Discovering connections between terahertz vibrations and elasticity underpinning the collective dynamics of the HKUST-1 metal–organic framework

Abstract

We employed a combination of theoretical and experimental techniques to study the metal–organic framework (MOF)-mechanics central to the paddle-wheel Cu₃(BTC)₂ porous structure, commonly designated as HKUST-1. Lattice dynamics of the hybrid framework at below 18 THz were measured by means of Raman and synchrotron far-infrared spectroscopy, and systematically correlated to collective vibrational modes computed from ab initio density functional theory (DFT). We have identified a number of intriguing low-energy framework vibration mechanisms, reminiscent of the ‘trampoline-like’ deformations and new oscillatory motions associated with Cu paddle-wheel ‘molecular rotors’. The three independent single-crystal elastic constants of the HKUST-1 (i.e. C₁₁, C₁₂ and C₄₄) were calculated using the DFT approach, taking into account the effects of dispersion corrections. We established the full elasticity solutions accompanying detailed deformation mechanisms that control its anisotropic mechanical properties, ranging from the Young's and shear moduli to linear compressibility and Poisson's ratio. Our results support the notion that the co-existance of soft modes and intrinsic shear distortions connected to the THz lattice dynamics dictate a range of anomalous elastic phenomena, for example: negative Poisson's ratio (auxeticity), negative thermal expansion (NTE), and exceedingly low shear moduli properties.

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1. Introduction

Metal–organic frameworks (MOFs)¹ are contemporary nanoporous solids with vast structural versatility,² giving rise to highly tunable physical and chemical functionalities.³ The current trend is rapidly shifting from purely chemical structure discovery to development of potential applications to afford commercial use in industry, engineering, and value-added products.⁴ In fact, research into potential applications of MOFs is no longer limited to gas separations, carbon capture and adsorption,⁵ but expanding into many innovative domains encompassing optoelectronics⁶ and lighting,⁷ biomedicine,⁸ heterogeneous catalysis⁹ and chemical sensing.¹⁰

The basic mechanical properties^11–13 and material toughness,^14,15 in conjunction with long-term structural, moisture and chemical stability of MOFs are central for manufacturing,¹⁶ shaping¹⁷ and scaling up, all of which underpinning successful deployment of aforementioned applications in many diverse technological settings. The emerging field of “MOF-mechanics”,^13,18,19 particularly concerning the effects of elevated pressure (i.e. hydrostatic compression) on the structural transformation and chemical response of MOFs is an area of intense study.^20,21 Tunability of MOF mechanical properties made possible through isoreticular design and construction,²² host–guest interactions,²³ and chemical modification²⁴ is another topic of current interest. Given the flexible nature of open frameworks²⁵ attributed to compliant metal coordination environments bridged by organic linkages, they offer exciting new opportunities to yield unusual elastic response (i.e. mechanical metamaterials)²⁶ which are rare in conventional materials, for example: negative linear compressibility (NLC), negative Poisson's ratio (NPR or auxeticity), and negative thermal expansion (NTE).^18,19,21 Large elastic anisotropy of MOFs are notable,^18,27 and this property could have important consequences on both mechanical and thermal stability determining the robustness of porous frameworks against structural destabilization,²⁸ irreversible plastic deformation^11,29 and amorphization.³⁰

Low-frequency lattice modes—called “terahertz (THz) vibrations”³¹ have recently been demonstrated to be prevalent in the family of zeolitic MOFs.³² These collective modes detected at THz frequencies have been identified to have a major influence for triggering a range of intriguing lattice dynamics, particularly gate opening and breathing phenomena dictating influx and release of absorbents, soft modes and shear-induced deformations that might be the source of framework destabilization and collapse. While the structural dynamics of certain MOFs have been studied before,³³ there is no detailed work connecting the behaviour of THz dynamics and elastic anisotropy to facilitate understanding of basic core mechanisms of MOFs. In this paper, we described the use of Raman and synchrotron far-infrared spectroscopy techniques in combination with ab initio density functional theory (DFT), to pinpoint and elucidate low-energy vibrational modes and associated deformation mechanisms detected in the paddle-wheel HKUST-1 [Cu₃(BTC)₂; BTC = benzene-1,3,5-tricarboxylic acid]³⁴ porous framework structure (Fig. 1).

