LOCALISATION AND DEFECTS IN PROPAGATIVE ORDERED STRUCTURES

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LOCALISATION AND DEFECTS IN PROPAGATIVE ORDERED STRUCTURES

A. Joets, R. Ribotta

To cite this version:

A. Joets, R. Ribotta. LOCALISATION AND DEFECTS IN PROPAGATIVE OR- DERED STRUCTURES. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-171-C3-180.

�10.1051/jphyscol:1989326�. �jpa-00229467�

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Colloque C3, supplément au n°3, Tome 50, Mars 1989 C3-171

LOCALISATION AND DEFECTS IN PROPAGATIVE ORDERED STRUCTURES

A. JOETS and R. RIBOTTA

Laboratoire de Physique des Solides, Bât. 510, université de Paris-Sud, F-91405 Orsay Cedex, France

Résumé : Les structures ordonnées propagatives produites dans la convection d'un fluide anisotrope constituent un modèle d'ondes non-linéaires. On étudie expérimentalement des phénomènes de localisation de l'amplitude et de la phase et l'on montre que deux types de localisation peuvent apparaître avec des échelles spatiales différentes. Dans un cas la structure convective ordonnée est confinée à l'intérieur de domaines isolés, dans lesquels la phase se propage à vitesse constante. Dans l'autre cas, à l'intérieur d'une structure homogène, l'ordre spatio-temporel peut être brisé par des modulations locales de ïa phase qui conduisent à des défauts de type dislocation et joint de grains.

Ces mécanismes de localisation peuvent être à l'origine d'une désorganisation progressive dans l'espace et dans le temps (complexité spatio-temporelle).

Abstract : The propagative structures found in the convection of an anisotropic fluid constitute a model for non-lineai waves. We present an experimental study of localization of the phase and of the amplitude and we find that localization effects may appear with much different space scales. In the first case the ordered structure of rolls is confined inside isolated domains inside which the phase propagates uniformly. In the second case, the space-time order can be broken by local modulations of the phase, which lead to defects analogous to dislocations and grain-boundaries. It is shown that these effects may give rise to a so-called spatio-temporal complexity.

Introduction

The convection of a liquid layer can present ordered and time-dependent states, in form of traveling waves / 1 - 3 / . In these states, the amplitude of the convection propagates with a constant velocity along the direction of the wavevector of the structure. Just as in the case of stationary states a transition to a spatio-temporal disorder may be achieved under the action of an appropriate control parameter, or may be observed under some conditions which are not yet understood. Traveling-wave convection is certainly a new prototype for non-linear waves / 4 / and although it is now extensively 3tudied in the frame of the Landau-Ginzburg and non-linear Schrodinger equations / 5 - 7 / , few experimental results were available concerning the precise modes of perturbation of the phase. More particularly, the defects of non-linear waves /8,9/ and their possible contribution to disorder have not up to now received much attention from experimentalists.

We present here an experimental study of localized perturbations of the phase and of the amplitude in traveling waves which appear in the convection of a nematic liquid crystal subjected to an AC electric field. The main feature of the liquid crystals which are anisotropic fluids, is that there exists a priviliged direction, which imposes a well-defined orientation to the convective rolls in the plane of the layer. Then the system can be considered as (spatially) quasi-unidimensional whenever it is stable against transverse perturbations. We find that this time-dependent state is not homogeneous but instead, large modulations of the phase and of the amplitude occur and make the basic state complex even at threshold. The localized modulations appear on very different space scales and may be stationary or transient. Firstly, the convective structure is at threshold, composed of an ensemble of isolated domains inside which the rolls are

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989326

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traveling with a constant velocity. Secondly, transient modulations of the phase of the rolls localized on small-scales can lead to singularities (or topological defects) such as sources and sinks. Another type of defect which corresponds to the sudden local appearance (or disappearance) of one spatial period, is analogous to a dislocation. These defects may either invade the whole space as the time goes, or they may trigger the nucleation of other defects, randomly in space and in time. They appear then as basic elements for a spatio-temporal complexity.

