Scaling anomalies in the sol-gel transition

Coagulating systems are characterized by the fact that aggregates of masses g and l react irreversibly to form a larger aggregate of mass g + l. The reaction rates K(g, l) for this process are assumed to depend only on the particle masses. This model is described by the corresponding kinetic equations. If the K(g, l) are assumed to grow fast enough, it happens that, at some finite time t(c), a fraction of the mass goes into a cluster of infinite size. Near and at the gel time, the mass spectrum c(g)(t) can be described in scaling terms, that is, there exists a divergent typical mass in terms of which the spectrum can be expressed, as well as two exponents describing the g-dependence of the mass spectrum. These exponents have been extensively studied, both numerically and in terms of the case in which K(g, l) = gl, which is exactly solvable. So far, the exact model strongly suggested the validity of certain general expressions for these exponents in terms of the homogeneity degree of the K(g, l). These, however, were not confirmed numerically. In the following, we study the exact model in the case in which the initial conditions display an algebraic tail, and show that, in this case, the scaling exponents can be calculated analytically and do not conform to the expressions initially suggested.