Numerical solution of Smoluchowski coagulation equation combined with Ostwald ripening

TL;DR

The paper tackles the simultaneous evolution of coagulation and Ostwald ripening during the concluding stage of phase transformation by solving a Smoluchowski-type integro-differential system with a migration–diffusion term in particle-volume space. It advances a computationally efficient numerical method based on low-rank representations of the coagulation kernel and finite-difference discretization, enabling fast time stepping and accurate capture of long-time behavior. The key finding is that, for various initial particle-volume distributions, the system converges to a universal particle-volume distribution at large times, consistent with recent analytical results for the coupled process; this universal form connects to the Schumann and Lifshitz–Slyozov distributions in the limiting pure-coagulation and pure- Ostwald ripening cases. The work provides a practical, scalable tool (SmoluchowskiSolver) for studying coupled coagulation and Ostwald ripening and sets the stage for extending to more complex kernels, fragmentation, and stochastic approaches, with potential impact on materials design and crystallization processes.

Abstract

The processes of simultaneous coagulation and Ostwald ripening of particles in the concluding stage of phase transformation are considered. We solve the integro-differential system of Smoluchowski-type kinetic and mass balance equations using a computationally efficient numerical algorithm based on low-rank matrices. We compare our numerical solutions for different initial particle-volume distributions with the universal distribution function for combined coagulation and Ostwald ripening. Our calculations confirm the tendency of a particulate ensemble to the universal particle-volume distribution to be approached asymptotically after a sufficiently long time, no matter what the initial particle-volume distribution might be.

Figures (6)

• Figure 1: Comparison of analytical and numerical solutions for $\gamma=1$, $\kappa=0.2$, $\chi=10^{-2}$, $\Delta_{o}=0.2$, $\Phi(0,0)=1$, $b_{o}=1$ and $c_{s}=10$ mol m$^{-3}$CNSNS. For numerical calculations the following parameters have also been chosen: ${H=20}$, $M=4000$, $M_{\xi}=M$, $T=0.5$, $M_{\tau}=20000$ and $d=5$.
• Figure 2: Comparison of analytical and numerical solutions for $\gamma=0.5$, $\kappa=0.2$, $\chi=10^{-1}$, $\Delta_{o}=0.2$, $\Phi(0,0)=1$, $b_{o}=1$ and $c_{s}=10$ mol m$^{-3}$CNSNS. For numerical calculations the following parameters have also been chosen: $H=20$, $M=4000$, $M_{\xi}=M$, $T=1.0$, $M_{\tau}=40000$ and $d=5$.
• Figure 3: Numerical solutions for $\Phi$ and $\xi\Phi$ with various initial particle-volume distributions at $T=10$ versus the universal distribution function (\ref{['e12']}).
• Figure 4: Numerical solutions for $\Phi$ and $\xi\Phi$ with various initial particle-volume distributions at $T=20$ versus the universal distribution function (\ref{['e12']}).
• Figure 5: Numerical solutions for $\Phi$ and $\xi\Phi$ with various initial particle-volume distributions at $T=50$ versus the universal distribution function (\ref{['e12']}).