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].