Capillary Driven Transport in Soft Solids

Soft solids, such as polymeric gels, are elastic materials that can be significantly deformed by capillary forces which act at the interface. The coupling between elasticity and capillarity is known as elastocapillarity and is useful to wide ranging applications, from drop pinchoff of bioinks for 3-D printing tissue scaffolds, to generating droplet patterns on microfluidic devices. In this dissertation, we develop mathematical models of elastocapillary driven motions in soft solids. The focus is on understanding the relevant physics in such complex phenomena, while also recovering the limiting cases of classical fluid mechanics and solid mechanics theories. This dissertation investigates two broad classes of dynamic elastocapillary effects. In the first part, we study the effect of the intrinsic surface tension of a soft solid which can drive the dynamic evolution of the material interface. This involves the study of elastic analogues of classical hydrodynamic instabilities, namely, drop oscillations, the Plateau-Rayleigh instability of a cylindrical interface, and the buoyancy-driven Rayleigh-Taylor instability. For each system, we derive a dispersion relationship that depends upon the respective physics and compute the spectrum and associated mode shapes. In the second part, we analyze the motion of a liquid droplet on a soft substrate and show how the substrate elasticity affects the wetting dynamics. We show how elastocapillary deformation can lead to spontaneous transport on a substrate with variable thickness, in a manner similar to cellular durotaxis, and estimate the droplet speed by balancing the viscoelastic dissipation with the driving wetting force. We then consider the canonical case of spontaneous drop spreading on a deformable substrate where both the drop shape and the solid interface continuously evolve. We construct a solution based on the lubrication approximation to describe the coupled interaction between the solid and fluid fields, which recovers the key features of soft spreading observed in previous experiments