I investigate the formation, accumulation, and suppression of dead lithium in solid electrolyte cells by using a multiphysics phase-field framework. A solid electrolyte cell is attractive because it can offer high energy density, improved thermal stability, reduced leakage risk, and better safety than conventional liquid-electrolyte systems. However, the solid electrolyte cell is not immune to lithium dendrite growth. During repeated charge and discharge, lithium dendrites may not dissolve completely, and the residual metallic lithium loses electronic contact with the electrode. This disconnected lithium becomes dead lithium. In a solid electrolyte cell, dead lithium gradually accumulates, reduces capacity, increases impedance, and can eventually threaten safety. My central aim is to understand how force, heat, and electrochemical fields act together in a solid electrolyte cell and how their coupling changes dendrite dissolution and dead lithium area.
I construct the model around a phase-field description in which an order parameter \(\xi\) distinguishes the solid electrolyte phase from the metallic lithium phase. When \(\xi=0\), the local state is the solid electrolyte phase. When \(\xi=1\), the local state is metallic lithium. Intermediate values describe the diffuse interface. I include lithium atoms, lithium ions, anions, and electrostatic potential. I then couple heat transfer through a temperature-dependent effective diffusion coefficient and mechanical deformation through elastic free energy. The solid electrolyte cell is therefore treated as a force-thermal-electrochemical system rather than a purely electrochemical one.
Phase-Field Formulation for a Solid Electrolyte Cell
I write the total Gibbs free energy of the solid electrolyte cell as the sum of gradient, chemical, electrostatic, and elastic contributions:
$$ G = \int_V \left[f_{\mathrm{grad}}(\xi) + f_{\mathrm{ch}}(\xi,c_i) + f_{\mathrm{elec}}(\xi,c_i,\phi) + f_{\mathrm{els}}(\xi)\right] dV. $$
The gradient energy density controls the diffuse interface width and interface anisotropy. In my implementation, I use the standard square-gradient form,
$$ f_{\mathrm{grad}}(\xi)=\frac{1}{2}\kappa |\nabla \xi|^2, $$
with the interface energy coefficient written as
$$ \kappa = \kappa_0\left[1+\delta\cos(\omega\theta)\right]. $$
Here, \(\kappa_0\) is the base gradient energy coefficient, \(\delta\) is the anisotropic strength, \(\omega\) is the anisotropic mode, and \(\theta\) is the angle between the interface normal and a coordinate axis. The chemical free energy density is
$$ f_{\mathrm{ch}}(\xi,c_i)=g(\xi)+c_0RT_0\sum_i \frac{c_i}{c_0}\ln\frac{c_i}{c_0}, $$
where the double-well function is
$$ g(\xi)=W\xi^2(1-\xi)^2. $$
In these expressions, \(W\) is the barrier height, \(c_0\) is the standard lithium volumetric concentration, \(R\) is the gas constant, and \(T_0\) is the reference ambient temperature. The electrostatic energy density is
$$ f_{\mathrm{elec}}(\xi,c_i,\phi)=\sum_i F c_i z_i \phi, $$
where \(F\) is Faraday’s constant, \(z_i\) is the valence of species \(i\), and \(\phi\) is the electrostatic potential. The elastic energy density is
$$ f_{\mathrm{els}}(\xi)=\frac{1}{2}C_{ijkl}\varepsilon^E_{ij}\varepsilon^E_{kl}. $$
I interpolate the elastic stiffness through the phase field. The elastic tensor is
$$ C_{ijkl}=\frac{E}{2(1+\nu)}(\delta_{il}\delta_{jk}+\delta_{ik}\delta_{jl})+\frac{E\nu}{(1+\nu)(1-2\nu)}\delta_{ij}\delta_{kl}. $$
The Young’s modulus and Poisson’s ratio in the solid electrolyte cell are expressed as
$$ E=E_e h(\xi)+E_s[1-h(\xi)], $$
$$ \nu=\nu_e h(\xi)+\nu_s[1-h(\xi)], $$
and the interpolation function is
$$ h(\xi)=\xi^3(6\xi^2-15\xi+10). $$
The elastic strain tensor is obtained by removing the eigenstrain contribution from the total strain:
$$ \varepsilon^E_{ij}=\varepsilon^T_{ij}-\lambda_i h(\xi)\delta_{ij}, $$
$$ \varepsilon^E_{kl}=\varepsilon^T_{kl}-\lambda_k h(\xi)\delta_{kl}. $$
Here, \(\lambda_i\) is the Vegard strain coefficient. This elastic contribution is important in a solid electrolyte cell because the solid electrolyte mechanically constrains lithium deposition and stripping. The mechanical field changes stress concentration, and stress concentration can alter the point at which a dendrite necks, fractures, or detaches.
