Selected for a Viewpoint in Physics PHYSICAL REVIEW E 80, 051116 ͑2009͒

Experimental realization of directed criticality in turbulent liquid crystals

Kazumasa A. Takeuchi,1,2,* Masafumi Kuroda,1 Hugues Chaté,2 and Masaki Sano1,† 1Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 2Service de Physique de l’État Condensé, CEA-Saclay, 91191 Gif-sur-Yvette, France ͑Received 24 July 2009; published 16 November 2009͒ This is a comprehensive report on the between two turbulent states of electroconvection in nematic liquid crystals, which was recently found by the authors to be in the ͑DP͒ ͓K. A. Takeuchi et al., Phys. Rev. Lett. 99, 234503 ͑2007͔͒. We further investigate both static and dynamic critical behaviors of this phase transition, measuring a total of 12 critical exponents, 5 scaling functions, and 8 scaling relations, all in full agreement with those characterizing the DP class in 2+1 dimen- sions. Developing an experimental technique to create a seed of topological-defect turbulence by pulse laser, we confirm in particular the rapidity symmetry, which is a basic but nontrivial consequence of the field- theoretic approach to DP. This provides a clear experimental realization of this outstanding truly out-of- equilibrium universality class, dominating most phase transitions into an absorbing state.

DOI: 10.1103/PhysRevE.80.051116 PACS number͑s͒: 64.60.Ht, 64.70.mj, 47.27.Cn, 05.45.Ϫa

I. INTRODUCTION ments have been performed in situations where DP-class transitions would be theoretically expected, but they have Absorbing states, i.e., states which systems may fall into always yielded mixed and/or partial results with limited ac- but never escape from, and phase transitions into them are curacy ͓7,11–22͔͑Table I͒. This lack of fully convincing ex- expected to be ubiquitous in nature. For instance, spreading perimental realizations, in contrast with the wealth of nu- or contamination processes such as fires or epidemics exhibit merical results, has been found surprising and a matter of such transitions when the propagation rate changes: initially concern in the literature ͓1,28͔. ͑ ͒ active infected regions eventually disappear forever at low Recently, though, studying a STI regime occurring in tur- rates, i.e., the absorbing state is reached, while they can be bulent liquid crystals, we found a transition whose complete ͑ ͒ sustained pandemic regime for fast-enough propagation. set of static critical exponents matches those of the DP class Examples abound far beyond hundreds of numerical models, ͓29͔. The goal of the present paper is to provide a compre- describing, e.g., catalytic reactions, granular flows, and cal- hensive report on this transition including not only a more cium dynamics in living cells, to name but a few, have been complete description of experiments designed to investigate ͓ ͔ shown to exhibit such absorbing phase transitions 1,2 . Such the static critical behavior but also new experiments giving phase transitions also naturally arise from general problems access to dynamic critical behavior. ͓ ͔ ͓ ͔ such as synchronization 3,4 , self-organized criticality 5 , The paper is organized as follows. We first illustrate the ͑ ͓͒ ͔ ͓ ͔ spatiotemporal intermittency STI 6 , and depinning 7,8 . coarse-grained dynamics of the turbulent regime of electro- The vast majority of these transitions share the same critical convection studied and our basic experimental setup ͑Sec. ͑ ͒ ͓ ͔ behavior, that of the “directed percolation” DP class 1,2 , II͒, with further details on image analysis given in the Ap- as long as they are continuous. pendix. To characterize the critical behavior, we perform Deep theoretical issues underpin this situation. Whereas three series of experiments: ͑a͒ steady-state experiment un- universality is well understood for systems in thermody- der constant applied voltages ͑Sec. III͒, ͑b͒ critical-quench namic equilibrium, this is still not the case for systems driven experiment starting from fully active initial conditions ͑Sec. out of equilibrium, where even the relevant ingredients de- IV͒, and ͑c͒ critical-spreading experiment starting from a termining the class are not well understood. In this context, single active seed, prepared with an experimental technique absorbing phase transitions are central because of their genu- using pulse laser, developed in this work ͑Sec. V͒. In Sec. ine nonequilibrium character since absorbing states directly VI, our results are summarized and we discuss why clear imply violation of the detailed balance. Janssen and Grass- DP-class critical behavior is observed rather easily in our ͓ ͔ berger 9,10 conjectured that the DP universality class con- system, contrary to many other experiments performed in the tains all continuous transitions into a single effective absorb- past in this context. ing state in the absence of any extra symmetry or conservation law. The DP class thus appears as the simplest and most common case as testified by overwhelming numeri- II. EXPERIMENTAL SETUP cal evidence. We work on the electrohydrodynamic convection of nem- However, the situation has been quite different in experi- atic liquid crystals, which occurs when a thin layer of liquid ments. Over the last 20 years and more, a number of experi- crystal is subjected to an external voltage strong enough to trigger the Carr-Helfrich instability ͓30,31͔. This is a priori a suitable system to study critical behavior, thanks to its pos- *[email protected] sibly large aspect ratio, fast response time, and easy control- †[email protected] lability. We focus on the transition between two turbulent

1539-3755/2009/80͑5͒/051116͑12͒ 051116-1 ©2009 The American Physical Society TAKEUCHI et al. PHYSICAL REVIEW E 80, 051116 ͑2009͒

TABLE I. Summary of critical exponents measured in earlier experiments. Number in parentheses is the range of error given by the corresponding authors.

a ͑1+1͒D system Size ␤ ␯Ќ ␯ʈ ␮Ќ ␮ʈ Refs.

Annular Rayleigh-Bénard 22 0.5 1.9͑1͒ 1.9 ͓11͔ Annular Rayleigh-Bénard 35 0.5 0.5 1.7͑1͒ 2.0͑1͓͒12͔ Linear Rayleigh-Bénard 25.7 0.30͑5͒ 0.50͑5͒ 0.50͑5͒ 1.6͑2͒ 2.0͑2͓͒12͔

Interface roughening ␯Ќ /␯ʈ =0.63͑4͓͒7͔ Viscous fingering 60 0.45͑5͒ 0.64͑2͒ 0.61͑2͓͒13͔ Vortices of fluid 15 0.5 1.7 ͓14͔ Taylor-Deen 90 1.30͑26͒ 0.64, 0.53 0.73 1.67͑14͒ 1.74͑16͓͒15͔ Taylor-Couette 70 1 0.4 1.4–2.5 ͓16͔

Granular flow ␯ʈ −␯Ќ =1 ͓18,23͔ Torsional Couette 0.30͑1͒ 0.53͑5͓͒19͔ Ferrofluidic spikes 108 0.30͑5͒ 1.1͑2͒ 0.62͑14͒ 1.70͑5͒ 2.1͑1͓͒20͔ Lateral heat convection in annulus 114 0.27͑3͒ 0.30͑4͒ 0.75͑3͓͒21͔

DPb 0.276 1.097 1.734 1.748c 1.841c ͓24͔

␯Ќ /␯ʈ =0.633, ␯ʈ −␯Ќ =0.637

͑2+1͒D system Size ␤ ␯Ќ ␯ʈ ␮Ќ ␮ʈ Refs.

Liquid columns 169 0.56͑5͓͒22͔ DSM1-DSM2 ͑present paper͒d 2.7ϫ106 0.59͑4͒ 0.75͑6͒ 1.29͑11͒ 1.08͑18͒ 1.60͑5͒

DPb 0.583͑3͒ 0.733͑3͒ 1.295͑6͒ 1.204͑2͒c 1.5495͑10͒c ͓25,26͔ aNumber of effective degrees of freedom indicated by the corresponding authors. bSome exponents are obtained using scaling relations. Errors are then estimated by the law of propagation of error. cSee also the remark ͓27͔. dOnly some of the measured exponents are shown ͑see Table II for the complete list͒.

