Method for upscaling from microstructure to continuum using mesoscale, heterogeneous homogenization

Abstract

To provide a method for upscaling a microstructure so as to generate a coarsened heterogeneous spatial distribution of porosity and a set of porosity dependent constitutive relationships.SOLUTION: A method for upscaling from a microstructure to a continuum comprises the steps of: receiving a three-dimensional (3D) microstructure model, bulk material properties, and / or porosity for each of battery components of an anode, a cathode and a separator; calculating a coarsened porosity model with emergent properties from the battery component microstructures as a function of the porosity; calculating Bruggeman coefficients for each battery component sub region from the effective ionic conductivity, electric and thermal conductivity, and ionic diffusivity; and creating a heterogeneous mesoscale 3D battery model by combining the anode, cathode and separator materials into a single cell structure and separately partitioning each into coarse voxels to create a 3D model of porosity.SELECTED DRAWING: Figure 18

Description

Detailed Description of the Invention

[0001] [CROSS REFERENCE TO RELATED APPLICATIONS] This application claims the benefit of U.S. Provisional Patent Application No. 63 / 592,325, filed October 23, 2023, entitled "Method for Electrochemical Modeling of Lithium-Ion Batteries by Microstructure-to-Continuum Upscaling Using Mesoscale Heterogeneous Homogenization," which is incorporated by reference in its entirety. [Field of the Invention] The present invention relates to sustainable energy storage using lithium-ion batteries and associated electrochemical simulations, and more specifically to a framework for determining the impact of heterogeneity on battery performance. [Background of the invention] In the scientific literature, there are two types of electrochemical models for lithium-ion batteries, both of which have drawbacks related to capturing performance and ageing drivers in a practical way.

[0002] (1) Continuum modeling, such as finite element (FE) analysis, where elements are in the range of millimeters with degrees of freedom to describe homogenized material properties (chemical composition and modeled microstructural behavior). While this provides the ability to simulate full-cell batteries, continuum modeling either uses generic empirical relationships to model microstructural behavior or requires laboratory measurements that are difficult to obtain.

[0003] (2) Microstructural modeling, where sub-micrometer resolution can capture the real microstructural behavior but is limited to small sample sizes. Microstructural modeling provides accurate microstructural behavior instead of generic empirical relationships and reduces the need for laboratory data that are difficult to obtain. However, simulating full-cell behavior with microstructural modeling is not practical due to the large number of degrees of freedom required.

[0004] The accuracy resulting from the heterogeneity of properties captured by microstructural modeling is important for performance, aging and degradation simulations, but the model size required for full-cell level simulations is not feasible with current computing power. Therefore, there is currently no complete solution based on a microstructural modeling approach. Existing continuum models typically assign uniform material properties to each material (anode, cathode, separator), which (incorrectly) assumes that the existing heterogeneity can be ignored. Therefore, it is necessary to upscale the heterogeneity from a microstructural model to a continuum model. [Summary of the Invention] Embodiments of the present invention provide a method to couple fine-resolution battery microstructural characterization (micrometer scale) with coarse-resolution battery electrochemical simulation (millimeter scale) within reasonable and practical coarse-resolution model sizes while preserving the effects of fine heterogeneities on performance, aging, and degradation.

[0005] The microstructure is upscaled to generate a coarsened heterogeneous spatial distribution of porosity and a set of constitutive relationships that depend on the porosity. A three-dimensional (3D) microstructural model, bulk material properties, and / or porosity are received for each battery component: anode, cathode, and separator. From the battery component microstructure, a coarsened porosity model with emergent properties is calculated depending on the porosity. The Bruggeman coefficients for each battery component subregion are calculated from the effective ionic conductivity, electrical conductivity, thermal conductivity, and ionic diffusivity. A heterogeneous mesoscale 3D battery model is created by combining the anode, cathode, and separator materials into one cell structure and partitioning each separately into coarse voxels to create a 3D model of the porosity.

[0006] Other systems, methods and features of the invention will be or become apparent to one with skill in the art upon examination of the following figures and detailed description. All such additional systems, methods and features are intended to be included herein, be within the scope of the invention, and be protected by the accompanying claims. [Brief description of the drawings]

[0007] The accompanying drawings are included to provide a further understanding of the invention, and are incorporated in and constitute a part of this specification. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the invention. The drawings illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention. [Figure 1] 1 is a flowchart outlining a workflow, an overview of a first exemplary embodiment of a simulation framework. [Diagram 2] FIG. 1 shows two contrasting examples of flexibility. [Diagram 3] FIG. 1 illustrates an exemplary process for fitting Bruggeman coefficients / constitutive relationships. [Figure 4] Here is a series of four plots showing examples of computable constitutive relations. [Diagram 5] FIG. 1 illustrates base case and homogeneous case of material properties. [Figure 6] Voltage / OCV vs. SOC curves at 3C (top) and 6C (bottom) comparing basic and homogeneous cells. [Figure 7] 1A-1C illustrate an example of a coarsening process for obtaining heterogeneous and homogeneous battery cells. [Figure 8] FIG. 8 is a detail of FIG. 7 showing the relative porosity of the cathode, separator, and anode. [Figure 9] 1 is a plot showing voltage / OCV vs. SOC curves at 3C (top) and 6C (bottom) comparing homogeneous and heterogeneous cells. [Figure 10]1 shows plots of anodic plating potential at the separator surface (EPOTLPL) versus SOC for 3C and 6C. [Figure 11] Shown is the distribution of plating potential versus porosity (top), a diagram of where plating occurs (center), and the distribution of plating potential in heterogeneous and homogeneous cells (bottom). [Figure 12] FIG. 11 is a contour plot of the plating potential at the anode surface near the separator (EPOTLPL) at the end of the charge phase comparing homogeneous and heterogeneous cells. [Figure 13] FIG. 1 is an explanatory diagram of the porosity distribution of a heterogeneous cell and an optimized cell. [Figure 14] Plots of Voltage / OCV vs. SOC curves at 2.4C (top) and 4.8C (bottom) comparing heterogeneous and optimized cells. [Figure 15] 1 shows plots of the anodic plating potential (EPOTLPL) on the separator surface versus SOC at 2.4C and 4.8C comparing the heterogeneous and optimized cells. [Figure 16] The distribution of EPOTLPL versus porosity (top) and the distribution of EPOTLPL in the heterogeneous and optimized cells (bottom) are shown. [Figure 17] FIG. 11 Contour plot of EPOTLPL at the anode surface near the separator at the end of the charging phase comparing the heterogeneous and optimized cells. [Figure 18] 1 is a flow chart of an exemplary embodiment of a method for upscaling a microstructure to a continuum. [Figure 19] FIG. 1 is a schematic diagram illustrating an example of a system for performing the functions of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0008] The following definitions are useful in interpreting the terms applied to the features of the embodiments disclosed herein and are intended solely to define the elements within the present disclosure.

