The evolution of energy storage technology is increasingly focused on achieving higher safety and energy density. In this context, the solid-state battery has emerged as a pivotal next-generation candidate, promising a significant leap over conventional liquid electrolyte systems. Its core advantage lies in replacing the flammable organic liquid electrolyte with a solid-state electrolyte (SSE), which inherently offers superior thermal stability and mechanical robustness. This fundamental shift not only mitigates severe safety hazards like leakage and thermal runaway but also enables the theoretical use of high-capacity lithium metal anodes, paving the way for dramatically increased energy density. Consequently, research into solid-state battery systems has intensified, aiming to overcome the remaining technical hurdles that impede their widespread commercialization.
Among the most critical challenges is the formation and growth of lithium dendrites. Even within the constrained environment of a solid-state battery, lithium tends to deposit non-uniformly during charging, forming needle-like or mossy metallic protrusions. These dendrites pose a dual threat: firstly, they can mechanically penetrate the solid electrolyte separator, leading to an internal short circuit and catastrophic failure; secondly, they are intrinsically unstable during the subsequent discharge (stripping) process. Incomplete stripping of these metallic structures leads to the formation of isolated, electrochemically inactive lithium fragments, commonly termed “dead lithium.” These fragments become electronically disconnected from the anode current collector and can no longer participate in the charge-discharge cycle. Over repeated cycling, the continuous accumulation of dead lithium constitutes a primary mechanism for irreversible capacity loss and rapid cell degradation, severely limiting the cycle life of the solid-state battery.
Understanding and mitigating dead lithium formation is thus paramount for the development of durable solid-state batteries. While experimental techniques provide crucial insights, they often struggle to capture the intricate, coupled processes occurring at the Li/SSE interface at the micro- and meso-scales. Computational modeling, particularly the phase-field method, has proven to be an indispensable tool for elucidating these complex phenomena. The phase-field method elegantly describes the evolution of microstructures by using a continuous order parameter field, eliminating the need to track sharp interfaces explicitly. This makes it exceptionally suitable for simulating the morphologically complex processes of lithium deposition, dendritic growth, and dissolution in a solid-state battery.
Previous phase-field studies have significantly advanced our understanding, often by coupling electrochemical kinetics with a secondary physical field. Models incorporating mechanics (electro-chemo-mechanical) have highlighted the critical role of stress, showing how stress concentrations at dendrite roots can accelerate local stripping and lead to fracture and isolation. Similarly, models coupling electrochemistry with heat transfer (thermo-electrochemical) have demonstrated that temperature gradients influence ion transport and reaction kinetics, thereby affecting deposition uniformity and dead lithium formation. However, the real operating environment of a solid-state battery is a symphony of simultaneously interacting forces: electrochemical driving forces, significant mechanical stresses (from stack pressure and Li plating/swelling), and non-isothermal conditions (due to joule heating and reaction enthalpies). A holistic simulation framework that integrates force, thermal, and electrochemical (FTE) couplings within a unified phase-field model is therefore essential for a more accurate and predictive understanding of dead lithium evolution in a solid-state battery.
Furthermore, while external operational parameters like temperature, pressure, and voltage have been explored, a systematic investigation into the influence of intrinsic material parameters within the phase-field model—such as lithium ion diffusivity, interfacial kinetic coefficient, and crystallographic anisotropy—on dead lithium formation remains less charted. These parameters are directly linked to the properties of the solid electrolyte and the lithium metal, and their optimization is key to designing better materials for a solid-state battery.
This article presents a comprehensive phase-field study focused on dead lithium formation during the stripping process in a model solid-state battery. The core of our work is the development and implementation of a multiphysics phase-field model that fully couples mechanical deformation, heat transfer, and electrochemical reactions. Using this FTE-coupled framework, we systematically investigate: 1) The individual and combined effects of incorporating thermal and mechanical fields on dendrite dissolution kinetics and final dead lithium morphology. 2) The impact of varying external operational conditions, specifically temperature and applied stack pressure, within the multiphysics context. 3) The influence of key electrochemical and materials parameters, including diffusion coefficient, interfacial mobility, and anisotropic strength, on the propensity for dead lithium formation. Through this multifaceted analysis, we aim to provide deeper mechanistic insights and quantitative guidance for strategies to suppress dead lithium and enhance the longevity of the solid-state battery.
