Adaptive finite element simulation of currents at microelectrodes to a guaranteed accuracy. ECE and EC2E mechanisms at channel microband electrodes

We extend the work in the accompanying paper (K. Harriman et al., Electrochem. Commun. 2 (2000) 567) on the use of adaptive finite element methods to simulate the current for a steady state E reaction mechanism at a channel microband electrode to the more complex ECE mechanism, and the non-linear EC₂E mechanism. We again use the standard Galerkin approach for the diffusion dominated (low-flow) case, and the streamline diffusion finite element method (SDFEM) for convection-dominated (high-flow) case, and compare our results with previous numerical and analytical approximations. We give a general discussion on the implications of our results for numerical simulation at microelectrodes.     

Simulation of electrophoretic separations by the flux-corrected transport method

Electrophoretic separations at typical experimental electric field strengths have been simulated by applying the flux-corrected transport (FCT) finite difference method to the transient, one-dimensional electrophoresis model. The performance of FCT on simulations of zone electrophoresis (ZE), isotachophoresis (ITP), and isoelectric focusing (IEF) has been evaluated. An FCT algorithm, with a three-point, central spatial discretization, yields numerical solutions without numerical oscillations or spurious peaks, which have plagued previously-published second-order solutions to benchmark ZE and ITP problems. Moreover, the FCT technique captures sharp zone boundaries and IEF peaks more accurately than previously-published, first-order upwind schemes.     

Analytical and Numerical Modeling Methods for Electrochemical Impedance Analysis of Single Cells on Coplanar Electrodes

Impedance measurements provide basic electrical properties and are used to analyze the characteristics of electrochemical materials for biomedical applications. The extracellular fluid (ECF) in microfluidic devices greatly affects the accuracy of impedance measurements of cells. When a single cell is placed in large amounts of ECF, the electric current mostly passes through the ECF, not the cell. Hence, this work presents a modeling method that is demonstrated in numerical and analytical solutions for eliminating the effect of ECF in coplanar impedance sensors. The proposed modeling method uses fundamental formulas of circuits that include the electrical parameters of the ECF, cytoplasm, and cell membrane. Equivalent circuit models for the coplanar impedance sensor are established to simulate the impedance as well as the measured ones for excitation frequencies in the range of 11–101 kHz. According to the calculation result using the proposed modeling method, the cytoplasm resistance, membrane capacitance, medium resistance, and medium capacitance of HeLa (human cervix adenocarcinoma) cell are 13.5 kΩ, 122.6 pF, 27.9 kΩ, and 337.7 pF, respectively. Moreover, the electric current distribution in the coplanar impedance sensor is investigated using finite element method (FEM) simulation software. The variation in the impedance during measurements with the simultaneous application of an alternating‐current (AC) voltage amplitude of 0.4 V_pp in the fluid volume range of 9–144 µL is also studied.     

Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.     

Electrical discharge in a nanometer-sized air/water gap observed by atomic force microscopy

We report a method to initiate and investigate electrical discharges of ambient air/water molecules in a nanometer-sized gap. Our methodology is based on a typical atomic force microscopy (AFM) setup, in which a cylinder discharge gap of < or =5 nm could be configured between the AFM probe and substrate. We observed highly localized stochastic nanoexplosions in which the discharge probability is dominated by the electric field, material-specific surface reactions, and humidity. AFM results, coupled with the boundary element method (BEM), finite element method (FEM), and method of characteristics (MOC) simulations, further revealed the generation of transient shock waves in the nanoscale discharge. The propagation of shock fronts significantly facilitates the radial expansion of the ionized particles, leading to the formation of microscale patterns on selected substrates. Our findings provide an initial understanding of nanoscale discharge and could be relevant to a few applications including nano/microstructuring, microelectronics, and plasma-assisted depositions.     

