In the multiphysics picture illustrated in Section 2.1, positive ions are the only moving species, and the relocation of vacancies is the outcome of the ionic transport1. Denote with c _α the concentration (i.e. the number of moles per unit volume) of a generic species α and with h ⃗ α the mass flux in terms of moles, i.e. the number of moles of species α measured per unit area per unit time. The mass balance equations characterize the chemo-diffusive transport of four different species within the solid electrolyte. They read: ∂ c Li 0 ∂ t = − w , (6a) ∂ c n − ∂ t = w , (6b) ∂ c Li + ∂ t + d i v h ⃗ Li + = w − y , (6c) ∂ c Li int + ∂ t + d i v h ⃗ Li int + = y , (6d) where the overall rate of the charge carrier generation is given by the mass action laws (3). The set of 4 mass balance Eqs 6a, 6b, 6c, 6d, contains five unknowns, the concentrations c _Li0 , c _n⁻ , c Li + , and c Li int + plus the electric potential ϕ, which is constitutively related to the mass fluxes h ⃗ α . In the present paper, the additional required equation is Ampère’s law (with Maxwell’s correction), which reads: d i v ∂ D ⃗ ∂ t + F h ⃗ Li + + h ⃗ Li int + = 0 . (7)

To conclude the set of balance equations, we introduce the usual balance of forces in small strains: d i v σ + b ⃗ = 0 ⃗ . (8)

The anode is a foil of pure metal lithium, a continuous reservoir of ions. Balance equations depict the conservation of charge and linear momentum, in the two unknowns electric potential ϕ and displacements u ⃗ : d i v i ⃗ = 0 , (9a) d i v σ + b ⃗ = 0 ⃗ . (9b)

During intercalation in the cathode, the lithium keeps its ionic nature “shielded” by its own electron. Such a species will be denoted with Li^⊕ to distinguish it from mobile charges Li⁺ in the electrolyte. Balance equations characterize the chemo-diffusive migration transport of Li^⊕, the electronic motion, and mechanical stress: ∂ c Li ⊕ ∂ t − d i v h ⃗ Li ⊕ = 0 , (10a) d i v i ⃗ c a = 0 , (10b) d i v σ + b ⃗ = 0 ⃗ . (10c)

The constitutive laws for the electric displacement D ⃗ , the fluxes of the species h ⃗ α and the stress tensor σ shall be defined. The electric displacement field is related to the electric potential ϕ constitutively by: D ⃗ = − ε | e l ∇ ϕ , (11) where ɛ | _el  = ɛ | _r  ɛ |₀ is the electric permittivity and its value is ɛ | _r times the electric permittivity in vacuum, denoted with ɛ |₀ = 8.85 × 10^–12 C/(V m). A linear dependence on the gradient of the electrochemical potential μ ̄ α is taken for the mass flux h ⃗ α of positive ions α = L i + , L i int + in the absence of convection: h ⃗ α = − D | α c α R T ∇ μ ̄ α , (12a) equipped with the thermodynamic specification μ ̄ α = μ α 0 + R T 1 + ln c α + z α F ϕ , (12b) where R is the universal gas constant, D | _α is the diffusivity of species α, μ α 0 is a reference value of the chemical potential, and T is the absolute temperature. Fick’s law (12a) takes the Nernst-Planck form h ⃗ α = − D | α ∇ c α − D | α F R T c α ∇ ϕ . (12c)

LiPON is an inorganic solid electrolyte, a glass ceramic which assumes an amorphous state instead of a regular crystalline structure. Its brittle, mechanical response is here described as for an homogeneous and isotropic material in small strains: σ = K e l t r ε 1 + 2 G e l d e v ε , (13) where K _el , G _el are the bulk and shear modulus respectively. The governing equations for the solid electrolyte at all points x ⃗ and times t come out incorporating constitutive Eqs 11–13 into the balance Eqs 6–8.

For the anode, an Ohmic behaviour models the electron flow i ⃗ a n = − k a n ∇ ϕ , (14) where k _an represents the electric conductivity. Although we are well aware that complex visco-plastic models exist for the lithium metal, see Anand and Narayan (2019); Sedlatschek et al. (2021) among others, in this note we limit ourselves to investigate the elastic response of the foil, i.e. σ = K a n t r ε 1 + 2 G a n d e v ε , (15) with K _an , G _an the bulk and shear modulus respectively.

