Nonlinear Electrostatics. the Poisson-Boltzmann Equation
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A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action
Article A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action Ivan Kennedy 1,2,*, Harold Geering 2, Michael Rose 3 and Angus Crossan 2 1 Sydney Institute of Agriculture, University of Sydney, NSW 2006, Australia 2 QuickTest Technologies, PO Box 6285 North Ryde, NSW 2113, Australia; [email protected] (H.G.); [email protected] (A.C.) 3 NSW Department of Primary Industries, Wollongbar NSW 2447, Australia; [email protected] * Correspondence: [email protected]; Tel.: + 61-4-0794-9622 Received: 23 March 2019; Accepted: 26 April 2019; Published: 1 May 2019 Abstract: A convenient practical model for accurately estimating the total entropy (ΣSi) of atmospheric gases based on physical action is proposed. This realistic approach is fully consistent with statistical mechanics, but reinterprets its partition functions as measures of translational, rotational, and vibrational action or quantum states, to estimate the entropy. With all kinds of molecular action expressed as logarithmic functions, the total heat required for warming a chemical system from 0 K (ΣSiT) to a given temperature and pressure can be computed, yielding results identical with published experimental third law values of entropy. All thermodynamic properties of gases including entropy, enthalpy, Gibbs energy, and Helmholtz energy are directly estimated using simple algorithms based on simple molecular and physical properties, without resource to tables of standard values; both free energies are measures of quantum field states and of minimal statistical degeneracy, decreasing with temperature and declining density. We propose that this more realistic approach has heuristic value for thermodynamic computation of atmospheric profiles, based on steady state heat flows equilibrating with gravity. -
LECTURES on MATHEMATICAL STATISTICAL MECHANICS S. Adams
ISSN 0070-7414 Sgr´ıbhinn´ıInstiti´uid Ard-L´einnBhaile´ Atha´ Cliath Sraith. A. Uimh 30 Communications of the Dublin Institute for Advanced Studies Series A (Theoretical Physics), No. 30 LECTURES ON MATHEMATICAL STATISTICAL MECHANICS By S. Adams DUBLIN Institi´uid Ard-L´einnBhaile´ Atha´ Cliath Dublin Institute for Advanced Studies 2006 Contents 1 Introduction 1 2 Ergodic theory 2 2.1 Microscopic dynamics and time averages . .2 2.2 Boltzmann's heuristics and ergodic hypothesis . .8 2.3 Formal Response: Birkhoff and von Neumann ergodic theories9 2.4 Microcanonical measure . 13 3 Entropy 16 3.1 Probabilistic view on Boltzmann's entropy . 16 3.2 Shannon's entropy . 17 4 The Gibbs ensembles 20 4.1 The canonical Gibbs ensemble . 20 4.2 The Gibbs paradox . 26 4.3 The grandcanonical ensemble . 27 4.4 The "orthodicity problem" . 31 5 The Thermodynamic limit 33 5.1 Definition . 33 5.2 Thermodynamic function: Free energy . 37 5.3 Equivalence of ensembles . 42 6 Gibbs measures 44 6.1 Definition . 44 6.2 The one-dimensional Ising model . 47 6.3 Symmetry and symmetry breaking . 51 6.4 The Ising ferromagnet in two dimensions . 52 6.5 Extreme Gibbs measures . 57 6.6 Uniqueness . 58 6.7 Ergodicity . 60 7 A variational characterisation of Gibbs measures 62 8 Large deviations theory 68 8.1 Motivation . 68 8.2 Definition . 70 8.3 Some results for Gibbs measures . 72 i 9 Models 73 9.1 Lattice Gases . 74 9.2 Magnetic Models . 75 9.3 Curie-Weiss model . 77 9.4 Continuous Ising model . -
Entropy and H Theorem the Mathematical Legacy of Ludwig Boltzmann
ENTROPY AND H THEOREM THE MATHEMATICAL LEGACY OF LUDWIG BOLTZMANN Cambridge/Newton Institute, 15 November 2010 C´edric Villani University of Lyon & Institut Henri Poincar´e(Paris) Cutting-edge physics at the end of nineteenth century Long-time behavior of a (dilute) classical gas Take many (say 1020) small hard balls, bouncing against each other, in a box Let the gas evolve according to Newton’s equations Prediction by Maxwell and Boltzmann The distribution function is asymptotically Gaussian v 2 f(t, x, v) a exp | | as t ≃ − 2T → ∞ Based on four major conceptual advances 1865-1875 Major modelling advance: Boltzmann equation • Major mathematical advance: the statistical entropy • Major physical advance: macroscopic irreversibility • Major PDE advance: qualitative functional study • Let us review these advances = journey around centennial scientific problems ⇒ The Boltzmann equation Models rarefied gases (Maxwell 1865, Boltzmann 1872) f(t, x, v) : density of particles in (x, v) space at time t f(t, x, v) dxdv = fraction of mass in dxdv The Boltzmann equation (without boundaries) Unknown = time-dependent distribution f(t, x, v): ∂f 3 ∂f + v = Q(f,f) = ∂t i ∂x Xi=1 i ′ ′ B(v v∗,σ) f(t, x, v )f(t, x, v∗) f(t, x, v)f(t, x, v∗) dv∗ dσ R3 2 − − Z v∗ ZS h i The Boltzmann equation (without boundaries) Unknown = time-dependent distribution f(t, x, v): ∂f 3 ∂f + v = Q(f,f) = ∂t i ∂x Xi=1 i ′ ′ B(v v∗,σ) f(t, x, v )f(t, x, v∗) f(t, x, v)f(t, x, v∗) dv∗ dσ R3 2 − − Z v∗ ZS h i The Boltzmann equation (without boundaries) Unknown = time-dependent distribution -
