Fast Charging of Lithium‐Ion Batteries: A Review of Materials Aspects

Advanced Energy MaterialsVolume 11, Issue 33 2101126 ReviewOpen Access Fast Charging of Lithium-Ion Batteries: A Review of Materials Aspects Manuel Weiss, Institute of Physical Chemistry, Justus Liebig University Giessen, Heinrich-Buff-Ring 17, 35392 Giessen, Germany Center for Materials Research (LaMa), Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, GermanySearch for more papers by this authorRaffael Ruess, Institute of Physical Chemistry, Justus Liebig University Giessen, Heinrich-Buff-Ring 17, 35392 Giessen, Germany Center for Materials Research (LaMa), Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, GermanySearch for more papers by this authorJohannes Kasnatscheew, Helmholtz-Institute Münster, IEK-12, Forschungszentrum Jülich GmbH, Correnstraße 46, 48149 Münster, GermanySearch for more papers by this authorYehonatan Levartovsky, Department of Chemistry and BINA, BIU Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 5290002 IsraelSearch for more papers by this authorNatasha Ronith Levy, Department of Materials Science and Engineering, Technion – Israel Institute of Technology, Haifa, 3200003 IsraelSearch for more papers by this authorPhilip Minnmann, Institute of Physical Chemistry, Justus Liebig University Giessen, Heinrich-Buff-Ring 17, 35392 Giessen, Germany Center for Materials Research (LaMa), Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, GermanySearch for more papers by this authorLukas Stolz, Helmholtz-Institute Münster, IEK-12, Forschungszentrum Jülich GmbH, Correnstraße 46, 48149 Münster, GermanySearch for more papers by this authorThomas Waldmann, ZSW – Zentrum für Sonnenenergie- und Wasserstoff-Forschung Baden-Württemberg, Helmholtzstrasse 8, 89081 Ulm, GermanySearch for more papers by this authorMargret Wohlfahrt-Mehrens, ZSW – Zentrum für Sonnenenergie- und Wasserstoff-Forschung Baden-Württemberg, Helmholtzstrasse 8, 89081 Ulm, GermanySearch for more papers by this authorDoron Aurbach, Department of Chemistry and BINA, BIU Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 5290002 IsraelSearch for more papers by this authorMartin Winter, Helmholtz-Institute Münster, IEK-12, Forschungszentrum Jülich GmbH, Correnstraße 46, 48149 Münster, Germany MEET Battery Research Center, Institute of Physical Chemistry, University of Münster, Correnstraße 46, 48149 Münster, GermanySearch for more papers by this authorYair Ein-Eli, Corresponding Author eineli@technion.ac.il Department of Materials Science and Engineering, Technion – Israel Institute of Technology, Haifa, 3200003 Israel Grand Technion Energy Program (GTEP), Technion – Israel Institute of Technology, Haifa, 3200003 Israel E-mail: eineli@technion.ac.il, juergen.janek@phys.chemie.uni-giessen.deSearch for more papers by this authorJürgen Janek, Corresponding Author juergen.janek@phys.chemie.uni-giessen.de orcid.org/0000-0002-9221-4756 Institute of Physical Chemistry, Justus Liebig University Giessen, Heinrich-Buff-Ring 17, 35392 Giessen, Germany Center for Materials Research (LaMa), Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany E-mail: eineli@technion.ac.il, juergen.janek@phys.chemie.uni-giessen.deSearch for more papers by this author Manuel Weiss, Institute of Physical Chemistry, Justus Liebig University Giessen, Heinrich-Buff-Ring 17, 35392 Giessen, Germany Center for Materials Research (LaMa), Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, GermanySearch for more papers by this authorRaffael Ruess, Institute of Physical Chemistry, Justus Liebig University Giessen, Heinrich-Buff-Ring 17, 35392 Giessen, Germany Center for Materials Research (LaMa), Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, GermanySearch for more papers by this authorJohannes Kasnatscheew, Helmholtz-Institute Münster, IEK-12, Forschungszentrum Jülich GmbH, Correnstraße 46, 48149 Münster, GermanySearch for more papers by this authorYehonatan Levartovsky, Department of Chemistry and BINA, BIU Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 5290002 IsraelSearch for more papers by this authorNatasha Ronith Levy, Department of Materials Science and Engineering, Technion – Israel Institute of Technology, Haifa, 3200003 IsraelSearch for more papers by this authorPhilip Minnmann, Institute of Physical Chemistry, Justus Liebig University Giessen, Heinrich-Buff-Ring 17, 35392 Giessen, Germany Center for Materials Research (LaMa), Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, GermanySearch for more papers by this authorLukas Stolz, Helmholtz-Institute Münster, IEK-12, Forschungszentrum Jülich GmbH, Correnstraße 46, 48149 Münster, GermanySearch for more papers by