Smoluchowski's equation and the θ-exponent for branched polymers

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Abstract
Bonding of large clusters by surface reactions can be modelled by Smoluchowski's coagulation equation with coagulation rates Kij=(ij)omega with omega =(d-1)/d in d-dimensional systems. It is shown that the cluster size distribution for large clusters well below the gelation transition has the form ck approximately k- theta xi k (k to infinity ) where theta =2 omega . Results are compared with those from lattice theories and from the Flory-Stockmayer theory. For the models Kij=1/2(imu jnu +inu jmu ) with mu , nu ij=ijomega +jiomega with -11/2(1+ omega ) and for Kij=ij one has theta =5/2.

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