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

Abstract

The present article deals with the theoretical study on electrophoresis of hydrophobic and dielectric spherical fluid droplets possessing uniform surface charge density. Unlike the ideally polarizable liquid droplet bearing constant surface ζ-potential, the tangential component of the Maxwell stress is nonzero for dielectric fluid droplets with uniform surface charge density. We consider the continuity of the tangential component of total stress (sum of the hydrodynamic and Maxwell stresses) and jump in dielectric displacement along the droplet-to-electrolyte interface. The typical situation is considered here for which the interfacial tension of the fluid droplet is sufficiently high so that the droplet retains its spherical shape during its motion. The present theory can be applied to nanoemulsions, hydrophobic oil droplets, gas bubbles, droplets of immiscible liquid suspended in aqueous medium, etc. Based on weak field and low charge assumptions and neglecting the Marangoni effect, the resultant electrokinetic equations are solved using linear perturbation analysis to derive the closed form expression for electrophoretic mobility applicable for the entire range of Debye−Huckel parameter. We further deduced an alternate approximate expression for electrophoretic mobility without involving exponential integrals. Besides, we have derived analytical results for mobility pertaining to various limiting cases. The results are further illustrated to show the impact of pertinent parameters on the overall electrophoretic mobility.

pH-Tunable electrokinetic movement of droplets

“…We compare our results to theoretical descriptions of EP mobilityof hydrophobic spherical fluid droplets to further scrutinize the behavior of the droplets. 32 We define the positive direction for motion in the same direction as the electric field, meaning that if a positive velocity is reported the motion was from the anode (positive electrode) to the cathode (negative electrode). For all experiments it is assumed that surfactant monomers interact with the PDMS or droplets through their hydrophobic tails, so their head group is exposed to the bulk solution.…”

“…In order to further understand the observed droplet motion with changing pH, we compare our experimental results to previously published analytical descriptions of droplet motion under different conditions. 32,33 P. Gopmandal et al 32 developed an analytical expression for the EP mobility of hydrophobic and dielectric spherical fluid droplets. Their equation can be simplified as follow when κa ≫ 1:…”

“…In this equation, σ (C m −2 ) is the surface charge density of the droplet, η e ad η d (Pa s) are viscosity of the electrolyte and droplet, ε e and ε d (F m −1 ) are the permittivity of the electrolyte and droplet, a (m) is the droplet radius, κ (1 m −1 ) is the inverse of the Debye length and λ eff (m) is the effective slip length, which for super hydrophobic droplets far greater than droplet radius. 32…”

“…44,45 As mentioned, droplet EP is more complex than particle EP due to the specific characteristics of droplets such as mobile surface charges, non-rigidity as well as the slip velocity at the liquid-liquid interface 31 and it has been shown that the general equations for particle EP fail to describe droplet EP. 32,44 This implies that even though we measured surface charge of the droplets, we only used the measurements to infer the effective charge (positive or negative) and not the magnitude of the charge for the interpretation of our EK mobility. However, in combination with the theoretical predictions, we are able to describe the droplet movement as a function of pH and get good agreement between the expected and measured mobility.…”

“…Further details about these differences are given in our review paper on mechanistic studies of droplets' EP. 31 Recently, different analytical expressions for the EP mobility of droplets for different conditions such as hydrophobic and dielectric droplets with uniform surface charge density 32 and weakly charged liquid droplets with slip surface have been obtained. 33 These equations show that the EP mobility of droplets is proportionate to their surface charge and size, but is highly dependent on the system characteristics such as viscosity and permittivity of both fluids.…”