Effective Diffusion Tensor Research Articles

Characterizing PEM fuel cell catalyst layer properties from high resolution three-dimensional digital images, Part II: Oxygen effective diffusion and proton effective conductivity tensors

The oxygen effective diffusion and proton effective conductivity tensors of a cathode catalyst layer (CCL) are computed using a two-scale approach on CCL microstructure three-dimensional images obtained by focused ion beam-scanning electron microscopy (FIB-SEM) after reconstruction of the ionomer distribution. The results show that the obtained tensors are not isotropic. It is also found that some off-diagonal components are not negligible in the case of the proton effective conductivity tensor. The simulations confirm that Knudsen diffusion has a significant impact on oxygen diffusion. The results are comparable to those obtained in previous works on synthetic images as regards the oxygen effective diffusion through-plane component but differ as regards the proton conductivity through-plane component. Contrary to previous works using synthetic images which tend to underestimate the proton conductivity, they are globally in better agreement with experimental data. The results also show that the effective proton conductivity is heterogeneous at the scale of the computational domains typically considered in several previous works due to the variability in the ionomer distribution at this scale. The impact of liquid water in the primary pores on the proton transport is explored and found to have a significant impact.

Diffusion calculations on reconstructed bentonite microstructures with anion exclusion effects.

Due to their prevalence in the lithosphere and their high capability of sorbing pollutants, smectite clays play a foreground role in environmental pollution studies, waste management, and soil science. In complementarity with existing approaches at the molecular or macroscopic scales, real microstructures have been employed to investigate ionic transport by diffusion through montmorillonite and water-saturated Wyoming bentonite at intermediate scales ranging between the nanometer and the micrometer. The coupled solute transport and electrostatic phenomena investigated at the nanopore scale are upscaled using the homogenization of porous media approach. Homogenization computations rely on a hierarchical description of bentonite that acknowledges the existence of pores networks at different scales. At the scale of montmorillonite layers, digitized TEM images have been employed to simulate the diffusion of ionic solutes by considering electrostatic interactions in the vicinity of the negatively charged clay platelets' surface. Finite element microstructures are created after extraction of the contours of the layers using dedicated image processing algorithms. Local electric potential distribution, anion exclusion, and cation inclusion are displayed by ion distribution maps. The effective diffusion tensor and the transport equation obtained through volume averaging are then used to simulate diffusion at the scale of a Wyoming bentonite sample composed of clay gels of variable density, solid grains, and micropores. Qualitative comparisons were made with existing diffusion data, and a particular attention is given to the anisotropy of the diffusion tensors at both the mesoscopic and macroscopic scales.

Brownian Motion in an N-Scale Periodic Potential

We study the problem of Brownian motion in a multiscale potential. The potential is assumed to have N+1 scales (i.e. N small scales and one macroscale) and to depend periodically on all the small scales. We show that for nonseparable potentials, i.e. potentials in which the microscales and the macroscale are fully coupled, the homogenized equation is an overdamped Langevin equation with multiplicative noise driven by the free energy, for which the detailed balance condition still holds. This means, in particular, that homogenized dynamics is reversible and that the coarse-grained Fokker–Planck equation is still a Wasserstein gradient flow with respect to the coarse-grained free energy. The calculation of the effective diffusion tensor requires the solution of a system of N coupled Poisson equations.

The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects

In this work, a two-dimensional time-fractional subdiffusion model is developed to investigate the underlying transport phenomena evolving in a binary medium comprised of two sub-domains occupied by homogeneous material. We utilise an unstructured mesh control volume method to validate the model against a derived semi-analytical solution for a class of two-layered problems. This generalised transport model is then used to perform computational homogenisation on various two-dimensional heterogenous porous media. A key contribution of our work is to extend the classical homogenisation theory to accommodate the new framework and show that the effective diffusivity tensor can be computed once the cell problems reach steady state at the microscopic scale. We verify the theory for binary media via a series of well-known test problems and then investigate media having inclusions that exhibit a molecular relaxation (memory) effect. Finally, we apply the generalised transport model to estimate the bound water diffusivity tensor on cellular structures obtained from environmental scanning electron microscope (ESEM) images for Spruce wood and Australian hardwood. A highlight of our work is that the computed diffusivity for the heterogeneous media with molecular relaxation is quite different from the classical diffusion cases, being dominated at steady-state by the material with memory effects.

