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DOE OSTI2017

Diffusion of New Products with Recovering Consumers

We consider the diffusion of new products in the discrete Bass-SIR model, in which consumers who adopt the product can later “recover” and stop influencing their peers to adopt the product. To gain insight into the effect of the social network structure on the diffusion, we focus on two extreme cases. In the “most-connected” configuration where all consumers are inter-connected (complete network), averaging over all consumers leads to an aggregate model, which combines the Bass model for diffusion of new products with the SIR model for epidemics. In the “least-connected” configuration where consumers are arranged on a circle and each consumer can only be influenced by his left neighbor (one-sided 1D network), averaging over all consumers leads to a different aggregate model which is linear, and can be solved explicitly. We conjecture that for any other network, the diffusion is bounded from below and from above by that on a one-sided 1D network and on a complete network, respectively. When consumers are arranged on a circle and each consumer can be influenced by his left and right neighbors (two-sided 1D network), the diffusion is strictly faster than on a one-sided 1D network. This is different from the case of non-recovering adopters, where the diffusion on one-sided and on two-sided 1D networks is identical. We also propose a nonlinear model for recoveries, and show that consumers’ heterogeneity has a negligible effect on the aggregate diffusion.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Efficient Reformulation of Linear and Nonlinear Solid-Phase Diffusion in Lithium-ion Battery Models using Symmetric Polynomials: Mass Conservation and Computational Efficiency

Lithium-ion batteries are typically modeled using porous electrode theory coupled with various transport and reaction mechanisms, along with suitable discretization or approximations for the solid-phase diffusion equation. The solid-phase diffusion equation represents the main computational burden for typical pseudo-2-dimensional (p2D) models since these equations in the pseudo r -dimension must be solved at each point in the computational grid. This substantially increases the complexity of the model as well as the computational time. Traditional approaches towards simplifying solid-phase diffusion possess certain significant limitations, especially in modeling emerging electrode materials which involve phase changes and variable diffusivities. A computationally efficient representation for solid-phase diffusion is discussed in this paper based on symmetric polynomials using Orthogonal Collocation and Galerkin formulation (weak form). A systematic approach is provided to increase the accuracy of the approximation (p form in finite element methods) to enable efficient simulation with a minimal number of semi-discretized equations, ensuring mass conservation even for non-linear diffusion problems involving variable diffusivities. These methods are then demonstrated by incorporation into the full p2D model, illustrating their advantages in simulating high C-rates and short-time dynamic operation of Lithium-ion batteries.

25 ENERGY STORAGE↗

DOE OSTI2019

A First-Principles Study of the Vacancy-Mediated Silver Impurity Diffusion in fcc Nickel

In this report, ab initio techniques (Density Functional Theory, DFT) are used to calculate the vacancy-mediated self-diffusion coefficient and impurity diffusion coefficient of silver (Ag) in pure fcc nickel (Ni). Prediction of diffusion coefficients is one of the most stringent tests of the ability of DFT calculations, because of the complex interplay of lattice constant, vacancy formation enthalpy, solute-vacancy binding enthalpy, and vibrational frequencies. The Ag diffusion and Ni self-diffusion coefficients calculated in the present work compare very favorably to previous experimental self-diffusion and Ag diffusion data. Extension of the methodology to nickel alloys is envisioned in future work.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2023

Ab Initio Simulations of Tritium Diffusion in Al 2 O and Intermetallic Al 12 (TM) 2.34 Aluminide Coating Phases

