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Continuum simulations for microscale 3D batteries
Journal Pre-proof Continuum Simulations for Microscale 3D Batteries Kim McKelvey, Marc Brunet Cabré, Aislan Esmeraldo Paiva PII:
S2451-9103(20)30015-6
DOI:
https://doi.org/10.1016/j.coelec.2020.01.008
Reference:
COELEC 502
To appear in:
Current Opinion in Electrochemistry
Received Date: 15 January 2020 Accepted Date: 20 January 2020
Please cite this article as: McKelvey K, Cabré MB, Paiva AE, Continuum Simulations for Microscale 3D Batteries, Current Opinion in Electrochemistry, https://doi.org/10.1016/j.coelec.2020.01.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Continuum Simulations for Microscale 3D Batteries Kim McKelvey*, Marc Brunet Cabré, Aislan Esmeraldo Paiva School of Chemistry, Trinity College Dublin, Dublin 2, Ireland. *[email protected]
Abstract: Continuum simulations provide a cost-effective approach to analyze the effect of geometric structure and dimension in microscale batteries. This allow us to explore how the introduction of 3D vertical structure into the battery design can be used to increase both areal capacity and power density. As we highlight here, continuum simulations contribute towards new insights into optimized geometric parameters, understanding performance of working 3D microscale batteries, and investigations into different physical processes occurring in nanoscale batteries.
TOC:
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Microscale rechargeable batteries that combine high areal energy and power density into a small volume (< 0.1 cm3) for use in micro-sensors, medical devices, microelectromechanical systems, and other powered integrated circuits are a target for battery development.1–4 Although the development of new chemistries and materials offers one approach to achieve both high power and energy density in microscale batteries, an alternative approach is the development of new geometric configurations.5–12 In a planar 2D battery geometry both electrodes (the anode and cathode) are planar and placed parallel to each other, and consequently the energy and power density of the battery are linked. Efforts to boost energy density by increasing the thickness of the electrodes reduce power density by increasing the distance over which ions must travel during charge/discharge, as illustrated in Figure 1 A. However, by carefully controlling the battery geometry through the introduction of 3D structure, energy and power density can be decoupled, allowing for the development of high energy and power density microscale batteries using existing materials and chemistries. 3D battery architecture introduces vertical structure to one or both of the anode and cathode.6 Vertical structure allows the electrodes to be arranged so that the ionic transport distance remains short for high power density while allowing large volume anodes and/or cathodes for high energy density. The introduction of 3D structure has the additional benefit of reducing the local current density at a given Crate, which improves performance by reducing charge transport overpotentials and degradation pathways. A range of different 3D structures have been proposed such as those shown in Figure 1 B and C. Often a solid-state electrolyte is used because these can be cast into complex 3D shapes, reduce the potential of shorting between electrodes, and reduce the risk of the electrolyte leaking.13,14 In addition, a solid-state electrolyte simplifies the fabrication of the battery by avoiding the need to use bulky spacers especially when space is of a premium. However, the introduction of 3D structure into microscale batteries has proven to be challenging,15–17 with only a limited number of functioning batteries reported primarily because of limitations in fabricating the required precise micro and nanoscale structures.18–20 Computational approaches to simulate the performance of 3D microscale batteries have proven invaluable in understanding how 3D microstructures can be used to increase battery performance. Unfortunately, 1D battery models and/or equivalent circuits do not capture the 3D nature of the battery geometry and therefore do not always give meaningful information on the performance of a 3D battery. Hence, continuum simulation approaches that model the transport of ions in 2D or 3D dimensions are needed to evaluate how 3D geometry influences battery performance. Here we will discuss recent developments in continuum simulations of microscale 3D batteries and highlight important lessons.
