In this work, I investigate the formation and accumulation of dead lithium in solid-state cells by using a phase-field model that couples mechanical, thermal, and electrochemical fields. Solid-state cells have attracted intense attention because they offer high energy density, improved safety, and better thermal stability compared with conventional liquid-electrolyte batteries. However, lithium dendrites still grow in solid-state cells, and repeated charge–discharge cycles can lead to the incomplete dissolution of these dendrites. The residual metallic lithium loses electrical contact with the electrode and becomes dead lithium, which no longer participates in electrochemical reactions. As dead lithium accumulates, the performance and cycle life of the solid-state cell degrade, and in severe cases, dendrite penetration can cause short circuits. Therefore, understanding how multiphysics coupling affects the dissolution of lithium dendrites and the formation of dead lithium is essential for improving solid-state cell reliability.
I use a phase-field method implemented in a finite-element framework to simulate the evolution of the lithium metal–solid electrolyte interface. The model incorporates gradient energy, chemical free energy, electrostatic energy, and elastic energy. By coupling heat transfer and mechanics into the electrochemical phase-field model, I capture the coupled behavior of temperature, stress, ion concentration, and electric potential. The simulations reveal that the dissolution cut-off time and the dead lithium area change significantly when the heat transfer model or the mechanical field is included. In addition, increasing the temperature, applying a low external pressure, or applying a high external pressure reduces the dead lithium area. For electrochemical parameters, decreasing the diffusion coefficient, increasing the interfacial mobility, and reducing the anisotropy strength all effectively reduce the dead lithium area. These findings provide guidance for suppressing dead lithium in solid-state cells.
The solid-state cell considered here uses a polymer solid electrolyte as the baseline system. The phase-field model introduces an order parameter \(\xi\) that distinguishes the solid electrolyte phase from the lithium metal phase. When \(\xi = 0\), the region represents the solid electrolyte; when \(\xi = 1\), the region represents metallic lithium; and intermediate values describe the diffuse interface. The model also tracks the concentrations of lithium atoms, lithium ions, and anions, denoted by \(c_i\) for \(i = \mathrm{Li}, \mathrm{Li}^+, \mathrm{A}^-\), as well as the electrostatic potential \(\varphi\). The total Gibbs free energy of the system is written as a volume integral:
$$G = \int_V \left[ f_{\mathrm{grad}}(\xi) + f_{\mathrm{ch}}(\xi, c_i) + f_{\mathrm{elec}}(\xi, c_i, \varphi) + f_{\mathrm{els}}(\xi) \right] dV.$$
The gradient energy density accounts for the interfacial energy and is given by
$$f_{\mathrm{grad}}(\xi) = \frac{1}{2} \kappa |\nabla \xi|^2,$$
where the coefficient \(\kappa\) depends on the orientation through
$$\kappa = \kappa_0 \left[ 1 + \delta \cos(\omega \theta) \right].$$
Here, \(\kappa_0\) is the gradient energy coefficient, \(\delta\) is the anisotropy strength, \(\omega\) is the anisotropy mode number, and \(\theta\) is the angle between the interface normal and the coordinate axis. The local chemical free energy density is expressed as
$$f_{\mathrm{ch}}(\xi, c_i) = g(\xi) + c_0 R T_0 \sum_i \frac{c_i}{c_0} \ln \frac{c_i}{c_0},$$
with the double-well function
$$g(\xi) = W \xi^2 (1 – \xi)^2.$$
In these expressions, \(W\) 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
$$f_{\mathrm{elec}}(\xi, c_i, \varphi) = \sum_i F c_i z_i \varphi,$$
where \(F\) is the Faraday constant and \(z_i\) is the valence of species \(i\). The elastic energy density is
$$f_{\mathrm{els}}(\xi) = \frac{1}{2} C_{ijkl} \varepsilon^E_{ij} \varepsilon^E_{kl},$$
and the elastic tensor \(C_{ijkl}\) is constructed from the Young’s modulus and Poisson’s ratio of the electrode and electrolyte phases. The Young’s modulus is interpolated as
$$E = E_e h(\xi) + E_s [1 – h(\xi)],$$
and the Poisson’s ratio is interpolated as
$$v = v_e h(\xi) + v_s [1 – h(\xi)],$$
where \(h(\xi) = \xi^3 (6\xi^2 – 15\xi + 10)\) is a smooth interpolation function. The elastic strain tensor is related to the total strain and the Vegard strain coefficients. This formulation allows the mechanical field to influence the phase evolution directly through the elastic energy term.
