= Solution
The change from a positive surface potential to a negative potential outside the compact layer is consistent with specific phosphate <adsorption> that overcompensates the positive oxide charge. The negative compact-layer charge then has a diffuse excess of positive <counterions>. This <charge reversal> is an interfacial-chemistry interpretation, not something a single monotone double-layer profile with an unchanged positive boundary charge could generate by screening alone. Surface potential and <zeta potential> at a slip plane are distinct observables.
The <Stern model> separates the specifically adsorbed compact layer from a diffuse layer. In a spherical diffuse layer, a <Poisson-Boltzmann equation> description is
$$
\frac1{r^2}\frac{d}{dr}\!\left(r^2\frac{d\psi}{dr}\right)
=-\frac1\varepsilon\sum_i z_i e n_{i,\infty}
\exp\!\left(-\frac{z_i e\psi}{k_BT}\right).
$$
At sufficiently small $|z_ie\psi|/k_BT$, the <Debye–Hückel approximation> gives $\nabla^2\psi=\kappa^2\psi$. If $a_s$ is the outer compact-layer radius, its decaying solution is
$$
\psi(r)=\psi(a_s)\frac{a_s}{r}\exp[-\kappa(r-a_s)].
$$
For a locally flat surface, the geometric factor tends to unity. Thus the far-field negative potential approaches zero from below over the <Debye–Hückel screening length>. Neither $+120$ nor $-70\ \mathrm{mV}$ is safely in the low-potential linear regime at room <temperature>, so the simple exponential is a far-field approximation, not a globally valid fit. The phosphate layer requires <adsorption> chemistry or an equivalent specified boundary model.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2019/paper-1-charge-reversal.png]
{title=Original illustrative compact-layer charge reversal followed by a negative screened far-field potential}
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