Directed Percolation Criticality in Turbulent Liquid Crystals

We experimentally investigate the critical behavior of a phase transition between two topologically different turbulent states of electrohydrodynamic convection in nematic liquid crystals. The statistical 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 directed percolation class in $2+1$ dimensions. This constitutes the first clear and comprehensive experimental evidence of an absorbing phase transition in this prominent nonequilibrium universality class.

Figure 1

Spatiotemporal intermittency between DSM1 and DSM2. (a) Sketch of a DSM2 with its many entangled disclinations, i.e., loops of singularities in orientations of liquid crystals. (b) Snapshot taken at 35.153 V. Active (DSM2) regions appear 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/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, namely, 34.858, 34.876, and 34.900 V. The diagrams are shown in the range of $1206\mu \mathrm{m}\times 899\mu \mathrm{m}$ (the whole observation area) in space and 6.6 s in time.

Figure 2

Variation of the average DSM2 fraction $\overline{\rho }$ with $V$ in the steady state. Inset: same data in logarithmic scales, with vertical error bars indicating the standard deviation of the time series $\rho (t)$. Fluctuations of $V$ are negligible except for the first data point, where the standard deviation is shown by a horizontal error bar. Solid lines are fitting curves.

Figure 3

Histograms of inactive (DSM1) lengths $l_x$, $l_y$ and duration $\tau$ in the steady state near criticality. Regions discarded in the fitting are painted light in color. Dashed lines indicate the estimated algebraic decay at threshold.

Figure 4

Critical-quench experiments. (a) Decay of $\rho (t)$ after the quench, for $V=34.86\mathrm{V}$, $34.88\mathrm{V},\dots ,35.16\mathrm{V}$ from the bottom left to the top right. The data for $V=35.04\mathrm{V}$ (showing the longest scaling) are indicated by a thicker line. (b) Scaling plot of data in (a), with $V_c$, $\alpha$, and ${\nu }_{\parallel }$ values estimated from the experiment [Eqs. (5, 6)]. The dashed curve indicates the DP universal scaling function $f(\zeta )$ obtained numerically from the process sketched in Fig. 1d (so-called contact process). A collapse of similar quality is obtained when using DP-class exponent values.