Teses / dissertações: "Équations de Poisson-Nernst Planck"

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Paragot, Paul. "Analyse numérique du système d'équations Poisson-Nernst Planck pour étudier la propagation d'un signal transitoire dans les neurones". Electronic Thesis or Diss., Université Côte d'Azur, 2024. http://www.theses.fr/2024COAZ5020.

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Les questions neuroscientifiques concernant les dendrites incluent la compréhension de leur plasticité structurale en réponse à l'apprentissage et la manière dont elles intègrent les signaux. Les chercheurs visent à élucider ces aspects pour améliorer notre compréhension de la fonction neuronale et de ses complexités. Cette thèse vise à offrir des perspectives numériques concernant la dynamique du voltage et des ions dans les dendrites. Notre objectif est de modéliser l'excitation neuronale dans les dendrites. Nous abordons la dynamique ionique suite à l'afflux de signaux nerveux. Pour les simuler précisément, nous résolvons le système d'équations Poisson-Nernst-Planck (PNP). Le système PNP est reconnu comme le modèle standard pour caractériser le phénomène d'électrodiffusion des ions dans les électrolytes, y compris les structures dendritiques. Ce système non linéaire présente des défis en modélisation et en calcul en raison de la présence de couches limites rigides (BL). Nous proposons des schémas numériques basés sur la méthode des volumes finis Discrete Duality Finite Volumes (DDFV) pour résoudre le système PNP. Elle permet un raffinement local du maillage au niveau de la BL, en utilisant des maillages généraux. Cette approche facilite la résolution du système sur un domaine 2D représentant la géométrie des dendrites. Nous utilisons des schémas numériques préservant la positivité des concentrations ioniques. Chapitres 1 et 2 présentent le système PNP et la méthode DDFV ainsi que ses opérateurs discrets. Le chapitre 2 présente un couplage "linéaire" des équations et explore son schéma numérique associé. Ce couplage a des problèmes de convergence, où nous illustrons ses limites à travers des résultats numériques. Le chapitre 3 introduit un couplage "non linéaire", permettant une résolution numérique précise du système PNP. Les deux couplages sont effectués avec la méthode DDFV. Dans le chapitre 3, nous illustrons la convergence d'ordre 2 en espace. Nous simulons un cas test impliquant la BL. Nous appliquons le schéma DDFV à la géométrie des épines dendritiques en 2D et discutons nos simulations en les comparant avec des simulations en 1D de la littérature. Nous introduisons également deux configurations originales de dendrites, fournissant des informations sur la manière dont les épines dendritiques s'influencent mutuellement, révélant l'étendue de leur influence mutuelle. Nos simulations montrent la distance de propagation de l'influx ionique lors des connexions synaptiques. Dans le chapitre 4, nous résolvons le système PNP sur un système multi-domaines 2D composé d'une membrane, d'un milieu interne et d'un milieu externe. Cette approche permet la modélisation de la dynamique du voltage de manière plus réaliste, et aide également à vérifier la cohérence des résultats du chapitre 3. Nous utilisons le logiciel FreeFem++ pour résoudre le système PNP dans ce contexte. Nous présentons des simulations correspondant aux résultats du chapitre 3, démontrant la sommation linéaire dans une bifurcation dendritique. Nous étudions la sommation des signaux en ajoutant des entrées à la membrane d'une branche dendritique. Nous identifions un seuil d'excitabilité où la dynamique du voltage est significativement influencée par le nombre d'entrées. Nous offrons également des illustrations numériques de la BL à l'intérieur du milieu intracellulaire, observant de petites fluctuations. Ces résultats sont préliminaires, visant à fournir des informations pour comprendre la dynamique dendritique. Le chapitre 5 présente un travail collaboratif mené lors du Cemracs 2022. Nous nous concentrons sur un schéma de volumes finis composite où nous visons à dériver les équations d'Euler avec des termes sources sur des maillages non structurés
Neuroscientific questions about dendrites include understanding their structural plasticityin response to learning and how they integrate signals. Researchers aim to unravel these aspects to enhance our understanding of neural function and its complexities. This thesis aims at offering numerical insights concerning voltage and ionic dynamics in dendrites. Our primary focus is on modeling neuronal excitation, particularly in dendritic small compartments. We address ionic dynamics following the influx of nerve signals from synapses, including dendritic spines. To accurately represent their small scale, we solve the well-known Poisson-Nernst-Planck (PNP) system of equations, within this real application. The PNP system is widely recognized as the standard model for characterizing the electrodiffusion phenomenon of ions in electrolytes, including dendritic structures. This non-linear system presents challenges in both modeling and computation due to the presence of stiff boundary layers (BL). We begin by proposing numerical schemes based on the Discrete Duality Finite Volumes method (DDFV) to solve the PNP system. This method enables local mesh refinement at the BL, using general meshes. This approach facilitates solving the system on a 2D domain that represents the geometry of dendritic arborization. Additionally, we employ numerical schemes that preserve the positivity of ionic concentrations. Chapters 1 and 2 present the PNP system and the DDFV method along with its discrete operators. Chapter 2 presents a "linear" coupling of equations and investigate its associated numerical scheme. This coupling poses convergence challenges, where we demonstrate its limitations through numerical results. Chapter 3 introduces a "nonlinear" coupling, which enables accurate numerical resolution of the PNP system. Both of couplings are performed using DDFV method. However, in Chapter 3, we demonstrate the accuracy of the DDFV scheme, achieving second-order accuracy in space. Furthermore, we simulate a test case involving the BL. Finally, we apply the DDFV scheme to the geometry of dendritic spines and discuss our numerical simulations by comparing them with 1D existing simulations in the literature. Our approach considers the complexities of 2D dendritic structures. We also introduce two original configurations of dendrites, providing insights into how dendritic spines influence each other, revealing the extent of their mutual influence. Our simulations show the propagation distance of ionic influx during synaptic connections. In Chapter 4, we solve the PNP system over a 2D multi-domain consisting of a membrane, an internal and external medium. This approach allows the modeling of voltage dynamics in a more realistic way, and further helps checking consistency of the results in Chapter 3. To achieve this, we employ the FreeFem++ software to solve the PNP system within this 2D context. We present simulations that correspond to the results obtained in Chapter 3, demonstrating linear summation in a dendrite bifurcation. Furthermore, we investigate signal summation by adding inputs to the membrane of a dendritic branch. We identify an excitability threshold where the voltage dynamics are significantly influenced by the number of inputs. Finally, we also offer numerical illustrations of the BL within the intracellular medium, observing small fluctuations. These results are preliminary, aiming to provide insights into understanding dendritic dynamics. Chapter 5 presents collaborative work conducted during the Cemracs 2022. We focus on a composite finite volume scheme where we aim to derive the Euler equations with source terms on unstructured meshes

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Cartailler, Jérôme. "Asymptotic of Poisson-Nernst-Planck equations and application to the voltage distribution in cellular micro-domains". Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066297/document.

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Dans cette thèse j’étudie l’impact de la géométrie de micro et nano-domaines biologiques sur les propriétés d'électrodiffusion, ceci à l'aide des équations aux dérivées partielles de Poisson-Nernst-Planck. Je considère des domaines non-triviaux ayant une forme cuspide ou elliptique. Mon objectif est de développer des modèles ainsi que des méthodes mathématiques afin d'étudier les caractéristiques électriques de ces nano/micro-domaines, et ainsi mieux comprendre comment les signaux électriques sont modulés à ces échelles. Dans la première partie j’étudie le voltage à l'équilibre pour un électrolyte dans un domaine borné, et ayant un fort excès de charges positives. Je montre que le premier temps de sortie dans une boule chargée dépend de la surface et non du volume. J’étudie ensuite la géométrie composées d'une boule à laquelle est attachée un domaine cuspide. Je construis ensuite une solution asymptotique pour le voltage dans les cas 2D et 3D et je montre qu’ils sont donnés au premier ordre par la même expression. Enfin, j’obtiens la même conclusion en considérant une géométrie formée d'une ellipse, dont je construis une solution asymptotique du voltage en 2D et 3D. La seconde partie porte sur la modélisation de la compartimentalisation électrique des épines dendritiques. A partir de simulations numériques, je mets en évidence le lien entre la polarisation de concentration dans l'épine et sa géométrie. Je compare ensuite mon modèle à des données de microscopie. Je développe une méthode de déconvolution pour extraire la dynamique rapide du voltage à partir des données de microscopie. Enfin j’estime la résistance du cou et montre que celle-ci ne suit pas la loi d'Ohm
In this PhD I study how electro-diffusion within biological micro and nano-domains is affected by their shapes using the Poisson-Nernst-Planck (PNP) partial differential equations. I consider non-trivial shapes such as domains with cusp and ellipses. Our goal is to develop models, as well as mathematical tools, to study the electrical properties of micro and nano-domains, to understand better how electrical neuronal signaling is regulated at those scales. In the first part I estimate the steady-state voltage inside an electrolyte confined in a bounded domain, within which we assume an excess of positive charge. I show the mean first passage time in a charged ball depends on the surface and not on the volume. I further study a geometry composed of a ball with an attached cusp-shaped domain. I construct an asymptotic solution for the voltage in 2D and 3D and I show that to leading order expressions for the voltage in 2D and 3D are identical. Finally, I obtain similar conclusion considering an elliptical-shaped domain for which I construct an asymptotic solution for the voltage in 2D and 3D. In the second part, I model the electrical compartmentalization in dendritic spines. Based on numerical simulations, I show how spines non-cylindrical geometry leads to concentration polarization effects. I then compare my model to experimental data of microscopy imaging. I develop a deconvolution method to recover the fast voltage dynamic from the data. I estimate the neck resistance, and we found that, contrary to Ohm's law, the spine neck resistance can be inversely proportional to its radius

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Abdul, Samad Feras. "Polarisation provoquée : expérimentation, modélisation et applications géophysiques". Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066049/document.

