Method of magnifying from microstructure to continuum using mesoscale heterohomogenization

Abstract

A method of magnifying from a microstructure to a continuum using mesoscale heterohomogenization is disclosed. The microstructure is amplified to generate a coarsened heterogeneous spatial distribution of porosity and a set of constitutive relationships dependent on porosity. Three-dimensional (3D) microstructure models, bulk material characteristics, and / or porosity are received for anode, cathode, and diaphragm battery assemblies. From the cell assembly microstructure, a coarsened porosity model with emergence characteristics is calculated as a function of the porosity. The Brugman coefficient of each cell assembly sub-region is calculated based on effective ionic conductivity, conductivity, thermal conductivity, and ion diffusivity. A heterogeneous mesoscale 3D battery model is created by combining an anode material, a cathode material, and a separator material into a single battery structure, and dividing each of the anode material, the cathode material, and the separator material into crude voxels, respectively, to create a 3D model of porosity.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the benefit of U.S. Provisional Patent Application No. 63 / 592,325, filed on October 23, 2023, entitled “Method for electrochemistry modeling of Li-ion Battery by upscaling from microstructure to continuum using a mesoscale, heterogeneous homogenization,” which is incorporated herein by reference in its entirety. Technical Field

[0003] The present invention relates to sustainable energy storage using lithium-ion batteries and related electrochemical simulations, and more particularly to a framework for determining the impact of heterogeneity on battery performance. Background Art

[0004] In the scientific literature, there are two types of electrochemical models used for lithium-ion batteries, both of which have practical drawbacks related to capturing performance and aging drivers:

[0005] (1) Continuum modeling, such as finite element (FE) analysis, where elements are on the millimeter scale and their degrees of freedom account for homogenized material properties (chemical composition plus modeled microstructural behavior). While this provides the ability to simulate entire unit cells, continuum modeling uses general empirical relationships for modeled microstructural behavior or requires laboratory measurements that may be difficult to obtain, and

[0006] (2) Microstructural modeling, where submicron resolution can capture the true microstructural behavior, but only for small sample sizes. Microstructural modeling provides accurate microstructural behavior rather than general empirical relationships and requires less hard-to-obtain laboratory data. However, simulating the behavior of the entire battery using microstructural modeling is no longer feasible due to the unmanageable number of degrees of freedom required.

[0007] The accuracy gained from capturing property heterogeneity through microstructural modeling is crucial for performance, aging, and degradation simulations, but the model size required for full-cell-level simulations is impractical to archive using current computing power. Consequently, a complete solution based on microstructural modeling approaches currently does not exist. Existing continuum models typically assign uniform material properties to each material (anode, cathode, and separator), assuming (incorrectly) that any existing heterogeneity can be ignored. Therefore, the heterogeneity from the microstructural model needs to be amplified into the continuum model. Summary of the Invention

[0008] Embodiments of the present invention provide a method that connects fine-resolution battery microstructure characterization (micrometer scale) and coarse-resolution battery electrochemical simulation (millimeter scale) while keeping the impact of fine-resolution heterogeneity on performance, aging and degradation within a reasonable and feasible coarse-resolution model size.

[0009] The microstructure is scaled up to generate a coarse heterogeneous spatial distribution of porosity and a set of porosity-dependent constitutive relations. A three-dimensional (3D) microstructure model, bulk material properties and / or porosity are received for anode, cathode and separator battery components. Based on the porosity, a coarse porosity model with emergent properties is calculated from the battery component microstructure. The Bruggeman coefficient of each battery component sub-region is calculated based on the effective ionic conductivity, electrical conductivity, thermal conductivity and ion diffusivity. A heterogeneous mesoscale 3D battery model is created by combining the anode material, cathode material and separator material into a single battery structure and dividing each of the anode material, cathode material and separator material into coarse voxels to create a 3D model of porosity.

[0010] Other systems, methods, and features of the present invention will be or become apparent to one of ordinary skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, and features be included within this description, be within the scope of the invention, and be protected by the accompanying claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] The accompanying drawings are included to provide a further understanding of the present invention and are incorporated into and constitute a part of this specification. The components in the drawings are not necessarily drawn to scale, but emphasis is placed upon clearly illustrating the principles of the present invention. The accompanying drawings illustrate embodiments of the present invention and, together with the description, serve to explain the principles of the present invention.

[0012] Figure 1 is a flowchart outlining an exemplary first embodiment of a workflow of a simulation framework.

[0013] Figure 2Two comparative examples of tortuosity are shown.

[0014] Figure 3 An exemplary procedure for fitting the Bruegmann coefficients / constitutive relations is shown.

[0015] Figure 4 is a series of four graphs illustrating examples of computable constitutive relations.

[0016] Figure 5 The basic situation of material properties and homogeneity are introduced.

[0017] Figure 6 Voltage / OCV versus SOC curves are shown at 3C (top) and 6C (bottom), comparing a basic cell to a homogenized cell.

[0018] Figure 7 Examples of the roughening process to obtain heterogeneous and homogeneous battery cells are shown.

[0019] Figure 8 yes Figure 7 Detail of the cathode, separator, and anode showing the relative porosity.

[0020] Figure 9 Graphs are shown showing voltage / OCV versus SOC at 3C (top) and 6C (bottom) comparing homogeneous cells to heterogeneous cells.

[0021] Figure 10 The graphs of the anodic precipitation potential (EPOTLPL) at the separator surface versus SOC at 3C and 6C are shown.

[0022] Figure 11 Shown are the distribution of precipitation potential versus porosity (top), a graphical illustration of where precipitation occurs (center), and the distribution of precipitation potential in heterogeneous and homogeneous cells (bottom).

[0023] Figure 12 Shown are contour plots of the precipitation potential at the anode surface close to the separator (EPOTLPL) at the end of the charging phase, comparing a homogeneous cell to a heterogeneous cell.

[0024] Figure 13 is a graphic illustration of the porosity distribution of heterogeneous batteries and optimized batteries.

