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Li-ion battery performance is strongly influenced by their cathodes' properties and consequently by the 3D microstructure of the particles the cathodes are comprised of. During calendaring and cycling, cracks develop within cathode particles, which may affect performance in multiple ways. On the one hand, cracks reduce internal connectivity such that electron transport within cathode particles is hindered. On the other hand, intra-particle cracks can increase the cathode reactive surface. Due to these contradictory effects, it is necessary to quantitatively investigate how battery cycling effects cracking and how cracking in-turn influences battery performance. Thus, it is necessary to characterize the 3D particle morphology with structural descriptors and quantitatively correlate them with effective battery properties. Typically, 3D structural characterization is performed using image data. However, informative 3D imaging techniques are time-consuming, costly and rarely available, such that analyses often have to rely on 2D image data. This paper presents a novel stereological approach for generating virtual 3D cathode particles that exhibit crack networks that are statistically equivalent to those observed in 2D sections of experimentally measured particles. Consequently, more easily available 2D image data suffices for deriving a full 3D characterization of cracked cathodes particles. In future research, the virtually generated 3D particles will be used as geometry input for spatially resolved electro-chemo-mechanical simulations, to enhance our understanding of structure-property relationships of cathodes in Li-ion batteries.

A stochastic 3D modeling approach for the nanoporous binder-conductive additive phase in hierarchically structured cathodes of lithium-ion batteries is presented. The binder-conductive additive phase of these electrodes consists of carbon black, polyvinylidene difluoride binder and graphite particles. For its stochastic 3D modeling, a three-step procedure based on methods from stochastic geometry is used. First, the graphite particles are described by a Boolean model with ellipsoidal grains. Second, the mixture of carbon black and binder is modeled by an excursion set of a Gaussian random field in the complement of the graphite particles. Third, large pore regions within the mixture of carbon black and binder are described by a Boolean model with spherical grains. The model parameters are calibrated to 3D image data of cathodes in lithium-ion batteries acquired by focused ion beam scanning electron microscopy. Subsequently, model validation is performed by comparing model realizations with measured image data in terms of various morphological descriptors that are not used for model fitting. Finally, we use the stochastic 3D model for predictive simulations, where we generate virtual, yet realistic, image data of nanoporous binder-conductive additives with varying amounts of graphite particles. Based on these virtual nanostructures, we can investigate structure-property relationships. In particular, we quantitatively study the influence of graphite particles on effective transport properties in the nanoporous binder-conductive additive phase, which have a crucial impact on electrochemical processes in the cathode and thus on the performance of battery cells.

A new cohesive phase-field (CPF) interface fracture model is proposed on the basis of the Euler-Lagrange equation of the phase-field theory and the interface fracture energy check w.r.t. that of the cohesive zone model. It employs an exponential function for the interpolation of fracture energy between the bulk phase and the interface, while the effective interface fracture energy $\tilde{G}_i$ is derived in such a way that the integrated phase-field fracture energy across the diffusive interface region remains consistent with the sharp interface fracture energy $G_i$ defined in the classical cohesive zone model. This consistency is the key to ensure that the numerical results remain insensitive to the choice of length-scale parameters, particularly the regularized interface thickness $L$ and the regularized fracture surface thickness $b$. By employing this energy consistency check, various CPF interface models in the literature are reviewed. Besides the length-scale insensitivity, the proposed CPF interface model offers further advantages. Thanks to the fact that the exponential interpolation function can be obtained conveniently from the relaxation solution of an Allen-Cahn equation, the proposed CPF model is advantageous over other models with high flexibility in handling structures containing complicated interface topology. In order to demonstrate this merit and to check the length-scale insensitivity in multiphysics context, the proposed CPF interface model is employed further to derive a thermodynamically consistent chemo-mechanical model relevant to Lithium-ion battery materials. Finite element simulation results of the concurrent bulk and interface fracture in polycrystalline electrode particles, reconstructed from images with segmented interfaces, confirm the expected computational advantages and the length-scale insensitivity in chemo-mechanical context.

