The Effect of Microstructure on the Galvanostatic Discharge of Graphite Anode Electrodes in LiCoO 2 -Based Rocking-Chair Rechargeable Batteries

Transcription

1 A /2009/ /A896/9/$25.00 The Electrochemical Society The Effect of Microstructure on the Galvanostatic Discharge of Graphite Anode Electrodes in LiCoO 2 -Based Rocking-Chair Rechargeable Batteries Madeleine Smith, a R. Edwin García, a, *,z and Quinn C. Horn b, * a School of Materials Engineering, Purdue University, West Lafayette, Indiana 47906, USA b Exponent Failure Analysis Associates, Natick, Massachusetts 01760, USA By starting from experimentally determined cross sections of rechargeable lithium-ion batteries, the effect of microstructure on the galvanostatic discharge of a LiCoO 2 LiC 6 cell was numerically modeled. Results demonstrate that when small graphite particles are part of a population with large particle sizes, diminished macroscopic power densities develop and limit the response of the entirety of the cell. Small particle-size populations electrochemically interact with large particle-size populations and lead to a macroscopic capacity loss, compared to cells with uniform particle size. Such capacity loss is a result of the lithium exchange between small and large anode particle-size populations, instead of the lithium exchange between electrode particles of opposite polarity. The analysis suggests that graphite particles of size smaller than the average value dominate the macroscopic electrical response of the device, for the induced localized lithium depletion leads to an increase in the polarization losses of the anode. Lithium depletion in the anode starts in the small particles, is followed by particles of complex morphology and rough surfaces, and continues with the depletion of large particles embedded in a fast diffusion environment The Electrochemical Society. DOI: / All rights reserved. Manuscript submitted December 31, 2008; revised manuscript received August 6, Published September 18, Rechargeable lithium-ion batteries have undergone a great deal of progress through the introduction of materials and engineering approaches over the past 20 years. These improvements have boosted their reliability and power density 1-3 and have proven that the distribution of particle aggregates of active material within the space of the cell determines the performance of batteries in the limit of large discharge rates, high energy densities, and low salt concentrations For high power density applications, particle particle electrochemical interactions dominate the response of the cell and promote the appearance of irreversible physical mechanisms. Microstructural processes such as interfacial side reactions, oxidation or reduction of the electrolyte, dissolution and replating of current collector metals, phase transformations within the electroactive materials, salt precipitation, and the formation of conductive dendrites within the separator, are several technologically relevant microphysical processes that diminish the lifetime of a rechargeable lithium-ion battery. 14,15 Recently developed cross-sectional characterization techniques in rocking-chair rechargeable lithium-ion batteries suggest that battery degradation mechanisms are localized microstructural events that are a consequence of particle configurations introduced during the processing of the composite electrodes. 16 The resulting particle configurations favor the occurrence of large localized variations in the cell s electrochemical behavior, leading to localized lithium accumulation or depletion. Moreover, available ex situ microstructural measurements of nanosized particulate electrode particles demonstrate that the benefits of using smaller lengthscales are lost when considering the sum of the contributions to polarization of a cell from the different macroscopic components. 17 The rechargeable battery characterization community currently demands in situ, realtime analysis to correlate the local response of the conforming materials to the macroscopic galvanostatic response. Characterization of such events has proved challenging due to the reactive nature of the lithium-ion chemistry. 18 In this context, the development of a numerical method that spatially resolves the time-dependent electrochemical behavior of a rechargeable lithium-ion battery allows the analysis of the peculiarities of specific microstructural features, e.g., interface or particle proximity effects, spatial correlations of crystallographic orientations, morphological texture, or defects such as pores and crack distributions. Such methods are ideal for analyzing events that are * Electrochemical Society Active Member. z redwing@purdue.edu only observed under extreme discharge conditions and are favored in localized microstructural sites. Numerical models aimed to simulate the galvanostatic discharge in rechargeable lithium-ion batteries have undergone a great deal of progress in the past two decades. Early descriptions were developed by West et al., 7 Atlung et al., 19,20 and Knutz et al. 21 Doyle et al. produced a mean-field description that incorporates aspects of microstructure into models that homogenize the spatial distribution of particulate active material and porosity. 22,23 This model is very successful and has been extended by several authors. Microstructural cell design was widely explored by Doyle, Newman, and others, 5,8-11,22 while side reactions and capacity limiting mechanisms were modeled by Darling and Newman 24 and Arora et al. 12 Hellweg considers the effects of porosity gradients, 6,25 and thermal effects have been incorporated into this formulation by Guo and Wang. 26,27 Most recently, the battery community has come to realize the importance of thermal effects on the galvanic response of battery architectures and has developed useful mean-field descriptions that include the effects of microstructure The current state-of-the-art on three-dimensional modeling of rechargeable lithium-ion batteries has focused over the past five years on the detailed description of the electrochemical, thermal, and mechanical states of individual and isolated particles of simple geometries, such as spherical and ellipsoidal In spite of their great success, such descriptions are not always amenable to incorporate the intricate details, randomness, and the effects that occur in real battery architectures. García and co-workers contributed to this field by removing the mean-field approximation used by previous authors, thus resolving the local particle particle electrochemical interactions of the active material phases under isothermal conditions. 35,36 Existing analytical and numerical descriptions assume the shape of particles of active material as prototypical spherical or ellipsoidal. While such approximation has allowed the analysis of the average response of electrode materials, recent cell characterizations demonstrate that the morphology of the particles greatly differs from the assumed ellipsoidal ideal. 16 This paper applies a numerical method that was experimentally validated and compared against existing models in previous publications 35,36 to simulate the effects of randomness of shape, morphology, and packaging that result by starting from a real LiCO 2 LiC 6 rocking-chair rechargeable battery cross section see Fig. 1. The impact on the local electrochemical fields is detailed; the effects of the external surfaces, particle interactions, and the effect of randomness on the lifetime of the device is discussed.

