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ceramic electrolyte interface: The role of interfacial resistance and surface defects
Journal of Power Sources 441 (2019) 227186
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Modeling of lithium electrodeposition at the lithium/ceramic electrolyte interface: The role of interfacial resistance and surface defects Giovanna Bucci a, b, *, Jake Christensen a a b
Robert Bosch LLC, 384 Santa Trinita Ave, Sunnyvale, CA, 94085, USA Massachusetts Institute of Technology, Department of Materials Science and Engineering, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA
H I G H L I G H T S
G R A P H I C A L A B S T R A C T
� Cathodic current is magnifed at cracklike defects on the electrolyte surface. � Due to confinement, the lithium inside a crack undergoes a large compressive stress. � Such stress generates a driving force that diverts deposition away from the crack. � The stress required to stop plating at crack tips grows with the surface resistance. � Ultimately, establishing good wetting conditions is critical to the cell reliability. A R T I C L E I N F O
A B S T R A C T
Keywords: Solid-state batteries Lithium anode Dendrites Interfaces
Inorganic solid electrolytes, paired with Li-metal anodes, could result in high energy density yet safe rechargeable lithium batteries. To enable Li-metal anodes, the ceramic separator needs to be mechanically robust to occlude the path for Li-dendrites growth and prevent cell shorting. Using a continuum model, we explore the role of surface geometry and interfacial resistance on Li deposition. The model calculates the intensification of the cathodic current due to crack-like defects on the electrolyte surface. The driving force for plating at the crack tip increases with the crack length. As the crack grows the current at its tip grows, making the system unstable. However, due to the stress-potential coupling, the plating rate decays as the pressure rises in the Li-filament inside the crack. High interfacial resistance leads to more uniform plating but it also causes high over potentials. A larger stress is required to compensate such overpotentials and stop plating at the crack tip. If the pressure at the crack tip overcomes the fracture toughness of the electrolyte material the defect propagates across the separator. Our results show that controlling the lithium-electrolyte interfacial properties, such as defect size and interfacial resistance, is critical for solid-state battery durability and charging performance.
1. Introduction Recently discovered solid electrolytes with high ionic conductivity enable the possibility of fabricating all-solid-state lithium batteries with
a power performance comparable to that of liquid-electrolyte batteries [1,2]. Engineering of the electrode/electrolyte interfaces and long term stable cycling remain a challenge for most solid-state lithium batteries [1,3–5]. When combined with a Li-metal anode, solid electrolytes could
* Corresponding author. Robert Bosch LLC, 384 Santa Trinita Ave, Sunnyvale, CA, 94085, USA. E-mail address: [email protected] (G. Bucci). https://doi.org/10.1016/j.jpowsour.2019.227186 Received 15 May 2019; Received in revised form 12 September 2019; Accepted 19 September 2019 Available online 17 October 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.
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Journal of Power Sources 441 (2019) 227186
provide the opportunity to stop Li-dendrite penetration and avoid cell shorting. To succeed, solid-electrolytes must resist fracture driven by lithium electrodeposition. A systematic understanding of Li-dendrites formation and growth is still necessary to guide the development of mechanically reliable materials. In the event of a short, the risk of fire and explosion should be mitigated because solid electrolytes are typi cally less flammable than organic liquid electrolytes. Multiple research groups have recently reported cases where ceramic solid electrolytes paired with a Li metal anode experience a short circuit event [6–9]. To enable Li-metal anodes, a solid electrolyte needs to be mechanically robust to occlude the path for Li-dendrites growth. Most of the literature in the field of solid electrolytes has focused on improving conductivity of varying material compositions. Ceramic solid electro lytes are been evaluated also for their mechanical properties. Dense LLZO (Li7La3Zr2O12) type materials prepared by hot pressing techniques have been mechanically tested by ultrasound spectroscopy and inden tation methods and are found to exhibit an elastic modulus of 150 GPa and fracture toughness of 0.86 1.63 MPa m0.5 [10,11]. High density sulfide electrolyte materials (xLi2-yP2S5, briefly LPS) prepared by hot pressing techniques show elastic moduli of 18 25 GPa [12]. Despite showing macroscopic deformability, LPS materials are brittle in nature as demonstrated by the fracture toughness measurements of 0.23 MPa m0.5. For reference, elastic modulus of polycrystalline Li has been reported to range from 1.9 to 9.8 GPa [13–15], and its yield strength from 0.41 to 0.89 MPa [15,16]. At the nanoscale Li-metal may exhibit yield stresses that are greater than bulk lithium [17–19]. The combination of properties that makes the electrolyte mechani cally reliable against Li-protrusions have not been clearly identified. Several experiments contradict the shear-modulus criterion proposed by Monroe and Newman for prevention of Li dendrites [20]. The Monroe-Newman model was developed for polymer, not ceramic elec trolytes, and it describes dendrite initiation as a perturbation growing on a perfectly smooth and flat interface. Its application has been extended to other classes of materials, including brittle ceramic electrolytes, where the physics of the lithium/electrolyte interface behaves quite differently. Because the shear modulus is not a fracture property of the electrolyte layer it is unlikely to be the determining factor for brittle ceramic electrolytes. Porz and coauthors suggest that stabilization of ceramic electrolyte interfaces against lithium metal penetration requires minimizing interfacial defects [21]. In Ref. [21], the surface defect population was intentionally varied, and galvanostatic electrodeposi tion experiments were used to investigate the role of surface morphology. Although many authors have demonstrated “extended cycling” with Li metal cells (even using liquid electrolytes), the unique achievement of thin film batteries is the ability to cycle hundreds to thousands of times while utilizing close to 100% of the lithium [22] (other cells only cycle a small fraction of a thick Li foil). However, thin film battery areal ca pacity is too low to be of interest for automotive applications [23]. Surface treatments to improve Li wetting on the ceramic electrolyte have consistently shown improvements in the cycling performance [24–29]. However, current density and cycle-life must be further increased to demonstrate relevance to vehicle electrification. Atomically smooth interfaces can be easily fabricated in the case of vapor-deposited LiPON [22]. Conversely, sharp surface flaws are likely to form during processing of low toughness materials such as LPS. Also, polishing of garnet solid electrolyte is expected to introduce surface defects whose depth depends on the type of abrasive [21]. Defects can be loosely defined as any non-uniformity at the macro scopic level, including large-scale porosity or cracks, or at the micro scopic level, such as pore channels, grain boundaries, and impurity precipitates. Here we aim to investigate the role of macroscopic defects on the current density distribution. We simulate the presence of sharp crack-like defects on the electrolyte’s surface and we calculate the cathodic current distribution as a function of the defect’s geometry and of the average interfacial resistance. Analysis of the grain boundary
