Origamic Metal-Organic Framework Toward Mechanical Metamaterial

The study addresses whether origami tessellation principles can be realized at the molecular scale within a crystalline framework to impart unusual mechanical responses, such as deployability and negative Poisson’s ratio. Origami design has influenced diverse technologies and spans length scales from macro to nano. Known tessellations (e.g., Miura-ori, double corrugation surface (DCS), Ron Resch, waterbomb, Yoshimura, square twist) confer extreme kinematics and metamaterial properties. While origami-inspired architectures exist at larger scales, translating tessellations into molecular materials remains challenging. Metal-organic frameworks (MOFs) offer tunable nodes and linkers with inherent flexibility enabling structural transformations; however, dynamic behavior has largely been interpreted via topology rather than explicit origami geometry. This work proposes and tests a MOF whose 2D sheet conforms to a DCS-like tessellation, seeking to reveal and validate origami-governed folding at the molecular level.

Prior work shows origami patterns can program deployability and mechanical metamaterial properties, including negative Poisson’s ratio and tunable thermal expansion. Common tessellations (Miura-ori, DCS, square twist) can share repeating patterns yet differ in folding due to mountain/valley assignments. MOFs are known for flexibility (breathing, swelling) arising from deformable building blocks and have shown anomalous responses (negative thermal expansion, negative linear compressibility, NPR) often rationalized by topological models. However, explicit geometric analysis using origami tessellations has been limited. Reports of flexible porphyrinic MOFs and 2D frameworks underscore solvent-dependent stacking and thermal responses. The concept of meta-MOFs highlights framework metamaterial behavior, motivating an origami-tessellation-based interpretation for hidden dynamics in MOFs.

• Synthesis: PPF-301 synthesized solvothermally from Zn(NO3)2·6H2O and 5,10,15,20-tetrakis[4-carboxymethyleneoxyphenyl] porphyrin (TCMOPP) in DMF/EtOH (3:1) with HNO3 at 80 °C for 24 h, followed by cooling to yield purple plate-like crystals. Porphyrin metallates to Zn during assembly; Zn paddlewheel SBUs form with DMF axially coordinated.
• Structural characterization: Synchrotron PXRD (PAL 2D SMC) confirmed phase and isostructure. Temperature-dependent synchrotron SCXRD (100–380 K, 20 K steps) in sealed capillaries with mother liquor retained crystallinity; data processed with HKL3000; structures solved/refined with SHELXL; disordered solvent handled via PLATON SQUEEZE. Thermal expansion coefficients computed using PASCal.
• Spectroscopy and analysis: FT-IR identified coordinated DMF; 1H NMR quantified solvent content (TCMOPP:DMF ≈ 1.08:5.04); TGA assessed stability (to ~700 K). Gas sorption: N2 at 77 K (non-porous); CO2 adsorption at 195/273/298 K measured.
• Electron microscopy: HR-TEM on as-synthesized and solvent-exchanged (DMF, EtOH) samples; FFT and Fourier filtering used to visualize tessellation; compared to a mathematical origami model.
• Origami geometric model: The 2D layer simplified into tiles (A–D) and nodal points (aryloxy O atoms). Defined folding angles θ1, θ2 and in-plane distances d1, d2; derived relationships between θ1–θ2 and θ1–d1/d2 from DCS tessellation geometry; compared against SCXRD-derived parameters.
• Molecular origin: Quantified dihedral and bond angles of aryloxy pivots (φA, φB; C–C–O–C dihedral; α = C–O–C). Computed potential energy surface for isolated aryloxy group across φ and α; compared to CSD distributions to assess energetic feasibility of observed changes.
• Mechanical properties: DFT calculations (VASP) to optimize structure and compute total energies; elastic tensor via ElaStic; elastic property visualization and directional extrema via ELATE, yielding Young’s modulus, shear modulus, linear compressibility, and Poisson’s ratio including NPR directions. Built a deployable mechanism model consistent with DCS folding-unfolding of the corrugated sheet.