Density functional theory simulations have been carried out to investigate the diffusion of interstitial tritium in Al 12 (TM) 2.34 and Al 2 O bulk phases. In Al 12 (TM) 2.34 the transition metal (TM) sites are occupied on average by 58.93 at.% Fe, 18.52 at.% Cr, and 22.54 at.% Ni. While the insertion of interstitial tritium in Al 2 O lead to a strong disordering of the structure, we only investigated interstitial tritium in Al 12 (TM) 2.34 and its iron end-member (i.e., Al 12 Fe 2.34 ). Nine diffusion pathways have been investigated along the a-, b-, and c -axis of Al 12 (TM) 2.34 . The direction of fastest diffusion for interstitial tritium is found to be along the b-axis, with a calculated diffusion coefficient of ≈10 -10 m 2 .s -1 at 600 K, which is at least one order of magnitude faster than those previously calculated for interstitial tritium in other Al-rich phases such as Fe2Alx, Fe4Al13, and FeNiAl5 for which D T =10 -11 m 2 .s -1 , ≈10 -12 m 2 .s -1 , and D T ≈10 -13 m 2 .s -1 respectively. The comparison of the energy landscape between Al 12 (TM) 2.34 and Al 12 Fe 2.34 for the fastest diffusion pathways in each axis direction, found that some pathways are not sensitive to transition metal mixing, while other are more affected, leading to higher energy barrier in Al 12 Fe 2.34 in part due to a more energetically favorable formation of Fe—T bond compared to Ni—T. However, we found that both materials have similar diffusion coefficients as the fastest diffusion pathways occurs along pathways that are not very sensitive to transition metal mixing.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Determination of the Diffusion Coefficients of Binary CH4 and C2H6 in a Supercritical CO2 Environment (500–2000 K and 100–1000 atm) by Molecular Dynamics Simulations

The self-diffusion coefficients of carbonaceous fuels in a supercritical CO2 environment provide transport information that can help us understand the Allam Cycle mechanism at a high pressure of 300 atm. The diffusion coefficients of pure CO2 and binary CO2/CH4 and CO2/C2H6 at high temperatures (500 K~2000 K) and high pressures (100 atm~1000 atm) are determined by molecular dynamics simulations in this study. Increasing the temperature leads to an increase in the diffusion coefficient, and increasing the pressure leads to a decrease in the diffusion coefficients for both methane and ethane. The diffusion coefficient of methane at 300 atm is approximately 0.012 cm2/s at 1000 K and 0.032 cm2/s at 1500 K. The diffusion coefficient of ethane at 300 atm is approximately 0.016 cm2/s at 1000 K and 0.045 cm2/s at 1500 K. The understanding of diffusion coefficients potentially leads to the reduction in fuel consumption and minimization of greenhouse gas emissions in the Allam Cycle.

Energy & Fuels↗

DOE OSTI2015

Viscous organic aerosol particles in the upper troposphere: diffusivity-controlled water uptake and ice nucleation?

Here, new measurements of water diffusion in secondary organic aerosol (SOA) material produced by oxidation of α-pinene and in a number of organic/inorganic model mixtures (3-methylbutane-1,2,3-tricarboxylic acid (3-MBTCA), levoglucosan, levoglucosan/NH 4 HSO 4 , raffinose) are presented. These indicate that water diffusion coefficients are determined by several properties of the aerosol substance and cannot be inferred from the glass transition temperature or bouncing properties. Our results suggest that water diffusion in SOA particles is faster than often assumed and imposes no significant kinetic limitation on water uptake and release at temperatures above 220 K. The fast diffusion of water suggests that heterogeneous ice nucleation on a glassy core is very unlikely in these systems. At temperatures below 220 K, model simulations of SOA particles suggest that heterogeneous ice nucleation may occur in the immersion mode on glassy cores which remain embedded in a liquid shell when experiencing fast updraft velocities. The particles absorb significant quantities of water during these updrafts which plasticize their outer layers such that these layers equilibrate readily with the gas phase humidity before the homogeneous ice nucleation threshold is reached. Glass formation is thus unlikely to restrict homogeneous ice nucleation. Only under most extreme conditions near the very high tropical tropopause may the homogeneous ice nucleation rate coefficient be reduced as a consequence of slow condensed-phase water diffusion. Since the differences between the behavior limited or non limited by diffusion are small even at the very high tropical tropopause, condensed-phase water diffusivity is unlikely to have significant consequences on the direct climatic effects of SOA particles under tropospheric conditions.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2016