Fig. 1 Crystal structure of HKUST-1 looking down the (a) <100>, (b) <110> and (c) <111> directions and (d) <100> direction displaying the Solvent Accessible Volume (SAV yellow surfaces). Colour code used: copper, orange; carbon grey; oxygen red; hydrogen white.

2. Experimental & theoretical methods

2.1 Experimental methodology

Infrared (IR) absorption spectra of HKUST-1 were recorded at the Multimode InfraRed Imaging and Microspectroscopy (MIRIAM) Beamline B22 at Diamond Light Source. Diamond synchrotron was operating in standard user mode, hence 900 bunches filling pattern for a total circulating current of 300 mA with top-up every 10 min (with electron beam lifetime of 18 hours). The spectrum was obtained under vacuum using a Bruker Vertex 80 V Fourier Transform Infrared (FTIR) interferometer equipped with a DLaTGS detector, operating at a resolution of 1 cm⁻¹ and utilising a 6 μm Mylar broad-band multilayer beamsplitter. The specific beamsplitter used allowed for excellent spectra to be obtained in the region <600 cm⁻¹ (∼20 THz). The IR spectra were acquired in ca. 2 minutes for best signal-to-noise ratio. In addition, the data set was recorded at room temperature via single-bounce Attenuated Total Reflection (ATR). Specifically, HKUST-1 powder was placed on top of a diamond ATR crystal and held in position by pressure applied on a high-density polyethylene (HDPE) disk. The Synchrotron IR beam therefore enters the ATR crystal from the bottom at an angle of 45° and is totally reflected at the interface of the crystal and the sample, then redirected by a 45° mirror to a detector. The absorption spectrum is thus measured as a modulation of this reflected light due to the evanescent wave penetrating the specimen at the diamond surface.

Raman spectra were recorded using a Bruker Senterra confocal microscope with a 50× objective with high numerical aperture 0.8. Operating with a 532 nm excitation laser at 50 mW and via a 50 μm aperture setting, a resolution 6–9 cm⁻¹ was dictated by the 1200 lines per mm grating and the spectral range of ∼4000 cm⁻¹ to 40 cm⁻¹ by the edge filter. Spectra were detected by an Andor deed depleted Si CCD Peltier cooled at −80 °C.

2.2 Computational methodology

We performed ab initio density functional theory (DFT) calculations using the periodic code CRYSTAL14.³⁵ Gaussian-type atom-centred all-electron basis sets were used consisting of 2880 basis functions for all of the HKUST-1 calculations (see ESI†). The ferromagnetic cubic geometry of HKUST-1 was optimised at the B3LYP and B3LYP-D levels of theory by relaxation of both the lattice parameters and atomic coordinates. We also calculated the elastic properties of the antiferromagnetic phase³⁶ for completeness, and the values differed by less than 1% on average with the greatest difference being 2%. To ensure the optimised structures were accurate the convergence criteria of the route mean square (RMS) of the gradient and the displacement were increased, from the default values of 0.0003 a.u. and 0.0012 a.u. to 0.0002 a.u. and 0.0004 a.u., respectively. In addition, the self-consistent field (SCF) convergence threshold was also improved from 10^–7 a.u. to 10^–10 a.u on the total energy and tighter tolerances for the evaluation of two-electron integrals were used.

