Partially Cooperative Phase Conversions and Noninteger Avrami Exponents

Partially cooperative phase transformations, with characteristically sigmoidal conversion curves, are commonly observed, but rigorous analytical solutions are widely familiar only for fully cooperative and fully noncooperative conversions (exp(−Kt4) and exp(−kt), respectively, in three dimensions). The JMAK formula, exp(−κtn) with noninteger Avrami exponent n, has been used to fit data for partially cooperative conversions, but this approach has only been empirical and so far seems to lack theoretical derivation and support. We show that the Ishibashi–Takagi modification of Avrami theory rigorously accounts for partial cooperativity that arises from the competition between random volume filling by newly formed nuclei of finite volume and cooperative domain growth. The imperfect cooperativity and finite initial domain volume are accounted for by a prenucleation growth time t0, resulting in conversion curves of the form exp(Kt04) exp(−K(t + t0)4), with K depending on nucleation and growth rates as in fully cooperative Avrami theory. The validity of the analytical theory, which solves the Finke–Watzky problem of competing nucleation and growth and can be cast in terms of two rate constants, has been confirmed by numerical simulations of domain growth with finite initial domain volume on a lattice with the nucleation rate varying over nearly 5 orders of magnitude. The first-order kinetics exponential decrease in the limit of no cooperativity is correctly recovered for large t0. Between the random and fully cooperative limits, the partially cooperative conversion curves resemble, but are not exactly matched by, empirical Avrami exp(−κtn) with a noninteger exponent or Finke–Watzky curves. A cooperativity parameter C = exp(−(4K)1/4t0) ranging between 0 and 100% is introduced and related to the empirical Avrami exponent.