The temporal evolution of the phase-field order parameter follows
$$ \frac{\partial \xi}{\partial t} = -L_\sigma\left(f’_{\mathrm{ch}}(\xi)+f’_{\mathrm{grad}}(\xi)+f’_{\mathrm{els}}(\xi)\right) – L_\eta h'(\xi)\left\{\exp\left[\frac{(1-\alpha)nF\eta_\alpha}{RT_0}\right]-\frac{c_{\mathrm{Li}^+}}{c_0}\exp\left[-\frac{\alpha nF\eta_\alpha}{RT_0}\right]\right\}. $$
In this equation, \(L_\sigma\) 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. During discharge, I introduce an activity factor \(f_d\). The evolution equation then becomes
$$ \frac{\partial \xi}{\partial t} = -f_d L_\sigma\left(f’_{\mathrm{ch}}(\xi)+f’_{\mathrm{grad}}(\xi)+f’_{\mathrm{els}}(\xi)\right) – f_d L_\eta h'(\xi)\left\{\exp\left[\frac{(1-\alpha)nF\eta_\alpha}{RT_0}\right]-\frac{c_{\mathrm{Li}^+}}{c_0}\exp\left[-\frac{\alpha nF\eta_\alpha}{RT_0}\right]\right\}, $$
where
$$ f_d=f_{\mathrm{step}}\left(-\frac{\phi_s}{\phi_b}\right). $$
I use \(f_d=1\) for active lithium and \(f_d=0\) for dead lithium. The variable \(\phi_s\) is the electrolyte potential, and \(\phi_b\) is the reference potential. This switch is essential for distinguishing between lithium that remains electrochemically connected to the electrode and lithium that has become dead lithium in the solid electrolyte cell.
Lithium-ion transport in the solid electrolyte is described by a modified diffusion equation:
$$ \frac{\partial c_{\mathrm{Li}^+}}{\partial t} = \nabla\cdot\left[D_{\mathrm{eff}}\nabla c_{\mathrm{Li}^+} + \frac{D_{\mathrm{eff}}c_{\mathrm{Li}^+}}{RT_0}nF\nabla\phi\right] – \psi\frac{d\xi}{dt}. $$
Here, \(\psi\) is a reaction-rate constant, and \(D_{\mathrm{eff}}\) is the effective diffusion coefficient. When heat transfer is included, the effective diffusion coefficient becomes temperature dependent:
$$ D_{\mathrm{eff}} = A\exp\left[-r c_{\mathrm{Li}^+} + \frac{E_\alpha}{R}\left(\frac{1}{T}-\frac{1}{T_0}\right)\right]. $$
The thermal boundary flux is written as
$$ -h_R(T-T_0)-\varepsilon_R\sigma_R(T^4-T_0^4), $$
where \(h_R\) is the convective heat-transfer coefficient, \(\varepsilon_R\) is the emissivity, and \(\sigma_R\) is the Stefan-Boltzmann constant. The potential field satisfies
$$ \nabla\cdot(\sigma_{\mathrm{eff}}\nabla\phi)=FC_s\frac{\partial\xi}{\partial t}, $$
with the effective conductivity interpolated as
$$ \sigma_{\mathrm{eff}}=\sigma_e h(\xi)+\sigma_s[1-h(\xi)]. $$
This set of equations allows me to describe how a solid electrolyte cell evolves under coupled force, thermal, and electrochemical driving forces.