[0009] As used in this disclosure, "Abaqus" refers to a software suite for finite element analysis and computer-aided engineering that was first released in 1978. Abaqus is a suite of engineering analysis software packages used to simulate the physical response of structures and solids to various environmental conditions. Abaqus is used by engineers to simulate complex problems in various industries. As used in this disclosure, "Newman model" refers to a theoretical electrochemical framework utilized in Abaqus to simulate the charging and discharging of battery cells. Abaqus provides a framework to simulate battery multiphysics that, in combination with thermal, swelling, and elasticity theories, aids in the design and optimization of battery cells. For example, the Newman model can be used to calculate concentration, potential, temperature, displacement, and pore pressure as degrees of freedom.

[0010] As used in this disclosure, "Bruggeman relationship" and "Bruggeman index" refer to the relationship between porosity and tortuosity of a granular porous medium. The Bruggeman relationship is an inverse power law of porosity versus an index called the Bruggeman index, which is typically assumed to have a value of 0.5.

[0011] As used in this disclosure, "continuum modeling" (finite element (FE) analysis) refers to a method of numerically solving partial differential equations associated with degrees of freedom (concentrations, electric potentials, etc.) defined in a continuum that cannot be solved analytically. A continuum model is a representation of a process or structure that changes gradually between two defined points. These points may be tangible, e.g., numerical or time, or they may be intangible. Exemplary embodiments herein refer to the Abaqus software suite for continuum modeling.

[0012] As used in this disclosure, "microstructural modeling" refers specifically to calculating transport properties taking into account the explicit presence of porous media at a scale where individual pores / solid phases are distinct and separated, and the effects of microstructure can be added to the pure material transport properties to obtain effective transport properties. Exemplary embodiments herein use DigitalROCK software for digital pore-scale simulation of microstructure material properties, providing, for example, pore space analysis, MICP curves, absolute permeability, relative permeability, capillary desaturation, tortuosity, ionic conductivity and diffusivity, and electrical and thermal conductivity.

[0013] As used in this disclosure, "battery" or "battery cell" refers to an energy storage device, such as a lithium ion battery.

[0014] As used in this disclosure, "C-rate" refers to the multiple of the current required to charge a battery in one hour.

[0015] As used in this disclosure, "tortuosity" refers to a measure of the shortest path through a material in a given direction.

[0016] As used in this disclosure, "grid" or "computational grid" generally refers to a geometrically defined area of ​​a battery cell, typically used for simulations of single layers and / or multiple layers, such as the anode, cathode, and separator, rather than the entire layer structure.

[0017] Reference will now be made in detail to embodiments of the invention, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numbers are used in the drawings and the description to refer to the same or like parts.

[0018] "The design and optimization of battery cells is an expensive, time-consuming and challenging process that typically relies on laboratory testing of the various components, materials and processes that lead to the final battery cell. The process of replicating these components, including not only the cell design but also the microstructure and chemical composition of the materials, can speed up the optimization process of batteries according to relevant quantities such as energy density and charge rate, as well as required metrics such as performance, aging and degradation."

[0019] Exemplary embodiments herein provide an efficient way to effectively upscale battery microstructures (provided by DigitalROCK) to full continuum models (provided in Abaqus), thereby enabling efficient modeling of full lithium-ion battery cells that account for heterogeneity without introducing an unfeasible number of degrees of freedom. Exemplary embodiments combine the advantages of microstructural characterization and continuum modeling while minimizing the disadvantages of each. As shown in FIG. 1, embodiments take a representative 3D microstructural model 110 and corresponding material properties 120 of each electrode / separator in a battery cell as input to a heterogeneity homogenization process 130. For example, the representative 3D microstructural model 110 and corresponding material properties 120 may be provided from DigitalROCK. The heterogeneous homogenization process 130 produces two results: a coarsened, heterogeneous spatial distribution of porosity (mesoscale 3D coarsened porosity model 140) and a set of constitutive relations 150 for other properties (e.g., ionic diffusivity and conductivity, electrical and thermal conductivity, saturation versus pressure, specific surface area, and linear elastic modulus) that depend on porosity. The heterogeneous homogenization process 130 is described in further detail below with reference to FIG. 18.

