anisotropie élastique, ^{comme les} polymères nématiques. ^{Les résultats} concernant l’ancrage symétrique ^sont présentés ^en detail : les conséquen-

ces d’un ancrage asymétrique ^sont examinées brièvement. Il semble qu’il ^{ne soit pas} possible d’adapter ^les

résultats obtenus ici par ^une transformation de symétrie ^utilisée précédemment ^{au cas de} nématiques proches

d’une phase smectique.

Abstract.

The Oseen-Frank theory ^{of curvature} elasticity of nematic liquid crystals ^{is used to} investigate

the occurrence of magnetic field-induced static periodic domains for weak anchoring of the nematic director at the sample boundaries. Planar and oblique director orientations are studied for different inclinations of a

magnetic ^field applied ^{in a} plane ^{normal to} the initial director orientation. Comparison with results for rigid anchoring shows that weak anchoring ^can profoundly influence the formation of the periodic distortion. These results _may be relevant to nematics of high ^elastic anisotropy, ^{such as} polymer nematics. The results are stated in some detail for symmetric anchoring ; consequences of asymmetric anchoring ^are briefly ^{examined. It}

appears that it may ^not be possible ^to adapt results of this work to the case of nematics exhibiting ^{a smectic} phase ^{via a} simple symmetry transformation used earlier.

Classification

Physics ^Abstracts

02.60

36.20

61.30

62.30

68.45

1. Introduction.

The Oseen-Frank continuum theory ^of elasticity [1- 2] has been successful in describing ^the anisotropic

elastic behaviour of nematic liquid crystals (for

reviews on the subject ^{see, for} instance, [3-6]). ^In

this theory the average local orientation of nematic molecules is described by ^{a unit vector n called the}

director. The elastic free energy, which is quadratic

in the spatial gradients ^of n, depends upon three curvature elastic constants Kl, K2 ^and K3 ^which correspond, respectively, ^to splay, ^{twist and bend}

distortions of the director field. A fourth constant,

K4, appears with the nilpotent part of the free energy which determines the surface torques without affec-

ting ^the equations ^of equilibrium of the nematic director. A uniform orientation, no, ^{of n between}

two flat plates ^{is a} configuration of minimum free energy for ^a nematic ; ^the stabilizing ^elastic torque opposes the effect of any disturbing ^{influence which}

tends to create small changes ^{in n} away from no.

The orientation of n can be controlled by suitably treating sample boundaries with which the nematic

comes into contact ([7] ^{is a recent review on the}

interfacial properties of nematics with references to earlier work). Different kinds of surface treatment can lead to different magnitudes ^of the director

anchoring energy ^at the surfaces.

An externally applied magnetic ^{field H can also}

affect n in a nematic sample by creating ^a torque ^via the diamagnetic susceptibility anisotropy X a. In ^a nematic with Xa > 0, ^{n tends to} align parallel ^{to H.}

When H lies in a plane ^{normal to} no ^the director field

experiences ^no torque. ^{However, H can} couple ^with

fluctuations in the director field and give ^{rise to a} destabilizing torque. ^When I H I - Hc, ^{called the}

Freedericksz threshold, ^the stabilizing ^elastic torque

can balance the destabilizing magnetic ^{torque and}

the director field remains unperturbed. ^When I HI> Hc, .the ^{elastic torque can no longer balance}

the magnetic torque ^{as the} uniformly aligned ^director

field has ^a higher total free energy than the distorted state ; thus the director field undergoes ^{a deforma-}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470100182900

tion. When I H I 2:: Hc, ^the distortion is generally

uniform in the sample plane.

The Freedericksz transition is a useful tool for the determination of the elastic constants of a nematic.

If _{ya is} known, different elastic constants can be evaluated by choosing different directions of H in a

plane ^{normal to} no. ^In particular, ^when no ^is parallel

to the sample boundaries and H normal to them, ^the

deformation for I H I 2:: Hc ^{is a pure} splay ^{and this} configuration ^is helpful ^for measuring Kl. Recently, Lonberg ^and Meyer [8], ^while studying ^the splay

Freedericksz geometry using ^a polymer ^nematic,

discovered a spatially periodic ^{structure above a well}

defined field threshold instead of a homogeneous splay deformation. Such a structure, ^with spatial periodicity ^{in the} sample plane, would involve ^a mixture of splay and twist deformations. Using ^the

continuum theory ^{in the} small deformations limit and the rigid anchoring hypothesis, Lonberg ^and Meyer ^{showed that the} periodic deformation (PD)

threshold is less than the splay Freedericksz thres- hold for nematics with Kl > 3.3 K2; in ^{such a} material, ^{the total} free energy associated with ^a pure

splay distortion can be further minimized when ⁿ escapes ^out of the (H, no) plane ^{via a twist.} Apart

from being ^{a novel} observation, the result of [8]

demonstrates the difficulty involved in measuring Kl in nematics having K, > K2 [9].

Using ^the rigid anchoring hypothesis ^{and the}

linearized continuum theory ^{it can be shown} [10]

that PD becomes less favourable than the homoge-

neous deformation (HD) ^when no ^is oblique ^relative

to the sample boundaries or when H is obliquely applied ^{in a} plane ^{normal to} no (as considered, ^for instance, by Deuling ^{et al.} [11]). Over certain ranges of the director tilt 0 relative to the sample ^boundaries

or of the field tilt angle w away from the sample normal, HD may have ^a lower threshold than PD.

These configurations may be of help ⁱⁿ determining Ki ^{of materials with} K, > K2. It is also shown that ^a symmetry transformation makes it possible ^{to write}

down results for the case K2 > K, using ^results

calculated for K, > K2. The situation K2 >> K, ^may

occur in the vicinity ^{of a} nematic-smectic transition

[12].