The temporal evolution of the phase-field parameter \(\xi\) is governed by
$$\frac{\partial \xi}{\partial t} = – L_\sigma \left( f’_{\mathrm{ch}}(\xi) + f’_{\mathrm{grad}}(\xi) + f’_{\mathrm{els}}(\xi) \right) – L_\eta h'(\xi) \left\{ \exp \left[ \frac{(1-\alpha)nF\eta_\alpha}{R T_0} \right] – \frac{c_{\mathrm{Li}^+}}{c_0} \exp \left[ -\frac{\alpha nF\eta_\alpha}{R T_0} \right] \right\},$$
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. The diffusion of lithium ions in the solid electrolyte follows a modified Fick’s law:
$$\frac{\partial c_{\mathrm{Li}^+}}{\partial t} = \nabla \cdot \left[ D_{\mathrm{eff}} \nabla c_{\mathrm{Li}^+} + \frac{D_{\mathrm{eff}} c_{\mathrm{Li}^+}}{R T_0} nF \nabla \varphi \right] – \psi \frac{d\xi}{dt},$$
where \(\psi\) is a constant related to the lithium-ion reaction rate. The effective diffusion coefficient \(D_{\mathrm{eff}}\) couples the heat transfer model through
$$D_{\mathrm{eff}} = A \exp \left[ -r c_{\mathrm{Li}^+} + \frac{E_\alpha}{R} \left( \frac{1}{T} – \frac{1}{T_0} \right) \right],$$
with \(A\), \(r\), and \(E_\alpha\) being the pre-exponential factor, fitting factor, and activation energy, respectively. The temperature field \(T\) is obtained from the heat transfer model, and the boundary flux includes convective and radiative contributions:
$$-h_R (T – T_0) – \varepsilon_R \sigma_R (T^4 – T_0^4),$$
where \(h_R\) is the convective heat transfer coefficient, \(\varepsilon_R\) is the emissivity, and \(\sigma_R\) is the Stefan–Boltzmann constant. The electric potential is solved from the Poisson equation:
$$\nabla \cdot \left( \sigma_{\mathrm{eff}} \nabla \varphi \right) = F C_s \frac{\partial \xi}{\partial t},$$
where \(C_s\) is the solid-phase lithium concentration, and the effective conductivity is \(\sigma_{\mathrm{eff}} = \sigma_e h(\xi) + \sigma_s [1 – h(\xi)]\), with \(\sigma_e\) and \(\sigma_s\) being the electrode and electrolyte conductivities, respectively.
For the finite-element model, I use a two-dimensional rectangular domain of \(8\,\mu\mathrm{m} \times 8\,\mu\mathrm{m}\). The lower boundary serves as the lithium metal anode and the initial nucleation site for the lithium dendrite. The upper boundary is assigned an initial voltage of \(0.1\,\mathrm{V}\) and an initial lithium-ion concentration. The solver time is set to \(90\,\mathrm{s}\) with a time step of \(1\,\mathrm{s}\). The maximum element size is \(0.05\,\mu\mathrm{m}\), the maximum element growth rate is \(1.3\), and the curvature factor is \(0.3\). The key parameters used in the simulations are summarized in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Gradient energy coefficient | \(\kappa_0\) | \(1 \times 10^{-10}\,\mathrm{J\,m^{-1}}\) |
| Anisotropy strength | \(\delta\) | 0.1 |
| Anisotropy mode | \(\omega\) | 4 |
| Barrier height | \(W\) | \(3.75 \times 10^5\,\mathrm{J\,m^{-3}}\) |
| Standard volume concentration | \(c_0\) | \(1 \times 10^3\,\mathrm{mol\,m^{-3}}\) |
| 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 | \(v_e\) | 0.42 |
| Electrolyte Poisson’s ratio | \(v_s\) | 0.3 |
| Vegard strain coefficients | \(\lambda_i\) | \(-0.866 \times 10^{-3}\), \(-0.773 \times 10^{-3}\), \(-0.529 \times 10^{-3}\) |
| Interfacial mobility | \(L_\sigma\) | \(1 \times 10^{-6}\,\mathrm{m^3\,J^{-1}\,s^{-1}}\) |
| Reaction constant | \(L_\eta\) | 0.5 s\(^{-1}\) |
| Symmetry factor | \(\alpha\) | 0.5 |
| Solid-phase lithium concentration | \(C_s\) | \(7.64 \times 10^4\,\mathrm{mol\,m^{-3}}\) |
| Electrode conductivity | \(\sigma_e\) | \(1 \times 10^7\,\mathrm{S\,m^{-1}}\) |
| 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\(^{-1}\) |