this authorThomas Waldmann, ZSW – Zentrum für Sonnenenergie- und Wasserstoff-Forschung Baden-Württemberg, Helmholtzstrasse 8, 89081 Ulm, GermanySearch for more papers by this authorMargret Wohlfahrt-Mehrens, ZSW – Zentrum für Sonnenenergie- und Wasserstoff-Forschung Baden-Württemberg, Helmholtzstrasse 8, 89081 Ulm, GermanySearch for more papers by this authorDoron Aurbach, Department of Chemistry and BINA, BIU Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 5290002 IsraelSearch for more papers by this authorMartin Winter, Helmholtz-Institute Münster, IEK-12, Forschungszentrum Jülich GmbH, Correnstraße 46, 48149 Münster, Germany MEET Battery Research Center, Institute of Physical Chemistry, University of Münster, Correnstraße 46, 48149 Münster, GermanySearch for more papers by this authorYair Ein-Eli, Corresponding Author eineli@technion.ac.il Department of Materials Science and Engineering, Technion – Israel Institute of Technology, Haifa, 3200003 Israel Grand Technion Energy Program (GTEP), Technion – Israel Institute of Technology, Haifa, 3200003 Israel E-mail: eineli@technion.ac.il, juergen.janek@phys.chemie.uni-giessen.deSearch for more papers by this authorJürgen Janek, Corresponding Author juergen.janek@phys.chemie.uni-giessen.de orcid.org/0000-0002-9221-4756 Institute of Physical Chemistry, Justus Liebig University Giessen, Heinrich-Buff-Ring 17, 35392 Giessen, Germany Center for Materials Research (LaMa), Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany E-mail: eineli@technion.ac.il, juergen.janek@phys.chemie.uni-giessen.deSearch for more papers by this author First published: 19 July 2021 https://doi.org/10.1002/aenm.202101126AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the 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Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract Fast charging is considered to be a key requirement for widespread economic success of electric vehicles. Current lithium-ion batteries (LIBs) offer high energy density enabling sufficient driving range, but take considerably longer to recharge than traditional vehicles. Multiple properties of the applied anode, cathode, and electrolyte materials influence the fast-charging ability of a battery cell. In this review, the physicochemical basics of different material combinations are considered in detail, identifying the transport of lithium inside the electrodes as the crucial rate-limiting steps for fast-charging. Lithium diffusion within the active materials inherently slows down the charging process and causes high overpotentials. In addition, concentration polarization by slow lithium-ion transport within the electrolyte phase in the porous electrodes also limits the charging rate. Both kinetic effects are responsible for lithium plating observed on graphite anodes. Conclusions drawn from potential and concentration profiles within LIB cells are complemented by extensive literature surveys on anode, cathode, and electrolyte materials—including solid-state batteries. The advantages and disadvantages of typical LIB materials are analyzed, resulting in suggestions for optimum properties on the material and electrode level for fast-charging applications. Finally, limitations on the cell level are discussed briefly as well. 1 Introduction 1.1 Aims, Scope, and Framework A dozen senior battery and supercapacitor expert scientists, and approximately 30 Ph.D. students and postdoctoral fellows from both Israel and Germany gathered in 2019, in the frame of the 4th German-Israeli Battery School (GIBS 4) in Berlin, Germany. The Berlin workshop was focused on in-depth discussions on four “hot subjects,” including the following topics: 1) How will the far future of electrochemical power sources be after the lithium era, if ever? 2) Will the future of portable power sources be based on liquid or solid electrolytes? 3) Fuel cells versus battery technologies—complementary or competitors? And lastly, the 4th topic of fast charging—a reality or just a dream? Here, we bring to the readers the outcome of Group 4 discussions that continued over the last two years on fast charging and materials aspects from a physicochemical point of view. 1.2 An Overview More energy in shorter time at lower cost and increased safety—battery research has always been striving for improvement. Significant progress has been made in the field of lithium-ion batteries (LIBs) since their commercialization in 1991.[1, 2] LIBs store more energy, meaning their specific energy could be significantly increased by alternative cathode materials, reaching, for example, 421 W h kg−1 using LiNixCoyMn1−x−yO2 (NCM) compared to 279 W h kg−1 of the original LiCoO2 (LCO).