The single-crystal elastic constants of the elasticity matrix (tensor) were calculated using the numerical first derivative of the analytic cell gradients.³⁷ These values correspond to the independent elastic stiffness coefficients, C_ij.³⁸ The unique coefficients were obtained via the method of deforming the optimised structure, using a three-point formula, in the symmetrically required directions of both positive and negative amplitudes. These particular deformations correspond to tensile and compressive strains required to obtain the elastic response. The magnitude of each individual strain deformation is defined as 1%, ensuring the response is in the purely elastic region. The mechanical properties were calculated via tensorial analysis implemented in Mathematica³⁹ and the ElAM code.⁴⁰

The vibrational frequencies at the gamma point were obtained by slightly displacing the optimised structure, with respect to the 3 N atomic coordinates, to allow for numerical differentiation of the analytical first derivatives, hence producing the mass-weighted Hessian matrix.⁴¹ The intensities of the IR-active modes were calculated via numerical differentiation, using the Berry phase approach.⁴² More specifically, the atomic Born tensors were evaluated via the derivation of the dipole moment with respect to the atomic displacements. This equates to the polarization differences between the optimised and the displaced structures and is equivalent to the time-integrated transient current that flows through the material during the vibrations. The intensities of the Raman-active modes were calculated analytically via Coupled Perturbed Hartree–Fock (CPHF) theory⁴³ and corrected for the experimental parameters of a laser frequency of 532 nm and a temperature of 293 K.

3. Results and discussion

3.1 Vibrational motions present in HKUST-1

The vibrational modes located in the mid-infrared (MIR) region are simply due to the common chemical vibrational motions of any system containing organic moieties. These motions are well characterised and understood and are the result of bond stretching and bending. Even in hybrid materials, such as MOFs, the variations from the isolated organic linkers are minor. However, the low energy vibrations located in the far-infrared (FIR) region are not as trivial to explain, and this is due to them no longer originating from isolated molecular motions but instead involve the collective motion of the framework (lattice vibrations).³² In HKUST-1, particularly interesting lattice vibrations were detected to lie in the region of under 600 cm⁻¹ (<18 THz) and involve contributions varying from Cu-paddle-wheel deformations, indicative of possible molecular rotor potentials,⁴⁴ to organic linker movements which could explain the phenomenon of negative thermal expansion (NTE) experimentally confirmed in HKUST-1.⁴⁵

Herein we have compared the experimental and theoretical IR-active and Raman-active vibrational motions identified in the region less than 18 THz. As can be seen from Fig. 2, the agreement is very good for both IR and Raman spectra. To gain a full physical understanding of the lattice dynamics of HKUST-1, we calculated the theoretical vibrational spectra using DFT, at the B3LYP level of theory with and without corrections for dispersion. B3LYP has been augmented with the Grimme DFT-D2⁴⁶ empirical atom–atom pairwise -C₆^ij/R_ij⁶ dispersion correction as modified for solids.⁴⁷ Due to the high cubic symmetry present in HKUST-1 there are not that many unique vibrational modes in this important region, specifically we determined respectively 12 and 25 unique IR and Raman active motions located under 600 cm⁻¹. It is worth mentioning, however, that many of the modes are triply degenerate, due to the symmetry of the framework and in addition, there are also 44 unique non-optically active vibrational motions in the same spectral region. In the next section, we will summarise all of the IR and Raman active modes and highlight some of the non-optically active modes that are of significant interest to the framework elastic properties and its anisotropy discussed after.

Fig. 2 Comparison of experimental and theoretical far-infrared (FIR) and Raman spectra in the 0–600 cm⁻¹ (0–18 THz) region. An empirical 10 cm⁻¹ FWHM Lorentzian line shape was applied to the DFT spectra to aid in comparison with the experimental data and a bar representation is present for the low intensity IR region below 6 THz.

