Computational Mechanics Model to Evaluate the Structural Maintenance of Catenated Hydrogen‐Bonded Organic Frameworks

ABSTRACT

Hydrogen‐bonded organic frameworks (HOFs) have recently been highlighted as next‐generation structural materials owing to their lightweight nature, mechanical flexibility, and chemical selectivity. However, despite extensive research efforts, the understanding of the structural behavior of nanometer‐sized HOFs remains confined to empirical observations. Using molecular dynamics, we uncover how HOF lattices respond mechanically from energy gradients and deformation tests. This highlights that catenation acts as a key source for reduced atomic fluctuations, effective shear redistribution, and emerging auxetic deformation under in‐plane loading. In particular, we demonstrate the robustness of HOF substrates modeled after biomolecular exoskeletons, proposing an engineering perspective on computational methodologies for advancing the structural design of porous organic materials.

This study presents a computational framework treating crystalline HOFs as adaptive atomic‐scale interfaces. Catenated HOFs preserve lattice stability under load while enabling auxetic deformation and reversible nonlinearity. The results provide direct evidence that supramolecular interactions govern macroscopic resilience and structural adaptability in HOF materials.

Abbreviations

σ

virial stress tensor (Pa)

σ _kl

stress in k direction exerted on l plane (Pa)

σ _vM

von Mises shear stress (Pa)

ρ

Bulk number density of atoms (m⁻³)

ε _vM

von mises shear strain

ε _kl

strain in k direction exerted on l plane

E_angle

angle bending energy (J)

E_bond

bond stretching energy (J)

E_cat

catenation stacking energy (J)

E_Coul

coulombic energy (J)

E_ev

valence band energy (J)

E_hb

hydrogen binding energy (J)

E_lp

lone‐pair energy (J)

E_mol

intrinsic molecular energy (J)

E_over

energy penalty term (J)

E_spc

energy from special terms (J)

E_sys

total system energy (J)

E_tors

dihedral torsion energy (J)

E_vdW

van der Waals energy (J)

F

interatomic force (N)

HOF

hydrogen‐bonded organic framework

ij

atom indices

MD

molecular dynamics

m

mass of atom (kg)

N

number of atoms

NVT

canonical ensemble

NPT

isothermal–isobaric ensemble

PBC

periodic boundary condition

p

population density

PE

potential energy

P

pressure (Pa)

q

partial charge (C)

r

displacement between atoms (m)

r

distance from atom position (m)

ReaxFF

reactive force field

T

temperature (K)

vdW

van der Waals

v

velocity of atom (m s⁻¹)

V

system volume (m³)

1. Introduction

Porous materials are attractive because they combine low density, high surface area, and adjustable pore structures [1]. These properties allow them to perform tasks that condensed, solid materials cannot. In mechanical engineering, porous lattices are used for energy absorption, damping, and structural lightweighting [2], since their voids help spread stress and prevent local failure [3, 4]. In chemistry and environmental engineering, pores increase surface‐to‐volume ratios [5], enabling efficient catalysis [6], selective adsorption [7], and ion or gas transport [8]. In biomedical fields, porous scaffolds influence how cells attach and grow [9, 10]. Traditionally, porous systems have been studied in metallic foams [11], zeolites [12], and template‐grown organic crystals [13]. A newer class, hydrogen‐bonded organic frameworks (HOFs), has emerged as highly ordered porous crystals [14]. Their lattice is stabilized by H binding and π–π stacking, creating unique opportunities for designing porous architectures with adaptable structures [15]. Understanding how these frameworks behave mechanically is now an important challenge.

HOFs form a distinct class of supramolecular crystals. Unlike metal organic frameworks or covalent organic frameworks, which rely on strong coordination or covalent bonds [16]. HOFs assemble mainly through directional hydrogen bonds, but their frameworks are also stabilized by π–π stacking [17], charge‐assisted electrostatic interactions [18], and van der Waals (vdW) forces [19]. These noncovalent forces are weaker, yet collectively stable, giving HOFs the ability to assemble and disassemble under different conditions [20]. Their nanoscale pores can be tuned by changing the geometry of the organic building block [21], altering the degree of catenation (overlapping networks) [22], or substituting functional groups that participate in H bonding [23]. This tunability allows HOFs to form ordered pores with precise dimensions, while still remaining mechanically flexible. This is particularly significant in materials engineering because pore size [24], connectivity [25], and framework elasticity [26] can now be controlled at the molecular scale. Thus, HOFs are not only promising for chemical storage and sensing but also as testbeds for linking nanostructure with mechanical response.

Despite their chemical versatility, HOFs are poorly understood from a mechanical perspective. Most research has focused on empirical water uptake [27], ion and proton separation [28], or molecular recognition [29], showing their macroscopic promise in sensors, filters, and responsive devices, respectively. Only recently has it become possible to observe that HOF pores can reversibly tilt and recover to chemical and photo‐stimuli [30, 31, 32]. Yet little is known about how HOFs deform, relax, or reorganize under applied loads. This gap exists because their unit cells are only a few nanometers in size, and current experimental techniques cannot easily capture their real‐time atomic motions [33, 34]. Still, understanding their mechanical adaptability is critical. Their ability to reorganize while preserving crystallinity is what makes them unique compared to other frameworks. If we could directly exploit this adaptive kinematic behavior, HOFs might serve as shock absorbers [35], auxetic metamaterials [36], or flexible membranes [37]. Thus, advancing knowledge of their structural and mechanical behaviors is necessary to translate HOFs from chemical curiosities into robust materials platforms.

