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NASA NTRS1986

Monte Carlo study of a model of diffusion-controlled reactions

The Monte Carlo method is used to perform averages over sink configurations in the present study of diffusion-controlled reactions occurring between solute particles and immobile spherical sinks. In order to determine the average steady state solute concentration profile in a locally perturbed solution for sink volume fractions phi of less than 0.3, the diffusion equation in the monopolar plus dipolar approximation of diffusive couplings between the sinks is solved numerically. The Monte Carlo method is shown to be the most accurate and efficient in the phi = 0.001-0.1 region, where a system of only 25 sinks suffices and the monopolar approximation alone is sufficiently accurate.

Beenakker, C. W. J.↗

NASA NTRSDate not supplied

Zircon (U-Th)/He Impact Crater Thermochronometry and the Effects of Shock Microstructures on Helium Diffusion Kinetics

Accurate and precise age determination of impact cratering events remains challenging and often contentious; less than half of all known craters are regarded as accurately and precisely dated. Zircon (U-Th)/He (ZHe) dating of impactites can be employed to date medium to large impact structures as ZHe ages can be fully reset in minutes at T >1000°C, a plausible scenario in the central melt pool. In contrast, complete resetting of ZHe at 200-300°C, encountered near the crater margins or due to post-impact hydrothermal overprinting, may take >103-6 years. To test the reliability of ZHe impact dating, we have quantified the effects of shock-induced microstructures on helium diffusion kinetics in well-characterized variably shocked zircon. We investigated samples from two impact structures, the Chicxulub multi-ring crater and Ries complex crater, to compare diffusion kinetics from structures with different size, age, and hydrothermal system longevity. Shock microstructures were characterized by backscattered-electron imaging prior to determining the He diffusion kinetics by prograde and retrograde fractional-release experiments via light-bulb furnace with incremental step-heating (10°C) from 300°C to 600°C. Next, we examine the internal interconnectivity and sizes of the diffusion domains by EBSD. While we found that zircon with few shock microstructures exhibited no marked deviation from helium diffusion kinetics of undamaged zircon, zircon grains with planar microstructures and granular textures are characterized by a dramatic decrease in helium retentivity due to the reduction in the effective domain size and the introduction of interconnected fast diffusion pathways. A subset of grains were ZHe dated and showed that less deformed grains yielded a weighted mean age within error of the accepted impact ages, while the grains with planar microstructures or granular textures gave systematically younger ages. These new diffusion data and ZHe ages demonstrate that highly shocked grains are unsuitable for ZHe impact crater dating. Therefore, detailed characterization of impact-induced microstructures is critical for determining accurate ZHe impact ages and offers the possibility of investigating post-impact hydrothermal circulation.

Zircon↗

DOE OSTI2016

Oxygen and Perfusion Kinetics in Response to Fractionated Radiation Therapy in FaDu Head and Neck Cancer Xenografts Are Related to Treatment Outcome

Purpose: To test whether oxygenation kinetics correlate with the likelihood for local tumor control after fractionated radiation therapy. Methods and Materials: We used diffuse reflectance spectroscopy to noninvasively measure tumor vascular oxygenation and total hemoglobin concentration associated with radiation therapy of 5 daily fractions (7.5, 9, or 13.5 Gy/d) in FaDu xenografts. Spectroscopy measurements were obtained immediately before each daily radiation fraction and during the week after radiation therapy. Oxygen saturation and total hemoglobin concentration were computed using an inverse Monte Carlo model. Results: First, oxygenation kinetics during and after radiation therapy, but before tumor volumes changed, were associated with local tumor control. Locally controlled tumors exhibited significantly faster increases in oxygenation after radiation therapy (days 12-15) compared with tumors that recurred locally. Second, within the group of tumors that recurred, faster increases in oxygenation during radiation therapy (day 3-5 interval) were correlated with earlier recurrence times. An area of 0.74 under the receiver operating characteristic curve was achieved when classifying the local control tumors from all irradiated tumors using the oxygen kinetics with a logistic regression model. Third, the rate of increase in oxygenation was radiation dose dependent. Radiation doses ≤9.5 Gy/d did not initiate an increase in oxygenation, whereas 13.5 Gy/d triggered significant increases in oxygenation during and after radiation therapy. Conclusions: Additional confirmation is required in other tumor models, but these results suggest that monitoring tumor oxygenation kinetics could aid in the prediction of local tumor control after radiation therapy.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