regimes, called dynamic scattering modes 1 and 2 ͑DSM1 ler, composed of heating wires and Peltier elements con- and DSM2͒, observed successively upon increasing the root- trolled by a proportional-integral-derivative feedback loop mean-square amplitude of the voltage V at relatively low with a lock-in amplifier ͓Fig. 2͑b͔͒. Windows of the thermo- frequencies ͓31,32͔. The difference between DSM1 and controller are made of sapphire in order to improve the spa- DSM2 lies in their density of topological defects in the di- tial homogeneity of the temperature. Throughout each series rector field ͓Fig. 1͑a͔͒. In the DSM2 state, a large quantity of of experiments, the cell temperature is maintained at 26.0 °C these defects, called disclinations, are present ͓34͔. They with fluctuations typically of a few mK, unless otherwise elongate and split constantly under the shear due to the fluc- stipulated, measured by three thermistors placed at different tuating turbulent flow around. In DSM1, on the other hand, positions close to the cell. disclinations are present but kept smaller than the critical We observe the electroconvection through the transmitted light from a handmade stabilized light source made of light- size and disappear immediately. Their density thus remains emitting diodes, recorded by a charge-coupled device cam- very low. The many disclinations in DSM2 lead to the loss of era. The observed region is a central rectangle of size 1217 macroscopic nematic anisotropy and to a lower light trans- ϫ911 ␮m2 ͑inset of Fig. 2͒. Since there is a minimum linear mittance than in DSM1. size of DSM2 domains, d/ͱ2 ͓32͔, we can roughly estimate Our basic experimental setup is shown in Fig. 2. The the number of effective degrees of freedom at 1650ϫ1650 sample cell is made of two parallel glass plates spaced by a Ϸ2.7ϫ106 for the convection area and 143ϫ107Ϸ1.5 ␮ polyester film of thickness d=12 m. Both inner surfaces ϫ104 for the observation area. Note that the meaningful fig- are covered with transparent electrodes of size 14 ure is that of the total system size, which is at least four 2 ϫ14 mm , coated with polyvinyl alcohol, and then rubbed orders of magnitude larger than in earlier experimental stud- in order that molecules are planarly aligned in the x direc- ies ͑Table I͒. In the following, we vary V and fix the fre- tion, defined thereby. The cell is filled with quency at 250 Hz at roughly one third of the cutoff fre- N-͑4-methoxybenzylidene͒-4-butylaniline ͑MBBA͒͑purity quency 820Ϯ70 Hz which separates the conducting and the Ͼ99.5%, Tokyo Chemical Industry͒ doped with 0.01 wt % dielectric regimes of electroconvection ͓30,31͔. of tetra-n-butylammonium bromide. The temperature of the Disclinations being topological defects, the spontaneous cell is kept constant carefully by a handmade thermocontrol- nucleation of DSM2 in a DSM1 domain is in principle for-

051116-2 EXPERIMENTAL REALIZATION OF DIRECTED-… PHYSICAL REVIEW E 80, 051116 ͑2009͒

(a) (b) (c)

DSM2

DSM1

DSM1 100µm

(d) (e) y [c] [r] x [r] [c] [c] t

time [r] [r] [r] 34.858 V 34.876 V 34.900 V

FIG. 1. ͑Color online͒ Spatiotemporal intermittency between DSM1 and DSM2. ͑a͒ Sketch of a DSM2 domain with many entangled disclinations, i.e., loops of singularities in orientations of liquid crystal. Blue dashed curves in the closeup indicate contour lines of equal alignment. ͑b͒ Snapshot taken at 35.153 V. Active ͑DSM2͒ patches appear darker than the absorbing DSM1 background. See also movie S1 in Ref. ͓33͔. ͑c͒ Binarized image of ͑b͒. See also movie S2 in Ref. ͓33͔. ͑d͒ Sketch of the dynamics: DSM2 domains ͑gray͒ stochastically contaminate ͓c͔ neighboring DSM1 regions ͑white͒ and/or relax ͓r͔ into the DSM1 state but do not nucleate spontaneously within DSM1 regions ͑DSM1 is absorbing͒. ͑e͒ Spatiotemporal binarized diagrams showing DSM2 regions for three voltages near the critical point. The diagrams are shown in the range of 1206ϫ899 ␮m2 ͑the whole observation area͒ in space and 6.6 s in time. bidden. It is indeed an essentially unobservable rare event by our eyes, so we automated it, based on the facts that except along the edges of the electrode and for very high DSM2 domains have lower transmittance, longer time corre- voltages far from the range investigated in the present paper. lations, and a minimum area of d2 /2 ͓32͔͑see Appendix for Therefore, the fully DSM1 state serves as an absorbing state. details͒. A typical result is shown in Fig. 1͑c͒ and in movie On the other hand, DSM2 domains introduced externally or S2 in Ref. ͓33͔. Figure 1͑e͒ displays spatiotemporal diagrams present initially in the system can remain sustained in the obtained this way, showing how DSM2 patches evolve in the bulk for large enough voltages but eventually disappear for steady state. This supports the qualitative dynamics illus- voltages lower than a certain threshold Vc. Closely above Vc, trated in Fig. 1͑d͒ and indeed looks like the directed perco- a regime of STI is observed, with DSM2 patches moving ͓ ͑ ͒ lation of, say, water in a porous medium under gravitational around on a DSM1 background Fig. 1 b and movie S1 in field. Ref. ͓33͔͔. The basic dynamics of the observed STI is illus- trated in Fig. 1͑d͒: active DSM2 patches evolve in space time ͑ essentially by contamination of neighboring inactive ab- III. STEADY-STATE EXPERIMENT sorbing͒ DSM1 regions and by relaxation into the DSM1 state. This suggests an absorbing phase transition induced by We first observe STI in the steady state under constant change in rates of both elementary processes ͓6͔, which are voltage V in the range of 34.858 VՅVՅ39.998 V. The functions of the applied voltage here. The order parameter ␳ voltage for the onset of steady roll convection ͑Williams do- V. Spatiotemporal distributions of DSM2 8.95= ءis then simply the ratio of the surface occupied by active main͒ is V DSM2 regions to the whole area. patches are recorded over the period 1000 sϽTϽ8000 s, Prior to any analysis, we must distinguish DSM2 domains which is longer than 103 correlation times defined from the from DSM1. This binary reduction can be easily performed fluctuations of the order parameter ␳͑t͒.

(a) (b) LED bridge diffuser (top) thermistors thermocontroller V power supply d =12µm cell amplifier power supply heating wire objective waveform Peltier (cooler) lock-in amp. PC 10x generator (side) fins PID controller half mirror CCD cell lenses to eyepiece unit sapphire windows

FIG. 2. ͑Color online͒ Schematic diagram of the experimental setup in its entirety ͑a͒ and for the thermocontroller ͑b͒. LED: light- emitting diode, CCD: charge-coupled device camera, PC: computer, PID: proportional-integral derivative. See text for details.

051116-3 TAKEUCHI et al. PHYSICAL REVIEW E 80, 051116 ͑2009͒

1 ͑ ͒ ␤ ͑ ͒ ͑ ͒ Vc = 34.856 4 V, = 0.59 4 , 2

0.8 0 where the numbers in parentheses indicate the range of errors 10 in the last digits ͓36͔. Our estimate ␤=0.59͑4͒ is in good DP 0.6 -1 agreement with the value for ͑2+1͒-dimensional DP, ␤ 10 ρ ρ =0.583͑3͓͒25,26͔.

0.4 -2 We then measure the distributions of the sizes l and dura- 10 tions ␶ of inactive ͑DSM1͒ regions or intervals between two 0.2 -4 -2 0 ͑ ͒ 10 10 10 neighboring active DSM2 patches. Histograms are made ε V 2 V 2 V 2 separately for each spatial direction, l and l , to take account =( - c )/ c x y 0 of the anisotropy of DSM1 ͓37,38͔. Care is also taken to 34 35 36 37 38 39 40 41 compensate missing intervals due to the finite observation Voltage (V) window: since an interval of size l may not be captured within a frame of size L, i.e., either edge of the interval may FIG. 3. ͑Color online͒ The average DSM2 fraction ¯␳ as a func- not be in the frame, with probability l/L, the unbiased dis- tion of V in the steady state. Inset: same data in logarithmic scales. ͑ ͒ ␳͑ ͒ tributions N1 l are estimated from the observed ones by Error bars indicate the standard deviation of fluctuations in t and ͑ ͒ ͑ ͒/͑ / ͓͒ ͔ V. Blue dashed lines are fitting curves, obtained by regression of all N1 l =Nobs l 1−l L 39 . the data points except for the four highest voltages. The results are shown in Fig. 4. We find that the DSM1 distributions decay algebraically within the observed length Figure 3 shows the time-averaged order parameter ¯␳, i.e., and time scale for voltages very close to criticality, while for the average fraction of DSM2. It shows that the transition is higher voltages they start to decay exponentially from certain continuous and that ¯␳ scales algebraically near the critical characteristic length or time scales, as expected for continu- point. Fitting those data in the algebraic regime ͑all the ous absorbing phase transitions ͓2͔. The observed power-law ␮ ␮ points except for the four highest voltages in Fig. 3͒ with the ͑ ͒ϳ − Ќ ͑␶͒ϳ␶− ʈ decays are fitted as N1 l l and N1 with usual scaling form ͓35͔ ␮ ͑ ͒ ␮ ͑ ͒ ␮ ͑ ͒ ͑ ͒ ␳ ϳ͑ 2 2͒␤ ͑ ͒ x = 1.08 18 , y = 1.19 12 , ʈ = 1.60 5 , 3 ¯ V − Vc , 1 ␤ ␮ ␮ ␮ the critical voltage Vc and the critical exponent are found where x and y indicate the exponent Ќ measured in the x to be and y directions, respectively. These exponents are directly (a) (b) (c)

