Experimental realization of directed percolation criticality in turbulent liquid crystals

This is a comprehensive report on the phase transition between two turbulent states of electroconvection in nematic liquid crystals, which was recently found by the authors to be in the directed percolation (DP) universality class [K. A. Takeuchi et al., Phys. Rev. Lett. 99, 234503 (2007)]. 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 $2+1$ dimensions. 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.

Figure 1

(Color online) Spatiotemporal intermittency between DSM1 and DSM2. (a) 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. (b) Snapshot taken at 35.153 V. Active (DSM2) patches appear darker than the absorbing DSM1 background. See also movie S1 in Ref. [33]. (c) Binarized image of (b). See also movie S2 in Ref. [33]. (d) Sketch of the dynamics: DSM2 domains (gray) stochastically contaminate [c] neighboring DSM1 regions (white) and/or relax [r] into the DSM1 state but do not nucleate spontaneously within DSM1 regions (DSM1 is absorbing). (e) Spatiotemporal binarized diagrams showing DSM2 regions for three voltages near the critical point. The diagrams are shown in the range of $1206\times 899\mu {\text{m}}^2$ (the whole observation area) in space and 6.6 s in time.

Figure 2

(Color online) Schematic diagram of the experimental setup in its entirety (a) and for the thermocontroller (b). LED: light-emitting diode, CCD: charge-coupled device camera, PC: computer, PID: proportional-integral derivative. See text for details.

Figure 3

(Color online) The average DSM2 fraction $\overline{\rho }$ as a function of $V$ in the steady state. Inset: same data in logarithmic scales. Error bars indicate the standard deviation of fluctuations in $\rho \left(t\right)$ and $V$. Blue dashed lines are fitting curves, obtained by regression of all the data points except for the four highest voltages.

Figure 4

(Color online) Histograms of (inactive) DSM1 lengths $l_x,l_y$ and duration $\tau$ in the steady state in double-logarithmic scales (a)–(c) and in semilogarithmic scales (d)–(f). The applied voltages are in ascending order from bottom right to top left (a)–(c) and from top right to bottom left (d)–(f), respectively. Dashed lines show the estimated algebraic decay at criticality.

Figure 5

(Color online) Correlation length ${\xi }_x$, ${\xi }_y$ and correlation time ${\xi }_{\parallel }$ in the steady state as functions of the deviation from criticality $\epsilon \equiv \left(V^2-V_{\text{c}}^2\right)/V_{\text{c}}^2$. Dashed lines are guides for the eyes.

Figure 6

(Color online) Distributions of (active) DSM2 domain sizes in the steady state. (a) and (b) Histograms of DSM2 lengths $l_x$ in the $x$ direction (a) and of DSM2 duration $\tau$ (b). The applied voltages are in ascending order from bottom left to top right. (c) Characteristic length and time scales $l_{x0},l_{y0},{\tau }_0$, defined from the exponential decay in the histograms. The range of errors is smaller than the symbol size. Dashed line indicates the critical voltage $V_{\text{c}}$. (Inset) Ratios of $l_{x0}$ and $l_{y0}$ to ${\tau }_0$ (same symbols as in the main panel).

Figure 7

(Color online) Decay of the order parameter $\rho \left(t\right)$ on critical quenching. (a) $\rho \left(t\right)$ versus $t$ for $V=34.86,34.88,\dots ,35.16\text{V}$ from bottom left to top right. The curve for $V=35.04\text{V}$ (showing the longest power-law regime) is indicated by a thick line. (b) Same data with rescaled axes $t{\left|\epsilon \right|}^{{\nu }_{\parallel }}$ and $\rho \left(t\right)t^{\alpha }$, showing data collapsing. For $V_{\text{c}}$, $\alpha$, and ${\nu }_{\parallel }$, we use values measured in the experiment [Eqs. (7, 9)], but a collapse of similar quality is obtained also with DP-class exponent values. The dashed curve indicates the DP universal scaling function $F_{\rho }^{\text{DP}}\left(x\right)$ obtained numerically from the contact process.

Figure 8

(Color online) Autocorrelation function $C\left(t,t_0\right)$ at $V=V_{\text{c}}$ measured in the critical-quench experiments. Raw data in the inset are collapsed with rescaled axes $t/t_0$ and $C\left(t,t_0\right)t_0^b$. Dashed line shows the estimated asymptotic algebraic decay.

Figure 9

(Color online) Local persistence probability $P_{\text{l}}\left(t\right)$ in the critical-quench experiments. The initial time is $t_0=2.14\text{s}$ for the main panel and $t_0=0.81\text{s}$ 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.

Figure 10

(Color online) Schematic diagram of the experimental setup for the critical-spreading experiment. UV: ultraviolet; ND: neutral density. See text for details.

Figure 11

(Color online) Probability of DSM2 nucleation induced by five successive pulses of laser. Laser is linearly polarized along or perpendicularly to the mean direction $\mathit{n}$ of the molecules (squares and circles, respectively). Error bars denote the 95% confidence intervals assuming the binomial distribution. (Inset) Typical spatiotemporal diagram of nucleated DSM2 cluster for $V=36.65\text{V}>V_{\text{c}}$. Scales are $500\times 500\mu {\text{m}}^2$ in space and 30 s in time.

Figure 12

(Color online) Results of the critical-spreading experiments. (a)–(c) Survival probability $P_{\text{s}}\left(t\right)$, volume $V\left(t\right)$, and mean square radius $R^2\left(t\right)$ of clusters started from a single DSM2 nucleus, for $V=36.25,36.29,\dots ,36.65\text{V}$ from bottom to top. The data for $R^2\left(t\right)$ are mostly overlapping. The curves for $V=36.45\text{V}=V_{\text{c}}$ are drawn with thick lines. [Inset of (a)] The ratio of the survival probability $P_{\text{s}}\left(t\right)$ from the critical-spreading experiments to the DSM2 fraction $\rho \left(t\right)$ from the critical-quench experiments both at criticality. The dashed line is a guide for the eyes, indicating roughly the asymptotic ratio of $P_{\text{s}}\left(t\right)/\rho \left(t\right)$. (d)–(f) Same data with axes scaled after the expected scalings [Eqs. (15, 16, 17)]. We use values of $V_{\text{c}}$ and critical exponents measured in the experiments. The dashed curves indicate the DP universal scaling functions $F_{\text{s}}^{\text{DP}}\left(x\right),F_{\text{v}}^{\text{DP}}\left(x\right),F_{\text{r}}^{\text{DP}}\left(x\right)$ obtained numerically from the contact process.