rectilinear low-frequency shear of homogeneously aligned nematic liquid crystals

16
Rectilinear Low-Frequency Shear of Homogeneously Aligned Nematic Liquid Crystals Author(s): S. J. Hogan, T. Mullin and P. Woodford Source: Proceedings: Mathematical and Physical Sciences, Vol. 441, No. 1913 (Jun. 8, 1993), pp. 559-573 Published by: The Royal Society Stable URL: http://www.jstor.org/stable/52321 . Accessed: 07/05/2014 18:02 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings: Mathematical and Physical Sciences. http://www.jstor.org This content downloaded from 169.229.32.136 on Wed, 7 May 2014 18:02:52 PM All use subject to JSTOR Terms and Conditions

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Page 1: Rectilinear Low-Frequency Shear of Homogeneously Aligned Nematic Liquid Crystals

Rectilinear Low-Frequency Shear of Homogeneously Aligned Nematic Liquid CrystalsAuthor(s): S. J. Hogan, T. Mullin and P. WoodfordSource: Proceedings: Mathematical and Physical Sciences, Vol. 441, No. 1913 (Jun. 8, 1993), pp.559-573Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52321 .

Accessed: 07/05/2014 18:02

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings:Mathematical and Physical Sciences.

http://www.jstor.org

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Page 2: Rectilinear Low-Frequency Shear of Homogeneously Aligned Nematic Liquid Crystals

Rectilinear low-frequency shear of homogeneously aligned nematic liquid crystals

BY S. J. HOGAN1, T. MULLIN2 AND P. WOODFORD3

'Department of Engineering Mathematics, Queen's Building, University Walk, Bristol BS8 I TR, U.K.

2Department of Atmospheric, Oceanic and Planetary Physics, University of Oxford, Parks Rd, Oxford OXI 3PU, U.K.

3UFB Mathe'matiques de la Descision, Universite' de Paris IX Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France

A thin film of nematic liquid crystal (ZLI 1085) is sandwiched between two horizontally mounted glass blocks, whose faces have been treated to align the molecules of the liquid parallel to the plane of the blocks. By moving one block relative to the other in its own plane, the liquid crystal is subjected to an oscillatory linear shear. Above a certain frequency-dependent amplitude, mechanical Williams domains of alternating bright and dark stripes are observed perpendicular to the direction of shear. A theoretical analysis of this phenomenon is carried out to provide predictions for both the thickness of the stripes and the critical amplitude as a function of frequency. Good agreement is found between the experimental and theoretical results.

1. Introduction

Liquid crystals are fluids that can exhibit anisotropy in their optical, electrical, and magnetic properties. In everyday use, they are found in displays ranging from watches to calculators and televisions. These uses can be considered as essentially static since little or no fluid flow is involved. Indeed the length scales in these applications would suggest that any flow phenomenon would be linear. But since a liquid crystal is fundamentally non-newtonian in character, some interesting dynamical behaviour is also to be expected. A good review of these instabilities can be found in the book by Blinov (1983).

In this paper we consider a nematic liquid crystal subject to an oscillatory linear shear. The crystal is sandwiched between two horizontally mounted glass blocks. At the boundary, the long thin molecules of the crystal line up parallel to the plane of the specially treated glass surfaces, providing the so-called homogeneous alignment throughout the fluid layer.

When one plate is moved in an oscillatory manner in its own plane, the molecules in the interior of the crystal remain horizontally oriented on average while moving back and forth. When the amplitude of oscillation is increased above a certain critical value, which is a function of frequency, a pattern of alternating bright and dark stripes (mechanical Williams domains) are observed under normal room lighting. The pattern is fixed with the stripes running perpendicular to the shear direction. The

Proc. R. Soc. Lond. A (1993) 441, 559-573 ( 1993 The Royal Society

Printed in Great Britain 559

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560 S. J. Hogan, T. Mullin and P. Woodford

separation of the stripes is usually the same order as the gap between the plates. Some very carefully controlled experiments with the crystal ZLI 1085 bear out the theoretical conclusions, with good agreement between the two sets of results.

