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# GEN PHYS PHYS 23

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Date Created: 10/22/15

PHYSICAL REVIEW E VOLUME 58 NUMBER 5 Rayleigh Renard convection in a homeotropically aligned nematic liquid crystal Leif Thomas1 Werner Pesch2 and Guenter Ahlers1 1Department of Physics and Center for Nonlinear Science University of Califomia at Santa Barbara Santa Barbara California 93106 2Institut ir T hearetische Physik Universitat Bayreuth Bayreuth Germany Received 22 April 1998 We report experimental results for convection near onset in a thin layer of a homeotropically aligned nematic liquid crystal heated from below as a function of the temperature difference AT and the applied vertical magnetic eld H When possible these results are compared with theoretical calculations The experi ments were done with three cylindrical cells of aspect ratios radiusheight Iquot 106 62 and 50 over the eld range 8ShEHHFS80 HF209 126 and 93 G are the Freedericksz elds for the three cells We used the Nusselt number N effective thermal conductivity to determine the critical Rayleigh number RC and the nature of the transition We analyzed digital images of the ow patterns to study the dynamics and to determine the mean wave numbers of the convecting states For 11 less than a codimensiontwo eld ha 46 the bifurcation is subcritical and oscillatory with traveling and standingwave transients Beyond hCt the bifurca tion is stationary and subcritical until a tricritical eld ht 572 is reached beyond which it is supercritical We analyzed the patterns to obtain the critical wave number kc and for hlthct the Hopf frequency we In the subcritical range we used the early transients towards the niteamplitude state for this purpose The bifurca tion sequence as a function of 11 found in the experiment con rms the qualitative aspects of the theoretical predictions Even quantitatively the measurements of RC kc and QC are reproduced surprisingly well con sidering the complexity of the system However the value of hCt is about 10 higher than the predicted value and the results for kc are systematically below the theory by about 2 at small 11 and by as much as 7 near ha At ha kc is continuous within the experimental resolution whereas the theory indicates a 7 disconti nuity The theoretical tricritical eld 111151 is somewhat below the experimental one The fully developed ow above RC for hlthCt has a very slow chaotic time dependence that is unrelated to the Hopf frequency For hctlthltht the subcritical stationary bifurcation also leads to a chaotic state The chaotic states persist upon reducing the Rayleigh number below RC ie the bifurcation is hysteretic Above the tricritical eld 11 we nd a bifurcation to a time independent pattern which within our resolution is nonhysteretic However in this eld range there is a secondary hysteretic bifurcation that again leads to a chaotic state observable even slightly below RC We discuss the behavior of the system in the high eld limit and show that at the largest experimental eld values RC and kc are within 6 and 1 respectively of their values for an in nite eld NOVEMBER 1998 S1063651X9800511X PACS nurnbers 6130v 4754r 4720Bp I INTRODUCTION Convection in a thin horizontal layer of an isotropic uid heated from below by a heat current Q is well known as RayleighBe39nard convection RBC 12 When the uid is a nematic liquid crystal NLC this phenomenon is altered in interesting ways 3 MC molecules are long rodlike ob jects that are orientationally ordered relative to their neigh bors but whose centers of mass have no positional order 45 The axis parallel to the average orientation is called the director n By con ning the NLC between two properly treated parallel plates 6 one can obtain a sample with Lmi form planar parallel to the surfaces ie in the xy plane or homeotropic perpendicular to the surfaces or parallel to the zaxis alignment of n The alignment can be reinforced by the application of a magnetic eld H parallel to the intended direction of n This is so because the diamagnetic suscepti bility is anisotropic usually being larger in the direction par allel to the long axis of the molecules The phenomena that occur near the onset of convection depend on the orientation of n and H 3 In this paper we are concerned with a hori 1063651X985855885131500 PRE E zontal layer of a hameatrapically aligned MC in a vertical magnetic eld H Hez and heated from below In that case QQez is parallel to n when the system is in the conduction state At a critical temperature difference ATH the uid will Lmdergo a transition from conduction to convection The precise value of AT5H the nature of the bifurcation at AT5H and the patternformation phe nomena beyond AT5H are expected to depend in interest ing ways upon H 7710 A feature common to the homeotropic MC and to an isotropic uid heated from below is that the system is isotro pic in the horizontal plane Thus the convection pattern may form with no preference being given to a particular horizon tal axis Lmless the experimental apparatus introduces an asymmetry In both cases the convection is driven by the buoyancy force However the mechanism in the NLC case is more involved 779 The usual destabilization due to the thermally induced density gradient is opposed by the stiff ness of the director eld which is coupled to and distorted by any ow Since relaxation times of the director eld are much longer than thermal relaxation times it is possible for 5885 1998 The American Physical Society 5886 director a 39 and J or velocity 3 to be out of phase as they grow in amplitude The existence of two very different time scales and this phase shift typi cally lead to an oscillatory instability also known as over stability ie the bifurcation at which these timeperiodic perturbations acquire a positive growth rate is a Hopf bifur cation 7811 This case is closely analogous to convection in binary uid mixtures with a negative separation ratio 1213 In that case concentration gradients oppose convec tion and concentration diffusion has the slow and heat dif fusion the fast time scale As in the binary mixtures the Hopf bifurcation in the NLC case is subcritical 910 For the NLC the fully developed nonlinear state no longer is time periodic Instead the statistically stationary state above the bifurcation is one of spatiotemporal chaos with a typical time scale that is about two orders of magnitude slower than the theoretical inverse Hopf frequency 14 However it is pos sible to actually measure the Hopf frequency by looking at the growth or decay of small perturbations that are either deliberately introduced 9 or that occur spontaneously when the system is close to the conduction state and near the bi furcation point A linear stability analysis of this system was carried out by several investigators 815717 A very detailed analysis was presented by Feng Decker Pesch and Kramer FDPK 10 These authors also provided a weakly nonlinear analy sis which allowed the distinction between