effect on the local current density will be treated in a future publication. The reminder of the paper is organized as follows: In section 2 we introduce the boundary value problem solved with a in-house finite element code. In section 3 we describe the simulation results and discuss their interpretation in view of published experimental trends. In section 4 we propose a self-limiting mechanism to prevent solid-state cell failure due to Li-dendrite penetration. 2. Statement of the boundary-value problem A finite element model was implemented to simulate Li-ion con duction in a solid-electrolyte layer. The model aims to calculate the current density of Li-electrodeposition on the electrolyte layer as a function of the geometry and resistance of the interface. The cathode overpotential is the driving force for lithium deposition and it is calculated by solving the electrochemical problem described below. We consider the homogeneous domain sketched in Fig. 1 as representing a solid Li-ion conducting material. Steady-state conduction across the electrolyte is modeled by the Laplace equation as in Eq. (1), with the potential of lithium φ treated as independent variable. A fixed current density at the positive electrode interface Ωþ (right edge in Fig. 1a). In Eq. (3) the constant i ¼ i=κ is the ratio between the current density and the electrolyte bulk conductivity κ. The kinetics at the negative electrode interface Ω is described by a linear approximation of the Butler-Volmer equation1 (see Eq. (2)). Δϕ ¼ 0
(1)
on Ω
rϕ⋅n ¼ hðϕ
ϕLi Þ
on ∂Ω
on ∂Ωþ
rϕ⋅n ¼ i
(2) (3)
The model assumes uniform wetting at the negative electrode interface. In other words, the boundary ∂Ω in Eq. (2) includes both the flat region and the defect. In the Robin boundary condition (Eq. (2)) imposed at the negative electrode, φLi is the potential of lithium in the Limetal phase. The convective coefficient h represents the ratio between surface and bulk conductivity according to the expression h ¼ ðρκÞ
1
ρ¼
RT þ ρfilm i0 F
(4)
In Eqq.(4) ρ is the total resistance of the interface, defined as the sum of the kinetic and film resistance. The latter is the resistance caused by a decomposition or solid electrolyte interphase (SEI) layer. The kinetic resistance depends on the exchange current density i0 , the Faraday constant F, the gas constant R, and the absolute temperature T. Solving the problem stated above provides the overpotential that drives lithium electrodeposition at the positive electrode interface. Elliptic boundary value problems as in Eq. (1) are characterized by a singularity in the solution around non-smooth regions of the boundary [30]. In the proximity of the singular point at the crack tip (i.e., as x→0 in Fig. 1b), the leading term of the singularity x1=α sinðθ =αÞ depends on the angle απ describing the local geometry. The singularity becomes more pronounced as the angle απ increases. Because of the not-convexity of the boundary, that is α > 1, even the first derivative of the solution is unbounded. In the limit where απ →2π , the solution behaves like x1=2 sinðθ =2Þ. From the mathematical character of the problem it follows that the electric field tends to become singular in proximity of a sharp boundary feature. The singularity is alleviated by the Robin boundary condition in Eq. (2) because an infinite overpotential is necessary to
1
In all the cases considered here, the difference between the overpotential at the crack tip and the overpotential away from the defect is well within the limit of applicability of such approximation. It follows that the relative current peak at the crack tip is accurately predicted by the linear approximation of the Butler-Volmer equation. 2
G. Bucci and J. Christensen
Journal of Power Sources 441 (2019) 227186
Fig. 1. a) Cartoon representation of the system stud ied in Eqq.(1)–(3). The domain consists of a solidelectrolyte layer, with applied uniform current at the positive electrode. Linearized Butler-Volmer kinetics is assumed at the negative electrode and the cathodic current is calculated for a given geometry of the interface. b) The plot shows the electric field ahead of the crack tip. The finite kinetics of the interface sup press the singularity of the electric field at the crack tip, while in the case of zero interfacial resistance the electric field becomes unbounded.
drive an infinite current density if the interfacial kinetics is finite.
1 μm below the surface and its radius of curvature is about 0.4 nm. The chosen values of crack depth in the simulations are within the range of observed defect sizes for highly polished LLZTO samples [21]. Typical separators used with liquid electrolytes are at most 25 μm thick [33]. For solid electrolytes, a thickness less than 50 μm remains a target to enable cell energy densities competitive with conventional Li-ion technology [34]. A recently published review discuss challenges of manufacturing thin ceramic separators [35]. Of particular interest is the solution at the corners highlighted in red and yellow in Fig. 2b. The analyses show that the current density has a positive peak at the concave corner (marked in red in Fig. 2b) and a negative peak at the convex corners (marked in yellow in Fig. 2b). In Fig. 2c, the same color code is used to plot the current density peaks normalized with respect to the remote value (i.e., the current density away from the defect). According to Fig. 2c, a larger deviation of the
3. Results Numerical solutions to the system of equations (1)–(3) were carried out with an in-house finite element code in two spatial dimensions. The results presented here illustrate the interplay between morphology and kinetics at the anode interface. The finite element grid is managed with the deal.II finite-element library [31,32]. A representative mesh of the solid-electrolyte domain is shown in Fig. 2a. The mesh is highly refined in the surrounding of the crack to improve the accuracy of the solution in that region. The crack-length in Fig. 2a and b corresponds to 1=20 of the electrolyte’s thickness W and the radius of curvature at the tip is set to W=51200. If we assume, for instance, the electrolyte layer to be 20 μm wide the crack tip reaches
Fig. 2. a) Finite element mesh used to discretize the ceramic electrolyte domain and calculate the potential. Of particular interest is the anode over potential as it determines the lithium electrodeposition rate. The mesh is highly refined around the sharp crack introduced at the anode surface. b) An extract of the mesh around the crack region is shown to highlight the main points of interest. The crack tip is marked in red and the convex corners are marked in yellow. In the baseline case we assume the crack length to be 1/20 of the electrolyte-layer’s width W. c) The current density undergoes a positive peak at the crack tip (red marks) and a negative peak at the cor ners marked in yellow. The current density values reported on the vertical axis are normalized with respect to the current density computed away from the defect. The current peaks diverge as the surface resistance decreases (see horizontal axis), while the bulk electro lyte conductivity is kept constant at 0.1 mS /cm in the simulations. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 3
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Journal of Power Sources 441 (2019) 227186