Molecular dynamics (MD) simulations now make it possible to explore these questions directly. By modeling all atoms in the system, MD tracks how H bonds, π‐stacking, and electrostatic forces govern stability and deformation [38, 39]. For porous crystals like HOFs, this is especially powerful, as it can reveal how catenation [40], pore geometry [41], and intermolecular forces [42] control load transfer and mechanical response [43]. These computational studies have addressed organic crystals, examining lattice vibrations, thermal stability, or defect formation. But very few have investigated HOFs under mechanical strain. Motivated by this, the present work fills this gap by using MD simulations to systematically analyze their anisotropic deformation. By connecting microstructural rearrangements to stress‐strain behavior, this study provides fundamental insights into how HOFs adapt under load. The computational results aim to guide the rational design of HOF networks for engineering applications where porosity and mechanical adaptability must coexist.

HOFs vary in their network assembly pathways and symmetry depending on the types of molecules and the interactions between them. The recent BioHOF [44] was selected as a representative model for our MD simulations because it combines structural simplicity with robust, well‐defined hydrogen‐bonded interactions, allowing clear interpretation of stress and strain propagation through the lattice. Furthermore, its tetrakis tetrahedral geometry forms a diamond‐like network with consistent spatial lattice constants, making it an ideal system for benchmarking microstructural and mechanical behavior in crystalline HOFs. This provides a clear example of highly ordered crystalline features in a solid manifesting as an organic framework.

2. Results and Discussion

2.1. Fundamental Effect of Catenation to Build Organic Crystal

We perform atomic modeling and MD simulations of HOF to investigate the role of catenation stacking in maintaining inherent structure. The crystallographic information of HOF was obtained from the Cambridge Crystallographic Data Centre (deposit no.: 2169825) [44]. The tetrakis building blocks, each containing amidinium and carboxylate terminals in equal proportion on four phenyl branches, self‐assemble into a diamond lattice. This assembly is stabilized by two hydrogen bonds formed between the terminal groups. Specifically, the H and O atoms from the coplanar CN₂H₄ ⁺ and CO₂ ⁻ terminals exhibit a 1.51 Å proximity, consistent with the reported strong hydrogen bond length of 1.56 Å [45], as shown in Figure 1a. In this study, the distance between the acceptor to H was used as hydrogen bond distance. Based on the conventional donor and acceptor notation for H bonds, our simulations yielded an N–O distance of 2.76 Å. This value is in excellent agreement with the 2.77 and 2.63 Å, respectively, reported in the crystallographic analysis [44] and density functional theory calculation [46]. In this system, the $R_2^2$(8) motif contains two donors and two acceptors. The intermolecular geometry is tightly packed because the tetrakis units assemble into a nearly ideal tetrahedral network. Therefore, shorter hydrogen bonding contacts occur than those typically observed in R₂ ²(8) systems, often measured on the order of ∼2 Å [45]. Such a compact bonding arrangement is also an intrinsic characteristic of the diamond‐like tetrahedral packing of this HOF. All molecular structures were visualized using the Open Visualization Tool software [47].

FIGURE 1.

Preparation of full‐catenation HOF network. (a) Chemical structure of the tetrakis molecules assembled with terminal H bonds. Atomic geometry of the H bonding is shown as inset. (b) Crystallographic unit cell consisting of four amidinium and carboxylate tetrakis molecules. The square symbol (□) indicates same cell lengths in the x‐ and y‐directions. (c) Isometric view showing the variation of HOF catenations. Two catenation networks are distinguished by yellow and cyan. (d) Convergence of the unit cell densities by using NPT dynamics ensemble. (e) Residual stress components after the thermal relaxation steps.

Due to the extended length of the phenyl branches (4.35 Å), the assembled HOF contains large voids between building blocks. These voids can be further occupied by identical HOF networks, a process known as catenation (or interpenetration), which increases the overall density of the material. The tetrakis HOF considered here can accommodate up to two interpenetrated networks stacked along the x‐, y‐, or z‐directions. For consistency in structural description, catenation is restricted to the z‐direction. The representative unit domain of the catenated networks is a tetragonal cell with side lengths of 20.36 Å in both the x‐ and y‐directions (in‐plane) and 15.65 Å along the z‐direction (height), as shown in Figure 1b. Periodic boundary condition (PBC) was applied to every direction to model the infinite HOF bulk.

Although a single HOF unit can experimentally accommodate up to 48 water molecules [44, 48], these guests are light, highly mobile, and do not establish a long‐range crystalline arrangement within the voids (Figure S1). Their adsorption primarily induces local crumpling of the framework rather than irreversible damage to the global HOF phase, and the characteristic hydrogen‐bonding network remains intact even in hydrated conditions. However, forcibly removing adsorbed water after such local distortions develop may artificially trigger phase transitions within the limited simulation box. To address this issue, the present study focuses on the intrinsic structural and mechanical response of the pristine HOF lattice, where the crystalline phase is fully preserved. We aim to analyze the stress and strain pathways inherent to the dry, exoskeletal HOF composed solely of tetrakis building blocks.

