Electrophoresis of a highly charged fluid droplet in dilute electrolyte solutions: Analytical Hückel‐type solution

Abstract

An analytical formula is presented here for the electrophoresis of a dielectric or perfectly conducting fluid droplet with arbitrary surface potentials suspended in a very dilute electrolyte solution. In other words, when the Debye length (κ −1 ) is very large, or κa≪1, where κ is the electrolyte strength and a stands for the droplet radius. This formula can be regarded as an extension of the famous Hückel solution valid for weakly charged rigid particles to arbitrarily charged fluid droplets. The formula reduces successfully to the ones obtained by Booth for a dielectric droplet, and Ohshima for a perfectly conducting droplet, both under Debye-Hückel approximation valid for weakly charged droplets. Moreover, the formula is valid for a gas bubble and a rigid solid particle as well. Classic results obtained by Hückel for a rigid particle are reproduced as well. We found that for a dielectric droplet, the more viscous the droplet is, the faster it moves regardless of its surface potential, contrary to the intuition based on the purely hydrodynamic consideration. For a perfectly conducting liquid droplet, on the other hand, the situation is reversed: The less viscous the droplet is, the faster it moves. The presence or absence of the spinning electric driving force tangent to the droplet surface is found to be responsible for it. As a result, an axisymmetric exterior vortex flow surrounding the droplet is always present for a dielectric liquid droplet, and never there for a conducting liquid droplet.

Electrophoresis of Dielectric and Hydrophobic Spherical Fluid Droplets Possessing Uniform Surface Charge Density

“…Next we have derived several closed form expressions of electrophoretic mobility under various limiting situations. Under the Hückel limit (i.e., for κa → 0), the mobility expression () reduces to the following simplified form If we further consider the droplet is perfectly dielectric, the above expression () reduces to following simplified form Further considering λ = 0, () reduces to The above expression is applicable for perfectly dielectric hydrophilic droplet under the Hückel limit, and () correctly merges with the results obtained by Tsai et al for the droplet with equivalent surface potential ζ = σa /ε e . Next we consider the liquid droplet as perfectly conducting.…”

“…Further considering λ = 0, (45) reduces to ( ) The above expression is applicable for perfectly dielectric hydrophilic droplet under the Huckel limit, and ( 46) correctly merges with the results obtained by Tsai et al 30 for the droplet with equivalent surface potential ζ = σa/ε e . Next we consider the liquid droplet as perfectly conducting.…”

“…For a positively charged droplet, the mobility enhances with the rise in viscosity of the droplet. The underlying mechanism is already discussed in Wu et al 29 and Tsai et al 30 for the dielectric droplet under Huckel limit (κa ≪ 1). We observed that such a specific pattern in mobility with respect to viscosity of droplet remains unaltered for the entire range of Debye− Huckel parameter κa.…”

“…A similar expression is derived by Ohshima et al 20 and Tsai et al, 30 applicable for the perfectly conducting hydrophilic droplet under the Huckel limit with equivalent surface potential ζ = σa/ε e .…”

“…Besides, Wu et al 29 studied numerically the electrophoresis of highly charged dielectric fluid droplet considering dielectric constant of the droplet is much smaller than that of the aqueous solution. In a recent study, Tsai et al 30 provided a closed form expression for electrophoretic mobility of perfectly dielectric and perfectly conducting fluid droplets under the Huckel limit (i.e., κa → 0). They have observed that the mobility of the dielectric fluid droplet enhances with the rise in its viscosity.…”