Understanding structure-property relationships of Li-ion battery cathodes is crucial for optimizing rate-performance and cycle-life resilience. However, correlating the morphology of cathode particles, such as in NMC811, and their inner grain architecture with electrode performance is challenging, particularly, due to the significant length-scale difference between grain and particle sizes. Experimentally, it is currently not feasible to image such a high number of particles with full granular detail to achieve representivity. A second challenge is that sufficiently high-resolution 3D imaging techniques remain expensive and are sparsely available at research institutions. To address these challenges, a stereological generative adversarial network (GAN)-based model fitting approach is presented that can generate representative 3D information from 2D data, enabling characterization of materials in 3D using cost-effective 2D data. Once calibrated, this multi-scale model is able to rapidly generate virtual cathode particles that are statistically similar to experimental data, and thus is suitable for virtual characterization and materials testing through numerical simulations. A large dataset of simulated particles with inner grain architecture has been made publicly available.

Many post-processing methods improve forecasts at individual locations but remove their correlation structure, which is crucial for predicting larger-scale events like total precipitation amount over areas such as river catchments that are relevant for weather warnings and flood predictions. We propose a method to reintroduce spatial correlation into a post-processed forecast using an R-vine copula fitted to historical observations. This method works similarly to related approaches like the Schaake shuffle and ensemble copula coupling, i.e., by rearranging predictions at individual locations and reintroducing spatial correlation while maintaining the post-processed marginal distribution. Here, the copula measures how well an arrangement compares with the historical distribution of precipitation. No close relationship is needed between the post-processed marginal distributions and the spatial correlation source. This is an advantage compared to Schaake shuffle and ensemble copula coupling, which rely on a ranking with no ties at each considered location in their source for spatial correlations. However, weather variables such as the precipitation amount, whose distribution has an atom at zero, have rankings with ties. To evaluate the proposed method, it is applied to a precipitation forecast produced by a combination model with two input forecasts that deliver calibrated marginal distributions but without spatial correlations. The obtained results indicate that the calibration of the combination model carries over to the output of the proposed model, i.e., the evaluation of area predictions shows a similar improvement in forecast quality as the predictions for individual locations. Additionally, the spatial correlation of the forecast is evaluated with the help of object-based metrics, for which the proposed model also shows an improvement compared to both input forecasts.

We investigate the influence of crystallographic twins on the elastoplastic response of $\gamma$-TiAl intermetallics via full-field FFT-based computations. We first introduce a hierarchical stochastic model, which is used to simulate synthetic polycrystalline microstructures containing twin grains with certain morphologies, and apply it to generate representative volume elements. Second, we develop a Fourier-based method with regularization for solving the effective and local mechanical response of polycrystalline media using the M\'eric-Cailletaud crystal plasticity constitutive law. Numerical results show that, across configurations of twinning, the corresponding average effective response is similar. Although differences were quantified, the effect of twins on the yield stress is negligible in practice (less than $1\%$).

This paper introduces a novel method for characterizing fracture mechanisms in composite materials using 3D image data gained by computed tomography (CT) measurements. In mineral liberation, the understanding of these mechanisms is crucial, particularly whether fractures occur along the boundaries of mineral phases (intergranular fracture) and/or within mineral phases (transgranular fracture). Conventional techniques for analyzing fracture mechanisms are focused on globally comparing the surface exposure of mineral phases extracted from image measurements before and after fracture. Instead, we present a virtual reassembling algorithm based on image registration techniques, which is applied to 3D data of composite materials before and after fracture in order to determine and characterize the individual fracture surfaces. This enables us to conduct a local quantitative analysis of fracture mechanisms by voxelwise comparing adjacent regions at fracture surfaces. A quantitative analysis of fracture mechanisms is especially important in the context of geometallurgical recycling processes. As primary deposits are decreasing worldwide, the focus is shifting to secondary raw materials containing low concentrations of valuable elements such as lithium. To extract these elements, they can be enriched as engineered artificial minerals in the slag phase of appropriately designed cooling processes. The subsequent liberation through comminution processes, such as crushing, is essential for the extraction of valuable minerals. A better understanding of crushing processes, especially fracture mechanisms in slags, is crucial for the success of recycling. The reassembling algorithm presented in this paper is evaluated through a simulation study, followed by an application to a naturally occurring ore and a slag resulting from a recycling process.