[0025] Figure 14 Plots of voltage / OCV versus SOC are shown at 2.4C (top) and 4.8C (bottom), comparing the heterogeneous cell to the optimized cell.

[0026] Figure 15Plots of the anodic precipitation potential at the separator surface (EPOTLPL) versus SOC at 2.4C and 4.8C are shown, comparing the heterogeneous cell with the optimized cell.

[0027] Figure 16 The distribution of EPOTLPL relative to porosity (top), the distribution of EPOTLPL in heterogeneous cells and optimized cells (bottom) are shown.

[0028] Figure 17 Shown are contour plots of the EPOTLPL at the anode surface close to the separator at the end of the charging stage, comparing the heterogeneous cell with the optimized cell.

[0029] Figure 18 is a flow chart of an exemplary embodiment of a method for scaling a microstructure to a continuum.

[0030] Figure 19 is a schematic diagram illustrating an example of a system for performing the functions of the present invention. DETAILED DESCRIPTION

[0031] The following definitions are used to explain terms applied to features of the embodiments disclosed herein and are only meant to define elements within the present disclosure.

[0032] As used in this disclosure, “Abaqus” refers to a software suite originally released in 1978 for finite element analysis and computer-aided engineering. Abaqus is an engineering analysis software package used to simulate the physical response of structures and solids to various environmental conditions. Engineers use Abaqus to simulate complex problems in a variety of industries. As used in this disclosure, “Newman model” refers to a theoretical electrochemical framework used by Abaqus to simulate the charging and discharging of battery cells. In combination with thermal, swelling, and elastic theories, Abaqus provides a framework for simulating the multi-physics of batteries to aid in the design and optimization of battery cells. The Newman model can be used, for example, to calculate concentration, potential, temperature, displacement, and pore pressure as degrees of freedom.

[0033] As used in this disclosure, "Brugmann relation" and "Brugmann index" refer to the relationship between porosity and tortuosity of granular porous media. The Brugmann relation is an inverse power law of porosity and an exponent (called the Brugmann index, which is typically assumed to be 0.5).

[0034] As used in this disclosure, "continuum modeling" (finite element (FE) analysis) refers to a method for numerically solving partial differential equations associated with defined degrees of freedom (concentrations, potentials, etc.) in a continuum, which are not feasible to solve analytically. A continuum model is a representation of a process or structure that changes gradually between two defined points. These points can be tangible points (e.g., numerical points or time points) or intangible points. Exemplary embodiments herein relate to the Abaqus software suite for continuum modeling.

[0035] As used in this disclosure, "microstructure modeling" refers to the calculation of transport properties, etc., taking into account the explicit presence of porous media at a scale where individual pores / solid phases are distinguishable and separable, and obtaining effective transport properties that add the effects of microstructure to the pure material transport properties. Exemplary embodiments herein use DigitalROCK software for digital pore-scale simulation of microstructural material properties, providing, for example, pore space analysis, MICP curves, absolute permeability, relative permeability, capillary desaturation, tortuosity, ionic conductivity and diffusivity, electrical conductivity, and thermal conductivity.

[0036] As used in this disclosure, a "battery" or "battery cell" refers to an energy storage device, such as a lithium-ion battery or the like.

[0037] As used in this disclosure, "C-rate" refers to the multiple of the current required to charge a battery in 1 hour.

[0038] As used in this disclosure, "tortuosity" refers to a measure of the shortest path through a material in a given direction.

[0039] As used in this disclosure, "mesh" or "computational mesh" generally refers to a geometrically defined region of a battery cell, typically located at one and / or multiple layers, such as anode, cathode, and separator, used to simulate instead of the entire layered structure.

[0040] Reference will now be made in detail to the embodiments of the present invention, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numerals are used in the drawings and the description to refer to the same or like parts.

[0041] The design and optimization of battery cells is an expensive, time-consuming, and difficult process that typically relies on laboratory testing of the different components, materials, and processes that produce the final battery cell. Depending on the desired metric (e.g., performance, aging, and degradation) and related quantities (e.g., energy density or charging speed), processes that replicate these components (including cell design as well as material microstructure and chemistry) can accelerate the battery optimization process.

[0042] The exemplary embodiments herein provide an efficient way to scale up the battery microstructure (provided by DigitalROCK) to the entire continuum model (provided to Abaqus), thereby enabling efficient modeling of the entire lithium-ion battery cell taking into account heterogeneity without introducing an infeasible number of degrees of freedom. The exemplary embodiments combine the benefits of microstructure characterization and continuum modeling while minimizing their respective disadvantages. Figure 1 As shown, the embodiment takes a representative 3D microstructural model 110 and corresponding material properties 120 of each electrode / membrane in the battery cell as input to a heterogeneous homogenization process 130. For example, the representative 3D microstructural model 110 and corresponding material properties 120 can be provided by DigitalROCK. The heterogeneous homogenization process 130 generates two results: a coarsened heterogeneous spatial distribution of porosity (mesoscopic scale 3D coarsened porosity model 140), and a set of porosity-dependent constitutive relations 150 for other properties (e.g., ion diffusivity and electrical conductivity, electrical and thermal conductivity, saturation versus pressure, specific surface area, linear elastic coefficient). Reference is made below to Figure 18 The heterogeneous homogenization process 130 is described in further detail.

[0043] The coarsened porosity model 140 and associated constitutive relations 150 are received as input to a continuum electrochemical solver 160 (e.g., Abaqus), which includes a 3D Newman model simulator to produce a 3D Newman model 165, wherein performance and aging metrics 170 can be calculated from the degrees of freedom of the model 165. A complete microstructure optimization cycle can be achieved by repeating this sequence of steps while varying the electrode / separator microstructure. This complete workflow accounts for the required heterogeneity in performance and aging within a feasible computational model size. An embodiment provides a simulation framework 1000 that accounts for the heterogeneity of properties throughout the battery cell simulation, which fully captures the correct local and global behavior of these metrics, particularly for performance and aging predictions.

