DOCSLIB.ORG

Double-Layer Capacitance on a Rough Metal Surface

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Double-Layer Capacitance on a Rough Metal Surface PHYSICAL REVIEW E VOLUME 53, NUMBER 6 JUNE 1996 Double-layer capacitance on a rough metal surface L. I. Daikhin School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel A. A. Kornyshev* and M. Urbakh† Institut fu¨r Energieverfahrenstechnik, Forschungszentrum Ju¨lich GmbH (KFA), 52425 Ju¨lich, Germany ~Received 30 November 1995! An expression for the double layer capacitance of rough metal-electrolyte, metal-semiconductor, or semiconductor-electrolyte interfaces is derived which shows the interplay between the Debye length and the lengths characterizing roughness. Different dependencies of the capacitance, as compared to the flat interface, on the concentration of charge carriers in electrolyte or semiconductor are predicted. Examples of the typical roughness spectra are considered. The cases of Euclidean roughness show weak dependence on the particular form of the roughness spectrum, being sensitive only to its main parameters: the random mean square height of roughness and correlation length. A method is proposed for the in situ characterization of surface rough- ness: the measurement of surface roughness with a ‘‘Debye ruler,’’ based on the conventional measurements of the double layer capacitance. @S1063-651X~96!05305-3# PACS number~s!: 68.45.2v, 41.20.Cv I. INTRODUCTION problem, giving rise to different functional dependencies on electrolyte concentration and potential. The Gouy-Chapman theory of electrolyte plasma near a In this article we show how the competition between the flat charged wall @1,2#, which appeared a decade earlier than Debye length and the correlation length of roughness modi- the Debye theory of bulk electrolytes @3#, is the basis of fies the Gouy-Chapman result. It is obvious, a priori, that the many successful constructions in electrochemistry @4#, col- limiting value of capacitance at short Debye lengths should loid science @5#, biophysics @6#, and semiconductor science follow Eq. ~1! but with S replaced by Sreal5RS. In the limit and technology @7#. In the low voltage limit, the Gouy- of long Debye lengths the roughness would not be mani- Chapman theory gives a transparent result for the space fested in the capacitance which would obey the native Eq. charge capacitance ~1!. How does the crossover between these two limits occur? One may expect to recover the whole curve, modifying Eq. C5CGC[«kS/4p, ~1! ~1!, 21 ˜ where k is the Gouy ~5Debye! length, « the dielectric C5R~k!CGC , ~18! constant of the solvent, and S the area of the flat interface. As it should be, the capacitance is inversely proportional to where the roughness function˜ R~k! varies between ˜R~0!51 the separation between the charge and counter charge, in and ˜R(`)5R.1. The problem for the theory is, then, to find plasma provided by the Debye length. this function. For the case of a weak roughness, we derive A long period in electrochemistry was associated with the the general expression for the roughness function, establish studies performed on the liquid mercury drop electrode, and its limiting behavior, and study the cases of different surface later on Ga, InGa, and GaTl alloys @8#. Providing an ideally morphologies ~sinusoidal corrugation, random Gaussian smooth interface between the metal and electrolyte, liquid roughness, and self-affine fractal structures!. electrodes allowed a set of classical results in electrocappil- In electrolyte solutions the Debye length can easily be lary phenomena, adsorption and electrochemical kinetics. varied by changing the electrolyte concentration with no Studies on solid electrodes ~Cd, Pb, Bi, Sn, Cu! faced prob- effect on surface roughness. Experimental data on ˜ lems associated with a nonsmooth character of the interface R(k)5C/CGC may then be used for probing the roughness @9#. The concept of the geometrical roughness factor of the metal surface in contact with electrolytes. The same R5Sreal/S, i.e., the ratio of the true surface to the apparent idea can be put into the basis of an evaluation of the rough- surface ~flat cross-section area! became common @9#. How- ness of metal-semiconductor or semiconductor-electrolyte ever, in many cases, experimental data cannot be rationalized interfaces. Debye length in semiconductors can be varied by in terms of this parameter only. The latter is not surprising. radiation or temperature-induced excitation of charge carriers Roughness may be responsible for additional characteristic into the conduction band @10#~doping will require a creation lengths, which may compete with other typical lengths in the of a new junction with an uncontrollable effect on the struc- ture of the contact!. The idea of a competition between the characteristic *Corresponding author. Electronic address: rbe016@djukfa11 scales of roughness and the optical and acoustic wavelengths †Permanent address: School of Chemistry, Tel Aviv University, was explored in the classical works of Rayleigh @11# and 69978 Tel Aviv, Israel. Electronic address: [email protected] Fano @12#, and in the subsequent studies in interfacial optics 1063-651X/96/53~6!