2 A897 +e Li, and the rate of transfer of lithium ions from the electrolyte to the active material phase is controlled by the Butler Volmer relation 38 I S = J nˆ = i o e a F/RT e c F/RT 4 The functional form of i o is i o = Fk r n S n Li an c Li 5 Figure 1. a corresponds to as-received digitized cross section of a typical LiCoO 2 LiC 6 rocking-chair rechargeable lithium-ion battery, defined in this paper as microstructure T. Upper composite layer corresponds to graphite anode, middle layer to electrolyte separator, and bottom layer to LiCoO 2 cathode electrode. b shows the digitized and segmented image: Green phase corresponds to anode carbon particles, red phase to cathode lithium cobalt oxide electrode particles, and black background to polymeric electrolyte. Electrical conductivity of separator was set to insulate the transport of electrons. Theoretical Framework For a system composed of lithium ions and electrons, in the limit of zero local charge accumulation, / t = 0, the transport of lithium ions is given by the set of partial differential equations 37 0= + L n 1 n t = D Li n + L with L the electromigration coefficient of charged lithium. 37 L = D LizFn 3 RT The electromigration coefficient is directly related to the electrical conductivity through the relation L = /zf the Nernst Einstein equation, as discussed by several authors. 8,37-39 Here, the electrostatic potential,, corresponds to a scalar potential satisfying the Maxwell equation, E = 0, i.e., Ampere s law, in the absence of magnetic fields. The electric field is related to the electrostatic potential through the expression E = and naturally satisfies the equation at all times in the absence of rapidly varying magnetic fields. 37,40 The lithium concentration, n, corresponds to the number of moles of lithium per unit volume dissolved at every volume element of material. Inside a particle of active material, Eq. 1 reduces to the classic diffusion and charge continuity equations by setting z = 0 in an electrically conductive medium. When a particle of active material is embedded in an electrolyte, an electrostatic potential gradient across the two abutting phases develops. At the interface of these two phases, charge redistribution takes place together with thermal and chemical fluctuations, which in turn produce charge images in the electrode particle. 38,41 The resultant charge distribution generates a region in space that demands electrical work through the recombination of a lithium ion and an electron to traverse the interface, and results in a macroscopic electrostatic potential difference with respect to a lithium reference. A local deviation from equilibrium of the surface electrostatic potential, i.e., an overpotential, constitutes the driving force for the reaction to take place. Such reaction is given by Li + 2 where k r is the reaction rate constant and n S is the solubility limit of lithium in the electrode. The first term in the second factor of the right side of Eq. 4 corresponds to the forward rate of the anodic process. The second term corresponds to the rate of the backward cathodic reaction. nˆ is the unit normal vector to the local surface of the cathodic or anodic particle. The exchange of ions across the interface brings back the potential to equilibrium. Physically, Eq. 4 and 5 describes the rate of lithium exchange at the particle electrolyte interface. At equilibrium, the forward and backward rates of reaction cancel each other, = 0. However, if an overpotential 0 is imposed, a net mass and charge flux is induced. In this paper, Eq. 3 is solved in its full form at every interfacial point, regardless of the complexity of the interfacial topology. It is also assumed that the interfacial overpotential = R i I S i.e., all the deviations from equilibrium of the potential at the particle electrolyte interface are due to charge recombination at the electrode electrolyte interface. R i is defined as the contact or interfacial ohmic resistance, i = c corresponds to the cathode and i = a is the anode. I S is the position-dependent surficial electronic current i.e., the current density at the particle electrolyte interface and is specified by the local charge flux that results from solving the electrochemical interactions of the particles. The interfacial resistance is a result of the local voltage drop during the intercalation process. The value of this quantity depends on the local chemistry of the interface and is sensitive to the processing of the device. For this model, the value of the interfacial resistance is fitted from experimental discharge curves by starting from typical values reported in the literature and further comparing the resultant macroscopic response against the experimental one. R i was specified in this paper by using data available from the literature. 12,22,23,35,36 The interfacial or surficial electrostatic potential, interface, is given by interface = U n R i I S Thus, in the limit of very small discharge rates, the interfacial potential, interface = U n Li, i.e., corresponds to the equilibrium potential, and as the discharge rate increases, the interfacial polarization losses play an increasingly dominant role. In this paper, U n Li corresponds to the lithium-dependent equilibrium potential, as introduced by Doyle et al. 22,39 The computation of the positiondependent equilibrium potential, U n Li, and the surficial current, I S, enables the direct calculation of the interfacial electrostatic potential. The spatial voltage distribution that results from solving Eq. 1-7 is a consequence of the local electrochemical inhomogeneities of the driving forces between differing electrically active phases, as they exchange lithium with its surroundings, and the transport limitations of each particle. The calculated voltage distribution allows quantification of the electrical interactions between the anode and cathode phases and establishment of correlations to the corresponding local lithium distribution. Furthermore, the voltage distribution that develops at the ohmic contacts that abut the anode and cathode electrode composites is a result of all the microstructural interactions in the electrodes and ultimately embodies the macroscopic voltage of the cell. 6 7

