Ingress of Li into Solid Electrolytes: Cracking and Sparsely Filled Cracks

The growth of Li dendrites in a solid electrolyte is commonly idealized by a pressure‐filled crack. Recent observations in both garnet and sulfide electrolytes show that sparsely filled cracks exist prior to shorting of the cell, thereby invalidating this assumption. Herein, a variational principle that uses the Onsager formalism to couple Li deposition into the crack, elastic deformation of the electrolyte, and cracking of the electrolyte with the electrochemical driving forces and dissipation within the electrolyte and interfaces is developed. Consistent with observations, it is shown that Li ingress and cracking occur together for garnet electrolytes, but the cracks are sparsely filled. This sparse filling is a direct consequence of the mismatch between the elastic opening of the cracks and the deposition of Li into the cracks across the crack flanks. An increase in the resistance of Li ingress into the tips of Li filaments results in crack propagating ahead of the Li filaments, as observed for sulfide electrolytes. In such cases, the cracks are largely dry. The results provide a framework to model Li ingress into solid electrolytes and explain why the observations are qualitatively so different from dendrites in liquid electrolytes.


Introduction
Solid-state batteries (SSBs) with lithium (Li) metal anodes offer the possibility of enhanced energy density and enhanced safety.[3] However, fully dense ceramic electrolytes can suffer from short circuits due to Li by the high fluid pressure generated by the electric overpotential across the crack flanks. [21,24,25]In a slightly different vein, Shishvan et al. [26,27] assumed that dendrites possess the same geometry as parallel sided-edge dislocations, with penetrating Li metal forming the "extra half-plane" of the edge dislocation.A defining feature of all these models is that the crack advance and Li/Na metal deposition within the crack proceed in unison: the metal completely fills the crack and the plating out of metal at the crack tip drives crack propagation.
The idealization of a dendrite as a pressurized crack or a macrodislocation is called into question by recent X-ray computed tomographic (X-ray CT) observations and associated measurements.Bruce and co-workers [28] reported X-ray CT observations which clearly show that cracks propagate in Argyrodite-type LPS (Li 6 PS 5 Cl) electrolytes as empty, dry cracks: Li metal is only present at the crack mouth near the plating electrode.They reported that a through-crack develops after 25 min in an electrolyte of thickness 1 mm when the imposed current is i ∞ ¼ 1.5 mAcm À2 , but the cell does not immediately short.This was subsequently confirmed for LPS electrolytes by the X-ray CT observations of Shearing et al. [29] reproduced in Figure 1a: the electrolyte cracked through its thickness, but Li (shaded in red) is present only near the plating electrode.In fact, at the time of through-cracking of the electrolyte, only about 20% of the volume of the crack is occupied by Li.In contrast, for the case of an LLZO electrolyte, through-cracking and shorting occur almost simultaneously (and at a shorting time on the order for 2-10 mins which is much less than that for LPS [10] ).This suggests that the above-described pressurized crack models are applicable for LLZO.However, using nano-X-ray CT methods Shearing et al. [30] observed that,while Li penetration and cracking occur approximately in unison for LLZO, the cracks within LLZO are only sparsely filled (Figure 1b).Collectively, these observations suggest that there remains a lack of understanding of the mechanisms by which dendrites propagate into solid electrolytes; current modeling frameworks are not reconciled with experimental observations.All models to date employ a quasi-equilibrium approximation which governs the pressure/state of the Li within the crack.This assumption inherently results in the pressurized and filled crack prediction which seems inconsistent with observations.Here we shall use the Onsager formalism to develop a nonequilibrium, kinetic framework for the penetration of Li into ceramic electrolytes.The shape, filling, and propagation of the crack by Li filaments/deposits are natural outcomes of the solution of the variational principle.We shall show that, unlike in liquid electrolytes where elasticity of the electrolyte plays no role, sparsely filled cracks are to be expected due to the mismatch between Figure 1.a) Optical micrograph and corresponding X-ray CT reconstruction of the crack in LPS showing that while the electrolyte has completely cracked the Li is only present near the mouth of the crack at the plating electrode interface.Modified from ref. [29] with the image on the left indicating the two planes over which the zoom-ins are shown.b) Scanning electron microscope (SEM) image and X-ray CT reconstruction of a crack within LLZO showing that the crack is partially filled with Li.Data reconstructed from measurements taken in ref. [30].
the elastic opening of the crack and the electrochemical forces that govern the filling of cracks.