A Molecular Modeler's Guide to Statistical Mechanics
A Molecular Modeler’s Guide to Statistical Mechanics Course notes for BIOE575 Daniel A. Beard Department of Bioengineering University of Washington Box 3552255 [email protected] (206) 685 9891 April 11, 2001 Contents 1 Basic Principles and the Microcanonical Ensemble 2 1.1 Classical Laws of Motion . 2 1.2 Ensembles and Thermodynamics . 3 1.2.1 An Ensembles of Particles . 3 1.2.2 Microscopic Thermodynamics . 4 1.2.3 Formalism for Classical Systems . 7 1.3 Example Problem: Classical Ideal Gas . 8 1.4 Example Problem: Quantum Ideal Gas . 10 2 Canonical Ensemble and Equipartition 15 2.1 The Canonical Distribution . 15 2.1.1 A Derivation . 15 2.1.2 Another Derivation . 16 2.1.3 One More Derivation . 17 2.2 More Thermodynamics . 19 2.3 Formalism for Classical Systems . 20 2.4 Equipartition . 20 2.5 Example Problem: Harmonic Oscillators and Blackbody Radiation . 21 2.5.1 Classical Oscillator . 22 2.5.2 Quantum Oscillator . 22 2.5.3 Blackbody Radiation . 23 2.6 Example Application: Poisson-Boltzmann Theory . 24 2.7 Brief Introduction to the Grand Canonical Ensemble . 25 3 Brownian Motion, Fokker-Planck Equations, and the Fluctuation-Dissipation Theo- rem 27 3.1 One-Dimensional Langevin Equation and Fluctuation- Dissipation Theorem . 27 3.2 Fokker-Planck Equation . 29 3.3 Brownian Motion of Several Particles . 30 3.4 Fluctuation-Dissipation and Brownian Dynamics . 32 1 Chapter 1 Basic Principles and the Microcanonical Ensemble The first part of this course will consist of an introduction to the basic principles of statistical mechanics (or statistical physics) which is the set of theoretical techniques used to understand microscopic systems and how microscopic behavior is reflected on the macroscopic scale. -
Entropy: from the Boltzmann Equation to the Maxwell Boltzmann Distribution
Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution A formula to relate entropy to probability Often it is a lot more useful to think about entropy in terms of the probability with which different states are occupied. Lets see if we can describe entropy as a function of the probability distribution between different states. N! WN particles = n1!n2!....nt! stirling (N e)N N N W = = N particles n1 n2 nt n1 n2 nt (n1 e) (n2 e) ....(nt e) n1 n2 ...nt with N pi = ni 1 W = N particles n1 n2 nt p1 p2 ...pt takeln t lnWN particles = "#ni ln pi i=1 divide N particles t lnW1particle = "# pi ln pi i=1 times k t k lnW1particle = "k# pi ln pi = S1particle i=1 and t t S = "N k p ln p = "R p ln p NA A # i # i i i=1 i=1 ! Think about how this equation behaves for a moment. If any one of the states has a probability of occurring equal to 1, then the ln pi of that state is 0 and the probability of all the other states has to be 0 (the sum over all probabilities has to be 1). So the entropy of such a system is 0. This is exactly what we would have expected. In our coin flipping case there was only one way to be all heads and the W of that configuration was 1. Also, and this is not so obvious, the maximal entropy will result if all states are equally populated (if you want to see a mathematical proof of this check out Ken Dill’s book on page 85). -
Boltzmann Equation II: Binary Collisions
Physics 127b: Statistical Mechanics Boltzmann Equation II: Binary Collisions Binary collisions in a classical gas Scattering out Scattering in v'1 v'2 v 1 v 2 R v 2 v 1 v'2 v'1 Center of V' Mass Frame V θ θ b sc sc R V V' Figure 1: Binary collisions in a gas: top—lab frame; bottom—centre of mass frame Binary collisions in a gas are very similar, except that the scattering is off another molecule. An individual scattering process is, of course, simplest to describe in the center of mass frame in terms of the relative E velocity V =Ev1 −Ev2. However the center of mass frame is different for different collisions, so we must keep track of the results in the lab frame, and this makes the calculation rather intricate. I will indicate the main ideas here, and refer you to Reif or Landau and Lifshitz for precise discussions. Lets first set things up in the lab frame. Again we consider the pair of scattering in and scattering out processes that are space-time inverses, and so have identical cross sections. We can write abstractly for the scattering out from velocity vE1 due to collisions with molecules with all velocities vE2, which will clearly be proportional to the number f (vE1) of molecules at vE1 (which we write as f1—sorry, not the same notation as in the previous sections where f1 denoted the deviation of f from the equilibrium distribution!) and the 3 3 number f2d v2 ≡ f(vE2)d v2 in each velocity volume element, and then we must integrate over all possible vE0 vE0 outgoing velocities 1 and 2 ZZZ df (vE ) 1 =− w(vE0 , vE0 ;Ev , vE )f f d3v d3v0 d3v0 . -
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. -
Boltzmann Equation
Boltzmann Equation ● Velocity distribution functions of particles ● Derivation of Boltzmann Equation Ludwig Eduard Boltzmann (February 20, 1844 - September 5, 1906), an Austrian physicist famous for the invention of statistical mechanics. Born in Vienna, Austria-Hungary, he committed suicide in 1906 by hanging himself while on holiday in Duino near Trieste in Italy. Distribution Function (probability density function) Random variable y is distributed with the probability density function f(y) if for any interval [a b] the probability of a<y<b is equal to b P=∫ f ydy a f(y) is always non-negative ∫ f ydy=1 Velocity space Axes u,v,w in velocity space v dv have the same directions as dv axes x,y,z in physical du dw u space. Each molecule can be v represented in velocity space by the point defined by its velocity vector v with components (u,v,w) w Velocity distribution function Consider a sample of gas that is homogeneous in physical space and contains N identical molecules. Velocity distribution function is defined by d N =Nf vd ud v d w (1) where dN is the number of molecules in the sample with velocity components (ui,vi,wi) such that u<ui<u+du, v<vi<v+dv, w<wi<w+dw dv = dudvdw is a volume element in the velocity space. Consequently, dN is the number of molecules in velocity space element dv. Functional statement if often omitted, so f(v) is designated as f Phase space distribution function Macroscopic properties of the flow are functions of position and time, so the distribution function depends on position and time as well as velocity. -
FORCES and INTERACTIONS BETWEEN NANOPARTICLES for CONTROLLED STRUCTURES by PAUL R. MARK a Dissertation Submitted to the Graduate
FORCES AND INTERACTIONS BETWEEN NANOPARTICLES FOR CONTROLLED STRUCTURES by PAUL R. MARK A dissertation submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Materials Science and Engineering Written under the direction of Dr. Laura Fabris And approved by ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ New Brunswick, New Jersey January, 2013 ABSTRACT OF THE DISSERTATION Forces and Interactions Between Nanoparticles for Controlled Structures PAUL R MARK Dissertation Director: Dr. Laura Fabris In recent years, structured nanomaterials have started to demonstrate their full potential in breakthrough technologies. However, in order to fulfill the expectations held for the field, it is necessary to carefully design these structures depending upon the targeted application. This tailoring process suggests that a feedback between theory and experiment could potentially allow us to obtain a structure as near optimum as possible. This thesis seeks to describe the theory and experiment needed to understand and control the interactions among nanoparticles to build a functional device for the efficient conversion of sunlight into energy. This thesis will discuss a simulation built from the existing theories explaining nanoparticle interactions -
Arxiv:1706.04027V2
Interactions between Silica Particles in the Presence of Multivalent Coions Biljana Uzelac, Valentina Valmacco,∗ and Gregor Trefalt† Department of Inorganic and Analytical Chemistry, University of Geneva, Sciences II, 30 Quai Ernest-Ansermet, 1205 Geneva, Switzerland (Dated: July 25, 2017) Forces between charged silica particles in solutions of multivalent coions are measured with col- loidal probe technique based on atomic force microscopy. The concentration of 1:z electrolytes is systematically varied to understand the behavior of electrostatic interactions and double-layer prop- erties in these systems. Although the coions are multivalent the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory perfectly describes the measured force profiles. The diffuse-layer potentials and regulation properties are extracted from the forces profiles by using the DLVO theory. The de- pendencies of the diffuse-layer potential and regulation parameter shift to lower concentration with increasing coion valence when plotted as a function of concentration of 1:z salt. Interestingly, these profiles collapse to a master curve if plotted as a function of monovalent counterion concentration. I. INTRODUCTION in the case of multivalent coion systems. Force pro- files across solutions containing multivalent counterions Interactions between charged objects in electrolyte so- are usually exponential, which is typical for double-layer lutions are important for many biological systems [1], and forces. Interestingly, their coion counterparts invoke non- in processes such as paper making [2], waste water treat- exponential and soft long-ranged forces, which can be ment [3], ceramic processing [4], ink-jet printing [5], par- well described with the PB theory [21, 27]. Extreme case ticle design [6], and concrete hardening [7]. -
Heteroaggregation and Homoaggregation of Latex Particles in the Presence of Alkyl Sulfate Surfactants
colloids and interfaces Article Heteroaggregation and Homoaggregation of Latex Particles in the Presence of Alkyl Sulfate Surfactants Tianchi Cao 1,2, Michal Borkovec 1 and Gregor Trefalt 1,* 1 Department of Inorganic and Analytical Chemistry, University of Geneva, Sciences II, 30 Quai Ernest-Ansermet, 1205 Geneva, Switzerland; [email protected] (T.C.); [email protected] (M.B.) 2 Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06520, USA * Correspondence: [email protected] Received: 19 October 2020; Accepted: 6 November 2020; Published: 13 November 2020 Abstract: Heteroaggregation and homoaggregation is investigated with time-resolved multi-angle dynamic light scattering. The aggregation rates are measured in aqueous suspensions of amidine latex (AL) and sulfate latex (SL) particles in the presence of sodium octyl sulfate (SOS) and sodium dodecyl sulfate (SDS). As revealed by electrophoresis, the surfactants adsorb to both types of particles. For the AL particles, the adsorption of surfactants induces a charge reversal and triggers fast aggregation close to the isoelectric point (IEP). The negatively charged SL particles remain negatively charged and stable in the whole concentration range investigated. The heteroaggregation rates for AL and SL particles are fast at low surfactant concentrations, where the particles are oppositely charged. At higher concentrations, the heteroaggregation slows down above the IEP of the AL particles, where the particles become like-charged. The SDS has higher affinity to the surface compared to the SOS, which induces a shift of the IEP and of the fast aggregation regime to lower surfactant concentrations. Keywords: colloidal stability; heteroaggregation; DLVO 1. Introduction Aggregation of particles in suspensions is important in many practical applications, such as ceramic processing, paint fabrication, or drug formulation [1–4]. -
Understanding Adhesion: a Means for Preventing Fouling
ELSEVIER Understanding Adhesion: A Means for Preventing Fouling R. Oliveira • Adhesion of particulate materials is an important step in the formation Engenharia Biol6gica, of fouling. Because the size of such materials is generally less than 1 tzm, Unicersidade do Minho, the phenomenon can be described in terms of colloid chemistry. Accord- Braga, Portugal ingly, the net force of interaction between foulants and the surface has been described in terms of DLVO theory (van der Waals attraction and electrostatic double-layer repulsion). However, those forces are sometimes not sufficient to describe the formation of fouling. Recent works have made it possible to calculate the effect of hydrophobic interactions and steric forces, which can also be taken into account. In aqueous media, the various types of interactions can be strongly affected by the pH, the ionic strength, the type of ions, and the presence of polymeric molecules. The objective of this work is to give a general overview of the basic physicochemical factors playing a role in fouling and to outline some practical aspects related to the theoretical reasoning to help prevent or at leasl[ mitigate fouling. © Elsevier Science Inc., 1997 Keywords: adhesion, DLVO theory, surface free energy, hydrophobic interactions, steric forces, ionic strength, ion bridging INTRODUCTION tant roles. In bifouling, the adhesion of microorganisms can be strongly dependent on their external appendages. The accumulation of inorganic particles, microorganisms, The net effect is a balance between all possible interac- macromolecules, and corrosion products on heat ex- tions. A knowledge of the roles of the main variables changer surfaces gives rise to the so-called fouling phe- affecting the interactions outlined above is of great impor- nomenon.