Elasto-capillary coalescence of multiple parallel sheets

Abstract

We analyse two-dimensional clamped parallel elastic sheets which are partially immersed in liquid as a model for elasto-capillary coalescence. In the existing literature this problem is studied via minimal energy analysis of capillary and elastic energies of the post-coalescence state, yielding the maximal stable post-coalescence bundle size. Utilizing modal stability analysis and asymptotic analysis, we studied the stability of the configuration before the coalescence occurred. Our analysis revealed previously unreported relations between viscous forces, body forces, and the instability yielding the coalescence, thus undermining a common assumption that coalescence will occur as long as it will not create a bundle larger than the maximal stable post-coalesced size. A mathematical description of the process creating the hierarchical coalescence structure was obtained and yielded that the mean number of sheets per coalesced region is limited to the subset 2 N where N is the set of natural numbers. Our theoretical results were illustrated by experiments and good agreement with the theoretical predictions was observed.

Elastocapillary coalescence of plates and pillars

“…The dynamics of coalescence in these two cases shows how the number of plates per cluster varies in a step-like manner, very similar to the experimental data reported by Pokroy et al [7] and Gat & Gharib [20]. All these features also agree qualitatively well with our own experimental observations shown in figure 1a,b.…”

“…Therefore, we approximate each plate as a rigid element with a localized bending response in an elastic hinge at the base [12]. This simplifies our analysis relative to the case of inhomogeneous bending and buckling (electronic supplementary material, appendix A) of individual plates [20,24]. The hinge elastic constant can be approximately derived from the bending response of a short cantilever by a transverse force F at its free end, with showing that the menisci can be pinned on both tips, or slip down from one or both tips.…”

“…(ii) Coupling capillary coalescence to drying dynamics For an evaporation dominated situation, the rate at which the liquid volume in each cell is reduced depends on the local surface area of the air-liquid interface and thus on deflection angles of the adjacent plates. Consequently, the drying dynamics is coupled to the evolution of the geometric configuration, and new instabilities associated with inhomogeneous cell volumes are expected, which cannot be captured by either energy minimization [8,18,19] or renormalization analysis [20].…”

Continuum dynamics of elastocapillary coalescence and arrest

“…Prior investigations of the elastocapillary coalescence of these systems have been primarily focused on understanding the final size of the bundle using energy minimization [4,6]. Exceptions include a phenomenological continuum mean-field theory [7] for arrested coarsening, and a discrete theory [8] for regular, ordered hierarchical aggregation. Here, we complement these studies using a long-wavelength continuum theory that imposes no hierarchical pattern ap r i o r i and is derived from a physically consistent microscopic picture that accounts for spatial/temporal variations driven by inhomogeneities in drying and eventually arrested by elasticity.…”

“…Beyond the linear regime, we modify the functional form of the free energy U (φ) by adding a semi-phenomenological term into (8). Since the system can eventually separate into two phases of tips or gaps in the absence of an elastic field, the energy density should have two minima at φ = ±1, suggesting a double-well potential a c (φ 2 /2 − φ 4 /4), as in the Cahn-Hilliard equation [11].…”