Theoretical Framework and Numerical Implementation
The phase-field model describes the system using a set of continuum field variables. The primary order parameter, \(\phi\), distinguishes the lithium metal phase (\(\phi = 1\)) from the solid-state electrolyte phase (\(\phi = 0\)), with the interface represented by a smooth variation between these values. The concentration fields for lithium ions (\(c_{Li^+}\)) and anions (\(c_{A^-}\)) in the electrolyte, and the electrostatic potential (\(\psi\)) are also defined across the domain. The total Gibbs free energy of the system, \(G\), is formulated as the integral of energy density contributions over the volume \(V\):
$$
G = \int_V \left[ f_{\text{grad}}(\phi) + f_{\text{chem}}(\phi, c_i) + f_{\text{elec}}(\phi, c_i, \psi) + f_{\text{elast}}(\phi) \right] dV
$$
Each term captures a specific physics essential for the solid-state battery environment:
1. Gradient Energy (\(f_{\text{grad}}\)): This term penalizes the formation of sharp interfaces, controlling the interface energy and width. It incorporates crystallographic anisotropy, which dictates preferred growth directions of lithium dendrites.
$$
f_{\text{grad}}(\phi) = \frac{1}{2} \kappa(\theta) |\nabla \phi|^2, \quad \text{with} \quad \kappa(\theta) = \kappa_0 [1 + \delta \cos(m\theta)]
$$
where \(\kappa_0\) is the gradient energy coefficient, \(\delta\) is the anisotropic strength, \(m\) is the symmetry mode number (e.g., 4 for cubic symmetry), and \(\theta\) is the angle between the interface normal and a reference axis.
2. Chemical Free Energy (\(f_{\text{chem}}\)): This term describes the bulk thermodynamic energies of the phases and the mixing entropy of the ionic species. A double-well potential \(g(\phi)\) stabilizes the pure phases.
$$
f_{\text{chem}}(\phi, c_i) = g(\phi) + c_0 R T_0 \sum_i \frac{c_i}{c_0} \ln\left(\frac{c_i}{c_0}\right), \quad g(\phi) = W \phi^2 (1-\phi)^2
$$
Here, \(W\) is the height of the energy barrier, \(c_0\) is a reference concentration, \(R\) is the gas constant, and \(T_0\) is the reference temperature.
3. Electrostatic Energy (\(f_{\text{elec}}\)): This term accounts for the energy of ions in the local electric field.
$$
f_{\text{elec}}(\phi, c_i, \psi) = F \psi \sum_i z_i c_i
$$
where \(F\) is Faraday’s constant and \(z_i\) is the charge number of species \(i\).
4. Elastic Strain Energy (\(f_{\text{elast}}\)): Critically important for the solid-state battery, this term captures the mechanical energy due to elastic deformation. Lithium deposition induces large volumetric strain (Vegard strain). The stress field evolves with the phase field and couples back to the chemical potential, influencing phase transformation kinetics.
$$
f_{\text{elast}}(\phi) = \frac{1}{2} \mathbf{C}_{ijkl}(\phi) \varepsilon_{ij}^{el} \varepsilon_{kl}^{el}
$$
The elastic stiffness tensor \(\mathbf{C}(\phi)\) and the eigenstrain \(\varepsilon^{eig}\) are interpolated between the Li metal and SSE properties using the phase field. The elastic strain is \(\varepsilon^{el} = \varepsilon^{total} – \varepsilon^{eig}(\phi)\).
Kinetic Equations and Multiphysics Couplings
The temporal evolution of the system is governed by a set of coupled partial differential equations derived from variational principles.