Problem Definition and a Variational Principle for Crack Growth
The penetration of Li into Li-stuffed ceramic single-ion conductor electrolytes occurs from the Li metal electrode that is being plated.There exists significant controversy in the literature on the geometry of the dendrite/Li filament (is it a crack or a dislocation-like defect?) and whether the fractured electrolyte is fully filled with Li or not.Here, we shall develop a variational principle where the geometry of the dendrite/crack and the extent of filling are outcomes of the coupled electromechanical kinetics.The key outcomes of the analysis will include: 1) the conditions under which dendrites/cracks grow and 2) the mode of growth of the dendrites/cracks.For consistency of terminology throughout this study, we shall refer to the defect within the electrolyte as a crack and the Li within this crack as the Li filament.We do this to avoid confusion with the term dendrite which typically has been used to describe a defect that is fully filled with Li.
In line with many experiments that investigate crack growth in solid electrolytes, we consider a symmetric cell as shown in Figure 2a with an external power source imposing a voltage difference Φ p between the two electrodes.A pre-existing crack of length a 0 in the single-ion conductor electrolyte is assumed to emanate from the interface of the plating Li electrode (inset of Figure 2a) and we develop a kinetic framework for investigating the response of this cell in terms of the growth of this pre-existing crack.Such cracks which are typically on the order of the grain size are well-known to exist in all engineering ceramics and hence it is reasonable to assume that the problem is related to the conditions for the growth of cracks rather than the nucleation of defects.We shall make three simplifying assumptions that are commonly employed for problems of this type: 1) The electrolyte remains electroneutral such that every Li þ cation is paired with an immobile anion within the single-ion conductor electrolyte.Thus, the Li þ occupancy in the electrolyte is spatially uniform and flux is driven purely by the electric potential gradient.A charged Debye layer on the order of 1 nm or less might form at the electrode/electrolyte interface but this is accounted in the measured electrode/electrolyte interface resistance; 2) The molar volume Ω e of Li within the electrolyte is assumed to be zero [21,26] because the Li in Li-stuffed ceramic electrolytes lies within a rigid ceramic skeleton that does not deform upon removal/addition of a Li atom; and 3) The electrolyte remains thermodynamically stable at the operating temperatures and pressures with any stability of the electrolyte with respect to the Li metal electrode accounted for by interface resistance between the electrode and electrolyte.
In addition, it will be shown that the Li filaments within the cracks are less than 100 nm in thickness.Recent micropillar compression tests [31][32][33] suggest that Li pillars in the micrometer length scale have a uniaxial yield strength σ Y above 100 MPa.Direct measurements [34] of stresses in the electrolytes around    cracks also suggest stresses on the order of 150 MPa.Taken together, this strongly suggests that Li within the crack is much stronger than bulk Li that creeps at a stress on the order of 1 MPa at room temperature. [32]Given the high stresses that Li filaments can sustain within cracks, we shall idealize the Li within the filaments as rigid.
Our aim is to analyze the growth of cracks and Li filaments within the electrolyte and to do so we define a system that occupies a domain V, as shown in Figure 2a.This system excludes the external power supply but encompasses the entire electrolyte volume V SE and an infinitesimal layer of each electrode.Both metal electrodes are maintained at fixed electrical potentials by the external power supply and this sets known boundary conditions for the system being analyzed.While this is the simplest system that can be analyzed for investigating the growth of cracks, it is nevertheless a complex open system with fluxes of Li þ ions across left and right boundaries of the system that are adjacent to the stripping and plating electrolyte/electrode interfaces S S and S P , respectively, a change in the total Li content in the system as the Li filaments grow due to Li þ ions being deposited into the crack, a net flux of electrons from the plating electrode to neutralize the Li þ ions being deposited into the crack, and storage of elastic energy within the electrolyte associated with opening of the crack.