Linking pore-scale and basin-scale effects on diffusive methane transport in hydrate bearing environments through multi-scale reservoir simulations

We explore the gas hydrate-generating capacity of diffusive methane transport induced by solubility gradients due to pore size contrasts in lithologically heterogeneous marine sediments. Through the use of 1D, 2D, and 3D reactive transport simulations, we investigate scale-dependent processes in diffusion-dominated gas hydrate systems. These simulations all track a sand body, or series of sands, surrounded by clays as they are buried through the gas hydrate stability zone. Methane is sourced by microbial methanogenesis in the clays surrounding the sand layers. In 1D, simulations performed in a Lagrangian reference frame demonstrate that gas hydrate in thin sands (3.6 m thick) can occur in high saturations (upward of 70%) at the edges of sand bodies within the upper 400 meters below the seafloor. Diffusion of methane toward the center of the sand layer depends on the concentration gradient within the sand: broader sand pore size distributions with smaller median pore sizes enhance diffusive action toward the sand’s center. Incorporating downhole log- and laboratory-derived sand pore size distributions, gas hydrate saturations in the center of the sand can reach 20% of the hydrate saturations at the sand’s edges. Furthermore, we show that hydrate-free zones exist immediately above and below the sand and are approximately 5 m thick, depending on the sand-clay solubility contrast. A moving reference frame is also adopted in 2D, and the angle of gravity is rotated relative to the grid system to simulate a dipping sand layer. This is important to minimize diffusive edge effects or numerical diffusion that might be associated with a dipping sand in an Eulerian grid system oriented orthogonal to gravity. Two-dimensional simulations demonstrate the tendency for gas hydrate to accumulate downdip in a sand body because of greater methane transport at depth due to larger sand-clay solubility contrasts. In 3D, basin-scale simulations illuminate how convergent sand layers in a multilayered system can compete for diffusion from clays between them, resulting in relatively low hydrate saturations. All simulations suggest that when hydrate present in clays dissociates with burial, the additional dissolved methane is soaked up by nearby sands preserving high hydrate saturations.

Nole, Michael↗

DOE OSTI2015

A High-Order Nonlinear Diffusion Acceleration for the SN Equations Discretized with the Discontinuous FEM I: Theory and Numerical Results

Recently, nonlinear diffusion acceleration (NDA) methods have received widespread attention in the radiation transport community because of their inherent efficiency (Willert 2014, Wang 2014, Park 2012). However, NDA is in general only conditionally stable \cite{Larsen2014}, while diffusion synthetic acceleration methods (DSA) are unconditionally stable if suitable discretization methods for the transport and diffusion systems are chosen \cite{Larsen1982}. The presented work introduces a new generalized NDA method for solving the transport equation discretized with $S_N$ (discrete ordinates) and discontinuous Finite Elements. In contrast to common coarse-mesh finite difference methods (CMFD), where the diffusion system is discretized using constant shape functions, the presented NDA method discretizes the diffusion system using the same (higher order) finite element basis functions as those for the transport system. This paper focuses on the solution of the steady-state transport equation with the low-order diffusion equation employing the same mesh and the same spatial shape functions, but the NDA methods currently under development for the Rattle$S_N$ake radiation transport solver\cite{Wang2013} based on the MOOSE\cite{Gaston2009} framework is more general: it allows for the diffusion system to be discretized using an arbitrary group structure and arbitrary mesh nested in their $S_N$ system counterparts, and arbitrary combinations of FEM basis functions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Hydraulically opened cone vertical tube diffuser with slanted anti-siphon hole

A diffuser for a water heater fill tubes having a tube wall with an outside diameter. The diffuser includes an elongated flexible diffuser body for positioning at the outlet end of the fill tube. The diffuser body has a sealing end for sealing with an outside surface of the fill tube wall closest to the inlet end of the fill tube, and an open end for positioning at an end closest to the outlet end of the fill tube. The open end has a diameter larger than an outside diameter of the tube wall, thereby creating an diffuser water outlet opening between the diffuser body and the tube wall for redirecting radial water flow emanating from the fill tube water outlet opening(s) toward the diffuser outlet opening. A fill tube assembly for a water heater, a water heater, and a method for heating water are also disclosed.

