The Kramers-Moyal expansion for electrochemical stochastic diffusion

It is proved that there is a general stochastic equation, according to which any random process in the transient mode can be presented by spatially homogeneous Kramers-Moyal expansion. In the electrochemical stochastic diffusion, an integral of the fluctuation component of electrode potential over the time plays the role of spatial coordinate. Based on these two facts, we derived a spatially homogeneous Kramers-Moyal expansion for the propagator of electrochemical stochastic diffusion. By using the limiting transition to long observation times, we obtained a time and spatially homogeneous asymptotic Kramers-Moyal expansion for the propagator of asymmetric non-Gaussian electrochemical stochastic diffusion. Under the conditions of Gaussian electrochemical noise, the asymptotic Kramers-Moyal expansion turns into the Einstein stochastic diffusion equation. The method of determining time and spatially homogeneous asymptotic Kramers-Moyal expansion for the propagator of asymmetric non-Gaussian electrochemical stochastic diffusion may be useful in the stochastic theory of slow electrochemical discharge and in the electrochemical noise diagnostics.     

Electrochemical symmetric stochastic diffusion

Symmetric stochastic diffusion in an equilibrium electrochemical ac circuit is studied theoretically. The electrochemical circuit included double layer capacitance and slow discharge resistance. Electrochemical analog of the stochastic Einstein formula is found. An equation is obtained for the excess of electrochemical stochastic diffusion. It is shown that an excess of electrochemical stochastic diffusion increases at high relaxation times in proportion to the observation time. It is found that excess is related to correlation between the phenomenon of electrochemical stochastic diffusion and the central limiting theorem.     

The fractal theory of electrochemical diffusion noise: Correlations of the third and fourth order

Based on the Langevin linear stochastic equation, the correlations of the 3rd and 4th order for thermal fluctuations of the electrode potential are studied in an electrochemical ac circuit involving an electric double layer capacitance, a resistance of steady-state diffusion, and a Warburg impedance. The presence of the noisy Warburg impedance in the ac circuit makes the Langevin linear stochastic equation fractal. The analogy with the steady-state diffusion noise and with the noise of the barrierless-activationless slow discharge is used. Equations for bispectrum and trispectrum of electrode-potential activation are shown. It is demonstrated that the intensity of bispectrum and trispectrum is determined exclusively by the noise of the steady-state diffusion resistance if one of frequency arguments in the polyspectrum is zero. It is found that in an electrochemical ac circuit containing the noisy Warburg impedance, the asymptotics of establishment of equilibrium values of asymmetry and excess of electrode-potential fluctuations (thermalization) obeys the power law rather than the exponential law. Furthermore, the excess thermalization proceeds faster as compared with asymmetry thermalization. The performed theoretical analysis of correlations of the 3rd and 4th order of the fractal noise of electrochemical diffusion is of practical interest. For instance, the concepts of the fractal electrochemical noise can be used in the noise diagnostics of devices of electrochemical power engineering and in the noise methods for studying corrosion systems.     

Theory of the first encounter with the boundary by a stochastic diffusion process in an equilibrium electrochemical <Emphasis Type="Italic">RC</Emphasis>-circuit

The methodology of electrochemial impedance is used for finding the characteristic function of the random time of the first encounter with the boundary by a process of electrochemical stochastic diffusion in an equilibrium ac circuit containing a double layer capacitance and a noisy charge-transfer resistance. The Nyquist diagram of the characteristic function suggests that the method of the first random encounter with the boundary by electrochemical stochastic diffusion may prove to be useful in the noise diagnosis of objects and devices of electrochemical power engineering and also in comparative studies of electrochemical corrosion processes.     

非恒稳含时电极过程中涨落-耗散的电化学效应——恒电势滴汞电极体系的随机热力学模型

以滴汞电极体系为模型,对非恒稳恒电势动态不可逆电极过程中的耗散-涨落效应进行了系统的研究.基于滴汞电极体系的电化学特征,提出了一个简化的含时随机热力学模型,从而可能对这类重要的含时物理化学过程进行涨落和耗散的定量分析.借助该简化的模型,成功地建立了恒电势滴汞电极过程的基本随机热力学公式,由此推出耗散-涨落效应的理论极谱曲线.在滴汞电极生长缓慢及扩散步骤严重滞后情况下,含时的滴汞电极过程将趋于在有效扩散层厚度演化的慢流型上的准定态过程.在这种准定态近似下,具体分析了涨落对极谱曲线的影响.结果表明,在涨落影响可以忽略的近平衡区,从耗散-涨落导出的极谱方程与从平衡态Nernst公式导出的极谱方程完全吻合.还计算了一个涨落诱导的极谱曲线偏离的典型范例.     

