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.     

Application of holographic interferometric microscopy to the study of diffusion in the polyvinylpyrrolidone–iodine system

Interferometric studies of the diffusion of molecular iodine into polyvinylpyrrolidone (PVP) are described. The diffusion process is observed by means of a holographic microscope which, with a camera, permits holograms to be recorded during the polymer–iodine interaction. The holograms are subsequently used to produce interferograms that monitor the diffusion process as a function of time. Measurements taken from the interferograms indicate that the diffusion mechanism is not purely Fickian and that the diffusing boundary of iodine is characteristic of case II diffusion. The interferograms yield values for average diffusion coefficients as a function of sample thicknesses that range from 2.32 to 9.53 × 10^?10 cm²/sec.     

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.     

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.     

分子马达的三态周期跳跃模型

研究了在一维三态周期跳跃模型下分子马达的定向运动.对于给定的任意初始分布,得出了与时间有关的几率分布的解析表达式,包括到达稳态之前的所有的瞬态过程,由此可获得马达在各个时刻的漂移速度v、扩散系数D以及描述马达随机性质的随机参数r(randomness parameter ).同时还计算了马达到达稳态所需要的特征时间.并把计算结果同实验进行了比较.     

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.     

A Deep Eutectic Solvent‐based Amperometric Sensor for the Detection of Low Oxygen Contents in Gaseous Atmospheres

The possibility of adopting deep eutectic solvents (DESs) instead of room temperature ionic liquids (RTILs) in membrane‐free electrochemical gas probes was estimated by first evaluating the performance of ethaline as electrochemical medium. This very easily prepared DES was chosen as prototype since it displays high conductivity and fairly modest viscosity, comparable with those of RTILs usually adopted in electrochemical measurements. Its electrostability window at Au, Pt and GC electrodes was first detected, together with diffusion coefficients displayed in this medium by ferrocene in the range 2.0–26.5 °C, it being adopted as prototype analyte in view of its well known electrochemical behavior and high enough solubility in ethaline. These diffusion coefficients were then used to infer viscosity values of ethaline at all temperatures considered, by exploiting the Stokes‐Einstein equation. Even though ferrocene diffusion coefficients turned out to be remarkably lower than those displayed in usual aprotic solvents, they were fairly higher than those usually found in electrochemical measurements conducted in RTILs, thus pointing out that the use of DESs as solvents adhering to electrode surfaces for assembling electroanalytical gas sensors could be advantageous. On these bases, a conveniently assembled DES‐based probe was tested for the electrochemical detection of low oxygen contents in cooled atmospheres. The quite satisfactory results found indicated that the drawback affecting DESs, consisting in the low values of diffusion coefficients displayed by dissolved analytes, can be overcome by using thin enough DES layers and resorting to a high sensitive detection approach such as amperometry under flow conditions. In fact, good sensitivities were found at all temperatures considered (2.0–26.5 °C), accompanied by a low detection limit (ca. 0.1 % v/v).     

The electrochemical behavior of a system with a limited number of molecules

In this paper, based on Einstein relationship between diffusion and random walk, the electrochemical behavior of a system with a limited number of molecules was simulated and explored theoretically. The transition of the current vs time responses from discrete to continuous was clearly obtained as the number of redox molecules increased from 10 to 10⁶. By correlation analysis between the simulation results and the results of analytical expressions, a quantized extent parameter was proposed to investigate the underlying rules of these discrete signals, which looked stochastic. The results revealed that this parameter would be useful to describe such systems.     

The effect of partial charge transfer on the electrochemical turbulent noise

Theoretical analysis of the effect of electrode potential on the spectral density of random alternating current emerged in electrochemical cell under the action of turbulent pulsations of the electrolyte solution velocity is carried out. An impedance model of metal electrode dissolution reaction, including two adsorption stages, is suggested, with allowance for the oxidized ion diffusion in electrolyte solution. It is known that in terms of the Ershler-Randles model, at low frequencies the experimentally measured slope of bilogarithmic frequency dependence of spectral density equals 3, which is characteristic of the diffusion control; at high frequencies the slope equals 4, which is characteristic of the kinetic control. It is shown that for the model of impedance of the two-stage adsorption oxidation process, in the middle segment of the spectrum the local slope must decrease down to 2, provided the first oxidation stage, which proceeds within the inner electrical double layer, is slow; the local slope must increase up to 6 (or 5, for diffusion control), provided the second oxidation stage (the partially oxidized ion desorption to solution) is slow. The “height” and “width” of the slope local changes appeared explicitly depending on the parameters of the partial charge transfer. This makes the turbulent noise method somewhat superior to the impedance method in the studying of the above-specified reaction type.     

Diffusion of two different water models and thermal conductivity in a protein—water system

The diffusion constant of dynamics simulation data evaluated by the time-dependent displacement or the velocity autocorrelation function provides equivalent results. The diffusion constant (D) increases with the cutoff distance for electrostatic energy. An extended version of the TIP 3P water model provides a proper value of D at small cutoff distance (8.5 Å); the SPC/E water model requires a larger cutoff distance (11.0 Å). Considering Langevin dynamics with a total friction γ, the Einstein relation (D ∼ 1/γ) is valid for large enough friction (γ > 5 ps⁻¹) only. Heating of a protein-water system by stochastic dynamics at the boundary is studied in detail. The calculated thermal conductivity of water agrees with experiment. The thermal conductivity of a protein molecule is about a factor of two smaller. © 1996 John Wiley & Sons, Inc.     

