Aggregate growth driven by monomer transfer

We propose a solvable aggregate growth model with the rate kernelK(i;j;l)∝i^μj^νl^? (??0), at which a monomer transfers from the A aggregate of size i to the A aggregate of size j only with the help of a B aggregate of size l. By means of the mean-field rate equation approach, we obtain the analytical solutions of the aggregate size distributions in several different cases. For a symmetrical system with μ = ν, the aggregate size distributionak(t) approaches the conventional scaling form in the μ<3/2 case; while for an asymmetrical system with μ≠ν, ak(t) takes the scaling form only in the case of μ<ν and μ+ν<2. As for the case with μ+ν>2 (μ≠ν) or μ>3/2 (μ = ν), the system may have a gelationlike transition. Moreover, we also study the kinetic scaling behaviour of the model with the generalized kernelK(i;j;l)~(i^μj^ν+i^νj^μ)l^?. The aggregate size distribution obeys a scaling law only in the μ+ν<3 case; while in other cases, the system may undergo a gelationlike transition after a sufficiently long time.