Approach to universal self-similar attractor for the levelling of thin liquid films

Abstract

We compare the capillary levelling of a random surface perturbation on a thin polystyrene film with a theoretical study on the two-dimensional capillary-driven thin film equation. Using atomic force microscopy, we follow the time evolution of samples prepared with different initial perturbations of the free surface. In particular, we show that the surface profiles present long term self-similarity, and furthermore, that they converge to a universal self-similar attractor that only depends on the volume of the perturbation, consistent with the theory. Finally, we look at the convergence time for the different samples and find very good agreement with the analytical predictions.

Symmetry plays a key role in the erasing of patterned surface features

“…and where we introduced the self-similar variable U = RT −1/4 . Note that the function φ can be written in terms of hypergeometric functions [31,52,54,58]. The solution to any summable initial perturbation Z(R, 0) = Z 0 (R) is simply given by the convolution (G * Z 0 )(R, T ).…”

“…In that sense, the attractor functions are referred to as universal attractors. If volume is added to the reference flat film by the initial perturbation, M 0 = 0, the profile d(x, t) will converge to the function F 0 (u) in finite time [31]. If no volume is added by the initial perturbation, then the profile will converge to the first term with a non-zero prefactor in Eq.…”

“…Since η and h 0 are obvious factors controlling the dynamics of the film, they have been scaled out using a normalized time. Here, we use the convergence time t c , which was previously defined [31] as the time when the asymptotic power-law behavior for the amplitude equals the initial amplitude of the perturbation. Convergence times can been computed theoretically for the three present configurations using a similar definition (see Theoretical Methods), and the experimentally determined convergence times are given in Fig.…”

Invariant Fast Diffusion on the Surfaces of Ultrastable and Aged Molecular Glasses

“…Previous studies have shown that profiles following Eq. 2 are self-similar after a transient time [13,[32][33][34][35]. Fig.…”

Relaxation of a highly deformed elastic filament at a fluid interface

“…Thus we emphasize here that the collapse into a symmetric shape is a consequence of being in a deeply nonlinear regime. The linear regime of our problem is similar to another curvature-driven problem -that of the relaxation of a perturbed liquid-air interface [20,21] -in that they flow to attracting set of shapes at long times. We show this explicitly by simulating five different asymmetric initial-conditions each with a straight portion attached to a semi-circular portion as in fig.…”