Brief Description of the Variational Principle
Here we briefly describe the salient features of the variational principle with full details and mathematical derivations provided in Section S1, Supporting Information.The kinetic variational principle makes use of the Onsager [35,36] formalism wherein the rate of loss of potential energy Π of the system drives the dissipation within the system.We treat the electroneutral single-ion conductor electrolyte as an isotropic linear dielectric with permittivity ϵ SE and a conductivity κ for Li þ ions.The electrolyte is also an isotropic linear elastic solid with shear modulus G and Poisson's ratio ν such that the stiffness tensor is , where δ ij is the Kronecker delta and the stress σ ij is related to the strain ε ij ≡ ðu i,j þ u j,i Þ=2 via σ ij ¼ ℂ ijkl ε kl where u i is the displacement field within the electrolyte.While the formulation is general and can be employed for a 3D system with an arbitrary crack shape, here for simplicity we shall describe the formulation in a 2D plane strain setting in the x 1 À x 2 plane with a crack growing in the x 1 -direction constrained to remain straight.Consider the system at a general time t such that a crack of length a and Li filament of length ζ exists within the electrolyte of thickness L and width W, as shown in Figure 2a.Initially, at t = 0, the filament is of infinitesimal thickness and equal to the initial crack length a 0 , that is, ζ ¼ a 0 .The voltage difference Φ p is imposed at t = 0.The origin of the co-ordinate system is located at the mouth of the crack on the electrolyte/plating electrode interface and the x 1 -direction is perpendicular to the interface, as shown in Figure 2a.We shall assume that an electron-conducting pathway always exists over the entire length of the filament.
Contributions to the rate of change Π of a potential energy arise from: 1) the rate of change ṄLi of Li content within the system; 2) the rate of change of electrical energy associated with the electric field E i and electric displacement D i where D i ¼ ε SE E i ; 3) the current density j i of Li þ ions across the left and right system boundaries S L and S R shown in Figure 2a and the current density j el i of electrons from the plating electrode; 4) the rate of change U : elas of elastic energy of the electrolyte; and 5) the changes in the surface energy due to fracture of the electrolyte in terms of the fracture energy γ SE and the adhesion energy γ adh associated with the formation of new Li/electrolyte interfaces on the crack flanks.Π follows as (Section S1.3, Supporting Information) where μ el and μ Li þ are the chemical potentials of the electrons and Li þ ions, respectively, that leave/enter the system at electrode/ electrolyte interfaces with outward unit normal n i , μ 0 is the chemical potential of Li metal, and F is the Faraday constant.Note that boundary integrals over the top and bottom free surfaces of the electrolyte S T and S B are included as D i is specified on these surfaces while tractions specified over the entire surface while the requirement that the system remains charge neutral relates ṄLi to the electron flux via Z In order to set up the Onsager principle, we also need to consider the sources of dissipation in the system and thereby define a dissipation potential D: The four sources of dissipation within the system are: 1) Li þ flux through the solid electrolyte of conductivity κ, 2) the resistance Z 0 for the Li þ flux across the electrode/electrolyte interfaces, 3) the resistance Z f for the Li þ flux across the crack flank, and 4) the resistance R Tip for Li þ to acquire an electron at the tip of the Li filament as it forms a new Li/electrolyte interface on the crack flanks; see Figure 2b.Consequently, the dissipation potential is defined by (Section S1.3, Supporting Information) where j ¼ j i n i and S Γ is the portion of the crack flanks over which the Li filament is present (inset of Figure 2a) with n i the outward unit normal to the electrolyte.In addition, J Tip is the Li þ current to the tip of the Li filament through each of the two crack flanks (symmetry dictates that the currents across the two flanks are equal), while conservation of Li dictates that the filament extends by a rate ζ ¼ 2J Tip =ðFρ m δ Tip Þ where ρ m is the molar density of Li and δ Tip is the crack opening at the tip of the filament (inset of Figure 2a).Note that Z 0 and Z f are assumed to be constants, implying a linearized form of Butler-Volmer kinetics across the electrode/electrolyte interfaces and the crack flanks.
The rate of change of potential energy ( 1) is combined with the dissipation potential ( 4) to define an augmented potential where φ is the electric potential and, ûi is the time derivative of the displacement field at constant crack length, as defined in Section S1.3, Supporting Information.The Onsager principle [35] demands that the system evolves such that the fields satisfy δ Ψ ¼ 0 at every instant.This requirement delivers the strongly coupled electromechanical governing partial differential equations and associated boundary conditions for the problem, as derived in Section S1.3, Supporting Information.
The governing equation for the electric field within the electrolyte V SE is the electroneutral Gauss law given by φ,ii ¼ 0 where φ ¼ φ þ U and U is the open-circuit potential between the electrolyte and Li metal.The boundary conditions associated with this Laplace equation are D i n i ¼ j i n i ¼ 0 on the lateral boundaries S T and S B (Figure 2a), where j i ¼ Àκ φ,i while the flux of Li þ through the electrode/electrolyte interfaces satisfies In contrast, on the crack flank S Γ where the Li filament is present, the flux boundary conditions involve the tensile stress σ n across the interface via where n i is the unit normal to the crack flank pointing into the crack.This boundary condition couples the solution of the electric field within the electrolyte to the mechanical problem.
In addition, the electric field/potential within the electrolyte needs to admit a current sink of magnitude at the Li filament tip of thickness δ Tip located at ζ (Figure 2b): this current sink lengthens the Li filament.Here, φTip is the value of φ in the vicinity of the filament tip within the electrolyte.This boundary condition requires special consideration as the existence of the sink implies a singularity in φ, and a superposition methodology is used to solve for this singularity, as detailed in Section S2, Supporting Information.Similarly, δ Ψ ¼ 0 delivers the mechanical balance laws within the electrolyte, that is, the static equilibrium equation σ ij,j ¼ 0 along with the associated boundary conditions.These boundary conditions are as follows: 1) the crack opening displacement δ Li at time t along S Γ is given by mass conservation as where j f follows from ( 6) and τðx 1 Þ is the time at which the filament tip ζ is first located at x 1 and 2) on all other electrolyte boundaries, that is,S S , S P , S T , S B and the crack flank S C where the Li filament is absent, the tractions vanish such that T i ¼ 0 .Additionally, T 1 ¼ 0 over S Γ (i.e., zero shear tractions between the Li filament and the electrolyte).The solution of σ ij,j ¼ 0 with these boundary conditions delivers the mechanical fields within the electrolyte for a given crack length a.It is emphasized that this mechanical solution requires a solution coupled with Gauss's law.In addition, the Onsager principle delivers the time evolution of crack length via the Griffith [37] condition where is the elastic strain energy within the electrolyte.Details of the derivations are given in Section S1.3, Supporting Information with material parameters for the LLZO electrolyte shown in Table 1.This highly coupled electromechanical formulation not only predicts the CCD i CCD but also the kinetics of crack and Li filament growth, including the shapes of the crack and Li filament, for operation of the cell at a current density > i CCD .Details of the numerical scheme to solve these coupled equations are provided in Section S2, Supporting Information.We emphasize here that while the above formulation has been presented for a symmetric cell, the predictions are not restricted to a symmetric cell especially when presented in terms of the cell current.This is similar to the case of the symmetric cells used in experiments where measurements of Li filament growth when presented in terms of cell current apply equally to a nonsymmetric cell.

