Chapter 1: Fundamentals of MOF Mechanics & Structure–Mechanical Property Relationships

Metal–organic frameworks (MOFs) and porous coordination polymers (PCPs) are terminologies¹  used to describe a vast and continuously expanding family of nanoporous ‘hybrid’ materials.²  They are inorganic–organic compounds,³  constructed by the molecular self-assembly of metal nodes (ions or clusters) and multitopic organic linkers,⁴  resulting in the formation of a plethora of network topologies and chemical structures.⁵  This bottom-up methodology when coupled with rational design could be harnessed to engineer three-dimensional (3-D) extended frameworks^6,7  as well as two-dimensional (2-D) layered structures,⁸  held together by coordination bonds oriented in specific directions. Figure 1.1 shows a few exemplars of MOF structures. Notably, the open framework structures of MOFs afford a precise cavity size and well-defined channel architecture, with pore dimensions ranging from ∼2 Å to >1 nm. The most intensively studied MOF/PCP materials are intrinsically crystalline – they possess long-range order and many are highly porous – exhibiting an internal surface area typically of the order of ∼1000 m² g⁻¹.⁷  Increasingly, research in the field of MOF/PCP materials has expanded beyond the primary domain of ordered crystalline phases⁹  to further encompass frameworks containing topological disorder and amorphous structures, such as MOF monoliths and hybrid glasses.¹⁰  Further still, the porous MOF structure can also serve as a ‘host’ structure to accommodate and protect luminescent ‘guest’ molecules confined in its pore cavities, yielding a Guest@MOF ‘composite’ system with tuneable photophysical and photochemical properties.¹¹ 

By virtue of the many pathways available for combining a multiplicity of organic and inorganic building units, the resultant chemical structures and functionalities of MOFs are particularly varied.^12,13  In principle, one could design, tune, and engineer an unusual combination of physical and chemical properties, which cannot be achieved in purely organic and inorganic materials alone. Indeed, a growing number of potential technological applications have been proposed. Many MOF structures are substantially more porous than any commercially available nanoporous sorbent materials, such as zeolites, silica gels, and activated carbons. Unsurprisingly, the first practical uses identified for MOFs thus encompassed gas separations and storage,¹⁴  catalysis,¹⁵  and CO₂ capture,¹⁶  which may be perceived as ‘classical’ applications for the more conventional porous materials quoted above. In contrast, more innovative applications have been proposed where MOF serves as the ‘active material’ for integration into electroluminescent devices¹⁷  and optoelectronics,¹⁸  smart sensors,^19,20  dielectrics,²¹  and for accomplishing functions linked to energy harvesting, conversion and transfer.^22,23  Encouragingly, a broad range of promising MOF applications visible on the horizon has attracted the attention of scientists, engineers, and technologists from a wide spectrum of disciplines, with a strong motivation to bridge the gap between fundamental research and real-world applications.

Much of the earlier research efforts on MOF/PCP and hybrid framework materials are focussed on synthesis, chemical characterisation, and adsorption related properties. However, practical applications require an in-depth understanding of the basic mechanical behaviour, not only of single crystals and microcrystalline powders, but also of polycrystalline thin films and coatings, bulk monoliths and pellets, mixed-matrix membranes and nanocomposites, including bespoke Guest@MOF systems.^24–27  For instance, the sensitivity of a reusable mechanochemical sensor comprising a MOF film adhered to the surface of a silicon microcantilever (substrate) scales with the elastic coupling of stress-and-strain in a MOF structure upon analyte sorption/release, whilst its multicyclic operation depends on the mechanical resilience of the film-to-substrate adhesion.²⁸  A second exemplar involves the deployment of MOFs as catalysts in reactors, where the thermomechanical stability of the porous frameworks subject to pressure, temperature, and humid conditions is vital for circumventing excessive framework deformation or stress-induced structural collapse that will gradually degrade performance over time. In yet another scenario, the fluorescence response of mechanochromic Guest@MOF materials¹¹  is heavily dependent on the interaction of nanoconfined fluorophores with imposed stresses and strains, for accurate calibration to allow force monitoring and optical stress sensing.²⁹ 

In essence, elementary knowledge and precise control of structure–mechanical property relationships are central to the fabrication of advanced devices, for enabling component manufacturing and pellet shaping in industry, while ensuring mechanical durability for a sustained long-term performance in a multitude of commercial and consumer settings.

Underpinning all of these fundamental questions and practical challenges is a rapidly expanding field of research, aptly termed ‘MOF Mechanics’, which is concerned with an array of topical problems on mechanical phenomena encompassing: elasticity, structural anisotropy and stability, yielding and plastic deformation, time-dependent viscous effects, high-pressure response, interfacial cohesion and debonding, framework dynamics and mechanical dissipation, crack initiation and propagation leading to fracture (rupture of chemical bonds). This book will present to the reader the experimental and theoretical studies in the New Science of MOF Mechanics, exemplified by key topics and cutting-edge research addressing the broad range of mechanical behaviour inherent in hybrid framework materials. To this end, the chapters in the book are organised as follows.

• Experimental studies to measure the Young’s modulus, hardness, mechanical anisotropy, interfacial adhesion and fracture toughness of crystals, monoliths, films and membranes by means of nanoindentation techniques are presented in Chapter 1 (Tan). This discussion is extended further to cover terahertz framework dynamics, soft modes, and shear deformation mechanisms that could destabilise the porous framework structures. Ashby-style materials selection charts are constructed using the latest available data to reveal general trends in the mechanical properties of MOF materials.

• In Chapter 2 (Marmier), the systematic characterisation of single-crystal elastic constants and the resultant anisotropic mechanical response are explored in detail, drawing from the theoretical and experimental data available to date. Special emphasis is given to highlight unusual framework mechanics responsible for anomalous physical phenomena, such as the negative Poisson’s ratio, negative linear compressibility, and negative thermal expansion. This chapter concludes with a critical treatment on the pertinent questions of framework ‘flexibility’ and proposed mechanisms.

• Computational modelling of MOF mechanics is covered in Chapter 3 (Rogge), where the readers will be exposed to state-of-the-art methodologies for constructing an atomistic model of an extended hybrid framework, and subsequently for simulating its structural response subject to directional stresses or hydrostatic pressure. Theoretical studies give new insights into complex mechanisms surrounding the elastic, plastic, and phase transformations of framework structures, which cannot be obtained by experiments alone.

• Chapter 4 (Moggach and Turner) focusses on the high-pressure deformation of MOF structures under hydrostatic compression via diamond anvil cells. Uniquely, the application of high-pressure X-ray crystallography reveals the evolution of pressure-induced structural deformations, phase transformations, and guest-mediated phenomena attributed to flexible frameworks.

• Rate effects and absorption of mechanical energy by flexible MOF structures are presented in Chapter 5 (Sun and Jiang). The pressure-stimulated liquid intrusion mechanism of hydrophobic MOFs is discussed through a combination of experimental and theoretical studies, elucidating the effect that deformation strain rate (quasistatic, medium, high) has on framework materials performance. Finally, mechanical energy dissipation, by means of non-intrusion mechanisms such as phase transition and pore collapse, is considered.

When a solid material is subjected to a small mechanical force, the relationship between stress (σ = P/A, applied force P divided by area A subject to the force) and strain (ε = ΔL/L₀, change in length ΔL divided by its initial length L₀) is linear — this purely elastic behaviour obeys Hooke’s law, such that σ ∝ ε. The material is in equilibrium, where any mechanical deformation or shape change experienced by the solid is reversible upon removal of the applied force. Crucially, in this linear elastic regime, the stress level must not exceed the yield strength (σ_Y) of the material, σ < σ_Y, beyond which its mechanical response will become irreversible and may turn nonlinear, as illustrated in Figure 1.2(a). This important concept of yield strength is related to the hardness (H), which is discussed in Sections 1.4 and 1.8.1.

Within linear elasticity, the mechanical properties of materials can be described by the Young’s modulus, Poisson’s ratio, shear modulus, and bulk modulus. The Young’s modulus or elastic modulus, E, is defined as the ratio of stress to strain, as follows:

hence, σ = Eε, strictly for unidirectional loading conditions only (i.e., uniaxial tension or uniaxial compression). E is a measure of the mechanical ‘stiffness’ of the framework structure, this corresponds to the slope of the stress vs. strain curve located below the yield point (Figure 1.2(a)). The elastic moduli are typically expressed in units of N m⁻² or Pascal (Pa). Conversely, the reciprocal of the stiffness property is called the mechanical ‘compliance’, S = 1/E or E⁻¹, with units of Pa⁻¹.

As shown in Figure 1.2(a and b), when the elastic solid is uniaxially deformed in the ‘longitudinal’ direction (ε_i), there will be a resultant lateral strain generated in the ‘transverse’ direction (ε_j). The Poisson’s ratio, ν, can be determined from the ratio of lateral strain to axial strain:

The negative sign ensures that the value of ν is always positive for a conventional material: the upper bound is 0.5 for an incompressible solid such as a rubbery polymer, while the lower bound is 0 for a fully compressible material like a foam. However, framework materials can exhibit a negative Poisson’s ratio (termed ‘auxetic’), with this anomalous mechanical phenomenon considered in detail in Chapter 2.

The shear modulus, G, is a measure of the framework’s ‘rigidity’ or torsional stiffness of material subject to an angular distortion, as depicted in Figure 1.2(c). It is defined as the ratio of shear stress to shear strain:

The bulk modulus, K, is a measure of the ‘volumetric stiffness’ (i.e., volumetric stress divided by volumetric strain, ΔV/V₀) of the material subject to a hydrostatic pressure, p, as illustrated in Figure 1.2(d). In this definition, the negative sign accounts for the volumetric contraction experienced in compression, so that the value of K remains positive:

The inverse of bulk modulus is called the compressibility, β = K⁻¹. Chapter 4 is dedicated to the study of the mechanical behaviour of MOF materials under hydrostatic compression, and it will give a rigorous treatment to this subject matter, including characterisation of framework structures with negative compressibility.

For an isotropic solid material, whose mechanical properties are not changing with direction, there are only two independent elastic constants to be established because the elastic properties (E, ν, G, K) are interrelated as follows:

For an anisotropic solid, however, the mechanical properties determined will be varying with direction of the applied loading. In this situation, the generalised Hooke’s law in 3-D can be applied to determine the strains developed along the three orthonormal axes of Figure 1.3(c), when the anisotropic solid is subject to a triaxial stress state (σ₁, σ₂, σ₃):

Equation 1.8

For the case of a biaxial stress state depicted in Figure 1.3(b), eqn (1.8) can be simplified by letting σ₃ = 0, thereby corresponding to a 2-D plane stress.

Some mechanical properties of MOFs have been studied by employing the instrumented indentation testing (IIT) method, which is commonly called ‘nanoindentation’.³⁰  Nanoindentation is normally performed using a Berkovich diamond indenter tip (three-sided pyramidal probe), with which the Young’s modulus (E) and hardness (H) of a small number of MOF materials in the form of single crystals,^31–33  monoliths,^34,35  thin films,^36,37  amorphous particles and glasses^38,39  have been characterised to date.

