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At least 1,063 records · Page 59

Synthesis, Characterization and Experimental Studies of Nano Zn–Al–Fe{sub 3}O{sub 4} Blended Alginate/Ca Beads for the Adsorption of Rhodamin B

In the present study, adsorptive removal of rhodamin-B (RMN-B) from aqueous solution was investigated by using nano Zn–Al–Fe{sub 3}O{sub 4} incorporated sodium alginate (SA) beads. SA is a natural polysaccharide, non toxic and biodegradable material. The incorporation of Zn–Al–Fe{sub 3}O{sub 4} into SA was done by ionic polymerization method using CaCl{sub 2}. A batch mode study was used for the adsorption of RMN-B onto Zn–Al–Fe{sub 3}O{sub 4}/ alginate beads. The adsorption efficiency of RMN-B was investigated by the following modeling equations like Freundlich, Langmuir, Pseudo-first order, Pseudo-second order, Intraparticle diffusion, Gibbs free energy, Enthalpy and Entropy. The predicted experimental results of various modeling equations were implied that the process follow Langmuir and pseudo-second order equations. The intraparticle diffusion parameters indicate that the diffusion between RMN-B and nano beads were predominant in throughout of process. The surface morphology of nano beads of before and after adsorption were show that RMN-B is strongly adhered on whole surface of the nano material. The FTIR spectrum indicates that the RMN-B strongly adsorbed on the surface and TGA study reveals that the alginate/Zn–Al–Fe{sub 3}O{sub 4}/Ca material has an excellent temperature withstand capacity.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

Galactic civilizations - Population dynamics and interstellar diffusion

A model is developed of the interstellar diffusion of galactic civilizations which takes into account the population dynamics of such civilizations. The problem is formulated in terms of potential theory, with a family of nonlinear partial differential and difference equations specifying population growth and diffusion for an organism with advantageous genes that undergoes random dispersal while increasing in population locally, and a population at zero population growth. In the case of nonlinear diffusion with growth and saturation, it is found that the colonization wavefront from the nearest independently arisen galactic civilization can have reached the earth only if its lifetime exceeds 2.6 million years, or 20 million years if discretization can be neglected. For zero population growth, the corresponding lifetime is 13 billion years. It is concluded that the earth is uncolonized not because interstellar spacefaring civilizations are rare, but because there are too many worlds to be colonized in the plausible colonization lifetime of nearby civilizations, and that there exist no very old galactic civilizations with a consistent policy of the conquest of inhabited worlds.

Newman, W. I.↗

Stagnation point heat transfer augmentation due to free stream turbulence

A model has been proposed for the momentum eddy diffusivity induced by free stream turbulence intensity and integral length scale. The eddy diffusivity model is applied to the stagnation point of a cylinder situated in a uniform crossflow in the presence of free stream turbulence. A numerical solution of the governing momentum and energy equations with the proposed eddy diffusivity model yielded results for the skin friction coefficient and the Nusselt number. The numerical predictions of the present work are compared with experimental data and the agreement between the two is seen to be very good.

Gorla, R. S. R.↗

The effects of diffusion on population gratings and the stability of ring lasers

The effects of diffusion on the coupling between counterpropagating modes in a two-level medium are studied in the rate equation limit. It is shown that diffusion can greatly decrease saturation gratings in low-pressure gases. Among many other effects which rely on spatial hole burning, this can alter the stability of bidirectional solutions of the third-order Lamb equation model of a ring laser.

