Orientational oscillations of stripe patterns induced by frustrated drifts

7 July 1997

PHYSICS

Physics Letters A 23 1 ( 1997) 185-

EJ_.WVIER

LETTERS

A

190

Orientational oscillations of stripe patterns induced by frustrated drifts Rainer Schmitz a, Walter Zimmermann a FORUM Modellierung und lnstitutfir b Mu-Plunck-lnstitutfr Received

a*b

Festkiirperforschung, Forschungszentrum Jiilich, D-52425 Jiilich, Germany Physik Komplexer Sysreme, D-01187 Dresden, Germuny

1 July 1996; revised manuscript received 28 February 1997; accepted for publication Communicated by A.R. Bishop

16 April I997

Abstract It is shown that frustrated drifts in two-dimensional and anisotropic pattern forming systems may lead to orientational oscillations of stripe patterns. Frustrated drifts may occur for instance in electroconvection in nematic liquid crystals with a spatially periodic variation of the so-called pretilt angle of the director field. The prediction of the orientational oscillations of stripe patterns is based on a symmetry adapted model. The related experimental situation is sketched where the phenomenon should be observable. 0 1997 Published by Elsevier Science B.V. PACS: 47.20. - k; 5.45. + b; 6 I .30

1. Introduction The understanding of pattern formation has been put forward during the recent years by a surge of exciting experimental and theoretical works [I]. A number of these works focus on effects of periodic and random inhomogeneities in pattern formation. Periodic ones may lead to commensurate-incommensurate transitions and random inhomogeneities may induce localization of wave patterns [2-l I]. Both types of inhomogeneities are studied in various fields of condensed matter physics too. A class of inhomogeneities which breaks the reflection symmetry locally (e.g. local drifts) affects transitions to spatially homogeneous states and bifurcations to spatially periodic patterns in a different manner. According to such symmetry breaking perturbations stationary periodic patterns may become time dependent, as demonstrated by calculations for 0375-960 I /97/$I7.00 0 PI/ SO375-9601(97)0030

I997 Published t-0

a model equation and for thermal convection with undulated container boundaries [4,6,5]. This phenomenon has been called Hopf bifurcation by frustrated drifts and does not occur for homogeneous phase transitions e.g. near thermal equilibrium, because the interplay between the wavelength of the pattern and a typical length scale of the inhomogeneities is essential for the occurrence of time-dependent solutions. Here we combine three complementary aspects of inhomogeneities in pattern formation. (1) Pretilt boundary conditions for the director field [ 121 break the reflection symmetry in the plane of the convection layer and lead to drifting patterns in electroconvection in nematic liquid crystals [IX- I61 as shown recently by theoretical considerations and by experiments. (2) Spatially varying drifts may force timedependent patterns [5,6]. (3) Digitizing the electrodes at the confining boundaries leads to an interesting

by Elsevier Science B.V. All rights reserved.

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interaction behavior between patterns of neighboring regions [3,171. According to the technological state of the art in display manufacturing [ 181 and new optical alignment techniques [19] it seems possible to prepare cell boundaries with a spatially periodic variation of the pretilt angle between the nematic director and the confining boundary. Such a spatially periodic pretilt angle, which is assumed in our theoretical investigation, breaks simultaneously the continuous translation symmetry in the convection cell as well as the local reflection symmetry. Hence, above a critical value of the pretilt angle a Hopf bifurcation by frustrated drifts and a time-dependent behavior of the convection patterns is expected [6]. This conclusion is based on an analysis of an amplitude equation which is adapted to the respective symmetry of a plausible experiment. The time dependence of the solutions may also be manifest in oscillations of the orientation of the stripe patterns, which has not been reported before. Our calculations described below are rather independent of a special system. We only assume a two-dimensional anisotropic pattern forming system with a broken translational and locally broken reflection symmetry. Nevertheless, to be specific we always have a well defined experimental design for electroconvection in nematic liquid crystals in mind as indicated in Figs. l-3.