[0044] Simulations of these embodiments can be incorporated into the fabrication of battery cells, for example, by providing parameter inputs directly to battery manufacturing equipment, such as particle size distribution, the amount of additives (such as carbon binder domains), and the target porosity of the electrode achieved by controlling the pressure applied during calendaring. Furthermore, simulations represent the behavior of real electrodes with structured heterogeneity and provide geometric inputs during relevant manufacturing processes, such as the spacing of ablation or microchannels.

[0045] Exemplary embodiments provide a mesoscale model of a battery cell that preserves heterogeneity but does not introduce an infeasible number of degrees of freedom. The embodiments homogenize subregions of the anode, cathode, and separator to introduce a non-uniform porosity / saturation distribution, but do not completely abandon the assumptions inherent in the industry standard Newman model (i.e., particles and electrolyte can be assumed to be present at all material points). To homogenize the material properties, effective properties are calculated based on parameters related to the properties of a particular microstructure (e.g., porosity and saturation). The embodiments use these constitutive relations as input to an Abaqus electrochemical simulation, where the analyzed battery cell has a non-uniform porosity distribution determined based on the average porosity of the subvolume of the material being scanned. Through this workflow, the distribution, and therefore the material microstructure, can be adjusted or optimized to meet performance targets or limit the negative effects of aging. Details of this process are described below, along with some representative results.

[0046] The exemplary embodiment starts by receiving fully resolved microstructural 3D models 110 and material properties 120 of the anode, cathode, and separator components to establish accurate mesoscopic scales for continuum modeling in Abaqus. These 3D models can be obtained directly from real materials, for example, by focused ion beam scanning electron microscopy (FIB-SEM) or X-ray microtomography 3D imaging. Alternatively, these materials can be generated procedurally by simulating the calendering process on a variety of particles in Abaqus and then procedurally introducing additives, such as those present in the carbon binder domain (CBD), or they can be generated procedurally by customized battery microstructure generation tools such as MATBOX from NREL (see Allen, JM, Chang, J., Usseglio-Viretta, FLE et al., “A Segregated Approach for Modeling the Electrochemistry in the 3-D Microstructure of Li-Ion Batteries and Its Acceleration Using Block Preconditioners,” J. Sci. Comput. 86, 42 (2021). Examples of fully resolved microstructures can be found in Figure 7 (Left side).

[0047] Furthermore, given that all material constitutive relations 150 are at least implicitly dependent on the bulk properties 120 of the material being used, exemplary embodiments incorporate bulk material properties such as ionic conductivity, diffusivity, and electrical conductivity. These properties can be obtained from the literature (see Ecker et al., 2014, for example, the article published by Madeleine Ecker et al., "Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery: I. Determination of Parameters", J. Electrochem. Soc. 162A1836 (2015), and the article published by Madeleine Ecker et al., "Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery: II. Model Validation", J. Electrochem. Soc. 162A1849 (2015)), or by molecular modeling and simulation, for example using Materials Studio.

[0048] The exemplary embodiment creates a mesoscale 3D model 140 by coarsening an original 3D microstructural material model 110 based on spatially varying porosity. In the original fine-resolution model, each element was mostly solid (zero porosity) or mostly pores (100% porosity), while in the newly coarsened porosity model, each element has the average porosity of all small elements in the same region. The method achieves this by calculating the average porosity in a subregion of each material and then systematically assigning this porosity to each element of the finite element mesh, for example, through a script. For example, if the original image were to be decomposed into cubes with a side length of 50 pixels, each element could correspond to the average porosity in the corresponding 50-pixel cube region of the original scan.

[0049] For porous / heterogeneous media, the most effective properties resulting from homogenization of the material microstructure can be calculated from the bulk material properties and the material porosity (or alternatively, the solid volume fraction). Given the use of these effective properties in the Newman model, the effective properties fully reflect the material microstructure to obtain results that accurately reflect those materials at hand. Examples of effective material properties that can be calculated from porosity are the effective ionic conductivity and effective diffusivity of electrolytes, which are calculated in a similar manner from porosity, tortuosity, and bulk electrolyte conductivity / diffusivity. Equations 2 and 3 are functional expressions relating the effective ionic conductivity to the bulk ionic conductivity (Equation 2) and the effective diffusivity to the bulk ionic diffusivity (Equation 3) for orthotropic materials. As Figure 2 As shown, tortuosity along a given direction is a measure of the shortest path through the material, and tortuosity itself can be considered a function of porosity through the well-established, albeit empirical, Brugmann relation,

[0050]

[0051] Among them, τ e is the tortuosity of the electrolyte along a given direction, ε e is the porosity and α is the Bruggemann index in a given direction. A Bruggemann index of 0.5 is usually assumed, which corresponds to a packed uniform sphere. However, for real materials, the index can differ significantly from this value.

[0052]

[0053]

[0054] Equation 2 provides the anisotropic effective ionic conductivity, and Equation 3 provides the anisotropic effective diffusivity as a function of porosity via the Brugmann relation. Here, K e is the bulk ionic conductivity, D e is the bulk diffusivity.

[0055] like Figure 3 To calculate the Brugmann indices, the original image of each material is decomposed into multiple subvolumes, as shown in Figure 2. A method for calculating the effective diffusivity is then run on each subvolume, for example, using a simulation code, to calculate a single effective diffusivity for each subvolume, along with the porosity within the subvolume. The effective diffusivity is then plotted against the porosity, and the Brugmann indices can then be fitted based on the bulk diffusivity and constitutive relations given by Equations 2 and 3.

[0056] Given that the Brugmann coefficient corresponds to the tortuosity of the microstructure, the same exponent can be used to describe the effective ionic conductivity curve. Although the equation describing the relationship between effective properties and porosity does not always follow the Brugmann relationship (for example, in the case of the effective thermal conductivity of electrode materials), the same strategy is used to generate porosity-dependent curves for all effective quantities.