Williams (1963) domains were first discovered by the application of an electric field to a thin layer of liquid crystal. They have also been observed during experiments on elliptical shearing by Guazzelli (1991). The work of Kozhevnikov (1986) is related to ours in that he examined theoretically and experimentally the results of oscillatory shear when the liquid crystal molecules are aligned perpendicular to the plates. He too observed mechanical Williams domains. A different form of instability in an identical situation but on a much larger length scale has been discovered by Clark et al. (1981) who found alternating bands of opposing twist running parallel to the shear direction.

This paper is organized as follows. The theoretical analysis is given in ??2 and 3. In ? 4, the experimental arrangements are discussed in detail. In ? 5, the two approaches are compared and discussed, together with topics for further work. The main conclusions are given in ?6.

2. Governing equations

The equations describing the motion of a nematic liquid crystal are well-known (see the review articles by Stephen & Straley (1974, ?VI) and Leslie (1979, ?C)).

Assuming incompressibility, we have

V v = 0, (2.1)

where v is the macroscopic velocity of the liquid crystal. In addition, by conservation of momentum, we can expect an equation of the form

Dv. OP Dt = x+

- (2.2)

where the density of the liquid crystal is denoted by p (assumed constant), P is the pressure and oij is an asymmetric tensor to be given later.

The final equation describing the motion of a liquid crystal has no counterpart in isotropic fluids. As mentioned in the introduction, liquid crystals possess anisotropy in some of their material properties. In the nematic case, this stems from the long thin molecules generally aligning up in one direction usually denoted by n, called the director. The magnitude of n is not important and so we take

n = 1. (2.3)

In most cases, any relative motion in the layers of fluid will disturb any alignment and the director will rotate. By conservation of director angular momentum, we have

D12 IS2 = r?+rv, (2.4)

Dt e

where I is the director moment of inertia per unit volume and Q is the local director angular velocity given by

Dn = n A Dt. (2.5)

The two terms on the right-hand side of equation (2.4) are torques on the director (per unit volume) due to elastic (Fe) and viscous (Fv) forces. No electromagnetic

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Homogeneously aligned nematic liquid crystals 561

torques are present in the current investigation. The derivation of the form of these torques is not given here (see, for example, de Gennes 1974). Suffice it to say that it can be shown that

Fe = n A h, (2.6) where h, the molecular field, is given by

hi i (8ni,j),t (2.7)

and W is the elastic energy, used here in the form of the one-constant approximation 3

W = 1K E (Vn )2 (2.8) i=1

and K is the material elastic constant. In addition

FV=-yln A N-7y2n AA n, (2.9)

where Yil 1Y2 are rotational viscosities, N is the rate of change of the director relative to the background fluid given by

Dn =Dt -wAn, (2.10)

where w = 'V A v and A is the rate of strain tensor given by

?j =1(vijy+V,i). 2.1

At the low frequencies we are considering, the left-hand side of (2.4) is much smaller than either torque on the right-hand side, and so our final equation reduces to the 'balance of torques'

Fe+Fv = 0, which can be rewritten using equations (2.6) and (2.9) as

nAh = y1nAN+y2nAAfn. (2.12)

The stress tensor a takes the form

0- > = Ol n, npAkpni ni + a2Ni n, + a3Nj ni + a4Aij + (x5Aiknk nj + X6Ajknk (2.13)

(see, for example, Leslie 1979). The coefficients acx (i = 1-6) are called the Leslie coefficients and are subject to various constraints, although not all need be positive. We follow Parodi (1969) who showed that only five of these coefficients are independent and that

cc2?+ c3-= x6-c/s5 (2.14)

We note that )Yl = ?X3 - C21 )'2 =X ?6- OX5 (2.15)

3. Theoretical analysis

The approach adopted here is similar to that of Kozhevnikov (1986). From figure 1, we take a set of cartesian axes fixed in the upper plate, with the x-axis parallel to the shear direction and the z-axis perpendicular to the layer. The y-axis is then parallel to the Williams domains. With this configuration, a fully two-dimensional approach can be made and thus we set

v = (vx,0, vZ) (3.1)

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562 S. J. Hogan, T. Mullin and P. Woodford

glass blocks v

ZT ,x b

flat toA/5 r 000oooooooooo ,l3sin ot a

linear bearing r

light

front surfaced mirror Figure 1. Sketch of the cell and coordinate axes.