Phase Field Evolution: The dynamics of the Li/SSE interface is described by a modified Allen-Cahn equation, where the driving force includes contributions from chemical, gradient, elastic, and electrochemical overpotential terms.
$$
\frac{\partial \phi}{\partial t} = -M_{\phi} \left( \frac{\delta f_{\text{chem}}}{\delta \phi} + \frac{\delta f_{\text{grad}}}{\delta \phi} + \frac{\delta f_{\text{elast}}}{\delta \phi} \right) – L_{\eta} h'(\phi) \left\{ \exp\left[\frac{(1-\alpha)F \eta}{RT}\right] – \frac{c_{Li^+}}{c_{Li^+}^0} \exp\left[-\frac{\alpha F \eta}{RT}\right] \right\}
$$
Here, \(M_{\phi}\) is the interfacial mobility, \(L_{\eta}\) is the electrochemical reaction coefficient, \(\alpha\) is the charge transfer coefficient, \(\eta\) is the local overpotential, and \(h(\phi)=\phi^3(10-15\phi+6\phi^2)\) is an interpolation function. A switching function \(f_d = f_{step}(-\psi/\psi_b)\) is introduced to deactivate the reaction term for regions identified as dead lithium (\(\phi\) islands disconnected from the anode).
Lithium Ion Transport: The conservation of lithium ions in the electrolyte is governed by a modified diffusion equation incorporating drift due to the electric field and a source/sink term from the electrochemical reaction.
$$
\frac{\partial c_{Li^+}}{\partial t} = \nabla \cdot \left[ D_{eff}(\phi, T) \nabla c_{Li^+} + \frac{D_{eff}(\phi, T) F}{RT} z_{Li^+} c_{Li^+} \nabla \psi \right] – \frac{\partial c_{Li, solid}}{\partial t}
$$
Thermal Coupling: The heat transfer equation is solved to obtain the temperature field \(T(\mathbf{x},t)\):
$$
\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (k(\phi) \nabla T) + \dot{q}
$$
The heat source \(\dot{q}\) includes Joule heating and reaction heat. Crucially, temperature affects multiple parameters: the effective diffusivity \(D_{eff}\) follows an Arrhenius law, \(D_{eff} = D_0 \exp(-E_a / RT)\), and the ionic conductivity and reaction kinetics are also temperature-dependent. This creates a two-way thermo-electrochemical coupling.
Mechanical Equilibrium: Assuming quasi-static mechanical equilibrium, the stress field satisfies:
$$
\nabla \cdot \boldsymbol{\sigma} = 0, \quad \text{with} \quad \boldsymbol{\sigma} = \mathbf{C}(\phi) : \varepsilon^{el}
$$
The evolving phase field \(\phi\) changes the eigenstrain distribution, which in turn alters the stress field. The calculated stress influences the chemical potential driving the phase field evolution via the \(\delta f_{\text{elast}}/\delta \phi\) term, establishing a two-way chemo-mechanical coupling.
Electrostatics: The potential distribution is solved using a charge conservation equation:
$$
\nabla \cdot ( \sigma_{eff}(\phi, T) \nabla \psi ) = F \frac{\partial c_{Li, solid}}{\partial t}
$$
where \(\sigma_{eff}\) is the effective electrical conductivity, interpolated between the metallic Li and ionic-conducting SSE.