The demand for lithium, as well as other critical resources, needed for electrochemical energy storage is expected to grow significantly in the future. Slags obtained from pyrometallurgical recycling represent a promising resource of valuable materials, among them lithium and rare earth elements found in artificial minerals particulate phases. This study investigates the flotation separation of engineered artificial minerals (EnAMs) in slags, such as lithium aluminate and gehlenite as valuable and gangue phases, respectively. Flotation experiments are carried out in a Partridge-Smith cell using oleic acid (OA) as a benchmark surfactant. Particle characterization is performed using SEM-based Mineral Liberation Analysis (MLA), which provides particle discrete information. From this information, bivariate Tromp functions based on non-parametric kernel density estimation are computed to characterize the separation behavior with respect to particle descriptors. This approach enables investigating the influence of particle size and shape on separation behavior of EnAMs. Furthermore, these results allow for the optimization of flotation experiments for enriching Li-bearing EnAMs.

Particle separation is typically investigated regarding one particulate property only. Virtually all separation processes, however, act on various particle properties in different ways. Modern particle analytical modalities enable a statistically meaningful multidimensional particle characterization. Within this study, individual particle fractions of magnesite, calcite and fluorite (-71 $\mu m$) are processed via the turbulent cross-flow separator cascade Cyclosizer (M16, MARC Technologies Pty Ltd), consisting of 5 hydrocyclones, thus producing 5 different product streams. Particle characterization is achieved via dynamic image analysis from which information on the particle shape and size is obtained. Using this data, bivariate Tromp functions are computed, which show the combined effect of the particle descriptors of roundness and area-equivalent diameter on the separation behavior. While the first cyclones recover predominantly coarse particles with high roundness values, fine particles with varying roundness are recovered in the latter cyclones.

Froth flotation predominantly separates particles according to their differences in wettability. However, other particle properties such as size, shape or density significantly influence the separation outcome as well. Froth flotation is most efficient for particles within a size range of about $20-200\mu m$, but challenges arise for very fine or coarse particles that are accompanied by low recoveries and poor selectivity. While the impact of particle size on the separation behavior in flotation is well-known by now, the effect of particle shape is less studied and varies based on the investigated zone (suspension or froth) and the separation apparatus used. A multidimensional perspective on the separation process, considering multiple particle properties, enhances the understanding of their collective influence. In this paper the two-dimensional case is studied, i.e., a parametric modeling approach is applied to determine bivariate Tromp functions from scanning electron microscopy-based image data of the feed and the separated fractions. With these functions it is possible to characterize the separation behavior of particle systems. Using a model system of ultrafine ($<10\mu m$) particles, consisting of differently shaped glass particles with different wettability states as the floatable and magnetite as the non-floatable fraction, allows for investigating the influence of particle size, shape and wettability, on the separation. In this way, the present paper contributes to a better understanding of the complex interplay between certain property vectors for the case of ultrafine particles. Furthermore, it demonstrates the benefits of using multivariate Tromp functions for evaluating separation processes, and points out the limitations of SEM based image measurements by means of mineral liberation analysis (MLA) for the studied particle size fraction.

A stochastic 3D microstructure model for polycrystals is introduced which incorporates two types of twin grains, namely neighboring and inclusion twins. They mimic the presence of crystal twins in $\gamma$-TiAl polycrystalline microstructures as observed by 3D imaging techniques. The polycrystal grain morphology is modeled by means of Voronoi and -- more generally -- Laguerre tessellations. The crystallographic orientation of each grain is either sampled uniformly on the space of orientations or chosen to be in a twinning relation with another grain. The model is used to quantitatively study relationships between morphology and mechanical properties of polycrystalline materials. For this purpose, full-field Fourier-based computations are performed to investigate the combined effect of grain morphology and twinning on the overall elastic response. For $\gamma$-TiAl polycrystallines, the presence of twins is associated with a softer response compared to polycrystalline aggregates without twins. However, when comparing the influence on the elastic response, a statistically different polycrystalline morphology has a much smaller effect than the presence of twin grains. Notably, the bulk modulus is almost insensitive to the grain morphology and exhibits much less sensitivity to the presence of twins compared to the shear modulus. The numerical results are consistent with a two-scale homogenization estimate that utilizes laminate materials to model the interactions of twins.