and n = (cos q, O, sin q), (3.2)

where qO(x, z, t) = 0 corresponds to the positive x-axis. At the plates themselves

q = 0, vx = 0, v, = O on z = 0, (3.3) qS = O, vx = w/Icoswt, vz = 0 on z =-h, (3.4)

where /1 is the amplitude of oscillation and w its frequency. In addition the viscosity coefficients acl and OC3 are assumed to be much smaller than the other acc and are ignored. Thus from equations (2.14) and (2.15)

Yl =-2 = (5-(6 =-72 Y- (3.5)

To solve equation (2.2), the components of the stress tensor a must first be calculated. Thus

=XX =-c2Cosqsin (Do +Wj4w) 4> x +?(ac+a6)[> cXos2 +Asinoscoss , (3.6)

XZ =-(X 2sin2(Do +W) +c4A

+a5[, sinqGcosqs+Asin2] + .6 a$z sin Gcos +A sn2qs5, (3.7)

7X, = 2cos 26(Dto w 2A

+ a, [- sin 55 cOS 5 +A GS ]+ a6 [a sin 0 cOS 0 +A sin2 ] (3.8)

oZz = 2sin0 cosq(I+w)?+c4 z?(a5 +?c6)[aYzsin2qS+Asinqcosq , (3.9)

where A = A =A A =2(>a+a ) and w= (O,w,O),

where w = -I Iv v

2 az axf

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Homogeneously aligned nematic liquid crystals 563

Using these expressions, the two non-zero components of equation (2.2) become

Dv dP [~a2V aA P D + a = (X4 [a + aA

ax ax az Do +W 5 cos 20 + a sin 2q5 Dt a+

sin qS cos a

I ?w) + sin2o a('5 + W

+ a2V A 2 2avZ (a3 A ax Dt ) a z az az?ax

+sin2q5[A ao aVxaos l aVZ+A

+25 ~ ~ ~ ~~_ ?q av~ \

az ax ax 2 0aZ aXJ (aa Cos2 +

aA sin2 0 + cos 2 avx a A aV a(1 -6-xy + [az aa0+aavzcoY+( xaz ax)

(2.12) r s t(3.10)

___ ___ __ I a x o a V + O

+-V = 2sin55 0 sav-aq5 1 / a2v aA12 az ax ax 2 a~xaz ax and

Dy aP 2VZ ?aAl P /+yj=X4 [ z2 aXj

Proc.~ D R. Soc LodA(93

os2 q5 w +os w) +ao singS cos2q! _0 Dt aA a

(a __S __ o+ snoooa(o+

a2VZ in 0+ A

Cs2 +cos 2q5(z o +j a

asx25 cosq5+ax az + a5~~a

+si2o A'Oavz__

' 02 aA~

ax az dz 2 kax az az J (a2V Z+aA\ sin2Ocos2 q(avx ax+-Aaz)

a ' ~ Z2 IaX; in2ax2 a

giving the first and second of the general governing equations. In addition equation (2.12) reduces to

__o-K2 = y sinq5os5(jz>cos av aV+y os2 5avz -y sin2 q5aVx, (3.12)

giving the third of the general equations of the problem. Equations (3.10), (3.1 1) and (3.12) have been derived for finite values of qS. Before

proceeding, let us take qS = In-0. The angle 0 is now measured from the vertical z- axis. If this value of q5 is substituted into equations (3.10), (3.1 1) and (3.12) and the

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564 S. J. Hogan, T. Mullin and P. Woodford

limit 0 < 1 taken, Kozhevnikov's (1986) equations (2) are recovered (subject to minor disagreements which we attribute to misprints). Returning to the present set of equations, it is assumed that 0 is small but that terms proportional to the product of 0 and the velocity or its gradients should be retained.

Equation (3.12) then becomes

'Y DO_KV2,0 =-2 y avx +y VZ (3.13)

Equation (3.10) becomes

Dv +P _2V ta2v a2v DX+aw

= (c4+ (X5+ CX6) OX2 + X2+ Dt Ox ax z2 Ox azj

+ ?X5 - o (5sa8VzX + az(0 --X) + Y6[ 8z(0 8vZ + a(0 dzzl (3.14)

and equation (3.11) gives

Dv aP a Do av 2 (v a2v Dt Oz DtJd2 axzz

+ a5 a az + z(0 az ) + 6 a ax + z(0 a)] (3.15)

where I = (O4+0C'5-y). (3.16)

