Directed percolation criticality in turbulent liquid crystals
Kazumasa A. Takeuchi, Masafumi Kuroda, Hugues Chaté, Masaki Sano
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Directed percolation criticality in turbulent liquid crystals
Kazumasa A. Takeuchi, ∗ Masafumi Kuroda, Hugues Chat´e, and Masaki Sano † Department of Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan Service de Physique de l’ ´Etat Condens´e, CEA-Saclay, 91191 Gif-sur-Yvette, France (Dated: October 24, 2018)We experimentally investigate the critical behavior of a phase transition between two topologi-cally different turbulent states of electrohydrodynamic convection in nematic liquid crystals. Thestatistical properties of the observed spatiotemporal intermittency regimes are carefully determined,yielding a complete set of static critical exponents in full agreement with those defining the directedpercolation class in (2+1) dimensions. This constitutes the first clear and comprehensive exper-imental evidence of an absorbing phase transition in this prominent non-equilibrium universalityclass.
PACS numbers: 47.52.+j, 68.18.Jk, 05.70.Jk, 64.70.Md
Transitions into an absorbing state, i.e. a state fromwhich a system can never escape, arise from simple mech-anisms expected to be widespread in nature. Examplesabound in a wide variety of situations in physics andbeyond [1], ranging from spreading processes like epi-demics and forest fires, to spatiotemporal chaos, catalyticreactions, and calcium dynamics in cells, etc. More-over, a host of problems such as synchronization [4], self-organized criticality [5], and wetting [6], can be mappedonto them. Having no equilibrium counterparts, absorb-ing phase transitions are central in the ongoing search forthe relevant ingredients determining universality out ofequilibrium.Over the past 25 years or so, it has been well estab-lished both in theory and in simulations that the vastmajority of absorbing phase transitions share the samecritical behavior, constituting the so-called directed per-colation (DP) universality class [1]. This is not surprisingsince the DP class corresponds to the simplest case of asingle effective absorbing state in the absence of any extrasymmetry or conservation law, as conjectured by Janssenand Grassberger [2, 3] and demonstrated by hundreds ofnumerical models [1].However, the situation is quite different at the experi-mental level. Searching for evidence of DP critical behav-ior, a string of experiments have been performed [7], butthe results remained unsatisfactory: they could not yielda complete set of critical exponent values in agreementwith those defining the DP class (Table S1 [8]). In viewof this state of affairs, the importance of finding at leastone fully convincing experimental realization of DP-classscaling has been stressed [9]. The main difficulty prob-ably stems from the fact that one must exclude long-range interactions [1], work with macroscopic systems totame quenched disorder [10], and study them over longenough scales so that the critical behavior is unambigu-ously measured. We have overcome these difficulties andreport here on a firm experimental observation of DPcriticality. We chose to work within electrohydrodynamic (EHD)convection regimes, which occur when a thin layer of ne-matic liquid crystals is subjected to an external voltagestrong enough to trigger the Carr-Helfrich instability [11].One advantage of this system is that very large aspect ra-tios can easily be realized, and that typical timescales aresmall (of the order of ten milliseconds). We focused on atransition between two topologically different turbulentstates, called dynamic scattering modes 1 and 2 (DSM1and DSM2), observed successively as V , the amplitude ofthe voltage, is increased [11, 12]. The difference betweenthe two states lies in their density of topological defectsin the director field (Fig. 1a). In the DSM2 state, thesedefects, called disclinations, are created and elongatedconsiderably due to shear, leading to the loss of macro-scopic nematic order and to a lower light transmittance.In DSM1, they are hardly elongated and their densityremains very low.Our quasi two-dimensional cell is made of two par-allel glass plates separated by a polyester film spacerof thickness d = 12 µ m. The inner surfaces are cov-ered with transparent electrodes of size 14 mm ×
14 mm,coated with polyvinyl alcohol and then rubbed in or-der to orient the liquid crystals planarly in the x direc-tion. The cell is filled with N -(4-methoxybenzylidene)-4-butylaniline (MBBA) doped with 0.01 wt.% of tetra- n -butylammonium bromide, maintained at temperature25 ◦ C with a standard deviation of 2 × − ◦ C, and illu-minated by a stabilized light source made of white LEDs.A CCD camera records the light transmitted through theplates. The observed region is a central rectangle of size1217 µ m × µ m. Since there is a minimum linear size ofDSM2 domains, namely d/ √ × ×
107 for the observa-tion area. Note that the meaningful figure is that of thetotal system size, which is at least four orders of mag-nitude larger than in earlier experimental studies [7]. Inthe following, we vary V and fix the frequency at 250 Hz, FIG. 1: (Color online) Spatiotemporal intermittency between DSM1 and DSM2. (a) Sketch of a DSM2 with its many entangleddisclinations, i.e. loops of singularities in orientations of liquid crystals. (b) Snapshot taken at 35 .
153 V. Active (DSM2) regionsappear darker than the absorbing DSM1 background. See also Movie S1 [8]. (c) Binarized image of (b). See also Movie S2 [8].(d) Sketch of the dynamics: DSM2 domains (gray) stochastically contaminate [c] neighboring DSM1 regions (white) and/orrelax [r] into the DSM1 state, but do not nucleate spontaneously within DSM1 regions (DSM1 is absorbing). (e) Spatiotemporalbinarized diagrams showing DSM2 regions for three voltages near the critical point, namely 34 . . .