current density emerges as the interfacial resistance decreases. For re sistances of at least 100 Ωcm2 the cathodic current and, therefore, the rate of lithium deposition are predicted to be insensitive to the defect. Introducing a uniform resistive layer generally reduces the sensitivity to the surface the morphology. The result shown in Fig. 2c applies to a highly conductive solid electrolyte, in fact we assumed a bulk conductivity κ ¼ 0:1 mS / cm. The interplay between bulk and surface kinetics is represented by the convective coefficient h in Eq. (2). If conduction across the electrolyte is fast, the interface acts as a kinetic bottleneck unless the interfacial resistance ρ is extremely low. Viceversa, if the bulk conductivity of the electrolyte is at least two orders of magnitude smaller than κ ¼ 0:1 mS / cm we expect a sharp current peak at the crack tip for any practical value of surface resistance. The yellow markers in Fig. 2c indicate that current density has a minimum at the convex corners of the interface. During charge, this helps preventing Li accumulation in proximity to the crack and helps lithium flow out of the crack. During discharge, a lower stripping rate is expected in the region around each surface defect. Generally, an external pressure is required to continuously reestablish a uniform contact at the Li-metal/electrolyte interface. If voids are allowed to form at the interface the anodic current at the points of contact becomes higher than the average applied value. Non-uniform current density can lead to additional voids formation. This cascade effect can be avoided if the external pressure is large enough to plastically deform the Li-film and restore interfacial contact. The results discussed so far are specific to the geometry illustrated in Fig. 2. Next, we analyze the sensitivity of these results with respect to the crack’s length and the radius of curvature at the crack tip. Both features of the defect’s geometry are expressed in relation to the width of the electrolyte layer W. Fig. 3a shows the current density peak at the crack tip for three values of crack length: one smaller (W/40) and one larger (W/5) than the default value W/20. As expected, deeper defects promote larger localization of the current. Again, if we assume a 20μm wide electrolyte layer, the current density at the tip of a 4 μm crack is expected to be five times larger than the average applied current. Because the driving force for electrodeposition increases with crack length, crack propagation would accelerate as the initial flaw extends across the electrolyte layer. The electro-chemical model discussed so far is appropriate to describe the condition of surface defects not yet filled with deposited lithium. This situation occurs at the beginning of charge or when new surface is being exposed by a propagating crack (cases a and c illustrated in Fig. 4). Also, the newly exposed surface around the crack tip tends to have lower film resistance. This further concentrates the current until a stable passivating layer is allowed to form. In general, the interface morphology affects the local kinetics but also the local mechanical stress. Compared to a smooth flat interface, the lithium deposited within cracks, voids, etc. undergoes larger pressure due to the confinement (case b in Fig. 4). Such pressure is eventually responsible for extending the crack, once the fracture toughness of the ceramic electrolyte is overcome. Before the defect extends, the pressure build-up alters the overpotential locally and reduces the cathodic cur rent at the crack tip and along the crack facets. This intermediate stage of coupled electro-chemo-mechanics is analyzed in Section 4. Here we focus on the initial stage of electrodeposition when lithium is only partially filling an existing or growing defect and the electro-chemistry is approximately decoupled from the mechanics. At this stage, a uniform passivating interfacial layer or a layer of glassy material with low defects concentration are beneficial to establishing uniform kinetics and avoiding Li-protrusions. These can be some of the advantages in using LiPON as electrolyte or as coating for other ceramic materials. The relationship between the crack length and the current-density peak at the crack tip is illustrated in Fig. 3b for the cases of lowest interfacial resistance (ρ ¼ 0:1 Ω cm2 and ρ ¼ 1:0 Ω cm2). The slope of the curve decreases with increasing surface resistance and it becomes
Fig. 3. a) The sensitivity of the current density at the crack tip with respect to the crack lengths is illustrated. In the legend, the crack length is expressed in relationship to the electrolyte-layer’s width W. When the solution is not dominated by the interfacial kinetics (i.e., for values of surface resistance � 10 Ω cm2) the current is very sensitive to the defect size. b) The simulation results indicate that the current peak at the crack tip scales approximately linearly with the crack length. As the surface resistance decreases from ρ ¼ 0:1 Ω cm2 to ρ ¼ 0:1 Ω cm2, the slope of the curve decreases indicating a lower sensitivity to the defect size.
negligible at ρ � 100 Ω cm2. The radius of curvature of the crack tip is related to the απ angle marked in Fig. 1b and, therefore, to the singularity of the electric field. As shown in Fig. 5a, the local current density peak generally increases with decreasing radius of curvature R. However, the sensitivity of the solution to R is not particularly pronounced and it is attenuated by the finite interfacial kinetics. In our simulations we considered geometries characterized by radii R � W/51200. For thin ceramic electrolytes of width W ¼ 20 μm this correspond to radii �0.39 nm. The continuum approximation and the definition of tip curvature become meaningless below this scale. These three data-points in Fig. 5b show an approximately linear relationship between the crack-tip radius and the peak value of the current. The slope in Fig. 5b is specific to the case of lowest interfacial resistance (ρ ¼ 0:1 Ω cm2). Such slope becomes less negative as ρ increases and the current is less sensitive to the local geometry of the tip. In all the analyses presented above, high interfacial resistance ap pears to be beneficial in evening the current density and avoid localized 4
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Journal of Power Sources 441 (2019) 227186
Fig. 4. Cartoon representation of the mechanism for electrodeposition-induced crack propagation. The accretion of lithium in the direction orthogonal to the crack fac ets tends to open the crack. At the same time, the constrain applied by a rigid solid electrolyte material causes the Li-filament to undergo hydrostatic pressure (case b). The pressure build-up can alter the local poten tial and slow down, or eventually stop, electrodeposition. The pressure at the crack tip is also responsible for propagating the crack (case c). For low toughness electrolyte materials, crack propagation can occur before a strong coupling between mechanics and electrochemistry is established. The stress-potential coupling is responsible for diverting electrodeposition away form the crack region. Two conditions associated with crack growth cause the local rate of elec trodeposition to temporarily rise: i) the hy drostatic stress at the crack tip decreases, and ii) the newly exposed surface tends to have lower film resistance.
electrodeposition. This is in apparent contradiction with experimental efforts directed towards lowering interfacial impedance in order to enable long-term stable cycling with lithium metal [24–29]. However, sluggish ion transport in the interfacial regions causes inhomogeneity in potential-current distributions. This scenario is different from the model assumption of uniform interfacial kinetics. Non-uniform contact is clearly detrimental, as it leads to electrical hot spots and the formation of sharp Li protrusions puncturing the electrolyte. If we modeled the interface with a random distribution of resistance values ρ, we would predict a noisy current signal. The amplitude of the current peaks and their spatial distribution would depend on the character of the ρ dis tribution. In general, those peaks would overlay with the current peak at the crack tip, causing additional stress localization. Such a model would simply reinforce the point that surface heterogeneity, including cracks, needs to be avoided.
