^{1,2}, Bingyu Zhao

^{3}, Bijan Berenji

^{1,2}and Tom Chou

^{2,4}

### Abstract

We analyze a fully stochastic model of heterogeneous nucleation and self-assembly in a closed system with a fixed total particle number M, and a fixed number of seeds N s. Each seed can bind a maximum of N particles. A discrete master equation for the probability distribution of the cluster sizes is derived and the corresponding cluster concentrations are found using kinetic Monte-Carlo simulations in terms of the density of seeds, the total mass, and the maximum cluster size. In the limit of slow detachment, we also find new analytic expressions and recursion relations for the cluster densities at intermediate times and at equilibrium. Our analytic and numerical findings are compared with those obtained from classical mass-action equations and the discrepancies between the two approaches analyzed.

This work was supported by the National Science Foundation through Grant Nos. DMS-1021850 (M.R.D.) and DMS-1021818 (T.C.). M.R.D. was also supported by an ARO MURI Grant No. W1911NF-11-10332, while T.C. was also supported by ARO Grant No. 58386MA.

I. INTRODUCTION

II. MASS-ACTION KINETICS

III. MASTER EQUATION FOR HETEROGENEOUS SELF-ASSEMBLY

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

## Figures

A schematic of the heterogeneous self-assembly process in a closed system. The open hexagons represent seed particles on which the monomers (filled circles) aggregate. In this example, the total mass, the number of seed particles, and the maximum cluster size are M = 30, N s = 6, and N = 6, respectively.

A schematic of the heterogeneous self-assembly process in a closed system. The open hexagons represent seed particles on which the monomers (filled circles) aggregate. In this example, the total mass, the number of seed particles, and the maximum cluster size are M = 30, N s = 6, and N = 6, respectively.

State space for a self-assembling system consisting of M = 5 total monomers, N s = 2 seeds, and a maximum cluster size of N = 3. In this example, σ = M/(N s N) = 5/6.

State space for a self-assembling system consisting of M = 5 total monomers, N s = 2 seeds, and a maximum cluster size of N = 3. In this example, σ = M/(N s N) = 5/6.

Mean cluster sizes ⟨n k (t)⟩ obtained from averaging 105 KMC simulations of the stochastic process in Eq. (9) with N = 5, N s = 10, and ɛ = 10−5. The dashed curves represent solutions from BD equations for comparison. (a) M = 5 corresponding to σ = 0.1, (b) M = 15 corresponding to σ = 0.3, and (c) M = 25 (σ = 1/2).

Mean cluster sizes ⟨n k (t)⟩ obtained from averaging 105 KMC simulations of the stochastic process in Eq. (9) with N = 5, N s = 10, and ɛ = 10−5. The dashed curves represent solutions from BD equations for comparison. (a) M = 5 corresponding to σ = 0.1, (b) M = 15 corresponding to σ = 0.3, and (c) M = 25 (σ = 1/2).

The difference δ k (t) ≡ ⟨n k (t)⟩ − c k (t) for k = 0, 1, 2, 3, 4, 5. In this example, M = 25, N = 5, and N s = 10.

The difference δ k (t) ≡ ⟨n k (t)⟩ − c k (t) for k = 0, 1, 2, 3, 4, 5. In this example, M = 25, N = 5, and N s = 10.

Expected metastable and equilibrium cluster numbers calculated as a function of M using numerical methods and combinatoric algorithms. Here, as in Fig. 3 , N s = 10, N = 5. For metastable concentrations, (a) and (b), the differences between the mean-field and exact results are small except for the largest cluster sizes k = 4, 5. For comparison, is shown by the dashed grey curve in (b). (c) and (d) show and , respectively. The differences between and are more subtle but are generally most noticeable for σ ∼ 0.1, 0.9. All densities are symmetric in k↔N − k about M = N s N/2, with , to order ɛ, forming an isosbestic point at M = N s N/2.

Expected metastable and equilibrium cluster numbers calculated as a function of M using numerical methods and combinatoric algorithms. Here, as in Fig. 3 , N s = 10, N = 5. For metastable concentrations, (a) and (b), the differences between the mean-field and exact results are small except for the largest cluster sizes k = 4, 5. For comparison, is shown by the dashed grey curve in (b). (c) and (d) show and , respectively. The differences between and are more subtle but are generally most noticeable for σ ∼ 0.1, 0.9. All densities are symmetric in k↔N − k about M = N s N/2, with , to order ɛ, forming an isosbestic point at M = N s N/2.

Plot of Δ(t) (N = 5, N s = 10) for M = 5, 25, 45. These curves were generated from KMC simulation of the stochastic process and numerical evaluation of Eqs. (1) for c k (t). Note that the error in the metastable regime (1 ≪ t ≪ ɛ−1) is largest for M = 45, while at equilibrium (t ≫ ɛ−1) the errors are largest for M ≈ 5, 45.

Plot of Δ(t) (N = 5, N s = 10) for M = 5, 25, 45. These curves were generated from KMC simulation of the stochastic process and numerical evaluation of Eqs. (1) for c k (t). Note that the error in the metastable regime (1 ≪ t ≪ ɛ−1) is largest for M = 45, while at equilibrium (t ≫ ɛ−1) the errors are largest for M ≈ 5, 45.

The overall error of mass-action kinetics. (a) The averaged error in the metastable regime Δ*. (b) The averaged error Δeq in the equilibrium limit t ≫ ɛ−1.

The overall error of mass-action kinetics. (a) The averaged error in the metastable regime Δ*. (b) The averaged error Δeq in the equilibrium limit t ≫ ɛ−1.

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