Spherical symmetric diffusion problem

We introduce a general and versatile MS Windows application for solving the spherically symmetric diffusion problem, involving up to two coupled spherically symmetric Smoluchowski equations. The application is based on a modular, configurable, user-friendly graphical interface, in which input parameters are introduced through a graphical representation of the system of partial differential equations and output attributes can be obtained graphically during propagation. The numerical algorithm consists of finite differencing in space and Chebyshev propagation in time; it includes an implementation of virtual gridding, which enhances the accuracy of calculating boundary conditions and steep potentials. The program has b een checked against a wide collection of analytical solutions and applied to an experimentally open problem in excited-state proton-transfer to solvent. © 1996 by John Wiley & Sons, Inc.     

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 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 diffusion potential in the gas phase

Potentiometric based electrochemical measurement of diffusion potential at a junction between two flowing flame plasma gases is described. A flame electrochemical cell was constructed using a specially designed burner, which supports two individual flames, each fed by separate premixed methane/oxygen/nitrogen streams. The two flames were in intimate contact, creating a flowing fluid gaseous junction. By aspirating metal salt solutions into these premixed feed gases, the concentration gradient at the interface between the two flames may be controlled. A measurable electrochemical diffusion potential was formed at this junction, the magnitude of which was dependent on the concentration ratio of charged species with different mobilities. In our flame electrolyte, the dominant charged species were atomic or molecular cations and electrons, which have a difference in mobilities of approximately three orders of magnitude. A two-electrode system, in conjunction with a high impedance electrometer was used to measure the potential difference across the flame electrochemical cell. The measured potential difference was analysed using theory developed for the liquid junction potentials by the Henderson equation.     

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.     

Electrochemical determination of diffusion coefficients of selected metals in dimethylsulfoxide

Diffusion coefficients of lead, zinc, thallium and gadolinium in dimethylsulfoxide have been measured electrochemically by means of the complete Koutecky equation. Instantaneous currents were measured on first drops from a D.M.E. and calculations were made by an iterative computer technique. Half-wave potentials, polarographic wave reversibility, and diffusion control data are reported.     

Electrochemical properties of Li symmetric solid-state cell with NASICON-type solid electrolyte and electrodes

All-solid-state phosphate symmetric cells using Li₃V₂(PO₄)₃ for both the positive and negative electrodes with the phosphate Li_1.5Al_0.5Ge_1.5(PO₄)₃ as the solid electrolyte were proposed. Amorphous Li_1.5Al_0.5Ge_1.5(PO₄)₃ was added into the electrode to increase the interface area between the active materials and the electrolyte. Any other phases were not formed at the electrode/electrolyte interface even after hot pressing at 600 °C. The discharge capacity was 92 mAh g^? 1 at 22 µA cm^? 2 at 80 °C, and 38 mAh g^? 1 at 25 °C, respectively. Symmetric cell configuration leads to simplify the fabrication process for all-solid-state batteries and will reduce manufacturing costs.     

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.     

Single molecule diffusion and the solution of the spherically symmetric residence time equation

The residence time of a single dye molecule diffusing within a laser spot is propotional to the total number of photons emitted by it. With this application in mind, we solve the spherically symmetric "residence time equation" (RTE) to obtain the solution for the Laplace transform of the mean residence time (MRT) within a d-dimensional ball, as a function of the initial location of the particle and the observation time. The solutions for initial conditions of potential experimental interest, starting in the center, on the surface or uniformly within the ball, are explicitly presented. Special cases for dimensions 1, 2, and 3 are obtained, which can be Laplace inverted analytically for d = 1 and 3. In addition, the analytic short- and long-time asymptotic behaviors of the MRT are derived and compared with the exact solutions for d = 1, 2, and 3. As a demonstration of the simplification afforded by the RTE, the Appendix obtains the residence time distribution by solving the Feynman-Kac equation, from which the MRT is obtained by differentiation. Single-molecule diffusion experiments could be devised to test the results for the MRT presented in this work.     

