Constant Diffusion Coefficient Research Articles

A new dynamic Monte Carlo method satisfying n-particle diffusion equation with position-dependent diffusion coefficient, free energy, and intermolecular interactions.

A dynamic Monte Carlo (MC) method recently proposed by us [Nagai et al., J. Chem. Phys. 156, 154506 (2022)] to describe single-particle diffusion of a molecule in a heterogeneous space with position-dependent diffusion coefficient and free energy is generalized here to n-particle dynamics, where n molecules diffuse in heterogeneous media interacting via their intermolecular potential. Starting from the master equation, we give an algebraic proof that the dynamic MC transition probabilities proposed here produce particle trajectories that satisfy the n-particle diffusion equation with position-dependent diffusion coefficient D0i(ri), free energy F1i(ri), and intermolecular interactions Vij(ri, rj). The MC calculations based on this method are compared to molecular dynamics (MD) calculations for two-dimensional heterogeneous Lennard-Jones test systems, showing excellent agreement of the long-distance global diffusion coefficient between the two cases. Thus, the particle trajectories produced by the present MC transition probabilities satisfy the n-particle diffusion equation, and the diffusion equation well describes the long-distance trajectories produced by the MD calculations. The method is also an extension of the conventional equilibrium Metropolis MC calculation for homogeneous systems with a constant diffusion coefficient to the dynamics in heterogeneous systems with a position-dependent diffusion coefficient and potential. In the present method, interactions and dynamics of the real systems are coarse-grained such that the calculation cost is drastically reduced. This provides an approach for the investigation of particle dynamics in very complex and large systems, where the diffusing length is of sub-micrometer order and the diffusion time is of the order of milliseconds or more.

Mechanistic Investigation of Enzyme Triggered Release from a Xyloglucan Matrix Tablet for Controlled Colonic Drug Delivery

The objective of this study was to investigate the mechanisms underlying drug release from a controlled colonic release (CCR) tablet formulation based on a xyloglucan polysaccharide matrix and identify the factors that control the rate of release for the purpose of fundamentally substantiating the concept and demonstrating its robustness for colonic drug delivery. Previous work demonstrated in vitro limited release of 5-aminosalicylic acid (5-ASA) and caffeine from these tablets in small intestinal environment and significant acceleration of release by xyloglucanase, an enzyme of the colonic microbiome. Targeted colonic drug delivery was verified in an animal study in vivo. In the present work, interaction of the xyloglucan matrix tablets with aqueous dissolution media containing xyloglucanase was found to lead to the spontaneous formation of a hydrated highly viscous gummy layer at the surface of the matrix which had a reduced drug content compared to the underlying regions and persisted with a nearly constant thickness that was inversely correlated to the enzyme concentration throughout the duration of the release process. Enzymatic hydrolysis of xyloglucan was determined to take place at the surface of the matrix leading to matrix erosion and a relation for the rate of enzymatic reaction as a function of bulk enzyme concentration and the concentration of dissolved xyloglucan in the gummy layer was derived. A mathematical model was developed encompassing aqueous medium ingress, matrix metamorphosis due to xyloglucan dissolution and matrix swelling, enzymatic hydrolysis of the polysaccharide and concomitant drug release due to matrix erosion and simultaneous drug diffusion. The model was fitted to data of reducing sugar equivalents in the medium reflecting matrix erosion and released drug amount. Enzymatic reaction parameters and reasonable values of medium ingress velocity, xyloglucan dissolution rate constant and drug diffusion coefficient were deduced that provided an adequate approximation of the data. Erosion was shown to be the overwhelmingly dominant drug release mechanism while the role of diffusion marginally increased at low enzyme concentration and high drug solubility. Changing enzyme concentration had a rather weak effect on matrix erosion and drug release rate as demonstrated by model simulations supported by experimental data, while xyloglucan dissolution was slow and had a stronger effect on the rate of the process. Therefore, reproducible colonic drug delivery not critically influenced by inter- and intra-individual variation of microbial enzyme activity may be projected.

