Self-Similarity and Energy Dissipation in Stepped Polymer Films

The surface of a thin liquid film with nonconstant curvature is unstable, as the Laplace pressure drives a flow mediated by viscosity. We present the results of experiments on one of the simplest variable curvature surfaces: a stepped polymer film. Height profiles are measured as a function of time for a variety of molecular weights. The evolution of the profiles is shown to be self-similar. This self-similarity offers a precise measurement of the capillary velocity by comparison with numerical solutions of the thin film equation. We also derive a master expression for the time dependence of the excess free energy as a function of the material properties and film geometry. The experiment and theory are in excellent agreement and indicate the effectiveness of stepped polymer films to elucidate nanoscale rheological properties.Comment: 5 pages, 4 figures, article accepted for publication in Physical Review Letter

Capillary leveling of stepped films with inhomogeneous molecular mobility

A homogeneous thin polymer film with a stepped height profile levels due to the presence of Laplace pressure gradients. Here we report on studies of polymeric samples with precisely controlled, spatially inhomogeneous molecular weight distributions. The viscosity of a polymer melt strongly depends on the chain length distribution; thus, we learn about thin-film hydrodynamics with viscosity gradients. These gradients are achieved by stacking two films with different molecular weights atop one another. After a sufficient time these samples can be well described as having one dimensional viscosity gradients in the plane of the film, with a uniform viscosity normal to the film. We develop a hydrodynamic model that accurately predicts the shape of the experimentally observed self-similar profiles. The model allows for the extraction of a capillary velocity, the ratio of the surface tension and the viscosity, in the system. The results are in excellent agreement with capillary velocity measurements of uniform mono- and bi-disperse stepped films and are consistent with bulk polymer rheology.Comment: Accepted for publication in Soft Matter, Themed Issue on "The Geometry and Topology of Soft Materials

Capillary-driven flow induced by a stepped perturbation atop a viscous film

Thin viscous liquid films driven by capillarity are well described in the lubrication theory through the thin film equation. In this article, we present an analytical solution of this equation for a particular initial profile: a stepped perturbation. This initial condition allows a linearization of the problem making it amenable to Fourier analysis. The solution is obtained and characterized. As for a temperature step in the heat equation, self-similarity of the first kind of the full evolution is demonstrated and a long-term expression for the excess free energy is derived. In addition, hydrodynamical fields are described. The solution is then compared to experimental profiles from a model system: a polystyrene nanostep above the glass transition temperature which flows due to capillarity. The excellent agreement enables a precise measurement of the capillary velocity for this polymeric liquid, without involving any numerical simulation. More generally, as these results hold for any viscous system driven by capillarity, the present solution may provide a useful tool in hydrodynamics of thin viscous films.Comment: Accepted for publication in Physics of Fluid

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

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.Comment: 7 pages, 5 figure

Liquid droplets act as "compass needles" for the stresses in a deformable membrane

We examine the shape of droplets atop deformable thin elastomeric films prepared with an anisotropic tension. As the droplets generate a deformation in the taut film through capillary forces, they assume a shape that is elongated along the high tension direction. By measuring the contact line profile, the tension in the membrane can be completely determined. Minimal theoretical arguments lead to predictions for the droplet shape and membrane deformation that are in excellent agreement with the data. On the whole, the results demonstrate that droplets can be used as probes to map out the stress field in a membrane

Characterization of the optical and X-ray properties of the northwestern wisps in the Crab Nebula

We have studied the wisps to the north-west of the Crab pulsar as part of a multi-wavelength campaign in the visible and in X-rays. Optical observations were obtained using the Nordic Optical Telescope in La Palma and X-ray observations were made with the Chandra X-ray Observatory. The observing campaign took place from 2010 October until 2012 September. About once per year we observe wisps forming and peeling off from (or near) the region commonly associated with the termination shock of the pulsar wind. We find that the exact locations of the northwestern wisps in the optical and in X-rays are similar but not coincident, with X-ray wisps preferentially located closer to the pulsar. This suggests that the optical and X-ray wisps are not produced by the same particle distribution. Our measurements and their implications are interpreted in terms of a Doppler-boosted ring model that has its origin in magne- tohydrodynamic (MHD) modelling. While the Doppler boosting factors inferred from the X-ray wisps are consistent with current MHD simulations of pulsar wind nebulae (PWN), the optical boosting factors are not, and typically exceed values from MHD simulations by about a factor of 3.Comment: 11 pages, 12 figure

Sequentializing cellular automata

We study the problem of sequentializing a cellular automaton without introducing any intermediate states, and only performing reversible permutations on the tape. We give a decidable characterization of cellular automata which can be written as a single sweep of a bijective rule from left to right over an infinite tape. Such cellular automata are necessarily left-closing, and they move at least as much information to the left as they move information to the right