[0057] The effective properties can be calculated as a function of an additional independent variable: saturation. Changes in saturation affect the amount of electrolyte in the medium, which affects the connectivity of the electrolyte and therefore the effective properties affected by the presence of the electrolyte. Additionally, the capillary pressure versus saturation curve can be calculated using a multiphase fluid flow lattice Boltzmann solver (e.g. PowerFLOW), which is used, for example, in FE implementations of the 3D Newman model in Abaqus to calculate the evolution of pore pressure with changes in saturation in fully coupled electrochemical-thermo-mechanical-pore pressure simulations. Although in Figure 4 The range of valid material properties described herein as part of the workflow is shown in , but in alternative embodiments the method can be extended to other properties and domains, for example, saturation versus pore pressure, porosity versus active surface area, porosity versus elastic modulus, covariation of all properties with saturation, etc.

[0058] The complete set of partial differential equations for the electrochemical, thermal and mechanical behavior of the battery is implemented in the Newman model 165. The degrees of freedom of the model include the lithium concentration in the liquid and solid phases of the electrodes and separator, the electric potential and ionic potential in the solid and liquid phases, respectively, and the temperature. A 3D model with anode, cathode and separator regions 140 is received as input, and effective properties are typically calculated based on the assumption of uniformly compressed spheres. However, these properties can be allowed to vary arbitrarily after the porosity field is input (for example, the porosity field calculated in the heterogeneous homogenization step). All other material properties associated with porosity through constitutive relations 150 can be used with an arbitrarily variable porosity field as an alternative to and / or in addition to the traditional relations.

[0059] The embodiments focus on two metrics 170 of performance and aging calculated from the results of Newman model simulations 165. For performance, the method focuses on the internal resistance of the battery cell, which can be calculated as the difference between the voltage divided by the current and the OCV divided by the current. For aging, the method focuses on the deposition potential, which is the difference between the solid potential and the electrolyte potential in the region of the anode near the separator. Lithium deposition only occurs when this potential is negative, with more deposition associated with lower potentials. Lithium deposition is a negative aging effect that removes available mobile lithium from the charge / discharge process and converts the mobile lithium into immobile solid metallic lithium, which slowly grows into crystalline structures called dendrites.

[0060] Contrary to typical assumptions in battery modeling, the Bruggmann exponents of battery materials are not necessarily isotropic, not necessarily close to 0.5, and vary between solids and electrolytes. Figure 5 The cells shown here only differ in the Bruggmann coefficients assigned to the solid and electrolyte phases. Under identical boundary conditions / loads, it was determined that models that implement the common assumption of a Bruggmann coefficient of 0.5 for both phases significantly overestimate the performance of the cell microstructure. In particular, Figure 6 As shown, in a representative homogeneous cell, the internal resistance of the cell is significantly higher, where the internal resistance of the cell can be determined by the difference between the open circuit voltage (OCV) divided by the current and the measured voltage divided by the current (the current is the same constant in both cases). The difference between the cells is only magnified when the discharge current is doubled (i.e., when the cell is charged at 6C instead of 3C, where 1C is the current required to fully discharge the cell in 1 hour). At some points in the 6C discharge simulation, the difference in internal resistance is as high as 100%. For a plot of OCV / voltage versus state of charge at 3C and 6C during the discharge phase, see Figure 6 It can be concluded that valid properties that actually reflect the microstructure at hand are crucial for simulation accuracy. The workflow of the embodiment enables efficient calculation of these material properties, which cannot be found in any available electrochemical simulation software.

[0061] The computed constitutive relations are a consequence of the heterogeneity of the microstructure of the battery material under consideration. However, in typical battery modeling applications, a single initial porosity / saturation is assigned to the battery material, which completely ignores the ongoing contribution of heterogeneity to the performance and aging of the battery cell. The workflow of the embodiment assigns heterogeneously distributed porosity and saturation at different coarsening levels. Since the most effective material properties are functions of these variables, this is equivalent to assigning a different effective ionic conductivity, diffusivity, etc. to each element in the grid. This enables the creation of heterogeneous mesoscale battery cells, where enough homogenization has been performed so that the Newman model can still be effectively utilized, but the heterogeneous distribution of porosity is maintained. This is achieved after full homogenization (see Figure 7 and Figure 8 Therefore, the mesoscopic scale battery (heterogeneous battery hereafter) retains the heterogeneity of the original scan of the microstructure, but has a computational efficiency closer to that of a fully homogenized battery.

[0062] The same effective material properties are used when simulating the charge / discharge cycles of heterogeneous and homogeneous cells; these constitutive relations are calculated from the material microstructure images using the above workflow. Figure 9As shown, only subtle performance differences exist between homogeneous and heterogeneous cells, although it is expected that these differences may be more pronounced at higher C-rates as more material properties are considered as a function of spatially varying variables such as porosity. Figure 9 Plots of OCV and voltage versus SOC are shown for 3C and 6C, respectively. In addition to the effective properties following the Brugmann relation, a constitutive relation for thermal conductivity is also included in this analysis.

[0063] The stark difference between heterogeneous and homogeneous cells becomes apparent when considering the various aging mechanisms that batteries are expected to undergo. One of these phenomena is called lithiation, in which metallic lithium can form on the surface of anode particles, typically in the area bordering the separator. This process is generally assumed to be irreversible and removes available active surface area and cyclable lithium from future charge / discharge cycles. Experimental evidence suggests that different regions of the battery will precipitate to different extents, and simulations have also shown that the spatial extent of precipitation is related to porosity (see, for example, “Quantifying Heterogeneities / Degradation During Fast Charge,” presented by Andrew Colclasure at the DOE Vehicle Technologies Profram 2020 Annual Value Review and Peer Assessment Conference, June 1–4, 2020, and “Understanding the Impact of Local Heterogeneities During Fast Charge,” presented by Matthew Keyser, Kandler Smith, Hakim Iddir, et al. at the U.S. Department of Energy Vehicle Technologies Office Annual Value Review, Arlington, Virginia, June 10–13, 2019). Therefore, when heterogeneity is included in the model, different aging behaviors in regions with different local porosity can be captured.