NASA NTRSDate not supplied

Surface Radiation Budget (SRB) Release 2 Shortwave Monthly Data in Native Format (SRB_REL2_SW_MONTHLY)

The SRB data include the average upward and downward fluxes, photosynthetically active radiative flux, aerosol and cloud optical depth, cloud fraction, and solar zenith angle at three hourly intervals for each day for the entire globe between 07/01/1983 and 10/31/1995. These parameters were derived with the Shortwave algorithm of the NASA World Climate Research Programme /Global Energy and Water-Cycle Experiment (WCRP/GEWEX) Surface Radiation Budget (SRB) Project. [Location=GLOBAL] [Temporal_Coverage: Start_Date=1983-07-01; Stop_Date=1998-07-26] [Spatial_Coverage: Southernmost_Latitude=-90; Northernmost_Latitude=90; Westernmost_Longitude=-180; Easternmost_Longitude=180] [Data_Resolution: Latitude_Resolution=1 degree; Longitude_Resolution=Ranges from 1 degree in the tropics to 120 degrees at the poles.; Temporal_Resolution=monthly; Temporal_Resolution_Range=monthly].

DIRECT/DIFFUSE SOLAR RADIATION↗

NASA NTRSDate not supplied

Surface Radiation Budget (SRB) Release 2 Shortwave 3 hourly Data in Native Format (SRB_REL2_SW_3HRLY)

The SRB data include the average upward and downward fluxes, photosynthetically active radiative flux, aerosol and cloud optical depth, cloud fraction, and solar zenith angle measured at three hourly intervals for each day for the entire globe between 07/01/1983 and 10/31/1995. These SW surface radiative parameters were derived with the Shortwave algorithm of the NASA World Climate Research Programme/Global Energy and Water-Cycle Experiment (WCRP/GEWEX) Surface Radiation Budget (SRB) Project. [Location=GLOBAL] [Temporal_Coverage: Start_Date=1983-07-01; Stop_Date=1998-07-26] [Spatial_Coverage: Southernmost_Latitude=-90; Northernmost_Latitude=90; Westernmost_Longitude=-180; Easternmost_Longitude=180] [Data_Resolution: Latitude_Resolution=1 degree; Longitude_Resolution=Ranges from 1 degree (tropics and subtropics) to 120 degrees (the poles).; Temporal_Resolution=3 hourly; Temporal_Resolution_Range=3 hourly].

ALL SKY SW SURFACE DOWNWARD DIFFUSE FLUX↗

DOE OSTI2024

Non-locality of mean scalar transport in two-dimensional Rayleigh–Taylor instability using the macroscopic forcing method

The importance of non-locality of mean scalar transport in two-dimensional Rayleigh–Taylor Instability (RTI) is investigated. The macroscopic forcing method is utilized to measure spatio-temporal moments of the eddy diffusivity kernel representing passive scalar transport in the ensemble averaged fields. Presented in this work are several studies assessing the importance of the higher-order moments of the eddy diffusivity, which contain information about non-locality, in models for RTI. First, it is demonstrated through a comparison of leading-order models that a purely local eddy diffusivity is insufficient to capture the mean field evolution of the mass fraction in RTI. Therefore, higher-order moments of the eddy diffusivity operator are not negligible. Models are then constructed by utilizing the measured higher-order moments. It is demonstrated that an explicit operator based on the Kramers–Moyal expansion of the eddy diffusivity kernel is insufficient. An implicit operator construction that matches the measured moments is shown to offer improvements relative to the local model in a converging fashion.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2003

Algorithms for the Fractional Calculus: A Selection of Numerical Methods

Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.

