Solid-state batteries represent a transformative advancement in energy storage technology due to their enhanced safety, high energy density, and superior cycle stability compared to conventional liquid electrolyte systems. However, the growth of lithium dendrites remains a critical challenge that compromises the performance and safety of solid-state batteries. Dendrite penetration through solid electrolytes can lead to internal short circuits, capacity fade, and thermal runaway. Understanding and mitigating dendrite formation is thus paramount for the commercialization of next-generation solid-state batteries. In this study, we employ a phase field model coupled with mechanical, thermal, and electrochemical phenomena to investigate the inhibition of lithium dendrite growth through optimized nanoskeleton structures and artificial separator morphologies. The phase field method effectively captures the complex interfacial dynamics and dendrite evolution under various conditions, providing insights into design strategies for robust solid-state battery architectures.
The phase field model integrates multiple physical fields to simulate dendrite growth in solid-state batteries. The evolution of the phase field variable $\xi$, which distinguishes the electrode ($\xi=1$) and electrolyte ($\xi=0$) phases, is governed by the following equation:
$$ \frac{\partial \xi}{\partial t} = -L_\sigma \left( f_{ch}’ + \kappa \nabla^2 \xi + f_{els}'(\xi) \right) – L_\eta h'(\xi) \left\{ \exp\left[ \frac{(1-\alpha)nF\eta_\alpha}{RT_0} \right] – \frac{c_{Li^+}}{c_0} \exp\left[ \frac{-\alpha nF\eta_\alpha}{RT_0} \right] \right\} $$
Here, $L_\sigma$ is the interfacial mobility, $L_\eta$ is the electrochemical reaction constant, $f_{ch}$ is the chemical energy density, $\kappa$ is the gradient energy coefficient incorporating anisotropy, $f_{els}$ is the elastic energy density, $h(\xi)$ is an interpolation function, $\alpha$ is the symmetry factor, $n$ is the charge transfer number, $\eta_\alpha$ is the overpotential, $c_{Li^+}$ is the lithium ion concentration, $c_0$ is the standard concentration, $F$ is Faraday’s constant, $R$ is the gas constant, and $T_0$ is the ambient temperature. The chemical energy density is expressed as a double-well potential:
$$ f_{ch} = W \xi^2 (1-\xi)^2 $$
where $W$ is the energy barrier height. The gradient energy term accounts for anisotropic effects:
$$ \kappa = \kappa_0 \left[ 1 + \delta \cos(\omega \theta) \right] $$
with $\kappa_0$ as the base gradient coefficient, $\delta$ as the anisotropy strength, $\omega$ as the anisotropy mode number, and $\theta$ as the angle between the interface normal and the coordinate axis. The elastic energy density $f_{els}$ couples mechanical stresses:
$$ f_{els}(\xi) = \frac{1}{2} C_{ijkl} \varepsilon_{ij}^E \varepsilon_{kl}^E $$
where $C_{ijkl}$ is the elasticity tensor, and $\varepsilon_{ij}^E$ is the elastic strain tensor. The effective Young’s modulus $E$ and Poisson’s ratio $\nu$ are interpolated between electrode and electrolyte phases:
$$ E = E_e h(\xi) + E_s \left[1 – h(\xi)\right] $$
$$ \nu = \nu_e h(\xi) + \nu_s \left[1 – h(\xi)\right] $$
The lithium ion transport follows a modified Fick’s law, incorporating thermal effects:
$$ \frac{\partial c_{Li^+}}{\partial t} = \nabla \cdot \left[ D_{eff} \nabla c_{Li^+} + \frac{D_{eff} c_{Li^+}}{RT_0} nF \nabla \varphi \right] – \chi \frac{d\xi}{dt} $$
where $D_{eff}$ is the effective diffusion coefficient, $\varphi$ is the electric potential, and $\chi$ is a constant related to the reaction rate. The temperature-dependent diffusion coefficient is given by:
$$ D_{eff} = A \exp\left[ -r c_{Li^+} + \frac{E_\alpha}{R} \left( \frac{1}{T} – \frac{1}{T_0} \right) \right] $$
The electric potential is solved using Poisson’s equation:
$$ \nabla \cdot (\sigma_{eff} \nabla \varphi) = F C_s \frac{\partial \xi}{\partial t} $$
where $\sigma_{eff}$ is the effective electrical conductivity, and $C_s$ is the solid-phase lithium concentration.
To investigate the role of nanoskeletons in dendrite suppression, we introduce an additional phase field variable $\psi$ to represent the nanoskeleton phase ($\psi=1$). The chemical energy density is extended as:
$$ f_{ch} = \sum_i c_i \mu_i + \left\{ W \xi^2 (1-\xi)^2 + W_1 \psi^2 (1-\psi)^2 + W_2 \xi^2 \psi^2 + RT \left( c_{Li^+} \ln \frac{c_{Li^+}}{c_0} + c_{Am^-} \ln \frac{c_{Am^-}}{c_0} \right) \right\} $$
The mechanical and transport properties are modified to include the nanoskeleton phase. The effective Young’s modulus becomes:
$$ E = E_e h(\xi) + \left[1 – h(\xi)\right] \left\{ E_n h(\psi) + \left[1 – h(\psi)\right] E_s \right\} $$
and the effective diffusion coefficient is:
$$ D_{eff} = D_e \left[ h(\xi) + h(\psi) \right] + D_s \left[ 1 – h(\xi) – h(\psi) \right] $$
We simulated two common nanoskeleton architectures: nanotube arrays and hierarchical porous structures. The nanotube array consists of uniformly spaced cylindrical obstacles, while the hierarchical structure features a multiscale porous network. The results demonstrate that nanoskeletons effectively redirect dendrite growth and reduce stress concentrations.