In this notation equation (2.1) can be written

avx + IVz = ?. (3.17)

It is now possible to eliminate P from equations (3.14) and (3.15) by cross differentiation. Then on differentiation of the subsequent equation with respect to z and using the identity

z[az( Dt! )- t KDtZ)J = a (V2vx) +- [vv 2v - v V2v] (3.18)

the following equation is obtained

LP a V2D - _ v K ]3 V 2 _

where aX dz3Zax2dz? +a 2La20 l?% a3-azV[VX-vXvVz], (3.19)

where the operator D is given by

D = yax4+ OX az+1az4 (3.20) ,yaO4 y a2aOZ2 yaOZ4

This procedure is identical in approach to that used by Rayleigh in his study of convection in a fluid layer heated from below and Taylor in his work on Couette flow between two rotating concentric cylinders (see Acheson 1990). Equations (3.13), (3.17) and (3.19) are the main governing equations in what follows, together with boundary conditions (3.3) and (3.4).

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Homogeneously aligned nematic liquid crystals 565

(a) Basic oscillatory flow and frequency constraints

The approach to this problem is guided by the experimental observation that a basic oscillatory flow becomes unstable above a certain amplitude to produce a steady pattern of convection cells. We therefore first present the derivation of the mathematical description of the base flow.

In the case of no motion, all the molecules are lined up parallel to the plate surface. Then for suberitical amplitudes of oscillation, no observable change takes place throughout the cell in light transmitted through the liquid crystal device and hence it is reasonable to assume that the molecules remain close to this orientation. Thus let the first-order solutions for qS and v be given as

5 = 51(t), v = (vlx(z, t), 0, 0). (3.21)

Equation (3.13) reduces to 051/0t = 0 and hence

01 = 0 (3.22)

from the boundary conditions. Equation (3.17) is satisfied identically and equations (3.14) and (3.15), in the absence of a pressure gradient, reduce to

dslx g 82v1x (3.23)

The quantity y is now seen to take on the role of a coefficient of viscosity in these circumstances, as the layers of molecules slide over one another.

If the frequency of oscillation is low, so that

<< y/ph2 (3.24)

then the solution to equation (3.23) which satisfies the boundary conditions is

V1x(z, t) = (/3(1 -z/h) cos wt. (3.25)

Thus the unperturbed flow between the plates at suberitical amplitudes is modulated Couette flow, with the molecules remaining parallel to the plates.

There are other constraints on w, namely

K yh pc2/y, (3.26)

where c is the speed of the sound in the liquid crystal. The first of these inequalities ensures that the viscous moments of the flow dominate the elastic moments (Woodford 1991). The second inequality ensures that the incompressibility assumption (equation (2.1)) is satisfied (Landau & Lifshitz 1986, ?43). It should be pointed out that the experiments in ?4 satisfied the above constraints on w.

(b) Governing equations for the mean flow and perturbation

In the experiments, when the oscillation amplitude fl was increased above a certain frequency-dependent level, the domain structure appeared. Mathematically this is represented by considering small perturbations to the first order solutions 0V and vlx of the previous sub-section. Thus

qS(x, z, t) = 01(t) + 500(x, z) + 0q(x, z, t), (3.27)

v(x, z, t) = (v1X(Z, t) + vox(x, z) + vx,(x z, t), 0, vO(x, z) + v (x, z, t)). (3.28)

Perturbations with subscript 0 correspond to the steady pattern visible to the observer. Primed perturbatiens represent co-existing oscillations.

The goal is to produce an equation for the perturbation q0. To this end, the

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566 S. J. Hogan, T. Mullin and P. Woodford

solutions (3.27) and (3.28) must be substituted into equations (3.13), (3.17) and (3.19). Discarding the elastic moments of q', as mentioned earlier, equation (3.19) becomes, to first order in the perturbations,

P a V2 _ &v' ? [p- V20 + 11V l?d'V132 [ atV ] X Y aX82 aZ 5 Ox]

= aX5aP2 az [7 v xV2v +)1dza +--[.vIxV2v/]. (3.29)

Similarly equation (3.13) becomes

[ t- x -[ V2550 + OXz = - aO 8850+aa6 (330 at -ax __x_x ax a

and equation (3.17) becomes

av dvX,a' dv dv +X +Z+dOx ?Oz = 0. (3.31)

ax az ax az

To separate these equations into steady and varying parts, an average over one oscillation of the plate is taken using the operator

<f(x, z, t)> = 2 X f(x, z, t) dt. (3.32)

Note that <v=> = v> = 0 by construction and that <v,x> = K51> = 0 as shown in the last sub-section.