Effects of nonequilibrium on velocity and plate height in reactive capillary electrophoresis

Models for velocity and plate height for reactive CE are developed under the formalism of generalized nonequilibrium theory, as described by Giddings. The resultant equations are consistent with chromatographic theory and validated with an independent stochastic simulation. Moreover, unlike prior methods for CE, this model allows calculation of thermodynamic equilibrium constants and kinetic rate constants from a single, undistorted peak. The theoretical development shows that velocity is directly dependent on the equilibrium constant and is independent of the rate constant. On the other hand, plate height varies little with equilibrium constant and is inversely proportional to rate constant. The ability to evaluate equilibrium constants from velocity and rate constants from plate height is most greatly influenced by electric field strength and mobility difference. The accuracy in calculated equilibrium constants is limited by mobility difference; however, the accuracy in rate constants is limited by plate height and equilibrium constant.     

Dual Fluctuation–Dissipation Relationships between Tri- and Bispectra of the Brownian Motion

A fluctuation–dissipation analysis of nonlinear noise accompanying Brownian motion (BM) in electric systems is carried out. It is shown that, for symmetrical BM, there exists a linear set of dual fluctuation–dissipation relationships linking an equilibrium trispectrum to a derivative of a bispectrum with respect to a direct electric current (or voltage), taken at equilibrium. Analysis of an electrochemical model of BM in the form of a symmetrical slow discharge yields a proportionality coefficient.     

Analytical theories of transport in concentrated electrolyte solutions from the MSA

Ion transport coefficients in electrolyte solutions (e.g., diffusion coefficients or electric conductivity) have been a subject of extensive studies for a long time. Whereas in the pioneering works of Debye, Hückel, and Onsager the ions were entirely characterized by their charge, recent theories allow specific effects of the ions (such as the ion size dependence or the pair association) to be obtained, both from simulation and from analytical theories. Such an approach, based on a combination of dynamic theories (Smoluchowski equation and mode-coupling theory) and of the mean spherical approximation (MSA) for the equilibrium pair correlation, is presented here. The various predicted equilibrium (osmotic pressure and activity coefficients) and transport coefficients (mutual diffusion, electric conductivity, self-diffusion, and transport numbers) are in good agreement with the experimental values up to high concentrations (1-2 mol L(-1)). Simple analytical expressions are obtained, and for practical use, the formula are given explicitly. We discuss the validity of such an approach which is nothing but a coarse-graining procedure.     

Chapman's stochastic relations for diffusiophoresis as exhibited by particles suspended in an imperfect gaseous mixture of nonuniform composition

Attention is redirected to Chapman's elementary stochastic treatment of motion induced in suspended particles in consequence of diffusion in the surrounding fluid medium. Whereas current treatments of this induced motion (diffusiophoresis) by kinetic theory relate to ideal gaseous systems, the application of Chapman's stochastic analysis is not thus restricted. The implications are examined of the hypothesis that for an imperfect gaseous mixture the induced motion of diffusiophoresis is entirely attributable to the gradient of pressure automatically set up by diffusion in an enclosed space.     

Optical, dynamic, and electro-optical properties of poly(N-acryloyl-11-aminoundecanoic acid) in solutions

The optical, electro-optical, and dynamic characteristics of poly(N-acryloyl-11-aminoundecanoic acid) in organic solvents and of the sodium salt of its monomer in water were studied via the methods of flow birefringence, equilibrium and nonequilibrium electric birefringence, and dynamic light scattering. It is shown that, in aqueous solutions, the monomer forms coarse particles of both symmetric and asymmetric shapes. The linear dimensions of these particles are estimated from the data of translational and rotational diffusion. Polymer macromolecules in organic solvents feature negative anisotropy of optical polarizability. Contributions of optical microform and macroform effects to the observed flow birefringence are analyzed in detail. The intrinsic optical anisotropy of the monomer unit of the polymer, which correlates well with the corresponding values for comb-shaped polymers of a similar structure, is estimated. It is shown that polymer molecules lack marked intrinsic permanent macromolecular dipoles and that electric birefringence in their solutions is associated with macroscopic induced dipole moments that appear during orientation of the dipole moments of polar groups in side chains of the polymer under application of an electric field.     