[3] Additionally, the use of solid-state batteries (SSBs) exclusively consisting of solid components may enable the application of lithium metal anodes (LMAs), which can offer higher energy density compared to batteries with graphite anodes.[4] Furthermore, SSBs may also improve the mechanical stability of the cell,[5] making it safer—especially when using flexible polymer electrolytes (PEs). However, a major challenge for widespread adoption of electric vehicles is the charging speed of the batteries used, that is, LIBs take too long to refill compared to traditional combustion engine powered vehicles. Based on that experience of refueling (500–800) km of range at a gas station in just five minutes, customers expect similar practice from electric vehicles.[6] Therefore, charging to 80% state of charge (SOC) within 15 min is targeted by the US Advanced Battery Consortium (USABC).[7] In this review, we analyze how this target transforms into requirements for materials and components on the cell level. To achieve fast-charging capabilities, the power density PV of utilized battery cells has to be increased, which comes at the cost of reduced energy density WV. Therefore, there are always trade-offs between wide range and fast charging. Kinetic models of battery cells show that overpotentials exist in every part of the battery cell. From transport of lithium ions and electrons in the electrodes, charge transfer across phase boundaries to transport through the electrolyte, polarization effects limiting the charging rate lead to Li metal plating, limited utilization of active material, and temperature increase.[8] Current-state high-voltage DC chargers can deliver peak powers of up to 350 kW.[9, 10] The Porsche Taycan with 93.4 kW h battery allows a maximum charging power of 270 kW, while the average during charging is 187 kW.[9-11] Thus, recharging from 5% to 80% SOC takes 23 min. For comparison, the Tesla Model 3 with 75 kW h battery is recharged to 80% SOC in 27 min using Tesla's own Supercharger delivering a peak power of 250 kW in its third generation. Thus, the average charging power is about 130 kW, with the maximum value only reached for five minutes during the initial 20%.[12] Of course, bigger battery packs will charge longer at given charging power. Thus, the charger has to be improved as well for long-range vehicles with big battery packs.[13] These state-of-the-art parameters are still significantly below what is required to reach the USABC goal for extreme fast charging (XFC): recharging within 15 min.[7] In the following, we take a closer look at the materials applied to reach these values in order to identify the rate-limiting steps. Considering the example of Tesla, LiNixCoyAl1−x−yO2 (NCA) cathodes are used[14] in combination with graphite anodes. We assume a 100 kW h battery pack providing 500 km of driving range. With a volume of 400 L at system level and 200 L at cell level, this pack reaches an energy density WV of 500 W h L−1 at cell level. Using the average voltage of 3.7 V, the charge density QV is thus 135 A h L−1. If we assume an electrode thickness of 200 µm (neglecting current collectors and separator), the resulting QA arises as 2.7 mA h cm−2, thus approximately 3 mA h cm−2. The required current density for charging is therefore 3 mA cm−2 at 1C or 12 mA cm−2 (4C), which would be needed to reach the XFC goal of 15 min charging time. In view of research on fast charging, a few key steps have been identified as rate-limiting: a) diffusion of lithium ions within the anode active material, b) diffusion of lithium ions in the cathode active material (CAM), c) lithium-ion transport in the electrolyte phase (liquid or solid), and d) charge-transfer kinetics at the phase boundaries. In this case, we define charge transfer as the whole process of transport between electrolyte and electrode, thus it includes desolvation in the case of liquid electrolytes, the actual charge transfer across the electrolyte–electrode interface, and—for the presence of an interphase—also the ion transport through this interphase, which goes along with two charge-transfer processes across the electrolyte–interphase and the interphase–electrode boundary, respectively. The influence of lithium-ion transport in the electrolyte is rather small within the separator, but inside the porous electrodes it plays a major role in the fast-charging ability of a given battery cell. From the materials perspective, lithium plating at the graphite anode and lithium diffusion in the CAM are primarily rate-limiting. Essentially, slow diffusion of lithium in the liquid electrolyte and the active materials causes the true rate-limiting steps. Morphology, shape, and orientation of active material particles can improve the limiting influence of lithium diffusion in the solid-state, which explains, for example, the recent trend to single crystalline CAM.