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Figure 1: A. In planer electrode designs thick electrodes allow high areal energy density but with reduced power density, while thin electrodes allow high power density but at the cost of areal energy density. A 3D electrode design allows for both high energy density and power density. Reproduced with modification from Front. Mech. Eng. 2017, 12(4): 459–476. B. Possible 3D electrode geometries, including arrays of interdigitated electrodes, interdigitated plate electrodes, cylindrical electrodes imbedded in the second electrode, and aperiodic architectures. Reproduced with modification from Chem. Rev. 2004, 104, 44634492. C. A 3D trench battery design. Reproduced with modification from Adv. Mater. 2007, 19, 4564–4567.
Simulation approaches: Continuum simulations, typically the finite element method (FEM) but also the finite volume method (FVM),21–23 in which a physical system is assumed to be continuous and homogeneous and so can be described by a continuous differential equation are typically used to simulate the transport of ions across the battery during both charge and discharge, and therefore model the primary operation of the battery.24,25 Almost exclusively Li-ion 3D batteries have been considered. Simulations are usually constructed in a dedicated simulation software,26 with the most popular being COMSOL Multiphysics due to its dedicated electrochemistry and battery module.27 In the simulation software the geometry and the physical equations that govern ion transport across the cell are defined, as well as various mesh, solver and simulation parameters. The simulation describes the transport of ions across the battery and into the electrodes during the charge-discharge cycle. The electrochemical reactions that occur at the electrode/electrolyte interface of a common Li-ion battery being:
where C is the negative battery electrode material (typically graphite) while MO is the positive cathode of the battery (typically a lithium metal oxide such as LiCoO2 (LCO) or LiFePO4 (LFP)). Three aspects of ion 3
transport are usually considered, transport within the electrodes, transport across the electrolyte and transport of ions across the electrolyte/electrode interface.23 Transport of ions within the electrodes is typically described by a diffusion process, Fick’s laws of diffusion are often utilized.23 Transport across the electrolyte is commonly described by Nernst-Plank equation with electroneutrality. Although as discussed by Notten et al the assumption of electroneutrality need not be taken, and instead the electro-magnetics can be explicitly incorporated via an electro-quasi-static formulation of Maxwell’s equations.28 Charge transport at the electrolyte/electrode boundaries is frequently described by the Butler-Volmer equation. Electron transport is not always explicitly included in the model with the electrical conductivity of the electrodes assumed to be sufficiently high that this aspect of the behavior can safely be excluded. As we discuss below, recent developments have coupled ion transport with other physical phenomena that occur, for instance the mechanical expansion/contraction of the electrode or electrolyte materials upon changes in lithiation or explicitly incorporating electrical double layers at the electrode/electrolyte boundaries of electrodes.29 Continuum simulations enable coupling of multiple types of physics relatively simply, especially within the COMSOL Multiphysics software package. The geometry of the 3D battery is reconstructed in silico, either based on an idealized geometry, or reconstructed from scanning electron microscopy images or focused ion-beam/scanning electron microscopy (FIB-SEM) tomography. It is often tempting to construct the full 3D geometry for the simulation; however, this should be carefully considered as a fully 3D model can consume large amounts of computational resources. In the authors experience, all possible effort should be made to reduce dimension as much as possible of the simulated geometry by using symmetries present in the electrode design, such as translational symmetries or rotational symmetries. For example, as discussed below and shown in Figure 3, a 3D battery design with interdigitated pillars can be reduced to a 2D axial symmetric geometry with a minimum loss of accuracy but a large saving of computation time.30 In fact, if the width of the electrolyte is sufficiently small compared to the lateral size of the electrode a 1D simulation geometry can give a good approximation for a conformal 3D battery design, as was recently demonstrated by Pearse et al.31
Geometry Optimization: Due to difficulties in the fabrication of working microscale batteries, simulation approaches have primarily been used as a tool to explore the effect of 3D battery geometry on battery performance. For instance previous research explored how the geometric dimensions, as well as electrolyte conductivity, in 3D batteries geometries (such as trench, interdigitated and 2.5 D geometries) can affect battery performance.25,32–35 A good recent example of this approach is the work of Zadin et al who used FEM simulations with porous electrode theory and Newman’s concentrated solution theory to simulate the electrochemical behavior of a 3D interlocking pillar Li-ion battery with LiCoO2 cathode and graphite anode.36 The effect of changing the pillar height and interpillar distance was explored as is shown in Figure 2, and an optimal geometry with 10 µm interpillar distances and 70 µm pillar heights for this battery composition was suggested. Simulations highlight that increasing the height of the 3D geometry (for instance pillar height or trench depth) increases areal capacity but also this comes at the cost of inducing inhomogeneous current 4