[0020] The coarsening porosity model 140 and associated constitutive relations 150 are received as input to a continuum electrochemical solver 160, such as Abaqus, which includes a 3D Newman model simulator that generates a 3D Newman model 165, from which performance and aging metrics 170 can be calculated. This sequence of steps can be repeated while modifying the electrode / separator microstructure to achieve a full microstructure optimization cycle. This complete workflow takes into account the necessary heterogeneity for performance and aging, within a feasible computational model size. The present embodiment provides a simulation framework 1000 that accounts for the heterogeneity of properties in a full battery cell simulation, fully capturing the correct local and global behavior of these metrics, especially performance and aging.

[0021] Simulations of these embodiments can be integrated into battery cell manufacturing, providing parameter inputs directly to the battery manufacturing equipment, such as particle size distribution, amount of additives such as carbon binder domains, and target electrode porosity through control of applied calendar pressure. Additionally, the simulations represent the behavior of real electrodes with structured heterogeneity, providing geometric inputs during the associated manufacturing process, such as ablation and microchannel spacing.

[0022] Exemplary embodiments provide a mesoscale model of a battery cell that maintains heterogeneity without introducing an unfeasible number of degrees of freedom. The embodiments homogenize subregions of the anode, cathode, and separator to introduce heterogeneous porosity / saturation distributions without completely abandoning the assumptions inherent in the industry-standard Newman model (i.e., that particles and electrolyte can be assumed to be present at all material points). To homogenize the material properties, effective properties are calculated as a function of parameters related to the properties of a particular microstructure, such as porosity and saturation. In this embodiment, these constitutive relations are used as input for an Abaqus electrochemical simulation, and the analyzed battery cell has a heterogeneous porosity distribution determined from the average porosity of the scanned material subvolumes. In this workflow, the distribution, and therefore the microstructure of the material, can be tuned or optimized to achieve performance goals or to suppress the adverse effects of aging. The details of this process are provided below, along with some representative results.

[0023] In an exemplary embodiment, we first receive fully resolved microstructural 3D models 110 and material properties 120 of the anode, cathode, and separator components to establish accurate mesoscales for continuum modeling in Abaqus. These 3D models can be obtained directly from the actual materials, for example, by FIB-SEM (Focused Ion Beam Scanning Electron Microscopy) or X-ray microtomography 3D imaging. Alternatively, these materials can be procedurally generated by simulating the calendering process for various particles in Abaqus and then procedurally introducing additives such as those present in the carbon binder domain (CBD), or using custom 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)). For an example of a fully resolved microstructure, see Figure 7 (left).

[0024] Furthermore, given that all material constitutive relations 150 depend, at least implicitly, on the bulk properties 120 of the materials used, the exemplary embodiment receives bulk material properties such as ionic conductivity, diffusivity, electrical conductivity, etc. These properties can be obtained from the literature (see, e.g., Ecker et al., 2014; Madeleine Ecker et al. “Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery: I. Determination of Parameters” J. Electrochem. Soc. 162 A1836 (2015), and Madeleine Ecker et al. “Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery: II. Model Validation” J. Electrochem. Soc. 162 A1849 (2015)) or via molecular modeling and simulation using, e.g., Materials Studio.

[0025] In an exemplary embodiment, the mesoscale 3D model 140 is created by coarsening the original 3D microstructured material model 110 based on spatially varying porosity. In the original fine resolution model, each element is mostly solid (zero porosity) or porous (100% porosity), but in the new coarsened porosity model, each element has the average porosity of all smaller elements in the same region. This is achieved by calculating the average porosity of each material subregion and systematically assigning these porosities to individual elements of the finite element mesh via script. For example, if the original image is divided into cubes with sides of 50 pixels, each element corresponds to the average porosity of the corresponding 50 pixel cubic region of the original scan.

[0026] For porous / non-homogeneous media, most effective properties resulting from homogenization of the material microstructure can be calculated from the bulk material properties and the porosity (or alternatively the solid volume fraction) of the material. Given that these effective properties are used in the Newman model, the effective properties appropriately reflect the material microstructure in order to obtain results that accurately reflect the material at hand. An example of an effective material property that can be calculated via porosity is the effective ionic conductivity and effective diffusivity of the electrolyte, which are calculated in a similar manner depending on the porosity, tortuosity, and bulk electrolyte conductivity / diffusivity. Equations 2 and 3 are functional expressions that relate the effective ionic conductivity (Equation 2) to the bulk ionic conductivity, and the effective diffusivity (Equation 3) to the bulk ionic diffusivity in orthotropic materials. As shown in Figure 2, the tortuosity in a given direction, which is a measure of the shortest path through the material, can itself be considered as a function of porosity through the empirical but well-established Bruggeman relationship.

[0027]

number

[0028] where τ e is the tortuosity of the electrolyte in a given direction, ε e is the porosity,

[0029]

number

[0030] is the Bruggeman exponent in a given direction. The Bruggeman exponent is usually assumed to be 0.5, which corresponds to a uniform packing of spheres. However, in real materials, the exponent can vary considerably from this value.

[0031]

number

[0032]

number

[0033] Equation 2 provides the anisotropic effective ionic conductivity, and Equation 3 provides the anisotropic effective diffusivity as a function of porosity via the Bruggeman relationship. Here, K e is the bulk ionic conductivity, and D e is the bulk diffusivity.

[0034] To calculate the Bruggeman exponent, the raw image of each material is divided into several subvolumes, as shown in Figure 3. A method for calculating effective diffusivity is then run on each subvolume, for example via a simulation code, and a single effective diffusivity is calculated for each subvolume along with the porosity within the subvolume. The effective diffusivity can then be plotted against the porosity, and the Bruggeman exponent can be fitted from the bulk diffusivity and the constitutive relationship given in Eqs. 2 and 3.