| Symbol | Meaning in the solid electrolyte cell model |
|---|---|
| \(\xi\) | Phase-field order parameter: 0 for solid electrolyte, 1 for metallic lithium |
| \(c_{\mathrm{Li}^+}\) | Lithium-ion concentration in the solid electrolyte cell |
| \(\phi\) | Electrostatic potential |
| \(T\) | Local temperature |
| \(D_{\mathrm{eff}}\) | Effective lithium-ion diffusion coefficient |
| \(L_\sigma\) | Interfacial mobility |
| \(L_\eta\) | Electrochemical reaction constant |
| \(\delta\) | Anisotropic strength |
| \(\sigma_{\mathrm{eff}}\) | Effective electrical conductivity |
| \(f_d\) | Activity factor: active lithium versus dead lithium |
Finite Element Implementation
I implement the multiphysics phase-field model in a finite element environment. The computational domain is a rectangular region \(8\,\mu\mathrm{m}\times8\,\mu\mathrm{m}\). The lower boundary represents the lithium-metal anode and the initial nucleation site of the dendrite. The upper boundary carries an initial voltage of \(0.1\,\mathrm{V}\) and an initial lithium-ion concentration. I run the simulation for \(90\,\mathrm{s}\) with a time step of \(1\,\mathrm{s}\). The maximum mesh element size is \(0.05\,\mu\mathrm{m}\), the maximum element growth rate is \(1.3\), and the curvature factor is \(0.3\). These settings allow the diffuse interface of the solid electrolyte cell to evolve without excessive numerical diffusion.
| Parameter | Symbol | Value |
|---|---|---|
| Gradient energy coefficient | \(\kappa_0\) | \(1\times10^{-10}\,\mathrm{J\,m^{-1}}\) |
| Anisotropic strength | \(\delta\) | 0.1 |
| Anisotropic mode | \(\omega\) | 4 |
| Barrier height | \(W\) | \(3.75\times10^5\,\mathrm{J\,m^{-3}}\) |
| Standard volumetric concentration | \(c_0\) | \(1\times10^3\,\mathrm{mol\,m^{-3}}\) |
| Reference 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 | \(\lambda_i\) | \(-0.866\times10^{-3}\), \(-0.773\times10^{-3}\), \(-0.529\times10^{-3}\) |
| Interfacial mobility | \(L_\sigma\) | \(1\times10^{-6}\,\mathrm{m^3\,J^{-1}\,s^{-1}}\) |
| Reaction constant | \(L_\eta\) | 0.5 s\(^{-1}\) |
| Symmetry factor | \(\alpha\) | 0.5 |
| Solid lithium concentration | \(C_s\) | \(7.64\times10^4\,\mathrm{mol\,m^{-3}}\) |
| Electrode conductivity | \(\sigma_e\) | \(1\times10^7\,\mathrm{S\,m^{-1}}\) |
| Electrolyte conductivity | \(\sigma_s\) | 0.1 S m\(^{-1}\) |
| Electrode specific heat | \(c_{pe}\) | 1200 J kg\(^{-1}\) K\(^{-1}\) |
| Electrolyte specific heat | \(c_{ps}\) | 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 | \(h_R\) | 10 W m\(^{-2}\) K\(^{-1}\) |
Coupling Heat Transfer into a Solid Electrolyte Cell Model
I first examine the role of heat transfer in a solid electrolyte cell. Most dead-lithium studies focus on liquid electrolytes, single fields, or two-field coupling. Here I compare a force-electrochemical phase-field model with a force-thermal-electrochemical phase-field model. The comparison isolates the effect of the heat-transfer model. When heat transfer is absent, the lithium dendrite morphology changes only slightly relative to the coupled case. However, the von Mises stress distribution changes more strongly. In the coupled solid electrolyte cell model, the stress difference between the primary trunk and lateral branches becomes larger. This change affects how the dendrite necks and detaches.
During discharge, I apply a voltage of \(0.01\,\mathrm{V}\) at the lower boundary to simulate lithium stripping. The active lithium factor \(f_d\) determines whether a phase-field region can dissolve. For the uncoupled heat-transfer model, the dissolution cutoff time is \(49\,\mathrm{s}\), and the dead lithium area is \(0.036\,\mu\mathrm{m}^2\). For the coupled heat-transfer model, the cutoff time is \(43\,\mathrm{s}\), and the dead lithium area is \(0.058\,\mu\mathrm{m}^2\). Thus, coupling heat transfer increases the dead lithium area by about \(61.1\%\). The shorter dissolution cutoff time is one reason: lithium stripping stops earlier, so less material is removed. The root region also remains larger in the coupled heat-transfer solid electrolyte cell because the stress concentration at the root is reduced. A reduced root stress concentration means the root is less suppressed during growth, so it occupies a larger area before dissolution.