The Critical Current Density
The above equations considerably simplify for the case of a nonpropagating crack and can then be used to deduce the CCD i CCD .
At an imposed cell current density magnitude i ∞ ≤ i CCD , the preexisting crack fills with Li and a steady-state is attained where Li þ Poisson's ratio of LLZO ν 0.2 [43]   Surface adhesion energy between LLZO/Li γ adh 0.22J m À2 [11]   Surface energy of LLZO γ SE 0.84 J m À2 [11]  flux across the crack flanks is switched-off, that is, from (6) σ n ¼ ÀFρ m φ.As the pre-existing crack is assumed to contain lithium along its entire length, there can be no lithium flux into its tip until the crack is forced to propagate.Moreover, with no Li flux into the crack, the solution of Gauss's law within the electrolyte is trivial as the Li þ flux within the electrolyte is 1D in the x 1 -direction.Thus, φ is given by where i ∞ ¼ Φ P =ð2Z 0 þ ðL=κÞÞ is the magnitude of the cell current density (i.e., the current per unit electrode/electrolyte interface area) and L is the thickness of the electrolyte.The calculation of i CCD then is simplified considerably for the case of an initial crack of length a 0 that emanates perpendicular from the plating electrode/electrolyte interface.From ( 6) and ( 11) it follows that the crack opening traction varies as where Given this traction distribution along the crack flanks, for the practical case of a 0 =L !0, there exists a closed-form solution for (9) given in Tada et al. [38] At i ∞ ¼ i CCD , the relation in ( 9) is satisfied, and therefore the crack and filament propagate, when where γ SE=Li ¼ γ SE À γ adh is the surface energy of an electrolyte/ Li interface.Re-arranging (13), a closed-form expression for i CCD follows as Note that in (13-14) we have employed γ SE=Li rather than γ SE as at incipient crack propagation at i CCD we anticipate the crack and Li filament to propagate in unison since γ SE=Li < γ SE (γ adh > 0; see Table 1).The predicted variation ( 14) of i CCD with electrode/ electrolyte interface resistance Z 0 is included in Figure 2c for selected choices of the initial crack length a 0 and using material parameters representative of the LLZO electrolyte, as listed in Table 1.In line with numerous observations, the model predicts a reduction in i CCD with increasing Z 0 and of course with increasing initial crack length a 0 : The reduction in i CCD with increasing Z 0 follows from the fact that a higher Z 0 induces higher crackopening tractions T n due to a higher electric potential within the electrolyte relative to the plating electrode; see Equation (12).Equation ( 14) succinctly captures some variables that affect i CCD and can be used to guide design.For example, we observe that i CCD increases with decreasing κ and Z 0 which are both variables that can be adjusted by via the electrochemistry of the electrolyte.Similarly, i CCD increases with the modulus G and the surface energy γ SE=Li but these material parameters are typically difficult to influence by processing routes.On the other hand, there is a strong influence of initial flaw size a 0 on i CCD (Figure 2c).It is well established that a 0 in engineering ceramics scales with the grain size but improved sintering routes in processing are expected to reduce a 0 and our model will then predict higher values of i CCD .We emphasize here that our analysis assumes a perfect Li plating electrode in which no voids have grown at the electrode/electrolyte interface.[15][16] These voids aresuppressed by stack pressure.Since our current analysis assumes a defect-free Li metal electrode, there is no dependency of stack pressure in this analysis.

Growth of Cracks in LLZO
Cracks grow in LLZO such that they are sparsely filled with Li [30] but given the very small shorting times of the cells, [10] it is reasonable to assume that the cell shorts soon after the crack fully penetrates the entire thickness of the electrolyte.Thus, the Li filament and crack extend in unison in LLZO with ζ % ȧ.In our formulation, this is achieved by setting R Tip ¼ 0 and we shall investigate the effect of R Tip > 0 in Section 4. The justification for the choice R Tip ¼ 0 in LLZO is based on the fact that density functional theory (DFT) calculations [39] show that LLZO surfaces have a much smaller bandgap compared to the bulk and thus trap a significant number of excess electrons.This implies that as Li þ is deposited near the tip of the filament (Figure 2b), it easily acquires electrons, resulting in negligible resistance to the plating of Li metal at the filament tip.[30] In engineering ceramics, flaws/cracks on the order of grain size commonly exist.Given that the grain sizes of LLZO sintered at temperatures in the range 1100 À 1200 o C are typically in the range 15 À 30 μm, [40] in the calculations, we assume an initial flaw of length a 0 ¼ 25 μm that emanates normally from the plating Li electrode/electrolyte interface and initially only contains an infinitesimally thin layer of Li.The electrode/electrolyte resistance is taken to be Z 0 ¼ 5 Ωcm 2 [9] and in the absence of direct measurements we assume that the crack flank/Li filament resistance Z f ¼ Z 0 .The calculations have been conducted by imposing a potential difference Φ P across the electrodes and we report the results in terms of the imposed cell current density magnitude i ∞ ¼ Φ P =ðL=κ þ 2Z 0 Þ, which is the current in the absence of the growth of the Li filament: the large width W of the electrolyte employed in the calculations implies that the growth of the Li filament has a negligible effect upon the average cell current i ∞ .

The Effect of Mismatch Between Elastic Crack Opening and Li Flux into the Crack
For cracks of initial length a 0 ¼ 25 μm, the CCD i CCD ¼ 0.71 mAcm À2 according to Equation (14).Now consider the case of an imposed current density of i ∞ ¼ 1 mAcm À2 > i CCD such that the crack and accompanying Li filament both extend.Predictions of the length of the crack and Li filament at three selected times t are shown in Figure 3.In Figure 3a, we include distributions of normalized current density ĵ ≡ ffiffiffiffiffi j i j i p =i ∞ while the corresponding distributions of the σ 22 stress component are shown in Figure 3b (also see Video S1, Supporting Information).First consider Figure 3a which includes a quiver plot to show the direction of the Li þ flux within the electrolyte and the shape of the crack (the opening of the crack has been magnified by two orders of magnitude so that the shape/profile is visible).Note that the Li filament fills the crack, as depicted by shading the crack black.The crack profiles in Figure 3a,b indicate that the crack opening is maximum near the midlength of the crack.(In contrast, the maximum opening along the length of an internally pressurized crack is typically at the crack mouth.)Throughout its loading history, the crack-opening rate is positive over the entire length of the crack, that is, the crack continues to thicken monotonically (i.e., open) consistent with observations in ref. [29].Thus, the bulged shape of the crack-opening profile necessarily arises from the spatial variation of the Li þ flux into the crack.
The spatiotemporal distributions of ĵ included in Figure 3a show that far ahead of the crack tip the current density is spatially uniform and 1D within the electrolyte with ĵ ¼ 1. Nearer the crack, the Li þ flux across the crack flanks implies that the flux field is no longer 1D with the current density achieving a maximum around the crack tip.While the flux magnitude decreases toward the plating electrode/electrolyte interface it is difficult to infer directly the current density j f into the crack from these distributions of ĵ.Predictions of the normalized crack flank current density j f =i ∞ are shown in Figure 3c at three selected times and clearly show that the crack flank flux is a maximum at the tip and monotonically decreases toward the crack mouth at the plating electrode interface.Qualitatively we would expect such a dependence as the overpotential φ in ( 6) is expected to decrease with decreasing x 1 ; see for example the dependence (11) of φ on x 1 in Figure 3. Predictions of the evolution of the a) normalized current density ĵ ≡ ffiffiffiffi ffi j i j i p =i ∞ and b) stress σ 22 within the electrolyte for a cell with R Tip ¼ 0 subjected to an imposed current density i ∞ ¼ 1 mAcm À2 .In (a) we include a quiver plot to show the direction of Li þ flux within the electrolyte.Debonding of the Li from the crack flanks is not permitted.The opening of the crack is magnified Â200 so that the crack profile is clearly visible.The time t and current crack length a are indicated for each of the three time instants (also see Video S1, Supporting Information).Corresponding predictions of the c) normalized crack flank current density j f =i ∞ and d) crack flank tractions T n .Note that a positive value of T n contributes to opening the crack whereas a negative one does the opposite.
the absence of flux into the growing Li filament.Now recall from (8) that the crack opening depends upon both the current density j f and the time interval over which the crack faces exist.The bulged opening arises from the fact that while j f decreases toward the plating electrode interface (and thereby tends to decrease δ Li ), τ also decreases with τ ¼ 0 for x 1 ≤ a 0 and this tends to increase δ Li .
The σ 22 stress distributions included in Figure 3b clearly show the high stresses associated with the crack tip stress intensity factor.More intriguing is the change in sign of σ 22 along the crack flank.Recall that σ 22 is related to the tractions on the crack flank by T n ¼ Àσ ij n j n i where n i is the unit normal to the crack flank pointing into the crack.Note that the definition of T n means that its positive sense is in the direction of the inward normal to the electrolyte.It follows that T n ¼ Àσ 22 j x 2 ¼0, x 1 ≤a where a is the current length of the crack and a negative T n implies that the crack flank tractions are attempting to close rather than open the crack.These crack flank tractions are plotted in Figure 3d for the three times t shown in Figure 3b.Early in the loading history when the crack has propagated to a ¼ 50 μm, the tractions remain positive indicating that the Li that fluxes into the crack is tending to open the crack over its entire length.With increasing time (and crack length a), while the qualitative form of the T n distribution remains similar, important quantitative differences emerge.The traction T n becomes negative near the crack mouth and tip, indicating that the Li within the crack is attempting to close the crack at these locations.In this analysis, we have assumed that the crack opening is set by ( 8) and thus equal to the time-integrated total Li þ flux at any location along the crack.The calculations show that with the crack-opening set by the time-integrated Li þ flux, elasticity of the electrolyte then requires the Li within the crack to close rather then open the crack at some locations and here the crack flank tractions become negative.This is most clearly seen in terms of the crack profile where the crack opening is lower near the crack mouth compared to that at midlength of the crack-this lower crack mouth opening is a direct consequence of the negative T n near the crack mouth.
Note that the spatial gradients in Li þ flux along the length of a crack exist for dendrites in liquid electrolytes.However, negative dendrite flank tractions do not develop in liquid electrolytes as liquids deforms with a vanishing deviatoric stress.Thus, dendrites in liquid electrolytes can take mushroom-like shapes with them bulging away from the plating electrode interface [41] but yet maintain compressive Li stresses (positive pressure) within the dendrite.

Debonding of Crack Flank/Li Filament
The finite shear modulus of a solid electrolyte implies that the negative crack flank tractions shown above are commonplace if we assume that the crack opening is set by the Li þ flux into the crack at every location along its flank.These negative tractions are on the order of at-least 25 MPa (Figure 3d) and the interface between the electrolyte and the Li filament needs to sustain a tensile traction of this order if this crack/Li filament propagation mode is to exist.The adhesive strength between Li and LLZO is on the order of a few kPa [42] and thus a better approximation is that the Li/electrolyte interface along the crack flank can only sustain compressive tractions.We therefore modify the formulation summarized in Section 2 to include the constraint that T n ðx 1 Þ ≥ 0. Specifically, we modify the boundary conditions for the stress equilibrium equation σ ij,j ¼ 0 over S Γ to be a contact-type boundary condition, that is, over is an outcome of the solution of the elasticity problem; see Section S3, Supporting Information for further details.
The temporal evolution of the debonding of the Li filament from the electrolyte is a complex phenomenon that ultimately results in a complicated structure of a partially filled crack as Figure 4. Structure of the Li filament within the crack near the crack mouth for the case of a cell subjected to an imposed current density i ∞ ¼ 1 mAcm À2 with R Tip ¼ 0. a) The Li within the crack at three time instants with t ¼ 155 s corresponding to the instant when debonding is about to commence at the crack mouth.Subsequently we show the Li filament geometry within the crack with the shading indicating the empty regions and Li deposited before and after detachment from the crack flanks.Note that the x 2 -axis scale is in nanometers such that the crack opens by less than 100 nm when it has propagated to a length a % 350 μm.b) The corresponding evolution of the crack flank tractions T n that are not permitted to be negative.observed via X-ray CT for LLZO. [30]To illustrate this, we consider the case analyzed in Section 3.1 and first illustrate the process by which the Li filament detaches from the electrolyte that results in partial filling of the crack near the crack mouth.The shape of the crack near its mouth is shown in Figure 4a at three selected times, with the crack lengths also indicated.At t ¼ 155 s, the crack is fully filled with Li over the region shown but the corresponding spatial distribution of T n shown in Figure 4b suggests that the crack flanks are about to debond from the Li filament at the mouth ðx 1 ¼ 0Þ as T n ðx 1 ¼ 0Þ ¼ 0. When debonding occurs we assume that an atomistically thin layer of Li is left adhered to the electrolyte flanks so that further Li þ flux into the crack can continue with the fluxing Li þ acquiring an electron from this layer that is connected to the plating electrode.This continued flux gives a Li deposition rate δLi < δ with the traction being maintained at T n ¼ 0 at x 1 ¼ 0. In fact, with increasing time, the debonding progresses toward the crack tip (i.e., x 1 > 0) and the expected Li filament structure within the crack and the corresponding traction distribution are plotted in Figure 4 at t ¼ 180 s.At this instant, T n ¼ 0 for x 1 < 20 μm and this portion of the crack is partially filled with Li.In Figure 4a, we shaded the crack volume to show three separate regions: 1) Li deposited before detachment is shaded grey; 2) Li deposited after detachment is shaded pink; and 3) a void or empty region is shaded blue.We emphasize that we have assumed that the Li within the crack is rigid.This implies that we are unable to predict the precise Li filament morphology within the partially filled crack.The Li morphology shown in Figure 4a assumes that the Li does not deform and thus the voided region is shown in Figure 4a between the Li that was initially deposited (grey in Figure 4a) and the Li that is deposited after detachment occurred (shaded pink in Figure 4a).Nevertheless, it is clear from Figure 4a that the extent of the empty region is largest at x 1 ¼ 0 (see structure at time t ¼ 220 s).This is a direct consequence of the fact that detachment first occurred at x 1 ¼ 0 and that the crack-opening rates are the largest at the crack mouth.
The geometry of the Li filament within the partially filled crack shown in Figure 4a is inaccurate due to the assumption that the Li is rigid.However, our aim here is not to describe the Li filament geometry in detail but rather quantify the level of filling of the crack and its effect on crack propagation.Keeping this in mind, we define a filling ratio c of the crack with its rate of change given by where f ⋅ g denotes the Macaulay brackets and ( 15) is integrated at locations x 1 along S Γ with initial conditions This coarse graining, whereby we only calculate the difference between the crack opening and deposition of Li, not only has the advantage of numerical convenience but also makes no assumptions about the geometry of the Li filament within the partially filled crack.Predictions using this approach are included in Figure 5 (for the case analyzed for Figure 3) with flux distributions shown in Figure 5a and the corresponding stresses σ 22 in Figure 5b (also see Video S2, Supporting Information).Two key differences are apparent: 1) in line with typical crack-opening profiles, the opening increases monotonically behind the crack tip with the maximum opening at the mouth (x 1 ¼ 0), and 2) the stresss σ 22 do not change sign along the crack flanks.This is further clarified via the T n distributions shown in Figure 5c where we see that the tractions vanish near the mouth and tip of the crack, that is, at locations where T n < 0 in Figure 3d.At these locations where T n ¼ 0, the crack is partially filled with Li, as seen in Figure 5d, where we show the spatial distribution of the crack-filling ratio c for two selected crack lengths a.
The predictions that cracks in LLZO are partially filled with Li are consistent with recent X-Ray CT observations [30] (see also Figure 1b) and our model gives a physical understanding of the phenomenon.When debonding of the Li filament from the electrolyte is permitted not only do partially filled cracks develop but negative tractions that tend to close the crack do not develop.These negative tractions tend to reduce the tip stresses (and stress intensity factor) and the absence of these negative tractions is expected to enhance crack growth rates.Predictions of the temporal evolution of the crack length a with time are included in Figure 6a for three imposed current densities i ∞ for the two sets of simulations, that is, 1) no debonding and 2) debonding permitted between the Li and the electrolyte along the crack flank.Clearly, when debonding is prevented, the growth of the crack slows down due to crack closing tractions (negative T n ) being imposed on the crack flanks near the tip and mouth of the crack.Predictions assuming no debonding and with debonding permitted of the average crack velocity ȧ h i (over the interval 150 μm ≤ a ≤ 250 μm) as a function of i ∞ are included in Figure 6b along with corresponding measurements from Kazyak et al. [10] The predictions assuming full adhesion and no debonding underestimate the crack velocity while predictions that allow debonding with the crack flank tractions constrained to T n ≥ 0 agree remarkably well with the measurements.Thus, the predictions of partially filled cracks are not only consistent with direct tomographic observations of the Li filament structure within cracks but are also consistent with more macroscopic measurements of the growth rate of cracks.We emphasize here that the growth rate of the cracks is primarily governed by the current i ∞ with the electrolyte conductivity κ mainly influencing the cell voltage required to impose this current.

Growth of Dry Cracks
Ning et al. [28] and Hao et al. [29] used a combination of in-situ X-ray CT (see also Figure 1a) and spatially mapped X-ray diffraction to demonstrate that the crack tip in sulphide electrolytes outruns the Li filament such that cracks traverse the entire electrolyte with Li absent from the crack over a finite length behind the crack tip.Thus, cracking of the electrolyte can occur before the cell short circuits.In our model, we attribute the difference in behaviour of LLZO and sulfide electrolytes to a difference in the magnitude of the Li filament tip resistance R Tip .DFT calculations [39] have shown that the surface bandgap for sulfide electrolytes is large compared to the bulk and therefore we do not anticipate an accumulation of excess electrons on the surface of the fractured LPS.Consequently, we anticipate that a substantial Figure 6.a) Predictions of the temporal evolution of the crack length a with R Tip ¼ 0 for three values of the imposed current density i ∞ .Results are shown for the cases when debonding is not permitted and when debonding of the Li from the crack flanks is allowed such that T n ≥ 0. b) Comparison between predictions and measurements [10] of the average crack speed a : over the interval 150 μm ≤ a ≤ 250 μm as a function of i ∞ .The predictions for the both the debonding allowed and prevented cases are included.resistance exists for Li þ to acquire electrons near the Li filament tip (see Figure 2b) and expect that, unlike LLZO, R Tip ≫ 0 for sulphide electrolytes.While direct measurements of R Tip have not been reported in the literature, we summarize the results of calculations here to show the effect of R Tip > 0 on the growth of the crack and Li filament.The aim is to show the effect of R Tip upon the mechanism of crack and Li filament propagation, and thus it suffices to keep all parameters fixed at the values for LLZO (Table 1) and only change R Tip for the calculations reported in this section.
Predictions of the growth of the crack/Li filament initiating from an initial crack of length a 0 ¼ 25 μm are included in Figure 7 (also see Video S3, Supporting Information).with the choice of R Tip ¼ 22 Ωcm (all loading parameters are the same as for the cases in Figure 3 and 5 and we restrict attention to the case when debonding is permitted such that T n ≥ 0).The high value of R Tip results in two major differences from the corresponding R Tip ¼ 0 results in Figure 5: 1) The Li filament and crack propagate simultaneously until t % 90 s (a ¼ ζ % 50 μm) but subsequently the crack tip accelerates ahead of the Li filament with the near-crack-tip region being devoid of Li.This is consistent with observations reported in ref. [29]; see Figure 3e.Thus, while the stress concentration persists near the crack tip, the flux concentration is no longer at the crack tip but further back and located at the Li filament tip.This is in contrast to the R Tip ¼ 0 case and 2) The crack propagation rate is lower compared to the R Tip ¼ 0 case.The crack propagates to a ¼ 500 μm in 250 and 315 s for R Tip ¼ 0 and R Tip ¼ 22 Ωcm, respectively.
To understand the lower crack propagation rate, note that shorter Li filaments imply that the crack opening tractions are now exerted over a smaller fraction of the crack flank (compare Figure 7c with Figure 5c).Thus, the magnitudes of the tractions need to be higher to allow the crack to propagate and these higher tractions requires greater crack openings, i.e. thicker Li filaments.To show this we compare the crack opening profiles for the R Tip ¼ 0 case from Figure 5 and the R Tip ¼ 22 Ωcm case in Figuer 7d.Until a % 50 μm the crack and Li filament propagate in unison and the crack profiles are similar in both cases, but for a > 50 μm the crack openings are larger for R Tip ¼ 22 Ωcm compared to those for R Tip ¼ 0. These larger openings induce larger crack flanks tractions (compared Figure 5c and 7c).The larger required crack opening take longer to be built-up by the Li þ fluxes, and this reduces the crack propagation rate.

Effect of the Filament Tip Resistance
Predictions of the temporal evolution of the crack length a and Li filament length ζ are included in Figure 8a for three values of the filament tip resistance R Tip and an imposed current density i ∞ ¼ 1 mAcm À2 .As discussed above, the crack and Li filament propagate in unison for R Tip ¼ 0. However, for the higher values of R Tip not only does the crack tip propagate more slowly but the divergence between the temporal evolution of a and of ζ also increases with increasing R Tip .An immediate consequence is that the time to crack the electrolyte and the time at which the cell shorts will differ at the higher values of R Tip .This is clearly observed in refs.[28,29], where X-ray imaging shows that the electrolyte has cracked but the cell has not shorted at the time of complete fracture of the electrolyte.
To quantify this difference, we define two failure times: 1) t crack for the cracking of the electrolyte, i.e., the time when a ¼ L and 2) the shorting time t short when ζ ¼ L. Note that for time t > t crack the electrolyte has fractured with tractions T n ¼ 0 on the crack flanks and the solution of the subsequent filament growth problem is a pure electrochemical solution with no mechanical coupling.Predictions of t crack and t short are included in Figure 8b as a function of the imposed current density i ∞ for R Tip ¼ 0 and R Tip ¼ 22 Ωcm.For R Tip ¼ 0, t crack ¼ t short so that cracking of the electrolyte is synonymous with shorting of the cell in line with observations for LLZO.On the other hand, shorting occurs significantly after cracking of the electrolyte in the R Tip ¼ 22 Ωcm case and this is qualitatively consistent with observations for sulfide electrolytes.We thus argue that the differences of cracking and shorting times between LLZO and sulfide electrolytes are related to the fact that excess electrons accumulate on LLZO surfaces but not sulfides.This is parameterised via R Tip in our analysis.In all cases, the cracking and shorting times decrease with increasing i ∞ as the flux of Li into the crack increases as i ∞ increases.However, for the sulphide electrolytes the tip resistance R Tip also strongly influences the growth rate of the Li filaments and cracks: the growth rate of both Li filaments and cracks decreases with increasing R Tip for a given i ∞ .