The reactive force field (ReaxFF) [49] was used to reproduce the chemical bond order, H bonding, and structural characteristics of organic materials. The potential parameters set is particularly specialized to reproduce the lattice stacking of molecules of ∼10 Å size [50]. The energy contributions to the ReaxFF potential are classified as follows:

$$E_{sys}=E_{bond}+E_{over}+E_{angle}+E_{tors}+E_{vdW}+E_{Coul}+E_{spc}$$ (1)

where E_sys is the total energy of the molecular system, E_bond , E_angle , and E_tors are the energy contributed by the bond stretching, angle bending, and dihedral torsion, E_over is the energy penalty to prevent the unrealistic positioning of the atom, E_vdW and E_Coul are the long‐range nonbond van der Waals (vdW) and Coulombic interactions, E_spc is the special terms consisting of the effects of lone‐pair (E_lp ), sp² conjugation, valence band (E_ev ), and H binding (E_hb )

We used the Large‐scale Atomic/Molecular Massively Parallel Simulator (Sandia Laboratory, USA) [51] code to capture the position and velocity trajectories of every atom. 1 fs of timestep was used to iterate the velocity Verlet integration [52]. The partial charge (q) of each atom was computed by using the charge equilibration algorithm with a resolution of 10⁻⁶ e, which is self‐embedded in the ReaxFF potential. To focus on the network‐scale structural behavior of the HOF, the terminal groups were represented by averaged partial charges reflecting delocalized H bonds. Geometry optimization was first conducted by using the conjugate gradient method with a convergence criterion of 10⁻⁶ eV Å⁻¹. Note that the purpose of geometry optimization is to primarily find a local minimum of potential energy configuration by adjusting atomic positions. 2 ns of canonical (NVT; N, number of atoms, V, system volume, and T, temperature are maintained constant) dynamics ensemble at 0.1 K and 3 ns of isothermal–isobaric (NPT; N, P, system pressure, and T are maintained constant) dynamics ensemble at 10 K and 1 atm conditions were followed to thermally relax the molecular structure. The Nosé–Hoover thermostat [53] and Berendsen barostat [54] algorithms, respectively, were used to control the system temperature and pressure per every 100 fs of timestep.

During the MD simulation, the virial stress tensor (σ) [55] exerted on the unit cell was calculated as follows:

$$\mathit{\sigma }=\frac{1}{V}\left[-\sum _i^N{m}_i\left({\mathbf{v}}_i\otimes {\mathbf{v}}_i\right)+\frac{1}{2}\sum _i^N\sum _{j\ne i}^N\left({\mathbf{r}}_{ij}\otimes {\mathbf{F}}_{ij}\right)\right]$$ (2)

where m_i and v _i are the mass and velocity of atom i, r _ij and F _ij are the displacement and force between the interpair atoms i and j. The virial stress obtained at the near 0 K temperature is mainly contributed by the potential term (r _ij ⊗F _ij ), as the kinetic effect (m_i v _i ⊗v _i ) from thermal vibration becomes negligible. Thus, the stress averaged by only the statical displacement and force is explicitly assumed as a continuum Cauchy stress [56, 57].

To highlight the role of catenation, we prepared an additional molecular model in which one HOF network was removed from the unit cell (half catenation) and subjected it to the same thermal relaxation process. The full and half catenation structures obtained from the NPT ensemble are shown in Figure 1c. Note that the CCDC unit cell contains only two network clusters because the periodic boundaries group the tetrakis molecules into two interconnected HOF networks. When the cell is expanded to 1 × 1 × 3n supercell, the clusters link through the periodic images according to their least‐common‐multiple connectivity pattern, thereby reconstructing all six catenated networks observed experimentally (Figure S2). If the replication in the z‐direction is not a multiple of three, the degree of catenated networks remains two. This nonlinear dependence suggests that the degree of catenation should be interpreted with care, as it may vary with the finite sizes of material fabricated. In this study, we removed one network from the 1 × 1 × 1 unit cell to consider an extreme case in which catenation is maximally reduced, enabling a clear comparison with the fully catenated HOF.

The fully catenated HOF, containing two interpenetrated networks, preserved a uniform diamond‐like lattice, whereas the single‐network HOF was severely distorted in the absence of catenation. The unit cell lengths of the half‐ and full‐catenation models converged to 17.41 Å × 17.41 Å × 13.38 Å and 20.31 Å × 20.31 Å × 15.61 Å, respectively (Figure S3). It is noteworthy that the quarter volume of the thermally equilibrated cell (1.61 nm³) is consistent with the previous density functional theory calculation (1.63 nm³) [46]. In the full HOF, the uniform lattice maintained its initial dimensions with negligible change. The experimental density of the HOF structure at 100 K (1.01 g cm⁻³) [44] matched with the full catenation model (1.02 g cm⁻³), as shown in Figure 1d. Hence, the temperature effect on the organic crystal is negligible in a cryogenic environment. While in the half‐catenation model, the distortion reduced the unit cell volume by more than 16% compared to the initial state. Catenation also influenced the residual stress distribution within the HOF material. After sequential NVT and NPT stabilization, the internal residual stress of the fully catenated structure converged to nearly zero, as shown in Figure 1e. By contrast, the half‐catenation structure retained an in‐plane residual stress of ±0.14 GPa. The energy, cell lengths, and second‐order virial stress components showed consistent convergence. The time integration equilibration reached the system a stable thermodynamic state with minimal residual stress, so that all subsequent deformation could be attributed solely to the applied strain.