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Les mécanismes produisant les signaux observés par la méthode de Polarisation Provoquée (PP) sur une large gamme de fréquences (entre 1 mHz et 20 kHz) ne sont pas encore complètement identifiés. Deux sujets particuliers ont été abordés dans cette thèse. L'origine du signal observé à haute fréquence (>1 kHz) a été analysé en effectuant des mesures de la résistivité complexe sur de l'eau à différentes salinités. Les résultats montrent des écarts sur la phase par rapport à la réponse attendue à haute fréquence. Ils dépendent du type d'électrode de mesures et de la conductivité du milieu. Un modèle basé sur un circuit électrique équivalent a été proposé pour modéliser ces effets. Nous avons aussi exploré le mécanisme responsable de la polarisation en présence de grains semi-conducteurs en analysant la dépendance de temps de relaxation. La réponse spectrale d'un milieu sableux saturé a été étudiée en variant la concentration et le type de l'électrolyte, la taille, le type et la quantité de grains semi-conducteurs insérés. En utilisant la méthode des éléments finis, nous avons entrepris une simulation numérique 2D basée sur la solution des équations de Poisson-Nernst-Planck dans le domaine temporel et spectral. Les résultats expérimentaux sont conformes à ceux issus de la simulation numérique et montrent une décroissance comparable du temps de relaxation avec l'augmentation de la concentration de l'électrolyte. Finalement, une campagne géophysique de terrain sur un site paléo-miniers contenant des grains semi-conducteurs complète l'approche de laboratoire. Des mesures de PP temporelles permettent de délimiter les zones de scories sur le site et d'en estimer le volume
The physical mechanisms responsible for the induced polarization response over the frequency range (from 1 mHz to 20 kHz) are not completely understood. In particular, within the framework of this thesis, two subjects have been addressed. The origin of the signal observed at high frequency (HF) (>1 kHz) was analyzed by carrying out Spectral IP measurements on tap water samples. A phase deviation from the expected response has been observed at HF. The resulted deviation in phase appears to be dependent on the measuring electrode type (potential electrodes) and conductivity of the medium. A model based on an equivalent electrical circuit and designed to represent HF response, has been proposed to correct these effects. The mechanism responsible for the polarization in a medium containing semi-conductor grains has been investigated by analyzing the dependence of the relaxation time. We carried out experimental measurements on a sand medium containing different types of semi-conductors. The spectral response was studied by varying the concentration and type of the electrolyte, the size and content of semi-conductor grains. By using the finite element method, a 2D numerical simulation based on Poisson-Nernst-Planck equations was performed in time and frequency domains. The experimental results are qualitatively in accordance with numerical simulation. It showed a comparable decrease in the relaxation time when increasing the electrolyte concentration. Finally, field measurements on a paleo-mining site containing semi-conductor grains have been acquired. Time-domain IP measurements allowed us to define the zones of slag in the site and led to estimate the slag volume

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Abdul, Samad Feras. "Polarisation provoquée : expérimentation, modélisation et applications géophysiques". Electronic Thesis or Diss., Paris 6, 2017. http://www.theses.fr/2017PA066049.

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Les mécanismes produisant les signaux observés par la méthode de Polarisation Provoquée (PP) sur une large gamme de fréquences (entre 1 mHz et 20 kHz) ne sont pas encore complètement identifiés. Deux sujets particuliers ont été abordés dans cette thèse. L'origine du signal observé à haute fréquence (>1 kHz) a été analysé en effectuant des mesures de la résistivité complexe sur de l'eau à différentes salinités. Les résultats montrent des écarts sur la phase par rapport à la réponse attendue à haute fréquence. Ils dépendent du type d'électrode de mesures et de la conductivité du milieu. Un modèle basé sur un circuit électrique équivalent a été proposé pour modéliser ces effets. Nous avons aussi exploré le mécanisme responsable de la polarisation en présence de grains semi-conducteurs en analysant la dépendance de temps de relaxation. La réponse spectrale d'un milieu sableux saturé a été étudiée en variant la concentration et le type de l'électrolyte, la taille, le type et la quantité de grains semi-conducteurs insérés. En utilisant la méthode des éléments finis, nous avons entrepris une simulation numérique 2D basée sur la solution des équations de Poisson-Nernst-Planck dans le domaine temporel et spectral. Les résultats expérimentaux sont conformes à ceux issus de la simulation numérique et montrent une décroissance comparable du temps de relaxation avec l'augmentation de la concentration de l'électrolyte. Finalement, une campagne géophysique de terrain sur un site paléo-miniers contenant des grains semi-conducteurs complète l'approche de laboratoire. Des mesures de PP temporelles permettent de délimiter les zones de scories sur le site et d'en estimer le volume
The physical mechanisms responsible for the induced polarization response over the frequency range (from 1 mHz to 20 kHz) are not completely understood. In particular, within the framework of this thesis, two subjects have been addressed. The origin of the signal observed at high frequency (HF) (>1 kHz) was analyzed by carrying out Spectral IP measurements on tap water samples. A phase deviation from the expected response has been observed at HF. The resulted deviation in phase appears to be dependent on the measuring electrode type (potential electrodes) and conductivity of the medium. A model based on an equivalent electrical circuit and designed to represent HF response, has been proposed to correct these effects. The mechanism responsible for the polarization in a medium containing semi-conductor grains has been investigated by analyzing the dependence of the relaxation time. We carried out experimental measurements on a sand medium containing different types of semi-conductors. The spectral response was studied by varying the concentration and type of the electrolyte, the size and content of semi-conductor grains. By using the finite element method, a 2D numerical simulation based on Poisson-Nernst-Planck equations was performed in time and frequency domains. The experimental results are qualitatively in accordance with numerical simulation. It showed a comparable decrease in the relaxation time when increasing the electrolyte concentration. Finally, field measurements on a paleo-mining site containing semi-conductor grains have been acquired. Time-domain IP measurements allowed us to define the zones of slag in the site and led to estimate the slag volume

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Lefebvre, Xavier. "Etude des modèles de transfert en nanofiltration : application du modèle hybride basé sur les équations de Nernst-Planck étendues par le développement du logiciel de simulation "nanoflux"". Montpellier 2, 2003. http://www.theses.fr/2003MON20082.

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Moreau, Antoine. "Calcul des propriétés homogénéisées de transfert dans les matériaux poreux par des méthodes de réduction de modèle : Application aux matériaux cimentaires". Thesis, La Rochelle, 2022. http://www.theses.fr/2022LAROS024.

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Cette thèse propose de coupler deux outils préexistant pour la modélisation mathématique en mécanique : l’homogénéisation périodique et la réduction de modèle, afin de modéliser la corrosion des structures de béton armé exposées à la pollution atmosphérique et au sel marin. Cette dégradation est en effet difficile à simuler numériquement, eu égard la forte hétérogénéité des matériaux concernés, et la variabilité de leur microstructure. L’homogénéisation périodique fournit un modèle multi-échelle permettant de s’affranchir de la première de ces deux difficultés. Néanmoins, elle repose sur l’existence d’un volume élémentaire représentatif (VER) de la microstructure du matériau poreux modélisé. Afin de prendre en compte la variabilité de cette dernière, on est amenés à résoudre en temps réduit les équations issues du modèle multi-échelle pour un grand nombre VER. Ceci motive l’utilisation de la méthode POD de réduction de modèle. Cette thèse propose de recourir à des transformations géométriques pour transporter ces équations sur la phase fluide d’un VER de référence. La méthode POD ne peut, en effet, pas être utilisée directement sur un domaine spatial variable (ici le réseau de pores du matériau). Dans un deuxième temps, on adapte ce nouvel outil à l’équation de Poisson-Boltzmann, fortement non linéaire, qui régit la diffusion ionique à l’échelle de la longueur de Debye. Enfin, on combine ces nouvelles méthodes à des techniques existant en réduction de modèle (MPS, interpolation ITSGM), pour tenir compte du couplage micro-macroscopique entre les équations issues de l’homogénéisation périodique
In this thesis, we manage to combine two existing tools in mechanics: periodic homogenization, and reduced-order modelling, to modelize corrosion of reinforced concrete structures. Indeed, chloride and carbonate diffusion take place their pores and eventually oxydate their steel skeleton. The simulation of this degradation is difficult to afford because of both the material heterogenenity, and its microstructure variability. Periodic homogenization provides a multiscale model which takes care of the first of these issues. Nevertheless, it assumes the existence of a representative elementary volume (REV) of the material at the microscopical scale. I order to afford the microstructure variability, we must solve the equations which arise from periodic homogenization in a reduced time. This motivates the use of model order reduction, and especially the POD. In this work we design geometrical transformations that transport the original homogenization equations on the fluid domain of a unique REV. Indeed, the POD method can’t be directly performed on a variable geometrical space like the material pore network. Secondly, we adapt model order reduction to the Poisson-Boltzmann equation, which is strongly nonlinear, and which rules ionic electro diffusion at the Debye length scale. Finally, we combine these new methods to other existing tools in model order reduction (ITSGM interpolatin, MPS method), in order to couple the micro- and macroscopic components of periodic homogenization

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Herda, Maxime. "Analyse asymptotique et numérique de quelques modèles pour le transport de particules chargées". Thesis, Lyon, 2017. http://www.theses.fr/2017LYSE1165/document.

