Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

We consider a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through membrane channels. Improving the classical Poisson–Nernst–Planck models where ion species are treated as point charges, this model includes ionic interaction due to finite sizes of ion species modeled by hard sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. We examine the problem and investigate the ion size effect on the current–voltage (I–V) relations numerically, focusing on the case where two oppositely charged ion species are involved and only the hard sphere components of the excess chemical potentials are included. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I–V curves. This is accomplished through a numerical implementation of the analytical strategy introduced by Ji and Liu in [Poisson–Nernst–Planck systems for ion flow with density functional theory for hard-sphere potential: I–V relations and critical potentials. Part I: Analysis, J. Dyn. Differ. Equ. (to appear)]. The second task is to numerically detect two critical potential values V _c and V ^c .The existence of these two critical values is first realized for a relatively simple setting and analytical approximations of V _c and V ^c are obtained in the above mentioned reference. We propose an algorithm for numerical detection of V _c and V ^c without using any analytical formulas but based on the defining properties and numerical I–V curves directly. For the setting in the above mentioned reference, our numerical values for V _c and V ^c agree well with the analytical predictions. For a setting including a nonzero permanent charge in which case no analytic formula for the I–V relation is available now, our algorithms can still be applied to find V _c and V ^c numerically.     

Chemo-electro-mechanical modeling of ionic-strength-sensitive hydrogel: Influence of Young’s modulus

In this paper, a chemo-electro-mechanical model is presented which considers the characteristics of the three phases of the ionic-strength-sensitive hydrogel, including the solid polymeric network matrix, interstitial fluid and ionic species. It is termed the multi-effect-coupling ionic-strength-stimulus (MECis) model and composed of Poisson–Nernst–Planck system for diffusion of the chemical ionic species in the interior hydrogel and external solution, associated with the fixed charge equation for simulating the interaction between the fixed charges and mobile ions, and the mechanical equilibrium equation to characterize the deformation behavior of the solid polymeric network matrix. The simulation results of the MECis model are examined by comparing with the experimental data published in open literature. It is demonstrated that the present MECis model could simulate well the responsive behavior of the ionic-strength-sensitive hydrogel quantitatively. The parameter study is conducted by the MECis model for analysis of the influence of the Young’s modulus on the characteristics of the smart hydrogel, and it is concluded that the present model can be employed as a good platform for design and optimization of the smart hydrogel-based BioMEMS.     

Pore-Scale Modeling of Electrokinetics in Geomaterials

Pore-scale finite-volume continuum models of electrokinetic processes are used to predict the Debye lengths, velocity, and potential profiles for two-dimensional arrays of circles, ellipses and squares with different orientations. The pore-scale continuum model solves the coupled Navier–Stokes, Poisson, and Nernst–Planck equations to characterize the electro-osmotic pressure and streaming potentials developed on the application of an external voltage and pressure difference, respectively. This model is used to predict the macroscale permeabilities of geomaterials via the widely used Carmen–Kozeny equation and through the electrokinetic coupling coefficients. The permeability results for a two-dimensional X-ray tomography-derived sand microstructure are within the same order of magnitude as the experimentally calculated values. The effect of the particle aspect ratio and orientation on the electrokinetic coupling coefficients and subsequently the electrical and hydraulic tortuosity of the porous media has been determined. These calculations suggest a highly tortuous geomaterial can be efficient for applications like decontamination and desalination.

     

Modeling the Chlorides Transport in Cementitious Materials By Periodic Homogenization

In this work, we develop a macroscopic model for diffusion–migration of ionic species in saturated porous media, based on periodic homogenization. The prior application is chloride transport in cementitious materials. The dimensional analysis of Nernst–Planck equation lets appear dimensionless numbers characterizing the ionic transfer in porous media. Using experimental data, these dimensionless numbers are linked to the perturbation parameter ${\varepsilon}$ . For a weak-imposed electrical field, or in natural diffusion, the asymptotic expansion of Nernst–Planck equation leads to a macroscopic model coupling diffusion and migration at the same order. The expression of the homogenized diffusion coefficient only involves the geometrical properties of the material microstructure. Then, parametric simulations are performed to compute the chloride diffusion coefficient through different complexity of the elementary cell to go on as close as possible to experimental diffusion coefficient of the two cement pastes tested.     