Diethelm, K.↗

DOE OSTI2015

Selective Metal Exsolution in BaFe 2-y My(PO 4 ) 2 (M = Co 2+ , Ni 2+ ) Solid Solutions

The 2D-Ising ferromagnetic phase BaFe 2+ 2 (PO 4 ) 2 shows exsolution of up to one-third of its iron content (giving BaFe 3+ 1.33 (PO 4 ) 2 ) under mild oxidation conditions, leading to nanosized Fe 2 O 3 exsolved clusters. Here we have prepared BaFe 2–y My(PO 4 ) 2 (M = Co 2+ , Ni 2+ ; y = 0, 0.5, 1, 1.5) solid solutions to investigate the feasibility and selectivity of metal exsolution in these mixed metallic systems. For all the compounds, after 600 °C thermal treatment in air, a complete oxidation of Fe 2+ to Fe 3+ leaves stable M 2+ ions, as verified by 57 Fe Mössbauer spectroscopy, TGA, TEM, microprobe, and XANES. Furthermore, the size of the nanometric α-Fe 2 O 3 clusters coating the main phase strongly depends on the y M metal concentration. For M-rich phases the iron diffusion is hampered so that a significant fraction of superparamagnetic α-Fe2O3 particles (100% for BaFe 0.5–x Co 1.5 (PO 4 ) 2 ) was detected even at 78 K. Although Ni 2+ and Co 2+ ions tend to block Fe diffusion, the crystal structure of BaFe 0.67 Co 1 (PO 4 ) 2 demonstrates a fully ordered rearrangement of Fe 3+ and Co 2+ ions after Fe exsolution. We found that the magnetic behaviors of the Fe-depleted materials are mostly dominated by antiferromagnetic exchange, while Co 2+ -rich compounds show metamagnetic transitions reminiscent of the BaCo 2 (PO 4 ) 2 soft helicoidal magnet.

Alcover, Ignacio Blazquez↗

DOE OSTI2023

Self-consistent hopping theory of activated relaxation and diffusion of dilute penetrants in dense crosslinked polymer networks

We generalize a microscopic statistical mechanical theory of the activated dynamics of dilute spherical penetrants in glass-forming liquids to study the influence of crosslinking in polymer networks on the penetrant relaxation time and diffusivity over a wide range of temperature and crosslink fraction (f n ). Our calculations are relevant to recent experimental studies of a nm-sized molecule diffusing in poly-(n-butyl methacrylate) networks. The theory predicts the penetrant relaxation time increases exponentially with the glass transition temperature, T g (f n ), which grows roughly linearly with the square root of fn due to the coupling of local hopping to longer-range collective elasticity. Moreover, T g is also found to be proportional to a geometric confinement parameter defined as the ratio of the penetrant diameter to the mean network mesh size. The decoupling ratio of the penetrant and Kuhn segment alpha times displays a complex non-monotonic dependence on f n and temperature that is well collapsed based on the variable T g (f n )/T. A model for the penetrant diffusion constant that combines activated relaxation and entropic mesh confinement is proposed, which results in a significantly stronger suppression of mass transport with degree of effective supercooling than predicted for the penetrant alpha time. Furthermore, this behavior corresponds to a new network-based type of “decoupling” of diffusion and relaxation. In contrast to the diffusion of larger nanoparticles in high temperature rubbery networks, our analysis in the supercooled regime suggests that for the penetrants studied the mesh confinement effects are of secondary importance relative to the consequences of crosslink-induced slowing down of activated hopping of glassy physics origin.

Chemistry↗

DOE OSTI2022

Assessment of diffuse-interface methods for compressible multiphase fluid flows and elastic-plastic deformation in solids