3.2 Distinct regions of THz vibrational modes of HKUST-1

Altogether HKUST-1 has 81 unique vibrational motions located under 600 cm⁻¹, all of which are summarised in the ESI. The majority are due to the deformational movements of either the organic linker moieties or the inorganic Cu-centric paddle wheel units but can be grouped into distinctive spectral regions, and there are many interesting motions located in the low-energy region under 100 cm⁻¹. Specifically, the vibrational modes can be characterised into the following regions, working from the higher energy modes downwards:

• above 530 cm⁻¹ (>16 THz) are Cu–O stretching motions;

• modes lying within 400–530 cm⁻¹ (ca. 12–16 THz) are in-plane and out-of-plane aromatic ring deformations;

• 100–400 cm⁻¹ modes (ca. 3–12 THz) are various combinations of ‘paddle-wheel’ deformations and linker rotational motions. These oscillatory deformations mainly consist of Cu–O stretching, O–Cu–O bending, and an interesting Cu–Cu buckling motion similar to vibrational buckling motions in small organic molecules.⁴⁸ Another interesting observation in this region, are modes where the symmetry of the displacements result in a swelling or ‘breathing’ appearance of the Cu-based paddle-wheel unit. These particular motions are all Raman-active and located around 3.4 and 6.4 THz, (112, 214 and 215 cm⁻¹), with the lowest energy example exhibiting the strongest structural deformation effect;

• modes under 100 cm⁻¹ (<3 THz) are associated with collective framework vibrations, including ‘trampoline-like’ motions, paddle-wheel rotors, and symmetric collective motions that could explain the source of anomalous mechanical behaviour.

3.3 Noteworthy collective vibrations

We have discovered 8 unique collective vibrations of significant interest each originating from three specific classes of motion. The first interesting motion can be described as ‘trampoline-like’ deformation and is depicted in Fig. 3. There are four vibrational modes demonstrating this particular motion, located at 58, 81, 94 and 98 cm⁻¹ (from 1.7 to 3 THz) corresponding to the Raman, IR, non-optically active, and Raman-active modes, respectively. The IR and non-active motions are primarily just the movement of the organic linker in a trampoline-like fashion but the Raman-active ones are more distinct. This is due to the symmetry of the mode located at 98 cm⁻¹, resulting in a breathing effect of the organic–inorganic cluster. In addition, the lower-energy Raman active mode at 58 cm⁻¹ also shows the second class of motion, involving a rotation of the paddle-wheel unit simultaneously (Fig. 4). This particular combination of motions are analogous to modes that have been suggested experimentally to drive negative thermal expansion (NTE).⁴⁵ The previous work⁴⁵ showed thermal ellipsoids suggesting the ‘trampoline-like’ motion at a temperature range of 100–404 K and also proposed that the liberation of the aromatic rings and twisting of the ‘paddle-wheel’ unit also contribute to the expansion upon thermal stimulus.⁴⁵ All of these low-energy deformations have been demonstrated in full in our work, with the indication that NTE can be driven by a combination of THz vibrational motions. Such NTE phenomenon is possibly also linked to the other non-active modes (located at 16, 20, 63 and 78 cm⁻¹), showing translational and rotating motions of the organic linker and Cu-based paddle-wheel which are all located in the important low-energy (THz) region, see summary in Table 1. It is also worth noting that similar motions have been suggested to be the source of NTE in other less complex MOF materials, such as the cubic MOF-5.⁴⁹

Fig. 3 Low-energy collective vibration, located at 81 cm⁻¹ (2.4 THz), viewed down the (a) <111> and (b) <110> directions, showing a ‘trampoline-like’ motion linked to negative thermal expansion (NTE). Videos of the motions are available in the ESI.†

Fig. 4 Low-energy collective vibration, located at 58 cm⁻¹ (1.7 THz), viewed down the (a) <110> and (b) <011> directions, showing a rotor motion of the Cu-based paddle-wheel moiety. Videos of the motions are available in the ESI.†

Table 1 Notable collective vibrational modes of HKUST-1

Mode		Activity	IRREP^a	Description
(cm^−1)	(THz)
a IRREP = Irreducible representation; Non = non-optically active mode.
16.32	0.49	Non	F2u	Strong paddle wheel deformation and translational motion (symmetric cluster rotation)
20.43	0.61	Non	Eu	Strong paddle wheel deformation and translational motion
58.07	1.74	Raman	F2g	Strong paddle wheel rotation with organic linker trampoline-like motion
63.17	1.89	Non	F2u	Paddle wheel deformation and translational motion with organic linker rotating
78.28	2.35	Non	Bg	Paddle wheel deformation with organic linker rotating (symmetric cluster rocking)
81.47	2.44	IR	F1u	Organic linker trampoline-like motion
94.15	2.82	Non	Bu	Organic linker trampoline-like motion
98.41	2.95	Raman	Ag	Organic linker trampoline-like motion (symmetric cluster breathing)

