Gollub, Jerry P.

Lowe, Mary

English

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Haverford College Haverford Scholarship Faculty Publications Physics 1985 Pattern Selection near the Onset of Convection: The Eckhaus Instability Mary Lowe Jerry P. Gollub Haverford College Follow this and additional works at: https://scholarship.haverford.edu/physics_facpubs Repository Citation Lowe, Mary, and Jerry P. Gollub. "Pattern selection near the onset of convection: the Eckhaus instability." Physical review letters 55.23 (1985): 2575-2578. This Journal Article is brought to you for free and open access by the Physics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. VOLUME 55, NUMBER 23 PHYSICAL REVIEW LETTERS 2 DECEMBER 1985Pattern Selection near the Onset of Convection: The Eckhaus InstabilityMary Lowe and J. P. GollubPhysics Department, Haverford College, Haverford, Pennsylvania 19041, and Physics Department,University ofPennsylvania, Philadelphia, Pennsylvania 19104(Received 26 August 1985)%e present an experimental study of the space and time evolution of the Eckhaus instability, ageneral mechanism of pattern selection for spatially periodic patterns in nonlinear systems. Using aconvecting liquid crystal layer, we observed long-wavelength modulations leading to the nucleationof new roll pairs. The development of this process is studied by time-resolved spatial Fourieranalysis and compared with predictions based on an amplitude equation.PACS numbers: 47.20. +mThe onset of thermal convection in a thin fluid maybe characterized by a parabolic linear stability curve inthe parameter space formed by the temperature differ-ence across the layer and the wave number of thegrowing disturbance. However, the phenomenon iscomplicated by the fact that two stability curves aretangent at the critical wave number A,, : the linear sta-bility curve (below which a small-amplitude periodicflow will decay) and the boundary of the Eckhaus in-stability. The latter causes a flow pattern whose wavenumber is too far from the critical value to go unstablethrough the development of slow spatial modulations.This leads to the nucleation or elimination of rollpairs, and eventually produces a new pattern with awave number closer to the critical value. Thus, theEckhaus instability is an important mechanism of pat-tern selection that can lead to a substantial change inthe wave number. This process is quite general, sincea similar phenomenon occurs in any translationally in-variant system where a normal bifurcation produces aperiodic structure in one space direction. 'The Eckhaus instability is difficult to observe, bothbecause it is often masked by secondary instabilities, 3and because lateral boundaries limit the range of stablewave numbers. 4 Nevertheless, the stability boundaryhas been measured in circular Couette flows and buck-ling beams6; limited observations have also been madeon hydrodynamic surface waves and on Rayleigh-Benard convection rolls in an annular container. 7 Inthis paper, we present the first direct observations ofthe space and time evolution of the Eckhaus instabili-ty. In earlier experimental studies5 6 the Eckhausboundary was approached gradually from the stableside. In that case, the instability occurs at zero wavenumber so that the spatial modulations observed in thepresent work could not be seen.We use electrohydrodynamic (rather than thermal)convection to obtain a sample containing at least 150rolls. This is sufficiently large that finite-size effectsshould be irrelevant on the time scale of the experi-ment. By controlling the initial wave number of theroll pattern and the layer depth, we are able to makeprecise measurements of the stability boundaries andthe time evolution of various spatial Fourier com-ponents of the pattern. The stability curve and thewave number of the secondary flow are compared withpredictions based on an amplitude equation, an expan-sion in powers and derivatives of a slowly varyingfield.The working fluid is a nematic liquid crystal [N-p-methoxybenzylidine-p-butylaniline (MBBA)] confinedbetween two