Brueckner doubles coupled cluster method with the polarizable continuum model of solvation

We present the theory and implementation for computing the (free) energy and its analytical gradients with the Brueckner doubles (BD) coupled cluster method in solution, in combination with the polarizable continuum model of solvation (PCM). The complete model, called PTED, and an efficient approximation, called PTE, are introduced and tested with numerical examples. Implementation details are also discussed. A comparison with the coupled-cluster singles and doubles CCSD-PCM-PTED and CCSD-PCM-PTE schemes, which use Hartree-Fock (HF) orbitals, is presented. The results show that the two PTED approaches are mostly equivalent, while BD-PCM-PTE is shown to be superior to the corresponding CCSD scheme when the HF reference wave function is unstable. The BD-PCM-PTE scheme, whose computational cost is equivalent to gas phase BD, is therefore a promising approach to study molecular systems with complicated electronic structure in solution.     

Modeling adhesion forces between deformable bodies by FEM and Hamaker summation

A new numerical approach is presented for predicting adhesion forces of particles at flat and rough surfaces. The new hybrid method uses the finite element method (FEM) for the determination of elastic and plastic particle deformation combined with numerical Hamaker summation. In the numerical approach, the influence of the plastic deformation can be fully included. We show how the adhesion force depends on the contact geometry and the material properties. For easy comparison with other models, the force-displacement behavior of the systems is presented. The numerical approach is supported by atomic force microscopy (AFM) measurements. The experimentally observed adhesion force hysteresis is described very well by the new approach. Although calculations in this article are focused on spherical particles, our approach can be extended to particles of arbitrary shapes.     

A parallel finite element simulator for ion transport through three‐dimensional ion channel systems

A parallel finite element simulator, ichannel, is developed for ion transport through three‐dimensional ion channel systems that consist of protein and membrane. The coordinates of heavy atoms of the protein are taken from the Protein Data Bank and the membrane is represented as a slab. The simulator contains two components: a parallel adaptive finite element solver for a set of Poisson–Nernst–Planck (PNP) equations that describe the electrodiffusion process of ion transport, and a mesh generation tool chain for ion channel systems, which is an essential component for the finite element computations. The finite element method has advantages in modeling irregular geometries and complex boundary conditions. We have built a tool chain to get the surface and volume mesh for ion channel systems, which consists of a set of mesh generation tools. The adaptive finite element solver in our simulator is implemented using the parallel adaptive finite element package Parallel Hierarchical Grid (PHG) developed by one of the authors, which provides the capability of doing large scale parallel computations with high parallel efficiency and the flexibility of choosing high order elements to achieve high order accuracy. The simulator is applied to a real transmembrane protein, the gramicidin A (gA) channel protein, to calculate the electrostatic potential, ion concentrations and I – V curve, with which both primitive and transformed PNP equations are studied and their numerical performances are compared. To further validate the method, we also apply the simulator to two other ion channel systems, the voltage dependent anion channel (VDAC) and α‐Hemolysin (α‐HL). The simulation results agree well with Brownian dynamics (BD) simulation results and experimental results. Moreover, because ionic finite size effects can be included in PNP model now, we also perform simulations using a size‐modified PNP (SMPNP) model on VDAC and α‐HL. It is shown that the size effects in SMPNP can effectively lead to reduced current in the channel, and the results are closer to BD simulation results. © 2013 Wiley Periodicals, Inc.     

3-D transient electrophoretic motion of a spherical particle in a T-shaped rectangular microchannel

This paper considers the electrophoretic motion of a spherical particle in an aqueous electrolyte solution in a T-shaped rectangular microchannel, where the size of the channel is close to that of the particle. This is a complicated transient process where the electric field, the flow field, and the particle motion are coupled together. A theoretical model was developed to investigate the influences of the applied electric potentials, the zeta potentials of the channel and the particle, and the size of the particle on the particle motion. A direct numerical simulation method using the finite element method is employed. This method employs a generalized Galerkin finite element formulation that incorporates both equations of the fluid flow and equations of the particle motion into a single variational equation where the hydrodynamic interactions are eliminated. The ALE method is used to track the surface of the particle at each time step. The numerical results show that the electric field in the T-shaped microchannel is influenced by the presence of the particle, and that the particle motion is influenced by the applied electric potentials and the zeta potentials of the channel and the particle. The path of the particle motion is dominated by the local electric field and the ratio of the zeta potential of the channel to that of the particle. The particle's velocity is also dependent on its size in a small channel.     

