This document discusses mathematical models used to calculate transmembrane potential in cell walls exposed to electric fields. It begins by introducing the topic and its importance in bioelectrical physics simulations. The document then presents three generations of models with increasing complexity:
1) The first generation model assumes a spherical cell geometry and derives Schwann's equation to calculate transmembrane potential as a function of the applied electric field and position on the cell surface.
2) The second generation model improves on this by using an ellipsoidal cell geometry, leading to a more complex solution involving elliptic integrals.
3) The third generation aims to model pulsed electric fields by considering the cell's potential to decay exponentially after a pulse, like a capacitor
Finite Element Method Linear Rectangular Element for Solving Finite Nanowire ...theijes
This paper concerned with the solution of finite nanowire superlattice quantum dot structures with a cylindrical cross-section determine by electronic states in various type of layers in terms of wave functions between structures containing the same number of barriers and wells (asymmetrical) or containing a different number (symmetrical). The solution is considered with the Finite element method with different base linear rectangular element to solve the one electron Ben Daniel-Duke equation. The results of numerical examples are compared for accuracy and efficiency with the finite difference method of this method and finite element method of linear triangular element. This comparison shows that good results of numerical examples.
This document discusses computer simulations of the structure and thermodynamics of colloidal solutions interacting through Yukawa or Lu-Marlow potentials. It presents:
1) A new attractive potential proposed by Lu and Marlow that takes into account particle size and is proportional to the inverse sixth power of distance for large separations.
2) Use of this potential and a repulsive electrostatic potential in a variational method to calculate theoretical structure factors, finding good agreement with experimental data.
3) Choice of hard spheres as a reference system and use of the Gibbs-Bogoliubov inequality to obtain an upper bound for the free energy of the colloidal system.
Numerical simulation of electromagnetic radiation using high-order discontinu...IJECEIAES
In this paper, we propose the simulation of 2-dimensional electromagnetic wave radiation using high-order discontinuous Galerkin time domain method to solve Maxwell's equations. The domains are discretized into unstructured straight-sided triangle elements that allow enhanced flexibility when dealing with complex geometries. The electric and magnetic fields are expanded into a high-order polynomial spectral approximation over each triangle element. The field conservation between the elements is enforced using central difference flux calculation at element interfaces. Perfectly matched layer (PML) boundary condition is used to absorb the waves that leave the domain. The comparison of numerical calculations is performed by the graphical displays and numerical data of radiation phenomenon and presented particularly with the results of the FDTD method. Finally, our simulations show that the proposed method can handle simulation of electromagnetic radiation with complex geometries easily.
A study of pion condensation within NJL model (Wroclaw, 2009)Roberto Anglani
1. The study investigates pion condensation in two-flavor quark matter using the Nambu–Jona-Lasinio model of QCD at finite density. It examines how the electric charge neutrality condition and explicit symmetry breaking via quark masses influence the onset of charged pion condensation.
2. It finds that the equality between the electric chemical potential and the in-medium pion mass persists as a threshold for a second-order phase transition to the pion condensed phase, even when pions are composite particles. Furthermore, the pion condensate is extremely fragile to symmetry breaking via current quark masses, and is ruled out for masses over 10 keV.
3. The study aims to
The document describes a simulation of the optical bandgap properties of particle arrays under different configurations. The simulation studied how the bandgap structure of a rhombohedral array of nanoparticles is affected by changing the particle arrangement (square lattice vs. triangular lattice), material (silicon, vanadium, graphite, polystyrene), and other parameters. Results from the simulations in MATLAB and COMSOL are presented, showing shifts in the bandgap regions between the different configurations. The goal of the simulation was to understand how to control and tune an optical structure's bandgap across the visible light spectrum.
Computational electromagnetics in plasmonic nanostructuresAliakbarMonfared1
This document summarizes computational methods for simulating the optical response of plasmonic nanostructures, specifically focusing on finite-difference time-domain (FDTD), finite element method (FEM), discrete dipole approximation (DDA), and boundary element method (BEM). It discusses these numerical approaches from both theoretical and practical perspectives. FDTD is described as explicitly calculating electric and magnetic fields on a grid over time to solve Maxwell's equations. Its applications include simulating optical properties and enhancing solar cell efficiency by tuning nanoparticle geometry. The document provides several examples of research using FDTD for these purposes.
Finite Element Method Linear Rectangular Element for Solving Finite Nanowire ...theijes
This paper concerned with the solution of finite nanowire superlattice quantum dot structures with a cylindrical cross-section determine by electronic states in various type of layers in terms of wave functions between structures containing the same number of barriers and wells (asymmetrical) or containing a different number (symmetrical). The solution is considered with the Finite element method with different base linear rectangular element to solve the one electron Ben Daniel-Duke equation. The results of numerical examples are compared for accuracy and efficiency with the finite difference method of this method and finite element method of linear triangular element. This comparison shows that good results of numerical examples.
This document discusses computer simulations of the structure and thermodynamics of colloidal solutions interacting through Yukawa or Lu-Marlow potentials. It presents:
1) A new attractive potential proposed by Lu and Marlow that takes into account particle size and is proportional to the inverse sixth power of distance for large separations.
2) Use of this potential and a repulsive electrostatic potential in a variational method to calculate theoretical structure factors, finding good agreement with experimental data.
3) Choice of hard spheres as a reference system and use of the Gibbs-Bogoliubov inequality to obtain an upper bound for the free energy of the colloidal system.
Numerical simulation of electromagnetic radiation using high-order discontinu...IJECEIAES
In this paper, we propose the simulation of 2-dimensional electromagnetic wave radiation using high-order discontinuous Galerkin time domain method to solve Maxwell's equations. The domains are discretized into unstructured straight-sided triangle elements that allow enhanced flexibility when dealing with complex geometries. The electric and magnetic fields are expanded into a high-order polynomial spectral approximation over each triangle element. The field conservation between the elements is enforced using central difference flux calculation at element interfaces. Perfectly matched layer (PML) boundary condition is used to absorb the waves that leave the domain. The comparison of numerical calculations is performed by the graphical displays and numerical data of radiation phenomenon and presented particularly with the results of the FDTD method. Finally, our simulations show that the proposed method can handle simulation of electromagnetic radiation with complex geometries easily.
A study of pion condensation within NJL model (Wroclaw, 2009)Roberto Anglani
1. The study investigates pion condensation in two-flavor quark matter using the Nambu–Jona-Lasinio model of QCD at finite density. It examines how the electric charge neutrality condition and explicit symmetry breaking via quark masses influence the onset of charged pion condensation.
