Generalized Einstein relations from a three-temperature theory of gaseous ion transport

Generalized Einstein relations between mobility and diffusion coefficients are derived within the framework of a recently developed three-temperature kinetic theory of gaseous ion transport. A previously assumed connection between diffusion and differential mobility is firmly established within low-order approximations of the kinetic theory. Equations are obtained for the approximate calculation of the unobservable parallel and perpendicular ion temperatures, and for two higher moments of the ion velocity distribution function that appear as correction terms in the generalized Einstein relations. The present theory is tested on inverse-power potentials and on several alkali ion-noble gas systems, and is compared to two previous semi-empirical generalized Einstein relations. Simple procedures are recommended for the estimation of the parameters and correction terms that occur, to permit approximate calculation of gaseous ion diffusion coefficients from mobility data without the need of a computer or specific knowledge of the ion-neutral interaction potential.     

Determination of the effective charge of individual colloidal particles

An optical method is presented that allows simultaneous determination of the diffusion constant and electrophoretic mobility of individual charged particles with radius down to 0.2 mum. By this method the size dependency of the effective charges and zeta potentials of individual particles can be investigated, as well as interparticle interactions and Brownian motion in confined geometries. The diffusion constant and mobility are determined from the power spectrum of the particle speed in a sinusoidal electrical field. The accuracy of the method was tested on PMMA spheres of known size in water. Experiments have been carried out on charged pigment particles with low concentration in a nonaqueous medium containing a charging agent. The mobility is found to be independent of the particle size.     

Electrophoresis of randomly charged particles

The technique of "rotational electrophoresis" has been developed recently to measure the charge nonuniformity on Brownian particles in a suspension. Measurements of the angular velocity in electrophoresis are interpreted with electrokinetic theory to give a standard deviation of zeta potential among N regions on a particle. A key question is how various molecular mechanisms of charging could lead to charge nonuniformity on a length scale of O (100 nm). Simple statistical physics is used to show that although Poisson charging with univalent charges cannot explain recent measurements showing significant charge nonuniformity, random charging with multivalent charges can better explain the data. In addition, the result shows that a small number of charged "visualization" molecules can be adsorbed to a particle without impacting the experimental technique of rotational electrophoresis.     

Diffusion processes with inertial effects and with boundary conditions — A solution to the Wang and Uhlenbeck problem

The problem of Wang and Uhlenbeck on the imposition of boundary conditions for space in the Fokker-Planck-Kramers equation is solved for Brownian motion under uniform and gravitational potentials. These cases with the full consideration of inertial effects lead to a modified diffusion equation with time-dependent diffusion coefficients determined by the initial condition of the velocity distribution. Moreover, the former case is applied to the rate theory for the diffusion limited reaction in liquids and new results have been obtained especially for the short time behavior where inertial effects play an important role.     

Heterogeneity of diffusion inside microbial biofilms determined by fluorescence correlation spectroscopy under two-photon excitation

Fluorescence correlation spectroscopy (FCS) under two-photon excitation was applied successfully to characterize the penetration and diffusion capabilities of fluorescent probes (latex beads and fluorescein isothiocyanate-dextran) of different size and electrical charge in two models of monomicrobial biofilms with low (Lactococcus lactis biofilm) or high (Stenotrophonas maltophilia biofilm) contents of extracellular polymeric substance (EPS). FCS measurements performed on each biofilm can show deviation from Brownian diffusion, depending on the local structure of the biofilm and the fluorophore size. In this case, we fitted the data to an anomalous diffusion model and determined apparent diffusion coefficients, which can be 50 times smaller than the values in aqueous solutions. This result was interpreted as steric hindrance of the diffusion of the fluorescent particles within the biofilm that can lead to a total inhibition as observed particularly in the mushroom-like structure of the S. maltophilia biofilm. Alternatively, mechanisms for the absence of FCS signal behavior were related to attractive electrostatic interactions between cationic particles and negatively charged bacteria or to specific interactions between dextrans and EPS of the biofilm matrix.     

Deposition of charged aerosol nanoparticles in diffusion batteries

The deposition of weakly charged aerosol nanoparticles onto fibers in a diffusion battery designed to measure the diffusion coefficients of particles is considered. The fiber collection efficiancies as functions of particle size and charge are determined by the numerical solution of the equation of convective diffusion in a system of parallel uncharged fibers located normal to a flow. It is shown of effect of the single charge of nanoparticles produced by a differential mobility analyzer on their deposition is negligible and may be ignored when calibrating diffusion batteries.     

Method of measuring charge distribution of nanosized aerosols

In this paper, we present the development of a method to accurately measure the positive and negative charge distribution of nanosized aerosols using a tandem differential mobility analyzer (TDMA) system. From the series of TDMA measurements, the charge fraction of nanosized aerosol particles was obtained as a function of equivalent mobility particle diameter ranging from 50 to 200 nm. The capability of this new approach was implemented by sampling from a laminar diffusion flame which provides a source of highly charged particles due to naturally occurring flame ionization process. The results from the TDMA measurement provide the charge distribution of nanosized aerosols which we found to be in reasonable agreement with Boltzmann equilibrium charge distribution theory and a theory based upon charge population balance equation (PBE) combined with Fuchs theory (N.A. Fuchs, Geofis. Pura Appl. 56 (1963) 185). The theoretically estimated charge distribution of aerosol particles based on the PBE provides insight into the charging processes of nanosized aerosols surrounded by bipolar ions and electrons, and agree well with the TDMA results.     

