Monte Carlo results for random coagulation

Abstract

This paper gives a Monte Carlo algorithm for stochastic coagulation when the coagulation kernel K_ij = f_ig_j + f_jg_i. For a monodisperse system with M = 10⁶ monomers, each realization takes 7 min on a CDC 7600. On the basis of Monte Carlo experiments performed with this algorithm, the following conclusions are inferred. If w_g = w_g(t) = the gel weight fraction = the proportion of monomers contained in the largest cluster at time t, then, as M → ∞, for K_ij = (ij)^ω:

$$\text{w ≦}\text{1}\text{2}\text{, w}{}_{\mathrm{g}}\text{→ 0}$$

;

$$\text{1}\text{2}\text{< w ≦ 1;}$$

w_g approaches a well-defined limit, nonzero after a critical time ([i]and [ii]support a conjecture of Ziff); and

$$\text{w > 1, w}{}_{\mathrm{g}}\text{(t) → 1, t > 0}$$

. (This last result was known to Domilovskii et al. in 1978.) For ω < 1, the Smoluchowski coagulation equation is the limit of random coagulation. For $\text{1}\text{2}\text{< ω < 1}$, this equation gives rise to mass nonconservation; the mass deficit w_g(t) is contained in a single cluster. For K_ij = [(f − 2)i + 2][(f − 2)j + 2], due to correlations, the coagulation equation is not the limit of random coagulation. Instead, its Flory version, incorporating gel reactivity, is the appropriate limit.