/6192~8!/$10.0053 6192 © 1996 The American Physical Society 53 DOUBLE-LAYER CAPACITANCE ON A ROUGH METAL SURFACE 6193 @13–15#. Similar methods were used in the theory of friction between consecutive peaks and valleys on the rough surface. @16# and quartz microbalance @17#. The interplay between the We will also assume that h,k21. roughness spectra and diffusion lengths was intensively in- Solving Eq. ~2!, it is convenient to Fourier transform the vestigated in the context of the anomalous frequency depen- potential and the surface profile function from tangential co- dence of the electrochemical impedance @18–21# and diffu- ordinates R5(x,y) to the corresponding wave vectors sion to the surface @22,23#. The competition between the K5(Kx ,Ky)asf~K!5*dRf~R!exp~2iK•R!. Equation ~2! roughness scale and the Debye length has been explored then transforms to @24–28# in the context of the surface stability and surface 2 forces. However, the features of the space charge ~double d 2K22k2 f~K,z!50. ~4! layer! capacitance, the key quantity in electrochemistry, were H dz2 J not considered. We address mainly solid metal-liquid electrolyte systems, According to the Rayleigh approximation @11# the solution but the basic results can be extended on other electrified of Eq. ~4! in the half space z,j(x,y), has the form interfaces, such as metal-semiconductor, semiconductor- K,z 5A K exp 2q z , 5 electrolyte, chargeable biological interfaces, and metal-solid f~ ! ~ ! ~ K ! ~ ! electrolyte contacts @29#. 2 2 1/2 where qK5(k 1K ) . This approximation neglects the terms proportional exp(qKz) which would be important in- II. BASIC EXPRESSION FOR CAPACITANCE side deep protrusions and grooves in the metal surface, ig- nored in our consideration. The boundary condition 3 leads A. Boundary problem for potential ~ ! to the integral equation on the prefactor A~K!, Consider a rough metal surface in contact with an electro- lyte. We take the z axis pointing towards the electrolyte and dK A~K! dR exp@2q j~x,y!# describe the interface by the equation z5j(x,y). The plane E ~2p!2 E K z50 is chosen such that the average value of the function 2 j(x,y) over the surface is equal to zero. 3exp@2i~K82K!•R#5~2p! f0d~K8!, ~6! In the Gouy-Chapman theory, the distribution of the elec- trostatic potential f~r! in the electrolyte is described by the where d~K! is the two-dimensional Dirac d function. nonlinear Poisson-Boltzmann equation. As a first step we Since we are bound to the case of weak roughness, restrict our consideration by its linearized version, valid for u¹j~x,y!u!1, hk!1, ~7! low electrode potentials f,kBT/e: the standard perturbation technique @11–17,20,23,25# may be ~¹22k2!f~r!50. ~2! applied to find A~K!. The first exponential in Eq. ~6! is ex- For a 1-1 binary electrolyte solution, the Debye length, panded into the series, 21 2 1/2 k 5(«kBT/8pne ) , where n is the electrolyte concentra- tion, « the dielectric constant of the solvent, e charge of E dR exp@2qKj~x,y!#exp@2i~K82K!•R# electron, T the temperature, and kB the Boltzmann constant @for multivalent ions n must be replaced by the ‘‘ionic 2 2 .~2p! d~K2K8!2qKj~K82K! strength’’ I5n(1/2)(z i ni where zi and ni are the valence and the stoichiometric coefficients of the ion of sort i#. For 1 dK 2 9 nondegenerate semiconductors with one sort of charge carri- 1 qK 2 j~K82K2K9!j~K9!, ~8! 21 2 1/2 2 E ~2p! ers, k 5(«kBT/4pne ) where n is the density of charge carriers and « is the high frequency dielectric constant of the and Eq. ~6! may be solved by iterations. In order to deter- semiconductor. An expression for k exists for solid electro- mine the first nonvanishing correction to the capacitance lytes with the same type of dependence on mobile ions con- caused by roughness, we must find A~K! up to the second centration and temperature @30#. order in h. Hence The solution of Eq. ~2! must satisfy the boundary condi- tion which fixes the potential at the metal-electrolyte inter- A~K!5A0~K!1A1~K!1A2~K!, ~9a! face where f x,y,z5j~x,y! 5f ~3! „ … 0 2 A0~K!5~2p! f0d~K!, ~9b! relative to the zero level in the bulk of the electrolyte: f~z `!50. A1~K!5f0q0h~K!, ~9c! ) 1 dK8 B. Perturbation theory A ~K!5 f q h~K2K8!h~K8!~2q 2q !. 2 2 0 0 E ~2p!2 K8 0 Consider weakly rough surfaces for which h, the charac- ~9d! teristic size of roughness in the z direction, is less than the tangential one, l.
