Boundary Conditions in Diffusional Growth and Sedimentation

In many applications, transport of particles can be described by the diffusion equation, or its convective-diffusion generalizations, in part of threedimensional space. In particular, in surface deposition or in growth of aggregates or sediments, particles in the suspension outside the surface or aggregate boundary can be considered as diffusing under the influence of applied forces, such as the double-layer interactions or gravity. However, the diffusion equation cannot be used at the substrate or inside the aggregate. One convenient and widely used approach has been to impose boundary conditions at the surface, to supplement and completely define the diffusion problem outside it. This short survey describes some theoretical results on the use of such boundary conditions and their physical interpretation. In order to define the geometry of the problem, let us consider Figure 1 (provided as a separate GIF file). We will assume that the surface is planar, at x = 0, and for simplicity use the one-dimensional version of the diffusion equation whenever this causes no confusion. Thus, for x > 0, the coordinate x measures the distance away from the substrate or growing aggregate. Some important problems lack planar symmetry. For instance, in the growth of colloid particles from small sub-units, the aggregates are typically spherical, and the radial diffusion equation must be used. Here, we assume that suspended particles at x > 0 move according to the diffusion equation: