I have conducted a comprehensive phase-field investigation of dead lithium formation in solid-state batteries, focusing on the coupling of mechanical, thermal, and electrochemical fields. The accumulation of dead lithium in solid-state batteries is a critical degradation mechanism that limits cycle life and safety. Unlike conventional liquid-electrolyte systems, solid-state batteries employ solid electrolytes with high mechanical strength, which can suppress lithium dendrite growth but also introduce complex stress and thermal interactions. In this work, I develop a multiphysics phase-field model that integrates heat transfer and mechanics into the electrochemical phase-field framework. I systematically analyze how the presence of thermal and mechanical fields alters dendrite dissolution, dead lithium area, and dissolution cut-off time. Furthermore, I explore the effects of external pressure, ambient temperature, diffusion coefficient, interfacial mobility, and anisotropic strength. The results provide quantitative insights into mitigating dead lithium in solid-state batteries.
Solid-state batteries have attracted significant attention due to their high energy density, improved safety, and thermal stability. However, the formation of lithium dendrites and subsequent dead lithium remains a major challenge. Dead lithium refers to metallic lithium that loses electrical contact with the electrode during stripping, typically due to uneven or incomplete dissolution. In solid-state batteries, the accumulation of dead lithium reduces capacity and can eventually lead to short circuits. Therefore, understanding the multiphysics coupling effects on dead lithium formation is essential for optimizing solid-state battery design.
1. Governing Equations and Multiphysics Coupling
I begin by formulating the Gibbs free energy of the system, which includes gradient energy, chemical free energy, electrostatic energy, and elastic energy. The total Gibbs free energy is expressed as:
$$ G = \int_V \left[ f_{\text{grad}}(\xi) + f_{\text{ch}}(\xi, c_i) + f_{\text{elec}}(\xi, c_i, \phi) + f_{\text{els}}(\xi) \right] dV $$
where \(\xi\) is the phase-field order parameter, \(c_i\) represents concentrations of lithium atoms, lithium ions, and anions, and \(\phi\) is the electrostatic potential. The gradient energy density is given by:
$$ f_{\text{grad}}(\xi) = \frac{1}{2} \kappa \nabla^2 \xi $$
with \(\kappa = \kappa_0 [1 + \delta \cos(\omega \theta)]\), where \(\kappa_0\) is the gradient energy coefficient, \(\delta\) is the anisotropic strength, \(\omega\) is the anisotropy mode, and \(\theta\) is the angle between the interface normal and the coordinate axis. The chemical free energy density is:
$$ f_{\text{ch}}(\xi, c_i) = g(\xi) + c_0 R T_0 \sum_i \frac{c_i}{c_0} \ln \frac{c_i}{c_0} $$
where \(g(\xi) = W \xi^2 (1 – \xi)^2\) is a double-well potential, \(W\) is the barrier height, \(c_0\) is the standard lithium volume concentration, \(R\) is the gas constant, and \(T_0\) is the ambient temperature. The electrostatic energy density is:
$$ f_{\text{elec}}(\xi, c_i, \phi) = \sum_i F c_i z_i \phi $$
and the elastic energy density is:
$$ f_{\text{els}}(\xi) = \frac{1}{2} C_{ijkl} \varepsilon_{ij}^E \varepsilon_{kl}^E $$
where \(C_{ijkl}\) is the elastic tensor, computed from the Young’s modulus and Poisson’s ratio of the electrode and electrolyte phases. The elastic strain tensors are:
$$ \varepsilon_{ij}^E = \varepsilon_{ij}^T \lambda_i h(\xi) \delta_{ij}, \quad \varepsilon_{kl}^E = \varepsilon_{kl}^T \lambda_i h(\xi) \delta_{kl} $$
with \(\varepsilon_{ij}^T\) being the total strain, \(\lambda_i\) the Vegard strain coefficient, and \(h(\xi) = \xi^3 (6\xi^2 – 15\xi + 10)\) the interpolation function.
The temporal evolution of the phase-field parameter \(\xi\) is governed by:
$$ \frac{\partial \xi}{\partial t} = – L_\sigma \left( f’_{\text{ch}}(\xi) + f’_{\text{grad}}(\xi) + f’_{\text{els}}(\xi) \right) – L_\eta h'(\xi) \left\{ \exp\left[ \frac{(1-\alpha)nF\eta_\alpha}{RT_0} \right] – \frac{c_{\text{Li}^+}}{c_0} \exp\left[ -\frac{\alpha nF\eta_\alpha}{RT_0} \right] \right\} $$
where \(L_\sigma\) is the interfacial mobility, \(L_\eta\) is the electrochemical reaction constant, \(\alpha\) is the symmetry factor, \(n\) is the charge transfer number, and \(\eta_\alpha\) is the overpotential. Lithium ion diffusion in the solid electrolyte follows Fick’s law with an effective diffusion coefficient \(D_{\text{eff}}\):
$$ \frac{\partial c_{\text{Li}^+}}{\partial t} = \nabla \cdot \left[ D_{\text{eff}} \nabla c_{\text{Li}^+} + \frac{D_{\text{eff}} c_{\text{Li}^+}}{R T_0} nF \nabla \phi \right] – \psi \frac{d\xi}{dt} $$
where \(\psi\) is a reaction rate constant. When the heat transfer model is coupled, the effective diffusion coefficient becomes temperature-dependent:
$$ D_{\text{eff}} = A \exp\left[ -r c_{\text{Li}^+} + \frac{E_\alpha}{R} \left( \frac{1}{T} – \frac{1}{T_0} \right) \right] $$
with \(A\), \(r\), and \(E_\alpha\) being the pre-exponential factor, fitting factor, and activation energy, respectively. The boundary flux accounts for heat convection and radiation. The electric potential evolution is described by the Poisson equation:
$$ \nabla \cdot \left( \sigma_{\text{eff}} \nabla \phi \right) = F C_s \frac{\partial \xi}{\partial t} $$
where \(\sigma_{\text{eff}} = \sigma_e h(\xi) + \sigma_s [1 – h(\xi)]\) is the effective conductivity, and \(C_s\) is the solid-phase lithium concentration.
Table 1 summarizes the key parameters used in my simulations. I employed a rectangular domain of \(8\,\mu\text{m} \times 8\,\mu\text{m}\) with a lithium metal anode and initial nucleation site at the bottom boundary. The upper boundary was set to an initial voltage of 0.1 V and a fixed lithium ion concentration. The simulation time was 90 s with a time step of 1 s. The maximum mesh element size was 0.05 \(\mu\)m.
| Parameter | Symbol | Value |
|---|---|---|
| Gradient energy coefficient | \(\kappa_0\) | \(1 \times 10^{-10}\) J·m\(^{-1}\) |
| Anisotropic strength | \(\delta\) | 0.1 |
| Anisotropy mode | \(\omega\) | 4 |
| Barrier height | \(W\) | \(3.75 \times 10^5\) J·m\(^{-3}\) |
| Standard volume concentration | \(c_0\) | \(1 \times 10^3\) mol·m\(^{-3}\) |
| Ambient 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 coefficients | \(\lambda_i\) | \(-0.866 \times 10^{-3}\), \(-0.773 \times 10^{-3}\), \(-0.529 \times 10^{-3}\) |
| Interfacial mobility | \(L_\sigma\) | \(1 \times 10^{-6}\) m\(^3\)·J\(^{-1}\)·s\(^{-1}\) |
| Reaction constant | \(L_\eta\) | 0.5 s\(^{-1}\) |
| Symmetry factor | \(\alpha\) | 0.5 |
| Solid-phase lithium concentration | \(C_s\) | \(7.64 \times 10^4\) mol·m\(^{-3}\) |
| Electrode conductivity | \(\sigma_e\) | \(1 \times 10^7\) S·m\(^{-1}\) |
| Electrolyte conductivity | \(\sigma_s\) | 0.1 S·m\(^{-1}\) |
| Electrode specific heat capacity | \(c_{pe}\) | 1200 J·kg\(^{-1}\)·K\(^{-1}\) |
| Electrolyte specific heat capacity | \(c_{ps}\) | 133 J·kg\(^{-1}\)·K\(^{-1}\) |
| Electrode thermal conductivity | \(\lambda_e\) | 1.04 W·m\(^{-1}\)·K\(^{-1}\) |
| Electrolyte thermal conductivity | \(\lambda_s\) | 0.45 W·m\(^{-1}\)·K\(^{-1}\) |
| Convective heat transfer coefficient | \(h\) | 10 W·m\(^{-2}\)·K\(^{-1}\) |
2. Effect of Heat Transfer Coupling on Dead Lithium
I first examined the influence of coupling a heat transfer model into the mechanical-electrochemical phase-field framework. The comparison was between a mechanical-electrochemical model and a mechanical-thermal-electrochemical model. In both cases, the lithium dendrite morphology after 90 s of growth showed minor differences. However, the von Mises stress distribution changed significantly. In the coupled heat transfer model, the stress concentration at the dendrite root was reduced, leading to a different dissolution behavior.
I simulated the dissolution process by applying a voltage of 0.01 V at the bottom boundary. The driving force for phase-field evolution during dissolution is modified by an activity factor \(f_d = f_{\text{step}}(-\phi_s / \phi_b)\), where \(\phi_s\) is the electrolyte potential and \(\phi_b\) is a reference potential. When \(f_d = 1\), the lithium is active; when \(f_d = 0\), it becomes dead lithium. Table 2 summarizes the dissolution cut-off times and dead lithium areas for the two models.
| Model | Dissolution cut-off time (s) | Dead lithium area (\(\mu\)m\(^2\)) |
|---|---|---|
| Mechanical-electrochemical | 49 | 0.036 |
| Mechanical-thermal-electrochemical | 43 | 0.058 |
The coupled heat transfer model resulted in a shorter dissolution cut-off time and a larger dead lithium area. This is attributed to the reduced stress concentration at the root, which allowed a larger root area to form during growth. At 10 s of dissolution, the mechanical-electrochemical model showed a clear tendency for the main trunk to detach from the root, while the coupled model exhibited less detachment. The temperature field thus alters the stress distribution and the subsequent dead lithium formation in solid-state batteries.
I then investigated the effect of external pressure when the heat transfer model is coupled. Applying a pressure of 5 MPa at the top boundary shifted the stress concentration to the middle of the main trunk. Table 3 compares the dead lithium areas under 5 MPa for the two models.
| Model (5 MPa) | Root von Mises stress (MPa) | Dissolution cut-off time (s) | Dead lithium area (\(\mu\)m\(^2\)) |
|---|---|---|---|
| Mechanical-electrochemical | 20.5 | 32 | 0.097 |
| Mechanical-thermal-electrochemical | 14.3 | 40 | 0.053 |
With external pressure, the coupled heat transfer model reduced the dead lithium area by 45.3%. I further varied the external pressure to 5, 10, and 20 MPa in the multiphysics model. Table 4 presents the results.
| External pressure (MPa) | Dissolution cut-off time (s) | Dead lithium area (\(\mu\)m\(^2\)) |
|---|---|---|
| 5 | 40 | 0.053 |
| 10 | 32 | 0.073 |
| 20 | 30 | 0.059 |
As the external pressure increases, the dendrite height decreases and the stress difference between the main trunk and side branches increases. The dead lithium area first decreases slightly at low pressure, then increases at moderate pressure due to stress concentration and fracture, and finally decreases again at high pressure because the dendrite becomes denser and dissolution is more complete. This non-monotonic behavior highlights the complex interplay between mechanical stress and thermal effects in solid-state batteries.
3. Effect of Mechanical Field Coupling on Dead Lithium
I next examined the influence of coupling a mechanical field into the thermal-electrochemical model. The elastic free energy \(f_{\text{els}}\) directly participates in the Gibbs free energy, affecting the phase-field evolution. The high mechanical strength of the solid electrolyte becomes apparent, making the lithium dendrite shorter and thicker. Table 5 compares the thermal-electrochemical and mechanical-thermal-electrochemical models at ambient temperature.
| Model | Dissolution cut-off time (s) | Dead lithium area (\(\mu\)m\(^2\)) | Core temperature (K) |
|---|---|---|---|
| Thermal-electrochemical | 28 | 0.098 | 301.8 |
| Mechanical-thermal-electrochemical | 43 | 0.058 | 299.5 |
The mechanical field reduces the dead lithium area by 40.8% and lowers the core temperature. This is because the mechanical field slows the lithium stripping rate, allowing more complete dissolution. The temperature distribution also changes due to the reduced dead lithium area and less heat concentration.
I then varied the ambient temperature to study its effect on dead lithium with and without the mechanical field. At 353 K, the uncoupled mechanical field model had a dissolution cut-off time of 47 s and a dead lithium area of 0.064 \(\mu\)m\(^2\), while the coupled model had 67 s and 0.041 \(\mu\)m\(^2\), respectively. Compared to the baseline at 293 K, the dead lithium area decreased by 34.6% (uncoupled) and 29.3% (coupled). At 273 K with the mechanical field, the dissolution cut-off time was 33 s and the dead lithium area was 0.085 \(\mu\)m\(^2\). Table 6 summarizes the temperature effects.
| Ambient temperature (K) | Mechanical field | Dissolution cut-off time (s) | Dead lithium area (\(\mu\)m\(^2\)) |
|---|---|---|---|
| 293 | No | 28 | 0.098 |
| 293 | Yes | 43 | 0.058 |
| 353 | No | 47 | 0.064 |
| 353 | Yes | 67 | 0.041 |
| 273 | Yes | 33 | 0.085 |
Higher temperatures promote more uniform lithium deposition and faster diffusion, reducing dead lithium. The mechanical field stabilizes the dendrite structure, making it less sensitive to temperature changes. During dissolution, stress accumulates at the root because the nucleation site is fixed, causing a decreasing stripping rate. For example, at 353 K with the mechanical field, the root dissolution area was 47.9% in the first 10 s, 16.2% from 10 to 30 s, and only 11.5% from 30 to 67 s. This non-uniform stripping is a key factor in dead lithium formation in solid-state batteries.
4. Effect of Electrochemical Parameters on Dead Lithium
4.1 Diffusion Coefficient
I investigated the effect of the diffusion coefficient on dendrite growth and dead lithium. When the diffusion coefficient is increased, lithium ions aggregate more readily, accelerating dendrite growth and producing taller trunks and longer side branches. When decreased, secondary dendrites become more numerous, and the root becomes narrower. Table 7 shows the dissolution results for different diffusion coefficients.
| Diffusion coefficient | Dissolution cut-off time (s) | Dead lithium area (\(\mu\)m\(^2\)) |
|---|---|---|
| Increased | 51 | 0.090 |
| Decreased | 33 | 0.044 |
A decreased diffusion coefficient leads to a smaller lithium deposition amount and a smaller dead lithium area, despite a shorter dissolution cut-off time. This suggests that reducing the diffusion coefficient can effectively mitigate dead lithium in solid-state batteries.
4.2 Interfacial Mobility
Interfacial mobility controls the evolution rate of the lithium/electrolyte interface. I simulated two extreme values: \(1 \times 10^{-5}\) and \(1 \times 10^{-7}\) m\(^3\)/(J·s). The baseline was \(1 \times 10^{-6}\) m\(^3\)/(J·s). Table 8 summarizes the results.
| Interfacial mobility (m\(^3\)/(J·s)) | Dissolution cut-off time (s) | Dead lithium area (\(\mu\)m\(^2\)) |
|---|---|---|
| \(1 \times 10^{-5}\) | 68 | 0.004 |
| \(1 \times 10^{-6}\) (baseline) | 43 | 0.058 |
| \(1 \times 10^{-7}\) | 84 | 0.024 |
Both increasing and decreasing the interfacial mobility reduce the dead lithium area compared to the baseline. Higher mobility promotes uniform lithium deposition and fewer secondary dendrites, leading to整体 detachment without isolated lithium. Lower mobility slows stripping, allowing more complete dissolution. Thus, tuning interfacial mobility is a viable strategy for suppressing dead lithium in solid-state batteries.
4.3 Anisotropic Strength
Anisotropic strength determines the directional growth tendency of lithium dendrites. I simulated values of 0.15, 0.05, and 0.001. At 0.001, the growth rates of the main trunk and side branches are similar, resulting in a more isotropic morphology. As the anisotropic strength increases, the main trunk grows faster and secondary dendrites proliferate. Table 9 compares the dead lithium areas for 0.15 and 0.05.
| Anisotropic strength | Dissolution cut-off time (s) | Dead lithium area (\(\mu\)m\(^2\)) |
|---|---|---|
| 0.15 | 41 | 0.056 |
| 0.05 | 45 | 0.051 |
| 0.10 (baseline) | 43 | 0.058 |
Increasing the anisotropic strength from 0.1 to 0.15 reduces the dead lithium area by 3.4%, while decreasing it to 0.05 reduces the area by 12.0%. A lower anisotropic strength leads to a more uniform dissolution and less residual lithium at the root. Therefore, reducing anisotropic strength is a more effective approach to minimize dead lithium in solid-state batteries.