[15] On the electrode level, the active particle size distribution, tortuosity, and porosity are relevant, since diffusion-based lithium transport on the electrode scale is strongly influenced by those parameters in anodes and—to less extent—in cathodes.[16-22] For the latter, lithium-ion mobility greatly depends on the SOC,[22-26] since the crystal structure and the sequence of diffusion jumps typically change with the lithium content. Though, a high ionic conductivity on the material level does not necessarily translate to a fast-chargeable electrode if the tortuosity is high, for example. Going further on the size scale to the cell level, the relevance of engineering aspects—thermal management and the applied charging protocol, for example—takes precedence over physicochemical properties of the materials involved. Thus, this will only be discussed briefly in this review. The physicochemical basics of LIBs with focus on kinetics will be summarized in Section 2. The origin of different overpotentials is discussed by means of electrochemical potential profiles. Thereby, the role of a small active particle size is noted, which allows for full utilization of the active material. Furthermore, differences between non-phase-transformation and phase-transformation electrodes (along with conversion-type ones as a special case thereof) will be highlighted, concluding that the evaluation of diffusion phenomena is challenging in the latter type and that conversion electrodes are much less suitable for fast-charging applications.[27-29] In addition to these electrode-related overpotentials, charge transfer at interfaces and interface degradation leading to interphase formation also have to be considered at both anode and cathode.[30-34] For liquid electrolytes (LEs), concentration polarization is responsible as the decisive factor for electrolyte overpotentials, showing that the limiting current can be a major problem for thick electrodes in particular. Solid electrolytes (SEs) offer increased charge-carrier concentrations and a lithium-ion transference number near unity, thus concentration polarization does not occur.[35] Hence, the rate is not affected by current limitation due to depletion of lithium ions in the electrodes, but rather by the generally limited (effective) ionic conductivity of SEs. Furthermore, SEs are suggested to enable the use of a lithium metal anode. As a consequence, lithium plating would not be fatal anymore but rather desired, in the case that dendrite growth can effectively be suppressed. Subsequently, these concepts will be translated to recent materials’ applications in the following sections. For the anode (Section 3), studies showing the superior fast-charging performance of materials with low diffusion barriers are presented. The benefits of small particles are highlighted as well, but also the pitfalls of increased degradation because of higher surface area. We present an overview of different anode materials and discuss their advantages and disadvantages. Following with the cathode in Section 4, the SOC dependence of the cathode overpotential is established using experimental data of state-of-the-art NCM material. In addition, the dependence of rate performance on lithium diffusivity is highlighted, that is, faster charging is possible with increasing lithium diffusion coefficient, by comparing to other CAMs. Section 5 focuses on electrolytes, both on liquid and solid electrolytes. Experimental studies are presented, which show that ionic conductivity in the separator liquid electrolyte is not rate-determining. At the anode side, transport within the liquid electrolyte (in the porous electrode) becomes rate-limiting, however. At high current densities, reactions are confined at the anode parts close to the separator, leading to a severe local potential drop and lithium plating eventually. It will also be elaborated in Section 5 that charge transfer at the electrode interfaces may become critical. The mechanical properties of solid electrolytes are particularly critical, as contact loss and significant increase in charge-transfer resistance may result from missing plasticity. In any case, the electrode microstructure—together with the active particle morphology and microstructure—is crucial and needs to be tailored to allow for high current densities and fast charging. High temperatures can be applied to overcome this shortcoming, as shown using experimental studies. In fact, preheating the EV battery in cold climate either internally or by external means to allow for faster charging is common practice.