6 slope -1.08(18) 6 slope -1.19(12) 6 10 10 10 slope -1.60(5)

4 4 4 10 10 10 )

) 35.303V ) τ

x 35.153V 35.153V y ( l l

35.050V 1

( ( 35.050V

1 34.982V 1 34.982V 2 2 N 2 N 10 34.930V N 10 34.930V 10 34.900V 34.900V 34.900V 34.876V 34.876V 34.876V 34.873V 34.873V 34.873V 0 34.858V 0 34.858V 0 34.858V 10 10 10 1 2 3 1 2 3 -1 0 1 2 10 10 10 10 10 10 10 10 10 10 lx (µm) ly (µm) τ (s)

6 (d) 6 (e) 6 (f) 10 10 10 34.90V 35.05V 35.80V 34.90V 35.05V 35.80V 34.90V 35.05V 35.80V 34.93V 35.15V 36.30V 34.93V 35.15V 36.30V 34.93V 35.15V 36.30V 34.98V 35.30V 37.00V 34.98V 35.30V 37.00V 34.98V 35.30V 37.00V 35.50V 38.00V 35.50V 38.00V 35.50V 38.00V 4 39.00V 4 39.00V 4 10 40.00V 10 40.00V 10 ) ) ) x τ y l ( l ( 1 ( 1 1 N N

2 N 2 2 10 10 10

0 0 0 10 10 10 0 500 1000 0 500 1000 0 20 40 60 80 lx (µm) ly (µm) τ (s)

FIG. 4. ͑Color online͒ Histograms of ͑inactive͒ DSM1 lengths lx ,ly and duration ␶ in the steady state in double-logarithmic scales ͑a͒–͑c͒ and in semilogarithmic scales ͑d͒–͑f͒. The applied voltages are in ascending order from bottom right to top left ͑a͒–͑c͒ and from top right to bottom left ͑d͒–͑f͒, respectively. Dashed lines show the estimated algebraic decay at criticality.

051116-4 EXPERIMENTAL REALIZATION OF DIRECTED-… PHYSICAL REVIEW E 80, 051116 ͑2009͒

(a) (b) (c) 3 3 2 10 10 10 slope -0.75(6) slope -0.78(9)

slope -1.29(11) 2 2 1 10 10 10 ) m) m slope -1 slope -1 (s) µ µ ( ( || ξ x y ξ 1 ξ 1 0 10 10 10

0 0 -1 10 10 10 -3 -2 -1 0 -3 -2 -1 0 -3 -2 -1 0 10 10 ε 10 10 10 10 ε 10 10 10 10 ε 10 10

FIG. 5. ͑Color online͒ Correlation length ␰x, ␰y and correlation time ␰ʈ in the steady state as functions of the deviation from criticality ␧ϵ͑ 2 2͒/ 2 V −Vc Vc. Dashed lines are guides for the eyes.

͑ ͒ ͑ connected to the fractal df of the DSM2 clusters tions. Fitting Eq. 6 to the data in scaling regions the first ␮ ␰ ␰ measured in the corresponding direction as =df +1. The five points for x , y and all the points except the first two for critical exponents ␮Ќ and ␮ʈ for DP can be estimated using ␰ʈ͒, we obtain scaling relations with the order parameter exponent ␤ and the ␯ = 0.75͑6͒, ␯ = 0.78͑9͒, ␯ʈ = 1.29͑11͒. ͑7͒ correlation length and time exponents ␯Ќ and ␯ʈ, namely ͓2͔, x y ␯DP ␮Ќ =2−␤/␯Ќ, ␮ʈ =2−␤/␯ʈ. ͑4͒ They are in good agreement with the DP values Ќ DP =0.733͑3͒ and ␯ʈ =1.295͑6͓͒25,26͔. In addition, the facts DP DP They give ␮ =1.204͑2͒ and ␮ʈ =1.5495͑10͒ for ͑2+1͒ di- ␰ ␰ Ќ that no significant anisotropy is found between x and y and mensions ͓26͔. Although the existence of slight discrepancies that they can be even shorter than the cell depth d=12 ␮m −2 of order 10 in Eq. ͑4͒ is suggested from numerical studies ͓Figs. 5͑a͒ and 5͑b͔͒ suggest that distributions of DSM2 ͓40,41͔, our estimates in Eq. ͑3͒ agree with values expected patches are practically not influenced by the anisotropy of for the DP class at any rate. DSM1 ͓37,38͔ and by the existence of coherent roll structure Moreover, the DSM1 distributions allow estimating cor- of width roughly d behind DSM1 ͓38͔. ␰ ␰ relation length and time scales, Ќ and ʈ, respectively, from On the other hand, distributions of sizes of active ͑DSM2͒ their exponential tails shown in Figs. 4͑d͒–4͑f͒. Fitting dis- patches do not show any long-range correlations even close tributions with an empirical form ͓11,20͔ to criticality ͓Figs. 6͑a͒ and 6͑b͔͒. It implies that the local ␮ /␰ ͑ ͒ N ͑l͒,N ͑␶͒ϳ͑Al− + B͒e−l , ͑5͒ recession of DSM2 into DSM1 sketched in Fig. 1 d is in- 1 1 deed always present. The effective relaxation rate can be with powers ␮Ќ and ␮ʈ fixed at the estimates in Eq. ͑3͒,we directly estimated just as an inverse of the characteristic time ␰ ␰ ␶ obtain the results shown in Fig. 5. Both Ќ and ʈ show 0 from the exponential tail of the temporal DSM2 distribu- ͑ ͒ ␶ algebraic divergence, tion, shown in Fig. 6 c . The characteristic time 0 increases

2 2 −␯Ќ 2 2 −␯ʈ linearly with V except for high voltages, where it is not sim- ␰Ќ ϳ͑V − V ͒ , ␰ʈ ϳ͑V − V ͒ , ͑6͒ c c ply determined from the dynamics of individual patches be- near criticality as expected, except for the first two points in cause of the saturation of the DSM2 fraction ͑Fig. 3͒.In Fig. 5͑c͒, which deviate from the presumably due particular, it exhibits no sign of criticality. All these observa- to finite length of movies ͑120 s͒ used to count the distribu- tions are consistent with, e.g., the dynamics of the so-called

6 (a)6 (b) (c) 10 10 1.2 80 40.00V 37.00V 40.00V 37.00V 60 l /τ 39.00V 35.80V 39.00V 35.80V 0 0 38.00V 35.15V 38.00V 35.15V 0.9 30 4 34.858V 4 34.858V 60 10 10

) V ) x 0 m) l τ

( 34 36 38 µ ( (s) 0.6 (

2 40 2 0 τ 0

N 2 2 N 10 10 l 0.3 τ 0 20 l x0 l 0 0 y0 10 10 0 V 0 0 500 1000 0 20 40 34 c 36 38 lx (µm) τ (s) Voltage (V)

FIG. 6. ͑Color online͒ Distributions of ͑active͒ DSM2 domain sizes in the steady state. ͑a͒ and ͑b͒ Histograms of DSM2 lengths lx in the x direction ͑a͒ and of DSM2 duration ␶ ͑b͒. The applied voltages are in ascending order from bottom left to top right. ͑c͒ Characteristic length ␶ and time scales lx0 ,ly0 , 0, defined from the exponential decay in the histograms. The range of errors is smaller than the symbol size. Dashed ͑ ͒ ␶ ͑ ͒ line indicates the critical voltage Vc. Inset Ratios of lx0 and ly0 to 0 same symbols as in the main panel .