Experimental study and electromechanical model analysis of the nonlinear deformation behavior of IPMC actuators

This paper develops analytical electromechanical formulas to predict the mechanical deformation of ionic polymer–metal composite(IPMC) cantilever actuators under DC excitation voltages. In this research, IPMC samples with Pt and Ag electrodes were manufactured, and the large nonlinear deformation and the effect of curvature on surface electrode resistance of the IPMC samples were investigated experimentally and theoretically. A distributed electrical model was modified for calculating the distribution of voltage along the bending actuator. Then an irreversible thermodynamic model that could predict the curvature of a unit part of an IPMC actuator is combined with the electrical model so that an analytical electromechanical model is developed. The electromechanical model is then validated against the experimental results obtained from Pt-and Ag-IPMC actuators under various excitation voltages. The good agreement between the electromechanical model and the actuators shows that the analytical electromechanical model can accurately describe the large nonlinear quasi-static deflection behavior of IPMC actuators.     

Electrically polarized mixtures of porous bodies

In this paper we propose a mathematical model of a mixture between a porous material and a polarized fluid in the mechanics of complex bodies. We use a variational approach to derive the global and microstructural balance laws for each phase of the mixture and for the mixture as a whole. Such balances differ from those generally proposed in theory of mixtures. We consider an open system and we account for external sources. Moreover, via the Coleman and Noll procedure we point out that the transport equation for the matter in a polarized mixture due to electrical field is governed by the Nernst–Planck equation.     

A coordinate transformation method for numerical solutions of the electric double layer and electroosmotic flows in a microchannel

The electric double layer (EDL) and electroosmotic flows (EOFs) constitute the theoretical foundations of microfluidics. Numerical solution is one of the effective means of analysis in microfluidics. In general, it is difficult to obtain an accurate numerical solution of complex EOFs because of multiphysical interactions and locally high gradients. In this paper, a new coordinate transformation method is proposed to numerically solve the Poisson–Boltzmann, Navier–Stokes and Nernst–Planck equations to study the EDL and complex EOFs in a microchannel. A series of numerical examples is presented including cases of a homogeneous, discontinous wall electric potential and a locally high wall potential. A systematic comparison of numerical solutions with and without the coordinate transformation is carried out. The numerical results indicate that the coordinate transformation effectively decreases the gradient of the electric potential, ion concentration and electroosmotic velocity in the vicinity of the solid wall, and greatly improves the stability and convergency of the solution. In a transformed coordinate system with a coarse grid, the numerical solutions can be as accurate as those in the original coordinate system with a refined grid. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation and design optimization of an electrohydrodynamic pump for dielectric liquids

With an ever increasing demand for more effective heat sinks, liquid based electronic cooling has become a new prospect in the field. The present study introduces an electrohydrodynamic (EHD) pump with a simple design for dielectric liquids which have potential applications for electronic cooling. The pump consists of an eccentrically sandwiched wire electrode placed at the horizontal centerline between two parallel flat-plate electrodes. The EHD flow of dielectric liquid induced by the space charge generated due to the Onsager effect was obtained by the numerical solution of the Poisson–Nernst–Planck equations for ion transport and the Navier–Stokes equations for fluid flow. Good agreement obtained in the comparison of the numerical and the experimental results of velocity for the centrally sandwiched wire electrode case confirmed the validity of the numerical results. For a fixed voltage, the pump flow rate depends on the eccentricity of the wire electrode with respect to the plate electrodes and also with the electrode dimensions. By using the Taguchi method an optimum design for the EHD pump is obtained considering the wire electrode diameter, the flat plate electrode length and the eccentricity (the horizontal distance between the centers of wire and flat-plate electrodes) as the design parameters for fixed channel dimensions.     

Effects of geometric modulation and surface potential heterogeneity on electrokinetic flow and solute transport in a microchannel

A numerical investigation is performed on the electroosmotic flow (EOF) in a surface-modulated microchannel to induce enhanced solute mixing. The channel wall is modulated by placing surface-mounted obstacles of trigonometric shape along which the surface potential is considered to be different from the surface potential of the homogeneous part of the wall. The characteristics of the electrokinetic flow are governed by the Laplace equation for the distribution of external electric potential; the Poisson equation for the distribution of induced electric potential; the Nernst–Planck equations for the distribution of ions; and the Navier–Stokes equations for fluid flow simultaneously. These nonlinear coupled set of governing equations are solved numerically by a control volume method over the staggered system. The influence of the geometric modulation of the surface, surface potential heterogeneity and the bulk ionic concentration on the EOF is analyzed. Vortical flow develops near a surface modulation, and it becomes stronger when the surface potential of the modulated region is in opposite sign to the surface potential of the homogeneous part of the channel walls. Vortical flow also depends on the Debye length when the Debye length is in the order of the channel height. Pressure drop along the channel length is higher for a ribbed wall channel compared to the grooved wall case. The pressure drop decreases with the increase in the amplitude for a grooved channel, but increases for a ribbed channel. The mixing index is quantified through the standard deviation of the solute distribution. Our results show that mixing index is higher for the ribbed channel compared to the grooved channel with heterogeneous surface potential. The increase in potential heterogeneity in the modulated region also increases the mixing index in both grooved and ribbed channels. However, the mixing performance, which is the ratio of the mixing index to pressure drop, reduces with the rise in the surface potential heterogeneity.     

