Elastocapillary instability under partial wetting conditions: bending versus buckling

The elastocapillary instability of a flexible plate plunged in a liquid bath is analysed theoretically. We show that the plate can bend due to two separate destabilizing mechanisms, when the liquid is partially wetting the solid. For contact angles $\theta_e > \pi/2$, the capillary forces acting tangential to the surface are compressing the plate and can induce a classical buckling instability. However, a second mechanism appears due to capillary forces normal to surface. These induce a destabilizing torque that tends to bend the plate for any value of the contact angle $\theta_e > 0$. We denote these mechanisms as "buckling" and "bending" respectively and identify the two corresponding dimensionless parameters that govern the elastocapillary stability. The onset of instability is determined analytically and the different bifurcation scenarios are worked out for experimentally relevant conditions.Comment: 12 pages, 13 figure

Capillary buckling of a thin film adhering to a sphere

We present a combined theoretical and experimental study of the buckling of a thin film wrapped around a sphere under the action of capillary forces. A rigid sphere is coated with a wetting liquid, and then wrapped by a thin film into an initially cylindrical shape. The equilibrium of this cylindrical shape is governed by the antagonistic effects of elasticity and capillarity: elasticity tends to keep the film developable while capillarity tends to curve it in both directions so as to maximize the area of contact with the sphere. In the experiments, the contact area between the film and the sphere has cylindrical symmetry when the sphere radius is small, but destabilises to a non-symmetric, wrinkled configuration when the radius is larger than a critical value. We combine the Donnell equations for near-cylindrical shells to include a unilateral constraint with the impenetrable sphere, and the capillary forces acting along a moving edge. A non-linear solution describing the axisymmetric configuration of the film is derived. A linear stability analysis is then presented, which successfully captures the wrinkling instability, the symmetry of the unstable mode, the instability threshold and the critical wavelength. The motion of the free boundary at the edge of the region of contact, which has an effect on the instability, is treated without any approximation