subcritical and supercritical bifurcations Quantitative bifurcation diagrams were predicted for the nematic liquid crystal MBBA N p methodxylbenzylidinepbutylaniline 1n the present work we repeated and slightly extended the calculations for the material 5CB 4npentyl4 cyanobiphenyl see below used in our experiments Since the material parameters of MBBA and 5GB are similar we found qualitatively the same bifurcation sequences as a function of the eld Here we outline brie y the theoretical results and their relationship to our experimental results As the magnetic eld is increased a subcritical Hopf bi furcation line is expected to terminate at H a in a codimensiontwo point CTP beyond that the perturbations which rst acquire a positive growth rate are at zero fre quency Close to but beyond the CTP this stationary bifurca tion is predicted to also be subcritical At an even higher eld H a tricritical point TCP is predicted to exist beyond which the primary bifurcation is expected to become super critical To our knowledge there are no predictions about the patterns that should form beyond either the subcritical Hopf bifurcation below the CTP or the subcritical stationary bifur cation between the CTP and the TCP Although there are no explicit predictions of the patterns for H gtH in analogy to isotropic uids one might expect convection rolls above the supercritical bifurcation unless nonBoussinesq effects 1819 yield a transcritical bifurcation to hexagons The phenomena described above were previously ex plored only partially by experiment Except for recent mea surements at relatively small elds 14 the experiments have been qualitative or semiquantitative 920 In the present paper we report the results of an extensive sytem atic experimental investigation of this system which covered a wide range of magnetic elds H In agreement with previous work 920 we nd a subcritical Hopf bifurcation at rela LEIF THOMAS WERNER PESCH AND GUENTER AHLERS PRE g tively small H that terminates in a codimensiontwo point CTP The CTP is located at a slightly higher eld than the theoretical prediction We measured the Hopf frequency wH from visualizations of the spontaneous small amplitude early transients just above ATE Except for the in uence of the small shift of the CTP we found wEH to be in quantitative agreement with the theory From these transients we also determined the critical wave vector kH and found it to be typically a few percent smaller than the theoretical value The reason for this small differ ence between theory and experiment is not known As re ported previously 14 we found the convecting nonlinear state for AT above ATE to be one of spatiotemporal chaos STC Except at very small H its characteristic wave num ber was smaller than k and insensitive to H and AT A longtime average of the structure factor of this state was consistent with the expected rotational invariance of the sys tem Depending on H the lower limit ATS at which this chaotic state made its hysteretic transition back to the con duction state was found to be 10 to 25 below RH The convective heat transport was consistent with that of an iso tropic uid with an average conductivity given by AWE 2 AH3 where A and AH are the conductivities per pendicular and parallel to n respectively This suggests a thorough randomization of the director orientations by the ow Beyond the CTP we found a subcritical stationary bifur cation as had been predicted 10 The niteamplitude state that evolved was also a state of STC but beyond a certain eld value greater than H a it had distinctly different proper ties from the chaotic state at lower H This difference was clearly evident from a discontinuity as a function of H of the characteristic wave number of the nonlinear state which was larger at the larger elds There is no theoretical guid ance for the interpretation of these experimentally observed phenomena The range H 2H was investigated for two cells with as pect ratios T 615 and 501 We refer to them as cells 5 and 6 respectively for details see Sec 111 below We found a primary bifurcation to a state with a hexagonal ow pattern Within our resolution this bifurcation was nonhysteretic and the Nusselt number N grew continuously from zero The appearance of hexagons rather than rolls is attributable to nonBoussinesq effects 1819 which occur when the up down symmetry is lifted by variations of the uid properties over the cell height Theoretically the bifurcation to hexa gons is transcritical and there should also be hysteresis as sociated with it However the hysteresis is often so small that it is unobservable even though hexagons are found over a substantial range 21 At suf ciently large ATATE the hexagons become unstable with respect to rolls 19 Since ATE decreases with the cell thickness d Nd 3 the range of stability of hexagons should depend on the thickness of the uid sample However for the thinner uid layer of cell 5 the existence range of hexagons was limited by a different secondary instability and grew from zero very near the TCP to EEATATE 71 201 at the highest elds available to us At this stability limit a hysteretic transition yielded the cha otic state and stationary rolls were never found With de creasing AT the chaotic state persisted down to ATS some PRE Q what smaller than ATE For the thicker uid layer of cell 6 typically ATE was about 2 C and hexagons were found only up to 520015 even at high elds For egt0015 a pattern of rolls not exhibiting STC was observed as ex pected for a weakly nonBoussinesq system At even higher 5 and consistent with the measurements in cell 5 a hysteretic secondary bifurcation again yielded the chaotic state II PARAMETER DEFINITIONS AND VALUES The quantitative aspects of the instabilities are determined by four dimensionless parameters that are formed from com binations of the uid properties 22 They are 10 the Prandtl number 42 039 1 MM the ratio between the directorrelaxation time and the heat diffusion time a 2K 7 l 2 33 the Rayleigh number agpdgAT a 2K 3 4 H and the dimensionless magnetic eld h HHF 4 with the Freedericksz eld 7139 33 HFTE pm 5 In these equations on is one of the viscosity coef cients KH is the thermal diffusivity parallel to if k33 is one of the elastic constants of the director eld XE is the anisotropy of the diamagnetic susceptibility a is the isobaric thermal ex pension coef cient and g is the gravitational acceleration The time scale of transients and pattern dynamics is mea sured in terms of the thermal diffusion time Iv dZKH 6 Both h andR are easily varied in an experiment and may be regarded as two independent control parameters The avail ability of h in addition to R makes it possible to explore an entire line of instabilities The parameters F a and 1 are essentially xed once a particular N39LC and temperature range have been chosen and even between different N39LCs there is not a great range at our disposal For 5CB at 256 the material and mean temperature used in this work we have 0263 and F461 The value of 1 is typically sev eral minutes