The constitutive equations result more involved for the cathode, since swelling-shrinking upon intercalation are accounted for in the definition of the small strain tensor ɛ and of the chemical potential μ Li ⊕ . Intercalation in the cathode active material entails an interaction between electro-chemistry and mechanics, bringing coupling terms in the chemical potential μ Li ⊕ = μ Li ⊕ 0 + R T 1 + ln c Li ⊕ − p Ω Li ⊕ , (16) and in the De Saint Venant–Kirchhoff constitutive model of the stress σ = K c a t r ε − ε s 1 + 2 G c a d e v ε − ε s , (17) with the pressure p = t r σ / 3 (18) and ε s = Ω Li ⊕ 3 c Li ⊕ − c Li ⊕ 0 1 .

The Fickian flux of lithium Li^⊕ is defined as usual: h ⃗ Li ⊕ = − D | Li ⊕ c Li ⊕ R T ∇ μ Li ⊕ . (19) The electron flow in the active and conductive material obeys Ohm’s law (14) with parameters that refer to the cathode. The governing equations for the electrodes incorporate Eqs 14, 15 into Eqs 9a, 9b for the anode and Eqs 14, 17, 19 into Eqs 10a, 10b, 10c for the cathode.

Initial and boundary conditions are made compatible with thermodynamic equilibrium at t = 0, tuning the current density i _bat (t) in time. Figure 5A shows the applied current density i _bat (t), modeled via a logistic curve, as a function of time. After a short steep transient, the current reaches a plateau, which corresponds to a steady current density |i _bat | = 0.035 [A/m ²]. It is kept fixed for about 40 h. The cell is discharged afterwards. In Figure 5B, the charge/discharge curves are plotted against the extracted charge: the voltage of the cell increases up to its maximum value 4.2 [V] while charging. The discharge process starts afterwards, the voltage reduces and a voltage drop becomes evident when the saturation concentration of the lithium in the cathode (the limiting factor) is approached.

Charging and discharging cause the deposition and remotion of lithium at the anode surface in contact with the solid electrolyte. No degradation models are introduced, hence the hysteresis diagram of Figure 5B is replicated during further cycles. The open circuit potential (OCP) used in the simulations has been calculated analytically, following the approach proposed in Purkayastha and McMeeking (2012).

The electric potential profile ϕ(x) in the battery at different times is depicted in Figure 6A, in the current configuration. The blue curve corresponds to the initial condition, when no current is applied. The black one to the maximum value after charging. At full charge, the electric potential discontinuities at the interfaces make Butler-Volmer currents vanishing. Based on the measured battery OCP, at full charge state ϕ L , t f − ϕ 0 , t f = 4.2 V .

Complete discharging, when the concentration of lithium Li^⊕ inside the cathode is close to the saturation limit of the cathode c Li ⊕ sat , is captured by the red line. Cathodic saturation is the limit factor for the battery operation (see also Magri et al. (2022) for an extensive discussion on limiting factors in electrochemical cells, induced by materials and architectures).

The evolution of lithium concentration in the current configuration of the cell at different time steps is given in Figure 7A. Note the interface motion due to deposition. The anode is an unlimited reservoir of lithium. The concentration of lithium (i.e. its density) does not change while the thickness does. The blue curve corresponds to the initial condition, at thermodynamic equilibrium, with the concentration of lithium inside the cathode equal to the saturation limit c Li ⊕ sat . The black curve reports the concentration of lithium during charge, when the potential reaches the limit value of 4.2 [V], corresponding to the end of the charge phase; the red curve depicts the final state at discharging, when the saturation limit at the interface between the electrolyte and the cathode is reached. Both black and red plots refer to states out of equilibrium. Although the C-rate is small, it is evident how lithium intercalates inside the cathode and accumulates near the electrolyte/cathode interface in the discharge process. Vice versa, the removal of lithium from the anodic foil is visible in charge.

Figure 7B magnifies Figure 7A and highlights the evolution of the lithium concentration in the electrolyte. Since two ionic concentrations are concurrently present, only their sum ( c Liˆ + + c Li int + ) has been plotted. Before the current is inverted, lithium concentration decreases in the left half of the electrolyte and increases by the cathode. During discharging, the opposite is seen.

The concentrations of species c_{Li_{0}}, c n − , c L i + and c L i int + are plotted separately in Figure 8A. The concentration profiles, after a transition period, tend to linearize. Notice the reciprocal exchange between the Li⁺ and L i int + , ruled by reaction 2, and how the (negative) uncompensated charges are filled by hopping lithium. Finally, the lithium concentrations in the solid electrolyte of Li⁺, L i int + and their sum at the interfaces with the electrodes as a function of time are reported in Figure 8B. The continuous lines correspond to the values estimated at the anodic interface, while the dashed curves correspond to the values at the cathode interface. The concentrations show an increment at the anode interface and a decrement at the cathode interface during discharging; before the current is inverted, the trend of the concentration was the opposite.