2.2. Atomistic Factor to Set Robust Lattice Structure

To elucidate the cause of the stable catenation of HOF networks, we measured the potential energy gradient around the assembly unit of the amidinium and carboxylate tetrakis molecules [58]. Specifically, one virtual H atom with q = +0.1e partial charge was positioned around the tetrakis molecules with a spatial resolution of 0.5 Å. The reference value of potential energy was measured at locations where the virtual H and tetrakis were far enough apart that they could not interact with each other. The static energy components were calculated at various locations by using the ReaxFF, and the difference from the absolute potential energy was obtained. By doing this, the potential well can be found where external atoms feel energetically favorable to approach [59].

The calculated Coulombic and vdW energy gradients are mapped in Figure 2a. Pronounced electric potential wells are observed near the backbones of the tetrakis molecules, while the equilibrium boundary of the vdW potential extends across the entire molecular surface. These attractive sites serve as the most favorable points for the approach of neighboring tetrakis molecules. The Coulombic interactions align the central molecular axes, whereas the vdW forces drive the stacking of phenyl branches. Upon stacking HOF networks, the Coulombic and vdW energy gradients within the voids are reduced by 0.005 and 0.001 eV Å⁻¹, respectively (Figure S4). A more detailed decomposition of the energy components revealed the mechanism of catenation. If the HOF network doubles due to interpenetration, the total potential energy (PE) is expected to scale linearly and approximately double. However, as shown in Figure 2b, several energy components, including the overall PE (Figure S5), increased by more than a factor of two. In particular, the lone‐pair, double bond, H bond, and Coulombic contributions arose from the stacking of phenyl rings, generating an additional interaction network beyond the amidinium–carboxylate binding. This enhanced interaction framework supported the single HOF network, enabling the tetrakis molecules to more effectively preserve their tetrahedral (i.e., torsional) geometry compared to the freestanding condition.

FIGURE 2.

Microscopic factor to build rigid catenation structure. (a) 3D mapping of the Coulombic (left) and vdW (right) energy changes near the tetrakis molecule assembly unit. Every scale bar is 10 Å. (b) Multiplication of the magnitude of energy components due to catenation. (c) Population density of the atoms captured from full and half catenation structures. (d) Comparison of the volume fractions of the atomic Voronoi regions. (e) Lognormal distribution curves of the atomic vibration amplitudes. The atom position data were captured during the last 10 ps of NVT relaxation.

The geometry of catenation stacking was characterized by using the population density (p), which is estimated by integrating the radial distribution function as follows:

$$p\left(r\right)=\frac{4\pi r^2〈\rho \left(r\right)〉}{N\rho }dr$$ (3)

where ρ is the bulk number density of atoms, 4πr ² is the volume integration of the spherical shell region of thickness of dr, 〈ρ(r)〉 is the local number density of atoms at a distance of r, and the angular brackets denote ensemble average. Thus the p function counts the number density of neighbor atoms from the specific position as a variable of radius. The population density profiles of the neighbor atoms of the full and half structures are compared in Figure 2c. The peaks for interatomic bonds (Table S1) and H bond (1.50 Å) lengths remained at the same position. Since the catenation stacking is induced by the nonbond interactions, we focused on a much farther regime over the interatomic bonding. There exists a peak near 5.91 Å where the profile of only the full catenation structure elevates upward. This peak is consistent with the nonbond stacking distance of tilted phenyl rings [60], verifying the catenation structure.

Due to catenation stacking, atoms within the HOF adopt a highly regular geometric arrangement. Visualization of the Voronoi regions surrounding each atom in the fully catenated structure reveals a repeating tessellation pattern, as shown in Figure 2d. The void fraction followed a hierarchical distribution: the central C atom occupied the smallest fraction (0.72%), followed by the catenation‐related nonbonded C atoms (3.84%), and the amidinium and carboxylate terminal groups (6.72%). This progressive increase highlights the systematic packing density within the framework. The tightly packed atomic configuration plays a crucial role in stabilizing atomic positions. As shown in Figure 2e, the vibration amplitudes of backbone, branch, and terminal atoms were compared between full and half catenation during thermal relaxation. The amplitudes were significantly reduced in the fully packed structure. Among all sites, the backbone C, with the largest moment of inertia, exhibited the least displacement, followed by the H‐bonded terminal groups, while the nonbonded phenyl rings showed the greatest motion. These results demonstrate that a more densely coupled interatomic network effectively suppresses fundamental atomic movements.

2.3. Structural Maintenance During Uniaxial Deformation

We now evaluate the structural maintenance performance of bulk HOF under the uniaxial tension and compression. We analyze the mechanical anisotropy in the parallel and orthogonal directions to the catenation stacking to elucidate the underlying mechanism. Specifically, the thermally equilibrated unit cell was uniaxially stretched or compressed at an engineering strain rate of 10⁷ s⁻¹ in the x‐ (ε _xx ) or z‐direction (ε _zz ). Such extreme condition in MD simulation is to investigate the atomistic mechanism in nanoscale materials at a small temporal regime [61, 62]. The loading tests were performed at 10 K. The orthogonal pressure was maintained at 1 atm to consider the Poisson effect [63]. The deformation was applied until the engineering strain reached ±0.2. Afterwards, the thermal relaxation process was performed by using 2 ns of NVT dynamics ensemble at 0.1 K. The deformed cell boxes were then unloaded in the opposite directions by using the same and dynamics ensemble conditions.