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The second intriguing class of collective dynamics already mentioned above is the strong paddle-wheel rotor shown to be linked to the Raman-active mode at 58 cm⁻¹ (1.7 THz). This particular motion is of high interest as it could suggest the possibility of utilising the mechanism for potential ‘molecular’ rotor applications.⁵⁰ This is a topical area of research with some examples in MOFs starting to appear.^44,51 However, most work done on MOFs has concentrated on the use of the organic linkers which is dissimilar from the rotation attributed to the metal–containing units, such as the rotor dynamics identified here regarding the paddle-wheel of HKUST-1.

The final exciting vibrational motion revealed from our work is less trivial to describe in words, but videos of its structural dynamics can be seen in the ESI† to aid in the conceptualisation. In principle, the motions fully involve the collective framework and the organic–inorganic cluster, briefly mentioned when discussing the Raman-active mode at 98 cm⁻¹ (∼3 THz). There are other modes in HKUST-1 whose dynamics are reminiscent of this motion, although they are more due to the symmetry of the mode resulting in the effect of synchronous cluster dynamics. Examples other than the Raman-active mode discussed, are a non-active cluster rocking motion located 78 cm⁻¹ (2.3 THz) and this is accompanied by another non-active higher energy cluster rocking motion at 138 cm⁻¹ (∼4 THz). However, the most exciting cluster-linked motion is the non-active mode at 16 cm⁻¹, which is the lowest energy vibrational mode of HKUST-1 found at ca. 0.5 THz. This particular motion involves a combination of rocking and translational dynamics resulting in the rotation of the linker–paddle-wheel 4-noded cluster. The rotational movement of the cluster could reveal the nature of the mechanism behind anomalous elastic behaviour present in the material, specifically the directional auxetic response (negative Poisson's ratio), discussed in more detail later in this work.

3.4 Single-crystal elastic constants of HKUST-1

The vibrational motions discussed so far can be intrinsically linked to the mechanical behaviour of HKUST-1, especially its elastic response. We therefore, have calculated the single-crystal elastic constants using DFT and via tensorial analysis can obtain the related mechanical properties, i.e. Young's modulus, shear modulus, linear compressibility and Poisson's ratio. The link between the elastic constants and the mechanical response is from Hooke's law being related to stress σ and strain ε (both second-rank tensors) in the following equation:³⁸
where C_ijkl represents the symmetric fourth-rank elasticity tensor connecting stress to strain. HKUST-1 only has three unique elastic constants (see Table 2), due to having a cubic symmetry and the symmetric elasticity matrix (stiffness tensor) therefore is:
The diagonal coefficient C₁₁ represents the framework stiffness along each of the three principal crystal axes, i.e. a, b, and c (due to them being equal) under uniaxial strain. The other diagonal coefficient C₄₄ is the shear stiffness measuring angular distortion against a shear deformation. The off-diagonal coefficient C₁₂ signifies the tensile–tensile coupling between any two orthonormal axes. Importantly, the mechanical properties discussed in the next section are calculated from the inversion of the elasticity matrix, known as the compliance matrix (S) in accordance to:

S_ijkl = C_ijkl⁻¹

Table 2 Single-crystal elastic stiffness coefficients (C_ij) of HKUST-1 computed from the DFT

Functional	C 11	C 12	C 44
B3LYP	27.719	25.722	5.399
B3LYP-D	27.539	25.476	5.273

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3.5 Framework elastic anisotropy under uniaxial and angular deformations

