In recent years, solid-state batteries have emerged as a pivotal focus in energy storage research due to their enhanced safety, high energy density, and extended cycle life. Unlike conventional liquid electrolytes, solid-state electrolytes offer superior mechanical strength, which can mitigate the growth of lithium dendrites—a phenomenon that compromises battery efficiency, lifespan, and safety. However, complete suppression of dendrite formation remains challenging. This study employs a phase field model coupled with mechanical, thermal, and electrochemical interactions to investigate how morphological optimizations in nanoskeletons and artificial separators influence lithium dendrite growth in solid-state batteries. By simulating various nanostructures and separator configurations, we identify key parameters that enhance dendrite suppression, contributing to the development of more reliable solid-state battery systems.
The phase field method is a powerful computational tool for modeling microstructural evolution in materials, including dendrite formation in batteries. In this work, we extend a previously established phase field framework to incorporate multiphysics interactions. The evolution of the phase field variable $\xi$, representing the lithium metal phase, is governed by the following equation:
$$ \frac{\partial \xi}{\partial t} = -L_\sigma \left( f_{ch}’ + \kappa \nabla^2 \xi + f_{els}'(\xi) \right) – L_\eta h'(\xi) \left{ \exp \left[ \frac{(1-\alpha) n F \eta_\alpha}{R T_0} \right] – \frac{c_{Li^+}}{c_0} \exp \left[ \frac{-\alpha n F \eta_\alpha}{R T_0} \right] \right} $$
Here, $L_\sigma$ denotes the interfacial mobility, $L_\eta$ is the electrochemical reaction constant, and $h(\xi) = \xi^3 (6\xi^2 – 15\xi + 10)$ is an interpolation function. The chemical energy density $f_{ch}$ is expressed as a double-well potential $W \xi^2 (1-\xi)^2$, where $W$ represents the energy barrier for lithium ion migration. The gradient energy coefficient $\kappa$ accounts for anisotropic effects: $\kappa = \kappa_0 [1 + \delta \cos(\omega \theta)]$, with $\kappa_0$, $\delta$, $\omega$, and $\theta$ denoting the base gradient coefficient, anisotropy strength, anisotropy mode number, and interface normal angle, respectively. The elastic energy density $f_{els}(\xi)$, coupled to mechanical effects, is given by:
$$ f_{els}(\xi) = \frac{1}{2} C_{ijkl} \varepsilon_{ij}^E \varepsilon_{kl}^E $$
where $C_{ijkl}$ is the elasticity tensor, and $\varepsilon_{ij}^E$ represents the elastic strain components. The effective Young’s modulus $E$ and Poisson’s ratio $\nu$ are interpolated as $E = E_e h(\xi) + E_s [1 – h(\xi)]$ and $\nu = \nu_e h(\xi) + \nu_s [1 – h(\xi)]$, respectively, with subscripts $e$ and $s$ referring to electrode and solid electrolyte properties. Lithium ion diffusion follows Fick’s law, modified for electrochemical potential:
$$ \frac{\partial c_{Li^+}}{\partial t} = \nabla \cdot \left[ D_{eff} \nabla c_{Li^+} + D_{eff} \frac{c_{Li^+}}{R T_0} n F \nabla \varphi \right] – \chi \frac{d\xi}{dt} $$
The effective diffusion coefficient $D_{eff}$ incorporates temperature dependence: $D_{eff} = A \exp \left[ -r c_{Li^+} + \frac{E_\alpha}{R} \left( \frac{1}{T} – \frac{1}{T_0} \right) \right]$, where $A$ is a pre-exponential factor, $E_\alpha$ is the activation energy, and $r$ is a fitting parameter. The electric potential $\varphi$ is solved using Poisson’s equation:
$$ \nabla \cdot (\sigma_{eff} \nabla \varphi) = F C_s \frac{\partial \xi}{\partial t} $$
with $\sigma_{eff} = \sigma_e h(\xi) + \sigma_s [1 – h(\xi)]$ representing the effective electrical conductivity. Key parameters used in our simulations are summarized in Table 1.