density across the electrode boundaries. Inhomogeneous current density contributes to local under and/or over utilization of the electrode materials as different parts of the electrode become lithiated/delithiated to different degrees.30 With an electrolyte with limited ionic conductivity, which is often exacerbated in the case of solid-state electrolytes, current density will be highest at the closest point between the electrodes. Often design of a 3D battery geometry requires a compromise between increasing the height of the 3D structures and increasing areal capacity and while keeping the current density within acceptable limits. Note that for typical battery the materials volumetric capacity of the electrodes is not equal, with the cathode typically being lower than the anode. Therefore, in a 3D battery design the larger volume electrode should be made of the material with the smallest volumetric capacity.36 In fact simple geometric calculations (which do not involve continuum simulation) can be used to calculate the expected capacity of a 3D battery design. Comparing experimentally data from 3D pyrolyzed photoresist pillars tested in a half cell configuration Dunn et al showed that simple geometric calculations of battery capacity of different designs was consistent with the experimentally measured values, although significant capacity fade was observed within a few charge/discharge cycles.37 As discussed, continuum simulations can be used to optimize 3D geometry dimensions, but these predictions need to be verified against experimental data. Physical parameters, such as diffusion coefficients or charge transfer coefficients, used in simulations are not always known with precision and therefore the simulation results can vary to a large degree. In addition, a number of other physical effects that can influence battery performance, such as expansion or contraction of electrode due to lithiation, effect of electrical double layers, thermal effects, formation of solid-electrolyte interphase (SEI), side reactions and degradation of the electrodes or electrolyte are often not included in the simulations.
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Figure 2: A. Cell potential and state of charge in the cathode and anode of an interdigitated pillar 3D battery at 8 different points of a charge and discharge cycle. B. Simulation of depth of lithiation in the cathode of an interdigitated pillar geometry with different pillar heights. Reproduced with modification from Electrochimica Acta 209 (2016) 138–148.
Performance Analysis: A second recent use of continuum simulations for 3D batteries has been as a tool to analyze the performance of a functioning battery. Recreating the operation of the 3D battery in silico allows for the analysis of how various components contribute to battery performance. While several 3D batteries have been reported,38 such as the fabrication of a 3D Li-ion micro battery using an imprinted microelectrode array and layer by layer stacking,18 these reports have not always coupled experimental results with simulation work. A recent example of this approach is our work simulating an all solid-state Li-ion battery with an array of pillars onto which the LCO cathode was deposited, followed by a LIPON solid state electrolyte and a 2D Si anode.30 Experimentally the performance of the 3D battery lagged what was expected, especially at higher C-rates. Therefore, a 2D-axial symmetric FEM simulation centered on a pillar, as shown in Figure 3, was constructed. Li-ion transport across the solid-state electrolyte was described using PoissonNernst-Plank with electroneutrality, transport of Li in the electrode was described using Fick’s law of diffusion, and ion transport at the electrolyte-electrode boundaries was described using the ButlerVolmer equation. As it is clear from the scanning electron micrograph of the battery cross section in Figure 3 A the fabrication of the battery resulted in large voids in the electrolyte. Once these voids in the electrolyte were incorporated into the simulation a good match between simulated charge/discharge response and experimentally measured response was obtained. From the simulated response the poor performance, especially at higher C-rates, could be linked to inhomogeneous current density and associated underutilization of the cathodic material. Without the aid of the continuum simulations the cause of the underperformance of the battery at high C-rates would be challenging to be obtained. 6
Another use of continuum simulations to understand the effect of 3D structure on battery performance has been the reconstruction of 3D porous cathode/electrolyte geometry from focused ionbeam/scanning electron microscopy (FIB-SEM) tomography.39,40 Finsterbusch et al used continuum simulations and compared the simulated response to experimental data, which suggested limitations in the Li transport across the cathode/electrolyte interface at low temperatures.39
Figure 3: A. 2D axial symmetric simulation geometry obtained from a SEM cross section of a working 3D solid-state Li-ion battery. Note the void in the LIPON electrolyte that is incorporated into the simulation geometry. B. Simulated battery operation, highlighting the non-uniform underutilization of the cathode that leads to sub-optimal battery performance. Reproduced with modification from ACS Appl. Mater. Interfaces 2016, 8, 32385−32391.