electrolyte layer. Furthermore, fracture propagation tends to accelerate over time because the current density at the tip grows linearly with the crack length. Due to the coupling between potential and stress, a self-limiting process can stop Li electrodeposition inside defects. In general, the po tential along the interface is altered by a local stress field. Inside a crack, compressive stress rises the electrochemical potential of lithium with respect to its stress-free state. As more layers are being deposited, the metal become more and more compressed within the confinement of a crack, and the local rate of electrodeposition progressively decreases in response to the mechanically shifted voltage. The coupling between mechanical stress and the local cell voltage can be exploited to pro gressively divert Li deposition away from the crack tip. Following Larch�e and Cahn [36], our previous work [21,37], we use the following expression to calculate the pressure required to shift lithium voltage enough to locally suppress electrodeposition
4. Self-limiting mechanism to avoid crack propagation
p¼
In the previous sections, we analyzed conditions that locally magnify the cathodic current and cause non-uniform deposition. Even if the rate of Li-electrodeposition is constant along the interface, the material deposited within cracks is likely to generate much larger stress than the metal plated away from the defect (see cartoon in Figs. 4 and 6a). A local current density of 1 mA cm 2 corresponds to a Li-metal deposition rate of 10 μ3 s 1. The low yield strength of lithium metal suggests that Li may plastically flow out of shallow cracks as more lithium is being deposited along the crack facets. At the same time, the Li-metal trapped inside the crack undergoes increasing hydrostatic pressure. In fact, the hydrostatic pressure is decoupled from the shearing stress that causes a material to plastically deform. Part of the pressure and, therefore, of the elastic energy, can be released by deforming the crack flanks. However, the main stress-relieving mechanism is likely to consist in propagating the crack, particularly with high stiffness low fracture toughness ceramic electrolytes. Some similarities are apparent with the problem of hy draulic fracturing, with the difference that the influx of lithium comes from ahead of the crack causing more severe stress concentration. At practical current densities, Li-protrusions are likely to grow from surface defects in order to accommodate the deposited volume of lithium. We speculate that diverting Li deposition away from defects is the only route to preventing Li-dendrites penetration. As long as the driving force for electrodeposition along the crack’s flanks is active it is likely to generate enough stress to propagate the fracture across a brittle
ϕtip F : βΩ
(5)
In Eq. (5), p is the average hydrostatic pressure experienced by the lithium filling the crack; Ω ¼ is Li partial molar volume (1:3⋅10 5 m3 =mol), β ¼ is the volume expansion coefficient (�1; here set to 1), F¼ Faraday constant. The surface accretion of lithium may be associated with local negative volume change of the electrolyte surface layers. The factor β can be used to scale down the volume generated via electrodeposition with respect to Li partial molar volume. Furthermore, newly deposited lithium layers and the SEI layer may alter the kinetic and mechanical properties of the interface. However, such effects are not taken into account in Eq. (5) and in the discussion that follows. According to Eq. (5), pressure scales with the anode overpotential φtip calculated at the crack tip. The value of ϕtip , in turns, depends on both the surface and the ohmic resistance. For a given electrolyte ma terial with a distribution of surface defects of a certain size, the pressure at the crack tip grows with the interfacial resistance. The relationship between the limiting pressure and the surface kinetics is illustrated in Fig. 6b for various crack lengths. The pressure required to stop elec trodeposition is predicted to be directly proportional to the surface resistance ρ, given that the electrolyte bulk conductivity is fixed at κ ¼ 0:1 mS / cm. The sensitivity to the crack length is apparent in the region of Fig. 6b where the bulk kinetics is dominant, i.e., for values of ρ � 1 Ω cm2. For less conductive electrolyte materials (κ < 0:1 mS / cm) the sensitivity to the defect size extends to a wider range of surface 5
G. Bucci and J. Christensen
Journal of Power Sources 441 (2019) 227186
Fig. 5. a) The sensitivity of the current density at the crack tip with respect to the radius of curvature of the crack tip is illustrated. In the legend, the crack length is expressed in relationship to the electrolyte’s width W. For a 20 μm wide electrolyte layer, 0.4 nm is the smallest radius considered in the simula tions. b) the relative peak in the current density at the crack tip was found to scale linearly with the radius of curvature of the crack tip. The relationship highlighted refers to the case with surface resistance ρ ¼ 0:1 Ω cm2.
Fig. 6. a) Cartoon representation of the mechanism for electrodepositioninduced crack propagation. The accretion of lithium in the direction orthog onal to the crack facets tends to open the crack. At the same time, the constrain applied by a rigid solid electrolyte material causes the Li-filament to undergo hydrostatic pressure. Given Li low yield stress, shearing can promote plastic flow of Li out of the crack. At the same time, the pressure build-up can alter the local potential and slow down, or eventually stop, electrodeposition. b) The limiting hydrostatic pressure required to divert Li electro-deposition away from the crack tip is plotted vs. the surface resistance at the anode interface (for a fixed bulk conductivity κ ¼ 0:1 mS / cm). The requirement on the electrolyte fracture toughness becomes less stringent as the kinetics of the inter face improves.
resistances. Keeping the crack tip pressure below the critical fracture stress is necessary to prevent further fracturing. This can be achieved by improving the mechanical properties of the electrolyte (high fracture toughness), improving the electrochemical properties (higher conduc tivity, lower surface resistance), decreasing the defect size (material processing), and/or decreasing the current density. For a given system, consisting of materials and interfaces, there is critical current density that generates enough pressure at the crack tip to fracture the electrolyte.
stress (due to the evolving geometry) does not significantly alter the interfacial kinetics. This condition is met at the beginning of charge and when new surface is exposed by the growth of an existing defect. The model calculates the intensification of the cathodic current due to crack-like defects on the electrolyte surface. While non-uniform elec trodeposition is generally detrimental to stable cycling, current locali zation at the crack tip is critical because additional Li volume needs to be accommodated in an already confined space. Due to the interplay between bulk and surface kinetics, if conduction across the electrolyte is fast, the interface kinetics contributes to lowering the sensitivity to the surface morphology. Conversely, sensi tivity to the surface morphology increases in poorly conductive elec trolytes. If the rate-limiting step for electro-deposition is due to the interfacial resistance, the current becomes insensitive to surface defects. Conversely, with a negligible surface overpotential, the cathodic current strongly depends on the curvature of the interface, and it is magnified by
5. Conclusions Sharp flaws are likely to form during processing of ceramic electro lyte materials, particularly low toughness sulfide materials. Recent ex periments [21] suggest that macroscopic defects can compromise the stability of all-solid-state cells with Li-metal anodes, providing a pathway for Li-dendrite penetration. Here we analyze the sensitivity of the current density distribution with respect to the morphology and resistance of the interface. First we propose an electrochemical model that focuses on the early stage of Li deposition when the mechanical 6
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Journal of Power Sources 441 (2019) 227186
the presence of sharp features. Such sensitivity decays as surface cracks become filled with lithium and the stress-potential coupling alters the current density distribution. Due to confinement, the lithium inside a crack undergoes a large compressive stress. Such stress generates an additional driving force that diverts deposition away from the crack in favor of flat regions of the interface. In the absence of stress, the driving force for electrodeposition at the crack tip scales with the crack length. As the crack grows the current at its tip grows, making the system unstable. However, the coupling be tween electrochemistry and mechanics contributes to smoothing of the cathodic current. Mechanical confinement causes pressure to rise within the Li metal filling a crack. Due to the stress-potential coupling, such pressure can divert electrodeposition away from the crack tip. The stress conditions that hinder Li-electrodeposition at the crack tip need to be established before the fracture stress is reached and the crack propagates. High interfacial resistance leads to more uniform electrodeposition prior to filling of the crack, but it also causes high overpotentials. A larger stress is required to compensate such overpotentials and stop electrodeposition at the crack tip. Once the pressure at the crack tip overcomes the fracture toughness of the electrolyte material the defect propagates across the electrolyte layer. Therefore establishing good Li wetting conditions is equivalent to relaxing the requirement on the electrolyte fracture toughness. This result restates the need for engi neering of electrode/electrolyte interfaces as a necessary condition for all-solid state battery performance and durability. Treatments to control the distribution of defects on the solid-electrolyte surface and to improve wetting with Li-metal will be critical for the success of high energy solidstate batteries.