[0064] While the actual mechanics of lithium deposition (i.e., the growth of lithium on the particle surface) are not explicitly modeled, the point at which lithium deposition becomes thermodynamically favorable can be determined. Lithium deposition is only possible when the potential difference between the solid and liquid in the anode is negative. Figure 10This difference at the junction closest to the separator is plotted against the state of charge in both heterogeneous and homogeneous cells during charge cycles at 3C and 6C. For the homogeneous cell, only the average difference at each point is reported, as it is essentially constant for the homogeneous cell, but for the heterogeneous case, the average potential of all points close to the separator, as well as the average value only in regions with unusually high or low porosity, is plotted to show the entire potential range. Clearly, precipitation occurs at a slightly higher state of charge in the heterogeneous cell than in the homogeneous cell. Consequently, the potential difference is higher in the heterogeneous cell at each state of charge, even at the extremes.

[0065] The most striking differences emerge when examining the distribution of the potential difference (hereafter referred to as the electrochemical potential for lithiation (EPOTLPL)) and how this distribution relates to the porosity distribution. Simulations clearly show that the EPOTLPL is porosity-dependent, as expected. Furthermore, the EPOTLPL exhibits a broad distribution in heterogeneous cells, while the distribution in homogeneous cells can be simplified to a Dirac delta function. Figure 11 These distributions are shown, along with graphs showing the model region from which the data were taken. The top graph shows the distribution of precipitation potential versus porosity, the center graph shows a graphical illustration of where precipitation occurs, and the bottom graph plots the distribution of precipitation potential in heterogeneous and homogeneous cells.

[0066] Since higher precipitation potentials are associated with less precipitation, this suggests that different anode regions close to the separator will precipitate to different extents, whereas the homogeneous model predicts the same degree of precipitation in all cell regions at a given distance from the separator. Figure 12 Depicted are contour plots of the precipitation potential at the anode surface closest to the separator (EPOTLPL) at the end of the charging phase.

[0067] Given that predicting the extent to which lithium plating occurs is crucial for predicting the long-term behavior of a battery, it is desirable to understand which areas of the battery are more prone to plating than other areas. The workflow of the exemplary embodiment enables this determination, which is not possible with other battery simulation software.

[0068] By tuning the porosity distribution, subtle performance differences between heterogeneous and homogeneous models can be amplified. Experimental evidence (Yari et al., 2020) suggests that certain particle distributions that result in a certain porosity distribution promote better ion transport within the battery. For example, introducing high-porosity channels into anode and cathode materials by laser ablation or micro-drilling can lead to better transport properties (see Dunlap et al., 2022). These lines of reasoning motivate the development of optimized batteries with artificial porosity distributions, in which high-porosity "highways" span the thickness of the anode and cathode, but the average porosity of the anode, cathode, and separator remains equal to the porosity of the corresponding regions of the previously considered unoptimized heterogeneous battery, e.g. Figure 13 Furthermore, the constitutive relations and material properties are equivalent between the models.

[0069] Figure 14 The voltage / OCV curves versus SOC at 2.4C and 4.8C are plotted for the heterogeneous and optimized cells. For these cells, significant differences in internal resistance between the cells were observed, even at a discharge current of 2.4C. As expected, this difference became more pronounced as the C rate increased. In addition, there was a significant difference in EPOTLPL between the optimized and heterogeneous cells, where the optimized cell was observed to precipitate at a state of charge 8% higher than that of the heterogeneous cell, as shown in Figure 5. Figure 15 Therefore, the EPOTLPL in the optimized cell is generally higher than that of the heterogeneous cell at all points in the charge cycle. Figure 16 This fact is further emphasized by the distribution of EPOTLPL in the anode close to the separator at the end of the charging phase shown, where the distribution of the optimized cell is clearly biased towards higher EPOTLPL, meaning that overall less precipitation should occur during charging. Figure 17 As shown in the corresponding contour plot in , it is clear that the high EPOTLPL regions corresponding to less precipitation are concentrated around the high porosity “highway”, as can be expected from the correlation between porosity and EPOTLPL mentioned earlier.

[0070] Figure 181 is a flow chart of an exemplary embodiment of a method 130 for scaling up a microstructure (e.g., provided by DigitalROCK) to a continuum modeler (e.g., Abaqus) for subsequent modeling of, for example, a lithium-ion battery. It should be noted that any process description or block in the flow chart should be understood to represent a module, segment, portion of code, or step comprising one or more instructions for implementing the specific logical functions in the process, and that alternative implementations are included within the scope of the present invention, wherein, depending on the functionality involved, the functions may be performed out of the order shown or discussed, including substantially simultaneously or in reverse order, as would be understood by a person having appropriate skill in the art of the present invention.

[0071] As shown in block 1810, a 3D microstructure 110 ( Figure 1 ), the battery components are, for example, anodes, cathodes, separators, and / or CBDs. For example, the 3D microstructure can be obtained by X-ray microtomography or by a program. As shown in block 1820, bulk and / or pure material properties 120 of the battery components are received ( Figure 1 ), bulk and / or pure material properties such as diffusivity, thermal conductivity, electrical conductivity (solids), ionic conductivity (electrolytes), elastic properties, etc.