Elastocapillary Phenomena in Soft Elastic Solids

Soft elastic solids play an important role in a wide range of applications such as in tissue scaffolds to grow artificial organs, in wearable contact lenses, as adhesives, in soft robotics and even as prototypical models to understand the mechanics of growth and morphology of organs. For a soft elastic material like hydrogel with its shear modulus in the range of tens of pascals, its surface tension also contributes to the mechanics of its deformation in addition to its elasticity. As opposed to a hard solid that is very difficult to deform, for the case of these soft solids, even a weak force like gravity can bring about significant deformation. Many of these aspects of the deformation and behavior of these ultrasoft materials are still not very well understood. Thus, the objectives of this dissertation were to understand the role of elastocapillarity (i.e, joint roles of solid surface tension and elasticity) and elastobuoyancy (i.e, joint roles of gravity and elasticity) that manifest in such solids. In this dissertation, we studied the role elastocapillarity in adhesion-induced instability in thin elastic films bonded to rigid substrates and also in surface oscillation modes of soft gel spheres set to vibration; the elastobuoyancy effect; elasticity mediated interaction of particles in soft solids as well as on thin films supported over a pool of liquid. We also presented some new results on how soft spherical gels undergo restricted spreading on rigid substrates with varying surface energies. In the first section, we studied how a thin confined layer of a soft elastic film loses adhesion from a rigid substrate by forming interfacial instabilities when a tensile stress is applied to it. We performed experiments to quantify the characteristic lengthscale of the patterns formed and found that they were significantly larger than the wavelengths of purely elastic instabilities. A linear stability analysis of the elastic field equations by taking into account the role of surface tension showed that the amplification of the wavelength is due to the role of elastocapillarity where the surface tension, elasticity, and film thickness contribute jointly in a non-trivial way. In addition, we found experimentally as well as theoretically that the stress required to adhesively fracture these films is much larger than Griffith’s fracture stress for stiffer elastic films, which is also due to the effect of elastocapillarity. We also studied the surface fluctuation of sessile hydrogel spheres subjected to mechanically-induced Gaussian white noise to understand the role of elastocapillarity in their oscillation modes. An important finding of this study is that they give a direct evidence that the surface tension of these elastic hydrogels is almost like that of water, which is the integral solvent in the swollen network of the polymeric gel. In the subsequent section, we introduced the new phenomenon of Elastobuoyancy. When a rigid sphere is placed on the surface of an ultrasoft hydrogel, it plunges into the soft substrate to an equilibrium depth where the elastic strain energy of the surrounding medium balances its weight. We refer to this state of the sphere as ‘Elastobuoyant’. By performing systematic experiments where we varied the sphere size and the elasticity of the substrate, we obtained scaling laws of the depth as a function of the radii, elastic modulus and the spheres buoyant weight, which were also supported by asymptotic analyses of the same. Following the section on elastobuoyancy, we reported a new set of principles to design self-assembly of particles by using the combined roles of surface tension, elasticity, and gravity in soft substrates. We used three different systems to study this elastic interaction macroscopically: (i) elastobuoyant assembly of particles suspended inside a soft elastic gel, (ii) elastocapillary assembly of particles floating on the surface of soft gels analogous to capillary attraction of objects on the surface of liquids, and (iii) assembly of particles on the surface of thin elastic membranes supported over a viscous liquid. In the second last chapter in this thesis, we presented some results on how soft elastic gel spheres spread on rigid substrates with different surface energies. Our observations indicate that their contact angles are slightly greater than those of equivalent liquid drops on similar substrates. The contact angles of these gel spheres increase as a function of elasticity and decrease when surface energy increases. We derived an expression for the excess elastic tension in the gel spheres at the crack tip by using an approach that is similar to estimating the viscous dissipation at the contact line during spreading of liquids. By using a general constitutive law where the elastic energy is not limited to the square of the strains, the singularity at the crack tip is artificially removed thereby forcing the gel to assume a liquid-like behavior. Our experimental results agreed reasonably well with the model. In the last chapter, we summarized the doctoral research and presented suggestions for future investigations. There are several appendices in this thesis that have interesting observations from partially completed projects that need additional research and analysis in the future

Linear models for thin plates of polymer gels

Within the linearized three-dimensional theory of polymer gels, we consider a sequence of problems formulated on a family of cylindrical domains whose height tends to zero. We assume that the fluid pressure is controlled at the top and bottom faces of the cylinder, and we consider two different scaling regimes for the diffusivity tensor. Through asymptotic-analysis techniques we obtain two plate models where the transverse displacement is governed by a plate equation with an extra contribution from the fluid pressure. In the limit obtained within the first scaling regime the fluid pressure is affine across the thickness and hence it is determined by its instantaneous trace on the top and bottom faces. In the second model, instead, the value of the fluid pressure is governed by a three-dimensional diffusion equation

Soft wetting with (a)symmetric Shuttleworth effect

The wetting of soft polymer substrates brings in multiple complexities as compared to the wetting on rigid substrates. The contact angle of the liquid is no longer governed by Young's law, but is affected by the substrate's bulk and surface deformations. On top of that, elastic interfaces exhibit a surface energy that depends on how much they are stretched -- a feature known as the Shuttleworth effect (or as surface-elasticity). Here we present two models by which we explore the wetting of drops in the presence of a strong Shuttleworth effect. The first model is macroscopic in character and consistently accounts for large deformations via a neo-Hookean elasticity. The second model is based on a mesoscopic description of wetting, using a reduced description of the substrate's elasticity. While the second model is more empirical in terms of the elasticity, it enables a gradient dynamics formulation for soft wetting dynamics. We provide a detailed comparison between the equilibrium states predicted by the two models, from which we deduce robust features of soft wetting in the presence of a strong Shuttleworth effect. Specifically, we show that the (a)symmetry of the Shuttleworth effect between the "dry" and "wet" states governs horizontal deformations in the substrate. Our results are discussed in the light of recent experiments on the wettability of stretched substrates