Imbibition of a liquid droplet on a deformable porous substrate

We consider the imbibition of a liquid droplet into a deformable porous substrate. The liquid in the droplet is imbibed due to capillary suction in an initially dry and undeformed substrate. Deformation of the substrate occurs as the liquid fills the pore space. In our model, a pressure gradient in the liquid across the developing wet substrate region induces a stress gradient in the solid matrix which in turn leads to an evolving solid fraction and hence deformation. For axisymmetric droplets, we assume that the imbibition and substrate deformation at a given radial position are one-dimensional (in the vertical direction). The coupling to the droplet geometry leads to axisymmetric configurations for the deformed wet substrate. We show that the model chosen to describe the dynamics of the liquid droplet, based in this case on existing models developed for droplet spreading on rigid porous substrates, has little influence on the resultant swelling or shrinking of the substrate—these general trends can be effectively predicted by a one-dimensional imbibition and deformation model—but does strongly influence the details of the wet substrate shape. We characterize these predictions and in some cases can obtain analytical solutions for the evolution.