Infiltration of liquid droplets into porous media: Effects of dynamic contact angle and contact angle hysteresis

We present a detailed theory for infiltration, which accounts for a general model for the dynamic contact angle between the droplet and the porous medium as well as contact angle hysteresis, and analyze the resulting equations of motion. The theory shows that infiltration of droplets into dry porous media involves three phases due to contact angle hysteresis: (1) An increasing drawing area (IDA) phase during which the interface between the droplet and the porous medium increases, (2) a constant drawing area (CDA) phase during which the contact line of the droplet remains pinned, and (3) a decreasing drawing area (DDA) phase. The theory is based on the following assumptions: (1) The droplet has the shape of a spherical cap, (2) the porous medium consists of a bundle of vertical tubes of same size, and (3) the pressure within the droplet is uniform. We find that infiltration always consists of a cascade process formed by the IDA, CDA, and DDA phases, where the entire process may begin or end in any of the three phases. The entire process is formulated with four nondimensional parameters: Three contact angles (initial, advancing, and receding) and a porous permeability parameter. A comparison of our theory to experimental data suggests that one should use different parameterizations for the dynamic contact angle models of the IDA and DDA phases. In general, the IDA and DDA phases are described by integro-differential equations. A numerical-solution approach is presented for solving the dynamic equations for infiltration. The total time of infiltration and the time dependence of drawing area are critically affected by the occurrence of the IDA, CDA, and DDA phases as well as by the permeability. With ordinary differential equations (ODEs), we are able to approximate the IDA phase and to describe exactly infiltration processes that start out with the CDA or DDA phase. (C) 2008 Elsevier Ltd. All rights reserved.