Colloidal Coagulation Kinetics 2D — Population Balance

Electrolyte & Particles

Electrolyte concentration 50 mM

Initial particle count 150

Temperature 298 K

Live State

Particles remaining

150

Elapsed time

0.0 s

Stability ratio W

—

Mean cluster size

1.00

Regime: slow coagulation — sticking is rare, most collisions bounce apart.

Panels

Cluster-size population N_k (log scale) Simulated total count history Mean-field projection from now (if held fixed)

How it works

This is a genuinely 2D-native representation of the same physics as the 3D version — but instead of animating individual colliding spheres, it directly integrates the discrete Smoluchowski population-balance equation (PBE), the master equation the 3D theory box only quotes in its size-independent limit. Every cluster size k (1, 2, 3, … monomers) gets its own population N_k(t), and the whole size distribution evolves every frame:

dN_k/dt = ½ Σ_{i+j=k} K_ij N_i N_j  −  N_k Σ_i K_ki N_i
K_ij = (k_r/W) · ¼(i^⅓+j^⅓)(i^-⅓+j^-⅓)     K_11 = k_r/W

The shape factor is the real size-dependent perikinetic (Brownian) coagulation kernel — it reduces to exactly k_r/W for two monomers, so the constant-kernel formula in the 3D theory box is the k=1 special case of the full kernel used here. The left panel plots the live cluster-size histogram N_k on a log scale; the right panel plots the running total particle count (solid) against a dashed mean-field curve projected from the current instant using the closed-form solution N(t') = N/(1+½K₁₁·N·t') — watch the solid trace bend to follow the dashed projection whenever you leave the sliders untouched, and peel away from it the moment you change electrolyte concentration or temperature.

Electrolyte concentration — same DLVO-barrier model as the 3D version: past ≈150 mM the simulated critical coagulation concentration collapses the barrier and W → 1.
Initial particle count — sets N₀ in every size bin's starting condition (all mass begins in the monomer bin, N₁(0)=N₀).
Temperature — raises the diffusion-limited rate constant k_r(T) ∝ T, exactly as Stokes–Einstein predicts for the 3D engine's per-particle diffusion coefficient.

Total monomer mass M(t) = Σ k·N_k(t) is conserved by the merge rule (each collision moves mass from two bins into their sum, never destroying or creating monomers) — the same volume-conserving r_new=(r₁³+r₂³)^(1/3) rule the 3D engine uses, just expressed as bin bookkeeping instead of sphere radii. Real-world relevance: this population-balance form is exactly what's solved numerically in water-treatment flocculation design and aerosol/smoke coagulation modelling, where tracking the full size distribution — not just the total count — determines filter loading and settling behaviour.