I first examine the effect of coupling a heat transfer model into the mechanical–electrochemical phase-field model. In a solid-state cell, the temperature field can alter the diffusion coefficient, the interfacial reaction rate, and the stress distribution. The simulations show that the overall morphology of the lithium dendrite changes only slightly when the heat transfer model is added, but the von Mises stress distribution changes noticeably. The primary dendrite trunk and the lateral branches exhibit larger stress differences in the coupled model. When I simulate the dissolution process by applying a discharge voltage of \(0.01\,\mathrm{V}\) at the bottom boundary, the dissolution cut-off time and the dead lithium area differ between the two models. Without heat transfer coupling, the dissolution cut-off time is \(49\,\mathrm{s}\) and the dead lithium area is \(0.036\,\mu\mathrm{m}^2\). With heat transfer coupling, the cut-off time decreases to \(43\,\mathrm{s}\) and the dead lithium area increases to \(0.058\,\mu\mathrm{m}^2\), an increase of about 61.1%. This increase is attributed to the shorter dissolution time and the larger residual root area. The temperature field changes the stress distribution, reducing stress concentration at the root and allowing a larger root area to form during growth. This larger root area is more difficult to dissolve completely, leading to more dead lithium.
When an external pressure of \(5\,\mathrm{MPa}\) is applied, the stress concentrates mainly in the middle trunk of the lithium dendrite. In the uncoupled heat transfer model, the maximum root stress reaches \(20.5\,\mathrm{MPa}\), while in the coupled model it is \(14.3\,\mathrm{MPa}\). The dissolution cut-off times are \(32\,\mathrm{s}\) and \(40\,\mathrm{s}\), respectively, and the dead lithium areas are \(0.097\,\mu\mathrm{m}^2\) and \(0.053\,\mu\mathrm{m}^2\). The coupled heat transfer model reduces the dead lithium area by about 45.3% under \(5\,\mathrm{MPa}\). This indicates that when external pressure is present, the temperature field helps to mitigate root stress concentration and reduces dead lithium formation. I further investigate different external pressures of \(5\), \(10\), and \(20\,\mathrm{MPa}\). As the external pressure increases, the height of the lithium dendrite decreases, and the stress difference between the trunk and lateral branches becomes larger. The dissolution cut-off times are \(40\), \(32\), and \(30\,\mathrm{s}\), and the dead lithium areas are \(0.053\), \(0.073\), and \(0.059\,\mu\mathrm{m}^2\). Thus, with increasing pressure, the dead lithium area first decreases, then increases, and finally decreases again. Low pressure reduces dead lithium because the dendrite morphology becomes more compact and the deposited area decreases. Moderate pressure increases root stress concentration and promotes fracture, increasing dead lithium. High pressure makes the dendrite more compact and slows down stripping, leading to more complete dissolution and a lower dead lithium area.
Next, I investigate the effect of coupling the mechanical field into the thermal–electrochemical phase-field model. The mechanical field is essential for solid-state cells because the solid electrolyte has high mechanical strength. When the mechanical field is coupled, the elastic free energy participates directly in the phase evolution, and the lithium dendrite becomes shorter and thicker. Without the mechanical field, the lithium deposition area is larger, the dissolution cut-off time is \(28\,\mathrm{s}\), and the dead lithium area is \(0.098\,\mu\mathrm{m}^2\). With the mechanical field, the cut-off time is longer and the dead lithium area is lower. The mechanical field reduces the stripping rate of lithium at the dendrite surface, allowing more complete dissolution. The temperature field also changes: without the mechanical field, the core temperature at the cut-off time is \(301.8\,\mathrm{K}\), while with the mechanical field it is \(299.5\,\mathrm{K}\). The larger dead lithium area in the uncoupled case causes more heat concentration.