• Structure and tessellation: PPF-301 is a 2D porphyrinic MOF comprising Zn paddlewheel SBUs and Zn-metallated TCMOPP linkers forming a corrugated sheet with an hourglass/DCS-like tessellation when simplified by nodal aryloxy O atoms. Four tile types (A–D) map to SBUs, linkers, and hollows.
• Thermal response (100–380 K): Unit-cell volume increases by ~5.2% on heating; 2D area S of the sheet increases by ~2.0%; interlayer spacing increases by ~3.1%; layer thickness d3 decreases by ~2.6% (negative thermal expansion of the 2D layer). Colossal thermal expansion along principal axis X3: αX3 = 170(3) MK⁻1. Tile areas A–D change minimally (−2.5% to +0.5%), indicating expansion driven by folding kinematics rather than tile dilation.
• Origami mechanics validation: Folding angles increase with temperature: θ1 by ~2.9° and θ2 by ~3.9% (SCXRD). Experimental θ1–θ2 and θ1–d1/d2 relationships match the DCS-based geometric model: d1 = √[(l′ sin α cos θ1 − 1)² + l² cos²α + l² sin²α sin²θ1] (definitions in Supplementary). Thus, flattening upon heating corresponds to increased folding angles, consistent with DCS deployment.
• Molecular pivot origin: Changes in aryloxy dihedrals correlate with folding: φA increases by ~1.0°, φB by ~2.4°. PES of isolated aryloxy group shows a shallow well; observed φ and α lie near low-energy regions populated in CSD, enabling folding with low energetic penalty.
• Mechanical properties (DFT/elasticity): Young’s modulus E ranges from 3.79 to 20.05 GPa (anisotropy Ax ≈ 5.30). Poisson’s ratio exhibits a negative minimum νmin = −0.107 along u = (−0.766, 0.438, 0.471) and v = (−0.314, 0.385, −0.868). Linear compressibility spans βmin = −1.90 TPa⁻1 to βmax = 128.11 TPa⁻1; shear modulus G ranges 1.50–6.40 GPa. NPR arises from counter-rotation of SBU and linker skeletons that unfold folded regions.
• Stability and porosity: Thermally stable to ~700 K; non-porous to N2 at 77 K; CO2 uptakes: 4.26 mmol/g (195 K), 1.96 mmol/g (273 K), 1.77 mmol/g (298 K).
• Solvent effects and robustness: HR-TEM Fourier-filtered images confirm the DCS-like tessellation persists after solvent exchange (DMF, EtOH), indicating the folding mechanism is intrinsic to the sheet geometry despite solvent-dependent stacking and degree of folding.

The findings demonstrate that a MOF can embody a recognizable origami tessellation (DCS) at the molecular scale, and that its thermal and mechanical responses are governed by tessellation kinematics rather than simple topological flexibility. By correlating SCXRD temperature evolution with a precise geometric model (θ1–θ2, θ1–d1/d2) and identifying aryloxy dihedral/bond angles as pivot points with shallow energetic penalties, the work bridges molecular structure and origami mechanics. The observed negative Poisson’s ratio and colossal thermal expansion along a principal axis establish PPF-301 as a mechanical metamaterial whose response arises from sheet deployment. Persistence of the tessellation across different solvent treatments supports the robustness of the folding mechanism. This approach reframes analysis of flexible MOFs from purely topological to geometrically tessellated frameworks, expanding design routes for programmable mechanical behavior.

This study introduces PPF-301, a 2D porphyrinic MOF whose corrugated layer realizes a DCS origami tessellation, exhibiting temperature-driven folding/unfolding consistent with origami kinematics. The dynamic motion originates in flexible aryloxy pivot angles of the linker, enabling negative layer thickness thermal expansion, colossal principal-axis thermal expansion, and a negative Poisson’s ratio. The work establishes origami tessellations as actionable design principles for “origamic MOFs,” a distinct class of mechanical metamaterials. Future directions include: tailoring pivot flexibility by substituting aryloxy with −CH2−, −S−, or −NH− groups to tune folding amplitude; fabricating controlled nanosheets to enhance and directly observe 2D layer motion; exploring solvent and crystal size effects systematically; and leveraging stimulus-driven folding to modulate metal–metal distances for potential 2D spin qubit frameworks and molecular quantum information applications.

• The magnitude and linearity of thermal responses are influenced by interlayer solvents and stacking, leading to non-linear temperature trends and potentially masking larger intrinsic sheet motions.
• Observed folding is constrained by bulk crystal stacking; larger deployability may require isolated nanosheets or thinner crystals.
• Mechanical property predictions rely on DFT-based elasticity for an idealized structure; experimental validation under mechanical loading at the single-crystal/nanosheet level remains to be performed.
• Gas sorption indicates limited porosity; generality to porous, guest-responsive origamic MOFs needs further exploration.