The structural deformations of the full catenation HOF cell are displayed in Figure 3a and Figure S6. The two HOF networks, distinguished by yellow and cyan, maintained their respective topologies under every deformation process. In other words, no new bonds were formed or broken (Table S2). Only the size of HOF networks varied depending on the directions of deformation. Under stretching along the x‐direction, the frameworks expanded in‐plane but contracted along the height direction. Conversely, x‐directional compression caused the networks to elongate vertically with in‐plane shrinking. When stretched along the z‐direction, the networks experienced severe in‐plane distortion and elongated 1.5 times to their original height. Under the z‐directional compression, the frameworks spread laterally on the xy plane. Notably, whenever in‐plane shrinkage occurred (i.e., z‐directional compression and x‐directional tension), the inherent voids between the HOF networks disappeared.

FIGURE 3.

Uniaxial deformation of unit cell of HOF networks. (a) Molecular snapshots captured after the deformation tests. The front view of 3 × 3 × 3 replication model is given to display the representative unit cell. The Connolly surface algorithm [64] with a probe radius of 3 Å is used to compute the void region in the HOF structure. Changes in the system microstates; orthogonal strains, unit cell density, and void volume during the (b) x‐ and (c) z‐directional uniaxial strains are applied. The schematic deformation modes are shown as inset.

For detailed analysis, the statistical properties of the HOF structure were quantified. First, we examine the tension and compression responses in the x‐direction, as shown in Figure 3b. The y‐direction cell lengths, which lie in the same in‐plane as x, were stretched and compressed simultaneously with the x‐strain, resulting in a negative Poisson's ratio that is typically observed in zigzag or entangled mesh structures, such as the HOF network studied here. Notably, under x‐direction tension, the y‐length increased nearly 1:1 with the x‐length, allowing the structure to expand while preserving a square geometry, whereas the z‐cell length contracted or expanded in the opposite direction of the x‐strain. Remarkably, the Poisson's ratio in the z‐direction reached 0.84 during x‐direction elongation, far exceeding the values of conventional metals (∼0.3) and rubbers (∼0.5) [65], which highlights the exceptional sensitivity of HOF networks to external load and ease of deformation. Such anisotropic change in cell dimensions also caused significant variations in density and internal void volume: under x‐direction tension, the density decreased while the void space more than doubled, whereas under compression, the density increased and the voids diminished, disappearing completely at a compressive strain of −0.1.

Next, we examine the response to z‐directional deformation, as shown in Figure 3c. Along the catenation stacking direction, the tension and compression induced symmetric compression and stretching, respectively, in the x‐ and y‐directions. The most striking feature, compared to the in‐plane response, is that the Poisson's ratios in both x and y are positive and exceed 1.0. This indicates that the proposed HOF structure can amplify deformations to more than 200% of the areal ratio by transferring load from the catenation axis into the in‐plane directions. During z‐direction deformation, the density of the unit cells changed quadratically with the square of the x‐ and y‐lengths, increasing under contraction and decreasing under expansion (Figure S7). Notably, at tensile strain of +0.05, the cell density rose above 1.28 g cm⁻³ due to the instantaneous agglomeration of HOF networks as a compact fiber, which closed internal voids thereafter. Conversely, under the z‐directional compression, the HOF networks expanded laterally in the in‐plane direction, and the internal voids increased accordingly.

It is important to clarify the difference between the compressive deformation presented in the simulations and the in‐plane tilting behavior reported in the previous experiments. In the experiments, water naturally evaporated from the square void inside the finite size HOF sample. Exclusion of water transformed the void between the tetrakis framework to be planarly tilted into the rhombic geometry [44]. On the other hand, in our MD simulations, the PBC unit cell was used for mechanical tests, which represents an infinite bulk without any free surfaces. Because atoms at the boundary must interact consistently with their periodic images, it is difficult to reproduce non‐periodic singularities such as rotation or swirling. In addition, the Berendsen barostat constrains cell angles, limiting shear‐induced distortions. For these reasons, the MD simulations captured an idealized, defect‐free in‐plane deformation mode of the tetrakis HOF. We anticipate that increasing the supercell size and thereby introducing additional internal degrees of freedom will allow localized deformation modes to emerge under load [66], including tilting or buckling.

2.4. Deformation Mechanism of Molecular Structure Components

The microstructure analysis was further carried out to better understand the novel behavior of HOF structure. As no H bonds are broken or new voids are created, this HOF structure is topologically isomorphic and differentiable in time and space. In this regard, the following atomic von Mises strain (${\epsilon }_{vM}^i$) can be derived based on the continuum finite strain theory.

$$\begin{array}{ccc}\hfill {\epsilon }_{vM}^i& & =\hfill \\ & & \sqrt{\frac{{\left({\epsilon }_{xx}-{\epsilon }_{yy}\right)}^2+{\left({\epsilon }_{yy}-{\epsilon }_{zz}\right)}^2+{\left({\epsilon }_{zz}-{\epsilon }_{xx}\right)}^2}{6}+{\epsilon }_{xy}^2+{\epsilon }_{yz}^2+{\epsilon }_{zx}^2}\hfill \end{array}$$ (4)

where ε _kl is the k directional (x, y, or z) strain component exerted on l plane (x, y, or z). The cutoff radius of 4 Å is used to construct the volume per each atom, which is larger than any other maximum radii of the vdW interaction. Thus, the atomic strain is obtained as the shift ratio of the entire volume domain due to the displacement of atom. Unlike calculating a displacement, using atomic strain allows us to detect where local deformation is concentrated in the tetrakis molecule.

