Electrostatic interaction between two spherical colloidal particles

Explicit exact analytic expressions are obtained in the form of infinite series for the potential distribution and the potential energy of the electrostatic interaction for the system of two dissimilar spheres in an electrolyte solution on the basis of the linearized Poisson—Boltzmann equation without recourse to Derjaguin's approximation. The leading term of the expression for the interaction energy (the zeroth order approximation) corresponds to the interaction energy that would be obtained if both spheres were ion-penetrable spheres (“soft” spheres). This term is a screened Coulomb interaction due to a simple linear superposition of the unperturbed potentials of the respective spheres, which is proportional to the product of their unperturbed surface potentials. The first-order approximation corresponds to the interaction energy that would be obtained if either sphere were a soft particle (the other being hard). The first-order correction term consists of two sub-terms, each of which is proportional to the square of the unperturbed surface potential of either sphere and does not depend on the unperturbed surface potential of the other sphere, can be interpreted as the interaction between the soft sphere and its image with respect to the hard sphere. This image interaction is attractive if the surface potential of the hard sphere is constant and repulsive if the surface charge density of the sphere is constant. It is shown that Derjaguin's method as well as its extension to the interaction of unequal spheres by Hogg, Healy and Fuerstenau (HHF) is quite a good approximation.     

Relative Boltzmann entropy, evolution equations for fluctuations of thermodynamic intensive variables, and a statistical mechanical representation of the zeroth law of thermodynamics

Generalized thermodynamics or extended irreversible thermodynamics presumes the existence of thermodynamic intensive variables (e.g., temperature, pressure, chemical potentials, generalized potentials) even if the system is removed from equilibrium. It is necessary to properly understand the nature of such intensive variables and, in particular, of their fluctuations, that is, their deviations from those defined in the extended irreversible thermodynamic sense. The meaning of temperature is examined by means of a kinetic theory of macroscopic irreversible processes to assess the validity of the generalized (or extended) thermodynamic method applied to nonequilibrium phenomena. The Boltzmann equation is used for the purpose. Since the relative Boltzmann entropy has been known to be intimately related to the evolution of the aforementioned fluctuations in the intensive thermodynamic variables, we derive the evolution equations for such fluctuations of intensive variables to lay the foundation for investigating the physical implications and evolution of the relative Boltzmann entropy, so that the range of validity of the thermodynamic theory of irreversible processes can be elucidated. Within the framework of this work, we examine a special case of the evolution equations for the aforementioned fluctuations of intensive variables, which also facilitate investigation of the molecular theory meaning of the zeroth law of thermodynamics. We derive an evolution equation describing the relaxation of temperature fluctuations from its local value and present a formula for the temperature relaxation time.     

Memory effect in friction on a particle caused by a system of fixed or moving scatterers with power law potential

The memory function for friction on a particle caused by a system of fixed or moving scatterers is evaluated for power law interaction. For a dilute system the study extends the steady-state calculation based on the Boltzmann equation to the case of frequency dependence due to the dynamics of the scattering process. For a dense gas an Enskog approximation can be used. The power law potential leads to scaling behavior of the dynamical friction coefficient as a function of reduced mass, coupling coefficient, and energy.     

On the Nonlocal Electron Kinetics in s- and p-Striations of DC Glow Discharge Plasmas: I. Electron Establishment in Striation-like Fields

The nonlocal behavior of the electrons in strongly modulated and period-averaged electric fields typical of s- and p-striations in neon glow discharge plasmas is investigated by numerically solving the axially inhomogeneous electron Boltzmann equation. A good agreement between the period lengths measured in the striations and those obtained from the spatially periodic electron relaxation in the period-averaged field of the striations is found confirming the close relation of both phenomena. The s- and p-striations represent the fundamental and first harmonics of the inherent periodic electron relaxation. Furthermore, starting from different boundary conditions the establishment of the velocity distribution function and of selected macroscopic quantities of the electrons into unique periodic states under the action of strongly modulated striation-like fields is investigated. It is shown that the same damping processes that cause in homogeneous fields a relaxation into homogeneous states lead to unique periodic states in strongly modulated fields.     

