Size effect in Quincke rotation: a numerical study

This paper deals with the Quincke rotation of small insulating particles. This dc electrorotation of insulating objects immersed in a slightly conducting liquid is usually explained by looking at the action of the free charges present in the liquid. Under the effect of the dc electric field, the charges accumulate at the surface of the insulating particle which, in turn, acquires a dipole moment in the direction opposite to that of the field and begins to rotate in order to flip its dipole moment. In the classical Quincke model, the charge distribution around the rotor is supposed to be purely superficial. A consequence of this assumption is that the angular velocity does not depend on the rotor size. Nevertheless, this hypothesis holds only if the rotor size is much larger than the characteristic ion layer thickness around the particle. In the opposite case, we show thanks to numerical calculations that the bulk charge distribution has to be accounted for to predict the electromechanical behavior of the rotor. We consider the case of an infinite insulating cylinder whose axis is perpendicular to the dc electric field. We use the finite element method to solve the conservation equations for the positive and the negative ions coupled with Navier-Stokes and Poisson equations. Doing so, we compute the bulk charge distribution and the velocity field in the liquid surrounding the cylinder. For sufficiently small cylinders, we show that the smaller the cylinder is, the smaller its angular velocity is when submitted to a dc electric field. This size effect is shown to originate both in ion diffusion and electromigration in the charge layer. At last, we propose a simple analytical model which allows calculating the angular velocity of the rotor when electromigration is present but weak and diffusion can be neglected.