^{1,a)}

### Abstract

In the classical analysis of electrophoresis, particle motion is a consequence of the interfacial fluid slip that arises inside the ionic charge cloud (or Debye screening layer) surrounding the particle surface when an external field is applied. Under the assumptions of thin Debye layers, weak applied fields, and zero polarizability, it can be shown that the electrophoretic velocity of a collection of particles with identical zeta potential is the same as that of an isolated particle, unchanged by interactions [L. D. Reed and F. A. Morrison, “Hydrodynamic interaction in electrophoresis,” J. Colloid Interface Sci.54, 117 (1976)]. When some of these assumptions are relaxed, nonlinear effects may also arise and result in relative motions. First, the perturbation of the external field around the particles creates field gradients, which may result in nonzero dielectrophoretic forces due to Maxwell stresses in the fluid. In addition, if the particles are able to polarize, they can acquire a nonuniform surface charge, and the action of the field on the dipolar charge clouds surrounding them drives disturbance flows in the fluid, causing relative motions by induced-charge electrophoresis. These two nonlinear effects are analyzed in detail in the prototypical case of two equal-sized ideally polarizable spheres carrying no net charge, using accurate boundary-element simulations, along with asymptotic calculations by the method of twin multipole expansions and the method of reflections. It is found that both types of interactions result in significant relative motions and can be either attractive or repulsive depending on the configuration of the spheres.

I. INTRODUCTION

II. PROBLEM FORMULATION

III. BOUNDARY-ELEMENT CALCULATIONS

A. Electric problem

B. Flow problem

IV. TRAJECTORIES AND FAR-FIELD FLOW DISTURBANCE

V. CONCLUDING REMARKS

### Key Topics

- Dielectrophoresis
- 47.0
- Electrophoresis
- 21.0
- Surface charge
- 19.0
- Electric fields
- 17.0
- Kinematics
- 11.0

## Figures

Problem geometry. We consider two equal-sized spheres of radius separated by a vector and placed in an external electric field .

Problem geometry. We consider two equal-sized spheres of radius separated by a vector and placed in an external electric field .

Typical unstructured mesh used in the boundary-element calculations of Sec. III. The mesh was obtained using the algorithm of Loewenberg and Hinch (Ref. 33) and is composed of 1280 six-point curved triangular elements, corresponding to 3842 quadrature points.

Typical unstructured mesh used in the boundary-element calculations of Sec. III. The mesh was obtained using the algorithm of Loewenberg and Hinch (Ref. 33) and is composed of 1280 six-point curved triangular elements, corresponding to 3842 quadrature points.

Dimensionless coefficients (a) , (b) , and (c) in the general expression (18) for the DEP force as functions of . The plots show results from the boundary-element calculations of Sec. III A, from the method of twin multipole expansions (Refs. 30 and 36) (cf. the Appendix) in which 50 terms were retained in the expansions, and from the method of reflections (Refs. 28 and 29) [Eqs. (26)–(28)].

Dimensionless coefficients (a) , (b) , and (c) in the general expression (18) for the DEP force as functions of . The plots show results from the boundary-element calculations of Sec. III A, from the method of twin multipole expansions (Refs. 30 and 36) (cf. the Appendix) in which 50 terms were retained in the expansions, and from the method of reflections (Refs. 28 and 29) [Eqs. (26)–(28)].

Dimensionless coefficients (a) , (b) , (c) , and (d) in the general expressions (19) and (20) for the DEP linear and angular velocities and as functions of . The plots show results from the boundary-element calculations of Sec. III B, from the method of twin multipole expansions^{31} (cf. the Appendix), in which 20 terms were retained in the expansions, and from the method of reflections [Refs. 28 and 29] [Eqs. (36)–(39)].

Dimensionless coefficients (a) , (b) , (c) , and (d) in the general expressions (19) and (20) for the DEP linear and angular velocities and as functions of . The plots show results from the boundary-element calculations of Sec. III B, from the method of twin multipole expansions^{31} (cf. the Appendix), in which 20 terms were retained in the expansions, and from the method of reflections [Refs. 28 and 29] [Eqs. (36)–(39)].

Dimensionless coefficients (a) , (b) , (c) , and (d) in the general expressions (19) and (20) for the ICEP linear and angular velocities and as functions of . The plots show results from the boundary-element calculations of Sec. III B, from the method of twin multipole expansions (Ref. 31) (cf. the Appendix), in which 20 terms were retained in the expansions, and from the method of reflections (Refs. 28 and 29) [Eqs. (40)–(43)].

Dimensionless coefficients (a) , (b) , (c) , and (d) in the general expressions (19) and (20) for the ICEP linear and angular velocities and as functions of . The plots show results from the boundary-element calculations of Sec. III B, from the method of twin multipole expansions (Ref. 31) (cf. the Appendix), in which 20 terms were retained in the expansions, and from the method of reflections (Refs. 28 and 29) [Eqs. (40)–(43)].

Dimensionless coefficients (a) , (b) , (c) , and (d) in the general expressions (19) and (20) for the total linear and angular velocities and (including both DEP and ICEP) as functions of . The plots show results from the boundary-element calculations of Sec. III B, from the method of twin multipole expansions (Ref. 31) (cf. the Appendix), in which 20 terms were retained in the expansions, and from the method of reflections (Refs. 28 and 29) [Eqs. (44)–(47)].

Dimensionless coefficients (a) , (b) , (c) , and (d) in the general expressions (19) and (20) for the total linear and angular velocities and (including both DEP and ICEP) as functions of . The plots show results from the boundary-element calculations of Sec. III B, from the method of twin multipole expansions (Ref. 31) (cf. the Appendix), in which 20 terms were retained in the expansions, and from the method of reflections (Refs. 28 and 29) [Eqs. (44)–(47)].

Radial component of the relative velocity between the two spheres scaled by the velocity scale as a function of the angle made between the external field and the line of centers of the two spheres . The radial velocity was calculated using Eq. (48) for (or, equivalently, ). The plot shows the velocities arising from DEP and ICEP, as well as the total velocity when both effects are present.

Radial component of the relative velocity between the two spheres scaled by the velocity scale as a function of the angle made between the external field and the line of centers of the two spheres . The radial velocity was calculated using Eq. (48) for (or, equivalently, ). The plot shows the velocities arising from DEP and ICEP, as well as the total velocity when both effects are present.

Typical trajectories of two spheres undergoing DEP and ICEP in an electric field . The two spheres are attracted in the direction of the field, pair up, and then separate in the transverse direction. The arrows indicate the direction of motion.

Typical trajectories of two spheres undergoing DEP and ICEP in an electric field . The two spheres are attracted in the direction of the field, pair up, and then separate in the transverse direction. The arrows indicate the direction of motion.

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