Effects of a high homologous temperature, T _H

At room temperature, 22 °C, lithium is at a T _H of 0.65. Sustained at T ≥ 22 °C, lithium is in a continuous state of annealing, reducing its dislocation density and increasing its average grain size. At these temperatures, lithium is also within the hot-working regime, where recovery processes occur simultaneously during plastic deformation. The net effect is that at T ≥ 22 °C lithium can sustain large plastic strains with little or no significant strain hardening. The mechanism by which softening simultaneously takes place in most body-centered cubic (BCC) metals is dynamic recovery, which occurs by the formation of well-developed, fine (approximately 1–10 μm) subgrain dislocation structures enabled by accelerated diffusion processes. Specifically, it is the ease of cross slip, climb, and dislocation unpinning that enables the annihilation of dislocation pairs during deformation and the creation of the recovery substructure. At a fixed temperature, the driving force for the diffusive recovery of pure metallic lithium is the amount of stored strain energy. Although the recovery rate depends primarily on temperature and the amount of strain, the strain rate can be a contributing factor as well. As recently reported by LePage, data from tensile experiments performed at temperatures ranging from 198 to 398 K (0.44 ≤ T _H ≤ 0.88) and strain rates spanning from 4 × 10⁻⁵ to 2 × 10⁻² s⁻¹ clearly illustrate the effects of temperature, strain rate, and dynamic recovery in bulk lithium [Reference LePage, Chen, Kazyak, Chen, Sanchez, Poli, Arruda, Thouless and Dasgupta13]. Only at the lowest temperature, strain hardening is observed. In addition, Masias et al. have reported similar experimental observations from recent tensile tests performed on lithium at room temperature [Reference Masias, Felten, Garcia-Mendez, Wolfenstine and Sakamoto14]. Whereas the uniaxial compression data presented by Masias does show work hardening, it is not immediately clear to what extent the experimental observation may be controlled by the effect of friction at the interface between the test specimen and the compression platens [Reference Masias, Felten, Garcia-Mendez, Wolfenstine and Sakamoto14].

In the context of high T _H and dynamic recovery, two important outcomes arise from the LePage and Masias results in relation to the stress-directed flow of lithium in a battery [Reference LePage, Chen, Kazyak, Chen, Sanchez, Poli, Arruda, Thouless and Dasgupta13, Reference Masias, Felten, Garcia-Mendez, Wolfenstine and Sakamoto14]. First, it is expected that the dislocation density in both the bulk lithium and lithium-filled defects at the SSE interface is quite low, regardless of the strain required to accommodate any shape change associated with initially filling an interfacial defect with metallic lithium. At sustained T ≥ 22 °C, continuous annealing will further diminish the dislocation density and effectively eliminate any potential strain rate effects. The important implication is that at T ≥ 22 °C, deformation mechanisms that rely on dislocation glide at the interface or within an interfacial defect may not be an effective means of alleviating pressure, particularly at small length scales where the probability of finding a dislocation may be quite low.

Effects of an externally applied compressive stack pressure, σ_sp

An externally applied σ_sp (nominally ∼7σ_y for bulk polycrystalline lithium when it is initially applied) is often used to promote adhesion and mitigate the formation of voids at the lithium/SSE interface [Reference Wang, Wolfenstine and Sakamoto12, Reference Dudney25]. Although σ_sp can clearly drive the flow of metallic lithium, it does so only to the extent that friction at the interface and the equilibrium aspect ratio a/t will allow, where a is the radius of contact between a flat circular SSE disk and the lithium source layer (anode) and t is the thickness of the lithium layer [Reference Dieter37]. Explicit assessment of the most deleterious stress conditions are directly enabled by the hot-forming literature, which provides a relevant source for considering the macroscopic aspects of this large length scale deformation process without requiring detailed knowledge regarding how the stack pressure is physically applied [Reference Dieter37].

As pointed out by Wang, a natural consequence of σ_sp is a zone of hydrostatic compression within the lithium anode [Reference Wang, Wolfenstine and Sakamoto12]. Accurate analytical descriptions of the resulting state of stress at the interface are illusive, as the stress components depend significantly on the friction conditions (Coulombic, sticking or an interface shear strength or a combination thereof) and the ratio of a/t. Factors such as, but not limited to, physical lateral constraints and the elastic properties of the SSE and current collectors may further complicate the analysis as well. Despite these complexities, important observations regarding the effects of stack pressure can be made subject to a number of simplifying assumptions. Here, we assume the SSE and the current collector on opposite sides of the lithium anode behave as rigid compression platens, that the metallic lithium mirrors the axial symmetry of the SSE disk and that under σ_sp, if the metallic lithium plastically deforms, it does so homogeneously by a combination of dislocation glide and climb through intragranular dislocation mechanisms with a stress exponent n ≥ 3. Under these conditions, the compressive principal stresses in the metallic lithium are σ_z and σ_r = σ_θ (due to axial symmetry), where the subscripts z and r denote, respectively, the direction normal to the lithium/SSE interface and the radial direction, r, outward from the axis of symmetry (σ_r is a compressive hoop stress due to the friction at the interface). It follows then that at equilibrium the average value of σ_z = σ_sp (energy balance) and the radially dependent hydrostatic stress, σ_H = (σ_z + 2σ_r)/3. Before electrochemical cycling, an equilibrium ratio of a/t is established in a manner that reflects balancing between the externally applied stack pressure and friction at the interface. Stresses in the lithium at the outer edge of the anode (radial position, r = a) are σ_z ∼ σ_y, σ_r ∼ 0, and σ_H ∼ σ_y/3, where σ_y = ∼0.5 MPa. At the center of the disk (r = 0), σ_z, σ_r, and σ_H peak in a manner that depends significantly on the friction conditions and the equilibrium aspect ratio of a/t [Reference Dieter37].

