Asakura-Oosawa model ($U_{\mathrm{dep}}(R)=-{\Pi }_p\pi (4R_e+R)(2R_e-R{)}^2/12$)

The Asakura-Oosawa model treats dilute nonadsorbing polymer coils as mutually interpenetrable ideal particles excluded from a layer around each colloid. For colloid radius $a$ and polymer exclusion thickness $R_g$, two exclusion spheres of radius $R_e=a+R_g$ overlap when $2a\le R<2R_e$. Adding their two spherical cap volumes gives $V_{\mathrm{ov}}=\pi (4R_e+R)(2R_e-R{)}^2/12$. Multiplication by $-{\Pi }_p$ gives the displayed pair free energy, with zero depletion interaction beyond $2R_e$. The range is controlled by polymer size and the depth by reservoir polymer concentration. At larger polymer concentration, polymer interactions and many-body overlap effects require care.