Macro-scale Model Research Articles

Homogenization theory of ion transportation in multicellular tissue

The transport of ions in biological tissues plays a crucial role in studying many biological and pathological problems. Certain multi-cellular structures, such as smooth muscles on vessel walls, can be considered as periodic bi-domain structures consisting of intracellular and extracellular spaces separated by semipermeable membranes. To model these structures, macro-scale models are proposed based on an electro-neutral (EN) microscale model with nonlinear interface conditions, using an unfolding operator. The membranes are treated as combinations of capacitors and resistors, while also taking into account the connectivity of the intracellular space. If the intracellular space is fully connected and forms a syncytium, the macroscale model is a bidomain nonlinear coupled partial differential equations system. Otherwise, if the intracellular cells are not connected, the macroscale model for the intracellular space is an ordinary differential system with source/sink terms from the connected extracellular space. The first-order error estimates for these models are achieved with proper regularity assumptions.

Modelado micromecánico del comportamiento no lineal de laminados angle-ply ante diferentes estados de tensión plana

The non-linear behaviour ocurring in symmetrical angle-ply laminates formed by unidirectional layers of carbon fiber reinforced polymeric matrix (CFRP), known as pseudo-ductility, has been previously studied numerically and experimentally under different uniaxial, biaxial and bending loads. This phenomenon is associated with different mechanisms which affect the mechanical behaviour of the material: large permanent strains, fiber reorientation and matrix cracking. The present study is based on a microscale approach through simulations based on the Finite Element Method (FEM), with the aim of predicting the homogenized response of these laminates subjected to multiaxial in-plane. The micromechanical model is based on a Representative Volume Element (RVE) where Periodic Boundary Conditions (PBC) are applied, together with a non-linear material model of the matrix and a cohesive interface between fibre and matrix. In addition, the implementation of the model is updated to be able to reproduce any plane stress state in a [±45]S laminate, without the need to add other meso- or macroscale models.

Predictive Modeling of Electrodeposition in a Single Pore Flow-Through Electrode: From Electronucleation to Coating Thickness Uniformity

The emergence of advanced manufacturing methods capable of producing porous three-dimensional structures has expanded the design space for next-generation functional components. The ability to fabricate ordered 3D foams for use in electrocatalysis reactors has increased the need for controlled deposition of catalytic metals onto porous support materials, such as carbon. However, there is a lack of clear design guidelines for electrodeposition onto 3D substrates, due to the geometric complexity and multi-scale nature of the problem. Furthermore, electro-nucleation phenomena are often overlooked in macro-scale models of current distribution during deposition. Here, a graphite flow-through electrode (FTE) is used as a model system for copper deposition within a single pore. Potential distributions and electro-nucleation phenomena are coupled in a continuum level model by incorporating nucleation size and density as a function of overpotential, determined experimentally using in situ atomic force microscopy (AFM). The model predictions are validated by measuring the coating uniformity in the pore using micro-computed X-ray tomography (μCT). A scaling analysis comprising dimensionless parameters such as the Wagner number is presented. The simplified scaling relationship framework can guide the electrodeposition process and electrode design to optimize plating of porous substrates under fluid flow conditions.

Nonfused Dimethoxybenzene Electron Acceptors in Organic Solar Cells: From Molecular Design to Structure–Performance Relationship