Multiphysics: Thirdly there has been an expansion of continuum simulations to incorporate additional physical phenomena, such as coupling ion-transport across the battery to mechanical deformation or the incorporation of electrical double layers at the electrode/electrolyte interface.29,41,42 Recent work by Grazioli et al explored the coupling of ionic conduction and mechanical deformation in a solid polymer electrolyte of a 3D trench geometry.43 Coupling between the ionic species and mechanical deformation in the solid-state electrolyte resulted in a 15 % increase in cell conductivity. As shown in Figure 4 the mechanical stress induced within the electrolyte could be significant, beyond the elastic limit and suggests that damage may occur during regular cell cycling. Grazioli et al also defined two scalar performance indicators, cell conductivity and uniformity index, by which to compare the performance across different 3D cell designs. This simple development should allow comparison of battery performance across different geometries, an approach that so far has not been explored. Finally, a range of physical parameters need to be defined for each simulation. As a first approximation it is often assumed that these parameters are constant and do not change during the simulation. However recent work by Notten et al has highlighted that changes in the diffusion coefficient with local Li-ion 7
concentration can more accurately fit experimental results for an all-solid-state thick-film Li-ion battery.44 This suggests that we need to carefully consider how simulation parameters are defined and their variation during simulations.
Figure 4: Simulation of coupled electrochemical and mechanical performance in a trench 3D battery architecture. A. Ionic concentration distribution and B. pressure distribution at the point of maximum ionic concentration in the solid polymer electrolyte in two cell designs (1 µm radii and 5 µm radii tip corners). Reproduced with modification from Electrochimica Acta 296 (2019) 1142e1162.
Summary: Continuum simulations, most often finite element method simulations, of the ionic transport during the charge/discharge cycles of 3D microscale batteries has proven vital in understanding how 3D geometry affects battery performance. This approach has a bright future as a tool to aid the development of microscale batteries, providing a cost effective and quick method to evaluate and optimize electrode structure. However, simulations need to be validated against experimental data because the complexity and the number of assumptions (such as physical parameter being constant, neglecting other physical effects) that are needed to construct simulations. Looking forward from an engineering perspective, the development of battery simulations needs to include the physical limitations in the fabrication process, which will place additional constraints on geometry optimization. For instance, the optimum 3D geometry for a battery might not be the one that maximizes areal capacity and current density, but the 3D geometry which optimizes capacity, current density and fabrication costs.
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Rademaker, T. J.; Akkermans, G. R. A.; Danilov, D. L.; Notten, P. H. L. On the Deviation of ElectroNeutrality in Li-Ion Battery Electrolytes. J. Electrochem. Soc. 2014, 161 (8), E3365. https://doi.org/10.1149/2.0411408jes. ** (43) Grazioli, D.; Verners, O.; Zadin, V.; Brandell, D.; Simone, A. Electrochemical-Mechanical Modeling of Solid Polymer Electrolytes: Impact of Mechanical Stresses on Li-Ion Battery Performance. Electrochimica Acta 2019, 296, 1122–1141. https://doi.org/10.1016/j.electacta.2018.07.234. Demonstrating a multiphysics approach to 3D battery simulations. ** (44) Kazemi, N.; Danilov, D. L.; Haverkate, L.; Dudney, N. J.; Unnikrishnan, S.; Notten, P. H. L. Modeling of All-Solid-State Thin-Film Li-Ion Batteries: Accuracy Improvement. Solid State Ion. 2019, 334, 111–116. https://doi.org/10.1016/j.ssi.2019.02.003. Simulations with diffusion coefficients that change with concentration.