Lawandy, N. M.↗

Simulation of 2-dimensional viscous flow through cascades using a semi-elliptic analysis and hybrid C-H grids

A semi-elliptic formulation, termed the interacting parabolized Navier-Stokes (IPNS) formulation, is developed for the analysis of a class of subsonic viscous flows for which streamwise diffusion is neglible but which are significantly influenced by upstream interactions. The IPNS equations are obtained from the Navier-Stokes equations by dropping the streamwise viscous-diffusion terms but retaining upstream influence via the streamwise pressure-gradient. A two-step alternating-direction-explicit numerical scheme is developed to solve these equations. The quasi-linearization and discretization of the equations are carefully examined so that no artificial viscosity is added externally to the scheme. Also, solutions to compressible as well as nearly compressible flows are obtained without any modification either in the analysis or in the solution process. The procedure is applied to constricted channels and cascade passages formed by airfoils of various shapes. These geometries are represented using numerically generated curilinear boundary-oriented coordinates forming an H-grid. A hybrid C-H grid, more appropriate for cascade of airfoils with rounded leading edges, was also developed. Satisfactory results are obtained for flows through cascades of Joukowski airfoils.

Ramamurti, R.↗

Kinetic Monte Carlo Simulation of Hydrogen Diffusion in Tungsten

Nuclear fusion is considered as the most promising sustainable energy source. If achieved, it could provide electricity to the mankind for millions of years. During the reaction, the fusion ash is continuously removed from the magnetic gap, the divertor region, and the divertor facing component is subjected to intense plasma bombardment. It is proposed to use tungsten as the divertor facing material, because it has low physical sputtering yield and superior thermomechanical properties. However, the high mobility of hydrogen in tungsten and the high bombardment flux bring the hydrogen isotopes deep into the tungsten tile. The presence of hydrogen in tungsten tends to weaken the gram boundaries and produce surface blistering. The ejection of tungsten particulates into the plasma core will cool the reaction and cease the fusion reactions. Therefore, it is important to have a detailed understanding of the hydrogen diffusion in tungsten bulk. The KMC method is a stochastic computer simulation to predict the time evolution of a certain system. Typically, at a given time, the system will evolve through a few possible paths with known probabilities. In each step, the KMC method analyzes the entire system and randomly determines the system evolving events based on all transition rates of all possible events and advance the physical time. This process is repeated till the simulation of the entire physical time is completed. In previous work, the authors developed a kinetic Monte Carlo (KMC) algorithm and program to simulate hydrogen diffusion process on tungsten reconstructed surface. In this article, this KMC method is extended to model the hydrogen diffusion in tungsten bulk. The diffusion coefficients are calculated and fitted to the Arrhenius equation. The result matches the experiment measurement very well. KMC diffusion simulation needs the locations of the hydrogen trapping sites, diffusion paths and the corresponding diffusion energy barriers. The hydrogen interstitial sites in tungsten crystal have been thoroughly studied by density function theory (DFT) and molecular dynamics (MD). Johnson and Carter used DFT to determine the preferred location of the hydrogen interstitial sites m tungsten crystal, which are the tetrahedral sites (T-sites) in body-centered-cubic (bcc) lattice surfaces. All T-sites are located on the surfaces of a lattice cell. There are four T-sites per surface and twelve T-sites per lattice cell. The hydrogen atoms diffuse by traveling from one T-site to one of its nearest four T-site neighbors. Thus, each T-site is surrounded by two neighbors on the same surface, and one neighbor on each of the two adjacent surfaces perpendicular to the surface holding the center T-site. The diffusion barrier between two neighboring T-sites was calculated to be 0.38 eV. Knowing the diffusion path, energy barrier, and the crystal structure, the KMC simulation is readily to be performed. Based on other researchers' DFT calculations, the hydrogen stable interstitial site in tungsten bulk is determined as the tetrahedral site on the surface of the bee lattice. Each T-site has 4 neighbors and the energy barrier between two adjacent T-sites are estimated to be 0.38 eV. Based on those facts and the KMC theory, a MATLAB program is developed to model the hydrogen diffusion in tungsten bulk. In the MATLAB program, substrate temperature, hydrogen concentration, physical time to be simulated, and the number of cycles are all customizable. To compare the KMC determined diffusion coefficients with experiment results in the literature, the simulations are performed for temperature 300 K - 2500 K and low hydrogen concentration (0.1%), mimicking the experiment condition. The calculated diffusion coefficients match the reference values measured by the experiment under low hydrogen concentration condition. The results are also fitted to the Arrhenius equation. The pre-exponential factor and the average activation energy in Arrhenius equation are obtained. The good agreement between simulation and experiment proves that the developed KMC algorithm truly reflect the well-established theory of interstitial hydrogens in tungsten. The developed program could easily be used to model the diffusion process and predict the diffusion coefficients under other concentrations and temperatures. This work is part of the ongoing development of hydrogen diffusion in tungsten integrating surface diffusion (previous work), bulk diffusion (this work), and adsorption/desorption on surface (future work)