but depends on the thickness of the uid layer It is given in the next section for each of our cells The critical value R501 of R and the uid parameters determine the critical temperature difference ATE for a sample of a given thickness d The realistic experimental requirement RAYLEIGHBENARD CONVECTION IN A 5887 that ATE a few C dictates that the sample thickness should be a few mm Typical values of HF are 10 to 20 G Thus modest elds of a kGauss or so are adequate to explore the entire range of interest In order to evaluate R from ATE hHHF and the theoretical values for Rhk5h and 195k we used the material properties given in Ref 22 We followed closely the calculational methods of FDPK In order to ensure a suf cient resolution of any boundary layers we used Cheby cheff modes in the Galerkin method no more than 20 were required III EXPERIMENTAL APPARATUS AND SAMPLE PREPARATION The apparatus used in this work was described previously 2314 We made measurements using three circular cells of different thicknesses identi ed as cells 4 5 and 6 24 The thickness and radius were d 394 mm r419 mm for cell 4 d660 mm r406 mm for cell 5 and d 888 mm r445 mm for cell 6 The corresponding ra dial aspect ratios TErd were 106 615 and 501 The uid was 5CB All experiments were performed at a mean tem perature of 256 C The vertical thermal diffusion time was Iv 139 383 and 694 s and HF was 201 126 and 934 G for cells 4 5 and 6 respectively Despite the longer time scales involved for experiments in the thicker cells cells 5 and 6 had an advantage over cell 4 due to the smaller eld strenng and temperature differences required to perform the measurements To ensure homeotropic alignment near the surfaces of the top and bottom plates of both cells a surface treatment with lecithin 614 was applied Defectfree homeotropic samples were prepared by apply ing a magnetic eld while cooling the bath and thus the sapphire top plate of the sample from above the isotropic nematic transition temperature TM to TltTNI During this process the bottom plate naturally lagged behind and thus an adverse density gradient existed In the nematicisotropic twophase region even the relatively small thermal gradients associated with small cooling rates induced convection 25 When the cooling was too rapid and the eld too small this led to a nematic sample with defects which remained frozen By using cooling rates of 1 Chour in the presence of a eld of h 17 over the temperature interval 36 to 34 C TM 351 C and annealing at 34 C for an hour or two the defects healed and a defectfree homeotropic sample could be prepared Further cooling could then be at least ten times as rapid without introducing new defects because the thresh old for convection in the nematic phase is large Before each experimental run the procedure was repeated The critical temperature differences for the onset of con vection were determined from heattransport measurements These are usually expressed in terms of the Nusselt number N2 ffAH 7 where A is the conductivity of the homeotropically aligned sample 23 and Ae EiQdAT 8 is the effective conductivity and contains contributions from diffusive heat conduction and from hydrodynamic ow ad 5888 LEIF THOMAS WERNER PESCH AND GUENTER AHLERS 105 I I 1 1 095 o Nusselt Number 09 h 085 l I I I 22 24 26 28 3 32 I I I I I 13 0 12 o 11 I I 1 4amp000000 coo Nusselt Number 09 I I 42 44 46 48 5 AT oC FIG 1 Examples of the Nusselt number as a function of AT for cell 5 The upper gure is for h 15 and the lower one is for h 50 Open circles were taken with increasing and solid circles with decreasing AT The transitions between conduction and convection are indicated by the arrows 52 54 56 58 vection Measurements of N were made by determining the heat current Q required to hold A T constant At each A T the heat current and temperature of the bath and bottom plate were measured at l min intervals for three to ve hours when typically all transients had died out In addition to heat ow measurements we also visualized the convective ow patterns The homeotropic samples were translucent even for d as large as several mm It was just about possible to see features of the bottom plate in typical ambient lighting Any director distortion by convection rolls or domain walls generated opaque regions with enhanced diffuse scattering which were easily visible It should be kept in mind that the optical signal in the images has a com plicated relationship to the hydrodynamic ow elds and that quantitative information about velocity or temperature eld amplitudes could not be obtained Such quantities as the wave vector of the patterns or frequencies of traveling convectionrolls could of course be determined quantita tively The samples were illuminated from above by a circular uorescent light Digital images were taken from above by a video camera which was interfaced to a computer Typically 50 to 200 images were averaged to improve the signalto noise ratio Averaged images were divided by an appropriate reference image to reduce the in uence of lateral variations in illumination and of other optical imperfections Some im ages were processed further by ltering in Fourier space IV RESULTS A Nusselt numbers and critical Rayleigh numbers Figure 1 shows N for cell 5 as a function of AT for two eld strengths h 15 and h 50 The open circles were ob tained with increasing and the solid circles with decreasing IdlE 3000 2500 EC 2000 1500 39 39 39 39 0 200 400 600 800 100 h2 FIG 2 Critical Rayleigh numbers for the onset of convection as a function of hz Open and lled symbols were obtained in cells 4 and 5 respectively The line is the theoretical prediction AT For the lower elds hlt20 N decreased below one when convection started This can be understood because the convecting sample has a distorted director with a component perpendicular to The contribution from this component to the conductivity corresponds to AL which is less than the conductivity A of the homeotropic case 23 It turns out that for small elds the direct hydrodynamic contribution to the heat ux is smaller than the decrease in the heat ux due to the director distortion by the ow For the higher elds h gt35 N remained above one in the convecting state Thus with the higher elds the hydrodynamic contribution to the heat ux is greater than the decrease in the heat ux due to any distortion of the director Both examples in Fig l dem onstrate the predicted and previously observed 91420 hys teretic nature of the bifurcation ie as AT was decreased the conduction state was reached at a value of AT equal to A T Slt A T C From data like those in Fig 1 critical temperature differ ences AT C were determined with an uncertainty of less than 1 The corresponding Rayleigh numbers are shown in Fig 2 as a function of W The open circles were obtained in cell 4 the lled ones in cell 5 The good agreement between the two data sets con rms the expected scaling of the eld with H F It also shows that using the uid properties at the mean temperature does not lead to systematic errors in RC