[7] However, an increased temperature for better kinetics is always accompanied by accelerated degradation and thus shorter lifetime. Therefore, we briefly discuss thermal management systems along with optimized charging protocols and other measures utilized on the cell level to enable fast-charging applications in Section 6. By highlighting the rate-limiting aspects of different battery components on the material level and suggesting optimal properties required for fast-charging applications, we hope to stimulate further research on this crucial topic, which thus might lead to better market adaptation of electric vehicles in the future. While certain aspects of anode and cathode materials are comparable, they are specific enough to justify the separation of their discussion. 2 Physicochemical Basics of Fast Charging Fast charging of batteries requires high current densities that cause high overpotentials, which occur at the different components in the battery. If these overpotentials exceed certain limits, a physicochemical reaction takes place and the battery is likely to fail. The most prominent mechanism that leads to cell failure is lithium plating at the negative electrode,[36-38] typically graphite. Degradation processes can also occur at the cathode, namely oxygen evolution at oxide cathode active materials such as NCM and other layered oxides.[39] The voltage of a battery cell V is given by the difference between the electrochemical potentials of electrons (Fermi levels) at the anode μ ˜ e − a and cathode μ ˜ e − c , respectively, according to V = − 1 F μ ˜ e − c − μ ˜ e − a = V oc + Δ φ c + Δ φ a (1) Hereby, F is Faraday's constant, μ ˜ i is the electrochemical potential of species i (in this case either electrons or Li+ ions), which in turn is the sum of the chemical potential of this species μi and its electrical potential ziFϕ with the charge number zi, given by μ ˜ i = μ i + z i F φ (2) During battery operation, lithium ions are transported within the cell and, therefore, μ ˜ Li + is lifted from its equilibrium value as schematically shown in Figure 1.[40] Such transport needs to be driven by an overpotential, which is induced in the cell via a lifted μ ˜ e − c or μ ˜ e − a at the cathode or anode contacts, respectively. The total overpotential at each electrode in the battery Δφc or Δφa can be considered as the sum of the overpotentials needed to drive the solid-state diffusion inside the respective active cathode or anode material ( Δ φ AM c or Δ φ AM a ), the overpotential to drive the charge transfer between electrode and electrolyte ( Δ φ CT c or Δ φ CT a ), and the overpotential to drive ionic transport in the electrolyte phase ( Δ φ EL c or Δ φ EL a ) within a porous electrode: Δ φ c = Δ φ AM c + Δ φ CT c + Δ φ EL c (3) Δ φ a = Δ φ AM a + Δ φ CT a + Δ φ EL a (4) Figure 1Open in figure viewerPowerPoint Schematic representation of the electrochemical potential profiles of electrons μ ˜ e − and lithium ions μ ˜ Li + in a battery cell during charging (solid lines) and in equilibrium (dashed lines). A higher electrochemical potential of electrons at the anode μ ˜ e − a or lower at the cathode μ ˜ e − c (relative to the OCV case) needs to be applied to drive the transport of lithium ions through the cell with the corresponding overpotentials Δϕa and Δϕc, respectively. The overpotentials themselves drop at the different components in the battery, namely the electrolyte ( Δ φ EL c and Δ φ EL a ), the interface between electrolyte and electrode ( Δ φ CT c and Δ φ CT a ), and inside the electrode active material ( Δ φ AM c and Δ φ AM a ). In the active materials, the electronic conductivities are mostly significantly higher than the ionic conductivities.[41] Therefore, any overpotentials Δ φ AM c or Δ φ AM a are caused primarily by ionic transport. This leads to a gradient of μ Li + inside the active material because of the solid-state diffusion that is driven by an applied voltage during charging or by the cell voltage during discharging. Solid-state diffusion is further elaborated in Section 2.1.1. At the interfaces between the electrodes and the electrolyte, charge transfer takes place, which also requires some driving force Δ φ CT that further alleviates μ ˜ e − .[40] As most CAMs in LIBs are high-voltage materials that exceed the stability window of typical electrolytes, side-reactions at the interface will occur that lead to the formation of a solid electrolyte interphase (SEI, or CEI for “cathode–electrolyte interphase”).