Electrokinetics of metal cylinders

We study theoretically the rotation induced on an uncharged metal nanocylinder immersed in an electrolyte by AC electric fields. We consider the rotation of the cylinder when subjected to a rotating electric field (electrorotation) and the orientation of the cylinder in an AC field with constant direction (electro-orientation). The cylinder rotation is due to two mechanisms: the electric field interaction with the induced dipole on the particle and the hydrodynamic stress on the particle originated by the induced-charge electro-osmotic (ICEO) flow around the particle. The cylinder rotation induced by the ICEO mechanism can be calculated by using the Lorentz reciprocal theorem, while the rotation due to the induced dipole is calculated from the cylinder polarizability. We employ 3D numerical computations using finite elements for the general case as well as analytical methods for slender cylinders. Both calculations use the thin-double-layer approximation. We compare the results for slender cylinders of both methods showing good agreement. The electro-orientation (EOr) due to dipole torque aligns the axis of slender cylinders with the applied field, but aligns the axis of short cylinders perpendicularly to the field. The EOr due to ICEO torque always aligns the axis of cylinders with the field. The rotation induced by ICEO torque tends to disappear for frequencies of the applied field much greater than the characteristic frequency for charging the double-layer capacitance of the metal-electrolyte interface.

Figure 1

Metal cylinder subjected to an ac electric field.

Figure 2

Comparison between analytical and numerical polarizabilities for $\beta =0.04$. Polarizabilities $A$ and $B$ as a function of nondimensional frequency $\tilde{\omega }=\omega C_{\text{DL}}a/\sigma$.

Figure 3

Imaginary part of polarizabilities as a function of nondimensional frequency $\tilde{\omega }=\omega C_{\text{DL}}a/\sigma$ for slenderness $\beta =0.5$ (a) and $\beta =0.04$ (b). Also plotted is $\mathrm{Im}[A+B]$ proportional to the angular velocity of rotation in ROT.

Figure 4

Real part of polarizabilities as a function of nondimensional frequency $\tilde{\omega }=\omega C_{\text{DL}}a/\sigma$ for slenderness $\beta =0.5$ (a) and $\beta =0.04$ (b). Also plotted is $\mathrm{Re}[A-B]$ proportional to the angular velocity of rotation in EOr.

Figure 5

EOr and ROT angular velocities due to dipole torque for a cylinder with $\beta =1$ and a cube as functions of nondimensional frequency $\tilde{\omega }$.

Figure 6

Angular velocity due to ICEO flow as a function of frequency for cylinders with $b/a=0.04,0.5,1.0$ and a cube.

Figure 7

Angular velocities due to dipole torque, ${\tilde{\mathrm{\Omega }}}_{\mathrm{DEP}}$, and to ICEO around the nanowire, ${\tilde{\mathrm{\Omega }}}_{\mathrm{ICEP}}$, as a function of frequency for $\beta =0.04$.

Figure 8

Real part of polarizabilities $A$ and $B$ as a function of frequency for $\beta =4.0$. Also plotted is $\mathrm{Re}[A-B]$, which is proportional to the EOr angular velocity.

Figure 9

Imaginary part of polarizabilities $A$ and $B$ as a function of frequency for $\beta =4.0$. Also plotted is $\mathrm{Im}[A+B]$, which is proportional to the ROT angular velocity.

Figure 10

Angular velocities as a function of frequency in EOr and ROT due to dipole torque and to ICEO flow for a cylinder with $\beta =4.0$.

Figure 11

Reduced polarizability ${\alpha }_{z}3ln(a/b)/4\pi \varepsilon a^3$ versus reduced frequency $\omega ln(a/b)C_{\text{DL}}b/\sigma$ for $b/a=0.0025$. The slender-body approximation is shown for the first and second order in $1/ln(a/b)$.

Figure 12

Boundary conditions on planes of symmetry for electrical and hydrodynamic problems.