= Solution
<DLVO theory> adds attractive <Van der Waals forces> and <electric double layer> repulsion to obtain an effective <colloid> pair potential. For two identical spheres of <radius> $b$ at a surface gap $h\ll b$, a useful schematic form is
$$
U(h)\simeq-\frac{A_Hb}{12h}+B e^{-\kappa_Dh},\qquad B>0.
$$
Here $A_H$ is the <Hamaker constant>. For a symmetric monovalent electrolyte with <number density> $n$ of each ionic species, $\kappa_D^2=2ne^2/(\epsilon k_BT)$. The exponential is a simplified screened repulsion; its amplitude depends on <electric potential> at the surface, <charge regulation> and geometry. Constant-charge and constant-potential <boundary conditions> need not produce the same close-approach law.
A primary attractive <minimum> near contact may be separated from dispersed configurations by an <energy> barrier; a shallower secondary <minimum> can produce loose reversible aggregation. A barrier much larger than $k_BT$ can give long-lived kinetic stabilization without making the dispersed state the global free-energy <minimum>. Adding salt shortens the <Debye–Hückel screening length> and can lower the barrier, allowing <colloidal coagulation>. Changing <pH> can alter surface ionization, so screening alone does not determine stability.
One can vary salt <concentration> while holding particle <concentration>, <temperature> and <pH> fixed, then follow <particle aggregation> using <light scattering> or <microscopy>. A solvent-only control and an independently measured surface-charge indicator help distinguish a change of screening from <adsorption> or <charge reversal>. Classical <DLVO theory> omits <steric stabilization>, <hydration forces> and strong ionic correlations; the formal attraction diverging at contact is cut off by molecular-scale physics.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2013/paper-1-dlvo-landscape.png]
{title=Original schematic pair potentials showing how a shorter screening length lowers a repulsive barrier}
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