2. It finds that the equality between the electric chemical potential and the in-medium pion mass persists as a threshold for a second-order phase transition to the pion condensed phase, even when pions are composite particles. Furthermore, the pion condensate is extremely fragile to symmetry breaking via current quark masses, and is ruled out for masses over 10 keV.
3. The study aims to
The document describes a simulation of the optical bandgap properties of particle arrays under different configurations. The simulation studied how the bandgap structure of a rhombohedral array of nanoparticles is affected by changing the particle arrangement (square lattice vs. triangular lattice), material (silicon, vanadium, graphite, polystyrene), and other parameters. Results from the simulations in MATLAB and COMSOL are presented, showing shifts in the bandgap regions between the different configurations. The goal of the simulation was to understand how to control and tune an optical structure's bandgap across the visible light spectrum.
Computational electromagnetics in plasmonic nanostructuresAliakbarMonfared1
This document summarizes computational methods for simulating the optical response of plasmonic nanostructures, specifically focusing on finite-difference time-domain (FDTD), finite element method (FEM), discrete dipole approximation (DDA), and boundary element method (BEM). It discusses these numerical approaches from both theoretical and practical perspectives. FDTD is described as explicitly calculating electric and magnetic fields on a grid over time to solve Maxwell's equations. Its applications include simulating optical properties and enhancing solar cell efficiency by tuning nanoparticle geometry. The document provides several examples of research using FDTD for these purposes.
DIELECTROPHORETIC DEFORMATION OF ERYTHROCYTES ON TRANSPARENT INDIUM TIN OXIDE...Larry O'Connell
1. The document presents a protocol for patterning transparent indium tin oxide (ITO) electrodes and investigates their efficacy for dielectrophoretically deforming erythrocytes compared to existing gold electrodes.
2. ITO electrodes were fabricated through a multi-step reactive ion etching process involving photoresist patterning and chromium/nickel masking layers. The transparent ITO electrodes allow full visualization of adhered cells, unlike opaque gold electrodes.
3. Erythrocytes suspended in a glucose solution were dielectrophoretically deformed at increasing electric field strengths on the ITO electrodes. Images were captured at each voltage step to measure cell deformation without occlusion from the electrodes.
»Over the last two decades several patents and research papers have reported purported practical methods to extract useful energy from the vacuum. I describe the inventions and analyze the underlying physics. From an analysis based on first principles it is clear that most of the inventions have fundamental errors and cannot work. The basic concept of harvesting zero-point energy remains viable, and at least one patented concept might work.
The vacuum is filled with a high density of zero-point energy, in the form of modes (vibrational patterns) of electromagnetic field. Over the last eight decades it has become clear that this zero-point field (ZPF) vacuum energy is not simply a mathematical formalism, but produces demonstrable effects on physical systems. Along with that realization has come the desire to extract energy from the ZPF.
One set of methods use nonlinear elements to convert the ZPF into a usable form. A rectifier (used to convert AC to DC) is a strongly nonlinear element. One patent makes use of antennas to capture the ZPF. This energy is then rectified and used. Another set of inventions simply rectify fluctuations (noise) in electronic elements as an extraction method. Using a detailed balance argument, I show that these methods cannot work.
Another set of patents describe using a Casimir cavity to mechanically extract energy from the ZPF. A Casimir cavity consists of two closely space reflecting plates that exclude ZPF electromagnetic modes having wavelengths larger than twice the gap spacing. The result is that the imbalance in the density of the ZPF inside and outside the cavity causes the plates to be attracted to each other. This attractive potential can be used, but only once. To produce power continuously, a method must be devised to form a reciprocating Casimir engine. The patents purport to switch off the Casimir attraction while the plates are pulled apart, so that they can repeatedly accelerate together and produce power. This approach is shown to be fundamentally flawed, and cannot produce power continuously.
A recently issued patent describes a method by which vacuum energy is extracted from gas flowing through a Casimir cavity. According to stochastic electrodynamics, the electronic orbitals in atoms are supported by ambient ZPF. When the gas atoms are pumped into a Casimir cavity, where long-wavelength ZPF modes are excluded, the electrons spin down into lower orbitals, releasing energy. This energy is harvested in a local absorber. When the electrons exit the Casimir cavity, they are re-energized to their original orbitals by the ambient ZPF. The process is repeated to produce continuous power. This method does not suffer from the fundamental flaws of the other approaches, and might work.«
This document discusses a study on the electromagnetic activity produced by oscillations of microtubules in cells. Microtubules are composed of electrically polar subunits that could generate electric fields when they vibrate mechanically. The study derives the electromagnetic field produced by an oscillating electric dipole to model the microtubule subunits. It then models microtubule networks in dividing and non-dividing cells and finds that the asymmetric network in a dividing cell produces a field that decays more slowly with distance. However, the calculated field intensities are very low and difficult to detect without sophisticated methods.
This document outlines a graduate student's thesis work on multi-scale modeling of micro-coronas. There are wide variations in both time and length scales involved in plasma modeling, from picoseconds to hours/days and from molecular to macroscopic scales. A multi-scale modeling technique of domain decomposition is proposed, using microscopic models locally where needed and macroscopic models for the rest. The goals are to develop a modeling tool that can span micro- to macro-scales and simulate plasmas in complex geometries. Challenges include bridging between scales and incorporating particle and fluid models.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The document studies the window effect of rectangular electrical pulses on the membrane potential of an osteoblast cell modeled as a dielectric. It presents an equivalent circuit model of the cell subjected to time domain electric fields such as rectangular pulses. The model includes separate charging time constants and relaxation times for the inner and outer cell membranes. Simulation results show that shorter pulse durations selectively affect the inner membrane more than the outer membrane, while longer pulses can fully charge both membranes. The window effect provides insight into bioelectric phenomena like electroporation and nanopore formation, with applications to cancer treatment.
Potential Energy Surface Molecular Mechanics ForceField Jahan B Ghasemi
This document provides an overview of potential energy surfaces (PES) in computational chemistry. It defines a PES as the relationship between a molecule's energy and its geometry. A PES is an n-dimensional surface that relates potential energy to n degrees of freedom in a molecule. Key points made include:
1) A PES allows visualization of how energy changes with molecular structure. Minima correspond to stable structures like reactants and products, while transition states are saddle points along the reaction coordinate.
2) Slices of multidimensional PES can be plotted against one or two geometric parameters to qualitatively represent the full hypersurface.
3) Stationary points on a PES satisfy dE/dq
This document summarizes a research paper that proposes using dielectrophoresis in a microfluidic device to separate live and dead biological cells. It describes how an applied non-uniform electric field can induce dipole moments in cells, causing them to be attracted to either high or low field regions depending on their dielectric properties. The document outlines the design of a microfluidic device with a 3D electrode structure intended to exploit these differences and separate live and dead mammalian cells based on their dielectric behavior over 50-70 kHz frequencies.
