HelmholtzSmoluchowski

Helmholtz–Smoluchowski is a fundamental relation in electrokinetics that describes the electrophoretic mobility of a charged particle moving through a Newtonian fluid under an applied electric field. The result combines ideas about the electric double layer at a charged interface with low Reynolds number hydrodynamics and is widely used to relate measurable particle motion to interfacial electrostatics.

The mobility μ depends on the thickness of the electric double layer relative to the particle size. In the thin double layer limit (κa ≫ 1, where κ is the inverse Debye length and a is the particle radius), the electrophoretic velocity v_E = μ E with μ = ε ζ / η. Here ε is the permittivity of the suspending medium, η its dynamic viscosity, and ζ is the zeta potential. In the opposite, thick double layer limit (κa ≪ 1), μ becomes (2 ε ζ) / (3 η). In general, μ can be written as μ = (ε ζ / η) f(κa), where f(κa) approaches 1 for large κa and 2/3 for small κa.

Applications of the Helmholtz–Smoluchowski relation include interpreting electrophoresis and electro-osmosis experiments, estimating zeta potentials from mobility measurements, and guiding analyses in colloid science and microfluidics. The expression assumes a rigid, nonconducting particle with a constant zeta potential, a Newtonian fluid at low Reynolds number, and a steady electric field with a diffuse electric double layer.

Historically, the concept reflects contributions from Hermann von Helmholtz and Marian Smoluchowski, who developed the theory of the electric double layer and its fluid-dynamic consequences, culminating in the named equation that bears their joint legacy.