Computation of oxygen diffusion properties of the gas diffusion medium -microporous layer assembly from the combination of X-ray microtomography and focused ion beam three dimensional digital images

The effective diffusion properties of a gas diffusion layer (GDL) made of the assembly of a fibrous gas diffusion medium (GDM) and a microporous layer (MPL) are characterized from numerical simulations on reconstructed images obtained from combining micro X-ray computed tomography and focused ion beam-scanning electron microscopy (FIB-SEM). Using a multiscale approach, the MPL effective diffusion tensor is characterized from the FIB-SEM reconstructed 3D images considering Knudsen and Fickian diffusions. The GDM-MPL assembly effective through-plane diffusion coefficient is computed via a mixed approach combing a continuum description for the MPL and the explicit consideration of the MPL cracks and GDM fibers. The impact of cracks in the MPL is thus evaluated. A diffusive resistance model is developed to evaluate the impact of the MPL penetration into the GDM and the impact of the GDM compressibility.

Novel connectivity tensor for a systematic assessment of topology and anisotropy of real membranes and microporous structures

In this work, an original and complete systematic topological assessment of anisotropy, connectivity and tortuosity of real 3D porous structures is presented. Several novelties are introduced here, like: the definition of the Average Effective Diffusivity Tensor (AEDT), allowing us to assess the desired topological parameters, the novel concept of connectivity tensor based on the inverse of the (diagonal) tortuosity tensor, the development of a new procedure of digital reconstruction of real 3D structures for CFD purpose, and a novel methodology for the connectivity tensor calculation, which avoids time-dependent diffusivities. After a detailed explanation of methodology and related validation using well-known 3D structures, we analyse two different real digitally-reconstructed 3D structures: LTL zeolite and clinoptilolite. As key-result, the internal voids distribution is characterised in terms of several existing and new topological parameters, all obtained from the mathematical properties of tortuosity and connectivity tensors. Among them, particular useful are the here-defined directionality factors, which indicate the preferential diffusional paths. The proposed approach is relatively easy and can be effectively applied not only to characterise existing materials, but also to design novel ones (in form of catalysts, selective membranes and/or adsorbents) with desired and predictable topological and mass transport properties.

Stirring by anisotropic squirming

We consider a fluid stirred by the locomotions of squirmers through it and generalize the stochastic hydrodynamic model proposed by Thiffeault and Childress, Phys. Lett. A (2010) and Lin et al., J. Fluid Mech. (2011) to the case in which the swimmers move in anisotropically random directions. A non-diagonal effective diffusivity tensor is derived with which the diffusive preference of a passive particle along any given direction can be computed to provide more details of the phenomena beyond scalar statistics. We further identify a fraction from the orthogonal decomposition of the drift-induced particle displacement to distinguish the underlying nonlinear mixing mechanism for different types of swimmers. Numerical simulations verify the analytical results with explicit examples of prescribed, anisotropic stirring motions. We also connect our formulation to several measures used in clinical medical research such as diffusion tensor imaging where anisotropic diffusion has a significant consequence.

Multiscale modelling of diffusion and enzymatic reaction in porous electrodes in Direct Electron Transfer mode

This work is dedicated to a multi-scale modelling of coupled diffusion and reaction in a porous micro-electrode operating in the Direct Electron Transfer mode. The pore-scale physico-electrochemical unsteady model is developed considering the oxygen reduction, catalyzed by an enzyme coating the pores of the electrode, coupled to the diffusion of oxygen and mass balance of enzymes. This model is formally upscaled to obtain an original closed unsteady macroscopic model operating at the electrode scale, together with the associated closure providing the effective diffusivity tensor. A validation of this model is carried out from a comparison with the solution of the initial 3D pore-scale governing equations considering the bilirubin oxydase as the catalyst. The relevance and accuracy of the macroscale model are proved allowing a considerable simulation speedup. It is further employed to successfully predict experimental voltammetry results obtained with porous gold electrodes functionnalized with a bilirubin oxidase mutant (BOD S362C). This model represents a breakthrough by providing an operational simple way of understanding and further optimizing porous electrodes functioning in DET mode.

Homogenization of enhancing thin layers

This paper derives an explicit formula for the effective diffusion tensor by using the solutions to some effective cell problems after homogenizing Road effective boundary conditions (EBCs). The concept of Road EBCs was proposed recently by H. Li and X. Wang, and in this paper, we extend the effective conditions on closed curves to those on patterns, especially on the included nodes. We also prove that homogenization process commutes with the derivation of Road EBCs. By analyzing the effective diffusion tensor, we obtain several rules for maximizing its trace with given Road-effective-diffusivity/scale and length/scale in each cell and define a notion of balanced patterns. Moreover, we give an estimate of the trace of the effective diffusion tensor of patterns satisfying some connectedness as Road-effective-diffusivity/scale goes to infinity.

Residence time distribution of passive scalars in magnetic nanofluid Poiseuille flow under uniform rotating magnetic fields

The diffusive and convective transport mechanisms that govern the residence time distribution (RTD) of magnetic nanofluid (ferrofluid) capillary Poiseuille flow under an external uniform rotating magnetic field (RMF) were experimentally and numerically investigated. The measured RTDs are reduced and approach plug flow under RMF excitation by shortening the elution time, stiffening the breakthrough rise, and delaying the breakthrough time as the RMF frequency and/or magnetic nanoparticle concentration increase(s). Ferrohydrodynamic (FHD) simulations predict the inception of secondary convective azimuthal flow but fail to explain the impact of the magnetic field on RTD evolution under the assumption of isotropic diffusion transport. The origin of anisotropy in scalar diffusive transport was elucidated by analyzing the rotational properties of magnetic nanoparticles with respect to ferrofluid vorticity and RMF arrangement. Therefore, independent measurement of an anisotropic effective diffusion tensor for RMF-excited quiescent ferrofluids facilitates improved FHD simulations of RTD without compromising the model’s predictive power.

Numerical stochastic homogenization by quasilocal effective diffusion tensors

This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The resulting effective deterministic model is given through a quasilocal discrete integral operator, which can be further compressed to an effective partial differential operator. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.

A model of solute diffusion in unsaturated double-porosity medium by homogenization

Solute diffusion is a key process in many fields like for example material science or environmental engineering. Diffusion mechanism in porous media is often described by Fick’s law. However, we could not use this law for nonstandard diffusion behaviors occurring in cases of heterogeneous media. The conception of double-porosity medium can be applied to a class of such media. The double-porosity medium is characterized by two distinct pore sizes: macro-porosity domain and micro-porosity domain, respectively, having the contrasted hydraulic properties. This paper presents the development of a macroscopic model for the solute diffusion in unsaturated double-porosity medium, by using homogenization method. The obtained macroscopic model is a system of two equations coupling on the interface of the macro- and micro-porosity domain for diffusion. This model contains the effective diffusion tensor representing for the entire medium. The developed model is verified by comparing with the reference solution of the fine scale model through a 3D numerical example of hydrogeology problem.

Estimation of radon diffusivity tensor for fractured rocks in cave mines using a discrete fracture network model

This study develops a numerical model for predicting radon effective diffusivity tensor for fractured rocks using a two dimensional discrete fracture network (DFN) model. This is motivated by the limitations of existing techniques in predicting the radon diffusion coefficient for the fractured zones of cave mines. These limitations include access to the fractured zones for the purpose of conducting field studies as well as replication of the degree of fracturing in these zones for laboratory studies. The caving of a rock mass involves the fracturing and breaking of intact and naturally fractured rock, which creates migration pathways for radon gas trapped within uranium-rich rock. Therefore, this study develops a stochastic DFN model with equations derived from radon transport to predict diffusivity. Our simulation results reveal the establishment of a representative elementary volume (REV) for diffusivity tensor; approximately equal principal and cross diffusivity magnitudes for each of the DFN domain; a range of diffusivity with porosity (calculated based on the fractures in the domain); and a significant effect of fracture density on diffusivity tensor. These results are essential in developing proactive measures for mitigation of radon gas in cave mines.

