Pinning and Avalanches in Hydrophobic Microchannels

Rare events appear in a wide variety of phenomena such as rainfall, floods, earthquakes, and risk. We demonstrate that the stochastic behavior induced by the natural roughening present in standard microchannels is so important that the dynamics for the advancement of a water front displacing air has plenty of rare events. We observe that for low pressure differences the hydrophobic interactions of the water front with the walls of the microchannel put the front close to the pinning point. This causes a burstlike dynamics, characterized by series of pinning and avalanches, that leads to an extreme-value Gumbel distribution for the velocity fluctuations and a nonclassical time exponent for the advancement of the mean front position as low as 0.38.

Figure 1

Experimental setup.

Figure 2

(a) Experimental results of the mean front position $h(t)\sim t^{\nu }$ in a hydrophobic microchannel for two different pressure differences. (b) Numerical results for $h(t)$ in a hydrophobic microchannel for two different pressure differences. (c) Asymptotic value of the exponent as a function of the applied pressure difference.

Figure 3

(a) Experimental evolution of the front velocity $v(t)$ ($\mu \mathrm{m}/\mathrm{s}$) at small pressure difference. (b) Distribution of experimental velocity fluctuations in the variable $x$ (see text) fitted to both a BHP-Gumbel and a Gaussian distribution. (c) Numerical evolution of the front velocity $v(t)$. (d) Distribution of numerical velocity fluctuations fitted to both a BHP-Gumbel and a Gaussian distribution.

Figure 4

(a) Microchannel with roughness in the form of steps and the water front that has left an air phase trapped between the structures that form the roughness. (b) (left) Detail of the fakir state, (right) detail of the contact line at different times.

Figure 5

Numerical results independently computed for the left- and right-hand sides of Eqs. (4, 5). Agreement with an effective Darcy’s law is found.