Mean Electrostatic Potential Research Articles

Numerical solution of a modified Poisson-Boltzmann equation for 1 : 2 and 2 : 1 electrolytes in the diffuse layer

A modified Poisson-Boltzmann (MPB) equation for an unsymmetrically charged electrolyte in the diffuse part of the electric double layer at a plane charged wall is solved numerically using a quasi-linearization procedure. Computations are carried out for 1 : 2 and 2 : 1 restricted primitive model electrolytes with no imaging and for a metallic wall and the results compared with the classical Gouy-Chapman-Stern theory. Except for negligible surface charge, the system with a divalent counter ion is the most sensitive to any change in its physical parameters. In general the MPB mean electrostatic potential, in contrast to the Gouy-Chapman-Stern potential, is not a monotonic decreasing function. The asymptotic behaviour of the MPB equation implies charge oscillations above a critical electrolyte concentration (≳0·23 M) while below this concentration imaging or surface charge-ion interactions can produce a charge inversion. Charge separation is found for no surface charge with a metallic wall. The point ion lim...

Renormalized plasma turbulence theory: A quasiparticle picture

A general renormalized statistical theory of Vlasov turbulence is given which proceeds directly from the Vlasov equation and does not assume prior knowledge of sophisticated field-theoretic techniques. Quasiparticles are the linear excitations of the turbulent system away from its instantaneous mean (ensemble-averaged) state or background; the properties of this background state ''dress'' or renormalize the quasiparticle responses. It is shown that all two-point responses (including the dielectric) and all two-point correlation functions can be completely described by the mean distribution function and three fundamental quantities. Two of these are the quasiparticle responses: the propagator and the potential source: which measure, respectively, the separate responses of the mean distribution function and the mean electrostatic potential to functional changes in an external phase-space source added to Vlasov's equation. The third quantity is the two-point correlation function of the incoherent part of the phase-space density which acts as a self-consistent source of quasiparticle and potential fluctuations. This theory explicitly takes into account the self-consistent nature of the electrostatic-field fluctuations which introduces new effects not found in the usual ''test-particle'' theories. Explicit equations for the fundamental quantities are derived in the direct interaction approximation. Special attention is paid to the two-point correlations and themore » relation to theories of phase-space granulation.« less

Distribution of ions in electrolyte solutions and plasmas. Part 3

Using the expression for the binary distribution function (Fji) obtained in Part 2, the potential of mean force between two ions (Wji) and the mean electrostatic potential (ψj) are determined for dilute solutions to terms of the order ε3 for symmetrical and ε2 for unsymmetrical electrolytes. The reason for the failure of the assumption Wji=eiψj=ejψi beyond the first-order in ε is discussed and it is shown that Wji=eiψi+(εn) where n= 3 for symmetrical and 2 for unsymmetrical electrolytes. The degree of error in both linearized and non-linearized forms of the Poisson–Boltzmann equation is determined, and it is shown that, although of the same order, the error is more severe in the linearized equation. The Kirkwood closure Fjik=FjiFjkFik is examined and it is shown that in the limit t→ 0 Fjik=(FjiFjkFik)[1 +(εn)] where again n= 3 for symmetrical and 2 for unsymmetrical electrolytes. By solving the third equation in the BBGY hierarchy, differential equations are obtained for these higher terms. The ground is prepared for the later investigation of concentrated electrolyte solutions.

Theory of the electric double layer using a modified poisson–boltzman equation

A modified Poisson–Boltzmann equation in diffuse double-layer theory which includes estimates for the fluctuation and exclusion volume terms is solved numerically for a 1 : 1 restricted-model electrolyte. Two estimates of the exclusion-volume term are considered at three electrolyte concentrations with varying surface charge when image terms are neglected. Details are given of the numerical calculation and double-layer properties such as the mean electrostatic potential, the ion–wall distribution functions and the differential capacitance are presented. Comparisons are made with the Gouy–Chapman–Stern theory and with some preliminary hypernetted chain (HNC) calculations.