| Model for the solid electrolyte cell | Dissolution cutoff time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) | Observed trend |
|---|---|---|---|
| Without coupled heat transfer | 49 | 0.036 | Smaller dead lithium area, slightly later cutoff |
| With coupled heat transfer | 43 | 0.058 | Larger dead lithium area, earlier cutoff |
| Relative change | -12.2% | +61.1% | Heat coupling increases residual dead lithium |
I then study how the coupling of heat transfer changes the effect of external pressure. Previous work on force-electrochemical models has shown that dendrite growth and stripping depend on the applied stack pressure. In a solid electrolyte cell, pressure is not a minor boundary condition; it directly changes the mechanical stress state. I apply an external pressure of \(5\,\mathrm{MPa}\) and compare the uncoupled and coupled heat-transfer cases. Without heat coupling, the maximum root stress reaches about \(20.5\,\mathrm{MPa}\). With heat coupling, the maximum root stress is about \(14.3\,\mathrm{MPa}\). The coupled thermal field reduces the root stress concentration. The dissolution cutoff times are \(32\,\mathrm{s}\) and \(40\,\mathrm{s}\), respectively. The dead lithium areas are \(0.097\,\mu\mathrm{m}^2\) and \(0.053\,\mu\mathrm{m}^2\). Therefore, when the solid electrolyte cell is under \(5\,\mathrm{MPa}\) external pressure, coupling heat transfer reduces the dead lithium area by about \(45.3\%\).
| Condition under 5 MPa | Maximum root stress (MPa) | Dissolution cutoff time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) |
|---|---|---|---|
| Without heat-transfer coupling | 20.5 | 32 | 0.097 |
| With heat-transfer coupling | 14.3 | 40 | 0.053 |
| Change from heat coupling | Lower by 30.2% | Longer by 25.0% | Lower by 45.3% |
I next vary the external pressure in the force-thermal-electrochemical solid electrolyte cell model. I choose \(5\,\mathrm{MPa}\), \(10\,\mathrm{MPa}\), and \(20\,\mathrm{MPa}\). As the external pressure increases, the dendrite height decreases, and the stress difference between the primary trunk and lateral branches becomes larger. The root stress also increases with pressure. The dissolution cutoff times are \(40\,\mathrm{s}\), \(32\,\mathrm{s}\), and \(30\,\mathrm{s}\) for \(5\), \(10\), and \(20\,\mathrm{MPa}\), respectively. The dead lithium areas are \(0.053\), \(0.073\), and \(0.059\,\mu\mathrm{m}^2\). The trend is nonmonotonic: the dead lithium area first decreases at low pressure, then increases at intermediate pressure, and then decreases again at high pressure. At low pressure, the root stress is lower than in the uncoupled case, so the effect of pressure on dead lithium formation is moderate. At intermediate pressure, root stress concentration becomes more severe, and the trunk is more likely to detach from the root, increasing dead lithium. At high pressure, the lithium deposit becomes denser and more uniform, while stripping becomes slower and more complete. This competition reduces the dead lithium area again.
| External pressure (MPa) | Dissolution cutoff time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) | Dominant mechanism in the solid electrolyte cell |
|---|---|---|---|
| 5 | 40 | 0.053 | Low pressure reduces root stress; moderate dead lithium |
| 10 | 32 | 0.073 | Root stress concentration promotes detachment |
| 20 | 30 | 0.059 | Dense deposition and slower stripping improve uniformity |
Coupling the Mechanical Field into a Solid Electrolyte Cell Model
The mechanical field is essential for a solid electrolyte cell because the solid electrolyte provides mechanical suppression of dendrite growth. I compare a thermal-electrochemical phase-field model with a force-thermal-electrochemical phase-field model. When the mechanical field is absent, the elastic free energy is not included in the total Gibbs free energy, so the solid electrolyte does not mechanically resist lithium deposition in the same way. In the uncoupled-mechanics case, the lithium deposition area is larger. The dissolution cutoff time is \(28\,\mathrm{s}\), and the dead lithium area is \(0.098\,\mu\mathrm{m}^2\). When the mechanical field is coupled, the dissolution cutoff time becomes longer, and the dead lithium area becomes smaller. I obtain a dead lithium area of \(0.058\,\mu\mathrm{m}^2\), which corresponds to a reduction of about \(40.8\%\). The mechanically coupled solid electrolyte cell also shows a lower internal core temperature when dissolution stops. The uncoupled-mechanics case has a core temperature of about \(301.8\,\mathrm{K}\), while the coupled-mechanics case reaches about \(299.5\,\mathrm{K}\). The larger dead lithium area in the uncoupled case causes stronger local heat concentration.