Neuronal modeling of patch-clamp data is based on approximations which are valid under specific assumptions regarding cell properties and morphology. Certain cells, which show a biexponential capacitance transient decay, can be modeled with a two-compartment model. However, for parameter-extraction in such a model, approximations are required regarding the relative sizes of the various model parameters. These approximations apply to certain cell types or experimental conditions and are not valid in the general case. In this paper, we present a general method for the extraction of the parameters in a two-compartment model without assumptions regarding the relative size of the parameters. All the passive electrical parameters of the two-compartment model are derived in terms of the available experimental data.
Abstract
Neuronal modeling of patch-clamp data is based on approximations which are valid under specific assumptions regarding cell properties and morphology. Certain cells, which show a biexponential capacitance transient decay, can be modeled with a two-compartment model. However, for parameter-extraction in such a model, approximations are required regarding the relative sizes of the various model parameters. These approximations apply to certain cell types or experimental conditions and are not valid in the general case. In this paper, we present a general method for the extraction of the parameters in a two-compartment model without assumptions regarding the relative size of the parameters. All the passive electrical parameters of the two-compartment model are derived in terms of the available experimental data. The experimental data is obtained from a DC measurement (where the command potential is a hyperpolarizing DC voltage) and an AC measurement (where the command potential is a sinusoidal stimulus on a hyperpolarized DC potential) performed on the cell under test. Computer simulations are performed with a circuit simulator, XSPICE, to observe the effects of varying the two-compartment model parameters on the capacitive transients of the current response. Our general solution for the parameter-estimation of a two-compartment model may be used to model any neuron, which has a biexponential capacitive current decay. In addition, our model avoids the need for simplifying and perhaps erroneous approximations. Our equations may be easily implemented in hardware/software compensation schemes to correct the recorded currents for any series resistance or capacitive transient errors. Our general solution reduces to the results of previous researchers under their approximations.
This document presents a correction and extension of previous work on calculating next nearest neighbor coupling in side-coupled linear accelerators. It derives analytical expressions for calculating the next nearest neighbor coupling coefficient (kk) based on magnetic dipole interactions between cavities. The approach calculates the dipole moment induced in the coupling slot by the field in the accelerating cavities, the interaction energy between the two dipoles, and employs perturbation theory. The calculated value of kk is typically less than 1%, which is then verified by comparison to experimental data from cold models.
Sergey seriy thomas fermi-dirac theorySergey Seriy
This document presents modern ab-initio calculations based on Thomas-Fermi-Dirac theory with quantum, correlation, and multishell corrections. It summarizes extensions made to the statistical model by including additional energy terms to account for quantum corrections, exchange energy, and correlation energy. This leads to a quantum- and correlation-corrected Thomas-Fermi-Dirac equation involving a density term, kinetic energy density term, and modified potential function. Solving this quartic equation in the electron density provides a way to determine the electron density distribution as a function of distance from the nucleus.
This document summarizes the design, fabrication, and testing of a microfluidic chip prototype for manipulating particles using dielectrophoresis (DEP). Finite element modeling was used to simulate the electric field distributions around quadrupole and comb electrode geometries. A prototype was fabricated containing these electrode designs in two separate microchannels. Silica microspheres were successfully manipulated within the chip using positive and negative DEP sequences, concentrating particles in the electrode areas. Testing demonstrated the potential of this technique for manipulating and separating microparticles in integrated microfluidic devices.
This document provides an introduction to the physics of dielectrics. It discusses electric dipoles, polarization, and the dielectric constant. It defines permanent and induced dipole moments. It introduces the electric field E and displacement field D inside dielectrics. Polarization P is introduced as the difference between D and E, and is proportional to the dielectric susceptibility χ and permittivity ε. Different types of polarization including electronic, atomic, orientation and ionic polarization are also described.
This document summarizes a student project that used the finite difference method and the Laplace equation to model the electrostatic properties of a non-symmetrical surface. The student created an Excel model of an infinitely long magnetic strip surrounded by a conducting box. The model was used to calculate potential, electric field, surface charge density, and capacitance per unit length for different node amounts. The results showed higher potential and charge density near the strip, and flux lines directed towards the box edges rather than another plate. Overall, the model behaved similarly to a parallel-plate capacitor except for non-symmetrical flux lines.
Numerical Simulation of 퐒퐢ퟏ−퐱퐆퐞퐱 Thin Film Solar Cell Using AMPS - 1DIOSR Journals
This document describes a numerical simulation of thin film silicon-germanium solar cells using the AMPS-1D simulation program. The simulation varied the germanium concentration in the intrinsic layer between 0-100% to determine the optimal band gap for maximum efficiency. The results showed that a germanium concentration of 90% produced the highest efficiency of 19.68% for device thicknesses under 6 micrometers. Above 6 micrometers, pure silicon performed equally well or better than silicon-germanium alloys.
1 ECE 6340 Fall 2013 Homework 8 Assignment.docxjoyjonna282
1
ECE 6340
Fall 2013
Homework 8
Assignment: Please do Probs. 1-9 and 13 from the set below.
1) In dynamics, we have the equation
E j Aω= − −∇Φ .
(a) Show that in statics, the scalar potential function Φ can be interpreted as a voltage
function. That is, show that in statics
( ) ( )
B
AB
A
V E dr A B≡ ⋅ = Φ −Φ∫ .
(b) Next, explain why this equation is not true (in general) in dynamics.
(c) Explain why the voltage drop (defined as the line integral of the electric field, as
defined above) depends on the path from A to B in dynamics, using Faraday’s law.
(d) Does the right-hand side of the above equation (the difference in the potential
function) depend on the path, in dynamics?
Hint: Note that, according to calculus, for any function ψ we have
dr dx dy dz d
x y z
ψ ψ ψ
ψ ψ
∂ ∂ ∂
∇ ⋅ = + + =
∂ ∂ ∂
.
2) Starting with Maxwell’s equations, show that the electric field radiated by an impressed
current density source J i in an infinite homogeneous region satisfies the equation
( )2 2 iE k E E j Jωµ∇ + = ∇ ∇⋅ + .
Then use Ampere’s law (or, if you prefer, the continuity equation and the electric Gauss
law) to show that this equation may be written as
( )2 2 1 i iE k E J j J
j
ωµ
σ ωε
∇ + = − ∇ ∇⋅ +
+
.
2
Note that the total current density is the sum of the impressed current density and the
conduction current density, the latter obeying Ohm’s law (J c = σE).
Explain why this equation for the electric field would be harder to solve than the equation
that was derived in class for the magnetic vector potential.