A general fourth-order mesoscopic multiple-relaxation-time lattice Boltzmann model and its macroscopic finite-difference scheme for two-dimensional diffusion equations

In this work, we first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the corresponding macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain a macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis on the finite-difference scheme and the mesoscopic MRT-LB model, and find that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis.

An inverse problem for nonlocal reaction-diffusion equations with time-delay

The reaction-diffusion equations have been studied in various aspects of nature, such as heat transfer and biological dynamic modelling. In this paper, we study a nonlocal reaction-diffusion model with time delay associated with the density and diffusion behaviour of biological populations. Based on our previous conclusions and numerical strategy of the direct problem, we continue to study the inverse problem about the diffusion coefficient as well as the parameters in the birth function, and also perform numerical simulations with error analysis. In addition, we further generalize the constant diffusion coefficient to the position-dependent diffusion rate, which is intended to improve the generalizability to multiple organisms diffusion in nature.

Computational study on the influence of variable diffusion coefficients on non-fourier transport in shear-rate-dependent fluid with solute particles

Heat transfer in fluids occurs in numerous industrial processes. It is a fact that diffusion coefficients during heat transfer in fluids do not remain constant. Therefore, consideration of the constant diffusion coefficient is not a good approximation. In view of this, generalized modelling of heat and mass transfer in fluids of variable properties is done, and these generalized problems are solved by finite element method (FEM). After pre- and post-calculations, numerical runs are executed to observe the influence of variation in conductance on transport mechanisms in the fluid. The impact of other parameters is also studied. The outcomes are recorded in the form of graphs and numerical data. Ohmic dissipation results in an increase in the temperature of the fluid for variable and constant viscosity. The thermal relaxation time has shown significant effects on the flow and thickness of the thermal boundary layer region. The temperature of the fluid decreases with thermal relaxation time. The thermal conductance of the fluid varies with an increase in temperature. Once the temperature rises, the ability of the fluid to conduct heat is increased, due to which more heat from the heated surface diffuses into the fluid. Hence, heat flux decreases.The behavior of the variable viscosity of the fluid on the diffusion of momentum into the fluid is studied, and it is now known that the diffusion of wall momentum into the fluid is compromised because of a decrease in the viscosity, which is inversely proportional to the temperature. Hence, the diffusion of wall momentum into the fluid in the isothermal case is greater than that in the temperature-varying case.

MATHEMATICAL MODELING OF DIFFUSION BOUNDARY PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS USING MATLAB AND C++ FOR NUMERICAL AND ANALYTICAL SOLUTIONS

The article examines a second-order parabolic partial differential equation of a three-dimensional (3D) non-stationary boundary problem with constant diffusion coefficients and periodic boundary conditions in the x and y directions. The method for reducing the (3D) non-stationary boundary problem to the corresponding one-dimensional (1D) non-stationary boundary problem using periodic boundary conditions in the x and y directions is discussed. The stationary (analytical) solution of the obtained (1D) stationary boundary problem is also obtained. The numerical solutions of the 1D boundary problem are obtained using the Matlab package "pdepe" and the C++ programming language. As a practical application of the developed mathematical model, the article discusses calculating the concentration of heavy metal Ca in a peat layer based on the obtained experimental data (measurements).

Application of image-based diffusion coefficients in multi-scale simulation of coalbed methane reservoirs

Gas diffusion in coalbed methane (CBM) reservoirs is a multi-scale process. It can be a dominant factor in long-term gas production performance. Producing coalbed methane requires a detailed understanding of diffusion behaviours and their contribution to production. This paper develops a core-scale model based on micro-CT images. Our model considers free and adsorbed gas diffusion, sorption, and viscous flow. It also includes the effect of swelling, gas moisture, and effective stress. Following that, it is used to simulate gas flow on micro-CT images. The simulation results quantify the influence of effective stress on gas diffusion in coal. The core-scale model is then coupled to the large-scale simulation, where we compare the gas production curves against the result based on the assumption of a constant diffusion coefficient. This model can be used to calculate the effective diffusion coefficient from a micro-CT image and to predict the production performance at the reservoir scale reliably using a coupled multi-scale approach.