This work describes three diffuse-interface methods for the simulation of immiscible, compressible multiphase fluid flows and elastic-plastic deformation in solids. The first method is the localized-artificial-diffusivity approach of Cook, Subramaniam et al., and Adler and Lele, in which artificial diffusion terms are added to the individual phase mass fraction transport equations and are coupled with the other conservation equations. The second method is the gradient-form approach that is based on the quasi-conservative method of Shukla et al., in which the diffusion and sharpening terms (together called regularization terms) are added to the individual phase volume fraction transport equations and are coupled with the other conservation equations. The third approach is the divergence-form approach that is based on the fully conservative method of Jain et al., in which the regularization terms are added to the individual phase volume fraction transport equations and are coupled with the other conservation equations. In the present study, all three diffuse-interface methods are used in conjunction with a four-equation, multicomponent mixture model, in which pressure and temperature equilibria are assumed among the various phases. The primary objective of this work is to compare these three methods in terms of their ability to: maintain constant interface thickness throughout the simulation; conserve mass, momentum, and energy; and maintain accurate interface shape for long-time integration. The second objective of this work is to consistently extend these methods to model interfaces between solid materials with strength. To assess and compare the methods, they are used to simulate a wide variety of problems, including (1) advection of an air bubble in water, (2) shock interaction with a helium bubble in air, (3) shock interaction and the collapse of an air bubble in water, and (4) Richtmyer–Meshkov instability of a copper–aluminum interface. The current work focuses on comparing these methods in the limit of relatively coarse grid resolution, which illustrates the true performance of these methods. In conclusion, this is because it is rarely practical to use hundreds of grid points to resolve a single bubble or drop in large-scale simulations of engineering interest.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1983

Numerical solution of the problem of flame propagation by the use of the random element method

A numerical, grid-free algorithm is presented for one-dimensional reaction-diffusion model of laminar flame propagation in premixed gases. It is based on the random element method we developed for the analysis of diffusional processes. The effect of combustion is taken into account by applying the principle of fractional steps to separate the process of diffusion, modeled by the random walk of computational elements, from the exothermic effects of chemical reaction, monitoring their strength. The validity of the algorithm is demonstrated by application to flame propagation problems for which exact solutions exist. The flame speed evaluated by its use oscillates around the exact value at a relatively small amplitude, while the temperature and species concentration profiles are self-correcting in their convergence to the exact solution. A satisfactory resolution is obtained by the use of quite a small number of computational elements which automatically adjust their distribution of fit sharp gradients.

Ghoniem, A. F.↗

DOE OSTI2015

Diffusion in Jammed Particle Packs

Using random walk simulations we explore diffusive transport through monodisperse sphere packings over a range of packing fractions Φ in the vicinity of the jamming transition at Φ c . Various diffusion properties are computed over several orders of magnitude in both time and packing pressure. Two well-separated regimes of normal “Fickian” diffusion, where the mean squared displacement is linear in time, are observed. The first corresponds to diffusion inside individual spheres, while the latter is the long-time bulk diffusion. The intermediate anomalous diffusion regime and the long-time value of the diffusion coefficient are both shown to be controlled by particle contacts, which in turn depend on proximity to Φ c . The time required to recover normal diffusion t * scales as (Φ–Φ c ) –0.5 and the long-time diffusivity D∞~(Φ–Φ c ) 0.5 , or D ∞ ~1/ t *. Here, it is shown that the distribution of mean first passage times associated with the escape of random walkers between neighboring particles controls both t * and D∞ in the limit Φ→Φ c .

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2016

An a priori DNS study of the shadow-position mixing model

In this study, the modeling of mixing by molecular diffusion is a central aspect for transported probability density function (tPDF) methods. In this paper, the newly-proposed shadow position mixing model (SPMM) is examined, using a DNS database for a temporally evolving di-methyl ether slot jet flame. Two methods that invoke different levels of approximation are proposed to extract the shadow displacement (equivalent to shadow position) from the DNS database. An approach for a priori analysis of the mixing-model performance is developed. The shadow displacement is highly correlated with both mixture fraction and velocity, and the peak correlation coefficient of the shadow displacement and mixture fraction is higher than that of the shadow displacement and velocity. This suggests that the composition-space localness is reasonably well enforced by the model, with appropriate choices of model constants. The conditional diffusion of mixture fraction and major species from DNS and from SPMM are then compared, using mixing rates that are derived by matching the mixture fraction scalar dissipation rates. Good qualitative agreement is found, for the prediction of the locations of zero and maximum/minimum conditional diffusion locations for mixture fraction and individual species. Similar comparisons are performed for DNS and the IECM (interaction by exchange with the conditional mean) model. The agreement between SPMM and DNS is better than that between IECM and DNS, in terms of conditional diffusion iso-contour similarities and global normalized residual levels. It is found that a suitable value for the model constant c that controls the mixing frequency can be derived using the local normalized scalar variance, and that the model constant a controls the localness of the model. A higher-Reynolds-number test case is anticipated to be more appropriate to evaluate the mixing models, and stand-alone transported PDF simulations are required to more fully enforce localness and to assess model performance.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2019

