Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2019 / Paper 1 / Section A(iii) / 5

charge stabilization of colloids supplies repulsion when like-charged surfaces and their electric double layers approach. steric stabilization uses attached or adsorbed molecular layers whose overlap costs conformational and osmotic free energy. Combining both gives electrosteric stabilization; a solvated coating must remain attached and well solvated. A viscous medium slows encounters but does not by itself create an equilibrium repulsive barrier.

DLVO theory adds screened electrostatic repulsion to Van der Waals force. A representative close-gap form isfor equal spheres of radius $a$, with Hamaker constant $A_H$ and inverse Debye–Hückel screening length $\kappa$. The formula is schematic near molecular contact, where hydration, surface roughness and finite ion size matter. A repulsive barrier many $k_{B}T$ high produces kinetic stability even if the primary attractive minimum is deeper than the dispersed state. Entry into that primary minimum generally gives strongly bound aggregation; a shallow secondary minimum can give reversible flocculation.

Adding electrolyte increases $\kappa$, shortens the repulsion and can remove the barrier, enabling fast diffusion-limited aggregation. The critical coagulation concentration is the approximate salt concentration at which this rapid-coagulation regime begins under specified conditions. It is not a universal property of the salt alone. Test the prediction using time-resolved dynamic light scattering or turbidity at controlled particle concentration, pH and temperature; check that added ions do not specifically adsorb or change the surface chemistry.

In the high-surface-potential Schulze-Hardy rule, the critical coagulation concentration scales approximately as $z^{-6}$ with the counterion valence. HenceThe more highly charged counterion therefore requires about eleven times less concentration in this idealized estimate. This assumes the cations are the counterions of a negatively charged surface, the surface-potential regime is appropriate, and the only changing mechanism is electrostatic stabilization. Specific adsorption and hydrolysis of aluminium ions can invalidate this scaling; matching ionic strengths instead is not the same prediction.