even for cell 4 where AT C is over 10 0C One sees that RC is quadratic in h at small h as is expected because the system should be invariant under a change of the eld direction The solid line follows the theoretical prediction and agreement between theory and experiment is excellent Results for RC over our full experimental eld range are shown as a function of h in Fig 3 Here we include data taken with cell 6 as open squares The data for RC reveal a sharp maximum at h443 We interpret this eld value as the codimensiontwo point hct and indicate it in Fig 3 by the dashed vertical line The solid line in the gure is the theo retical prediction for RC evaluated for the properties of our sample For the entire range hlthct the theoretical result is in excellent agreement with the data However the theory gives hct4l8 which is slightly lower than the experimen tal value Above hct the measurements of RC are systemati cally larger than the calculation although the largest discrep ancy is only about 4 The triangles in Fig 3 show the lower limit of existence the saddlenode Rayleigh number RS of the nite FREQ 4000 3000 2000 1000 0 FIG 3 Critical Rayleigh numbers RC and saddlenode Rayleigh numbers RS over the experimentally accessible eld range The open circles lled circles and open squares are RC for cells 4 5 and 6 respectively The open and lled triangles are RS for cells 4 and 5 respectively The dashed line indicates the location of the codimensiontwo point as found experimentally The solid line is the theoretical prediction for RC The plusses at h35 were ob tained with short equilibration times and cell 5 see text amplitude convecting state as determined from data like those in Fig 1 They suggest that the tricritical bifurcation is located near h59 which is larger than the theoretically calculated value he 51 However we will return later to the best estimate of h Measurements similar to those shown in Fig 3 were made by Salan and FemandezVela 20 SF using the nematic liquid crystal MBBA Their results are shown in Fig 4 to gether with the theoretical curve for that case 26 The data and the curve illustrate that there are signi cant quantitative differences between the bifurcation lines of different nemat ics In Fig 4 the experimental points lie on average about 25 above the theoretical curve and the lower hysteresis limit is further below the bifurcation line than we found for 5CB The equilibration times after each temperature step used by SF were 30 min which is a factor of six to ten shorter than those of our experiments In addition the temperature steps of SF were a factor of two larger than ours yielding a difference in the average rate of change of the temperature of a factor of 12 or more Looking for an explanation of the difference between the experimental and theoretical RC re vealed in Fig 4 we conducted one run with equilibration times similar to those of SF but using our 5CB sample It gave the plusses in Fig 3 As can be seen these results do not differ signi cantly from the data taken with our usual 4000 3000 2000 1000 FIG 4 Results for RC and RS from Ref 20 obtained with IVIBBA The theoretical curve for RC was computed from typical uid properties of IVIBBA 26 RAYLEIGH BENARD CONVECTION IN A 5889 FIG 5 A sequence of images ofthe travelling or standing wave transients for h32 and 60015 in cell 5 The number in each image corresponds to the time in units of tv that elapsed since ow rst became visible A time series of the pixel intensity was taken at the location marked in the top left image longer equilibration times Thus we have no explanation for the difference between the SF data for R and the theoretical curve However the agreement between our runs with the different equilibration times implies that our usual experi mental procedure yielded quasistatic results B Hopf frequency and critical wave vector Once the critical Rayleigh numbers were measured a de tailed analysis of the patterns could be undertaken At rst we will characterize the Hopf bifurcation for h below hot Since in that eld range the bifurcation is subcritical we had to use the smallamplitude transients to determine we and kc Figure 5 shows images from cell 5 that are characteristic of these patterns They were taken at the times in units of iv 383 s indicated in each gure after the pattern initially became visible This typically occurred around 1 h after 6 was raised from below zero to around 0015 Inspection of successive images revealed that the transients could be either traveling or standing waves sometimes with both occurring at different locations in the same cell In the top left image of Fig 5 a location is indicated at which a time series of the pixel intensity was acquired This time series is shown in Fig 6 along with its corresponding power spectrum The length of the time series was limited by the rapid growth of the pattern to its niteamplitude steady state Because of this only a small number of periods could be obtained before the niteamplitude state was reached Thus to avoid errors associated with incommensurate sampling the data were windowed before its Fourier transform was evaluated This process was repeated at several pixel locations in the cell The signal from the second harmonic was often found to be stronger than that from the fundamental Thus it was used to calculate the frequency The frequencies at different loca tions generally were within 1 of each other and were av eraged to determine the critical Hopf frequency we 5890 Pixel Intensity Arb Scale Power Arb Scale 0 1O 20 30 40 Frequency w FIG 6 The upper gure is the time series of the pixel intensity at the location shown in the top left image of Fig 5 The lower gure is the power spectrum of that time series The mean fre quency calculated from the solid circles was used to determine the Hopf frequency The dependence upon h of the measured we is compared with theory in Fig 7 The arrow indicates the location of the theoretical codimensiontwo point while the dashed line rep resents the experimental determination of het As can be seen away from the codimensiontwo point the agreement with the measurements is excellent In accordance with theory the experimental we changes discontinuously to zero at het above which the bifurcation is stationary By evaluating the Fourier transforms of images such as those in Fig 5 the critical wave number ke of the patterns could be measured The transforms were based on the central parts of the images by using the lter function Wrl cos7Trr02 for rltr0 and Wr O for rgtr0 Here re was set equal to 85 of the sample radius Time averag ing the square of the modulus of the transforms over the length of the time series yielded the structure factor ltSk O I I l 0 1 0 20 30 4O 50 60 FIG 7 The Hopf frequency we as a function of h Open and lled circles are for cells 4 and 5 respectively Open squares are for cell 6 The solid line is the theoretical prediction for we The dashed line arrow indicates the location of the codimensiontwo point as found experimentally theoretically LEIF