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Highlights: Optimization of 3D battery geometry and dimensions via continuum simulations. A tool for qualitative analysis of experimental microscale 3D batteries. Extension to multiphysics simulations, coupling ionic transport with other physical processes.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S2451-9103(20)30015-6
DOI:
https://doi.org/10.1016/j.coelec.2020.01.008
Reference:
COELEC 502
To appear in:
Current Opinion in Electrochemistry
Received Date: 15 January 2020 Accepted Date: 20 January 2020
Please cite this article as: McKelvey K, Cabré MB, Paiva AE, Continuum Simulations for Microscale 3D Batteries, Current Opinion in Electrochemistry, https://doi.org/10.1016/j.coelec.2020.01.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Continuum Simulations for Microscale 3D Batteries Kim McKelvey*, Marc Brunet Cabré, Aislan Esmeraldo Paiva School of Chemistry, Trinity College Dublin, Dublin 2, Ireland. *[email protected]
Abstract: Continuum simulations provide a cost-effective approach to analyze the effect of geometric structure and dimension in microscale batteries. This allow us to explore how the introduction of 3D vertical structure into the battery design can be used to increase both areal capacity and power density. As we highlight here, continuum simulations contribute towards new insights into optimized geometric parameters, understanding performance of working 3D microscale batteries, and investigations into different physical processes occurring in nanoscale batteries.
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Microscale rechargeable batteries that combine high areal energy and power density into a small volume (< 0.1 cm3) for use in micro-sensors, medical devices, microelectromechanical systems, and other powered integrated circuits are a target for battery development.1–4 Although the development of new chemistries and materials offers one approach to achieve both high power and energy density in microscale batteries, an alternative approach is the development of new geometric configurations.5–12 In a planar 2D battery geometry both electrodes (the anode and cathode) are planar and placed parallel to each other, and consequently the energy and power density of the battery are linked. Efforts to boost energy density by increasing the thickness of the electrodes reduce power density by increasing the distance over which ions must travel during charge/discharge, as illustrated in Figure 1 A. However, by carefully controlling the battery geometry through the introduction of 3D structure, energy and power density can be decoupled, allowing for the development of high energy and power density microscale batteries using existing materials and chemistries. 3D battery architecture introduces vertical structure to one or both of the anode and cathode.6 Vertical structure allows the electrodes to be arranged so that the ionic transport distance remains short for high power density while allowing large volume anodes and/or cathodes for high energy density. The introduction of 3D structure has the additional benefit of reducing the local current density at a given Crate, which improves performance by reducing charge transport overpotentials and degradation pathways. A range of different 3D structures have been proposed such as those shown in Figure 1 B and C. Often a solid-state electrolyte is used because these can be cast into complex 3D shapes, reduce the potential of shorting between electrodes, and reduce the risk of the electrolyte leaking.13,14 In addition, a solid-state electrolyte simplifies the fabrication of the battery by avoiding the need to use bulky spacers especially when space is of a premium. However, the introduction of 3D structure into microscale batteries has proven to be challenging,15–17 with only a limited number of functioning batteries reported primarily because of limitations in fabricating the required precise micro and nanoscale structures.18–20 Computational approaches to simulate the performance of 3D microscale batteries have proven invaluable in understanding how 3D microstructures can be used to increase battery performance. Unfortunately, 1D battery models and/or equivalent circuits do not capture the 3D nature of the battery geometry and therefore do not always give meaningful information on the performance of a 3D battery. Hence, continuum simulation approaches that model the transport of ions in 2D or 3D dimensions are needed to evaluate how 3D geometry influences battery performance. Here we will discuss recent developments in continuum simulations of microscale 3D batteries and highlight important lessons.