[11]
[12] [13] [14] [15] [16]
[17]
[18]
[19]
[20] [21]
Acknowledgments [22]
This work was funded in part by the US Department of Energy, Advanced Research Projects Agency for Energy (ARPA-E), IONICS Pro gram (Award No. DE-AR0000775).
[23] [24]
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Contents lists available at ScienceDirect
Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
Modeling of lithium electrodeposition at the lithium/ceramic electrolyte interface: The role of interfacial resistance and surface defects Giovanna Bucci a, b, *, Jake Christensen a a b
Robert Bosch LLC, 384 Santa Trinita Ave, Sunnyvale, CA, 94085, USA Massachusetts Institute of Technology, Department of Materials Science and Engineering, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA
H I G H L I G H T S
G R A P H I C A L A B S T R A C T
� Cathodic current is magnifed at cracklike defects on the electrolyte surface. � Due to confinement, the lithium inside a crack undergoes a large compressive stress. � Such stress generates a driving force that diverts deposition away from the crack. � The stress required to stop plating at crack tips grows with the surface resistance. � Ultimately, establishing good wetting conditions is critical to the cell reliability. A R T I C L E I N F O
A B S T R A C T
Keywords: Solid-state batteries Lithium anode Dendrites Interfaces
Inorganic solid electrolytes, paired with Li-metal anodes, could result in high energy density yet safe rechargeable lithium batteries. To enable Li-metal anodes, the ceramic separator needs to be mechanically robust to occlude the path for Li-dendrites growth and prevent cell shorting. Using a continuum model, we explore the role of surface geometry and interfacial resistance on Li deposition. The model calculates the intensification of the cathodic current due to crack-like defects on the electrolyte surface. The driving force for plating at the crack tip increases with the crack length. As the crack grows the current at its tip grows, making the system unstable. However, due to the stress-potential coupling, the plating rate decays as the pressure rises in the Li-filament inside the crack. High interfacial resistance leads to more uniform plating but it also causes high over potentials. A larger stress is required to compensate such overpotentials and stop plating at the crack tip. If the pressure at the crack tip overcomes the fracture toughness of the electrolyte material the defect propagates across the separator. Our results show that controlling the lithium-electrolyte interfacial properties, such as defect size and interfacial resistance, is critical for solid-state battery durability and charging performance.
1. Introduction Recently discovered solid electrolytes with high ionic conductivity enable the possibility of fabricating all-solid-state lithium batteries with
a power performance comparable to that of liquid-electrolyte batteries [1,2]. Engineering of the electrode/electrolyte interfaces and long term stable cycling remain a challenge for most solid-state lithium batteries [1,3–5]. When combined with a Li-metal anode, solid electrolytes could
* Corresponding author. Robert Bosch LLC, 384 Santa Trinita Ave, Sunnyvale, CA, 94085, USA. E-mail address: [email protected] (G. Bucci). https://doi.org/10.1016/j.jpowsour.2019.227186 Received 15 May 2019; Received in revised form 12 September 2019; Accepted 19 September 2019 Available online 17 October 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.
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Journal of Power Sources 441 (2019) 227186
provide the opportunity to stop Li-dendrite penetration and avoid cell shorting. To succeed, solid-electrolytes must resist fracture driven by lithium electrodeposition. A systematic understanding of Li-dendrites formation and growth is still necessary to guide the development of mechanically reliable materials. In the event of a short, the risk of fire and explosion should be mitigated because solid electrolytes are typi cally less flammable than organic liquid electrolytes. Multiple research groups have recently reported cases where ceramic solid electrolytes paired with a Li metal anode experience a short circuit event [6–9]. To enable Li-metal anodes, a solid electrolyte needs to be mechanically robust to occlude the path for Li-dendrites growth. Most of the literature in the field of solid electrolytes has focused on improving conductivity of varying material compositions. Ceramic solid electro lytes are been evaluated also for their mechanical properties. Dense LLZO (Li7La3Zr2O12) type materials prepared by hot pressing techniques have been mechanically tested by ultrasound spectroscopy and inden tation methods and are found to exhibit an elastic modulus of 150 GPa and fracture toughness of 0.86 1.63 MPa m0.5 [10,11]. High density sulfide electrolyte materials (xLi2-yP2S5, briefly LPS) prepared by hot pressing techniques show elastic moduli of 18 25 GPa [12]. Despite showing macroscopic deformability, LPS materials are brittle in nature as demonstrated by the fracture toughness measurements of 0.23 MPa m0.5. For reference, elastic modulus of polycrystalline Li has been reported to range from 1.9 to 9.8 GPa [13–15], and its yield strength from 0.41 to 0.89 MPa [15,16]. At the nanoscale Li-metal may exhibit yield stresses that are greater than bulk lithium [17–19]. The combination of properties that makes the electrolyte mechani cally reliable against Li-protrusions have not been clearly identified. Several experiments contradict the shear-modulus criterion proposed by Monroe and Newman for prevention of Li dendrites [20]. The Monroe-Newman model was developed for polymer, not ceramic elec trolytes, and it describes dendrite initiation as a perturbation growing on a perfectly smooth and flat interface. Its application has been extended to other classes of materials, including brittle ceramic electrolytes, where the physics of the lithium/electrolyte interface behaves quite differently. Because the shear modulus is not a fracture property of the electrolyte layer it is unlikely to be the determining factor for brittle ceramic electrolytes. Porz and coauthors suggest that stabilization of ceramic electrolyte interfaces against lithium metal penetration requires minimizing interfacial defects [21]. In Ref. [21], the surface defect population was intentionally varied, and galvanostatic electrodeposi tion experiments were used to investigate the role of surface morphology. Although many authors have demonstrated “extended cycling” with Li metal cells (even using liquid electrolytes), the unique achievement of thin film batteries is the ability to cycle hundreds to thousands of times while utilizing close to 100% of the lithium [22] (other cells only cycle a small fraction of a thick Li foil). However, thin film battery areal ca pacity is too low to be of interest for automotive applications [23]. Surface treatments to improve Li wetting on the ceramic electrolyte have consistently shown improvements in the cycling performance [24–29]. However, current density and cycle-life must be further increased to demonstrate relevance to vehicle electrification. Atomically smooth interfaces can be easily fabricated in the case of vapor-deposited LiPON [22]. Conversely, sharp surface flaws are likely to form during processing of low toughness materials such as LPS. Also, polishing of garnet solid electrolyte is expected to introduce surface defects whose depth depends on the type of abrasive [21]. Defects can be loosely defined as any non-uniformity at the macro scopic level, including large-scale porosity or cracks, or at the micro scopic level, such as pore channels, grain boundaries, and impurity precipitates. Here we aim to investigate the role of macroscopic defects on the current density distribution. We simulate the presence of sharp crack-like defects on the electrolyte’s surface and we calculate the cathodic current distribution as a function of the defect’s geometry and of the average interfacial resistance. Analysis of the grain boundary