[0035] Considering that the Bruggeman coefficient corresponds to the tortuosity of the microstructure, the same exponent can be used to describe the effective ionic conductivity curve. Although the equations describing the relationship between effective properties and porosity do not necessarily follow the Bruggeman relationship (e.g., in the case of the effective thermal conductivity of electrode materials), this same strategy can be adopted to construct porosity dependence curves for all effective quantities.

[0036] 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 electrolyte connectivity and therefore may affect effective properties affected by the presence of electrolyte. Additionally, capillary pressure-saturation curves can be calculated using a multiphase flow lattice Boltzmann solver such as PowerFLOW, which is used to calculate the change in pore pressure with changes in saturation, for example in fully coupled electrochemical-thermal-mechanical-pore pressure simulations in the FE implementation of the 3D Newman model in Abaqus. The range of effective material properties described herein as part of this workflow is shown in Figure 4, but in alternative embodiments, the approach can be extended to other properties and fields, such as saturation vs. pore pressure, porosity vs. active surface area, porosity vs. elastic modulus, covariance of all properties with saturation, etc.

[0037] A 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 this model include the concentration of lithium in the liquid and solid phases of the electrodes and separator, the electric and ionic potentials of the solid and liquid phases, respectively, and the temperature. A 3D model with the anode, cathode and separator regions 140 is taken as input and conventional effective properties are calculated based on the assumption that the spheres are uniformly packed. However, these properties can be arbitrarily varied following the input porosity field, as calculated in the heterogeneous homogenization step. Along with the arbitrarily varying porosity field, all other material properties related to the porosity via constitutive relations 150 can be used instead of and / or in addition to the conventional relations.

[0038] In an embodiment, the method focuses on two metrics 170, performance and aging, calculated from the results of the Newman model simulation 165. For performance, the method focuses on the internal resistance of the battery cell, which can be calculated as the difference between the voltage and the OCV divided by the current. For aging, the method focuses on the plating potential, which is the difference between the solid potential and the electrolyte potential in the region of the anode close to the separator. Lithium plating only occurs when this potential is negative, and more plating correlates with lower potential. Lithium plating is a negative aging effect that takes available mobile lithium from the charge / discharge process and converts it into immobile solid metallic lithium that slowly grows as crystalline structures called dendrites.

[0039] Contrary to common assumptions in battery modeling, the Bruggeman exponents of battery materials are not necessarily isotropic, nor are they necessarily close to 0.5, nor are they the same between solid and electrolyte. Consider the cell shown in Figure 5. The only difference is the Bruggeman coefficients assigned to the solid and electrolyte. Under identical boundary conditions / loads, models implementing the common assumption of a Bruggeman coefficient of 0.5 for both phases were found to significantly overestimate the performance of the battery microstructure. In particular, the internal resistance of the cell, which can be determined by the difference between the open circuit voltage (OCV) and the measured voltage divided by the current (which is the same constant in both cases), is clearly higher for a typical homogeneous cell, as shown in Figure 6. It is only when the discharge current is doubled (i.e., when the battery is charged at 6C instead of 3C, where 1C is the current that needs to be applied constantly to fully discharge the battery in 1 hour) that the difference between the cells is amplified. At certain points in the 6C discharge simulation, the difference in internal resistance is more than 100%. See Figure 6 for plots of OCV / voltage versus state of charge at 3C and 6C during the discharge phase. From this, it can be concluded that effective properties that actually reflect the microstructure at hand are important for the accuracy of the simulation. The workflow of the embodiment allows for efficient calculation of these material properties, which are not found in available electrochemical simulation software.

[0040] The calculated constitutive relations are a result of the heterogeneity of the microstructure of the battery material considered. However, in conventional battery modeling applications, the battery material is assigned a single initial porosity / saturation, which completely ignores the remaining contribution of the heterogeneity to the performance and aging of the battery cell. In the present embodiment workflow, we assign heterogeneous distributions of porosity and saturation at different coarsening levels. This is equivalent to assigning each element in the mesh a different effective ionic conductivity, diffusivity, etc., since most effective material properties are functions of these variables. This allows us to create a heterogeneous mesoscale battery cell. In this case, sufficient homogenization is performed so that the Newman model can still be effectively utilized, but the heterogeneous distribution of porosity is maintained. This is not achieved in the case of full homogenization (see Figures 7-8). In this way, the mesoscale cell (hereafter referred to as the heterogeneous cell) maintains the heterogeneity of the original scan of the microstructure, but with computational efficiency much closer to the fully homogenized cell.

[0041] The same effective material properties were used when simulating the charge-discharge cycles of the heterogeneous and homogeneous cells. These constitutive relations were calculated from the material microstructure images by the workflow procedure described above. Overall, there are only subtle differences in performance between the homogeneous and heterogeneous cells, as shown in Figure 9, but it is expected that these differences may become more pronounced at higher C-rates as more material properties are considered as a function of spatially varying variables such as porosity. Figure 9 shows plots of OCV and voltage vs. SOC at 3C and 6C, respectively. In addition to the effective properties following the Bruggeman relationship, the constitutive relationship for thermal conductivity is included in this analysis.