THOMAS WERNER PESCH AND GUENTER AHLERS 1311133 05 8k arb units 0 2 4 6 8 10 12 K FIG 8 The azimuthal average of the timeaveraged structure factor Sk of the travelling standing wave transients at h 32 for cell 5 The mean wavenumber calculated from the solid circles was used as the critical wave number Figure 8 shows the azimuthal average ltSkgt of ltSkgt for the run at h 32 We used a weighted average of the three points nearest the peak of the second harmonic of ltSkgt to calculate ke Figure 9 displays the results for ke for all h together with the theoretical analysis For hlthet the measured critical wave number of the transients is systematically smaller than the theoretical one When the codimensiontwo point is ap proached the experimental wave numbers make a smooth rather than discontinuous transition to those associated with the stationary bifurcation whereas the theory predicts a 7 discontinuity of ke at het The reason for these discrepancies is as yet unknown Above het the agreement between the experimental and theoretical wave numbers is excellent As shown explicitly for Re in Fig 2 Re we and ke are proportional to h2 for small h C Nonlinear state below the tricritical eld het Because of the subcritical nature of the bifurcation for h lthet a niteamplitude state develops directly at onset The time dependence and spatial structure of this state are very different from that of the smallamplitude transient state The rst two rows of Fig 10 show typical images of the patterns from cell 5 that are characteristic of the fully developed ow They are from a single experimental run with constant exter nal conditions They were taken at the times indicated in FIG 9 The characteristic wave numbers of the observed pat terns as a function of h Circles The wave number of the small amplitude transients Triangles The wave number kn of the fully developed spatially and temporally chaotic ow as measured close to the onset of such ows Open and lled symbols were obtained in cells 4 and 5 respectively FIG 10 Top two rows a sequence of images taken with con stant external conditions h50 60014 for cell 5 The time elapsed since 6 was raised from below zero in units oftv 383s is given in the top left corner of each image Bottom row The struc ture factor of two of the images shown above and the average of the structure factor of 75 images spanning a time interval of 7241 The structure factor was obtained using a Hanning window and thus is dominated by the patterns near the cell center each image in units of iv 383 s which had elapsed since 6 had been raised from below zero to 0014 The convection rolls of the fully developed ow show an irregular time de pendence with typical time scales around 100 times longer than the inverse Hopf frequencies of the transients One can see that the chaotic behavior is associated with the for mation of defects and the continuous reorientation of the convection rolls This continuous reorientation of the rolls is evident in the rightmost image in the bottom row of Fig 10 labeled Avg It shows the time average of the structure factor The average involved 75 images taken over a total time period of 724tU over three days It is seen to contain contributions at all angles consistent with the idea of a sta tistically stationary process of nonperiodic pattern evolution and with the expected rotational symmetry of the system Similar results for cell 4 have been shown previously 14 When h was increased above hot the nature of the pattern at rst did not change noticeably For instance as evident in Fig 9 the characteristic wave number of the pattern as de noted by the triangles remained close to 34 for h S 55 Over this eld range the patterns of the fully developed ow look similar to those illustrated in Fig 10 ie they exhibit spa tiotemporal chaos It is instructive to examine the transients that lead from the small amplitude to the niteamplitude statistically sta tionary state This is done in Fig 11 Here the number in each image gives the time in units of iv which had elapsed since AT was raised slightly 14 above ATC At t47 smallamplitude transients like those in Fig 5 are evident in part of the cell By t 94 these had lled the cell and grown to a saturated amplitude At this stage they formed nearly RAYLEIGH BENARD CONVECTION IN A 5891 FIG 11 A temporal succession of images during the transient leading from conduction to convection when AT was raised slightly above ATC for cell 5 The eld was h50 The numbers are the elapsed time in units of tv since the threshold was exceeded straight parallel rolls with a wave number that was smaller than kc However these straight rolls turned out to be un stable to a zigzag instability In the end this instability led to the spatially and temporally disordered pattern as shown in Fig 10 Thus we see that a secondary instability led to a chaotic state rather than to a new timeindependent pattern This phenomenon most likely is similar to the one encoun tered in very early experiments on spatiotemporal chaos us ing liquid helium 2728 where ordinary RB convection became chaotically time dependent most likely because the secondary skewedvaricose instability 29 was crossed Heattransport measurements of the fully developed ow are shown in Fig 12 for several eld values as a mction of 6 They illustrate the evolution with h of the hysteretic na ture of the bifurcation As can be seen also in Fig 3 the hysteresis I esl increased with h for h lthct from about 10 at the low elds to nearly 25 close to the codimension two point Above this point the hysteresis decreased and sug gested the existence of a tricritical point near 12259 see below for more detail about the tricritical region When the Nusselt numbers in Fig 12 are plotted against the Rayleigh number R they fall on or approach a single curve independent of h This suggests that the convection in the chaotic state is suf ciently vigorous to achieve nearly complete randomization of the director orientations regardless of h In that case one would expect that the sys tem should behave approximately like an isotropic uid with an averaged conductivity xan 2AL xH3 Thus we plot in Fig 13 a modi ed Nusselt number JV given by the ratio of the effective conductivity of the convecting state to xan as a function of Ravg where RVg is computed using KavganpCP in Eq 3 rather than KH At all but the highest elds where the primary bifurcation is supercritical the data reach the common curve At small Ravg this curve extrapolates to JV 1 near Ravg 1708 the cross in the g ure which is the critical Rayleigh number of an isotropic uid An analogous behavior has been observed in binary mixture convection with negative separation ratios 1 30 where the bifurcation is also subcritical In that case the con vective ow achieves thorough mixing of the concentration eld and N approaches a curve that is independent of 1 In both cases the mixing achieved by the ow can persist be cause of the existence of a slow time scale namely that of 5892 14 45 39 39 mam 8 12 kw g 1 0000 quot9339 Z I g 14 35 