effect on the local current density will be treated in a future publication. The reminder of the paper is organized as follows: In section 2 we introduce the boundary value problem solved with a in-house finite element code. In section 3 we describe the simulation results and discuss their interpretation in view of published experimental trends. In section 4 we propose a self-limiting mechanism to prevent solid-state cell failure due to Li-dendrite penetration. 2. Statement of the boundary-value problem A finite element model was implemented to simulate Li-ion con duction in a solid-electrolyte layer. The model aims to calculate the current density of Li-electrodeposition on the electrolyte layer as a function of the geometry and resistance of the interface. The cathode overpotential is the driving force for lithium deposition and it is calculated by solving the electrochemical problem described below. We consider the homogeneous domain sketched in Fig. 1 as representing a solid Li-ion conducting material. Steady-state conduction across the electrolyte is modeled by the Laplace equation as in Eq. (1), with the potential of lithium φ treated as independent variable. A fixed current density at the positive electrode interface Ωþ (right edge in Fig. 1a). In Eq. (3) the constant i ¼ i=κ is the ratio between the current density and the electrolyte bulk conductivity κ. The kinetics at the negative electrode interface Ω is described by a linear approximation of the Butler-Volmer equation1 (see Eq. (2)). Δϕ ¼ 0
(1)
on Ω
rϕ⋅n ¼ hðϕ
ϕLi Þ
on ∂Ω
on ∂Ωþ
rϕ⋅n ¼ i
(2) (3)
The model assumes uniform wetting at the negative electrode interface. In other words, the boundary ∂Ω in Eq. (2) includes both the flat region and the defect. In the Robin boundary condition (Eq. (2)) imposed at the negative electrode, φLi is the potential of lithium in the Limetal phase. The convective coefficient h represents the ratio between surface and bulk conductivity according to the expression h ¼ ðρκÞ
1
ρ¼
RT þ ρfilm i0 F
(4)
In Eqq.(4) ρ is the total resistance of the interface, defined as the sum of the kinetic and film resistance. The latter is the resistance caused by a decomposition or solid electrolyte interphase (SEI) layer. The kinetic resistance depends on the exchange current density i0 , the Faraday constant F, the gas constant R, and the absolute temperature T. Solving the problem stated above provides the overpotential that drives lithium electrodeposition at the positive electrode interface. Elliptic boundary value problems as in Eq. (1) are characterized by a singularity in the solution around non-smooth regions of the boundary [30]. In the proximity of the singular point at the crack tip (i.e., as x→0 in Fig. 1b), the leading term of the singularity x1=α sinðθ =αÞ depends on the angle απ describing the local geometry. The singularity becomes more pronounced as the angle απ increases. Because of the not-convexity of the boundary, that is α > 1, even the first derivative of the solution is unbounded. In the limit where απ →2π , the solution behaves like x1=2 sinðθ =2Þ. From the mathematical character of the problem it follows that the electric field tends to become singular in proximity of a sharp boundary feature. The singularity is alleviated by the Robin boundary condition in Eq. (2) because an infinite overpotential is necessary to
1
In all the cases considered here, the difference between the overpotential at the crack tip and the overpotential away from the defect is well within the limit of applicability of such approximation. It follows that the relative current peak at the crack tip is accurately predicted by the linear approximation of the Butler-Volmer equation. 2
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Journal of Power Sources 441 (2019) 227186
Fig. 1. a) Cartoon representation of the system stud ied in Eqq.(1)–(3). The domain consists of a solidelectrolyte layer, with applied uniform current at the positive electrode. Linearized Butler-Volmer kinetics is assumed at the negative electrode and the cathodic current is calculated for a given geometry of the interface. b) The plot shows the electric field ahead of the crack tip. The finite kinetics of the interface sup press the singularity of the electric field at the crack tip, while in the case of zero interfacial resistance the electric field becomes unbounded.
drive an infinite current density if the interfacial kinetics is finite.
1 μm below the surface and its radius of curvature is about 0.4 nm. The chosen values of crack depth in the simulations are within the range of observed defect sizes for highly polished LLZTO samples [21]. Typical separators used with liquid electrolytes are at most 25 μm thick [33]. For solid electrolytes, a thickness less than 50 μm remains a target to enable cell energy densities competitive with conventional Li-ion technology [34]. A recently published review discuss challenges of manufacturing thin ceramic separators [35]. Of particular interest is the solution at the corners highlighted in red and yellow in Fig. 2b. The analyses show that the current density has a positive peak at the concave corner (marked in red in Fig. 2b) and a negative peak at the convex corners (marked in yellow in Fig. 2b). In Fig. 2c, the same color code is used to plot the current density peaks normalized with respect to the remote value (i.e., the current density away from the defect). According to Fig. 2c, a larger deviation of the
3. Results Numerical solutions to the system of equations (1)–(3) were carried out with an in-house finite element code in two spatial dimensions. The results presented here illustrate the interplay between morphology and kinetics at the anode interface. The finite element grid is managed with the deal.II finite-element library [31,32]. A representative mesh of the solid-electrolyte domain is shown in Fig. 2a. The mesh is highly refined in the surrounding of the crack to improve the accuracy of the solution in that region. The crack-length in Fig. 2a and b corresponds to 1=20 of the electrolyte’s thickness W and the radius of curvature at the tip is set to W=51200. If we assume, for instance, the electrolyte layer to be 20 μm wide the crack tip reaches
Fig. 2. a) Finite element mesh used to discretize the ceramic electrolyte domain and calculate the potential. Of particular interest is the anode over potential as it determines the lithium electrodeposition rate. The mesh is highly refined around the sharp crack introduced at the anode surface. b) An extract of the mesh around the crack region is shown to highlight the main points of interest. The crack tip is marked in red and the convex corners are marked in yellow. In the baseline case we assume the crack length to be 1/20 of the electrolyte-layer’s width W. c) The current density undergoes a positive peak at the crack tip (red marks) and a negative peak at the cor ners marked in yellow. The current density values reported on the vertical axis are normalized with respect to the current density computed away from the defect. The current peaks diverge as the surface resistance decreases (see horizontal axis), while the bulk electro lyte conductivity is kept constant at 0.1 mS /cm in the simulations. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 3
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Journal of Power Sources 441 (2019) 227186
