Electro-osmosis

Electro-osmosis is fluid motion driven by an applied electric field acting on mobile electric charge near a surface. A charged interface attracts a diffuse population of counterions in an electric double layer. A field component tangent to the interface exerts a force density ${\rho }_e\mathbf{E}$ on that mobile charge; momentum transfer to the surrounding solvent drives flow opposed by viscous stress. In a capillary electrophoresis measurement, this flow advects the sample as well as its electrical drift relative to the fluid. Locally, the measured axial velocity is $v_{\mathrm{observed}}=u_{\mathrm{eo}}+{\mu }_{\mathrm{ep}}E$, where $u_{\mathrm{eo}}$ is the electroosmotic fluid velocity and ${\mu }_{\mathrm{ep}}$ is the electrophoretic mobility. ^[ref]

A homogeneous charged channel can add a constant electro-osmotic drift to the ideal electrophoretic velocity at fixed field. It changes the coefficient of a linear length-versus-entry-time law without generating a new length exponent. Charge-dependent adsorption, heterogeneous trapping or cancellation of drift followed by diffusion lies outside that simple claim; wall charge cannot be excluded as an indirect cause in every real experiment.

With $\psi (0)=\zeta$, $\psi \prime (d)=0$ and no slip at either wall, linear screening gives $\psi /\zeta =\cosh [\kappa (d-x)]/\cosh (\kappa d)$. The velocity is $v/U=1-(1-\mathrm{sech}\kappa d)x/d-\cosh [\kappa (d-x)]/\cosh (\kappa d)$. This electrically neutral-wall boundary assumes no external normal field; it differs from a grounded wall.

Zero wall charge constrains the jump in normal electric displacement, while fixing a wall potential is a separate electrical condition. Taking the external normal field to vanish gives a zero normal derivative at an ideal uncharged wall. A grounded wall may carry induced charge even though its imposed potential is zero.

Differentiating the screened-channel velocity gives $x_{\max }=d-{\kappa }^{-1}\mathrm{arsinh}[(\cosh \kappa d-1)/(\kappa d)]$. For $\kappa d\gg 1$, $x_{\max }\simeq \log (\kappa d)/\kappa$. The outer flow declines almost linearly to the passive no-slip wall rather than forming a symmetric plug.

In a half-space with no pressure gradient, Stokes balance and Poisson electrostatics give $\eta v\prime \prime =\epsilon E\psi \prime \prime$. No slip at the wall and zero bulk shear imply $v=\epsilon E(\psi -\zeta )/\eta$. The bulk velocity is $U=-\epsilon \zeta E/\eta$; it is not the fluid speed at the no-slip wall.