051116-5 TAKEUCHI et al. PHYSICAL REVIEW E 80, 051116 ͑2009͒

(a) contact process ͓42͔͓depicted in Fig. 1͑d͔͒, a prototypical 0 10 model showing a DP-class transition ͓1,2͔, indicating that such process indeed governs the coarse-grained dynamics of slope -0.48(5) our STI regime in liquid crystal turbulence. Similarly, the spatial distributions of DSM2 sizes yield

) -1 t

͓ ͑ ͔͒ ( characteristic length scales lx0 and ly0 Fig. 6 c . Although 10 they appear to be also dominated by the relaxation process, ρ /␶ ͓ 35.06V as suggested from an effectively constant ratio l0 0 inset of Fig. 6͑c͔͒, they clearly show anisotropy ͓Fig. 6͑c͔͒,asop- 35.04V posed to the correlation lengths ␰ and ␰ estimated from -2 35.02V x y 10 -1 0 1 2 3 DSM1-size distributions ͓Figs. 5͑a͒ and 5͑b͔͒. This implies 10 10 10 10 10 that the contamination process of DSM2 is indeed driven by t (s) the anisotropic fluctuating shear flow of surrounding DSM1. 1 (b) The larger effective contamination rate in x is in line with the 10

fact that the turbulent structure of DSM1 remains mainly in 0 the x-z plane ͓37͔ but at odds with the global elliptic shape of 10

growing DSM2 nuclei, longer in y, observed for higher volt- -1 ) ages ͓32,43͔. Both would be explained if we assume that α 10 (s

turbulent flow behind is faster and more correlated in the y α -2 t 10 )

direction, but further studies are necessary on this point. t (

ρ -3 10 IV. CRITICAL-QUENCH EXPERIMENT -4 10 -4 -2 0 A typical experiment performed usually on numerical 10 10 10 ν models showing absorbing phase transitions is the critical t |ε | || (s) decay of active patches from fully active initial conditions ͓ ͔ 1,2 . In such critical-quench experiments, correlation length, FIG. 7. ͑Color online͒ Decay of the order parameter ␳͑t͒ on and time grow in time, and, as long as they remain much critical quenching. ͑a͒ ␳͑t͒ versus t for V smaller than the system size, scaling estimates are free from =34.86,34.88, ... ,35.16 V from bottom left to top right. The curve finite-size effects. for V=35.04 V ͑showing the longest power-law regime͒ is indi- ␯ Experiments are performed as follows: we first apply cated by a thick line. ͑b͒ Same data with rescaled axes t͉␧͉ ʈ and ͑ӷ ͒ ␳͑ ͒ ␣ ␣ ␯ 60 V Vc to the cell and wait until it is entirely invaded by t t , showing data collapsing. For Vc, , and ʈ, we use values DSM2 domains. We then suddenly decrease the applied volt- measured in the experiment ͓Eqs. ͑7͒ and ͑9͔͒, but a collapse of age to a value in the range of 34.86 VՅVՅ35.16 V, i.e., similar quality is obtained also with DP-class exponent values. The DP͑ ͒ near Vc, and observe the time decay of activity for 900 s. We dashed curve indicates the DP universal scaling function F␳ x repeat this ten times for each V and average the results over obtained numerically from the contact process. this ensemble. during a given set of experiments. We also confirmed that V We first measure the decay of the order parameter ␳͑t͒ c here is consistent with a threshold roughly estimated from ͓ ͑ ͔͒ ␳͑t͒ decays expo- after the quench Fig. 7 a . As expected, steady state just before the critical-quench experiments. Our nentially with a certain characteristic time for lower voltages, estimate of the critical exponent ␣ is again in good agree- converges to a finite value for higher voltages, and in be- ment with the DP value ␣DP=0.4505͑10͓͒26͔. V tween at =35.04 V, it decays algebraically over the whole Furthermore, the scaling form of Eq. ͑8͒ implies that the observation time. A simple scaling ansatz implies the follow- time series ␳͑t͒ for different voltages collapse on a single ing functional form for ␳͑t͒ in this case: ␣ ␯ curve F␳͑x͒ when ␳͑t͒t is plotted as a function of t͉␧͉ ʈ. Our

−␣ 1/␯ʈ ͓ ͑ ͔͒ ␳͑t͒ϳt F␳͑␧t ͒, ␣ = ␤/␯ʈ, ͑8͒ data do collapse reasonably well Fig. 7 b , where the upper and lower branches correspond to VϾV and VϽV , respec- ␧ϵ͑ 2 2͒/ 2 c c where V −Vc Vc is the deviation from criticality and tively. It is compared and found in good agreement with the ͑ ͒ F␳ x is a universal scaling function. From the slopes of the universal scaling function F␳͑x͒ of the DP class ͓dashed algebraic regimes for the three V values closest to the thresh- curve in Fig. 7͑b͔͒, calculated numerically from the old, we estimate ͑2+1͒-dimensional contact process. It shows that the decay V = 35.04͑1͒ V, ␣ = 0.48͑5͒. ͑9͒ of DSM2 patches is governed by this DP universal function c except for the very early stage where it is influenced by Note that Vc measured here is slightly higher than in the microscopic features of the liquid crystal as expected. ͑ ͒ steady-state experiments. In fact, the roll convection onset We also measure the autocorrelation function C t,t0 de- V was also higher. We believe this is because of fined as 8.96= ءV possible slight shift in the controlled temperature and also of C͑t,t ͒ = ͗␳͑r,t͒␳͑r,t ͒͘ − ͗␳͑r,t ͒͗͘␳͑r,t͒͘, ͑10͒ the aging of our sample, a well known property of MBBA, 0 0 0 ͗ ͘ during the days which separated the two sets of experiments. where ¯ denotes average in space and over ensembles. ͑ ͒ On the other hand, no measurable shift of Vc was detected During the critical decaying process, C t,t0 is not a function

051116-6 EXPERIMENTAL REALIZATION OF DIRECTED-… PHYSICAL REVIEW E 80, 051116 ͑2009͒

0 0 10 10 -1 10 slope -1.55(7)

) -2 0 10 t

-1 , t -3 10 ( 10 C )