39 39 1 g 12 e quot40quot 5 quot39 Z I Q00039 39 39 1 l0 eeee 14 I 39 E 12 ooif o 0 1 0 O O O O 93 9 90000 14 15 39 39 f 12 J 10ooo oo eoe 39 9011 o3 o2 o1 o 01 ATATC1 FIG 12 Nusseltnumber measurements for cell 5 illustrating the variation of the size of the hysteresis loop between conduction and convection with h The number in the upper left corner of each plot is the eld h Open circles were taken with increasing and solid circles with decreasing AT The arrows show the values of es ERS Rc 1 director or concentration relaxation Further support for the idea that the chaotic ow in some respects can be approximated by isotropic uid convection is found in Fig 9 where for hS 55 the wave vectors triangles are independent of h and much closer to the critical value 1650 3117 than to the critical values kCh of theanisotropic system circles in Fig 9 Exact agreement with klcso would of course not be expected even for a genuine isotropic uid because of the nite ow amplitude and various wave numberselection processes Lastly we note that an extrapolation of the data in Fig 13 to N 1 and RM 1708 yields an initial slope E of N 1 1Ran1708 1 of about 06 For a laterally in nite sys tem of straight rolls in an isotropic uid with a large Prandtl number one expects S 1 2 143 31 However experiments in nite systems with modest aspect ratios 32 have always yielded smaller values usually in the range of 06 to 1 Par ticularly when many defects are present as in our case one would expect the heat transport to be suppressed relative to that of a perfect straightroll structure D Tricritical region and beyond This section is devoted to the phenomena that occur near the tricritical eld hh At rst the Nusselt numbers and LEIF THOMAS WERNER PESCH AND GUENTER AHLERS 1311133 2 18 A A A 8 0 916 I A00 w 393 A 00 f 0quot A09 3 g 00 314 o g9 12 at 1 x 4 2000 3000 4000 5000 6000 Ravg FIG 13 Nusseltnumber measurements for cell 5 normalized by the averaged conductivity xanE2i H3 as a function of the Rayleigh number 12an computed with KanEAanpCp The data are for h15 open circles 25 solid circles 35 open squares 45 solid squares 50 open triangles and 55 solid tri angles The diamonds are for h64 which is in the supercritical region Here the solid diamonds are for the hexagons or rolls which form supercritically and the open ones are for the chaotic nite amplitude state which forms via a secondary bifurcation see Fig 19 below The cross corresponds to the critical Rayleigh number RC 1708 of an isotropic uid the patterns are described and the corresponding bifurcation diagram is given Further subsections deal with the precise determination of h and with hexagons observed near thresh old for hgth I Nusselt numbers and patterns From the measurements of the Nusselt number see Fig 12 there is clear evidence of a tricritical eld ht above which the primary bifurcation is supercritical For instance for h60 measurements of N revealed no hysteresis at the primary bifurcation and within our resolution N grew con tinuously from one beyond ATC This is exempli ed for cell 5 and h63 in Fig 14 The open solid circles correspond to the stable states reached by increasing decreasing AT 33 This behavior of N stands in sharp contrast to that shown in Fig 12 for lower elds Besides the Nusselt number the analysis of the patterns gives important additional insight in particular with respect 108 I 1 1 1 b 1061 o 104 0 39 Nusselt Number 102 o I 1nqcoeof3 001 0 001 002 003 004 e FIG 14 Nusseltnumber measurements for h63 and cell 5 illustrating the supercritical nature of the bifurcation characteristic of the high elds Open circles were taken with increasing and solid circles with decreasing AT 1311133 FIG 15 Bifurcation diagram in the region of the Rh plane close to the tricritical point Solid circles primary bifurcation Solid triangles RS Solid squares hysteretic secondary bifurcation to chaotic convection Shaded areas labeled kn 34 36 and 39 cor respond to chaotic regimes with different mean wave numbers The wedgeshaped area labeled RollsHex shows the parameter range over which timeindependent convection is stable For cell 5 the pattern is hexagonal in this entire region For cell 6 the pattern is hexagonal in this region for GS 0015 For larger 6 but still in this region it consists of timeindependent rolls to secondary transitions In Fig 15 all the available informa tion has been condensed in a bifurcation diagram for the vicinity of the tricritical point At rst we will focus on the wedge labeled RollsHex where timeindependent convection is stable For both cell 5 and 6 a seemingly supercritical primary bifurcation led to a hexagonal pattern For cell 5 this pattern is shown in Fig 16 The range of 6 over which the hexagons were stable differed in the two cells In cell 5 hexagons remained stable up to Rnh solid squares in Fig 15 for the entire range of h At Rnh a transition to a spatially and temporally chaotic roll pattern with a lower characteristic wave number occurred For hlt7 5 this transition was distinguished by a jump in N as well as the onset of time dependence of N as illustrated in Fig 17 The upper gure gives the steadystate N and shows the jump at enERn RC 1 The lower gure is the time series of N obtained in the same run Here 6 was held FIG 16 Image of the hexagonal ow in cell 5 for h65 and e 001 The pattern was essentially the same over the entire exis tence range of hexagons RAYLEIGHBFNARD CONVECTION IN A 5893 o o L o o D Q E 3 2 E 11 8 Z 0 o o E o o 19 0 001 002 003 004 005 8 Nusselt Number time 5 hrs FIG 17 Nusseltnumber measurements for h 61 in cell 5 The upper gure illustrates the dependence of the Nusselt number on e The dotted line indicates 6 where the transition from hexagons to rolls occurred when AT was increased The lower gure illustrates the dependence of the Nusselt number on time Time is measured in units of 5 h ie the time between steps in e constant for a 5 h period at each of the eleven successively increasing values The data show that N is steady below and time dependent above 6h For 1227 5 the discontinuity in N was no longer pronounced but a transition to time depen dence still occurred at 6h Thus depending on the eld either of these two indicators was used to determine the lo cation of 6h In the thicker cell 6 a transition from hexa gons to rolls occurred near 620015 independent of eld strength This transition was not associated with a measur able change in the wave number a jump in N or a time dependence of N A further increase of 6 again led to a transition at 6h from steady rolls to the chaotic state con sistent with the cell 6 experiments The results for 6h obtained in cells 5 and 6 are shown in Fig 18 as circles and triangles respectively 015 01 8n 005 I O 55 60 65 70 75 80 h FIG 18 Values of 6h where a transition to a spatially and temporally chaotic state occurred Circles and triangles were ob tained in cells 5 and 6 respectively The lines represent a t of a quadratic polynomial in hhn 1 to the data The ts extrapolate to zero at hn583i07 for cell 5 and hn58i3 for