Calculation of Combined Diffusion Coefficients from the Simplified Theory of Transport Properties

The aim of this study is to check if it is possible to use the combined diffusion coefficients introduced by Murphy at equilibrium in a two-temperature model (electron temperature T_e different from that of heavy species T_h), such as that defined by Devoto and Bonnefoi for transport properties. On the one hand, the two-temperature (2-T) theory of transport properties was established by Devoto and Bonnefoi by separating electrons and heavy species because of their mass difference. Their simplified theories allow the calculation of transport coefficients (except diffusion) out of thermal equilibrium, but it has to be noted that when T_e tends toward T_h, the results are those obtained with an equilibrium calculation. On the other hand, Murphy's combined diffusion coefficients describe the diffusive mixing of two nonreactive ionized gases at equilibrium. First, the exact combined diffusion coefficients of Murphy are calculated for an Ar–N₂ (50 wt.%) mixture at atmospheric pressure. Expressions of combined diffusion coefficients are then obtained by using the simplified theory of Bonnefoi at thermal equilibrium. The results of the calculation of combined diffusion coefficients from the simplified theory of transport properties, assuming Te=Th, are compared with those of Murphy at equilibrium. It is shown that large discrepancies occur as soon as the ionization degree is over 10%. These results prove that the simplified 2-T theory of transport coefficients cannot be used for the treatment of diffusion, probably because the mass flux of electrons is no longer constrained. Thus, a new theory of transport coefficients has to be developed, taking into account the coupling of electrons and heavy species.     

Influence of varying electroosmotic flow on the effective diffusion in electric field gradient separations

Recently the use electric field gradient focusing (EFGF) to enhance focusing of proteins has been proposed and explored to provide significant improvement in separation resolution. The objective of EFGF is to focus proteins of specific electrophoretic mobilities at distinct stationary locations in a column or channel. This can be accomplished in a capillary by allowing the electric potential to vary in the streamwise direction. Because the electric field is varying, so also is the electrokinetic force exerted on the proteins and the electroosmotic velocity of the buffer solution. Due to the varying electric field, the Taylor diffusion characteristics will also vary along the column, causing a degradation of peak widths of some proteins, dependent on their equilibrium positions and local velocity distributions. The focus of this paper is an analysis that allows characterization of the local Taylor diffusion and resulting protein band peak width as a function of the local magnitude of the EOF relative to the average fluid velocity for both cylindrical and rectangular channels. In general the analysis shows that as the ratio of the local electroosmotic velocity to the average velocity deviates from unity, the effective diffusion increases significantly. The effectiveness of EFGF devices over a range of protein diffusivities, capillary diameters, flow velocities, and electric field gradient is discussed.     

Theory and practice of a powerful technique for electrochemical investigation of solid solution electrode materials

The theory and practice of a solid-state electrochemical technique which allows measurement of the chemical diffusion coefficient and partial conductivities of the mobile species in a mixed ionic-electronic conductor, as well as the equilibrium partial thermodynamic quantities, are described. The theory incorporates nonideal thermodynamic behavior of the carriers as well as cross terms in the flux equations. Once the cell is assembled, all parameters can be measured as a function of composition without any further physical manipulation, so that the experiments are well suited to automatic control.     

Electrochemical noise generated by anodic dissolution or diffusion processes

A model of the noise generated by electrochemical reactions and by diffusion is proposed. The elementary fluctuations are supposed to be the particle fluxes which are Poisson white noise. This model is successfully used to describe the experimental stochastic behaviour of two cases of non-equilibrium electrochemical interfaces: the noise generated by anodic dissolution of iron in acidic medium and that by diffusion of a reacting species in the bulk of the electrolyte.     