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Radial diffusion in magnetodiscs

The orbits of charged particles in magnetodiscs are considered. The bounce motion is assumed adiabatic except for transits of a small equatorial region of weak magnetic field strength and high field curvature. Previous theory and modeling have shown that particles scatter randomly in pitch angle with each passage through the equator. A peaked distribution thus diffuses in pitch angle on the time scale of many bounces. It is argued in this paper that spatial diffusion is a further consequence when the magnetodisc has a longitudinal asymmetry. A general expression for DLL, the diffusion of equatorial crossing radii, is derived. DLL is evaluated explicitly for ions in Jupiter's 20-35 radii magnetodisc, assumed to be represented by Connerney et al.'s (1982) Voyager model plus a small image dipole asymmetry. Rates are energy, species, and space dependent but can average as much as a few tenths of a planetary radius per bounce period.

Birmingham, T. J.↗

A transport equation for the scalar dissipation in reacting flows with variable density: First results

Although the different regimes of premixed combustion are not well defined, most of the recent developments in turbulent combustion modeling are led in the so-called flamelet regime. The goal of these models is to give a realistic expression to the mean reaction rate (w). Several methods can be used to estimate (w). Bray and coworkers (Libby & Bray 1980, Bray 1985, Bray & Libby 1986) express the instantaneous reaction rate by means of a flamelet library and a frequency which describes the local interaction between the laminar flamelets and the turbulent flowfield. In another way, the mean reaction rate can be directly connected to the flame surface density (Sigma). This quantity can be given by the transport equation of the coherent flame model initially proposed by Marble & Broadwell 1977 and developed elsewhere. The mean reaction rate, (w), can also be estimated thanks to the evolution of an arbitrary scalar field G(x, t) = G(sub O) which represents the flame sheet. G(x, t) is obtained from the G-equation proposed by Williams 1985, Kerstein et al. 1988 and Peters 1993. Another possibility proposed in a recent study by Mantel & Borghi 1991, where a transport equation for the mean dissipation rate (epsilon(sub c)) of the progress variable c is used to determine (w). In their model, Mantel & Borghi 1991 considered a medium with constant density and constant diffusivity in the determination of the transport equation for (epsilon(sub c)). A comparison of different flamelet models made by Duclos et al. 1993 shows the realistic behavior of this model even in the case of constant density. Our objective in this present report is to present preliminary results on the study of this equation in the case of variable density and variable diffusivity. Assumptions of constant pressure and a Lewis number equal to unity allow us to significantly simplify the equation. A systematic order of magnitude analysis based on adequate scale relations is performed on each term of the equation. As in the case of constant density and constant diffusivity, the effects of stretching of the scalar field by the turbulent strain field, of local curvature, and of chemical reactions are predominant. In this preliminary work, we suggest closure models for certain terms, which will be validated after comparisons with DNS data.

Mantel, T.↗

Stochastic partial differential equations in turbulence related problems

The theory of stochastic partial differential equations (PDEs) and problems relating to turbulence are discussed by employing the theories of Brownian motion and diffusion in infinite dimensions, functional differential equations, and functional integration. Relevant results in probablistic analysis, especially Gaussian measures in function spaces and the theory of stochastic PDEs of Ito type, are taken into account. Linear stochastic PDEs are analyzed through linearized Navier-Stokes equations with a random forcing. Stochastic equations for waves in random media as well as model equations in turbulent transport theory are considered. Markovian models in fully developed turbulence are discussed from a stochastic equation viewpoint.