The von Mises strains of HOF structure during the deformation are captured in Figure 4a. The strain was calculated from the displacement relative to the thermally relaxed state. Under the compression in the x‐direction, the strain was minimal as the relative positions and microstructure showed minor change. When stretched in the x‐direction, however, a high shear strain developed along the H bonds to prevent network failure as the branches expanded laterally. Under the z‐directional compression, the shear strain became concentrated at the ends of the H bonds, similar to that of x‐directional stretching. At the early stage of z‐directional stretching, the shear strain was distributed throughout the entire branch. Under the larger deformation, the HOF network aggregated into a single compact structure. During this aggregation process, the backbone structure was twisted in the z‐direction, thus the von Mises strain exceeded 0.5 at the phenyl ring branches.

FIGURE 4.

Microstructural changes of HOF networks. (a) Magnified view of the catenation stack under the x‐ (top) and z‐directional (bottom) deformation. The atoms are colored according to their von Mises strain values. Every scale bar is 10 Å. Atomic structure properties during the deformation; (b) length of the H bonding (number of H bonds is shown as an inset), (c) dihedral angle between the phenyl ring branches, and (d) nonbond stacking distance between the catenation networks.

We further examined key microstructural factors associated with the large deformation. First, the terminal hydrogen bond length was measured during deformation, as shown in Figure 4b. The bond remained stable at its initial length of 1.50 Å but elongated by up to 1.3%, reaching 1.52 Å during stretching in the z‐direction. Number of hydrogen bonds provides a direct measure of the structural stability of the HOF structure. We computed the number of neighbor hydrogen‐bonded pairs between the CN₂H₄ ⁺ and CO₂ ⁻ terminals during the mechanical loading. The initial set of 32 hydrogen bonds remained constant over the compression and tensile tests up to ±0.2 strain, indicating that the HOF lattice maintains network topology under substantial mechanical loading. Second, the central dihedral angle of the tetrakis molecules was analyzed, as shown in Figure 4c. The angle was maintained at 102.7° during x‐directional tension and z‐directional compression, while a stepwise decrease to below 101° was observed under x‐directional compression and z‐directional tension. Finally, the stacking distance between interpenetrated HOF networks was analyzed, as shown in Figure 4d. The inter‐network distance rapidly decreased to less than 4 Å during z‐directional stretching, whereas the spacing between branches gradually increased under the compression. In the x‐directional deformation, the maximum strain‐induced displacement did not exceed 0.5 Å. To sum up, catenation structure produces a structure that becomes increasingly adherent under tensile loading, to the extent of distorting the molecules when pulled in the overlapping direction. Furthermore, a sufficient nonbond spacing between the HOF networks is well preserved to effectively absorb and dissipate compressive loads.

2.5. Macroscopic Monitoring of Mechanical Stress Dissipation

In this section, we reveal the mechanical behavior to maintain the porous structure and find the stress flow inside the tetrakis molecules under deformation. First, the strain‐stress behaviors of the HOF unit cell are displayed in Figure 5a. During tension and compression in the x‐direction, the residual stress response deviates gradually from the stabilized reference point. Under tension, the stress decreases below 0.02 GPa up to a strain of +0.1 and then rises to 0.24 GPa. Under compression, it increases to 0.29 GPa at a strain of −0.1 before dropping sharply to 0.58 GPa. In the z‐direction under compression, no initial stress response is observed, followed by a decrease to −0.28 GPa beyond a strain of +0.1. Across all three deformation modes, microstructural changes are accommodated by voids in the catenation stack, allowing most of the internal stress to dissipate. In contrast, during the z‐directional stretching, a notable stress reduction occurs well before a strain of +0.1, after which a sharp rise leads to tensile stresses exceeding 15 GPa. This behavior arises because, under tensile loading along the catenation direction, the HOF network behaves as a solidified mass due to agglomeration among phenyl branches. Despite this nonlinear load response, the system closely follows the loading curve and fully recovers upon unloading. Thus, HOFs, with their robust internal phase structures, exhibit remarkable resilience while responding sensitively to external loads.

FIGURE 5.

Anisotropic mechanical behaviors of HOF unit cell. (a) Strain‐stress curves of the HOF unit cell during the x‐ and z‐directional deformation tests. The loading and unloading profiles are indicated by solid and dashed lines, respectively. (b) Deformation process of the catenation HOF structure under the z‐directional loading and unloading. The atoms are colored according to their von Mises stress values. Every scale bar is 10 Å. (c) Change in the PE components measured at the catenation stacking and intramolecular regimes.

To detect the atomic region of stress concentration, the atomic von Mises shear stress (${\sigma }_{vM}^i$) was calculated by combining six stress components as follows:

$$\begin{array}{ccc}\hfill {\sigma }_{vM}^i& & =\hfill \\ & & \sqrt{\frac{{\left({\sigma }_{xx}-{\sigma }_{yy}\right)}^2+{\left({\sigma }_{yy}-{\sigma }_{zz}\right)}^2+{\left({\sigma }_{zz}-{\sigma }_{xx}\right)}^2}{2}+3\left({\sigma }_{xy}^2+{\sigma }_{yz}^2+{\sigma }_{zx}^2\right)}\hfill \end{array}$$ (5)

where σ _kl is the k directional (x, y, or z) stress component exerted on l plane (x, y, or z). The von Mises stress mapping of the HOF networks during the z‐directional loading and unloading is shown in Figure 5b.

First, let us observe the stretching (loading) and compression (unloading) process. From the initial state, applying tension up to +0.1 in the z‐direction stretched the HOF lattice and reduced the catenation distance, causing the networks to draw closer and adhered. This adhered structure arranged the angles and positions of the phenyl rings in a more ordered configuration, which helps relieve stress at the terminal H bonds. As the tensile strain increases further, the entire network, now acting as a single mass, tilted up and down, distributing high stress throughout the tetrakis molecule. In this stabilized state, the structure carried a residual stress of more than 6 GPa (Figure S8). When compressed again in the z‐direction, the shortened catenation distance returned to its original spacing, and the network separates. The residual stress was not released all at once but lingered along the central backbone (Figure S9). This uneven distribution creates an imbalance that induces buckling at the hydrogen‐bond sites of the tetrakis molecule.

Next, let us consider the reversed sequence of compression (loading) and stretching (unloading). Applying compression along the z‐direction does not significantly narrow the catenation distance; instead, the molecule distorted as the phenyl rings slide against one another. Once compression is complete, the catenation distance widens during the relaxation, which helps relieve stress concentrated around the backbone of tetrakis. However, this relief comes with a cost: the uneven stress across the molecule left the tetrakis molecule unstable and prone to distortion. Even under a small tensile strain during the unloading, the catenations slid diagonally, gradually dissipating the remaining stress. After the full unloading, the structure regained its initial diamond‐like lattice, completing the cycle of deformation and recovery.

For better understanding of such nonlinear stress in the HOF networks, the ReaxFF energy components were classified as below.

$$E_{cat}=E_{lp}+E_{ev}+E_{hb}$$ (6)

$$E_{mol}=E_{bond}+E_{angle}+E_{tors}$$ (7)

where E_cat and E_mol are the PE contributions to the catenation stacking and intermolecular deformation, respectively. In MD simulations, ReaxFF evaluates both bonded and non‐bonded interactions for every atomic configuration and computes the corresponding energy terms. Then, the total ReaxFF energy becomes the sum of all individual components. This allows us to monitor not only the total energy but also each contribution listed in Equation (1). A bond is treated as formed when the interatomic distance is identified and is typically shorter than 1.6 Å [49]. Under this condition, the bonded terms E_bond , E_angle , and E_tors become active and represent intramolecular deformation energy. When the interatomic distance exceeds 1.6 Å, these bonded terms decay toward zero, and non‐bonded interactions begin to dominate the energy landscape. The main long‐range contributions include E_lp , E_ev , and E_hb . By summing only these non‐bonded components, we can isolate the energy associated with intermolecular stacking and assembly. This classification also enables identification of the force‐field contributions since each energy term corresponds to a potential gradient in space. Such formulation is to systematically determine whether the surface or body domain of the tetrakis molecule dominances the mechanical stress flow [67].

The change in the PE contributions is thus classified during the z‐directional deformation tests, as shown in Figure 6c. Loading in the z‐direction significantly increased the strain energy stored within the molecule itself. At the same time, the interaction energy between the catenation networks decreased, allowing the HOF lattice to remain energetically stable even under global deformation (Figure S10). However, because the catenation was formed between the relatively mobile phenyl rings, the system did not follow the same recovery path during unloading. As the tensile strain decreased toward +0.1 or below, there is a point where the energy stored in the catenation stackings rapidly drops. At this stage, separation occurred between the tetrakis molecules, and the resulting non‐uniform residual stress distorted the molecular configuration. In contrast, during the z‐direction compression, the redistribution of energy between catenations and intermolecular forces was more uniform, keeping the total energy at a consistent level. This balanced energy exchange enabled the tetrakis framework to progressively recover its molecular geometry and return to the initial state during the unloading.

FIGURE 6.

Structural robustness of anisotropic HOF surfaces. (a) Thermally relaxed HOF surface models with the z‐ and x‐directional catenation stackings. (b) Schematic illustration of the HOF exoskeleton surrounding a large enzyme molecule. (c) Change in the surface morphologies view from the z‐ (top) and x‐directions (bottom) before and after the thermal relaxation. (d) Convergence of the PE level in two surface models. Spatial distribution profiles of the (e) density and (f) uniaxial stress measured from the fixed bottom.

2.6. Extended Structure Maintenance in Large HOF Surface Model

We further analyzed the large‐scale HOF structure with catenation to explore its applicability to practical engineering fields. As shown in Figure 6a, the HOF models replicated twelve and ten times in the z‐ and x‐directions were individually prepared. PBC was applied in every cell box boundary. The system energy of the replicated models was first minimized by using the conjugate gradient method with a convergence criterion of 10⁻⁶ eV Å⁻¹. Then they were thermally relaxed by using 2 ns of NVT dynamics ensemble at 0.1 K and 3 ns of NPT dynamics ensemble at 10 K and 1 atm. Next, a gap of 70 Å was inserted to dissociate the catenation stacking. The motion of atoms within a 10 Å thickness at one side of the HOF structure was restricted during the time integration, which is to represent the subsurface bulk. The other side of the HOF substrate was designated as a free surface. Such protocol was to model the representative domain of HOF exoskeleton that encases enzymes and biomolecules [68], as illustrated in Figure 6b. Finally, 800 ps of NVT dynamics ensemble at 0.1 K was used to the Newtonian atoms to equilibrate the surface structure. Each HOF network connected by the catenation stacking remained robust throughout the z‐ and x‐directional replications after the surface equilibration. Note that over ten replications are required in the catenation stacking (z) direction for the PE level of substrate reaches the periodic bulk (Figure S11).