2. System Electroconvection in nematic liquid crystals is a well characterized pattern forming system [ 12,201,

Fig. I. A sketch of an electroconvection cell filled with a nematic liquid crystal. Without an applied voltage across the cell one has the so-called planar alignment of the nematic director, n (indicated by the bars), which is in our geometry parallel to the +I direction and parallel to the confining top and bottom plates. When the amplitude of an applied ac-voltage crosses some critical value then convective motion sets in and the director field is periodically deformed as indicated in this sketch.

Letters

A 231 (1997)

185-190

Fig. 2. Sketch of a convection cell with an angle f 8 between the director field and the top and bottom plate, respectively. For such a director configuration the convection rolls are drifting near onset [ 1% 161. (The small periodic deformations of the director field induced by the convective flow, as indicated in Fig. I, are discarded in this sketch.)

where convection patterns, as indicated in Fig. 1, occur above a critical amplitude of the applied acvoltage. For the most often investigated configuration the orientation field, the director n (n = - n), is parallel to the confining upper and lower plates. In this case the first instability leads often to stationary and spatially periodic convection rolls. A pretilt angle between the director, n, and the confining plates is essential for the functioning of commercial displays [ 181, whereby its effects on electroconvection were investigated recently [ 13- 161. Here we assume for simplicity homogeneous pretilt angles f3 at both boundaries as depicted in Fig. 2. It is also easy to see that for this configuration the reflection symmetry is broken with respect to the x direction. As a consequence the convection rolls are drifting as shown theoretically [13-151 and experimentally [ 161, whereby the drift velocity depends on the sign of the angle 0 and its modulus [ 13-151. Here we consider a situation where the pretilt angles between the nematic liquid crystal and the confining boundaries change periodically either along the x or along the y direction, cf. Fig. 3. In the first case the reflection and the continuous translation symmetry with respect to the x direction are broken simultaneously. In the second case the reflection symmetry with respect to the x direction and the translation symmetry in y direction and broken.

3. Model Since we are mainly interested in qualitative consequences of the broken symmetries, we do not solve the equations of motion of electroconvection of nematic liquid crystals [12,21] for a periodic pretilt

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For this approach it is assumed that the envelope A(x, y, t) is slowly varying on the typical wavelengths A = 2rr/q, of the pattern [25]. Eq. (2) has to be modified to describe the effects of a spatially varying pretilt angle. The leading contributions of the pretilt angle are a spatially dependent part of the control parameter E and a spatially periodic “frequency” i M( X, y) corresponding to a periodic drift [6]. Since the dynamics of the pattern comes from the space-dependent frequency iM( X, y) we focus in this work on the following form of the equation,

a)

b)

a,A=[E+iM(x,

y) +

A -

IAI*]A.

(3)

For a constant M( X, y) = + o it is easy to see that i

A = F exp(iot) is a simple solution of Eq. (31, with F = 6. After a transformation via Eq. (l), this

Fig. 3. The homogeneous. ment of the periodically The arrows rolls in each

pretilt angle indicated in Fig. 2 was assumed to be As an example of an inhomogeneous pretilt aligndirector we assume a pretilt angle, 0, varying either in x direction (a) or periodically in y direction (b). indicate the virtual drift direction of the convection area of the x-y plane.

angle at the boundaries. Instead, we start with a so-called amplitude equation for a supercritical bifurcation in electroconvection [22] and generalize it in a manner to model the essential effects of spatially periodic pretilt angles. The convection rolls occurring in electroconvection at threshold may be of normal or oblique orientation with respect to the director orientation at the boundaries [23,24]. For normally oriented rolls the linear modes of the whole convection structure can be formally written in the form, u( x, y, z, r) = [ A( X, y, t)e’qc.’ + c.c.] u,( z, q,).

(1)

The components of U(X, y, z, t> describe the physical variables of the system, such as the flow field, the electric field and the director distortion of the pattern and (I,,( z, q,> describes the variation of those fields across the fluid layer. qC = 271-/h is the wave number of the stationary pattern at threshold (cf. Figs. 1 and 2). The amplitude equation for the envelope A(x, y, t> of the periodic pattern in Eq. (1) is of the following resealed form [22], a,~=

[t.+

A - I ~1’1~.

