 Solution

The literal $u(r)=-C_4/r^4$ curve is monotone increasing from $-\infty$ at $r\to 0^{+}$ toward zero from below as $r\to \infty$:Taken literally at a zero gap, it predicts an unbounded collapse energy and force. That extrapolation is outside the far-field dipole approximation and cannot describe actual touching matter.

Finite atomic/ionic dimensions, electronic overlap and Pauli repulsion supply short-distance constraints. Solvent and hydration layers, steric coatings and surface chemistry can add further contributions. An illustrative regularized model is $u_{\mathrm{toy}}=A/r^{12}-C_4/r^4$ with $A>0$; it has a finite minimum at $r=(3A/C_4{)}^{1/8}$ and becomes repulsive at shorter distances. This is a schematic regularization, not a derived complete ion–sphere contact law.

Separately, correcting the center-distance convention to $R=a+r$ already removes this particular $r=0$ divergence from the far-field expression, but does not make a dipole approximation accurate at contact. Near-field multipoles and microscopic repulsion still require a better model. Calibrated colloidal probe technique measurements can test where a long-range force model ceases to agree with actual approach curves.^[ref]