[30-33] Therefore, the charge transfer between active materials and electrolyte can involve several intermediate steps and may lead to significant overpotentials.[30-33, 40] The charge transfer is discussed in Section 2.1.2. Also the transport of ions in the electrolyte needs to be driven by an overpotential Δ φ EL and adds up to the total overvoltage.[40] Here, we have to distinguish typical liquid electrolytes with comparably low ion concentrations and transference number ( t Li + < 1) and solid electrolytes with high ion concentrations and t Li + ≅ 1.[40] Transport in the electrolyte occurs via both diffusion (driven by ∇ μ Li + ) and migration (driven by ∇φ). Thereby, migration in liquid electrolytes is typically neglected whereas transport in the solid electrolyte is exclusively caused by migration.[40, 42] These mechanisms are further discussed in Sections 2.2.1 and 2.2.2, respectively. 2.1 Electrode Overpotentials 2.1.1 Solid-State Diffusion Single-Phase Intercalation Electrodes The overpotential that is required to drive the solid-state diffusion in an intercalation-type electrode is determined by the gradient of the chemical potential of the ions ( z Li + = 1) Δ φ AM c = − 1 F μ Li + x = L − μ Li + x = 0 (5)under the assumption that electronic conductivity is significantly higher than ionic conductivity inside the electrode. Under this condition, the chemical diffusion coefficient D ˜ Li of the neutral component lithium (Li0) is only controlled by the mobility of the lithium ions, and we use the symbol D ˜ Li + to denote this. Formally, however, in every electrochemical experiment D ˜ Li is evaluated. The effect of mixed electronic and ionic conduction on the electrode polarization is discussed in detail by Usiskin and Meier.[43] For simplicity, we consider here the overpotential at the cathode, and the analogue anode case is added below. In Equation (5), Δ φ AM is described by the difference between μ Li + at the surface of the active material (x = L) and at the center of the active material particle (x = 0) in the case of spherical particle type (radius L) electrode materials. Nernst's equation connects the chemical potentials with the respective activities of Li+ ions ( a Li + ) according to Δ φ AM c = − R T F ln a Li + L − ln a Li + 0 = − R T F Δ ln a Li + (6) Hereby, R is the gas constant, T the temperature, and F Faraday's constant. If now only a small concentration gradient Δ c Li + as compared to the total concentration of Li+ ions in the electrode c Li + is established, Equation (6) can be written as[44] Δ φ AM c = − R T F ∂ ln a Li + ∂ ln c Li + Δ ln c Li + = − R T F W Δ c Li + c Li + (7) Equation (7) links the overpotential with the gradient of Li+ ions inside the electrode via the thermodynamic enhancement factor W = ∂ ln ( a Li + ) ∂ ln ( c Li + ) .[44] W can be determined from coulometric titration of an electrode under the assumption that the electrode is not undergoing any phase transformation during lithium intercalation.[44] The anode overpotential results similarly as Δ φ AM a = R T F ∂ ln a Li + ∂ ln c Li + Δ ln c Li + = R T F W Δ c Li + c Li + (8) The concentration profile of Li+ can be calculated via Fick's second law ∂ c Li + ∂ t = D ˜ Li + ∂ 2 c Li + ∂ x 2 (9)with the solid-state chemical diffusion coefficient of Li+ ions in the intercalation electrode D ˜ Li + . We note that Equation (9) can only be applied for small concentration gradients, that is, under the assumption of constant D ˜ Li + inside the active material. To solve the differential Equation (9), the following boundary conditions (Equations (10)–(12)) can be assumed, which represent the galvanostatic charging with current I from t = 0 onward.[45] The current induces a higher concentration of Li+ ions at the electrode's surface (initial concentration c Li + 0 ), which then leads to an expansion of the Nernstian diffusion layer over time throughout the volume of the electrode. c Li + = c Li + 0 0 ≤ x ≤ L ; t = 0 (10) − D ˜ Li + ∂ c Li + ∂ x = I A F x = L ; t > 0 (11) ∂ c Li + ∂ x = 0 x = 0 ; t ≥ 0 (12) With A being the surface area of the electrode, this leads to the solution[45] c Li + = c Li + 0 + I t F A L + I L F A D ˜ Li + [ 3 x 2 − L 2 6 L 2 − 2 π 2 ∑ n = 1 ∞ ( − 1 ) 2 n 2 exp ( − n 2 π 2 D ˜ Li + L 2 t ) cos ( n π x L ) ] (13) We note that Equation (12) generally only applies for a flat plate; for other geometries, such as cylinders or spheres, geometry-related constants have to be included,[43] but the differences in the equations are only minor, which implies that the flat plate geometry describes the system sufficiently well. The concentration polarization can now be calculated as the difference in concentrations between the surface (x = L) and the center of the spheri