Lattice model of living polymerization. I. Basic thermodynamic properties

A Flory-Huggins type lattice model of living polymerization is formulated, incorporating chain stiffness, variable initiator concentration r, and a po lymer-solvent interaction chi. Basic equilibrium properties [average chain length L, average fraction of associated monomers Phi, specific heat C-P, e ntropy S, polymerization temperature T-p, and the chain length distribution p(N)] are calculated within mean-field theory. Our illustrative calculatio ns are restricted to systems that polymerize upon cooling [e.g., poly(alpha -methylstyrene)], but the formalism also applies to polymerization upon hea ting (e.g., sulfur, actin). Emphasis is given to living polymer solutions h aving a finite r in order to compare theory with recent experiments by Gree r and co-workers, whereas previous studies primarily focused on the r --> 0 (+) limit where the polymerization transition has been described as a secon d order phase transition. We find qualitative changes in the properties of living polymer solutions for nonzero r: (1) L becomes independent of initia l monomer composition phi(m)(0) and temperature T at low temperatures [L(T much less than T-p)similar to 2/r], instead of growing without bound; (2) t he exponent describing the dependence of L on phi(m)(0) changes by a factor of 2 from the r --> 0(+) value at higher temperatures (T greater than or e qual to T-p); (3) the order parametertype variable Phi develops a long tail with an inflection point at T-p; (4) the specific heat maximum C-P(*) at T -p becomes significantly diminished and the temperature range of the polyme r transition becomes broad even for small r [r similar to O(10(-3))]. Moreo ver, there are three characteristic temperatures for r > 0 rather than one for r --> 0: a "crossover temperature" T-x demarking the onset of polymeriz ation, an r-dependent polymerization temperature T-p defined by the maximum in C-P (or equivalently, the inflection point of Phi), and a "saturation t emperature" T-s at which the entropy S of the living polymer solution satur ates to a low temperature value as in glass-forming liquids. A measure of t he "strength" of the polymerization transition is introduced to quantify th e "rounding" of the phase transition due to nonzero r. Many properties of l iving polymer solutions should be generally representative of associating p olymer systems (thermally reversible gels, colloidal gels, micelles), and w e compare our results to other systems that self-assemble at equilibrium. ( C) 1999 American Institute of Physics. [S0021-9606(99)50539-4].