3) Show that magnetic field radiated by an impressed current density source satisfies the
equation
2 2 iH k H J∇ + = −∇× .
Explain why this equation for the magnetic field would be harder to solve than the
equation that was derived in class for the magnetic vector potential.
4) Show that in a homogenous region of space the scalar electric potential satisfies the
equation
2 2
i
v
c
k
ρ
ε
∇ Φ + Φ = − ,
where ivρ is the impressed (source) charge density, which is the charge density that goes
along with the impressed current density, being related by
i ivJ jωρ∇⋅ = −
Hint: Start with E j Aω= − −∇Φ and take the divergence of both sides. Also, take the
divergence of both sides of Ampere’s law and use the continuity equation for the
impressed current (given above) to show that
1 ii v
c c
E J
j
ρ
ωε ε
∇⋅ = − ∇⋅ = .
Note: It is also true from the electric Gauss law that
vE
ρ
ε
∇⋅ = ,
but we prefer to have only an impressed (source) charge density on the right-hand side of
the equation for the potential Φ. In the time-harmonic steady state, assuming a
homogeneous and isotropic region, it follows that ρv = ρvi. That is, there is no charge
3
density arising from the conduction current. (If there were no impressed current sources,
the total charge density would therefore be ze ...
This document presents a dielectric covered hairpin probe for measuring electron density in reactive plasmas. Covering the hairpin probe with a thin dielectric layer, such as quartz, protects the probe surface from contamination and allows it to operate at higher densities by reducing energy losses. The document develops a model to account for the additional capacitance introduced by the dielectric layer. Experimental tests in an argon plasma verify that electron density measurements made with the covered probe, when corrected using the new model, agree well with measurements from an uncovered reference probe. The dielectric covering improves the performance of the hairpin probe in reactive plasmas.
Kaytlin Brinker modeled proton exchange membrane fuel cells by developing equations to describe the charge distribution in the Stern layer and diffuse layer of the membrane. Her results showed that including the charge imbalance in the membrane is important for calculating fuel cell properties accurately. For future work, she plans to use molecular modeling software to simulate proton diffusion in Nafion under varying hydration conditions and predict transport properties from a theoretical diffusion model.
This report discusses an experiment on flow visualization of electro-kinetic force chemical mechanical planarization (EKF-CMP). Straight and radial electrode specimens were created in glass chambers to observe flow patterns under various voltages and solution conditions using video and simulations. Results showed that lower gap lengths and radial electrode orientation produced faster fluid flow. While solution pH had little effect, EKF-CMP was found to significantly improve material removal rate over conventional CMP, making it suitable for creating refined 3D stacked wafers needed to satisfy Moore's Law. In conclusion, the experiment validated that external voltage application enhances CMP performance.
DIELECTROPHORETIC DEFORMATION OF ERYTHROCYTES ON TRANSPARENT INDIUM TIN OXIDE...Larry O'Connell
1. The document presents a protocol for patterning transparent indium tin oxide (ITO) electrodes and investigates their efficacy for dielectrophoretically deforming erythrocytes compared to existing gold electrodes.
2. ITO electrodes were fabricated through a multi-step reactive ion etching process involving photoresist patterning and chromium/nickel masking layers. The transparent ITO electrodes allow full visualization of adhered cells, unlike opaque gold electrodes.
3. Erythrocytes suspended in a glucose solution were dielectrophoretically deformed at increasing electric field strengths on the ITO electrodes. Images were captured at each voltage step to measure cell deformation without occlusion from the electrodes.
»Over the last two decades several patents and research papers have reported purported practical methods to extract useful energy from the vacuum. I describe the inventions and analyze the underlying physics. From an analysis based on first principles it is clear that most of the inventions have fundamental errors and cannot work. The basic concept of harvesting zero-point energy remains viable, and at least one patented concept might work.
The vacuum is filled with a high density of zero-point energy, in the form of modes (vibrational patterns) of electromagnetic field. Over the last eight decades it has become clear that this zero-point field (ZPF) vacuum energy is not simply a mathematical formalism, but produces demonstrable effects on physical systems. Along with that realization has come the desire to extract energy from the ZPF.
One set of methods use nonlinear elements to convert the ZPF into a usable form. A rectifier (used to convert AC to DC) is a strongly nonlinear element. One patent makes use of antennas to capture the ZPF. This energy is then rectified and used. Another set of inventions simply rectify fluctuations (noise) in electronic elements as an extraction method. Using a detailed balance argument, I show that these methods cannot work.
Another set of patents describe using a Casimir cavity to mechanically extract energy from the ZPF. A Casimir cavity consists of two closely space reflecting plates that exclude ZPF electromagnetic modes having wavelengths larger than twice the gap spacing. The result is that the imbalance in the density of the ZPF inside and outside the cavity causes the plates to be attracted to each other. This attractive potential can be used, but only once. To produce power continuously, a method must be devised to form a reciprocating Casimir engine. The patents purport to switch off the Casimir attraction while the plates are pulled apart, so that they can repeatedly accelerate together and produce power. This approach is shown to be fundamentally flawed, and cannot produce power continuously.
A recently issued patent describes a method by which vacuum energy is extracted from gas flowing through a Casimir cavity. According to stochastic electrodynamics, the electronic orbitals in atoms are supported by ambient ZPF. When the gas atoms are pumped into a Casimir cavity, where long-wavelength ZPF modes are excluded, the electrons spin down into lower orbitals, releasing energy. This energy is harvested in a local absorber. When the electrons exit the Casimir cavity, they are re-energized to their original orbitals by the ambient ZPF. The process is repeated to produce continuous power. This method does not suffer from the fundamental flaws of the other approaches, and might work.«
This document discusses a study on the electromagnetic activity produced by oscillations of microtubules in cells. Microtubules are composed of electrically polar subunits that could generate electric fields when they vibrate mechanically. The study derives the electromagnetic field produced by an oscillating electric dipole to model the microtubule subunits. It then models microtubule networks in dividing and non-dividing cells and finds that the asymmetric network in a dividing cell produces a field that decays more slowly with distance. However, the calculated field intensities are very low and difficult to detect without sophisticated methods.
This document outlines a graduate student's thesis work on multi-scale modeling of micro-coronas. There are wide variations in both time and length scales involved in plasma modeling, from picoseconds to hours/days and from molecular to macroscopic scales. A multi-scale modeling technique of domain decomposition is proposed, using microscopic models locally where needed and macroscopic models for the rest. The goals are to develop a modeling tool that can span micro- to macro-scales and simulate plasmas in complex geometries. Challenges include bridging between scales and incorporating particle and fluid models.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The document studies the window effect of rectangular electrical pulses on the membrane potential of an osteoblast cell modeled as a dielectric. It presents an equivalent circuit model of the cell subjected to time domain electric fields such as rectangular pulses. The model includes separate charging time constants and relaxation times for the inner and outer cell membranes. Simulation results show that shorter pulse durations selectively affect the inner membrane more than the outer membrane, while longer pulses can fully charge both membranes. The window effect provides insight into bioelectric phenomena like electroporation and nanopore formation, with applications to cancer treatment.