The photovoltaic mechanism of asymmetric non-fullerene acceptors (NFAs) has not been sufficiently understood, although it is developing as a forceful method to design effective organic solar cells. This article systematically investigated and designed several configurations of nonfused acceptors at the micro level. Skeletons were constructed by a central 1,4-dimethoxybenzene core and flanked by different dithieno-pyrroles and thiophene derivates as electron-donating groups; the end-capped groups were composed of two 1,1-dicyanomethyl-3-indanone. Through theoretical calculation, one found that the designed DBPT series possessed narrower gaps, broadened the absorption to the near-infrared region, larger electron-accepting power, and separated the distance of holes and electrons. They also yielded enhancement mobility, which may relate to the significant dipole moment and electronic coupling. Furthermore, we constructed the device models by using PBDB-T as the donor. The heterojunction model results show that the optimized blends of PBDB-T/DBPT-4F achieved more intermolecular fragment charge amounts, more CT states, and smaller binding energy, showing a significant charge separation/recombination ratio. Then, under the macro-scale model, the device PBDB-T/DBPT-4F has a better power conversion efficiency of 14.69%, owing to a large simulated JSC of 18.82 mA cm–2, consistent with the trend of better interfacial transfer property. This work provided the global description of the molecular light response and the microscopic physicochemical mechanism of the intermolecular charge transfer process, which also sheds light on the structure–property relationship of unfused asymmetric NFAs and enables the strategy to develop valuable OSCs.

Multiscale framework for prediction of residual stress in additively manufactured functionally graded material

Additive manufacturing (AM) of functionally graded material (FGM) is of great interest for alleviating the abrupt material discontinuity and steep gradient of physical properties across the interface between dissimilar materials. This study proposes a multiscale framework based on the inherent strain method to accurately predict residual stress distribution in FGMs processed via AM using directed energy deposition (DED). First, mesoscale representative volume elements for different chemical compositions of austenitic and ferritic steels were modeled with actual process parameters and individual material properties to calculate the inherent strains. Subsequently, the corresponding inherent strains of each chemical composition were continuously mapped to a macroscale model using the lumping layer method to obtain the residual stress distribution. Five types of specimens with different interlayers and scanning strategies were prepared using DED. The contour method, neutron diffraction, and deep hole drilling were used to determine the residual stress distribution. The results indicate that the residual stress distribution in the FGM predicted by the proposed multiscale model is consistent with the experimental measurements. Based on the proposed model, the effects of the interlayers and scanning strategies on the residual stress and distortion of the FGM were investigated. The proposed model allows for a better understanding of the effect of the composition gradient on the residual stress and enables accurate prediction of residual stresses in FGM structures.

Lamellar Spacing Modelling for LPBF Aluminum Parts

The high cooling rates reached during metal additive manufacturing (MAM) generate microstructures very different from those obtained by other conventional manufacturing methods. Therefore, research about the modeling of this type of microstructure is of great interest to the MAM community. In this work, the prediction of the lamellar spacing of an AlSi10Mg sample manufactured by laser powder bed fusion (LPBF), is presented. A multiscale approach is used, combining a CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry) model to predict the material properties, with a macroscale model of the sample manufacturing and with a microscale model to predict the microstructure. The manufacturing and metallographic characterization of the sample is also included. The results prove that the multiscale strategy followed is a valid approximation to simulate this type of manufacturing process. In addition, it is shown that the use of a generic simulation software focused on metal casting processes can be useful in predicting the lamellar spacing of the microstructure manufactured by LPBF. Finally, the relationship between the cooling rate and the resulting lamellar spacing has been established for this AlSi10Mg under the specific manufacturing conditions considered.

Study of Formation and Growth Behavior of CO<sub>2</sub> Hydrate Films

The objective of this study is to elucidate the formation and growth behavior of CO2 hydrate. Based on hydrate film thickness measurements, the process of hydrate film development was classified into "formation" and "growth". A macro-scale thickness prediction model based on mass transport was constructed for these processes. Molecular dynamics calculations were also introduced and combined with the macro-scale prediction model to construct a model capable of predicting film thickness.

Membrane Electrode Assembly Degradation Modeling of Proton Exchange Membrane Fuel Cells: A Review

Proton exchange membrane fuel cells (PEMFCs) have been recognized as a promising power generation source for a wide range of automotive, stationary, and portable electronic applications. However, the durability of PEMFCs remains as one of the key barriers to their wide commercialization. The membrane electrode assembly (MEA) as a central part of a PEMFC, which consists of a proton exchange membrane with a catalyst layer (CL) and gas diffusion layer (GDL) on each side, is subject to failure and degradation in long-running and cycling load conditions. The real-time monitoring of the degradation evolution process through experimental techniques is challenging. Therefore, different numerical modeling approaches were proposed in the literature to assist the understanding of the degradation mechanisms in PEMFCs. To provide modeling progress in the addressed field, this paper briefly discusses the different degradation mechanisms occurring in the MEA. In particular, we present a detailed review of MEA degradation modeling research work, with special attention paid to the physical-based models (mechanistic models). Following the most recent relevant literature, the results showed that the combination of microstructure component models with macro-scale comprehensive PEMFC models provides a better understanding of degradation mechanisms when compared to single-scale degradation models. In this sense, it is concluded that in order to develop an accurate and efficient predictive degradation model, the different relevant scales ranging from nano- to macro-sized scales should be considered, and coupling techniques for multiscale modeling have to be advanced. Finally, the paper summarizes the degradation models for different MEA components. It is highlighted that the GDL chemical degradation models that describe damage accumulation are relatively limited. The paper provides a useful reference for the recent developments in the MEA degradation modeling of PEMFCs.

A Hierarchy of Network Models Giving Bistability Under Triadic Closure

Triadic closure describes the tendency for new friendships to form between individuals who already have friends in common. It has been argued heuristically that the triadic closure effect can lead to bistability in the formation of large-scale social interaction networks. Here, depending on the initial state and the transient dynamics, the system may evolve towards either of two long-time states. In this work, we propose and study a hierarchy of network evolution models that incorporate triadic closure, building on the work of Grindrod, Higham, and Parsons [Internet Mathematics, 8, 2012, 402--423]. We use a chemical kinetics framework, paying careful attention to the reaction rate scaling with respect to the system size. In a macroscale regime, we show rigorously that a bimodal steady-state distribution is admitted. This behavior corresponds to the existence of two distinct stable fixed points in a deterministic mean-field ODE. The macroscale model is also seen to capture an apparent metastability property of the microscale system. Computational simulations are used to support the analysis.

Non-isothermal direct bundle simulation of SMC compression molding with a non-Newtonian compressible matrix

Compression molding of Sheet Molding Compounds (SMC) is a manufacturing process in which a stack of discontinuous fiber-reinforced thermoset sheets is formed in a hot mold. The reorientation of fibers during this molding process can be either described by macroscale models based on Jeffery's equation or by direct mesoscale simulations of individual fiber bundles. In complex geometries and for long fibers, direct bundle simulations outperform the accuracy of state-of-the-art macroscale approaches in terms of fiber orientation and fiber volume fraction. However, it remains to be shown that they are able to predict the necessary compression forces considering non-isothermal, non-Newtonian and compaction behavior. In this contribution, both approaches are applied to the elongational flow in a press rheometer and compared to experiments with 23% glass fiber volume fraction. The results show that both models predict contributions to the total compression force and orientation reasonably well for short flow paths. For long flow paths and thick stacks, complex deformation mechanisms arise and potential origins for deviation between simulations models and experimental observations are discussed. Furthermore, Jeffery's basic model is able to predict orientations similar to the high-fidelity mesoscale model. For planar SMC flow, this basic model appears to be even better suited than the more advanced orientation models with diffusion terms developed for injection molding.

Multiscale modelling of curing-induced z-pin/composite interfacial cracks

The curing process generates thermal residual stress in z-pinned composites, leading to partial or complete cracking along z-pin/composite interfaces. In this study, we applied a novel multiscale framework to evaluate the delamination behaviors of z-pinned composites considering the effects of z-pin/composite interfacial cracks. Unit cell models were used to represent the typical microstructures and micromechanisms of z-pinned composites with diverse interfacial conditions and to predict z-pin pullout behaviors. The derived z-pin bridging laws are then used to simulate the behaviors of individual z-pins in macroscale models. The load-versus-displacement relationships of z-pinned composites under mode-Ⅰ loading conditions are predicted using macroscale models and then verified experimentally. A parametric study reveals that interfacial cracks can affect the failure behaviors of z-pinned composites by changing the microscale bridging mechanisms of individual z-pins. It is theoretically feasible to customise z-pin/composite interfacial bonding properties, z-pin surface roughness, and curing temperature to optimise the delamination properties of z-pinned composites.

Modelling of 3D-printed bio-inspired Bouligand cementitious structures reinforced with steel fibres

Latencies and limitations exist in the simulation of bio-inspired structures or highly complex geometries manufactured by digital concrete fabrication. This study develops a computational framework to portray the anisotropic material attributes and predict the crack patterns of hardened 3D printed fiber-reinforced concrete beams. Bio-inspired Bouligand structures printed with four pitch angles of 0˚, 15˚, 30˚, and 45˚ are adopted to validate the numerical model. Extensive experiments, including uniaxial compressive tests, splitting tensile tests, and three-point bending tests, are conducted. The material model is attained from the samples extracted from the printed concrete blocks with and without steel fibres after a 28-day curing age for an accurate representation of the design and process. The micromechanical modelling approach is employed to obtain the homogeneous materials for the macro-scale model. The finite element (FE) analysis provides sound agreement with the experiment results of both stress–strain curves and crack patterns. In addition, under bending loading, the 3D printed beams with a unidirectional printing pattern or a small pitch angle perform better than those with larger pitch angles.

Upscaling the Poisson–Nernst–Planck equations for ion transport in weakly heterogeneous charged porous media

The Poisson–Nernst–Planck (PNP) equations govern the continuum level description of ions in electrolytes and especially the impact of charged surfaces. In numerous applications such surfaces are complex, varying on a small lengthscale compared to the overall scale of the system, often prohibiting the direct prediction of the osmotic swelling pressures induced by ion behaviours in Debye layers near surfaces. With periodicity, upscaling techniques can be readily used to determine the behaviour of the swelling pressure on large lengthscales without solving the PNP equations on the complex domain, though generalising to cases where the periodicity is only approximate is more challenging. Here, we generalise a method by Bruna and Chapman (2015) for upscaling a non-periodic diffusion equation to the PNP equations. After upscaling, we find a rational derivation of the swelling pressure closely resembling the classical, though phenomenological, use of Donnan membrane theory predictions for the swelling pressure in cartilage, together with a novel contribution driven by heterogeneous fixed (surface) charges. The resulting macroscale model is also shown to be thermodynamically consistent, though its comparison with a recent upscaled models for swelling pressure in cartilage mechanics emphasises the need to understand how macroscale models depend on differing upscaling techniques, especially in the absence of perfect periodicity.

Study of structural factors of structure-resolved filter media on the particle loading performance with microscale simulation

Microstructure strongly affects the macroscopic properties of filter media. Many previous works studied the filter loading behavior with the single fiber or macroscale model using unresolved fiber networks, but the influence of pore-scale structure was not thoroughly investigated. In this paper, the dynamic dust loading process in structure-resolved filter media was studied with microscale simulations. The distribution of trapped particles in the filter media in depth and cake filtration regimes was recorded, and the evolution of pressure drop and filtration efficiency was analyzed in depth. This work systematically explored how structural factors such as filter thickness, solidity (i.e., packing density), bulk, and fiber orientation affect the dust holding capacity (DHC) of the filter media. We also investigated better composite filters based on various designed filters with multilayer, uniform, or gradient structures. The results showed that the filter L-bot with a finer fiber layer at the bottom had the highest dust loading capacity, 63.98% higher than the filter L-top with an inverted structure, followed by the gradient filter GI-lin and GI-exp with open pore structure near the inlet. Our work presents a strategy for designing innovative depth filter media with effective and prolonged service life. Our detailed simulation with structure-resolved fiber network and particle deposits provides a fundamental understanding of the dust loading process.