The elastic anisotropy of HKUST-1 can be witnessed when examining the 3-D directional Young's and shear moduli (see Fig. 6 and Table 3). Firstly the Young's modulus is simply the ratio of uniaxial stress over strain and can be obtained via tensorial analysis of the compliance matrix through the following equation, depending on a unit vector u:

Fig. 5 Low-energy collective vibration, located at 16 cm⁻¹ (0.5 THz), viewed down the (a) <100> and (b) <110> directions, showing a cluster rotation mechanism, linked to auxeticity. Videos of the motions are available in the ESI.†

Fig. 6 (a) Young's modulus representation surface E(ϑ, φ) in 3-D spherical coordinates, along with (b) shear modulus representation surface G(ϑ, φ, χ). The accompanying structure–property relationships can be seen for (c) E_min is along the <100> axes and E_max is along the <111> axes. (d) G_min is along the <110> axes and G_max is along the <100> axis.

Table 3 Mechanical properties of HKUST-1

Elastic property		B3LYP	B3LYP-D
Young's modulus, E (GPa)	E max	15.15	14.81
	E min	2.96	3.05
	A E = Emax/Emin	5.12	4.86
Shear modulus, G (GPa)	G max	5.40	5.27
	G min	1.00	1.03
	A G = Gmax/Gmin	5.40	5.12
Linear compressibility, β (TPa^−1)	β	12.63	12.74
Poisson's ratio, ν	ν max	1.21	1.19
	ν min	−0.31	−0.28
Zener anisotropy, A		5.41	5.11
Acoustic velocities(km s^−1)	max	6.18	6.14
	min	5.62	5.59

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Both the minimum and maximum Young's moduli of HKUST-1 can be easily explained from a structure–property relationship. The origin of E_max (15 GPa) is related to compression in the direction of the planarity of the aromatic rings. The link between deforming aromatic rings of the organic linker and the direction of E_max has also been detected for ZIF and MIL framework structures.^18,52 In the case of HKUST-1 these maxima are located along the <111> axes. Another intriguing uniaxial property is the minimum Young's moduli, E_min (see Fig. 6). The minimum is located along the <001> axes and can be related to only partial reinforcement of the linker–paddle-wheel–linker (L–PW–L) moiety oriented along each axes. The L-PW-L unit does not provide as much mechanical rigidity as the single-linker reinforcement that has been recently found in our work on the Zr-based MIL-140 series and hence we see an E_min of just 3 GPa.⁵²

Moving on to the angular rigidity of HKUST-1, we evaluate the shear modulus which can similarly be straightforwardly explained via structure–property relationships. Unlike the uniaxial directions of the 3-D Young's modulus surface, the shear modulus depends on an additional unit vector v and hence the representation shown in Fig. 6, depicts the maximum (blue) and minimum response (green) resulting from angular deformations in the respective directions. As per the single unit vector equation shown earlier, the shear modulus is also derived from the compliance matrix but via the following two unit vector equation:

Due to the cubic symmetry the maximum and minimum values (G_max and G_min) are located in multiple directions, with G_max lying along each principal crystallographic axes <100> whereas G_min is present in the <110> direction. This logically is due to the nodes inclusive of flexible organic components allowing for deformation upon shear strain more easily than the inorganic-based paddle-wheel on its own. This effect is illustrated very clearly in Fig. 6, where the different nodes are highlighted.

Before moving on to more complex elastic behaviour, it is worth noting that due to the cubic space group of HKUST-1, the linear compressibility (β) response of the framework upon hydrostatic compression is isotropic, unlike all of the other mechanical properties discussed in this work (Table 3). The magnitude of β is calculated to be 12.74 TPa⁻¹ at the B3LYP-D level of theory using the following equation:

β(u) = u_iu_jS_ijkl

3.6 Auxetic behaviour and Poisson's ratio (ν)

An interesting elastic property of HKUST-1 is the Poisson's ratio, where in certain crystalline directions a negative value has been predicted, indicating an auxetic response. As per the shear modulus discussed before, the Poisson's ratio depends on two unit vectors (u, v) and hence the 3-D representation (Fig. 7) is again limited to the maxima and minima in each direction. This instance however, due to the auxetic directions, the maximum is depicted as blue and minimum as green and red for positive and negative values, respectively. The value of the Poisson's ratio represents the ratio of the transverse strain to the axial strain under uniaxial deformation and is obtained from the following equation:

Fig. 7 (a) Poisson's ratio representation surface ν(ϑ, φ, χ), where blue signifies the maximum Poisson's ratio and green and red denote the positive and negative minimum Poisson's ratios respectively. (b) Demonstration of the plausible mechanism associated with cluster rotational dynamics responsible for the auxetic response.

