Figure 3.
Properties of the largest cluster (LC) in a symmetric system consisting of an equal number of A and B molecules, each with valency m. (A and B) The weight fraction of the largest cluster, WLC, where colored plots correspond to different numbers of molecules (green, 103 molecules; blue, 5 × 103 molecules; and red, 104 molecules), and the weight fraction of the gel phase, Wg, from the analytical theory (9) (black lines). Molecules A and B have identical valency m = 3 (A) or m = 6 (B).The results of the simulations of WLC converge toward the theory (black line) for pA > pA,c. (C) Appearance of rings in the largest cluster is abrupt at pA,c, indicated by the lines at pA = 0.2 for valency m = 6 and at pA = 0.5 for valency m = 3; the color code is as in A and B. (D) Average number of bonds in each molecule A in the largest cluster. When this quantity is >2, there must be rings, and as in C, this condition is robustly met when pA > pA,c. Each color corresponds to a different valency, m = 6 to m = 2 from top to bottom (and left to right for pA,c). For each valency, simulations for 103, 5 × 103, and 104 molecules are plotted as dotted, dot-dashed, and dashed lines, respectively, and it is clear that the number of molecules does not significantly affect the results, especially past the pA,c. All results in A–D are the means of 50 simulations for systems with 103 and 5 × 103 molecules, and 10 simulations for the system with 104 molecules. To see this figure in color, go online.