transparent conductive electrodes with anadjustable separation d = 20—120 p, m. A potentialdifference V larger than V, ( = 6 V ac) induces a one-dimensional roll pattern with critical wave number k,(on the order of —,' d) similar to that resulting from theRayleigh-Benard instability. We define a controlparameter e —= ( V —V, )/ V, and a dimensionless wavenumber 0 =—(k —k, )/k, . The nematic director isaligned in a particular horizontal direction by surfacetreatment, and the rolls are always perpendicular tothis direction. This alignment has the effect ofsuppressing two-dimensional instabilities that mightcompete with the Eckhaus mode.The experiment was performed by creating a rollpattern with a fixed wavelength of 200 p, m, using asmall spatially periodic field produced by an interdigi-tized electrode. s The periodic field is then eliminatedwithout a change in e, and the pattern is allowed toevolve to a steady state that depends on the (variable)layer thickness. If the initial conditions are located inthe Eckhaus unstable region of the phase diagram (seeFig. 2), the initially periodic pattern develops a long-wavelength modulation, as shown in Fig. 1. [See alsothe light intensity measurements in Fig. 3(a).] Thepattern is visible because the convection is associatedwith a periodic tilt of the director out of the horizontalplane. This produces a spatially varying index of re-fraction that focuses transmitted light.We have measured the Eckhaus stability boundaryeE ( 0 ) for 0 ( 0, as shown in Fig. 2. The Eckhausmodulations occur only to the left of eE(0). If the in-itial conditions are chosen to lie to the right of eE(0)(closer to 0=0), the wave number still evolves to-1985 The American Physical SocietyVOLUME SS, NUMBER 23 PHYSICAL REVIEW LETTERS 2 DECEMBER 198S0. 1000. 0800. 060G. 0409QS O. 020O. 000-0 30 -0 25 -0 20 -G. 15 -G. 10 -0 05 0 DG15QSFIG. 2. Measured stability boundaries as a function of thedistance t. above onset and the dimensionless wave numberQ. The solid line is a parabolic fit to the data for the linearstability curve eL (Q), while the dashed line is the predictionfor the Eckhaus boundary ~E(Q) = 3~t. (Q).FIG. 1. Development of the Eckhaus instability from aspatially periodic pattern for a=0.052. New ro11 pairs nu-cleate in the regions of weak optical contrast, leading to ahigher wave number. Only a small part of the sample isshown. In the initial pattern, ~Q~=0. 194. Refer to Fig.3(a) for a graph of the optical intensity vs position.In order to study the evolution of the pattern selec-tion process that results from the Eckhaus instability,we utilized time-resolved spatial Fourier analysis ofthe digitized optical intensity patterns. After the Eck-haus instability develops, the intensity function I(x)has clear modulations, as shown in Fig. 3(a), in whicheach rapid oscillation corresponds to one roll. The in-tensity is related (to lowest order) to the square of thetilt angle of the director field. The spectrum thereforeexhibits a peak at 2kt, where kt is the initial waveward Q = 0, but by a different process involving dislo-cation motion. This latter process is slower by at leastan order of magnitude.We also found the linear stability boundary et (Q),where the decay time of a small-amplitude periodicflow diverges when approached from below. To mea-sure eL, (Q), a small-amplitude periodic state isprepared by external forcing at e & 0. (By our keepinge negative, the growth of other modes is suppressed. )We then remove the forcing and increase e abruptly to0 & e & ez (Q). The decay rate of the convective am-plitude vanishes linearly with e at eL(g), thus allow-ing a determination of this boundary by extrapolationto zero decay rate. The solid line in Fig. 2 is a parabol-ic fit to the experimental data for the linear stabilitycurve. [Although a precise comparison is not possibleon the basis of published computations, et. (g) isreasonably consistent vrith numerical results forMBBA. 