Emrimental Set -UD

The experimental set -up is the classical one for the study of Electro-Hydro-Dynamic (EHD) instabilities in the planar geometry. A layer of nematic liquid crystal (Merck Phase V) of negative dielectric anisotropy ea is sandwiched between two glass plates coated with semi-transparent electrodes.

The nematic liquid crystal is characterized by a long-range order of the orientation of the molecules. The local direction of the molecular alignement, which is represented by a unit vector n (the director)/lO/, is made homogeneous in the plane of the plates along, say, the x-axis (the x and y axes are in the plane of the layer). An AC voltage is applied between the plates, so that the electric field E is across the layer, along the z-axis. The applied constraint is measured by the dimensionless parameter e = (V2-V2th)/V2th, where Vth is the voltage at threshold. The frequency f of the excitation is an additional parameter, which is kept fixed in each experiment. The thickness of the sample is typically LZ= 50 ^jm and the lateral dimensions are Lx=

2.5cm and L = 1.5cm, so that the aspect ratio in the wavevector direction is of order 500. Therefore the layer can be considered as infinite as compared t o the scale of the most unstable modes (typically the Y wavelength X of the roll structure is X 5 2Lz = 100jm). The sample is enclosed in a container thermally regulated by a water bath and the temperature is kept constant within at least 0.1'C around 21'C for more than 5 hours.

It is known that EHD instabilities develop inside a well defined range of frequencies between DC and the cut-off frequency fc (conduction regime)/lO,ll/. We briefly recall that, in the classical picture / l l / , the destabilizing mechanism is due to the anisotropic drag of the ionic charges present in the medium under the action of the Coulomb force pE (p is the charge density). This drag creates a viscous torque which can reinforce the initial perturbation of the alignment. Because the driving force pE is efficient only for frequencies f lower than the inverse of the relaxation time of the charges T ( ~ 0 . 0 5 s ) , this destabilizing mechanism does not work beyond some cut-off frequency fc, which is then the upper limit of the

"conduction regime"/l2/. For each experiment, the frequency f is fixed and the voltage is continuously increased by small steps bV (bV N 0.025V) fiom the zero value. Above some critical value Vth which depends on f (typically Vth r 10 V for f r fc/2), the rest state bifurcates to a convective state. The symmetry and the time-dependence of the structure at threshold depends strongly of the two control parameters V and f 1131. We are interested here in the high-frequency part of the conduction regime, where the rolls at threshold are always aligned normaly to the molecular alignment n (i.e. along the y axis), and propagative /3/. Up to e r 0.4, no significant pertubation in the alignment of the rolls is detected.

Therefore, in this voltage range, the pattern is quasi-unidimensional and is characterized by a unique wavevector k parallel to x. The period

X

Because of the coupling between the velocity gradients and the molecular orientation, any convective flow periodic in space induces a periodic modulation of the optical axis of the crystal measured by its tilt angle d(x) over the horizontal x,y plane. Incoming light along ^l; with extraordinary polarization is periodically focalized along lines parallel to y

/lo/,

(3-5 pm in diameter) are immersed inside the layer in order to allow us to trace the trajectories of the fluid particles. The focal lines correspond t o the vertical (up and down) motion. The optical pattern in the focal plane is recorded through a CCD 512x512 pixel camera and analyzed by digital image processing. A line y=yo, parallel to the x-axis, is selected inside the image and its profile gives a measure of the transmitted intensity I(x) along this line. Successive images are periodically recorded with a time interval bt (typically

&=0.03 sec) and the successive profiles (for the same yo) are plotted one above each other. The resulting figure is a space-time diagram, where the intensity I is represented as a function of the coordinate x and of the time t (Fig.1). T h e intensity profile I(x) is a periodic succession of peaks separated by X = 2 4 k , the period of the convection. The peaks which correspond t o a up (or down) motion inside the layer, are aligned along lines ("peak lines"), of slope dt/dx = u-1, where u is the velocity u of the whole roll pattern in the x

direction (Fig 1). Peak lines parallel to the t axis would indicate a purely stationary pattern. In a uniform motion (progressive wave), the space-time diagram is a perfectly ordered 2-D structure of oblique lines.

