= Solution
The literal $u(r)=-C_4/r^4$ curve is monotone increasing from $-\infty$ at $r\to0^+$ toward zero from below as $r\to\infty$:
$$
u'(r)=4C_4/r^5>0.
$$
\b[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 $u_{\rm toy}=A/r^{12}-C_4/r^4$ 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.<direct measurement of colloidal forces using an atomic force microscope by William A. Ducker and others>[1991, pp.239–241, sphere–plane force measurements and short-range deviations]{ref}
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