I then study the influence of ambient temperature after coupling the mechanical field. At \(353\,\mathrm{K}\), without the mechanical field, the dissolution cut-off time is \(47\,\mathrm{s}\) and the dead lithium area is \(0.064\,\mu\mathrm{m}^2\). With the mechanical field, the cut-off time is \(67\,\mathrm{s}\) and the dead lithium area is \(0.041\,\mu\mathrm{m}^2\). Compared with the baseline cases, the dead lithium area decreases by 34.6% and 29.3%, respectively. The reduction is smaller when the mechanical field is coupled, indicating that the mechanical field enhances structural stability and reduces the sensitivity to temperature changes. At \(273\,\mathrm{K}\) with the mechanical field, the cut-off time is \(33\,\mathrm{s}\) and the dead lithium area is \(0.085\,\mu\mathrm{m}^2\). The stress evolution during dissolution shows that the lateral branches and secondary dendrites dissolve first, and then stress concentrates at the root. The root dissolution rate decreases over time because the nucleation point at the root hinders contraction. For example, at \(353\,\mathrm{K}\) with the mechanical field, the root dissolution area decreases from 47.9% in the first \(10\,\mathrm{s}\) to 16.2% between 10 and \(30\,\mathrm{s}\), and finally to 11.5% between 30 and \(67\,\mathrm{s}\). This decreasing trend indicates that the root becomes increasingly difficult to dissolve, which is a limitation of the current model.
I also examine the effects of electrochemical parameters on dead lithium formation in solid-state cells. The diffusion coefficient describes the migration ability of lithium ions. When the diffusion coefficient increases, lithium ions aggregate more, and the dendrite grows faster. The primary trunk becomes taller and the side branches longer, while secondary dendrites are fewer and the root width is larger. When the diffusion coefficient decreases, secondary dendrites become more numerous, and the trunk strips faster during dissolution. The dissolution cut-off times are \(51\,\mathrm{s}\) for increased diffusion coefficient and \(33\,\mathrm{s}\) for decreased diffusion coefficient. The dead lithium areas are \(0.090\,\mu\mathrm{m}^2\) and \(0.044\,\mu\mathrm{m}^2\), respectively. Although a smaller diffusion coefficient shortens the cut-off time, the total lithium deposition is smaller, so the residual dead lithium area is also smaller. Increasing the diffusion coefficient increases the dead lithium area by 55.2%, while decreasing it reduces the dead lithium area by 24.1%.
The interfacial mobility is another key kinetic parameter in the phase-field model. It determines the evolution rate of the lithium metal–solid electrolyte interface and affects dendrite morphology, growth rate, and stability. When the interfacial mobility increases, secondary dendrites are suppressed, and the dendrite surface becomes smoother. When it decreases, the dendrite grows faster and the surface becomes rougher. I simulate interfacial mobilities of \(1 \times 10^{-5}\,\mathrm{m^3\,J^{-1}\,s^{-1}}\) and \(1 \times 10^{-7}\,\mathrm{m^3\,J^{-1}\,s^{-1}}\). With higher mobility, lithium deposition is more uniform and dendrite growth is slower. With lower mobility, primary dendrites grow faster and secondary dendrites are more numerous. The dissolution cut-off times are \(68\,\mathrm{s}\) and \(84\,\mathrm{s}\), and the dead lithium areas are \(0.004\,\mu\mathrm{m}^2\) and \(0.024\,\mu\mathrm{m}^2\). Both are lower than the baseline case with \(1 \times 10^{-6}\,\mathrm{m^3\,J^{-1}\,s^{-1}}\). Higher mobility leads to more uniform stripping and fewer isolated lithium fragments, while lower mobility allows more complete dissolution. Thus, increasing interfacial mobility is an effective strategy to reduce dead lithium in solid-state cells.
The anisotropy strength controls the directional growth tendency of the lithium dendrite. When the anisotropy strength increases, the tip growth rate becomes faster, and the trunk grows more rapidly than the side branches. When the anisotropy strength is very small, the growth rates of the trunk and side branches are similar. I simulate anisotropy strengths of \(0.15\), \(0.05\), and \(0.001\). For \(0.15\), the primary trunk grows much faster, and secondary dendrites grow rapidly. For \(0.05\), the difference between trunk and side branches is smaller. For \(0.001\), the growth is nearly isotropic. The dissolution cut-off times for \(0.15\) and \(0.05\) are \(41\,\mathrm{s}\) and \(45\,\mathrm{s}\), and the dead lithium areas are \(0.056\,\mu\mathrm{m}^2\) and \(0.051\,\mu\mathrm{m}^2\). Compared with the baseline anisotropy strength of \(0.1\), increasing it to \(0.15\) reduces the dead lithium area by 3.4%, while decreasing it to \(0.05\) reduces the dead lithium area by 12.0%. The larger reduction at lower anisotropy is due to the narrower root and more numerous weak points, which lead to more isolated lithium but a smaller total residual area. Therefore, reducing the anisotropy strength is more effective for minimizing dead lithium in solid-state cells.