Potential Energy Surface Molecular Mechanics ForceField Jahan B Ghasemi
This document provides an overview of potential energy surfaces (PES) in computational chemistry. It defines a PES as the relationship between a molecule's energy and its geometry. A PES is an n-dimensional surface that relates potential energy to n degrees of freedom in a molecule. Key points made include:
1) A PES allows visualization of how energy changes with molecular structure. Minima correspond to stable structures like reactants and products, while transition states are saddle points along the reaction coordinate.
2) Slices of multidimensional PES can be plotted against one or two geometric parameters to qualitatively represent the full hypersurface.
3) Stationary points on a PES satisfy dE/dq
This document summarizes a research paper that proposes using dielectrophoresis in a microfluidic device to separate live and dead biological cells. It describes how an applied non-uniform electric field can induce dipole moments in cells, causing them to be attracted to either high or low field regions depending on their dielectric properties. The document outlines the design of a microfluidic device with a 3D electrode structure intended to exploit these differences and separate live and dead mammalian cells based on their dielectric behavior over 50-70 kHz frequencies.
Neuronal modeling of patch-clamp data is based on approximations which are valid under specific assumptions regarding cell properties and morphology. Certain cells, which show a biexponential capacitance transient decay, can be modeled with a two-compartment model. However, for parameter-extraction in such a model, approximations are required regarding the relative sizes of the various model parameters. These approximations apply to certain cell types or experimental conditions and are not valid in the general case. In this paper, we present a general method for the extraction of the parameters in a two-compartment model without assumptions regarding the relative size of the parameters. All the passive electrical parameters of the two-compartment model are derived in terms of the available experimental data.
Abstract
Neuronal modeling of patch-clamp data is based on approximations which are valid under specific assumptions regarding cell properties and morphology. Certain cells, which show a biexponential capacitance transient decay, can be modeled with a two-compartment model. However, for parameter-extraction in such a model, approximations are required regarding the relative sizes of the various model parameters. These approximations apply to certain cell types or experimental conditions and are not valid in the general case. In this paper, we present a general method for the extraction of the parameters in a two-compartment model without assumptions regarding the relative size of the parameters. All the passive electrical parameters of the two-compartment model are derived in terms of the available experimental data. The experimental data is obtained from a DC measurement (where the command potential is a hyperpolarizing DC voltage) and an AC measurement (where the command potential is a sinusoidal stimulus on a hyperpolarized DC potential) performed on the cell under test. Computer simulations are performed with a circuit simulator, XSPICE, to observe the effects of varying the two-compartment model parameters on the capacitive transients of the current response. Our general solution for the parameter-estimation of a two-compartment model may be used to model any neuron, which has a biexponential capacitive current decay. In addition, our model avoids the need for simplifying and perhaps erroneous approximations. Our equations may be easily implemented in hardware/software compensation schemes to correct the recorded currents for any series resistance or capacitive transient errors. Our general solution reduces to the results of previous researchers under their approximations.
This document presents a correction and extension of previous work on calculating next nearest neighbor coupling in side-coupled linear accelerators. It derives analytical expressions for calculating the next nearest neighbor coupling coefficient (kk) based on magnetic dipole interactions between cavities. The approach calculates the dipole moment induced in the coupling slot by the field in the accelerating cavities, the interaction energy between the two dipoles, and employs perturbation theory. The calculated value of kk is typically less than 1%, which is then verified by comparison to experimental data from cold models.
Sergey seriy thomas fermi-dirac theorySergey Seriy
This document presents modern ab-initio calculations based on Thomas-Fermi-Dirac theory with quantum, correlation, and multishell corrections. It summarizes extensions made to the statistical model by including additional energy terms to account for quantum corrections, exchange energy, and correlation energy. This leads to a quantum- and correlation-corrected Thomas-Fermi-Dirac equation involving a density term, kinetic energy density term, and modified potential function. Solving this quartic equation in the electron density provides a way to determine the electron density distribution as a function of distance from the nucleus.
This document summarizes the design, fabrication, and testing of a microfluidic chip prototype for manipulating particles using dielectrophoresis (DEP). Finite element modeling was used to simulate the electric field distributions around quadrupole and comb electrode geometries. A prototype was fabricated containing these electrode designs in two separate microchannels. Silica microspheres were successfully manipulated within the chip using positive and negative DEP sequences, concentrating particles in the electrode areas. Testing demonstrated the potential of this technique for manipulating and separating microparticles in integrated microfluidic devices.
This document provides an introduction to the physics of dielectrics. It discusses electric dipoles, polarization, and the dielectric constant. It defines permanent and induced dipole moments. It introduces the electric field E and displacement field D inside dielectrics. Polarization P is introduced as the difference between D and E, and is proportional to the dielectric susceptibility χ and permittivity ε. Different types of polarization including electronic, atomic, orientation and ionic polarization are also described.
This document summarizes a student project that used the finite difference method and the Laplace equation to model the electrostatic properties of a non-symmetrical surface. The student created an Excel model of an infinitely long magnetic strip surrounded by a conducting box. The model was used to calculate potential, electric field, surface charge density, and capacitance per unit length for different node amounts. The results showed higher potential and charge density near the strip, and flux lines directed towards the box edges rather than another plate. Overall, the model behaved similarly to a parallel-plate capacitor except for non-symmetrical flux lines.
Numerical Simulation of 퐒퐢ퟏ−퐱퐆퐞퐱 Thin Film Solar Cell Using AMPS - 1DIOSR Journals
This document describes a numerical simulation of thin film silicon-germanium solar cells using the AMPS-1D simulation program. The simulation varied the germanium concentration in the intrinsic layer between 0-100% to determine the optimal band gap for maximum efficiency. The results showed that a germanium concentration of 90% produced the highest efficiency of 19.68% for device thicknesses under 6 micrometers. Above 6 micrometers, pure silicon performed equally well or better than silicon-germanium alloys.
1 ECE 6340 Fall 2013 Homework 8 Assignment.docxjoyjonna282
1
ECE 6340
Fall 2013
Homework 8
Assignment: Please do Probs. 1-9 and 13 from the set below.
1) In dynamics, we have the equation
E j Aω= − −∇Φ .