Using equations (3.29)-(3.31) and averaging, it can be shown that

K d3_ 20 /VO d30Y aV1 \ p/d V2/ ya aX aZ aXfo+Dvo= - daZ2 a / \ai [vlxVVz]), (3.33)

V +8 = KVlx87), (3.34)

avOx+8Voz = O. (3.35) ax az The products on the right-hand side of equations (3.33) and (3.34) do not vanish because both qS' and v' vary with t.

These averaged equations (3.33)-(3.35) are now subtracted from their unaveraged counterparts equations (3.29)-(3.31). Assuming that the relative variation of all first order oscillatory products from their mean is small and noting that the second term on the right-hand side of equation (3.29) can be neglected in comparison with the first term (Woodford 1991), the following equations for the primed quantities are obtained

[yv a Dt x a axdz2 az (3.36)

80'_8Vz = -v x80?o(3.37)

avX' + avzl = O. (3.38) ax az Proc. R. Soc. Lond. A (1993)

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Homogeneously aligned nematic liquid crystals 567

(c) The equation for the steady perturbation 00 The velocities v0x and v0z can be eliminated from equations (3.33)-(3.35) to give,

using equation (3.25) for vlx,

K(+4 = h ZKcost> V4p (

Z h) ax2 V2<vz cos t>

+ ,B/D L(' -z) a <q' cos wt>1. (3.39)

The two averaged quantities on the right-hand side of this equation must now be expressed in terms of q50. On multiplying by cos wt and averaging, equation (3.36) becomes

PlK) v2<VI sin wot> - DK<v cos wt> = -/ 2h (3.40) ly 2haX aZ2'

When the multiplicative factor is sin wt, a similar procedure reduces equation (3.36) to

P V2<v cosw ot> + D<v Isin wt> = 0. (3.41)

Therefore [D2 + () V<] Kcos wt>>= (h?D a (3.42) [ y J 2h OXdaZ2

F pi2 4w2pf 032 and + +y VJ <v' sin wt> = -2hf xz2 (3.43)

Similar treatments of equation (3.37) and (3.38) produce

<0K'cos wt> =---K <v sin wt>, (3.44) (0 O

-<V1 Cos wt> = - <vx cos wt>, (3.45)

a <v' sin wt> = -a <Kv' sin wt>. (3.46)

Then equations (3.42)-(3.46) provide evolution equations for three important averages by cross-differentiation, to give

[ pN2 031 __ D2+ V4 <VI COS ot> =-2h - D (3.47)

[D2 + (0) V4 <vl sin wt> = - ?hyV2 a35z (3.48)

[D2 + ( 2) V41 <qS' cos wt> = - (3.49)

These equations are now combined with equation (3.39). Using the identity

[D2+ ()V4 ][D<K cos ot>-P a V2<v/ Cos Ot>] = 0 (3.50)

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568 S. J. Hogan, T. Mullin and P. Woodford

we eventually find after lengthy manipulations

2-LD2+(Pi)2 V41(D?+d4P5_4-(wf) LV dx2l+ (3.51) This twelfth-order partial differential equation is central to what follows. It is to be compared with equation (6) of Kozhevnikov (1986) for the case of normally oriented liquid crystals.

(d) Solution procedure

Equation (3.51) can be solved by noting the physical result that the domains are periodic in the x direction, have no y-dependence and the molecules align themselves parallel to the plates at z = 0, h that is

00 Iz=O,h = 0. (3.52) These considerations imply that an eigensolution of the form

00(x, z) =A cos kx sinpz (3.53) can be found where p = i/h. (3.54)

The wavenumber k in the x direction will be determined below using a minimization argument. On substituting equation (3.52) into equation (3.51) and rearranging, it can be shown that K

I(g2( p)=(2KIp) 2 /(-2/ h) F() (3.55) where = k2/p2 = k2h2/R2 (3.56)

and F([) -[(G?2) [G2 +cd 1)2]j (3.57)