3 A898 Table I. Rechargeable lithium-ion battery summary of material parameters. Definition Name Value Units A Battery cross-sectional area cm 2 D a Lithium diffusivity in Li y C 6 particles cm 2 /s D m Lithium-ion diffusivity in electrolyte/carbon mixture cm 2 /s D s Lithium-ion diffusivity in electrolyte in separator cm 2 /s D p Lithium diffusivity of in LiCoO 2 particles cm 2 /s R c Electrode contact resistance k ar Reaction rate constant at anode particle/electrolyte interface cm/s k cr Reaction rate constant at cathode particle/electrolyte interface cm/s n T Normalization concentration value for lithium mol/dm 3 n ao Normalized initial concentration of lithium in anode 0.72 n co Normalized initial concentration of lithium in cathode 0.18 T,a Electric conductivity of Li y C 6 particles S/cm T,m Electric conductivity of electrolyte/carbon mixture S/cm T,s Ionic conductivity of electrolyte in separator S/cm T,c Electric conductivity of LiCoO 2 particles S/cm a Anodic empirical constant 0.5 c Cathodic empirical constant 0.5 a Anode thickness 75 m c Cathode thickness 75 m s Separator thickness 28 m Porosity of electrolyte/carbon mixture 0.3 a Initial electrostatic potential in anode 0 V c Initial electrostatic potential in cathode 4.2 V Battery Abstraction and Numerical Setup The formulation described in the previous paragraph was implemented by modifying the Object Oriented Finite Element Analysis program. 42,43 A thorough description of the basic finite-element implementation can be found in any standard finite-element textbook. 44 The procedure consists in discretizing the spatial domain into regions, elements, that are described by Eq. 1 and 2 in the bulk regions, and by Eq. 4-7 at the boundaries between the active phases and the abutting electrolyte. 45 The analyzed battery microstructure is summarized in Fig. 1. The lithium-ion electrode microstructure was obtained by scanning electron microscopic characterization of crosssectioned commercial cells prepared by Exponent, Inc., using a proprietary technique. 16,46,47 The anode composite consists of LiC 6 particles, and the cathode consists of LiCO 2 particulates, embedded in an electrolytic medium. 23,39 A nonaqueous carbonate solvent mixture and single lithium salt dispersed in an inert polymer matrix are assumed. All the material properties and utilized initial conditions are summarized in Table I. Adobe Photoshop was used to clean up and digitally enhance the image of the battery cross section. The cathode particles were colored red and the anode particles were colored green. The image contrast between the particles of active material and electrolyte matrix was increased to facilitate the identification of each feature. The author s discretion was used to identify those microstructural features of the cathode and anode particles that were indeed part of the cross section. Thus, the simulated topology of the different transport paths e.g., the electrolyte carbon mixture paths abutting the active material phase, the geometry of the separator, and the shape associated with each particle of electrode material of the two-dimensional section is reconstructed in the computer as close as possible to those provided by the microstructure. The resultant microstructure was labeled microstructure T and meshed using an initial grid of linear triangular elements. A finite-element mesh based on the Galerkin method was then refined and annealed to match the boundaries of the underlying microstructural features by applying adaptive meshing techniques developed by Langer et al. 42,43 This procedure resulted in a two-dimensional, triangular finite-element mesh with 48,134 nodes and 95,852 elements see Fig. 2. This mesh was partitioned into individual meshes, each representing an individual phase i.e., the positive electrode phase, electrolyte phase, and the negative electrode phase, a subdomain. Each subdomain is locally coupled to the abutting phase only through Eq. 4 and solved using the methods defined by Zienkiewicz et al. 44,45 The resultant coupled system of ordinary differential equations was solved by using single time-stepping algorithms, as described by Zienkiewicz et al. and pioneered by Liniger The described approach greatly improves the condition number of the system and allows taking time steps two orders of magnitude larger than the ones taken through Figure 2. Triangular finite-element mesh resulted from meshing microstructure T, as was used in this paper. The mesh comprises 95,852 elements, 48,134 nodes, and 144,402 degrees of freedom. Selected insets exemplify mesh detail in refined regions. 42 Sections a and b correspond to zoomed-in regions of the anode. Section b corresponds to a selected zoomed-in region in the cathode. See text for further details.