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Cette thèse est consacrée à l'étude mathématique de quelques modèles d'équations aux dérivées partielles issues de la physique des plasmas. On s'intéresse principalement à l'analyse théorique de différents régimes asymptotiques de systèmes d'équations cinétiques de type Vlasov-Poisson-Fokker-Planck. Dans un premier temps, en présence d'un champ magnétique extérieur on se concentre sur l'approximation des électrons sans masse fournissant des modèles réduits lorsque le rapport me{mi entre la masse me d'un électron et la masse mi d'un ion tend vers 0 dans les modèles. Suivant le régime considéré, on montre qu'à la limite les solutions vérifient des modèles hydrodynamiques de type convection-diffusion ou sont données par des densités de type Maxwell-Boltzmann-Gibbs, suivant l'intensité des collisions dans la mise à l'échelle. En utilisant les propriétés hypocoercives et hypoelliptiques des équations, on est capable d'obtenir des taux de convergence en fonction du rapport de masse. Dans un second temps, par des méthodes similaires, on montre la convergence exponentielle en temps long vers l'équilibre des solutions du système de Vlasov-Poisson-Fokker-Planck sans champ magnétique avec des taux explicites en les paramètres du modèles. Enfin, on conçoit un nouveau type de schéma volumes finis pour des équations de convection-diffusion non-linéaires assurant le bon comportement en temps long des solutions discrètes. Ces propriétés sont vérifiées numériquement sur plusieurs modèles dont l'équation de Fokker-Planck avec champ magnétique
This thesis is devoted to the mathematical study of some models of partial differential equations from plasma physics. We are mainly interested in the theoretical study of various asymptotic regimes of Vlasov-Poisson-Fokker-Planck systems. First, in the presence of an external magnetic field, we focus on the approximation of massless electrons providing reduced models when the ratio me{mi between the mass me of an electron and the mass mi of an ion tends to 0 in the equations. Depending on the scaling, it is shown that, at the limit, solutions satisfy hydrodynamic models of convection-diffusion type or are given by Maxwell-Boltzmann-Gibbs densities depending on the intensity of collisions. Using hypocoercive and hypoelliptic properties of the equations, we are able to obtain convergence rates as a function of the mass ratio. In a second step, by similar methods, we show exponential convergence of solutions of the Vlasov-Poisson-Fokker-Planck system without magnetic field towards the steady state, with explicit rates depending on the parameters of the model. Finally, we design a new type of finite volume scheme for a class of nonlinear convection-diffusion equations ensuring the satisfying long-time behavior of discrete solutions. These properties are verified numerically on several models including the Fokker-Planck equation with magnetic field

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Minton, Geraint Philip. "Modelling the static and dynamic behaviour of electrolytes : a modified Poisson-Nernst-Planck approach". Thesis, University of Manchester, 2014. https://www.research.manchester.ac.uk/portal/en/theses/modelling-the-static-and-dynamic-behaviour-of-electrolytes-a-modified-poissonnernstplanck-approach(de9671fd-feb5-4870-b0a9-ad6a28ff953d).html.

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In this thesis a method is presented for the modelling the effects of the excluded volume (ion-ion) and ion excess polarisability (ion-solvent) interactions in an electrolyte at a smooth planar electrode. The impact of these interactions is studied in terms of the equilibrium state of single and mixed electrolytes, the dynamic response of single electrolytes to a time-dependent applied potential, and their effect on the reaction rate, for both steady and time-dependent applied potentials. For reacting systems, the reaction rate is modelled using a modified form of the Frumkin-Butler-Volmer equation, in which the interactions are explicitly accounted for. At equilibrium, the model offers improvement over models which only account for the excluded volume interaction, in terms of both the predicted electrolyte structure and the electrical properties of the electrode. For example accounting for the polarisability interaction is shown to limit and then reverse the growth in the differential capacitance at the point of zero charge as the bulk concentration increases, an effect is not seen when only the excluded volume interaction is accounted for. Another example is for mixed electrolytes, in which accounting for the polarisability interaction leads to better agreement with experimental data regarding the composition of the double layer. For the response of an electrolyte to a potential step, the two interactions both lead to peaks in the time taken to reach equilibrium as a function of the potential. The effect of the domain length on the equilibration time is qualitatively discussed, together with the differences between the two interaction models. The response to a time-dependent potential is analysed through simulated electrochemical impedance spectroscopy and consideration of the capacitance dispersion effect. Between this and the potential step response data it is shown that the interactions themselves have little direct effect on the dynamic processes beyond the way in which they limit the ion concentrations in the double layer and alter the differential capacitance of the system. The investigation of the effect of the ion interactions on the reaction rate shows that both terms can either increase or decrease the rate, relative to a system with no interactions, depending on the details of the reaction and the applied potential. This is linked to the changes in the electric field within the double layer, which are caused by the interactions, and how this affects the reactant flux in that region. In terms of simulated EIS, deviations are observed relative to the equivalent circuit for the system, the reasons for which are discussed.

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Lim, Jong Il. "Transient finite element analysis of electric double layer using Nernst-Planck-Poisson equations with a modified stern layer". Texas A&M University, 2006. http://hdl.handle.net/1969.1/4703.

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Finite element analysis of electric double layer capacitors using a transient nonlinear Nernst-Planck-Poisson (NPP) model and Nernst-Planck-Poisson-modified Stern layer (NPPMS) model are presented in 1D and 2D. The NPP model provided unrealistic ion concentrations for high electrode surface potential. The NPPMS model uses a modified Stern layer to account for finite ion size, resulting in realistic ion concentrations even at high surface potential. The finite element solution algorithm uses the Newton-Raphson method to solve the nonlinear problem and the alpha family approximation for time integration to solve the NPP and NPPMS models for transient cases. Cubic Hermite elements are used for interfacing the modified Stern and diffuse layers in 1D while serendipity elements are used for the same in 2D. Effects of the surface potential and bulk molarity on the electric potential and ion concentrations are studied. The ability of the models to predict energy storage capacity is investigated and the predicted solutions from the 1D NPP and NPPMS models are compared for various cases. It is observed that NPPMS model provided realistic and correct results for low and high values of surface potential. Furthermore, the 1D NPPMS model is extended into 2D. The pore structure on the electrode surface, the electrode surface area and its geometry are important factors in determining the performance of the electric double layer capacitor. Thus 2D models containing a porous electrode are modeled and analyzed for understanding of the behavior of the electric double layer capacitor. The effect of pore radius and pore depth on the predicted electric potential, ion concentrations, surface charge density, surface energy density, and charging time are discussed using the 2D Nernst-Planck-Poissonmodified Stern layer (NPPMS) model.

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Neuen, Christian P. T. [Verfasser]. "Numerical Simulation of Ion Migration with Particle Dynamics and the Heat-Poisson-Nernst-Planck System / Christian P. T. Neuen". Bonn : Universitäts- und Landesbibliothek Bonn, 2016. http://d-nb.info/1124540164/34.

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Pods, Jurgis Jonas [Verfasser], e Peter [Akademischer Betreuer] Bastian. "Electrodiffusion Models of Axon and Extracellular Space Using the Poisson-Nernst-Planck Equations / Jurgis Jonas Pods ; Betreuer: Peter Bastian". Heidelberg : Universitätsbibliothek Heidelberg, 2014. http://d-nb.info/1179925513/34.

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Pods, Jurgis [Verfasser], e Peter [Akademischer Betreuer] Bastian. "Electrodiffusion Models of Axon and Extracellular Space Using the Poisson-Nernst-Planck Equations / Jurgis Jonas Pods ; Betreuer: Peter Bastian". Heidelberg : Universitätsbibliothek Heidelberg, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:16-heidok-171286.

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Filbet, Francis. "Contribution à l'analyse et la simulation numérique de l'équation de Vlasov". Nancy 1, 2001. http://docnum.univ-lorraine.fr/public/SCD_T_2001_0068_FILBET.pdf.

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Ce travail est consacré à l'étude de quelques problèmes de la physique des plasmas: le transport de particules chargées et l'étude des collisions. Dans un premier temps, plusieurs méthodes eulériennes pour la discrétisation de l'équation de Vlasov, modélisant le transport des particules, sont proposées. L'originalité de ces méthodes est l'utilisation d'un maillage ou d'une grille de l'espace des phases pouvant aller jusqu'à six dimensions. Une démonstration rigoureuse de la convergence et des estimations d'erreurs sont d'abord présentées pour un schéma simplifié. Puis des schémas d'ordre plus élevé sont proposés et appliqués à la physiques des faisceaux. Leur précision permet de mettre en évidence des phénomènes très fins comme la formation de halos. Ensuite, des schémas déterministes appliquées à l'opérateur de Landau, qui décrit les collisions binaires dans un plasma, sont proposés. Des tests numériques permettent de comparer les différentes méthodes et mettent en évidence l'effet des collisions dans l'évolution du plasma. Dans la dernière partie, le problème d'existence de solutions pour le modèle de Vlasov-Darwinen dimension trois est traité. Pour cela, des méthodes classiques sur les équations cinétiques et des résultats surles problèmes elliptiques sont utilisés. Enfin, la convergence du système de Vlasov-Darwin vers Vlasov-Poisson est prouvée.

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14

WANG, ZHENG. "HIERARCHICAL APPROACH TO PREDICTING TRANSPORT PROPERTIES OF A GRAMICIDIN ION CHANNEL WITHIN A LIPID BILAYER". University of Cincinnati / OhioLINK, 2003. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1069794237.

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15

MATSUNO, NOBUNAKA. "THEORETICAL AND EXPERIMENTAL STUDIES OF ION TRANSPORT THROUGH BIOLOGICAL MEMBRANE CHANNELS". University of Cincinnati / OhioLINK, 2003. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1060886930.

