The pursuit of higher energy density and enhanced safety in electrochemical energy storage has propelled solid-state batteries (SSBs) to the forefront of battery research. Replacing flammable liquid electrolytes with solid counterparts offers a promising path to mitigate thermal runaway risks. However, the persistent challenge of lithium dendrite growth and the subsequent formation of “dead lithium” remain critical barriers to their long-term cyclability and safety. Dead lithium refers to isolated, electrochemically inactive metallic lithium that loses electronic contact with the electrode during stripping (discharge) cycles. Its accumulation leads to irreversible capacity loss, increased cell impedance, and in severe cases, internal short circuits if dendrites penetrate the electrolyte. While the superior mechanical strength of solid electrolytes can physically suppress dendrite propagation compared to liquid systems, incomplete dissolution during stripping still leads to dead lithium formation over repeated cycling.
Experimental investigations have proposed various strategies, from stress modulation to novel electrode architectures, to inhibit dead lithium. Concurrently, the phase-field method has emerged as a powerful computational tool to simulate the complex interplay between electrochemistry, mechanics, and transport phenomena during lithium deposition and stripping. Most existing phase-field studies on dead lithium focus on liquid electrolytes or are limited to two-physics couplings, such as chemo-mechanical or thermo-electrochemical models. A comprehensive understanding within a unified solid-state battery framework, incorporating the coupled effects of stress, temperature, and electrochemistry, is still evolving. Furthermore, the systematic investigation of how intrinsic electrochemical parameters influence dead lithium morphology is relatively scarce. This work aims to address these gaps by developing and utilizing a multiphysics phase-field model that couples mechanical, thermal, and electrochemical fields to simulate dead lithium formation in a solid-state battery environment. We analyze the individual and combined effects of these physical fields and delve into the impact of key electrochemical parameters on the final dead lithium area.
Theoretical Framework and Numerical Model
Phase-Field Formulation and Free Energy Functional
The core of the phase-field model is the introduction of a non-conserved order parameter, $\xi(\mathbf{x}, t)$, which distinguishes the different phases within the solid-state battery system: $\xi = 0$ represents the solid electrolyte phase, $\xi = 1$ represents the metallic lithium electrode phase, and the diffuse interface ($0 < \xi < 1$) represents the interphase region. The system’s thermodynamics is described by the total Gibbs free energy functional $G$:
$$ G = \int_V \left[ f_{\text{grad}}(\xi) + f_{\text{ch}}(\xi, c_i) + f_{\text{elec}}(\xi, c_i, \phi) + f_{\text{els}}(\xi) \right] dV $$
where $f_{\text{grad}}$, $f_{\text{ch}}$, $f_{\text{elec}}$, and $f_{\text{els}}$ are the gradient, chemical, electrostatic, and elastic energy densities, respectively. The concentrations $c_i$ (with $i = \text{Li}, \text{Li}^+, \text{A}^-$ for lithium atom, lithium ion, and anion) and the electrostatic potential $\phi$ are other field variables.
The gradient energy density penalizes sharp interfaces and incorporates crystallographic anisotropy, crucial for dendritic growth:
$$ f_{\text{grad}}(\xi) = \frac{1}{2} \kappa |\nabla \xi|^2, \quad \text{with} \quad \kappa = \kappa_0 [1 + \delta \cos(\omega \theta)] $$
Here, $\kappa_0$ is the gradient energy coefficient, $\delta$ is the anisotropic strength, $\omega$ is the mode number (e.g., 4 for cubic symmetry), and $\theta$ is the angle between the interface normal and a reference axis.
The chemical free energy density consists of a double-well potential $g(\xi)$ that stabilizes the bulk phases and an entropic mixing term:
$$ f_{\text{ch}}(\xi, c_i) = W \xi^2 (1-\xi)^2 + c_0 R T_0 \sum_i \frac{c_i}{c_0} \ln \left( \frac{c_i}{c_0} \right) $$
where $W$ is the energy barrier height, $c_0$ is a reference concentration, $R$ is the gas constant, and $T_0$ is the reference temperature.
The electrostatic energy density accounts for the interaction between charged species and the electric field:
$$ f_{\text{elec}}(\xi, c_i, \phi) = \sum_i F z_i c_i \phi $$
where $F$ is Faraday’s constant and $z_i$ is the valence of species $i$.