[0042] The differences between heterogeneous and homogeneous batteries become clearer when considering the various aging mechanisms that batteries are expected to undergo. One of these phenomena, known as lithium plating, can result in the formation of metallic lithium on the surface of the anode particles, usually in the area that abuts the separator. This process is generally assumed to be irreversible, removing available active surface area and cyclable lithium from future charge-discharge cycles. Experiments have shown that different regions of the battery experience different degrees of plating, and simulations have shown that the spatial extent of plating correlates with porosity (see, e.g., Andrew Colclasure “Quantifying Heterogeneities / Degradation During Fast Charge” Presentation, DOE Vehicle Technologies Profram 2020 Annual Merit Review and Peer Evaluation Meeting, June 1-4, 2020, and Matthew Keyser, Kandler Smith, Hakim Iddir, et.al. “Understanding Impact of Local Heterogeneities During Fast Charge” Presentation, USDepartment of Energy Vehicle Technologies Office Annual Merit Review, Arlington VA, June 10-13, 2019). Thus, including heterogeneity in the model can capture the different aging behavior in regions with different local porosity.

[0043] Although the actual mechanism of lithium plating, i.e., the growth of lithium on the particle surface, is not explicitly modeled, the point at which lithium plating becomes thermodynamically favorable can be determined. Lithium plating is possible only when the potential difference between the anode solid and the liquid is negative. Figure 10 shows plots of this potential difference at the integral point closest to the separator versus state of charge for heterogeneous and homogeneous cells for 3C and 6C charge cycles. For the homogeneous cell, it is essentially constant, so only the average difference at each point is reported, whereas in the heterogeneous case, the average potential of all points close to the separator is plotted along with the average value only in the regions of anomalously high or low porosity to show the full range of potentials. It is clear that plating occurs at a slightly higher state of charge in the heterogeneous cell than in the homogeneous cell. Thus, at each state of charge, the potential difference is higher in the heterogeneous cell, even at the extremes.

[0044] The most striking difference emerges when examining the distribution of the potential difference (hereafter referred to as the lithium plating potential (EPOTLPL)) and how this distribution correlates with the porosity distribution. As expected, it is evident from the simulations that the EPOTLPL correlates with the porosity. Furthermore, the EPOTLPL has a broad distribution in the heterogeneous cells, whereas the distribution in the homogeneous cells can be simplified to a delta function. Figure 11 shows these distributions along with a diagram showing the region of the model where the data were obtained. The top diagram shows the distribution of plating potential versus porosity, the middle diagram shows an illustration of where plating occurs, and the bottom diagram plots the plating potential distribution in the heterogeneous and homogeneous cells.

[0045] A higher plating potential correlates with less plating, indicating that different regions of the anode closer to the separator have different degrees of plating, whereas the homogeneous model predicts that all regions of the cell at a given distance from the separator have the same degree of plating. Figure 12 shows a contour plot of the plating potential at the end of the charge phase (EPOTLPL) at the face of the anode closest to the separator.

[0046] Given that predicting the extent to which lithium plating occurs is critical to predicting the long-term behavior of a battery, it is desirable to understand which areas of a cell are more susceptible to plating than others. The workflow of the exemplary embodiments allows for this determination, which is not available in other battery simulation software.

[0047] Subtle differences in performance between heterogeneous and homogeneous models may be amplified by tuning the porosity distribution. Experimental evidence (Yari et.al, 2020) suggests that distributions of particles resulting in certain porosity distributions facilitate ion transport within the battery. For example, introducing high-porosity channels into anode and cathode materials by laser ablation or micro-drilling may result in better transport properties (see Dunlap et al, 2022). These inferences motivate the development of optimized cells with artificial porosity distributions, where high-porosity “highways” span the thickness of the anode and cathode, yet the average porosity of the anode, cathode, and separator remains equal to that of their respective regions in the non-optimized heterogeneous cells considered previously, as shown in Figure 13. Furthermore, constitutive relations and material properties are comparable between the models.

[0048] Figure 14 shows the voltage / OCV vs. SOC curves at 2.4C and 4.8C for the heterogeneous and optimized batteries. For these cells, a significant difference in the internal resistance between the cells is observed even at a discharge current of 2.4C. As expected, this difference becomes more pronounced with increasing C-rate. Furthermore, there is a dramatic difference in the EPOTLPL between the optimized and heterogeneous cells, as shown in Figure 15, where the optimized cells are observed to plate at a state of charge that is 8% higher than the state of charge at which the heterogeneous cells plate. Thus, the EPOTLPL of the optimized cells is typically higher than the heterogeneous cells at all points in the charge cycle. This fact is further highlighted by the EPOTLPL distribution at the anode near the separator at the end of the charge phase shown in Figure 16. In this case, the distribution of the optimized cells is clearly biased towards higher EPOTLPL, implying that overall less plating should occur during charging. As shown in the corresponding contour plots in Figure 17, it is clear that the areas of high EPOTLPL corresponding to low plating are concentrated around the high porosity “highways”, as expected from the correlation between porosity and EPOTLPL discussed earlier.

[0049] 18 is a flow chart of an exemplary embodiment of a method 130 for upscaling a microstructure (e.g., provided by DigitalROCK) in a continuum modeler (e.g., Abaqus) for subsequent modeling of, for example, a lithium-ion battery. It should be noted that the process descriptions or blocks in the flow chart should be understood as representing modules, segments, portions of code, or steps that include one or more instructions for implementing certain logical functions in the process, and that alternative implementations are within the scope of the invention in which functions may be performed in an order different from that shown or described, including substantially simultaneously or in reverse order, depending on the functionality involved, as would be understood by a person reasonably skilled in the art of the invention.

