Beyond the single smooth defect: the open extensions of the Raphaël–de Gennes single-defect theory of contact-line depinning dynamics

In 1989 Raphaël and de Gennes solved the dynamics of a contact line interacting with one 'smooth' (Gaussian) defect at a time inside a thin capillary and found two inequivalent near-threshold laws: at imposed force the time-averaged velocity obeys $\bar U\sim\epsilon^{1/2}$ with $\epsilon=(F-F_m)/F_m$ the relative overforce above the defect's maximum pinning force $F_m$, while at imposed velocity $U\sim\bar\epsilon^{3/2}$ with $\bar\epsilon=(\bar F-F_U)/F_U$ the relative excess of the mean force above the quasistatic unhooking force $F_U$. The paper explicitly restricted itself to this case and flagged what it left open: the many-defect situation in which the line couples simultaneously to numerous defects (with the expectation, not a derivation, that statistical averaging suppresses the fixed-force/fixed-velocity distinction), sharp 'mesa' defects whose disanchoring proceeds by a qualitatively different mechanism (coalescence of two portions of the line), topographical 'bumps' conjectured to obey the same scaling laws, and the experimental confirmation of the two laws in well-controlled single-defect capillary or fiber setups. Periodic-heterogeneity theories (Joanny and Robbins 1990 and later homogenization work), the functional-renormalization-group depinning program and its experiments, and the recent multiscale Langevin framework of Perrin et al. (which itself uses periodic arrays of identical Gaussian defects as a mean-field surrogate) cover adjacent regimes; single-defect experiments tested static pinning and post-depinning line-shape relaxation but not the force–velocity laws; the dynamics of depinning from a single mesa defect and the topographical extension remain essentially untreated.

Background

When a liquid partially wets a flat solid, the curve where the liquid–gas interface meets the solid is called the contact line, and the angle at which the interface meets the solid is the contact angle. On an ideal, chemically uniform surface a single equilibrium angle $\theta_0$ is selected by Young's law (balance of the three interfacial tensions). On any real surface the quasi-static angle is instead bracketed between two limits, a receding angle $\theta_r$ and an advancing angle $\theta_a$: the phenomenon of contact angle hysteresis. Its origin is pinning: localized surface defects — chemical ones (small patches of different wettability, i.e. different local equilibrium angle) or topographical ones (bumps, grooves) — exert forces on the contact line and can hold it stationary even under a finite driving force. When the drive slightly exceeds what the defects can sustain, the line advances by successively anchoring on and tearing off defects; the detachment event is called depinning, or 'disanchoring' in this literature when it refers to leaving a single defect.

The static theory of pinning is due to Joanny and de Gennes (1984). They showed that on length scales between the defect size $d$ and a macroscopic cutoff $L$ the contact line behaves as an elastic string: pinching it at one point produces a restoring force that is Hookean, with spring constant $k=\pi\gamma\theta_0^2/\ln(L/d)$, where $\gamma$ is the liquid–gas surface tension. A localized defect is then characterized by the maximum pinning force $F_m$ it can exert on the line. On an ideal homogeneous surface the dynamics is linear: for a small deviation of the contact angle from equilibrium, the velocity of the line is proportional to the unbalanced Young force (the net interfacial-tension force per unit length that appears when the local angle departs from its equilibrium value), the proportionality being set by viscous dissipation in the liquid wedge near the line. What the static theory could not describe is the dynamics of motion near the pinning threshold on a defective surface.