For a solid-state battery, the elastic energy density is critical due to the large mechanical stresses generated by lithium deposition (large volumetric change). It is given by:
$$ f_{\text{els}}(\xi) = \frac{1}{2} \mathbf{C}_{ijkl}(\xi) \varepsilon^E_{ij} \varepsilon^E_{kl} $$
The elastic tensor $\mathbf{C}_{ijkl}$ and the elastic strain $\varepsilon^E_{ij}$ are interpolated across the interface using the function $h(\xi) = \xi^3(6\xi^2 – 15\xi + 10)$, which ensures a smooth transition. The elastic strain is related to the total strain and the eigenstrain (or Vegard strain) $\lambda_i$ induced by lithium concentration/phase change: $\varepsilon^E_{ij} = \varepsilon^T_{ij} – \lambda_i h(\xi) \delta_{ij}$.
Kinetic Equations: Evolution of Fields
The temporal evolution of the phase field $\xi$ is governed by the Allen-Cahn equation, where the driving force includes contributions from chemical, gradient, elastic, and electrochemical reaction terms:
$$ \begin{aligned}
\frac{\partial \xi}{\partial t} &= – L_{\sigma} \left( \frac{\partial f_{\text{ch}}}{\partial \xi} + \frac{\partial f_{\text{grad}}}{\partial \xi} + \frac{\partial f_{\text{els}}}{\partial \xi} \right) \\
&\quad – L_{\eta} h'(\xi) \left\{ \exp\left( \frac{(1-\alpha)nF\eta_{\alpha}}{RT_0} \right) – \frac{c_{\text{Li}^+}}{c_0} \exp\left( -\frac{\alpha nF\eta_{\alpha}}{RT_0} \right) \right\}
\end{aligned} $$
Here, $L_{\sigma}$ is the interfacial mobility, $L_{\eta}$ is the electrochemical reaction coefficient, $\alpha$ is the charge transfer symmetry factor, $n$ is the number of electrons transferred, and $\eta_{\alpha}$ is the activation overpotential. A switching function $f_d = f_{\text{step}}(-\phi_s/\phi_b)$ is incorporated during the stripping stage to deactivate regions that lose electronic contact, defining them as dead lithium ($f_d=0$).
The transport of lithium ions in the solid electrolyte is described by a modified Fick’s law, accounting for both diffusion and migration under the electric field:
$$ \frac{\partial c_{\text{Li}^+}}{\partial t} = \nabla \cdot \left[ D_{\text{eff}} \nabla c_{\text{Li}^+} + \frac{D_{\text{eff}} c_{\text{Li}^+}}{RT_0} nF \nabla \phi \right] – \psi \frac{d\xi}{dt} $$
where $\psi$ is a constant related to the reaction rate. The effective diffusion coefficient $D_{\text{eff}}$ is the key parameter for coupling the heat transfer model, as it follows an Arrhenius-type temperature dependence:
$$ D_{\text{eff}} = A \exp\left[ -r c_{\text{Li}^+} + \frac{E_a}{R} \left( \frac{1}{T} – \frac{1}{T_0} \right) \right] $$
Here, $A$, $r$, and $E_a$ are material parameters, and $T$ is the spatially and temporally evolving temperature field obtained from solving the heat equation. The boundary heat flux is defined as $-h_R (T – T_0) – \varepsilon_R \sigma_R (T^4 – T_0^4)$, incorporating both convective and radiative cooling.
The electrostatic potential is solved using Poisson’s equation, with conductivity interpolated across phases:
$$ \nabla \cdot \left( \sigma_{\text{eff}} \nabla \phi \right) = F C_s \frac{\partial \xi}{\partial t}, \quad \sigma_{\text{eff}} = \sigma_e h(\xi) + \sigma_s [1 – h(\xi)] $$
where $C_s$ is the solid lithium concentration, and $\sigma_e$ and $\sigma_s$ are the conductivities of the electrode and electrolyte, respectively.
Finite Element Implementation and Parameters
The coupled system of partial differential equations is solved using the finite element method in COMSOL Multiphysics 6.2. The computational domain is a 2D rectangle (8 µm × 8 µm) representing a section of the solid-state battery near the anode. The bottom boundary serves as the lithium metal anode, with an initial nucleation site for a dendrite seed. A constant voltage is applied at the top boundary to drive deposition/stripping, with an initial lithium ion concentration. The mesh is finely resolved near the interface region to capture the dendrite morphology accurately. The simulation involves a two-step process: a 90-second potentiostatic lithium deposition, followed by a potentiostatic stripping phase until dissolution ceases. The key material and model parameters used in this study for the PEO-based solid polymer electrolyte system are summarized in the table below.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Gradient energy coefficient | $\kappa_0$ | $1.0 \times 10^{-10}$ | J m⁻¹ |
| Anisotropic strength | $\delta$ | 0.1 | – |
| Energy barrier height | $W$ | $3.75 \times 10^{5}$ | J m⁻³ |
| Reference temperature | $T_0$ | 293 | K |
| Electrode Young’s modulus | $E_e$ | 7.8 | GPa |
| Electrolyte Young’s modulus | $E_s$ | 1.0 | GPa |
| Interfacial mobility | $L_{\sigma}$ | $1.0 \times 10^{-6}$ | m³ J⁻¹ s⁻¹ |
| Reaction coefficient | $L_{\eta}$ | 0.5 | s⁻¹ |
| Electrode conductivity | $\sigma_e$ | $1.0 \times 10^{7}$ | S m⁻¹ |
| Electrolyte conductivity | $\sigma_s$ | 0.1 | S m⁻¹ |
| Activation energy for diffusion | $E_a$ | Varies | J mol⁻¹ |
| Convective heat transfer coefficient | $h_R$ | 10 | W m⁻² K⁻¹ |
Results and Discussion: Influence of Multiphysics Coupling
Effect of Thermal Field Coupling on Dead Lithium
First, we isolate the effect of the thermal field by comparing the standard chemo-mechanical model (CM) with the fully coupled thermo-chemo-mechanical model (TCM) for the solid-state battery. While the final dendrite morphology after growth shows minor visual differences, the internal stress ($\sigma_{vM}$) distribution is significantly altered by the temperature field. The thermal gradients and temperature-dependent properties (like $D_{\text{eff}}$) modify the local reaction and diffusion rates, thereby influencing stress generation during deposition.
During the stripping phase, the dissolution “cut-off” time (when active stripping ceases) and the final dead lithium area differ between the two models. The TCM model generally exhibits a shorter cut-off time but a larger dead lithium area compared to the CM model under baseline conditions. This suggests that while thermal effects may accelerate the overall stripping kinetics initially, they also promote non-uniform dissolution, leading to earlier necking and isolation of lithium segments, particularly at the dendrite root. The von Mises stress evolution reveals that in the TCM model, stress concentration at the dendrite root is less severe during the early stripping phase, which slows down the root’s dissolution rate relative to the dendrite trunk. This results in a more substantial residual lithium base that becomes dead lithium.
Effect of External Pressure under Thermal Coupling
Applying external pressure is a viable strategy to mechanically suppress dendrites in a solid-state battery. We investigate its effect within the TCM framework. Applying a uniform external pressure (e.g., 5 MPa) on the top boundary significantly changes the stress state. The von Mises stress intensifies along the main dendrite trunk. Crucially, the presence of the thermal field moderates the stress increase at the root compared to a non-thermal CM model under the same pressure.
Simulations with varying external pressures (5, 10, and 20 MPa) in the TCM model reveal a non-monotonic relationship between pressure and dead lithium area. The results are summarized below:
| External Pressure (MPa) | Dissolution Cut-off Time (s) | Dead Lithium Area (µm²) | Primary Morphological Change |
|---|---|---|---|
| 0 (Baseline TCM) | 43 | 0.058 | — |
| 5 | 40 | 0.053 | Reduced root stress concentration. |
| 10 | 32 | 0.073 | Increased root stress, promoting fracture. |
| 20 | 30 | 0.059 | Denser morphology, slower but uniform stripping. |
A low pressure (5 MPa) can slightly reduce dead lithium by modifying the initial dendrite morphology to be denser without causing excessive root fracture. A medium pressure (10 MPa) increases root stress concentration, leading to premature fracture and a higher dead lithium area. A high pressure (20 MPa), while causing the highest root stress, compacts the dendrite so significantly that its overall morphology is more robust and strips more uniformly and slowly, ultimately reducing the dead lithium area again. This complex interplay highlights the necessity of the multiphysics model for optimizing external pressure in a solid-state battery.