Wigner function approach to the quantum Brownian motion of a particle in a potential

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.     

Influence of resonant charge transfer on ion diffusion and generalized Einstein relations.

A three-temperature theory of ion diffusion, and related generalized Einstein relations (GER) connecting diffusion and mobility, are shown to remain valid even in the presence of resonant charge transfer (RCT) which strongly distorts the ion velocity distribution function electric fields. Extreme models of RCT for which exact solutions are known, are used for testing. A previous semi-empirical parametrization scheme for the GER is refined to include the case of RCT and compared with the calculations of Sinha, Lin, and Bardsley for the system He⁺ in He. Good results are obtained for both the diffusion coefficients and the anisotropic ion temperatures as a function of electric field strength.     

Characterization of mineral oxide charging in apolar media

This paper presents an investigation of the charging behavior of mineral oxide particles dispersed in apolar media. There are a growing number of applications that seek to use electrostatic effects in apolar media to control particle movement and improve aggregation stability. Progress is limited, however, by incomplete knowledge of the mechanism(s) of particle charging in these systems. It has been shown in a number of cases that the acid-base properties of both the particles and the surfactants used to stabilize charge play key roles. A mechanism for acid-base charging has previously been established for mineral oxides in aqueous systems, where the surface hydroxyl groups act as proton donors or receivers depending on the pH of the surrounding solution. In water, the pH at which the surface charge density is zero, i.e., the point of zero charge (PZC), can be used to characterize the acid-base nature of the mineral oxide particles. The current work explores the possible extension of this charging behavior to apolar systems, with the key difference that the surface hydroxyl groups of the mineral oxides react with the surfactant molecules instead of free ions in solution. The apolar charging behavior is explored by measuring the electrophoretic mobility of a series of mineral oxides dispersed in a solution of Isopar-L and AOT, a neutral surfactant in water. The electrophoretic mobility of the particles is found to scale quantitatively, with respect to both sign and magnitude, with their aqueous PZC value. This provides support for the theory of acid-base charging in apolar media and represents a method for predicting and controlling particle charge of mineral oxides dispersed in apolar media.     

Analysis of particle-wall interactions during particle free fall

In this study, the vertical motion of a particle in a quiescent fluid falling toward a horizontal plane wall is analyzed, based on simplified models. Using the distance between the particle and wall as a parameter, the effects of various forces acting on the particle and the particle motion are examined. Without the colloidal and Brownian forces being included, the velocity of small particles is found to be approximately equal to the inverse of the drag force correction function used in this study as the particle approaches the near-wall region. Colloidal force is added to the particle equation of motion as the particle moves a distance comparable to its size. It is found that the particle might become suspended above or deposited onto the wall, depending on the Hamaker constant, the surface potentials of the particle and wall, and the thickness of the electrical double layer (EDL). For strong EDL repulsive force and weaker van der Waals (VDW) attractive force, the particle will become suspended above the wall at a distance at which the particle velocity is zero. This location is referred to as the equilibrium distance. The equilibrium distance is found to increase with increased in EDL thickness when a repulsive force barrier appears in the colloidal force interaction. For the weak EDL repulsive force and strong VDW attractive force case, the particle can become deposited onto the wall without the Brownian motion effect. The Brownian jump length was found to be very small. Many Brownian jumps would be required in a direction toward the wall for a suspended particle to become deposited.     

Diffusion in Confined Geometries

Diffusive transport of particles or, more generally, small objects, is a ubiquitous feature of physical and chemical reaction systems. In configurations containing confining walls or constrictions, transport is controlled both by the fluctuation statistics of the jittering objects and the phase space available to their dynamics. Consequently, the study of transport at the macro‐ and nanoscales must address both Brownian motion and entropic effects. Herein we report on recent advances in the theoretical and numerical investigation of stochastic transport occurring either in microsized geometries of varying cross sections or in narrow channels wherein the diffusing particles are hindered from passing each other (single‐file diffusion). For particles undergoing biased diffusion in static suspension media enclosed by confining geometries, transport exhibits intriguing features such as 1) a decrease in nonlinear mobility with increasing temperature or also 2) a broad excess peak of the effective diffusion above the free diffusion limit. These paradoxical aspects can be understood in terms of entropic contributions resulting from the restricted dynamics in phase space. If, in addition, the suspension medium is subjected to external, time‐dependent forcing, rectification or segregation of the diffusing Brownian particles becomes possible. Likewise, the diffusion in very narrow, spatially modulated channels is modified via contact particle–particle interactions, which induce anomalous sub‐diffusion. The effective sub‐diffusion constant for a driven single file also develops a resonance‐like structure as a function of the confining coupling constant.     