(a) Show that in statics, the scalar potential function Φ can be interpreted as a voltage
function. That is, show that in statics
( ) ( )
B
AB
A
V E dr A B≡ ⋅ = Φ −Φ∫ .
(b) Next, explain why this equation is not true (in general) in dynamics.
(c) Explain why the voltage drop (defined as the line integral of the electric field, as
defined above) depends on the path from A to B in dynamics, using Faraday’s law.
(d) Does the right-hand side of the above equation (the difference in the potential
function) depend on the path, in dynamics?
Hint: Note that, according to calculus, for any function ψ we have
dr dx dy dz d
x y z
ψ ψ ψ
ψ ψ
∂ ∂ ∂
∇ ⋅ = + + =
∂ ∂ ∂
.
2) Starting with Maxwell’s equations, show that the electric field radiated by an impressed
current density source J i in an infinite homogeneous region satisfies the equation
( )2 2 iE k E E j Jωµ∇ + = ∇ ∇⋅ + .
Then use Ampere’s law (or, if you prefer, the continuity equation and the electric Gauss
law) to show that this equation may be written as
( )2 2 1 i iE k E J j J
j
ωµ
σ ωε
∇ + = − ∇ ∇⋅ +
+
.
2
Note that the total current density is the sum of the impressed current density and the
conduction current density, the latter obeying Ohm’s law (J c = σE).
Explain why this equation for the electric field would be harder to solve than the equation
that was derived in class for the magnetic vector potential.
3) Show that magnetic field radiated by an impressed current density source satisfies the
equation
2 2 iH k H J∇ + = −∇× .
Explain why this equation for the magnetic field would be harder to solve than the
equation that was derived in class for the magnetic vector potential.
4) Show that in a homogenous region of space the scalar electric potential satisfies the
equation
2 2
i
v
c
k
ρ
ε
∇ Φ + Φ = − ,
where ivρ is the impressed (source) charge density, which is the charge density that goes
along with the impressed current density, being related by
i ivJ jωρ∇⋅ = −
Hint: Start with E j Aω= − −∇Φ and take the divergence of both sides. Also, take the
divergence of both sides of Ampere’s law and use the continuity equation for the
impressed current (given above) to show that
1 ii v
c c
E J
j
ρ
ωε ε
∇⋅ = − ∇⋅ = .
Note: It is also true from the electric Gauss law that
vE
ρ
ε
∇⋅ = ,
but we prefer to have only an impressed (source) charge density on the right-hand side of
the equation for the potential Φ. In the time-harmonic steady state, assuming a
homogeneous and isotropic region, it follows that ρv = ρvi. That is, there is no charge
3
density arising from the conduction current. (If there were no impressed current sources,
the total charge density would therefore be ze ...
This document presents a dielectric covered hairpin probe for measuring electron density in reactive plasmas. Covering the hairpin probe with a thin dielectric layer, such as quartz, protects the probe surface from contamination and allows it to operate at higher densities by reducing energy losses. The document develops a model to account for the additional capacitance introduced by the dielectric layer. Experimental tests in an argon plasma verify that electron density measurements made with the covered probe, when corrected using the new model, agree well with measurements from an uncovered reference probe. The dielectric covering improves the performance of the hairpin probe in reactive plasmas.
Kaytlin Brinker modeled proton exchange membrane fuel cells by developing equations to describe the charge distribution in the Stern layer and diffuse layer of the membrane. Her results showed that including the charge imbalance in the membrane is important for calculating fuel cell properties accurately. For future work, she plans to use molecular modeling software to simulate proton diffusion in Nafion under varying hydration conditions and predict transport properties from a theoretical diffusion model.
This report discusses an experiment on flow visualization of electro-kinetic force chemical mechanical planarization (EKF-CMP). Straight and radial electrode specimens were created in glass chambers to observe flow patterns under various voltages and solution conditions using video and simulations. Results showed that lower gap lengths and radial electrode orientation produced faster fluid flow. While solution pH had little effect, EKF-CMP was found to significantly improve material removal rate over conventional CMP, making it suitable for creating refined 3D stacked wafers needed to satisfy Moore's Law. In conclusion, the experiment validated that external voltage application enhances CMP performance.
1. Mathematical Process Structuring Cellular Transmembrane Potential Model
M. Stefani
Old Dominion University, Norfolk, Virginia, USA
The focus of this paper is to discuss the various formal mathematical models used in the calcu-
lations for transmembrane potential present in cell walls during exposure to electric fields. This
has direct application in the field of bioelectrical physics where the theoretical effect known as elec-
troporation is caused by these transmembrane potentials. Therefore, in order to structure realistic
simulations these potentials must be accurately calculated using apropos assumptions, approxima-
tions and boundaries. As always, mathematics has supplied the tools to create a useful model, but
the proper tools must be chosen for the right formulations. The distillation of such a model is to be
presented in this paper.
Keywords: Transmembrane, Potenital, Model
I. INTRODUCTION
Within the field of bioelectrical physics, there is a par-
ticular interest in the effects of electric fields on biological
tissue. One such theorized effect is known as electropo-
ration. Electroporation being an increase of permeabil-
ity in a cell membrane cause by a transmembrane po-
tential (TMP). These potentials are induced by pulsed
external electric fields around the cell. The interaction
of these fields with the conductive mediums within and
without the cell, cause a strong electromagnetic interac-
tion in the immediate vicinity of the membrane. Then
through molecular interactions, it causes pores to form
in the bi-lipid layers (the fundamental structures of cell
walls). This effect takes place on a scale that makes direct
visualization all but impossible. We can make molecular
dynamic models for very small sections of a bi-lipid layer
or experiments can be done on full cells, but directly mea-
suring the poration is not currently possible and neither
is complete accuracy in measuring the TMP+
.
Therefore, models to calculate the possible TMP are
extremely important in these studies that have far reach-
ing effects in the field of medicine, biology, and applied
physics. In this paper, different ”generations” of possible
models will be presented and discussed in terms based on
Dr. Adam’s ”idealization of the process of mathematical
modeling” presented in FIG.1
Each generation of model will be an iteration passed
through this feedback loop probing its strengths and
weaknesses in order to build the next model until an ac-
ceptable level of complexity and accuracy is reached.
II. APPROACHING THE PROBLEM
A. General solution to the Laplace Equation
Fundamentally, the desired model will calculate the
potential for any point on the membrane. In order to do
this reasonably, a solution for all points in space is re-
quired. Electrostatically, this means the potential can be
described by a function that solves the Laplace equation.