4 A899 Figure 3. Central inset shows the original digitized battery microstructure T. Extracted sections 1 5 and the locations in T are highlighted. Each section samples the response of a specific population of electrode particles in the cell. Section 2* assesses the effect of sampling neighboring subpopulations of particles from T with respect to section 2. standard means. The tolerance for convergence was set to and the tolerance of the Newton solver was set to Each subdomain resulted in a system of linear equations that took on the order of floating-point operations and two Newton steps to converge for each individual time step on a 40 CPU PowerPC G5 computer cluster. Each simulation takes on the order of 5 h of wall time for a 1C discharge. A full analysis associated with the stability and convergence of the described model can be found elsewhere. 50 The described model resolves the spatial details of the particle particle electrochemical interactions of both positive and negative electrode materials. The transport properties associated with the topology of the underlying carbon electrolyte mixture in both cathode and anode were averaged out because the percolating nanostructure that comprises this nanocomposite is at least two orders of magnitude smaller than the smallest electrode particle. The assigned average properties were specified to be homogeneous throughout the matrix of both electrodes, thus establishing a natural electrical network that connects the active material particles and the back contact. Similarly, the electrolyte in the separator region was assumed structureless. Used lithium diffusivity as well as ionic and electronic conductivity values correspond to those used by Doyle et al., 22 Chiang and Hellweg, 25 and Doyle. 39 After analyzing the local and macroscopic response of the microstructure shown in Fig. 1b, the digitized microstructure was divided into five sections, each labeled 1 through 5 see Fig. 3. While an infinite number of possibilities exist to divide microstructure T, the performed sectioning was made without any bias on how each individual region performs. Each section was meshed using an initial grid size of Each individual mesh was refined and annealed to match the underlying microstructure by applying the same meshing techniques described in the previous paragraph. The generated mesh had on the order of 13,000 nodes and 27,000 elements. Simulations were performed in each subsection of the microstructure to assess the impact of each individual section and microstructural feature on the overall initial battery. For all meshes, the time step of the implemented finite difference scheme was set to 40 s. The cutoff voltage was set to 3 V, and a 1C current density corresponds to ma/cm 2 for section 1. The capacity of sections 2 5 was normalized to this value in the rest of this paper. The proposed approach enables the possibility to assess the contributions of each section of the cell on the macroscopic performance relative to each other and allows the systematic exploration of the interactions of individual populations of electrode particulates by carefully sampling sections that are removed or inserted, and thus assess if they become the controlling feature s of the simulated domain. In comparison, while a similar analysis can be readily performed by starting from computer-generated battery representations, the inherent randomness of particle shape and morphology that results from processing a real battery microstructure would be lost. The analysis presented in the section below incorporates all the aforementioned details and rationalizes the response of the entire cell microstructure T by comparing the differences in behavior of individual sections. Results and Discussion Full battery analysis. Figure 4 summarizes the lithium distribution in microstructure T at different instants as the battery galvanostatically discharges. In this figure, and for the remainder of this paper, the position-dependent lithium concentration has been normalized by the solubility limit of lithium in graphite, i.e., n T = mol/dm 3. In accordance to what is known in the literature, simulation results performed over the entire battery cross section demonstrate that the smallest graphite particles are completely depleted of lithium as early as 1500 s after the battery initiates the galvanostatic discharge process e.g., at locations,, and, while the largest anode particles have 50% of normalized lithium concentration to continue the discharge, after 3600 s e.g., at location. The location of the smallest particles inside the composite negative electrode plays a very important role in specifying the rate at which they delithiate: Particles surrounded by a high conductivity environ- Figure 4. Time evolution of lithium distribution of microstructure T during a 1C discharge. Lithium concentration is normalized by the solubility limit of lithium in graphite, n T = mol/dm 3. In the anode, small anode particles, such as those illustrated by and, and particles of complex geometry, such as those illustrated in region, deintercalate faster than large graphite particles or particles of spherical shape. In contrast, the deintercalation in clusters of large particles, such as, is slower. In the cathode, all these effects are suppressed because of the slow interfacial kinetics and small lithium diffusivity in the cathode.