Simulation Setup and Parameters
The model is implemented using the finite element method. A two-dimensional domain representing a cross-section of the anode | SSE interface is used. A single lithium dendrite nucleus is placed at the bottom boundary (anode). The simulation involves two stages: a growth stage (charging) where a negative overpotential is applied at the anode to drive Li deposition, followed by a dissolution/stripping stage (discharging) where a positive overpotential is applied. The evolution is tracked until the stripping current diminishes, and the remaining isolated \(\phi\) regions are quantified as dead lithium area. Key parameters for the model, representative of a polymer-based solid-state battery system, are summarized in Table 1.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Gradient Energy Coefficient | \(\kappa_0\) | 1.0 × 10-10 | J m-1 |
| Anisotropic Strength | \(\delta\) | 0.1 | – |
| Barrier Height | \(W\) | 3.75 × 105 | J m-3 |
| Interfacial Mobility | \(M_{\phi}\) | 1.0 × 10-6 | m3 J-1 s-1 |
| Reaction Coefficient | \(L_{\eta}\) | 0.5 | s-1 |
| Li+ Diffusivity in SSE | \(D_{SSE}\) | 1.0 × 10-12 | m2 s-1 |
| Young’s Modulus (Li Metal) | \(E_{Li}\) | 7.8 | GPa |
| Young’s Modulus (SSE) | \(E_{SSE}\) | 1.0 | GPa |
| Poisson’s Ratio (Li Metal) | \(\nu_{Li}\) | 0.42 | – |
| Poisson’s Ratio (SSE) | \(\nu_{SSE}\) | 0.30 | – |
| Thermal Conductivity (SSE) | \(k_{SSE}\) | 0.45 | W m-1 K-1 |
| Activation Energy for Diffusion | \(E_a\) | 0.4 | eV |
| Reference Temperature | \(T_0\) | 293 | K |
Results and Discussion: The Multiphysics Influence on Dead Lithium
1. Effect of Coupling the Thermal Field
We first isolate the impact of thermal coupling by comparing simulations from a coupled Force-Electrochemical (FE) model and the full Force-Thermal-Electrochemical (FTE) model. In both cases, mechanical coupling is present. The initial dendrite morphology after the growth stage shows only subtle differences. However, the von Mises stress distribution reveals a significant change: in the FTE model, the stress contrast between the primary dendrite trunk and the side branches is more pronounced.
During the stripping stage, a critical divergence emerges. The dissolution process ceases earlier in the FTE model (at ~43s) compared to the FE model (~49s). Consequently, the final area of dead lithium is larger in the FTE simulation. This can be attributed to two interrelated factors: First, the temperature rise in the FTE model (due to Joule heating) accelerates the local dissolution kinetics, potentially leading to a more rapid “pinch-off” of dendrite arms before they can be fully retracted. Second, the altered stress distribution in the FTE model, particularly a reduced stress concentration at the dendrite root, slows down the dissolution rate at the critical root region. While this root stabilization might be beneficial, the overall shorter process time leads to a larger fraction of incompletely dissolved lithium, primarily in the form of broken side branches. This underscores that the thermo-mechanical interplay in a solid-state battery critically dictates the kinetics of stripping and the final dead lithium morphology, which would be missed in an isothermal mechanical model.
2. Combined Effect of Temperature and External Pressure
The operation of a solid-state battery typically involves applying a stack pressure to maintain intimate contact between the solid layers. We investigated this within our FTE framework. Applying a uniform external pressure of 5 MPa during both growth and stripping stages significantly alters the outcome. In the FE model, the pressure induces high stress concentrations at the dendrite root, making it a preferential site for rapid stripping and fracture, leading to a large increase in dead lithium area. In the FTE model, however, the same external pressure results in a markedly lower dead lithium area.
The explanation lies in the modified stress state due to thermal expansion. The FTE model predicts a more moderate stress build-up at the root. When external pressure is applied, the combined stress does not reach the critical level for premature root fracture as in the isothermal case. Instead, the pressure promotes a denser, more uniform initial Li deposition (during growth) and a more controlled, complete stripping process, ultimately reducing dead lithium. We performed a parameter study on the external pressure (\(P_{ext}\)) within the FTE model. The results, summarized in Table 2, show a non-monotonic trend:
| External Pressure (MPa) | Stripping Cut-off Time (s) | Dead Lithium Area (µm²) | Primary Observation |
|---|---|---|---|
| 0 | 43 | 0.058 | Baseline |
| 5 | 40 | 0.053 | Denser deposit, less fracture. |
| 10 | 32 | 0.073 | Increased root stress, more fracture. |
| 20 | 30 | 0.059 | Highly dense deposit, slow/complete stripping. |
At low pressure (5 MPa), benefits are seen. At medium pressure (10 MPa), increased mechanical driving force for root fracture dominates, raising dead lithium. At high pressure (20 MPa), the morphology is so dense and uniform that stripping, although slower, is very thorough, reducing dead lithium again. This complex, non-linear interplay highlights the necessity of the multiphysics model for optimizing the stack pressure in a practical solid-state battery.