Such space-time diagrams can be recorded over times up to some hours and are then particularly convenient t o characterize states that are slowly varying in time. The accuracy of the measurements is limited by the spatial resolution of the individual peaks and by the spatial homogeneity of the structure (absence of moving defects). In our experiments we have been able t o measure velocities u as low as 1.5 10-5X/sec. in uniform motion, for times up t o 104 sec.

Amplitude localization on larne scales: t h e domains of travelinrr rolls

The working frequency is fixed a t some value inside the high frequency part of the conduction regime: f ^N 0.8 fC = 600 HZ (here fc ^N 750 Hz). The voltage is smoothly increased from zero up to a threshold Vth = 15 V, a t which a convective ordered state develops continuously. The pattern appears inhomogeneous in space and consists of isolated domains of elliptical-like shape, inside which a periodic structure of rolls aligned along y translates uniformily in the x direction (Fig.1). The typical value of the propagation velocity is u N 0.2 Xlsec. = 10 p l s e c . The lateral dimensions of a domain are typically lx = 3d, 1 = 4d. These domains are randomly spread in the plane of the layer and no deformation of the molecular alignment is detected in the space between them. They remain stable in space, thus indicating a Y nearly zero group velocity. In this traveling wave structure, the convective variables behave like A(x)cos(kx-ut), where w = k.u. It is found that, for a typical domain, the amplitude A(x) has a lump shape that can fairly well be fitted t o a sech*(x/lx) function, which is typical of a soliton-like profile /3/.

Space-time diagrams are plotted for increasing values of the voltage and u is found t o be finite a t threshold, while the amplitude of the instability measured by the transmitted light intensity I ^N 1/12 (where

1/1

a#/&)

increases continuously from zero, indicating an apparent direct bifurcation. Simultaneously, the domains increase their extension in both directions, so that they get closer t o each other and connect completely as the constraint parameter r ::0.8. For r > 0.8, the velocity u rapidly falls down t o zero and one recovers a quasi-stationary and homogenous state. As the frequency is decreased to some value fl = 530 Hz, the velocity measured a t threshold decreases sharply and continuously, while the size of the domains diverges.

The velocity falls by two orders of magnitude over a range Af 2 10-20 Hz, and t h e homogeneous state of Normal Rolls are recovered. Up t o now these Normal Rolls were believed t o be a pure stationary state. In fact, we have recently found that the Normal Rolls also propagate but with a much lower velocity (u r 10-4 X/sec.)(Joets A. and Ribotta R., t o appear in the Proceedings of the 12th International Liquid Conference, Freiburg, 1988).

When they are represented in the space-time diagram, the traveling waves constitute a n ordered spatio-temporal structure characterized by a phase

+

Fig. 1. Time dependence of the optical density profile along x showing motion to left (travelling rolls). The domain contains five wavelengths A (f = 600Hz, E = 0.3).

Fig. 2. spatio-temporal "dislocationr1. At t = 4sec., two traveling rolls (= one period A) suddenly disappear.

There is a symmetry between the left and right traveling rolls (k H -k, or u H -u) which corresponds in the x,t space to a symmetry a w -a in the angle of the peak lines over the t axis, a = arctg u. We note that this space-time pattern is analogous to the (spatial) structure of the stationary Oblique Rolls which are symmetrically tilted over the y axis 1141. It is then interesting t o compare the topology and the role of the defects in each of these two states. For instance, we have shown that the motion and the particular topology of the dislocations and of the grain boundaries in the Oblique Rolls play an important role in the stability of the structure 1151.