Quantifying the relationship between geometric descriptors of microstructure and effective properties like permeability is essential for understanding and improving the behavior of porous materials. In this paper, we employ a previously developed stochastic model to investigate microstructure-property relationships of nonwovens. First, we show the capability of the model to generate a wide variety of realistic nonwovens by varying the model parameters. By computing various geometric descriptors, we investigate the relationship between model parameters and microstructure morphology and, in this way, assess the range of structures which may be described by our model. In a second step, we perform virtual materials testing based on the simulation of a wide range of nonwovens. For these 3D structures, we compute geometric descriptors and perform numerical simulations to obtain values for permeability as an effective material property. We then examine and quantify the relationship between microstructure morphology and permeability by fitting parametric regression formulas to the obtained data set, including but not limited to formulas from literature. We show that for structures which are captured by our model, predictive power may be improved by allowing for slightly more complex formulas., Comment: 20 pages, 9 figures

During the mechano-fusion process for dry particle coating, (hetero)-aggregates are formed consisting of host particles which are coated by smaller guest particles. During this process, the latter are exposed to intense particle-particle interactions and particle-wall impacts, which lead to deformation of the guest particles original shape. These deformations on the nano- and microscale can heavily influence the effective macroscopic properties of the resutling coated particles. We present a method to quantify the shape deformation of guest particles during mechano-fusion based on measurements acquired by atomic force microscopy before and after mechano-fusion. To this end, we reconstruct the 3D shape of guest particles by means of an ellipsoidal fit, constrained by the known volume of the guest particles. Using these reconstructed shapes, we can quantify the degree of deformation by comparing the aspect ratios of the ellipsoidal fits before and after mechano-fusion. Such a quantification enhances the understanding of how process-related parameters influence the geometric descriptors of the involved particles, which in turn impact the overall macroscopic properties of the material.

Wide-angle light scattering (WALS) offers the possibility of a highly temporally and spatially resolved measurement of droplets in spray-based methods for nanoparticle synthesis. The size of these droplets is a critical variable affecting the final properties of synthesized materials such as hetero-aggregates. However, conventional methods for determining droplet sizes from WALS image data are labor-intensive and may introduce biases, particularly when applied to complex systems like spray flame synthesis (SFS). To address these challenges, we introduce a fully automatic machine learning-based approach that employs convolutional neural networks (CNNs) in order to streamline the droplet sizing process. This CNN-based methodology offers further advantages: it requires few manual labels and can utilize transfer learning, making it a promising alternative to conventional methods, specifically with respect to efficiency. To evaluate the performance of our machine learning models, we consider WALS data from an ethanol spray flame process at various heights above the burner surface (HABs), where the models are trained and cross-validated on a large dataset comprising nearly 35000 WALS images.

All-solid-state batteries (ASSBs) offer a promising route to safer batteries with superior energy density compared to conventional Li-ion batteries (LIBs). However, the design of the composite cathode and optimization of the underlying microstructure is one of the aspects requiring intensive research. Achieving both high energy and power density remains challenging due to limitations in ionic conductivity and active material loading. Using structure-resolved simulations, we investigate the potential of perforated and layered electrode designs to enhance ASSB performance. Design strategies showing significant performance increase in LIBs are evaluated regarding their application to ASSBs. Composite cathodes with solid electrolyte channels in the structure do not significantly increase cell performance compared to unstructured electrodes. However, the design with a two-layer cathode proves promising. The layered structure effectively balances improved ionic transport due to increased solid electrolyte fraction at the separator side and substantial active material loading through increased active material fraction at the current collector side of the cathode. Our research highlights key challenges in ASSB development and provides a clear direction for future studies in the field., Comment: 46 pages, 15 figures

The structural characterization of hetero-aggregates in 3D is of great interest, e.g., for deriving process-structure or structure-property relationships. However, since 3D imaging techniques are often difficult to perform as well as time and cost intensive, a characterization of hetero-aggregates based on 2D image data is desirable, but often non-trivial. To overcome the issues of characterizing 3D structures from 2D measurements, a method is presented that relies on machine learning combined with methods of spatial stochastic modeling, where the latter are utilized for the generation of synthetic training data. This kind of training data has the advantage that time-consuming experiments for the synthesis of differently structured materials followed by their 3D imaging can be avoided. More precisely, a parametric stochastic 3D model is presented, from which a wide spectrum of virtual hetero-aggregates can be generated. Additionally, the virtual structures are passed to a physics-based simulation tool in order to generate virtual scanning transmission electron microscopy (STEM) images. The preset parameters of the 3D model together with the simulated STEM images serve as a database for the training of convolutional neural networks, which can be used to determine the parameters of the underlying 3D model and, consequently, to predict 3D structures of hetero-aggregates from 2D STEM images. Furthermore, an error analysis is performed to evaluate the prediction power of the trained neural networks with respect to structural descriptors, e.g. the hetero-coordination number., Comment: 37 pages, 16 figures, 5 tables

