The governing equations for nucleus size and concentration with boundary conditions for the concentration at the nucleus and at infinity.
Showing u[r/R CNT, τ] at six times evenly spaced from τ = 0 (red) to τ = 3.4 (blue). The black curved arrow depicts the moving interface and changing concentration at the surface of the nucleus. As the nucleus dissolves it drives the surrounding concentration upward, which in turn accelerates the rate at which the nucleus dissolves. Note that the nucleus dissolves even though it starts in an enriched zone. The two figures above the graph depict the nucleus (red) in the local concentration field at two different times.
Showing the nucleus size as a function of dimensionless time. As the classically pre-critical nucleus dissolves, the increasing local concentration gradient accelerates the dissolution. The numerical solution to the coupled differential equations becomes unstable at the point where the trajectory terminates.
Showing u[r/R CNT, τ] at six times evenly spaced from τ = 0.0 (red) to τ = 6.0 (blue). The black curved arrow depicts the moving interface and changing concentration at the surface of the nucleus. As the nucleus grows the local equilibrium concentration drops causing the nucleus to absorb even more of the surrounding solutes. This diffusion coupled growth process drives the classically pre-critical nucleus over the nucleation barrier. The two figures above the graph depict the nucleus (red) in the local concentration field at two different times.
Showing the nucleus size as a function of dimensionless time. The classically pre-critical nucleus is driven over the barrier as the concentration profile in Fig. 4 relaxes.
Nucleus size as a function of dimensionless time starting from an enriched profile with the initial gradient favoring growth (a = 0.75). The nucleus grows and then begins to dissolve again at the turning point R*.
Schematic showing an initial concentration profile and the profile at later times around a classically pre-critical nucleus of initial size R 0. If the local concentration gradient initially directs solutes toward the nucleus, then there must be an “initial super-enrichment” zone (green) with a concentration above even the elevated concentration at the nucleus surface. The red shaded area depicts an additional zone of “dynamical super-enrichment” as described in the text. The size R 2 is the maximum in the initial concentration profile as a function of distance from the center of the nucleus. The last concentration profile shown, when the nucleus reaches the turning point size R *, depicts the situation where all super-enrichment has been exhausted and the concentration cannot drive further growth of the nucleus. After reaching R *, the nucleus will begin to dissolve again.
The separatrix as estimated using Eq. (17) with ℓ = 4.0. The figure also shows points on the separatrix found by solving Eqs. (6)–(9) numerically for different values of a, ℓ, and R 0/R CNT. Nucleus sizes and concentration profiles below the curve (shaded gray region) are pre-critical and points above the curve are post-critical. The classical scenario is a classically critical nucleus (R 0 = R CNT) in a uniformly supersaturated concentration profile (a = 0). The classical scenario is just one point on the separatrix when concentration fluctuations are considered. The inset figures depict classically pre-critical, classically critical, and classically post-critical nuclei in dark gray with the surrounding concentration profiles in light gray that result in transition state configurations. Results for ℓ = 2, 4, and 8 approximately collapse onto a single curve with the dimensionless parameter aℓ 3 representing the initial degree of superenrichment and with R 0/R CNT representing the initial nucleus size.
The fate of a nucleus in the present model depends on super-enrichment and super-deletion in the initial concentration profile and on the existence of a turning point R * between R 0 and R CNT. The turning point, when it exists, can be estimated using Eq. (17) or more accurately by solving the coupled ODE/PDE model for c(r, t) and R(t).
Definitions of parameters in the coupled nucleation – diffusion model.
Definitions of scaled variables in the coupled nucleation – diffusion model.
Values of R 0/R CNT, a, and ℓ on the separatrix found by solving Eqs. (6)–(9). As shown in Fig. 8, the separatrix approximately collapses to a simpler relationship between R 0/R CNT and aℓ3, a measure of the total super-enrichment around the nucleus. The symbol *** indicates that it was not possible with ℓ = 2 and with c(r,0) > 0 for r > R 0 to find the separatrix using the family of initial conditions in Eq. (10).
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