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Figure 1: A. In planer electrode designs thick electrodes allow high areal energy density but with reduced power density, while thin electrodes allow high power density but at the cost of areal energy density. A 3D electrode design allows for both high energy density and power density. Reproduced with modification from Front. Mech. Eng. 2017, 12(4): 459–476. B. Possible 3D electrode geometries, including arrays of interdigitated electrodes, interdigitated plate electrodes, cylindrical electrodes imbedded in the second electrode, and aperiodic architectures. Reproduced with modification from Chem. Rev. 2004, 104, 44634492. C. A 3D trench battery design. Reproduced with modification from Adv. Mater. 2007, 19, 4564–4567.
Simulation approaches: Continuum simulations, typically the finite element method (FEM) but also the finite volume method (FVM),21–23 in which a physical system is assumed to be continuous and homogeneous and so can be described by a continuous differential equation are typically used to simulate the transport of ions across the battery during both charge and discharge, and therefore model the primary operation of the battery.24,25 Almost exclusively Li-ion 3D batteries have been considered. Simulations are usually constructed in a dedicated simulation software,26 with the most popular being COMSOL Multiphysics due to its dedicated electrochemistry and battery module.27 In the simulation software the geometry and the physical equations that govern ion transport across the cell are defined, as well as various mesh, solver and simulation parameters. The simulation describes the transport of ions across the battery and into the electrodes during the charge-discharge cycle. The electrochemical reactions that occur at the electrode/electrolyte interface of a common Li-ion battery being:
where C is the negative battery electrode material (typically graphite) while MO is the positive cathode of the battery (typically a lithium metal oxide such as LiCoO2 (LCO) or LiFePO4 (LFP)). Three aspects of ion 3
transport are usually considered, transport within the electrodes, transport across the electrolyte and transport of ions across the electrolyte/electrode interface.23 Transport of ions within the electrodes is typically described by a diffusion process, Fick’s laws of diffusion are often utilized.23 Transport across the electrolyte is commonly described by Nernst-Plank equation with electroneutrality. Although as discussed by Notten et al the assumption of electroneutrality need not be taken, and instead the electro-magnetics can be explicitly incorporated via an electro-quasi-static formulation of Maxwell’s equations.28 Charge transport at the electrolyte/electrode boundaries is frequently described by the Butler-Volmer equation. Electron transport is not always explicitly included in the model with the electrical conductivity of the electrodes assumed to be sufficiently high that this aspect of the behavior can safely be excluded. As we discuss below, recent developments have coupled ion transport with other physical phenomena that occur, for instance the mechanical expansion/contraction of the electrode or electrolyte materials upon changes in lithiation or explicitly incorporating electrical double layers at the electrode/electrolyte boundaries of electrodes.29 Continuum simulations enable coupling of multiple types of physics relatively simply, especially within the COMSOL Multiphysics software package. The geometry of the 3D battery is reconstructed in silico, either based on an idealized geometry, or reconstructed from scanning electron microscopy images or focused ion-beam/scanning electron microscopy (FIB-SEM) tomography. It is often tempting to construct the full 3D geometry for the simulation; however, this should be carefully considered as a fully 3D model can consume large amounts of computational resources. In the authors experience, all possible effort should be made to reduce dimension as much as possible of the simulated geometry by using symmetries present in the electrode design, such as translational symmetries or rotational symmetries. For example, as discussed below and shown in Figure 3, a 3D battery design with interdigitated pillars can be reduced to a 2D axial symmetric geometry with a minimum loss of accuracy but a large saving of computation time.30 In fact, if the width of the electrolyte is sufficiently small compared to the lateral size of the electrode a 1D simulation geometry can give a good approximation for a conformal 3D battery design, as was recently demonstrated by Pearse et al.31