In this expression

G(C) - lb(a+a7)+ g(3.58)

and a = (pwo/,yp2)2 = (p(wh2/,yi2)2. (3.59)

The solution for the amplitude /? is given in equation (3.55) but the value of k (or equivalently ~) corresponding to instability needs to be determined. In fact the minimum value of F(C) is required in order to attain the threshold for instability. This occurs at ~ = 0 which must be evaluated numerically for each value of o (or equivalently a) for a given liquid crystal. In this way, the threshold amplitude /i0 is given by 80 = (2K/p)2(Z2/wh)F(~O). (3.60)

This solution will be discussed in detail in ?5 but already it appears that /80 is proportional to the inverse of the frequency and to the inverse of the depth of the liquid crystal layer. From equations (3.24), (3.26) and (3.59), the solution (3.60) can be seen to be valid for the range

(Kpl,y2)2 << a << (V/,y2)2. (3.61)

4. Experimental apparatus

The apparatus consists of two glass blocks of dimensions 10 cm x 8 cm x 1 cm mounted on an INVAR frame and spaced apart using three micrometers which are accurate to + 0.5 ,m. The faces which sandwich the liquid crystal material are optically flat to A/5 over the whole area. The lower block is mounted on linear

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Homogeneously aligned nematic liquid crystats 569

-~-- shear

Figure 2. Williams domains observed in the experiment.

bearings and is connected to a large electromagnetic vibrator which is in turn driven by an electronic oscillator and amplifier. By this means, oscillatory shear is applied along the long axis of the lower block. The entire system is mounted on a vibration isolating table inside an air temperature controlled cabinet that is accurate to + 0.1 I?C. The amplitude of the applied vibration was measured both by a linear displacement device and by using the Michelson interferometer technique suggested by Ben-Yosef et al. (1974). A sketch of the cell is given in figure 1.

The nematic liquid crystal used was a proprietory Merck brand ZLI 1085 which has a positive OC3. This material was aligned using rubbed PVA on the glass surface. The direction of rubbing was along the long axes of the blocks and no perceptible difference was found in their results when either parallel or anti-parallel rubbing was used. The cells were capillary filled and the alignment was checked using cross polars. The appearance of the Williams domains instabilities could be observed directly from the scattered incident light and in detail through a microscope. Their properties were also investigated using the diffraction pattern formed by an incident laser beam. Velocity estimates were made by observing the motion of I ,um spheres through the microscope. A typical example of the domains is given in figure 2.

The results shown in figure 3 are the observed critical shear amplitudes for the appearance of the instability plotted as a function of the applied frequency for a 50 ,um layer. We have investigated the thickness range 5-50 ,um experimentally and details of the variety of instabilities found are given in Mullin (1992). The main point to be made here is that the curve has the same qualitative shape for all the thicknesses of the layer.

The curve in figure 3 gives estimates of the critical amplitude for the first appearance of the Williams domain instability. These estimates were obtained by fixing the frequency and then increasing the amplitude in small steps until the instability is first observed. Each instability has a rapid growth rate associated with

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570 S. J. Hogan, T. Mullin and P. Woodford

E 140-

-100 1

U

c 60-

I , I I , . , . I

40 120 200 frequency / Hz

Figure 3. Experimental observations of threshold amplitude as a function of frequency for a cell gap of 50 tim.

E80-

Mc 60--

0 I 40 I l 20 100 180

frequency /Hz

Figure 4. Experimental observations of roll structure wavelength as a function of frequency.

it so that determination of a critical value, although necessarily subjective, is relatively straightforward. In addition, there is no evidence of hysteresis in either transition and so the critical value can be determined by either the appearance or disappearance of the instability.

We have drawn smooth curves through the experimental points in figure 3 to show the overall gross dependence of critical amplitude on applied frequency. However, the Williams domains also exhibit large scale collective features and are organized in patches which have length scales of the order of centimetres. The number of these macrostructures is dependent in a systematic way on the frequency of the applied shear and the thickness of the layer. This in turn means that in the very narrow parameter ranges where there are nonlinear interactions between competing numbers of patches there are sharp changes in the slope. These features are beyond the scope of present study and the interested reader is referred to Mullin (1992) for further details.