Load more

Recommended publications
  • Microfluidics
    MICROFLUIDICS Sandip Ghosal Department of Mechanical Engineering, Northwestern University 2145 Sheridan Road, Evanston, IL 60208 1 Article Outline Glossary 1. Definition of the Subject and its Importance. 2. Introduction 3. Physics of Microfluidics 4. Future Directions 5. Bibliography GLOSSARY 1. Reynolds number: A characteristic dimensionless number that determines the nature of fluid flow in a given set up. 2. Stokes approximation: A simplifying approximation often made in fluid mechanics where the terms arising due to the inertia of fluid elements is neglected. This is justified if the Reynolds number is small, a situation that arises for example in the slow flow of viscous liquids; an example is pouring honey from a jar. 3. Ion mobility: Velocity acquired by an ion per unit applied force. 4. Electrophoretic mobility: Velocity acquired by an ion per unit applied electric field. 5. Zeta-potential: The electric potential at the interface of an electrolyte and substrate due to the presence of interfacial charge. Usually indicated by the Greek letter zeta (ζ). 1 6. Debye layer: A thin layer of ions next to charged interfaces (predominantly of the opposite sign to the interfacial charge) due to a balance between electrostatic attraction and random thermal fluctuations. 7. Debye Length: A measure of the thickness of the Debye layer. 8. Debye-H¨uckel approximation: The process of linearizing the equation for the electric potential; valid if the potential energy of ions is small compared to their average kinetic energy due to thermal motion. 9. Electric Double Layer (EDL): The Debye layer together with the set of fixed charges on the substrate constitute an EDL.
    [Show full text]
  • Screened Electrostatic Interaction of Charged Colloidal Particles in Nonpolar Liquids Carlos Esteban Espinosa
    Screened Electrostatic Interaction of Charged Colloidal Particles in Nonpolar Liquids A Thesis Presented to The Academic Faculty by Carlos Esteban Espinosa In Partial Fulfillment of the Requirements for the Degree Master of Science in Chemical Engineering School of Chemical & Biomolecular Engineering Georgia Institute of Technology August 2010 Screened Electrostatic Interaction of Charged Colloidal Particles in Nonpolar Liquids Approved by: Dr. Sven H. Behrens, Adviser School of Chemical & Biomolecular Engineering Georgia Institute of Technology Dr. Victor Breedveld School of Chemical & Biomolecular Engineering Georgia Institute of Technology Dr. Carson Meredith School of Chemical & Biomolecular Engineering Georgia Institute of Technology Date Approved: 12 May 2010 ACKNOWLEDGEMENTS First, I would like to thank my adviser, Dr. Sven Behrens. His support and orienta- tion was essential throughout the course of this work. I would like to thank all members of the Behrens group that have made all the difference. Dr. Virendra, a great friend and a fantastic office mate who gave me im- portant feedback and guidance. To Qiong and Adriana who helped me throughout. To Hongzhi Wang, a great office mate. I would also like to thank Dr. Victor Breedveld and Dr. Carson Meredith for being part of my committee. Finally I would like to thank those fellow graduate students in the Chemical Engineering Department at Georgia Tech whose friendship I am grateful for. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS .......................... iii LIST OF TABLES ............................... vi LIST OF FIGURES .............................. vii SUMMARY .................................... x I INTRODUCTION ............................. 1 II BACKGROUND .............................. 4 2.1 Electrostatics in nonpolar fluids . .4 2.1.1 Charge formation in nonpolar fluids . .4 2.1.2 Charge formation in nonpolar oils with ionic surfactants .
    [Show full text]
  • The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration
    The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration Alexander M. Smith*,a, Alpha A. Lee*,b and Susan Perkin*,a aDepartment of Chemistry, Physical & Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, U.K. bSchool of EnGineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA Corresponding author emails: [email protected] [email protected] [email protected] 1 ABSTRACT According to classical electrolyte theories interactions in dilute (low ion density) electrolytes decay exponentially with distance, with the Debye screeninG lenGth the characteristic length-scale. This decay length decreases monotonically with increasing ion concentration, due to effective screening of charges over short distances. Thus within the Debye model no long-range forces are expected in concentrated electrolytes. Here we reveal, using experimental detection of the interaction between two planar charged surfaces across a wide range of electrolytes, that beyond the dilute (Debye- Hückel) regime the screening length increases with increasing concentration. The screening lengths for all electrolytes studied – including aqueous NaCl solutions, ionic liquids diluted with propylene carbonate, and pure ionic liquids – collapse onto a single curve when scaled by the dielectric constant. This non-monotonic variation of the screening length with concentration, and its generality across ionic liquids and aqueous salt solutions, demonstrates an important characteristic of concentrated electrolytes of substantial relevance from biology to energy storage. TOC Image: 2 Electrolytes are ubiquitous in nature and in technology: from the interior of cells to the oceans, from supercapacitors to nanoparticle dispersions, electrolytes act as both solvent and ion conduction medium.
    [Show full text]
  • Electrophoresis of Charged Macromolecules
    Electrophoresis of Charged Macromolecules Christian Holm Institut für Computerphysik, Universität Stuttgart Stuttgart, Germany 1! Charge stabilized Colloids! The analytical description of charged colloidal suspensions is problematic:! n " Long ranged interactions: electrostatics/ hydrodynamics! n " Inhomogeneous/asymmetrical systems! n " Many-body interactions! Alternative! : the relevant microscopic degrees of freedom are simulated! via Molecular Dynamics! ●Explicit" particles (ions) with charges ε ●Implicit" solvent approach, but hydrodynamic interactions of the solvent are included via a Lattice-Boltzmann algorithm Test of LB implementation for Poiseuille! Simulation box of size 80x40x10. Velocity profile for a Poiseuille flow in a channel, which is tilted by 45◦ relative to the Lattice-Boltzmann node mesh. Computed using ESPResSo. Profile of the absolute fluid velocity of the Poiseuille flow in the 45◦ tilted channel. Red crosses represent simulation data, the blue line is the theoretical result and the dashed blue line represents the theoretical result, using the channel width as a fit parameter 3! EOF in a Slit Pore! Simulation results for a water system. Solid lines denote simulation results, the dotted lines show the analytical results for comparison. Red stands for ion density in particles per nm3, blue stands for the fluid velocity in x-direction, green denotes the particle velocity. All quantities in simulation units. 4! Colloidal Electrophoresis! local force balance FE = FDrag leads to stationary state ν FE F = Z E − Zeff
    [Show full text]
  • Electrical Double Layer Interactions with Surface Charge Heterogeneities
    Electrical double layer interactions with surface charge heterogeneities by Christian Pick A dissertation submitted to Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy Baltimore, Maryland October 2015 © 2015 Christian Pick All rights reserved Abstract Particle deposition at solid-liquid interfaces is a critical process in a diverse number of technological systems. The surface forces governing particle deposition are typically treated within the framework of the well-known DLVO (Derjaguin-Landau- Verwey-Overbeek) theory. DLVO theory assumes of a uniform surface charge density but real surfaces often contain chemical heterogeneities that can introduce variations in surface charge density. While numerous studies have revealed a great deal on the role of charge heterogeneities in particle deposition, direct force measurement of heterogeneously charged surfaces has remained a largely unexplored area of research. Force measurements would allow for systematic investigation into the effects of charge heterogeneities on surface forces. A significant challenge with employing force measurements of heterogeneously charged surfaces is the size of the interaction area, referred to in literature as the electrostatic zone of influence. For microparticles, the size of the zone of influence is, at most, a few hundred nanometers across. Creating a surface with well-defined patterned heterogeneities within this area is out of reach of most conventional photolithographic techniques. Here, we present a means of simultaneously scaling up the electrostatic zone of influence and performing direct force measurements with micropatterned heterogeneously charged surfaces by employing the surface forces apparatus (SFA). A technique is developed here based on the vapor deposition of an aminosilane (3- aminopropyltriethoxysilane, APTES) through elastomeric membranes to create surfaces for force measurement experiments.
    [Show full text]
  • Open Rxg276 Final Thesis 2017.Pdf
    The Pennsylvania State University The Graduate School Department of Chemical Engineering CHEMICALLY-DRIVEN TRANSPORT OF COLLOIDAL PARTICLES AND MOLECULES A Dissertation in Chemical Engineering by Rajarshi Guha 2017 Rajarshi Guha Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2017 The dissertation of Rajarshi Guha was reviewed and approved* by the following: Darrell Velegol Distinguished Professor of Chemical Engineering Committee Co-Chair Dissertation Co-Advisor Manish Kumar Professor of Chemical Engineering Committee Co-Chair Dissertation Co-Advisor Ayusman Sen Distinguished Professor of Chemistry Dissertation Co-Advisor Thomas K. Wood Professor of Chemical Engineering Special Signatory Dissertation Co-Advisor Michael Hickner Professor of Materials Science and Engineering, Chemical Engineering Philip Savage Professor of Chemical Engineering Head of the Department of Chemical Engineering *Signatures are on file in the Graduate School iii ABSTRACT Chemically-driven transports are ubiquitous in nature and govern the working principles of several biochemical processes. Understading the mechanisms involved in such transport processes would enable us to answer and potentially solve many problems in separation, diagnostics, energy, electronics and communications. In this dissertation, I primarily addressed chemically-driven transports which arise from artificial and natural concentration gradients. The two major chemically-driven transports discussed in this dissertation are electrokinetic
    [Show full text]
  • Nanofluidics: a Pedagogical Introduction Simon Gravelle
    Nanofluidics: a pedagogical introduction Simon Gravelle To cite this version: Simon Gravelle. Nanofluidics: a pedagogical introduction. 2016. hal-02375018 HAL Id: hal-02375018 https://hal.archives-ouvertes.fr/hal-02375018 Preprint submitted on 21 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Nanofluidics: a pedagogical introduction Simon Gravelle 01 MARCH 2016 1 Generalities reasonable expectation. Moreover, one can notice that most of the biological processes involving fluids 1.1 What is nanofluidics? operate at the nano-scale, which is certainly not by chance [8]. For example, the protein that regulates Nanofluidics is the study of fluids confined in struc- water flow in human body, called aquaporin, has got tures of nanometric dimensions (typically 1−100 nm) sub-nanometric dimensions [15, 16]. Aquaporins are [1, 2]. Fluids confined in these structures exhibit be- known to combine high water permeability and good haviours that are not observed in larger structures, salt rejection, participating for example to the high due to a high surface to bulk ratio. Strictly speak- efficiency of human kidney. Biological processes in- ing, nanofluidics is not a new research field and has volving fluid and taking place at the nanoscale attest been implicit in many disciplines [3, 4, 5, 6, 7], but of the potential applications of nanofluidics, and con- has received a name of its own only recently.