Rendall, Joseph D.↗

DOE OSTI2018

Diffusion Behavior of Mn and Si Between Liquid Oxide Inclusions and Solid Iron-Based Alloy at 1473 K

In order to clarify the changes in the composition of oxide inclusions in steel, the effect of the metal and oxide composition on the reaction between solid Fe-based alloys and liquid multi-component oxides was investigated using the diffusion couple method at 1473 K. The measured concentration gradients of Mn and Si in the metal indicated that Mn diffused into the metal from the oxide, while the diffusion of Si occurred in the opposite direction. In addition, the MnO content in the oxide decreased with heat treatment time, while the SiO{sub 2} content increased. The compositional changes in both phases indicated that the Mn content in the metal near the interface increased with heat treatment with decreasing MnO content in the oxide. Assuming local equilibrium at the interface, the calculated [Mn]{sup 2}/[Si] ratio at the interface in equilibrium with the oxide increased with increases in the MnO/SiO{sub 2} ratio in the oxide. The difference in the [Mn]{sup 2}/[Si] ratios between the interface and the metal matrix increased, which caused the diffusion of Mn and Si between the multi-component oxide and metal. By measuring the diffusion lengths of Mn and Si in the metal, the chemical diffusion coefficients of Mn and Si were obtained to calculate the composition changes in Mn and Si in the metal. The calculated changes in Mn and Si in the metal agreed with the experimental results.

36 MATERIALS SCIENCE↗

DOE OSTI2018

A reduced thermal diffusion model for H and H 2

Here this work details the development of a new, reduced thermal diffusion model. The proposed model derives from the thermal diffusion model of Chapman and Cowling (1970). In its derivation, a set of mixture-averaged like approximations are made, which results in the number of operations being reduced from $\mathscr{O}$($\mathcal{n}^2$) to sub-linear, where n is the number of species in the chemical model. With these approximations, the new, reduced model thermal diffusion coefficients can be calculated independently for each species. The model is validated against multicomponent thermal diffusion cases using multiple fuel and diluent mixtures at various pressures, temperatures, and equivalence ratios. The resulting reduced model thermal diffusion fluxes agree well with the multicomponent values, with a multiplicative scaling factor identified using a least squares regression. Unstretched laminar flame speeds are compared using the multicomponent and newly developed models. Finally, an a posteriori comparison in a turbulent configuration shows excellent agreement of both the mean and fluctuations of the thermal diffusion coefficients.

42 ENGINEERING↗

DOE OSTI2014

A hybrid transport-diffusion model for radiative transfer in absorbing and scattering media

A new multi-scale hybrid transport-diffusion model for radiative transfer is proposed in order to improve the efficiency of the calculations close to the diffusive regime, in absorbing and strongly scattering media. In this model, the radiative intensity is decomposed into a macroscopic component calculated by the diffusion equation, and a mesoscopic component. The transport equation for the mesoscopic component allows to correct the estimation of the diffusion equation, and then to obtain the solution of the linear radiative transfer equation. In this work, results are presented for stationary and transient radiative transfer cases, in examples which concern solar concentrated and optical tomography applications. The Monte Carlo and the discrete-ordinate methods are used to solve the mesoscopic equation. It is shown that the multi-scale model allows to improve the efficiency of the calculations when the medium is close to the diffusive regime. The proposed model is a good alternative for radiative transfer at the intermediate regime where the macroscopic diffusion equation is not accurate enough and the radiative transfer equation requires too much computational effort.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