5 A900 Figure 5. Voltage distribution evolution of microstructure T during a 1C discharge. Here, the position-dependent electrostatic potential compounds the polarization and ohmic losses. As a result of the complete deintercalation of smaller size particles and complex morphology particles such as those illustrated by, the macroscopic anodic voltage increases with respect to a pure lithium reference. The observed behavior is undesirable in the vicinity of the current collector because such polarization contribution translates to a macroscopic drop on the instantaneous power density. ment transport lithium ions faster away from the region, e.g., small graphite particles directly facing the cathode, such as at, and small anode particles embedded in a medium with tortuous electrolyte paths, e.g., small particles surrounded by large anode particles, such as at, or small graphite particles facing the back ohmic contact, such as at, promote local lithium accumulation in the electrolyte because lithium ions have to diffuse around larger particles to leave the anode region and reach the separator cathode region, such as at. In contrast, large anode particles facing the cathode do not necessarily delithiate first, as a result of the greater amount of stored charge. Furthermore, clusters of large anode particles, such as the one illustrated by region, delithiate at a slower rate because the smaller amount of lithium available outside the anode in the vicinity of the anode-electrolyte interface suppresses the rate of lithium deintercalation see Eq. 4. Graphite particles that present a complex morphology or have a rough surface are the next group of particles to become depleted of lithium e.g., region. In such particles, the effectively larger area that the negative electrode particles present to the immediate surroundings enables faster deintercalation rates. Moreover, local lithium depletion is enhanced in those edges and corners where lithium is removed faster. The trend is reversed as the particles become more spherical in morphology. The number of lithium ions expelled into the electrolyte by large anode particles is maintained for longer periods of time because the total amount of charge stored in a large particle of graphite is greater than the total amount of useful charge stored in smaller particles. Moreover, as particle size increases, lithium transport limitations in the graphite particles become evident, particularly close to those particle interface regions in the anode where lithium deintercalation rate is higher. Regardless of its size, once anode graphite particles become completely devoid of lithium, they become obstacles for lithium-ion diffusion in the electrolyte to those electrode particles that remain electrically active and are still deintercalating lithium into the electrolyte matrix. Macroscopically, the depleted, inactive particles increase the resistance of the cell and diminish the delivered instantaneous power density. Such effects can be readily suppressed by engineering the particle size and morphology to be monodispersed and uniformly distributed, as demonstrated in previous publications. 35,36 For the utilized material parameters and analyzed microstructure, the electrochemical effects observed in the anode are not present in the cathode. Here, the low lithium diffusivity of LiCoO 2 induces lithium accumulation at the electrolyte cathode particle interface, which limits the interfacial intercalation rate. For the utilized material parameters, and throughout the discharge sequence, lithium is inserted in the cathode particles; however, lithium accumulation is favored in the electrolyte region because lithium accumulation in the cathode is limiting. Based on the observed trends, it is expected that lithium intercalation would be improved if smaller cathode particles were used or if a different positive electrode chemistry with improved lithium diffusivity was utilized. Figure 5 shows that after 3600 s of discharge, the weighted average contributions to polarization from lithium-depleted particles in contact with the anode current collector interface region greatly contribute to a macroscopic drop in the potential delivered by the entire cell. However, a large volume fraction of anode particles exist in the anode electrode composite e.g., region that store sufficient charge to maintain the operation of the power source. The unused material contributes to a decrease in the macroscopic capacity and energy density of the cell. The predicted discharge response of microstructure T shows that clusters of small particles in the anode deintercalate completely at an earlier time, thus they constitute the high power density regions. Moreover, during the recharging sequence, the same clusters of small particles are the first to reach the lithium solubility limit. The complete charge/discharge occurring in small electrode particles is highly desirable for high power density applications because they guarantee complete material utilization per particle; however, when such regions interact electrochemically with active material regions with relatively sluggish electrochemical kinetics such as the processes occurring in electrode materials where the particle-size distribution differs from monodispersed, the combined power density response is detrimental to the overall performance of the cell. This behavior is a result of the large potential that develops in the anode current collector, which in turn renders the charge stored in the large particles unavailable. Such a response leads to an apparent loss of the macroscopic capacity of the cell. In addition, the kinetics of intercalation observed in electrode composites whose active material particles have capricious shapes induces localized lithium depletion/ accumulation in the edges and corners of the particles. Overall, the average response of a negative electrode particle of microstructure