Small -scale localization of the phase and defects of non-linear waves

In order to study the most simple cases of localized states we choose the experimental conditions so as to keep the basic state of progressive wave stable against any transverse disturbance (i.e. of wavevector along y). This is realized either by selecting elongated domains of small extension in the direction of the roll axis, or by reducing with thin electrodes the width of the layer along y (without imposing solid boundaries).

Accurate measurements of the velocity, using the space-time diagram, indicate that, quite often and even close t o threshold, the propagation velocity u is, apparently a t random in time, not uniform inside a progressive wave. Instead, small local modulations of u may suddenly appear a t some place. It is found that there the spatial period X ⁼ 2r/k may vary by 5 t o 10 % within one time period T = 2 r l w . The location of this perturbation is generally random. Two effects are found following this phase modulation.

1)- The dislocations:

In the first case the local wavevector k reaches a limit value k beyond which the state is

loc 1

unstable. The stable state is recovered by either creating or expelling one period if k

<

loc 1 loc 1

respectively. Then the local phase modulation relaxes diffusively and the progressive wave appears more uniform in space. There is no neat change in the average wavevector k. In the space-time representation (see Fig.2), this sudden change appears similar to the dislocation found in the Oblique Rolls structure. In particular, the deformation field around the "core" is unsymmetrical and it contains a small element of a wave with a much smaller velocity. Such a topology is also found for the real dislocations in the Oblique Rolls, where the core contains a small portion of a roll with almost opposite tilt 1151. This situation, where only one dislocation is created a t a time is obviously met when the local modulation is produced in the contact between two half-waves with a slight difference 6u in velocity (here 6u/u is of order 0.1). Here the phase modulation has a kink shape (see Fig. 3a). This is reminiscent of the so-called "misfit dislocations"

that are involved in crystal polygonization (tilt boundaries). The dislocations can also be periodically created along the t axis if the two half-waves keep traveling with a different velocity (the phase or velocity modulation although limited in space is then continuous in time). Another and more interesting case is when only a small part of the wave with a slightly different velocity is sandwiched inside a uniform wavetrain. Then, two opposite local modulations of the spatial period are created one with a compression of the front the other one with a dilation (symmetric phase profile, Fig. 3b). A pair of dislocations of opposite

"sign" would then be created, leaving invariant the wavenumber. However this situation has not been found up to now.

In any case the local phase modulation gives rise directly to a shock of the non-linear waves /4/

and the sudden jump in the local wavevector is a consequence of the finite bandwidth of the unstable modes. In effect, the shock moves the local wavevector towards t h e limit of the allowed band where it becomes unstable against the Eckhaus-Benjamin-Feir instability 116-181. The shock induces locally a dilation or a compression of some rolls and this may bring the system into the the unstable band inside the neutral limit. Then the system is pulled back to the interior of the stable band after the creation or the

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Fig. 3. a) In the contact between two half-waves with a slight difference in velocity, a phase modulation A + = +-(ut+kx) with a kink shape is created before the spatio-temporal dislocation.

b) A compression and a dilatation of the rolls along x correspond to a symmetrical phase modulation.

I n that situation, two dislocations of opposite sign would be created.

Fig. 4. Space-time diagram for a source defect. Pairs of rolls are periodically created in time. The expulsed rolls start from the defect core with a damped oscillating motion.

annihilation of a period A, respectively.

