The application of percolation theory to the compaction of pharmaceutical powders

Based on the concepts of percolation theory the compaction process is interpreted as a site-bond percolation phenomenon. The combination of the Heckel equation with an equation derived earlier yields a simple relationship between the tensile strength σ_t, or the deformation hardness P and the relative density. This mathematical model is identical with the fundamental law of percolation theory i.e., X = s(p − p_c)^q with X= system property equivalent to tensile strength or deformation hardness, S = scaling factor, p = site occupation or bond formation probability corresponding to the relative density p,p_c = percolation threshold and critical exponent q = 1 according to percolation on a Bethe lattice. In the case of the tensile strength σ_t and the deformation hardness p, two percolation thresholds p_c(1) and p_c(2) corresponding to p_c(1) and p_c(2) could be identified. The relative density p_c(1) which is close to the relative poured or relative tapped density can be interpreted as a bond percolation threshold. The particles are bonded by weak interparticulate forces and form only loose compacts as used for filling of capsules. The relative density p_c(2) is the relative density where the first stable pharmaceutical compact is achieved which can no longer be disintegrated mechanically into its primary particles. It is of special interest that the above equation is also valid for the elastic modulus with the only percolation threshold p_c(1) = p_c(1).