T corresponds to the one delivered by electrode materials with spherical or ellipsoidal morphology. However, i the extremes of localized behavior occurring in corners and edges in composite electrodes, ii the effects of randomness that results from mixing the particles during battery assembly on the local intercalation, and iii the details that emerge from accounting for the particle particle electrochemical interactions all define the locations in the anode where plating and lithium dendrites initiate. For microstructure T, because the smallest particles become completely delithiated first during discharge and become the first to reach the solubility limit during the recharge sequence, it is expected that dendrites are favored to nucleate and grow from these locations. This effect is not observed in large particles facing the cathode. Additionally, the performed spatially resolved simulation shows that during the first five min of battery discharge, the sudden depletion of lithium from the surface of the anode particles induces a local lithium accumulation into the surrounding electrolyte. Such electro-

6 A901 Figure 6. Lithium-ion distribution for sections 1 5 of microstructure T. Lithium concentration is normalized by the solubility limit of lithium in graphite, n T = mol/dm 3. While the observed deintercalation/ intercalation trends are common in regions 1 5, the specific state of charge of each electrode particle differs from the one delivered by microstructure T as a result of the local electrochemical interactions that arise from clusters of particles removed from the simulated battery system. In each section, the local state of charge is specified by the local particle particle electrochemical interactions. For example, particle A in section 2 interacts primarily with B and the cathode as a result of the large electrochemical potential difference. However, in section 2*, particle B is not present, thus enabling the interactions with particles C, D, E, and F. Similarly, the cluster of particles deintercalates more lithium when it is part of section 2* than when it is part of section 1. This is because the electrochemical environment of the particles in the isolated section favors lithium transport upward for section 2* and favors lithium transport downward in section 1. Values of normalized lithium concentration are shown in selected locations. chemically induced chemical shock induces a local voltage drop. As the battery continues to discharge, the lithium stored at the core of the large particles replenishes the surface of the anode particles and enables electrical recovery at the back contact. Finally, the present simulation shows that lithium concentration gradients are present during galvanostatic discharge inside the graphite particles; however, they are very shallow and much less pronounced than the lithium concentration differences that exist between neighboring large and small anode particles. c Individual region analysis. While it is clear that every region in the cell contributes to specify the macroscopic voltage of the battery, it is not clear to what degree each region or microstructural feature in isolation controls the galvanostatic response of the entire cell, to what degree the particle particle electrochemical interactions dominate the apparent macroscopic capacity of the cell, or in which way specific details define the lifespan of the device. To assess the contribution of every subregion to the response of microstructure T, the device was split into five subsections regions 1 5. A simulation was performed for every region in isolation from the surrounding abutting ones. The state of charge in each section at the end of a 1C discharge is summarized in Fig. 6 and is compared against microstructure T. The response of each electrode particle in each section is similar to the one observed when it is part of microstructure T: Smaller graphite particles will delithiate first, followed by particles with greater total area exposed to the electrolyte, trailed by large particles and spherical particles. Moreover, those sections that sample many small particles sections 2, 4, and 5 from microstructure T completely deintercalate before those sections that sample c The shallow lithium gradients in the anode particles are a result of the large characteristic diffusion distance, L = Da t = 600 m, which is 10 times larger than the average graphite particle size, 10 m. large graphite particles sections 1 and 3. Results suggest that isolated sections with smaller particle size deliver high power densities, as demonstrated experimentally by currently pursued processing approaches. Important differences in behavior are predicted to occur when each section is discharged in isolation from the surrounding electrochemical interactions that microstructure T imposes. Figures 7 and 8 summarize the local time-dependent electrochemical response of section 1 and show that individual particles closer to the top and bottom edges of the cell deintercalate 20% less lithium as compared to when they are part of microstructure T. Figure 6 illustrates such an example. Here, as a result of its electrochemical isolation, particle G remains unaffected by the extracted electronic current when discharged as part of section 1 alone; however, when integrated into section 2*, the average lithium concentration of the same particle corresponds to n = mol/dm 3, because of the additional interactions with cluster. Finally, when particle G is part of microstructure T, it interacts with the surrounding neighbors and its lithium concentration at the end of the discharge is n = mol/dm 3. The largest contribution to the power density of section 1 is attributed to the small graphite particles facing the cathode layer, e.g., at particle C. As compared to microstructure T, the electrostatic potential of the anode of section 1 is higher on the small anode particles located