An extension of the localized artificial diffusivity method for immiscible and high density ratio flows

The localized artificial diffusivity (LAD) method is widely regarded as the preferred multi-material regularization scheme for the compact finite difference method, because it is conservative, easy to implement, and generally robust for a wide range of multi-material problems. However, traditional LAD methods face significant challenges when applied to flows with large density ratios and when maintaining thermodynamic equilibrium across material interfaces. These limitations arise from the formulation of the artificial diffusivity flux and the reliance on enthalpy diffusion for interface regularization. Additionally, traditional LAD methods struggle to ensure stability under large density ratio conditions, fail to maintain a finite interface thickness, and are therefore unsuitable for modeling immiscible interfaces. Here, in this work, we discuss the origins of these issues in traditional LAD methods and propose modifications which enable the simulation of large density ratio and immiscible flows. The proposed method targets the artificial diffusion fluxes at gradients and ringing in the volume fraction, rather than the mass fraction in traditional methods, to consistently regularize large density ratio interfaces. Furthermore, the proposed method introduces an artificial bulk density diffusion term to enforce equilibrium conditions across interfaces. To address the challenge of modeling immiscible flows, a conservative diffuse interface term is incorporated into the formulation to ensure a finite interface thickness. Specific consideration is taken in the design of the method to ensure that these crucial properties are maintained for N -material flows. The effectiveness of the proposed method is demonstrated through a series of canonical test cases, and its accuracy is validated by comparison with experimental data on micro-bubble collapse in water. These results highlight the method’s robustness and its ability to overcome the limitations of traditional LAD approaches.

Artificial diffusivity↗

DOE OSTI2023

Modeling of fission gas diffusion and release for Gd 2 O 3 doped UO 2

Uranium dioxide (UO 2 ) is the primary nuclear fuel in light water reactors, and its excess neutronic reactivity can be controlled by adding burnable absorbers, such as Gd 2 O 3 . This burnable absorber has a large neutron absorption cross-section, lowering the high reactivity of the reactor's initial fuel load. However, there needs to be more understanding of how added Gd 2 O 3 influences the properties of UO 2 under irradiation. To understand the behavior of defects and fission gas in the UO 2 /Gd 2 O 3 system under irradiation, we use cluster dynamics modeling supported by density functional theory calculations. First, we calculate the formation energies of Gd point and cluster defects, and evaluate the temperature-dependent defect concentrations using the defect formation energies and entropies. We show that Gd is soluble in UO 2 , introducing a negative charge in the system. Using this information, we adapted the cluster dynamics code Centipede to model the influence of Gd on U self-diffusion and Xe diffusion in UO 2 with 10 wt% Gd 2 O 3 . Also, we analyzed the Xe diffusion as a function of Gd 2 O 3 concentration, showing that the Xe diffusivity is decreased, which means that the athermal diffusivity due to electronic stopping persists at higher temperatures. In conclusion, the decrease in Xe diffusion means that more Xe stays in the matrix, decreasing the Xe release, and lowering its influence of fission gas release on the thermomechanical properties of UO 2 .

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2017

The equilibrium-diffusion limit for radiation hydrodynamics

The equilibrium-diffusion approximation (EDA) is used to describe certain radiation-hydrodynamic (RH) environments. When this is done the RH equations reduce to a simplified set of equations. The EDA can be derived by asymptotically analyzing the full set of RH equations in the equilibrium-diffusion limit. Here, we derive the EDA this way and show that it and the associated set of simplified equations are both first-order accurate with transport corrections occurring at second order. Having established the EDA’s first-order accuracy we then analyze the grey nonequilibrium-diffusion approximation and the grey Eddington approximation and show that they both preserve this first-order accuracy. Further, these approximations preserve the EDA’s first-order accuracy when made in either the comoving-frame (CMF) or the lab-frame (LF). And while analyzing the Eddington approximation, we found that the CMF and LF radiation-source equations are equivalent when neglecting O(β 2 ) terms and compared in the LF. Of course, the radiation pressures are not equivalent. It is expected that simplified physical models and numerical discretizations of the RH equations that do not preserve this first-order accuracy will not retain the correct equilibrium-diffusion solutions. As a practical example, we show that nonequilibrium-diffusion radiative-shock solutions devolve to equilibrium-diffusion solutions when the asymptotic parameter is small.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2013