To summarize the quantitative effects, Table 2 compares the dead lithium area under different conditions. Table 3 lists the dissolution cut-off times and dead lithium areas for the main parameter variations.
| Condition | Dissolution cut-off time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) | Relative change |
|---|---|---|---|
| Uncoupled heat transfer, baseline | 49 | 0.036 | — |
| Coupled heat transfer, baseline | 43 | 0.058 | +61.1% |
| Uncoupled heat transfer, 5 MPa | 32 | 0.097 | — |
| Coupled heat transfer, 5 MPa | 40 | 0.053 | −45.3% |
| Coupled heat transfer, 10 MPa | 32 | 0.073 | — |
| Coupled heat transfer, 20 MPa | 30 | 0.059 | — |
| Uncoupled mechanical field | 28 | 0.098 | — |
| Coupled mechanical field | — | 0.058 | −40.8% |
| 353 K, uncoupled mechanical field | 47 | 0.064 | −34.6% |
| 353 K, coupled mechanical field | 67 | 0.041 | −29.3% |
| 273 K, coupled mechanical field | 33 | 0.085 | — |
| Increased diffusion coefficient | 51 | 0.090 | +55.2% |
| Decreased diffusion coefficient | 33 | 0.044 | −24.1% |
| Interfacial mobility \(1 \times 10^{-5}\) | 68 | 0.004 | — |
| Interfacial mobility \(1 \times 10^{-7}\) | 84 | 0.024 | — |
| Anisotropy strength 0.15 | 41 | 0.056 | −3.4% |
| Anisotropy strength 0.05 | 45 | 0.051 | −12.0% |
| Parameter | Change | Effect on dead lithium area |
|---|---|---|
| Heat transfer model | Coupled | Increases by 61.1% (baseline) |
| External pressure (5 MPa) | Low pressure | Decreases by 45.3% when heat transfer is coupled |
| External pressure (10 MPa) | Moderate pressure | Increases compared with 5 MPa |
| External pressure (20 MPa) | High pressure | Decreases compared with 10 MPa |
| Mechanical field | Coupled | Decreases by 40.8% |
| Temperature | Increase to 353 K | Decreases by 29.3%–34.6% |
| Temperature | Decrease to 273 K | Increases |
| Diffusion coefficient | Increase | Increases by 55.2% |
| Diffusion coefficient | Decrease | Decreases by 24.1% |
| Interfacial mobility | Increase | Strong decrease |
| Interfacial mobility | Decrease | Moderate decrease |
| Anisotropy strength | Increase to 0.15 | Decreases by 3.4% |
| Anisotropy strength | Decrease to 0.05 | Decreases by 12.0% |
The results demonstrate that multiphysics coupling strongly influences dead lithium formation in solid-state cells. The heat transfer model affects the stress distribution and the dissolution kinetics, while the mechanical field stabilizes the dendrite structure and reduces the dead lithium area. Temperature plays a dual role: higher temperatures enhance diffusion and uniform deposition, reducing dead lithium, but excessively high temperatures may alter the solid electrolyte interphase and increase dead lithium through side reactions. External pressure has a non-monotonic effect: low pressure reduces dead lithium by compacting the dendrite, moderate pressure increases root stress and fracture, and high pressure promotes dense deposition and slow stripping, reducing dead lithium again. Among electrochemical parameters, decreasing the diffusion coefficient, increasing the interfacial mobility, and reducing the anisotropy strength are effective strategies to reduce dead lithium in solid-state cells.
In conclusion, I have developed a multiphysics phase-field model to study dead lithium in solid-state cells under coupled mechanical, thermal, and electrochemical fields. The model reveals that the dissolution cut-off time and dead lithium area are sensitive to the coupling of heat transfer and mechanics. The mechanical field reduces the dead lithium area by about 40.8%, and the heat transfer model can either increase or decrease the dead lithium area depending on the applied pressure. Increasing temperature reduces dead lithium, while low and high external pressures also reduce dead lithium compared with moderate pressure. For electrochemical parameters, smaller diffusion coefficients, higher interfacial mobilities, and lower anisotropy strengths all reduce dead lithium. These findings provide a systematic understanding of how multiphysics coupling and electrochemical parameters affect dead lithium in solid-state cells, and they offer practical guidance for designing more durable solid-state cells with suppressed dead lithium accumulation.