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16

Kubeil, Clemens. "Zum Einfluss elektrochemischer Doppelschichten auf den Stofftransport in nanoskaligen Elektrolytsystemen:". Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-218829.

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Es besteht enormes Interesse den Stofftransport in nanoskaligen Systemen zu verstehen und selektiv zu steuern, um analytische und synthetische Anwendungen zu entwickeln, aber auch um die physiologischen Prozesse lebender Zellen zu entschlüsseln. Im Rahmen dieser Arbeit wurde der Einfluss der elektrochemischen Doppelschicht an ausgewählten nanoskaligen Elektrolytsystemen untersucht. Die Gleichrichtung von Ionenströmen (engl. Ionic Current Rectification ICR) in Nanoporen mit einer Oberflächenladung äußert sich in einer gekrümmten Strom-Spannungs-Kurve. Die Überlappung von innerem und äußerem Potential ist dabei hinsichtlich der Ionenverteilung und somit der Porenleitfähigkeit einander verstärkend oder gegenläufig. Auf Grundlage dieses Mechanismus wurde die Gleichrichtung bei einem sehr großen Verhältnis von Porenöffnung zu Debye-Länge erklärt. Ferner wurde mittels der eingeführten relativen Leitfähigkeit κ´ die verschiedenen Leitfähigkeitszustände in Abhängigkeit der Elektrolytkonzentration und Temperatur sichtbar gemacht und Implikationen für Sensoranwendungen wie z.B. dem resistiven Pulszähler zur Partikelanalyse abgeleitet. Es wurde ein numerisches Modell basierend auf dem Poisson-Nernst-Planck-Gleichungssystem entwickelt, um die Translokation eines Nanopartikels durch eine konische Nanopore bei einer geringen Leitsalzkonzentration zu beschreiben. Neben dem klassischen Volumenausschluss-Effekt tritt zusätzlich ein Gleichrichtungseffekt (ICR-Effekt) in der Pore auf. Eine Analyse zur Entflechtung von Partikelgröße und Partikelladung aus der Pulshöhe und Pulsform wurde erfolgreich durchgeführt. Wie der Stofftransport durch eine Oberflächenladung auf dem umgebenden Material einer Nanoelektrode beeinflusst wird, wurde anhand des voltammetrischen Verhaltens diskutiert. An sehr kleinen Elektroden (< 10 nm) ist demnach der Einfluss der elektrochemischen Doppelschicht auf die Strom-Spannungs-Kurve besonders groß und kann auch bei Vorliegen eines hohen Leitsalzüberschusses nicht vernachlässigt werden. In leitsalzfreien Elektrolyten sind die gefundenen Effekte so deutlich, dass sie auch an größeren Elektroden experimentell zweifelsfrei festgestellt worden sind
There is an enormous interest in understanding and selectively controlling the material transport in nanoscale systems to develop analytical and synthetic applications, but also to decipher the physiological processes of living cells. Within this thesis, the influence of the electrochemical double layer on selected nanoscale electrolyte systems was studied. Ionic Current Rectification (ICR) in nanopores carrying a surface charge manifests itself in a non-linear current-voltage-curve. The overlap of interior and exterior potential is cumulative or opposing with regard to the ion distribution and therefore the pore conductivity. Based on this mechanism, ICR for very large ratios of pore size and Debye length was explained. Furthermore, the different conducting states as a function of electrolyte concentration and temperature were visualized by introducing the relative conductivity κ´ and hence implications for sensor applications such as the resistive pulse sensor have been deduced. A numerical model based on the Poisson-Nernst-Planck-equations was developed to describe the translocation of a nanoparticle through a conical nanopore at a low electrolyte concentration. An additional rectification effect (ICR effect) occurs in the pore beside the conventional volume exclusion effect. An analysis was successfully performed to deconstruct the particle size and particle charge from the pulse height and shape. The material transport is affected by a surface charge on the shrouding material of nanoelectrodes as it was discussed by means of the voltammetric behaviour. The influence of the electrochemical double layer on the current-voltage-curve is particularly large at very small electrodes (< 10 nm) and cannot be neglected even at a high excess of supporting electrolyte. The observed effects were pronounced in unsupported electrolytes, so that they could be clearly detected experimentally at even larger electrodes

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17

Backhaus, Karsten. "Das dielektrische Verhalten der Öl-Papier-Isolierung bei Belastung mit hoher Gleichspannung". Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-228992.

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Basierend auf den physikalischen Eigenschaften der unterschiedlichen ölintrinsischen und injizierten Ladungsträger wird ein neues Leitfähigkeitsmodell für Isolieröl und -papier für die Belastung mit hoher Gleichspannung aufgestellt. Das Modell wird mit der Wahl geeigneter Randbedingungen für das elektrische Feld und der Teilchenströme auf die Poisson-Nernst-Planck-Gleichung übertragen. Es steht damit ein Werkzeug zur Verfügung, das dielektrische Verhalten der Öl-Papier-Isolierung zu modellieren, dessen Parameter auf den physikalischen Ladungsträgereigenschaften wie Mobilität und Diffusion basieren. Mit dessen Hilfe werden sowohl die nichtlineare Leitfähigkeit als auch das Durchschlagverhalten als deren Extrapolation feldstärkeabhängig erklärt.

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18

Mohajeri, Arash. "Effective diffusion coefficients for charged porous materials based on micro-scale analyses". Connect to thesis, 2009. http://repository.unimelb.edu.au/10187/5780.

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Estimation of effective diffusion coefficients is essential to be able to describe the diffusive transport of solutes in porous media. It has been shown in theory that in the case of uncharged porous materials the effective diffusion coefficient of solutes is a function of the pore morphology of the material and can be described by their tortuosity (tensor). To estimate the apparent diffusion coefficients, the values of tortuosity and porosity should be known first. In contrast with calculation of porosity, which can be easily obtained, estimation of tortuosity is intricate, particularly with increasing micro-geometry complexity in porous media. Moreover, many engineering materials (e.g, clays and shales) are characterized by electrical surface charges on particles of the porous material which can strongly affect the diffusive transport properties of ions. For these materials, estimation of effective diffusion coefficients have been mostly based on phenomenological equations with no link to underlying microscale properties of these charged materials although a few recent studies have used alternative methods to obtain the diffusion parameters.
In the first part of this thesis a numerical method based on a recently proposed up-scaled Poisson-Nernst-Planck type of equation (PNP) and its microscale counterpart is employed to estimate the tortuosity and thus the effective and apparent diffusion coefficients in thin charged membranes. Beside this, a new mathematical approach for estimation of tortuosity is applied and validated. This mathematical approach is also derived while upscaling of micro-scale Poisson-Nernst-Planck system of equations using the volume averaging method. A variety of different pore 2D and 3D micro-geometries together with different electrochemical conditions are studied here. To validate the new approaches, the relation between porosity and tortuosity has been obtained using a multi-scale approach and compared with published results. These include comparison with the results from a recently developed numerical method that is based on macro and micro-scale PNP equations.
Results confirm that the tortuosity value is the same for porous media with electrically uncharged and charged particles but only when using a consistent set of PNP equations. The effects of charged particles are captured by the ratio of average concentration to effective intrinsic concentration in the macroscopic PNP equations. Using this ratio allows to consistently take into account electro-chemical interactions of ions and charges on particles and so excludes any ambiguity generally encountered in phenomenological equations.
Steady-state diffusion studies dominate this thesis; however, understanding of transient ion transport in porous media is also important. The last section of this thesis briefly introduces transient diffusion through bentonite. To do so, the micro Nernst-Planck equation with electro-neutrality condition (NPE) is solved for a porous medium which consists of compacted bentonite. This system has been studied before in another research using an experimental approach and the results are available for both transient and steady-state phases. Three different conditions are assumed for NPE governing equations and then the numerical results from these three conditions are compared to the experimental values and analytical phenomenological solution. The tortuosity is treated as a fitting parameter and the effective diffusion coefficient can be calculated based on these tortuosity values. The results show that including a sorption term in the NPE equations can render similar results as the experimental values in transient and steady state phases. Also, as a fitting parameter, the tortuosity values were found varying with background concentration. This highlights the need to monitor multiple diffusing ion fluxes and membrane potential to fully characterize electro-diffusive transport from fundamental principles (which have been investigated in first part of this thesis) rather than phenomenological equations for predictive studies.
This research has lead to two different journal articles submissions, one already accepted in Computers and Geotechnics (October 22, 2009, 5-yrs Impact Factor 0.884) and the other one still under review.

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19

Conway, Eamon. "Mathematical modelling of ionic transport through Nanopores". Thesis, Queensland University of Technology, 2019. https://eprints.qut.edu.au/134168/1/Eamon_Conway_Thesis.pdf.

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The transport of ionic species through nanopores is important in determining the underlying behaviour of electrolytes on the nanoscale; the understanding of which has important applications in the development of bio-molecular sensors and nanofluidic diodes. Importantly, this thesis has developed a novel mathematical model of ionic transport through a nanopore that can be quantitatively compared to experimental results. In doing so, we have been able to further understand the mechanisms behind ionic transport and explain previously unexplained experimental results.

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20

Bokun, G. S., Ya G. Groda, R. N. Lasovsky e V. S. Vikhrenko. "Charge Distribution Around Nanoscale Nonhomogeneities in Solid State Ionics". Thesis, Sumy State University, 2015. http://essuir.sumdu.edu.ua/handle/123456789/42717.