Effect of Mechanical Field Coupling on Thermal Response
Conversely, we examine the impact of adding the mechanical field to a thermo-electrochemical (TC) model, creating the full TCM model. The inclusion of elasticity via $f_{\text{els}}(\xi)$ directly influences the phase-field evolution, making the growing dendrite shorter and thicker due to the mechanical inhibition from the solid electrolyte. This morphological change has direct consequences for dead lithium.
During stripping, the TCM model (with mechanics) shows a longer dissolution cut-off time and a significantly smaller dead lithium area compared to the TC model (without mechanics). The mechanical constraint slows down the surface recession rate of lithium, leading to more thorough and uniform stripping. Consequently, the thermal field’s core temperature rise during stripping is also lower in the TCM model because less heat is generated from the Joule effect and reaction in the smaller, more constrained dead lithium region.
Effect of Ambient Temperature under Mechanical Coupling
With the mechanical field active, we vary the ambient temperature $T_0$ to study its isolated effect. Increasing temperature (e.g., to 353 K) accelerates ion diffusion and interface kinetics. This leads to a more uniform lithium deposition initially and a more complete stripping process later. As a result, the dead lithium area decreases. Decreasing temperature (e.g., to 273 K) has the opposite effect, increasing the dead lithium area. However, comparing the percentage reduction in dead lithium area when temperature increases, the reduction is less pronounced in the TCM model than it would be in a model without mechanics. This indicates that the mechanical field stabilizes the system, making the solid-state battery‘s dead lithium formation less sensitive to temperature fluctuations—a valuable insight for operational management.
The stress evolution during stripping at different temperatures reveals a common feature: as stripping proceeds, stress concentrates at the dendrite root. This root stress buildup progressively slows down the dissolution rate at that critical point, often leaving behind the root as part of the dead lithium. This is a key failure mechanism highlighted by the model.
Results and Discussion: Influence of Electrochemical Parameters
Beyond external fields, the intrinsic electrochemical parameters within the phase-field model itself play a decisive role in dead lithium formation. We systematically vary three key parameters from their baseline values.
Diffusion Coefficient ($D_{\text{eff}}$)
The effective diffusion coefficient governs lithium ion transport in the electrolyte. We modify the pre-exponential factor $A$ to change $D_{\text{eff}}$ uniformly. A lower $D_{\text{eff}}$ leads to a steeper concentration gradient at the interface, promoting the growth of more, finer secondary branches and a narrower primary trunk. During stripping, these fragile secondary branches dissolve quickly, and the narrow trunk easily necks off, but because the total amount of deposited lithium is less, the final dead lithium area is reduced. A higher $D_{\text{eff}}$ enables faster ion supply, leading to a taller, smoother dendrite with fewer secondaries and a wider base. While it strips more slowly, the larger initial deposit and the robust root can lead to a larger residual dead lithium area if necking occurs.
| Diffusion Coefficient Change | Dendrite Growth Morphology | Dead Lithium Area (µm²) | Change from Baseline |
|---|---|---|---|
| Increased ($\uparrow D$) | Taller, smoother, wider root. | 0.090 | +55.2% |
| Baseline ($D_0$) | Moderate branching. | 0.058 | 0% |
| Decreased ($\downarrow D$) | More branched, narrower trunk. | 0.044 | -24.1% |
Interfacial Mobility ($L_{\sigma}$)
The interfacial mobility controls how fast the phase boundary moves in response to a thermodynamic driving force. A higher $L_{\sigma}$ (e.g., $1 \times 10^{-5}$ m³ J⁻¹ s⁻¹) results in a very smooth, compact dendrite with minimal branching, as the interface quickly relaxes to minimize surface energy. During stripping, this compact structure dissolves uniformly as a whole, leaving virtually no isolated fragments; dead lithium area is minimized. A lower $L_{\sigma}$ (e.g., $1 \times 10^{-7}$ m³ J⁻¹ s⁻¹) leads to a highly branched, fractal-like dendrite because the interface cannot smooth out quickly. While this structure has a large surface area and might seem prone to creating dead lithium, its stripping process is very slow, allowing most branches to dissolve completely from the tip backwards, also resulting in a low dead lithium area, though the process takes much longer.