| Model for the solid electrolyte cell | Dissolution cutoff time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) | Internal core temperature (K) |
|---|---|---|---|
| Without mechanical coupling | 28 | 0.098 | 301.8 |
| With mechanical coupling | 43 | 0.058 | 299.5 |
| Change from mechanical coupling | Longer by 53.6% | Lower by 40.8% | Lower by 2.3 K |
I also examine how ambient temperature affects dead lithium before and after coupling the mechanical field. Temperature is a major variable in a solid electrolyte cell because it changes lithium-ion diffusivity, interfacial reaction rates, and stress evolution. I simulate an ambient temperature of \(353\,\mathrm{K}\). Without mechanical coupling, the dissolution cutoff time is \(47\,\mathrm{s}\), and the dead lithium area is \(0.064\,\mu\mathrm{m}^2\). With mechanical coupling, the dissolution cutoff time is \(67\,\mathrm{s}\), and the dead lithium area is \(0.041\,\mu\mathrm{m}^2\). Relative to the lower-temperature baselines, the dead lithium area decreases by about \(34.6\%\) without mechanical coupling and by about \(29.3\%\) with mechanical coupling. The smaller reduction after mechanical coupling shows that the mechanical field stabilizes the dendrite structure and reduces its sensitivity to temperature.
When the ambient temperature is lowered to \(273\,\mathrm{K}\) in the mechanically coupled solid electrolyte cell, the dissolution cutoff time becomes \(33\,\mathrm{s}\), and the dead lithium area increases to \(0.085\,\mu\mathrm{m}^2\). This result agrees with earlier observations that lower temperatures favor dead lithium formation. At lower temperature, lithium-ion diffusion is slower, concentration gradients are stronger, and stripping is less uniform. The solid electrolyte cell therefore retains more dead lithium after the dissolution cutoff.
| Mechanical coupling | Ambient temperature (K) | Dissolution cutoff time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) |
|---|---|---|---|
| Without | 353 | 47 | 0.064 |
| With | 353 | 67 | 0.041 |
| With | 273 | 33 | 0.085 |
During dissolution in the mechanically coupled solid electrolyte cell, the lateral branches and secondary dendrites dissolve first because they have larger exposed interfacial area. After those parts dissolve, stress accumulates near the root. The nucleation point is fixed at the root, so root shrinkage is mechanically constrained. This constraint slows root stripping and leaves residual lithium. For the \(353\,\mathrm{K}\) case, the root-area dissolution fraction is about \(47.9\%\) during the first \(10\,\mathrm{s}\). From \(10\) to \(30\,\mathrm{s}\), the fraction drops to about \(16.2\%\). From \(30\) to \(67\,\mathrm{s}\), only about \(11.5\%\) of the initial root area dissolves. I express the interval dissolution fraction as
$$ \Delta A_{\mathrm{root}}^{(t_1,t_2)}=\frac{A_{\mathrm{root}}(t_1)-A_{\mathrm{root}}(t_2)}{A_{\mathrm{root}}(0)}\times100\%. $$
| Time interval (s) | Root-area dissolution fraction | Interpretation for the solid electrolyte cell |
|---|---|---|
| 0 to 10 | 47.9% | Fast initial stripping of exposed root material |
| 10 to 30 | 16.2% | Slower stripping as root stress accumulates |
| 30 to 67 | 11.5% | Very slow final stripping; dead lithium remains |
Electrochemical Parameter Effects in a Solid Electrolyte Cell
After examining thermal and mechanical coupling, I study three electrochemical parameters that influence dead lithium in a solid electrolyte cell: diffusion coefficient, interfacial mobility, and anisotropic strength. These parameters are not merely numerical constants. They control lithium transport, interface kinetics, and dendrite morphology. Their variation changes the competition between complete stripping and dead-lithium formation.