(2)

solution corresponds to a traveling wave u a cos( qc x 5 w t). For the frequency M(x, y) we are using throughout this work there is simple harmonic behavior either in x direction or in y direction, corresponding to the two pretilt configurations given in Fig. 3, respectively: M(X) = 2G cos( kx),

(4a)

M( y) = 2G cos( ky).

(4b)

Since the spatial variation of M( X, y) is incorporated on the level of the amplitude equation we have to assume that M(x, y) is slowly varying on the scale of the pattern, k =z q,. The solutions of Eq. (3) can be roughly classified into those which are symmetric, A,(x) = A,(x + A,) with respect to a translation by A, = r/k or antisymmetric, A,(x) = - A,(x + A,). A,(x) correspond to the harmonic and A,(x) correspond to solutions being sub-harmonic with respect to the modulation Mx, y).

4. Threshold

and eigenmodes

The modulation M(x. y) affects already the bifurcation from the trivial state, u = A = 0, and may switch it from a stationary bifurcation into a Hopf bifurcation [6], depending on the amplitude G. For the homogeneous limit, M(x, y) = 0, the linear part of Eq. (3) is solved by A = F explcrt +

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i(Qx + my)] and the neutral stability condition, Re( (+> = 0, leads to an expression for the neutral curve, Em= Q* + P*. &(Q, P> separates the range where A = 0 is linear stable from the range where A = 0 becomes unstable. The absolute threshold is at cc = 0, Q,, PC = 0, where A = 0 is unstable beyond. Assuming a periodic forcing M( y> in y direction the linear stability of A = 0 can be calculated by a Floquet-Bloch ansatz A( x, y, t) = exp[ LTt + i( Qx + Py)]Cy= _ NF1 exp(ilky/2) + c.c. for finite values of the modulation amplitude G, whereas for small values of G a systematic perturbation expansion can be employed. The linear eigenvalue problem has to be solved numerically in general, however, for small values of G analytical expressions for the threshold can be obtained with a perturbational approach [26]. For such a perturbation expansion one transforms Eq. (3) with v = a, + i w and A = exp(a t)B into an equation for B. The resulting equation for B can be solved for small values of G = SC by a perturbation expansion E( 6, k) = &(O)(k) + 8&(‘)(k) + S *E(*)(k) 6J( 6 k) = &I”‘(k) + s*&*)(k) + and B;rj’~B,(;)f6B,(x)+G’B,(~)f . . . . *‘.’ Ordering the equation for B with respect to powers of 6 one obtains at the orders S ’ and S * solubility conditions for determining the frequency and threshold shifts for the harmonic solutions, F,, = 2G2/k2

+ 0( G4),

and for the sub-harmonic E,

=

k*/4

o = 0,

(5)

solutions,

+ G*/2 k* + 0( G4),

w=G+O(G3).

(6) Both thresholds hit each other at a special modulation strength G,, and one has ^a rcodimension-twopoint (CTP). For G > G,, = k‘{8/3 the Hopf bifurcation of the sub-harmonic solution has the lowest threshold, E, < E,,, with a Hopf frequency w of the order of the modulation amplitude, G. Thus time-dependent solutions occur immediately beyond threshold for G > G,,. Up to G a GcTp the analytical expressions (5) and (6) agree rather good with a fully numerical solution of the eigenvalue problem. (Hence we skip the numerical curves.) The harmonic and sub-harmonic solutions of the linear part of Eq. (3) take the following qualitative

Letters

A 231 (19971

185-190

G = 0.03

M(Y)

& = 0.025

k=1/4

Fig. 4. Five snap shots, (l)-(5), of U(X, y, t) are shown during one half of a period of the orientational oscillation of a stripe pattern. Part (6) has to be compared with part (3) and shows that the “defect lines” in the second half of a period occur at a different location. The temporally periodic bending of the convection rolls is a consequence of a periodic “frequency”, M(y), which mimics a the periodic pretild indicated in Fig. 3b. The horizontal “defect lines” have a periodicity which is subharmonic to the modulation M(y).

form after transformation [26]: u,, = F,[cos(

into real space via Eq. (1)

q, x) f (2G/k2)

+O(G2)]W, u, = [ F, cos( ky/2)

cos( ky) sin( q,x)

qc),

(7a)

cos( q, x + uCt)

+Fz sin( ky/2)

cos(q,x-

+O(G)]Uo(z,

4,).

mCt) (7b)

The amplitudes F, are constants and w, = f G. The linear sub-harmonic solution is a superposition of two counter-propagating traveling waves, which are manifest in the nonlinear regime as orientational oscillations of a periodic pattern (see Fig. 4). The solutions of Eq. (3) for an x-dependent M(x), as defined in Eq. (4a). are simply obtained by replacing the y dependence in the related expressions by an x dependence and the system behaves quasi-one-dimensionally [6].

