Projected velocity and streamlines in the laboratory frame for same-sized particles under stationery condition (force balance with no acceleration). The white bars in the figures denote the length scale of 750 nm. (a) P ₂ projection of the electrophoretic velocity component along the electric field direction normalized by electrophoresis velocity v ∞ = μ E E shows very good far-field 1/r ³ behavior of the simulated velocity field, indicated by the red dashed line. Yellow dashed line shows the position of particle surface. Blue line shows the P ₂ projection of the Smoluchowski potential flow velocity field expressed by Eq. (11b). It is clear that the projected 1/r ³ behavior stops at a small distance away from the solid surface. The peak position of the project result is delineated by the black dashed line, corresponding to the white dashed curve in panel (b). (b) Streamlines for the negatively charged PS sphere with a = 0.75 μ m , (with λ _D=96.1 nm, κ a=7.8, σ = 7000 e / μ m 2 ) plotted in the lab frame. Here the colors indicate the magnitude of the velocity, the interface between the inner and outer flow fields (the reference surface) is denoted by white dashed curve. (c) Streamlines for a particle of the same radius as (a), acted on by a constant external body force to moves at same velocity as μ E E ∞ . This represents the Stokes flow field in the lab frame. The contrast with the electrophoretic flow field shown in panel (b) is clear. (d) Streamline for the flow field obtained from a charged particle under the same electric field strength as in panel (b) but fixed by a reverse external (non-electrical) force acting on the particle. This flow field can also be obtained approximately by the superposition of the electrophoretic flow field in panel (b) and the Stokes flow field in panel (c) with a reversed velocity. It should be noted that the external (non-electrical) force required to immobilize the particle is not the same as the electrical force, thereby explaining the difference in the drag coefficients. The difference in the two forces accounts for the non-zero flow field close to the particle.