Leidenfrost Effect as a Directed Percolation Phase Transition

Volatile drops deposited on a hot solid can levitate on a cushion of their own vapor, without contacting the surface. We propose to understand the onset of this so-called Leidenfrost effect through an analogy to nonequilibrium systems exhibiting a directed percolation phase transition. When performing impacts on superheated solids, we observe a regime of spatiotemporal intermittency in which localized wet patches coexist with dry regions on the substrate. We report a critical surface temperature, which marks the upper bound of a large range of temperatures in which levitation and contact coexist. In this range, with decreasing temperature, the equilibrium wet fraction increases continuously from zero to one. Also, the statistical properties of the spatiotemporally intermittent regime are in agreement with that of the directed percolation universality class. This analogy allows us to redefine the Leidenfrost temperature and shed light on the physical mechanisms governing the transition to the Leidenfrost state.

Figure 1

(a) Schematic phase diagram for levitation and contact. Both states can be (locally) observed for substrate temperatures between $T_L^{-}$ and $T_L^{+}$ where they coexist. (b) Sketch of the dynamics of directed bond percolation. (c) Ethanol drops with equilibrium radius $R$ and velocity $U$ impact a sapphire substrate with temperature $T$. We record side views and use total internal reflection (TIR) imaging to measure the thickness $h(r,t)$ of the gas film squeezed between the liquid and the solid (sketch not to scale).

Figure 2

(a) TIR snapshots for the impact of a drop with radius $R=1.1\mathrm{mm}$ and impact velocity $U=0.65\mathrm{m}/\mathrm{s}$ (i.e., $We=17$) for an imposed superheat $\mathrm{\Delta }T=87\mathrm{K}$. Contact nucleates at $t=0.01\mathrm{ms}$ (red arrow) and spreads until it starts receding ($t=0.07\mathrm{ms}$). (b) Spatiotemporal binarized representations of the wet regions for three superheat below ($\mathrm{\Delta }T=69\mathrm{K}$), close to ($\mathrm{\Delta }T=87\mathrm{K}$), and above ($\mathrm{\Delta }T=98\mathrm{K}$) the critical point. We show a square of $0.77\mathrm{mm}\times 0.77\mathrm{mm}$ within the observation area during 5 ms (time goes downwards).

Figure 3

(a) Temporal evolution of the wet fraction $\rho (t)$ during an impact with $R=1.1\mathrm{mm}$, $U=0.65\mathrm{m}/\mathrm{s}$ (i.e., $We=17$), and $\mathrm{\Delta }T=87\mathrm{K}$. After an initial transient, the wet fraction reaches a plateau value, $\overline{\rho }$. (b) Mean wet fraction $\overline{\rho }$ in the steady state as a function of the imposed superheat $\mathrm{\Delta }T$. The vertical and horizontal error bars represent the standard deviation of $\rho (t)$ and the accuracy of the thermocouple, respectively. (c) Same data plotted as a function of $ϵ=(\mathrm{\Delta }{T}_c-\mathrm{\Delta }T)/\mathrm{\Delta }{T}_c$ and on a log-log scale. The lines are fits of the form $\overline{\rho }\sim {ϵ}^{\beta }$ where $\beta$ is the DP exponent in ($2+1$) dimensions (dashed red line) and the best fit exponent (solid green line).

Figure 4

(a),(b) Distributions of the duration $\tau$ and lateral length $\ell$ of the dry intervals in the steady state for $\mathrm{\Delta }T=87\mathrm{K}$, close to the critical superheat. The distributions of $\ell$ and $\tau$ decay algebraically in good agreement with the DP power-law scalings (red dashed lines). (c),(d) Collapse of the temporal (c) and spatial (d) distributions in (a) and (b) using the directed percolation power laws: ${\tau }^{{\mu }^{\Vert }}N(\tau )\sim f(\tau {ϵ}^{{\nu }_{\Vert }})$ and ${\ell }^{{\mu }^{\perp }}N(\tau )\sim f(\ell {ϵ}^{{\nu }_{\perp }})$ where $f$ is an unknown function.

Figure 5

Survival probability $P_s(t,\mathrm{\Delta }T)$ of a contact nucleated at a single location during an impact for three different superheat $\mathrm{\Delta }T>\mathrm{\Delta }{T}_c$. The dashed lines are fits of the form $\exp [-(t-t_0)/{\tau }^{*}(\mathrm{\Delta }T)]$, where $t_0$ is a constant and ${\tau }^{*}(\mathrm{\Delta }T)$ is the characteristic lifetime of contact.