Model of spontaneous droplet transport on a soft viscoelastic substrate with nonuniform thickness

Dynamic wetting of droplets on soft solids has many industrial and biological applications which require an understanding of the underlying fluid transport mechanism. Here we study the case of a droplet on a viscoelastic substrate of variable thickness which is known to give rise to a spontaneous droplet transport. This phenomenon is known as droplet durotaxis and has been observed experimentally. Here we develop a model assuming a small linear gradient in substrate thickness to reveal the physical mechanism behind this transport phenomena. We show the variable thickness causes an asymmetric deformation along the drop contact line, which causes a variation in the contact angle. This generates a net driving force on the drop, causing it to move in the direction of higher thickness. The resulting drop velocity is determined by balancing the work done by the moving drop with the viscoelastic dissipation of the substrate (viscoelastic braking) and computed from a self-consistent model. We find our results to be in qualitative agreement to previously reported experimental findings.