Exhaustion of nucleation in a closed system

We determine the distribution of cluster sizes that emerges from an initial phase of homogeneous aggregation with conserved total particle density. The physical ingredients behind the predictions are essentially classical: Supercritical nuclei are created at the Zeldovich rate, and before the depletion of monomers is significant, the characteristic cluster size is so large that the clusters undergo diffusion-limited growth. Mathematically, the distribution of cluster sizes satisfies an advection partial differential equation (PDE) in “size space.” During this creation phase, clusters are nucleated and then grow much larger than the critical size, so nucleation of supercritical clusters at the Zeldovich rate is represented by an effective boundary condition at zero size. The advection PDE subject to the effective boundary condition constitutes a “creation signaling problem” for the evolving distribution of cluster sizes during the creation era. Dominant balance arguments applied to the advection signaling problem show that the characteristic time and cluster size of the creation era are exponentially large in the initial free-energy barrier against nucleation, $G_{*}$. Specifically, the characteristic time is proportional to $e^{(2∕5)G_{*}∕k_{B}T}$ and the characteristic number of monomers in a cluster is proportional to $e^{(3∕5)G_{*}∕k_{B}T}$. The exponentially large characteristic time and cluster size give a posteriori validation of the mathematical signaling problem. In a short note, Marchenko [JETP Lett. 64, 66 (1996)] obtained these exponentials and the numerical prefactors $2∕5$ and $3∕5$. Our work adds the actual solution of the kinetic model implied by these scalings, and the basis for connection to subsequent stages of the aggregation process after the creation era.

Figure 1

Characteristics of the reduced PDE. The flux $n^{1∕3}r$ is constant along the characteristics. The density of the curves is proportional to the solution $r(n,t)$.

Figure 2

Zeldovich nucleation rate $j$ as a function of time. After a time of $O\left(1\right)$, the super-saturation decreases slightly, and the nucleation rate gets turned off. The dashed line is an approximation for large time $t⪢1$, showing superexponential decay.

Figure 3

Density of cluster sizes, $r(n,t)$, for various values of $t$.

Figure 4

Errors in $j$ and $R$, together with a line of slope $-3∕2$ in a log-log plot. The $x$ axis is the logarithm (base 2) of the number of points used, and the $y$ axis is the (natural) logarithm of the error.