Note that, in the system with triangle-shaped monomers, the attachment order for the first assembly step following nucleation is larger, $\gamma =4$, compared to all subsequent assembly steps, where $\gamma =3$.

Since cluster growth is by the addition of monomers, stochastic effects result in some interface roughness (akin to Eden growth [87]) leading to a larger surface-to-volume ratio than given by $p_s=a\sqrt{s}$ and, hence, to an effective growth exponent $\omega >1/2$. Since this affects only the system with hexagon-shaped monomers, we do not discuss this additional effect here further and simply approximate $\omega$ by $1/2$ for hexagon-shaped monomers.

Unlike for hexagon-shaped building blocks, in the case of square and triangular building blocks, the prefactor $a$ additionally includes the multiplicity $q_s$ of the seed-forming reaction pathways. For example, the multiplicity considers that the seed can typically grow in two directions unless it forms directly at the corner of an edge.

The combinatorial factors $f_s$ for clusters of small size $s$ typically deviate from the asymptotic form $a{s}^{\omega }$. Note, however, that, for increasing target size $S$, the approximation becomes ever more accurate, as the asymptotic form of $f_s$ applies to an increasingly large fraction of all assembly steps.

In general, $\beta >\alpha$ holds if $\sigma >\gamma +1$ and $\omega <3/2$, meaning that this must be true, in general.

The latter is particularly pronounced in the system with hexagonal monomers, where, due to the absence of a domino effect that fills up the edges of a cluster after an initial seed has formed, the typical morphologies of the cluster exhibit some interface roughness during growth. Note that the assembly time obtained from integrating the effective model in which we use $\omega =1/2$ coincides with the analytic estimate ${\theta }_{\mathrm{th}}=1/2$ for the system with hexagonal building blocks. Furthermore, using a value of $\omega$ slightly larger than $1/2$ in Eq. (17) (which would be plausible due to the roughening effect) consistently results in a value of $\theta$ smaller than $1/2$ as observed from the simulation data. For example, the measured value $\theta \approx 0.4$ is obtained with $\omega =0.6$ according to Eq. (17).

Note that some of the approximations used here actually become exact in the limit of large $S$. For example, neglecting the diffusion term and higher-order terms in the advection equation is irrelevant in the limit $S\to \infty$, as their contribution becomes negligible against the leading-order advection term [see Eqs. (c4) and (c5)], showing that the advective contribution to cluster growth equals the square of the diffusive contribution. Similarly, neglecting the nucleation term in Eq. (22) becomes increasingly irrelevant as $S$ increases.

We remark that the precondition for the applicability of our theory is that the growth rate of large clusters consistently scales as ${\tilde{r}}_{s\to s+1}\sim s^{\omega }$ with some arbitrary growth exponent $\omega$.

There are either no or only a small number of successive lower-order processes for the assembly steps immediately following nucleation. Hence, the growth rates for small clusters behave a bit differently, as we discuss below, specifically for triangle-shaped building blocks. In the other cases, however, good agreement with the simulation data can be obtained even if we do not treat the initial assembly steps separately and assume the same growth rate for them as for larger clusters.

Note also that explicit boundaries violate the equivalence between homogeneous and heterogeneous systems (cf. Appendix pp10-s1) due to the distinct properties of the boundary species.