The different stages of nanoindentation testing are depicted in Figure 1.4. Initially, at loading stage (1), the diamond indenter tip slowly pushes into the sample surface, typically to reach a maximum surface penetration depth of around 1–2 µm. Shallower indents of a depth of several hundred nanometres, or less, can be achieved, but this will require a sharp indenter tip (end radius ≲ 50 nm). Subsequently, the compressed sample is held at a constant maximum load (2), to overcome time-dependent effects (e.g., creep). Finally, the tip is slowly withdrawn from the deformed region, yielding the unloading test segment (3). The changing indenter load (P) and vertical displacement (h) data are continuously recorded during nanoindentation testing, to yield an indentation load–depth (P–h) curve. This P–h curve is subsequently used to compute the values of E and H by employing the Oliver and Pharr (O&P) method,⁴⁰  outlined below. For a detailed treatment of basic nanoindentation theory and further data analysis techniques, the reader may consult critical reviews available in the literature.^30,41–44 

Nanoindentation experiments may be performed under either load-controlled (quasi-static) or displacement-controlled (dynamic) modes, the latter is known also as the ‘continuous stiffness measurement’ (CSM).⁴¹  These techniques differ by the way the elastic contact stiffness, S, is derived. Under quasi-static testing, the elastic contact stiffness is calculated from the slope of the unloading segment in the P–h curve, thus S = dP/dh (Figure 1.4). As such, the E and H values are obtained only at the maximum indentation depth, h_max. In CSM testing, however, the E and H values can be determined continuously, as a function of the surface penetration depth during the loading segment. This is made possible by superimposing a small sinusoidal displacement at a specific excitation frequency (e.g., 2 nm at 45 Hz)⁴¹  onto the primary loading signal (Figure 1.4 inset), and the dynamic response of the system is utilised to compute the changing magnitude of S with indentation depth, h.

The elastic contact stiffness (S) is later used to calculate the reduced modulus, E_r:⁴⁵ 

where the constant β varies with the geometry of the indenter tip: for example, β = 1 for a spherical tip, and β = 1.034 for a Berkovich tip. The contact area established under load, A, is a function of the contact depth, h_c. This is given by a tip area function A(h_c) as exemplified by the five-term polynomial in eqn (1.10), where C_n are constants obtained by curve fitting.⁴⁰  The calibration procedure involves indentation of an isotropic material of known Young’s modulus, which is typically a polished sample of fused silica (E = 72 GPa).

The reduced modulus, E_r, is a function of the Young’s moduli and Poisson’s ratios of the sample (E_s, ν_s) and the indenter (E_i, ν_i):

It follows that the Young’s modulus of the sample, E_s, can be determined using the following expression:

For a diamond indenter probe, E_i = 1141 GPa and ν_i = 0.07. When the stiffness of the sample is significantly lower than the indenter stiffness, E_s ≪ E_i, eqn (1.12) can be approximated by E_s = E_r(1 − ν_s²). The calculation of the Young’s modulus of the test sample (E_s) therefore will require knowledge of its Poisson’s ratio (ν_s), when the latter is an unknown a value may be chosen based on these guidelines: glasses and ceramics (ν_s ∼ 0.2), metals (∼0.3), polymers (∼0.45), rubbery elastomers (∼0.5). Although for stiffer materials like metals, the sensitivity of calculated E_s to the input value of ν_s is weak,³⁰  for relatively low stiffness MOF-type materials (E ≲ 10 GPa),²⁴  the error does become more significant.⁴⁶  The situation is further complicated by the fact that the Poisson’s ratio of MOFs can span a wide range of values (Chapter 2) and, may vary due to their mechanical anisotropy, as described below (Section 1.4).

Indentation hardness, H, also termed ‘nanohardness’, quantifies the plastic deformation of the material beyond the yield point (further details in Section 1.8.1). Based on the O&P method, H is calculated by dividing the applied load by the projected area of contact:

Therefore, the hardness determined from the maximum indentation depth is given by P_max/A(h_max). Akin to stress, the unit for nanohardness is Pascal (Pa).

The O&P method⁴⁰  was derived on the basis that the test sample is homogeneous and elastically isotropic, which is true for many polycrystalline materials that exhibit an approximately isotropic mechanical response. However, this is clearly not the case even for most single crystals with a cubic symmetry.⁴⁷  Discrepancies for cubic (Al, Cu, β-brass) and hexagonal single crystals (Zn) of metals caused by elastic anisotropy have been studied in 1994 by Vlassak and Nix,⁴⁸  who demonstrated that eqn (1.11) still holds if the elastic modulus terms are redefined as the ‘indentation modulus’, M, as follows:

where M_s ≡ M_{hkl} = E_s/(1 − ν_s²), designating the indentation modulus oriented normal to the {hkl} facet of a single crystal.

In 1998, Hay et al.⁴⁹  adopted the indentation modulus methodology to probe hexagonal single crystals of β-silicon nitride, which are highly anisotropic, and found that the Young’s modulus was underestimated by ∼20% in the stiffest direction, whereas in the most compliant direction it was overestimated by ∼10% if the indentation results were not corrected for anisotropic effects. In 2004, Bei et al.⁵⁰  applied this approach to measure the anisotropic Young’s moduli of Cr₃Si intermetallics, where they reported consistent results from Berkovich nanoindentation versus ultrasonic testing. However, it was not until 2009 that Tan et al.³²  demonstrated its implementation for studying the anisotropic mechanical behaviour of two polymorphic Cu-based MOFs, both with a dense structure, one 3-D, the other 2-D. The single-crystal nanoindentation results show a strong structure–mechanical property correlation along specific crystallographic orientations. For example, the stiffest crystal facet of the 3-D framework (M_{100} ∼ 93 GPa) is oriented normal to the underlying metal–oxygen–metal (M–O–M) chains, thereby conferring a stiff ‘backbone structure’. In contrast, it was established that the most compliant facet of the 2-D framework (M_{010} ∼ 35 GPa) is oriented normal to the stacking of hydrogen-bonded layers. Furthermore, the elastic anisotropy of these framework crystals was found to be large (stiffness variation >60%) compared to the hardness anisotropy (hardness variation of ∼12%).³²  Interestingly, this parallels the findings for metallic single crystals that despite large anisotropy of yield stress show only small hardness anisotropy (ca. 13–20%), with this phenomenon being attributed to the complex plastic strain field generated under the indenter.⁴⁸  While the dislocation mechanisms responsible for the plastic deformation of metals are well known, the plastic deformation mechanisms for MOFs and hybrid frameworks are currently not well understood (Section 1.8).

To yield reliable and reproducible nanoindentation measurements of MOF materials, it is generally important to pay attention to the following points in relation to sample preparation. First, the sample to be probed must be secured on a much stiffer ‘substrate’ material, such as an epoxy mount or a metal stub to eliminate any compliance issues associated with underlying substrate. Second, the sample surface must be microscopically flat for accurate contact area determination (accuracy of A(h_c) has a major effect on E and H calculations) and to minimise the overall experimental scatter. A smooth surface with a mean roughness of ∼10 nm can be achieved by cold mounting the crystals or monoliths in an epoxy resin, followed by careful grinding using a non-penetrating lubricant (e.g., water for hydrophobic MOFs or glycerol for water-sensitive samples), and then by polishing with an increasingly finer grade of diamond suspensions to yield the final smooth surface. It is vital to ensure that the chosen surface preparation steps do not significantly alter the properties of the sample surface, which may be caused by several factors, such as surface contamination, guest infiltration into MOF pores, chemical degradation, residual stress, and subsurface cracking.

Two representative studies, where the structure–mechanical property relationships of MOFs have been established with these nanoindentation techniques are discussed in more detail in the following sections.

The elastic moduli of the single crystals of a family of zeolitic imidazolate framework (ZIF) materials, encompassing ZIF-4, ZIF-7, ZIF-8, ZIF-9, ZIF-20, ZIF-68, and ZIF-zni, have been systematically characterised by single-crystal nanoindentation.⁵¹  As shown in Figure 1.5(a), the chemical structure, crystal symmetry and network topology of the samples are distinctively different, so are their framework density, porosity and solvent accessible volume (SAV). The P–h curves of the seven ZIF structures (Figure 1.5(b)), when indented to a maximum depth of 1 µm, revealed a diverse mechanical response in terms of the attained maximum load, slope at initial unloading, and the extent of elastic recovery indicated by the residual depth (h_f) after complete unload. Because the nanoindentation measurements were conducted in CSM mode, the elastic modulus (E) can be determined as a function of indentation depth (Figure 1.5(d)). It can be seen that the E values are unique to each framework structure, and they remain relatively constant beyond the initial contact (h > 100 nm). For example, ZIF-8, ZIF-20, and ZIF-68, which have a more porous structure (SAV ∼ 50%) exhibit a lower structural stiffness in the range of E = 3–4 GPa. In contrast, the densest framework, ZIF-zni (SAV ∼ 12%), has elastic moduli of ca. 8–9 GPa, further revealing the mechanical anisotropy associated with its two different crystal facets, where E₍₀₀₁₎ is higher than E₍₁₀₀₎.

The nanoindentation results of ZIFs show that the underpinning framework architecture constructed from a varied combination of organic and inorganic blocks has a major influence on the mechanical behaviour of the resultant porous frameworks. Not only the network topology plays a role, but the stereochemistry of the imidazolate-type linkers (e.g., bulkiness and stiffness) and porosity are equally important for determining the mechanics of the resultant frameworks. For instance, while ZIF-7, ZIF-8, and ZIF-9 all have identical sodalite (sod) topology, the sterically bulky benzimidazolate (bIm) ligands in ZIF-7 and ZIF-9 confer a greater stiffness value of E ∼ 6 GPa, compared with ZIF-8, whose E ∼ 3 GPa due to its less bulky 2-methylimidazolate (mIm) ligands. Moreover, as shown in Figure 1.5(e), an elastic modulus versus framework density correlation of the form of E ∝ ρ² has been proposed to describe the elasticity trend for the ZIF family of materials. Likewise, an elastic modulus versus SAV relationship of ZIFs has been established, as presented in the Figure 1.5(e) inset, showing an inverse correlation of exists between structural stiffness and porosity.

Single-crystal nanoindentation has been employed to measure the elastic properties of a family of multiferroic MOFs adopting the ABX₃ perovskite topology, see Figure 1.6. Four isostructural frameworks of dimethylammonium metal formate:^53,54  [(CH₃)₂NH₂][M(HCOO)₃] where A = [(CH₃)₂NH₂]⁺ and B is M²⁺ (= Ni²⁺, Co²⁺, Zn²⁺, or Mn²⁺), have been systematically studied. The material is a ‘dense’ framework structure because the dimethylammonium cation occupies the A site, so there is no porosity left in the otherwise open framework to accommodate additional guest molecules. Nanoindentation measurements were performed on the {012}-oriented facets of the pseudo-cubic crystals (Figure 1.6(a)), where the P–h curves and Young’s moduli of the four isostructural frameworks as a function of indentation depth are shown in Figure 1.6(b).⁵²  The determined moduli of the dense MOF perovskites lie in the range of E = 19–25 GPa comparable to the Young’s modulus of a metal-free hybrid perovskite (E ∼ 15 GPa)⁵⁵  and the moduli of several inorganic–organic halide perovskites (E ∼ 12–19 GPa for APbX₃),⁵⁶  but they are around one order of magnitude higher than the nanoporous ZIF structures⁵¹  discussed in Section 1.4.2.

The stiffness of the four hybrid perovskite frameworks rises in accordance with the sequence Mn²⁺ ≈ Zn²⁺ < Co²⁺ < Ni²⁺. As the compounds are isostructural, the differing mechanical properties can be linked to the different divalent metal cations (M²⁺) forming the MO₆ octahedral sites. Figure 1.6(c) shows the approximately linear correlation between the Young’s moduli and the ligand field stabilisation energy (LFSE) connecting the four cations. However, a straightforward correlation to the cation radius was not found, albeit a general trend of the variation in the elastic moduli with octahedral bond distance can be seen in Figure 1.6(d). Consequently, it was proposed from a mechanical stability standpoint that, a higher LFSE in the octahedral environment bestows a greater resistance to mechanical deformation at the MO₆ sites (metal nodes), thereby increasing the elastic modulus of the overall framework structure.⁵²  A follow-on nanoindentation study reported by Li et al.⁵⁷  on two analogous MOF perovskites further revealed that the Young’s moduli increase with an increase in the number of hydrogen bonding interactions established between the A-site molecular cation and the negatively charged framework.