2 ) - The grain boundaries:

Usually it is found that when the phase modulation is symmetric, a pair of trains of dislocations is created, instead of a pair of dislocations of opposite sign. The regular stacking of dislocations is in fact similar to a grain (tilt) boundary because it separates two domains of waves with opposite sense of propagation. The two waves which can exist are of the form:

L = A(x,t) expi(wt+kx) for the left-going wave and R = B(x,t) expi(&-kx) for the right-going one, where A and B are complex functions of x and t. When L and R emerge in opposite direction from the defect one has a source and when L and R meet at it, one has a sink. In other words one has a source (a sink) when L stands on the left (right) side and R on the right (left) side of the defect. These two topological defects reflect, in fact, the non-linear nature of the waves. It is obvious that the sink and the source are singularities that result directly from the compressive and the dilative shocks respectively. The topology of the grain boundaries is quite complex around the core, where the phase as well as the amplitude of the waves have a damped oscillatory behavior in time along the wave direction. One may here too, notice the striking similarity with the topology of the real grain-boundaries in the Oblique Rolls 1151. In fact, the defect can be seen as a superposition of the right R and left going L waves with a mutual penetration limited in space by a non-linear damping. Then, the singularity inside the core represents the oscillatory amplitude of a standing wave and the damped oscillatory part is a constant amplitude wave L modulated by a decreasing contribution of R (on the left side of the defect, for instance) as shown on Fig. CThe location of the core of the grain boundary is defined by the position of its symmetry axis, parallel to the t-axis. I t corresponds to a separatrix between two rolls, and the nature of the separatrix (upwards or downwards motion) alternates periodically in time with a period T close to that of the two waves (T, and Tg may be different). Consequently the vorticity of the two rolls besides the separatrix is periodically inverted and the core of the defect is a "local standing wave". The phase and the amplitude of the rolls are strongly modulated in the core of the grain boundary and around it. Some of these features can be related to the fact that the allowed wavevector of a roll structure lies inside a finite band of values (limited by the Eckhaus-Benjamin-Feir instability). The amplitude of the convection decreases exponentially to zero when the wavevector passes beyond (resp. below) the upper limit (resp. lower limit) of the band of stable wavevectors. Because of the propagation, the diameter of the two rolls located next to the core is increasing with time. When their local wavenumber reaches the lower limit, the corresponding mode becomes unstable and is exponentially damped. At this time, we observe that the amplitude of the convection tends to zero and a new roll pair emerges with an growing amplitude. At every cycle of the oscillation the amplitude passes by a maximum when the roll diameter has its optimum value. The expulsion of a new pair in the core of the source acts now on the neighbouring rolls as a forcing to which the system responds as a damped oscillator (see Fig.4).

There are now two grain boundaries with opposite "charge": a source and a sink and there is an apparent full symmetry between them. An important question concerns the mutual interaction of these two defects. Since they enclose a newly formed wave (say R) with opposite velocity, a repelling indicates that R invades the domain of L. An attraction leads to the disappearance of R. Then it is clear that the stability of these two defects governs the spatio-temporal order. Preliminary studies would indicate that the stability depends on local phase modulations rather than on a global constraint (i.e. on large scales) (Ribotta R. and Joets A., to appear in the Proceedings of the Conference "New trends in nonlinear dynamics and pattern forming systems", Cargese, 1988). A grain boundary can move erratically in time and in this case it steps

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aside by multiples of half a spatial period AL/2 or AB/2. This displacement can be repeated a t every time period TL or T E such as t o appear as a uniform propagation with a velocity w (Fig 5). A remarkable case is when w = u,, or u . The defect can also move with w > u as it is shown on Fig 5, where it propagates

B L,B

with a velocity w = 120 p l s e c while the wave velocity is around 60 pnlsec. A moving defect (sink or source) is indeed the front of a shock wave.

Recently Coullet et al. have shown numerically that topological such as the grain boundary can be described in the frame of a complex isotropic 2-D Landau-Ginzburg equation 181. The defect is then described as a kink-type defect connecting two inhomogeneous (in space) solutions of the progressive wave type. It is likely that the phase modulations that give rise t o singularities can be accounted for by such an envelope equation as well. Indeed, recent numerical simulations performed by us, using a 1-D equation agree with some of our experimental results (Ribotta R. and Joets A., in preparation).