$$ \text{Dead Lithium Area} \propto \frac{1}{\sqrt{L_{\sigma}}} \quad \text{(for extreme values)} $$
This suggests that both very high and very low interfacial mobility can be beneficial for reducing dead lithium, albeit through different morphological and kinetic pathways. Optimizing this parameter is crucial for the interfacial design of a solid-state battery.
Anisotropic Strength ($\delta$)
The anisotropic strength dictates the preference for growth along certain crystallographic directions. A high $\delta$ (e.g., 0.15) leads to strong directional growth, producing dendrites with long primary trunks and pronounced, rapidly growing secondary arms. A low $\delta$ (e.g., 0.05) weakens this preference, resulting in a more isotropic, mossy growth with less distinction between primary and secondary branches. An extremely low $\delta$ (≈ 0.001) leads to essentially isotropic, globular growth.
Regarding dead lithium, reducing $\delta$ from the baseline (0.1) to 0.05 decreases the dead lithium area. The more isotropic morphology, while potentially depositing more lithium, appears to strip more uniformly without creating pronounced weak points for necking. Increasing $\delta$ to 0.15 also slightly reduces the dead lithium area compared to the baseline, despite creating more vulnerable slender branches. This is because the primary trunk becomes more dominant, and the root, though narrow, may be more strongly connected until the very end of stripping. However, this high-anisotropy structure is likely more dangerous in terms of short-circuit risk during growth.
| Parameter | Strategy to Reduce Dead Lithium | Mechanism | Potential Trade-off |
|---|---|---|---|
| Diffusion Coefficient ($D$) | Decrease $D$ (within operable limits). | Reduces total deposition, promotes finer structures that fully dissolve. | Increases cell polarization, lowers power density. |
| Interfacial Mobility ($L_{\sigma}$) | Increase $L_{\sigma}$ significantly. | Promotes smooth, compact deposition that strips uniformly. | May require specific interface engineering. |
| Anisotropic Strength ($\delta$) | Decrease $\delta$. | Promotes isotropic, mossy growth that strips more uniformly. | Mossy growth may reduce Coulombic efficiency and fill porosity. |
Conclusions
This work presents a comprehensive multiphysics phase-field study on dead lithium formation in solid-state batteries, coupling mechanical, thermal, and electrochemical fields. The primary conclusions are:
- Multiphysics Coupling is Essential: The inclusion or exclusion of the thermal or mechanical field significantly alters the stress distribution, dissolution kinetics, and final dead lithium morphology. Models lacking these couplings may not capture key degradation mechanisms in solid-state batteries.
- External Pressure has a Non-linear Effect: Under thermo-chemo-mechanical coupling, the relationship between external pressure and dead lithium area is non-monotonic. Low (5 MPa) and high (20 MPa) pressures can reduce dead lithium, while a medium pressure (10 MPa) may increase it due to promoted root fracture, highlighting the need for precise pressure optimization.
- Mechanical Field Stabilizes Against Temperature: Coupling the mechanical field makes the system less sensitive to ambient temperature variations regarding dead lithium formation, as it constrains morphology and slows stripping, leading to more complete dissolution.
- Electrochemical Parameters are Powerful Levers: Intrinsic model parameters offer direct pathways to influence dead lithium:
- Reducing the diffusion coefficient decreases dead lithium area by limiting total deposition.
- Increasing the interfacial mobility is highly effective, promoting smooth deposits that strip uniformly.
- Reducing the anisotropic strength favors isotropic growth, which also leads to more uniform stripping and less dead lithium.
These insights, derived from the coupled multiphysics model, provide guidelines for the holistic design of solid-state batteries—from optimizing stack pressure and operating temperature to engineering electrolyte properties and electrode interfaces to intrinsically suppress dead lithium and enhance cycle life.