Flux-dependent anisotropic pellet diffusivity in particle-resolved CFD simulations of fixed beds

The implementation of multicomponent diffusion in porous catalyst particles into resolved-particle fixed bed computational fluid dynamics (CFD) presents challenges. Usually an isotropic effective diffusivity is employed for each species, which includes molecular diffusion, Knudsen diffusion and viscous flow. To obtain this quantity, constant and isotropic molar flux ratios are assumed. In fixed bed models, pellets composed of isotropic materials are situated in non-isotropic fluid-phase gradients in the bed, which call the assumptions into question. In this paper we present an extended model using a non-isotropic effective diffusivity tensor, which allows evaluation of the flux assumptions. The method is illustrated for resolved particle steady-state CFD simulations of ethylene oxidation in small test beds of spheres and four-hole cylinders. Comparisons to the original constant molar flux ratio model are made, which show that the flux changes affect reaction, species and temperature profiles in the particles, and are stronger in particles with longer diffusion paths. Larger differences between results from the two diffusion models are also seen when the effective diffusivity is decreased by reducing porosity and increasing tortuosity, typical of a reaction system where deactivation increases by carbon deposition.

Old and New Approaches Predicting the Diffusion in Porous Media

An accurate quantitative description of the diffusion coefficient in a porous medium is essential for predictive transport modeling. Well-established relations, such as proposed by Buckingham, Penman, and Millington–Quirk, relate the scalar diffusion coefficient to the porous medium’s porosity. To capture the porous medium’s structure in more detail, further models include fitting parameters, geometric, or shape factors. Some models additionally account for the tortuosity, e.g., via Archie’s law. A validation of such models has been carried out mainly via experiments relating the proposed description to a specific class of porous media (by means of parameter fitting). Contrary to these approaches, upscaling methods directly enable calculating the full, potentially anisotropic, effective diffusion tensor without any fitting parameters. As input only the geometric information in terms of a representative elementary volume is needed. To compute the diffusion–porosity relations, supplementary cell problems must be solved numerically and their (flux) solutions must be integrated. We apply this approach to provide easy-to-use quantitative diffusion–porosity relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, or random porous media. As a discretization method, we use the discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We furthermore compare the obtained diffusion–porosity relations with the well-established relations mentioned above and also with the well-known Voigt–Reiss or Hashin–Shtrikman bounds. We discuss the ranges of validity and further provide the explicit relations between the diffusion and surface area and comment on the role of a tortuosity–porosity relation.

A multiscale model for carbon adsorption of BTX compounds: Comparison of volume averaging theory and experimental measurements

In this work, the method of volume averaging is applied for the mathematical modeling of transport and adsorption of benzene, toluene, and xylene in a packed bed of activated particles. One benefit of this approach is that it allows one to directly incorporate measured microscale information into macroscale models for predicting the effective transport and adsorption process. This work is novel in that it combines two levels of upscaling and a nonlinear adsorption process. The first level of upscaling develops an effective model for the mass transport and reaction in an activated carbon particle; within the particle, only diffusion and reaction are considered because of the very small pore sizes. The second level of upscaling develops the effective model for a collection of carbon particles forming the porous medium contained in a fixed-bed reactor; here, convection, diffusion, and dispersion are considered. The resulting model resembles a classical mobile-immobile representation of the transport and adsorption process.As part of the upscaling process, we develop the homogenized transport equations and their associated effective parameters using two different averaging volume support scales (i.e., at two disparate length scales). The effective parameters are all diffusion or dispersion tensors. These include (1) the effective diffusion tensor defining diffusion in the homogenized carbon particle, (2) the effective diffusion tensor for the immobile phase in the two-region representation of the medium, and (3) the effective hydrodynamic dispersion tensor (which included diffusion and dispersion) for the mobile region of the porous catalyst bed. Each of these effective parameters are determined by numerically solving closure problems over an idealized spatially periodic model of a porous medium. One novel feature of these particular closure problems is that they describe nonlinear adsorption at the microscale, which is a problem that is not currently represented in the literature.Once derived, the two-scale, two-equation mobile-immobile model was applied to predict experimentally-measured concentration breakthrough curves from packed bed columns with activated carbon from coconut shell as the adsorbent to the removal of petrochemical contaminants (BTX) by adsorption. There were no adjustable parameters in this modeling effort; the only modeling choice was whether the mass transfer coefficient should be computed from the correlations of Wakao and Funazkri (1978) for BTX components. The equations from closure problems and Darcy’s scale of transport of this work were discretized using the finite volumes method and the solutions are found numerically through of a computational code and some packages from the free software OpenFOAM®, version 2.2.x.This work has been selected by the Editors as a Featured Cover Article for this issue.