b -4 0 10 ) 10 (s -1 0 1 2 0

10 10 10 10 t

-2 - b 0 t - t0 (s) -2 t 10 t 10 ( ) slope -2.5(3) l -2 0 10 P t , t

( -3 t=0 C 10 2.5 s 34.86V 10 -4 34.92V 5.8 s 35.04V 13.4 s 35.12V 31.4 s 35.16V -4 10-1 100 101 10 -4 0 1 10 10 10 -1 0 1 t / t0 10 10 10 t - t 0 (s) ͑ ͒ ͑ ͒ FIG. 8. Color online Autocorrelation function C t,t0 at V ͑ ͒ ͑ ͒ =Vc measured in the critical-quench experiments. Raw data in the FIG. 9. Color online Local persistence probability Pl t in the / ͑ ͒ b inset are collapsed with rescaled axes t t0 and C t,t0 t0. Dashed line critical-quench experiments. The initial time is t0 =2.14 s for the shows the estimated asymptotic algebraic decay. main panel and t0 =0.81 s for the inset. The applied voltages are in ascending order from top to bottom. Dashed lines are guides for the ͑ ͒ / eyes with the same slope. of t−t0 inset of Fig. 8 but scaled rather by t t0, a feature sometimes referred to as “aging” in the literature ͓44͔. Our ͑ ͒ data can be collapsed Fig. 8 using the expected scaling scopic nonuniversal time. Typical results are shown for t0 ͑ ͒ form =2.14 s in Fig. 9. The local persistence probability Pl t is ␭ / found to converge to a finite constant for VϽV and to decay ͑ ͒ϳ −b ͑ / ͒ ͑ ͒ϳ − C z͑ → ϱ͒ ͑ ͒ c C t,t0 t0 FC t t0 , FC x x x , 11 Ͼ exponentially for V Vc, as expected from numerical studies ͓ ͔ ͑ ͒ with of absorbing phase transitions 47–51 . At criticality, Pl t ͑ ͑ ͒ ␭ / ͑ ͒ ͑ ͒ decays algebraically again beyond some time of the order of b = 0.9 1 , C z = 2.5 3 . 12 ͒ t0 with DP ͑ ͒ Both are in agreement with DP values b =0.901 2 and ␪ ͑ ͒ ͑ ͒ ␭DP/ DP ͑ ͒ l = 1.55 7 . 14 C z =2.583 14 , estimated from the scaling relations ͓ ͔ ␪ ͑ ͒ 45 The value of l for 2+1 -dimensional DP is still a matter of ␤/␯ ␭ / ͑␤ ␯ ͒/␯ ͑ ͒ debate, with different estimates in past numerical studies: b =2 ʈ, C z =1+ + ds Ќ ʈ, 13 ␪DP ͑ ͓͒ ͔ ␪DPտ ͓ ͔ ␪DP l =1.50 1 48 , l 1.6 50 , and very recently l ͑ ͒ ͓ ͔ where ds =2 is the spatial . =1.611 7 , found with an improved algorithm 51 . Our The dynamic aspect of critical behavior can also be char- value, which is rather robust with respect to changing the acterized using first-passage quantities ͓46͔. One of these is initial conditions and the initial time ͑Fig. 9͒, is in agreement ͑ ͒ the local persistence probability Pl t , defined as the prob- with all numerical estimates to our accuracy. ability that the local state at a given point in space has not changed until time t, which typically shows a power-law V. CRITICAL-SPREADING EXPERIMENT ␪ ͑ ͒ϳ − l decay Pl t t . The nontrivial local persistence exponent ␪ In order to complete the characterization of the dynamic l is known to be in general independent of usual critical exponents such as ␤,␯Ќ ,␯ʈ and far less is known about its critical behavior of the DSM1-DSM2 transition, we per- universality mainly due to the fact that persistence is a quan- formed critical-spreading experiments, which start from a tity involving an infinite-point . single seed of active DSM2 region. This allows us to mea- In the context of DP, local persistence is measured from sure other critical exponents, such as the one governing the ͑ϱ͒ the probability that initially inactive “sites” do not become scaling of the probability Ps that a cluster starting from a ͑ϱ͒ϳ␧␤Ј active up to time t. ͑The persistence of activity is always single active seed survives forever: Ps , known to dominated by the local relaxation into the inactive state and serve as another order parameter characterizing absorbing shows only exponential decay similar to Fig. 6.͒ Initial con- phase transitions ͓1,2͔. ditions are typically set to be random in numerical studies The two exponents ␤ and ␤Ј are known to be equal to ͓47–50͔, which is, however, impossible in this experiment at each other for the DP class, thanks to an extra symmetry present. We therefore consider, as “initial condition” for the linked to time-reversal, the so-called rapidity symmetry ͓ ͔ ␣ ␦ ␦ local persistence configurations chosen at some moment t0 1,2 . This symmetry also implies that = , where is de- during the critical quench and measure the persistence prob- fined from the time decay of the survival probability at ͑ ͒ ͑ ͒ϳ −␦ ability Pl t−t0 . threshold, Ps t t . Since the scaling regime for the order parameter decay We stress, however, that the rapidity symmetry and the observed in Fig. 7 starts only after times of typically 1–2 s, resulting scaling relations do not hold generically in absorb- ␤ we further limit ourselves to t0 values larger than this micro- ing phase transitions. Although a value of Ј consistent with

051116-7 TAKEUCHI et al. PHYSICAL REVIEW E 80, 051116 ͑2009͒

mirror 1 n ⊥ E Nd:YAG laser laser thermally-insulated n || E UV-pass chamber 0.8 laser filter LED ty

lens diffuser bili a

thermocontroller V b 0.6 cell amplifier on pro i 0.4 lens objective eat ND filters 10x waveform l generator uc

N 0.2 dichroic mirror t mirror filters half mirror 0 CCD 34 36 38 40 42 to eyepiece unit PC lenses Voltage (V)

FIG. 10. ͑Color online͒ Schematic diagram of the experimental FIG. 11. ͑Color online͒ Probability of DSM2 nucleation induced setup for the critical-spreading experiment. UV: ultraviolet; ND: by five successive pulses of laser. Laser is linearly polarized along neutral density. See text for details. or perpendicularly to the mean direction n of the molecules ͑squares and circles, respectively͒. Error bars denote the 95% confidence intervals assuming the binomial distribution. ͑Inset͒ Typical spa- DP was suggested from the critical scaling of hysteresis tiotemporal diagram of nucleated DSM2 cluster for V=36.65 V loops in the DSM1-DSM2 transition ͓52͔, it remains impor- ϾV . Scales are 500ϫ500 ␮m2 in space and 30 s in time. tant to assess all spreading exponents as accurately and in- c dependently as possible in order to complete our character- ization of DP-class critical behavior and check directly the B. Results rapidity symmetry. We perform the critical-spreading experiments with the technique above. For each measurement we emit ten succes- A. Technique for nucleating DSM2 sive pulses polarized along the molecules ͑in the x direction͒, with which DSM2 is always nucleated for voltages of inter- In contrast to numerical simulations, it is not easy, in ex- est. In order to make reliable statistics for the survival prob- periments, to prepare an initial, single, localized seed of ability, it is necessary to repeat experiments at least hundreds DSM2 in an otherwise homogeneous DSM1 system. We de- of times for each voltage. This led us to improve further the veloped an experimental technique for nucleating a DSM2 temperature control of the cell, both in short and long time patch artificially using a pulse laser. scales. This is achieved by placing the thermocontroller in a The experimental setup for the critical-spreading experi- thermally insulated chamber ͑Fig. 10͒, made of plastic foam ments is schematically shown in Fig. 10. We emit 4–6 ns stage, wall, and ceiling, whose temperature inside is kept pulses of Nd:YAG laser ͑MiniLase II 20 Hz, New Wave constant by circulating constant-temperature water. The cell Research͒, focused by an objective lens ͑times ten, numerical temperature is finally stabilized at 25.0 °C with fluctuations aperture of 0.30, UPlanFLN, Olympus͒, into the cell. Using at most 2 mK over 2 weeks. This allows us to repeat the its third harmonic at 355 nm, around which MBBA has a ͓ ͔ experiment 563–567 times for each voltage in the range of broad absorption band 53,54 , and reducing its energy to Յ Յ 9 36.25 V V 36.65 V. The roll convection onset and the ء roughly 0.3 nJϷ2ϫ10 eV at the cell, we can indeed critical voltage roughly measured in steady state were V nucleate DSM2 from the absorbing DSM1 state ͑inset of Fig. =8.55 V and V Ϸ36.5 V, respectively. 11͒ without any observable damage to the sample. Figure 11 c We measure not only the survival probability P ͑t͒ but shows the probability of the DSM2 nucleation induced by s also the volume V͑t͒ and the mean square radius R2͑t͒ of emitting five successive laser pulses at 20 Hz. The nucleation DSM2 clusters, averaged over all the repetitions, even if the probability increases with voltage and is significantly higher cluster dies before time t. The following relations are then for laser polarized along the mean director field. This con- expected from the scaling ansatz ͓1,2͔: firms that nucleation is indeed brought about by the laser ␦ /␯ ͑ ͒ϳ − ͑␧ 1 ʈ͒ ␦ ␤ /␯ ͑ ͒ absorption of MBBA since its absorbance is higher along the Ps t t Fs t , = Ј ʈ, 15 long axis of the molecule ͓53,54͔. Moreover, the electronic ␪ /␯ structure of MBBA reveals that the ultraviolet absorption ͑ ͒ϳ ͑␧ 1 ʈ͒ ␪ ͑ ␯ ␤ ␤ ͒/␯ ͑ ͒ V t t Fv t , = ds Ќ − − Ј ʈ, 16 band stems from the C-N and C-C bonds between the aniline and benzyliden rings and is strongly coupled with twist 2 ␨ 1/␯ʈ R ͑t͒ϳt F ͑␧t ͒, ␨ =2/z =2␯Ќ/␯ʈ, ͑17͒ angles there ͓54͔. We therefore speculate that absorption of r ͑ ͒ ͑ ͒ ͑ ͒ ultraviolet laser pulses might lead to a sudden conformation where Fs x ,Fv x ,Fr x are universal scaling functions. change in the molecular structure, creating a topological de- The experimental results are shown in Fig. 12. Except for fect and thus triggering a DSM2 nucleation. R2͑t͒ where all the data are almost overlapping, which is also

051116-8 EXPERIMENTAL REALIZATION OF DIRECTED-… PHYSICAL REVIEW E 80, 051116 ͑2009͒

0 (a) 4 (b) 5 (c) 10 10 10 slope -0.46(5) slope 1.15(9) 4 ) )

2 10 2 ) t m m ( -1 4 3 slope 0.22(5) µ s P (t)/ρ(t) 10 10 s µ P )(

3 t 3 )( ( t 10 ( V

2 2

t (s) R 1 -1 1 3 -2 10 10 10 2 2 10 10 10 -1 1 3 -1 0 1 2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 t (s) t (s) t (s)