It is important to be able to see the bigger picture and visualise how the mechanical properties of MOFs and other hybrid framework materials are compared with conventional engineering materials, such as metals, polymers, and ceramics. To this end, an Ashby-style plot (materials property chart) may be constructed like the one presented in Figure 1.7, obtained by curating the latest Young’s modulus (E) vs. hardness (H) datasets from the literature. Chiefly, the E–H domain associated with the hybrid framework materials straddles the borders between the metallic, polymeric (organic), and ceramic (inorganic) materials. Within the hybrid framework domain itself, it can be seen that the upper bound is populated by ‘dense’ frameworks that are mechanically stiffer and harder (higher E and H), whilst the lower bound is occupied by porous MOFs and open framework structures which are structurally more compliant and softer (lower E and H). There is an ‘intermediate’ zone, populated by a family of hybrid inorganic–organic perovskites (HOIPs), particularly the multiferroic MOFs with ABX₃ topology and halide perovskites APbX₃ described above. Overall, this chart sheds light on the immense tuneability of the mechanical properties of hybrid frameworks as a whole. In fact, this outcome is unsurprising in light of the vast scope to combine a multitude of organic and inorganic moieties to yield products of different architectures.

The elastic properties of a MOF single crystal, even that with a cubic symmetry, are inherently anisotropic. Accordingly, the magnitude of the elastic constants (E, G, and ν) are orientationally dependent for a single crystal, and thus its mechanical response will change with the direction of loading (Figure 1.3(c)). MOF crystals with a lower crystal symmetry are well known for their extreme elastic anisotropy, which may give rise to anomalous mechanical phenomena, such as a negative Poisson’s ratio (auxeticity), negative thermal expansion (NTE), and negative linear compressibility (NLC); these topics are covered in depth in Chapters 2 and 4. Hitherto, the majority of results on the elastic anisotropy of MOFs have been derived from computational modelling studies, such as density functional theory (DFT)^69–71  and molecular dynamics (MD)^72,73  simulations (Chapter 3), complemented by a more limited set of experiments that have precisely measured the elastic stiffness tensor⁴⁶  and NLC of MOF crystals.^74,75  This section details a few exemplars in the field.

The elastic stiffness tensor (C_ij, see definitions in Chapter 2 with the Voigt notation) of a single crystal of a prototypical ZIF material with sodalite topology, termed ZIF-8, has been measured by means of laser Brillouin scattering experiments.⁴⁶  Due to its cubic symmetry, ZIF-8 possesses three independent elastic coefficients, the values of which are C₁₁ = 9.52 GPa, C₁₂ = 6.86 GPa, and C₄₄ = 0.97 GPa. The stiffness tensor for a cubic ZIF-8 crystal can be cast into a 6 × 6 matrix:

Equation 1.15

Herewith, the coefficient C₁₁ (= C₂₂ = C₃₃ because of cubic symmetry) designates the stiffness along the orthonormal a, b, and c principal crystal axes, respectively, when subject to uniaxial strain. The shear coefficient C₄₄ (= C₅₅ = C₆₆) is the stiffness against an angular distortion, when subject to a shear strain. Finally, the stiffness coefficient C₁₂ (= C₁₃ = C₂₃ = C₂₁ = C₃₁ = C₃₂) corresponds to tensile–tensile coupling between any two orthonormal axes. It follows that the C_ij values of ZIF-8 obey the fundamental stability criteria of a cubic crystal: C₁₁ > |C₁₂|, C₁₁ + 2C₁₂ > 0, and C₄₄ > 0.⁴⁷  Subsequently, the degree of elastic anisotropy for a cubic crystal can be characterised by the Zener ratio A,⁷⁶  where . For ZIF-8, A = 0.73, thus it is moderately anisotropic, noting that A is unity for an isotropic material.⁴⁶ 

Tensorial analysis of C_ij (see Section 2.2 of Chapter 2) is useful for analysing the direction-dependent E and ν, with the complete picture revealing the elastic behaviour of a single crystal of ZIF-8 illustrated in Figure 1.8. In terms of Young’s modulus, the representation surface of E exhibits protuberances along the ⟨100⟩ cube axes, corresponding to the highest stiffness of E_{max (100)} ∼ 3.8 GPa oriented normal to the 4-membered rings (4MRs) of the sodalite framework. Moreover, its lowest stiffness is E_{min (111)} ∼ 2.8 GPa, oriented normal to the 6-membered rings (6MRs). It follows that the Young’s modulus anisotropy of ZIF-8 is given by E_max/E_min = 1.35. Furthermore, an intermediate stiffness of E₍₁₁₀₎ ∼ 3 GPa is present along the ⟨110⟩ face diagonal of the cubic unit cell. Nanoindentation experiments conducted on the above three selected crystal facets are consistent with the results obtained from Brillouin scattering, see Table 1.1. Additionally, theoretical DFT calculations also produced a reasonably good agreement to the experiments (Table 1.1). Crucially, the theoretical results revealed that the underpinning deformation mechanism during uniaxial loading is controlled by the tension/compression of the Zn–N chemical bonds (while imidazolate rings remain rigid), accommodated by bending of the bond angles in the N–Zn–N tetrahedra and Zn–mIm–Zn bridging linkages (Figure 1.8(d)).

The complete anisotropic shear response of ZIF-8 is shown in Figure 1.9 for the 3-D representation surfaces of both the minimum and the maximum shear moduli. Noteworthy is the exceedingly low shear modulus of ZIF-8, where G_min (= C₄₄) ≲ 1 GPa, suggesting a low framework rigidity against structural distortion when subject to shear stresses applied in antiparallel directions of the plane of the 4MRs. In fact, this configuration corresponds to a structurally compliant 4-noded framework, which is susceptible to collapse under shear deformation. DFT reveals that the low shear resistance is due to the pliant ZnN₄ tetrahedra, allowing the framework to distort through the bending of the N–Zn–N and Zn–mIm–Zn bond angles (Figure 1.9(c)). In contrast, the maximum shear modulus is G_max = 1.3 GPa and lies on the planes of the 6MRs, giving better resistance to angular distortion subject to shear stress.

Turning to the Poisson’s ratio, the 3-D representation surfaces of ν_max and ν_min are depicted in Figure 1.10. More specifically, the lower and upper bounds are 0.33 ≤ ν ≤ 0.54. Interestingly, the exceptionally low shear modulus of ZIF-8 is accompanied by a high value of Poisson’s ratio, which surpasses the normal maximum of 0.5 for incompressible rubbery solids (isotropic).

Further to knowing the single-crystal properties, for practical reasons generally it would be of interest to determine the isotropic aggregate properties of a ‘bulk’ polycrystalline solid. To this end, averaging methods such as the Voigt–Reuss–Hill (VRH) scheme can be employed to estimate the mechanical properties of crystal aggregates based on the stiffness and compliance tensors (see Section 2.2.1.2 of Chapter 2). This is exemplified in Table 1.2 for ZIF-8. The obtained values correspond to that of a textureless polycrystalline material; this hypothetical solid comprising ZIF-8 crystals will exhibit isotropic elastic properties.

Research about the role of defects on the function of MOFs is becoming increasingly relevant in the field.^77,78  In this context, it is important to understand how structural defects may impact anisotropic mechanical properties. For example, Möslein et al.⁷⁹  demonstrated the use of tip force microscopy (TFM) to map the local stiffnesses of ZIF-8 nanocrystals harvested from different growth times (see Figure 1.11(a)), where a lower mean stiffness and higher anisotropy in Young’s modulus distribution were observed for the defective single crystals. Further insights can be obtained from DFT simulations of ZIF-8 with structural defects, by comparing the elastic constants (C_ij) of ideal frameworks with those containing ‘missing zinc’ and ‘missing mIm linker’ defects. It was found that the defective structures give rise to a higher degree of mechanical anisotropy, because the maximum and minimum values of Young’s moduli are greatly modified by structural imperfection, as depicted in Figure 1.11(b). The theoretical models allow one to pinpoint the structure–property relationships of a defective structure, where the organic and inorganic connectivity, or indeed the lack of it (see Figure 1.11(c)), alters the mechanical response in very specific directions. Equally, averaged polycrystalline properties from the VRH scheme suggest there will be distinct and sometimes unpredictable changes to the isotropic behaviour stemming from framework defects (Table 1.2). Another DFT study considers the role of missing clusters on the mechanical anisotropy of UiO-66,⁸⁰  the findings of which are presented in Section 3.3.1 of Chapter 3.

Theoretical DFT calculations of the elastic behaviour of a HKUST-1 (Cu₃BTC₂, BTC = benzene-1,3,5-tricarboxylate) single crystal reported by Ryder et al.⁸¹  demonstrated a highly anisotropic mechanical response, as depicted in Figure 1.12. Because of its cubic symmetry, HKUST-1 has three independent elastic constants: C₁₁ = 27.7 GPa, C₁₂ = 25.7 GPa, and C₄₄ = 5.4 GPa, where the C_ij values were computed using a DFT B3LYP functional. The 3-D representation surfaces of E, G, and ν all show large elastic anisotropies. We first consider its Young’s modulus, where the extremum values are: E_max⟨111⟩ ∼ 15 GPa oriented along the cube body diagonals, while E_min⟨100⟩ ∼ 3 GPa directed along the principal cube axes (Figure 1.12(a)). Therefore, the anisotropic ratio for the Young’s modulus is E_max/E_min = 5, which is notably greater than that of ZIF-8 discussed above. Turning to the shear modulus, the DFT predicted values are G_max⟨100⟩ = 5.4 GPa and G_min⟨100⟩ = 1 GPa. The maximum and minimum resistance against shear-induced structural distortion can thus be understood by examining how the shear deformation acts on the arrangement of nodal ‘hinges’ (Cu paddlewheel clusters) connecting the ‘rigid’ BTC linkers, see Figure 1.12(b).