The rigid anchoring hypothesis, corresponding ^to

infinite director anchoring energy ^at the sample boundaries, is conducive to a relatively simple understanding of field induced director deforma- tions. Still, ^{in a real} situation, ^the anchoring energy is finite. Rapini ^and Papoular [13] proposed ^a simple

form of the anchoring energy ^and showed that the Freedericksz threshold decreases with the anchoring

energy. This picture ^was generalized by Guyon ^and

Urbach [14] ^{to include more} general deformations.

In a recent paper, Oldano [15] has included weak

anchoring ^{of one} of the director fluctuations in PD.

He has shown that weak anchoring ^can considerably

affect the domain of occurrence of PD. He has also

independently pointed ^{out the} symmetry transforma- tions of the governing equations which enable one to

write down results for K2 > K, using results obtained for K, > K2.

In this communication the linear perturbations approach ^{used in} [8, 10, 15] is extended to the study

of the effect of weak anchoring ^on the relative

occurrence of PD and HD. The results are compared

with those for rigid anchoring. ^The general ^{form of}

the anchoring energy used in [14] ^is employed. ^In

section 2, the differential equations ^{are derived and}

the boundary conditions stated. In section 3, ^the planar ^director configuration is studied in the splay

Freedericksz geometry. ^{The case of an} oblique ^field

with planar ^director orientation is presented ⁱⁿ

section 4. In section 5 oblique initial director orienta- tion is considered with H along ^a symmetry ^direction.

Section 6 lists the limitations of the mathematical model used in this work.

2. Mathematical model, differential equations, ^boun- dary conditions and solution.

The mathematical model is similar to that employed

earlier ([10], Fig. 1). ^{The nematic} is confined between flat plates ^{z = ± h and} uniformly ^oriented

in the xy plane making ^an angle 0 ^{with x axis. The}

uniform orientation

Fig. 1.

^{Planar director} orientation ; equal anchoring strengths ; Bp

Bo

^{B; H along z. RH is} the ratio of

HLM’ the PD threshold and HHF9 ^{the HD} threshold ; _pc ^is the dimensionless domain wave vector at PD threshold ;

RK

KIf K2. ^{(a) Plot of RH vs. RK and (b) plot o fpc vs. RK}

for difterent anchoring strengths ^B

(1) ^oo (rigid anchoring) (2) ^103-’ (3) ^103-2’ (c) ^{Plot of} RH ^{vs. vB and} (d) plot ^of pc vs. B for different materials having RK

(1) ¹⁵ (2) ¹⁰ (3) ^5. Strong anchoring ^{of the 0’ and} 0’ perturba-

tions and high RK favours the occurrence of PD (Sect. 3.1).

also coincides with the easy axis ^at either surface. A

magnetic ^field

is applied ^{in a} plane ^{normal to} no, making angle w

with the xz plane. ^Under the action of linear

perturbations 0’(x,y,z) ^and 4o’(x,y,z) ^the

director field becomes

To second order in the fluctuations and their spatial derivatives the total free energy is

where V is the sample volume, o- ^the sample surface, Fv the volume energy density (whose expression ^is given ⁱⁿ [10]), ^and F, ^the surface energy density. F u ^{is a sum of two} parts each of which is proportional ^to

the square of ^one of the perturbations away from the easy axis ^at the sample ^surface [13, 14].

Be, Be, B4,9 Bcp’ ^{which give a measure of the} anchoring strength, have the dimensions of surface tension. In

general ^{one can} have different anchoring strengths ^{at the two} boundaries for a given perturbation. ^{The form}

of F(,, ^also suggests ^{that the} sample ^{is thin} compared ^to linear dimensions along ^x and y and that ^we consider

a part ^{of the} sample far away from the edges. Thus, boundary conditions ^on surfaces normal to x and y are

ignored. ^On minimizing FT ^with respect to 0’ and )/ ^the Euler-Lagrange equations that result are

where 0’ IXY

0 20 ilax a y etc. ; 6 ⁼ z/h ; y - y/h ; x --+ xlh; equations (2) ^of [10] summarize the other definitions. The boundary conditions are given by

Thus, though K4 ^{does not} determine the equations ^of equilibrium (2), ^{it does enter} the surface torque ^and hence the boundary conditions. For rigid anchoring ^conditions

one gets

In a given situation it is also necessary ^to investigate ^the homogeneous Freedericksz threshold for the same set of parameters. This is done by assuming ^{that the} perturbations depend only ^{on §. Then,} equations (2, 3)

for weak anchoring ^become

For rigid anchoring ^the boundary conditions are again given by (4). ^{As can be seen,} K4 ^{does not affect the}

HD threshold.

Before going ^{over to the} method of solution certain assumptions ^are stated which are valid for the present ^work. (i) K4

^0; (ii) dependence ^{of 0’}

and 0’ ^{on x is} dropped ; (iii) it is assumed that

The effects of taking dependence of the fluctuations

on x and having asymmetrical anchoring ^are briefly

discussed in section 6.

The method of solution is as follows : the perturba-

tions are assumed to vary with y ^{as exp} ( i py ) ^where

p, which is real, is the normalized wave vector along

y. Equations (2) ^{now reduce to a set of} ordinary

differential equations ^{which are} solved with boun-

dary conditions (3) ^or (4) resulting ^{in a} compatibility

condition. (The series solution method, ^{used and}

described in [10] ^{is found to} be convenient for this

purpose.) ^{For a} given ^set of values of p, h, Ki, B 8’ B cJ>’ (J ^{and t/1, the magnetic field strength Hl}

is varied so that the compatibility equation ^{is satisfied}

for the lowest possible ^{value H’} ( p ) . ^{Now the}

neutral stability ^curve is obtained by calculating H’ ( p ) ^{as a function} of p. If this ^curve has a

minimum at H.J- = HLM ^{= H’} pc ) , ^{HLM is taken as}

the PD threshold and Pc ^as the dimensionless ^wave-

vector at threshold. For the same set of parameters,

equations (5) ^are solved with boundary ^conditions (6) ^or (4), leading ^{either to a} compatibility ^condition

or to a formula. The lowest value HHF ^of Hl ^{is the} homogeneous Freedericksz threshold for the given

case. If RH ⁼ HLM HHF ^{- 1,} then PD is taken to be

more favourable than HD. If, however, RH > 1, ^HD

is assumed to be more favourable than PD.