current density emerges as the interfacial resistance decreases. For re sistances of at least 100 Ωcm2 the cathodic current and, therefore, the rate of lithium deposition are predicted to be insensitive to the defect. Introducing a uniform resistive layer generally reduces the sensitivity to the surface the morphology. The result shown in Fig. 2c applies to a highly conductive solid electrolyte, in fact we assumed a bulk conductivity κ ¼ 0:1 mS / cm. The interplay between bulk and surface kinetics is represented by the convective coefficient h in Eq. (2). If conduction across the electrolyte is fast, the interface acts as a kinetic bottleneck unless the interfacial resistance ρ is extremely low. Viceversa, if the bulk conductivity of the electrolyte is at least two orders of magnitude smaller than κ ¼ 0:1 mS / cm we expect a sharp current peak at the crack tip for any practical value of surface resistance. The yellow markers in Fig. 2c indicate that current density has a minimum at the convex corners of the interface. During charge, this helps preventing Li accumulation in proximity to the crack and helps lithium flow out of the crack. During discharge, a lower stripping rate is expected in the region around each surface defect. Generally, an external pressure is required to continuously reestablish a uniform contact at the Li-metal/electrolyte interface. If voids are allowed to form at the interface the anodic current at the points of contact becomes higher than the average applied value. Non-uniform current density can lead to additional voids formation. This cascade effect can be avoided if the external pressure is large enough to plastically deform the Li-film and restore interfacial contact. The results discussed so far are specific to the geometry illustrated in Fig. 2. Next, we analyze the sensitivity of these results with respect to the crack’s length and the radius of curvature at the crack tip. Both features of the defect’s geometry are expressed in relation to the width of the electrolyte layer W. Fig. 3a shows the current density peak at the crack tip for three values of crack length: one smaller (W/40) and one larger (W/5) than the default value W/20. As expected, deeper defects promote larger localization of the current. Again, if we assume a 20μm wide electrolyte layer, the current density at the tip of a 4 μm crack is expected to be five times larger than the average applied current. Because the driving force for electrodeposition increases with crack length, crack propagation would accelerate as the initial flaw extends across the electrolyte layer. The electro-chemical model discussed so far is appropriate to describe the condition of surface defects not yet filled with deposited lithium. This situation occurs at the beginning of charge or when new surface is being exposed by a propagating crack (cases a and c illustrated in Fig. 4). Also, the newly exposed surface around the crack tip tends to have lower film resistance. This further concentrates the current until a stable passivating layer is allowed to form. In general, the interface morphology affects the local kinetics but also the local mechanical stress. Compared to a smooth flat interface, the lithium deposited within cracks, voids, etc. undergoes larger pressure due to the confinement (case b in Fig. 4). Such pressure is eventually responsible for extending the crack, once the fracture toughness of the ceramic electrolyte is overcome. Before the defect extends, the pressure build-up alters the overpotential locally and reduces the cathodic cur rent at the crack tip and along the crack facets. This intermediate stage of coupled electro-chemo-mechanics is analyzed in Section 4. Here we focus on the initial stage of electrodeposition when lithium is only partially filling an existing or growing defect and the electro-chemistry is approximately decoupled from the mechanics. At this stage, a uniform passivating interfacial layer or a layer of glassy material with low defects concentration are beneficial to establishing uniform kinetics and avoiding Li-protrusions. These can be some of the advantages in using LiPON as electrolyte or as coating for other ceramic materials. The relationship between the crack length and the current-density peak at the crack tip is illustrated in Fig. 3b for the cases of lowest interfacial resistance (ρ ¼ 0:1 Ω cm2 and ρ ¼ 1:0 Ω cm2). The slope of the curve decreases with increasing surface resistance and it becomes
Fig. 3. a) The sensitivity of the current density at the crack tip with respect to the crack lengths is illustrated. In the legend, the crack length is expressed in relationship to the electrolyte-layer’s width W. When the solution is not dominated by the interfacial kinetics (i.e., for values of surface resistance � 10 Ω cm2) the current is very sensitive to the defect size. b) The simulation results indicate that the current peak at the crack tip scales approximately linearly with the crack length. As the surface resistance decreases from ρ ¼ 0:1 Ω cm2 to ρ ¼ 0:1 Ω cm2, the slope of the curve decreases indicating a lower sensitivity to the defect size.
negligible at ρ � 100 Ω cm2. The radius of curvature of the crack tip is related to the απ angle marked in Fig. 1b and, therefore, to the singularity of the electric field. As shown in Fig. 5a, the local current density peak generally increases with decreasing radius of curvature R. However, the sensitivity of the solution to R is not particularly pronounced and it is attenuated by the finite interfacial kinetics. In our simulations we considered geometries characterized by radii R � W/51200. For thin ceramic electrolytes of width W ¼ 20 μm this correspond to radii �0.39 nm. The continuum approximation and the definition of tip curvature become meaningless below this scale. These three data-points in Fig. 5b show an approximately linear relationship between the crack-tip radius and the peak value of the current. The slope in Fig. 5b is specific to the case of lowest interfacial resistance (ρ ¼ 0:1 Ω cm2). Such slope becomes less negative as ρ increases and the current is less sensitive to the local geometry of the tip. In all the analyses presented above, high interfacial resistance ap pears to be beneficial in evening the current density and avoid localized 4
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Fig. 4. Cartoon representation of the mechanism for electrodeposition-induced crack propagation. The accretion of lithium in the direction orthogonal to the crack fac ets tends to open the crack. At the same time, the constrain applied by a rigid solid electrolyte material causes the Li-filament to undergo hydrostatic pressure (case b). The pressure build-up can alter the local poten tial and slow down, or eventually stop, electrodeposition. The pressure at the crack tip is also responsible for propagating the crack (case c). For low toughness electrolyte materials, crack propagation can occur before a strong coupling between mechanics and electrochemistry is established. The stress-potential coupling is responsible for diverting electrodeposition away form the crack region. Two conditions associated with crack growth cause the local rate of elec trodeposition to temporarily rise: i) the hy drostatic stress at the crack tip decreases, and ii) the newly exposed surface tends to have lower film resistance.
electrodeposition. This is in apparent contradiction with experimental efforts directed towards lowering interfacial impedance in order to enable long-term stable cycling with lithium metal [24–29]. However, sluggish ion transport in the interfacial regions causes inhomogeneity in potential-current distributions. This scenario is different from the model assumption of uniform interfacial kinetics. Non-uniform contact is clearly detrimental, as it leads to electrical hot spots and the formation of sharp Li protrusions puncturing the electrolyte. If we modeled the interface with a random distribution of resistance values ρ, we would predict a noisy current signal. The amplitude of the current peaks and their spatial distribution would depend on the character of the ρ dis tribution. In general, those peaks would overlay with the current peak at the crack tip, causing additional stress localization. Such a model would simply reinforce the point that surface heterogeneity, including cracks, needs to be avoided.