[0050] As indicated by block 1810, a 3D microstructure of a battery component 110 (FIG. 1), such as an anode, a cathode, a separator, and / or a CBD, is received. For example, the 3D microstructure may be obtained via X-ray microtomography or procedurally. As indicated by block 1820, bulk and / or pure material properties of the battery component 120 (FIG. 1), such as diffusivity, thermal conductivity, electrical conductivity (solid), ionic conductivity (electrolyte), and elastic properties, among others, are received.

[0051] As shown in block 1830, emergent properties from the microstructure of the battery components are calculated as a function of porosity. For example, microstructure images can be uploaded to DigitalROCK as 8-bit RAW files. Here, the porous media characterization capabilities of DigitalROCK can be used to calculate the emergent properties as a function of porosity by first segmenting the microstructure into several regions (so as to have representative values ​​at different porosities) and then calculating the material properties of each sub-region, e.g., via the diffusion solver in DigitalROCK. The results in DigitalROCK can be interpreted, e.g., with a python script. Here, the Bruggeman coefficients for effective ionic, electrical and thermal conductivity, as well as diffusivity, can be calculated for each battery component sub-region, as shown in block 1840. For example, the Bruggeman coefficients for effective ionic, electrical and diffusivity can be determined using curve fitting functions built into the ScyPy python script. For thermal conductivity, the generalized Bruggeman relationship is given as follows:

[0052]

number

[0053] is fitted to the data using fitting parameters A and a, and the bulk thermal conductivity of the electrolyte is added to the right hand side to obtain the thermal conductivity of the fluid when the solid volume fraction is zero. Abaqus allows the definition of orthotropic Bruggeman coefficients for the effective ionic conductivity and diffusivity in the electrolyte, so that only these values ​​are stored for the later Abaqus input file generation. For the effective electrical and thermal conductivity, tables can be generated based on the calculated curves and saved, for example, under the keywords ELECTRIC CONDUCTIVITY and CONDUCTIVITY, for inclusion in the Abaqus input file. Additional saturation dependencies for elastic properties vs. porosity, effective area vs. porosity, saturation vs. pore pressure, and all constitutive relations are also possible.

[0054] A heterogeneous mesoscale 3D battery model 140 (FIG. 1) is created, as shown in block 1850. For example, in DigitalROCK, the anode, cathode, and separator materials may also be combined into a single cell structure, as shown in block 1852. Each of the anode, cathode, and separator are separately divided into coarse voxels, as shown in block 1854. For example, starting from a single cell structure, the original 8-bit image may be coarsened to a 32-bit image, in the sense that the 8-bit image is broken down into larger voxels and each voxel is assigned an average porosity. Thus, 32 storage is required for accurate porosity reporting. For each cell region (anode, cathode, separator), the voxels are sized such that there can be a collection of unique voxels defined with a uniform voxel size throughout. Note that the x, y, and z dimensions of each voxel do not need to be the same, and this can be achieved by cropping the images used from each material appropriately before combining them. Preferably, the desired level of coarsening is determined before setting up the model so that the size of the model can be determined accordingly.

[0055] From the final coarsened image, the raw file can be converted to a txt file, e.g. via an executable that associates the porosity to each coarsened voxel via its position in the image. For this purpose, two executables are used depending on the endianness of the image. For a given grain size, the level of coarsening can be selected. An Abaqus input file can be created using the coarsened image and the constitutive relations. It is preferred that the desired material properties (apart from those calculated above) are known to the user, e.g. through experimental data or other sources.

[0056] In the following, an exemplary approach for generating a single battery cell with one-dimensional load is described. An input file is created for Abaqus, for example using the Python script mentioned above, based on the provided microstructure 3D model 110 (FIG. 1) and bulk / pure material properties 120 (FIG. 1). In the following, an example of such a script is detailed among other possible implementations. The user input for the script may be a text file containing parts of the input file that are not related to the material properties calculated by DigitalROCK, and the geometry definition is omitted. Additionally, the size of the anode and separator in pixels through the thickness can be provided along with the overall (pixel) dimensions of the cell, the scale factor (e.g., if the width of the cell is 14 pixels, but the physical size of the cell at that dimension is 140 micrometers, the scale factor would be 10-5), the number of cycles of the simulation (at least two cycles are recommended, since the first cycle unrealistically starts the simulation from an equilibrium state), the C-rate to be used in the simulation, a tag that determines whether the cell is homogeneous or heterogeneous, a txt file to be generated from the coarsened image, and an optional name for the input file to be generated. Elements created in Abaqus can be optionally defined with variable x, y and z dimensions (depending on the scale factor). By default, cubic elements are created with dimensions equal to the scale factor.

[0057] The geometry generation and space allocation in this example relies heavily on the NumPy python package. First, node coordinates are generated based on a NumPy array generated based on the number of voxels in each direction and the voxel size (default: scale factor). Then a multidimensional grid with these coordinates is generated using NumPy's meshgrid function, which are stacked and transposed so that each row of an object called "node_coordinates" represents the x, y, z coordinate of a node. Then, using the meshgrid function again on the array of cell sizes, each node is assigned a unique flattened index, which is parsed and converted into a dictionary where node coordinates are assigned to their corresponding node index. A grid is created corresponding to element indices as opposed to node indices. From this grid, an element node is procedurally assigned to each element depending on its 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 sets are defined by their positions in the thickness direction. Therefore, the thickness of the anode and separator voxels must be defined by the user. This involves first calculating the center of each element by averaging their y (thickness) coordinates. Now, a new list is also determined that contains only the y coordinate of each node, so that the cell's most extreme faces can be determined later. Then, via the voxel size, the pixel size of the anode and separator, and the center y coordinate of each element, the node sets for the anode, cathode, and separator are determined. That is, if the center of an element is below the most extreme edge of the anode, the element is included in the anode set, if it is above that point but below the most extreme edge of the separator, the element is included in the separator set, and otherwise in the cathode set. Then, the elements at the most extreme edges of the battery are included in the "ground" set (the elements at the most extreme edge of the anode) or in the "load" set (the elements at the most extreme edge of the cathode). Finally, a node set is generated for each of these elements by adding the node of each element to the list.

[0058] An array of zeros is generated with dimensions equal to the number of voxels in the cell, which will be the porosity array for each element. The element indices are defined to correspond directly to the porosity assignments in the previously generated txt file, so that porosity is assigned directly based on its corresponding location in the image. This array is then flattened to match the dimensions of the array containing the column of element node numbers. While optional, the next step illustrates the power of this general approach to mesh generation. When running test cases, it may be desirable to translate the geometry down in Y by the thickness of the cell and rotate 90 degrees. Because porosity is defined per element, elements are defined by node numbers, and node coordinates are assigned to node numbers based solely on their location relative to each other in the grid, when the grid of node coordinates is rotated or translated, the elements, element sets, and porosity assignments are translated together. Thus, by translating the y coordinates of all nodes down and rotating the entire array 90 degrees, all of the information can effectively be rearranged. At this point, the geometry is fully defined.

[0059] The assignment of the remaining node sets, relative to the origin and a reference point set on the sensor node located at the extreme end of the cathode in the lower corner opposite the origin in the x-direction, can be determined by looking up the desired coordinate in the list of node coordinates and returning the index of the desired coordinate. The function can be used to write elements and node sets in the style of an Abaqus input file. These node and element sets are written to an input file along with the information required by Abaqus regarding the type of element (QEC3D8 in this example), the surface definition of the extreme face, the material definition (assuming the material is named as described in the Abaqus electrochemistry documentation), and the orientation. The remainder of the input file is output according to a user-supplied text file and given material property curves, with the porosity defined according to the initial conditions.

[0060] After the geometry is procedurally generated, this information is output in an Abaqus input file along with the material property definition, the assignment of porosity for each element (mean values ​​are assigned to the set of elements if the anode, cathode, and separator are assumed to be homogeneous), and step information including the calculated charge rate according to the 1-C rate (contained in a text file provided by the user) and the number of steps according to the number of cycles defined by the user (again, one cycle at 1-C is provided by the user). Since the electrical conductivity is typically defined via a characteristic table of bulk electrical conductivity and the Bruggeman coefficients used in the electrolyte, and the simulation will not run unless this table is defined, any bulk electrical conductivity defined in this manner must be set to zero when generating the input file. This will result in the electrical conductivity defined by the Bruggeman relation being zero, and only electrical conductivity defined with the keyword ELECTRIC CONDUCTIVITY can be used. Since both this keyword and the (thermal) CONDUCTIVITY keyword do not allow a dependency on the solid volume fraction by default, the user subroutine USER FIELD is used to obtain the porosity together with the GETVRM function, which calculates the solid volume fraction as 1 minus the porosity and provides it as a field variable.

[0061] With the input files created, the user can run the model in Abaqus and the upscaling method is complete. Post-processing can then be done according to the user's specific needs.

[0062] The system for performing the functions detailed above may be a computer, an example of which is shown in the schematic diagram of FIG. 19. The system 2000 includes a processor 502, a storage device 504, a memory 506 in which software 508 defining the functions described above is stored, an input / output (I / O) device 510 (or peripherals), and a local bus, or local interface 512, for enabling communication within the system 2000. The local interface 512 may be, for example, but not limited to, one or more buses or other wired or wireless connections as known in the art. The local interface 512 may have additional elements omitted for simplicity, such as controllers, buffers (caches), drivers, repeaters, and receivers, to enable communication. Additionally, the local interface 512 may include address, control, and / or data connections to enable appropriate communication between the aforementioned components.

[0063] The processor 502 is a hardware device for executing software, particularly software stored in the memory 506. The processor 502 may be a custom-made or commercially available single-core or multi-core processor, a central processing unit (CPU), a coprocessor among multiple 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.

[0064] The memory 506 may include any one or 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, tape, CD-ROM, etc.). Additionally, the memory 506 may incorporate 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 various components are located remotely from one another but may be accessed by the processor 502.

[0065] The software 508 defines the functions performed by the system 2000 in accordance with the present invention. The software 508 in memory 506 may include one or more separate programs, each of which includes an ordered list of executable instructions for carrying out the logical functions of the system 2000, as described below. The memory 506 may include an operating system (O / S) 520. The operating system essentially controls the execution of programs in the system 2000 and provides scheduling, input / output control, file and data management, memory management, communication control and related services.

[0066] The I / O devices 510 may include input devices such as, for example, but not limited to, a keyboard, a mouse, a scanner, a microphone, etc. Additionally, the I / O devices 510 may also include output devices such as, for example, but not limited to, a printer, a display, etc. Finally, the I / O devices 510 may further include devices that communicate via both input and output, such as, but not limited to, a modulator / demodulator (modem; for accessing another device, system, or network), a radio frequency (RF) or other transceiver, a telephone interface, a bridge, a router, or other devices.

[0067] When system 2000 is operating, processor 502 is configured to execute software 508 stored in memory 506, to communicate data to and from memory 506, and to generally control the operation of system 2000 in accordance with software 508, as described above.

[0068] During operation of the functions of system 2000, processor 502 is configured to execute software 508 stored in memory 506, to communicate data to and from memory 506, and to generally control the operation of system 2000 in accordance with the software 508. Operating system 520 is read by processor 502, possibly buffered within processor 502, and then executed.