Simulation in electrochemistry using the finite element method: Part 1: The algorithm

The finite element method (FEM) is well known in engineering technology. Numerous commercial packages are available. Extensions using an algorithm with an adaptive grid make this method very useful for a great variety of problems. This paper shows the applicability of this powerful tool to electrochemical problems in a general manner. In a first step, the mathematical equations are generalised to handle flexibly different electrochemical mechanisms and electrode geometries. Then, the finite element method is applied to these formulations and electrochemical boundary conditions are introduced. The spatial and time discretisations are discussed and a new method for flux calculation is introduced. Essential advantages of the adaptive finite element (AFE) algorithm are its flexibility and its applicability to many types of electrochemical processes and methods.     

Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet

We study the effects of Marangoni stresses on the flow in an evaporating sessile droplet, by extending a lubrication analysis and a finite element solution of the flow field in a drying droplet, developed earlier. The temperature distribution within the droplet is obtained from a solution of Laplace's equation, where quasi-steadiness and neglect of convection terms in the heat equation can be justified for small, slowly evaporating droplets. The evaporation flux and temperature profiles along the droplet surface are approximated by simple analytical forms and used as boundary conditions to obtain an axisymmetric analytical flow field from the lubrication theory for relatively flat droplets. A finite element algorithm is also developed to solve simultaneously the vapor concentration, and the thermal and flow fields in the droplet, which shows that the lubrication solution with the Marangoni stress is accurate for contact angles as high as 40 degrees. From our analysis, we find that surfactant contamination, at a surface concentration as small as 300 molecules/microm(2), can almost entirely suppress the Marangoni flow in the evaporating droplet.     

Computational Study of the Surface‐Enhanced Raman Scattering from Silica‐Coated Silver Nanowires

Surface‐enhanced Raman scattering (SERS) is a popular vibrational spectroscopic technique that can have several applications in chemical and biological sensing. Within the last decade or so, our ability to chemically synthesize nanostructures has improved to the point that the rational design of a variety of SERS substrates is now viable. In this report, we describe a computational study using the finite element method (FEM) to investigate the effects of patchy silica coatings on silver nanowires. We found that varying the degree of silica coating on silver nanowires impacts the enhancement and may be explained through two processes. The first process is a consequence of changes in the dielectric environment surrounding the nanowire due to the silica. As additional layers of silica coat the nanowire, the localized surface plasmon resonance of the nanowire redshifts. The second process is a result of silica distorting the local electric field around the nanowire surface. Anisotropic silica coating can influence anticipated enhancement depending on its spatial localization with respect to excited plasmon modes in the nanowire. We propose that the design of nanostructures with patchy silica coatings can be a viable tool for increasing surface enhancement.     

Computation of dynamic adsorption with adaptive integral,finite difference,and finite element methods

Analysis of diffusion-controlled adsorption and surface tension in one-dimensional planar coordinates with a finite diffusion length and a nonlinear isotherm, such as the Langmuir or Frumkin isotherm, requires numerical solution of the governing equations. This paper presents three numerical methods for solving this problem. First, the often-used integral (I) method with the trapezoidal rule approximation is improved by implementing a technique for error estimation and choosing time-step sizes adaptively. Next, an improved finite difference (FD) method and a new finite element (FE) method are developed. Both methods incorporate (a). an algorithm for generating spatially stretched grids and (b). a predictor-corrector method with adaptive time integration. The analytical solution of the problem for a linear dynamic isotherm (Henry isotherm) is used to validate the numerical solutions. Solutions for the Langmuir and Frumkin isotherms obtained using the I, FD, and FE methods are compared with regard to accuracy and efficiency. The results show that to attain the same accuracy, the FE method is the most efficient of the three methods used.     