FIG. 1. Idealization of the process of mathematical modeling
2
Φ = 0 (1)
The Laplace equation has known general solutions for
different symmetries. A uniform electric field applied to
an idealized spherical cell is assumed. The general solu-
tion in this geometry is:
Φ(r, θ, φ) =
∞
l=0
l
m=−l
[Almrl
+ Blm
1
rl+1
]Ylm(θ, φ) (2)
Here, the terms A and B are solved in each region by
using matching conditions(given by potential theory) to
adjoining regions where E0 is the uniform field and σ is
the charge distribution.
lim
r→∞
(− Φ) = E0 (3)
Φi = Φo (4)
ˆn · (σoEo − σiEi) = 0 (5)
2. 2
The Yml terms in the general solution are the Spherical
harmonics. This solution can be simplified because of the
uniform field assumption which would cause an azimuthal
symmetry in the charge distribution on the cell. This
sets the m indices to 0 and this results in the well know
Legendre-polynomials. This gives the ”ready-to-apply”
general solution for the first generation model.
Φ(r, θ) =
∞
l=0
[Almrl
+ Blm
1
rl+1
]Pl(cos(θ)) (6)
III. FIRST GENERATION: SPHERICAL
SYMETRY
As FIG.2 shows, there are permittivity and conductiv-
ity factors to considered inside and outside the cell and
yet another possible value given by the membrane itself.
The first generation model requires some assumptions.
Namely, that the cell wall behaves as a non-conductive
shell and that the material outside and inside the medium
is conductive. Using the general solution, create a system
of equations describing the potential outside and inside
of the cell. Then, solve the system of equations by ap-
plying matching conditions that gives the complete set of
equations that describe the potential throughout space.
Then, solve for the TMP, which is the difference in po-
tential on the outside and inside of the shell.
∆Φ = Φo − Φi (7)
From this, the well established Schwans equation is de-
rived.
∆Φ =
3C
2K
ERcos(θ) (8)
C = λo[3dR2
λi + (3d2
R − d3
)(λm − λi)] (9)
K = R3
(λm+2λo)(λm+
1
2
λi)−(R−d)3
(λo−λm)(λi−λm)
(10)
This is the first generation solution to the potential.
R is the radius of the sphere, d is the thickness of the
cell wall and the λ terms are the various permittivitys.
This rather long expression tells us the potential will be
cos(θ) dependent and directly related to the strength of
the applied electric field.
A. ”Simple as posible, but not simpler”
This is good, but the solution is cumbersome and can
be simplified. According to Kotnik at el.(2000) when
FIG. 2. spherical model
applying the non-conductive assumption λm = 0, as well
as other physical values for λ, this expression C
K is very
accurately approximated to 1. This gives the final first
generation model.
∆Φ =
3
2
ERcos(θ)ˆr (11)
Using the magnitude of EF this calculates the TMP per-
pendicular to the surface with a θ dependence.
B. Errors, assumptions, changes: something is
wrong here, an ellipsoid belongs here
When using all physical values in C and K it changes
the first generation estimate about 0.1%. Meaning, these
details can be safely discarded in this leading term. How-
ever, one assumption made could cause a large difference
in the potential.
Cells aren’t actually perfectly spherical. Due to this,
most other researchers have advanced their models by
adapting to a spheroid. Following a path not traveled,
and perusing a better geometry for a more physically
accurate and generalizable model, this paper will use an
ellipsoid geometry.
IV. SECOND GENERATION: SPHERE TODAY,
GONE TOMORROW
Progressing to an ellipsoid has a much uglier solution.
This is because there is no general solution to the Laplace
equation for this geometry. Therefore, method used to
calculate the potential must adjusted. In order to solve
for an arbitrary geometry, the integrable form of the
laplace equation is required. Starting in ellipsoidal co-
ordinates, the final itegrable equation (Eq.14) is derived.
2
Φ =
4 ϕ(ξ)
(ξ − µ)(ξ − ν)
∂
∂ξ
ϕ(ξ)
∂Φ
∂ξ
= 0 (12)
ϕ(ξ)
∂Φ
∂ξ
= −
E
2
(13)
3. 3
converting ξ into surface element gives
Φ(ξ) =
∞
ξ
Eds
2 ϕ(s)
(14)
where
ϕ(s) = (a2
µ2
+ s)(b2
µ2
+ s)(c2
µ2
+ s) (15)
ξ = 0 on the surface of any scaled ellipse by a factor µ
described by
x2
a2
+
y2
b2
+
z2
c2
= µ2
(16)
From Eq.14 ”shells” of ellipsoidal shapes can be inte-
grated throughout space. Using this fundamental inte-
gral form of potential, the calculation for external poten-
tial can be integrated at µ = 1 and, from there, an inter-
nal potential can be calculated by integrating the scaling
factor µ over the thickness of the cell wall. The differ-
ence of these potentials, along with appropriate matching
conditions, will give the new TMP. Unfortunately, this is
were the model leaves the realm of neat analytic solu-
tions.
These integrals result in a function known as the el-
liptic integral of the first kind and is heavily dependent
on the magnitudes of the axis of the ellipsoid. However,
an approximate answer can be given by an expansion of
this function(Gradshteyn2007) and through this process
the potential for an ellipsoid is solved. The less than
attractive result is given by:
∆Φ = (E·ˆn)
a2
+ b2
+ c2
2
√
a2 − b2
x2
a4
+
y2
b4
+
z2
c4
F(a, b, c) (17)
F(a, b, c) sin−1
√
a2 − b2
a
+
a2
−c2
a2−b2 sin−1
(
√
a2−b2
a )3
6
+...
(18)
So stands the second generation model for TMP per-
pendicular to the ellipsoid surface.
There are several important considerations to make
when analyzing this model. One such consideration is
that this solution is true in the limit where x, y, z satis-
fying the elliptic equation describing the cell wall.
x2
a2
+
y2
b2
+
z2
c2
= 1 (19)
Another important note is that this model is dependent
on orientation of the EF. Its directional normal vector of
the EF will result in different final expressions when con-
verted to ellipsoidal coordinates. It is very pleasing to
note, that when a = b = c a sphere is restored and con-
verting the normal directional vector from the unifrom
EF to radial normal vector resurrects the Schwans equa-
tion (see appendix).
A. Errors, assumptions, changes: can’t have a
perfect model for an imperfect world
Geometrically, and physically this is a good model for
TMP in a constant electric field (EF). However, in the in-
troduction, it is mentioned that the poration is produced
by a pulsed EF*. These pulses can take many forms: Si-
nusoidal, Retangular, Triangular, ect. Our current model
assumes a uniform and constant EF. Physically, as the
EF intensity changed with these pulses, the TMP would
also change. It is possible to make a rough model of these
changes by basing a model on a rectangular pulse. Es-
sentially, this models a cell exposed to a direct current
EF that is suddenly turned off. We assume there will
only be rectangular pulses because other pulses require
unique solutions given by the Laplace transformation†
.