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The system of Nernst-Planck-Poisson equations is modified by including the gradient terms in the chemical potential expression. The gradient terms are important in the regions of significant inhomogeneities, e. g. near the interface boundaries. These modified equations are used for investigating the particle density distribution in the vicinity of interphase boundary of a solid electrolyte. The differential equation of the fourth order for the problem of contact between two solid phases is formulated. Its analytic solution which describes non monotonic distribution of charge in both phases is obtained. It is shown that the gradient component added in the transport equations makes a decisive contribution in the double layer region. The approach is further expanded to the system composed of bulk phases and the intergrain layer between them. The particle density distributions at different conditions are investigated. The quasy-one dimensional lattice model of the fuel cell is considered in the frame of kinetic Monte Carlo simulation. It is shown that the electrostatic interaction between ions makes a significant contribution to the activation energy of migration of the particles. On the other hand, the fluctuations of the energy barriers slightly increase the particle migration activation energy. It is found that at blocked electrodes in the near electrode regions electrical double layers are formed. The thickness of the electrical double layer is around few lattice constants.

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21

Dickinson, Edmund John Farrer. "Charge transport dynamics in electrochemistry". Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:e4acac56-7265-49ec-9a36-49b3ae6729ed.

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Electrolytic solutions contain mobile ions that can pass current, and are essential components of any solution-phase electrochemical system. The Nernst–Planck–Poisson equations describe the electrodynamics and transport dynamics of electrolytic solutions. This thesis applies modern numerical and mathematical techniques in order to solve these equations, and hence determine the behaviour of electrochemical systems involving charge transport. The following systems are studied: a liquid junction where a concentration gradient causes charge transport; an ideally polarisable electrode where an applied potential difference causes charge transport; and an electrochemical cell where electrolysis causes charge transport. The nanometre Debye length and nanosecond Debye time scales are shown to control charge separation in electrolytic solutions. At equilibrium, charge separation is confined to within a Debye length scale of a charged electrode surface. Non-equilibrium charge separation is compensated in solution on a Debye time scale following a perturbation, whereafter electroneutrality dictates charge transport. The mechanism for the recovery of electroneutrality involves both migration and diffusion, and is non-linear for larger electrical potentials. Charge separation is an extremely important consideration on length scales comparable to the Debye length. The predicted features of capacitive charging and electrolysis at nanoelectrodes are shown to differ qualitatively from the behaviour of larger electrodes. Nanoscale charge separation can influence the behaviour of a larger system if it limits the overall rate of mass transport or electron transfer. This thesis advocates the use of numerical methods to solve the Nernst–Planck–Poisson equations, in order to avoid the simplifying approximations required by traditional analytical methods. As this thesis demonstrates, this methodology can reveal the behaviour of increasingly elaborate electrochemical systems, while illustrating the self-consistency and generality of fundamental theories concerning charge transport.

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22

Volkov, Anton. "Ionic and electronic transport in electrochemical and polymer based systems". Doctoral thesis, Linköpings universitet, Fysik och elektroteknik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-135429.

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Electrochemical systems, which rely on coupled phenomena of the chemical change and electricity, have been utilized for development an interface between biological systems and conventional electronics. The development and detailed understanding of the operation mechanism of such interfaces have a great importance to many fields within life science and conventional electronics. Conducting polymer materials are extensively used as a building block in various applications due to their ability to transduce chemical signal to electrical one and vice versa. The mechanism of the coupling between the mass and charge transfer in electrochemical systems, and particularly in conductive polymer based system, is highly complex and depends on various physical and chemical properties of the materials composing the system of interest. The aims of this thesis have been to study electrochemical systems including conductive polymer based systems and provide knowledge for future development of the devices, which can operate with both chemical and electrical signals. Within the thesis, we studied the operation mechanism of ion bipolar junction transistor (IBJT), which have been previously utilized to modulate delivery of charged molecules. We analysed the different operation modes of IBJT and transition between them on the basis of detailed concentration and potential profiles provided by the model. We also performed investigation of capacitive charging in conductive PEDOT:PSS polymer electrode. We demonstrated that capacitive charging of PEDOT:PSS electrode at the cyclic voltammetry, can be understood within a modified Nernst-Planck-Poisson formalism for two phase system in terms of the coupled ion-electron diffusion and migration without invoking the assumption of any redox reactions. Further, we studied electronic structure and optical properties of a self-doped p-type conducting polymer, which can polymerize itself along the stem of the plants. We performed ab initio calculations for this system in undoped, polaron and bipolaron electronic states. Comparison with experimental data confirmed the formation of undoped or bipolaron states in polymer film depending on applied biases. Finally, we performed simulation of the reduction-oxidation reaction at microband array electrodes. We showed that faradaic current density at microband array electrodes increases due to non-linear mass transport on the microscale compared to the corresponding macroscale systems. The studied microband array electrode was used for developing a laccase-based microband biosensor. The biosensor revealed improved analytical performance, and was utilized for in situ phenol detection.

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23

Kramer, Stephan Christoph. "CUDA-based Scientific Computing". Doctoral thesis, Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2012. http://hdl.handle.net/11858/00-1735-0000-000D-FB52-0.

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24

Fortin, Pierre. "Algorithmique hiérarchique parallèle haute performance pour les problèmes à N-corps". Phd thesis, Université Sciences et Technologies - Bordeaux I, 2006. http://tel.archives-ouvertes.fr/tel-00135843.

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Cette thèse porte sur la méthode dite « méthode multipôle rapide » qui résout hiérarchiquement le problème à N-corps avec une complexité linéaire pour n'importe quelle précision. Dans le cadre de l'équation de Laplace, nous souhaitons pouvoir traiter efficacement toutes les distributions de particules rencontrées en astrophysique et en dynamique moléculaire.
Nous étudions tout d'abord deux expressions distinctes du principal opérateur (« multipôle-to-local ») ainsi que les bornes d'erreur associées. Pour ces deux expressions, nous présentons une formulation matricielle dont l'implémentation avec des routines BLAS (Basic Linear Algebra Subprograms) permet d'améliorer fortement l'efficacité de calcul. Dans la gamme de précisions qui nous intéresse, cette approche se révèle plus performante que les améliorations existantes (FFT, rotations et ondes planes), pour des distributions uniformes ou non.
Outre une nouvelle structure de données pour l'octree sous-jacent et des contributions algorithmiques à la version adaptative, nous avons aussi efficacement parallélisé notre méthode en mémoire partagée et en mémoire distribuée. Enfin, des comparaisons avec des codes dédiés justifient l'intérêt de notre code pour des simulations en astrophysique.

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25

Hsieh, Chia-Yu, e 謝佳佑. "On Poisson-Nernst-Planck Type Equations". Thesis, 2014. http://ndltd.ncl.edu.tw/handle/53108790841063399781.

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博士
國立臺灣大學
數學研究所
102
This thesis consists of three parts. The first part is devoted to the stability problem of boundary layer solutions to Poisson-Nernst-Planck (PNP) systems. PNP system has been widely used to describe the electron transport of semiconductors and the ion transport of ionic solutions, and plays a crucial role in the study of many physical and biological problems. PNP system with Robin boundary condition for the electrostatic potential admits a boundary layer solution as a steady state. We obtain some stability results by studying the asymptotic behavior for the steady state. In the second part, we study existence of solutions for a modified PNP system. Historically, from the Debye-Huckel theory, PNP systems are adopted for dilute ionic solutions. However, in biological system, ionic solutions are usually highly concentrated. The modified PNP system, which takes into account the relative drag from interaction of different ions, involves much more complicated nonlinear coupling between unknown variables. Compare with the original PNP system, it brings extra difficulties in analysis. We use Galerkin''s method and Schauder''s fixed-point theorem, and give an estimate of upper bounds of solutions to prove existence of local solutions. The third part deals with existence of solutions of a modified PNP equation with cross-diffusion. By using Galerkin''s method and Schauder''s fixed-point theorem, we obtain the local existence for this system. Moreover, we obtain the global existence by uniform in time L^2 estimates of solutions. We also consider a modified Keller-Segel system with similar modification and develop some local and global existence results.

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26

Lo, Jen-Ho, e 羅仁和. "Size-Modified Poisson-Nernst-Planck Model". Thesis, 2012. http://ndltd.ncl.edu.tw/handle/96805650568180052217.

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碩士
國立新竹教育大學
應用數學系碩士班
100
The Poisson-Nernst-Planck (PNP) model is a basic continuum model for simulating ionic flows in an open ion channel. The effects of finite particle size on electrostatics, density profile, and diffusion have been a long existing topic in the study of ionic solution [8]. The Poisson equation is derived from Coulomb's law in electrostatics and Gauss's theorem in calculus. The Nernst-Planck equation is equivalent to the convection-diffussion model. An entropy functional that accounts for the finite size effects of ions in electrolytes proposed by Borukhov et al. [1] for the Poisson-Boltzmann (PB) equation has been generalized by Lu and Zhou [8] to the PNP model. We obtain second-order convergent results for the finite size linear PNP model with exact solutions. For nonlinear finite size PNP model with exact solutions, the numerical errors are almost zero.

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27

Chen, Jyun-Yang, e 陳君羊. "A Study of Poisson-Nernst-Planck Equations for IonChannels". Thesis, 2010. http://ndltd.ncl.edu.tw/handle/53950929085511225337.

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碩士
國立臺灣大學
數學研究所
98
In this paper, we use the perturbation methods and numerical simulations to observe the behavior of the solution of Poisson-Nernst-Planck equations. First, we do the low-concentration limit case to derive the Goldman-Hodgkin-Katz formula which can be used to explain the occurrence of the nerve impulse. In addition, we obtain a sufficient condition such that current and voltage hold a linear relationship. And for the channel with different wall shapes, we find the numerical solutions. Furthermore, by the energetic variational approach, we derive the modifier Nernst-Planck equation corresponding different geometries of channel. By the modifier equation, we find an approximate solution when the wall shape function of ion channel is an exponential function.