No specific material has been chosen in this work which only attempts ^{to arrive at some} conclusions

regarding the relative occurrence of PD and HD. In

general ^{for a} given material and sample ^thickness

both HLM ^and HHF decrease when the anchoring ^is

slackened. However, ^{their rates of decrease are} different. In order to determine the relative occur- rence of PD and HD their absolute values are not

needed ; the ratio RH is sufficient. Hence, assuming

that all quantities ^are measured in cgs units, ^{it is}

assumed for convenience that

all angles ^are measured in radians. Thus, Kl, K3, Be, B. ^{are measured in terms of K2.} As K - 10-6 dyne

and B -10-3 _erg cm- 2 [7, Tables X and XI], ^the higher ^{limit for B - 104} in this work. The rigid anchoring ^limit actually corresponds ^{to the} special

case Bo = Bo

^{oo .} It, therefore, ^seems meaningful

to present the results in two parts, ^{one for} equal anchoring energies (B (J ⁼ B "’) ^{and the other for}

unequal anchoring energies (B (J =F ^{Bo though the}

latter case is the more realistic one.

3. Planar director orientation ; ^field along ^z.

(J ⁼ 0 ⁼ 0. ^For PD, equations (4) ^of [10] ^result.

Seeking solutions of the form ( 0’, 0’) = [f ( ) ^{cos py,} g ( ) sin py] it becomes clear that

the differential equations support ^two uncoupled

modes :

As Mode 2, which is associated with a much higher

elastic energy than Mode 1, ^{has a} higher ^threshold only ^{Mode 1} is considered. For weak anchoring ^the boundary conditions become

The boundary conditions at § = - I ^{do not} yield any

new condition and this is in conformity ^{with the} equations supporting ^two distinct modes. For HD, equations (5, 6) ^become

As the field couples only ^to 0’, 0’ ^{is not a} part ^of

HD. With 0’ - cos q§ , the Freedericksz threshold is

given by

where qc is the lowest root of the equation

Equations (11, 12) yield ^{results identical to those of} [13]. ^For rigid anchoring, qc = ?r/2 ^and (11) ^reduces

to the well known splay Freedericksz field.

3.1 EQUAL ^ANCHORING STRENGTHS ; B 0 = Bo

B.

Figure ^{la contains a} plot ^of RH ⁼ HLM HHF

versus RK ⁼ Kl K2 ^{for different anchoring strengths}

BB. Figure ^Ib gives ^the corresponding ^curves of pc,

the domain wave vector. For a given ^{B, as} RK

decreases from a high ^value RH increases. When

RK a lower limit RKL’ RH --+ ^1. Thus, ^{for fixed B,}

PD is favourable for RK > RKL and HD for

RK : RKL. When B is diminished keeping RK ^fixed (i.e. ^{for a} given material) RH increases. RKL ^increases

when B is decreased ; when the director anchoring slackens, ^PD occurs over a shorter range of

RK. ^From figure Ib it is found that Pc decreases with

RK ^{such that} Pc -+ ^{0 as} RK --+ RKL* ^{For a} given RK (fixed ’material) ^{lower the} anchoring strengths ^B,

lower is pc. Thus, ^weak anchoring may ^cause the domain size at PD threshold to enlarge. ^{This is} clearly ^a consequence of ^a greater ^{extent of relaxa-}

tion of the director at the boundaries caused by ^a slackening ^{of the} anchoring.

Figures lc and Id illustrate the variations of

RH ^and Pc’ respectively, ^{with B} for different RK (i.e.

for different materials). ^{For constant} RK, RH ^increa-

ses and Pc decreases when B is reduced ; ^when

B --+ a lower limit Bm, RH -+ ¹ and Pc --+ ^{0. Thus, for}

a given material, PD is favourable for B > Bm ^and

HD for B Bm. ^When RK ^is decreased, Bm ^is

enhanced. Thus, materials with higher RK ^{have a} larger B domain of existence than materials with lower RK. ^In brief, therefore, strong anchoring (large Bo = B 41 = ^{B) and high} RK = Kl K2 ^favour

the occurrence of PD relative to HD.

3.2 UNEQUAL ANCHORING STRENGTHS ; Bø:1=

Bol

^{This case is more} realistic than the previous

one. To appreciate ^{the roles} played by Bo ^and Bp ⁱⁿ

determining the relative occurrences of PD and HD,

B. ^{is fixed at a high value} (104). ^{Figures 2a, 2b}

show the variations of RH and Pc ^with RK ^for

different Bo* ^{It can be seen} that for the constant,

high Bo ^{(i) RH increases} and Pc decreases when Be ^is

reduced for a given ^material RK ; a slackening ^of

the (J’ anchoring expands the domain size of PD ; (ii) ^{for a fixed} Bog RH ^increases and Pc diminishes ^as

RK ^is decreased from a high ^{value ; when} RK -+ a

lower limit RKMI RH -+- 1 ^and Pc -+- 0; ^{thus, PD is}

more favourable for RK > RKM and HD for

RK : RKM; (iii) RKM ^{increases when} Be ^{is decrea-} sed ; ^{thus when} 0 ^is strongly anchored, ^a slackening .

of the 0’ anchoring curtails the RK range of ^occur-

rence of PD. These conclusions are similar to those of section 3.1.

