Fluid-sphere electrophoresis at low surface charge

We derive an analytical Debye–Hückel approximate solution of Hill’s fluid-sphere electrophoresis model, generalizing Booth’s theory with interfacial Damköhler ( Da ), Péclet ( Pe ) and Marangoni ( Ma ) numbers, among others. Booth’s theory holds generally for Da → ∞ , which corresponds to fast interfacial-exchange kinetics and/or vanishing interfacial-charge mobility—conditions that are difficult to realized in practice. For nano- and micro-scale air bubbles in water at pH = 7 , nano-bubbles with reduced radius κ a ≪ 1 are in a regime where the electrophoretic mobility is independent of Da . On the other hand, micro-bubbles with κ a ≫ 1 have a mobility that varies significantly with Da . Nano-bubbles have a higher interfacial potential than their micro-bubble counterparts. For spherical micro- and macro-air bubbles in water at pH = 7 , the surface potential varies with bubble size ( − ζ ≈ 7 –28 mV), notably smaller in magnitude than from Smoluchowski’s model (for rigid spheres) or Booth’s model (for fluid spheres). Our theory, which provides the first compelling interpretation of Alty’s (1924) air-bubble mobilities in water, suggests that a large body of ζ -potentials may need to be reevaluated in the context of finite interfacial charge mobility and exchange kinetics.