Homogeneous self-assembly and growth in a closed unit volume initiated with M = 30 free monomers. At a specific intermediate time 0 < t < t* in this depicted realization, there are six free monomers, four dimers, four trimers, and one cluster of size four. For each realization of this process, there will be a specific time t* at which a maximum cluster of size N = 6 in this example is first formed (blue cluster).
Mean first assembly times evaluated via the heuristic definition Eq. (7) (dashed line) and as a function of the detachment rate q i = q, for M = 7 N = 3 in panel (a) and for M = 9, N = 4 in panel (b). Here p i = p = 1. We also show the exact results (solid line) obtained via the stochastic formulation in Eq. (12) which we derive in Sec. III . Qualitative and quantitative differences between the two approaches arise, which become even more evident for N > 3, q → 0, as we shall later discuss. These discrepancies underline the need for a stochastic approach.
Allowed transitions in stochastic self-assembly starting from an all-monomer initial condition. In this simple example, the maximum cluster size N = 3. (a) Allowed transitions for a system with M = 7. Since we are interested in the first maximum cluster assembly time, states with n 3 = 1 constitute absorbing states. The process is stopped once the system crosses the vertical red line. (b) Allowable transitions when M = 8. Note that if monomer detachment is prohibited (q = 0), the configuration (0, 4, 0) (yellow) is a trapped state. Since a finite number of trajectories will arrive at this trapped state and never reach a state where n 3 = 1, the mean first assembly time T 3(8, 0, 0) → ∞ when q = 0.
Mean first assembly times for M = 7 and N = 3 in panel (a) and M = 8 and N = 3 in panel (b). Exact results derived in Eq. (12) are plotted as black solid lines, while red circles are obtained by averaging over 105 KMC trajectories. The dashed blue line shows the q → 0 approximation in Eq. (18) and the q → ∞ approximation in Eq. (23) .
Comparison of theory with simulations for N = 10, and several values of M. Symbols are derived from 104 KMC simulations for M = 50, 200, 1000. In panel (a) the dashed lines are obtained by plotting the curve T 10(M, 0, …, 0) = A/q where A is given by imposing passage through the first point to the left in the graph. Note that all other points align to the same curve. Solid lines are derived from Eq. (23) in the dominant path approximation. In panel (b) results from the hybrid approximation with r = 2 in Eq. (E2) are superimposed on the same data. Note the much better fit in the hybrid approximation as q → ∞, especially as M becomes larger.
First assembly times T N (M, 0, …, 0) as a function of M for q = 100 and several values of N in panel (a), and for N = 5 and several values of q in panel (b). The black dashed lines represent the dominant path approximation for large q in Eq. (23) , while the solid black line represents the hybrid approximation in Eq. (E2) for r = 2. We chose to plot only representative cases, not to clutter the graphics, but similar trends persist in panel (a) for N = 4, 6, 8 and in panel (b) for q = 10, 100. Note that the dominant path approximation ceases to be accurate for very large values of M and that the hybrid approximation provides a better fit as q → ∞.
Probability distributions for the first assembly time for N = 4 and M = 50 and for various values of q (a)–(f). The black bars are obtained as a normalized histogram of 104 KMC simulations. The dashed red and solid blue lines are the probability density functions estimated via the dominant path approximation in Eq. (D1) and via the hybrid approximation with r = 3 in Eq. (E2) , respectively. The detachment rate q increases as indicated in each subplot. Note that initially the distribution has a log-normal shape and later turns into an exponential. As predicted, the analytical estimate given by Eq. (D1) becomes accurate for q ⩾ M. Also note the change in scale and the broadening of the distribution as q increases.
First assembly time distributions for N = 8 and M = 200 for various values of q (a)–(f). The black bars are obtained as a normalized histogram of 104 KMC simulations. The dashed red and solid blue lines are the probability density functions estimated via the dominant path approximation in Eq. (D1) and via the hybrid approximation with r = 3 in Eq. (E2) respectively. The detachment rate q is successively increased in each subplot. Note that the distribution begins as a bimodal curve and acquires a log-normal shape, before turning to an exponential for larger q. As in Fig. 7 , the hybrid approximation becomes increasingly accurate as q increases.
First assembly time distributions for N = 8 and q = 100 for various values of the total mass M (a)–(f). The black bars are obtained as a normalized histogram of 105 KMC simulations. The dashed red and solid blue lines are the probability density functions estimated via the dominant path approximation in Eq. (D1) and via the hybrid approximation with r = 3 in Eq. (E2) , respectively. Total mass M increases as indicated in each subplot. Note that the distribution evolves from an exponential with decay rate given by Eq. (D3) , valid for q ⩾ M, towards a more log-normal shape. In this case, for very large M both dominant path and hybrid approximations fail.
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