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Here $r\gg {\lambda }_D$, so the thin-double-layer result givesand therefore $v=3.13\times 10^{-5}$ m s${}^{-1}$, along the field for the positive zeta potential.

In the opposite limit $r\ll {\lambda }_D$, the Hückel electrophoretic mobility is ${\mu }_e=2ϵ\zeta /(3\eta )$, giving $2.08\times 10^{-8}$ m${}^2$ V${}^{-1}$ s${}^{-1}$, two-thirds of the thin-layer value at the same $\zeta$. Both asymptotic expressions are independent of radius at fixed zeta potential; the crossover depends on $r/{\lambda }_D$. These classical approximations neglect double-layer relaxation and strong-potential corrections.

In electrophoresis, an applied electric field drives a charged particle and its associated diffuse ionic atmosphere relative to the liquid. Electric forcing is balanced by viscous stresses, giving a steady velocity $\mathbf{v}={\mu }_e\mathbf{E}$ in the linear regime. The relevant electrokinetic potential is the zeta potential at the slipping plane, rather than necessarily the solid surface potential.

For a thin electric double layer, the surrounding layer develops an electro-osmotic flow and the Smoluchowski electrophoretic mobility is ${\mu }_e=ϵ\zeta /\eta$. The sign of $\zeta$ fixes the direction. This transport is different from dielectrophoresis, which can move a neutral polarizable particle in a nonuniform field.

A charged colloid in a polar electrolyte is surrounded by an electric double layer. For a negatively charged surface, counterions are positive and co-ions negative; their distributions reverse for a positive surface. A compact Stern layer contains specifically adsorbed or closely approaching ions, followed by a diffuse layer where electrostatic attraction competes with thermal motion. Mobile charge integrated over the surroundings balances the particle charge in an overall neutral system.

For a mean-field dilute electrolyte, $n_i(r)=n_{i,\infty }\exp [-z_{i}e\psi (r)/(k_{B}T)]$ with Poisson's equation relates charge density and potential. At weak potential, the diffuse potential outside a sphere decays approximately as $\psi (r)\propto e^{-\kappa (r-a)}/r$. The inverse screening length obeys ${\kappa }^2=e^2{\sum }_i{z}_i^2{n}_{i,\infty }/(ϵk_{B}T)$: added salt shortens the Debye length. The zeta potential is the potential at the hydrodynamic slip plane, not necessarily the surface potential. The simple model neglects ion size, correlations and chemical charge regulation; a polar solvent with almost no dissociated ions need not provide a short screening length.

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.