Induced charge electrophoresis of a conducting cylinder in a nonconducting cylindrical pore and its micromotoring application

Induced charge electrophoresis of a conducting cylinder suspended in a nonconducting cylindrical pore is theoretically analyzed and a micromotor is proposed that utilizes the cylinder rotation. The cylinder velocities are analytically obtained in the Dirichlet and the Neumann boundary conditions of the electric field on the cylindrical pore. The results show that the cylinder not only translates but also rotates when it is eccentric with respect to the cylindrical pore. The influences of a number of parameters on the cylinder velocities are characterized in detail. The cylinder trajectories show that the cylinder approaches and becomes stationary at certain positions within the cylindrical pore. The proposed micromotor is capable of working under a heavy load with a high rotational velocity when the eccentricity is large and the applied electric field is strong.

Figure 1

Schematic illustration of the conducting cylinder suspended in the cylindrical pore. $(\pm a,0)$ are the two foci of the bipolar coordinates; ${\mathbf{e}}_{\tau }$ and ${\mathbf{e}}_{\sigma }$ are the unit vectors in the bipolar coordinates that normal and tangent to the cylinder surface, respectively; $\tau ={\tau }_i$ and ${\tau }_o$ indicate the surfaces of the cylinder and the cylindrical pore, respectively; $(acoth{\tau }_i,0)$ and $(acoth{\tau }_o,0)$ are the centers of the cylinder and the cylindrical pore, respectively, in the Cartesian coordinates; $R_i=a/sinh{\tau }_i$ and $R_o=a/sinh{\tau }_o$ are the radii of the cylinder and the cylindrical pore, respectively; and $\tilde{\varepsilon }$ is the distance between the centers of the cylinder and the cylindrical pore. The uniform electric field $-(E_{0}sin{\theta }_0{\mathbf{e}}_x+E_{0}cos{\theta }_0{\mathbf{e}}_y)$ is imposed, where the electric field phase angle ${\theta }_0$ defines the direction of the electric field.

Figure 2

Variation of the cylinder velocity $U_x/U_i$ with the radius ratio $R_r$ at different eccentricities $\varepsilon$. The ES component is 5 times amplified for a better observation.

Figure 3

Variation of the cylinder velocity $U_y/U_i$ with the radius ratio $R_r$ at different eccentricities $\varepsilon$.

Figure 4

Variation of the cylinder velocity $\mathrm{\Omega }{R}_i/U_i$ with the radius ratio $R_r$ at different eccentricities $\varepsilon$.

Figure 5

Cylinder velocity maps. $\tilde{\zeta }=0$, 0.2, and 1 in Figs. (a1),(b1), (a2),(b2), and (a3),(b3), respectively. Radius ratio $R_r=0.1$ and electric field phase angle ${\theta }_0=0$. The vector field indicates the cylinder translational velocity $\mathbf{U}=U_x{\mathbf{e}}_x+U_y{\mathbf{e}}_y$. The contour plot shows the cylinder rotational velocity $\mathrm{\Omega }$, where the positive and negative values indicate the counterclockwise and clockwise directions, respectively.

Figure 6

Variation of the rotational velocity of the micromotor with the load $M_l$ obtained from (a) the Dirichlet condition and (b) the Neumann condition, and (c) schematic diagram of the micromotor. Here the radius ratio $R_r=0.5$, the eccentricity $\varepsilon =0.5$, the electric field phase angle ${\theta }_0=7\pi /4$, the dimensionless $\zeta$ potential of the cylindrical pore $\tilde{\zeta }=0$, the cylinder radius $R_i=10\mu \mathrm{m}$, and the electric field strength $E_0=10\mathrm{kV}{\mathrm{m}}^{-1}$.