Observation of two-dimensional p-type dopant diffusion across a p{sup +}-InP/n{sup -}-InGaAs interface using scanning electron microscopy

Scanning electron microscopy (SEM) with potential calculations has been shown to be effective for the detection of p-type dopant diffusion, even across a Zn doped p{sup +}-InP/non-doped n{sup -}-InGaAs/n{sup +}-InP heterojunction. Heterojunction samples were observed using SEM and the electrostatic potential was calculated from Zn concentration profiles obtained by secondary ion mass spectrometry. The sensitivity of SEM for the potential was derived from the SEM observations and potential calculation results. The results were then used to investigate the dependence of the SEM contrast on the Zn diffusion length across the p{sup +}-InP/non-doped n{sup -}-InGaAs interface. Accurate dopant mapping was difficult when the Zn diffusion length was shorter than 30 nm, because the heterojunction affects the potential at the interface. However, accurate dopant mapping was possible when the Zn diffusion length was longer than 30 nm, because the factor dominating the potential variation was not the heterojunction, but rather Zn diffusion 30 nm distant from the interface. Thus, Zn diffusion further than 30 nm from a Zn-doped p{sup +}-InP/non-doped n{sup -}-InGaAs interface can be effectively detected by secondary electron (SE) imaging. SE imaging with potential calculations can be widely used for accurate dopant mapping, even at heterojunctions, and is, therefore, expected to be of significant assistance to the compound semiconductor industry.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Understanding copper diffusion in CuInSe 2 with first-principles based atomistic and continuum models

We investigate the diffusion of copper in CuInSe 2 using thermodynamic and kinetic models based on density functional theory calculations, attempting to reconcile large differences in reported experimental diffusivities. We find that observations of rapid chemical diffusion can be explained by large thermodynamic factors, which we calculate using a compositionally constrained model of intrinsic point defect formation. We further characterize how copper diffusion coefficients depend on material synthesis conditions and exhibit their variation across the CuInSe 2 secondary phase diagram. In doing so, we identify stable off-stoichiometries that are dominated by either vacancy- or interstitial-mediated diffusion mechanisms. These results are employed in the development of a continuum reaction–diffusion model, which we use to simulate experimental depth profiles.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

A Finite Difference informed Random Walk solver for simulating radiation defect evolution in polycrystalline structures with strongly inhomogeneous diffusivity

Diffusivity of species and defects on grain boundaries is usually several orders of magnitude larger than that inside grains. Such strongly inhomogeneous diffusivity requires prohibitively high computational demands for modeling microstructural evolution. Here, this paper presents a highly-efficient numerical solver, combining the Finite Difference method and Random Walk model, designed for accurately modeling strongly inhomogeneous diffusion within polycrystalline structures. The proposed solver, termed Finite Difference informed Random Walk (FDiRW), integrates a customized Finite Difference (cFD) scheme tailored for fast diffusion along thin grain boundaries represented by a single-layer of nodes. Numerical experiments demonstrate that the FDiRW solver achieves an impressive efficiency gain of 1560x compared to traditional Finite Difference methods while maintaining accuracy, making it feasible for personal computer machines to handle diffusional systems with strongly inhomogeneous diffusivity across static polycrystalline microstructures. The model has been successfully applied to simulate radiation defect evolution, showcasing its scalability to engineering scales in both length and time dimensions.

36 MATERIALS SCIENCE↗