Gelation in input-driven aggregation

TL;DR

This work analyzes input-driven irreversible aggregation within a mean-field Smoluchowski framework, focusing on how a constant monomer source shapes the long-time mass distribution and gelation behavior. It develops two complementary analytical schemes (Flory and Stockmayer) to treat the post-gel regime and contrasts their predictions, especially for the product kernel $K_{i,j}=ij$ where gelation occurs. The paper derives exact stationary distributions for mass-independent rates in binary and ternary cases, and provides detailed generating-function and moment analyses that quantify giant-component formation, mass transport, and mass distributions both pre- and post-gel. Numerical simulations using FFT-based convolutions validate the analytic results and demonstrate convergence toward the predicted Flory and Stockmayer limits across binary and ternary product-kinetic models. The findings illuminate gelation dynamics in input-driven systems and lay groundwork for higher-order aggregation and non-mean-field extensions.

Abstract

We investigate irreversible aggregation processes driven by a source of small mass clusters. In the spatially homogeneous situation, a well-mixed system is consists of clusters of various masses whose concentrations evolve according to an infinite system of nonlinear ordinary differential equations. We focus on the cluster mass distribution in the long time limit. An input-driven aggregation with rates proportional to the product of merging partners undergoes a percolation transition. We examine this process analytically and numerically. There are two theoretical schemes and two natural ways of numerical integration on the level of a truncated system with a finite number of equations. After the percolation transition, the behavior depends on the adopted approach: The giant component quickly engulfs the entire system (Flory approach), or a non-trivial stationary mass distribution emerges (Stockmayer approach). We also outline generalization to ternary aggregation.

Figures (9)

• Figure 1: Analytical and numerical results for the mass distributions in input-driven binary aggregation with mass-independent rates governed by Eqs. \ref{['Smol']}.
• Figure 2: Convergence of the concentrations $c_1(t)$, $c_2(t)$, $c_3(t)$ to analytical predictions \ref{['eq:exact_c123']} with $\lambda=0.01$. A few hundred equations suffices to obtain very accurate results for the concentrations of light clusters.
• Figure 3: Numerical results for stationary cluster mass distribution for binary and ternary aggregation for various values of the ratio $\lambda$ characterizing relative strength of the binary aggregation events. The number of equations in numerical integration is $2^{16}$.
• Figure 4: Numerical solutions of equations \ref{['ckt-23']} with $\lambda = 0.01$ approaches to the analytical prediction \ref{['ck-23_final']} for the steady state. The vertical lines correspond to the estimate $s_*= t^2$ of the crossover mass for $t=5$, $25$, $50$, $100$. The number of equations in numerical integration is $2^{18}$.
• Figure 5: The mass $g$ of the giant component vanishes when $t < t_g$. In the percolating phase, $g$ is represented by \ref{['gt']}.