In general, the Williams domains are parallel roll-like structures that are aligned orthogonal to the direction of the shear. However, at supercritical shear rates they also exhibit dislocations, a feature which is also found in electrohydrodynamic convection (Blinov 1983). The Williams domains have a characteristic length scale of approximately the gap width and show only a weak dependence on the external control parameters. We illustrate this in figure 4 where we show a measured wavelength of the roll structures as a function of frequency for a 50 gm cell. It should be noted that the variation of this length scale over the entire cell is better than 1 ,um showing that the two surfaces are very close to parallel.

Proc. R. Soc. Lond. A (1993)

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Page 14: Rectilinear Low-Frequency Shear of Homogeneously Aligned Nematic Liquid Crystals

Homoqeneou8ly aligned nematic liquid cry8tal8 571

5. Results

In this section the results of ?? 3 and 4 are compared. For a given liquid crystal in a cell of given gap-width oscillating at a known frequency w, the quantity ac is known, as are the coefficients of the quadratic function G((). It is therefore a straightforward matter to evaluate the point C = 0(ac) at which the minimum value of F(W) occurs. Then, as w (or h or any other parameter) is varied, a new value of ac is obtained and the minimization procedure repeated. Although this can be done analytically the whole process was performed numerically, using the NAG routine E04BBF to carry out the minimization of F.

The liquid crystal used in ?4 was ZLI 1085, whose Leslie coefficients are at this time unknown. Nevertheless Dr I. Sage (personal communication) kindly provided us with the Frank elastic constants in this case. In the usual notation K1 =

20 x 10-7 dynt, K2 = 9 x 10-7 dyn, K3 =17 x 10-7 dyn. Therefore the one-constant approximation of equation (2.8), namely K1 = K2= K3= K is seen to be not unreasonable. The only other source of information about this particular crystal appears to be the paper by Clark et al. (1981). Nevertheless their results are not sufficient to reconstruct the remaining required parameters. To proceed the calculations were performed using the data for MBBA near 25 ?C namely p = I gm cm-3, K = 6x 10-7 dyn, a1 = 0.07P, c2 = -0.78P, a3 = -0.01P, c4 = 0.83P, a5 = 0.46P and c6 = -0.36P (Stephen & Straley 1974).

The Onsager relation, equation (2.14), is seen to be approximately satisfied, with an error smaller than the maximum of the two coefficients which are ignored, namely al and C3. From equation (3.5) we take

y = 0.78P (5.1)

and from equation (3.16) y = 0.26P. (5.2)

In addition

(c4 + ?5)/ = 1=.65, y/y = 0.33. (5.3)

With these parameter values, ( = k0/p) and F(C0) are plotted in figure 5 as a function of ac. It can be seen that the dependence on ac for the low frequencies used in the experiments (i.e. X << 1) is very weak. In fact in this region

Co = k0/p /0.65 (5.4)

and hence an estimate can be obtained for the lateral extent of the domain size, L, namely

L = nlo/-% / 1.5h (5.5)

using equation (3.54). This value is slightly less than that found in the experiments (see figure 5). In contrast, Kozhevnikov (1986) found L / 2h for the normally oriented case.

In figure 6 equation (3.60) is plotted against the frequency for h = 50 ptm. It can be seen that the overall agreement in slope between figures 3 and 6 is excellent for 100 Hz < f < 200 Hz, but for lower frequencies, where the elastic moments become more important, the agreement is not so good. Nevertheless there remains an almost constant difference between the two figures at the higher frequencies. There

t I dyn = 1O' N.

Proc. R. Soc. Lond. A (1993)

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Page 15: Rectilinear Low-Frequency Shear of Homogeneously Aligned Nematic Liquid Crystals

572 S. J. Hogan, T. Mullin and P. Woodford

0.65251(a)

1.900

0.64754 41/2 1.896

0.6425 1.892

0 4 8 0 4 8 104cr 104cr

Figure 5. 2 and F(C0) as a function of c.

200

-= 100

0 100 200 f/Hz

Figure 6. Theoretical calculations of threshold amplitude as a function of frequency for h = 50 gm.

are several possible explanations for this difference. The most obvious reason is that the theoretical curve is drawn for MBBA whereas the experiments were performed using ZLI 1085. Yet the value of F is rather robust no matter what reasonable choices are made for the Leslie coefficients. The main effect on /l0 comes about through the change in K, see equation (3.60). If K were taken to be (say) 15 x 10-7 dyn (a single average of the available coefficients for ZLI 1085), then the theoretical values in figure 6 would increase by a factor of (15/6)1 - 1.6, making the agreement much worse. It could be argued that the theoretical curve could be used to estimate some of the Leslie coefficients, by curve-fitting to the experiments. This procedure is quite straightforward since the theoretical curve actually depends on only two parameter combinations, namely (a4 + OC5) and OC2. Nevertheless agreement could only be obtained in this way for a continuum of unphysical parameter values, with y < 0.