1 (d)4 (e)3 (f) 10 10 10 ) ) ζ θ /s

) 2 2 0 /s δ 10 2 10 m (s m µ µ ( δ ( t ζ t θ 0 ) -1 t - -

10 ) (

10 t t s ( ) P t 2 ( R V -2 -2 2 10 10 10 -4 -2 0 -4 -2 0 -4 -2 0 10 10 10 10 10 10 10 10 10 t |ε|ν || (s) t |ε|ν || (s) t |ε|ν || (s)

͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ FIG. 12. Color online Results of the critical-spreading experiments. a – c Survival probability Ps t , volume V t , and mean square radius R2͑t͒ of clusters started from a single DSM2 nucleus, for V=36.25,36.29, ... ,36.65 V from bottom to top. The data for R2͑t͒ are ͓ ͑ ͔͒ ͑ ͒ mostly overlapping. The curves for V=36.45 V=Vc are drawn with thick lines. Inset of a The ratio of the survival probability Ps t from the critical-spreading experiments to the DSM2 fraction ␳͑t͒ from the critical-quench experiments both at criticality. The dashed line is a ͑ ͒/␳͑ ͒ ͑ ͒ ͑ ͒ ͓ guide for the eyes, indicating roughly the asymptotic ratio of Ps t t . d – f Same data with axes scaled after the expected scalings Eqs. ͑ ͒ ͑ ͔͒ 15 – 17 . We use values of Vc and critical exponents measured in the experiments. The dashed curves indicate the DP universal scaling DP͑ ͒ DP͑ ͒ DP͑ ͒ functions Fs x ,Fv x ,Fr x obtained numerically from the contact process. typically the case in simulations of DP-class models, the data satisfactorily with those of the ͑2+1͒-dimensional contact show opposite curvatures below and above a certain voltage. process ͓dashed curves in Figs. 12͑d͒–12͑f͔͒. As in the data Seeking for the curve with the longest algebraic regime in collapse of ␳͑t͓͒Fig. 7͑b͔͒, the collapsed data show typical V͑t͒, which is statistically most reliable, we locate the critical time scales above which the corresponding quantities are voltage Vc at governed by the DP scaling functions. ͑ ͒ ͑ ͒ Vc = 36.45 2 V. 18 We then measure the three critical exponents ␦,␪,␨ from the VI. SUMMARY AND DISCUSSION algebraic regime for three voltages around Vc, yielding We have performed three series of experiments, namely, ␦ = 0.46͑5͒, ␪ = 0.22͑5͒, ␨ = 1.15͑9͒. ͑19͒ steady-state experiments, critical-quench experiments, and critical-spreading experiments, to characterize the critical be- All of them are in good agreement with the DP exponents, havior of the DSM1-DSM2 transition in liquid crystal turbu- ␦DP=0.4505͑10͒, ␪DP=0.2295͑10͒, and ␨DP=1.1325͑10͒ lence. Table II summarizes the main results. We have mea- ͓26͔. This confirms the scaling relations expressing the rapid- sured in total 12 critical exponents with reasonable accuracy ity symmetry. It is further confirmed by plotting the ratio of typically over a few orders of magnitude of power-law re- ͑ ͒ ␳͑ ͒ the survival probability Ps t to the DSM2 fraction t in the gimes. All of the measured exponent values agree within a critical-quench experiments ͓inset of Fig. 12͑a͔͒, which few percent with those defining the DP universality class shows that the two order parameters become asymptotically ͓36͔. Given that most of them are theoretically linked proportional to each other through scaling relations, we can equivalently say that we have experimentally confirmed those scaling relations, 8 in P ͑t͒ϳm2␳͑t͒͑20͒ s total, that connect the measured exponents ͑Table III͒. with the coefficient m2 Ϸ3.2. Among them, we have confirmed in particular the rapidity We also tested data collapse ͓Figs. 12͑d͒–12͑f͔͒, which symmetry ␣=␦, providing also the asymptotic amplitude of provides reasonable results given the limited statistical accu- the ratio between the two order parameters, m2 Ϸ3.2. More- racy of the data. The obtained scaling functions coincide over, we have also tested the expected scaling forms of Eqs.

051116-9 TAKEUCHI et al. PHYSICAL REVIEW E 80, 051116 ͑2009͒

TABLE II. Summary of the measured critical exponents ͑see the even weak disorder changes the asymptotic critical behavior remark ͓36͔ for the range of errors shown in the list͒. ͓56–58͔, but the characteristic length or time scale separating DP and disordered, so-called “activated” critical behavior, Exponent DSM1-DSM2 DPa grows fast with decreasing strength of disorder ͓56,57͔.Itis therefore important to work with systems made of macro- Density order parameter ␤ 0.59͑4͒ 0.583͑3͒ scopic units, where quenched disorder is expected to be neg- b Correlation length ␯Ќ 0.75͑6͒ 0.78͑9͒ 0.733͑3͒ ligible. We consider that the quenched disorder in our sys-

Correlation time ␯ʈ 1.29͑11͒ 1.295͑6͒ tem, which may take the form of inhomogeneities in the b c Inactive interval in space ␮Ќ 1.08͑18͒ 1.19͑12͒ 1.204͑2͒ electrodes or impurities in the sample, is also sufficiently c weak. Inactive interval in time ␮ʈ 1.60͑5͒ 1.5495͑10͒ Still, our DSM1-DSM2 transition seems to be the only ␣ ͑ ͒ ͑ ͒ Density decay 0.48 5 0.4505 10 fully convincing one. Three factors explain, in our view, why ␪ ͑ ͒ ͑ ͒d Local persistence l 1.55 7 1.611 7 our experiments provided such clear DP scaling laws. Aging in autocorrelator b 0.9͑1͒ 0.901͑2͒ ͑a͒ Large system size and fast response. As already men-

␭C /z 2.5͑3͒ 2.583͑14͒ tioned, one great advantage to work with electroconvection Survival probability ␦ 0.46͑5͒ 0.4505͑10͒ is that very large aspect ratios can easily be realized. The ␪ ͑ ͒ ͑ ͒ number of effective degrees of freedom of our cell is 2.7 Cluster volume 0.22 5 0.2295 10 ϫ106, which is orders of magnitude larger than any earlier ␨ ͑ ͒ ͑ ͒ Cluster mean square radius 1.15 9 1.1325 10 experiment ͑Table I͒. This considerably suppresses finite-size aReferences ͓25,26͔. effects and allows us to observe scaling on several orders of b For ␮Ќ and ␯Ќ, exponents measured in x and y directions are magnitude. Similarly, the typical microscopic time scales of shown in this order. liquid crystals are very short ͑of the order of 10 ms͒, provid- cSee also the remark ͓27͔. ing accurate statistics in reasonable laboratory time. dReference ͓51͔. ͑b͒ Almost perfectly absorbing state. The condition for being an absorbing state, i.e., that the system can never es- ͑8͒, ͑11͒, and ͑15͒–͑17͒ through data collapse and found cape once it entered, appears to be somewhat too strict from them in good agreement with numerically obtained DP uni- the experimental point of view. Indeed, spontaneous nucle- versal scaling functions. Reproducibility of the results was ation of the active state seems to have been present at least in also checked for three independent critical exponents some of the past experiments ͓1,20͔, which blurs the critical