[0069] It should be noted that if the system 2000 is implemented in software 508, the instructions for implementing the system 2000 may be stored on any computer-readable medium for use by or in connection with any computer-related device, system, or method. Such a computer-readable medium may correspond to either or both of the memory 506 or the storage device 504 in some embodiments. In the context of this specification, a computer-readable medium is an electronic, magnetic, optical, or other physical device or means that can store or store a computer program for use by or in connection 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 connection with a processor or other such instruction execution system, apparatus, or device. Although the processor 502 has been given as an example, such an instruction execution system, apparatus, or device may in some embodiments be any computer-based system, processor-containing system, or other system that can obtain instructions from and execute instructions from an instruction execution system, apparatus, or device. In the context of this specification, a "computer-readable medium" may be any means that can store, communicate, propagate, or transport a program for use by or in connection with a processor or other such instruction execution system, apparatus, or device.

[0070] Such computer-readable medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or propagation medium. More specific examples (non-exhaustive list) of computer-readable medium include an electrical connection with one or more wires (electronic), a portable computer diskette (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 disk read-only memory (CDROM) (optical). It should be noted that the computer-readable medium may even be a paper or other suitable medium on which the program is printed. Because the program may be captured electronically, for example, through optical scanning of the paper or other medium, and then compiled, interpreted, or processed in a suitable manner as required, and then stored in computer memory.

[0071] In alternative embodiments in which system 2000 is implemented in hardware, system 2000 may be implemented with any one or combination of technologies, such as discrete logic circuits having logic gates for implementing logical functions in response to data signals, application specific integrated circuits (ASICs) having appropriate combinatorial logic gates, programmable gate arrays (PGAs), field programmable gate arrays (FPGAs), and the like, each of which are well known in the art.

[0072] Advantageously, the above-described embodiments provide an efficient means of improving the accuracy of battery cell continuum model simulations without implementing an infeasible number of degrees of freedom or significantly modifying a well-known framework (Newman model). By moving away from common microstructural model assumptions such as a Bruggeman exponent of 0.5, the present embodiments allow for the proper capture of microstructural differences even between similar battery cells. For example, electrodes with similar porosity may have different degrees of tortuosity.

[0073] This embodiment provides a practical method to model aging mechanisms in full cells driven by local property extremes such as concentration, temperature, or plating potential. Performance captured by metrics such as internal resistance can be realized as a direct result of different microstructures even when the average porosity of each material (anode, separator, cathode) is the same.

[0074] This embodiment facilitates the optimization of electrode microstructure for both performance and aging, as shown in the detailed section of the optimized cell model that accurately captures the constitutive nature of the heterogeneity. The results point to the clear potential of microstructural optimization for battery cell performance.

[0075] The optimized cell and the heterogeneous cell are indistinguishable from each other in modeling a typical continuum battery because the porosity of the anode, cathode, and separator are the same between the models. Therefore, both are modeled as the homogeneous cell discussed above. Adjusting the size and location of these highways and further varying the porosity distribution may further improve the performance of the battery cell. Thus, in addition to improved accuracy through more accurate constitutive relationships, the above-described embodiments allow for optimization of the battery microstructure that is not possible with conventional battery simulation techniques.

[0076] It will be apparent to those skilled in the art that various modifications and variations can be made to the structure of the present invention without departing from the scope or spirit of the invention. In view of the foregoing, it is intended that the present invention cover the modifications and variations of this invention provided they come within the scope of the following claims and their equivalents.

Claims

1. A method of upscaling from microstructure that produces a coarsened, heterogeneous spatial distribution of porosity and a set of constitutive relations that are porosity dependent, comprising: receiving a three-dimensional (3D) microstructural model for each of a plurality of battery components, the battery components comprising an anode, a cathode, and a separator; receiving bulk material properties and / or porosity of each of the battery components; calculating a coarse-grained porosity model having emergent properties from a microstructure of the battery component as a function of the porosity; calculating a Bruggeman coefficient for each battery component subregion from the group of effective ionic conductivity, electrical and thermal conductivity, and ionic diffusivity; and generating a heterogeneous meso-scale 3D battery model, the generating step further comprising: combining the anode, cathode and separator materials into a unitary cell structure; and dividing the anode, cathode and separator individually into coarse voxels and creating a 3D model of the porosity.

2. 2. The method of claim 1, wherein the porosity of the received bulk material properties comprises a fine resolution model consisting of a plurality of elements, each element being characterized as either a solid with substantially zero porosity or a pore with 100% porosity.

3. calculating emergent properties from the microstructure of the battery component as a function of the porosity, calculating the average porosity of each material sub-region; assigning to each element of the coarsened porosity model the average porosity of all elements within a domain; and assigning said average porosity to individual elements of a finite element mesh.

4. The method of claim 1 , wherein the voxels are uniquely defined for each of the anode, cathode and separator with a uniform voxel size throughout.

5. The method of claim 1 , further comprising creating an Abaqus input file using the coarsened image and constructive relationships.

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

7. The method of claim 1 , further comprising the step of determining an effective characteristic as a function of the degree of saturation.

8. The method of claim 1 further comprising the steps of calculating a curve of capillary pressure versus saturation and calculating the change in pore pressure with changing saturation.

9. 7. The method of claim 6, further comprising simulating a 3D Newman model (165), wherein the continuum electrochemical solver comprises a 3D Newman model simulator.

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

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

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

14. The method of claim 1 , further comprising outputting the heterogeneous meso-scale 3D battery model to a file.

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

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