at the back contact after 3600 s of operation. The voltage distribution in the anode demonstrates that the central region of the negative electrode still possesses lithium for the discharge to proceed, thus on the order of 50% of the anode remained unutilized. The behavior of section 1 is controlled by the anode particles and the availability of paths to transport lithium ions to the cathode region. The effect of anode particle clusters is analyzed by assessing the impact of cluster on the electrochemical behavior of the entire cell and when considered as part of section 2. An additional analysis on section 2* is also performed, for it does not sample cluster and instead samples the surrounding large particles. The electrochemical response that is delivered by both sections at the end of a 1C discharge is summarized in Fig. 9. Here, particle C in section 2 interacts primarily with particle B. However, in section 2*, particle C is not present, thus B interacts with particles of similar size, such as particle D. The direct comparison demonstrates that clusters of small graphite particles are responsible for inducing a greater lithium accumulation in the electrolyte, on defining extremes of lithium accumulation, and on determining the specific location and mechanism through which a cell fails. Figure 10 summarizes the macroscopic voltage response as a function of the effective charge capacity delivered by microstructure T and sections 1 5. The macroscopic capacity of each cell section was normalized against the total charge delivered by section 1, Q = 2312 mah/m 2, for a 1C discharge. A comparison of the voltage response as a function of the normalized capacity for each section including section 2* demonstrates the correlations in the behavior of individual microstructural features in specific sections of microstructure T and compares them to the overall behavior. Figure 10 shows that the response of microstructure T is dominated by the galvanostatic response of clusters of small anode particles and the electrochemical response of those small particles that are closer to the anode back contact. In particular, clusters of small particles that are part of sections 2 and 4 identified in Fig. 4 as cluster control the galvanostatic response of the battery and specify the macroscopic power density of the entire cell. Thus, at a normalized capacity of 50% in Fig. 10a, cluster, becomes depleted of lithium. In Fig. 10b, sections 1, 2, 3, and 5 are unable to compensate for the polarization losses of section 4, leading to a macroscopic voltage drop and delivered macroscopic capacity of 60%, Q = 1445 mah/m 2, as compared to the one delivered by section 1 alone. When subjected to galvanostatic transients, the local electrochemical interactions allow the cell to recover. The time scale and extent of the observed recovery of the cell greatly depend on the

7 A902 Figure 7. Lithium distribution in section 1. The microstructural fields for a discharge rate of 1C are shown. Concentration values are normalized by mol/dm 3, the solubility limit of graphite. Section 1 is highlighted on the upper-right inset. Small particles such as at location A and elongated regions such as at location B deintercalate lithium faster to its surroundings as compared to the surrounding larger particles. Also, small particles closer to the cathode deintercalate first such as the one at location C, and large particles remain charged after 1 h at location D. The corresponding electrostatic potential distribution is shown in Fig. 8. Figure 8. Voltage distribution of section 1. Simulated region is highlighted in the upper-right inset. The individual contributions to polarization of those small anodic particles that are closer to the anode ohmic back A and B contact drive the power of the system down in spite of the low electrostatic potential at the core of the anode layer. ability of the depleted particles to exchange lithium with the surrounding electrode particles. Thus, an appreciable recovery of microstructure T, on the order of 100 mv, is observed as a result of the interactions of all the particles, while such recovery is only on the order of a few microvolts in the individual sections because of the fewer electrochemically active particles available to exchange lithium.

8 A903 Conclusion Figure 9. Electrochemical distribution for two subsections of microstructure T, sections 2 top and 2* bottom, as labeled in Fig. 6 at the end of a 1C discharge sequence. The cluster of particles, absent in section 2*, constitutes 30% of the anode section and greatly contributes in driving the deintercalation of the anode section 2. Both cell sections share more than 50% of the electrode material; however, both sections sample different morphologies and sizes that specify very different electrochemical fields at the end of the discharge process. The simulation shows that the poor transport properties of the cathode make the small size clusters of anode particles detrimental to the delivered macroscopic power density and promotes lithium accumulation in the electrolyte phase. Concentration values are normalized by mol/dm 3, the solubility limit of graphite. Figure 10. Normalized capacity curves of sections 1 5, as labeled in Fig. 6. The capacity of each cell was normalized to the charge delivered by section 1, Q 1 = 2312 mah/m 2. The macroscopic response of microstructure T is the weighted average contribution of each section plus the electrochemical interactions between the different phases. Thus, sections 1, 2 Q mah/m 2, 3 Q mah/m 2, and 5 Q 5 = 1595 mah/m 2 contribute with a relatively larger capacity; section 4, Q 4 = 1156 mah/m 2 contributes with polarization losses that ultimately limit the macroscopic power delivered by the entire section. For sections 2 and 3, the particle size is more homogeneous and thus delivers a sustained voltage response throughout the 1 h discharge. Therefore, when the