The theoretical Poisson’s ratios of HKUST-1 exhibit anomalous values of ν_max = 1.2 and ν_min = −0.3 (Figure 1.12(c)), which can be attributed to its high elastic anisotropy (A = 5.4). The negative Poisson’s ratio suggests that the single crystal is auxetic when loaded in specific directions (further analysis in Section 2.3.3 of Chapter 2), so the mechanism could be linked to the dynamics of the node-linkages that uncoil when the framework is stretched along the ⟨110⟩ axis, thereby resulting in expansion in the transverse ⟨11̄0⟩ direction. Furthermore, it is worth considering the (bulk) polycrystalline mechanical properties of HKUST-1, see Table 1.2, derived from the VRH averaging scheme. The Young’s modulus of a ‘bulk’ sample was found to be E_VRH ∼ 8 GPa, which is in close agreement to that determined from nanoindentation for isotropic monolithic HKUST-1 (E_monoHKUST-1 ∼9 GPa).³⁴  For the Poisson’s ratio of a bulk HKUST-1 sample, it is intriguing to see that it is positive (ν_VRH = 0.45) even though it can be auxetic as a single crystal (ν_min = −0.3). The findings thus reveal the importance of controlling the precise crystallographic orientation of the sample or polycrystalline film in order to achieve auxeticity, since a random distribution of grains will diminish this effect. However, the epitaxial growth of accurately oriented MOF films and coatings remains a challenging task, and this is still limited only to a few examples of MOFs.^36,82,83 

The mechanical behaviour of porous monolithic MOFs has been little explored to date. Tian et al. reported the enhanced methane adsorption capacity of a monolithic HKUST-1 (Cu₃BTC₂) MOF derived from a sol–gel synthesis route.³⁴  The monolith in fact comprises a polycrystalline aggregate of nanosized HKUST-1 crystals, with the efficient packing and densification route resulting in the formation of a porous solid with a record-breaking methane storage capacity of 259 cm³ (STP) cm⁻³ (versus the US Department of Energy (DOE) target of 263 cm³ (STP) cm⁻³). Figure 1.13 shows the nanoindentation results determined from a sample of HKUST-1 monolith (termed _monoHKUST-1) mounted on an epoxy resin, where 60 residual indents after unloading from 2 µm deep indentations can be observed. The Young’s modulus of _monoHKUST-1 was found to be E ∼ 9.3 GPa (taking ν = 0.433),³⁶  which is comparable with the stiffness value predicted from theoretical DFT (E = 8.1 GPa, see Table 1.2)⁸¹  that assumes a combined elastic response from a polycrystalline aggregate of HKUST-1 crystals. While the modulus of _monoHKUST-1 matches the conventional HKUST-1, the monolith hardness (H ∼ 460 MPa) is more than 130% higher than its conventional counterpart (H ∼ 200 MPa),³⁶  see the inset in Figure 1.13(c). The results suggest that the high bulk density of _monoHKUST-1 (ρ_monolith = 1.06 g cm⁻³ vs. ρ_crystal = 0.883 g cm⁻³) not only improves the volumetric adsorption capacity, but also enhances the mechanical durability of the densified MOF monolith. Of note, there is no sign of surface cracking detected in the vicinity of the residual indents, as evidenced from the AFM topographic image in Figure 1.13(a), indicating the good mechanical resilience of densified _monoHKUST-1. The findings also suggest that the sol–gel method could potentially be deployed to fabricate mechanically tough monolithic MOF materials aimed at practical engineering use.

Crystalline MOFs can experience structural collapse and framework densification to yield an amorphous phase when subject to an external stimulus, such as high temperature, hydrostatic pressure, or shear stress. Figure 1.14 depicts the crystal–amorphous transition of a ZIF-4 crystal that transforms into amorphous a-ZIF upon heating to 300 °C, and its subsequent recrystallisation to form crystalline ZIF-zni by heating to 400 °C. The recovered monolith shown in Figure 1.14(c) reveals two disparate phases (designated as ‘dark’ and ‘bright’), where nanoindentation was applied for differentiating between a-ZIF and ZIF-zni based on their mechanical properties. It can be seen in Figure 1.14(d) that, the Young’s modulus and hardness of the ‘dark’ phase (E ∼ 6.5 GPa, H ∼ 0.65 GPa) perfectly match those of a pristine monolithic a-ZIF material that is isotropic.³⁸  The disordered network topology of the amorphous phase implies mechanical isotropy, more precisely, a-ZIF has lost the elastic anisotropy of its ZIF-4 parent phase (where E_max/E_min ∼ 1.6 for a ZIF-4 crystal).⁸⁴  Remarkably, a-ZIF exhibits glass-like behaviour evident from the curved external surfaces and internal cavities present on the monolith (Figure 1.14(c)), indicating that the amorphisation process involves viscous flow during formation of a-ZIF at 300 °C. Subsequently, systematic nanoindentation studies on a family of ZIF melt-quenched glasses comprising a_gZIF-4, a_gZIF-62, a_gZIF-76, and a_gZIF-76-mbIm, have yielded Young’s moduli (in GPa) of 6.9, 6.6, 6.3, and 6.1, respectively. Likewise, the hardness values of these ZIF glasses are similar to that of a-ZIF, all of which are distributed in a narrow range of H = 0.66–0.68 GPa;³⁹  their properties can best be contrasted against other crystalline open frameworks shown in the Ashby-style plot in Figure 1.7.

Turning to the nanoindentation of the ‘bright’ phase that can be observed in Figure 1.14(c), the moduli and hardness values (E ∼ 9 GPa, H ∼ 1.2 GPa) were found to be in the range of the ZIF-zni crystal, which is anisotropic (Figure 1.14(b)). In fact, the recrystallised ZIF-zni is a polycrystalline monolith, comprising submicron grains of random orientation, as confirmed from its electron diffraction pattern. One can therefore expect that the nanoindentation results of the above specimen will lie within the upper and lower bounds of a single crystal of ZIF-zni (DFT predictions: E_max ∼ 12.3 GPa and E_min ∼ 4.7 GPa).⁸⁴ 

Precise measurement of the mechanical properties of nanocrystals (∼100 s of nm) and isolated micron-sized crystals (a few microns) cannot be accomplished using a conventional instrumented indentation technique because of the extremely small volume of samples involved. The latter technique requires a sample size of at least 50–100 µm ⁵⁹  for it to be securely mounted in epoxy and surface polished. The nanomechanical characterisation of individual nanocrystals can be achieved, for example, through the application of atomic force microscopy (AFM)-based nanoindentation techniques.

Zeng and Tan⁸⁵  demonstrated the application of a diamond-tipped stainless steel cantilever probe (with calibrated cube-corner indenter) for quantitative characterisation of fine-scale crystals of ZIF-8. Figure 1.15(a) shows the AFM topographic scan of the ZIF-8 nanocrystals with a size of around 500 nm, prepared by drop casting of its suspension onto a glass substrate. Representative P–h curves for a set of very shallow indents (h_max ∼ 10–30 nm) and another set of deeper indents (h_max < 80 nm) are shown in Figure 1.15(b and c), respectively, corresponding to applied loads in the range of ca. 2–12 µN. Large elastic recovery upon tip unloading is evident, especially for the set of shallow indents in contrast to the deeper ones that exhibit more permanent deformation after indenter unload. It was demonstrated that reliable nanoindentation curves in the few µN load range (for surface penetration depth of tens of nm) can be better obtained through a high unloading strain rate of ε· > 60 s⁻¹, see Figure 1.15(d). It can be seen that a Young’s modulus of ZIF-8 in the range of E ∼ 3–4 GPa can be determined, in agreement with conventional instrumented nanoindentation⁵¹  but the crystal size concerned is around 1000 times smaller by employing the AFM nanoindentation approach.⁸⁵  Whilst the results are promising, the wider applicability of this AFM-based indentation technique to probe a wide range of MOF nanocrystals should be systematically investigated.

The mechanical properties of polycrystalline thin films and surface coatings of MOF materials have also been studied by nanoindentation, the available data of which are summarised in Figure 1.7. Bundschuh et al.³⁶  applied a liquid epitaxy method to fabricate a {100}-oriented film comprising HKUST-1 crystals with a thickness of ca. 1 µm, grown on a gold-coated silicon substrate. The epitaxial film has a surface roughness of ca. 10–15 nm (Figure 1.16(a)), deemed to be suitable for nanoindentation study without additional surface preparation steps. However, the P–h curve of the film sample shown in Figure 1.16(b) does not conform to the standard parabolic response because of the strong influence of the stiffer and harder substrate located beneath the relatively thin sample of HKUST-1. The substrate effect becomes prevalent in Figure 1.16(c and d), revealing how the indentation modulus rises sharply as a function of indentation depth (in contrast to a relatively constant moduli when probing a single crystal or monolith, see Figure 1.14). A 10% value of the surface penetration depth was typically applied to minimise the substrate contribution,⁴²  corresponding to an indentation depth of ∼100 nm, where the indentation modulus was determined to be 11.4 ± 2 GPa. Subsequently, assuming isotropy and by taking a theoretical Poisson’s ratio of ν = 0.433,⁸⁶  the Young’s modulus of the HKUST-1 film was estimated as E_{(100) epitaxial film} = 9.3 GPa. Note that this is in contradiction to a theoretical DFT study that found that the E₍₁₀₀₎ of a single crystal of HKUST-1 is minimum, and its magnitude is three times lower (Figure 1.12(a)).⁸¹  The contributions from grain boundaries in epitaxial film combined with the hard substrate effect may explain the observed discrepancy, and indeed the wider spread of data depicted in Figure 1.7 for a number of MOF thin film samples indented to date. Interestingly, the magnitude of E_{(100) epitaxial film} matches that of a monolithic HKUST-1 sample comprising a random aggregate of nanocrystals: E_monoHKUST-1 = 9.3 ± 0.3 GPa,³⁴  which represents an isotropic response determined from a more substantial indentation depth of 2000 nm (see Figure 1.13).

This study highlights some outstanding challenges surrounding the reliable nanoindentation measurements of thin-film MOF samples, especially those possessing a thickness of ≲1 µm; future efforts are warranted in this area to explore the efficacy of AFM nanoindentation for characterising MOF thin films. Further discussions on the mechanics of MOF films and polycrystalline coatings are given in Section 1.8.3.

Many of the unusual mechanical features of MOFs and their unique functions observed on the macroscopic and microscopic scales can be explained by examining the terahertz (THz) vibrations of the underpinning molecular framework. Terahertz vibrational modes are collective lattice dynamics with an oscillation frequency of typically below 20 THz or ∼700 cm⁻¹ (1 THz = 10¹² Hz = 4.14 meV ≈ 33.3 cm⁻¹); indeed they are low-energy phonons that stem from the structural ‘flexibility’ of the MOF framework. Experimentally, the THz vibrations of MOFs have been measured by high-resolution inelastic neutron scattering (INS),^87–89  synchrotron radiation far-infrared (SR-FIR) spectroscopy,⁸⁷  Raman spectroscopy,^81,90  and terahertz-time domain spectroscopy (THz-TDS)⁹¹  techniques. Identification of the nature of the collective vibrational modes is challenging, because these low-energy vibrations belong to the collective dynamics of the entire framework. There are hence no characteristic frequencies associated with the standard functional groups that are commonplace to, for example, mid-infrared (MIR) spectroscopy, where the fingerprinting of diverse chemical moieties exists. Instead, the precise identification of the THz modes of MOF structures requires the application of theoretical simulations, such as quantum mechanical DFT calculations.^87,91,92  A combined experimental and theoretical methodology can offer rich insights into the THz modes of nanoporous frameworks. This approach sheds new light on how low-energy phonons may affect the thermal and mechanical stability of extended framework structures,^89,93,94  helping to reveal complex physicochemical phenomena through soft phonon modes⁹⁵  and rotor dynamics,⁹⁶  which may affect the performance of nanoconfined guest–host systems.^29,97 

The THz dynamics of ZIF structures have been relatively well researched compared with other families of MOF materials. Using SR-FIR experiments coupled with DFT calculations to investigate the far-infrared (FIR) modes of ZIF-4, ZIF-7, and ZIF-8, Ryder et al.⁸⁷  established that ZIFs exhibit two commonly shared collective dynamics under ∼21 THz, as shown in Figure 1.17(a). First, the vibrations in the range of 8–10 THz (265–325 cm⁻¹) arise from flexible ZnN₄ tetrahedra. Second, at around twice greater than the energy of the former band, the vibrational modes at 18–21 THz (600–700 cm⁻¹) originate from ring deformations of the imidazolate-type linkers. FIR absorption spectroscopy is a highly sensitivity probe for characterising vibrational modes linked to C, N, and H that prevail in ZIFs. When combined with a synchrotron light source, SR-FIR enables the collection of THz signals with a high signal-to-noise ratio down to 0.6 THz (∼20 cm⁻¹). Theoretically, DFT vibrational calculations suggest that notable lattice dynamics of porous frameworks are ubiquitous in the region below 3 THz (≲100 cm⁻¹). In practice, however, it can be seen that the intensity of FIR modes is weaker in this low wavenumber region, in contrast to the more intense energy loss signals measured via inelastic neutron scattering (INS).

Unlike optical techniques (e.g., IR and Raman), neutrons have no optical selection rules, meaning that in principle all transitions are active.⁹⁸  Furthermore, it is advantageous to use INS to study ZIFs (and MOFs in general)^89,97,99,100  due to its high sensitivity for detecting hydrogen modes, simply by leveraging the large incoherent neutron cross section of hydrogen. For the low wavenumber region below 100 cm⁻¹ a precise match between experiment and theory remains challenging for ZIFs, though a reasonable agreement between them can be observed (see Figure 1.17(b)). Improved peak-to-peak assignment between the INS and DFT spectra has been demonstrated in another study on a Zr-based MOF, termed MIL-140A, within the low-energy THz region of 0–250 cm⁻¹.⁹⁶  In the following section we shall consider lattice dynamics determined from DFT simulations for exemplifying several interesting mechanical and physical phenomena underpinning the functions of porous ZIF structures.

In the case of ZIF-8 as depicted in Figure 1.17(c), it is interesting to consider the collective lattice dynamics of a ‘soft mode’ at 0.57 THz (∼19 cm⁻¹) and a shear-induced deformation mode at 0.65 THz (∼22 cm⁻¹).⁸⁷  Importantly, these two sub-THz vibrations influence the elasticity of the porous network and potentially trigger structural destabilisation of the sodalite cage. The presence of a soft phonon mode may explain the occurrence of a mechanically-triggered phase transition under pressure¹⁰¹  and guest-induced structural response¹⁰²  evidenced in ZIF-8. When the framework is directionally stressed, the six-membered rings (6MR) distort to accommodate tensile and compressive strains propagated through the interconnected mIm–Zn–mIm linkages, permitting framework flexibility in accordance with the dynamics of the soft mode. Turning to the shearing mode, this involves the angular distortion of the four-membered rings (4MR) caused by a pair of antiparallel shear forces. The four-node geometrical configuration of the 4MR is mechanically unstable, making intrinsically susceptible to undergo a structural collapse under shear deformation.

Likewise, the porous structures of ZIF-4 and ZIF-7 can experience lattice distortions driven by the shearing dynamics in the low-frequency THz regime (≲1 THz), attributed to the flexibility of the 4MRs. Crucially, the crystalline structures of ZIF-4, ZIF-7, and ZIF-8 can be made amorphous via ball milling,^103,104  where the shear stress induced by the sliding motion of the impacting balls induces structural collapse and densification of the porous frameworks. It has been postulated that the stress-induced amorphisation of ZIFs, and more generally the structural destabilisation of MOFs, can be triggered by THz shear modes inherent in porous solids. Moreover, research has found that when the shear modes are not causing framework amorphisation, they play a vital role by facilitating lattice modes that alter specific functions. For instance, the soft modes at sub-THz frequencies in ZIF-4 enable the stretching and compression of 6MR apertures, which are accommodated by shearing of 4MR in the perpendicular orientation (Figure 1.18(a)). This mechanism offers a pathway for phase transitions observed in desolvated ZIF-4,¹⁰⁵  further modifying the accessible pore volume for enhanced gas adsorption.¹⁰⁶  Another example involves the sub-THz ‘breathing’ mode of ZIF-7 (Figure 1.18(b)), where the shearing of 4MR enables the twisting spiral motion of the bulky benzimidazolate (bIm) linkers surrounding the 6MR aperture. This kind of pore opening mechanism, akin to ‘gate-opening’ can be leveraged to regulate the level of CO₂ adsorption by ZIF-7.¹⁰⁷  In particular, the collective vibrational mode of the order of ∼1 THz associated with the dynamics of gate-opening/-closing of ZIF-8 (Figure 1.17(c)), is the key to understanding its unusual guest uptake characteristics,^101,108  chemical selectivity,¹⁰²  and basic mechanics^92,109  in the wider context. A recent study by Möslein et al. on large-cage ZIF-71 with the RHO topology,¹¹⁰  further elucidated that low-frequency THz dynamics (Figure 1.18(c)) play a key role in influencing the mechanical properties and various physical phenomena that control the functions of porous frameworks.

The HKUST-1 framework has a few distinct classes of low-energy lattice dynamics in the THz and sub-THz regimes, which are linked to the elastic behaviour of the porous structure.⁸¹  Figure 1.19(a) illustrates the ‘trampoline-like’ deformation modes viewed down the ⟨111⟩ and ⟨110⟩ crystallographic directions. There are four such vibrational motions located in the range of 1.7–3 THz, which are either IR, non-optically active, or Raman modes. The first two modes comprise the out-of-plane bending deformation of the benzene-1,3,5-tricarboxylate (BTC) organic linker that mimics the oscillatory dynamics of a trampoline. The lower-energy Raman-active mode at 1.7 THz is unique in the sense that it is accompanied by the simultaneous rotational motion of the Cu paddlewheel, see Figure 1.19(b). Significantly, such combined trampolining and molecular rotor dynamics offer the mechanism needed to afford negative thermal expansion (NTE), as experimentally observed in HKUST-1.¹¹¹  More precisely, in the temperature range of ca. 100–400 K the HKUST-1 framework experiences shrinkage (instead of expansion) when heated to yield a smaller unit cell volume. In the same way, similar lattice dynamics could be the source of NTE phenomenon in MOF-5 ^93,95  and other MOF structures constructed from pliant linkages.¹¹² 

Now, we consider a sub-THz vibration of HKUST-1 with synchronous cluster dynamics, where this mode is postulated to be the origin of the negative Poisson’s ratio (NPR) or auxetic phenomenon predicted by DFT. Specifically, the collective dynamics at 0.5 THz couple the rocking and translational motions, thereby compensating for the coordinated rotation of the linker–paddle-wheel clusters, as depicted in Figure 1.19(c). From a mechanical standpoint, this mechanism enables the organic–inorganic clusters to rotate in a spiral fashion upon uniaxial stretching in the axial direction, producing another elongation (instead of contraction) in the lateral direction, and thus exhibiting a auxetic response where ν_min = −0.3 for HKUST-1.⁸¹ 

Further to the dynamics of ZIFs and HKUST-1 materials, the study of THz vibrations has been extended to a family of structurally more flexible MOFs, such as DUT-8(Ni) and MIL-53(Al). Krylov et al.⁹⁰  employed Raman spectroscopy to probe the collective vibrational modes of the ‘rigid’ vs. ‘flexible’ forms of the pillared-layer DUT-8(Ni) framework, focussing on the low wavenumber region below 300 cm⁻¹ (≲10 THz). While the rigid DUT-8(Ni) nanocrystals can be desolvated without undergoing a phase transformation, the flexible DUT-8(Ni) macro crystals reversibly transform between a desolvated closed-pore (CP) phase and a large-pore (LP) phase upon adsorption of gas or liquid guest molecules. Figure 1.20(a) shows that the rigid framework of DUT-8(Ni) exhibits a persistent sub-THz band at 23 cm⁻¹ (0.69 THz), even after desolvation, corresponding to the LP phase. In contrast, the CP phase of flexible DUT-8(Ni) after desolvation exhibits a higher characteristic band at 59 cm⁻¹, although this 1.8 THz mode is absent in the as-synthesised version of flexible DUT-8(Ni).

Notable also is the width of the 59 cm⁻¹ Raman band in the CP form, which is significantly broader than the 23 cm⁻¹ band in the LP form, indicating an increase in the intermolecular π–π interactions within the crystal when it adopts a closed pore configuration (Figure 1.20(b)). In terms of its elastic properties, a high degree of mechanical anisotropy is expected for DUT-8(Ni) akin to its isostructural analogue termed DMOF-1, where the 3-D representation surface of the Young’s modulus of the latter is depicted in Figure 1.20(c). On the basis of the LP structure of DUT-8(Ni), it is anticipated that the mechanically stiff directions are oriented along the naphthalene dicarboxylate linkers pointing towards the {110} and {1̄00} axes, and in the {001} out-of-plane direction where the Ni₂(dabco) (dabco = 1,4-diazabicyclo[2.2.2]octane) chains are present. Clearly the mechanically pliant directions correspond to the a- and b-axes of the porous channels marked in Figure 1.20(b), where the soft phonon mode responsible for the breathing mechanism of DUT-8(Ni) LP is operational and has a major impact on the mechanical behaviour of the entire framework.

Hoffman et al.¹¹³  investigated the role of THz vibrations on the mechanics of the breathing behaviour and thermodynamic properties of flexible MIL-53(Al) using periodic DFT calculations. Like DUT-8(Ni) elucidated above, the structure of MIL-53(Al) can switch between the LP and CP phases upon guest adsorption or under mechanical stress.^114,115  The theoretical results reveal that the soft modes are vital as they help to stabilise the CP structure at low temperatures, and they could also incite LP-to-CP phase transformation. Figure 1.21 shows the volume–frequency characteristics of MIL-53(Al) in the low-frequency THz region, in which some collective motions are observed to undergo relatively large frequency variations ascribed to either molecular repulsion or internal strain effects. It is conceivable that such vibrational fluctuations could trigger phase transformation. Figure 1.21 shows the trampoline-like deformation motions of the linker fluctuate in the range of ca. 2–4 THz as the cell volume evolves from the LP-to-CP phase. The findings thus suggest that the THz mode concerned may be responsible for stimulating the breathing mechanism observed in MIL-53(Al). Furthermore, the theoretical results suggest that the THz vibrations have a pronounced effect on the thermodynamic properties (e.g., specific heat capacity, bulk modulus, thermal expansion coefficient), notable in such a way that even a small deviation in the predicted frequencies results in a large change in the resultant calculated properties. This study demonstrates that computational simulations are thus a powerful tool to use to gain insights into the detailed mechanical mechanisms of highly flexible MOF structures (Chapter 3), which might be intractable from experiments alone.