Polymer-based batteries offer potentially higher power densities and a smaller ecological footprint compared to state-of-the-art lithium-ion batteries comprising inorganic active materials. However, in order to benefit from this potential advantages, further research to find suitable material compositions is required. In the present paper, we compare two different electrode composites of poly(2,2,6,6-tetramethylpiperidinyloxy-4-ylmethacrylate) (PTMA) and CMK-8, one produced with and one without crosslinking the PTMA. The influence of both approaches on the corresponding electrodes is comparatively investigated using electrochemical measurements and statistical 3D microstructure analysis based on synchrotron X-ray tomography. A particular focus is put on the local heterogeneity in the coating and how the crosslinking influences the interaction between PTMA and CMK-8. It is shown that crosslinked PTMA--compared to its non-crosslinked counterpart--exhibits a more heterogeneous microstructure and, furthermore, leads to better surface coverage of CMK-8, larger pores and shorter transportation pathways through the latter. These changes improve the electrochemical properties of the electrode., Comment: 43 pages, 13 figures

Computed tomography (CT) can capture volumes large enough to measure a statistically meaningful number of micron-sized particles with a sufficiently good resolution to allow for the analysis of individual particles. However, the development of methods to efficiently investigate such image data and interpretably model the observed particle features is still an active field of research. When image data of particles exhibiting a wide range of shapes and sizes is considered, traditional image segmentation methods, such as the classic watershed algorithm, struggle to recognize particles with satisfying accuracy. Thus, more advanced methods of machine learning must be utilized for image segmentation to improve the validity of subsequent analyzes. Moreover, CT data does not include information about the mineralogical composition of particles and, therefore, additional SEM-EDS image data has to be acquired. In this paper, micro-CT image data of a particle system mostly consisting of zinnwaldite-quartz composites is considered. First, an image segmentation method is applied which uses deep convolutional neural networks, in particular an adaptation of the U-net architecture. This has the advantage of requiring less hand-labeling than other machine learning methods, while also being more flexible with the possibility of transfer learning. Then, fully parameterized models based on vine copulas are designed to determine multivariate probability distributions of descriptor vectors for the size, shape, texture and composition of particles -- allowing for the estimation and interpretable characterization of interdependencies between particle descriptors. For model fitting, the segmented three-dimensional CT data and co-registered two-dimensional SEM-EDS data are used.

From simulating galaxy formation to viral transmission in a pandemic, scientific models play a pivotal role in developing scientific theories and supporting government policy decisions that affect us all. Given these critical applications, a poor modelling assumption or bug could have far-reaching consequences. However, scientific models possess several properties that make them notoriously difficult to test, including a complex input space, long execution times, and non-determinism, rendering existing testing techniques impractical. In fields such as epidemiology, where researchers seek answers to challenging causal questions, a statistical methodology known as Causal Inference has addressed similar problems, enabling the inference of causal conclusions from noisy, biased, and sparse data instead of costly experiments. This paper introduces the Causal Testing Framework: a framework that uses Causal Inference techniques to establish causal effects from existing data, enabling users to conduct software testing activities concerning the effect of a change, such as Metamorphic Testing, a posteriori. We present three case studies covering real-world scientific models, demonstrating how the Causal Testing Framework can infer metamorphic test outcomes from reused, confounded test data to provide an efficient solution for testing scientific modelling software.