Space-time correlations of passive scalar in Kraichnan model

We consider the two-point, two-time (space-time) correlation of passive scalar R(r,τ) in the Kraichnan model under the assumption of homogeneity and isotropy. Using the fine-gird PDF method, we find that R(r,τ) satisfies a diffusion equation with constant diffusion coefficient determined by velocity variance and molecular diffusion. Its solution can be expressed in terms of the two-point, one time correlation of passive scalar, i.e., R(r,0). Moreover, the decorrelation of R^(k,τ), which is the Fourier transform of R(r,τ), is determined by R^(k,0) and a diffusion kernal.

(Invited) Multi-Site, Multi-Reaction Model for the Simulating Intercalation and Alloy Electrodes

We overview enabling research for the characterization and design of cells using intercalation and alloy electrodes. Specifically, we have recently developed and implemented a multi-site, multi-reaction (MSMR) model. We shall demonstrate the approach with various chemistries of interest today, and the creation of reduced order models to simply calculations.We first discuss some of the similarities and differences between the MSMR model and the original Doyle-Fuller-Newman model [1, 2], which we shall refer to as the DFN model, first published in 1993. This model represents an intercalation electrode using porous electrode theory, but it also includes a detailed description of transport of Li ions in the electrolyte, transport of lithium in the solid phase, and the kinetics of the intercalation reaction as a Butler-Volmer equation. The MSMR model is based on the same porous electrode theory used in the original DFN model, and it uses the same treatment of transport in the electrolyte. However, the MSMR model introduces significant changes in the representation of both transport in the solid phase and the kinetics of the intercalation reaction. The main reason for the changes in the MSMR model is because it hypothesizes that intercalated lithium consists of not just one species, but multiple species. There is significant evidence to support this point of view, both from cyclic voltammetry experiments as well as X-ray diffraction analyses [4].The DFN model, as originally described and often used today, represents solid phase diffusion as a linear equation with a constant diffusion coefficient. This representation is known to be highly inaccurate, for example, in the case of graphite, where measurements have shown that the diffusion coefficient can vary by several orders of magnitude depending on the mole fraction of intercalated lithium [3,4]. More sophisticated treatments of solid phase diffusion use the chemical potential as the driving force for diffusion and because the chemical potential can be written in terms of the OCV, this explains how the Open Circuit Voltage now plays a role in the diffusion equation. Since the MSMR model uses multiple species, the solid phase diffusion problem should also involve multiple species, but the entire system of multi-species diffusion equations simplifies to a single equation, if one assumes that the various species of lithium are locally equilibrated with each other within the active material particles. The details of this analysis are given in [6].The DFN model uses a Butler-Volmer equation to represent the kinetics of the intercalation reaction, whereas the MSMR model uses a Butler-Volmer equation for each species; the intercalation reaction rate thus becomes the sum of these multiple Butler-Volmer equations, which are formulated using activities instead of concentrations, as is done in the original DFN model.Finally, the MSMR model proposes a simple empirical relationship, based on a generalized Nernst equation, to describe the relationship between the mole fraction of each intercalated species and the OCV. This formulation provides a systematic way to derive analytical formulas to represent the OCV relationship x(U), where x is the total mole fraction of intercalated species and U is the corresponding OCV. Several examples [5,6] show that this formalism suffices to provide accurate fits of x(U) to experimental data for many different electrode materials in common use, and it also provides analytical expressions for the derivative dx/dU, which shows up in the expressions for the solid phase diffusion equation.In summary, the MSMR model attempts to give a more accurate representation of solid phase diffusion and a more complete representation of intercalation interfacial kinetics, and, in this context, our hope is that the MSMR model provides a useful complement to the original DFN model. M. Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc., 140(1993)1526.T. F. Fuller, M. Doyle, and J. Newman, J. Electrochem. Soc., 141(1994)1.D. Levi, E. A. Levi, and D. Aurbach, Journal of Electroanalytical Chemistry, 421, 89 (1997).D. R. Baker and M.W. Verbrugge, J. Electrochem. Soc., 159 (8) A1341-A1350 (2012).M. W. Verbrugge, D. R. Baker, and X. Xiao, J. Electrochem. Soc., 163(2016)A262-A271.M. W. Verbrugge, D. R. Baker, B. J. Koch, X. Xiao, and W. Gu, J. Electrochem. Soc., 164(2017)E3243-E3253.D. R. Baker, M. W. Verbrugge and W. Gu, J. Electrochem. Soc., 166 (4) A521-A531 (2019).

