Abstract. We derive a model of the coupled mechanical and electrochemical effects of polyelectrolyte gels. We assume that the gel, which is immersed in a fluid domain, is an immiscible and incompressible mixture of a solid polymeric component and the fluid. As the gel swells and de-swells, the gel-fluid interface can move. Our model consists of a system of partial differential equations for mass and linear momentum balance of the polymer and fluid components of the gel, the NavierStokes equations in the surrounding fluid domain, and the Poisson-Nernst-Planck equations for the ionic concentrations on the whole domain. These are supplemented by a novel and general class of boundary conditions expressing mass and linear momentum balance across the moving gel-fluid interface. Our boundary conditions include the permeability boundary conditions proposed in earlier studies. A salient feature of our model is that it satisfies a free energy dissipation identity, in accordance with the second law of thermodynamics. We also show, using boundary layer analysis, that the well-established Donnan condition for equilibrium arises naturally as a consequence of taking the electroneutral limit in our model.

We use a variational principle to derive a mathematical model for a nematic electrolyte in which the liquid crystalline component is described in terms of a second-rank order parameter tensor. The model extends the previously developed director-based theory and accounts for presence of disclinations and possible biaxiality. We verify the model by considering a simple but illustrative example of liquid crystal-enabled electro-osmotic flow around a stationary dielectric spherical particle placed at the center of a large cylindrical container filled with a nematic electrolyte. Assuming homeotropic anchoring of the nematic on the surface of the particle and uniform distribution of the director on the surface of the container, we consider two configurations with a disclination equatorial ring and with a hyperbolic hedgehog, respectively. The computed electro-osmotic flows show a strong dependence on the director configurations and on the anisotropies of dielectric permittivity and electric conductivity of the nematic, characteristic of liquid crystal-enabled electrokinetics. Further, the simulations demonstrate space charge separation around the dielectric sphere, even in the case of isotropic permittivity and conductivity. This is in agreement with the induced-charge electro-osmotic effect that occurs in an isotropic electrolyte when an applied field acts on the ionic charge it induces near a polarizable surface.

No abstract

Molecules of a nematic liquid crystal respond to an applied magnetic field by reorienting themselves in the direction of the field. Since the dielectric anisotropy of a nematic is small, it takes relatively large fields to elicit a significant liquid crystal response. The interaction may be enhanced in colloidal suspensions of ferromagnetic particles in a liquid crystalline matrixferronematics-as proposed by Brochard and de Gennes in 1970. The ability of these particles to align with the field and, simultaneously, cause reorientation of the nematic molecules, greatly increases the magnetic response of the mixture. Essentially the particles provide an easy axis of magnetization that interacts with the liquid crystal via surface anchoring.We derive an expression for the effective energy of ferronematic in the dilute limit, that is, when the number of particles tends to infinity while

We consider a gel as an immiscible mixture of polymer and solvent, and derive governing equations of the dynamics. They include the balance of mass and linear momentum of the individual components. The model allows to account for nonlinear elasticity, viscoelasticity, transport and diffusion. The total free energy of the system combines the elastic contribution of the polymer with the FloryHuggins energy of mixing. The system is also formulated in terms of the center of mass velocity and the diffusive velocity, involving the total and the relative stresses. This allows for the identification of special regimes, such as the purely diffusive and the transport ones. We also obtain an equation for the rate of change of the total energy yielding decay for special choices of boundary conditions. The energy law motivates the Rayleghian variational approach discussed in the last part of the article. We consider the case of a gel in a one-dimensional strip domain in order to study special features of the dynamics, in particular, the early dynamics. We find that the monotonicity of the extensional stress is a necessary condition to guarantee the propagation of the swelling interface between the gel and its solvent. Such monotonicity condition is satisfied for data corresponding to linear entangled polymers. However, for polyssacharide gels the monotonicity of the stress fails at a critical volume fraction, suggesting the onset of de-swelling. The weak elasticity is responsible for the loss of monotonicity of the stress. The analysis also suggests that type II diffusion is a hyperbolic phenomenon rather than a diffusive one. One goal is to compare the derivation method, assumptions and resulting equations with other models available in the literature, and determine their regimes of validity. The stress-diffusion coupling model by Yamaue and Doi is one main benchmark. We assume that the gel is non-ionic, and neglect thermal effects.

In this article, we analyze a model of the incipient dynamics of gel swelling, and perform numerical simulations. The governing system consists of balance laws for a mixture of nonlinear elastic solid and solvent yielding effective equations for the gel. We discuss the multiscale nature of the problem and identify physically realistic regimes. The mixing mechanism is based on the Flory-Huggins energy. We consider the case that the dissipation mechanism is the solid-solvent friction force. This leads to a system of weakly dissipative nonlinear hyperbolic equations. After addressing the Cauchy problem, we propose physically realistic boundary conditions describing the motion of the swelling boundary. We study the linearized version of the free boundary problem. Numerical simulations of solutions are presented too.

We propose a Landau-de Gennes variational theory fit to simultaneously describe isotropic, nematic, smectic- A , and smectic- C phases of a liquid crystal. The unified description allows us to deal with systems in which one, or all, of the order parameters develop because of the influence of defects, external fields and/or boundary conditions. We derive the complete phase diagram of the system, that is, we characterize how the homogeneous minimizers depend on the value of the constitutive parameters. The coupling between the nematic order tensor and the complex smectic order parameter generates an elastic potential which is a nonconvex function of the gradient of the smectic order parameter. This lack of convexity yields in turn a loss of regularity of the free-energy minimizers. We then consider the effect on an infinitesimal second-order regularization term in the free-energy functional, which fixes the optimal number of defects in the singular configurations.

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...