Colloidal transport in anisotropic porous media: Kinetic equation and its upscaling

We discuss three-dimensional suspension-colloidal-nano transport in anisotropic porous media. The Boltzmann kinetic equation captures stochastic particle velocity distribution on the micro-scale due to the stochastic nature of the pore-space geometry. We express the anisotropy by tensorial micro-scale transport coefficients and a direction-dependent velocity distribution. The Bhatnagar–Gross–Krook (BGK) form of Boltzmann’s equation for particle transport with a particle capture rate proportional to the particle velocity is formulated. We introduce an equivalent sink term into the kinetic equation instead of non-zero initial data, yielding an evolution system not requiring Cauchy’s data. Exact probabilistic homogenisation is performed by solution of a functional operator equation in Hilbert space of Fourier images. The homogenised macro-scale model has the form of a three-dimensional advective–diffusive–reactive equation, where the particle flow in all directions is slower than the carrier fluid speed. This delay is the collective result of particle flow and capture and is explained by preferential capture of faster particles. Closed expressions of upscaled transport coefficients as a function of microscale parameters are explicit. For the particular case of quasi-one-dimensional flow, where the average transverse flux is zero, we derive the upscaled equation and investigate the continuous and pulse colloidal injections.

A constitutive macroscale model for compressible magneto-active polymers based on computational homogenization data: Part II — Magnetic nonlinear regime

This paper presents a comprehensive, microstructure-guided framework for the macroscopic, continuum-based modeling of isotropic, compressible magneto-active polymers (MAPs). Based on the methodology along with the foundations that were laid in the first part of this paper series, we further extend the constitutive framework in this follow-up contribution towards the magnetic nonlinear regime. Key idea of the constitutive approach is an additive split of the material part of the total energy density function into three contributions associated with (i) an elastic ground-stress, (ii) a magnetically induced mechanical stress, and (iii) the magnetization, respectively. We propose suitable constitutive functions in an energy-based setting that allow to accurately capture occurring saturation effects. The constitutive macroscale model fulfills the poly-convexity condition for a large set of {F̄,B̄}, which implies Legendre–Hadamard ellipticity and thus ensures material stability. The accuracy and excellent predictive performance of the proposed phenomenological macroscale model is demonstrated by the direct comparison with the homogenized material response of the underlying microstructure for various load cases. This is supplemented by a detailed comparative study with previously developed macroscale models that points out the advantages and superior performance of the proposed formulation. Further characteristics and modeling capabilities are illustrated by the solution of some application-oriented boundary value problems along with the application of a micro-macro decoupling scheme. The main emphasis of the numerical studies lies on the investigation of the magnetostrictive and magnetorheological effect of compressible MAPs at the macroscale level.

Temporal scale-bridging of chemistry in a multiscale model: Application to reactivity of an energetic material

A heterogeneous multiscale method (HMM) to evolve chemistry across disparate temporal scales is presented. The method directly couples a particle-based, microscale coarse-grain chemistry model to a macroscale continuum finite element model of deformation. The macroscale model obtains both the material equation-of-state and instantaneous chemical reaction rates through evaluation of the microscale model subject to distinct boundary conditions. The approach is demonstrated through several simulations of the energetic material 1,3,5-trinitrohexahydro-s-triazine (RDX) subject to slow heating, including a simulation inspired by the scaled thermal explosion (STEX) experiment.