Diffusion equations and hard collisions in multiple scattering of charged particles

The processes of angular-spatial evolution of multiple scattering of charged particles are described by the Lewis (special case of Boltzmann) integro-differential equation. The underlying stochastic process for this evolution is the compound Poisson process with transition densities satisfying the Lewis equation. In this paper we derive the Lewis equation from the compound Poisson process and show that the effective method of the solution of this equation can be based on the idea of decomposition of the compound Poisson process into processes of soft and hard collisions. Formulas for transition densities of soft and hard collision processes are provided in this paper together with the formula expressing the general solution of the Lewis equation in terms of those transition densities.     

Characterization of the porous structure of carbon adsorbents with a wide size distribution of micropores

Our study using the nonlocal density functional theory (NDFT) showed that active coals might have a bidisperse microporous structure. The binomial equation of the theory of volume filling of micropores (TVFM) approximates well the nitrogen adsorption isotherms at relative pressures from 1 × 10⁻⁴ to 0.2. The dominant micropore sizes calculated in terms of the characteristic adsorption energy lie in the region of the maximum of the size distribution of micropores calculated by the NDFT method. The tentative micropore sizes can be determined from the modified second term of the TVFM equation. The Henry and BET equations describe very limited regions of the nitrogen adsorption isotherm on microporous active coals.     

Kinetic description of the expansion of a supersonic pulsed jet to vacuum: II. Mixtures of monoatomic gases

A model for the expansion of a pulsed jet of a mixture of monoatomic gases was advanced on the basis of the Boltzmann kinetic equation. The Grad method was used to solve the Boltzmann equation. The set of moment equations was analyzed by matching asymptotic expansions. The behavior of the velocity slip and temperature difference was analyzed as a function of the conditions in the source and the component-gas concentrations in the mixture. A functional relationship between these quantities and the form of the interaction potential has been established.     

Incorporation of the electron energy-loss straggling into the Fermi–Eyges equation

A modified Fermi–Eyges equation has been derived from the linear Boltzmann equation by including a term for describing electron energy-loss straggling. The solution has been obtained by the use of a generalized Eyges' method, yielding the electron energy distribution expressed with moments method in addition to Eyges' original solution. The first- and second-order approximations of the spectrum give the well-known continuous-slowing-down approximation (CSDA) and Gaussian distribution, respectively. Inclusion of the third-order moment in the spectrum yields the Vavilov distribution approximated with the Airy function. The higher order approximations can be evaluated numerically.     

An analysis of nonlinear ion transport problem including arbitrary valences of oxidized and reduced species

The nonlocal identification problem related to nonlinear ion transport model including diffusion and migration is studied. Ion transport is assumed to be superposition of diffusion and migration under the influence of an electric field. Mathematical modeling of the experiment leads to an identification problem for a strongly nonlinear parabolic equation with nonlocal additional condition. It is shown that the nonlocal identification problem can be reduced to the initial-boundary value problem for nonlinear parabolic equation. Iteration method for numerical solution of this problem is proposed. Numerical results and their interpretation are presented for wide class of materials, including various values of valences and diffusivities of oxidized and reduced species.     

Solving the finite-difference non-linear Poisson–Boltzmann equation

The Poisson–Boltzmann equation can be used to calculate the electrostatic potential field of a molecule surrounded by a solvent containing mobile ions. The Poisson–Boltzmann equation is a non-linear partial differential equation. Finite-difference methods of solving this equation have been restricted to the linearized form of the equation or a finite number of non-linear terms. Here we introduce a method based on a variational formulation of the electrostatic potential and standard multi-dimensional maximization methods that can be used to solve the full non-linear equation. © 1992 by John Wiley & Sons, Inc.     

局域平衡假设和反应-扩散过程

局域平衡假设作为不可逆过程热力学理论的基础通常被认为适用于一般条件下的物理化学过程。本文从玻耳兹曼方程和涨落的随机理论出发重新研究了局域平衡假设对反应—扩散过程的适用性。表明对于涉及非线性化学动力学的反应—扩散过程, 从随机理论得到的结果和局域平衡假设是不一致的。     

Approximate expression for the potential energy of double-layer interaction between two parallel similar plates with constant surface potential

An approximate equation for the electric potential distribution is obtained by linearizing the Poisson–Boltzmann with respect to the deviation of the electric potential from the surface potential. On the basis of the solution to this linearized Poisson–Boltzmann equation, an approximate expression is derived for the potential energy of the double layer interaction per unit area between two parallel similar plates at constant surface potential. It is found that this linearization approximation works quite well for small plate separations for all values of the surface potentials.     