900 V. Thediagrams are shown in the range of 1206 µ m × µ m (the whole observation area) in space and 6 . roughly one third of the cutoff frequency 820 ±
70 Hz.Closely above the threshold V c marking the appear-ance of DSM2, a regime of spatiotemporal intermittency(STI) is observed, with DSM2 patches moving around ina DSM1 background (Fig. 1b and Movie S1 [8]). Thissuggests an absorbing phase transition [13] with DSM1playing the role of the absorbing state (Fig. 1d). Prior toany analysis, we must distinguish DSM2 domains fromDSM1. This binary reduction can be easily performed byour eyes, so we automated it using the facts that DSM2regions look darker, have longer time correlation, andhave minimum area of d / ρ is just the ratio of the surface occupied by active(DSM2) regions to the whole area.We first observe the steady-state STI regime underconstant voltages V , in the range of 34 .
858 V ≤ V ≤ .
998 V. The voltage for the onset of steady roll convec-tion is V ∗ = 8 .
95 V. The time-average of ρ in the steadystate, ¯ ρ , is taken over the period 1000 sec < T < V , a clear signature of criticality (Fig. 2). The criticalvoltage V c and the critical exponent β are determined FIG. 2: (Color online) Variation of the average DSM2 fraction¯ ρ with V in the steady state. Inset: same data in logarithmicscales, with vertical errorbars indicating the standard devia-tion of the time series ρ ( t ). Fluctuations of V are negligibleexcept for the first data point, where the standard deviation isshown by a horizontal errorbar. Blue lines are fitting curves. by fitting these data to the usual form ¯ ρ ∼ ( V − V ) β .(Here, deviations from criticality are measured in termsof V instead of V by convention, since the dielectrictorque that drives the convection is proportional to V [11].) This yields the following estimates V c = 34 . , β = 0 . , (1)where the numbers in parentheses indicate the uncer-tainty in the last digits, which correspond here to a 95%confidence interval in the sense of Student’s t. Our esti- FIG. 3: (Color online) Histograms of inactive (DSM1) length l x , l y and duration τ in the steady state near criticality. Dashedlines indicate the estimated algebraic decay at threshold. mate β = 0 . β DP = 0 . N ( l ) and N ( τ ), the distributions ofthe sizes l and durations τ of the inactive (DSM1) re-gions. For instance the distribution N ( l x ) in the x di-rection is obtained by detecting all inactive segments inthe x direction for all y and t . We find that they de-cay algebraically at criticality up to a finite-size expo-nential cutoff (Figs. 3a-c). The power-law decay is fittedas N ( l ) ∼ l − µ ⊥ and N ( τ ) ∼ τ − µ k with µ x = 1 . , µ y = 1 . , µ k = 1 . , (2)where µ x and µ y indicate the exponent µ ⊥ measured inthe x and y direction, respectively. They are again ingood agreement with the DP values µ DP ⊥ = 1 . µ DP k = 1 . ν ⊥ and ν k via µ ⊥ = 2 − β/ν ⊥ and µ k = 2 − β/ν k ,which lead us to estimate ν x = 0 . , ν y = 0 . , ν k = 1 . . (3)They should be compared to the DP values ν DP ⊥ =0 . ν DP k = 1 . ν ⊥ are in reasonably good agreement with the DPvalue, while the agreement on ν k is less satisfactory. Oursecond set of experiments provides another, independentestimate of ν k .Setting the voltage to 60 V, i.e. much higher than V c ,we wait until the system is totally invaded by DSM2 do-mains. We then suddenly decrease V to a value in therange of 34 .
86 V ≤ V ≤ .
16 V, i.e. near V c , and observethe time decay of activity for 900 seconds. We repeat this10 times for each V value and average the results over thisensemble. In such “critical-quench experiments,” correla-tion lengths and times grow in time, and, as long as theyremain much smaller than the system size, the scalingestimates are free from finite-size effects. As expected, ρ ( t ) decays exponentially with a certain correlation timefor V ≤ . V = 35 . V ≥ .
06V (Fig. 4a). A simple scaling Ansatz impliesthe following functional form for ρ ( t ) in this case: ρ ( t ) ∼ t − α f ( ε t /ν k ) , α = β/ν k , (4)where ε = ( V − V ) /V is the deviation from criticalityand f ( ζ ) is a universal scaling function. From the slopesof the algebraic regimes for the three V values closest tothe threshold, we estimate V c = 35 . , α = 0 . +(8) − (5) . (5)Note that V c is slightly higher than in the steady-stateexperiments. In fact, the roll convection onset V ∗ =8 .
96 V is also higher. We believe this is because, duringthe few days which separated the two sets of experiments,our sample has aged, a well-known property of MBBA.On the other hand, no measurable shift was detectedduring a given set of experiments. Our α value is againin good agreement with the DP value α DP = 0 . β and α in Eqs. (1)and (5), and the scaling relation (4) yield ν k = 1 . +(14) − (21) , (6)which is in good agreement with the DP value ν DP k =1 . ν k lie onopposite sides of the DP value, with their confidence in-tervals overlapping this reference number. Furthermore,Eq. (4) implies that the time series ρ ( t ) for different volt-ages collapse on the universal function f ( ζ ) when ρ ( t ) t α is plotted as a function of t | ε | ν k . Our data do collapsereasonably well on the universal function of DP (Fig. 4b).Continuous absorbing phase transitions are character-ized by three independent (static) exponents [1]. So far,we have found those three independent algebraic scalinglaws (typically over two decades). The critical exponentvalues all agree within a few percent with those of theDP class in (2+1) dimensions. In addition, data collapseis satisfactorily achieved onto the universal scaling. Thisconstitutes the first complete and unambiguous experi-mental realization of a DP-class absorbing phase transi-tion [15]. Looking back at previous attempts to exhibit a FIG. 4: (Color online) Critical-quench experiments. (a) De-cay of ρ ( t ) after the quench, for V = 34 .
86 V, 34 .
88 V, · · · ,35 .
16 V from the bottom left to the top right. The data for V = 35 .