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28

Tseng, Chun-Chieh, e 曾俊傑. "Poisson–Nernst–Planck–Fermi Simulations of TRPV1 Ion Channel". Thesis, 2016. http://ndltd.ncl.edu.tw/handle/71670000375281264955.

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碩士
國立新竹教育大學
應用數學系碩士班
104
There are tens of thousands of channels within a typical animal cell in which each channel allows specific ions to pass through and control cellular membrane voltage difference. It requires a lot of time and uses expensive equipment to get the channel data in biological experiments. Since the channel is too small, biologists must use the special microscopy to observe the structure of TRPV1 channel. In this thesis, we simulate the cation transport through TRPV1 by using the Poisson-Nernst-Planck-Fermi (PNPF) model. The PNPF solvers for simulating biological ion channels and nanofluids include the steric effect of ions and water molecules with interstitial voids and the correlation effect of many ions with different valences. We combine the classical Scharfetter-Gummel (SG) method with simplified matched interface and boundary (SMIB) method so that it become a new method (SMIB-SG). The method can analyze molecular surfaces and singular charges of channel proteins and exhibit important features in flow simulations such as optimal convergence, efficient nonlinear iterations, and physical conservation. This method also allows water to pass through the channel. The PNPF currents are in accord with the experimental I-V (current-voltage) data and I-C (current-concentration) data of the TRPV1 channel with various calcium concentrations.

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29

Yen, Ching-Yu, e 嚴清宇. "A Surface Free Poisson-Nernst-Planck Model for Biological Systems". Thesis, 2012. http://ndltd.ncl.edu.tw/handle/46997575887401065258.

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碩士
國立新竹教育大學
應用數學系碩士班
100
Numerical methods are developed for solving the Poisson-Nernst-Planck model in which the electric permittivity is a continuous function of the Boltzmann distribution in terms of the van der Waals (vdW) potential. The vdW potential is expressed as a summation of all pairwise the Lennard-Jones interactions between ions in solvent and the atoms in a biomolecule (protein). The vdW potential has internal layer (potential wall) around the molecule. It is found that direct finite difference approximation of the vdW potential is unable to capture the potential wall but gives good approximation away from the wall. On the other hand, a splitting function method can yield a sharp wall but does not give good approximation away from the wall.

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30

Chen, Bo-Yun, e 陳博允. "Comparison of Hodgkin Huxley model and Poisson Nernst Planck equations". Thesis, 2019. http://ndltd.ncl.edu.tw/handle/sw4uru.

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碩士
國立臺灣大學
應用數學科學研究所
107
This thesis presents a dynamic simulation of intracellular and extracellular ionic concentrations and electric potential, then create an action potential, which is generated by a difference of the electrochemical potential between two sides of a cell membrane. Ion species including Sodium, Potassium and Chlorine. This simulation would involve Poisson-Nernst-Planck (PNP) system and Hodgkin–Huxley (HH) model. The former gives a standard model for describing behaviors of ionic diffusion and electrophoresis. The latter gives a transformation between mechanism of ion channels and a circuit. We want to combine and compare the results of these two models, then try to verify that the PNP equations can reduce to the HH model. In this study, methodologies are based on finite volume method and pseudospectral method for space discretization. After changing the semi-discrete scheme to a system of ODE by method of lines(MOL), we use ode15s solver on MATLAB to handle for time integration.

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31

Yang, Shiang-He, e 楊祥鶴. "A simplified second-order Poisson Nernst-Planck model for ion channel". Thesis, 2012. http://ndltd.ncl.edu.tw/handle/18008945036579320589.

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碩士
國立交通大學
應用數學系所
100
The Poisson-Nernst-Planck (PNP) model is a basic continuum model for simulating ionic flows in an open ion channel. It is one of commonly used models in theoretical and computational. The Poisson equation is derived from Coulomb's law in electrostatics and Gauss's theorem in calculus. The Nernst-Planck equation is equivalent to the convection-diffussion model. Many computation methods have been constructed for the solution of the PNP equations. However, we want to simplify the second order solver of proposed in the literature [24] but, we must to deal with some problems. For example, singular charges, nonlinear coupling and interface. First, we apply the decomposition method [5] proposed by Chern, Liu,and Wang to cope with the singular charges. Second, the matched interface and boundary (MIB) method [24] is used for the interface problem. Third, The initial guess are given by Poisson Boltzmann (PB) equation and two iterative schemes are utilized to deal with the coupled nonlinear equations. Finally, the real data of Gramicidin A (GA) channel protein is obtained from the protein data bank (PDB).

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32

Huang, Yung-Chih, e 黃永智. "Numerical Results of CaVab Channels by Poisson–Nernst–Planck–Fermi Theory". Thesis, 2017. http://ndltd.ncl.edu.tw/handle/94c8q8.

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碩士
國立清華大學
應用數學系所
105
Animal cells have numerous ion channels, each channel controls cellular membrane voltage diﬀerence and allows corresponding ions to pass through the channel by this way to keep the normal physiological function. Because the size of ion channels is too tiny, observing and recording its various data needs a lot of time and expensive equipment. Because of that, we simulate the cation transport through CaVab by using the Poisson-Nernst-Planck-Fermi (PNPF) model. The method can analyze molecular surfaces and singular charges of channel proteins and exhibit important features in ﬂow simula-tions such as optimal convergence, eﬃcient nonlinear iterations, and physical conservation. The PNPF currents are in accord with the experimental I-V (current-voltage) data of the CaVab channel with various voltage. Another simulation method using atomic PF theory adds binding sites in the channel with ions. After calculation, it is concluded that the ion will be stable or not at the binding site. It shows that the theory can describe a "knock-oﬀ" mech-anism of calcium selectivity and the transport of ions through the channel.

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33

Wu, Jian-Min, e 吳建民. "Numerical Results Of TRPV1 Channels By Poisson-Nernst-Planck-Fermi Theory". Thesis, 2017. http://ndltd.ncl.edu.tw/handle/w8kxhz.

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碩士
國立清華大學
應用數學系所
105
There are tens of thousands of channels within a typical animal cell in which each channel allows specific ions to pass through and control cellular membrane voltage difference. Since these channels are too small, biologists must observe the channel structure by expensive equipment like special microscope, and it takes a lot of time to get the channel data in biological experiments too. In this paper, we use the Poisson-Nernst-Planck-Fermi model to simulate the case where ions transport through TRPV1. We combine the classical Scharfetter-Gummel method with the simplified matched interface and boundary method so that it becomes a new method (SMIB-SG). The method can analyze molecular surfaces and singular charges of channel proteins and exhibit important features in flow simulations such as optimal convergence, efficient nonlinear iterations, and physical conservation. This method also allows water to pass through the channel. The PNPF currents are in accord with the experimental current-voltage data and current-concentration data of the TRPV1 channel with various calcium concentrations. In this thesis, we propose a mobility model that is based on a previous work and achieve better match to experimental data.

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34

Chen, Yen-Chen, e 陳彥辰. "Numerical Simulations of Calcium Channel by Poisson–Nernst–Planck–Fermi Model". Thesis, 2017. http://ndltd.ncl.edu.tw/handle/h4zp47.

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35

Chang, Wei-Chan, e 張偉楨. "Asymptotic Analysis of Steric Poisson-Nernst-Planck Model for Multi-Species Ions". Thesis, 2014. http://ndltd.ncl.edu.tw/handle/dqzb4n.

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碩士
國立臺灣大學
數學研究所
102
Poisson-Nernst-Planck system models ionic flow in ion channels. Differ from basic Poisson-Nernst-Planck, steric Poisson-Nernst-Planck accurately describes the behaviour of ions for size effect terms are added into steric Poisson-Nernst-Planck in order to make a description of the fact that ions complete to each other to hold exact spaces in ion channel. The goal in this paper is to analyse the asymptotic behaviour we may say asymptotic stability of steric Poisson-Nernst-Planck. In the text, We make use of the skill for eigenvalues including positive semi-definite pencil in linear algebra to simplify the difficulties in the analysis. More important thing is that the way in this paper is easier and extends more cases than one which is changing variable in former literature. On the other hand, we also write some programs to simulate various situations we encounter. Though we get lots of numerical results; however, all of them support the conclusions in the analysis. Furthermore, in the future, the further goal is to take advantage of steric Poisson-Nernst-Planck systems to modify Hodgkin-Huxley equations which describe the behaviour of neuron.

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36

Hu, Hong-Zhi, e 胡鴻志. "Poisson-Nernst-Planck-Fermi Nonlinear Solver for TRPV1 Channels with Bi-Conjugate Gradient Method". Thesis, 2018. http://ndltd.ncl.edu.tw/handle/57354n.

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37

Ciou, Mei-Ru, e 邱渼茹. "Asymptotic analysis of the steady-state solution of a Poisson-Nernst-Planck system with steric effects". Thesis, 2018. http://ndltd.ncl.edu.tw/handle/s73s6n.

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碩士
國立清華大學
應用數學系所
106
This article is mainly to explore the asymptotic behavior of the Poisson-Nernst-Planck system with steric effects. More specifically, we study the behavior of electrostatic potential energy and ion concentration at the inner and boundary when the Debye length approaches zero. The mathematical results show that this electrostatic potential and concentration tend to zero inside the region, and the slope at the boundary will be infinite, so the ions are mainly concentrated at the boundary.

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38

Lee, Chiun-Chang, e 李俊璋. "Limit Problems of Solutions for the Coupled Nonlinear Schrödinger Equations and Steady-state Solutions of the Poisson-Nernst-Planck Systems". Thesis, 2010. http://ndltd.ncl.edu.tw/handle/76911253477102290338.