| 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 |
| Environmental 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 |
| Vagard Strain Coefficients | $\lambda_i$ | $-0.866 \times 10^{-3}$, $-0.773 \times 10^{-3}$, $-0.529 \times 10^{-3}$ |
| Pre-exponential Factor | $A$ | $2.582 \times 10^{-9}$ m²/s |
| Solid Phase Lithium Concentration | $C_s$ | $7.64 \times 10^4$ mol/m³ |
| Standard Lithium Concentration | $c_0$ | $1 \times 10^3$ mol/m³ |
To analyze nanoskeleton effects, we introduce an additional phase field variable $\psi$, where $\xi = 0$ and $\psi = 1$ defines the nanoskeleton structure. The chemical energy density is modified as:
$$ f_{ch} = \sum_i c_i \mu_i + \left{ W \xi^2 (1-\xi)^2 + W_1 \psi^2 (1-\psi)^2 + W_2 \xi^2 \psi^2 + R T \left( c_{Li^+} \ln \frac{c_{Li^+}}{c_0} + c_{Am^-} \ln \frac{c_{Am^-}}{c_0} \right) \right} $$
where $W_1$ and $W_2$ represent energy barriers between the nanoskeleton and electrode/electrolyte phases. The effective Young’s modulus and diffusion coefficient are updated to $E = E_e h(\xi) + [1 – h(\xi)] \left{ E_n h(\psi) + [1 – h(\psi)] E_s \right}$ and $D_{eff} = D_e [h(\xi) + h(\psi)] + D_s [1 – h(\xi) – h(\psi)]$, respectively. We simulated two common nanoskeleton morphologies: nanotube array structures and hierarchical/multilevel porous frameworks. In nanotube arrays, uniformly spaced nanotubes guide lithium ion transport, while hierarchical structures feature densely packed porous skeletons that alter dendrite growth paths. Our results demonstrate that nanoskeletons with uniform surface roughness, such as triangular protrusions, reduce the primary dendrite height by up to 21.04% by promoting lateral ion distribution and dissipating growth energy. However, increasing roughness non-uniformity elevates dendrite height by 25.57%, indicating that optimal inhibition requires regular and homogeneous nanostructures.
For artificial separators, we introduce a phase field variable $\phi$ to represent the separator phase. The chemical energy density is expressed as:
$$ f_{ch} = W \xi^2 (1-\xi)^2 + W \phi^2 (1-\phi)^2 + M \xi^2 \phi^2 + c_{Li^+} \left( \mu_+ + R T \ln \frac{c_{Li^+}}{c_0} \right) + c_{Am^-} \left( \mu_- + R T \ln \frac{c_{Am^-}}{c_0} \right) $$
where $M$ denotes 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 evaluate their impact on dendrite suppression. Smaller pores (e.g., 0.3 μm) restrict longitudinal lithium ion movement, forcing dendrites to expend more energy on lateral growth and reducing primary dendrite height. Increasing separator thickness enhances mechanical resistance, but combining thickness optimization with reduced pore spacing yields superior results. For instance, a separator with 0.4 μm thickness and 0.4 μm pore spacing reduces dendrite height by 17.70%, compared to 6.95% for a 0.2 μm thickness and 0.5 μm pore spacing. This represents a 10.75% improvement over thickness-only optimization.
Furthermore, we propose a novel “tile”-shaped separator cross-section, featuring concave and convex curves, to direct lithium ion motion. Compared to conventional rectangular cross-sections, this morphology reduces primary dendrite height by 12.75% by encouraging downward lateral growth and dissipating energy through curved pathways. The concave regions accumulate lithium ions, while convex structures deflect growth angles, minimizing vertical penetration risks. Both upward and downward convex configurations show similar efficacy, with upward convex designs slightly outperforming. This morphological innovation highlights the potential of tailored separator geometries in enhancing solid-state battery performance.
In conclusion, our phase field simulations reveal that nanoskeleton roughness uniformity and artificial separator morphology critically influence lithium dendrite growth in solid-state batteries. Uniformly rough nanoskeletons suppress dendrite height by over 20%, while non-uniformity exacerbates growth. Similarly, optimizing separator thickness and pore size synergistically inhibits dendrites, and “tile”-shaped cross-sections offer additional suppression. These findings provide actionable strategies for designing next-generation solid-state batteries with improved safety and longevity. Future work will explore dynamic cycling conditions and experimental validation to further refine these models.