On regularization results for a two-dimensional nonlinear time-fractional inverse diffusion problem

Ill-posed inverse problems arise in a variety of practically important applications including image restoration, medicine, ecology as well as many other branches of pure and applied sciences. In this paper, we consider an ill-posed inverse problem for a two-dimensional nonlinear time-fractional inverse diffusion model involving the Caputo time-fractional derivative as follows:{Dtβu(x,y,t)=−a(x)(ux(x,y,t)+uy(x,y,t))+ℓ(x,y,t,u(x,y,t)),x>0,y>0,t>0,u(x,y,0)=0,x>0,y>0,limx→∞⁡u(x,y,t)=0,y>0,t>0,u(1,y,t)=u1(y,t),y>0,t>0, where β∈(0,1) is fixed, a is a space-dependent diffusion coefficient, ℓ is a nonlinear function satisfying a Lipschitz condition, and the data u1 is given approximately. We want to determine u(x,y,t) for 0≤x<1. There exists a vast literature on regularization results related to the considered problem. We are concerned with regularization results for the nonlinear case. The first regularization results in such a case were studied by Tuan, Hoan and Tatar in [17]. While these results apply to a constant diffusivity coefficient in the one-dimensional setting, the results of Vo and co-worker in [18], more generally, apply to the case of the space-dependent diffusion coefficient in the two-dimensional setting. In all these papers relatively strong a priori assumptions on the regularity of the exact solution are made. In this work, we weaken such a priori assumptions via two regularization strategies based on the integral equation method. We obtain several explicit error estimates including an error estimate of the Hölder-Logarithmic type for all x∈[0,1), which can be seen as the improvement and the generalization of error estimates presented by Zheng and Wei in [20,21]. Eventually, several numerical examples are presented for the purpose of illustrating the theoretical results.

Impact of hydrodynamic dispersion on mixing-induced reactions under radial flows

Mixing-induced reaction fronts play a key role in a range of subsurface processes. In many applications, reactive fronts develop under radial flows, where a reactant is injected and displaces another. Analytical solutions for reactive front dynamics under radial flows have been derived under the assumption of a constant diffusion coefficient. However, the impact of mechanical dispersion still remains unexplored. We investigate this question here by deriving approximate analytical expressions for the reaction front properties as a function of time, dispersion length and Péclet/Damköhler number, as well as from corresponding numerical simulations. Our results indicate that mechanical dispersion leads to a more advanced front and enhanced reaction rate, compared to the dispersion-free scenario. This leads to new scaling laws for the front position, width and reaction rate. We discuss the implications of these findings for field conditions over a range of temporal and spatial scales. Under most realistic scenarios, dispersion is expected to be dominant over diffusion, suggesting a broad relevance of these results.

Numerical Simulation of Hydrogen Diffusion in Cement Sheath of Wells Used for Underground Hydrogen Storage

The negative environmental impact of carbon emissions from fossil fuels has promoted hydrogen utilization and storage in underground structures. Hydrogen leakage from storage structures through wells is a major concern due to the small hydrogen molecules that diffuse fast in the porous well cement sheath. The second-order parabolic partial differential equation describing the hydrogen diffusion in well cement was solved numerically using the finite difference method (FDM). The numerical model was verified with an analytical solution for an ideal case where the matrix and fluid have invariant properties. Sensitivity analyses with the model revealed several possibilities. Based on simulation studies and underlying assumptions such as non-dissolvable hydrogen gas in water present in the cement pore spaces, constant hydrogen diffusion coefficient, cement properties such as porosity and saturation, etc., hydrogen should take about 7.5 days to fully penetrate a 35 cm cement sheath under expected well conditions. The relatively short duration for hydrogen breakthrough in the cement sheath is mainly due to the small molecule size and high hydrogen diffusivity. If the hydrogen reaches a vertical channel behind the casing, a hydrogen leak from the well is soon expected. Also, the simulation result reveals that hydrogen migration along the axial direction of the cement column from a storage reservoir to the top of a 50 m caprock is likely to occur in 500 years. Hydrogen diffusion into cement sheaths increases with increased cement porosity and diffusion coefficient and decreases with water saturation (and increases with hydrogen saturation). Hence, cement with a low water-to-cement ratio to reduce water content and low cement porosity is desirable for completing hydrogen storage wells.

Determination of constant diffusion coefficient for error function solution of diffusion equation using film-roll method

The error function solution of the diffusion equation with the constant surface concentration was derived by the heat kernel to determine the constant diffusion coefficient for diffusion satisfying the infinite dye-bath condition. For the sublimation diffusion of disperse dye in paste into PET film using the film-roll method, the constant surface concentration was determined from the concentration distribution, and the diffusion coefficients of each layer were obtained by the constant surface concentration from the error function solution. When comparing the diffusion coefficients between the layers and comparing the mean diffusion coefficients for different times at a specific temperature, the constant surface concentrations determined from the quadratic regression curves for concentration–distance plots were more appropriate than those determined from the steady-state concentration distributions, which was also confirmed by the plots of concentrations obtained from the error function solution. At a specific temperature, the average of the mean diffusion coefficients obtained by the constant surface concentration of the quadratic regression curve at three specific times matched well with the constant total-amount diffusion coefficient obtained by the slope of the linear regression line for the plot of total amount against square root of time, which was confirmed by their Arrhenius plots.

Nonparametric adaptive estimation for interacting particle systems

AbstractWe consider a stochastic system of interacting particles with constant diffusion coefficient and drift linear in space, time‐depending on two unknown deterministic functions. Our concern here is the nonparametric estimation of these functions from a continuous observation of the process on for fixed and large . We define two collections of projection estimators belonging to finite‐dimensional subspaces of . We study the ‐risks of these estimators, where the risk is defined either by the expectation of an empirical norm or by the expectation of a deterministic norm. Afterwards, we propose a data‐driven choice of the dimensions and study the risk of the adaptive estimators. The results are illustrated by numerical experiments on simulated data.

Reduced order modeling for elliptic problems with high contrast diffusion coefficients

We consider a parametric elliptic PDE with a scalar piecewise constant diffusion coefficient taking arbitrary positive values on fixed subdomains. This problem is not uniformly elliptic, as the contrast can be arbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) that is com- monly made on parametric elliptic PDEs. We construct reduced model spaces that approximate uniformly well all solutions with estimates in relative error that are independent of the contrast level. These estimates are sub-exponential in the reduced model dimension, yet exhibiting the curse of dimensionality as the number of subdomains grows. Similar es- timates are obtained for the Galerkin projection, as well as for the state estimation and parameter estimation inverse problems. A key ingredient in our construction and analysis is the study of the convergence towards limit solutions of stiff problems when diffusion tends to infinity in certain domains.

Fourth-order multiple-relaxation-time lattice Boltzmann model and equivalent finite-difference scheme for one-dimensional convection-diffusion equations.

In this paper, we first develop a fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is used. We also perform the Chapman-Enskog analysis to recover the CDE from the MRT-LB model. Then an explicit four-level finite-difference (FLFD) scheme is derived from the developed MRT-LB model for the CDE. Through the Taylor expansion, the truncation error of the FLFD scheme is obtained, and at the diffusive scaling, the FLFD scheme can achieve the fourth-order accuracy in space. After that, we present a stability analysis and derive the same stability condition for the MRT-LB model and FLFD scheme. Finally, we perform some numerical experiments to test the MRT-LB model and FLFD scheme, and the numerical results show that they have a fourth-order convergence rate in space, which is consistent with our theoretical analysis.

Convergence for an Immersed Finite Volume Method for Elliptic and Parabolic Interface Problems

In this article we analyze an immersed interface finite volume method for second order elliptic and&amp;nbsp;&amp;nbsp; parabolic interface problems. We show the optimal convergence of the elliptic interface problem in L^2 and energy norms.&#x0D; For the parabolic interface problem, we prove the optimal order in L^2 and energy norms for piecewise constant and variable diffusion coefficients respectively. Furthermore, for the elliptic interface problem, we demonstrate super convergence at element nodes when the diffusion coefficient&amp;nbsp; is a piecewise constant.&amp;nbsp; Numerical examples&amp;nbsp; are also provided to confirm the optimal error estimates.

Well‐posedness for a diffusion–reaction compartmental model simulating the spread of COVID‐19

This paper is concerned with the well‐posedness of a diffusion–reaction system for a susceptible‐exposed‐infected‐recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio‐temporal spread of the COVID‐19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions.

Revealing the hidden signature of fault slip history in the morphology of degrading scarps

Active faults accommodate tectonic plate motion through different slip modes, some stable and aseismic, others characterized by the occurrence of large earthquakes after long periods of inactivity. Although the slip mode estimation is of primary importance to improve seismic hazard assessment, this parameter inferred today from geodetic observations needs to be better constrained over many seismic cycles. From an analytical formulation developed for analyzing fault scarp formation and degradation in loosely consolidated material, we show that the final topographic shape generated by one earthquake rupture or by creep (i.e., continuous slip) deviates by as much as 10–20%, despite a similar cumulated slip and a constant diffusion coefficient. This result opens up the theoretical possibility of inverting, not only the cumulated slip or averaged slip rate, but also the number of earthquakes and their sizes from scarp morphologies. This approach is all the more relevant as the number of rupture events is limited. Estimating the fault slip history beyond a dozen earthquakes becomes very difficult as the effect of erosion on scarp morphology prevails. Our modeling also highlights the importance of trade-offs between fault slip history and diffusive processes. An identical topographic profile can be obtained either with a stable fault creep associated with rapid erosion, or a single earthquake rupture followed by slow erosion. These inferences, derived from the simplest possible diffusion model, are likely to be even more pronounced in nature.

Enhanced electrochemical redox kinetics of La0.6Sr0.4Co0.2Fe0.8O3 in reversible solid oxide cells

Due to its high conversion efficiency in fuel-to-power and power-to-fuel modes, reversible solid oxide cell (RSOC) is widely regarded as a promising device. In this study, La0.6Sr0.4Co0.2Fe0.7M0.1O3 (M=Fe, Mo, Nb) oxides are prepared and composited with Sm0.2Ce0.8O2-(Li0.67Na0.33)2CO3 to form reversible single component cell (RSCC). Among them, La0.6Sr0.4Co0.2Fe0.7Mo0.1O3 (LSCFM) shows the highest oxygen vacancy concentration, further promoting catalytic activity towards oxygen reduction reaction (ORR) and hydrogen oxidation reaction (HOR). Besides, Mo or Nb doping enhances the surface reaction rate constant (kchem) and oxygen chemical bulk diffusion coefficient (Dchem) at the temperature range of 550–700 °C. LSCFM shows the highest kchem of 39.3 × 10−5 cm s−1, approximate 3.5 times as 11.1 × 10−5 cm s−1 for La0.6Sr0.4Co0.2Fe0.8O3 (LSCF), and Dchem for LSCFM is 31.1 × 10−6 cm2 s−1, approximate 2.5 times as 12.4 × 10−6 cm2 s−1 for LSCF at 700 °C. Furthermore, the rate determining steps (RDS) of ORR and HOR for LSCFM-based cell are the reduction of oxygen atoms to O− and the charge transfer reaction, respectively. Under SOFC/SOEC reversible operation, the RSCC composed of LSCFM exhibits the best cell performance. In the solid oxide fuel cell mode, the maximum power density reaches 236.2 mW cm−2 and in the solid oxide electrolysis cell mode the current density at 1.3 V is -312.4 mA cm−2.