cell 6 5894 LEIF THOMAS WERNER PESCH AND GUENTER AHLERS o 12 3 e E o g e O o 11 00 w o w 0 O o 3 20 2 o o oo 0 1o oo 004 002 O 002 004 006 008 a FIG 19 Nusseltnumber measurements for h64 in cell 5 Open circles were taken with increasing and solid circles with de creasing AT The dotted line indicates Equot On the basis of the usual Landau equation for a tricritical bifurcation one would expect the hysteresis to grow gradu ally as h is reduced below h However within our resolu tion this was not the case and a hysteretic primary bi lrca tion to a chaotic state occurred immediately below kl Indeed the secondary bifurcation line enh for hgth met the primary bifurcation line at h within experimental reso lution as can be seen already in Fig 15 It is shown more explicitly in Fig 18 where the range 6 of timeindependent patterns vanishes near h 58 Fitting a quadratic polynomial in hhni 1 to enh yields hn 583i07 cell 5 and hn58i3 cell 6 for the eld where 6 vanishes Within error these values agree with the tricritical eld obtained from the slope of the Nusselt number see the next sectionThe bifurcation for hgth is supercriti cal but the amplitude at constant egt0 diverges as h ap proaches h from above Since secondary bifurcations occur at nite values of the amplitude we expect 6h to vanish at h Therefore the 6 measurements provide a relatively precise lower limit h 576 for the tricritical eld The secondary bifurcation at 6 is strongly hysteretic As illustrated in Fig 19 when 6 was decreased from 6 en a transition back to hexagons did not occur Instead the cha otic state persisted to values of e slightly below zero at which point the conduction state was reached see also the solid triangles in Fig 15 In a separate section we will come back to the hexagons 2 Determination of the tricritical point As h approaches the tricritical point from above the ini tial slope S1 of N for rolls is expected to diverge as 1h ihl For cell 6 we estimated S1102 from data for 6 20015 where rolls were observed At a given h S1 was determined by tting the polynomial N1S1VES2J62 9 with e ATA T07 1 to the data The parameters ATC S1 and S2 were adjusted in the t Figure 20 shows the depen dence of USU upon h as solid circles The tting procedure did not yield highly accurate values because the Nusselt data for elt 0015 had to be excluded thus the error bars for S1 are relatively large The line is a t of a quadratic polynomial in hh 1 to the results for 1S1rh This t indicates the tricritical point to be at h 572i 26 FIG 20 The dependence on h of the reciprocal of the initial slope S 1 of the Nusselt number as obtained from a t of the data to Eq The open circles are lSUI for cell 5 and the lled ones are lSl for cell 6 For cell 5 data over the range 0lt elt En were used and the pattern was hexagonal For cell 6 data over the range 0015lt elt En were used and the pattern was one of rolls The solid line represents a t of a quadratic polynomial in hhti l to the cell 6 SI data The eld h572i26 where lSl for cell 6 extrapolates to zero is interpreted to be the tricritical point The theoretical results for lS 1 are given by the dashed line They yield a tricritical point at hih51 The theoretical results for lSUI are given by the dashdotted line The location of the codimensiontwo point is given by the solid experimental and dashed theoretical short vertical lines In order to compare the theory with the measurements above h we calculated S1 using the properties of 5CB The results for 1S1 are shown as a dashed line in Fig 20 There is quite reasonable agreement with the experimental data for h260 particularly when it is considered that the initial slope of N in nite systems usually is smaller than the theoretical value for the in nite system However the theory yields a tricritical eld hgh51 which differs signi cantly from the experimental estimates We note that this difference is in the same direction as and somewhat larger than the corresponding one for the codimension two point We have no explanation for this difference 3 Hexagons In this section we will discuss in more detail the hexago nal patterns Hexagonal patterns at onset may be attributable to departures of the physical system from the Oberbeck Boussinesq OB approximation 1819 ie to a variations of the uid properties over the imposed temperature range For isotropic uids it has been shown that nonOB effects lead to hexagons at a transcritical hysteretic primary bifur cation 1921 Below onset for eaSeSO both hexagons and the conduction state are stable Above onset hexagons are stable for OS ES er For ers ES 51 hexagons and rolls are both stable while for e Eb only rolls are stable When the thickness of the llid layer is increased AT is reduced and thus departures from the OB approximation become smaller Thus the range of 6 over which hexagons are stable is reduced when the thickness of the uid layer is increased as seen in the experiment by comparing cells 5 and 6 A stability analysis of RBC with nonOB effects in a ho meotropically aligned NLC has not yet been carried out and would be very tedious Thus in order to obtain at least a qualitative idea of the expected range of stable hexagons we used the theoretical results for the isotropic uid with the 1311133 4000 3800 3600 II 3400 3200 3000 0 FIG 21 The critical Rayleigh number as a function of NW The solid line was calculated using the parameters of Ref 22 The open triangle is RC00 from Eq 13 The solid triangle is the many mode numerical result for R0700 The solid circles are from cell 5 and the open squares are from cell 6 The vertical bars indicate the location of the codimensiontwo point right bar and the tricritical point left bar The plusses crosses and dashed line show what happens to the data and the theoretical curve if the viscosity a4 is increased by 75 uid properties of 5CB The values of 60 etc are deter mined by a parameter 7 which was de ned by Busse l9 and is given by 7922079791 with yo App yl Aa2a y2Avv y3AA and y4ACpCp Here p is the density a the thermal expansion coef cient V the kinematic viscosity A the conductivity and C p the heat ca pacity The quantities A p etc are the differences in the val ues of the properties at the bottom hot and top cold end of the cell For A we used A and for V we used a4 2p The coef cients 7 in the equation for 79 are given by Busse 19 However here we use the more recent results 34 for large Prandtl numbers 7902676 791 6631 7922765 793 9540 and 794 6225 where 793 differs signi cantly from the earlier calculation At the elds where the hexagons were observed the tem perature difference across cell 5 cell 6 was close to 505 0C 207 0C At these temperature differences we ob tained 79 15 ea l7gtlt104 e15gtlt10 2 and e 53gtlt102 for cell 5 For cell 6 the values are 79 06 ea 28gtlt105 e25gtlt10 3 and 6 87gtlt10 3 From these estimates it follows that the hysteresis of size 6 is too small to be noticeable in either cell with our resolution The largest value of e at which hexagons could exist in cell 6 would be e 87gtlt 10 3 However we