'jWhen e is small, an amplitude equation is believedto provide a good description of the Eckhaus instabili-ty. Within this approximation, eL(g) is parabolic,and eE(Q) =3eL(Q). The dashed line in Fig. 2shows this prediction for eE(g), assuming eL (Q) isdetermined experimentally. The experimental dataagree satisfactorily with it.200"~ 80-LLjCLcn40'c)2k)] 200 2400PQS IT ION (P. lTi)2k)+ kE2k5600100 200 BOOWAVE NUM BER ( c rn )400FIG. 3. (a) Optical intensity as a function of position,transverse to the roll axis (t = 180 s). The Eckhaus instabil-ity is evident as a pronounced periodic spatial modulation.(e = 0.052 and~ Q ~ = 0.194 initially. ) (b) Correspondingpower spectrum, sho~ing the initia1 peak at 2kI, and theEckhaus sidebands at 2k~ + k~ and 2k~ + 2k'( = 2kf ).VOLUME 55, NUMBER 23 PHYSICAL REVIEW LETTERS 2 DECEMBER 198503,0.2i.kEk C 0. 1 "I//x(a)number of the pattern. Amplitude and phase modula-tion with wave number kE lead in principle to sum anddifference wave numbers so that there should be spec-tral components at 2kt + kz and 2kt + 2k+. In fact, weusually observe only the positive sidebands. The peakat 2kt —kE appears only for initial patterns located ex-tremely close to the Eckhaus boundary. This is con-sistent with theory, to which predicts a Q-dependentasymmetry in the power of the sidebands. [A smallpeak at kE is also visible in Fig. 3(b), as well as unla-beled peaks near 50 cm ' corresponding to roll pairs. ]The wave number kE of the Eckhaus modulation (inunits of k, ) is plotted as a function of Q in Fig. 4(a)for fixed e=0.052. The data indicate a trend towarddecreasing kE as Q approaches the Eckhaus boundaryQE(e), located at QE=0.13 for e=0.052. [Qz(e) isthe inverse of eE(Q), shown in Fig. 2.] The solid linerepresents the theoretical predictionto for the most rapidly growing mode: (kE/k, ) = Q —[3QE —Q ] /4Q .Actually, the system is unstable with respect to a bandof wave numbers 0& (kE/k, ) & 6(Q —QE). Thedashed line represents the upper bound. Although thescatter is substantial (and greater than the measure-ment error), the experimental data lie above the mostrapidly growing mode but within the unstable band.Although the various spectral peaks are close togeth-er, it is possible to follow their intensities Ak, definedto be the areas under the corresponding spectral peaks,as functions of time. An example is shown in Fig.4(b). The initial peak at 2kt decays partly because ofthe large initial flow amplitude (created by periodicforcing), and partly because of the growth of the Eck-haus peak at 2k&+kz. The latter initially grows ex-ponentially at a rate roughly proportional to the dis-tance (measured horizontally in Fig. 2) from the Eck-haus stability boundary. It subsequently decays as thepeak at 2kf(=2kt+2kz) becomes comparable in size.The latter eventually saturates and becomes the dom-inant peak in the spectrum of the final stable pattern.It is a result of the nucleation of a roll pair at eachwavelength I/kz of the modulation. We find that kf issurprisingly close to band center:~ kf —k, ~/ k, & 0.06.The time development of the nucleation of a rollpair is shown in detail by the optical intensity I(x) inFig. 5. Each full oscillation corresponds to a singleroll. In this example, the flow amplitude modulates tozero near 1500 p, m at t = 180 s after the initial periodicpattern is established. Then two new rolls appear(lowest panel in Fig. 5), thus increasing the wavenumber of the pattern. We presume, but have notchecked, that roll pairs would be eliminated (ratherthan nucleated) if Q & 0. The late stages of the evolu-tion are sensitive to the phase of the rolls, and there-fore cannot be described by an amplitude equation thatignores fast variations. We note that Kramer and Zim-merman2 have performed numerical simulations of theroll nucleation process based on the Swift-Hohenberg0.0:——0.00 0.10 0.20 0.30120300' (b),.200.100.00 100 200 300 400 500TIME ( sec )FIG. 4. (a) Wave number ke of the Eckhaus modulation(in units of k, ) from real-space (triangles) and spectral(crosses) measurements as a function of the dimensionlessinitial wave number Q, for e = 0.052. It vanishes at the Eck-haus stability boundary QE QL is the wave number o. f thelinear stability curve at a=0,052. The solid line is thetheoretical prediction for the most rapidly growing wavenumber, whereas the dashed line represents the upperbound of the unstable band. (b) Time evolution of thepower Ak of the spectral peaks at 2k~, 2k] + k~, and 2kf.