When Bo ^{is fixed at some} high ^value (lW _ 106)

the results for different B 41 ^{and RK are found to be}

rather dissimilar (Figs. ^2c, 2d) ^{to those of} figures 2a,

2b. It is found that for the given, high BB (i) ^both RH and Pc ^{decrease at a} given RK (i.e. ^{for a fixed} material), ^when B. ^{is reduced ; (ii) at constant} B., RH ^increases and pc diminishes when RK is decreased from a high ^{value ; when} RK -+ RKN, RH -+ ^{1 and}

p, -+ 0 showing that PD is favourable for RK > RKN

and HD for RK RKN ; (iii) RKN decreases when B .

is reduced ; ^{when the} 0’ anchoring ^is slackened, ^the RK ^{range of} occurrence of PD broadens ; this is in agreement with the results of Oldano [15]. ^In particular, ^when B (J - 106 Bel> - 101.5 ^{(Fig. 2c,}

curve 3), RKN ’ ^{2. This is close to} the limit 2 arrived at for rigid anchoring of 0’ and weak anchoring ^of cp’ ⁱⁿ [15]. ^A consequence ^of this is that for ^a fixed, high RK, P,, increases with B. ^{when Be takes a}

Fig. ^2.

Planar director orientation ; unequal anchoring strengths ; Bo :A BO; ^{H along z. (a) Plot of RH vs. RK and}

(b) plot of p, ^vs. RK ^for strong anchoring ^of 0’ (B (cp) = 1f) and different anchoring strengths ^of 6’; Be

(1) ¹⁰⁴ (2) ^103v (3) ^103.4 (4) ^103.2. (c) ^{Plot of} RH ^vs. RK ^for strong anchoring ^{of 0’ and different} anchoring strengths ^of 0’;

Be = 104 ; Bo = ^{(1) 103 (2) 102; for curve 3, Be} = 106, Bo

^{101.5. (e) Plot of RH vs.} Bel> ^{and (f) plot of Pc vs.} Bel> ^{at a}

fixed, high anchoring strength ^{of B’} (Be = 1w ) for different materials having RK

(1) ¹⁵ (2) ¹⁰ (3) ^5. (e) ^{Plot of} RH

vs. Bo ^and (f) plot of pc ^vs. Be ^{at a fixed,} high anchoring strength ^of 0’ (Bo =104 ) ^for different materials having

RK = (1’) ¹⁵ ( 2’ ) ¹⁰ (3’) 5. The results ^are in agreement ^with [15] ; strong anchoring ^{of 6’ and weak} anchoring ^of

0’ ^{favours PD relative to HD} (Sect. 3.2).

constant, high ^{value ; at} sufficiently ^low RK, Pc

diminishes when B. ^{is increased (Fig. 2d).}

Before attempting ^at giving ^{a tentative} explanation

of the above results, it will be helpful ^to study figures 2e, ^{2f which} depict variations of RH and Pc ^{with the} anchoring strengths ^for different materials RK .

Let Be ^{be fixed at a} high ^value (104). ^When B /> ^is

reduced, RH and pc ^{decrease for a constant} RK ; ^when B/> ^{tends to low values, RH} and Pc ^{tend to their} respective lower limits. A reduction of RH ^with B.

implies ^a diminution of the PD threshold HLM

relative to HHF. ^{This is,} again, ⁱⁿ agreement ^{with the} findings ^of [15].

The following qualitative ^reason may ^account for the above mentioned results : in the present ^case,

HD is associated with a single degree ^of freedom ; (J’, ^{which is} strongly ^{anchored with a fixed,} high BB.

PD is, however, associated with ^an additional degree

of freedom, 0’. ^When B , ^{is reduced, HLM decreases,}

though Bo ^is kept ^{constant for a} given ^material

RK . ^{But HHF, the HD threshold remains fixed at}

some value which is determined by Be alone ; HHF ^is

in dependent ^of B., (Eqs. (11), (12)). ^{This seems to}

account for a reduction in the PD threshold relative to the HD threshold when the 0’ anchoring ^is

slackened keeping ^0’ strongly ^anchored.

When 0’ ^is strongly ^{anchored at a fixed}

B/> (104), ^{RH and Pc} vary with Bo ^{for a} given

material RK ^{as depicted in figures 2e, 2f. When}

Bo ^is decreased, RH ^increases and Pc ^decreases.

When Bo --+ BBM, RH -+ 1 ^and pc -+ ^0. Thus, ^{for a} given material and constant, high B., ^PD is favoura- ble for BB > Bom ^{and HD for} Be BBM. When RK ^is enchanced, Bom diminishes ; ^{thus materials with} higher RK ⁼ Ki K2 have a broader BB ^{range of}

existence of PD.

A tentative explanation of these results may follow as stated below: it will serve to remember that we are dealing ^{with the case} RK ⁼ KIIK 2> ^I.

As shown earlier, HHF ^{does not} depend upon B

When BB ^{is reduced,} HHF ^{decreases and so does} HLM. ^{However, a} large, ^constant B. ^{implies that cp’,}

the degree ^of freedom of PD associated with the much smaller elastic constant K2, ^{is strongly ancho-}

red. (This ^{must be} contrasted with the previous

discussion where 0’, ^the degree ^{of freedom associa-}

ted with the much larger elastic constant, ^was

strongly anchored.) ^When Bo ^is decreased, HHF might ^{diminish more} rapidly ^than HLM ^{and this} might ^{account for} RH ^{1 as B -} BBM. ^As HHF ^{is the}

lowest possible ^{threshold field} with p ⁼ 0, ^{it is}

natural that pc -+ 0 ^when RH - ^{1. Thus,} it appears that HD may become ^more favourable than PD when 0 ’ ^is strongly anchored and the 0’ anchoring ^is

slackened. Thus the conclusions of this section are seen to be in agreement ^with the’ results of [15] ^and

strongly ^contrast the results of section 3.1, It is seen

that high RK ^{and weak} anchoring ^of 0’ ^{relative to}

the 8’ anchoring ^{are conducive to} the formation of PD.