3. Effect of Coupling the Mechanical Field
We next evaluate the importance of mechanical coupling by comparing the FTE model with a simpler Thermal-Electrochemical (TE) model. The absence of mechanical energy in the TE model leads to less constrained dendrite growth, resulting in a larger, more ramified deposit. During stripping, the TE model shows a faster cut-off time (28s vs. 43s in FTE) and a significantly larger dead lithium area. The elastic energy in the FTE model acts as a thermodynamic penalty on the creation of new interface area and influences the local chemical potential. This slows down the stripping kinetics, particularly at vulnerable tips and side branches, allowing for a more synchronous and complete dissolution that reduces the formation of isolated fragments. Furthermore, the temperature field in the FTE model shows a lower peak temperature compared to the TE model for the same electrical input, as the mechanical work done during Li deformation provides an additional energy dissipation pathway. This demonstrates that mechanical coupling is not merely additive but fundamentally alters both the electrochemical and thermal response of the solid-state battery system.
4. Combined Effect of Temperature and Mechanical Coupling
Varying the ambient operating temperature within the FTE model reveals expected but nuanced trends. Increasing the temperature from 293 K to 353 K accelerates ion transport and reaction kinetics. This leads to a more uniform Li deposition during growth and a more efficient stripping process, reducing the final dead lithium area. Conversely, lowering the temperature to 273 K slows dissolution, causing an earlier cut-off and a substantial increase in dead lithium. The key insight from the FTE model is the moderating role of mechanical coupling. The stress field provides a stabilizing influence that reduces the sensitivity of dead lithium formation to temperature swings compared to predictions from a pure TE model. This suggests that a mechanically robust solid electrolyte interface can help maintain performance consistency across a range of operating temperatures in a solid-state battery.
Influence of Electrochemical and Material Parameters
Beyond operational conditions, the intrinsic properties of the materials constituting the solid-state battery are paramount. Our phase-field framework allows us to probe the impact of key parameters embedded in the model.
1. Lithium Ion Diffusivity (\(D_{Li^+}\))
The diffusivity of Li+ in the solid-state electrolyte is a central property governing rate capability. We simulated cases with increased and decreased \(D_{Li^+}\) relative to the baseline. A higher \(D_{Li^+}\) facilitates rapid ion supply to the growing dendrite tip, leading to elongated, faster-growing primary branches with fewer side branches. During stripping, this morphology, while larger, tends to dissolve from the tip inward in a more coordinated manner. However, the larger initial volume often leaves behind a non-negligible dead root. A lower \(D_{Li^+}\) severely limits ion transport, leading to a denser, more branched “mossy” growth with many thin, fragile side branches. These thin branches are highly susceptible to being severed and becoming dead lithium during the stripping process. Quantitatively, decreasing \(D_{Li^+}\) increased the dead lithium area by over 50% in our simulations, highlighting that improving ionic conductivity in the SSE is crucial not just for rate performance but also for cycle life by promoting uniform stripping in a solid-state battery.
2. Interfacial Mobility (\(M_{\phi}\))
The interfacial mobility controls the kinetic rate at which the Li/SSE interface can move in response to a thermodynamic driving force. It is linked to the atomic attachment/detachment kinetics. A high \(M_{\phi}\) allows the interface to quickly adjust to minimize its energy, resulting in a smooth, rounded dendrite morphology with minimal side-branching. This smooth interface strips uniformly, leaving virtually no dead lithium. A low \(M_{\phi}\), on the other hand, implies sluggish interface kinetics. The system cannot smooth out instabilities, leading to a highly unstable, fractal-like dendritic growth. Although the stripping process for such a structure is very slow, the extreme fragility of the numerous fine filaments means that even slight dissolution leads to widespread fragmentation and a high dead lithium area. This parameter underscores the importance of the intrinsic interfacial properties between Li metal and the solid electrolyte in a solid-state battery; a “softer” or more compliant interface (higher mobility) is highly desirable.
