Theory of asymmetric electrochemical stochastic diffusion

Strict analysis of electrochemical strochastic diffusion due to asymmetric Brownian motion of electric charge in an equilibrium electrochemical ac circuit containing double electric layer capacitance and noisy Faradaic resistance is carried out. Cumulant analogs (for 3rd and 5th order correlations) of the Einstein formula are obtained. It is proved that equilibrium asymmetric (nongauss) stochastic diffusion is in agreement with the central limiting theorem of the probability theory. The Hurst exponent was found in the case of the nongauss components of the process of equilibrium stochastic diffusion. Apart from electrochemistry, the performed stochastic analysis of equilibrium electrochemical nongauss diffusion is also of general theoretical interest, including its application in the stochastic theory of asymmetric anomalous transport and strict theory of fluctuations at large deviations from equilibrium.     

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.     

An electrochemical cell for simultaneous electrochemical and spectroelectrochemical measurements under semi-infinite diffusion conditions and thin-layer conditions

A combination cell to accomplish simultaneous electrochemical and spectroelectrochemical measurements under both thin-layer and semi-infinite diffusion conditions is described and characterized. Fibre optics and a reflective electrode are used to couple the cell to the spectrophotometer. This allows different electrode materials to be used. Moreover, the cell is thermostatically controlled and equipped with a magnetic stirrer, and can be used for temperatures down to −40°C.The application of electrochemical pulse and sweep techniques are demonstrated. Dynamic spectroelectrochemical techniques such as linear sweep and cyclic voltabsorptometry (LSVA and CVA) as well as the corresponding derivative voltabsorptometric techniques derivative linear voltabsorptometry (DLSVA) and derivative cyclic voltabsorptometry (DCVA) are also applicable under thin-layer conditions. DLSVA and DCVA are the optical analogues of linear sweep and cyclic voltammetry. No epoxy cement or other sealing compounds are required and the solution comes in contact with only quartz, Teflon and the electrodes. Under aprotic conditions the cell response is in accordance with theory down to a solution thickness of 15 μm, where rapid exhaustive electrolysis intrinsic to thin-layer electrochemistry can be achieved in less than 1 s.The electrochemical and spectroelectrochemical characterization of the cell demonstrated that this design is very well suited to different electrochemical pulse and sweep techniques with simultaneous spectroscopic characterization of reaction products under finite and semi-infinite diffusion conditions in organic solvents. This offers the opportunity for cross-correlations of the electrochemical and spectroscopic information, which should lead to more reliable results. The adjustment of different thin-layer thicknesses is highly reproducible and the exchange of the solution inside the thin-layer cavity with the bulk solution after each thin-layer experiment can be easily performed. The electrochemical behaviour of the cell is in accordance with theory for cyclovoltammetric measurements under both bulk and thin-layer conditions. Derivative voltabsorptometric techniques are applicable and the response of the cell is in accordance with theory, particularly under finite diffusion conditions. The use of a bifurcated optical fibre bundle allows a more flexible experimental arrangement, and the application of a triple split bundle for the investigation of light-sensitive electron-transfer compounds [34,35] is in progress. The third end of the optical fibre bundle will be used to apply additional selective irradiation to convert electron-transfer-active photochromic compounds inside the thin-layer cavity depending on the redox state.An additional aspect of the current investigations is the application of the present cell for electrogenerated chemiluminescence (ECL) [36,37]. The highly reflecting electrode and the integrated stirrer are advantageous for this type of measurement. Finally, further work is in progress to utilize the integrated temperature control of the cell for spectroelectrochemistry at low temperatures, particularly with more unusual solvents like liquid sulphur dioxide [38] and liquid dimethylamine [39].     

A discrete stochastic model of diffusion controlled reactions. The rate constant

Diffusion controlled chemical reaction is modelled as random walk on a cubic lattice. The rate coefficient is shown to depend on the spatial distribution of the reactants.

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Series expansion methods for extracting ternary diffusion coefficients from Rayleigh interferometric data

Three series expansions are examined for use in extracting ternary diffusion coefficients from Rayleigh interferometric data for free diffusion experiments. Convergence comparisons show that, providing enough terms are used, two of the three are suitable, including one used earlier by Albright and Sherrill. A new simpler analysis, using only linear least squares, has been applied to simulated and real data with good results. Advantages and disadvantages of series methods are compared to a direct non-linear least squares analysis.     

Approximate rate constants for nonideal diffusion and their application in a stochastic model

Rate constants are presented for diffusion in dilute nonideal solutions with or without the presence of a spatially varying potential field. Expressions for the rate constants have been derived by earlier workers, and essentially the same derivation is reviewed and expanded in this paper to justify the expressions used for the rate constants. The diffusion rate constants are used in a random walker model to demonstrate how solution nonidealities can be captured accurately using this approach. Examples are presented of ideal solute diffusion as well as nonideal diffusion of nonelectrolytes and simple electrolytes in water. The use of the approach to simulate advection is described, and a possible strategy for extending the approach to more concentrated solutions is briefly discussed.     