4. Planar director orientation ; oblique ^field.

0 = 0; & :0 0. ^For PD, equations (7) ^of [10] ^are

obtained. The boundary conditions result from (3) by dropping x dependence ^{of the} perturbations. ^As compared ^{to section} 3, .p ^the angle ^made by ^{H with}

the xz plane ^{is an} additional parameter ^{on which} both HLM ^and HHF ^will depend. ^The oblique ^field (OF) introduces new field couplings between 0’ and

cp’ ^and destroys ^{the modal} symmetry which exists for PD when w

^0. Essentially, OF mixes the two

uncoupled modes of section 3 in different _propor- tions depending upon the field angle w and this will be found to produce ^even qualitative differences between results of this section and those of the

previous ^{one. HD is} governed by ^the equations

for weak anchoring. Equation (13) supports ^two independent modes of which that mode which ^corres-

ponds ^{to even} variations of 0’ and 0’ with 6 ^is

chosen. The HD threshold can be conveniently

determined by using the series solution method of

[10]. ^In the presence of OF, ^{HD is also associated}

with two degrees ^{of freedom ; as is clear from} (13), HHF is affected by ^both Be ^and B cf> . ^{This is again}

found to be responsible ^for introducing qualitative

dissimilarities between the conclusions of this section and those of section 3. For strong anchoring, ^{the HD}

threshold is given by the well known expression [5, 11],

In general, the values of different quantites go ^over to those calculated in section 3 quantitatively ^when w - 0.

4.1 EQUAL ^ANCHORING STRENGTHS ; Be Bo

B.

Figure 3 ^contains plots ^of RH HLM ( ’" ) HHF ^{( "’) and p,, ( 4/ )} calculated as func- tions of different parameters. ^In figures 3a-3d, .p ^has

been fixed at a small value (0.1). The variations of

RH and pc ^with RK ^{for different B} (Figs. 3a, 3b) ^and

the plots ^of RH and against ^{B for different} RK ^can

be discussed exactly ^as done in section 3.1 for t/1

⁰ (Fig. la-ld). ^The only point ^to be noticed is that as compared ^{to the case} 4r ⁼ 0, RH ^is higher ^and

Pc ^lower for gi

0.1; ^{when H} is moved away from

Fig. ^3.

Planar director orientation ; oblique ^field (OF) ; equal anchoring strengths B,

Bell

^{B. qi}

^{angle made by}

OF with the xz plane = 0.1. Dotted lines represent regions ^{where PD exists as a solution but has higher} threshold than HD. (a) ^{Plot of} RH ^vs. RK ^and (b) plot of pc ^vs. RK ^{for different} anchoring strengths B

(1) ^oo (rigid anchoring) (2) ¹⁰⁴ (3) ^103-5 (4) ^103.25. (c) ^{Plot of} RH ^{vs. B and} (d) plot of pc ^{vs. B} for different materials having RK

(1) ¹⁵ (2) ¹⁰ (3) ^5.

RK = 15 ; (e) ^{Plot of} RH ^{vs. qi and} (f) plot ^of pc vs. qi ; RK = 10 ; (g) ^{Plot of} RH ^{vs. qi and} (h) plot ^of pc vs. qi ^{for different} anchoring strengths ^B

(1) ^oo (2) ¹⁰⁴ (3) ^103-5 (4) ^103-25. The effect of the OF is detrimental ^to the formation of PD

(Sect. 4.1).

the xz plane, the domain size of PD increases ; ^the PD threshold also moves closer to HHF. Again, ^{for a}

fixed B, RKL(41 = 0.1 ) -RK(41 = 0 ) ; ^{thus OF}

has the effect of reducing ^the RK ^{range of occurrence}

of PD, ^all other parameters remaining ^{the same.}

Similarly, ^{for a} given ^material RK ,

Bm ( 41 ⁼ 0.1 ) > B. = 0 ) ; ^{OF causes reduction}

in the B range of existence of PD.

The above results can be understood as follows : OF does not affect the modal structure of HD.

Though 0’ ^{is an} additional degree of freedom as

compared ^{to the case w}

^{0, one can still} study ^the 0’ ^{even mode.} Equations (13) ^{show that a decrease}

in B diminishes HHF. ^{In the case of} PD, ^OF mixes up the independent Modes 1 and 2 of section 3. As Mode 2 has a much higher elastic energy than Mode 1 the mode mixing enhances the effective elastic energy and hence the threshold of PD relative to HD for a given ^{set of} parameters. ^This might ^{account for}

the reduction in the ranges of occurrence of PD relative to HD for different parameters.

Figures 3e-3h illustrate the variations of RH and p,

with the field angle w for different anchoring streng- ths, ^{B and for two} elastic ratios RK. ^{For fixed} RK ^and B, RH ^increases and Pc ^{decreases when w is increased}

from a low value. When w - 4’m, RH -+ ^{1. Thus, for}

a given material and fixed anchoring strengths B, ^PD

is favourable for w qf m ^{and HD} for w > qim. ^The

increase in the domain wave length ^{of PD as H is}

moved away from the ^xz plane ^is again evident. For

a given ^material RK , ^{’" m} diminishes with B ; ^a

reduction in the anchoring strengths curtails the range of existence of PD. This ^can be seen to be a

consequence ^{of more} of Mode 2 PD being ^mixed

with Mode 1 PD as H makes larger angles w with ^the

xz plane. ^{For a fixed} anchoring strength ^B, .pm

increases with RK ; the range of occurrence of PD broadens when the elastic ratio increases. In general, when 0 ^-+ qi., pc ^{does not} approach ⁰ (dotted ^lines

in Figs. 3e-3h). ^{This is} again ^{seen to be a conse-}

quence of the mode mixing by ^OF. As w increases,

more of Mode 2 gets mixed with Mode 1. As HD is a

combination of pure mode deformations, it is rather difficult to match 0’ and cp’ ^{of PD} (which ^are asymmetrical) with 0’ and cp’ ^{of HD} (which ^are symmetrical ^with respect ^to §).