十二烷基苯磺酸钠扩散系数的电化学测定

表面活性剂水溶液体系胶束扩散系数的测定是研究表面活性剂水溶液体系性质的重要方法之一。胶束扩散系数的测定已有准弹性光散射[1]、小角X-射线散射[2]、扩散-粘度[3]、极谱法[4]、循环伏安[5]等方法报导。但有关十二烷基苯磺酸钠(sod ium dodecy benzene sulfonate,SDBS)水溶     

Electrochemical diffusion potential in a flame plasma: theory and experiment

A flame doped with an appropriate additive to produce positive ions and free electrons is a quasineutral, weak, continuum plasma. When bounded by a metallic burner upstream and a metal plate downstream, the two electrodes and flame plasma can be viewed as a gas-phase electrochemical cell. When the ion (and electron) density varies continuously along the flame axis, an expression for the diffusion potential can be derived in terms of the concentration gradient. The familiar logarithmic dependence on the ion concentration is obtained. A plasma sheath develops at the metal plate electrode; it sustains a potential difference which can be modeled by a Boltzmann distribution of the electrons in the sheath. Since the plate has to be cooled in practice, the average sheath temperature is less than the flame temperature because the sheath occurs inside the thermal boundary layer which covers the plate electrode. Inevitably, the reduced sheath temperature affects the sheath voltage. Experimental measurements of the “cell” voltage are made for the two cases of a positive concentration gradient using a sodium plasma, and a negative gradient by doping the flame with methane. As predicted theoretically, the cell voltages have opposite signs. However, the magnitude of the cell voltage seems to depend significantly on the sheath temperature which appears to decrease steadily with increasing distance downstream from the burner. It is also possible that the measured cell voltages involve unknown surface contact potentials. When compared with solution concentration cells, gas-phase flame systems exhibit both similarities and differences.     

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.     

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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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.     

Flow injection analysis simulations and diffusion coefficient determination by stochastic and deterministic optimization methods

Stochastic and deterministic simulations of dispersion in cylindrical channels on the Poiseuille flow have been presented. The random walk (stochastic) and the uniform dispersion (deterministic) models have been used for computations of flow injection analysis responses. These methods coupled with the genetic algorithm and the Levenberg–Marquardt optimization methods, respectively, have been applied for determination of diffusion coefficients. The diffusion coefficients of fluorescein sodium, potassium hexacyanoferrate and potassium dichromate have been determined by means of the presented methods and FIA responses that are available in literature. The best-fit results agree with each other and with experimental data thus validating both presented approaches.     

Catalytic conversion reactions mediated by single-file diffusion in linear nanopores: hydrodynamic versus stochastic behavior

We analyze the spatiotemporal behavior of species concentrations in a diffusion-mediated conversion reaction which occurs at catalytic sites within linear pores of nanometer diameter. Diffusion within the pores is subject to a strict single-file (no passing) constraint. Both transient and steady-state behavior is precisely characterized by kinetic Monte Carlo simulations of a spatially discrete lattice-gas model for this reaction-diffusion process considering various distributions of catalytic sites. Exact hierarchical master equations can also be developed for this model. Their analysis, after application of mean-field type truncation approximations, produces discrete reaction-diffusion type equations (mf-RDE). For slowly varying concentrations, we further develop coarse-grained continuum hydrodynamic reaction-diffusion equations (h-RDE) incorporating a precise treatment of single-file diffusion in this multispecies system. The h-RDE successfully describe nontrivial aspects of transient behavior, in contrast to the mf-RDE, and also correctly capture unreactive steady-state behavior in the pore interior. However, steady-state reactivity, which is localized near the pore ends when those regions are catalytic, is controlled by fluctuations not incorporated into the hydrodynamic treatment. The mf-RDE partly capture these fluctuation effects, but cannot describe scaling behavior of the reactivity.     

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.     