Furthermore, the assumption that after the pulse, the
cell behaves much like a capacitor is made. Capacitors
in this situation will discharge its voltage exponentially.
The next model will tackle this complication.
V. THIRD GENERATION: THERE’S STILL
TIME
The cell is expected to behave as a capacitor that would
discharge exponentially. This kind of discharge can be
modeled by:
∆Φ(t) = ∆Φ[1 − exp(−
t
τ
)] (20)
Here, t is time and τ is the time constant of a capac-
itor. This constant can be derived from the physics of
capacitance in electrostatics.
τ =
R m(2λo + λi)
2dλoλi
(21)
m
d is the capacitance of the membrane. Combining
this multiplicative decaying factor with the second model
produces the final model. Do to the excessive length of
this expression, its explicit representation is omitted.
A. Errors, assumptions, changes: know when to
hold ’em, know when to fold ’em
The remaining errors of this model are extremely com-
plex to resolve. For example, a real world cell would not
have a uniform cell wall. To resolve this, finite element
analysis is required (no thank you). Another example is
the models limitation to a rectangular pulse. This can be
resolved by reducing the model back to a sphere (losing
the greatest strength of this model) and utilizing Laplace
transformation to account for the time varying electric
field.
4. 4
VI. DISCUSSION
Throughout the modeling process good approxima-
tions were made both model and method were scraped
between generations, as a result, a new realistic and pow-
erful model of transmembrane potential is created. The
strength of this new model is based in the more gener-
alized ellipsoid geometry when compared to past mod-
els based on spheroids. By varying the axis length, a
spheres, spheroid and even close approximations to rod-
shaped cells common in bacteria, can be modeled. Such a
model was not created before. The only limitation of this
model is the assumption of rectangular pulse. The next
step for this research is to create a program that will be
able to numerically calculate and graph these potentials,
given cell and experimental parameters.
VII. ACKNOWLEDGMENTS
Many thanks to maths, physics, and those who’ve
taught me.
VIII. APPENDIX: A VESTIGIAL ORGAN,
ONLY OF INTEREST IF THERE’S A PROBLEM
A. Further maths
• Ellipsoidal coordinates:
x2
=
(a2
+ ξ)(a2
+ µ)(a2
+ ν)
(a2 − b2)(a2 − c2)
(A-1)
y2
=
(b2
+ ξ)(b2
+ µ)(b2
+ ν)
(b2 − a2)(b2 − c2)
(A-2)
z2
=
(c2
+ ξ)(c2
+ µ)(c2
+ ν)
(c2 − a2)(c2 − b2)
(A-3)
• (ξ, µ, ν) orthogonal basis:
∂x
∂ξ
∂x
∂µ
+
∂y
∂ξ
∂y
∂µ
+
∂z
∂ξ
∂z
∂µ
= 0 (A-4)
∂x
∂µ
∂x
∂ν
+
∂y
∂µ
∂y
∂ν
+
∂z
∂µ
∂z
∂ν
= 0 (A-5)
∂x
∂ξ
∂x
∂ν
+
∂y
∂ξ
∂y
∂ν
+
∂z
∂ξ
∂z
∂ν
= 0 (A-6)
• Elliptical normal vector conversion:
ˆn =
x
a2 , y
b2 , z
c2
x2
a4 + y2
b4 + z2
c4
(A-7)
• Calculation of integral:
t = s
µ2
∆Φ =
∞
ξ
Eds
2 ϕ(s)
−
1
µ0
µ
∞
ξ
Edt
2 ϕ(t)
dµ (A-8)
where ξ is the greatest root of the function
f(µ, ξ) =
x2
a2µ2 + ξ
+
y2
b2µ2 + ξ
+
z2
c2µ2 + ξ
− 1 (A-9)
and
1 − µ0 = ∆µ = d
x2
a4
+
y2
b4
+
z2
c4
(A-10)
using this and factor P for matching conditions
yeilds
∆Φ = P
E
4
d
x2
a4
+
y2
b4
+
z2
c4
∞
ξ
ds
ϕ(s)
(A-11)
the indefinite elliptic integral is a special function,
elliptic integral of the first kind E.I.
ds
ϕ(s)
=
2E.I. sin−1
√
a2−b2
√
a2−s2
|a2
−c2
a2−b2
√
a2 − b2
+ const.
(A-12)
which expands as
E.I.[β|δ] = β +
δβ3
6
+
(−4δ + 9δ2
)β5
120
+ O[β7
] (A-13)
all of these elements put together, give Eq.17 and
Eq.18 in the paper.
• Ellipse returns to sphere:
a = b = c = r (A-14)
x2
+ y2
+ z2
= r2
(A-15)
expand sin−1
sin−1
√
a2 − b2
a
=
√
a2 − b2
a
+ ... (A-16)
choose E to be uniform in ˆz
ˆz · ˆn = rcos(θ) · ˆr (A-17)
substituting all of these new conditions
∆Φ = (Ercos(θ)ˆr)
3r2
2
√
a2 − b2
x2 + y2 + z2
r4
√
a2 − b2
r
+ 0 + ...
(A-18)
which simplifies to Schwan’s equation
∆Φ =
3
2
Ercos(θ)ˆr (A-19)
5. 5
B. Footnotes
* The EF must be pulsed because in order for poration
to occur. This is because the field strength must be very
large and if left constant the cell would simply rip apart.
† This is reasonable after reading Kotnik 1998 paper.
Rectangular pulses are also the most commonly used
pulse form.
+ Transmembrane potentials translate to physical
voltage across the membrane.
[1] H.P. Schwan. Electrical properties of tissue and cell sus-
pensions. Adv. Biol. Med. Phys. 5: 147-209, (1997)
[2] C. Grosse, H.P. Schwan. Cellular membrane potenitals in-
duced by alternating fields. Biophys. j. 63: 1632-1642,
(1992)
[3] T. Kotnik, D. Miklavcic, T. Slivnik. Time course of trans-
membrane voltage induced by time-varying electric fields-a
method for theoretical analysis and its application. Bio-
electrochem. Bioenerg. 45: 3-16, (1998)
[4] T. Kotnik, D. Miklavcic. Second-order model of membrane
electric field induced by alternating external electric fields.
IEEE Trans. Biomed. Eng. 47: 1078-1081, (2000)
[5] T. Kotnik. Resting and Induced Transmembrane Voltage.
Electroporation. Tech. Tre. 1: 7-13, (2014)
[6] I.S. Gradishteyn, I.M. Ryzhik. Table of Integrals, Series,
and Products Seventh Edithon 5: 619,621,859 (2007)