HelmholtzSmoluchowskiGesetz

The Helmholtz–Smoluchowski equation is a fundamental relation in electrokinetics that describes how a liquid near a charged surface moves under an applied electric field. It is used to characterize electroosmotic flow in capillaries and microchannels, as well as the electrophoretic mobility of colloidal particles in electrolyte solutions.

In its commonly cited form for a thin electric double layer, the fluid velocity u due to electroosmosis is u = - (ε ζ / η) E, where ε is the dielectric permittivity of the fluid (ε = εr ε0), η is the dynamic viscosity, ζ is the zeta potential of the surface, and E is the applied electric field. The corresponding mobility is μ_EO = u / E = - ε ζ / η. The negative sign reflects conventional directions chosen for E and flow. This relation holds when the double layer thickness is small compared to the characteristic dimension (κa >> 1) and under assumptions of Newtonian, incompressible, laminar flow with no slip at the boundary.

Origins and naming reflect contributions by Hermann von Helmholtz (electrostatics) and Marian Smoluchowski (hydrodynamics) to electrokinetic theory in the late 19th and early 20th centuries. The formula provides a practical link between measurable transport properties and interfacial charge characteristics, enabling estimation of zeta potentials from mobility data.

Applications include predicting and controlling electroosmotic flow in microfluidic devices, interpreting capillary electrophoresis measurements, and characterizing surface charge in colloidal suspensions. Limitations arise for thick double layers, non-Newtonian fluids, significant surface conduction, or complex surface chemistries where corrections are needed.