Machine-Learning Potentials Predict Orientation- and Mode-Dependent Fracture in Refractory Diborides

TL;DR

This work addresses the challenge of predicting fracture properties in brittle ceramics by developing machine-learning interatomic potentials (MLIPs) trained on ab initio data for TiB$_2$, ZrB$_2$, and HfB$_2$ (TMB$_2$). Using $K$-controlled molecular statics with atomistic cracked-plate models, the authors map orientation- and mode-dependent fracture pathways across six crack geometries and extrapolate macroscale initiation toughness $K_{Ic}^{inity}$ and fracture strength $\sigma_{ ext{max}}^{inity}$, finding intrinsic brittleness with $K_{Ic}^{inity}$ in the ~1.8–2.9 MPa$\, ext{m}^{1/2}$ range and $\sigma_{ ext{max}}^{inity}$ around 2.0 GPa. They further explore mixed Mode-I/II loading in TiB$_2$, revealing mode-dependent crack deflection onto pyramidal planes and establishing a practical mixed-mode framework with $K^{ ext{mix}}=\\sqrt{K_I^2+K_{II}^2}$ and related partitioning, supported by nanoindentation experiments showing oblique crack trajectories near 40 degrees. Comparisons with experiments highlight microstructural effects not captured in defect-free atomistic models, motivating future extensions to finite temperature and defect-inclusive microstructures. Overall, the MLIP-based approach provides a predictive, atomistic framework for orientation- and mode-dependent fracture in refractory ceramics with potential for broader applicability.

Abstract

Fracture toughness ($K_\mathrm{Ic}$) and fracture strength ($σ_\mathrm{f}$) are key criteria in the selection and design of reliable ceramics. However, their experimental characterization remains challenging -- especially for ceramic thin films, where size and interfacial effects hinder accurate and reproducible measurements. Here, machine-learning interatomic potentials (MLIPs) trained on \textit{ab initio} datasets of single crystal models deformed up to fracture are used to characterize transgranular cleavage in pre-cracked ceramic diboride TMB$_2$ (TM = Ti, Zr, Hf) lattices through stress intensity factor ($K$)-controlled loading. Mode-I simulations performed across distinct crack geometries show that fracture is primarily driven by straight crack extension along the original plane. The corresponding macroscale fracture-initiation properties ($K_\mathrm{Ic} \approx 1.7$-2.9 MPa$\cdot\sqrt{\text{m}}$, $σ_\mathrm{f} \approx 1.6$-2.4 GPa) are extrapolated using established scaling laws. Considering TiB$_2$ as a representative system, additional simulations explore loading conditions ranging from pure Mode-I (opening) to Mode-II (sliding). TiB$_2$ models containing prismatic cracks exhibit their lowest fracture resistance under mixed-mode conditions, where the crack deflects onto pyramidal planes--as confirmed by nanoindentation tests on TiB$_2$(0001) thin films. This study establishes $K$-controlled, MLIP-based simulations as predictive tools for orientation- and mode-dependent fracture in ceramics. The approach is readily extendable to finite temperatures for evaluating fracture behavior under conditions relevant to refractory applications.

Figures (8)

• Figure 1: Uniaxial tensile simulations at 300 K using MLIP-based molecular dynamics (ML-MD). (a) Maximum stress (theoretical tensile strength) sustained by TMB$_{2}$:s, (TM$=$Ti, Zr, Hf) during uniaxial tensile deformation along [0001] (square), $[10\overline{1}0]$ (vertical hexagon), and $[\overline{1}2\overline{1}0]$ (horizontal hexagon) directions. (b) Cleavage fracture after reaching the maximum stress point, with plane identification. The image on the right in (a) shows a diboride hexagonal lattice structure in the $\alpha$-phase, including orthogonal crystal axes. The shadowed boron hexagonal layer is aligned with the basal plane.
• Figure 2: Cracked-plate lattice geometries of hexagonal $\alpha$-structured diborides considered in this work. (a) (0001) crack surface with $[10\overline{1}0]$ (Nr.1) and $[\overline{1}2\overline{1}0]$ (Nr.2) crack-front directions. (b) $(\overline{1}2\overline{1}0)$ crack surface with $[10\overline{1}0]$ (Nr.3) and [0001] (Nr.4) crack-front directions. (c) $(10\overline{1}0)$ crack surface with [0001] (Nr.5) and $[\overline{1}2\overline{1}0]$ (Nr.6) crack-front directions.
• Figure 3: Bond breakage and volumetric strain distribution in cracked plate models subjected to Mode-I loading as a function of the stress intensity factor. Atomic configurations just before, shortly after, and well above $K_{Ic}$ for different crack geometries: (a, b) (0001), (c, d) $(\overline{1}2\overline{1}0)$, and (e, f) $(10\overline{1}0)$ planes. The simulation snapshots illustrate volumetric strain distributions in blue/red color scale.
• Figure 4: Comparison of extrapolated Mode-I fracture toughness $K_{Ic}^{\infty}$ (a) and fracture strength $\sigma_{\text{max}}^{\infty}$ (b) for TiB$_2$ (teal), ZrB$_2$ (orange), and HfB$_2$ (red), across all six crack geometries. The geometry indices $Nr.1$--$Nr.6$ are also colored in teal--orange--red, consistent with Fig. \ref{['Geo']}. Note that all crack geometries are modeled for all three materials. (c) Illustration of the influence of the plate area on the calculated fracture toughness (case of TiB$_2$ with crack geometry $Nr.1$). Note that the fracture mechanism remains qualitatively unchanged with increasing size of the supercell.
• Figure 5: Fracture mechanisms in Group-IV TMB$_2$ compounds following crack initiation ($K_I \gtrsim K_{Ic}$). Simulation snapshots are shown for one representative example per crack geometry: (a) $(0001)[10\overline{1}0]$ (Nr. 1), (b) $(\overline{1}2\overline{1}0)[10\overline{1}0]$ (Nr. 3), and (c) $(10\overline{1}0)[\overline{1}2\overline{1}0]$ (Nr. 6). Each panel compares TiB$_2$ (*-1), ZrB$_2$ (*-2), and HfB$_2$ (*-3) under pure Mode-I loading (the stress intensities $K_I$ are expressed in MPa$\cdot\sqrt{m}$). Each subpanel illustrates crack extension due to stress intensities well above $K_{Ic}$ ($K_I \gtrsim K_{Ic} + 0.5~\mathrm{MPa}\sqrt{\mathrm{m}}$), whereas the insets show magnifications of atomic configurations immediately after reaching $K_{Ic}$. Cracks are seen to either propagate along the initial plane or deflect depending on material and orientation. TM atoms (Ti, Zr, Hf) are depicted in dark gray; B atoms in light blue. The selected plate models have an area of $L^2 = 30$ nm $\times$ 30 nm = 900 nm$^2$.