observe hexagons to exist to nearly twice this value If the same is true for the thinner cell the hexagonroll transition attributable to nonOB effects should happen at a value of 45 greater than 6h for the eld range over which the experiments were performed Thus instead of leading to a time independent state as observed in cell 6 the hexagonroll transition is preceded by a transition to a state exhibiting spatiotemporal chaos at en The hexagonal pattern may be regarded as a superposition of three sets of rolls with amplitudes A i 139 123 corre sponding to the three basis vectors at angles of 1200 to each RAYLEIGHBFNARD CONVECTION IN A 5895 445 1 1 I 44 lt 435 43 I l l I o 104 h2 FIG 22 The critical wave number as a function of NW The solid line was calculated using the parameters of Ref 22 The open triangle is km from Eq 13 The solid triangle is the many mode numerical result for km The vertical bars indicate the loca tion of the codimensiontwo point right bar and the tricritical point left bar The dashed line shows what happens to the theoretical curve if the viscosity a4 is increased by 75 the data for kc remain unchanged other According to the Landau model 13 the steadystate amplitudes A are determined by 6141 bAAg11Ai g12A1Ag13A1A0 10 and the corresponding cyclic permutation for i23 Since all amplitudes are expected to be equal in hexagons A1 A2A3A one has eAbA2 g2g A30 11 where g is the selfcoupling coef cient g and g is the crosscoupling coef cient g12 g13 Because of the term 6A2 the bifurcation is transcritical and thus hysteretic How ever as we discussed above and as is shown for instance by the data in Fig 14 this effect is not resolved in the experi ment because the coef cient 9 which is determined by 79 is too small Thus we neglect the term 6A2 and have to a good approximation 3e N 13A2 2N 12 g g At the tricritical point g vanishes as g g0h ht How ever there is no reason why g should vanish also at h Thus one would expect the slope S U 3g 2g of N near 60 to remain nite at h and equal to 32g To test this idea we tted N for cell 5 over the 6 range where hexagons were observed ie up to en to an equation like Eq 9 Since data quite close to threshold could be used the results for S U are much more precise than those for cell 6 They are given in Fig 20 as open circles One can see that lSUZ is nonzero at h It extrapolates to zero near h 53 which is well below ht 57 226 Unfortunately the relatively large 5896 FIG 23 Time sequences of the spatially and temporally chaotic ow forR 4000 and at different values ofh for cell 5 The images in each row were taken at the eld indicated in the leftmost image Time increases from left to right Images were taken in one hour intervals The wave numbers of the patterns are h 47 kn 34 h 56 kn36 h 575 kn39 uncertainty of h and S1 prevents the accurate determination of At h we nd gN 32S1h03 With increasing h g also increases For instance the data in Fig 20 suggest that gN32S1hi 10311208 for h66 The experimental results for US 1 cannot agree quantita tively with the theory because we already know that h h is lower than the experimental value Nevertheless we calcu lated lSLh and give it in Fig 20 as the dashdotted line We see that the relationship between lSU2 and 1S1 is quite similar in theory and experiment E The high eld limit of RC and kc It is highly probable that there exists a high eld regime where the convection phenomena become independent of the eld since the director is then frozen in the homeotropic con guration It is instructive to examine whether the experi mental data extend to suf ciently high elds to fully reveal this behavior Building on the results of FDPK 10 one can show that in a onemode approximation the neutral curve in the limit h gtoo is given by AL Hkz 7T2 Rc gt 2181 a4 a4 a4 k22712722a1 2 2 2 k4b4 13 4 4 Here 771a4 a5 a22 and 772a3 a4 a62 are Miesowicz viscosity coef cients 4 and 110697 38 27123026 and 17150567739 This leads to Rcmkc 30906 and kcm4294 Using many modes one obtains numerically Rcmkc 30566 and kcm4328 close to the LEIF THOMAS WERNER PESCH AND GUENTER AHLERS FREQ onemode result One can see that only the viscosities enter into Ram and not the elastic constants This is so because the director is held rigid by the eld Ram is larger than the isotropic uid value because of the additional viscous inter action between the ow and the rigid director eld In the high eld limit we obtain 11261120R1 h20h 4 14 The coef cient R1 has not been calculated in detail but is proportional to k3 The fact that at order lh2 elastic con stants enter suggests the beginning of some director distor tion by the ow In Figs 21 and 22 we show the experimental data and theoretical results as a function of h 2 The one mode high eld limits Ram and kw are shown in the gures as open triangles The corresponding numerical manymode results are given as solid triangles The experimental data foch and kc are consistent with the expected dependence on h but respectively fall about 4 above and 1 below the calcula tion The data for RC and kc at the highest experimental eld h 80 are already within about 6 and 1 respectively of the in nite eld value Thus it seems unlikely that qualita tively new phenomena could be discovered by measurements at even higher elds We examined whether the small difference between the theory and the experiment could be removed by small adjust ments in the values of the uid properties We found that an increase by 75 of a4 yielded the plusses and crosses for the data in Fig 21 and the dashed lines in Figs 21 and 22 the data for kc in Fig 22 are not affected by changing a4 The adjustment of a4 produced an excellent t for both kc and RC However it spoiled the excellent agreement for RC along the oscillatory branch below the codimension two point shown in Fig 3 and did not signi cantly reduce the difference between calculation and experiment for kc at small h which is shown in Fig 9 Various other attempts to adjust the uid properties used in the theoretical calculations were unsuccessful in yielding improved overall agreement between theory and experiment over the entire eld range F Nonlinear states at high elds Beyond the codimension two point the niteamplitude ow was split into three regions distinguished by their char acteristic wave numbers kn3436 and 39 These regions are shown in Fig 15 Figure 23 shows some characteristic patterns Qualitatively the patterns appear similar each ex hibiting spatiotemporal chaos The transitions between these state depended on both h and R They were determined by measuring the change in kn as h was varied at a given Ray leigh number As already mentioned above to our knowl edge there are no theoretical predictions for these patterns ACKNOWLEDGMENTS One of us WP acknowledges support from a NATO Collaborative Research grant One of us GA is grateful to the Alexander von Humboldt Foundation for support The work in Santa Barbara was supported by the National Sci ence Foundation through Grant No DMR9419168

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