Diffusion in the presence of cylindrical obstacles arranged in a square lattice analyzed with generalized Fick-Jacobs equation

The generalized Fick-Jacobs equation is widely used to study diffusion of Brownian particles in three-dimensional tubes and quasi-two-dimensional channels of varying constraint geometry. We show how this equation can be applied to study the slowdown of unconstrained diffusion in the presence of obstacles. Specifically, we study diffusion of a point Brownian particle in the presence of identical cylindrical obstacles arranged in a square lattice. The focus is on the effective diffusion coefficient of the particle in the plane perpendicular to the cylinder axes, as a function of the cylinder radii. As radii vary from zero to one half of the lattice period, the effective diffusion coefficient decreases from its value in the obstacle free space to zero. Using different versions of the generalized Fick-Jacobs equation, we derive simple approximate formulas, which give the effective diffusion coefficient as a function of the cylinder radii, and compare their predictions with the values of the effective diffusion coefficient obtained from Brownian dynamics simulations. We find that both Reguera-Rubi and Kalinay-Percus versions of the generalized Fick-Jacobs equation lead to quite accurate predictions of the effective diffusion coefficient (with maximum relative errors below 4% and 7%, respectively) over the entire range of the cylinder radii from zero to one half of the lattice period.     

On the use of the hypothesis of local electroneutrality in colloidal suspensions for the calculation of their dielectric properties

The validity of the hypothesis of electroneutrality outside the double layer of a suspended particle with an applied ac electric field is analyzed. It is shown that the electrolyte solution remains electroneutral for distances greater than a few Debye lengths from the particle surface only when the diffusion coefficients of the two ion species are identical. On the contrary, in the general case, a volume charge density around the particle builds up, which extends to distances that are proportional to the square root of the effective diffusion coefficient value divided by the frequency. These distances can easily attain many particle radii. Numerical results for both uncharged and charged suspended particles are presented, and a correction to existing analytical expressions for the field-induced ion distributions around uncharged particles (J. Phys. Chem. 2004, 108, 8397) is given. While the charge densities far from the particle are usually very weak, it is shown that they strongly contribute to the dipole coefficient value and, therefore, to the calculated values of the permittivity and conductivity increments. The errors that would be committed if these charge densities were ignored, assuming local electroneutrality and determining the dipole coefficient at a few Debye lengths from the particle surface, are analyzed and shown to be substantial.     

锂离子在石墨负极材料中扩散系数的测定

锂离子电池是以各种碳材料为负极而起来的一 种新型电池,成功地解决了以 为负极瓣锂可充电电池的安全性问题,已经应用于锂离子电池的负极材料有石墨和石油焦炭,正在研究的负极材料有热解碳,石墨化碳纤维,硼炭或硼炭氮化合物以及锡基氧化物等[1],石墨的比容量要比石油焦炭的比容量高一倍左右,其理论比容量372mA.h.g^-1,但锂离子在石墨材料中的扩散系数比较低,限制了以其为负极材料的电池的大电流充放电能力,锂离子在电极材料中的扩散系数可以用多种电化学方法测量得到,主要有：电位间歇滴定方法（PITT）（Potentiostatic Intermittent Titratiobn Technique)^[2,3,4,6],恒电流间歇滴定法（GITT）（Galvanostatic Intermittent Titration Technology)^[6],电流脉冲松弛法（CPR）（Current Pulse Relaxation Method)^[3,6]和交流阻抗法（A－C Technology)^[4,5,6],GITT,CPR,A-C等方法测定锂离子扩散系数时,由于相变发生处dE/dy值不容易准确得到（相变时,dE/dy→0）,此时测得的扩散系数误差比较大,PITT方法测定锂离子扩展系统,不存在这个问题,能比较准确地测定整个嵌入组成范围内的锂离子扩散系数。     

Effect of electrostatic, hydrodynamic, and Brownian forces on particle trajectories and sieving in normal flow filtration

Particle deposition and fouling are critical factors governing the performance of microfiltration and ultrafiltration systems. Particle trajectories were evaluated by numerical integration of the Langevin equation, accounting for the combined effects of electrostatic repulsion, enhanced hydrodynamic drag, and Brownian diffusion. In the absence of Brownian forces, particles are unable to enter the membrane pores unless the drag associated with the filtration velocity can overcome the electrostatic repulsion. Brownian forces significantly alter this behavior, allowing some particles to enter the pore even at low filtration velocities. The average particle transmission, evaluated from the probability of having a particle enter the pore, increases with increasing filtration velocity due to the greater hydrodynamic drag force on the particle. These results provide important insights into particle behavior in membrane systems.     

On the theory of surfactant mobility in micellar systems

Specific features of surfactant diffusion in micellar systems are described in terms of mobility, i.e., the limiting velocity of a particle under the action of a unit force. Micellar solutions of nonionic and ionic surfactants are analyzed. A relation is established between average surfactant mobility and the mobilities of individual particles. Although micelles have a lower mobility than monomers have, the average mobility of surfactants is shown to increase rather than decrease upon micellization. In parallel, formulas describing diffusion coefficients are derived, with part of the formulas having been available in the literature.