| Parameter | Symbol | Value |
|---|---|---|
| Interfacial Mobility | $L_\sigma$ | $1 \times 10^{-6}$ m³/(J·s) |
| Reaction Constant | $L_\eta$ | 0.5 s⁻¹ |
| Energy Barrier Height | $W$ | $3.75 \times 10^5$ J/m³ |
| Gradient Energy Coefficient | $\kappa_0$ | $1 \times 10^{-10}$ J/m |
| Anisotropy Strength | $\delta$ | 0.1 |
| Anisotropy Mode Number | $\omega$ | 4 |
| Symmetry Factor | $\alpha$ | 0.5 |
| Ambient Temperature | $T_0$ | 293 K |
| Electrode Conductivity | $\sigma_e$ | $1 \times 10^7$ S/m |
| Electrolyte Conductivity | $\sigma_s$ | 0.1 S/m |
| Electrode Diffusion Coefficient | $D_e$ | $1.7 \times 10^{-15}$ m²/s |
| Electrolyte Diffusion Coefficient | $D_s$ | $2 \times 10^{-15}$ m²/s |
| 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 |
| Pre-exponential Factor | $A$ | $2.582 \times 10^{-9}$ m²/s |
| Solid Phase Li Concentration | $C_s$ | $7.64 \times 10^4$ mol/m³ |
| Standard Li Concentration | $c_0$ | $1 \times 10^3$ mol/m³ |
The impact of nanoskeleton surface roughness on dendrite growth was systematically evaluated. Uniform and regular surface textures, such as triangular protrusions, significantly suppressed primary dendrite height. In nanotube array structures, the maximum dendrite height decreased by 16.62%, while in hierarchical structures, the reduction reached 21.04%. This inhibition stems from the dispersion of ion flux and guidance of lateral dendrite growth. However, increasing roughness non-uniformity diminished these benefits. For irregular surface morphologies, the dendrite height increased by 17.87% in nanotube arrays and 25.57% in hierarchical structures, indicating heightened sensitivity to roughness distribution in multiscale frameworks.
Artificial separators play a crucial role in regulating ion transport and mechanical confinement in solid-state batteries. We introduced a phase field variable $\phi$ to model the separator phase, with the chemical energy density formulated as:
$$ f_{ch} = W \xi^2 (1-\xi)^2 + W \phi^2 (1-\phi)^2 + M \xi^2 \phi^2 + c_{Li^+} \left( \mu_+ + RT \ln \frac{c_{Li^+}}{c_0} \right) + c_{Am^-} \left( \mu_- + RT \ln \frac{c_{Am^-}}{c_0} \right) $$
where $M$ represents the energy barrier between the separator and electrode phases. We designed double-layer porous separators with varying pore sizes (0.3 μm, 0.4 μm, and 0.5 μm) and thicknesses (0.2 μm and 0.4 μm) to assess their influence on dendrite inhibition. Smaller pore sizes enhanced dendrite suppression by constraining longitudinal ion transport and increasing the energy required for dendrite propagation. For instance, reducing the pore size from 0.5 μm to 0.3 μm decreased the primary dendrite height by approximately 25%.
| Separator Configuration | Pore Size (μm) | Thickness (μm) | Dendrite Height Reduction |
|---|---|---|---|
| Baseline | 0.5 | 0.2 | 0% |
| Optimized Pore | 0.4 | 0.2 | 17.70% |
| Optimized Thickness | 0.5 | 0.4 | 6.95% |
| Combined Optimization | 0.4 | 0.4 | 24.65% |
Increasing separator thickness from 0.2 μm to 0.4 μm alone resulted in a modest height reduction of 6.95%. However, combining thickness increase with pore size reduction to 0.4 μm amplified the suppression to 17.70%, underscoring the synergy between geometric parameters. The mechanical resistance of thicker separators impedes both longitudinal and lateral dendrite growth, while smaller pores prolong the ion transport path, dissipating dendrite formation energy.
We further proposed a novel “tile”-shaped separator cross-section to enhance dendrite inhibition. This morphology features curved surfaces that guide lithium ions along specific trajectories, reducing vertical dendrite progression. The convex and concave segments alter ion distribution and stress fields, leading to a 12.75% decrease in primary dendrite height compared to conventional rectangular cross-sections. The ion concentration profiles confirm redirected flux toward the concave regions, minimizing localized dendrite initiation.
The phase field simulations reveal that optimized nanoskeleton and separator morphologies significantly mitigate dendrite risks in solid-state batteries. Uniform nanoskeleton roughness distributes mechanical stresses and ion fluxes, while tailored separator geometries regulate transport pathways. These findings provide design guidelines for high-performance solid-state battery architectures, emphasizing the importance of morphological control in enhancing safety and longevity. Future work will explore dynamic cycling conditions and multi-physics interactions to further advance solid-state battery technology.
In conclusion, the integration of nanoskeletons and artificial separators with optimized morphologies offers a promising approach to suppress lithium dendrite growth in solid-state batteries. The phase field model, incorporating mechanical, thermal, and electrochemical couplings, effectively captures the underlying mechanisms and enables predictive design. By advancing material architectures, we can unlock the full potential of solid-state batteries for sustainable energy storage applications.