epl draft Height fluctuations of a contact line: a direct measurement of the renormalized disorder correlator

We have measured the center-of-mass fluctuations of the height of a contact line at depinning for two different systems: liquid hydrogen on a rough cesium substrate and isopropanol on a silicon wafer grafted with silanized patches. The contact line is subject to a confining quadratic well, provided by gravity. From the second cumulant of the height fluctuations, we measure the renormalized disorder correlator ∆(u), predicted by the Functional RG theory to attain a fixed point, as soon as the capillary length is large compared to the Larkin length set by the microscopic disorder. The experiments are consistent with the asymptotic form for ∆(u) predicted by Functional RG, including a linear cusp at u = 0. The observed small deviations could be used as a probe of the underlying physical processes. The third moment, as well as avalanche-size distributions are measured and compared to predictions from Functional RG. A direct measurement of the fixed-point function ∆(u), the so-called renormalized disorder correlator, which plays a central role in the Functional RG theory (FRG) of pinned elastic systems, was recently proposed [1] and verified in an exact numerical determination of ground states for interfaces in various types of disorders [2]. The main idea is to put the elastic system in a quadratic potential well, which acts as a large-scale cutoff and makes the problem well-defined. The shift between the center of mass and the center of the well is proportional to the renormalized force and its fluctuations are the quantity computed in the FRG [1]. The results of [2] show a remarquable agreement in the statics between the measured ∆(u) and the 1and 2-loop predictions from the Functional RG [3–5]. These ideas and numerical tests have been extended to the depinning transition [6, 7] in the case of local elasticity, and to reaction diffusion models [8]. Finally, a first-principle calculation of the distribution of avalanches from the FRG was performed and verified by numerics [9–11]. An outstanding challenge is to test these predictions in experiments. The depinning of the contact line of a fluid on a disordered substrate has been studied experimentally [12–15] and its critical scaling established. Since gravity naturally creates a quadratic well, this raises the interesting possibility of measuring a FRG fixed-point function for depinning or, conversely, to learn more about the physical system using these (universal) fluctuations as a new probe. To do so, the capillary length, which provides the well, does not need to be larger than the measurement scale. In fact, the finite capillary length is used as an advantage. Consider a fluid in a large reservoir and its contact line (CL) on a plate, parametrized by (x, u(x)) within the plate (fig. 1). Its energy can be modeled as