Electroosmosis modulated peristaltic biorheological flow through an asymmetric microchannel: mathematical model

A theoretical study is presented of peristaltic hydrodynamics of an aqueous electrolytic non-Newtonian Jeffrey bio-rheological fluid through an asymmetric microchannel under an applied axial electric field. An analytical approach is adopted to obtain the closed form solution for velocity, volumetric flow, pressure difference and stream function. The analysis is also restricted under the low Reynolds number assumption (Stokes flow) and lubrication theory approximations (large wavelength). Small ionic Peclét number and Debye–Hückel linearization (i.e. wall zeta potential ≤ 25 mV) are also considered to simplify the Nernst–Planck and Poisson–Boltzmann equations. Streamline plots are also presented for the different electro-osmotic parameter, varying magnitudes of the electric field (both aiding and opposing cases) and for different values of the ratio of relaxation to retardation time parameter. Comparisons are also included between the Newtonian and general non-Newtonian Jeffrey fluid cases. The results presented here may be of fundamental interest towards designing lab-on-a-chip devices for flow mixing, cell manipulation, micro-scale pumps etc. Trapping is shown to be more sensitive to an electric field (aiding, opposing and neutral) rather than the electro-osmotic parameter and viscoelastic relaxation to retardation ratio parameter. The results may also help towards the design of organ-on-a-chip like devices for better drug design.     

Constitutive equations for an electroactive polymer

Ionic electroactive polymers can be used as sensors or actuators. For this purpose, a thin film of polyelectrolyte is saturated with a solvent and sandwiched between two platinum electrodes. The solvent causes a complete dissociation of the polymer and the release of small cations. The application of an electric field across the thickness results in the bending of the strip and vice versa. The material is modeled by a two-phase continuous medium. The solid phase, constituted by the polymer backbone inlaid with anions, is depicted as a deformable porous media. The liquid phase is composed of the free cations and the solvent (usually water). We used a coarse grain model. The conservation laws of this system have been established in a previous work. The entropy balance law and the thermodynamic relations are first written for each phase and then for the complete material using a statistical average technique and the material derivative concept. One deduces the entropy production. Identifying generalized forces and fluxes provides the constitutive equations of the whole system: the stress–strain relations which satisfy a Kelvin–Voigt model, generalized Fourier’s and Darcy’s laws and the Nernst–Planck equation.     

Karhunen–Loève Galerkin method with decimated sampling technique for the simulation of complex fluids defined in the phase space

In complex fluids, solute molecules with structural length scales much larger than atomic are dispersed in solvents of simple fluids such as water. The rheological properties of complex fluids are determined by dynamics of solute molecules which can be modeled by the Fokker–Planck equation defined in a six-dimensional phase space. In the present investigation, we devise a method of efficient simulation of complex fluid flows employing the Karhunen–Loève Galerkin (KLG) method. Adopting the decimated sampling of solvent flow fields, a reduced-order model for the Fokker–Planck equation is obtained, which can be employed for the the simulation of complex fluids with a decent computer time. As a specific example, we consider a flow of dilute polymeric liquids over a cylinder, whose constitutive equation is the FENE (finitely extensible nonlinear elastic) model. It is found that the KLG method with the decimated sampling technique yields accurate results at a computational cost less than a hundredth of that for the numerical simulation using the Fokker–Planck equation. The KLG method supplemented by the decimated sampling technique is an efficient method of coarse-graining for equations of complex fluids defined in the phase space.     

Moderately large flexural vibrations of composite plates with thick layers

In this paper moderately large amplitude vibrations of a polygonally shaped composite plate with thick layers are analyzed. Three homogeneous and isotropic layers with a common Poisson’s ratio are perfectly bonded and their arbitrary thickness and material properties are symmetrically disposed about the middle plane. Mindlin–Reissner kinematic assumptions are implemented layerwise, and as such model both the global and local response. Geometric nonlinear effects arising from longitudinally constrained supports are taken into account by Berger’s approximation of nonlinear strain–displacement relations. Overall cross-sectional rotations are defined and subsequently a correspondence of this complex problem to the simpler case of a homogenized shear-deformable nonlinear plate with effective stiffness and hard hinged boundary conditions is found. The nonlinear steady-state response of composite plates subjected to a time-harmonic lateral excitation is investigated and the phenomena of nonlinear resonance are studied and evaluated.     