4.2 UNEQUAL ^ANCHORING STRENGTHS ; B(J =1= Bcp.

-

Figures ^{4a, 4b} depict the variations of RH and

with RK for different BB ^{and fixed} B cp ( = 1W) ^{for a}

small value of the field angle ( 4f

0.1 ) . ^{One can}

remark a similarity in the behaviour of RH and

with that shown by figures 2a, ^2b. Quantitatively, ^it

is seen that for fixed Be ^and RK N RH ( 4f ⁼ 0.1 ) ^>

RH(.p=O) and that pc(qi = 0.1) Pc(.p=O).

Thus, ^{for a} given material and fixed strengths ^of

Fig. ^4.

Planar director orientation ; oblique field ; unequal anchoring strengths Bo o B,,; 0, inclination of OF with the xz plane, = 0.1. Dashed parts ^{of the curves} indicate that HLM > HHF; (a) ^{Plot of} RH ^vs. RK ^and (b) plot of p, ^vs. RK

for B. = 104; B 0

(1) ^103.7 (2) ^103.4 (3) ^103.2. (c) ^Plot of RH ^vs. RK ^and (d) plot of Pc ^vs. RK ^for Bo = 104; B 4> = ^{(1) 103} (2) ¹⁰² (3) l0us; _Bo = 106; (4) B0= 101.5 ^{(for case 4, Pc has not been drawn to} avoid further confusion). (e) ^{Plot of} RH

vs. B. ^{and (f) plot of pc vs.} B. for different materials ; RK

(1) ¹⁵ (2) ¹⁰ (3) 5 ; Bo ^{=104 for these curves.} (e) ^{Plot of} RH ^vs. B. ^and (f) plot of p,, ^vs. Bo ^for RK

(1’) ¹⁵ (2’) ¹⁰ (3’) 5 ; ^{For these} curves, B. = 1W. Figures ^{4c-4f are to be} compared ^with figures ^2c-2f. RK ^{= 15 ;} Bo ^{1W; (g) Plot of RH vs. 0 and (h) plot of pc} Ys. «/1 for BB

(1) ^10-’.75 (2) ^101.5 (3) ^{103.25 .} RK ^{= 15 ;} Bo = 104; (i) ^Plot of RH ^{vs. «/1 and} (j) plot of pc ^{Ys. «/1 for} B 4> = (1)103 ^{(2) 102.5 (3) 102.} Figures 4g-4j

are to be compared ^with figures 3e-3h for the equal anchoring ^case. Under the action of OF, RH and Pc ^{behave in a} qualitatively ^{different way as} compared ^{to the case of H} applied along ^z (Sect. 4.2).

anchoring, the domain size of PD expands ^{when H is}

moved away from the ^xz plane ; HLM ^{also increases}

relative to HHF. ^{For constant} anchoring strengths, RKM (t/1 = 0.1 ) > RKM (t/1 = 0 ) ; ^the RK range of

occurrence of PD shrinks when H is rotated away from the z axis.

Figures 4c, ^4d illustrate the behaviour of RH and

when RK is varied for strong anchoring ^of

0’(Bo - 104 - 106) ^{and different anchoring}

strengths ^of 0’; ^{the field} angle ^is again ^{fixed at a}

small value ( = 0.1 ) . ^{For a} given ^set ( Bog BO), ^RH

increases and pc decreases ^when RK is reduced from

a high value ; ^as RK -+ RKN9 RH -+ ^{1 and} pc -+ ^{0. On}

comparing ^{these curves} with those of figures 2c, ^{2d a} striking dissimilarity is observed. As long ^as

Bo - Bog RKN ( B02) RKN(B"’l) ^{for B.2 B."l’}

However, ^when B03 - ^Be, RKN ( B-03 )>-

RKN Bol). ^{(For the case 0 =} 0 it had been found,

in section 3.2, ^that RKN keeps diminishing ^with B.

for a given Bo ^{and that when} B. ^{Be, RKN approa-}

ches a lower limit 2 [15]. ^{In the} present ^{case, for} OF, RKN decreases with B. ^{only when} B. ^{is not very}

small compared ^to B,,; ^when B. becomes very small, RKN again ^{increases as} B. is reduced further).

To understand this more fully, ^a given ^material

RK ^is considered ; ^{one of the} anchoring strengths

is fixed and the other varied ; ^{the field} angle ^is again kept constant at 0.1 (Figs. 4e, 4f ; it is instructive to compare these with Figs. ^2e, 2f of section 3.2 drawn

for w

0). ^For strong anchoring ^of 0’, ^{the varia-}

tions of RH ^and pc ^with B. ^{are similar to those for}

t/J

^{0. The} quantitative difference is that for fixed

RK, B9, Bc/J’ RH = 0-1) - RH (4’ = 0) ^and pc ( 4, ⁼ 0.1 ) - pc ( = 0 ) . This also implies ^that Bom((p = 0.1) -Bom(ip =0) ; ^{thus, for a} given material, ^when 0 ’ ^is strongly anchored, ^the B 9 ^range

of occurrence of PD gets ^{curtailed when H is moved} away from the xz plane.

On fixing B 9 ^{at a} high ^{value and on} reducing B.

for a given ^material RK , ^{the behaviour of RH is}

found to be rather different from that depicted ⁱⁿ figure ^2e. Initially, ^when B 41 ^{is reduced, RH and}

diminish. However, ^when B. ^{attains a} sufficiently

low value, RH ^starts increasing ; finally ^when Bo --+ BOM9 ^{RH --+ 1 ; in the same limit, Pc attains a}

limiting ^value. Thus, ^for B. > BOM9 ^{PD is favourable}

while for B c/J B,6m, HD is favourable.This is in contrast to the results for w ^{= 0} (Figs. 2e) ^where

both RH ^and pc ^{tend to} lower limits when B, ^is

diminished to low values for a fixed, high B9. ^Thus

for OF, when 0’ is strongly ^anchored, RH ^{at small}

B cfJ ^{can be higher than that at large} B cfJ ; ^{this seems to}

be the source of the qualitative difference between

figures ^{2e, 2d and 4c, 4d.}

A qualitative ^reason for this variation of RH ^{can be} given ^as follows : when BcfJ ^{is reduced, both HLM and} HHF decrease. As the field angle ^{is not too} large, ^the

initial reduction in RH ^with B , ^can be understood as

in section 3.2 ; HLM ^{decreases more} rapidly ^than HHF. The idea is that in the region ^of large B., ^{as Bo}

is also large ^the additional elastic _energy OF2

contributed by Mode 2 PD on being mixed ^with

Mode 1 PD may ^not make much difference. Howe- ver, in the region ^{of low} B,,, ^{AF2 might be important}

enough ^to bring ^{down the rate} of decrease of

HLM ^as compared ^{to that of} HHF ^{and this} might

account for the increase in RH ^as B cfJ is reduced to low values.

Figures 4g-4j also illustrate the effect of OF for a

given material when H is applied ^{at different} angles ;

different magnitudes ^{of the} anchoring strengths ^are

also considered. In figures 4g, 4h, RH and Pc ^are plotted ^{as functions} of w ^for strong anchoring ^of 0’ and for different anchoring strengths Bo. ^For

constant Be, ^{as w is} increased from a low value, RH ^is

enhanced and Pc ^{reduced. When w -} qim, RH -+ ^1.

Thus, ^{for a fixed} Be, ^PD is favourable when w -- 4fm

and HD when w > t/1M. ^{At a} given ^t/J, RH ^diminishes and Pc increases when Bo ^{is enhanced.} t/JM gets reduced when Bo is decreased ; the w range of existence of PD, ^{for a} given material and strong 0’ anchoring, shrinks when the 0’ anchoring ^is

slackened. Thus, the message conveyed by figures 4g, 4h is very similar ^to that illustrated in figures ^3e-

3h.

When 0’ is firmly ^anchored and w varied for different BcfJ ^{(Figs. 4i,} 4j), RH and Pc behave rather

differently and this is more easily appreciated ^with

the help ^of figures 4e, ^{4f. For a fixed} B., there exists

a limit qic ^{such that when w} increases from a low value to qic, RH -+ ^1. t/J c decreases with B,,; ^{when 0’}

is strongly ^{anchored the w range of} occurrence of PD gets curtailed when the anchoring ^of cfJ’ is slackened.

The difference between figures 4g, ^{4h and} 4i, 4j

arises because when B. is enhanced RH ^either

increases or diminishes depending upon the value of 41.

The calculations corresponding ^to figures 4g-4j

have been repeated ^for RK ^{= 10 but as the curves are}

similar they ^{have not} been shown. The quantitative

differences are the following: ^{for fixed} B 0’ B cfJ ^and

4,7 RH ^is higher and Pc ^{lower as} compared ^{to the case} RK = 15. For constant B 0 ^and B cfJ’ t/J c ^{is smaller ;}

thus a decrease in RK ^causes the w range of existence of PD to shrink.

It is thus seen that the effect of making ^the applied

field oblique ^{can differ} qualitatively ^as compared ^to

the case in which H is applied ^{normal to the} plates.

Again, strong anchoring ^{of the} director, small w ^and high RK ^{seem to} favour the existence of PD.

5. Oblique ^director orientation ; symmetric ^field.

lies in the xz plane ^{and is normal to} no ⁼ (c 9’ 0, So) - ^The

differential equations governing ^{PD are} equations (10) ^of [10] ^which again support ^{the two} independent

Modes of section 3. As Mode 2 has ^a much higher

threshold than Mode 1, only the latter is studied.

For weak anchoring ^the boundary ^conditions become, ^{with the ansatz} of section 3,

HD is governed by

H couples only ^to 0’ and therefore HD is unaffected

by B.. ^In equations (11), (12), Kl ^is replaced by d2.

For rigid anchoring, again, qc = 7r/2.

Owing ^{to the} obliqueness of the director orienta-

tion, K3 ^{and 0 enter the} picture ^{as additional} parameters. ^{It is} convenient to define the ordered triad K ⁼ ( K,, ^K2, K3 ^with RK continuing ^{to repre-}

sent K1/ K2" ^{To study the effect of K3, a large value}

(= 15 ^{and a smaller value} 7) ^{have been}

chosen for it.

5.1 EQUAL ^ANCHORING STRENGTHS ; Be ⁼ Bo

^B.

The director tilt angle ^{is fixed at a low}

value ( 0

0.1 ) . ^The variations of RH and Pc ^with RK ^{are shown in} figures ^5a-5d for different anchoring strengths ^{B and} K3. The results are similar to those of figures la, Ib drawn for the planar ^{case, (J = 0.}

When RK ^is decreased, RH ^increases and Pc ^diminis-

hes till, ^as RK -+ RKL, RH - 1 ^and pc -+ ^{0. The}

quantitative differences between the figures ^{5 and 1}

are the following : (i) ^{For constant B and} RK, RH and Pc increase with K3 ; ^thus, when K3 ^{is enhanced,}

the PD domain size decreases and the PD threshold gets enhanced relative to the HD threshold. (ii) RKL increases with K3 ^{for a} given anchoring strength B ; thus enhancement of K3 reduces the RK range of existence of PD.

These effects might ^{be due to the} following

reason : K3 has the effect of enhancing the elastic energy associated with HD and PD. However, ^while

the increase for HD is associated with 0’ alone, ^the

increase in elastic energy ^{of PD occurs for 0’ and} 0’.