Diffusion Coefficient
I vary the diffusion coefficient around the baseline value used in the solid electrolyte cell model. When the diffusion coefficient increases, lithium ions distribute more uniformly in the electrolyte, but the total lithium deposition area also increases. The dendrite trunk becomes taller, and lateral branches become longer. Secondary dendrites are fewer because the concentration field is smoother. When the diffusion coefficient decreases, secondary dendrites become more numerous. These small secondary dendrites are fragile and detach easily during dissolution. In addition, the root region becomes narrower, which promotes separation of the trunk from the root.
For an increased diffusion coefficient, the dissolution cutoff time is \(51\,\mathrm{s}\), and the dead lithium area is \(0.090\,\mu\mathrm{m}^2\). For a decreased diffusion coefficient, the dissolution cutoff time is \(33\,\mathrm{s}\), and the dead lithium area is \(0.044\,\mu\mathrm{m}^2\). Relative to the baseline dead lithium area of \(0.058\,\mu\mathrm{m}^2\), increasing the diffusion coefficient raises the dead lithium area by about \(55.2\%\), while decreasing it lowers the dead lithium area by about \(24.1\%\). Even though a smaller diffusion coefficient produces faster stripping and a shorter cutoff time, the initial lithium deposit is smaller, so less dead lithium remains.
| Diffusion coefficient | Dissolution cutoff time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) | Change relative to baseline |
|---|---|---|---|
| Increased | 51 | 0.090 | +55.2% |
| Baseline | 43 | 0.058 | Reference |
| Decreased | 33 | 0.044 | -24.1% |
Interfacial Mobility
Interfacial mobility controls the kinetic rate of the lithium/electrolyte interface. In a solid electrolyte cell, interfacial mobility affects dendrite morphology, growth rate, and stability. When interfacial mobility increases, secondary dendrites are suppressed, and the lithium surface becomes smoother. When interfacial mobility decreases, dendrite growth is faster, and the surface becomes rougher.
I test interfacial mobilities of \(1\times10^{-5}\,\mathrm{m^3\,J^{-1}\,s^{-1}}\) and \(1\times10^{-7}\,\mathrm{m^3\,J^{-1}\,s^{-1}}\). For the higher mobility, the dissolution cutoff time is \(68\,\mathrm{s}\), and the dead lithium area is \(0.004\,\mu\mathrm{m}^2\). For the lower mobility, the dissolution cutoff time is \(84\,\mathrm{s}\), and the dead lithium area is \(0.024\,\mu\mathrm{m}^2\). Both values are lower than the baseline dead lithium area obtained with \(1\times10^{-6}\,\mathrm{m^3\,J^{-1}\,s^{-1}}\). The reason is different in the two cases. At high interfacial mobility, lithium deposition is more uniform, and the number of isolated secondary dendrites is smaller, so the deposit strips as a more coherent body. At low interfacial mobility, stripping is slower and more complete, so isolated lithium is less likely to remain. In both cases, the solid electrolyte cell produces less dead lithium than the baseline.
| Interfacial mobility (\(\mathrm{m^3\,J^{-1}\,s^{-1}}\)) | Dissolution cutoff time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) | Morphological effect in the solid electrolyte cell |
|---|---|---|---|
| \(1\times10^{-5}\) | 68 | 0.004 | Smooth deposition, few secondary dendrites |
| \(1\times10^{-6}\) | 43 | 0.058 | Baseline case |
| \(1\times10^{-7}\) | 84 | 0.024 | Slow stripping, more complete dissolution |
Anisotropic Strength
Anisotropic strength controls the directional preference of dendrite growth. In a solid electrolyte cell, anisotropic strength strongly influences the growth direction and the distribution of side branches. When anisotropic strength increases, tip growth becomes faster. When anisotropic strength decreases, the growth advantage of the trunk becomes less pronounced, and the morphology becomes more isotropic.
