Solid-state batteries have garnered significant attention in recent years due to their high energy density, enhanced safety, and superior thermal stability compared to conventional liquid electrolyte batteries. However, the formation of lithium dendrites and the subsequent accumulation of dead lithium—electrochemically inactive lithium metal that loses contact with the electrode—remain critical challenges. Dead lithium adversely affects battery performance, cycle life, and can lead to internal short circuits. While solid-state electrolytes with higher mechanical strength can suppress dendrite growth, incomplete dissolution during cycling results in dead lithium accumulation. This study employs a phase field method to simulate lithium dendrite dissolution and dead lithium formation under coupled mechanical-thermal-electrochemical fields in solid-state batteries. The model integrates Gibbs free energy formulations, evolution equations, and finite element analysis to explore the effects of temperature, external pressure, and electrochemical parameters on dead lithium morphology. Results indicate that coupling heat transfer or mechanical fields alters dissolution cutoff times and dead lithium area. Increasing temperature or applying specific external pressures reduces dead lithium area, while adjustments in diffusion coefficient, interfacial mobility, and anisotropic strength also significantly impact dead lithium formation.
The phase field method is a powerful computational tool for simulating microstructural evolution in materials, including lithium dendrite growth and dissolution in batteries. In solid-state batteries, the interplay between mechanical stress, thermal effects, and electrochemical reactions governs dendrite behavior. This work extends existing models by incorporating a multi-physics approach that couples mechanical, thermal, and electrochemical fields. The Gibbs free energy of the system is defined using an order parameter (\xi), where (\xi = 0) represents the solid electrolyte phase and (\xi = 1) denotes the lithium metal phase. The total Gibbs free energy (G) is expressed as:
$$G = \int_V \left[ f_{\text{grad}}(\xi) + f_{\text{ch}}(\xi, c_i) + f_{\text{elec}}(\xi, c_i, \varphi) + f_{\text{els}}(\xi) \right] dV$$
Here, (f_{\text{grad}}) is the gradient energy density, (f_{\text{ch}}) is the chemical free energy density, (f_{\text{elec}}) is the electrostatic energy density, and (f_{\text{els}}) is the elastic energy density. The gradient energy density accounts for interface energy and anisotropy:
$$f_{\text{grad}}(\xi) = \frac{1}{2} \kappa \nabla^2 \xi$$
where (\kappa = \kappa_0 [1 + \delta \cos(\omega \theta)]) includes the gradient energy coefficient (\kappa_0), anisotropic strength (\delta), modulus (\omega), and angle (\theta). The chemical free energy density incorporates a double-well potential:
$$f_{\text{ch}}(\xi, c_i) = g(\xi) + c_0 R T_0 \sum \frac{c_i}{c_{0i}} \ln \frac{c_i}{c_0}$$
with (g(\xi) = W \xi^2 (1 – \xi)^2), where (W) is the barrier height. The electrostatic energy density is given by:
$$f_{\text{elec}}(\xi, c_i, \varphi) = \sum F c_i z_i \varphi$$
where (F) is Faraday’s constant and (z_i) is the valence. The elastic energy density considers mechanical effects:
$$f_{\text{els}}(\xi) = \frac{1}{2} C_{ijkl} \varepsilon_{ij}^E \varepsilon_{kl}^E$$
The elastic tensor (C_{ijkl}) is defined as:
$$C_{ijkl} = \frac{E}{2(1 + \nu)} (\delta_{il} \delta_{jk} + \delta_{ik} \delta_{jl}) + \frac{E \nu}{(1 + \nu)(1 – 2\nu)} \delta_{ij} \delta_{kl}$$
where (E = E_e h(\xi) + E_s [1 – h(\xi)]) is Young’s modulus, (\nu = \nu_e h(\xi) + \nu_s [1 – h(\xi)]) is Poisson’s ratio, and (h(\xi) = \xi^3 (6\xi^2 – 15\xi + 10)) is an interpolation function. The elastic strain tensor includes Vegard strain coefficients (\lambda_i).
The evolution of the phase field parameter (\xi) is governed by:
$$\frac{\partial \xi}{\partial t} = -L_\sigma \left( f_{\text{ch}}'(\xi) + f_{\text{grad}}'(\xi) + f_{\text{els}}'(\xi) \right) – L_\eta h'(\xi) \left{ \exp \left[ \frac{(1 – \alpha) n F \eta_\alpha}{R T_0} \right] – \frac{c_{Li^+}}{c_0} \exp \left[ \frac{-\alpha n F \eta_\alpha}{R T_0} \right] \right}$$
Lithium ion diffusion follows Fick’s law with thermal coupling:
$$\frac{\partial c_{Li^+}}{\partial t} = \nabla \cdot \left[ D_{\text{eff}} \nabla c_{Li^+} + D_{\text{eff}} \frac{D_{\text{eff}} c_{Li^+}}{R T_0} n F \nabla \varphi \right] – \psi \frac{d\xi}{dt}$$
The effective diffusion coefficient (D_{\text{eff}}) incorporates temperature effects:
$$D_{\text{eff}} = A \exp \left[ -r c_{Li^+} + \frac{E_\alpha}{R} \left( \frac{1}{T} – \frac{1}{T_0} \right) \right]$$
The electric potential is solved using Poisson’s equation:
$$\nabla \cdot (\sigma_{\text{eff}} \nabla \varphi) = F C_s \frac{\partial \xi}{\partial t}$$
where (\sigma_{\text{eff}} = \sigma_e h(\xi) + \sigma_s [1 – h(\xi)]) is the effective electrical conductivity.
