Collective Buckling in Metal-Organic Framework Materials

TL;DR

The work addresses how collective buckling arises in MOFs by linking a single-linker soft-mode double-well potential to a dipolar coupling between linkers, forming a lattice model analyzed with mean-field theory. It provides MOF-5 parametrization via DFT, showing a strain-dependent ferrobuckling transition with a calculable $T_C$, and discusses a potential quantum crossover to a parabuckling state described by a transverse-field Ising model, which is negligible for MOF-5. The main contributions are a rigorous derivation of the buckling coordinate, a dipolar interaction framework, and a self-consistent mean-field treatment yielding actionable predictions for strain-tunable buckling in MOFs. This framework creates a quantitative bridge from microscopic linker dynamics to macroscopic phase behavior in framework materials, with potential applicability to more complex 3D MOF networks.

Abstract

We develop a framework to describe collective buckling in metal-organic frameworks (MOFs). Starting from the microscopic structure of a single organic linker, we define a buckling coordinate governed by an effective double-well potential. Coupling between linkers arises from dipole-dipole interactions, leading to a lattice Hamiltonian. We analyze the transition between ordered and disordered phases within a mean-field approximation and determine the critical temperature. As an example for our theory, we discuss the collective buckling instability for the prototypical cubic framework MOF-5 under different values of uniaxial strain. Our approach enables a quantitative description of collective buckling in framework materials.

Figures (6)

• Figure 1: Molecular buckling and its interactions in MOFs. (a) Schematic illustration of a MOF composed of inorganic building units (IBU), shown as tetrahedra, and molecular linkers shown as solid lines. The amplitude of the buckling is described by the scalar quantity $b$, the mode amplitude. The interaction of molecules, denoted by $J_{ij}$, is modeled as a dipole-dipole interaction. (b) Effective potential of molecular buckling. Parabolic behaviour (dashed line) describes straight molecules without buckling. The double-well potential (solid line) prefers molecular buckling.
• Figure 2: Distribution of linker angles for the 1,4-benzenedicarboxylate (bdc) linker of the metal-organic framework MOF-5, obtained from the Cambridge Structural Database (CSD) CSD. An angle of 180$^{\circ}$ corresponds to a straight unbuckled configuration, while smaller angles indicate increasing buckling. The plot stops at the angle 179.5$^{\circ}$ as the large majority of structures have a symmetry restricted angle of exactly 180$^{\circ}$.
• Figure 3: Illustration of the atomic displacement for the bdc linker of MOF-5. The relaxed configuration is shown on the left and in the center. The buckled configuration is shown on the right with atomic displacements $\mathbf{\delta x}_j$.
• Figure 4: Buckling of bdc molecules in MOF-5. (a) The crystal structure of MOF-5. Zn$_4$O clusters are coordinated by bdc molecules, forming a periodic three-dimensional cubic network. (b) Top view of a (100)-plane (c) Total energy versus the buckling of bdc molecules for various strain values. Under strain, the total energy follows the form of a double-well potential.
• Figure 5: Mean-field order parameter $m$ as function of temperature for different values of applied strain. Phase diagram in the temperature-strain plane, showing a transition between a ferrobuckling phase and a disordered phase.