Convective Current in a Four-Electrode Electrochemical Cell at Various Boundary Conditions at Anodes

The frequency dependence of the transfer function of an electrochemical cell is studied experimentally and theoretically in the frequency region 0.005 to 1 Hz under conditions of controlled convective diffusion, at various boundary conditions on the anodes. It is shown that the results are independent of the conditions on the anodes at frequencies exceeding a diffusion frequency. On the other hand, the effect of the boundary conditions al low frequencies is substantial. In particular, it is shown that it is feasible to design a cell with a conversion efficiency indefinitely increasing in the direction of lower frequencies.__________Translated from Elektrokhimiya, Vol. 41, No. 8, 2005, pp. 987–992.Original Russian Text Copyright © 2005 by Agafonov, Nesterov.     

Recent Advances in Voltammetry

Recent progress in the theory and practice of voltammetry is surveyed and evaluated. The transformation over the last decade of the level of modelling and simulation of experiments has realised major advances such that electrochemical techniques can be fully developed and applied to real chemical problems of distinct complexity. This review focuses on the topic areas of: multistep electrochemical processes, voltammetry in ionic liquids, the development and interpretation of theories of electron transfer (Butler–Volmer and Marcus–Hush), advances in voltammetric pulse techniques, stochastic random walk models of diffusion, the influence of migration under conditions of low support, voltammetry at rough and porous electrodes, and nanoparticle electrochemistry. The review of the latter field encompasses both the study of nanoparticle-modified electrodes, including stripping voltammetry and the new technique of ‘nano-impacts’.     

Review and theoretical analysis of ac–av methods for the investigation of hydrogen insertion: Part II. Entry side impedance, transfer function and transfer impedance formalism

The formalism related to alternating current–voltage methods is reviewed, completed and unified for hydrogen insertion in metals and alloys, in relation to permeation cell membranes with potentiometric or potentiostatic detection of hydrogen and electrodes with impermeable boundary conditions. The electrochemical system representation in terms of an electrical dipole or quadripole is first given to introduce the impedance on the entry side, the transfer impedances and the potential and current density transfer functions through the membranes, which can be used experimentally to investigate hydrogen insertion. Next, the surface process is presented, taking the direct (one-step) insertion mechanism into consideration for the sake of simplicity. Based on the surface process and diffusion formalisms, theoretical expressions are derived for the entry side impedance, transfer functions and transfer impedances, depending on the electrode or membrane type and the boundary conditions.     

Anomalous lineshapes and aging effects in two-dimensional correlation spectroscopy

Multitime correlation functions provide useful probes for the ensembles of trajectories underlying the stochastic dynamics of complex systems. These can be obtained by measuring their optical response to sequences of ultrashort optical pulse. Using the continuous time random walk model for spectral diffusion, we analyze the signatures of anomalous relaxation in two-dimensional four wave mixing signals. Different models which share the same two point joint probability distribution show markedly different lineshapes and may be distinguished. Aging random walks corresponding to waiting time distributions with diverging first moment show dependence of 2D lineshapes on initial observation time, which persist for long times.     

The influence of the transmission coefficient of the boundary on the diffusion in a solid surrounded by vacuum

The calculations leading to the differential equations of statistical kinematics inside a solid and near the boundary between a solid and vacuum are given. The influence of the transmission coefficient of the boundary on the concentration of the diffusing material in solids is taken into account by introducing the length of extrapolation expressed by the polynomial of arbitrary order depending on the needed accuracy. The obtained length of extrapolation is influenced not only by the diffusion parameters and the boundary’s characteristic feature but also by the initial distribution of concentration and by the time of diffusion. The presence of an external force is neglected. The numerical calculation for the planar diffusion of hydrogen in a palladium monocrystal is given as an example. The present method may be particularly useful for diffusion in thin layers and membranes, or when the detailed information on the distribution of diffusing material near the boundary between a solid and vacuum is needed. This revised version was published online in July 2006 with corrections to the Cover Date.     

An analysis of the potassium concentrations of soft drinks by HPGe gamma spectrometry

A pyrochemical processing has become one of the potential technologies for a future nuclear fuel cycle. An integrated multi-physics simulation and electrotransport model of a molten-salt electrolytic process are proposed and discussed with respect to the recovery of pure uranium when using thermochemical data. This study has been performed to provide information for diffusion boundary layers between the molten salt (KCl-LiCl) and electrode. The diffusion-controlled electrochemical model demonstrate a prediction of the electrotransport behaviors of LWR spent fuel as a function of the time up to the corresponding electrotransport satisfying a given applied current based on a galvanostatic electrolysis.     

Étude des comportements déterministe et stochastique de l'interface électrochimique à l'aide de corrélateurs numériques

It is shown that correlation methods are particularly suited for analysing the deterministic and stochastic behaviour of an electrochemical interface. The electrode impedance, which exhibits the deterministic behaviour, can be measured by correlation by means of a white noise; this method allows us to perform very fast measurements. The electrochemical noise, which exhibits the stochastic behaviour, is measured by a crosscorrelation method. Characteristics and performances of these methods are analysed in detail taking account of requirements of the polarization control. These methods are applied to the experimental study of some electrochemical interfaces.     

A macrokinetic model of redox sorption on metal–ion exchanger nanocomposites at electrochemical polarization

A conceptual macrokinetic model of redox sorption on metal–ion exchanger nanocomposites upon electrochemical polarization is formulated and a corresponding mathematical model is constructed. The solution to a multi-point boundary value problem for the concentration of a sorbed substance (oxygen) is given. The concentration front of the sorbed substance is characterized by a concentration gradient in the near-surface layer of the solution, by layers of the products of metal oxidation in the composite forming due to both external and internal diffusion transfer, and by chemical and electrochemical reactions at the interphase boundaries. A considerable reduction in the concentration gradient of the sorbate in layers of the products of oxidation of metal and the growth of the diffusion layer of the solution with polarizing currents weaker than the limiting diffusion current are noted.