A finite element model of an 8μm × 8μm domain is used, with a lithium metal anode at the bottom boundary and initial nucleation points for dendrites. The simulation time is 90 seconds with a 1-second step. Key parameters for the solid-state battery model are summarized in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Gradient Energy Coefficient | \(\kappa_0\) | 1×10-10 J/m |
| Anisotropic Strength | \(\delta\) | 0.1 |
| Anisotropic Modulus | \(\omega\) | 4 |
| Barrier Height | \(W\) | 3.75×105 J/m3 |
| Standard Volume Concentration | \(c_0\) | 1×103 mol/m3 |
| Environmental Temperature | \(T_0\) | 293 K |
| Electrode Young’s Modulus | \(E_e\) | 7.8 GPa |
| Electrolyte Young’s Modulus | \(E_s\) | 1 GPa |
| Electrode Poisson’s Ratio | \(\nu_e\) | 0.42 |
| Electrolyte Poisson’s Ratio | \(\nu_s\) | 0.3 |
| Vegard Strain Coefficient | \(\lambda_i\) | -0.866×10-3, -0.773×10-3, -0.529×10-3 |
| Interfacial Mobility | \(L_\sigma\) | 1×10-6 m3/(J·s) |
| Reaction Constant | \(L_\eta\) | 0.5 s-1 |
| Symmetry Factor | \(\alpha\) | 0.5 |
| Solid Lithium Concentration | \(C_s\) | 7.64×104 mol/m3 |
| Electrode Conductivity | \(\sigma_e\) | 1×107 S/m |
| Electrolyte Conductivity | \(\sigma_s\) | 0.1 S/m |
| Electrode Specific Heat | \(c_{pe}\) | 1200 J/(kg·K) |
| Electrolyte Specific Heat | \(c_{ps}\) | 133 J/(kg·K) |
| Electrode Thermal Conductivity | \(\lambda_e\) | 1.04 W/(m·K) |
| Electrolyte Thermal Conductivity | \(\lambda_s\) | 0.45 W/(m·K) |
| Convective Heat Transfer Coefficient | \(h\) | 10 W/(m2·K) |
The coupling of heat transfer and mechanical fields significantly influences dead lithium formation in solid-state batteries. When the heat transfer model is integrated into the mechanical-electrochemical phase field framework, the dissolution cutoff time decreases, and the dead lithium area increases by 61.1% compared to the model without heat transfer. This is attributed to altered stress distribution in lithium dendrites, where von Mises stress concentrations shift, affecting dissolution rates. Under an external pressure of 5 MPa, the model without heat transfer exhibits severe stress concentration at the dendrite roots, leading to a substantial increase in dead lithium area. In contrast, the coupled model shows a reduction in dead lithium area due to moderated stress effects. Further analysis under varying external pressures (5 MPa, 10 MPa, and 20 MPa) reveals that dissolution cutoff times decrease with increasing pressure, but dead lithium area follows a non-monotonic trend: it decreases initially, then increases, and decreases again at higher pressures. This behavior is linked to changes in dendrite morphology and stress-induced fracture mechanisms.
The inclusion of the mechanical field in the thermal-electrochemical phase field model enhances structural stability, resulting in a longer dissolution cutoff time and a 40.8% reduction in dead lithium area. Temperature distributions also shift, with lower core temperatures in the coupled model due to reduced heat concentration from smaller dead lithium areas. Variations in environmental temperature (e.g., 273 K and 353 K) demonstrate that higher temperatures reduce dead lithium area, as increased ion diffusion and interface reactions promote uniform dissolution. However, the mechanical field dampens the sensitivity to temperature changes, maintaining dendrite integrity. Notably, the dissolution rate at the dendrite root decreases over time due to nucleation point constraints, leading to residual lithium accumulation.
Electrochemical parameters play a crucial role in dead lithium formation in solid-state batteries. The diffusion coefficient (D_{\text{eff}}) affects lithium ion migration and dendrite morphology. Increasing the diffusion coefficient accelerates dendrite growth, resulting in taller primary stems and longer side branches, which increases dead lithium area by 55.2% due to incomplete dissolution. Conversely, decreasing the diffusion coefficient reduces lithium deposition and dead lithium area by 24.1%, despite a shorter dissolution cutoff time. Interfacial mobility (L_\sigma) governs interface evolution rates. Higher interfacial mobility (1×10-5 m³/(J·s)) promotes uniform lithium deposition and smoother dendrites, reducing dead lithium area to 0.004 μm². Lower interfacial mobility (1×10-7 m³/(J·s)) slows dissolution, leading to a dead lithium area of 0.024 μm². Both cases show improvements over the baseline, highlighting the importance of interface kinetics. Anisotropic strength (\delta) influences dendrite directionality. Reducing (\delta) to 0.05 decreases dead lithium area by 12.0%, as isotropic growth minimizes fragile side branches. Increasing (\delta) to 0.15 also reduces dead lithium area by 3.4%, but promotes more isolated lithium particles due to enhanced secondary branching.
In conclusion, the multi-physics phase field model provides insights into dead lithium behavior in solid-state batteries. Coupling heat transfer or mechanical fields alters dissolution dynamics and dead lithium area. Elevated temperatures and optimized external pressures reduce dead lithium accumulation. Adjustments in diffusion coefficient, interfacial mobility, and anisotropic strength offer effective strategies for mitigating dead lithium in solid-state batteries. Future work should explore additional factors, such as interface reactions and SEI layer changes at high temperatures, to further enhance solid-state battery performance and safety.