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博士
臺灣大學
數學研究所
98
In this thesis, we investigate two different types partial differential equations, one is the coupled nonlinear Schrödinger equations and the other is the renormalize Poisson-Boltzmann equations (the steady-state solutions of the Poisson-Nernst-Planck systems). Recently, a rich variety of dynamical phenomena and a turbulent relaxation have been observed in rotating Bose-Einstein condensates depicted by Gross-Pitaevskii equations coupled with rotating fields and trap potentials. The dynamical phenomena range from shock-wave formation to anisotropic sound propagation. The turbulent relaxation leads to the crystallization of vortex lattices. To see the dynamical phenomena and the turbulent relaxation of two-component rotating Bose-Einstein condensates, we study the incompressible and the compressible limits of two-component systems of Gross-Pitaevskii equations. Our arguments generalize the idea of [22] and define "H-function" a modulated energy functional which may control the propagation of densities and linear momentums under the effect of rotating fields and trap potentials. The Poisson-Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. We study a renormalized Poisson-Boltzmann (RPB) equation with a small dielectric parameter ∈2 and nonlocal nonlinearity which takes into consideration of the preservation of the total amounts of each individual ion. This equation can be derived from the original Poisson-Nernst-Planck (PNP) system. Under Robin type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviors of one dimensional solutions of RPB equations as the parameter $epsilon$ approaches to zero. In particular, we show that in case of electro-neutrality, i.e., (∑N1 κ=1 ακακ=∑N2 l=1 blβl), we prove that φ∈''s solutions of 1-D RPB equations may tend to a nonzero constant c at every interior point as $epsilon$ goes to zero. The value c can be uniquely determined by ακ, bl''s valences of ions, ακ, βl''s total concentrations of ions and the limit of φ∈''s at the boundary x=±1. In particular, when N1=1, N2=2, a1=b1=1 and b2=2, a precise formula of the value c and the ratio β1/β2 is given in (4.1.3). Such a result can not be found in conventional 1-D Poisson-Boltzmann (PB) equations. On the other hand, as (∑N1 κ=1 ακακ≠∑N2 l=1 blβl) (non-electroneutrality), solutions of 1-D RPB equations have blow-up behavior which also may not be obtained in 1-D PB equations.

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39

Kubeil, Clemens. "Zum Einfluss elektrochemischer Doppelschichten auf den Stofftransport in nanoskaligen Elektrolytsystemen:: Leitfähigkeit von Nanoporen und Voltammetrie an Nanoelektroden". Doctoral thesis, 2016. https://tud.qucosa.de/id/qucosa%3A30154.

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Es besteht enormes Interesse den Stofftransport in nanoskaligen Systemen zu verstehen und selektiv zu steuern, um analytische und synthetische Anwendungen zu entwickeln, aber auch um die physiologischen Prozesse lebender Zellen zu entschlüsseln. Im Rahmen dieser Arbeit wurde der Einfluss der elektrochemischen Doppelschicht an ausgewählten nanoskaligen Elektrolytsystemen untersucht. Die Gleichrichtung von Ionenströmen (engl. Ionic Current Rectification ICR) in Nanoporen mit einer Oberflächenladung äußert sich in einer gekrümmten Strom-Spannungs-Kurve. Die Überlappung von innerem und äußerem Potential ist dabei hinsichtlich der Ionenverteilung und somit der Porenleitfähigkeit einander verstärkend oder gegenläufig. Auf Grundlage dieses Mechanismus wurde die Gleichrichtung bei einem sehr großen Verhältnis von Porenöffnung zu Debye-Länge erklärt. Ferner wurde mittels der eingeführten relativen Leitfähigkeit κ´ die verschiedenen Leitfähigkeitszustände in Abhängigkeit der Elektrolytkonzentration und Temperatur sichtbar gemacht und Implikationen für Sensoranwendungen wie z.B. dem resistiven Pulszähler zur Partikelanalyse abgeleitet. Es wurde ein numerisches Modell basierend auf dem Poisson-Nernst-Planck-Gleichungssystem entwickelt, um die Translokation eines Nanopartikels durch eine konische Nanopore bei einer geringen Leitsalzkonzentration zu beschreiben. Neben dem klassischen Volumenausschluss-Effekt tritt zusätzlich ein Gleichrichtungseffekt (ICR-Effekt) in der Pore auf. Eine Analyse zur Entflechtung von Partikelgröße und Partikelladung aus der Pulshöhe und Pulsform wurde erfolgreich durchgeführt. Wie der Stofftransport durch eine Oberflächenladung auf dem umgebenden Material einer Nanoelektrode beeinflusst wird, wurde anhand des voltammetrischen Verhaltens diskutiert. An sehr kleinen Elektroden (< 10 nm) ist demnach der Einfluss der elektrochemischen Doppelschicht auf die Strom-Spannungs-Kurve besonders groß und kann auch bei Vorliegen eines hohen Leitsalzüberschusses nicht vernachlässigt werden. In leitsalzfreien Elektrolyten sind die gefundenen Effekte so deutlich, dass sie auch an größeren Elektroden experimentell zweifelsfrei festgestellt worden sind.
There is an enormous interest in understanding and selectively controlling the material transport in nanoscale systems to develop analytical and synthetic applications, but also to decipher the physiological processes of living cells. Within this thesis, the influence of the electrochemical double layer on selected nanoscale electrolyte systems was studied. Ionic Current Rectification (ICR) in nanopores carrying a surface charge manifests itself in a non-linear current-voltage-curve. The overlap of interior and exterior potential is cumulative or opposing with regard to the ion distribution and therefore the pore conductivity. Based on this mechanism, ICR for very large ratios of pore size and Debye length was explained. Furthermore, the different conducting states as a function of electrolyte concentration and temperature were visualized by introducing the relative conductivity κ´ and hence implications for sensor applications such as the resistive pulse sensor have been deduced. A numerical model based on the Poisson-Nernst-Planck-equations was developed to describe the translocation of a nanoparticle through a conical nanopore at a low electrolyte concentration. An additional rectification effect (ICR effect) occurs in the pore beside the conventional volume exclusion effect. An analysis was successfully performed to deconstruct the particle size and particle charge from the pulse height and shape. The material transport is affected by a surface charge on the shrouding material of nanoelectrodes as it was discussed by means of the voltammetric behaviour. The influence of the electrochemical double layer on the current-voltage-curve is particularly large at very small electrodes (< 10 nm) and cannot be neglected even at a high excess of supporting electrolyte. The observed effects were pronounced in unsupported electrolytes, so that they could be clearly detected experimentally at even larger electrodes.

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40

Backhaus, Karsten. "Das dielektrische Verhalten der Öl-Papier-Isolierung bei Belastung mit hoher Gleichspannung". Doctoral thesis, 2016. https://tud.qucosa.de/id/qucosa%3A30546.

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Basierend auf den physikalischen Eigenschaften der unterschiedlichen ölintrinsischen und injizierten Ladungsträger wird ein neues Leitfähigkeitsmodell für Isolieröl und -papier für die Belastung mit hoher Gleichspannung aufgestellt. Das Modell wird mit der Wahl geeigneter Randbedingungen für das elektrische Feld und der Teilchenströme auf die Poisson-Nernst-Planck-Gleichung übertragen. Es steht damit ein Werkzeug zur Verfügung, das dielektrische Verhalten der Öl-Papier-Isolierung zu modellieren, dessen Parameter auf den physikalischen Ladungsträgereigenschaften wie Mobilität und Diffusion basieren. Mit dessen Hilfe werden sowohl die nichtlineare Leitfähigkeit als auch das Durchschlagverhalten als deren Extrapolation feldstärkeabhängig erklärt.

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41

Bzainia, Amir. "Preparation and ionic transport properties of conductive polymers for dye-sensitized solar cells". Master's thesis, 2018. http://hdl.handle.net/10198/19585.