What Is the Fractional Laplacian? A Comparative Review with New Results

The fractional Laplacian in $\mathbb{R}^{d}$, which we write as (–Δ) α/2 with, α ϵ (0, 2) has multiple equivalent characterizations. Furthermore, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. Yet, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: “What is the fractional Laplacian?” Beginning from first principles, we compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Next, we present quantitative comparisons using a sample of state-of-the-art methods. We discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems: radial basis function collocation (for the Riesz fractional Laplacian) and nonharmonic lifting (for the spectral fractional Laplacian). In our numerical studies, we aim to compare different definitions on bounded domains using a collection of benchmark problems. We consider the fractional Poisson equation with both zero and nonzero boundary conditions, where the fractional Laplacian is defined according to the Riesz definition, the spectral definition, the directional definition, and the horizon-based nonlocal definition. We verify the accuracy of the numerical methods used in the approximations for each operator, and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Effect of sucrose on the oxidation-reduction conditions and retention of rhenium during vitrification of low-activity waste

Technetium speciation and solubility during nuclear waste vitrification are affected by oxygen partial pressure in the melt. Thus, reducing agents such as sucrose, which are routinely added to nuclear waste feeds to repress foaming, can significantly affect the volatilization of Tc and increase its retention in glass. Using Re as a surrogate for Tc, we measured the effect of sucrose addition to two low-activity waste melter feeds on the Re retention, oxygen partial pressure, and iron redox as functions of temperature. As expected, Re retention in glass increased as the sucrose content increased. Furthermore, this is attributed to sucrose reaction with nitrates and nitrites in the feed. Destruction of nitrates and nitrites reduces the salt phase fraction and thus likely promotes Re diffusion into the glass-forming melt. In agreement with available literature, we discuss that a modest increase in the sucrose content (compared to current practice) can significantly improve the Tc retention without reducing Fe 2+ to metal.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2020

Affine Approximation of Parametrized Kernels and Model Order Reduction for Nonlocal and Fractional Laplace Models

In this work, we consider parametrized problems driven by spatially nonlocal integral operators with parameter-dependent kernels. In particular, kernels with varying nonlocal interaction radius $\delta > 0$ and fractional Laplace kernels, parametrized by the fractional power $s\in(0,1)$, are studied. Furthermore, in order to provide an efficient and reliable approximation of the solution for different values of the parameters, we develop the reduced basis method as a parametric model order reduction approach. Major difficulties arise since the kernels are not affine in the parameters, singular, and discontinuous. Moreover, the spatial regularity of the solutions depends on the varying fractional power $s$. To address this, we derive regularity and differentiability results with respect to $\delta$ and $s$, which are of independent interest for other applications such as optimization and parameter identification. We then use these results to construct affine approximations of the kernels by local polynomials. Finally, we certify the method by providing reliable a posteriori error estimators, which account for all approximation errors, and support the theoretical findings by numerical experiments.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Predictive Integrated Modeling of Heated Tokamak Plasmas (Final Technical Report)

The research carried out by the Lehigh University fusion group during this reporting period: (1) Self-consistent predictive core-pedestal ITER scenario modeling; (2) Microtearing instabilities and electron thermal transport in standard and spherical tokamak discharges; (3) Neural network model of the Multi-Mode anomalous transport module; (4) Anomalous transport in KSTAR, JET, EAST, Alcator C-Mod and DIII-D discharges; (5) Theory for transport in magnetized plasmas; (6) Control-oriented modeling of the q-profile dynamic response in preparation for advanced scenario control in KSTAR and in EAST; (7) Modeling of edge harmonic oscillation in DIII-D QH mode discharges; (8) Elucidating plasma dynamics in Hasegawa-Wakatani turbulence by information geometry; (9) Non-linear Langevin and Fractional Fokker–Planck equations for anomalous diffusion by Levy stable processes; and (10) Investigation of the plasma shaping effects on the H-mode pedestal structure.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