Studies of electrochemical properties of compacted clays by concentration potential method

The development of concentration (membrane) potential upon step-wise change in salt concentration has been studied for diaphragms made of various strongly compacted clays (montmorillonite, illite, kaolinite) equilibrated with 0.1 M NaCl solution. Porous ceramic filters were used to confine the clays mechanically to be able to achieve high extent of compaction (dry density approximately 2000 kg/m3). A theoretical analysis has revealed that the relaxation pattern is primarily controlled by the properties of porous filters and only slightly depends on the clay properties. At the same time, quasi-stationary values of concentration potential are directly related to the electrochemical perm-selectivity of clay. This property has revealed considerable differences in the electrochemical behaviour of various clays used in this study. This has been attributed to the differences in the micro-structure of clays, in particular to the existence or nonexistence of the so-called interlayer water where cations may retain some mobility. It has also been shown that in clays with high electrochemical perm-selectivity, one can expect a strong increase in the diffusivity of cationic radio-tracers with decreasing ionic strength of equilibrium electrolyte solution. At the same time, low electrochemical perm-selectivity means no noticeable dependence of this kind. The correctness of this observation has been corroborated by the comparison of our findings with the literature data on the diffusion of cationic radio-tracers through compacted montmorillonite (high perm-selectivity) and kaolinite (low perm-selectivity). To check the self-consistency of our approach, we have also carried out sample measurements of diffusion of cationic and anionic radio-tracers through compacted illite. It has been found that the measured effective diffusion coefficients were in excellent agreement with the electrochemical perm-selectivity estimated for this clay from the measurements of concentration potential.     

Monte Carlo simulation of ion transport in non-linear ion mobility spectrometry

A program for Monte Carlo simulation of ion transport in non-linear ion mobility spectrometry, also known as field asymmetric ion mobility spectrometry (FAIMS) or differential mobility spectrometry (DMS), has been developed. Simulations are based on elastic collisions between the ions and the gas particles, and take into account the effects of flow dynamics and asymmetric electric fields. Using this program, the separation and diffusion of the ions moving in a planar DMS filtration gap are demonstrated. Ion focusing in a cylindrical filtration gap is also confirmed. A characteristic compensation voltage is found to provide insight for understanding separation in non-linear ion mobility spectrometry. The simulation program is used to study the characteristics of non-linear ion mobility spectrometry, the effect of the carrier gas flow, and the dependence of the compensation voltage and nonlinear mobility coefficient (α) on the applied asymmetric electric field.     

Transport in polymer-gel composites: response to a bulk concentration gradient

This paper examines the response of electrolyte-saturated polymer gels, embedded with charged spherical inclusions, to a weak gradient of electrolyte concentration. An electrokinetic model was presented in an earlier publication, and the response of homogeneous composites to a weak electric field was calculated. In this work, the influence of the inclusions on bulk ion fluxes and the strength of an electric field (or membrane diffusion potential) induced by the bulk electrolyte concentration gradient are computed. Effective ion diffusion coefficients are significantly altered by the inclusions, so-depending on the inclusion surface charge or zeta potential-asymmetric electrolytes can behave as symmetrical electrolytes and vice versa. The theory also quantifies the strength of flow driven by concentration-gradient-induced perturbations to the equilibrium diffuse double layers. Similarly to diffusiophoresis, the flow may be either up or down the applied concentration gradient.     

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。     

Range of Validity of a Simplified Model for Diffuse Charge Dynamics

The modelling of electrochemical processes often requires the solution of the Poisson‐Nernst‐Planck (PNP) equations. In complex geometries, such as porous electrodes, that is challenging due to the presence of disparate length scales, ranging from the Debye screening length (～nm) to the device length scale (～cm). To overcome this difficulty, one often assumes that the electric double layer (EDL) is at quasi‐equilibrium to construct a simplified model that accounts for ion diffusion in the electro‐neutral bulk of the electrolyte while replacing the EDLs with appropriate boundary conditions. Various researchers have demonstrated that such an approach is valid in the asymptotic limit of a thin EDL and moderate electrode potentials. In this note, we explore the range of validity of this approximation by considering a one‐dimensional electrolytic cell with blocking electrodes subjected to a step change and time‐periodic alternations in the electrodes’ potentials by calculating the errors associated with the approximate approach as functions of the EDL thickness and electric field frequency and intensity. Additionally, we delineate numerical instabilities associated with the numerical solutions of the bulk equations with the nonlinear boundary condition peculiar to this problem.