5. Nonlinear

solution

The bifurcation in electroconvection described by Eq. (2) is often supercritical [22]. A modification by an additional frequency iM(x, y> as in Eq. (3) does

R. Schmitz. W. Zimmermann/

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Physics Letters A 231 f 1997) 185- I90

not change the sign of the nonlinear coefficient and hence the bifurcation in presence of a small pretilt is expected to be supercritical, too. For the modulation M(y) as given in Eq. (4b), the numerically calculated sub-harmonic solution of Eq. (3) is depicted in Fig. 4 for six different times during the evolution of slightly more than one half of an oscillation cycle. Plotting the linear solutions in Eq. (7b) gives the same scenario. The harmonic time dependence, however, is only valid immediately above threshold. Keeping the modulation amplitude G fixed and increasing the control parameter E, then the time dependence of the solutions becomes more and more anharmonic and beyond some critical value &,,Jk), which depends on the modulation wave number k, the time-dependent solutions bifurcate into a stationary one (see also Ref. [26]). A modulation M(x) as given in Eq. (4a) corresponds to a periodic pretilt along the x direction and similar quasi-one-dimensional and time-dependent solutions are expected as discussed in Ref. [6]. While a single periodic modulation leads to periodic or stationary behavior (for smaller values of G) more complex modulations, for instance superpositions of harmonic modulations with different wave numbers, may lead to chaotic solutions behavior immediately above threshold.

magnetic field in x direction is applied [ 14,15,20], which would correspond to a reduction of the modulation amplitude G. In experiments on electroconvection another possibility of experimental variations of the parameter G is the variation of the frequency of the applied voltage [20,28].

References [I]

P. Manneville, (Academic

Dissipative

structures

Press, London,

M.C. Cross and P.C. Hohenberg, 85 I; in: Santa Fe Institute plexity,

Vol.

complex

Rev. Mod. Phys. 65 (1993) patterns in nonequilibrium

eds. P. Cladis

(Addision-Wesley,

turbulence

Studies in the sciences of com-

21, Spatio-temporal

systems,

and weak

1990);

New York,

and P. Palffy-Muhoray

1995).

[2] R.E. Kelly

and D. Pal. J. Fluid Mech.

[3] M. Lowe,

B.S. Albert

86 (1978)

and J.P. Gollub,

433.

J. Fluid

Mech.

173

( I9861 253; P. Coullet.

[41 G.

Phys. Rev. Lett. 56 (1986)

Hartung,

(1991)

[51 W.

2741.

Zimmermann

Studies poral

724.

F.H. Busse and 1. Rehberg, Phys. Rev. Lett. 66 and R. Schmitz,

in:

in the sciences of complexity,

patterns

in nonequilibrium

Cladis and P. Palffy-Muhoray

Santa

Vol.

complex

Fe Institute

21. Spatio-temsystems,

(Addison-Wesley,

eds. P.

New York,

1994) p. 273; R. Schmnz

and W. Zimmermann,

Phys. Rev. E 53 (1996)

5993.

l61 W.

Zimmemtann

and R. Schmitz,

Phys. Rev. E 53 (1996)

Rl321.

6. Conclusions

[71 M. [81 W.

Belzons

et al., Europhys.

Zimmermann,

Lett. 4 (1987) 909.

M. Serselberg

and F. Petruccione,

Phys.

Rev. E 48 ( 1993) 2699.