From measuring the difference of surface groove compared with the initial state, we observed the significant suppression of structural distortion in the catenation direction, as shown in Figure 6c. In the z‐directional surface model, a checkerboard‐like arrangement of void and tetrakis regions are clearly divided. Even after the thermal relaxation, the morphology of these regions remained well preserved (Figure S12). By contrast, the surface morphology in the x‐direction, which is connected by hydrogen bonds rather than catenation, was less uniform and less symmetrical. In this case, the tetrakis shifted more noticeably within the xy plane after the relaxation. Nevertheless, the horizontal void maintained its phase continuity without any interruption or agglomeration inside the substrate.

Such behavior was quantitatively analyzed to elucidate the role of catenation. First, the stabilization level of potential energy relative to the initial state during surface relaxation was examined, as shown in Figure 6c. The converged PE per single tetrakis molecule in the z‐direction, where catenation occurs, was decreased by −3.19 eV, which is more than seven times lower than that in the x‐direction (−0.45 eV). In other words, stacking at the surface height without direct stacking between tetrakis branches leads to a more stable structure.

Nevertheless, the HOF network, still connected by H bonds, remains structurally and mechanically robust. The spatial analysis of density along the depth direction clearly reveals the crystalline lattice characteristics of the HOF, as shown in Figure 6e. In the z‐direction, replication through catenation fills the voids between tetrakis units, resulting in a nearly constant density of approximately 1.03 g cm⁻³ across the entire depth. A notable feature is the decrease of approximately 12 Å in surface height after stabilization compared to the initial configuration. This can be attributed to catenation, where non‐bonding interactions between phenyl branches at the free surface eliminate residual tension and allow the branches to adopt a closer, more stable configuration. On the other hand, the x‐direction forms a truss‐like structure with periodically varying density through H bonding. The alternating arrangement of high‐ and low‐density branches, together with the central backbone of the tetrakis molecule, causes the density to fluctuate between 0.5 and 1.9 g cm⁻³. The resulting fluctuations of 10.21 Å correspond to the in‐plane lattice constant of tetrakis HOF, showing excellent agreement with the experimentally observed periodicity of 10.18 Å [44].

A uniform stress distribution was observed in the substrate along the robust lattice structure, as shown in Figure 6f. In the catenated z‐direction surface model, the maximum fluctuation in residual stress remained below 0.08 GPa. In contrast, the x‐direction surface model, connected by hydrogen bonds, exhibited a periodic alternation of tensile and compressive stresses that reflected the repeating molecular geometry of tetrakis. Specifically, stresses of 0.22 GPa and −0.17 GPa were distributed bidirectionally at the terminal hydrogen bonds and at the pivot points of the central backbone.

3. Conclusions

This study sets out to clarify the mechanical adaptability of HOFs, a property that has remained poorly understood despite their chemical versatility and promise as porous materials. Through the detailed MD simulations of the confined HOF unit crystal, we evaluated how catenation, hydrogen bonding, and molecular geometry govern the structural response of HOF lattices under external loading. The results reveal that catenation is not merely a structural by‐product but a key mechanism for stabilizing the lattice, reducing atomic vibration amplitudes, and distributing shear strain across the network. Specifically, the catenated frameworks behave collectively as a solidified mass, enabling large reversible strains, while in‐plane loading highlights auxetic‐like responses and energy dissipation through void maintenance.

Our analysis of residual stresses and energy redistribution demonstrates that HOF lattices possess a remarkable ability to maintain structural coherence under large deformation. Even when subjected to the nonlinear responses, such as during tensile loading, the networks resist collapse by distributing stress through H bonds and noncovalent stacking. The cooperative action of these interactions ensures that local distortions are absorbed while the overall lattice framework is preserved. Importantly, upon unloading, the structures consistently recover their original configuration, highlighting that resilience and stability are intrinsic to the internal phase architecture of HOFs. Such characteristics point to their potential as robust porous materials that can endure repeated loading cycles without significant degradation.

Extending the simulations to surface‐replicated domains further confirms that catenation not only strengthens lattice continuity but also prevents disruptive surface distortions, an essential feature for practical deployment. The capacity of HOFs to maintain ordered frameworks under stress suggests opportunities for designing adaptive materials that combine chemical selectivity with mechanical resilience under chemical or thermal stimuli. The proposed analytical method offers an adaptive tool to predict the microscopic origins of network assembly and mechanical behavior in emerging various HOFs. Future studies may explore how controlled catenation or functionalization can be used to tailor this balance, enabling applications in flexible devices, responsive membranes, or lightweight load‐bearing systems. These findings, therefore, provide a mechanical foundation for translating HOF chemistry into engineered architectures with long‐term stability and performance.

Author Contributions

B.G.: Conceptualization, methodology, investigation, visualization, writing – original draft. J.C.: Funding acquisition, supervision, validation, writing – review & editing.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting File 1: smll73140‐sup‐0001‐SuppMat.docx.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.