Numerical simulations of the equilibrium shape of liquid droplets on gradient surfaces

The equilibrium shape of liquid droplets on horizontal and inclined plates that have a surface energy gradient is simulated numerically by applying a finite element method based on the principle of energy minimum in the present study. The numerical results show that the liquid droplet shape changes with locations under the influence of the unbalanced surface tension created by the gradient surface. It is shown that the contact angle reaches the maximum value at the one end of the droplet (2D), but it becomes minimum at the other end; the triple-phase contact line deforms toward the region with a smaller contact angle. It is further shown that the length of the liquid droplet increases with an increase in the surface energy gradient on the surface. More interestingly, an inflexion point appears when the droplet length varies with the center contact angle of the droplet, where the liquid droplet just locates at the transition region from the hydrophilic side to the hydrophobic side. It shifts to the hydrophilic side with the increase in the surface energy gradient. On the inclined gradient surface, the gravity induces a significant deformation of the equilibrium droplet shape towards the bottom of the surface. And the surface energy gradient further enhances the deformation when the unbalanced surface tension is directed to the bottom of the surface. However, the droplet shrinks back when the unbalanced surface tension is opposite to the component of gravity.