Generic rigidity and accidental modes in metal-organic frameworks

TL;DR

MOFs are prone to mechanical instability and soft guest-induced motions, complicating predictive design. The authors develop a rigidity-matrix framework based on a constraint network with bond-stretching and bond-bending springs parameterized by $UFF4MOF$, linking topology to dynamics via the Maxwell–Calladine index $ \nu \equiv N_0 - N_{\mathrm{ss}} = dN_s - N_c$. Through large-scale analysis of 5,682 CoRE MOFs, they find most structures are formally over-constrained yet cluster near the isostatic threshold, with many accidental zero modes that threaten stability; in UiO-66, adding long-range auxiliary constraints lifts these modes into soft finite-frequency bands. The approach enables rapid, interpretable mechanical screening and suggests a design principle of near-criticality and connections to topological mechanics in porous crystals.

Abstract

Metal-organic frameworks (MOFs) combine high porosity with structural fragility, raising important questions about their mechanical stability. We develop a rigidity-based framework in which spring networks parameterized by UFF4MOF are used to construct rigidity and dynamical matrices. Large-scale analysis of 5,682 MOFs from the CoRE 2019 database shows that most frameworks are formally over-constrained yet cluster sharply near the isostatic threshold, revealing accidental geometric modes and placing many MOFs near mechanical instability. In the representative case of UiO-66, we show that auxiliary long-range constraints introduced by tuning the neighbor cutoff lift these modes into soft, flat, finite-frequency bands. The results show that rigidity-matrix analysis can rapidly identify MOFs likely to remain mechanically stable. This near-criticality mirrors behavior known from topological mechanics and points to a deeper design principle in porous crystals.

Figures (6)

• Figure 1: A: MOFs are typically obtained via solvothermal methods solvothermal, where metal ions or clusters and multidentate organic linkers self-assemble into extended crystalline networks. B: The resulting frameworks exhibit permanent porosity, high internal surface areas, and tunable structures and functionalities. C: Many MOFs are mechanically fragile and can degrade or collapse under external stress, highlighting the importance of identifying mechanically robust frameworks. D: MOFs have been investigated for gas storage, separations, heterogeneous catalysis, and chemical sensing.
• Figure 2: Zero-mode count $N_0$ as a function of the constraint cutoff parameter $\tau$, which controls which neighbors are included in the rigidity model. In our convention, larger $\tau$ reduces the coordination weight used by CrystalNNcrystalNN, removing constraints and increasing $N_0$. Each colored curve corresponds to a different MOF, and purple stars ($\ast$) mark the plateau $\tau$ values selected automatically by CrystalNN. For clarity, the UiO-66 curve is scaled by a factor of 10. The inset shows the FIGXAU framework: Color indicates the baseline bonding network, while the gray version illustrates the auxiliary bonds identified when $\tau$ is reduced.
• Figure 3: Phonon dispersions and inverse participation ratio (IPR) maps for representative materials. (A) ABIXOZ ($\nu/N_s \!=\! -1.89$) stable and over-constrained; (B) IKEBUV01 ($\nu/N_s \!=\! +0.075$) near-isostatic with delocalized soft modes; (C) UiO-66 ($\nu/N_s \!=\! -1.14$) moderately over-constrained with mixed localization behavior.
• Figure 4: (A) Benchmark phonon dispersion calculated via GULP, reproduced from Ref. UiO-66. (B) Corresponding result from our rigidity-based model ($N_0 = 238$) using the default $\tau$ setting. (C) Decreasing the cutoff to $\tau = 0.5$ introduces auxiliary constraints, reducing the zero mode count to $N_0 = 215$. (D) Further tuning to $\tau = 0.035$ eliminates all zero modes ($N_0 = 0$). The progression from (B) to (D) demonstrates how assigning finite stiffness to long-range auxiliary bonds lifts zero modes into soft, collective bands, progressively filling the low-frequency spectrum relative to the benchmark in (A).
• Figure 5: Rigidity statistics across the CoRE MOF dataset. (A) Scatter plot of $\nu$ for 500 representative MOFs, with the red dashed line marking the isostatic threshold ($\nu = 3$). (B) Histogram of the Maxwell–Calladine index density ($\nu/N$) for 5,182 MOFs at $\Gamma$. The majority of frameworks are formally over-constrained yet cluster sharply near the isostatic point. This concentration is dominated by accidental modes, which place many MOFs on the verge of mechanical instability.