Chow, P.-L.↗

Ocean Turbulence, III: New GISS Vertical Mixing Scheme

We have found a new way to express the solutions of the RSM (Reynolds Stress Model) equations that allows us to present the turbulent diffusivities for heat, salt and momentum in a way that is considerably simpler and thus easier to implement than in previous work. The RSM provides the dimensionless mixing efficiencies Gamma-alpha (alpha stands for heat, salt and momentum). However, to compute the diffusivities, one needs additional information, specifically, the dissipation Epsilon. Since a dynamic equation for the latter that includes the physical processes relevant to the ocean is still not available, one must resort to different sources of information outside the RSM to obtain a complete Mixing Scheme usable in OGCMs. As for the RSM results, we show that the Gamma-alpha s are functions of both Ri and Rq (Richardson number and density ratio representing double diffusion, DD); the Gamma-alpha are different for heat, salt and momentum; in the case of heat, the traditional value Gamma-h = 0.2 is valid only in the presence of strong shear (when DD is inoperative) while when shear subsides, NATRE data show that Gamma-h can be three times as large, a result that we reproduce. The salt Gamma-s is given in terms of Gamma-h. The momentum Gamma-m has thus far been guessed with different prescriptions while the RSM provides a well defined expression for Gamma-m(Ri,R-rho). Having tested Gamma-h, we then test the momentum Gamma-m by showing that the turbulent Prandtl number Gamma-m/Gamma-h vs. Ri reproduces the available data quite well. As for the dissipation epsilon, we use different representations, one for the mixed layer (ML), one for the thermocline and one for the ocean;s bottom. For the ML, we adopt a procedure analogous to the one successfully used in PB (planetary boundary layer) studies; for the thermocline, we employ an expression for the variable epsilon/N(exp 2) from studies of the internal gravity waves spectra which includes a latitude dependence; for the ocean bottom, we adopt the enhanced bottom diffusivity expression used by previous authors but with a state of the art internal tidal energy formulation and replace the fixed Gamma-alpha = 0.2 with the RSM result that brings into the problem the Ri, R-rho dependence of the Gamma-alpha; the unresolved bottom drag, which has thus far been either ignored or modeled with heuristic relations, is modeled using a formalism we previously developed and tested in PBL studies. We carried out several tests without an OGCM. Prandtl and flux Richardson numbers vs. Ri. The RSM model reproduces both types of data satisfactorily. DD and Mixing efficiency Gamma-h(Ri,Rq). The RSM model reproduces well the NATRE data. Bimodal epsilon-distribution. NATRE data show that epsilon (Ri < 1) approximately equals 10epsilon(Ri > 1), which our model reproduces. Heat to salt flux ratio. In the Ri much greater than 1 regime, the RSM predictions reproduce the data satisfactorily. NATRE mass diffusivity. The z-profile of the mass diffusivity reproduces well the measurements at NATRE. The local form of the mixing scheme is algebraic with one cubic equation to solve.

Canuto, V. M.↗

Interfacial layers in high-temperature-oxidized NiCrAl

The utility of Auger electron spectroscopy combined with ball cratering for depth analysis of oxide and diffusion layers produced in a Ni-14Cr-24Al alloy by oxidation in air at 1180 C for 25 hr is demonstrated. During postoxidation cooling, the oxide layers formed by this alloy spalled profusely. The remaining very thin oxide was primarily Cr2O3 with a trace of Ni. The underlying metal substrate exhibited gamma/gamma-prime and beta phases with a metallic interfacial layer which was similar to the bulk gamma/gamma-prime phase but slightly enriched in Cr and Al. These data are compared to electron microprobe results from a nominally identical alloy. The diffusion layer thickness is modelled with a simple mass balance equation and compared to recent results on the diffusion process in NiCrAl alloys.