(Here,~ Q ~ = 0.194 for the initial pattern. )180M Jg+U4 C T21900 ) 500 t 700 2100POSITION (p.m)FIG. 5. Detail of the optical intensity I(x) at varioustimes for fixed ~=0.052. The nucleation of a new roll pair(two extra full oscillations) at a minimum of the envelope ofI (x) is visible in the lower two panels. (Initially,~ Q ~= 0.194.)VOLUME SS, NUMBER 23 PHYSICAL REVIEW LETTERS 2 DECEMBER 198Smodel.Before concluding, we note severalfeatures of this process. First, the Eckhaus modula-tions are not perfectly coherent: The spacing betweenthe modulations fluctuates somewhat. Second, the ini-tial forcing could possibly affect the wave number ofthe subsequent modulations to a small extent. Third,there are cases in which the pattern evolution does notremain completely one dimensional. %'hen the initialconditions are close to eE(g), the Eckhaus instabilityoccurs along segments of the rolls. This process leavesdiffuse dislocations that climb rapidly along the rollaxis. Thus the Eckhaus instability leads smoothly intodislocation motion, which dominates the process ofpattern selection for ( g ) ( ( gE).We have presented a quantitative study of the spaceand time evolution of the Eckhaus instability, a gen-eral mechanism of pattern selection for periodic struc-tures. We find that the Eckhaus stability boundary isin good agreement (for e ( 0.1) with predictions basedon an amplitude equation. The wave number kE(g)of the modulations lies within the expected band but ishigher than the most rapidly growing wave number ob-tained from linear theory. As a result, the Eckhausinstability is somewhat more effective in driving thewave number of the pattern toward k, than the lineartheory would suggest. The question of whether thefinite initial forcing can significantly affect the instabil-ity is being addressed theoretically. ' The nonlinearstages of the evolution, especially the nucleation andgrowth of new rolls, are not yet quantitatively under-stood. This experiment demonstrates that the linearstability curve itself does not delineate the allowedperiodicities of stable convective flows.This work was supported by the National ScienceFoundation under Grants No. MSM-8310933 and No.DMR-8216718. We appreciate helpful discussionswith F. H. Busse, P. Coullet, S. Fauve, P. Hohenberg,A. Libchaber, L. Kramer, and Y. Sawada.&W. Eckhaus, Studies in Nonlinear Stability Theory,(Springer, Berlin, 1965); A. C. Newell and J. A. Whitehead,J. Fluid Mech. 38, 279 (1969); R. M. Clever and F. H.Busse, J. Fluid Mech. 65, 625 (1974); J. T. Stuart and R. C.DiPrima, Proc. Roy. Soc. London, Ser. A 362, 27 (1978);H. Riecke and H. Paap, to be published.2L. Kramer and W. Zimmerman, Physica (Amsterdam)16D, 221 (1985).3F. H. Busse, in Hydrodynamic Instabilities and the Transi-tion to Turbulence, edited by H. L. Swinney and J. P. Gollub(Springer-Verlag, New York, 1981), p. 97.4M. C. Cross, P. G. Daniels, P. C. Hohenberg, and E. D.Siggia, J. Fluid Mech. 127, 155 (1983); L. Kramer and P. C.Hohenberg, Physica (Amsterdam) 13D, 357 (1984).5M. A. Dominguez-Lerma, D. S. Cannell, and G. Ahlers,to be published; D. S. Cannell, M. A, Dominguez-Lerma,and G. Ahlers, Phys. Rev. Lett. 50, 1365 (1983).6M. Boucif, J. E. Wesfreid, and E. Guyon, J. Phys.(Paris), Lett. 45, L-413 (1984).7T. B. Benjamin, Proc. Roy. Soc. London, Ser. A 299, 59(1967); V. Croquette, private communication.sM. Lowe and J. P. Gollub, Phys. Rev. A 31, 3893 (1985);M. Lowe, J. P. Gollub, and T. C. Lubensky, Phys. Rev. Lett.51, 786 (1983).9P. A. Penz and G. W. Ford, Phys. Rev. A 6, 414 (1972).~OL, Kramer, private communication.

Pattern Selection near the Onset of Convection: The Eckhaus Instability

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Pattern Selection near the Onset of Convection: The Eckhaus Instability

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