Adaptative finite element simulation of currents at microelectrodes to a guaranteed accuracy. Application to a simple model problem

In this series of papers we consider the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the ‘edge-effect’. Our approach to overcoming this problem involves the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm. This allows us to calculate the current to within a prescribed tolerance. We begin by demonstrating the power of the method for a simple model problem — an E reaction mechanism at a microdisc electrode — for which the analytical solution is known. In this paper we give the background to the problem, and show how an a posteriori error bound can be used to drive an adaptive mesh-generation algorithm. We then use the algorithm to solve our model problem and obtain very accurate results on comparatively coarse meshes in minimal computing time. We give the technical details of the background theory and the derivation of the error bound in the accompanying paper.     

Numerical assessment of the analytical models used to determine flexural and shear moduli in I-Beams when the tensile and compressive moduli are different

The different compressive and tensile moduli of fibre reinforced composites have been considered in the analysis of the flexural and shear moduli of I-beams. Firstly, the neutral axis has been determined analytically and then, assuming that location of the neutral axis, the analytical flexural modulus of I-beams has also been obtained. In order to assess the proposed procedure, virtual pure bending and three-point bending tests at different spans have been carried out using the finite element method. The compressive and tensile moduli have been taken into account by defining two parts in the numerical models. The numerical flexural and shear moduli have been determined by reducing the data obtained in the virtual tests. Analytical and numerical results are in good agreement. Therefore, the flexural modulus determined by the proposed analytical approach can be introduced as a material property in the finite element method.     

Robust and high‐resolution simulations of nonlinear electrokinetic processes in variable cross‐section channels

We present a model and an associated numerical scheme to simulate complex electrokinetic processes in channels with nonuniform cross‐sectional area. We develop a quasi‐1D model based on local cross‐sectional area averaging of the equations describing unsteady, multispecies, electromigration‐diffusion transport. Our approach uses techniques of lubrication theory to approximate electrokinetic flows in channels with arbitrary variations in cross‐section; and we include chemical equilibrium calculations for weak electrolytes, Taylor–Aris type dispersion due of nonuniform bulk flow, and the effects of ionic strength on species mobility and on acid–base equilibrium constants. To solve the quasi‐1D governing equations, we provide a dissipative finite volume scheme that adds numerical dissipation at selective locations to ensure both unconditional stability and high accuracy. We couple the numerical scheme with a novel adaptive grid refinement algorithm that further improves the accuracy of simulations by minimizing numerical dissipation. We benchmark our numerical scheme with existing numerical schemes by simulating nonlinear electrokinetic problems, including ITP and electromigration dispersion in CZE. Simulation results show that our approach yields fast, stable, and high‐resolution solutions using an order of magnitude less grid points compared to the existing dissipative schemes. To highlight our model's capabilities, we demonstrate simulations that predict increase in detection sensitivity of ITP in converging cross‐sectional area channels. We also show that our simulations of ITP in variable cross‐sectional area channels have very good quantitative agreement with published experimental data.     

