Modeling of Nematic Electrolytes and Nonlinear Electroosmosis

Abstract

We derive a mathematical model of a nematic electrolyte based on the Ericksen-Leslie theory of liquid crystal flow. Our goal is to investigate the nonlinear electrokinetic effects that occur because the nematic matrix is anisotropic, in particular, transport of ions in a direction perpendicular to the electric field as well as quadratic dependence of the induced flow velocity on the electric field. The latter effect makes it possible to generate sustained flows in the nematic electrolyte that do not reverse their direction when the polarity of the applied electric field is reversed. From a practical perspective, this enables the design of AC-driven electrophoretic and electroosmotic devices. Our study of a special flow in a thin nematic film shows good qualitative agreement with laboratory experiments. 2261displacement. There is a growing interest in nonlinear electrokinetics, in which the flow velocities grow as the square of the applied field. Such a dependence allows one to use an AC field to drive stationary flows. In the case of isotropic electrolytes, the corresponding effects are the so-called AC electrokinetics (ACEK) [4] and induced-charge electrokinetics (ICEK) [5,6]. The spatial charge is induced on energized electrodes (the case of ACEK) or on the "floating" polarizable (metal) particles located in an externally applied electric field (the case of ICEK). In both effects, the electrokinetic velocities grow as E 2 , where one power of E induces the charge near the highly polarizable metal surface, while the second power of E drives these charges to trigger flows or to transport particles. Both ACEK and ICEK combined with broken symmetry of electrodes or particles can lead to an AC-driven pumping of the fluids or electrophoresis of free particles [5,6,7].The studies of the liquid crystal-enabled electrokinetics are a part of a much larger field of liquid crystal colloids that is currently experiencing a great deal of interest partially as a result of the progress in the field of nanotechnology. Recent experiments [8,9,10,11,12,13,14] demonstrate that when the isotropic electrolyte is replaced with an anisotropic electrolyte, a liquid crystal containing ions, the electrokinetic flows become strongly nonlinear, with the velocities growing as a square of the electric field. For such a flow, if the polarity of the applied field is reversed, the direction of the flow remains unchanged, enabling AC-driven electroosmosis and electrophoresis. The nonlinearity disappears as soon as the liquid crystal is melted into an isotropic phase. Despite the similarity in the quadratic field dependences of the flows, the mechanisms of the liquid crystal-enabled electrokinetics and the ACEK and ICEP effects in isotropic fluids are different, as discussed in [15]. Separation of charges and electrokinetics in isotropic electrolytes requires highly polarizable (metal) particles or interfaces; the isotropic electrolyte plays a supportive role, supplying the counterions. In the case of liquid crystal-enabled electrokinetics, the s...

Electro-osmosis in nematic liquid crystals

“…Note that in the case λ ε = 1, the solution (61) was already obtained in [12] under the initial assumption of isotropic viscosity η(y) ≡ 1/2. Our results show that although the assumption may seem oversimplified, it leads to qualitatively correct behavior of the flow (compare dashed and solid lines in the top row of Fig.…”

“…Although they do not interact with the nematic via potential forces, their motion with respect to the liquid crystal contributes to the dissipation [12] …”

“…A detailed discussion of the role of G and F can be found in [12]. Briefly, the solution (56) is correct as long as the following three conditions are satisfied: (a) The applied field E is strong enough to overcome thermal fluctuations on the length scale of the stripe's width, i.e., F 1 and diffusion can be neglected; (b) The applied field E is lower than E c =ceL/(ε 0 ε ⊥ ), i.e., G 1; and (c) The anisotropy parameters λ ε and λ σ are finite and bounded away from zero.…”

“…This setup is probably the simplest to analyze as the director patterns in the experiments were periodic, well defined (no topological defects), and homogeneous in the direction of the applied electric field. A mathematical model of this experiment was considered in [12] for the specific case of isotropic viscosity and dielectric permittivity of the nematic. In this paper, we propose a more general theory which incorporates the full set of nematic viscosities as well as dielectric anisotropy of the liquid-crystalline matrix that is also expected to trigger electrokinetic flows [9,11].…”

“…An alternative derivation can be found in [12]. Inspired by ideas [13][14][15], the authors established a system of governing equations from the local form of balance of linear and angular momentum.…”