Raphaël and de Gennes (1989) solved the simplest case of this dynamical problem: a contact line in a capillary so thin (radius smaller than the mean spacing between defects) that it meets only one defect at a time, with the defect taken to be 'smooth' — a Gaussian chemical inhomogeneity whose wettability varies gradually, of width $d$. They showed that two natural experiments are inequivalent. (a) Fixed force: the weight of a liquid column in a vertical capillary imposes a constant force $F$; writing $F=F_m(1+\epsilon)$ with $0<\epsilon\ll1$, the time-averaged velocity is $\bar U\sim\epsilon^{1/2}$, because the macroscopic motion is rate-limited by the slow passage of the line's tip across the defect. (b) Fixed velocity: a piston pushing a horizontal column imposes the speed $U$; the line then leaves the defect in an abrupt, burst-like event, and the time-averaged force $\bar F$ exceeds the quasistatic unhooking force $F_U$ (the force at which the line would leave the defect in an infinitely slow process); with $\bar\epsilon=(\bar F-F_U)/F_U$ one finds $U\sim\bar\epsilon^{3/2}$. The derivations rest on a separation of time scales — the line's tip (its most distorted point, at the defect) relaxes with time $t_2=1/(k\mu_m)$, where the tip mobility $\mu_m$ (velocity per unit force, set by the liquid viscosity $\eta$ and the defect width) describes the local motion — and, for the fixed-velocity case, on solving the detachment event through a Riccati equation (a first-order nonlinear differential equation, quadratic in the unknown function) whose exact solution is written in terms of Airy functions (the standard special functions of such near-turning-point problems). The inequivalence of the two protocols is a peculiarity of the single-defect situation: the velocity at fixed force is dominated by the long residence time near the defect, whereas the force at fixed velocity is dominated by the short, abrupt disanchoring event.

The authors stressed that this analysis is 'strictly restricted to the single defect problem (and to smooth defects)', and their concluding discussion marked out exactly what remains to be done: in practical situations the contact line couples simultaneously to many defects, and the resulting statistical averaging over pinning sites probably — but this was a conjecture, not a derivation — suppresses the distinction between the fixed-force and fixed-velocity regimes, yielding a single macroscopic force–velocity relation whose detailed dynamics had not been worked out; the opposite limit of 'mesa' defects (sharp steps, where the wettability changes abruptly rather than gradually) was expected to disanchor by a qualitatively different mechanism involving the coalescence of two portions of the contact line, and was left for future study; for mechanical (topographical) defects the same scaling laws were conjectured but not verified; and the predicted laws themselves awaited experimental confirmation in well-controlled single-defect setups such as thin capillaries or fibers. Later literature has built substantial neighboring edifices — periodic-heterogeneity dynamics, the statistical physics of depinning transitions, homogenization theories, and multiscale models of nanoscale defects — but none of them closes the question as posed.

Problem Statement

Determine the near-threshold macroscopic force–velocity relation of a viscous-liquid contact line moving over a nonideal surface in the regimes excluded by the Raphaël–de Gennes single-smooth-defect analysis, and test its predictions. The common objective is a defect-resolved dynamical theory of depinning — the counterpart, for each excluded situation, of the pair of laws $\bar U\sim\epsilon^{1/2}$ (imposed force, $\epsilon=(F-F_m)/F_m$) and $U\sim\bar\epsilon^{3/2}$ (imposed velocity, $\bar\epsilon=(\bar F-F_U)/F_U$) established for one smooth Gaussian defect in a thin capillary. Concretely, the source-flagged open components are: (i) the many-defect case, in which the line couples simultaneously to numerous defects with specified statistics, from the dilute regime of occasional simultaneous encounters to collective pinning — including whether statistical averaging collapses the inequivalence of the fixed-force and fixed-velocity protocols into a unique macroscopic force–velocity relation, and what the detailed dynamics are; (ii) the single sharp ('mesa') defect, whose disanchoring proceeds by coalescence of two portions of the contact line rather than the smooth unhooking analyzed for Gaussian defects; (iii) single topographical (bump) defects, for which it remains to be shown (or refuted) that the same scaling laws hold once the defect's effect on the local geometry and wetting boundary condition is derived rather than assumed; and (iv) the experimental confirmation or refutation of the two predicted scaling laws in a well-controlled single-defect setup, such as a thin capillary or a fiber. These components share one scientific objective — a predictive account of how defect populations and defect type set the force–velocity relation near depinning. The four components are independent: each is a separately posed open problem of the source. The many-defect component concerns a random defect population with specified statistics; a strictly periodic array of identical defects is only a mean-field surrogate for that population.

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