Numerical simulation of DNA sample preconcentration in microdevice electrophoresis

A numerical model is presented for the accurate and efficient prediction of preconcentration and transport of DNA during sample introduction and injection in microcapillary electrophoresis. The model incorporates conservation laws for the different buffer ions, salt ions, and DNA sample, coupled through a Gaussian electric field to account for the field modifications that cause electromigration. The accuracy and efficiency required to capture the physics associated with such a complex transient problem are realized by the use of the finite element-flux corrected transport (FE-FCT) algorithm in two dimensions. The model has been employed for the prediction of DNA sample preconcentration and transport during electrophoresis in a double-T injector microdevice. To test its validity, the numerical results have been compared with the corresponding experimental data under similar conditions, and excellent agreement has been found. Finally, detailed results from a simulation of DNA sample preconcentration in electrophoretic microdevices are presented using as parameters the electric field strength and the other species concentrations. The effect of the Tris concentration on sample stacking is also investigated. These results demonstrate the great potential offered by the model for future optimization of such microchip devices with respect to significantly enhanced speed and resolution of sample separation.     

Numerical simulation of bubble dynamics in a micro-channel under a nonuniform electric field

A numerical method is used to simulate the motion and coalescence of air bubbles in a micro-channel under a nonuniform electric field. The channel is equipped with arrays of electrodes embedded in its wall and voltages are applied on the electrodes to generate a specified electric field gradient in the longitudinal direction. In the study, the Navier-Stokes equations are solved by using the level set method handling the deformable/moving interfaces between the bubbles and the ambient liquid. Both the polarization Coulomb force and the dielectrophoresis force are considered as the force source of the Navier-Stokes equations by solving the Maxwell's equations. The flow field equations and the electric field equations are coupled and solved by using the finite element method. The electric field characteristics and the dynamic behavior of a bubble are analyzed by studying the distributions of the electric field and the force, the deformation and the moving velocity of the air bubble. The result suggests that the model of dispersed drops suspended in the immiscible dielectric liquid and driven by a nonuniform electric field is an effective method for the transportation and coalescence of micro-drops.     

Rapid concentration of deoxyribonucleic acid via Joule heating induced temperature gradient focusing in poly-dimethylsiloxane microfluidic channel

This paper reports rapid microfluidic electrokinetic concentration of deoxyribonucleic acid (DNA) with the Joule heating induced temperature gradient focusing (TGF) by using our proposed combined AC and DC electric field technique. A peak of 480-fold concentration enhancement of DNA sample is achieved within 40 s in a simple poly-dimethylsiloxane (PDMS) microfluidic channel of a sudden expansion in cross-section. Compared to a sole DC field, the introduction of an AC field can reduce DC field induced back-pressure and produce sufficient Joule heating effects, resulting in higher concentration enhancement. Within such microfluidic channel structure, negative charged DNA analytes can be concentrated at a location where the DNA electrophoretic motion is balanced with the bulk flow driven by DC electroosmosis under an appropriate temperature gradient field. A numerical model accounting for a combined AC and DC field and back-pressure driven flow effects is developed to describe the complex Joule heating induced TGF processes. The experimental observation of DNA concentration phenomena can be explained by the numerical model.     

Modelling mass analyzer performance with fields determined using the boundary element method

Computer modelling is widely used in the design of mass analysers to evaluate proposed designs and determine the effects of manufacturing imperfections. For quadrupole mass filters and ion traps, the models require accurate values of the electric field throughout the regions of the analyser in which ions travel. Most published results using models to predict mass analyser behaviour use electric fields computed with finite element (FE) or finite difference (FD) method. However, the boundary element method (BEM) is capable of achieving the same, or higher, accuracy with both computation times and memory requirements that are at least an order of magnitude less than those required by FE and FD methods. In this paper, electric field evaluation is performed using the BEM formulated in a manner described by previous workers; modifications to their method are described, which lead to higher accuracy field values. Simultaneous equation solution techniques are incorporated, which avoid solutions that are physically not realistic. The performance of linear quadrupole mass spectrometers with hyperbolic, circular and planar section electrodes has been determined using fields computed using these methods and compared with previous results obtained by alternative field computation techniques and with experiment. Behaviour of an ion trap mass spectrometer with circular symmetry has also been investigated. The results demonstrate that in each case using the BEM to determine the fields produces the observed behaviour. Copyright © 2010 John Wiley & Sons, Ltd.