    [Show full text]
  • Arxiv:2007.11114V2 [Cond-Mat.Soft] 18 Nov 2020 Tering and Device Clogging
    Reversible Trapping of Colloids in Microgrooved Channels via Diffusiophoresis under Steady-State Solute Gradients Naval Singh,1 Goran T. Vladisavljevi´c,1 Fran¸coisNadal,2 C´ecile 3 3 1, Cottin-Bizonne, Christophe Pirat, and Guido Bolognesi ∗ 1Department of Chemical Engineering, Loughborough University, Loughborough, LE11 3TU, United Kingdom 2Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, LE11 3TU, United Kingdom 3Institut Lumi`ere Mati`ere, UMR5306 Universit´eClaude Bernard Lyon 1 - CNRS, Universit´ede Lyon, Villeurbanne Cedex, 69622, France (Dated: November 20, 2020) The controlled transport of colloids in dead-end structures is a key capability that can enable a wide range of applications, such as bio-chemical analysis, drug delivery and underground oil recovery. This letter presents a new trapping mechanism that allows the fast (i.e., within a few minutes) and reversible accumulation of sub-micron particles within dead-end micro-grooves by means of parallel streams with different salinity level. For the first time, particle focusing in dead-end structures is achieved under steady-state gradients. Confocal microscopy analysis and numerical investigations show that the particles are trapped at a flow recirculation region within the grooves due to a combination of diffusiophoresis transport and hydrodynamic effects. Counterintuitively, the particle velocity at the focusing point is not vanishing and, hence, the particles are continuously transported in and out of the focusing point. The accumulation process is also reversible and one can cyclically trap and release the colloids by controlling the salt concentration of the streams via a flow switching valve. Particle transport in confined structures plays an im- instance, steady-state gradients of chemically reactive portant role in several technological applications, includ- solutes [6] can induce particle focusing at the location ing drug delivery, diagnostics, enhanced oil recovery, par- where uDP = 0.
    [Show full text]
  • Debye Length Dependence of the Anomalous Dynamics of Ionic Double Layers in a Parallel Plate Capacitor R
    Article pubs.acs.org/JPCC Debye Length Dependence of the Anomalous Dynamics of Ionic Double Layers in a Parallel Plate Capacitor R. J. Kortschot, A. P. Philipse, and B. H. Erne*́ Van ’tHoff Laboratory for Physical and Colloid Chemistry, Debye Institute for Nanomaterials Science, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands ABSTRACT: The electrical impedance spectrum of simple ionic solutions is measured in a parallel plate capacitor at small applied ac voltage. The influence of the ionic strength is investigated using several electrolytes at different concentrations in solvents of different dielectric constants. The electric double layers that appear at the electrodes at low frequencies are not perfectly capacitive. At moderate ionic strength, ion transport agrees with a model based on the Poisson−Nernst−Planck (PNP) equations. At low ionic strength, double layer dynamics deviate from the PNP model, and the deviation is well described by an empirical function with only one fit parameter. This deviation from the PNP equations increases systematically with increasing Debye length, possibly caused by the long-range Coulomb interaction. 1. INTRODUCTION This paper starts with a theoretical discussion of electrode polarization in section 2 and a description of our experimental The accumulation of charged species near an electrode, so-called fi electrode polarization, occurs in and complicates dielectric methods in section 3. Then data are presented and tted in spectroscopy measurements of, for instance, biological materials,1 section 4. Finally, a general discussion is made in section 5 and solids,2 and colloidal dispersions.3 Additionally, electrode polar- conclusions are drawn in section 6.
    [Show full text]
  • 10.626 Lecture Notes, Electrostatic Correlations
    V. Electrostatics Lecture 28: Electrostatic Correlations April 13, 2011 Notes by MIT Student (and MZB) 1. Mean-Field Theory Until now, we have developed continuum models for the dynamics of electrolytes at the macroscopic scale, much larger than typical length scales for intermolecular forces. In particular, we have used principles of nonequlibrium thermodynamics, to arrive at general models of interfacial voltage (Nernst equation), reaction kinetics (Butler-Volmer equation), and transport (Poisson-Nernst-Planck equations): ⎧ µ = k T ln(γ c i ) + z eφ ⎪ i B i i ⎪ Fi = −Mi ci∇µi ⎪ ⎨ ∂ci + ∇⋅ F = 0 ⎪ ∂t i ⎪ ⎪ −∇⋅ (ε∇φ) = ρ = ∑zieci ⎩ i For short-range interactions, such as chemical bonds or collision forces contributing to the diffusivity ( D = M / k T ), this formulation is easily justified, but for long-range interactions, i i B such as the electrostatic Coulomb potential, which decays very slowly with distance (like 1/r), it can be more problematic. Continuum models like the Poisson-Nernst-Planck equations are “mean-field approximations” which express the statistically averaged behavior of a large number of discrete ions in terms of only the mean concentrations ci (number/volume) and a particular definition of the mean potential φ. Each ion migrates in the mean electric field, E = −∇φ , which is self-consistently produced by the mean charge density, not by the discrete, fluctuating charges in the molecular system. Due to the long-range nature of electrostatic forces, however, discrete ion-ion interactions leading to “electrostatic correlations” are important, even in dilute electrolytes. 2. Bjerrum length What is the length scale below which electrostatic correlations are important? In very dense charged systems, it is the ion size, as in solvent free ionic liquids (see below).