Carbon-Binder Optimization for Lithium-Ion Battery Extreme Fast Charge

Battery performance is strongly correlated with electrode microstructure and weight loading of the electrode components. Among them are the carbon-black and binder additives that enhance effective conductivity and provide mechanical integrity. However, these both reduce effective ionic transport in the electrolyte phase and reduce energy density. Therefore, an optimal additive loading is required to maximize performance, especially for fast charging where ionic transport is essential. Such optimization analysis is however challenging due to the nanoscale imaging limitations that prevent characterizing this additive phase and thus quantifying its impact on performance.In this work [1], an additive-phase generation algorithm [2] has been developed to remedy this limitation and identify percolation threshold used to define a minimal additive loading (cf. figure, left). Impact of carbon-black binder (CBD) loading on electrochemical performance has been investigated through a wide array of numerical methods and experiments on NMC/graphite cells. These range from the representation, characterization, and homogenization of electrode microstructures, battery macroscale modeling, and impedance and rate capabilities measurements on various cell formats. The combined work provides information on connectivity/percolation of the solid network, effective solid conductivity and ionic diffusivity, interfacial area, cell capacity, lithium plating, impedance, and transport polarization at the beginning of life (BOL) [1]. Rate capability test demonstrates capacity improvement at fast charge at BOL (cf. figure, right), from 37% to 55%, respectively for high (10%wt) and low (4%wt) additive loading during 6C CC charging (and from 80% to 86% at the end of 10 min 6 CC-CV), in agreement with macroscale model. Improvements are attributed to a combination of lower cathode impedance, reduced electrode tortuosity and cathode thickness [1].[1] F. Usseglio-Viretta et. al., Carbon-binder Weight Loading Optimization for Improved Lithium-ion Battery Rate Capability, submitted to Journal of the Electrochemical Society [2] F. Usseglio-Viretta et. al., SoftwareX, 17, 100915 (2022), https://doi.org/10.1016/j.softx.2021.100915 Figure 1

Bending and stretching behavior of graphene structures using continuum models calibrated with modal analysis

Two-dimensional nanomaterials exhibit specific mechanical characteristics essential for commercialization and industrial applications, such as nano electro-mechanical systems. When analyzing a mechanical field, it is important to determine these materials' bending stiffness and stretching properties. The apparent bending stiffness and stretching of graphene structures at the macro-scale differ from theoretical predictions at the nano-scale. This discrepancy results from thermally generated dynamic waves in these atomic-based structures. Therefore, characterization methods based on atomistic static methods were unable to capture these effects. This study uses hybrid atomistic-continuum models to apply modal analysis to determine these key parameters. The proposed approach exploits the advantages of both atomistic and continuum models. Our approach is based on optimization techniques such as the simulated annealing algorithm. The unknown parameters of the continuum models, such as the bending stiffness and stretching, were acquired from molecular dynamics simulations and continuum mechanics resonance frequencies. The effects of diameter on the mechanical properties and value of different mode shapes of graphene nanostructures are studied using this method, which incorporates the effects of nano and macro-scale models. The presented approach provides insight into the mechanical vibrational characteristics of two-dimensional graphene structures at an atomic level.

Effects of Composite Electrode Structure on Performance of Intermediate-Temperature Solid Oxide Electrolysis Cell

The composite electrode structure plays an important role in the optimization of performance of the intermediate-temperature solid oxide electrolysis cell (IT-SOEC). However, the structural influence of the composite electrode on the performance of IT-SOEC is not clear. In this study, we developed a three-dimensional macroscale model coupled with the mesoscale model based on percolation theory. We describe the electrode structure on a mesoscopic scale, looking at the electrochemical reactions, flow, and mass transport inside an IT-SOEC unit with a composite electrode. The accuracy of this multi-scale model was verified by two groups of experimental data. We investigated the effects of operating pressure, volume fraction of the electrode phase, and particle diameter in the composite electrode on electrolysis reaction rate, overpotential, convection/diffusion flux, and hydrogen mole fraction. The results showed that the variation in the volume fraction of the electrode phase had opposite effects on the electrochemical reaction rate and multi-component diffusion inside the composite electrode. Meanwhile, an optimal range of 0.8–1 for the particle diameter ratio was favorable for hydrogen production. The analysis of IT-SOEC with composite electrodes using this multi-scale model enables the subsequent optimization of cell performance and composite electrode structure.