Geometry Optimization: Due to difficulties in the fabrication of working microscale batteries, simulation approaches have primarily been used as a tool to explore the effect of 3D battery geometry on battery performance. For instance previous research explored how the geometric dimensions, as well as electrolyte conductivity, in 3D batteries geometries (such as trench, interdigitated and 2.5 D geometries) can affect battery performance.25,32–35 A good recent example of this approach is the work of Zadin et al who used FEM simulations with porous electrode theory and Newman’s concentrated solution theory to simulate the electrochemical behavior of a 3D interlocking pillar Li-ion battery with LiCoO2 cathode and graphite anode.36 The effect of changing the pillar height and interpillar distance was explored as is shown in Figure 2, and an optimal geometry with 10 µm interpillar distances and 70 µm pillar heights for this battery composition was suggested. Simulations highlight that increasing the height of the 3D geometry (for instance pillar height or trench depth) increases areal capacity but also this comes at the cost of inducing inhomogeneous current 4
density across the electrode boundaries. Inhomogeneous current density contributes to local under and/or over utilization of the electrode materials as different parts of the electrode become lithiated/delithiated to different degrees.30 With an electrolyte with limited ionic conductivity, which is often exacerbated in the case of solid-state electrolytes, current density will be highest at the closest point between the electrodes. Often design of a 3D battery geometry requires a compromise between increasing the height of the 3D structures and increasing areal capacity and while keeping the current density within acceptable limits. Note that for typical battery the materials volumetric capacity of the electrodes is not equal, with the cathode typically being lower than the anode. Therefore, in a 3D battery design the larger volume electrode should be made of the material with the smallest volumetric capacity.36 In fact simple geometric calculations (which do not involve continuum simulation) can be used to calculate the expected capacity of a 3D battery design. Comparing experimentally data from 3D pyrolyzed photoresist pillars tested in a half cell configuration Dunn et al showed that simple geometric calculations of battery capacity of different designs was consistent with the experimentally measured values, although significant capacity fade was observed within a few charge/discharge cycles.37 As discussed, continuum simulations can be used to optimize 3D geometry dimensions, but these predictions need to be verified against experimental data. Physical parameters, such as diffusion coefficients or charge transfer coefficients, used in simulations are not always known with precision and therefore the simulation results can vary to a large degree. In addition, a number of other physical effects that can influence battery performance, such as expansion or contraction of electrode due to lithiation, effect of electrical double layers, thermal effects, formation of solid-electrolyte interphase (SEI), side reactions and degradation of the electrodes or electrolyte are often not included in the simulations.
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Figure 2: A. Cell potential and state of charge in the cathode and anode of an interdigitated pillar 3D battery at 8 different points of a charge and discharge cycle. B. Simulation of depth of lithiation in the cathode of an interdigitated pillar geometry with different pillar heights. Reproduced with modification from Electrochimica Acta 209 (2016) 138–148.
