Theory of droplet ripening in stiffness gradients

Liquid droplets embedded in soft solids are a new composite material whose properties are not very well explored. In particular, it is unclear how the elastic properties of the matrix affect the dynamics of the droplets. Here, we study theoretically how stiffness gradients influence droplet growth and arrangement. We show that stiffness gradients imply concentration gradients in the dilute phase, which transport droplet material from stiff to soft regions. Consequently, droplets dissolve in the stiff region, creating a dissolution front. Using a mean-field theory, we predict that the front emerges where the curvature of the elasticity profile is large and that it propagates diffusively. This elastic ripening can occur at much higher rates than classical Ostwald ripening, thus driving the dynamics. Our work shows how gradients in elastic properties control the arrangement of droplets, which has potential applications in soft matter physics and biological cells.