Physica D 222 (2006) 21–28 www.elsevier.com/locate/physd Scaling theory for gelling systems: Work in progress F. Leyvraz ∗ Centro de Ciencias F´ ısicas, UNAM, Apartado postal 48-3, CP 62251 Cuernavaca Morelos, Mexico Available online 28 September 2006 Abstract Gelation as it arises in the kinetic equations describing irreversible aggregation–the so-called Smoluchowski equations–is briefly reviewed. The scaling theory near the gel point, immediately before the transition, is presented. The equations presented earlier for the scaling function and the exponent τ are cast in a form more amenable to numerical study. An algorithm to solve them is described and applied to the kernels of the form K (x , y ) = x µ y ν + x ν y µ . A theory for the behaviour immediately after the gel transition is also presented and is found to be verified in one special exactly solved case (product kernel with power-law initial conditions). c  2006 Elsevier B.V. All rights reserved. Keywords: Gelation; Scaling theory; Smoluchowski equations 1. Introduction In many systems one finds that irreversible aggregation of “clusters” A(m) of mass m occurs and plays an important role. Particular instances are aerosol physics, where suspended particles coagulate due to van der Waals forces, polymer chemistry and astrophysics (where the clusters may be quite varied, going from galaxies to planetary systems) as well as chemical systems, such as polymers growing through condensation polymerization. This last example will be of particular importance, as it is the typical system for which gelation–the main subject of this paper–has been observed. In all these systems one is among other things interested in the cluster size distribution as a function of time. To obtain it one then relies on kinetic equations, derived from the following assumptions: let the reaction A(m) + A(m ′ ) −→ K (m,m ′ ) A(m + m ′ ) (1) occur at a rate K (m, m ′ ), that is, let any two clusters of size m and m ′ react on a time scale given by K (m, m ′ ) −1 . If one further assumes that no spatial correlations build up, in other words, that we may use a mean-field model in which the probability of encounter of two clusters of masses m 1 and m 2 is proportional ∗ Tel.: +52 55 56227779; fax: +52 55 56227775. E-mail address: f leyvraz2001@yahoo.com. to the product of the concentrations of such clusters, one obtains the following kinetic equations for the concentrations c(m, t ) of clusters of mass m: ˙ c(m, t ) = 1 2  ∞ 0 dm 1 dm 2 K (m 1 , m 2 ) c(m 1 , t )c(m 2 , t ) ×[δ(m − m 1 − m 2 ) − δ(m − m 1 ) − δ(m − m 2 )]. (2) These are an infinite set of coupled nonlinear ODEs, which represent a challenging problem. Few exact solutions are known, which are reviewed in [8]. Let us now describe what is meant by gelation. From the form of the reaction scheme (1) as well as from Eq. (2) it is clear that at the formal level the total mass M 1 (t ) M 1 (t ) =  ∞ 0 mc(m, t )dm (3) is conserved. It is well known, however, that this is not generally true. If K (m 1 , m 2 ) grows fast enough with the mass, a finite amount of mass can escape to infinity in finite time. In a rough and ready way, one can say that if we define the degree of homogeneity λ of the reaction kernel K (m 1 , m 2 ) by K (am 1 , am 2 ) = a λ K (m 1 , m 2 ) (4) then gelation is expected to occur whenever λ> 1. Let us first attempt to explain how this might come about and what 0167-2789/$ - see front matter c  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2006.08.011