Larson, L. A.↗

Stochastic modeling of catalytic processes in nanoporous materials: Beyond mean-field approach

Transport and reaction in zeolites and other porous materials, such as mesoporous silica particles, has been a focus of interest in recent years. This is in part due to the possibility of anomalous transport effects (e.g. single-file diffusion) and its impact in the reaction yield in catalytic processes. Computational simulations are often used to study these complex nonequilibrium systems. Computer simulations using Molecular Dynamics (MD) techniques are prohibitive, so instead coarse grained one-dimensional models with the aid of Kinetic Monte Carlo (KMC) simulations are used. Both techniques can be computationally expensive, both time and resource wise. These coarse-grained systems can be exactly described by a set of coupled stochastic master equations, that describe the reaction-diffusion kinetics of the system. The equations can be written exactly, however, coupling between the equations and terms within the equations make it impossible to solve them exactly; approximations must be made. One of the most common methods to obtain approximate solutions is to use Mean Field (MF) theory. MF treatments yield reasonable results at high ratios of reaction rate k to hop rate h of the particles, but fail completely at low k=h due to the over-estimation of fluxes of particles within the pore. We develop a method to estimate fluxes and intrapore diffusivity in simple one- dimensional reaction-diffusion models at high and low k=h, where the pores are coupled to an equilibrated three-dimensional fluid. We thus successfully describe analytically these simple reaction-diffusion one-dimensional systems. Extensions to models considering behavior with long range steric interactions and wider pores require determination of multiple boundary conditions. We give a prescription to estimate the required parameters for these simulations. For one dimensional systems, if single-file diffusion is relaxed, additional parameters to describe particle exchange have to be introduced. We use Langevin Molecular Dynamics (MD) simulations to assess these parameters.

36 MATERIALS SCIENCE↗

Investigations of turbulent scalar fields using probability density function approach

Scalar fields undergoing random advection have attracted much attention from researchers in both the theoretical and practical sectors. Research interest spans from the study of the small scale structures of turbulent scalar fields to the modeling and simulations of turbulent reacting flows. The probability density function (PDF) method is an effective tool in the study of turbulent scalar fields, especially for those which involve chemical reactions. It has been argued that a one-point, joint PDF approach is the one to choose from among many simulation and closure methods for turbulent combustion and chemically reacting flows based on its practical feasibility in the foreseeable future for multiple reactants. Instead of the multi-point PDF, the joint PDF of a scalar and its gradient which represents the roles of both scalar and scalar diffusion is introduced. A proper closure model for the molecular diffusion term in the PDF equation is investigated. Another direction in this research is to study the mapping closure method that has been recently proposed to deal with the PDF's in turbulent fields. This method seems to have captured the physics correctly when applied to diffusion problems. However, if the turbulent stretching is included, the amplitude mapping has to be supplemented by either adjusting the parameters representing turbulent stretching at each time step or by introducing the coordinate mapping. This technique is still under development and seems to be quite promising. The final objective of this project is to understand some fundamental properties of the turbulent scalar fields and to develop practical numerical schemes that are capable of handling turbulent reacting flows.

Gao, Feng↗

Diffusion Synthetic Acceleration for Heterogeneous Domains, Compatible with Voids

A standard approach to solving the S N transport equations is to use source iteration with diffusion synthetic acceleration (DSA). Although this approach is widely used and effective on many problems, there remain some practical issues with DSA preconditioning, particularly on highly heterogeneous domains. For large-scale parallel simulation, it is critical that both (a) preconditioned source iteration converges rapidly and (b) the action of the DSA preconditioner can be applied using fast, scalable solvers, such as algebraic multigrid (AMG). For heterogeneous domains, these two interests can be at odds. In particular, there exist DSA diffusion discretizations that can be solved rapidly using AMG, but they do not always yield robust/fast convergence of the larger source iteration. Conversely, there exist robust DSA discretizations where source iteration converges rapidly on difficult heterogeneous problems, but fast parallel solvers like AMG tend to struggle applying the action of such operators. Moreover, very few current methods for the solution of deterministic transport are compatible with voids. This paper develops a new heterogeneous DSA preconditioner based on only preconditioning the optically thick subdomains. The resulting method proves robust on a variety of heterogeneous transport problems, including a linearized hohlraum mesh related to inertial confinement fusion. Moreover, the action of the preconditioner is easily computed using O(1) AMG iterations, convergence of the transport iteration typically requires 2 to 5× fewer iterations than current state-of-the-art “full” DSA, and the proposed method is trivially compatible with voids. On the hohlraum problem, rapid convergence is obtained by preconditioning less than 3% of the mesh elements with five to ten AMG iterations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Assessment of integral models for non-Boussinesq lazy plumes using numerical simulations