Statistical mechanical theories of the electric double layer

A comparison is made of different recent developments in theories of the electric double layer in aqueous media, based on statistical mechanics. An important advance was made when Henderson, Blum and co-workers showed how the Ornstein-Zernike equations, which originally applied to bulk liquid mixtures, could be modified to yield the density and charge distributions in the diffuse layer next to a charged wall. The interface solution system was treated as a mixture in which the wall of the interface is a solute particle which is exceedingly large in size but small in concentration. In the case of electrolytes (systems with Coulombic forces) a correction was necessary to earlier formulations of the theory. Various approximations well-known in modern statistical mechanical theories of liquids were used by the above authors in the Ornstein-Zernike equations modified to apply at a plane interface. All these approximations make use of the primitive model of an electrolyte, in which ions are treated as charged hard spheres embedded in a continuous solvent medium with a uniform dielectric permittivity so that the ions (and their hydration shells) have the same permittivity as the bulk electrolyte. The hypernetted chain (HNC) approximation, which has been very successful for bulk electrolytes was shown to be equivalent to the non-linear Poisson-Boltzmann (P.B.) equation of Gouy and Chapman, when the effects of the ionic hard-sphere terms can be ignored. Another approach, the so-called mean spherical approximation (MSA), corresponds to the Gouy-Chapman-Stern (GCS) theory at small potentials (linearised P.B. or Debye-Huckel equation. In the GCS theory, the ions interact with the interfacial wall as hard-spheres but continue to interact with each other as point charges. In the MSA ion-size effects are taken into account in a consistent manner. Although the differential capacity of the charged interface is calculated by the MSA theory and compared with that derived from the Stern theory, there is an important difference. The inner region capacity is much too large in the MSA theory because the effective dielectric permittivity of the inner region is put equal to that bulk electrolyte. Monte Carlo calculations of the electric double layer, based on statistical mechanics, have been performed by Torrie and Valleau, who use the primitive model of the bulk electrolyte. Their concentration profiles in the diffuse layer are in good agreement even up to 1 M with the so-called modified Gouy-Chapman (MGC) theory, which introduces an inner region with the same permittivity as the bulk electrolyte. However the criticism mentioned above about the use of bulk permittivity applies also to these calculations. An entirely different approach to electric double layer theory has been used by the authors and co-workers. This develops a modified Poisson-Boltzmann (MPB) equation for the electrostatic potential in the diffuse layer, which is based on the classical work of Kirkwood in strong electrolyte theory. The exclusion volume term and the fluctuation term are re-introduced, again with a primitive model of the bulk electrolyte. Making use of a ‘closure’ procedure originally due to Loeb for handling the fluctuation terms, the classical P.B. equation is replaced by a non-linear differential-difference equation which is solved numerically. At moderate electrolyte concentrations (>1 M) the mean electrostatic potential and the distribution of ions in the diffuse layer exhibit damped oscillatory behaviour. The MSA shows similar oscillations which are attributed to ion-size effects, neglected in the P.B. equation. Physically the damped oscillatory behaviour means that there is a stratification of charge in the vicinity of the interface which depends primarily on the electrolyte concentration. At low surface charge and electrolyte concentration, the exclusion volume term has negligible effect compared to the fluctuation term but as the surface charge or electrolyte concentration is increased, the exclusion volume term becomes increasingly important. Comparison of the differential capacitance is made with preliminary calculations of Henderson, Blum and Smith, based on the HNC and identical behaviour is obtained at low surface charge. In both theories, the permittivity of the Spern inner region is equated to the bulk permittivity. Effects relating to solvent structure still remain to be incorporated. Solvent effects have been considered by Ninham and his school, using the Percus-Yevick and HNC equations, but these have been confined to non-coulombic solutes. Improvements in the model of the Stern layer can be incorporated into our theory based on the MPB equation. Also by adapting related work in the electrolyte bulk by one of the authors (C.W.) it is possible that some solvent effects can be incorporated into the MPB equation.

Numerical solution of a modified Poisson-Boltzmann equation in electric double layer theory

A modified Poisson-Boltzmann equation in the diffuse part of the electric double layer is solved using a quasi-linearisation technique. A damped oscillatory behaviour is demonstrated in the mean electrostatic potential and in the ion-wall distribution functions for a fixed wall potential at relatively high electrolyte concentrations.

Calculation of the mean electrostatic potential in the electric double layer using the mean spherical approximation

An analytical solution within the mean spherical approximation is given for the mean electrostatic potential and ion-wall distribution function for a restricted primitive model electrolyte in the vicinity of a plane charged wall.