[0072] As shown in box 1830, emergent properties are calculated from the battery component microstructure as a function of porosity. For example, a microstructure image can be uploaded into DigitalROCK as an 8-bit raw file. Here, the porous media characterization capabilities of DigitalROCK can be used to calculate emergent properties as a function of porosity as follows: the microstructure is first segmented into multiple regions (to have representative values ​​at different porosities), and then the material properties of each sub-region are calculated, for example, by DigitalROCK's diffusion solver. The results from DigitalROCK can then be interpreted, for example, by a python script. Here, for each battery component sub-region, Brugmann coefficients can be calculated for effective ionic conductivity, electrical conductivity, thermal conductivity, and diffusivity, as shown in box 1840. For example, the built-in curve fitting function in the ScyPy python script can be used to determine the Brugmann coefficients for effective ionic conductivity, electrical conductivity, and diffusivity. For thermal conductivity, the generalized Brugmann relationship has the form:

[0073]

[0074] It is fitted to the data using the fitting parameters A and a, where the bulk thermal conductivity of the electrolyte is added to the right to ensure that a solid volume fraction of zero yields the thermal conductivity of the fluid. Given that Abaqus can define orthotropic Brugmann coefficients for the effective ionic conductivity and diffusivity in the electrolyte, only these values ​​are stored for subsequent generation of the Abaqus input file. For the effective electrical and thermal conductivities, tables can be generated based on the calculated curves and stored for inclusion in the Abaqus input file, for example, with the keywords conductivity and conductivity. There may also be relationships between elastic properties and porosity, between effective area and porosity, between saturation and pore pressure, and additional saturation dependencies for all constitutive relations.

[0075] As shown in block 1850, a heterogeneous mesoscopic scale 3D battery model 140 ( Figure 1 ). For example, in DigitalROCK, anode materials, cathode materials, and separator materials can also be combined into a single battery structure, as shown in box 1852. Each of the anode, cathode, and separator is divided into coarse voxels, as shown in box 1854. For example, starting from a single battery structure, the original 8-bit image can be coarsened into a 32-bit image in the sense that the 8-bit image can be decomposed into larger voxels, where an average porosity is assigned to each voxel; therefore, 32-bit storage is required to achieve accurate porosity reporting. The voxel size is used for each battery region (anode, cathode, separator), where a unique set of voxels can always be defined with a uniform voxel size. It should be noted that the x, y, and z dimensions of each voxel do not need to be the same, which can be achieved when the images used from each material are cropped accordingly before combining. Preferably, the desired level of coarsening can be determined before building the model so that the size of the model can be determined accordingly.

[0076] Based on the final roughened image, the original file can be converted into a text file, for example, by an executable file that associates porosity with each roughened voxel by its position in the image. Two executable files are then used for this purpose, one to be used depending on the byte order of the image. The roughening level can be selected for a given grain size. The roughened image and constitutive relations can be used to create an Abaqus input file. Preferably, the user knows the desired material properties (in addition to those calculated above), for example, from experimental data or other sources.

[0077] An exemplary method for generating a single battery cell with a one-dimensional load is described below. For example, using the Python script described above, based on the provided microstructure 3D model 110 ( Figure 1 ) and bulk / pure material properties 120( Figure 1) creates an input file for Abaqus. Details of one example of such a script in other possible implementations are provided below. The user input to the script can be a text file containing the portion of the input file that does not pertain to the material properties calculated by DigitalROCK, omitting the geometry definitions. In addition, the pixel dimensions of the anode and separator in the thickness direction, as well as the (pixel) dimensions of the entire cell, a scaling factor (e.g., if the cell is 14 pixels wide, but the physical dimensions of the cell are 140 microns at this size, the scaling factor could be 10-5), the number of cycles to be simulated (at least two cycles are recommended, as it is impractical to simulate from the equilibrium state of the first cycle), the C-rate to be used in the simulation, a label to determine whether the cell is homogeneous or heterogeneous, a text file to be generated from the roughened image, and the desired name of the input file to be generated. Optionally, the variable x, y, and z dimensions (as a function of the scaling factor) can be defined for the elements to be created in Abaqus; by default, cubic elements are created with dimensions equal to the scaling factor.

[0078] This example's geometry generation and region assignment rely heavily on the NumPy Python package. First, node coordinates are generated from a NumPy array based on the number of voxels in each direction and the voxel size (default: scale factor). NumPy's mesh feature is then used to generate a multidimensional grid with these coordinates. These coordinates are stacked and transposed so that each row of the object named "node_coordinates" represents the x, y, and z coordinates of a node. Mesh's mesh feature is then used again on the cell's array of dimensions to assign each node a unique flattened index, which is then decomposed and converted into a dictionary, where the node coordinates are assigned to their corresponding node index. A grid is created that corresponds to the element index (rather than the node index). From this grid, element nodes are procedurally assigned to each element based on their position in the grid (the node grid has one more row in each direction than the element grid, so the assignment is determined accordingly). This grid is then transposed so that each row corresponds to a voxel and each column corresponds to one of the eight element nodes. After this, the element set is defined by its position in the thickness direction; therefore, the user does not need to define the anode and separator voxel thicknesses. This is done by first averaging the y (thickness) coordinates in each element and calculating the center of each element. Here, a new list containing only the y coordinates of each node is also determined so that the extreme faces of the battery can then be determined. Next, the anode, cathode, and diaphragm node sets are determined by the voxel size, pixel size, and center y coordinate of the anode and diaphragm. That is, if the center of an element is below the extreme edge of the anode, the element is included in the anode set, if the center of the element is above this point but below the extreme edge of the diaphragm, the element is included in the diaphragm set, otherwise the element is included in the cathode set. Elements at either end of the battery are then included in either the "ground" set (elements at the end of the anode) or the "load" set (elements at the end of the cathode). Finally, a node set is generated for each of these elements by adding the nodes from each element to the list.

[0079] A zero array is generated, which becomes a porosity array for each element, with dimensions equal to the number of voxels in the cell. The element indices are defined so that they correspond directly to the porosity assignments in the previously generated text file, thus directly assigning porosity based on the corresponding position of the element indices in the image. This array is then flattened to match the dimensions of the array containing the columns of element node numbers. While optional, the next step illustrates the power of this general approach to mesh generation: while running a test, you may want to translate the geometry down in the y-direction by the thickness of the cell and rotate the geometry 90 degrees. Since porosity is defined for each element, elements are defined by node number, and node coordinates can be assigned to node numbers based solely on their position relative to each other in the grid. If the grid of node coordinates is rotated or translated together, the elements, element collections, and porosity assignments are translated accordingly. Therefore, all information can be effectively rearranged by translating all node y coordinates down and rotating the entire array 90 degrees. At this point, the geometry is fully defined.

