Theory of nonlinear ac susceptibilities of dynamic electrorheological fluids under a dc electric field

By using a perturbation approach, we theoretically investigate nonlinear ac susceptibilities of dynamic electrorheological fluids under a dc electric field. We consider the effect of a finite volume fraction. It is shown that the dynamic effect arising from a shear flow plays an important role in the ac responses. Meanwhile, the effect of a finite volume fraction offers a correction. The collective relaxation time is also derived by using our recently-established dielectric dispersion spectral representation, and it is shown to generally increase with the volume fraction of particles. Some theoretical results are compared favorably with the experimental observations by others. 2006 Elsevier B.V. All rights reserved. An electrorheological (ER) fluid [1,2] contains polarizable particles suspended in a liquid of low dielectric constant. When an electric field is applied to such fluid, the particles will form chains and enhance anisotropy largely. In a realistic situation, the fluid flow exerts force and torque on the suspended particles, setting the particles in both translational and rotational motions. The torque puts on the particles will lead to the rotation of particles about their centers [3–10]. Experiments [7] showed that the induced interparticle forces between rotating ER particles are quite different from the values predicted by the existing theories that have not included the motion of particles. Wan et al. [8,9] theoretically investigated such kind of rotation and pointed out that the interaction forces between the ER particles is reduced by a factor that depends on the angular velocity of rotation. Most recently, this was partly demonstrated in experiments [11]. It has been shown that when a suspension consisting of (rotating) dielectric particles with angular frequency x1 having nonlinear characteristics is subjected to a dc field, the electric response will generally consist of ac fields at 0009-2614/$ see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.05.110 * Corresponding author. Fax: +86 21 65118743. E-mail address: [email protected] (J.P. Huang). various of frequencies like x1 and 3x1 [12–19]. In this work, we shall use a perturbation approach [13,20] to investigate the dynamic effect due to a shear flow on nonlinear ac responses in ER fluids under a dc electric field. In so doing, we shall include the effect of a finite volume fraction by investigating a dilute-limit system, which is beyond a single particle model in which this effect disappears naturally. We first consider a particle in the ER fluids under an applied dc electric field E0 1⁄4 E0 b Z. A particle suspended in the host medium will have a rotational motion about its center caused by a shear flow. We assume that the angular velocity caused by the shear flow is x1 b Y. Then, if we rotate with the particle, the particle will seem to be at rest, and it feels an electric field written as