Electrochemical evidences and consequences of significant differences in ions diffusion rate in polyacrylate-based ion-selective membranes

The origin and effect of surface accumulation of primary ions within the ion-selective poly(n-butyl acrylate)-based membrane, obtained by thermal polymerization, is discussed. Using a new method, based on the relation between the shape of a potentiometric plot and preconditioning time, the diffusion of copper ions in the membrane was found to be slow (the diffusion coefficient estimated to be close to 10(-11) cm(2) s(-1)), especially when compared to ion-exchanger counter ions--sodium cations diffusion (a diffusion coefficient above 10(-9) cm(2) s(-1)). The higher mobility of sodium ions than those of the copper-ionophore complex results in exposed ion-exchanger role leading to undesirably exposed sensitivity to sodium or potassium ions.     

The effect of stochastic barrier fluctuation on semiclassical transmission probability and Shannon entropy of a symmetric double well potential

Stochastic fluctuation of barrier height and width of a symmetric double well plays a very significant role in the corresponding dynamics by increasing the semiclassical transmission probability and Shannon entropy of the system. The population of the system has been observed to be spread into several under barrier states starting from the or [, where and are the wave functions describing the two lowest degenerate states] in presence of the stochastic fluctuation. This distribution over several states is manifested by steady increase in Shannon entropy. However, any arbitrary value of the stochastic fluctuation cannot increase the populations of the upper energy states and consequently no gain in the net value of Shannon entropy results. There is an optimum frequency for which the Shannon entropy passes through a maximum, which is also found out in this work. We have also calculated the semiclassical WKB like transmission probability as a function of time and it is clear that the random fluctuation of barrier causes the transmission probability to increase to a significant amount. As the total energy of the system remains below the potential barrier, this transmission probability is equivalent to tunneling probability. It has been clearly shown that if the fluctuation is made to be periodic (without changing the frequency and magnitude of the fluctuation) it cannot effect any significant change in the overall dynamics.     

Electrochemical studies of the lateral diffusion of TEMPO in the aqueous liquid/vapor interfacial region

Surface partitioning and lateral mobility of TEMPO (2,2,6,6-tetramethyl-1-piperidynyloxy free radical) in the aqueous liquid/gas interfacial region were investigated electrochemically with 100 nm wide, 1.0 cm long microband electrodes positioned at the air/water interface. For redox active amphiphiles such as TEMPO, the electrochemical current is the sum of the surface and solution components representing the diffusive transport of TEMPO in both domains as well as the dynamics of equilibration at the air/water interface. Interpretation of the recorded current-voltage curves was aided by a FEMLAB simulation code developed to analyze transport processes in this class of systems. TEMPO and TEMPO(+) partition constants (K(T), K(T)+) and solution diffusivities (D(sol), equal for the two species) were obtained experimentally yielding K(T) = 5.0 +/- 0.7 x 10(2) M(-1), K(T+) = 41 +/- 3 M(-1), and D(sol) = 7.7 +/- 0.35 x 10(-6) cm(2)/s. In view of the low value of K(T)+, TEMPO(+) was assumed not to partition to the air/water interface. We further assumed that the desorption rate constants (k(des)) of both TEMPO and TEMPO(+) were the same. Good fits between the recorded and simulated cyclic voltammograms were obtained using two correlated, adjustable parameters, k(des) and the TEMPO lateral, surface diffusion constant (D(surf)). Detailed analysis of the behavior of this class of systems was obtained for a broad range of D(surf) and k(des) values. In addition, a calibration curve of k(des) versus D(surf) was obtained. Assuming that TEMPO k(des) is in a likely range of 10-100 s(-1), its lateral diffusion constant is in the range of 7.9-3.6 x 10(-5) cm(2)/s. In view of our earlier work (Wu, D. G.; Malec, A. D.; Head-Gordon, M.; Majda, M. J. Am. Chem. Soc. 2005, 127, 4490-4496) showing that at the air/water interface TEMPO is unimmersed, and that its interactions with water are limited to hydrogen bonding with one or two water molecules, D(surf) can be related to the viscosity of the aqueous interfacial region.