Analytic solutions of gelation theory for finite, closed systems

摘要

The theory of gelation as formulated by Donoghue and Gibbs for polycondensing,fhyphen;functional units in a finite, closed system is solved analytically to obtain both exact formulas and the more useful asymptotic expansions of these. Both forms of solution are derived from exact contour integral representations that follow immediately from the basic generating function presented earlier. The two firsthyphen;order saddle points of the integrands of these representations, at agr; and at1/(fhyphen;1),coalesce at the critical extent of reactionagr;cto form a secondhyphen;order, or rsquo;rsquo;monkeyrsquo;rsquo; saddle. A known technique that is valid in such circumstances, as the more familiar method of steepest descents is not, is applied to derive asymptotic expansions that approximate the integrals uniformly well at all extents of reaction, rather than diverging atagr;cas the steepest descents expansions do. The uniform expansions are applied to finite systems to obtain both approximations to the mean polymer size distribution that accurately reproduce the bimodal distribution of coexisting sol and gel at extents of reaction aboveagr;c,rather than diverging at the molecular weight of the gel, and approximations to the moments of these distributions that are uniformly valid at all extents of reaction, rather than diverging atagr;c.The uniform expansions are applied to the infinite system in two limits. Taken in the limit asNrarr;infin;at constant extent of reaction, they reproduce all the wellhyphen;known results for the infinite system, and extend these somewhat. In this limit, the uniform expansions agree with the steepest descents expansions, which thus also produce the above results. However, because of the divergencies atagr;c,the common limiting expressions do not describe the gel. Taken in theNrarr;infin;limit with agr; and(khyphen;NWg) /N2/3fixed, the formula forkmmacr;k/N1/3,as given by the uniform expansions, is shown to give the correct analytic expression for the limiting ensemble average gel distribution. The gel fraction of a macroscopic system is shown from the new distribution to consist of a single molecule of limiting sizeNWg(agr;),the weight of gel as given by Stockmayer. The limiting weight average molecular weight of the system aboveagr;c,which is shown to belsqb;NWg(agr;)rsqb;2,is thus due entirely to the one gel molecule.