Travelling-Wave Electrophoresis, Electro-Hydrodynamics, Electro-Rotation, and Symmetry- Breaking of a Polarizable Dimer in Non-Uniform Fields

Abstract

A theoretical framework is presented for calculating the polarization, electro-rotation, travelling-wave dielectrophoresis, electro-hydrodynamics and induced-charge electroosmotic flow fields around a freely suspended conducting dimer (two touching spheres) exposed to non-uniform direct current (DC) or alternating current (AC) electric fields. The analysis is based on employing the classical (linearized) Poisson-Nernst-Planck (PNP) formulation under the standard linearized 'weak-field' assumption and using the tangent-sphere coordinate system. Explicit expressions are first derived for the axisymmetric AC electric potential governed by the Robin (mixed) boundary condition applied on the dimer surface depending on the resistance-capacitance circuit (RC) forcing frequency. Dimer electro-rotation due to two orthogonal (out-of-phase) uniform AC fields and the corresponding mobility problem of a polarizable dimer exposed to a travelling-wave electric excitation are also analyzed. We present an explicit solution for the non-linear induced-charge electroosmotic (ICEO) flow problem of a free polarized dimer in terms of the corresponding Stokes stream function determined by the Helmholtz-Smoluchowski velocity slip. Next, we demonstrate how the same framework can be used to obtain an exact solution for the electro-hydrodynamic (EHD) problem of a polarizable sphere lying next to a conducting planar electrode. Finally, we present a new solution for the induced-charge mobility of a Janus dimer composed of two fused spherical colloids, one perfectly conducting and one dielectrically coated. So far, most of the available electrokinetic theoretical studies involving polarizable nano/micro shapes dealt with convex configurations (e.g., spheres, spheroids, ellipsoids) and as such the newly obtained electrostatic AC solution for a dimer provides a useful extension for similar concave colloids and engineered particles.

Keywords: Janus mobility; dimer (touching spheres); electro-hydrodynamics; electro-rotation; electrophoresis; induced-charge electroosmosis; polarization; tangent-sphere coordinates; travelling wave.

Figure 1

Schematic description of the problem of a dimer that is composed of two geometrically identical spheres in free space as in Section 2, Section 3, Section 4 and Section 5 and Section 7. The dimer is subjected to: a uniform AC electric field acting in the z direction (Section 2, Section 3, Section 4 and Section 5 and Section 7), a uniform AC electric field acting in the x direction (Section 3), and a non-homogeneous axisymmetric travelling wave propagating along the z direction (Section 4). The two spheres are identically conductive in Section 2, Section 3, Section 4 and Section 5, and in Section 7 the lower sphere is coated by a thin dielectric layer.

Figure 2

The solution of Equation (8) of Section 2 for (a) $\Omega =0$, (b) $\Omega =0.5$, (c) $\Omega =1$, and (d) $\Omega =10$, and where the exact solution of $\Omega =0$ is given by Equation (9). The asymptotic solution is of Equation (10).

Figure 2

The solution of Equation (8) of Section 2 for (a) $\Omega =0$, (b) $\Omega =0.5$, (c) $\Omega =1$, and (d) $\Omega =10$, and where the exact solution of $\Omega =0$ is given by Equation (9). The asymptotic solution is of Equation (10).

Figure 3

Schematic description of the problem in Section 6 of a spherical particle next to a wall (z = 0), which is subjected to a uniform DC electric field acting in the z direction.

Figure 4

The (a) contours of the Stokes stream function and (b) velocity vectors around the spherical particle placed next to a wall at z = 0 of Section 6, and which is subjected to a uniform DC electric field acting in the z direction. The velocity-vector field modulus was adjusted for better viewing.