The effective diffusivity of ordered and freely evolving bubbly suspensions

We investigate the dispersion of a passive scalar such as the concentration of a chemical species, or temperature, in homogeneous bubbly suspensions, by determining an effective diffusivity tensor. Defining the longitudinal and transverse components of this tensor with respect to the direction of averaged bubble rise velocity in a zero mixture velocity frame of reference, we focus on the convective contribution thereof, this being expected to be dominant in commonly encountered bubbly flows. We first extend the theory of Kochet al.(J. Fluid Mech., vol. 200, 1989, pp. 173–188) (which is for dispersion in fixed beds of solid particles under Stokes flow) to account for weak inertial effects in the case of ordered suspensions. In the limits of low and of high Péclet number, including the inertial effect of the flow does not affect the scaling of the effective diffusivity with respect to the Péclet number. These results are confirmed by direct numerical simulations performed in different flow regimes, for spherical or very deformed bubbles and from vanishingly small to moderate values of the Reynolds number. Scalar transport in arrays of freely rising bubbles is considered by us subsequently, using numerical simulations. In this case, the dispersion is found to be convectively enhanced at low Péclet number, like in ordered arrays. At high Péclet number, the Taylor dispersion scaling obtained for ordered configurations is replaced by one characterizing a purely mechanical dispersion, as in random media, even if the level of disorder is very low.

Relativistic analysis of stochastic kinematics.

The relativistic analysis of stochastic kinematics is developed in order to determine the transformation of the effective diffusivity tensor in inertial frames. Poisson-Kac stochastic processes are initially considered. For one-dimensional spatial models, the effective diffusion coefficient measured in a frame Σ moving with velocity w with respect to the rest frame of the stochastic process is inversely proportional to the third power of the Lorentz factor γ(w)=(1-w^{2}/c^{2})^{-1/2}. Subsequently, higher-dimensional processes are analyzed and it is shown that the diffusivity tensor in a moving frame becomes nonisotropic: The diffusivities parallel and orthogonal to the velocity of the moving frame scale differently with respect to γ(w). The analysis of discrete space-time diffusion processes permits one to obtain a general transformation theory of the tensor diffusivity, confirmed by several different simulation experiments. Several implications of the theory are also addressed and discussed.

Effects of Mesoscopic Microstructural Features on the Effective Ionic Diffusivity of Solid Electrolytes for Li Batteries

The kinetics of ion transport in solid electrolytes is known to be highly sensitive to the topological characteristics of ion conduction pathways. The conduction mechanisms usually involve concurrent ionic diffusion along a variety of mesoscopic features of internal microstructures such as bulk grain, structural domains, and associated boundaries as well as their network. Due to the complexity of these structural features of the solid electrolytes during operation, it is significantly challenging to characterize the microstructure-ionic diffusion property relationship. For better mechanistic understanding of relevant conduction mechanisms, it is necessary to thoroughly explore the impacts of individual microstructural factors on the overall kinetic properties and performances. In this presentation, we will present our development of an efficient mesoscopic computational method for extracting the effective diffusivity of the solid electrolyte containing arbitrarily distributed microstructural inhomogeneities such as differently oriented multiple grains, several structural variants of phase domains, and grain/domain boundaries. Using two- and three-dimensional digital microstructures generated by phase-field simulations as well as the fundamental diffusivity tensors of reference phases and boundaries derived from atomistic calculations as inputs, the method enable us to efficiently compute the effective diffusivity tensor of the entire system. We will then discuss the applications of this framework to investigating the relationship between relevant individual microstructural features and effective diffusivity of highly conductive solid electrolytes (e.g., LLTO and LLZO) focusing on the impacts of topological features of grains, internal mesoscopic structures of high- and low-temperature phase domains, and grain/phase boundary networks. *This work of was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Computation of Quasi-Local Effective Diffusion Tensors and Connections to the Mathematical Theory of Homogenization

This paper aims at bridging existing theories in numerical and analytical homogenization. For this purpose the multiscale method of M\aalqvist and Peterseim [Math. Comp., 83 (2014), pp. 2583--2603]...