Figure 9A depicts the displacement of the anode/electrolyte and electrolyte/cathode interfaces as a function of time. They are induced by the thickening of the anode during charge (and vice versa during discharge), by the elastic deformation due to the lateral constraints, and by the swelling/shrinking of the cathode upon extraction/intercalation. Since the adopted model is one dimensional and no bulk forces are applied along the cell, the Cauchy stress is uniform. It is plotted in Figure 9B as a function of time.

The swelling-shrinking of the cathode upon intercalation/extraction can sharpen or reduce the stress in the cell, depending upon the cathodic change in volume with the lithium ions concentration. For LiCoO ₂, a loss of lithium causes a volumetric expansion, as reported by Mendoza et al. (2016) and Reimers and Dahn (1992). Delithiation induced swelling during battery charge is governed by the molar volume (three times the chemical expansion coefficient) Ω L i ⊕ = − 2.4 ⋅ 1 0 − 7 m 3 m o l − 1 Mendoza et al. (2016), where the negative value indicates expansion during delithiation.

Define as usual displacement as the difference between the location in the current configuration and the point position in the reference, i.e. u ⃗ X ⃗ = x ⃗ − X ⃗ . (26)

Such a field is shown in Figure 10 at different time steps, in the current configuration. The initial configuration (t = 0), assumed at thermodynamic equilibrium, is taken to be displacement and stress free (blue curve). To better figure out the motion scenario within the cell, consider the reference and the intermediate configuration. All points that belong to the anode in the initial (referential) state do no undergo any configurational change, i.e. their position is unchanged in the intermediate configuration. Consider a point P ⃗ in the LIPON electrolyte in the reference configuration. At a given time, the lithium plating causes a positive (rightward) displacement of the electrolyte/lithium foil interface, which corresponds to a rigid displacement at the point P ⃗ for the intermediate configuration. The current configuration is a competition between the configurational change of the anode and the mechanical response due to the compressive state of the cell induced by the boundary conditions. In view of the lithium plating during charge, the anode thickens and fixing the two ends of the cell generates a compressive state of stress. The intermediate configuration deforms (shortening) into the current configuration, inducing a leftward (negative) displacement to all points in the anode that existed at the beginning of the charging process. The displacements within the electrolyte turned out to be still rightward, i.e. positive. A similar outcome holds for the cathodic displacement field. They are depicted in Figure 10. The definition (26) cannot apply to the material that has been generated because of plating, since new volume has been created. For them, we can adopt a different definition of displacements, i.e. the difference between the location in the current configuration and the point position in the intermediate configuration, i.e. u ⃗ * X * ⃗ = x ⃗ − X * ⃗ (27)

The definitions (26) and (27) are identical for the anode, but different for electrolyte and cathode. We will thus base the notion of strain from definition (26) for the latter, and from definition (27) for the anode. It is trivial to show that the definition of all strain measures is consistent with the classical ones within each material. The same conclusion can therefore be inferred for the constitutive laws. Since we assumed that the response of all materials in the cell is linear elastic, a straight line depicts the displacement field in Figure 10. The displacement jump at the electrolyte/lithium foil interface corresponds to the effect of thickening due to plating, followed by the elastic deformation of the freshly deposited lithium. The blue line refers to the initial time step: since no external force is applied, no displacement is observed. Once the charging process takes place, an elastic displacement arises concurrently with deposition and cathode swelling. At the full charge, identified by the black line in Figure 10, the displacement (26) discontinuity ΔL _a = ΔL ^ela + ΔL ^dep at the electrolyte/lithium foil interface is clearly visible. Being constrained by u ⃗ ( L a + L e + L c ) = 0 ⃗ , the right end of the cell results motionless. After the current is inverted, lithium starts to be stripped from the anode surface, and the displacements decrease. Since the plated lithium cannot be fully extracted (unless the C-rate tends to zero) a residual displacement field, with its related stress, is seen at full discharge.

To emphasize the role of plating, the mechanics of charging/descharging has been plotted in Figure 11 neglecting the effects of deposition, as per anodic intercalation without swelling. The consequences on the electro-chemical response are small, but from a mechanical prospective, outcomes largely differ. Displacements are plotted in Figure 11A, to be compared with Figure 10. The extraction of lithium from the cathode causes an expansion of the latter, which induces compressive stresses due to the fully restrained boundary conditions at the two ends of the battery. The maximum displacements are reached on full charge, depicted with a black curve in Figure 11A, which is one order of magnitude smaller than the one with plating–see Figure 10. The stress arising from the mere cathodic expansion is very small compared to the one due to plating.