Related to the discussion above, a theoretical DFT study by Wang et al.¹¹⁶  revealed the dramatic effects that guest molecules have on modifying the elastic properties of a flexible MIL-53(Al) ‘host’ structure. The hydrated structure of MIL-53(Al) contains guest water molecules that are hydrogen bonded to the porous host, conferring a reinforced wine-rack ‘guest@host’ assembly, with augmented elastic moduli but reduced mechanical anisotropy. For example, for the LP phase, the vacant structure of MIL-53(Al) was predicted to exhibit E_{max(LP/empty)} = 94.4 GPa and E_{min(LP/empty)} = 0.9 GPa, in stark contrast to the water-occupied MIL-53(Al) structure with E_{max(LP/water)} = 75.4 GPa and E_{min(LP/water)} = 21.6 GPa. It follows that the anisotropy ratio of the Young’s moduli significantly fell from E_max/E_min ∼ 105 to just 3.5; with the model assuming that no phase transformation occurs upon water intrusion. Likewise, for the CP phase of MIL-53(Al), it was predicted that the hydrated structure has a maximum Young’s modulus of E_{max(CP/water)} = 126 GPa, which represents a major increase from the E_{max(CP/empty)} = 71 GPa of the vacant framework; accordingly the anisotropy ratio fell from around 44 to 28 upon hydration of the CP phase. The simulations reveal that hydrogen bonds from the adsorbed water molecules mechanically reinforce the initially porous framework and by doing so they not only suppress mechanical anisotropy but shift the pliant directions of the original structure.¹¹⁶  While direct comparison with experiments are currently not available to validate the above elastic calculations of MIL-53(Al), there are reported single-crystal nanoindentation measurements that have been performed on several desolvated vs. solvent-containing MOF structures that suggest a highly tuneable mechanical response as a function of the guest pore occupancy.^33,51 

Finally, to conclude this section we shall consider the molecular rotational dynamics of a zirconium-based framework, called MIL-140A [ZrO(O₂C–C₆H₄–CO₂)], where the rotor-like twisting motions of the organic linkers are prevalent.⁹⁶  Figure 1.22(a) summarises the cooperative THz dynamics of the type-A and type-B rotors observed under 3 THz (∼100 cm⁻¹), in which the hindered rotations of the C₆H₄ aromatic rings within the 1,4-benzenedicarboxylate (BDC) linkers can be classified into either the symmetric or asymmetric modes. Of great importance is the consequence that rotor motions have on the dynamic evolution of the SAV, which quantifies the accessible voids present in the framework. The rotational energy barriers have been estimated by DFT calculations for the φ = 0°→180° twist, revealing that the type-A rotational barrier is ∼283 meV per aromatic ring (Figure 1.22(b)). Moreover, the barrier for type-B rotors cannot be reliably obtained because of the steric hindrance of the overlapping linkers. As depicted in Figure 1.22(c), it is striking to see that the morphology of the SAV evolves continuously with the twist angle φ, oscillating between ∼25% and 28% (φ_min/max respectively), whereby the SAV equilibrium is ∼26% when φ = 0°. When φ_min = 160°, this angular motion confers a fully ‘gate-open’ geometry when the adjacent pores coalesce to form continuous 1-D channels along the c-axis. More broadly, this exemplar shows how entwined the low-energy THz rotational modes with the MOF mechanics, thereby driving structural transformation and altering the physicochemical response of porous frameworks.

The ‘hardness’ of a material is a measure of its mechanical resistance against localised plastic deformation. Generally, plasticity occurs (in metals) when the applied stress exceeds the yield strength of the material, when σ_applied ≥ σ_Y. As illustrated in Figure 1.2(a), plastic deformation produces a permanent plastic strain (ε_p), which is an irreversible deformation on a material that is sustained after complete removal of the applied force.

Typically, the hardness value is characterised by indentation-based techniques (e.g., Mohs, Vickers, Knoop, Shore, Rockwell, and Brinell hardness tests), or by employing a scratch test to assess the materials surface resistance towards abrasion. Mohs hardness is the earliest hardness scale that was introduced, comprising values of 1 to 10 established by ranking a series of progressively harder materials to resist scratching, namely (softest → hardest): talc, gypsum, calcite, fluorite, apatite, feldspar, quartz, topaz, corundum, and diamond.¹¹⁷  Of course, this is a crude estimate of the different degrees of hardness, expressed in a qualitative manner. A more quantitative approach, such as the Vickers ‘microhardness’ test, involves the use of a Vickers diamond indenter (four-sided pyramid) and indentation loads ranging from 100 mN to 10 N. This technique is commonly used for measuring the hardness of (bulk) engineering materials, such as metals, ceramics, polymers, and composites. The measured hardness is expressed in units of HV.

Because of the relatively small sample volumes involved compared with conventional solids, the characterisation of the hardness of MOF materials is often achieved by instrumented nanoindentation (Section 1.4). Using a Berkovich diamond tip (three-sided pyramid) and performed under a maximum load of just tens of mN or less.^{33,35,57,59,64}  The value of hardness derived from nanoindentation, also termed ‘nanohardness’, can be calculated from eqn (1.13): H = P/A, where P is the indentation load and A is the projected contact area under load (see Figure 1.4). The nanohardness is expressed in units of Pascal (Pa).

From the foregoing discussion, it should be clear that the determined hardness value is not a unique materials property (like the Young’s and shear moduli), but a function of the chosen test method and its accompanying test parameters, including indenter geometry, applied load, indentation depth, and formula/model used to compute the magnitude of hardness. It is therefore important to note that the H values generated from different techniques are not necessarily comparable, unless the main features of the tests are similar.²⁴  In this chapter, only the hardness measurements of MOFs derived from nanoindentation are considered. From the E–H materials selection chart (Figure 1.7), it can be seen that the hardness values of porous ZIFs, boron imidazolate framework (BIF), HKUST-1, a-ZIF, MOF glasses, and dense ZIF-zni typically lie between several hundreds of MPa and ∼1 GPa. In contrast, the hardness values of other dense hybrid frameworks (comprising inorganic and organic building blocks) are at least one order of magnitude higher, thereby lying mainly in the range of about 1–10 GPa.

More specifically, in the family of ZIF structures there is an inverse correlation between the hardness of single crystals and the SAV, as elucidated below.⁵¹  What can be considered as ‘soft porous crystals’ are the highly porous framework structures (SAV ∼50%) of ZIF-20, ZIF-68, and ZIF-8, the hardness values of which are: H = ∼250 MPa, ∼300–500 MPa, and ∼550 MPa, respectively. Lesser porous frameworks, such as ZIF-7 and ZIF-9 (both SAV ∼ 26%), while having the same sodalite (sod) topology as ZIF-8, have a greater hardness of H = ∼650–700 MPa due to the sterically bulkier bIm linkers of ZIF-7 and ZIF-9 vs. the mIm linkers in ZIF-8. The least porous ZIF is ZIF-zni (SAV = 12.2%), which to date remains the hardest compound in the ZIF family with H = ∼1.1 GPa. Surprisingly, the isostructural BIF-1-Li crystal (LiB(Im)₂, SAV = 5.3%) has a substantially lower hardness of ∼0.15 GPa, which is an order of magnitude lower than that of ZIF-zni.⁶²  From the relative deformation data of the different tetrahedra (LiN₄ > ZnN₄ > BN₄), it can be reasoned that the lithium coordination environment and adjoining Li–Im–B linkages are more susceptible to mechanical deformation compared with the zinc counterpart.

Hardness (H) and yield strength (σ_Y) of conventional materials are correlated through the Tabor relationship:¹¹⁹ 

where C is termed the ‘constraint factor’. The value of C is materials dependent, ranging from a value of ∼1.5 for brittle solids such as glasses (small E/σ_Y ratio) to ∼3 for ductile materials such as metals (large E/σ_Y ratio).¹²⁰  In the context of nanoindentation with a spherical indenter tip, it has been proposed that an identical relationship exists between hardness and the yield pressure (P_Y), such that H = C·P_Y. Unlike a Berkovich tip, the spherical geometry (typical tip radius in the order of 1 µm) generates a much greater contact area at the same penetration depth, thereby delaying the onset of plasticity as shown in Figure 1.23(a) for the comparative indentation tests conducted on a single crystal of ZIF-8. The ‘indentation stress–strain’ curve (Figure 1.23(b)), in the form of contact pressure (P_m) versus contact strain (a/R) can subsequently be derived from a spherical indentation experiment (for method, see ref. 121). The point of deviation from the Hertzian linear response indicates the yield pressure, P_Y. For ZIF-8, using this methodology it can be estimated that the magnitudes of P_Y lie in the range of 300–350 MPa and there is also sign of plastic anisotropy associated with the (100), (110), and (111) facets. Beyond the yield point, the indentation stress–strain curves appear to show a power-law hardening behaviour, given by P_m = (a/R)ⁿ, where n is the strain hardening exponent. While this kind of hardening curve is typically linked to dislocation entanglement prevalent in metallic materials, the underlying mechanism is currently unknown for inorganic–organic (hybrid) solids constructed from strongly directional coordination/covalent bonds.

Spherical indentation studies on dense 3-D inorganic–organic frameworks have shown considerably higher yield pressures of up to P_Y ∼ 2.3 GPa, and with constraint factors of C ∼ 2 and 2.4 depending on the indented crystal orientation.³²  Imaging techniques, such as cross-sectional transmission electron microscopy (TEM) revealed whether dislocation glides are present beneath the indented zone, which may accommodate plastic flows observed in a number of inorganic crystals that are intrinsically brittle in nature.^122,123  On the theoretical front, Banlusan et al.¹²⁴  have employed large-scale MD simulations to investigate the plastic deformation behaviour of a cubic MOF-5 crystal subject to uniaxial compression, see Figure 1.24. They show that irreversible deformation mechanism in the crystal is governed by slip of dislocations driven by compressive and shear stresses, leading to the formation and propagation of shear collapse bands shown in Figure 1.24(e–h). This theoretical study demonstrates that the activation of the multiple ⟨001⟩{100} slip systems controls the plastic deformation of a cubic framework crystal, where the organic–inorganic linkages are all oriented along the cubic axes. The flexibility of the metal clusters facilitates the rotation of the organic linkers to initiate structural yielding under a shear strain. It can be seen that the amount of pore collapse and structural densification caused by the application of compressive stress is very substantial beyond the yield point. The theoretical findings are very interesting, but there is still no direct experimental evidence of dislocation activity in MOFs for validating the large-strain predictions described above.

Recent developments on the study of the yield stress and hardening behaviour of ZIF monoliths show that by using a combination of Berkovich/cube-corner indentation and finite-element (FE) modelling (Figure 1.25(a)) it is possible to determine an improved value of yield strengths, where σ_Y = 200 MPa for a ZIF-8 monolith and σ_Y = 130 MPa for a ZIF-71 monolith.³⁵  The constraint factor was found to be C = 2.1, which is an intermediate value for a brittle and a ductile material. Beyond the elastic limit, FE simulations show that the monoliths experience very limited plastic hardening, supported by the lack of materials pile-up characterised by AFM in the vicinity of the residual indents (Figure 1.25(d)). Therefore, the use of an elastic-perfectly plastic constitutive model is sufficient to simulate the experimental indentation curves of the ZIF monoliths considered in this work. The nanograined microstructure of the ZIF monoliths was revealed by tip force microscopy (TFM), suggesting that grain boundary sliding is likely operational under stress (Figure 1.25(e)); this mechanism is also consistent with the absence of surface cracking when tested under a Berkovich indenter (see Section 1.8.4 on fracture toughness). This study also demonstrates the novel application of near-field infrared nanospectroscopy (nanoFTIR)¹²⁵  to characterise the local deformations across a residual indent of MOF at ∼20 nm spatial resolution, revealing the effect of stress concentration on the framework distortion/collapse and breakage of chemical bonds.³⁵ 

This section considers the plastic deformation behaviour of layered 2-D metal–organic nanosheets (MONs) when stressed beyond the elastic limit. The earliest example concerns the ‘pop-in’ phenomenon (defined as a displacement burst under a constant load)^126,127  observed during nanoindentation of a copper phosphonoacetate (CuPA) single crystal with a (dense) layered inorganic–organic framework architecture.³²  The pop-ins detected in the P–h curves arise from the breakage of the hydrogen bonds binding the adjacent 2-D layers together, but gave way under shear stresses acting on the 2-D planes. The AFM topography height image obtained from the plastically deformed region revealed the formation of shear bands, where the height of the slip steps was in the range of 10–50 nm.

Another exemplar concerns a dense 2-D nanosheet crystal constructed from a Mn 2,2-dimethylsuccinate (MnDMS) framework (Figure 1.26(a)), where the thickness of the monolayer is around 1 nm and the 2-D stack is held together by van der Waals interactions.⁶⁷  Spherical nanoindentation performed normal to the (100) and (010) crystal facets yielded substantial pop-in displacements caused by the shear-induced delamination of the weakly bonded 2-D layers, see Figure 1.26(b and c). Using the knowledge of critical load (P*) measured from the first pop-in event (i.e., the onset of plasticity), and by applying the Hertzian elastic contact theory (eqn (1.17)),¹²⁸  it is possible to estimate the magnitude of the critical resolved shear stress (τ_crit) to initiate splitting of the two adjacent layers in the 2-D framework, given by:

where E_r is the reduced modulus and R is the radius of the spherical indenter tip. It was reported that the values of critical resolved shear stress are as low as τ_crit = 0.24–0.39 GPa for the in-plane directions of the layered crystals of MnDMS.⁶⁷  In contrast, the 2-D layers of CuPA held together by hydrogen bonds have a distinctively higher value of τ_crit ∼ 1 GPa.³²  The above findings explain why micromechanical exfoliation by ultrasonication and mechanical shearing present an effective pathway by which to produce thin nanostructures of 2-D MOFs held together by weak interactions.