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.     

An accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems using gradient-based diffusion and tau-leaping

Stochastic simulation of reaction-diffusion systems enables the investigation of stochastic events arising from the small numbers and heterogeneous distribution of molecular species in biological cells. Stochastic variations in intracellular microdomains and in diffusional gradients play a significant part in the spatiotemporal activity and behavior of cells. Although an exact stochastic simulation that simulates every individual reaction and diffusion event gives a most accurate trajectory of the system's state over time, it can be too slow for many practical applications. We present an accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems designed to improve the speed of simulation by reducing the number of time-steps required to complete a simulation run. This method is unique in that it employs two strategies that have not been incorporated in existing spatial stochastic simulation algorithms. First, diffusive transfers between neighboring subvolumes are based on concentration gradients. This treatment necessitates sampling of only the net or observed diffusion events from higher to lower concentration gradients rather than sampling all diffusion events regardless of local concentration gradients. Second, we extend the non-negative Poisson tau-leaping method that was originally developed for speeding up nonspatial or homogeneous stochastic simulation algorithms. This method calculates each leap time in a unified step for both reaction and diffusion processes while satisfying the leap condition that the propensities do not change appreciably during the leap and ensuring that leaping does not cause molecular populations to become negative. Numerical results are presented that illustrate the improvement in simulation speed achieved by incorporating these two new strategies.     

Enhanced diffusion in conic channels by means of geometric stochastic resonance

Geometric stochastic resonance of Brownian particles diffusing across a converging conic channel subject to oscillating forces is studied in this paper. Conic channel geometries have been previously considered as a model for transport of particles in biological membranes, zeolites, and nanostructures. For this system, a broad excess peak of the effective diffusion above the free diffusion limit is exhibited over a wide range of frequencies, suggesting a synchronization effect in the confining geometry as particles respond to the periodic modulation of the external force. This indicates that the geometric stochastic resonance effect with unbiased ac forces can be exploited for improving the transport of particles in complex geometries.     

Spontaneous electrochemical transport reactions in ionic and ionic-electronic salt melts: The production of diffusion coatings

Extensive material concerning the interaction of metals in ionic and ionic-electronic salt melts is presented. The mechanism and character of the phenomenon of the directed spontaneous transport of metals by their ions in salt melts are established. Examples of application of this phenomenon for depositing diffusion coatings on metals and alloys (by aluminum, beryllium, boron, zinc, titanium, chromium, silicon, and so on, as well as by two-component coatings) are presented.     

Efficient stochastic simulation of simultaneous reaction and diffusion in a gas-liquid interface

A stochastic simulation of simultaneous reaction and diffusion is proposed for the gas-liquid interface formed in the surface of a gas bubble within a liquid. The interface between a carbon dioxide bubble and an aqueous solution of calcium hydroxide was simulated as an application example, taken from the integrated production of calcium carbonate. First Gillespie’s stochastic simulation algorithm was applied in separate reaction and diffusion simulations. The results from these simulations were consistent with deterministic solutions based on differential equations. However it was observed that stochastic diffusion simulations are extremely slow. The sampling of diffusion events was accelerated applying a group molecule transfer scheme based on the binomial distribution function. Simulations of the reaction-diffusion in the gas-liquid interface based on the standard Gillespie’s stochastic algorithm were also slow. However the application of the binomial distribution function scheme allowed to compute the concentration profiles in the gas-liquid interface in a fraction of the time required with the standard Gillespie’s stochastic algorithm.     

Ion diffusion and electrochemical capacitance in aligned and packed single-walled carbon nanotubes

Direct measurement of ion diffusion in aligned, densified single-walled carbon nanotube electrodes showed that the diffusion coefficient for transport of ions (KSCN in acetonitrile) parallel to the alignment direction of the nanotubes was close to the theoretical limit of perfectly straight pores, achieving a value 20 times larger than that of activated carbon electrodes (1 × 10(-5) vs 5 × 10(-7) cm(2)/s). In contrast, the diffusion coefficient for ion transport perpendicular to the alignment direction was an order of magnitude smaller (8 × 10(-7) cm(2)/s). As an example of the ramifications of this anisotropic diffusion phenomenon, the difference in performance of the aligned carbon nanotubes as electrochemical-capacitor electrodes was evaluated. At low discharge rates, the performances of the two orientations were identical, but as the discharge rate was increased, a more rapid decline in capacitance was observed for the perpendicular orientation (66 vs 14% decline in capacitance when the discharge current was increased from 0.01 to 1 A/g). Furthermore, the maximum power rating of the perpendicular electrode was lower than that of the parallel electrode (1.85 vs 3 kW/kg during operation at 1 V).     