I simulate anisotropic strengths of \(0.15\), \(0.05\), and \(0.001\) for growth. At \(0.001\), the primary trunk and side branches grow at similar rates, so their sizes are similar. As anisotropic strength increases, the primary trunk grows more rapidly and secondary dendrites on the trunk also grow faster. For dissolution, I compare \(0.15\) and \(0.05\). At \(0.15\), the dissolution cutoff time is \(41\,\mathrm{s}\), and the dead lithium area is \(0.056\,\mu\mathrm{m}^2\). At \(0.05\), the dissolution cutoff time is \(45\,\mathrm{s}\), and the dead lithium area is \(0.051\,\mu\mathrm{m}^2\). Relative to the baseline anisotropic strength of \(0.1\), increasing it to \(0.15\) reduces the dead lithium area by about \(3.4\%\), while decreasing it to \(0.05\) reduces the dead lithium area by about \(12.0\%\). Reducing anisotropic strength is therefore more effective for lowering dead lithium in the solid electrolyte cell than increasing it. A larger anisotropic strength creates more isolated lithium particles, but it also makes the trunk thinner and the root narrower. The total dead lithium area depends on the balance between root residue and detached particles.
| Anisotropic strength | Dissolution cutoff time (s) | Dead lithium area (\(\mu\mathrm{m}^2\)) | Change relative to \(\delta=0.1\) |
|---|---|---|---|
| 0.15 | 41 | 0.056 | -3.4% |
| 0.10 | 43 | 0.058 | Baseline |
| 0.05 | 45 | 0.051 | -12.0% |
Integrated Discussion for Solid Electrolyte Cells
The simulations show that dead lithium in a solid electrolyte cell is not controlled by a single field. The thermal field changes lithium-ion diffusion and the dissolution cutoff time. The mechanical field changes stress concentration and the mechanical stability of dendrites. The electrochemical parameters change deposition morphology, interface kinetics, and stripping completeness. Because these effects interact, a solid electrolyte cell must be analyzed as a coupled system.
I summarize the main coupling effects below. In general, coupling heat transfer into the force-electrochemical model shortens the dissolution cutoff time and increases dead lithium area. Coupling the mechanical field into the thermal-electrochemical model lengthens the dissolution cutoff time and decreases dead lithium area. External pressure has a nonmonotonic effect: low pressure can reduce dead lithium, intermediate pressure can increase it, and high pressure can reduce it again by making deposition denser and stripping slower. Increasing ambient temperature reduces dead lithium. Decreasing diffusion coefficient reduces dead lithium area. Increasing or decreasing interfacial mobility can both reduce dead lithium, but through different mechanisms. Reducing anisotropic strength reduces dead lithium more effectively than increasing it.
| Variable | Direction of change | Effect on dissolution cutoff time | Effect on dead lithium area in the solid electrolyte cell |
|---|---|---|---|
| Heat-transfer coupling | Added | Shorter | Increases |
| Mechanical coupling | Added | Longer | Decreases |
| External pressure | 5 MPa | 40 s | Low dead lithium area |
| External pressure | 10 MPa | 32 s | Higher dead lithium area |
| External pressure | 20 MPa | 30 s | Lower dead lithium area again |
| Ambient temperature | Increase to 353 K | Longer | Decreases |
| Ambient temperature | Decrease to 273 K | Shorter | Increases |
| Diffusion coefficient | Increase | Longer | Increases |
| Diffusion coefficient | Decrease | Shorter | Decreases |
| Interfacial mobility | Increase | Longer | Decreases strongly |
| Interfacial mobility | Decrease | Much longer | Decreases |
| Anisotropic strength | Increase to 0.15 | Shorter | Decreases slightly |
| Anisotropic strength | Decrease to 0.05 | Longer | Decreases more strongly |
I also note two limitations of the present solid electrolyte cell model. First, because the nucleation point is fixed at the root, root stripping shows a prolonged slow stage. The root retains more lithium than the trunk, and this residual lithium contributes to the final dead lithium area. Second, the model does not explicitly include high-temperature interfacial side reactions. At very high temperature, side reactions may modify the solid-electrolyte interphase structure and could change dead lithium accumulation. These effects should be added in future work on solid electrolyte cell degradation.