The failure mechanisms of a 2-D MOF nanosheet with atomic-sized pores, comprising a copper 1,4-benzenedicarboxylate (CuBDC) framework, have been studied by Zeng et al. utilising AFM-based nanoindentation experiments coupled with FE modelling.¹²⁹  In terms of its elastic–plastic properties in the through-thickness directions, the Young’s modulus is E ∼ 23 GPa, and the yield strength is estimated to be σ_Y ∼450 MPa. In this study, a power-law hardening behaviour was proposed beyond the yield point to simulate the P–h curves observed in AFM indentation (Figure 1.27). Of note, this study shows that it is possible to differentiate between the modes of mechanical failure that occur during nanoindentation, by carefully examining the distorted shapes of the indentation P–h curves. For CuBDC nanosheets, three distinctive plastic deformation modes have been proposed: mode I – interfacial slippage between nanosheets, mode II – fracture of the nanosheets, and mode III – interfacial delamination of the nanosheets. The failure mechanisms of the CuBDC nanosheets are summarised in Figure 1.27(c and d), where these mechanisms are potentially applicable to a broader family of van der Waals solids^130,131  and layered 2-D compounds such as covalent organic frameworks (COFs).^132,133 

Under the influence of mode I, the coplanar layers are pushed apart sideways by the nanoindenter as it penetrates deeper into the 2-D stack, where the van der Waals interactions are ruptured predominantly by shear-induced deformation. This kind of slippage failure results in the occurrence of a ‘pop-in’ phenomenon during the loading stage. In the case of mode II, the failure mechanism can be attributed to a stress build-up at the indenter tip prior to fracture, identifiable by the formation of anomalous ‘humps’ evident in the P–h curves (identifiable by a distinct rise in load, prior to its decline). For mode III, the structural bending mechanism causes the interlayer delamination of the 2-D stack during indenter loading, followed by the occurrence of the ‘pop-out’ and recovery phenomena during the unloading stage. All three mechanisms are operational at different stages of the indentation experiment, as evidenced from the AFM nanoindentation data shown in Figure 1.27(e). A vital implication of local nanoscale failure on the overall mechanical properties of the nanosheets can be recognised through the substantial decline of their Young’s moduli: from E ∼ 23 GPa (in the normal P–h curves), falling to ∼12 GPa as a result of delamination failure, and the stiffness further decreases to ∼4 GPa due to the slippage failure and fracture of the 2-D layers.

For device applications in the dielectrics, sensors, lighting and optoelectronics sectors, it is important to fabricate MOF materials in the form of high-quality thin films or well-adhered coatings onto a range of engineering substrates and components.¹³⁴  These are in fact polycrystalline films encompassing fine-scale MOF crystals prepared through a range of methods,¹³⁵  such as dip coating in a colloidal dispersion of preformed nanocrystals, seeded growth, layer-by-layer deposition (Figure 1.16), electrochemical methods, or more recently by employing more sophisticated techniques such as chemical vapour deposition (CVD) and lithography.¹³⁶  Mechanical characterisation of the properties of the polycrystalline thin film (apart from the E and H values) is not commonly reported in the literature, although the knowledge of surface adhesion (against delamination failure), interfacial behaviour, and fracture strength are central to the practical applications of technological devices.

Figure 1.28(a) shows the results of a nanomechanical study performed on a polycrystalline film comprising submicron crystals of ZIF-8, fabricated via dip coating on a glass substrate.⁸⁵  A cube-corner AFM diamond probe was used to generate a significantly deeper indentation up to h ∼ 200 nm (vs. typically tens of nm for AFM nanoindentation), with which the collective mechanical response of a polycrystalline film can be measured. By comparing the AFM height topographic images of the polycrystalline surface, obtained before and after indentation testing, these reveal a sliding mechanism accompanied by pile-up/sink-in from the interparticle slippage of adjacent nanocrystals caused by shear-induced glide. The P–h curves exhibit load build-up, pop-in, and subsequent load-drop phenomena that can be explained by the grain boundary sliding mechanism that occurs at the interfaces of the polycrystals. The data reveal that for a thin-film MOF coating prepared by dip coating/drop casting the sliding deformation or glide of the neighbouring nanocrystals could be triggered by a relatively small external load, on the order of just several micro-Newtons (µN).

The critical depth (h*) at when the first grain-boundary slippage occurs can be established from the experimental P–h curves plotted against an ideal function (assuming no interfacial sliding), as shown in Figure 1.28(b). Of note, it was found that an indentation depth of no greater than ∼1/3 of the size of the smallest individual nanocrystal (∼300 nm) is all that is required to trigger the first slippage for this thin-film sample. It is expected that this magnitude is dependent upon the fabrication routes employed as the interfacial response of grain boundaries should vary with the adhesion strength of the polycrystals, local packing pattern, and the type of the underlying substrate on which the crystals are deposited. Four different failure modes of the polycrystalline film, denoted as A to D are shown in Figure 1.28(c), from which the critical stresses corresponding to each mechanism can be estimated. Generally, mode A is attributed to grain boundary slippage, which occurs at a shallow deformation of ∼10 nm. Mode B is associated with polycrystalline fracture, where a failure strength of up to 1 GPa has been detected. Mode C is due to the accumulated compaction of the porous framework under compression, while mode D might be linked to the buckling of chemical bonds in the direction of loading.

Nanoscratch measurements under an instrumented nanoindenter have been applied by Buchan et al. on a polycrystalline HKUST-1 (Cu₃BTC₂) coating (Figure 1.29(a)), fabricated by electrochemical reaction on a pure copper substrate to study the film-to-substrate adhesion properties and failure mode.¹³⁷  Although increasing the reaction time of the electrochemical process increases the overall film thickness and coverage, it has the less desired effect of generating a higher surface roughness (on the order of a few µm) due to a larger average crystal size accompanied by secondary growth. The results obtained from nanoscratch tests are usually semi-quantitative as these measurements are affected by a combination of materials factors and test parameters chosen for a specific study. The ‘ramp-load’ test may be employed to identify the critical force needed to generate surface failure. An example is shown in Figure 1.29(b) for a film with a mean thickness of 4 µm, which resulted in an exposed substrate during a 100 µm scratch test (F_N(max) = 100 mN) using a Berkovich indenter tip. The first 20% of the scratch distance yielded a steeper surface penetration curve (green), but reduced in its gradient with distance. For the remaining 80% scratch distance, however, a linear displacement curve with a constant slope was obtained. After unloading, scanning of the film surface revealed the extent of elastic recovery and plastic deformation of the underlying copper substrate (orange curve). In this example, the film could survive only the initial 20% scratch distance corresponding to a critical normal load of 20 mN, as corroborated by the electron microscopy and optical profilometry images shown in Figure 1.29(c).

Apart from polycrystalline MOF films, the ramp-load nanoscratch methodology described above has also been implemented by Li et al.³⁹  to study the scratch resistance of a bulk MOF glass sample prepared using a melt-quenched ZIF material, termed as a_gZIF-62. The experiments used a conical diamond indenter with a spherical tip radius of ∼5 µm, traversing over a distance of 500 µm to produce a monotonically increasing normal load reaching a maximum value of F_N(max) = 50 mN. The onset of yielding was detected by comparing the surface profiles before and after scratch testing, see Figure 1.30(a). However, there was no sudden jump observed in the frictional coefficient (µ = F_T/F_N) curve of a_gZIF-62 as a function of scratch distance (Figure 1.30(b)), suggesting that there was no ductile fracture in the limit of the spherical scratch regime.¹³⁸  This is an interesting finding, nonetheless, the authors did not report images of the surface topography in the region of the scratch to corroborate the absence of any form of surface or subsurface fractures, thus presenting opportunities for future studies.

Another approach for nanoscratch testing is using a ‘pass-and-return’ wear test method, this is typically conducted at a lower load so that the film underneath the probe can survive a higher number of passes before delamination failure sets in. Figure 1.31 shows the results of such a cyclic wear test (constant F_N = 20 mN for 100 µm, 10 cycles) performed on a polycrystalline film of a Zr-based MOF called UiO-66 (for its crystal structure see Figure 1.1(g)), elucidating the differential surface damage experienced by the anodically- versus cathodically-deposited films grown on a zirconium substrate (foil) via an electrochemical method.¹³⁹  Stassen et al. demonstrated that while the cathodically-deposited film detaches completely from the substrate due to its poor cohesion strength, it can be seen that the anodically-deposited film has been compressed but still remains adhered to the substrate. The enhanced interfacial adhesion strength of the anodically-deposited film is attributed to the zirconium oxide film acting as a bridging layer between the UiO-66 nanocrystals and the metallic substrate. From a mechanical standpoint, this is an important finding as it holds the key to the fabrication of damage-tolerant MOF films required for practical applications.

Fracture toughness (K_c) is a measure of materials resistance to crack propagation starting from some pre-existing microscopic flaws, which act as stress raisers. This is a crucial materials property, especially for materials of limited ductility, as catastrophic failures occur at a stress level that is well below the yield strength of the material.

In the discipline of fracture mechanics, K_Ic denotes the mode-I fracture toughness value, determined under the ‘crack opening’ or tensile configuration in plane strain. Mode I is the most common fracture mode in practice. Modes-II and -III correspond to the sliding and shear loading configurations, respectively.¹⁴⁰  The SI unit for K_c is MPa m^1/2.

Only a few studies thus far have attempted to characterise either K_Ic or the associated fracture surface energy (γ_s) of MOFs and inorganic–organic framework materials, as can be seen in Table 1.3. The available K_Ic vs. E data are presented as an Ashby-style plot in Figure 1.32, showing the projected upper and lower bounds in relation to conventional engineering materials. The existing data suggest a relatively low fracture toughness value for MOFs and dense framework materials on the order of 0.1 MPa m^1/2, much below the magnitudes expected for brittle ceramics and glassy polymers. It is therefore imperative to study the toughness properties of MOF materials. On the one hand, it is critical to understand the mechanisms involved and on the other hand to be able to design framework materials with adequate robustness to withstand the rigour of practical applications.

For two polymorphic crystals of copper phosphonoacetate (CuPA) with a dense framework,³²  the Berkovich nanoindenter was employed to generate radial cracks under an applied load of 50–100 mN. The values of K_Ic were calculated in accordance with the Laugier’s expression given in eqn (1.18), which assumes a Palmqvist crack configuration:¹⁴¹ 

where P is the maximum indentation load, k is an empirical constant of the indenter shape (k = 0.016 for Berkovich;¹⁴² k = 0.057 for cube corner),¹⁴³ a is the half-diagonal length, l is the crack length and c = a + l (see Figure 1.33). From Figure 1.32, it can be seen that CuPA-1, which has an extended 3-D structure, exhibits a more pronounced toughness anisotropy compared with the layered architecture of CuPA-2. Moreover, it is striking to see that these two dense frameworks are located in the region near the lower limit for K_c, and both of which possess a toughness value (eqn. (1.19)) of G_c ≲0.001 kJ m⁻². Brittle solids such as glasses, silicon, oxides and carbides are also located close to this lower limit for K_Ic, but their toughness values are notably higher, with 0.002 ≲ G_c ≲ 0.2 kJ m⁻².