The effect of carbon black mixture composition on the structural and electrochemical characteristics of gas diffusion electrodes for electrosynthesis of hydrogen peroxide

Gas-diffusion electrodes containing polytetrafluoroethylene, acetylene black A437E and its mixtures with two types of furnace black (P702 and P268E) with different size of particles are studied. The electrode capacitance, the average diameter and volume of pores and their surface area are investigated as a function of the electrode composition. The kinetic characteristics of electrodes in the oxygen electroreduction are assessed. The slope of polarization curves in Tafel coordinates and the transfer coefficient value are shown to be virtually independent of the electrode composition. The preparative synthesis of Н₂О₂ is carried out.     

Two-parameter stochastic resonance in a model of electrochemical oxidation of formic acid on Pt

Stochastic resonance (SR) is shown in a two-parameter system, a model of electrochemical oxidation of formic acid on Pt. The driving current and the saturation coverage for carbon monoxide are two control parameters in this model. Modulation of an excitable focal stable state close to a Hopf bifurcation by a weak periodic signal in one parameter and noise in the other parameter is found to give rise to SR. The results indicate that the noise can enlarge a weak periodic signal and lead the system to be ordered. The scenario and novel aspects of SR in this system are discussed.     

On the diffusion of ferrocenemethanol in room-temperature ionic liquids: an electrochemical study

The electrochemical behaviour of ferrocenemethanol (FcMeOH) has been studied in a range of room-temperature ionic liquids (RTILs) using cyclic voltammetry, chronoamperomery and scanning electrochemical microscopy (SECM). The diffusion coefficient of FcMeOH, measured using chronoamperometry, decreased with increasing RTIL viscosity. Analysis of the mass transport properties of the RTILs revealed that the Stokes-Einstein equation did not apply to our data. The "correlation length" was estimated from diffusion coefficient data and corresponded well to the average size of holes (voids) in the liquid, suggesting that a model in which the diffusing species jumps between holes in the liquid is appropriate in these liquids. Cyclic voltammetry at ultramicroelectrodes demonstrated that the ability to record steady-state voltammograms during ferrocenemethanol oxidation depended on the voltammetric scan rate, the electrode dimensions and the RTIL viscosity. Similarly, the ability to record steady-state SECM feedback approach curves depended on the RTIL viscosity, the SECM tip radius and the tip approach speed. Using 1.3 μm Pt SECM tips, steady-state SECM feedback approach curves were obtained in RTILs, provided that the tip approach speed was low enough to maintain steady-state diffusion at the SECM tip. In the case where tip-induced convection contributed significantly to the SECM tip current, this effect could be accounted for theoretically using mass transport equations that include diffusive and convective terms. Finally, the rate of heterogeneous electron transfer across the electrode/RTIL interface during ferrocenemethanol oxidation was estimated using SECM, and k(0) was at least 0.1 cm s(-1) in one of the least viscous RTILs studied.     

Lattice microbes: High‐performance stochastic simulation method for the reaction‐diffusion master equation

Spatial stochastic simulation is a valuable technique for studying reactions in biological systems. With the availability of high‐performance computing (HPC), the method is poised to allow integration of data from structural, single‐molecule and biochemical studies into coherent computational models of cells. Here, we introduce the Lattice Microbes software package for simulating such cell models on HPC systems. The software performs either well‐stirred or spatially resolved stochastic simulations with approximated cytoplasmic crowding in a fast and efficient manner. Our new algorithm efficiently samples the reaction‐diffusion master equation using NVIDIA graphics processing units and is shown to be two orders of magnitude faster than exact sampling for large systems while maintaining an accuracy of ～ 0.1%. Display of cell models and animation of reaction trajectories involving millions of molecules is facilitated using a plug‐in to the popular VMD visualization platform. The Lattice Microbes software is open source and available for download at http://www.scs.illinois.edu/schulten/lm © 2012 Wiley Periodicals, Inc.     

The magnus expansion for the damped harmonic oscillator

It is shown that he time-evolution operator for the damped harmonic oscillator can be written as the exponential of an anti-Hermitean operator provided t is close enough to zero. The range of applicability of the Magnus expansion is discussed.     

苯胺的电化学聚合计时电流方程式

本文提出了苯胺的电化学聚合计时电流方程式, 并进行了验证。浓度为1.10mol.dm^-^3苯胺(内含2.2mol.dm^-^3HCl)溶液在铂圆盘电极上进行电化学聚合反应, 理论与实验相符。