This seems to account for the increase in RH ^{and a}

decrease in the domain size.

Fig. ^5.

Oblique ^director orientation ; symmetrical field ; equal anchoring strengths ; B 8

Bo

^{B. The initial}

director orientation with x axis

0

0.1. K3

^{7 for} (a, ^{b, e,} f) ^and K3 ^{= 15 for} (c, ^d, g, h). (a, c) ^{Plot of} RH ^vs. RK ^and (b, d) plot of pc ^vs. RK for different anchoring strengths B

(1) ^oo (rigid anchoring) (2) ^1W (3) ^{1()3 -’} (4) ^{103.21 .} (e, g)

Plot of RH ^{vs. B and} (f, h) plot of pc ^{vs. B} for different materials with RK

^{(1) 15 (2) 10 (3) 5. In general,} when ^is increased, ^{the PD} domain size diminishes. The ranges ^of existence of PD with respect ^to RK and B also shrink when K3 ^is

enhanced (Sect. 5.1).

Figures ^5e-5h, which illustrate the variations of

RH and Pc ^{with B, are similar to figures lc, Id ;} RH -+ ¹ and p, -+ ^{0 as B -} Bm. Predictably, ^the quantitative differences are the following : (i) ^For

constant B and RK, ^, RH and Pc increase with

K3. (i i ) ^{For fixed} RK, ⁹ Bm ^increases when K3 ^is

enhanced. Thus, ^{for a} given ^material, the B range of

occurrence of PD shrinks when K3 is increased.

Figures 6a-6h contain plots ^of RH and Pc against

the initial director tilt angle ^{0. For fixed K and} B, RH increases with 0 such that RH -+ 1 when 0 - eM;

thus, PD is favourable for 0 -- Om and HD for

0 > OM, ^{For constant K and 0,} RH ^{increases as B is}

diminished ; ^a predictable consequence of this is that for a given K, BM decreases when B is reduced ;

Le. the 0 range of existence of PD gets ^curtailed when the anchoring ^{on the} perturbations ^{is slacke-}

ned. For fixed RK ^and B, Om ^decreases when K3 ^is enhanced ; thus, ^an increase in K3 also reduces the 0 range of occurrence of PD. Om ^{increases with} RK ^for given K3 ^{and B ; PD exists over a} wider 0 range when RK ^{is enhanced.}

The behaviour of Pc with 0 ^{is found to be rather} different from the variation of pc with other parame- ters discussed in previous sections. For constant K and B, _pc ^{decreases to zero as 6 -} Om" ^However,

when 0 is increased from a low value (close ^{to the} planar orientation), Pc initially increases ; ^{i.e. the}

PD domain size first diminishes and then increases

as the director tilt angle ^is changed ^{from the} planar

orientation. This initial increase in p,, ^{becomes more} marked when (i) ^{B is} large ^{for a} given ^K, (ii) RK ^is large for constant B and K ^and (iii) K ^is

enhanced for fixed RK ^{and B. When 0’ and} 0’ ^are

not strongly anchored, PC diminishes continuously ^to

zero as (J (JM ^{even for} high K3 ^or RK. [It ^{must also}

be stated that when K3 is much smaller ( =1 ) , Pc does not increase with 0 even for high RK ^{or B ;}

these curves have not been shown.]

As was mentioned earlier, K3 ^{seems to contribute}

more towards the elastic energy of PD, ^{via the} additional term Wa = K3 ( S 9 cP:z) 2, ^{than to the HD}

elastic free energy. As 0 is increased, Wa ^becomes larger ; ^this might ^{account for the initial} increase in Peas ^{also the} enhancement in R H ; ^an increase in RH implies that the PD threshold increases faster than the HD threshold when 0 is enhanced. As HHF ^{is the}

lowest possible ^field threshold with p

0, ^{it is} natural that as 0 - (JM ^and R H -+ 1, Pc ^0.

5.2 UNEQUAL ANCHORING STRENGTHS ; § Bo:o Bol

- Figures 7a-7h illustrate the behaviour of RH ^and

Fig. ^6.

Oblique ^director orientation ; symmetric field ; equal anchoring strengths B 8

Bo

^{B; RK = 10 for (a, b, c,}

d) ^and RK ^{= 15 for} (e, f, g, h) ; K3

^{7 for} (a, ^{b, e,} f) ^and K3 ^{= 15 for} (c, d, g, h) ; ^B

(1) ^oo (rigid anchoring) (2) ¹⁰⁴ (3) ^103.5 (4) ^{103.25 in all} diagrams. (a, ^{c, e,} g) ^{Plots of} RH ^{vs. 0 and} (b, d, f, h) plots of Pc ^{vs. 0. A reduction in} K3 ^{and an}

increase in RK ^{and B seem to} favour the occurrence of PD. The PD domain size decreases and then increases when 0 is enhanced (Sect. 5.1).

Fig. ^7.

Oblique ^director orientation ; symmetric field ; unequal anchoring strengths ; B,:0 B.; initial director tilt 0

0.1. K3 = 7 ^for (a, b, ^e, f) ^and K3 ^{= 15 for (c, d, g,} h) ; Strong anchoring ^{of 0’} (Bo = 104) ; ^{(a, c) Plots of RH vs.}

RK ^and (b, d) plots of p, ^vs. RK for different anchoring strengths B 0

(1) ^103.7 (2) ^103.4 (3) 103.2. Strong anchoring ^{of 0’;}

B 0 = 104; (e, g) ^{Plots of} RH ^vs. RK ^and (f, h) plots of pr ^vs. RK ^{for different} anchoring strengths B _ ^{( 1 ) 103 (2) 102}

(3) ^{101.5.The results are similar to those of} figures 2a-2d which represent planar director orientation. When K3 ^is

enhanced, ^the RK range of occurrence of PD becomes narrower as compared ^{to the} planar ^case (Sect. 5.2).