particles in contact with the current collector of section 4 are depleted, a the voltage of microstructure T is impacted. Subsequent internal electrochemical interactions controlled by the depleted particles lead to a macroscopic loss of the power density of the cell. b The resultant macroscopic capacity of microstructure T is 60% of that delivered by section 1, i.e., Q T = 1433 mah/m 2. Simulations show that the response of a rechargeable lithium-ion battery is a combined result of particle-size distribution, clustering, surface roughening, and the interactions from each particle of every region of the cell. These interactions allow the cell to recover from sudden discharge transients, and as the size of the cell increases, the details of the spatial coupling of the particles in the cell are averaged out. The long-term macroscopic capacity of the cell is controlled by the extremes on the particle-size distribution, which induce electrical and chemical concentrations. For the analyzed microstructure and utilized material parameters, lithium in the anode particles is first depleted in smaller particles, followed by small particle clusters, particles of complex morphology, and finally, by particles that directly interact with these three populations and cathode particles across the electrolyte. The performed calculations suggest that lithium-ion accumulation is favored in those regions of the anode where small particle-size clusters form. Similarly, dendrite formation events that are a result of extreme electrochemical behavior are favored at the surface of small anode particles and particles of complex geometries facing the cathode. Overall, the realization of such events is entirely controlled by the way in which the material was processed and the configuration, morphology, and properties of each phase as it was set during cell assembly. Particle clustering and elongated graphite particles at the back of the anode are unfavored topologies, for they induce polarization losses that translate into a lower macroscopic charge. Purdue University assisted in meeting the publication costs of this article. References 1. C. A. Vincent, Solid State Ionics, 134, M. Wakihara, Mater. Sci. Eng. R., 33, R. M. Dell, Solid State Ionics, 134, S.-Y. Chung, J. T. Bloking, and Y.-M. Chiang, Nature Mater., 2, M. Doyle and Y. Fuentes, J. Electrochem. Soc., 150, A B. Hellweg, MS Thesis, Department of Materials Science and Engineering, Massachusetts Institute of Technology K. West, T. Jacobsen, and S. Atlung, J. Electrochem. Soc., 129, M. Doyle, J. Newman, A. S. Gozdz, C. N. Schmutz, and J.-M. Tarascon, J. Electrochem. Soc., 143, P. Arora, M. Doyle, A. S. Gozdz, and J. Newman, J. Power Sources, 88, G. S. Nagarajan, J. W. Van Zee, and R. M. Spotnitz, J. Electrochem. Soc., 145, M. W. Verbrugge and B. J. Koch, J. Electrochem. Soc., 150, A P. Arora, M. Doyle, and R. E. White, J. Electrochem. Soc., 146, J. W. Long, B. Dunn, D. R. Rolinson, and H. S. White, Chem. Rev. (Washington, D.C.), 104, S. Suresh, Fatigue of Materials, Cambridge University Press, Cambridge, UK Q. Horn and Y. Shao-Horn, J. Electrochem. Soc., 150, A Q. C. Horn and K. C. West, Abstract 318, The Electrochemical Society Meeting Abstracts, Chicago, IL, May 8, Y. Shao-Horn, S. Osmialowski, and Q. C. Horn, J. Electrochem. Soc., 149, A O. Crowther and A. C. West, J. Electrochem. Soc., 155, A S. Atlung, K. West, and T. Jacobsen, J. Electrochem. Soc., 126, S. Atlung, B. Z. Chistiansen, and T. Jacobsen, J. Electrochem. Soc., 131, B. C. Knutz, K. West, B. Z. Christiansen, and S. Atlung, J. Power Sources, 43 44, M. Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc., 140, M. Doyle, T. F. Fuller, and J. Newman, Electrochim. Acta, 39, R. Darling and J. Newman, J. Electrochem. Soc., 145, Y.-M. Chiang and B. Hellweg, Abstract 144, The Electrochemical Society Meeting Abstracts, Vol , San Francisco, CA, Sept 2 7, W. B. Gu and C. Y. Wang, J. Electrochem. Soc., 147, W. B. Gu and C. Y. Wang, in Thermal-Electrochemical Coupled Modeling of a Lithium-Ion Cell, R. A. Marsh, Z. Ogumi, J. Prakash, and S. Surampudi, Editor, PV 99-25, p. 748, The Electrochemical Society Proceedings Series, Pennington, NJ L. Rao and J. Newman, J. Electrochem. Soc., 144, L. Song and J. W. Evans, J. Electrochem. Soc., 147, K. F. Thomas and J. Newman, J. Electrochem. Soc., 150, A G. G. Botte, V. R. Subramanian, and R. E. White, Electrochim. Acta, 45, Y.-B. Yi, C.-W. Wang, and A. M. Sastry, J. Electrochem. Soc., 151, A X. Zhang, W. Shyy, and A. M. Sastry, J. Electrochem. Soc., 154, A

9 A X. Zhang, A. M. Sastry, and W. Shyy, J. Electrochem. Soc., 155, A R. E. García, Y.-M. Chiang, W. C. Carter, P. Limthongkul, and C. M. Bishop, J. Electrochem. Soc., 152, A R. E. García and Y.-M. Chiang, J. Electrochem. Soc., 154, A R. E. García, C. M. Bishop, and W. C. Carter, Acta Mater., 52, J. S. Newman, Electrochemical Systems, Prentice-Hall International, Englewood Cliffs, NJ C. M. Doyle, Ph.D. Thesis, University of California at Berkeley, Berkeley, CA J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York C. A. Vincent and B. Scrosati, Modern Batteries, An Introduction to Electrochemical Power Sources, John Wiley & Sons, New York S. A. Langer, W. C. Carter, and E. R. Fuller, OOF: Analysis of Real Material Microstructures S. A. Langer, E. R. Fuller, and W. C. Carter, Comput. Sci. Eng., 3, O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method, Vol.1, McGraw-Hill, London O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method, Vol.1, Chap. 11, McGraw-Hill, London Q. C. Horn and K. C. White, in International Meeting on Lithium Batteries, Tianjin, China, p Q. C. Horn and K. C. White, in Advanced Automotive Battery Conference, Tampa, FL, O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method, Vol.1, Chap. 17, McGraw-Hill, London W. Liniger, Technical Report RC2198, IBM Research R. E. García, Ph.D. Thesis, Massachusetts Institute of Technology, MA 2003.