3. Crystallographic Anisotropic Strength (\(\delta\))
This parameter governs the degree to which lithium prefers to grow along certain crystallographic directions. With strong anisotropy (\(\delta = 0.15\)), dendrites grow as sharp, needle-like primary trunks with well-defined side branches at specific angles. The focused growth leads to deep penetration. During stripping, these narrow trunks can develop localized “necking” points that easily break, but the total volume of deposited Li is more confined. With weak anisotropy (\(\delta = 0.05\)), growth is more isotropic, resulting in a compact, bush-like structure with less pronounced tips. This structure, while less penetrating, has a broader connection to the substrate and strips more uniformly, though it may retain a broader base. Reducing anisotropy consistently decreased the final dead lithium area in our simulations. This suggests that promoting polycrystalline or amorphous Li deposition, or using SSEs that induce isotropic growth, could be a beneficial strategy for mitigating failure in a solid-state battery. The results for key parameter variations are consolidated in Table 3.
| Parameter (Change from Baseline) | Effect on Dendrite Morphology | Effect on Stripping & Dead Lithium | Implication for Solid-State Battery |
|---|---|---|---|
| Diffusivity \(D_{Li^+}\) (Lower) | Mossy, highly branched, dense. | Fragile branches sever easily; Large increase in dead Li. | High SSE ionic conductivity is critical. |
| Interfacial Mobility \(M_{\phi}\) (Higher) | Smooth, rounded, minimal branching. | Uniform, complete dissolution; Drastic reduction in dead Li. | Fast interface kinetics promote stability. |
| Anisotropic Strength \(\delta\) (Lower) | Compact, bush-like, isotropic. | More uniform stripping from base; Reduction in dead Li. | Isotropic Li growth is preferred. |
Conclusions
In this work, we have developed and applied a comprehensive multiphysics phase-field model that fully couples mechanical stress, heat transfer, and electrochemical processes to investigate the formation of dead lithium during the stripping phase in a solid-state battery. The simulations provide several critical insights that underscore the complexity and interconnectedness of the phenomena governing cycle life.
First, the incorporation of additional physical fields beyond electrochemistry fundamentally alters the predicted behavior. Coupling the thermal field accelerates dissolution kinetics but can increase dead lithium area due to premature pinch-off, an effect modulated by the concomitant change in mechanical stress distribution. Conversely, coupling the mechanical field stabilizes the stripping process, slows down dissolution kinetics, and generally leads to a more complete removal of lithium, thereby reducing dead lithium. This demonstrates that models relying on single or dual couplings may draw incomplete or even misleading conclusions about performance limits of a solid-state battery.
Second, the analysis of external operational parameters within the multiphysics context reveals non-trivial optimal points. For instance, the effect of external stack pressure on dead lithium is non-monotonic, with benefits at low and very high pressures but a detrimental region at intermediate pressures due to stress-driven root fracture. Similarly, while higher temperatures universally promote less dead lithium, the stabilizing effect of mechanical coupling reduces the system’s sensitivity to temperature fluctuations.
Third, the study of intrinsic material parameters offers direct guidance for material design. To minimize dead lithium accumulation in a solid-state battery, the solid electrolyte should ideally possess: (i) high lithium-ion diffusivity to ensure uniform ion supply and prevent mossy growth; (ii) the electrode/electrolyte pair should exhibit high interfacial mobility (fast kinetics) to enable smooth, stable interface motion; and (iii) conditions that promote low crystallographic anisotropy in lithium deposition to avoid fragile, neck-prone dendritic structures.
This multiphysics phase-field framework serves as a powerful virtual laboratory for the solid-state battery. It enables the decoupling and detailed study of intertwined phenomena that are difficult to isolate experimentally. Future work will involve extending the model to three dimensions, incorporating more detailed descriptions of solid electrolyte fracture and interfacial degradation reactions, and exploring the effects of heterogeneous solid electrolyte microstructures. Such advanced modeling will be indispensable for accelerating the rational design of durable, high-performance solid-state batteries.