If application of the external stack pressure initiates plasticity, then t naturally decreases and conservation of volume forces the radial outward flow of lithium until the drop in pressure is balanced by friction at the interface, at which point the equilibrium value of a/t is established. Under subsequent electrochemical cycling, the required force balance only permits the compressive stack pressure to drive the flow of metallic lithium when a/t is less than the equilibrium value. Assuming the friction conditions do not change, the magnitude of a will reach a steady-state value that is determined by the maximum amount of lithium ever plated to the anode during cycling. Assuming t _max is not exceeded with further cycling, then the steady-state value of a is fixed (a _max) and although a _max/t increases and decreases with cycling, the ratio never dips below the equilibrium value. Subject to these constraints, once the equilibrium value of a _max/t is achieved, the resultant compressive stack pressure is not capable of driving dislocation-mediated flow or creep in lithium with a stress exponent n ≥ 3. The magnitude of the individual stress components σ_z, σ_r, and σ_H will change with radial position, but a basic energy balance (assuming equilibrium) dictates that the average value of σ_z = σ_sp regardless of a _max/t, and, thus, the average values of σ_r and σ_H can never be more than σ_sp. The important implication is that as a _max/t increases during discharge (the lithium anode becomes thin relative to the fixed contact radius, a _max, with the SSE), the radially dependent values of σ_z, σ_r, and σ_H in the lithium anode will increase, but only in such a way that the gradients in each component result in an average σ_z = σ_sp and, therefore, an average σ_r and σ_H ≤ σ_sp. We also note that as the radially dependent values of σ_z, σ_r, and σ_H increase with a _max/t, the average state of stress becomes increasingly hydrostatic as σ_r becomes similar in magnitude to σ_z.

Subject to the constraints and assumptions associated with the analysis described earlier, an important outcome is that the hydrostatic stress field in a lithium anode under an externally applied compressive stack pressure is not uniform. Moreover, the gradient in hydrostatic stress with radial position inherently increases with a _max/t during electrochemical cycling. We also note that regardless of the changes in a _max/t with cycling, the average hydrostatic stress will never be more than σ_sp, which is typically less than 5 MPa [Reference Dieter37]. As pointed out by Wang, significant decreases in the self-diffusion rates of sodium and indium have been experimentally observed under uniform hydrostatic compression ranging from 120 to 960 MPa [Reference Wang, Wolfenstine and Sakamoto12]. We note the self-diffusion coefficient of a pure metal subjected to a uniform hydrostatic stress decreases in a manner that reflects a concurrent drop in the equilibrium vacancy concentration, C _veq, [Eq. (22)] and is estimated to have the form shown in Eq. (23),

(22)$${C_{{\rm{veq}}}} = {C_{{\rm{vo}}}}\exp \left( {{{\Omega {\sigma _{\rm{H}}}} \over {{k_{\rm{B}}}T}}} \right)\quad ,$$

(23)$$D = {D_{\rm{o}}}\exp \left( {{{\Omega {\sigma _{\rm{h}}} - {{{Q_{\rm{d}}}} / {{N_{\rm{A}}}}}} \over {{k_{\rm{B}}}T}}} \right)\quad ,$$

where C _vo is the stress free equilibrium vacancy concentration (vacancies/m³), D _o is the pre-exponential term (m²/s), Q _d is the activation energy for self-diffusion (J/mol), and N _A is Avogadro’s number (atoms/mol) [Reference Dieter37]. Based on the input values in Table II, the decrease in the self-diffusion coefficient associated with a uniform hydrostatic compressive stress of 5 MPa is 2.5%. As such, it seems unlikely that a nonuniform stress field with an average compressive σ_H ≤ 5 MPa will have a significant impact on the self-diffusion coefficient of lithium unless the vacancy concentration drops well below the equilibrium value defined in Eq. (22). It is important to note that if the equilibrium vacancy concentration within a hydrostatic stress gradient is maintained via the action of vacancy sources, such as the incoherent lithium/SSE boundary, grain boundaries and/or dislocations, then the local flux of lithium is driven solely by the chemical potential gradient of the lithium atoms [Reference Larche and Cahn38, Reference Suezawa, Saitoh, Kojima and Kasuya39, Reference van Loo, Pieraggi and Rapp40]. The details of lithium diffusional creep with the relevant stress gradient length scale under discussion has been described in Ref. Reference Herbert, Hackney, Dudney, Thole and Phani9 and Reference Herbert, Hackney, Dudney, Thole and Phani10 and found to predict the observed length scale (indentation depth) and strain rate–dependent hardness of lithium.

TABLE II:

Variables and values.

^a Activation energy for vacancy formation = 0.9 eV/atom and density = 0.534 g/cm³ [Reference Suezawa, Saitoh, Kojima and Kasuya39].

Vacancy annihilation and nonequilibrium vacancy concentrations

In an attempt to rationalize the experimentally measured current densities at the point of SSE failure (short circuiting) in a symmetric lithium/LLZO/lithium cell, it was recently suggested that regardless of the type or chemistry of the SSE, the self-diffusion of lithium may be the controlling mechanism in determining the CCD at which failure of the SSE occurs [Reference Wang, Wolfenstine and Sakamoto12]. As argued by Wang, the flux of vacancies to the lithium/SSE interface may be insufficient to meet the demand of incoming lithium ions (particularly at high current densities), thereby enabling localized, nonplanar lithium plating to occur above the average plane of the lithium/SSE interface [Reference Wang, Wolfenstine and Sakamoto12]. We further note that Krauskopf et al. have recently considered the role of diffusional creep in determining the morphology of the lithium/SSE interface as well, but their analysis is applied to the process of stripping rather than plating [Reference Krauskopf, Hartmann, Zeier and Janek41].

As pointed out by Wang, regardless of whether the point effect of diffusion occurs at an interface defect or at a flat region of the interface or whether σ_sp ≥ 0, gradients in the hydrostatic stresses will influence the flux of vacancies [Reference Wang, Wolfenstine and Sakamoto12]. What is not immediately clear, however, is to what extent the lithium deposition at the interface diminishes the concentration of vacancies. Wang appears to argue that the vacancy concentration drops well below the equilibrium value defined in Eq. (22) based on the kinetics of vacancy annihilation as the lithium is plated at the lithium/SSE interface [Reference Wang, Wolfenstine and Sakamoto12]. However, most studies into the electromigration phenomenon, which has similar physics, have argued that the vacancy concentration is close to the equilibrium concentration [Eq. (22)] [Reference Blech42, Reference Blech and Herring43, Reference Shatzkes and Lloyd44, Reference Kirchheim and Kaeber45, Reference Clement and Lloyd46, Reference Tu47, Reference Ross48, Reference Kirchheim49, Reference Korhonen, Borgesen, Tu and Li50]. To address the efficacy of diffusion-based stress relaxation under conditions of a local increase in hydrostatic stress confined to small volumes, we submit the discussion should refer to relevant experimental results and applicable findings from the field of electromigration.

As previously described, nanoindentation experiments provide a unique and direct opportunity to consider the possibility of small length scale diffusional relaxation of a local hydrostatic stress in lithium at a rigid interface. As revealed by both analytical and numerical approximations such as the ECM and FEA, both spherical and pyramidal indentation of elastic and elastic plastic solids creates a zone of hydrostatic compression directly beneath the indenter tip [Reference Bower28, Reference Zarudi, Zhang, Cheong and Yu51]. According to FEA, the decay length of the hydrostatic stress field is approximately on the order of the indentation depth [Reference Zarudi, Zhang, Cheong and Yu51]. In this way, the indentation experiments provide a unique opportunity to examine how stress in the lithium could relax by diffusive flow under conditions of a local hydrostatic stress gradient at a nearly rigid interface. In relation to the operation of a battery, the indentation displacement rate, dh/dt, would be analogous to the local increase in the lithium plating rate at the SSE interface because of the point effect of diffusion [Eq. (4)], whereas the mean pressure the lithium surface can support is simultaneously measured through the hardness.

At small length scales (h ≤ ∼350 nm), the nanoindentation observations of Herbert et al. reveal the physics of lithium stress relaxation scale with the indenter tip velocity, which is taken to be analogous to the lithium displacement rate at a local “hot spot” at the lithium/SSE interface [Eq. (4)] [Reference Herbert, Hackney, Dudney, Thole and Phani9]. From the experimental study in Ref. Reference Herbert, Hackney, Dudney, Thole and Phani9, at h = 350 nm, the measured tip velocity dh/dt = 2.6 nm/s. The analogy with the local increase in current density through Eq. (4) gives |Δi| = |i|(Δ_o,a) = 1.9 mA/cm². Under these conditions, the mean pressure supported by lithium is measured at ∼40 MPa, which is nearly two orders of magnitude greater than σ_y for bulk lithium (∼0.5 MPa), indicating that dislocation plasticity is not operating under these conditions.

Although indeed a relatively large pressure, the measured H of 40 MPa only corresponds to at most 14% of the theoretical shear strength of lithium (rigid slip, i.e., ex nihilo dislocation nucleation). This observation assumes the shear stress, τ, required to initiate rigid slip ranges from G/(2π) to G/30, where the shear modulus, G = 4.25 GPa, τ_max = σ_y/2 (Tresca’s yield criteria), H = Cσ_y, and the constraint factor, C = 1. In addition, we note 40 MPa is 18 times lower than the minimum elastic stress limit predicted by Sneddon’s theoretical load–displacement relationship, thus negating the possibility of elastic stress relaxation [Reference Sneddon52]. This observation assumes a rigid indenter tip, E = 9 GPa, Poisson’s ratio, ν = 0.362, the equivalent cone angle, ψ = 70.32°, and the contact depth, h _c = h. The apparent paradox of insignificant dislocation-mediated plasticity at small length scales was resolved in Ref. Reference Herbert, Hackney, Dudney, Thole and Phani9 by correlating the H–h behavior at low indentation depths to the self-diffusion length scale and an analogue of the Nabarro–Herring creep mechanism (volume diffusion to the free surface).

Through the representative H–h data presented in Figs. 1(a) and 1(b), two distinctly different mechanisms of stress development are revealed as a function of h. At small length scales, H increases with depth in a manner that has been rationalized by atomic diffusion analogous to Nabarro–Herring creep, with an effective ‘grain size’ given by the contact radius (∼2.8h for the equivalent cone). As previously noted, H peaks at a value significantly greater than σ_y for bulk lithium, but much lower than the stress corresponding to either the theoretical shear strength or the pressure associated with purely elastic deformation. The precipitous drop in H immediately following the peak stress signals a distinct transition to a second deformation mechanism, hypothesized to be dislocation-mediated flow enabled by the operation of a dislocation multiplication source. Interpretation of the H–h behavior observed in BCC lithium is aided by direct comparison to the H–h behavior observed in BCC iron, which is shown in Fig. 1(b). As both figures show, the general trend in H versus h between lithium and iron is quite similar. Whereas the difference in stress is nominally expected, the difference in length scales (indentation depth) at the sudden decrease in H is astonishing. Interpretation of the H–h behavior of iron [Fig. 1(b)] is well documented [Reference Pharr, Herbert and Gao19, Reference Soman, Herbert, Aifantis and Hackney53]. In broad brushstrokes, the initial build up of pressure at small depths is uniquely related to the elastic properties of iron and the effective radius of the Berkovich indenter tip. The peak in H is representative of the onset of plastic flow, either by dislocation nucleation or propagation of existing dislocations. Immediately after the abrupt drop in pressure, the more gradual decrease in H with increasing depth is a phenomenon known as the indentation size effect, which is rationalized in the context of a strain gradient imposed by the indenter tip geometry. Although Herbert et al. have interpreted lithium’s H peak and subsequent depth dependence in a similar manner, the dramatic increase in the h/H ratio for the transition to dislocation-mediated flow in lithium is rationalized in the context of high T _H deformation, specifically diffusion creep stress relaxation (n = 1) at indentation depths below the sudden decrease in H [Reference Herbert, Hackney, Dudney, Thole and Phani9]. The point to be emphasized here is that the transition to dislocation-mediated flow as a stress relief mechanism in lithium can be frustrated if the stressed volume is ‘small’ in relation to the proximity of existing dislocations or a dislocation source.

Perhaps the simplest argument to demonstrate this concept is that of a pre-existing dislocation source threading through a constrained spherical volume in which a nonuniform internal shear stress is elevated relative to the surrounding volume. The existing dislocation can only bow out and multiply to relieve the localized pressure if τ > ∼3 Gb/R _sv, where R _sv is the radius of the spherical volume [Reference Kraft, Gruber, Mönig and Weygand17]. Note that because of the nonuniform stress field, the shear stress criteria require the separation distance between the dislocation pinning points (the dislocation line length) to be expressed as a fraction of R _sv, as the multiplication process requires sufficient room to physically operate. Here, we generously estimate the separation distance to be 2/3R _sv. If R _sv is analogous to a hemispherical cap enclosing a residual hardness impression, then the equivalent cone gives the radius of the spherical volume as h tan ψ. Assuming 350 nm represents the depth required to initiate the multiplication process, then R _sv = 978.6 nm. This suggests a pre-existing dislocation density greater than 3.3 × 10¹¹ m⁻² (a moderately cold worked metal) is required to provide a high probability that a dislocation is available for dislocation glide/climb–induced stress relief within the stressed volume. Similarly, El-Awady has proposed that the minimum shear stress for plasticity within a confined volume of 1 µm in diameter will correspond to a dislocation density on the order of 10¹² m⁻² [Reference El-Awady54]. At sustained T ≥ 22 °C, these dislocation densities in pure lithium are highly unlikely given the continuous annealing and recovery process that takes place at T _H ≥ 0.65.

The nanoindentation results suggest that diffusion creep via self-diffusion of lithium is active at small length scales although the displacement rate and stress are ‘large.’ However, as the length scale of the local deformation and strain rate continues to increase, so does the hydrostatic stress, at least until local dislocation plasticity is activated. Nevertheless, the indentation results can be rationalized through a simple increase in diffusion length rather than a dramatic decrease in the availability of vacancies, suggesting that diffusional creep remains a viable stress relief mechanism for lithium at room temperature even in the presence of a high frequency spatial variation in the hydrostatic stress.

The depletion of vacancies in a local region of hydrostatic compression has also been considered in some detail by the electromigration/line stress community beginning in the 1990s [Reference Blech42, Reference Blech and Herring43, Reference Shatzkes and Lloyd44, Reference Kirchheim and Kaeber45, Reference Clement and Lloyd46, Reference Tu47, Reference Ross48, Reference Kirchheim49]. Although there are some differences in the geometry and origin of the stress development between electromigration and lithium deposition under a rigid electrolyte, the physics are directly relevant to the lithium/SSE interface. A quantitative argument related to vacancy supply can be developed by considering a locally increased hydrostatic pressure developed at x = 0 having a hydrostatic stress variation about a local region of length scale, L, such that an equilibrium vacancy concentration varies with position as

(24)$${C_{{\rm{veq}}}}\left( x \right) = {C_{{\rm{veq}}}}\left( \infty \right)\exp \left( {{{\Omega {\sigma _{{\rm{H}}\;{\rm{max}}}}} \over {{k_{\rm{B}}}T}}\exp \left( { - {x \over {2L}}} \right)} \right)\quad ,$$

where C _veq(∞) is the equilibrium concentration of vacancies (vacancies/m³) far from x = 0 and σ_{H max} is the hydrostatic stress at x = 0. The detailed derivation of Eq. (24) is available in Appendix C. Taking C _veq(∞) = 5.7 × 10¹³ vacancies/m³ at T = 304.15 K and σ_{H max} = −50 MPa, Eq. (24) predicts the decrease in vacancy concentration at x = 0 to be ∼23%. Here, it is assumed σ_{H max} is 10 times the maximum average hydrostatic stress, noting the actual magnitude is unknown because of its dependence on the friction conditions and the ratio of a _max/t, 50 MPa corresponds to the upper limit of the maximum H measured by nanoindentation where the Nabarro–Herring type diffusional creep can still be used to rationalize the experimental observations.

Wang further suggests there is a deviation in vacancy concentration from the equilibrium distribution due to the plating of lithium at the lithium/SSE interface [Reference Wang, Wolfenstine and Sakamoto12]. This general idea has also been considered in the electromigration field where the vacancy annihilation correlates with electromigration-induced atomic fluxes into constrained volumes. However, Korhonen has suggested the stress effects on diffusivity are very limited in the electromigration phenomena [Reference Korhonen, Borgesen, Tu and Li50]. That is, electromigration would not be a problem plaguing the electronics industry if there were no vacancies because of rapid annihilation and, thus, no atomic substitutional diffusion. For a vacancy source such as a dislocation, grain boundary, surface, or interface at a distance d from the increase in local hydrostatic stress, with d > L, the following expression gives a linear approximation to the vacancy flux [vacancies/(m² s)] to the stressed volume,

(25)$$\eqalign{{J_{{\rm{Vacancy}}\;{\rm{supply}}}} \cr= M\left[ {{k_{\rm{B}}}T\left( {{{{C_{{\rm{veq}}}}\left( d \right) - {C_{\rm{V}}}\left( 0 \right)} \over d}} \right) - {C_{\rm{V}}}\left( 0 \right)\Omega {{{\sigma _{{\rm{H}}\;{\rm{max}}}}} \over L}} \right]\quad ,$$

where M is the mobility of vacancies, M = D/(k _BT) [(m² atom)/(J s)], C _veq(d) is the equilibrium concentration of vacancies at position d (vacancies/m³), and C _V(0) is the nonequilibrium vacancy concentration (vacancies/m³) at x = 0 (see Appendix C). As per Eq. (25), even as C _V(0) goes to zero (which would violate the laws of thermodynamics), there is still a flow of vacancies coming to (and atom flow away from) the stressed volume with a local increase in hydrostatic pressure. Although the complex relationship between σ_{H max} and L is not addressed here, Eqs. (24) and (25) indicate that flux increases significantly with large gradients in stress produced by the combination of a large magnitude of σ_{H max} and a small length scale over which the stress acts. In the cases of indentation and lithium plating, we also note the presence of the incoherent lithium/indenter and lithium/SSE interface provides an efficient source of vacancies such that the vacancy concentration remains close to that predicted by equilibrium [Reference Gleiter55]. Collectively, these observations suggest the supply of vacancies to ‘small’ stressed volumes at the lithium/SSE interface will not be significantly diminished by the local hydrostatic stress imposed by the compressive stack pressure, the higher levels of stress in locally deformed lithium as measured by nanoindentation or the deposition of lithium. In addition to bulk self-diffusion, the presence of the incoherent lithium/SSE interface must also be considered as a diffusion path. In this case, any atomic ledges of lithium at the interface provide unique pathways for pipe diffusion, which can be a conservative process that does not require vacancies and, thus, creates a fast transport path for the self-diffusion of lithium away from regions of relatively high stress at the interface [Reference Swinburne, Arakawa, Mori, Yasuda, Isshiki, Mimura, Uchikoshi and Dudarev56, Reference Liu, Li, Shao, Gong, Wang, McCabe, Jiang and Tomé57].

Elastic displacement at the epicenter for the point effect of diffusion (δ at x = 0)

As per Eqs. (8) and (9), it is apparent that all the stress components defined in Eqs. (16)–(20) depend linearly on the elastic normal displacement at the interface, δ. We now use the typical approach to diffusional creep that considers the difference in displacement velocity across a boundary to give the elastic displacement at the boundary. For a slowly changing chemical potential field,

(26)$${\nabla _{x,x}}{\theta _{\rm{a}}} + {\nabla _{y,y}}{\theta _{\rm{a}}} = 0\quad ,$$

and

(27)$${\theta _{\rm{a}}} = {\mu _{\rm{a}}}\left( {x,y} \right) - {\mu _{{\rm{ao}}}}\quad ,$$

with θ_a defined in Eq. (27) as the relative chemical potential, μ_a(x, y) as the spatial dependence of the lithium chemical potential due to a nonuniform stress, and μ_ao as the chemical potential of the lithium in a uniform (stress) system, all in units of J/atom. The solution consistent with the interface symmetry used here is

(28)$${\theta _{\rm{a}}} = \alpha \cos \left( {\omega x} \right)\exp \left( {\omega y} \right)\quad ,$$

where the boundary condition on the undulating interface is from Eq. (17) (Porz boundary condition)

(29)$${\mu _{\rm{a}}}\left( {{\rm{interface}}} \right) - {\mu _{{\rm{ao}}}} = - \Omega {\sigma _{yy}}\left( {{\rm{interface}}} \right) = \Omega {{9E\omega \delta } \over {16}}\quad ,$$

giving

(30)$$\alpha = \Omega {{9E\omega \delta } \over {16}}\quad ,$$

or similarly for the Wang [Reference Wang, Wolfenstine and Sakamoto12] boundary condition [Eq. (17)]

(31)$$\alpha = \Omega {{E\omega \delta } \over 2}\quad ,$$

where α is the magnitude of the chemical potential increase (J/atom) due to the elastic displacement, δ, as may be discerned by examining Eqs. (27) and (28). The flux of atoms normal to the surface on the lithium side of the boundary at the interface is

(32)$${J_{\rm{n}}}\left( {{\rm{Li}}} \right) = {M \over \Omega }\left( {{{{\rm{d}}{\theta _{\rm{a}}}\left( {{\rm{interface}}} \right)} \over {{\rm{d}}y}}} \right) = - {D \over {\Omega {k_{\rm{B}}}T}}\alpha \omega \;\cos \left( {\omega x} \right)\quad .$$

The flux of atoms normal to the surface on the SSE side of the boundary is given by either Eq. (10) or (14). Now, we can solve for the elastic displacement, δ, specifically at x = 0 on the undulating interface due to diffusional creep,

(33)$${{{\rm{d}}\delta } \over {{\rm{d}}t}} = \Omega {{\left| i \right|} \over q}\Delta \omega - {D \over {{k_{\rm{B}}}T}}\alpha \omega \quad .$$

It is then recognized that the magnitude of all the stress components depends on δ:

(34)$$\delta = {{16{k_{\rm{B}}}T\left| i \right|\Delta } \over {9DE\omega q}}\left[ {1 - \exp \left( { - {{9E\Omega D{\omega ^2}} \over {16{k_{\rm{B}}}T}}t} \right)} \right]\quad .$$

Under steady-state conditions (rate of lithium ions arriving at the interface equals the rate of lithium atoms diffusing away), the steady-state value of δ can be determined by replacing α in Eq. (33) with Eq. (30) or (31) and setting Eq. (33) equal to zero or by letting t go to infinity in Eq. (34). In this way, the derivation presented here provides an explicit description of the steady-state elastic displacement of lithium (at x = 0) created by the point effect of diffusion in relation to the average current density and the diffusional creep relaxation process. As per the proposed model, this elastic displacement is the physical source of the localized stress intensification acting at the lithium/SSE interface during plating.

Direct comparisons between the leading hypotheses describing the state of stress at the lithium/SSE interface

As pointed out by Swamy, the equilibrium state of electrochemically generated stress within a lithium-filled defect at the interface has an upper bound that, here, we describe in relation to the average current density at the interface [Reference Swamy, Park, Sheldon, Rettenwander, Porz, Berendts, Uecker, Carter and Chiang11]. From Eq. (34), it is apparent that after sufficient time is allowed to achieve equilibrium conditions at a fixed temperature, the steady-state elastic displacement, δ_SS, at the apex of the defect (x = 0) is uniquely related to the defect geometry (Δ and ω), the average current density at the interface, and the elastic modulus of the lithium-filled defect.

By relating Δ and ω to $\bar{h}$ and the maximum shear stress controlled by Nabarro–Herring creep to $\bar{H}$, Eq. (2) gives the critical elastic deflection, δ_critical, at the transition point from diffusion- to dislocation-mediated flow. Details of this derivation are provided in Appendix D. The enabling analogy is that 1/4 of the defect wavelength (π/(2ω), which corresponds to the defect geometry that is under compression, represents the indentation contact radius at the transition point. Using the equivalent cone geometry for the Berkovich indenter (ψ = 70.32°), the contact radius is 2.8h and, thus, $\bar{h} = {{\left( {2\pi } \right)} / {\left( {11.2\omega } \right)}}$. Because the constraint factor relating H and the flow stress is taken to be 1 (as per our hypothesis, there is no plastically deformed volume of material beneath the indenter i.e., elastically constrained), then the maximum shear stress required to initiate the transition is taken to be H/2 (Tresca’s maximum shear stress yield criterion), which is given by Eq. (18) when x = 0, and, therefore, H = 6/16Eωδ. Substituting for $\bar{H}$ and $\bar{h}$ in Eq. (2), δ then becomes the critical elastic deflection at the transition point from diffusion- to dislocation-mediated flow, giving δ_critical = 91.7/E. Derived in this way, δ_critical is expressed as a unique function of E because its dependence on the average current density at the interface is inherently incorporated through Eq. (2) and the constant term, 19.3 N/m. As such, the derivation is limited to the operating temperature of 31 °C. Given lithium’s large shear anisotropy factor (A = 8.4), 3 ≤ E ≤ 21 GPa, and, thus, δ_critical ranges from ∼30.6 to 4.4 nm depending on the crystallographic orientation of the lithium within the defect.

For the illustrative purposes of Fig. 2, here, we take the average planar current density to be 1 mA/cm² (a commonly observed CCD that causes failure), assume the sharpest defect geometry the model’s small strain assumption will allow (Δ = 1/ω and the ratio of defect wavelength to amplitude, N = λ/Δ, is minimized at 2π), and set δ = δ_critical = 10.2 nm (this assumes E = 9 GPa). In doing so, the defect geometry becomes fixed through Eqs. (D1) and (D2) such that ω = 9.124 × 10⁵ m⁻¹ and Δ = 1.096 × 10⁻⁶ m. Assuming $\bar{h}$ is representative of the critical length scale required to make the transition from diffusion- to dislocation-mediated flow, a rough estimate of the least volume required to activate the dislocation multiplication/nucleation mechanism is the smallest hemispherical cap that encloses the residual hardness impression at $\bar{h}$. Ignoring the strain rate effects and taking $\bar{h} = 350\;{\rm{nm}}$, the minimum activation volume is 1.3 μm³. By comparison, the fraction of defect under compression (−π/(2ω) ≤ x ≤ π/(2ω)) has a volume of (π³Δ)/(12ω²) = 3.4 μm³, which is ∼2.6 times larger than the smallest hemispherical cap enclosing a residual hardness impression at the average depth corresponding to the hypothesized transition from diffusion- to dislocation-mediated flow.

As predicted by Eqs. (16)–(20), Figs. 3(a) and 3(b) directly compare the elastic stress distribution at the interface in a manner that reflects several leading hypotheses for SSE failure. Figure 3(a) shows stresses consistent with the hypotheses and boundary conditions proposed by Porz, Herbert, and Wang for a lithium-filled defect at the interface [Reference Porz, Swamy, Sheldon, Rettenwander, Frömling, Thaman, Berendts, Uecker, Carter and Chiang8, Reference Herbert, Hackney, Dudney, Thole and Phani9, Reference Herbert, Hackney, Dudney, Thole and Phani10, Reference Swamy, Park, Sheldon, Rettenwander, Porz, Berendts, Uecker, Carter and Chiang11]. The defect geometry is that of Fig. 2, which is described earlier. Figure 3(b), on the other hand, shows stresses representative of the hypotheses and boundary conditions proposed by Wang for a flat interface [Reference Wang, Wolfenstine and Sakamoto12]. We note the SSE failure conditions described by Swamy are generally consistent with Wang’s flat interface hypothesis as well [Reference Swamy, Park, Sheldon, Rettenwander, Porz, Berendts, Uecker, Carter and Chiang11, Reference Wang, Wolfenstine and Sakamoto12]. For Figs. 3(a) and 3(b), the battery-relevant conditions yielding the predicted stresses are |i| = 1 mA/cm², E = 9 GPa, and T = 304 K.

Figure 3:

The predicted 2D stress components at (a) an undulating and (b) flat interface. For (a), the defect geometry is that of Fig. 2. For both (a) and (b), the elastic modulus of lithium and the average current density at the interface are taken to be 9 GPa and 1 mA/cm², respectively.

As Figs. 3(a) and 3(b) show, much of the predicted stress is well above σ_y for bulk polycrystalline lithium (∼0.5 MPa). Recalling the previously reported nanoindentation and micropillar experiments, both of which reveal lithium’s unexpectedly high strength at small length scales and high T _H, we submit the predicted stresses can be achieved on the basis of the hypothesis presented in section “Informed guidance from nanoindentation, past and future”, which in broad brushstrokes states that within a ‘small’ volume of lithium (∼3.4 μm³) at the lithium/SSE interface (nonplanar or planar), an increase in pressure due to preferential plating approaches the theoretical strength limit of lithium because diffusive flow mechanisms limit stress relief when the probability of finding a mobile dislocation or operable dislocation source within the locally stressed volume is low [Reference Herbert, Hackney, Dudney, Thole and Phani9, Reference Herbert, Hackney, Dudney, Thole and Phani10, Reference Xu, Ahmad, Aryanfar, Viswanathan and Greer16]. Of significant importance, we note that Fig. 3(a) indicates the possibility of dislocation-mediated flow outside the apex of the undulation, as the xy shear stresses reach a maximum between the apex and root of the undulation (x = ±π/(2ω)). At the apex (x = 0), it is apparent there is a significant deviatoric stress component in an orientation other than principal coordinate system of Fig. 2. The predicted gradients in stress also provide ample driving force for the self-diffusion of lithium away from the compressively stressed regions of the interface (−π/(2ω) ≤ x ≤ π/(2ω)). Across the interface, the corresponding stress in the SSE at the apex (x = 0) is a mode I tensile stress equal in magnitude to σ_xx. Figure 3(b), on the other hand, shows that for the elastic displacement imposed here, δ_critical, the resulting stress at the flat interface is entirely hydrostatic, indicating dislocation-mediated flow within the critically stressed volume at a flat interface is significantly more limited. The gradient in hydrostatic stress, however, still provides ample driving force for the diffusive flow of lithium away from the compressively stressed regions of the interface. As with the defect (nonplanar) interface, the corresponding stress in the SSE at the epicenter for the point effect of diffusion (x = 0) is a mode I tensile stress equal in magnitude to σ_xx, which for the flat interface is ∼2.7 times larger than that of the nonplanar interface.

Although the predicted stresses in Figs. 3(a) and 3(b) are quite large relative to σ_y, we note the small strain assumption imposed by the analysis method limits the model’s applicability to defects with a ratio of wavelength to amplitude, N = λ/Δ ≥ 2π [for Figs. 3(a) and 3(b), N = 2π]. As N decreases below this limit, the defect becomes sharper and, thus, it is expected that smaller aspect ratios will result in significantly higher magnitudes of σ_xx, σ_yy, and τ_xy. The significant outcome is that for sharper defects (smaller N), the mode I tensile stress acting in the SSE at the apex of the undulation could be considerably higher than the stress presented in Figs. 3(a) and 3(b). However, as previously noted, not every defect of this volume is necessarily subjected to such deleterious stresses. As per the hypothesis stated in section “Informed guidance from nanoindentation, past and future” and summarized previously, for a fixed current density, the pressure within a ‘small’ volume at the lithium/SSE interface could be alleviated by dislocation-mediated flow, as the stress relief will inherently depend on the probability of finding or not finding a mobile dislocation or dislocation source within close proximity to the stressed volume.

We further note that during charging, there is, of course, a thermodynamic limit to the magnitude of stress that can be generated in the lithium anode by an applied voltage above the equilibrium voltage, ΔV. The increase in equilibrium voltage due to stress is on the order of ΔV = (Ωσ_{H max})/q. As observed here, the maximum stress set by diffusional creep and dislocation plasticity is ∼50 MPa, thus, the corresponding ΔV = 6.7 mV.

Deformation mechanism map and the critical defect length scales corresponding to the maximum build-up of stress at the lithium/SSE interface

For the nonplanar interface defects, the culmination of the proposed analysis is the deformation mechanism map presented in Fig. 4, which is expressed for the following conditions: |i| = 1 mA/cm², N = 2π, E = 9 GPa, and T = 304 K. We note the maps recently reported by Barai et al. are similar in nature, but they are based on the bulk properties of lithium and, therefore, do not consider the length scale effects described here or the possibility of diffusional creep as the dominant stress relief mechanism [Reference Barai, Higa and Srinivasan58]. Nevertheless, the maps may be directly applicable to the previously described analysis of Narayan [Reference Narayan and Anand32].

Figure 4:

Deformation mechanism map and the corresponding shear stress within lithium filled defects at the lithium/SSE interface. The map is based on the following conditions: (1) the defect’s aspect ratio of wavelength to amplitude is 2π; (2) the average current density at the interface is 1 mA/cm²; (3) the elastic modulus of the lithium within the defect is 9 GPa; (4) the absolute temperature is 304 K.

The proposed map presented in Fig. 4 does not capture the element of time, rather it assumes sufficient time has elapsed such that δ is either δ_SS or δ_critical depending on the volume of the defect that is under compression. Details of the plot development are provided in Appendix D. The intersection of the δ_critical horizontal line with the linear function of δ_SS with V ^2/3 defines the defect volume that produces the maximum elastic displacement at the apex of the undulation and, thus, the greatest potential danger to interface failure by mode I or mixed mode fracture of the SSE. Based on the specified conditions, the volume of the defect under compression (−π/(2ω) ≤ x ≤ π/(2ω)) is ∼3.4 μm³, which is represented by the vertical dashed line. As previously described, the driving force for potential mode I failure of the SSE is the normal tensile stress balancing the compressive stress in lithium at the apex of the undulation, i.e., σ_xx = −15.7 MPa at x = 0, which is shown in Fig. 3(a).

The magnitudes of τ_SS and τ_critical [Eqs. (D3) and (D4)] are presented on the 2nd y-axis. The steady-state shear stress, τ_SS, can alleviate pressure within the defect by driving dislocation plasticity, but only if there is a mobile dislocation contained within the stressed volume or an active source that has sufficient room to physically operate (note the shear stress plotted in Fig. 4 does not occur in the same coordinate system as that depicted in Fig. 2). Alternatively, if the only means of stress relief within the defect is diffusion-mediated flow with a stress exponent n ≤ 3, then the magnitude of τ_SS at x = 0 is predicted on the 2nd y-axis. The intersection of the τ_SS and τ_critical curves show the most dangerous defect volume, as the intersection represents the defect volume at which the maximum stress occurs because of the kinetic limitations of diffusional creep relaxation. Increasing the defect volume beyond the intersection point, the predicted shear stress becomes τ_critical, as it denotes the shear stress required to trigger the transition from diffusion to dislocation-mediated flow. As indicated by the solid green line, τ_critical decreases with increasing volume, as given by

(35)$${\tau _{{\rm{critical}}}} = 2.361 \times {10^{ - 5}}{\left( {{V^{{2 / 3}}}} \right)^{{{ - 1} / 2}}}\quad .$$

In broad terms, the results presented in Fig. 4 illustrate that the length scale inhibition to dislocation plasticity revealed in Eq. (3) limits the possible stress relief mechanism to diffusional creep. The diffusion creep mechanism becomes less efficient as the defect size increases because of the increasing diffusion length, causing a larger mode I tensile stress in the SSE. Although there is a limitation to this effect in that increasing stresses will eventually lead to dislocation plasticity, the stress at which dislocation plasticity can alleviate pressure at small defect length scales may not fall below the threshold in stress required to cause brittle failure of the SSE.

As the aspect ratio N increases, the defect geometry changes such that the amplitude decreases and/or the wavelength increases. Regardless, the defect becomes more blunt. Under these conditions, the volume of the defect at the intersection point is independent of N. The key outcome is that both the maximum stress required to trigger the transition to dislocation-mediated flow [it now follows a different relation that that of Eq. (35)] and the maximum normal crack opening stress, −σ_xx, are lower and, thus, the probability of failure by fracture of the SSE is reduced as N increases. Accordingly, the most dangerous defects for a given |i| and E are relatively sharp defects, with small values of N. Considering the same values of |i| and N, a change in E brings a proportional change in δ_critical (section “Direct comparisons between the leading hypotheses describing the state of stress at the lithium/SSE interface”) such that for the same defect volume, a higher E requires less elastic displacement to trigger the transition to dislocation-mediated flow. At the intersection point, the magnitudes of τ_criticcal and −σ_xx are the same although less elastic displacement was required to reach these same levels of stress. Accordingly, the same size defect filled with lithium of a higher index of crystallographic orientation (higher E) will tolerate less elastic displacement before triggering the transition or potentially initiating fracture. Finally, for the same E and N, an increase in |i| shifts the defect danger zone to smaller volume defects. Under these conditions, the compressively stressed volume requires a higher shear stress to initiate the transition to dislocation-mediated plasticity and the lithium supports a higher magnitude of σ_xx, thus, increasing the risk of fracturing the SSE.

Interfacial stress relief from the SSE

Because the outcome of both interface conditions (flat versus undulating) is a mode I tensile stress acting at x = 0 in the SSE, it is clear the time- and potentially length scale–dependent properties of the SSE could contribute significantly to stress relief at the lithium/SSE interface. At present, very little is known about the time- and length scale–dependent properties of relevant SSEs. Interestingly, previous indentation experiments performed in LiPON indicate it has a significant propensity for time-dependent deformation [Reference Herbert, Tenhaeff, Dudney and Pharr59]. Motivated by LiPON’s well-documented ability to provide long and efficient cycling of lithium in thin film batteries and the potential role of the SSE in mitigating pressure at the interface, preliminary indentation creep experiments have been performed in a 20 μm-thick LiPON film at indentation depths of 400 and 650 nm and temperatures of 25, 40, and 60 °C [Reference Bates, Dudney, Neudecker and Evans35, Reference Dudney36]. Over a 60 s hold period at each depth and temperature, Fig. 5 shows the measured stress exponent for creep, n, as defined by the constitutive creep relation $\dot{\varepsilon } = A{\sigma ^n}$, is found to be ∼20 and independent of length scale and temperature over the ranges examined. Details of the experimental method used to determine n are described in the section “Experimental methods”. The implemented methodology is nearly identical to that previously reported by Berla et al., who examined the creep behavior of lithiated silicon and found n to be near 20 as well [Reference Berla, Lee, Cui and Nix60]. Significant test method refinements are currently under development in hopes of improving the precision of future measurements and, thus, more precise estimates of n. Regardless, the uncommonly high magnitude of n suggests that rather than the transport of individual atoms, molecular units of LiPON (3 Li⁺ and a unit of PO₄⁻³) are capable of local rearrangement by short-range diffusion and, thereby potentially capable of contributing directly to stress relief at the interface. Following the analysis proposed by Berla, the stress exponent of 20 corresponds to an activated volume of 112 ų, whereas a molecular unit of LiPON would occupy 81 ų in the fully dense glass [Reference Lacivita, Westover, Kercher, Phillip, Yang, Veith, Ceder and Dudney61]. This suggests the diffusive flow occurs on a length scale that is similar to that of a molecular unit of LiPON. Moreover, it suggests the properties of LiPON may be critical in relieving pressure at the lithium/SSE interface. By contrast, the substantially higher elastic modulus of ceramic SSEs (E _LLZO = ∼150 GPa versus E _LiPON = ∼77 GPa) suggests a combination of strong ionic and covalent bonding such that room temperature creep is less favorable and, therefore, the ceramic SSEs may be more susceptible to fracture than LiPON [Reference Yu, Schmidt, Garcia-Mendez, Herbert, Dudney, Wolfenstine, Sakamoto and Siegel7, Reference Herbert, Tenhaeff, Dudney and Pharr59]. Accordingly, the time-dependent behavior observed in LiPON may be one reason why the cyclability of the thin film SSE significantly outperforms both polycrystalline and single crystal superionic conductors such as LLZO.

Figure 5:

Lipon, the stress exponent for creep, n, as a function of temperature and length scale as determined by nanoindentation. The distances specified in each column indicate the indentation depth at which n was measured.