Concluding Remarks
A variational principle for modeling the ingress of Li into solid electrolytes has been developed by making use of Onsager formalism.The Onsager principle [35,36] allows us to couple together the elastic deformation and cracking of the electrolyte, the electrochemical driving forces in the cell, and the ion transport dissipative processes in the electrolyte bulk and across the electrolyte/electrode and electrolyte/crack interfaces.The kinetics of crack/Li filament propagation along with the shapes of the crack and Li filament are natural outcomes of the solution to the variational problem.
Unlike liquid electrolytes, in solid electrolytes, the elasticity of the solid implies that the crack cannot assume arbitrary shapes if the tractions along the Li filament/crack interface are to remain compressive.Consequently, the mismatch between the timeintegrated flux of Li þ ions into the crack and elastic opening of the crack can result in sparse filling of the cracks with Li, while the resistance to the ability of Li þ ions to acquire electrons at the Li filament tip implies that the crack can propagate ahead of the Li filament in some cases.The predictions are not only in good qualitative agreement with X-ray CT observations of the cracking of LLZO and LPS electrolytes but are also in excellent quantitative agreement with measured crack velocities.The framework provides a comprehensive understanding of the kinetics of cracking and Li filament propagation in solid electrolytes.It shows that the mismatch between the elastic crack opening and the flux of Li þ ions into the cracks is a maximum near the mouth of the crack at the interface with the plating electrode.Direct X-ray CT observations to measure the extent of filling of cracks as a function of position within the electrolyte have not been reported to date.Such observations are desirable and will serve to further test the fidelity of the modeling framework introduced here.

Figure 2 .
Figure2.a) Sketch of the symmetric cell analyzed with the system of volume V that is bounded by S ≡ S L ∪S R ∪S T ∪S B .The electrolyte of size L Â W has a volume labeled V SE and the plating and stripping electrode interfaces are labeled S P and S S , respectively.There exists an initial crack of length a 0 that is perpendicular to S P .b) Sketch showing the plating of the Li at the tip of the filament with a current J Tip along with the current density j f ¼ j i n i across the crack flanks along with crack surface S Γ along which the Li filament of length ζ is present with the remainder of the crack of length a being dry.The dry crack flanks are labeled S C c) Predictions of the CCD i CCD as a function of electrode/electrolyte interface resistance Z 0 for selected values of a 0 using properties representative of LLZO (Table1).

Figure 5 .
Figure 5.The effect of debonding between the Li filament and the crack flank on the predictions of the evolution of the a) normalized current density ĵ ≡ ffiffiffiffi ffi j i j i p =i ∞ and b) stress σ 22 within the electrolyte for a cell with R Tip ¼ 0 subject to an imposed current density i ∞ ¼ 1 mAcm À2 .In (a) we include a quiver plot to show the direction of the Li þ flux within the electrolyte.Debonding between the Li filament and crack flanks is assumed to occur such that the crack flank traction T n ≥ 0. The opening of the crack is magnified Â200 so that the crack profile is clearly visible.The time t and current crack length a are indicated for each of the three time instants (also see Video S2, Supporting Information).Corresponding predictions of the c) crack flank traction T n and d) filling fraction c along the crack.

Figure 7 .
Figure 7. Predictions of the evolution of the a) normalized current density ĵ ≡ ffiffiffiffi ffi j i j i p =i ∞ and b) stress σ 22 within the electrolyte for a cell with R Tip ¼ 22 Ωcm subjected to an imposed current density i ∞ ¼ 1 mAcm À2 .In (a) we include a quiver plot to show the direction of the Li þ flux within the electrolyte.Debonding is assumed to occur such that the crack flank traction T n ≥ 0. The opening of the crack is magnified Â200 so that the crack profile is clearly visible.The time t and current crack length a are indicated for each of the three time instants (also see Video S3, Supporting Information).c) Corresponding predictions of the crack flank traction T n and d) comparison of the crack profile for the R Tip ¼ 22 Ωcm case with the R Tip ¼ 0 case of Figure 5.

Figure 8 .
Figure 8. a) Predictions of the temporal evolution of the crack length a and Li filament length ζ for a cell subjected to an imposed current density i ∞ ¼ 1 mAcm À2 .Results are shown for three choices of the filament tip resistance R Tip .b) Predictions of the time t crack to crack the electrolyte and the time t short to short the cell as a function of i ∞ for two choices of R Tip .These predictions are for an electrolyte of thickness L ¼ 1 mm.

Table 1 .
Summary of material parameters for the LLZO electrolyte.