Pinning and Avalanches in Hydrophobic Microchannels M. Queralt-Martı ´n, 1 M. Pradas, 1,2 R. Rodrı ´guez-Trujillo, 3 M. Arundell, 3 E. Corvera Poire ´, 4, * and A. Herna ´ndez-Machado 1,† 1 Departament ECM, Facultat de Fı ´sica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain 2 Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, United Kingdom 3 Departament Electro ´nica, Facultat de Fı ´sica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain 4 Departamento de Fı ´sica y Quı ´mica Teo ´rica, Facultad de Quı ´mica, Universidad Nacional Auto ´noma de Me ´xico, Me ´xico DF 04510, Mexico (Received 21 September 2010; published 10 May 2011) 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. DOI: 10.1103/PhysRevLett.106.194501 PACS numbers: 47.61.k, 47.55.dr, 47.55.np Controlling the advancement of a fluid inside a micro- channel is one of the main issues in microfluidics. The huge amount of technological, biological, and medical applications found in the miniaturization of fluidic systems has required considerable effort in order to understand the physics of microfluidics [1–6]. As downsizing leads to an increased surface-to-volume ratio, surface properties of the microchannel, such as wetting and roughness, are of fun- damental importance since they would largely determine the large or low resistance to flow [4,7–12]. They will also determine, together with the driving pressure, the total or partial impregnation of the microchannel. For instance, for hydrophobic interactions and low pressures, a state can be reached in which the fluid does not totally impregnate the microchannel. This is known as the fakir state and has been observed in nanopatterned surfaces [4,7]. A different way of characterizing surface properties is to understand the underlying mechanisms in the dynamics of the advance- ment of a water front. For example, while a hydrophobic microchannel is being filled, one might conceive situations in which the water front experiences either low resistance to flow or high resistance to flow (as shown in the present work) depending on the type of disorder encountered by the front. We study the dynamics of a water front displacing air at constant pressure difference into a polydimethylsiloxane (PDMS) hydrophobic microchannel with the natural roughening present during its fabrication using a standard soft-lithography protocol [13–15], i.e., without a deliberate controlled nanopattern engraved. Our experimental results show that the advancement of the front at low pressure differences is characterized by a series of pinning and avalanches that leads to a dynamics of rare events. Extreme-value distributions deal with the stochastic be- havior of the extremes of independent and identically distributed random variables. The most common of these types of distributions is the Gumbel distribution [15–18]. It has been found to describe physical phenomena in fracture cracks, granular materials, fluctuations near a critical point, and imbibition of fluids in porous media [19–25]. We find that the velocity fluctuations during the advancement of the water front in a hydrophobic micro- channel follow a BHP-Gumbel distribution [26] that characterizes the rare events observed. The experimental setup is illustrated in Fig. 1 (see [15] for details). We track the interface position as a function of time and find that the mean front advances with a power law in time h  t  . Figure 2(a) shows hðtÞ for two different applied pressure differences. At low driving pressure dif- ferences, we obtain a nonclassical exponent of value as low as  ¼ 0:38. Detailed analysis of the data reveals a dynamics of pinning and avalanches. More precisely, FIG. 1 (color online). Experimental setup. PRL 106, 194501 (2011) PHYSICAL REVIEW LETTERS week ending 13 MAY 2011 0031-9007= 11=106(19)=194501(4) 194501-1 Ó 2011 American Physical Society