On the nanoporous ZIF monoliths studied by Tricarico et al.,³⁵  it was not possible to generate cracks using a Berkovich indenter due to the large included angle of ∼142° (face angle = 65.3°). Likewise, no cracks were observed for the HKUST-1 monolith tested using the Berkovich indenter,³⁴  as shown in Figure 1.13(a). For the ZIF-8 and ZIF-71 monoliths, the radial cracks required for the K_Ic analysis of eqn (1.18) can only be attained using a cube-corner indenter tip (face angle = 35.3°) to induce higher stresses for crack initiation/propagation. The different geometries of the indenter tips are compared in Figure 1.33(a). Between the two ZIF monoliths processed using a sol–gel method, Tricarico et al.³⁵  demonstrated that the ZIF-8 monolith is relatively easier to fracture. As shown in Figure 1.33(b), radial cracks can be seen propagating from all three corners of the cube-corner indent when subjected to a maximum indentation depth of 2 µm (P_max ∼ 5 mN). In contrast, under an identical test setup, a maximum depth of 5 µm was necessary to induce cracking on the ZIF-71 monolith. Notably, the fracture toughness value of the ZIF-71 monolith was determined to be K_Ic = 0.145 MPa m^1/2, which is almost twice as high as that of the ZIF-8 monolith (Table 1.3). Because the Young’s modulus of the ZIF-71 monolith (E = 1.67 GPa) is relatively lower than the monolith of ZIF-8 (3.18 GPa), the toughness of ZIF-71 (G_c ∼ 0.013 kJ m⁻²) is appreciably higher that of ZIF-8 (G_c ∼ 0.002 kJ m⁻²), see Figure 1.32. The fine-grained nanostructure of the ZIF-71 monolith capable of grain boundary sliding (Figure 1.25(e)) may contribute to improved toughness,³⁵  compared with a monolith of a melt-quenched a_gZIF-62 glass with a lower fracture toughness of 0.104 MPa m^1/2 and toughness of ∼0.001 kJ m⁻².¹⁴⁵  Noteworthy, the mode-I fracture toughness of glassy ZIF-62 was measured using the single-edge precracked beam (SEPB) method,¹⁴⁷  enabled by the availability of a millimetre-sized specimen fabricated via the melt-quenching and shaping of a ZIF-62 powder.

Figure 1.33(c) shows the varied fracture patterns evidenced in the ZIF-62 glass, where the sample was tested by Vickers microindentation employing a maximum load of 5 N. Stepniewska et al.¹⁴⁸  attributed the formation of the shear cracks to the weaker Zn–N coordinative bonding of the ZIF structure compared with other glass families containing stronger covalent, ionic, or metallic bonds. The crack deflection in the indent leading to the formation of large radial cracks was observed in the ZIF-62 glass. Intriguingly, in 1983 Lawn et al.¹⁴⁹  reported a similar shear-induced cracking pattern via the Vickers indentation of a soda-lime glass. In particular, the ‘shear faults’ (shear cracks) they reported are reminiscent of the patterns observed in Figure 1.33(c), emphasising that the narrow shear faults are not ‘slip bands’ characteristic of dislocation multiplications prevalent in metals. Of course, the latter mechanism is unlikely in ZIF-62 as it is amorphous. Instead, the origin of the shear cracks may be linked to the maximum shear trajectory surfaces that develop during the plastic indentation of brittle materials.¹⁵⁰  In depth research on the crack morphologies and failure mechanisms of MOF and framework materials is thus warranted to shed new light on this largely unexplored topic.

Reactive MD simulations by To et al.¹⁴⁵  revealed the breakage of the Zn–N bonds in ZIF-62 glass under uniaxial deformation, see Figure 1.34. The model introduced a precrack at the atomic scale (a flaw for stress concentration) prior to application of a tensile strain (ε) to propagate the crack. With increasing strain the crack propagates transversely to the direction of the axial loading, with the simulation terminated when all stretched bonds were ruptured across the crack to form two new macroscopic surfaces. The MD simulations yielded a fracture toughness of 0.097 MPa m^1/2, which is similar to that of the experimental value (∼0.1 MPa m^1/2). Subsequently, the surface energy for melt-quenched ZIF-62 glass was calculated from the Irwin formula in eqn (1.20), assuming that the material is in plane strain.

The theoretical value was found to be γ_s = 0.98 J m⁻², taking E = 4.1 GPa and ν = 0.395 derived from MD simulations. Using the experimental values (E = 5.2 GPa and ν = 0.343), the fracture surface energy was determined to be 0.82 J m⁻². However, the discrepancies between the experiments and simulations are considerable in terms of the magnitudes of strength (σ_max = 8 MPa (exp) vs. 703 MPa (MD)) and strain at maximum stress ( = 0.3% (exp) vs. 17.38% (MD)). Such discrepancies can be explained by surface flaws inherent to the physical sample, but are absent in the (idealised) theoretical sample that has only a prescribed flaw (Figure 1.34(b)); there is also a major difference in sample size between the experiments and simulations.¹⁴⁵ 

The foregoing discussion highlights the many opportunities as well as challenges faced in this emerging field. The research to date demonstrates that the fracture behaviour of glassy MOFs, bulk monoliths, single crystals, and composites of hybrid frameworks is very much an underexplored research territory. The basic mechanisms responsible for crack initiation, propagation, and catastrophic fracture thus must be systematically investigated to unravel structure–property relationships that will guide future work in improving the damage tolerance of framework materials. For example, Mahdi et al. systematically characterised the fracture energy of mixed-matrix membranes (MMMs) incorporating MOF nanoparticles as fillers, where the composite samples were tested under large-strain uniaxial tension until rupture. The results show that for the (glassy) Matrimid/ZIF-8 nanocomposite membrane, a nanoparticle loading of higher than ∼10 wt% significantly impaired the toughness properties of the membrane due to embrittlement effects.¹⁵¹  A similar trend was observed for a PVDF (polyvinyl difluoride)/ZIF-90 nanocomposite membrane.¹⁵²  When employing rubbery polyurethane (PU) as a matrix, it was found that the resultant PU/ZIF-8 nanocomposite also suffered from reduced ductility with increased concentration of MOF nanocrystals, but this effect set in only above 30 wt% of MOF filler when the hyperelastic matrix (initially rubbery) began to experience some embrittlement.¹⁵³  Such mechanical effects on a much wider family of polymer matrices,^154–156  hitherto, have not been systematically studied by means of experiments or theory.

Finally, we shall briefly touch upon the time-dependent mechanical properties of MOFs and hybrid framework materials, which are connected to phenomena such as viscoelasticity (at a small strain), stress relaxation (at a constant strain), creep (at a constant stress), and viscoplasticity (at a large strain). Research in this topic area is scarce in the context of MOFs, with only a few exemplars known to date. As they are constructed from organic and inorganic building blocks, it is expected that the mechanical response of hybrid framework structures and MOF-derived composites should exhibit some form of rate dependency. A better elucidation of the structure–property relationships underpinning rate-dependent MOF mechanics will be beneficial for real-world scenarios where engineering materials are commonly exposed to a combination of external stimuli in the thermo–mechanical domain (e.g., rate of deformation, temperature gradient, impact, friction, non-hydrostatic pressure). Crucially, combined thermomechanical loading can generate unexpected results, which are hard to predict by studying the effects of the individual components in isolation.

The time-dependent mechanical response and stress relaxation behaviour of MOF crystals¹⁰⁹  associated with the structural ‘flexibility’ and chemistry of porous framework are treated in greater detail in Section 5.4 of Chapter 5. Here, worth highlighting are the indentation strain–rate effects on the measured hardness of a family of melt–quenched ZIF glasses, reported by Li et al.³⁹  Using an indentation strain–rate jump method,¹⁵⁷  they determined the strain–rate sensitivity values (m, which reflects a material’s susceptibility to creep deformation under a constant stress) of glassy ZIFs to lie in the range of m = 0.0579–0.0757, akin to glassy polymers (m = 0.05–0.10).³⁹  This is an interesting development, as such a fine-scale indentation approach could readily be applied to probe the time-dependent deformation of a vast range of crystalline phases and non-amorphous MOF monoliths.

A few studies have characterised the dynamic modulus (E*) of mixed-matrix membranes incorporating MOF as a filler,^151–153  and of polycrystalline powders (e.g., MOF nanosheets)¹⁵⁸  subject to a cyclic loading under a small strain. This can be achieved using dynamic mechanical analysis (DMA) techniques, which are well established for measuring the viscoelasticity of polymers and soft matter across a range of temperatures and frequencies. The measured viscoelastic properties include the storage and loss moduli, E′ and E″, respectively, relaxation temperatures (T_α,β,γ), and the dissipation factor (loss tangent, tan δ), defined by:

where σ* and ε* are the time-varying oscillatory stress and strain (in sinusoidal waveform), respectively, while denotes the imaginary component of E*. E′ corresponds to the amount of elastic energy stored in the viscoelastic media that is recoverable, while E″ corresponds to energy dissipated by inelastic processes such as entropic motions and/or rotations of flexible structures. To illustrate the associated phenomena, let us consider the PU/ZIF-8 nanocomposite system¹⁵³  depicted in Figure 1.35. In a rubbery polymer matrix such as PU, above the glass transition temperature (T > T_g) a distinct scaling can be observed between the storage moduli and filler content rising systematically from 0 to 40 wt%. Conversely, at T_g the loss moduli systematically decline with increasing filler content, although the value of T_g itself is only marginally affected. The thermomechanical results reveal the intimate interactions between the MOF nanoparticles and their surrounding rubbery matrix; this pinning effect reduces chain mobility and impedes structural relaxation. The highly tuneable viscoelastic response demonstrated here is fascinating, and should be further explored to uncover how such a coupled thermomechanical response may be translated to other combinations of MOF–polymer nanocomposite systems.^26,159,160 

Chapter 1 introduced the readers to the core principles of ‘MOF mechanics’. Using representative examples, we discussed the basic ideas and elucidated the latest methodologies underpinning the study of the elasticity and anisotropy, as well as inelasticity and fracture, of MOFs, including the terahertz dynamics ubiquitous to framework structures. The most up-to-date mechanical properties charts are constructed and presented herein. These charts not only capture the latest findings that have been accomplished since the first review article on the mechanical properties of hybrid frameworks was published over a decade ago,²⁴  but also reveal directions for possible new discoveries in the field. The foundations gained from this opening chapter prepare the readers for the forthcoming topics, focussing on anomalous elasticity and framework flexibility (Chapter 2), the computational modelling of MOF mechanics (Chapter 3), high-pressure deformations (Chapter 4), rate effects and mechanical energy absorption (Chapter 5).

To conclude, what does the new science of MOF mechanics have to offer? In a nutshell, the research into the mechanical properties of MOFs is no longer just about determining the ‘mechanical stability’ of framework materials. Clearly, it has been proven to be a rich platform for conducting multi-faceted research that appeals to numerous types of scientists, engineers, and technologists. Chapter 1 represents only the tip of the iceberg – the aim is to inspire new research, with the quest for establishing how chemistry governs structure–mechanical property relationships, and to invite readers to explore uncharted territories via basic and applied research on MOF mechanics.

DUT

Dresden University of Technology

HKUST

Hong Kong University of Science and Technology

MIL

Materials Institute Lavoisier

MOF

Metal–Organic Framework

PCP

Porous Coordination Polymer

UiO

Universitetet i Oslo

ZIF

Zeolitic Imidazolate Framework