Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2019 / Paper 1 / Section B / 10 / iv

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 isAt sufficiently small $|z_{i}e\psi |/k_{B}T$, the Debye–Hückel approximation gives ${\nabla }^2\psi ={\kappa }^2\psi$. If $a_s$ is the outer compact-layer radius, its decaying solution isFor 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.

For an ideal dilute electrolyte with molar ionic strength $I=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$, measured in $\mathrm{mol}{\mathrm{L}}^{-1}$, the Debye–Hückel screening length isTaking water's relative permittivity as approximately $78.5$ at $25^{\circ }\mathrm{C}$, $I=0.10\mathrm{M}$ for NaCl. Therefore ${\kappa }^{-1}\simeq 0.96\mathrm{nm}$. This is an exponential screening length, not a hard outer edge of the diffuse layer. The factor of $1000$ converts molarity into $\mathrm{mol}{\mathrm{m}}^{-3}$.

A $0.10\mathrm{M}$ formula-unit concentration of fully dissociated $\mathrm{AlC}{\mathrm{l}}_3$ gives $0.10\mathrm{M}$ trivalent cations and $0.30\mathrm{M}$ chloride:At unchanged temperature and permittivity, the Debye–Hückel screening length scales as $I^{-1/2}$, giving ${\kappa }^{-1}=0.96/\sqrt{6}\simeq 0.39\mathrm{nm}$. It falls by a factor $\sqrt{6}$, not by three. This is a formal screening estimate: at this ionic strength, trivalent-ion activity, finite ion size, hydrolysis and specific adsorption make the ideal dilute model quantitatively limited.