Integral modelling of turbulent buoyant plumes is crucial for rapid predictions of plume characteristics. While the governing equations are typically derived using self-similarity and a Boussinesq approximation, these assumptions may not hold for plumes originating from finite-area sources with large density ratios. Here, this work evaluates the accuracy of integral-scale models for non-Boussinesq lazy plumes using high-fidelity numerical simulations of turbulent helium plumes. We analyse the plume kinematics by computing vertical fluxes, plume radius and radial profiles, establishing some disparities between common practice and physical accuracy. We identify how the definition of the plume radius changes the perception of the plume structure when the flow is not self-similar and derive a relationship between the flux-based and threshold-based definitions without requiring self-similarity. We then examine the plume dynamics by evaluating the source terms from the governing plume equations. Our results support neglecting diffusive and viscous effects but emphasise the importance of the mean pressure gradient, even in the self-similar regime. Two coefficients need to be modelled: the well-known entrainment coefficient and the lesser-known momentum correction coefficient, which is a correction required for the momentum equation to account for self-similar and slender approximations. The momentum correction coefficient is found to be approximately constant and slightly greater than the assumed value of 1. The standard entrainment coefficient models perform well up to a local Richardson number three times the asymptotic value but overpredict entrainment for larger Richardson numbers. We propose a correction using the known finite limit of entrainment at infinite Richardson number.

Meehan, Michael Alexander [Sandia National Laborat↗

A reaction–diffusion model for phenotypic evolution

We present a reaction–diffusion mathematical model for the evolutionary dynamics of phenotypic evolution. A detailed deduction of the equations is presented for the one-dimensional version, from which a more general model is proposed. Particular cases are studied using analytical approximations and numerical simulations. Results indicate that the approach proposed produces results that are coherent with mainstream models in evolutionary dynamics, suggesting that the reaction–diffusion model could be an alternative tool in the analysis of evolutionary dynamics.

97 MATHEMATICS AND COMPUTING↗

Diffusion-Limited Kinetics in Reactive Systems

A proper representation of chemical kinetics is vital to understanding, modeling, and optimizing many important chemical processes. In liquid and surface phases, where diffusion is slow, the rate at which the reactants diffuse together limits the overall rate of many elementary reactions. Commonly, the textbook Smoluchowski theory is utilized to estimate effective rate coefficients in the liquid phase. On surfaces, modelers commonly resort to much more complex and expensive Kinetic Monte Carlo (KMC) simulations. Here, in this study, we extend the Smoluchowski model to allow the diffusing species to undergo chemical reactions and derive analytical formulas for the diffusion-limited rate coefficients for 3D, 2D, and 2D/3D interface cases. With these equations, we are able to demonstrate that when species react faster than they diffuse they can react orders of magnitude faster than predicted by Smoluchowski theory, through what we term “the reactive transport effect”. We validate the derived steady-state equations against particle Monte Carlo (PMC) simulations, KMC simulations, and non-steady-state solutions. Furthermore, using PMC and KMC simulations, we propose corrections that agree with all limits and the computed data for the 2D and 2D/3D interface steady-state equations, accounting for unique limitations in the associated derived equations. Additionally, we derive equations to handle couplings between diffusion-limited rate coefficients in reaction networks. We believe these equations should make it possible to run much more accurate mean-field simulations of liquids, surfaces, and liquid–surface interfaces accounting for diffusion limitations and the reactive transport effect.

Johnson, Matthew Sean↗