[0080] The remaining node set assignments to be made involve setting reference points at the origin and sensor nodes, which can be located at the end of the cathode, at the bottom corner opposite the origin along the x-direction, which can be determined by looking up the desired coordinate in the node coordinate list and then returning the index of the desired coordinate. Functions can be used to write element and node sets in the style of an Abaqus input file. These node and element sets are then written to the input file along with the information required by Abaqus, which is related to the element type (in this example, QEC3D8), surface definition of the extreme surfaces, material definition (assuming the materials are named as described in the Abaqus electrochemistry documentation), and orientation. The rest of the input file is printed based on the text file provided by the user, the given material property curves, and the porosity defined according to the initial conditions.

[0081] After the geometry is generated procedurally, this information is printed to the Abaqus input file, along with the material property definitions, the porosity assignments for each element (where an average value is assigned to the set of elements if the anode, cathode, and separator are assumed to be homogeneous), and the step information. The charge rate is calculated based on the 1-C rate (included in a text file provided by the user), and the number of steps depends on the number of cycles defined by the user (again, a single cycle at 1-C will be provided by the user). The conductivity is typically defined by a properties table of the bulk conductivity and Bruggmann coefficient used in the electrolyte. The simulation will not run unless this table is defined, so the bulk conductivity defined in this way must be set to zero when generating the input file. This makes the conductivity defined by the Bruggmann relation zero, and only the conductivity defined by the keyword ELECTRIC CONDUCTIVITY is then used. Taking into account that by default neither this keyword nor the (thermal) CONDUCTIVITY keyword is allowed to depend on the solid volume fraction, the user subroutine "USER FIELD" is used together with the GETVRM function to obtain the porosity, where the solid volume fraction can be calculated as 1 minus the porosity and then provided as a field variable.

[0082] With the creation of the input file, the scale-up method is complete as the user is now able to run the model in Abaqus and subsequently perform post-processing based on the user's specific needs.

[0083] The present system for performing the functions described in detail above may be a computer, an example of which is shown in FIG. Figure 19 Schematic diagram of is shown. System 2000 includes processor 502, storage device 504, memory 506, input / output (I / O) device 510 (or peripheral device), local bus or local interface 512, and software 508 defining the functions mentioned above is stored in memory 506, and local bus or local interface 512 enables communication within system 2000. Local interface 512 can be, for example, but not limited to, one or more buses or other wired or wireless connections, as known in the art. Local interface 512 can have additional elements (for example, controller, buffer (cache), driver, repeater and receiver) to enable communication, and these elements are omitted for simplicity. In addition, local interface 512 may include address, control and / or data connections to enable appropriate communication between the components mentioned above.

[0084] Processor 502 is a hardware device for executing software, particularly software stored in memory 506. Processor 502 can be any custom or commercially available single-core or multi-core processor, a central processing unit (CPU), a secondary processor among several processors associated with the present system 2000, a semiconductor-based microprocessor (in the form of a microchip or chipset), a microprocessor, or generally any device for executing software instructions.

[0085] The memory 506 may include any one or a combination of volatile memory elements (e.g., random access memory (RAM, such as DRAM, SRAM, SDRAM, etc.) and non-volatile memory elements (e.g., ROM, hard drive, magnetic tape, CDROM, etc.). In addition, the memory 506 may include electronic, magnetic, optical, and / or other types of storage media. It should be noted that the memory 506 may have a distributed architecture in which components are located remotely from each other but can be accessed by the processor 502.

[0086] According to the present invention, software 508 defines the functions performed by system 2000. The software 508 in memory 506 may include one or more separate programs, each of which contains an ordered listing of executable instructions for implementing the logical functions of system 2000, as described below. Memory 506 may contain operating system (O / S) 520. The operating system generally controls the execution of programs within system 2000 and provides scheduling, input and output control, file and data management, memory management, communication control, and related services.

[0087] I / O devices 510 may include input devices such as, but not limited to, a keyboard, a mouse, a scanner, a microphone, etc. Furthermore, I / O devices 510 may also include output devices such as, but not limited to, a printer, a display, etc. Finally, I / O devices 510 may also include devices that communicate via input and output, such as, but not limited to, a modulator / demodulator (modem; used to access another device, system, or network), a radio frequency (RF) or other transceiver, a telephone interface, a bridge, a router, or other devices.

[0088] When the system 2000 is in operation, the processor 502 is configured to execute software 508 stored within the memory 506 , transfer data to and from the memory 506 , and generally control the operation of the system 2000 according to the software 508 , as explained above.

[0089] When the functionality of system 2000 is in operation, processor 502 is configured to execute software 508 stored in memory 506, transfer data to and from memory 506, and generally control the operation of system 2000 according to software 508. Operating system 520 is read by processor 502, possibly buffered within processor 502, and then executed.

[0090] When system 2000 is implemented in software 508, it should be noted that the instructions for implementing system 2000 may be stored on any computer-readable medium for use by, or in conjunction with, any computer-related device, system, or method. In some embodiments, such computer-readable medium may correspond to either or both of memory 506 and storage device 504. In the context of this document, a computer-readable medium is an electronic, magnetic, optical, or other physical device or apparatus that can contain or store a computer program for use by, or in conjunction with, a computer-related device, system, or method. The instructions for implementing the system may be embodied in any computer-readable medium for use by, or in conjunction with, a processor or other such instruction execution system, device, or apparatus. While processor 502 is mentioned by way of example, in some embodiments, such an instruction execution system, device, or apparatus may be any computer-based system, system containing a processor, or other system that can retrieve and execute instructions from an instruction execution system, device, or apparatus. In the context of this document, a "computer-readable medium" can be any means that can store, communicate, propagate, or transport the program for use by or in connection with a processor or other such instruction execution system, apparatus, or device.

