Colloidal particles undergo Brownian motion (Stokes–Einstein diffusion) and occasionally collide. Whether a collision sticks depends on the balance of van der Waals attraction and electrostatic double-layer repulsion (DLVO theory). Von Smoluchowski's rapid-coagulation law describes the diffusion-limited case where every collision sticks:
dN/dt = -k_r N² k_r = 8kT / (3η)
N(t) = N₀ / (1 + k_r N₀ t) (perikinetic coagulation)
When a repulsive energy barrier exists, only a fraction of collisions succeed. This is captured by the Fuchs stability ratio W, which multiplies the half-life by W ( k = k_r / W ):
W = 2R ∫ exp(V(h)/kT) / (2R+h)² dh (Fuchs 1934)
sticking probability per collision ≈ 1/W
Adding electrolyte compresses the electrostatic double layer (shorter Debye length), lowering the DLVO energy barrier. Past a sharp threshold — the critical coagulation concentration (ccc) — the barrier collapses and W → 1 (fast coagulation). The Schulze–Hardy rule states this threshold falls steeply with counter-ion charge z:
ccc ∝ z⁻⁶ (Al³⁺ coagulates ~700× more efficiently than Na⁺)
- Electrolyte concentration — sets the DLVO barrier height in this model; sweeping past ≈150 mM crosses the simulated critical coagulation concentration and the sticking probability jumps toward 1.
- Particles merge with volume conservation, r_new = (r₁³+r₂³)^(1/3) — matching real coalescing colloidal aggregates rather than simple overlap.
- Temperature raises the diffusion coefficient D = kT/(6πηr), speeding up Brownian collisions (and, physically, mildly raising k_r itself).
Real-world relevance: this is the same mechanism behind water-treatment flocculation (adding alum/FeCl₃ to clarify turbid water), milk curdling, blood clotting surface chemistry, and paint/ink pigment stability.