Induced-charge electrophoresis of ideally polarizable particle pairs

We study the planar two-dimensional relative induced-charge electrophoretic (ICEP) motion of a pair of identical, ideally polarizable circular cylindrical microparticles carrying no net charge and freely suspended in an unbounded electrolyte solution under a uniform steady (DC) external electric field acting in an arbitrary direction relative to the instantaneous orientation of their line-of-centers (LOC). Within the framework of the thin electric-double-layer (EDL) limit and sufficiently weak fields description of particle paths is obtained via integration of the quasisteady kinematic equations of motion based on the instantaneous geometric configuration. Owing to the inherent nonlinearity of the ICEP mechanism, interaction of the effects of external-field components parallel and perpendicular, respectively, to the LOC results in its rotation which, in turn, determines that particles undergo transient pairing eventually moving apart in the general direction perpendicular to the external field. Dielectrophoresis is demonstrated to only have a secondary effect on the resulting motion.

Figure 1

Schematic definition of geometric configuration and cylindrical bipolar coordinates ($\alpha$, β). Marked by the black circles are the pair of particles of radii $a^{\prime }$ centered at $\left(\pm b^{\prime },0\right)$ and subject to the uniform external field ${\mathit{E}}_{\mathbf{0}}^{\prime }$. The two families of nonintersecting circles correspond to curves of constant $\alpha$ (with $\pm {\alpha }_0$ pertaining to the pair of particles). The red arcs leaning on the common chord extending between $\left(\pm c^{\prime },0\right)$ describe the family of constant-β curves.

Figure 2

Electroosmotic streamline patterns around a pair of identical conducting cylinders whose centers are four radii apart. The red arrows mark the directions of the external field $\theta =0, \pi /2, \pi /6,$ and $\pi /3$ relative the line of centers.

Figure 3

Schematic representation of the translational and angular velocities of the cylinders associated with ${\mathrm{\Psi }}_1$ (a) and ${\mathrm{\Psi }}_2$ (b). Similarly to Ref. [20], in part (b) the transverse velocities of the cylinders $\mathit{\alpha }=\pm {\mathit{\alpha }}_0$ are depicted as $\pm {\widehat{\mathit{e}}}_{y}V, (V>0)$, respectively, with their common angular velocity $\mathit{\Omega }={\widehat{\mathit{e}}}_{\mathit{z}}\mathrm{\Omega }, (\mathrm{\Omega }<0),$ and ${\widehat{\mathit{e}}}_{\mathit{z}}$ a unit vector in the positive z direction [cf. Eq. (4.9)].

Figure 4

The surface $\mathit{S}$ bounding the fluid domain.

Figure 5

Variation with the distance r (scaled by particle radius) of the coefficients $\overline{A},\overline{C},$ and $\overline{B}$ and their corresponding asymptotes (solid and dashed lines, respectively).

Figure 6

The cylinder– pair relative motion: Paths (solid lines) and local speed (color code) of particle-center motion relative to (stationary) midpoint of line-of-centers under a uniform DC field parallel to $X$ axis. Coordinates are scaled by particle radius (hence white quarter unit circle inaccessible to particle center). Light-green curves present pairs of exact- and (moderately large separation) approximate-trajectories corresponding to common values of $r_{\min }$ ($=2,3$, and 4.4, respectively).

Figure 7

Variation with $r$ of $R_{\overline{A}},R_{\overline{C}},$ and $R_{\overline{B}}.$

Figure 8

The trajectory pattern of Fig. 6 with the curves marked in red presenting pairs of ICEP and corresponding dipolophoretic (DIP, combined ICEP and DEP) paths (solid and dashed lines, respectively), each pair emanating from the same point.