Bilinear kernel

Table 4. The types of monomers undergoing homopolymerization and corresponding bilinear kernels in the Smoluchowski coagulation equation [21]. The functional groups A react with each other in models 1 and 2 and with groups B only in models 3 through 5...

Of interest also are the results of applications of the Smoluchowski equation for systems with more complex aggregation physics than that provided by bilinear kernels. Leyvraz and Tschudi [32] conjectured that for the kernel ICy-fi) ) gelation occurs only when co > 1/2. The post-gelation behavior of a general system with a multiplicative kernel given by Eq. (60) has been analyzed by van Don-gen et al. [56]. By assuming that the distribution past the gel point could be expressed through that at t=tc, thus... 

The Smoluchowski coagulation equation has effectively been applied to model random irreversible homopolymerization of the monomer types presented in Table 4. As can be seen, in all cases the coagulation kernel has the bilinear form ... 

The presence of the second term, however, creates complications. The full convolution form of the equation must be kept, even at long times furthermore, the kernel for the convolution involves (bilinearly) the correlation function itself, for all kc k 0. Thus, we may no longer deal with a single wave vector to find the behavior of (Ak(t)A k) for a particular k it is necessary to know the correlation function for all k kc. This point could grieviously complicate the making of a small wave vector expansion of the laws of motion, since no matter how small we make k, we still have to deal with the whole range of k. We shall take the point of view, however, that kc is small enough that k can be considered small, and thus we shall perform k expansions just as we performed k expansions. 

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