Direct Derivation of Corrective Terms in SDE Through Nonlinear Transformation on Fokker–Planck Equation

This paper examines the problem of probabilistic characterization of nonlinear systems driven by normal and Poissonian white noise. By means of classical nonlinear transformation the stochastic differential equation driven by external input is transformed into a parametric-type stochastic differential equation. Such equations are commonly handled with Itô-type stochastic differential equations and Itô's rule is used to find the response statistics. Here a different approach is proposed, which mainly consists in transforming the Fokker–Planck equation for the original system driven by external input, in the transformed probability density function of the new state variable. It will be shown that the Wong–Zakay or Stratonovich corrective term and the hierarchy of correction terms in the case of Poissonian white noise arise in a natural way.     

MODELING CONTACTS OF IONIC POLYMER METAL COMPOSITES BASED TACTILE SENSORS

We report the first attempt to model the contacts of an ionic polymer metal composite（IPMC） based tactile sensor. The tactile sensor comprises an IPMC actuator, an IPMC sensor and the target to be detected. The system makes use of multiple contacts to work： the actuator comes into contact with the sensor and pushes the movement of sensor; the contact between the sensor and the object detects the existence and the stiffness of the target. We integrate modeling of various physical processes involved in IPMC devices to form a simulation scheme. An iteration and optimization strategy is also described to correlate the experimental and simulation results of an IPMC bending actuator to identify the two key parameters used in electromechanical transduction. Modeling the multiple contacts will aid the design and optimization of such IPMC based soft robotics.     

Dynamic behaviour of a thin laminated plate embedded with auxetic layers subject to in-plane excitation

The dynamic modelling of a simply-supported thin laminated plate subject to in-plane excitation is established based on the classic shear theory and von Kármán nonlinear theory. The method of multiple scales is used to determine an approximate solution for the system. According to solvability conditions, the nonlinear modulation equations arising from the principal parametric resonances are obtained and two possible nontrivial solutions are performed. To analyze the nonlinear dynamic response of the plate embedded with auxetic layers, 5-layered sandwich plate, in which two auxetic elastic layers are alternatively sandwiched between three positive Poisson’s ratio (PPR) elastic ones, is presented. The natural frequency of model (m, n) shows an increase with respect to the absolute value of Poisson’s ratio. Particularly, the amplitude-frequency responses of the laminated plate subject to principal parametric resonance are analyzed for different values of Poisson’s ratio. Moreover, it can be found that for model (m, n), there must be some certain value or interval of negative Poisson’s ratio (NPR), which, results in zero response effect, in other words, the in-plane excitation will be ineffective for this model when the Poisson’s ratio just lies at such a value or interval. Furthermore, it can also be observed that the certain interval of Poisson’s ratio becomes wider with the increase of damping.     

Stochastic averaging of energy harvesting systems

A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.     

A nonlinear mechanics model of bio-inspired hierarchical lattice materials consisting of horseshoe microstructures

Development of advanced synthetic materials that can mimic the mechanical properties of non-mineralized soft biological materials has important implications in a wide range of technologies. Hierarchical lattice materials constructed with horseshoe microstructures belong to this class of bio-inspired synthetic materials, where the mechanical responses can be tailored to match the nonlinear J-shaped stress–strain curves of human skins. The underlying relations between the J-shaped stress–strain curves and their microstructure geometry are essential in designing such systems for targeted applications. Here, a theoretical model of this type of hierarchical lattice material is developed by combining a finite deformation constitutive relation of the building block (i.e., horseshoe microstructure), with the analyses of equilibrium and deformation compatibility in the periodical lattices. The nonlinear J-shaped stress–strain curves and Poisson ratios predicted by this model agree very well with results of finite element analyses (FEA) and experiment. Based on this model, analytic solutions were obtained for some key mechanical quantities, e.g., elastic modulus, Poisson ratio, peak modulus, and critical strain around which the tangent modulus increases rapidly. A negative Poisson effect is revealed in the hierarchical lattice with triangular topology, as opposed to a positive Poisson effect in hierarchical lattices with Kagome and honeycomb topologies. The lattice topology is also found to have a strong influence on the stress–strain curve. For the three isotropic lattice topologies (triangular, Kagome and honeycomb), the hierarchical triangular lattice material renders the sharpest transition in the stress–strain curve and relative high stretchability, given the same porosity and arc angle of horseshoe microstructure. Furthermore, a demonstrative example illustrates the utility of the developed model in the rapid optimization of hierarchical lattice materials for reproducing the desired stress–strain curves of human skins. This study provides theoretical guidelines for future designs of soft bio-mimetic materials with hierarchical lattice constructions.