Mechanical instability of electrode-electrolyte interfaces in solid-state batteries

The interfacial contact between active material and solid electrolyte in a composite electrode limits the kinetics of all-solid-state batteries (ASSB). Despite the progress in processing techniques to improve cohesion in composite electrodes, the electrochemical reactions and mechanical stresses developed during battery operation affects interface properties. Here, we propose a one-dimensional radially symmetric analytical model based on the cohesive theory of fracture, to investigate the mechanical stability of interfaces in ASSB microstructures. Using the cohesive-energy approach, we analyze the delamination criterion and derive a stability condition for fracture propagation. Furthermore, we investigate the role of particle size and material properties on delamination, and we explore the effect of delamination on area-specific impedance. We report that delamination is induced when electrode particles undergo a volumetric change of about $7.5\%$ during (de)intercalation. Compliant electrolytes $(E<25\mathrm{GPa})$ are found to accommodate up to $25\%$ of particle volume change and delay the onset of delamination. The study identifies geometric regimes for mechanical stability. Such regimes are based on the relative size of the damage zone with respect to the particle radius. Finally, we demonstrate that delamination can significantly influence the total charge/discharge time if highly conductive electrolytes are employed. Overall, the analyses provide guidelines for engineering electrode-electrolyte interfacial properties by controlling particle size, material stiffness, and adhesive strength and length scale.

Figure 1

The electrode microstructure is idealized by an assembly of space-filling truncated octahedra. Each polyhedron embeds an electrode particle and it is treated as a representative volume. The external surface of the electrolyte shell is close to spherical, therefore we assume radial symmetry. The thickness of the electrolyte shell, with respect to the particle radius, depends on the volume ratio occupied by the particles. Typical values of active material loading are in the range 50–60% for commercial batteries. On the right the geometric factor $f=R_B/R_A$ is plotted as a function of active material loading.

Figure 2

Schematic representation of the 1D analysis carried out. A single particle is idealized as spherical in shape and it is embedded in a shell of solid electrolyte. The solid electrolyte material is visualized by an elastic spring and the interface by a cohesive traction-separation law. The spring is a schematic for illustrative purposes; the model is a continuum solid. The particle changes its volume following the change in stoichiometry. During battery charge Li deintercalates from the positive electrode causing, in most cases, the active material to contract. We do not model the individual electrode particles explicitly, but we regard their radius change as an imposed displacement $\underline{u}$. Constitutive laws for the solid electrolyte and the cohesive interface are sketched. The SE layer linear elastic response consists of a monotonic function of its total elongation (stretching of the spring). The interface is modeled with a linear traction separation law. The traction transmitted across the interface is a decreasing function of the displacement jump, i.e., the separation of the interface. The interface becomes fully debonded when a critical opening is reached and the cohesive force is null. If the solid-electrolyte material is elastoplastic, the emergence of plastic flow in proximity of the interface may bound the stress and prevent delamination to occur. The object of this study is the classification of regimes of no-fracture, stable crack growth and of sudden mechanical failure of the electrode-electrolyte interface.

Figure 3

(a) The contour plot on the left shows the relative volume contraction of electrode particles [according to Eq. (10)]. For most combinations of material properties, interfacial fracture energy, and electrolyte stiffness, 2.5% of radial strain (corresponding 7.5% volume change) is sufficient to nucleate fracture. Such a low Vegard's strain is typical for most Li-intercalating compounds. Compliant solid electrolyte materials are preferable to avoid delamination. Cohesive length is fixed at 1 nm. (b) The chemical strain required to fracture decreases with the length of the fracture process zone. The plot shows a generalization of the result in (a) illustrated in nondimensional form. The intercalation strain is normalized with respect to the interfacial cohesive length. The horizontal axis represents the ratio between cohesive stiffness (i.e., the slope of the traction separation law in Fig. 2) and the electrolyte Young's modulus. The cohesive stiffness is negative; here we consider the absolute value $2{\gamma }_{am\text{−}se}/\left[[u\right]{]}_R$.

Figure 4

The two plots illustrate the difference between mechanically (a) unstable and (b) stable interfaces in the plane defined by the relative crack opening and the relative displacement of the interface. The red curves represent the equilibrium configurations obtained by imposing the balance between the radial force in the electrolyte shell and the cohesive force at the interface. Before fracture occurs, the opening of the interface is zero and the first part of the equilibrium curve is vertical. After nucleation, the slope of the red curve depends on the relative stiffness of bulk SE material and interface according to the equation $u/R_A=-\left(1/A_{\mathrm{critical}}\right)\left(\left[[u\right]{]/\left[[u\right]]}_R\right)$ and the definition of $A_{\mathrm{critical}}$ as in Eq. (11). The black lines are constructed by imposing the compatibility with the external loading (i.e., the displacement due to the shrinking particle). The black line shifts up as the particle continues shrinking according to equation $\left[[u\right]{]/\left[[u\right]]}_R=\alpha c-u/R_A$ for increasing values of $\alpha c\ge {\left(\alpha c\right)}_{\mathrm{fracture}}$. In (a) there is no intersection between compatible and equilibrium curves after fracture nucleates. This is interpreted as a nonsmooth transition from a fully coherent to a fully incoherent electrode-electrolyte interface.

Figure 5

Regimes of stable vs unstable delamination at the electrode particle interface are represented. According to Eq. (12), fracture stability depends on three dimensionless parameters. The parameter on the horizontal axis is descriptive of the constitutive behavior of the SE material (Young's modulus $E_{se})$ and of the interface (cohesive strength $F_c)$. The particle's radius $A$ and cohesive length $\left[[u\right]{]}_R$ are the governing length scales for this problem. The contour lines define the stable particle size normalized by the cohesive length. Mechanical stability can also be achieved by increasing the cohesive length or decreasing the particle size. For a fixed volume ratio of active material, stability improves with the stiffness of the solid electrolyte. The dependence on the Poisson's ratio is weak, unless the SE material is close to being incompressible.

Figure 6

A plot of stress fields in the solid-electrolyte shell as a function of Possion's ratio (a) $\nu =0.15$ (b) $\nu =0.35$. The stress fields are developed during the charging process, where the embedded electrode particle shrinks in size. $E_{se}$ is the Young's modulus of solid electrolyte and ${\varphi }_{AM}$ is the volume ratio of active material.

Figure 7

If elastoplastic constitutive behavior is assumed for the solid electrolyte, the SE shell may undergo plastic flow in proximity to the interface. Depending on the electrolyte yield stress ${\sigma }_{\mathrm{yield}}$ and the interfacial fracture strength $F_c$, the system may release energy via plastic flow or fracture. The contour lines in Fig. 3 define the transition between these two modes according to Eq. (7). The result has a weak dependence on the volume fraction of active material, as highlighted by the shift of the contour lines. Such contour lines correspond, from right to left, to increasing volume ratios of active material (within the range $0.4–0.7)$. For any value of SE Poisson's ratio, plasticity will prevent fracture if the yield stress is not greater than half of the fracture strength.

Figure 8

Via random walk analysis we calculate the first passage from the center to the surface of a partially delaminated spherical particle. The delaminated surface is treated as a reflective boundary. The particle's surface is discretized with 2160 triangles and delaminated surfaces are either contiguous or noncontiguous collections of triangles [see panel (c)]. A large shift in both mean and standard deviation of the first passage times is observed when delaminated surfaces are contiguous [see panels (a) and (b)]. The average deintercalation (intercalation) time increases with delamination; however, the impact on the total charging (discharging) time may be small, if the particle radius is small compared to the electrode thickness or if the electrolyte is not very conductive.