Electric double layer

An electric double layer consists of an interfacial electric charge and the compensating distribution of mobile ions in the adjacent chemical solution. Counterions are enriched and coions depleted near a simple charged interface. A diffuse layer is described approximately by the Poisson-Boltzmann equation; a compact specifically adsorbed layer requires additional surface chemistry. Overlap of diffuse layers can produce a screened repulsion between like-charged colloids.

A surface can change its surface charge density as its local electric potential and solution composition change, for example through reversible ionization or ion adsorption. This couples the surface reaction equilibrium to the surrounding electric double layer. Fixed-charge and fixed-potential models are limiting boundary conditions rather than interchangeable descriptions. Salt or pH changes can therefore alter both electrostatic screening and the repulsion amplitude.

A dimensionless ratio comparing surface conduction with bulk conduction over a characteristic length. With pore radius r and surface conductance $K_s$, one convention is $\mathrm{Du}=K_s/(\sigma r)$; the cylindrical surface-to-bulk conductance ratio is twice this value.

A model separating a compact interfacial layer from a diffuse ion distribution. Specified adsorption and boundary conditions determine the potential at the outer compact-layer boundary; a diffuse-layer equation alone does not determine specific adsorption.

Overcompensation of an initial interfacial charge by adsorbed species or correlated ions, so the charge or potential inferred outside the compact interfacial region has the opposite sign. Screening of a fixed boundary charge alone is not the same mechanism.

The electrokinetic potential at a hydrodynamic slipping plane relative to bulk solution. It need not equal the potential at the physical solid surface.

The spatially extended ionic atmosphere screening a charged surface beyond the compact Stern layer.

An ion with the same charge sign as the surface or fixed charge being screened. In a simple diffuse layer it is depleted near a similarly charged surface.

The compact interfacial region associated with finite ion closest approach and specifically adsorbed charge. A diffuse electrolyte layer lies beyond it.

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.

A coion has electric charge of the same sign as the reference interface or particle. Its number density is depleted near that interface in the simplest Poisson-Boltzmann equation description. A counterion has the opposite sign. These roles depend on the reference charge: near a positively charged sphere in magnesium chloride, magnesium ions are coions and chloride ions are counterions.