5. Discussion and Summary of Findings
My multiphysics phase-field simulations reveal that the thermal and mechanical fields significantly affect dead lithium formation in solid-state batteries. The key findings are summarized in Table 10.
| Factor | Effect on dead lithium area | Mechanism |
|---|---|---|
| Coupling heat transfer | Increases by 61.1% (no pressure) | Shorter dissolution time, larger root area |
| External pressure 5 MPa with heat transfer | Decreases by 45.3% | Reduced root stress concentration |
| External pressure 10 MPa | Increases moderately | Stress concentration and fracture |
| External pressure 20 MPa | Decreases again | Denser dendrite, slower stripping |
| Coupling mechanical field | Decreases by 40.8% | Slower stripping, more complete dissolution |
| Higher temperature (353 K) | Decreases | Faster diffusion, uniform deposition |
| Lower temperature (273 K) | Increases | Slower diffusion, uneven stripping |
| Increased diffusion coefficient | Increases by 55.2% | More lithium deposition |
| Decreased diffusion coefficient | Decreases by 24.1% | Less lithium deposition |
| Increased interfacial mobility | Decreases significantly | Uniform deposition, fewer isolated lithium |
| Decreased interfacial mobility | Decreases significantly | Slower stripping, more complete dissolution |
| Increased anisotropic strength (0.15) | Decreases by 3.4% | Thinner trunk, narrower root |
| Decreased anisotropic strength (0.05) | Decreases by 12.0% | More uniform dissolution |
The coupling of heat transfer and mechanics introduces complex interactions. The heat transfer model increases dead lithium without external pressure but reduces it under low external pressure. The mechanical field generally reduces dead lithium by stabilizing the dendrite and slowing dissolution. Temperature elevation reduces dead lithium, but excessive temperature may cause interfacial side reactions that alter the solid electrolyte interphase and increase dead lithium. My model does not account for such side reactions, which is a limitation.
Electrochemical parameters offer additional levers. Reducing the diffusion coefficient, increasing interfacial mobility, and reducing anisotropic strength all reduce dead lithium. Among these, interfacial mobility has the most dramatic effect, reducing the dead lithium area to 0.004 \(\mu\)m\(^2\) at high mobility. However, extremely low interfacial mobility can increase the deposited area during growth, which may be undesirable. Thus, a balanced optimization is needed.
In solid-state batteries, the mechanical strength of the electrolyte is a double-edged sword. It suppresses dendrite growth but can also cause stress concentration and fracture, leading to dead lithium. My simulations show that applying moderate external pressure (e.g., 5 MPa) can reduce dead lithium, but high pressure (10 MPa) may increase it due to fracture. Very high pressure (20 MPa) again reduces dead lithium by densifying the dendrite. This non-monotonic trend suggests an optimal pressure window.
I also observed that the dissolution cut-off time is not always correlated with dead lithium area. For example, a shorter dissolution cut-off time can still yield a smaller dead lithium area if the initial lithium deposition is smaller. This highlights the importance of considering both growth and dissolution stages in solid-state battery modeling.
6. Conclusion
I have developed a multiphysics phase-field model that couples mechanical, thermal, and electrochemical fields to study dead lithium formation in solid-state batteries. The main conclusions are:
1. Coupling a heat transfer model into the mechanical-electrochemical framework changes the stress distribution and increases the dead lithium area by 61.1% without external pressure. Under 5 MPa external pressure, the coupled model reduces the dead lithium area by 45.3%.
2. The dead lithium area exhibits a non-monotonic trend with increasing external pressure: it decreases at low pressure, increases at moderate pressure, and decreases again at high pressure.
3. Coupling a mechanical field into the thermal-electrochemical model reduces the dead lithium area by 40.8% and lowers the core temperature. The mechanical field also makes the dendrite less sensitive to temperature changes.
4. Higher ambient temperatures reduce dead lithium, while lower temperatures increase it. However, the root dissolution rate decreases significantly over time due to the fixed nucleation site.
5. Reducing the diffusion coefficient decreases the dead lithium area by 24.1%, while increasing it increases the area by 55.2%.
6. Both increasing and decreasing the interfacial mobility reduce the dead lithium area, but extremely low mobility can increase the deposited area during growth.
7. Reducing the anisotropic strength from 0.1 to 0.05 reduces the dead lithium area by 12.0%, while increasing it to 0.15 reduces the area by 3.4%.
These findings provide quantitative guidelines for suppressing dead lithium in solid-state batteries through multiphysics design. Future work should incorporate interfacial side reactions and varying cycle numbers to further improve the predictive capability of the model.
In summary, my phase-field study demonstrates that the interplay of thermal, mechanical, and electrochemical fields is crucial for understanding and mitigating dead lithium in solid-state batteries. The results highlight the potential of optimizing external pressure, temperature, and electrochemical parameters to enhance the cycle life and safety of solid-state batteries.