Steady Moving Cracks in Drying Colloidal Films: Three Limits

The drying of dispersions containing colloidal particles is encountered in many applications such as in processing of ceramic materials, film formation in paints and coatings, and synthesis of photonic bandgap materials. In many cases, shrinkage stresses generated during drying fracture the film. While much of the previous work has focused on cracking in static cracks, there are very few studies on the dynamics of cracks in colloidal coatings. The problem is complex as one needs to account for the mechanics of the particle network, flow of the interstitial fluid and the dynamics of crack motion. Here, we adopt the constitutive relation proposed by Russel and co-workers for a saturated packing of colloidal spheres to derive analytical solution for the stress, displacement, and pressure fields near the crack tip for a steady moving crack. To simplify the problem, first we consider the two extreme cases, namely, the high speed limit (undrained case) where the crack motion is much faster than Darcy flow rate and the opposite extreme of very slow crack propagation i.e low speed limit (drained case). Next, we take the general case where crack-tip motion is comparable to that for the interstitial flow time. The results incorporate micro-structural details of the system including particle volume fraction and nature of packing, and particle's mechanical properties such as modulus and Poisson's ratio. While predicted results are in line with similar results obtained for brittle materials, the predicted crack speeds are at least an order of magnitude higher than those observed in experiments. This difference is purely depends on difference in inertial effects of particle and fluid phases near the crack tip.