␤,␮Ќ ,␮ʈ, using two other samples with different cell sizes behavior beyond some finite scales. and cutoff frequencies. Based on all these results, we con- On the other hand, our active state, DSM2, consists of clude that the DSM1-DSM2 transition constitutes an unam- topological defects, whose spontaneous formation is in prin- biguous experimental realization of an absorbing phase tran- ciple forbidden. Of course this may occur in practice, as sition in the DP universality class. suggested from the observation of vanishing hysteresis at the We now return to our initial remark concerning the sur- DSM1-DSM2 transition ͓52͔, but the nucleation rate remains prising scarcity of experimental realizations of DP-class tran- so low that we cannot directly observe it, constituting an sitions ͑Table I͒. almost perfectly absorbing state. One central difficulty lies in the necessity to avoid, as ͑c͒ Fluctuating absorbing state. In most earlier experi- much as possible, quenched disorder, which is known to be ments and numerical studies, the absorbing state has been relevant. It is theoretically known that such disorder does essentially fluctuation-free or laminar. This is indeed a natu- affect DP criticality and even destroys it for strong disorder ral choice suggested from the minimal theory of DP and ͓55͔. Recent theoretical and numerical studies show that causes of course no problem in numerical studies. In experi- ments, however, such absorbing states may typically cause TABLE III. Experimentally confirmed scaling relations. long-range effects through the rigidity of their laminar pat- tern and/or the propagation of solitonlike objects, which may Scaling relations LHS RHS DPa further reduce the effective system size and even break DP ͓ ͔ b scaling 1,2 . In contrast, our absorbing state, DSM1, is itself ␮Ќ =2−␤/␯Ќ ͑in x͒ 1.08͑18͒ 1.21͑8͒ 1.204͑2͒ a fluctuating turbulent state. Long-range interactions are then ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ in y 1.19 12 1.24 10 1.204 2 likely to be killed by the local turbulent fluctuations of ␮ ␤/␯ b ͑ ͒ ͑ ͒ ͑ ͒ ʈ =2− ʈ 1.60 5 1.54 5 1.5495 10 DSM1, which may have led to the observation of clean DP ␣=␤/␯ʈ 0.48͑5͒ 0.46͑5͒ 0.4505͑10͒ critical behavior. It is noteworthy to remark that the Chaté- b=2␤/␯ʈ 0.9͑1͒ 0.91͑10͒ 0.901͑2͒ Manneville coupled map lattice ͓59͔, a deterministic numeri- ␭ / ͑␤ ␯ ͒/␯ ͑ ͒ ͑ ͒c ͑ ͒ C z=1+ +ds Ќ ʈ 2.5 3 2.62 17 2.583 14 cal model for absorbing phase transitions with a nonchaotic ͑laminar͒ absorbing state, exhibits non-DP critical behavior ␦=␤/␯ʈ 0.46͑5͒ 0.46͑5͒ 0.4505͑10͒ ␪ ͑ ␯ ␤͒/␯ ͑ ͒ ͑ ͒c ͑ ͒ probably due to solitonlike objects propagating through lami- = ds Ќ −2 ʈ 0.22 5 0.25 11 0.2295 10 ␨ / ␯ /␯ ͑ ͒ ͑ ͒c ͑ ͒ nar regions. Interestingly, an elementary modification to ren- =2 z=2 Ќ ʈ 1.15 9 1.16 14 1.1325 10 der its absorbing state itself chaotic does lead to DP scaling aReferences ͓25,26͔. ͓60͔. bSee also the remark ͓27͔. In conclusion, we have experimentally found that the c The value of ␯x is used for ␮Ќ. DSM1-DSM2 transition in the electroconvection of nematic

051116-10 EXPERIMENTAL REALIZATION OF DIRECTED-… PHYSICAL REVIEW E 80, 051116 ͑2009͒ liquid crystals, showing an absorbing transition into spa- threshold Vc since DSM1 always appears first when the volt- tiotemporal intermittency, clearly belongs to the DP univer- age is applied. ͑c͒ Since DSM2 domains have lower trans- sality class. Both static and dynamic critical behaviors have mittance than DSM1, we extract the regions where the nor- been investigated with the help of the newly developed ex- malized intensity is less than a certain threshold, Ith. Here we perimental technique to create a seed of DSM2, confirming a chose Ith=1.5, determined so as to obtain a good agreement total of 12 critical exponents, 5 scaling functions, 8 scaling with direct visual observations particularly in movies. ͑d͒ relations, and in particular the rapidity symmetry, in full Taking into consideration that DSM2 domains move much agreement with those characterizing the DP class in 2+1 slower than the local intensity fluctuations in DSM1 ͑recall dimensions. We hope that this clear and comprehensive ex- that DSM1 is itself a turbulent state͒ and that the minimum perimental realization of a DP-class transition will trigger DSM2 area is d2 /2 ͓32͔, where d is the depth of the cell, we further studies of absorbing phase transitions and of related take the logical intersection ͑“AND” operator͒ of the three situations with genuinely nonequilibrium critical properties. successive images and then remove clusters with area et al. ͓ ͔ In this respect the recent works of Corté 61 and of smaller than d2 /2. Clusters are screened out in this way, and et al. ͓ ͔ Mangan 62 are especially encouraging, concerning their binarized images in the middle frame are used as final experiments and realistic situations, respectively, for absorb- images. In other words, the intersection is used only for com- ing transitions with a conserved field. parison with the minimum area. ͑e͒ Finally, we cut off the periphery of the image of width d/2ͱ2 since this region is ACKNOWLEDGMENTS biased in the step ͑d͒. The size of the binarized images re- ϫ ␮ 2 The authors are grateful to I. Dornic, F. Ginelli, J. A. duces to 1206 899 m , which roughly corresponds to ϫ Hoyos, S. Kai, and N. Oikawa for fruitful discussions. We 142 106 effective degrees of freedom. We confirmed that ͑ ͒ would also like to thank M. Henkel and G. I. Menon for the chosen threshold Ith=1.5 works well all over the range drawing our attention to aging and Ref. ͓41͔ and to local of voltages we investigate and that no DSM2 region is persistence, respectively. This work was supported in part by falsely detected when binarizing images of the fully DSM1 the Grant-in-Aid for Scientific Research ͑Grant No. phase. Typical results of the binarization are shown in Figs. ͑ ͒ ͑ ͒ ͓ ͔ 18068005͒ and by the JSPS. 1 c and 1 e and movie S2 in Ref. 33 , where we can con- firm that DSM2 domains are precisely detected. APPENDIX: BINARIZING IMAGES For the critical-spreading experiments presented in Sec. V, we have slightly modified the binarization algorithm in order Every analysis presented in the paper is performed using to detect DSM2 patches originating only from the prepared binarized images, where DSM2 domains are distinguished seed and not to miss them. To this end, we have reduced the from the absorbing DSM1 background. We give here a de- intensity threshold Ith to 1.2 and, instead, binarized only tailed description of the binarizing algorithm we used. within a target region, defined from positions of DSM2 The binarization is carried out in the following manner. patches in previous images ͑position of the seed for the first ͑a͒ We prepare three successive images taken at 15 frames/s image͒ and assumed maximal displacement of DSM2, which and remove the inhomogeneity of the incident light intensity. is chosen to be much larger than the actual displacement, ͑b͒ We then normalize the obtained intensity I of the three namely, d during successive two images taken at 15 frames/s ͗ ͘ images with respect to mean IDSM1 and standard deviation and extrapolated diffusively. We again confirmed that bina- ␦ IDSM1 of the DSM1 intensity at a given voltage, namely, rized images from a single set of parameters closely follow ͑ ͗ ͒͘/␦ Inorm= I− IDSM1 IDSM1. Note that we can separately mea- the actual evolution of DSM2 patches for all the voltages of sure the intensity of the fully DSM1 state even above the interest. A typical result is shown in the inset of Fig. 11.

͓1͔ H. Hinrichsen, Adv. Phys. 49, 815 ͑2000͒. Politi, and A. Torcini, Phys. Rev. E 68, 065102͑R͒͑2003͒. ͓2͔ M. Henkel, H. Hinrichsen, and S. Lübeck, Non-Equilibrium ͓9͔ H. K. Janssen, Z. Phys. B: Condens. Matter 42, 151 ͑1981͒. Phase Transitions, Vol. I: Absorbing Phase Transitions ͓10͔ P. Grassberger, Z. Phys. B: Condens. Matter 47, 365 ͑1982͒. ͑Springer, Dordrecht, 2008͒. ͓11͔ S. Ciliberto and P. Bigazzi, Phys. Rev. Lett. 60, 286 ͑1988͒. ͓3͔ L. Baroni, R. Livi, and A. Torcini, Phys. Rev. E 63, 036226 ͓12͔ F. Daviaud, M. Bonetti, and M. Dubois, Phys. Rev. A 42, 3388 ͑2001͒. ͑1990͒. ͓4͔ V. Ahlers and A. Pikovsky, Phys. Rev. Lett. 88, 254101 ͓13͔ S. Michalland, M. Rabaud, and Y. Couder, Europhys. Lett. 22, ͑2002͒. 17 ͑1993͒. ͓5͔ R. Dickman, M. A. Muñoz, A. Vespignani, and S. Zapperi, ͓14͔ H. Willaime, O. Cardoso, and P. Tabeling, Phys. Rev. E 48, Braz. J. Phys. 30,27͑2000͒. 288 ͑1993͒. ͓6͔ Y. Pomeau, Physica D 23,3͑1986͒. ͓15͔ M. M. Degen, I. Mutabazi, and C. D. Andereck, Phys. Rev. E ͓7͔ S. V. Buldyrev, A. -L. Barabási, F. Caserta, S. Havlin, H. E. 53, 3495 ͑1996͒. Stanley, and T. Vicsek, Phys. Rev. A 45, R8313 ͑1992͒. ͓16͔ P. W. Colovas and C. D. Andereck, Phys. Rev. E 55, 2736 ͓8͔ F. Ginelli, V. Ahlers, R. Livi, D. Mukamel, A. Pikovsky, A. ͑1997͒.