Performance Analysis: A second recent use of continuum simulations for 3D batteries has been as a tool to analyze the performance of a functioning battery. Recreating the operation of the 3D battery in silico allows for the analysis of how various components contribute to battery performance. While several 3D batteries have been reported,38 such as the fabrication of a 3D Li-ion micro battery using an imprinted microelectrode array and layer by layer stacking,18 these reports have not always coupled experimental results with simulation work. A recent example of this approach is our work simulating an all solid-state Li-ion battery with an array of pillars onto which the LCO cathode was deposited, followed by a LIPON solid state electrolyte and a 2D Si anode.30 Experimentally the performance of the 3D battery lagged what was expected, especially at higher C-rates. Therefore, a 2D-axial symmetric FEM simulation centered on a pillar, as shown in Figure 3, was constructed. Li-ion transport across the solid-state electrolyte was described using PoissonNernst-Plank with electroneutrality, transport of Li in the electrode was described using Fick’s law of diffusion, and ion transport at the electrolyte-electrode boundaries was described using the ButlerVolmer equation. As it is clear from the scanning electron micrograph of the battery cross section in Figure 3 A the fabrication of the battery resulted in large voids in the electrolyte. Once these voids in the electrolyte were incorporated into the simulation a good match between simulated charge/discharge response and experimentally measured response was obtained. From the simulated response the poor performance, especially at higher C-rates, could be linked to inhomogeneous current density and associated underutilization of the cathodic material. Without the aid of the continuum simulations the cause of the underperformance of the battery at high C-rates would be challenging to be obtained. 6
Another use of continuum simulations to understand the effect of 3D structure on battery performance has been the reconstruction of 3D porous cathode/electrolyte geometry from focused ionbeam/scanning electron microscopy (FIB-SEM) tomography.39,40 Finsterbusch et al used continuum simulations and compared the simulated response to experimental data, which suggested limitations in the Li transport across the cathode/electrolyte interface at low temperatures.39
Figure 3: A. 2D axial symmetric simulation geometry obtained from a SEM cross section of a working 3D solid-state Li-ion battery. Note the void in the LIPON electrolyte that is incorporated into the simulation geometry. B. Simulated battery operation, highlighting the non-uniform underutilization of the cathode that leads to sub-optimal battery performance. Reproduced with modification from ACS Appl. Mater. Interfaces 2016, 8, 32385−32391.
Multiphysics: Thirdly there has been an expansion of continuum simulations to incorporate additional physical phenomena, such as coupling ion-transport across the battery to mechanical deformation or the incorporation of electrical double layers at the electrode/electrolyte interface.29,41,42 Recent work by Grazioli et al explored the coupling of ionic conduction and mechanical deformation in a solid polymer electrolyte of a 3D trench geometry.43 Coupling between the ionic species and mechanical deformation in the solid-state electrolyte resulted in a 15 % increase in cell conductivity. As shown in Figure 4 the mechanical stress induced within the electrolyte could be significant, beyond the elastic limit and suggests that damage may occur during regular cell cycling. Grazioli et al also defined two scalar performance indicators, cell conductivity and uniformity index, by which to compare the performance across different 3D cell designs. This simple development should allow comparison of battery performance across different geometries, an approach that so far has not been explored. Finally, a range of physical parameters need to be defined for each simulation. As a first approximation it is often assumed that these parameters are constant and do not change during the simulation. However recent work by Notten et al has highlighted that changes in the diffusion coefficient with local Li-ion 7
concentration can more accurately fit experimental results for an all-solid-state thick-film Li-ion battery.44 This suggests that we need to carefully consider how simulation parameters are defined and their variation during simulations.
Figure 4: Simulation of coupled electrochemical and mechanical performance in a trench 3D battery architecture. A. Ionic concentration distribution and B. pressure distribution at the point of maximum ionic concentration in the solid polymer electrolyte in two cell designs (1 µm radii and 5 µm radii tip corners). Reproduced with modification from Electrochimica Acta 296 (2019) 1142e1162.
Summary: Continuum simulations, most often finite element method simulations, of the ionic transport during the charge/discharge cycles of 3D microscale batteries has proven vital in understanding how 3D geometry affects battery performance. This approach has a bright future as a tool to aid the development of microscale batteries, providing a cost effective and quick method to evaluate and optimize electrode structure. However, simulations need to be validated against experimental data because the complexity and the number of assumptions (such as physical parameter being constant, neglecting other physical effects) that are needed to construct simulations. Looking forward from an engineering perspective, the development of battery simulations needs to include the physical limitations in the fabrication process, which will place additional constraints on geometry optimization. For instance, the optimum 3D geometry for a battery might not be the one that maximizes areal capacity and current density, but the 3D geometry which optimizes capacity, current density and fabrication costs.
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Highlights: Optimization of 3D battery geometry and dimensions via continuum simulations. A tool for qualitative analysis of experimental microscale 3D batteries. Extension to multiphysics simulations, coupling ionic transport with other physical processes.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: