Finite Diffusion Research Articles

An Investigation of Chaotic Diffusion in a Family of Hamiltonian Mappings Whose Angles Diverge in the Limit of Vanishingly Action

The chaotic diffusion for a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by using the solution of the diffusion equation. The system is described by a two-dimensional mapping for the variables action, $I$, and angle, $\theta$ and controlled by two control parameters: (i) $\epsilon$, controlling the nonlinearity of the system, particularly a transition from integrable for $\epsilon=0$ to non-integrable for $\epsilon\ne0$ and; (ii) $\gamma$ denoting the power of the action in the equation defining the angle. For $\epsilon\ne0$ the phase space is mixed and chaos is present in the system leading to a finite diffusion in the action characterized by the solution of the diffusion equation. The analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures.

Monte Carlo Methods for Reactor Kinetic Simulations

The use of the Monte Carlo (MC) method for space-time reactor kinetics is expected to be much more accurate than the presently used deterministic methods largely based on few-group diffusion theory. However, the development of the MC method for space-time reactor kinetics poses challenges because of the vastly different timescales of neutrons and delayed neutron precursors and their vastly different populations that also change with time by several orders of magnitude. In order to meet these challenges in MC-based space kinetics, we propose various new schemes such as deterministic decay of precursors in each time step, adjustment of weights of neutrons and precursors for population control, use of mean number of secondaries per collision, and particle splitting/Russian roulette to reduce the variance in neutron power. The efficacy of these measures is first tested in a simpler point-kinetics version of the MC method against analytical or accurate numerical solutions of point-kinetics equations. The ideas are then extended to space-dependent MC kinetics and are validated against a transport theory/MC transient benchmark. We have also tested our methods by comparison with results of realistic space-time kinetics benchmarks/studies involving multiregion reactors, energy dependence, movement of control rods, and feedback—most of which are based on few-group diffusion theory treated by the finite difference method. To facilitate exact comparison with such benchmarks, we have implemented the schemes described above for space-time reactor kinetics based on finite difference diffusion MC, a method developed by us earlier in a different context.

Modeling multicomponent fuel droplet vaporization with finite liquid diffusivity using Coupled Algebraic-DQMoM with delumping

Multicomponent fuel droplet vaporization models for use in combustion CFD codes often prioritize computational efficiency over model complexity. This leads to oversimplifying assumptions such as single component droplets or infinite liquid diffusivity. The previously developed Direct Quadrature Method of Moments (DQMoM) with delumping model demonstrated a computationally efficient and accurate approach to solve for every discrete species in a well-mixed vaporizing multicomponent droplet. To expand the method to less restrictive cases, a new solution technique is presented called the Coupled Algebraic-Direct Quadrature Method of Moments (CA-DQMoM). In contrast to previous moment methods for droplet vaporization, CA-DQMoM solves for the evolution of two liquid distributions by coupling a monovariate, homogeneous DQMoM approach with additional algebraic moment equations, allowing for a more complex droplet vaporization model with finite rates of liquid diffusion to be solved with computational efficiency. To further decrease computational expense, an approximation that employs the same nodes for both distributions can be used in certain cases. Finally, a delumping technique is adapted to the finite diffusivity model to reconstruct discrete species information at minimal computational cost. The model is proven to be accurate relative to a full discrete component model for both a kerosene droplet comprised of 36 species and a multicomponent droplet of 200 species while maintaining the computational efficiency of continuous thermodynamics models. The combined accuracy and computational efficiency demonstrated by the CA-DQMoM with delumping model for a multicomponent fuel droplet with finite liquid diffusivity makes it ideal for incorporation into CFD models for complex combustion process.

“Virtual density” and traditional boundary perturbation theories: Analytic equivalence and numeric comparison

We compare and contrast “virtual density” perturbation theory with the traditional boundary perturbation theory developed by Pomraning, Larsen, and Rahnema in the context of diffusion theory. First, after reviewing that literature, we mathematically prove that virtual density perturbations and traditional boundary perturbations are precisely equivalent for arbitrary 1-D problems, which constitute non-uniform isotropic expansions. We also mathematically prove that these two perturbation theories are equivalent for 2-D boundary shift problems, which constitute non-uniform anisotropic expansions. Extension of this proof to swellings or 3-D problems is straightforward. We compare the two theories numerically for a series of alternating uranium and sodium 1-D slabs in finite difference diffusion, and we show that virtual density theory predicts reactivities much more accurately and efficiently than traditional boundary perturbation theory. Boundary perturbation theory is often very inaccurate on a coarse mesh but converges to the virtual density solution as the mesh becomes finer. We also compare the two theories for axial assembly swelling in an abbreviated FFTF benchmark with a coarse mesh. Here we find that reactivity coefficients obtained via virtual density perturbation theory agree with reference solutions to within 0.1%, while those obtained via boundary perturbation theory exhibit sporadic accuracy – sometimes in the range of 1–5% error, more frequently in the range 5–20% error, and occasionally well over 100% error in control rod assemblies. We conclude that although virtual density perturbation theory and boundary perturbation theory are analytically equivalent, boundary perturbations in diffusion theory are often thwarted in coarse mesh finite difference solutions due to inaccurate flux gradients along mesh cell surfaces in heterogeneous cores.

The “virtual density” theory of neutronics

Sustainable nuclear energy will likely require fast reactors to complement the current light water reactor paradigm. In particular, breed-and-burn sodium fast reactors (SFRs) offer a unique combination of fuel cycle and power density features. Unfortunately, large breed-and-burn SFRs are plagued by positive sodium void worth. In order to mitigate this drawback, one must quantify various sources of negative reactivity feedback, among which geometry distortions (bowing and flowering of fuel assemblies) are often dominant. These distortions arise mainly from three distinct physical phenomena: irradiation swelling, thermal swelling, and seismic events.Distortions are notoriously difficult to model, because they break symmetry and repeating patterns. Currently, no efficient and fully general method exists for computing neutronic effects of distortions. Computing them directly via diffusion would require construction of exotic meshes that seldom exist within the neutronics community. Many deterministic transport methods are geometrically flexible but tightly constrained by assumptions of symmetry and repeating patterns. Monte Carlo offers the only high-fidelity approach to arbitrary geometry, but resolving minute reactivities and flux shift tallies within large heterogeneous cores requires CPU years per case and is thus prohibitively expensive. Currently, the most widely-used methods consist of various approximations involving weighting the uniform radial swelling reactivity coefficient by the power distribution. These approximations agree fairly well with experimental data for flowering in some cores, but they are not fully general and cannot be trusted to work for arbitrary distortions in generic cores. Boundary perturbation theory, developed in the 1980s, is fully general and mathematically rigorous, but it is inaccurate for coarse mesh diffusion and has never been applied in industry.Our solution is the “virtual density” theory of neutronics, which alters material density (isotropically or anisotropically) instead of explicitly changing geometry. While geometry is discretized, material densities occupy a continuous domain; this allows density changes to obviate the greatest computational challenges of geometry changes. Although primitive forms of this theory exist in Soviet literature, they are only applicable to cases in which entire cores swell uniformly. Thus, we conceive a much more general and pragmatic form of virtual density theory to model non-uniform and localized geometry distortions via perturbation theory. In this work, we develop and implement virtual density theory entirely in the context of diffusion.In order to efficiently validate virtual density perturbation theory, we conceive the “virtual mesh” method for diffusion theory. This new method involves constructing a slightly perturbed “fake” mesh that produces correct first-order reactivity and flux shifts due to anisotropic swelling or expansion of individual mesh cells. First order reactivities computed on a virtual mesh agree with continuous energy Monte Carlo to within 1σ uncertainty.We validate virtual density theory via the virtual mesh method in 3-D coarse mesh models of the Fast Flux Test Facility (FFTF) and Jo¯yo¯ benchmarks using the MATLAB-PETSc-SLEPc (MaPS) multigroup finite difference diffusion code, which we developed for this purpose. We model a panoply of non-uniform anisotropic swelling scenarios, including axial swelling of individual assemblies, axial swelling of each mesh cell in proportion to its fission power, and radial core flowering with arbitrary axial dependence. In 3-D coarse mesh Cartesian cores with explicit coolant gaps, we model individual assembly motion, assembly row motion with arbitrary axial dependence, and assembly row “s-shape” bowing. In all cases, we find that virtual density perturbation theory predicts reactivity coefficients that agree with virtual mesh reference cases to within 0.01%. These reactivity coefficients are two to four orders of magnitude more accurate than those computed via boundary perturbation theory. In general, this virtual density perturbation method can precisely predict reactivity coefficients due to anisotropic swelling or expansion of any core region in any direction.

Cohesive zone modelling of hydrogen induced cracking on the interface of clad steel pipes

A coupled finite element hydrogen diffusion and cohesive zone modelling approach has been applied to simulate hydrogen induced fracture initiation in a hot rolled bonded clad steel pipe. The results are compared to experimental fracture mechanical testing in air and under in situ electrochemical hydrogen charging. A best fit to the experimental fracture initiation toughness value in air was achieved for an initial cohesive stiffness kn = 4·106 MPa/mm and a critical cohesive stress σc = 1210 MPa. For simulating under hydrogen influence, the hydrogen induced lowering of the cohesive energy was computed both in terms of the lattice concentration and the total concentration. Two different formulations for calculating the dislocation trap density were considered. The simulated results revealed that both hydrogen in lattice and hydrogen trapped at dislocations can be responsible for the observed hydrogen induced reduction in fracture initiation toughness. The choice of trap density formulation appeared significant only under the assumption that both lattice and trapped hydrogen infer an influence on the hydrogen induced lowering of the cohesive strength. Further effort is needed to provide a reliable description of the interface hydrogen content and distribution.

Application of numerical inverse method in calculation of composition-dependent interdiffusion coefficients in finite diffusion couples

The previously developed numerical inverse method was applied to determine the composition-dependent interdiffusion coefficients in single-phase finite diffusion couples. The numerical inverse method was first validated in a fictitious binary finite diffusion couple by pre-assuming four standard sets of interdiffusion coefficients. After that, the numerical inverse method was then adopted in a ternary Al-Cu-Ni finite diffusion couple. Based on the measured composition profiles, the ternary interdiffusion coefficients along the entire diffusion path of the target ternary diffusion couple were obtained by using the numerical inverse approach. The comprehensive comparisons between the computations and the experiments indicate that the numerical inverse method is also applicable to high-throughput determination of the composition-dependent interdiffusion coefficients in finite diffusion couples.

Coorbital thermal torques on low-mass protoplanets

Using linear perturbation theory, we investigate the torque exerted on a low-mass planet embedded in a gaseous protoplanetary disc with finite thermal diffusivity. When the planet does not release energy into the ambient disc, the main effect of thermal diffusion is the softening of the enthalpy peak near the planet, which results in the appearance of two cold and dense lobes on either side of the orbit, of size smaller than the thickness of the disc. The lobes exert torques of opposite sign on the planet, each comparable in magnitude to the one-sided Lindblad torque. When the planet is offset from corotation, the lobes are asymmetric and the planet experiences a net torque, the `cold' thermal torque, which has a magnitude that depends on the relative value of the distance to corotation to the size of the lobes $\sim\sqrt{\chi/\Omega_p}$, $\chi$ being the thermal diffusivity and $\Omega_p$ the orbital frequency. We believe that this effect corresponds to the phenomenon named `cold finger' recently reported in numerical simulations, and we argue that it constitutes the dominant mode of migration of sub-Earth-mass objects. When the planet is luminous, the heat released into the ambient disc results in an additional disturbance that takes the form of hot, low-density lobes. They give a torque, named heating torque in previous work, that has an expression similar, but of opposite sign, to the cold thermal torque.

A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

Abstract In this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.

Spin pumping into superconductors: A new probe of spin dynamics in a superconducting thin film

Spin pumping refers to the microwave-driven spin current injection from a ferromagnet into the adjacent target material. We theoretically investigate the spin pumping into superconductors by fully taking account of impurity spin-orbit scattering that is indispensable to describe diffusive spin transport with finite spin diffusion length. We calculate temperature dependence of the spin pumping signal and show that a pronounced coherence peak appears immediately below the superconducting transition temperature Tc, which survives even in the presence of the spin-orbit scattering. The phenomenon provides us with a new way of studying the dynamic spin susceptibility in a superconducting thin film. This is contrasted with the nuclear magnetic resonance technique used to study a bulk superconductor.

Magneto-thermal reconnection processes, related mode momentum and formation of high energy particle populations

In the context of a two-fluid theory of magnetic reconnection, when the longitudinal electron thermal conductivity is relatively large, the perturbed electron temperature tends to become singular in the presence of a reconnected field component and an electron temperature gradient. A finite transverse thermal diffusivity removes this singularity while a finite ‘inductivity’ can remove the singularity of the relevant plasma displacement. Then (i) a new ‘magneto-thermal’ reconnection producing mode, is found with characteristic widths of the reconnection layer remaining significant even when the macroscopic distances involved are very large; (ii) the mode phase velocities can be both in the direction of the electron diamagnetic velocity as well in the opposite (ion) direction. A numerical solution of the complete set of equations has been carried out with a simplified analytical reformulation of the problem. A sequence of processes is analyzed to point out that high-energy particle populations can be produced as a result of reconnection events. These involve mode-particle resonances transferring energy of the reconnecting mode to a superthermal ion population and the excitation of lower hybrid waves that can lead to a significant superthermal electron population. The same modes excited in axisymmetric (e.g. toroidal) confinement configurations can extract angular momentum from the main body of the plasma column and thereby sustain a local ‘spontaneous rotation’ of it.

Easy method for measurement of environmental impedance and superconducting phase fluctuations in one-dimensional arrays of Josephson junctions

We conduct microwave impedance measurements on a one-dimensional (1D) array of Josephson junctions to experimentally determine the Josephson inductance and shunt resistance of the constituent junctions. The effective Josephson energy provides an estimate of the environmental impedance, which is greatly increased due to phase fluctuations in the neighboring junctions. This enhancement is attributed to the charge solitons in the 1D system. In general, the environmental impedance is dominated by the junction’s normal resistance in the superconducting phase coherent regime, but overwhelmed by zero-bias resistance and differential resistance, respectively, in the Coulomb blockaded regime and in the phase fluctuating regime. The change in phase fluctuations owing to a dc bias agrees with the finite temperature phase diffusion model.

Elucidating effects of cell architecture, electrode material, and solution composition on overpotentials in redox flow batteries

An improved method for quantitative measurement of the charge transfer, finite diffusion, and ohmic overpotentials in redox flow batteries using electrochemical impedance spectroscopy is presented. The use of a pulse dampener in the hydraulic circuit enables the collection of impedance spectra at low frequencies with a peristaltic pump, allowing the measurement of finite diffusion resistances at operationally relevant flow rates. This method is used to resolve the rate-limiting processes for the V2+/V3+ redox couple on carbon felt and carbon paper electrodes in the vanadium redox flow battery. Carbon felt was limited by both charge transfer and ohmic resistance, while carbon paper was limited by charge transfer, finite diffusion, and ohmic resistances. The influences of vanadium concentration and flow field design also are quantified.

Hybrid Multi-level solvers for discontinuous Galerkin finite element discrete ordinate diffusion synthetic acceleration of radiation transport algorithms

This paper examines two established preconditioners which were developed to accelerate the solution of discontinuous Galerkin finite element method (DG-FEM) discretisations of the elliptic neutron diffusion equation. They are each presented here as a potential way to accelerate the solution of the Modified Interior Penalty (MIP) form of the discontinuous diffusion equation, for use as a diffusion synthetic acceleration (DSA) of DG-FEM discretisations of the neutron transport equation. The preconditioners are both two-level schemes, differing in the low-level space utilised. Once projected to the low-level space a selection of algebraic multigrid (AMG) preconditioners are utilised to obtain a further correction step, these are therefore “hybrid” preconditioners. The first preconditioning scheme utilises a continuous piece-wise linear finite element method (FEM) space, while the second uses a discontinuous piece-wise constant space. Both projections are used alongside an element-wise block Jacobi smoother in order to create a symmetric preconditioning scheme which may be used alongside a conjugate gradient algorithm. An eigenvalue analysis reveals that both should aid convergence but the piece-wise constant based method struggles with some of the smoother error modes. Both are applied to a range of problems including some which are strongly heterogeneous. In terms of conjugate gradient (CG) iterations needed to reach convergence and computational time required, both methods perform well. However, the piece-wise linear continuous scheme appears to be the more effective of the two. An analysis of computer memory usage found that the discontinuous piece-wise constant method had the lowest memory requirements.

English

The detailed mathematical study of the recent paper by Sajjadi, Hunt and Drullion (2014) is pre- sented. The mathematical developement considered by them, for unsteady growing monochro- matic waves is also extended to Stokes waves. The present contribution also demonstrates agree- ment with the pioneering work of Belcher and Hunt (1993) which is valid in the limit of the complex part of the wave phase speed \c_i \downarrow 0. It is further shown that the energy-transfer parameter and the surface shear stress for a Stokes wave reverts to a monochromatic wave when the second harmonic is excluded. Furthermore, the present theory can be used to estimate the amount of energy transferred to each component of nonlinear surface waves on deep water from a turbulent shear flow blowing over it. Finally, it is demonstrated that in the presence of turbulent eddy viscosity the Miles (1957) critical layer does not play an important role. Thus, it is concluded that in the limit of zero growth rate the effect of the wave growth arises from the elevated critical layer by finite turbulent diffusivity, so that the perturbed flow and the drag force is determined by the asymmetric and sheltering flow in the surface shear layer and its matched interaction with the upper region.

Neumann boundary controls for finite diffusion process

ABSTRACTThree types of boundary controls are considered: proportional control, feed-forward control and integrated-by-past-values control. They stabilise the diffusion process exponentially at the desired reference on the controlled boundary. The process is actuated through the Neumann boundary while on other boundaries it is given with the Dirichlet data and, in higher dimensions, also with the homogeneous Neumann boundary condition. Two systems, deterministic and stochastic, are compared as an ideal and real description of a physical system. Disturbance in the stochastic system is induced by a white noise on the controlled boundary. Physically, this noise represents different types of uncertainties like a side-reaction mass flux and other uncertainties that are not part of a deterministic model. At first the proportional control is analysed on the deterministic system and then it is represented in the feedback-integrated past controls form, and finally the developed controls are applied on the stochastic system, which boundary is disturbed with an unmeasured white noise. The regulation error that emerges in stochastic control is analysed and proved to be a zero-mean, bounded-variance Gaussian variable that is correlated temporarily. As an example of such a system, any electrochemical reduction or oxidation process where diffusion mass transfer is important can be considered.

Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity

Consider the dynamics of a thin film flowing down a heated substrate. The substrate heating generates a temperature distribution on the free surface, which in turn induces surface-tension gradients and corresponding thermocapillary stresses that affect the free surface and therefore the fluid flow. We study here the effect of finite substrate thermal diffusivity on the film dynamics. Linear stability analysis of the full Navier-Stokes and heat transport equations indicates if the substrate diffusivity is sufficiently small, the film becomes unstable at a finite wavelength and at a Reynolds number smaller than that predicted in the long-wavelength limit. This property is captured in a reduced-order system of equations derived using a weighted-residual integral-boundary-layer method. This reduced-order model is also used to compute the bifurcation diagrams of solution branches connecting the trivial flat film to travelingwaves including solitary pulses. The effect of finite diffusivity is to separate a simultaneous Hopf-transcritical bifurcation into its individual component bifurcations. The appropriate Hopf bifurcation then connects only to the solution branch of negative-hump pulses, with wave speed less than the linear wave speed, while the branch of positive-single-hump pulses merges with the branch of positive-two-hump pulses at a supercritical Reynolds number. In the regime where finite-wavelength instability occurs, there exists a Hopf-bifurcation pair connected by a branch of periodic solutions, whose period cannot be increased indefinitely. Numerical simulation of the reduced-order system shows the development of a train of coherent structures, each of which resembles a stationary positive-hump pulse, and, in the regime of finite-wavelength instability, wavelength selection and saturation to periodic traveling waves.

Radial Growth in 2D Revisited: The Effect of Finite Density, Binding Affinity, Reaction Rates, and Diffusion

Dendrite growth of metal patches on colloidal particles shows a variety of structures depending on the preparation conditions. The morphology of these patches suggests a cross‐over from a reaction to a diffusion limited growth, implicating diffusion on the particle surface. Interestingly, the morphological and optical characteristics of the patches continuously change between two limiting behaviors. To understand this growth process, extensive simulations are performed, studying the fractal dimension and the dynamics of growth of a patch on a particle of a finite size, as a function of the initial density and the binding affinity of diffusing tracers. Several important growth regimes are characterized that enable to optimize the pathway for the synthesis of optically active, patchy particles.

Tracer diffusion in a polymer gel: simulations of static and dynamic 3D networks using spherical boundary conditions

We have investigated an alternative to the standard periodic boundary conditions for simulating the diffusion of tracer particles in a polymer gel by performing Brownian dynamics simulations using spherical boundary conditions. The gel network is constructed by randomly distributing tetravalent cross-linking nodes and connecting nearest pairs. The final gel structure is characterised by the radial distribution functions, chain lengths and end-to-end distances, and the pore size distribution. We have looked at the diffusion of tracer particles with a wide range of sizes, diffusing in both static and dynamic networks of two different volume fractions. It is quantitatively shown that the dynamical effect of the network becomes more important in facilitating the diffusional transport for larger particle sizes, and that one obtains a finite diffusion also for particle sizes well above the maximum in the pore size distribution.

Characterization of Interface State Density of SiO2/SiC (000-1) Based on Oxygen Concentration at the Interface during Thermal Oxidation

Inferior interfacial electrical properties of thermally grown SiO2 films on SiC, compared to those on Si, is probably brought by carbon atoms in the substrates. Although few direct evidence f has been reported, it is quite natural to consider that some carbon atoms exist as impurities at the interface as long as we assume a finite diffusion coefficient for carbon atoms. It is well known that we cannot avoid formation of high density of interface states in dry oxide interface. Various processes for interface state reduction have been reported, such as FGA for instance. The mechanism is not clear, however, it is apparently different from termination of Si dangling bonds, which is the case for SiO2/Si, since FGA for SiO2/SiC requires much higher temperature than Si case. For reduction of interface states, it is important to quickly remove carbon atoms out of the film to the ambient. In this paper, we focus on the role of oxygen atoms, in addition to being oxidants, as a carbon removers by generating CO molecules. We report how the interface states density is related to oxygen concentration at the interface.The substrate in our experiments was 4H-SiC(000-1) epi-wafers with the miscut angle of 4°. After modified RCA cleaning, thermal oxidation was carried out at 900-1200°C in O2 ambient. The oxide thickness was characterized by spectroscopic ellipsometry. Then the Al gate electrodes were fabricated on top of the oxide films, and we characterized interface state density by C-V measurement. From the oxidation date, that is thickness vs. oxidation time, we have extracted the linear rate and the parabolic rate constants by fitting with the Deal-Grove model. Using these constants, we have calculated the oxygen concentration at the interface, which should be determined with oxidation rate at the interface and the flux of the oxygen diffusion from the ambient to the interface.Figure 1 is the oxide thickness vs. oxidation time with variation of oxidation temperature. The dotted lines are the fitting curves. The experimental data are well fitted by the Deal-Grove model, indicating that thermal oxidation of SiC can be basically explained by diffusion of oxidant species through oxide film and oxidation reaction at the interface, as same as for Si oxidation. Figure 2 is the calculated oxygen density at the interface as a function of oxide thickness. It is natural that the oxygen density decreases as the oxide becomes thicker due to decrease in oxygen diffusion to the interface. The decrease in the oxygen density for higher oxidation temperature is due to decrease in the oxygen solubility. Figure 3 shows the summary of the relation between interface state density and the calculated oxygen density at the interface. In this figure the interface state density was evaluated at 1.0 eV below the bottom of SiC conduction band, which is usually observed for C-face substrate and strongly affect the reliability properties, such as flatband voltage shift. It is clear that the interface state density decreases as the oxygen density increases. It could be explained by that oxygen atoms act as the carbon remover, if we assume that the interface states are carbon related. Decrease in the interface sate density for higher oxidation temperature can be explained by fact that temperature dependence of the rate of CO formation is larger than that of SiO2 formation, that is supplied oxygen atoms are more used for carbon removal at higher temperature. Figure 1