Double-layer interaction of two charged colloidal spherical particles of a concentrated dispersion in a medium of low dielectric constant. Part 2.—A cell model

The double-layer force between two neighbouring, charged, identical, spherical colloidal particles O1 and O2 of a concentrated dispersion in a hydrocarbon medium is investigated. Only one ionic species, constituting the counter-ion, is present in the hydrocarbon at a very low concentration. It is assumed that the surface charge on each particle is fixed and uniformly distributed over the surface. We seek the mean force between O1 and O2 averaged over all positions of counter-ions and other particles. Introducing a cell model, the two particles are enclosed by a symmetrical surface S0 such that the net charge contained by S0 is zero. The position of the surface S0 is determined by assuming that the normal derivative of the mean electrostatic potential is zero on it. Because the concentration of counter-ions is sufficiently small to give an almost uniform charge density in the hydrocarbon medium enclosed by S0, the volume contained in S0 equals twice the mean volume per particle in the dispersion. In order to isolate electrically the region contained within S0 as much as possible from the surroundings, an additional condition on S0 is that the potential variation on it be small. The potential satisfies the Poisson–Boltzmann equation in the hydrocarbon medium and the Laplace equation inside the particles, the solutions to these equations being expressed in terms of multipole expansions about the two particle centres. Up to quadrupole terms, the double layer force between the two particles is found to be attractive at separations corresponding to the mean separation between neighbouring particles in the dispersion, although this force becomes repulsive at smaller separations. The force calculations indicate that the electric double layer will not produce stability, in contrast to its role in the Deryaguin–Landau–Verwey–Overbeek theory for dilute aqueous suspensions.

Higher-order closures and potential problems in diffuse double layer and strong electrolyte solution theory

An nth-order closure is proposed between the higher-order fluctuation potentials of the mean electrostatic potentials and the potentials of the mean force. At the zeroth level the closure gives the Gouy-Chapman theory, at the first level the Loeb extension of the Gouy-Chapman theory and the Debye-Hückel theory, and at the second level the Outhwaite extension of the Debye-Hückel theory together with the equivalent problem in the diffuse double layer. Using the primitive model for the electrolyte with a plane uniformly charged electrode producing the double layer, a statistical mechanical analysis is carried out to determine the relations between the higher-order potentials of the mean force and the mean potentials. For the nth-order closure this gives a system of n + 1 differential equations in the diffuse double layer or n differential equations in the electrolyte bulk satisfied by the higher-order fluctuation potentials.

Comment on the second moment condition of Stillinger and Lovett

The second moment condition of Stillinger and Lovett is shown to be equivalent to a normalisation condition on the mean electrostatic potential of Debye and Hückel for an ideal solvent.

Negative Work Function of Thermal Positrons in Metals

Theoretical study shows that thermalized positrons are thrown out of a metal with an energy of the order of several eV. This phenomenon is shown to be closely related to the electron work function of metals. Since energy is emitted when positrons leave the metal surface, it is named "negative work function." The negative work function of thermal positrons is ${\ensuremath{\Phi}}^{p}\ensuremath{\simeq}\ensuremath{\Delta}{\ensuremath{\varphi}}^{e}\ensuremath{-}{\ensuremath{\mu}}_{c}^{p}+O(\frac{{N}_{p}}{N})$, where ${N}_{p}$ and $N$ are the total number of positrons and electrons, respectively, in the metal. $\ensuremath{\Delta}{\ensuremath{\varphi}}^{e}$ is the electrostatic potential across the metal surface due to the double layer taken from the electron work-function calculation, and ${\ensuremath{\mu}}_{c}^{p}$ is the correlation contribution to the positron chemical potential at the mean electrostatic potential.

Activity-measurement by the partition method.—I

In the light of Milner’s calculation of the mean electrostatic potential in chaos of ions, activity-measurement is of special interest at low concentrations aqueous, and still lower concentrations in non-aqueous solution, owing to the simple and peculiar dilution law which the theory predicts for such consentrations. The methods which have chiefly been relied on for such data—that of the Freezing-point depression and that of the concentration cell-—tend to lose recision at low concentrations, owing, in the former, to the smallness of the depression measured, and in the latter, to polarisation effects.