    [Show full text]
  • 1 Transport-Induced Inversion of Screening Ionic Charges In
    Transport-induced inversion of screening ionic charges in nano-channels Xin Zhu, Lingzi Guo, Sheng Ni, Xingye Zhang, Yang Liu* College of Information Science and Electronic Engineering Zhejiang University Hangzhou, China 310013 *E-mail: [email protected] Abstract: This work reveals a counter-intuitive but basic process of ionic screening in nano-fluidic channels. Steady-state numerical simulations and mathematical analysis show that, under significant longitudinal ionic transport, the screening ionic charges can be locally inverted in the channels: their charge sign becomes the same as that of the channel surface charges. The process is identified to originate from the coupling of ionic electro-diffusion transport and junction 2-D electrostatics. This finding may expand our understanding on ionic screening and electrical double layers in nano-channels. Furthermore, the charge inversion process results in a body-force torque on channel fluids, which is a possible mechanism for vortex generation in the channels and their nonlinear current- voltage characteristics. TOC Graphic 1 Ionic screening is a fundamental physical process in electrolytic solutions. In nano-fluidic channels, the formation of screening electrical double layers (EDLs) at channel surfaces plays a central role in their transport phenomena, including perm-selectivity1-3, concentration polarization4-8, and nonlinear conductance9-20, as well as their broad device applications21-26. In modeling ionic screening in nano-channels, the condition of local electroneutrality was often adopted by assuming complete screening in the channel transversal direction: the sum of the surface and ion charges along an arbitrary transversal line is zero7,8,13. Nevertheless, more detailed account of the electrostatics, particularly at the junctions of nano-channels and micro-reservoirs, have been found important when electrokinetic transport is significant 10,14,15,21.
    [Show full text]
  • Introduction to Plasma Physics
    Published by CERN in the Proceedings of the CAS-CERN Accelerator School: Plasma Wake Acceleration, Geneva, Switzerland, 23–29 November 2014, edited by B. Holzer, CERN-2016-001 (CERN, Geneva, 2016) Introduction to Plasma Physics P. Gibbon Forschungszentrum Jülich GmbH, Institute for Advanced Simulation, Jülich Supercomputing Centre, Jülich, Germany Abstract These notes are intended to provide a brief primer in plasma physics, intro- ducing common definitions, basic properties, and typical processes found in plasmas. These concepts are inherent in contemporary plasma-based accelera- tor schemes, and thus provide a foundation for the more advanced expositions that follow in this volume. No prior knowledge of plasma physics is required, but the reader is assumed to be familiar with basic electrodynamics and fluid mechanics. Keywords Plasma properties; 2-fluid model; Langmuir waves; electromagnetic wave prop- agation; dispersion relation; nonlinear waves. 1 Plasma types and definitions Plasmas are often described as the fourth state of matter, alongside gases, liquids and solids, a definition which does little to illuminate their main physical attributes. In fact, a plasma can exhibit behaviour characteristic of all three of the more familiar states, depending on its density and temperature, so we obviously need to look for other distinguishing features. A simple textbook definition of a plasma [1, 2] would be: a quasi-neutral gas of charged particles showing collective behaviour. This may seem precise enough, but the rather fuzzy-sounding terms of ‘quasi-neutrality’ and ‘collectivity’ require further explanation. The first of these, ‘quasi-neutrality’, is actually just a mathematical way of saying that even though the particles making up a plasma consist of free electrons and ions, their overall charge densities cancel each other in equilibrium.
    [Show full text]