Based on a model calculation we suggest an experimental configuration where a novel temporal periodic orientation oscillation of a stripe pattern is expected. The model suggested in Eq. (3) is motivated by a special design of electroconvection in nematic liquid crystals (cf. Figs. l-3). The amplitude equation (3) is also related to coupled oscillator systems with amplitude degrees of freedom [27] and one might speculate which of the phenomena for coupled oscillator systems can be modeled by experiments such as described in this work. Using double periodic functions M(y) then quasi-periodic and chaotic oscillations are also very likely. If in experiments with nematic liquid crystals a pretilt angle has been prepared one might ask how the drift velocity can be modified by external fields. A reduction of the drift velocity is expected when a

Dl

L.

Howle.

R.P. Behringer

and J. Georgiades,

Nature

362

( 1993) 230; D.M. Shattuck.

R.P. Behringer,

G.A. Johnson and J.G. Geor-

giadis, Phys. Rev. Lett. 75 (1995)

1101 P.

Coullet

and D. Walgraef,

W. Zimmermann

1934.

Europhys.

et al., Europhys.

A. Ogawa, W. Zimmermann,

Lett.

IO (1989)

Lett. 24 (1993)

K. Kawasaki

52.5;

217;

and T. Kawakatsu,

J. Phys. II (Paris) 6 (1996) 305.

[I II

0.

Steinbock.

P. Kettunen

and K. Showalter,

Science

269

(I 995) 1857.

[I21

P.G. de Gennes and J. Prost, The physics of liquid (Clarendon.

Il.31 V.A.

Rashunathan.

Mol. 0yst.

II41

Oxford,

Murthy

199 (1991)

in: NATO

fects, singularities

in nematic

A. Hertrich. t 1995) 733.

and N.V.

Madhusudana,

239.

AS1 Series C, Vol. liquid

crystals:

332. De-

mathematical

aspects, eds. J.M. Coron, J.M. Ghidaglia

Helein (Kluwer,

[I51

P.R.M.

Liq. Cryst.

W. Ztmmermann, and physical

crystals

1993).

Dordrecht,

A. Krekhov

and F.

1991) p. 401.

and W. Pesch, J. Phys. (Paris) II 5

R. Schmitz, W. Zimmermnnn/Physics

190

[16] 0. Ahmetshin, V. Delev and 0. Scaldin, Mol. Cryst. Liq. Cryst. 265 (1995) 315. [17] Y. Hidaka et al., J. Phys. Sot. Japan 61 (1992) 3950; 63 (1994) 1698; Mol. Cryst. Liq. Cryst. 265 (1995) 291. [18] B. Bahadur, Liquid crystals, applications and uses (World Scientific, Singapore, 1993). [19] M. Schadt, H. Seiberle and A. Schuster, Nature (London) 381 (1996) 212. [20] S. Kai and W. Zimmermamr, Prog. Theor. Phys. Suppl. 99

(I 989) 458; W. Zimmermann, Mat. Res. Bull. 16 (1991) 46; L. Kramer and W. Pesch, Ann. Rev. Fluid Mech. 27 (1995) 515. [21] H. Pleiner and H.R. Brand, in: Pattern formation in liquid crystals, eds. A. Buka and L. Kramer (Springer, Berlin, 1995). [22] E. Bodenschatz, W. Zimmermann (Paris) 49 (1988) 1875.

and L. Kramer,

J. Phys.

Letters A 231 (1997) 185-190 [23] R. Ribotta, A. Joets and L. Lei, Phys. Rev. Lett. 56 (1986) 1595. [24] W. Zimmermann and L. Kramer, Phys. Rev. Lett. 55 (1985) 402. [25] A.C. Newell and J.A. Whitehead, J. Fluid Mech. 38 (1969) 279; AC. Newell, T. Passot and J. Lega, Ann. Rev. Fluid Mech. 25 (1992) 399. [26] W. Zimmermann, Effects of inhomogeneities in pattern formation, Habilitationsschrift, RWTH Aachen (1994), unpublished; Degenerated Hopf-bifurcation and chaos induced by frustrated drifts, Physica D (1996), submitted for publication. [27] G.B. Ermentrout, Physica D 41 (1990) 219. [28] S. Rasenat, V. Steinberg and 1. Rehbeg, Phys. Rev. A 42 (1990) 5998.