3)-Phase modulations on large scales and quclsi-coherent modes:

A sink or a source is a topological singularity which "separates" in space two waves L and R. We have just seen that a defect can stay stable in space or it may move with a velocity w that can be higher than that of any of the two waves. A limit case is when this propagation velocity w tends t o infinity (in the x,t diagram the singular line would be an horizontal). Then a wave, say L, changes a t a sudden into its opposite R as shown in Fig 6. If this reversal occurs periodically in time and on a large space scale, one has a collective longitudinal mode of oscillation of the structure. Therefore there is an interesting apparent continuity between a pure mode of oscillation in time and the singularity (sink or source). A moving grain boundary is, in fact, a front which carries with a velocity w, a singularity and when w tends t o infinity the reversal of wave occurs coherently over the entire space. The topology of this latter case is quite analogous to that of the twin boundaries in the Oblique Rolls structure. There also exists, under some conditions, another collective mode of oscillation of the structure: the longitudinal compression-dilation (therefore named "time-dependent Eckhaus" mode) with a wavelength that can be as low as that of the basic structure of convection. Reports on its observation will be given elsewhere. If we now consider the more general case of 2-D structures, then the propagative state may become now unstable t o transverse modes that can be propagative such as the Busse-Clever oscillatory instability 1191 or t o transverse standing waves. This latter case is more often found and is the most dangerous one in the process of rapid transition to the chaos already reported as similar to a Martensitic Transformation in solids (Yang X.D. and Ribotta R., to be published).

4) - Space-time complen'ty:

The progressive wave (say R) is the basic state which can be unstable against either localized or homogeneous perturbative modes. Some of these perturbations may lead t o singularities such as the ones hereabove presented. The complexification of the basic state (disordering both in time and space) can be thought as the result of either.homogeneous modes or localized states as in the case of stationary states. In our experiments as the control parameter (the external constraint) is increased just above threshold, so that

E ^_U 0.05, defects appear randomly both in time and in space and their number increases with c. In the meantime grain boundaries reverse the direction of the propagation over space scales that can either grow or decrease t o zero whenever they mutually annihilate. On figure 6 are shown spatio-temporal grain boundaries and dislocations along with twin boundaries. Further increasing c makes the behaviour become unpredictable and the result is a fully chaotic motion with no apparent temporal correlations on small time scales. Usually the disordered state is reached for a typical E around 1.

It is likely that the phase modulations that give rise t o singularities can be accounted for by an envelope equation in one-D. Indeed, Bretherton and Spiegel 1201 have shown that localized states of phase

Fig. 5. Space-time diagram for a source moving to right with a velocity w higher than the propagation velocity u. As a result, the wave has ;eversed its sense of propagation.

Fig. 6. "Twin boundary". At t = 5sec. the wave reverses coherently its sense of propagation without apparent phase discontinuity.

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instability fluctuating in time are also solutions of this type of equation. Also Nozaki and Bekki have proposed that soliton-like solutions may be obtained from a Landau-Ginzburg equation 151. The development of topological singularities which in our case are triggered after a local phase modulation may be interpreted as a nucleation mechanism following an Eckhaus-Benjamin-Feir instability. Up t o now such an equation for the envelope has not yet been derived from the basic microscopic system of equations.

Nevertheless it is possible t o reproduce numerically most of the spatio-temporal effects hereabove described by use of the set of coupled phenomenological envelope equations (Ribotta R. and Joets A., to be publihed).

Conclusion

We have found that the traveling wave state of convection of a layer of an anisotropic fluid offers quite rich examples of localized states and of singularities in non-linear waves. Inhomogeneous states that correspond t o localization of the envelope occur either on large scales thus restricting the convective traveling state inside stable domains well limited in space, or on scales comparable t o the spatial period of the waves. In this latter case, the localized phase perturbations give rise to shock-like effects. Because of the finite bandwidth of the unstable state, topological singularities are created. We believe that these elementary defects can be described by the use of a Landau-Ginzburg equation and we have shown their possible role in the transition to the so-called spatio-temporal complexity.

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