electrolyte layer. Furthermore, fracture propagation tends to accelerate over time because the current density at the tip grows linearly with the crack length. Due to the coupling between potential and stress, a self-limiting process can stop Li electrodeposition inside defects. In general, the po tential along the interface is altered by a local stress field. Inside a crack, compressive stress rises the electrochemical potential of lithium with respect to its stress-free state. As more layers are being deposited, the metal become more and more compressed within the confinement of a crack, and the local rate of electrodeposition progressively decreases in response to the mechanically shifted voltage. The coupling between mechanical stress and the local cell voltage can be exploited to pro gressively divert Li deposition away from the crack tip. Following Larch�e and Cahn [36], our previous work [21,37], we use the following expression to calculate the pressure required to shift lithium voltage enough to locally suppress electrodeposition
4. Self-limiting mechanism to avoid crack propagation
p¼
In the previous sections, we analyzed conditions that locally magnify the cathodic current and cause non-uniform deposition. Even if the rate of Li-electrodeposition is constant along the interface, the material deposited within cracks is likely to generate much larger stress than the metal plated away from the defect (see cartoon in Figs. 4 and 6a). A local current density of 1 mA cm 2 corresponds to a Li-metal deposition rate of 10 μ3 s 1. The low yield strength of lithium metal suggests that Li may plastically flow out of shallow cracks as more lithium is being deposited along the crack facets. At the same time, the Li-metal trapped inside the crack undergoes increasing hydrostatic pressure. In fact, the hydrostatic pressure is decoupled from the shearing stress that causes a material to plastically deform. Part of the pressure and, therefore, of the elastic energy, can be released by deforming the crack flanks. However, the main stress-relieving mechanism is likely to consist in propagating the crack, particularly with high stiffness low fracture toughness ceramic electrolytes. Some similarities are apparent with the problem of hy draulic fracturing, with the difference that the influx of lithium comes from ahead of the crack causing more severe stress concentration. At practical current densities, Li-protrusions are likely to grow from surface defects in order to accommodate the deposited volume of lithium. We speculate that diverting Li deposition away from defects is the only route to preventing Li-dendrites penetration. As long as the driving force for electrodeposition along the crack’s flanks is active it is likely to generate enough stress to propagate the fracture across a brittle
ϕtip F : βΩ
(5)
In Eq. (5), p is the average hydrostatic pressure experienced by the lithium filling the crack; Ω ¼ is Li partial molar volume (1:3⋅10 5 m3 =mol), β ¼ is the volume expansion coefficient (�1; here set to 1), F¼ Faraday constant. The surface accretion of lithium may be associated with local negative volume change of the electrolyte surface layers. The factor β can be used to scale down the volume generated via electrodeposition with respect to Li partial molar volume. Furthermore, newly deposited lithium layers and the SEI layer may alter the kinetic and mechanical properties of the interface. However, such effects are not taken into account in Eq. (5) and in the discussion that follows. According to Eq. (5), pressure scales with the anode overpotential φtip calculated at the crack tip. The value of ϕtip , in turns, depends on both the surface and the ohmic resistance. For a given electrolyte ma terial with a distribution of surface defects of a certain size, the pressure at the crack tip grows with the interfacial resistance. The relationship between the limiting pressure and the surface kinetics is illustrated in Fig. 6b for various crack lengths. The pressure required to stop elec trodeposition is predicted to be directly proportional to the surface resistance ρ, given that the electrolyte bulk conductivity is fixed at κ ¼ 0:1 mS / cm. The sensitivity to the crack length is apparent in the region of Fig. 6b where the bulk kinetics is dominant, i.e., for values of ρ � 1 Ω cm2. For less conductive electrolyte materials (κ < 0:1 mS / cm) the sensitivity to the defect size extends to a wider range of surface 5
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Fig. 5. a) The sensitivity of the current density at the crack tip with respect to the radius of curvature of the crack tip is illustrated. In the legend, the crack length is expressed in relationship to the electrolyte’s width W. For a 20 μm wide electrolyte layer, 0.4 nm is the smallest radius considered in the simula tions. b) the relative peak in the current density at the crack tip was found to scale linearly with the radius of curvature of the crack tip. The relationship highlighted refers to the case with surface resistance ρ ¼ 0:1 Ω cm2.
Fig. 6. a) Cartoon representation of the mechanism for electrodepositioninduced crack propagation. The accretion of lithium in the direction orthog onal to the crack facets tends to open the crack. At the same time, the constrain applied by a rigid solid electrolyte material causes the Li-filament to undergo hydrostatic pressure. Given Li low yield stress, shearing can promote plastic flow of Li out of the crack. At the same time, the pressure build-up can alter the local potential and slow down, or eventually stop, electrodeposition. b) The limiting hydrostatic pressure required to divert Li electro-deposition away from the crack tip is plotted vs. the surface resistance at the anode interface (for a fixed bulk conductivity κ ¼ 0:1 mS / cm). The requirement on the electrolyte fracture toughness becomes less stringent as the kinetics of the inter face improves.
resistances. Keeping the crack tip pressure below the critical fracture stress is necessary to prevent further fracturing. This can be achieved by improving the mechanical properties of the electrolyte (high fracture toughness), improving the electrochemical properties (higher conduc tivity, lower surface resistance), decreasing the defect size (material processing), and/or decreasing the current density. For a given system, consisting of materials and interfaces, there is critical current density that generates enough pressure at the crack tip to fracture the electrolyte.
stress (due to the evolving geometry) does not significantly alter the interfacial kinetics. This condition is met at the beginning of charge and when new surface is exposed by the growth of an existing defect. The model calculates the intensification of the cathodic current due to crack-like defects on the electrolyte surface. While non-uniform elec trodeposition is generally detrimental to stable cycling, current locali zation at the crack tip is critical because additional Li volume needs to be accommodated in an already confined space. Due to the interplay between bulk and surface kinetics, if conduction across the electrolyte is fast, the interface kinetics contributes to lowering the sensitivity to the surface morphology. Conversely, sensi tivity to the surface morphology increases in poorly conductive elec trolytes. If the rate-limiting step for electro-deposition is due to the interfacial resistance, the current becomes insensitive to surface defects. Conversely, with a negligible surface overpotential, the cathodic current strongly depends on the curvature of the interface, and it is magnified by
5. Conclusions Sharp flaws are likely to form during processing of ceramic electro lyte materials, particularly low toughness sulfide materials. Recent ex periments [21] suggest that macroscopic defects can compromise the stability of all-solid-state cells with Li-metal anodes, providing a pathway for Li-dendrite penetration. Here we analyze the sensitivity of the current density distribution with respect to the morphology and resistance of the interface. First we propose an electrochemical model that focuses on the early stage of Li deposition when the mechanical 6
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the presence of sharp features. Such sensitivity decays as surface cracks become filled with lithium and the stress-potential coupling alters the current density distribution. Due to confinement, the lithium inside a crack undergoes a large compressive stress. Such stress generates an additional driving force that diverts deposition away from the crack in favor of flat regions of the interface. In the absence of stress, the driving force for electrodeposition at the crack tip scales with the crack length. As the crack grows the current at its tip grows, making the system unstable. However, the coupling be tween electrochemistry and mechanics contributes to smoothing of the cathodic current. Mechanical confinement causes pressure to rise within the Li metal filling a crack. Due to the stress-potential coupling, such pressure can divert electrodeposition away from the crack tip. The stress conditions that hinder Li-electrodeposition at the crack tip need to be established before the fracture stress is reached and the crack propagates. High interfacial resistance leads to more uniform electrodeposition prior to filling of the crack, but it also causes high overpotentials. A larger stress is required to compensate such overpotentials and stop electrodeposition at the crack tip. Once the pressure at the crack tip overcomes the fracture toughness of the electrolyte material the defect propagates across the electrolyte layer. Therefore establishing good Li wetting conditions is equivalent to relaxing the requirement on the electrolyte fracture toughness. This result restates the need for engi neering of electrode/electrolyte interfaces as a necessary condition for all-solid state battery performance and durability. Treatments to control the distribution of defects on the solid-electrolyte surface and to improve wetting with Li-metal will be critical for the success of high energy solidstate batteries.
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Acknowledgments [22]
This work was funded in part by the US Department of Energy, Advanced Research Projects Agency for Energy (ARPA-E), IONICS Pro gram (Award No. DE-AR0000775).
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