On the derivation and extension of the uniquac equation

The UNIQUAC equation is derived by phenomenological arguments based on a two-fluid theory. This derivation is free of the inconsistencies which arise when a lattice one-fluid theory is used to derive UNIQUAC or any other local-composition equation. The essential step in the derivation is the adoption of Wilson's assumption that local compositions can be related to overall compositions through Boltzmann factors.In UNIQUAC, the energy terms in the Boltzmann factors are assumed identical to those in the excess energy of mixing. When the assumption of identity is replaced by the much less stringent assumption of proportionality, a three-parameter UNIQUAC equation is derived. This equation often gives a better fit of experimental data whenever all three binary paramters are freely adjusted. Upon assigning a universal value to the third parameter, only marginal improvement is obtained.     

An idealized model for nonequilibrium dynamics in molecular systems

The nonequilibrium dynamics of highly nonlinear and multidimensional systems can give rise to emergent chemical behavior which can often be tracked using low-dimensional order parameters such as a reaction path. Such behavior cannot be readily surmised by stationary projected stochastic representations such as those described by the Langevin equation or the generalized Langevin equation (GLE). The irreversible generalized Langevin equation (iGLE) contains a nonstationary friction kernel that in certain limits reduces to the GLE with space-dependent friction. For more general forms of the friction kernel, the iGLE was previously shown to be the projection of a mechanical system with a time-dependent Hamiltonian [R. Hernandez, J. Chem. Phys. 110, 7701 (1999)]. In the present work, the corresponding open Hamiltonian system is shown to be amenable to numerical integration despite the presence of a nonlocal term. Simulations of this mechanical system further confirm that the time dependence of the observed total energy and the correlations of the solvent force are in precise agreement with the projected iGLE. This extended nonstationary Hamiltonian is thus amenable to the study of nonequilibrium bounds and fluctuation theorems.     

Electrostatic interaction between a cylinder and a planar surface

 An exact analytical expression for the potential energy of the electrostatic interaction between a plate-like particle 1 and a cylindrical particle 2 of radius a ₂ immersed in an electrolyte solution of Debye–Hückel parameter κ is derived on the basis of the linearized Poisson–Boltzmann equation without recourse to Derjaguin's approximation. Both particles may have either constant surface potential or constant surface charge density. In the limit of κa ₂→0, in particular, the interaction between a plate with zero surface charge density and a cylinder having constant surface charge density becomes identical to the usual image interaction between a line charge (a charged rod of infinitesimal thickness) and an uncharged plate. Received: 22 September 1998  Accepted in revised form: 27 January 1999     

Highly accurate and simple analytical approach to nonlinear Poisson–Boltzmann equation

A novel method is suggested to analytically solve a nonlinear Poisson–Boltzmann (NLPB) equation. The method consists chiefly of reducing the NLPB equation to linear PB equation in several segments by approximating a free term of the NLPB equation by piecewise linear functions, and then, solving analytically the linear PB equation in each segment. Superiority of the method is illustrated by applying the method to solve the NLPB equation describing a colloid sphere immersed in an arbitrary valence and mixed electrolyte solution; extensive test indicates that the resulting analytical expressions for both the electrical potential distribution Ψ (r) and surface charge density/surface potential relationship (σ/Ψ ₀) are characterized with two properties that mathematical structures are much simpler than those previously reported and application scope can be arbitrarily wide by adjusting the linear interpolation range. Finally, it is noted that the method is “universal” in that its applications are not limited to the NLPB equation.     

Comparison of reaction rates for helium plasmas calculated using nonequilibrium and equilibrium electron energy distribution functions

The electron energy distribution functions for helium plasmas have been calculated using the Boltzmann equation. Three characteristic temperatures of these distribution functions have been determined (from mean energy, the Einstein formula, and the logarithmic slope). The reaction rates for nonequilibrium and three equilibrium (corresponding to these three characteristic temperatures) distribution functions have been calculated and compared We have found that the use of equilibrium values for reaction rates of processes going from the ground state can lead to great errors in results, the use of equilibrium values for processes going from higher levels is possible for higher reduced electric fields, and there is no problem with using equilibrium values. for superelastic processes.     

The parameters of hydrogen chloride and hydrogen bromide direct-current glow-discharge plasmas

A comparative analysis of the parameters of HCl and HBr plasmas was performed by means of mathematical simulation, solving the Boltzmann kinetic equation in the one-component approximation. The specifics of the kinetics of the electron impact-induced processes and the composition of plasma charged species were analyzed.