Despite these limitations, the results provide a consistent picture of dead lithium formation in a solid electrolyte cell. Dead lithium forms when lithium stripping is incomplete or when parts of the dendrite lose electronic contact. The dissolution cutoff time alone is not sufficient to predict dead lithium area. The initial deposited area, root geometry, stress concentration, and local transport all matter. A shorter cutoff time can sometimes coincide with a smaller dead lithium area if the initial deposit is small. A longer cutoff time can sometimes coincide with a smaller dead lithium area if stripping is more uniform and complete. Therefore, optimizing a solid electrolyte cell requires simultaneous control of thermal, mechanical, and electrochemical conditions.
Conclusions
I have developed and applied a force-thermal-electrochemical phase-field model to study dead lithium in solid electrolyte cells. I simulated lithium dendrite growth and dissolution, compared different multiphysics couplings, and explored thermal, mechanical, and electrochemical parameter effects. My main conclusions are as follows.
1. When heat transfer is coupled into the force-electrochemical phase-field model, the stress distribution in the solid electrolyte cell changes. The dissolution cutoff time becomes shorter, and the dead lithium area increases by about \(61.1\%\). Under an external pressure of \(5\,\mathrm{MPa}\), the uncoupled heat-transfer case has stronger root stress concentration and a larger dead lithium area. Coupling heat transfer reduces the dead lithium area by about \(45.3\%\). When external pressure is varied from \(5\) to \(10\) to \(20\,\mathrm{MPa}\), the dead lithium area first decreases, then increases, and finally decreases again.
2. When the mechanical field is coupled into the thermal-electrochemical phase-field model, the dissolution cutoff time becomes longer, and the dead lithium area decreases by about \(40.8\%\). The internal core temperature is also lower than in the uncoupled-mechanics case. The mechanically coupled solid electrolyte cell is more stable against temperature variation. Increasing ambient temperature reduces dead lithium, while decreasing ambient temperature increases it. Root stripping slows considerably because the root is mechanically constrained at the nucleation point.
3. Electrochemical parameters strongly affect dead lithium in a solid electrolyte cell. Increasing the diffusion coefficient increases the dead lithium area by about \(55.2\%\), while decreasing it reduces the dead lithium area by about \(24.1\%\). Increasing or decreasing interfacial mobility both reduce dead lithium, but through different mechanisms: higher mobility smoothens deposition, while lower mobility makes stripping slower and more complete. Increasing anisotropic strength to \(0.15\) reduces dead lithium slightly, by about \(3.4\%\), whereas decreasing it to \(0.05\) reduces dead lithium by about \(12.0\%\). Reducing anisotropic strength is therefore more effective for suppressing dead lithium in the solid electrolyte cell.
4. The coupled results show that dead lithium in a solid electrolyte cell cannot be reliably predicted from the dissolution cutoff time alone. The initial deposition area, root geometry, stress concentration, thermal transport, and interface kinetics all contribute. The most favorable conditions in my simulations combine moderate or high temperature, carefully selected external pressure, reduced diffusion coefficient, tuned interfacial mobility, and reduced anisotropic strength. These conditions promote uniform lithium deposition, slower or more complete stripping, and lower residual dead lithium in the solid electrolyte cell.
| Main finding | Solid electrolyte cell condition | Dead lithium response |
|---|---|---|
| Heat-transfer coupling | Force-electrochemical plus heat | Dead lithium area increases |
| Mechanical coupling | Thermal-electrochemical plus mechanics | Dead lithium area decreases |
| Low external pressure | 5 MPa | Dead lithium area decreases relative to intermediate pressure |
| High external pressure | 20 MPa | Dead lithium area decreases again after an intermediate increase |
| Higher temperature | 353 K | Dead lithium area decreases |
| Lower temperature | 273 K | Dead lithium area increases |
| Lower diffusion coefficient | Slower lithium transport | Dead lithium area decreases |
| Higher interfacial mobility | Smoother interface kinetics | Dead lithium area decreases strongly |
| Lower interfacial mobility | Slower stripping | Dead lithium area decreases |
| Lower anisotropic strength | More isotropic growth | Dead lithium area decreases |
In summary, my phase-field study shows that dead lithium in a solid electrolyte cell is a multiphysics problem. The solid electrolyte cell can suppress dendrite growth mechanically, but dead lithium still accumulates when stripping is incomplete or when stress-driven detachment occurs. The coupled force-thermal-electrochemical framework provides a useful route for evaluating how operating conditions and material parameters can be tuned to reduce dead lithium and extend the life of a solid electrolyte cell.