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The spherical-particle polarization-contrast factor $\tilde{K}=({\widetilde{ϵ}}_p-{\widetilde{ϵ}}_m)/({\widetilde{ϵ}}_p+2{\widetilde{ϵ}}_m)$. Its real part controls ordinary time-averaged dielectrophoresis in the quasistatic dipole approximation.

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.

Positive dielectrophoresis acts up the gradient of the squared RMS field, towards stronger-field regions; negative dielectrophoresis acts down that gradient, towards weaker-field regions. The sign is the sign of $\Re \tilde{K}(\omega )$.

The force is not generally tangent to an electric field line. Field lines show the direction of the field itself, whereas $\nabla |{\mathbf{E}}_{\mathrm{RMS}}{|}^2$ describes how its strength changes in space. A spatially uniform field gives zero dipole-order dielectrophoretic force even when it is large. Reversing the field each half-cycle does not reverse this time-averaged force, because the relevant intensity-like quantity is quadratic.

Different materials have different frequency-dependent polarization and conduction. For an $e^{i\omega t}$ convention, a useful relative complex permittivity is ${\widetilde{ϵ}}_r={ϵ}_r-i\sigma /({ϵ}_0\omega )$. Substituting the particle and medium responses into $\tilde{K}$ determines whether each population undergoes positive or negative dielectrophoresis and with what magnitude.

Choose a frequency at which one population has $\Re \tilde{K}>0$ and another has $\Re \tilde{K}<0$. A nonuniform electrode field then sends them towards different regions, allowing collection in separate channels or trapping one while the other passes. Alternatively, different positive-force magnitudes can produce different deflections in a flowing suspension. A crossover where $\Re \tilde{K}=0$ can selectively suppress the force on one material.

The useful frequency depends on the surrounding medium as well as the particle. An unchanging, real-permittivity model would not by itself give frequency-selective separation; dielectric dispersion or conductivity contrast is needed. Particle size affects force magnitude, and heating, electro-osmotic flow and surface polarization can complicate a real separation.