An examination of the order of the terms omitted from the analysis indicates that their inclusion may not improve matters greatly. But a clue to the resolution of this problem lies in a closer examination of the flow prior to instability. As /? is increased a mean flow is set up in mid-cell (part of a larger mean circulation pattern) and this appears to act as a trigger for the instability at an earlier stage than predicted. Another reason could be that the suberitical flow is not of the form assumed in

Proc. R. Soc. Lond. A (1993)

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Page 16: Rectilinear Low-Frequency Shear of Homogeneously Aligned Nematic Liquid Crystals

Homogeneously aligned nematic liquid crystals 573

equation (3.21). The full theoretical treatment of these problems must be set aside for future work.

6. Conclusions

Mechanical Williams domains have been observed in a homogeneously aligned liquid crystal cell, arranged perpendicularly to the direction of the rectilinear low- frequency shear. The critical amplitude is a decreasing function of frequency. A theoretical analysis has been carried out in which the stability of modulated Couette flow (with the director remaining parallel to the plates) is examined. The resulting curve has an inverse dependence on frequency and on cell gap width. Comparison between theory and experiment is encouraging, particularly in the range 100 Hz < f < 200 Hz. The presence of an induced mean flow appears to trigger the instability at a lower amplitude than theoretically predicted.

S. J. H. is grateful to the SERC for support in the form of an Advanced Fellowship in Applied Mathematics. T. M. acknowledges the support of his research through the SERC Nonlinear Initiative. P. W. was supported by an SERC studentship while completing the MSc course in Mathematical Modelling and Numerical Analysis at the University of Oxford. The authors are grateful to D. Binks and N. Porter for assistance with some of the experiments and to K. Long for manufacturing the apparatus and preparing the cells.

References Acheson, D. J. 1990 Elementary fluid dynamics. Oxford: Clarendon. Ben-Yosef, N., Ginio, 0. & Weitz, A. 1974 Measurement and analysis of mechanical vibrations by

means of optical heterodyning techniques. J. Phys. E 7, 218-220. Blinov, L. M. 1983 Electro-optical and magneto-optical properties of liquid crystals. Wiley.

Clark, M. G., Saunders, F. C., Shanks, I. A. & Leslie, F. M. 1981 A study of flow alignment instability during rectilinear oscillatory shear of nematics. Molec. Cryst. Liq. Cryst. 70, 195-222.

de Gennes, P. G. 1974 The physics of liquid crystals. Oxford: Clarendon. Guazzelli, E. 1991 The motion of defects in convective structures of the elliptical shear instability

of a nematic. In Nematics (ed. J.-M. Coron, J.-M. Ghidaglia & F. Helein), NATO ASI Series C (Mathematical and Physical Sciences) vol. 332, pp. 141-171. Dordrecht: Kluwer.

Kozhevnikov, E. N. 1986 Domain structure in a normally oriented nematic liquid crystal layer under the action of low frequency shear. Sov. Phys. JETP 64, 793-796.

Landau, L. D. & Lifshitz, E. M. 1986 Theory of elasticity, 3rd edn. Oxford: Pergamon. Leslie, F. M. 1979 Theory of flow phenomena in liquid crystals. Adv. Liq. Cryst. 4, 1-81.

Mullin, T. 1992 Hydrodynamic instabilities in a nematic liquid crystal device under oscillatory shear. Preprint.

Parodi, 0. 1969 Stress tensor for a nematic liquid crystal. J. Phys., Paris 31, 581-584.

Stephen, M. J. & Straley, J. P. 1974 Physics of liquid crystals. Rev. Mod. Phys. 46, 617-704.

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Woodford, P. 1991 Oscillatory shears in nematic liquid crystals. MSc thesis, Oxford University, U.K.

Received 22 September 1992; accepted 27 November 1992

Proc. R. Soc. Lond. A (1993)

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