[0091] Such a computer readable medium can be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or propagation medium. More specific examples (a non-exhaustive list) of computer readable media may include the following: an electrical connection having one or more conductors (electronic), a portable computer floppy disk (magnetic), a random access memory (RAM) (electronic), a read-only memory (ROM) (electronic), an erasable programmable read-only memory (EPROM, EEPROM, or flash memory) (electronic), an optical fiber (optical), and a portable compact disc read-only memory (CDROM) (optical). It should be noted that the computer readable medium can even be the paper or another suitable medium on which the program is printed, as the program can be electronically captured, for example, by optically scanning the paper or other medium, and then compiled, interpreted, or otherwise processed in a suitable manner if desired, and then stored in a computer memory.

[0092] In an alternative embodiment where the system 2000 is implemented in hardware, the system 2000 may be implemented using any one or a combination of the following technologies, all of which are well known in the art: discrete logic circuits having logic gates for implementing logic functions based on data signals, application specific integrated circuits (ASICs) having appropriate combinations of logic gates, programmable gate arrays (PGAs), field programmable gate arrays (FPGAs), etc.

[0093] Advantageously, the above-described embodiments provide an effective means of improving the accuracy of battery cell continuum model simulations without having to implement an infeasible number of degrees of freedom or significantly alter the well-known framework (the Newman model). By moving away from common microstructural model assumptions, such as a Bruggmann index of 0.5, the embodiments are even able to correctly capture microstructural differences between similar battery cells; for example, electrodes with similar porosity may have different tortuosity values.

[0094] Embodiments provide a practical approach for modeling aging and degradation mechanisms throughout a battery driven by local extreme property values ​​(e.g., concentration, temperature, or precipitation potential). Performance captured by metrics such as internal resistance can be realized as a direct result of different microstructures, even those with the same average porosity in each material (anode, separator, cathode).

[0095] The examples further optimize the electrode microstructure for both performance and aging, as shown in the detailed section of the optimized battery model that accurately captures the structural properties of heterogeneity. This result points to the clear potential of microstructural optimization of battery cell performance.

[0096] In typical continuum battery modeling, the optimized cell and the heterogeneous cell are indistinguishable from each other because the anode, cathode, and separator materials have the same porosity between models. Therefore, both the optimized cell and the heterogeneous cell can be modeled as the previously considered homogeneous cell. By adjusting the size or location of these highways, or further changing the porosity distribution in other ways, it is possible to further improve the performance of the battery cell. Therefore, in addition to the increased accuracy from the more accurate constitutive relations, the above embodiments enable battery microstructure optimization that was not possible with previous battery simulation techniques.

[0097] It will be apparent to those skilled in the art that various modifications and variations may be made to the structure of the present invention without departing from the scope or spirit of the present invention. In view of the foregoing, it is intended that the present invention encompass modifications and variations of the present invention as long as these modifications and variations fall within the scope of the appended claims and their equivalents.

Claims

1. A method for upscaling from a microstructure to generate a coarsened heterogeneous spatial distribution of porosity and a set of porosity-dependent constitutive relations, comprising the following steps: receiving a three-dimensional 3D microstructure model of each of a plurality of battery components, wherein the battery components include an anode, a cathode, and a separator; receiving bulk material properties and / or porosity of each of the battery components; calculating a coarsened porosity model having emergent properties from the battery component microstructure according to the porosity; For each battery component sub-region, calculating a Brugmann coefficient from the group consisting of effective ionic conductivity, electrical conductivity, thermal conductivity, and ionic diffusivity; and Creating a heterogeneous mesoscopic 3D battery model also includes the following steps: combining anode materials, cathode materials, and separator materials into a single cell structure; and Each of the anode, cathode and separator was divided into coarse voxels to create a 3D model of porosity.

2. The method according to claim 1, wherein: The received porosity of the bulk material property comprises a fine resolution model including a plurality of elements, wherein each element is characterized as either a solid having substantially zero porosity or a pore having 100% porosity.

3. The method according to claim 2, wherein: Calculating emergent properties from the battery component microstructure based on the porosity, further comprising the steps of: Calculate the average porosity in the subregion for each material; Assigning to each element of the roughened porosity model the average porosity of all elements in the region; and Each element of the finite element mesh is assigned the average porosity.

4. The method according to claim 1, wherein: For each of the anode, cathode, and separator, the voxel is always uniquely defined as having a uniform voxel size.

5. The method of claim 1 further comprising the step of creating an Abaqus input file using the roughened image and the constitutive relations.

6. The method of claim 1 further comprising the step of receiving the roughened porosity model and associated constitutive relations as input to a continuum electrochemical solver.

7. The method according to claim 1, further comprising the steps of: Based on the saturation, the effective characteristics are determined.

8. The method according to claim 1, further comprising the steps of: Calculate the capillary pressure versus saturation curve; and Calculate the change in pore pressure as saturation changes.

9. The method according to claim 6, further comprising the step of simulating the 3D Newman model (165), wherein: The continuum electrochemical solver includes a 3D Newman model simulator.

10. The method of claim 9, further comprising the step of calculating performance and aging metrics (170) based on the degrees of freedom of the 3D Newman model (165).

11. The method of claim 10, further comprising the step of determining a point on the surface of the 3D Newman model (165) at which lithium deposition becomes thermodynamically favorable.

12. The method according to claim 11, further comprising the step of optimizing the microstructure of the anode material and the cathode material of the 3D Newman model.

13. The method of claim 12, further comprising the step of introducing a plurality of high porosity channels into the anode material and the cathode material of the 3D Newman model.

14. The method according to claim 1, further comprising the step of printing the heterogeneous mesoscopic-scale 3D battery model to a file.

15. The method of claim 14, further comprising the step of providing the heterogeneous mesoscopic scale 3D battery model file to a continuum modeler.

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