FIB/SEM tomography represents an indispensable tool for the characterization of three-dimensional nanostructures in battery research and many other fields. However, contrast and 3D classification/reconstruction problems occur in many cases, which strongly limits the applicability of the technique especially on porous materials, like those used for electrode materials in batteries or fuel cells. Distinguishing the different components like active Li storage particles and carbon/binder materials is difficult and often prevents a reliable quantitative analysis of image data, or may even lead to wrong conclusions about structure-property relationships. In this contribution, we present a novel approach for data classification in three-dimensional image data obtained by FIB/SEM tomography and its applications to NMC battery electrode materials. We use two different image signals, namely the signal of the angled SE2 chamber detector and the Inlens detector signal, combine both signals and train a random forest, i.e. a particular machine learning algorithm. We demonstrate that this approach can overcome current limitations of existing techniques suitable for multi-phase measurements and that it allows for quantitative data reconstruction even where current state-of the art techniques fail, or demand for large training sets. This approach may yield as guideline for future research using FIB/SEM tomography.

It is well known that the spatial distribution of the carbon-binder domain (CBD) offers a large potential to further optimize lithium-ion batteries. However, it is challenging to reconstruct the CBD from tomographic image data obtained by synchrotron tomography. In the present paper, we consider several approaches to segment 3D image data of two different cathodes into three phases, namely active material, CBD and pores. More precisely, we focus on global thresholding, a local closing approach based on EDX data, a k-means clustering method, and a procedure based on a neural network that has been trained by correlative microscopy, i.e., based on data gained by synchrotron tomography and FIB-SEM data representing the same electrode. We quantify the impact of the considered segmentation approaches on morphological characteristics as well as on the resulting performance by spatially-resolved transport simulations. Furthermore, we use experimentally determined electrochemical properties to identify an appropriate range for the effective transport parameter of the CBD. The developed methodology is applied to two differently manufactured cathodes, namely an ultra-thick unstructured cathode and a two-layer cathode with varying CBD content in both layers. This comparison elucidates the impact of a specific structuring concept on the 3D microstructure of cathodes., Comment: 35 pages and 19 figures

Electrolyte filling is a time-critical step during battery manufacturing that also affects the battery performance. The underlying physical phenomena during filling mainly occur on the pore scale and are hard to study experimentally. In this paper, a computational approach, i.e.\ the lattice Boltzmann method, is used to study the filling process and corresponding pore-scale phenomena in 3D lithium-ion battery cathodes. The electrolyte flow through the nanoporous binder is simulated using a homogenization approach. Besides the process time, the influence of structural and physico-chemical properties is investigated. Those are the particle size, the binder distribution, and the volume fraction and wetting behavior of active material and binder. Optimized filling conditions are discussed by capillary pressure-saturation relationships. It is shown how the aforementioned influencing factors affect the electrolyte saturation. Moreover, the amount of the entrapped residual gas phase and the corresponding size distribution of the gas agglomerates are analyzed in detail. Both factors are shown to have a strong impact on mechanisms that can adversely affect the battery performance. The results obtained here indicate how the filling process, the final electrolyte saturation, and potentially also the battery performance can be optimized by adapting process parameters and the electrode and electrolyte design.

The site-specific management of weeds in grassland is often challenging because different weed control strategies have different trade-offs regarding the required resources and treatment efficiency. So, the question arises whether a wide tractor-based system with section control or a small agricultural robot has a higher weed control performance for a given infestation scenario. For example, a small autonomous robot moving from one weed to the next might have much shorter travel distances (and thus lower energy and time costs) than a tractor-mounted system if the locations of the weeds are relatively isolated across the field. However, if the plants are highly concentrated in small areas so-called clusters, the increased width of the tractor-mounted implement could be beneficial because of shorter travel distances and greater working width. An additional challenge is the fact that there is no complete knowledge of the weed locations. Weeds may not have been detected, for example, due to their growth stage, occlusion by other objects, or misclassification. Weed control strategies must therefore also be evaluated with regard to this issue. Thus, in addition to the driving distance, other metrics are also of interest, such as the number of plants that were actually controlled or the size of the total treatment area. We performed this investigation for the treatment of the toxic Colchicum autumnale, which had been detected in drone images of extensive grassland sites. In addition to real data, we generated and analyzed simulated weed locations using mathematical models of stochastic geometry. These offer the possibility to simulate very different spatial distributions of toxic plant locations. Different treatment strategies were then virtually tested on this data using Monte Carlo simulations and their performance was statistically evaluated., Comment: 20 pages, 9 figures

A law of large numbers for the empirical distribution of parameters of a one-layer artificial neural networks with sparse connectivity is derived for a simultaneously increasing number of both, neurons and training iterations of the stochastic gradient descent., Comment: 12 pages, 4 figures