Solution (source code)

= Solution

One method is to give every particle a sufficiently large <surface charge density> of the same sign. For example, attach ionizable sulfonate groups covalently to the surface. Dissociation leaves negatively charged surface groups and releases mobile positive <counterions> into the water. A <surfactant> with a strongly bound head group can provide another charging route, provided it does not desorb during the experiment.

Each charged particle is surrounded by an <electric double layer>. As two such particles approach, overlap of these ionic clouds raises the electrostatic and ionic <free energy>, producing repulsion. In a dilute electrolyte, the range is set by the <Debye–Hückel screening length>,
$$
\lambda_D=\kappa^{-1},\qquad
\kappa^2=\frac{e^2}{\varepsilon_0\varepsilon_r k_BT}\sum_i z_i^2 n_i.
$$
For identical weakly charged spheres a representative <screened Coulomb pair potential> is
$$
U_{\mathrm{el}}(R)\simeq\frac{Q^2}{4\pi\varepsilon_0\varepsilon_r(1+\kappa a)^2R}
\,e^{-\kappa(R-2a)},\qquad R\geq2a.
$$
Its sum with the <Van der Waals force> attraction is the <DLVO theory> pair potential. Choose the charging level so that the repulsive barrier is many $k_BT$ high; particles then rarely reach the primary attractive well during the observation time. This is kinetic stabilization rather than removal of all attraction.

Deionized water is useful because low added <ionic strength> gives a long-ranged repulsion. It does not mean that the charged dispersion contains no <ions>: the surface's own <counterions> are unavoidable and contribute to screening. Surface dissociation, contamination and particle concentration therefore matter even without added salt. An alternative valid route is a dense, firmly attached layer of water-soluble <polyethylene glycol>, producing <steric stabilization> without relying on a large electrostatic barrier.