051116-11 TAKEUCHI et al. PHYSICAL REVIEW E 80, 051116 ͑2009͒

͓17͔ H. Téphany, J. Nahmias, and J. A. M. S. Duarte, Physica A shift in data collapsing is seen. ͑d͒ For values derived from an 242,57͑1997͒. equation, the range of error is estimated by the law of propa- ͓18͔ A. Daerr and S. Douady, Nature ͑London͒ 399, 241 ͑1999͒. gation of error. ͑e͒ For values cited from literature, we simply ͓ ͔ ͑ ͒ 19 A. Cros and P. Le Gal, Phys. Fluids 14, 3755 2002 . borrow the range of errors provided by the corresponding au- ͓ ͔ 20 P. Rupp, R. Richter, and I. Rehberg, Phys. Rev. E 67, 036209 thors. ͑2003͒. ͓37͔ G. Strangi, C. Versace, N. Scaramuzza, D. E. Lucchetta, V. ͓21͔ V. Lepiller, A. Prigent, F. Dumouchel, and I. Mutabazi, Phys. Carbone, and R. Bartolino, Phys. Rev. E 59, 5523 ͑1999͒. Fluids 19, 054101 ͑2007͒. ͓38͔ T. Nagaya, T. Takeda, and H. Orihara, J. Phys. Soc. Jpn. 68, ͓22͔ C. Pirat, A. Naso, J.-L. Meunier, P. Maïssa, and C. Mathis, 3848 ͑1999͒. Phys. Rev. Lett. 94, 134502 ͑2005͒. ͓ ͔ ͓ ͔ ͓23͔ H. Hinrichsen, A. Jiménez-Dalmaroni, Y. Rozov, and E. 39 This correction was not made in Ref. 29 . Although algebraic Domany, Phys. Rev. Lett. 83, 4999 ͑1999͒. decays in the distributions are not much influenced, this cor- ͓24͔ I. Jensen, J. Phys. A 32, 5233 ͑1999͒. rection is significant to have exponential decays in sufficiently ͓25͔ P. Grassberger and Y. C. Zhang, Physica A 224, 169 ͑1996͒. large scales to estimate the correlation length and time. ͓26͔ C. A. Voigt and R. M. Ziff, Phys. Rev. E 56, R6241 ͑1997͒. ͓40͔ B. Hede, J. Kertész, and T. Vicsek, J. Stat. Phys. 64, 829 ͑ ͒ ͓27͔ Scaling relations used for ␮Ќ and ␮ʈ ͓Eq. ͑4͔͒ may be slightly 1991 . violated in the order of 10−2 due to some intermittency effects ͓41͔ M. Henkel and R. Peschanski, Nucl. Phys. B 390, 637 ͑1993͒. in DP ͓40,41͔. ͓42͔ T. E. Harris, Ann. Probab. 2, 969 ͑1974͒. ͓28͔ P. Grassberger, in Nonlinearities in Complex Systems, edited ͓43͔ Note that parallel and perpendicular directions ͑x and y in our by S. Puri and S. Dattagupta ͑Narosa, New Delhi, 1997͒,p.61. notation͒ in the second reference in Ref. ͓32͔ are opposite. ͓29͔ K. A. Takeuchi, M. Kuroda, H. Chaté, and M. Sano, Phys. Rev. ͓44͔ M. Henkel, J. Phys.: Condens. Matter 19, 065101 ͑2007͒. Lett. 99, 234503 ͑2007͒. ͓45͔ F. Baumann and A. Gambassi, J. Stat. Mech.: Theory Exp. ͓30͔ E. Dubois-Violette, P. G. de Gennes, and O. Parodi, J. Phys. ͑2007͒, P01002. ͑Paris͒ 32, 305 ͑1971͒. ͓46͔ S. N. Majumdar, Curr. Sci. 77, 370 ͑1999͒, and references ͓31͔ P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, therein. 2nd ed. ͑Oxford University Press, Oxford, 1993͒. ͓47͔ H. Hinrichsen and H. M. Koduvely, Eur. Phys. J. B 5, 257 ͓32͔ S. Kai, W. Zimmermann, M. Andoh, and N. Chizumi, J. Phys. ͑1998͒. Soc. Jpn. 58, 3449 ͑1989͒; Phys. Rev. Lett. 64, 1111 ͑1990͒. ͓48͔ E. V. Albano and M. A. Muñoz, Phys. Rev. E 63, 031104 ͓33͔ J. Nehring, Phys. Rev. A 7, 1737 ͑1973͒. Note that topological ͑2001͒. defects observed there are obtained by switching off the ap- ͓49͔ G. I. Menon, S. Sinha, and P. Ray, Europhys. Lett. 61,27 plied voltage from DSM2 ͑see also Ref. ͓32͔͒. ͑2003͒. ͓34͔ See the supplementary material to Ref. ͓29͔, EPAPS Document ͓50͔ J. Fuchs, J. Schelter, F. Ginelli, and H. Hinrichsen, J. Stat. No. E-PRLTAO-99-056749, for movies S1 and S2. This mate- Mech.: Theory Exp. ͑2008͒, P04015. rial may be accessed from the reference section of the online ͓51͔ P. Grassberger, J. Stat. Mech.: Theory Exp. ͑2009͒, P08021. article, or via the EPAPS homepage http://www.aip.org/ ͓52͔ K. A. Takeuchi, Phys. Rev. E 77, 030103͑R͒͑2008͒. pubservs/epaps.html. ͓53͔ M. Mizuno and T. Shinoda, Mol. Cryst. Liq. Cryst. 41, 155 ͓35͔ The deviation from criticality is measured in terms of V2 in- ͑1978͒. stead of V by convention since the dielectric torque that drives ͓54͔ M. Mizuno and T. Shinoda, Mol. Cryst. Liq. Cryst. 69, 103 the electroconvection is proportional to V2 ͓30,31͔. The results ͑1981͒. ␳ϳ͑ ͒␤ ͓ ͔ do not change if ¯ V−Vc is used. 55 J. Hooyberghs, F. Iglói, and C. Vanderzande, Phys. Rev. Lett. ͓36͔ Confidence intervals of the critical exponents given in the 90, 100601 ͑2003͒. present paper are determined as follows. ͑a͒ For values esti- ͓56͔ J. A. Hoyos, Phys. Rev. E 78, 032101 ͑2008͒. mated from a finite number of independent data points ͓Eqs. ͓57͔ T. Vojta and M. Dickison, Phys. Rev. E 72, 036126 ͑2005͒;T. ͑2͒ and ͑7͔͒, 95% confidence intervals in the sense of student’s Vojta, A. Farquhar, and J. Mast, ibid. 79, 011111 ͑2009͒. t are provided. ͑b͒ For values estimated from a continuous ͓58͔ C. J. Neugebauer, S. V. Fallert, and S. N. Taraskin, Phys. Rev. curve ͓Eqs. ͑3͒, ͑9͒, ͑14͒, ͑12͒, and ͑19͔͒, range of fitting is E 74, 040101͑R͒͑2006͒; S. V. Fallert and S. N. Taraskin, ibid. varied within reasonable regions, and both the dependence on 79, 042105 ͑2009͒. fitting regions and the confidence interval from each fitting are ͓59͔ H. Chaté and P. Manneville, Physica D 32, 409 ͑1988͒. incorporated in the final estimate of error. When the threshold ͓60͔ H. Chaté ͑unpublished͒. is simultaneously determined ͓Eqs. ͑9͒ and ͑19͔͒, uncertainty ͓61͔ L. Corté, P. M. Chaikin, J. P. Gollub, and D. J. Pine, Nat. Phys. in the estimate of the threshold is also considered. ͑c͒ For 4, 420 ͑2008͒. values determined from data collapsing ͓Eq. ͑12͔͒, we provide ͓62͔ N. Mangan, C. Reichhardt, and C. J. Olson Reichhardt, Phys. the range of values of the exponent with which no systematic Rev. Lett. 100, 187002 ͑2008͒.

051116-12