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Dupla diplomação Universite Libre de Tunis (ULT)
This work aims to improve the components of the dye-sensitized solar cells (DSSCs) which are a type of photovoltaics that consist mainly of a photoanode, a counter electrode, a light sensitive molecule (sensitizer) and an electrolyte solution that regenerates the solar cell through a redox system. The improvement of the DSSC focuses on the material used for the counter electrode. Usually, it is made out from platinum sputtered on a conductive glass. However, platinum is an expensive metal that is hard to manipulate and can be corroded by the mediator. In this perspective, an alternative material for the expensive platinum is investigated which is the conductive polymer poly(3,4-ethylenedioxythiophene) (PEDOT). This polymer has a structure that promotes high ionic and electronic conductivities, and it can be doped with different anions (e.g. PSS, perchlorate). PEDOT was synthesized chemically and electrochemically. Its chemical structure was characterized by FTIR. The electrochemical behavior of PEDOT was assessed by cyclic voltammetry (CV). The catalytic activity of PEDOT towards the redox system proved to be higher than the activity of the platinum. In addition to the experimental approach, modelling of the ionic-electronic conductivity of PEDOT was performed based on the Nernst-Planck-Poisson and the Butler-Volmer formalisms, and the simulation outputs were fitted to the experimental data. In the last step, the conductive polymer PEDOT was used as a counter electrode to fabricate DSSCs. The cells were characterized through electrochemical impedance spectroscopy (EIS) and through current-voltage (J-V) curves. The based PEDOT cells demonstrated an efficiency of 8.1%, which was higher than the based-platinum solar cells (6.3%).
Ce travail vise à améliorer les composants des cellules solaires à pigment photo-sensibles (en anglais dye-sensiztized solar cells, DSSCs). Ce type de cellule photovoltaïque se compose principalement d'une photoanode, d'une contre électrode, d'une molécule sensible à la lumière (pigment) et d'une solution électrolyte qui régénère la cellule solaire à travers un système redox. L'amélioration de la DSSC se concentre sur le matériau utilisé dans la contre électrode. Généralement, cette dernière est faite de platine déposé sur un verre conducteur. Cependant, le platine est un métal coûteux qui est difficile à manipuler et peut être corrodé par le couple redox. Dans cette perspective, un matériau alternatif pour le platine est étudié et qui s’agit du polymère conducteur (3,4-éthylènedioxythiophène) (PEDOT). Ce polymère a une structure qui favorise la conductivité ionique et électronique, et il peut être dopé avec des anions différents (par exemple PSS, perchlorate). Le PEDOT a été synthétisé par la voie chimique ainsi que la voie électrochimique. Sa structure chimique a été caractérisée par FTIR. Le comportement électrochimique du PEDOT a été évalué par la voltamétrie cyclique (CV). L'activité catalytique du PEDOT vis-à-vis du système redox s'est avérée plus élevée que celle du platine. En plus de l'approche expérimentale, la modélisation de la conductivité ionique et électronique du PEDOT a été réalisée en se basant sur les formalismes de Nernst-Planck-Poisson et de Butler-Volmer. Les résultats de la simulation ont été ajustés aux données expérimentales. Dans la dernière étape, le polymère conducteur PEDOT a été utilisé comme une contre électrode pour fabriquer des DSSCs. Les cellules ont été caractérisées par la spectroscopie d'impédance électrochimique (EIS) et par les courbes de courant-tension (J-V). Les cellules à base de PEDOT ont montré une efficacité de 8,1 %, ce qui était plus élevé que les cellules solaires à base de platine (6,3 %).
Este trabalho visa melhorar os componentes das células solares sensíveis ao corante (DSSCs) que são um tipo de células fotovoltaicas que consistem principalmente em um foto-ânodo, um contra elétrodo, uma molécula sensível à luz (sensibilizante) e uma solução de eletrólitos que regenera a célula solar através de um sistema redox. A melhoria da DSSC centra-se no material usado para o contra elétrodo. O contra elétrodo é geralmente preparado através da deposição de uma camada fina de platina na superfície de um substrato de vidro condutor. No entanto, a platina é um metal caro que é difícil de manipular, pode ser corroído pelo redox e nessa perspetiva, materiais alternativos (polímeros condutores, grafite, nanotubos de carbono…) para a substituição da platina têm vindo a ser investigados. Nesta investigação foi escolhido o polímero condutor poli (3,4-etilenodioxitiofeno) (PEDOT) para substituir a platina na preparação do contra elétrodo. Este polímero possui uma estrutura que promove altas condutividades iônicas e eletrônicas, podendo ser dopado com diferentes aniões (por exemplo, PSS, perclorato). O PEDOT foi sintetizado quimicamente e electroquimicamente, a sua estrutura química foi caracterizada por espectroscopia de infravermelhos (FTIR) e o comportamento eletroquímico do PEDOT foi avaliado através de voltamétrica cíclica (CV). A atividade catalítica de PEDOT para o sistema redox provou ser maior do que a atividade da platina. Além da abordagem experimental, a modelagem da condutividade iônica-eletrônica do PEDOT foi realizada com base nos formalismos de Nernst-Planck-Poisson e Butler-Volmer, e os resultados da simulação foram ajustados aos dados experimentais. Na última etapa, o polímero condutor PEDOT foi usado como um contra elétrodo para fabricar DSSCs. As células foram caracterizadas por espectroscopia de impedância eletroquímica (EIS) e por curvas de corrente-tensão (J-V) e os resultados obtidos mostram que as DSSCs fabricadas, usando PEDOT como base apresentam uma eficiência de 8,1%, maior do que as células solares baseadas em platina de 6,3%.
يهدف هذا العمل إلى تحسين مكونات الخلايا الشمسية الصبغية (DSSCs) والتي تعد نوعًا من الخلايا الكهروضوئية ال ت ي تتكون أساسًا من مصعد )أنود(، مهبط )كتود(، صباغ حساس للضوء ومحلول إلكتروليتي لإعادة توليد الخلايا الشمسية من خلال نظام الأكسدة. يركز تحسين الخلايا على المواد المستخدمة في المهبط )كتود(، و الذ ي عادة ما يتم تصنيعه من البلاتين الموضوع على زجاج ناقل للكهرباء. يعد البلاتين معدنًا مكلفًا يصعب معالجته ويمكن تآكله بواسطة الإلكتروليت . لذا، تمت دراسة مادة بديلة للبلاتين باهظ الثمن وهي مبلمر موصل ) 4.3 -إيتيلين ديوكسي تيوفين( (PEDOT) . يحتوي هذا المبلمر على هيكل يعززبصفة عالية التوصيلات الأيونية والإلكترونية، ويمكن إشابته بأنيونات مختلفة . ت م تصنيع PEDOT كيميائيا وكهربائيا ث م تشخيصه بواسطة FTIR لمعرفة تركيبته الكيميائية. ت م تقييم السلوك الكهروكيميائي ل PEDOT بواسطة قياس الجهد الدوري . ت م إثبات أ ن النشاط التحفيزي ل PEDOT تجاه نظام الأكسدة أعلى من نشاط البلاتين. بالإضافة إلى المنهج التجريبي، ت م نمذجة التوصيلية الأيونية الإلكترونية ل PEDOT استنادًا على معادلات نرست-بلانك-بواسون وبتلر- فلمر، وت م مقارنة النتائج التجريبية بنتائج المحاكاة. أخيرا، ت م استخدام هذا المبلمر الموصل كمهبط )كتود( لتصنيع خلايا شمسية صبغية. ت م تقييم الخلايا من خلال مطيافية المعاوقة الكهروكيميائية (EIS) ومن خلال منحنيات التيار-توتر (J-V) . أظهرت الخلايا القائمة على PEDOT كفاءة 8.1 ٪، والتي كانت أعلى من الخلايا الشمسية القائمة على البلاتين ) 6.3 .)
Without funding, however I would not have the opportunity to accomplish this work. I gratefully acknowledge funding form the project “AIProcMat@N2020-Advanced Industrial Processes and Materials for a Sustainable Northern Region of Portugal 2020”, with the reference NORTE-01-0145-FEDER-000006, supported by “Norte Portugal Regional Operational Programa” (NORTE 2020), under the Portugal 2020 Partnership Agreement, through the European Regional Development Fund (ERDF) and of Project POCI-01-0145-FEDER-006984-Associate Laboratory LSRE-LCM funded by ERDF through COMPETE2020-Programa Operacional Competitividade e Internacionalização (POCI) and by national funds through FCT-Fundação para a Ciência e a Tecnologia.

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42

Kim, Younggy. "Ionic separation in electrodialysis : analyses of boundary layer, cationic partitioning, and overlimiting current". Thesis, 2010. http://hdl.handle.net/2152/ETD-UT-2010-08-1724.

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Electrodialysis performance strongly depends on the boundary layer near ion exchange membranes. The thickness of the boundary layer has not been clearly evaluated due to its substantial fluctuation around the spacer geometry. In this study, the boundary layer thickness was defined with three statistical parameters: the mean, standard deviation, and correlation coefficient between the two boundary layers facing across the spacer. The relationship between the current and potential under conditions of the competitive transport between mono- and di-valent cations was used to estimate the statistical parameters. An uncertainty model was developed for the steady-state ionic transport in a two-dimensional cell pair. Faster ionic separations were achieved with smaller means, greater standard deviations, and more positive correlation coefficients. With the increasing flow velocity from 1.06 to 4.24 cm/s in the bench-scale electrodialyzer, the best fit values for the mean thickness reduced from 40 to less than 10 μm, and the standard deviation was in the same order of magnitude as the mean. For the partitioning of mono- and di-valent cations, a CMV membrane was examined in various KCl and CaCl₂ mixtures. The equivalent fraction correlation and separation factor responded sensitively to the composition of the mixture; however, the selectivity coefficient was consistent over the range of aqueous-phase ionic contents between 5 and 100 mN and the range of equivalent fractions of each cation between 0.2 and 0.8. It was shown that small analytic errors in measuring the concentration of the mono-valent cation are amplified when estimating the selectivity coefficient. To minimize the effects of such error propagation, a novel method employing the least square fitting was proposed to determine the selectivity coefficient. Each of thermodynamic factors, such as the aqueous- and membrane-phase activity coefficients, water activity, and standard state, was found to affect the magnitude of the selectivity coefficient. The overlimiting current, occurring beyond the electroneutral limit, has not been clearly explained because of the difficulty in solving the singularly perturbed Nernst-Planck-Poisson equations. The steady-state Nernst-Planck-Poisson equations were converted into the Painlevé equation of the second kind (P[subscript II] equation). The converted model domain is explicitly divided into the space charge and electroneutral regions. Given this property, two mathematical formulae were proposed for the limiting current and the width of the space charge region. The Airy solution of the P[subscript II] equation described the ionic transport in the space charge region. By using a hybrid numerical scheme including the fixed point iteration and Newton Raphson methods, the P[subscript II] equation was successfully solved for the ionic transport in the space charge and electroneutral regions as well as their transition zone. Above the limiting current, the sum of the ionic charge in the aqueous-phase electric double layer and in the space charge region remains stationary. Thus, growth of the space charge region involves shrinkage of the aqueous-phase electric double layer. Based on this observation, a repetitive mechanism of expansion and shrinkage of the aqueous-phase electric double layer was suggested to explain additional current above the limiting current.
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