Electrokinetics of a particle attached to a fluid interface: Electrophoretic mobility and interfacial deformation

The electrophoretic mobility of a microparticle with a small $\zeta$ potential attached to a fluid interface is studied analytically in the limit of two fluids with vanishing viscosity and permittivity ratios under the assumption of a weak applied electric field. The analysis is based on perturbation expansions in the capillary number, the electric capillary number, and in the deviation of the contact angle on the particle surface from ${90}^{\circ }$. For the case of a very thin electric double layer (EDL), the Smoluchowski limit for the electrophoretic mobility is recovered. For arbitrary values of the EDL thickness, a spherical particle is considered. In that case the electrophoretic mobility is a function of the EDL thickness and the contact angle at the particle surface. Hydrophobic particles have a higher mobility than hydrophilic ones. The deformation of the fluid interface is shown to be separable into a dynamic and static part. The motion of the particle (dynamic part) does not contribute to the interfacial deformation in the case of a small capillary number, even for arbitrary EDL thickness. The static interfacial deformation due to electric stresses, known as electrodipping, is examined for spherical particles and shown to be independent of the first-order perturbation in the contact angle. The limits of validity of our theory and possible applications are discussed.

Figure 1

Schematic of a particle attached to a fluid interface. Without loss of generality, we assume the negatively charged particle to move with a velocity $U$ in the positive $z$ direction due to an constant applied electric field $E_{\infty }$. The contact angle of the fluid interface at the particle surface is denoted by $\mathrm{\Theta }$, and $a$ stands for a characteristic length scale. ${\mathrm{\Gamma }}_P, {\mathrm{\Gamma }}_I$, and ${\mathrm{\Gamma }}_{\infty }$, denote the particle, the fluid interface and the far field domain, i.e., a sphere with a radius of $r\to \infty$, respectively. The normal vectors corresponding to the three boundaries of the fluid domain are denoted by ${\mathit{n}}^I,{\mathit{n}}^P$, and ${\mathit{n}}^{\infty }$. The origin of the coordinate system lies in the plane of the undeformed fluid interface.

Figure 2

Schematic of the two equivalent systems considered when computing the electrophoretic mobility of a sphere attached to a flat fluid interface. The electrophoretic mobility of a sphere attached to a flat fluid interface is equal to a the mobility of two spherical caps fused at the three-phase contact line. The perturbation parameter $\beta$ represents the shift of the center of the sphere along the $x$ direction relative to the fluid interface. $a$ denotes the radius of the sphere.

Figure 3

Electrophoretic mobility of a spherical particle attached to a flat fluid interface as a function of the Debye parameter for different contact angles $\beta =cos\left(\mathrm{\Theta }\right)$.

Figure 4

Plot of the first-order interfacial deformation $\frac{\overline{h}\left(\overline{\rho }\right)}{\text{Ce}}=\overline{\mathfrak{h}}$ (in Ce) over the dimensionless distance $\overline{\rho }$ for different values of the dimensionless Debye length [see Eq. (48)]. Inset: Dipping depth [see Eq. (54)] of the particle as a function of the Debye length.

Figure 5

Comparison between the numerical results and the predictions of the theory. (a) Electrophoretic mobility as a function of the Debye parameter obtained with Eqs. (32) and (34) for different contact angles compared to numerical solutions. The lines indicate the analytical, the symbols the numerical results. (b) Electrophoretic mobility as a function of the contact angle for two different values of the Debye parameter. The lines indicate the analytical, the symbols the numerical results.

Figure 6

Computational domain used in the COMSOL Multiphysics simulations.