The minima values, v_min and the auxetic response of −0.3, are located in the <110> direction, with the corresponding transverse in the <11̄0> direction. The mechanism behind this particular response is non-trivial to explain but could be linked to the dynamics of the inorganic–organic clusters highlighted in Fig. 5, rotating upon being stretched to allow for the elongation in the transverse direction instead of the intuitive and more common response of contraction. The suggested mechanism shows the complex nature of the origins for anomalous elastic behaviour in MOFs, as new candidates for mechanical metamaterials.⁵³ This is also a good example of why these particular calculations and studies are vital to obtain an accurate structure–property relationship at the atomic level between the framework in question and highlighting the complex 3-D directionally dependent elastic properties.

3.7 Bulk modulus and polycrystalline mechanical properties

The stability under uniform hydrostatic compression of a given material can be quantified by the bulk modulus (K). In this work, we used the Voigt–Reuss–Hill (VRH) average as the theoretical value of the bulk modulus, which equates to the numerical average of the Voigt and Reuss methods that assume a uniform strain, and uniform stress respectively. The value obtained at the B3LYP level of theory was 26 GPa, which is in reasonable agreement with the literature.⁵⁴ However, as the calculated values are based on ideal polycrystalline materials assuming no residual solvent or defects, it is logical to expect experimental values would be relatively higher.

The other VRH averaged polycrystalline properties (Table 4) are the isotropic analogues of the mechanical response discussed above but with the added benefit of replicating more realistically the conditions of bulk materials used for many potential applications. For example, the values can be helpful to approximate the elastic properties of randomly oriented polycrystalline MOF coatings.^15,55 We established that the polycrystalline average Poisson's ratio of HKUST-1 is positive (ν_VRH = 0.45) despite exhibiting a significant degree of auxeticity (ν_min = −0.28) in single crystal form.

Table 4 Isotropic Voigt–Reuss–Hill (VRH) averaged elastic properties, corresponding to the polycrystalline state. The bulk (K), Young's (E) and shear (G) moduli are in GPa and the Poisson's ratio (ν) is dimensionless

DFT functional	K VRH	E VRH	G VRH	νVRH
B3LYP	26.39	8.10	2.80	0.45
B3LYP-D	26.16	8.07	2.78	0.45

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4. Conclusions

We demonstrated the connections that can be made between vibrational modes of a framework material and its mechanical properties. Ab initio density functional theory (DFT) was employed to calculate and explain the important vibrational motions, structural dynamics and elastic properties of HKUST-1. Importantly, the vibrational calculations provide insights into significant vibrational modes including ‘trampoline-like’ motions, paddle-wheel rotors, and soft modes^32,56 linked to symmetric collective mechanisms, which could explain the origin of auxeticity and negative thermal expansion (NTE) of HKUST-1 evidenced in certain crystallographic directions. In addition to the negative Poisson's ratio, a full analysis of the elastic properties was demonstrated explaining the structure–property relationships underlying the mechanics of the Cu-based MOF. Our theoretical approach coupled with experimental spectroscopy provide new understandings of the basic molecular mechanisms underlying lattice dynamics surrounding the elastic (in)stability of the paddle-wheel based framework material. It is also important to note that, the elucidated THz vibrational methodologies could be used to gain a deeper understanding of the fundamental mechanisms applicable to entirely new applications, for example: MOF host–guest interactions underpinning uptake-release sensing, molecular-scale ‘structural health’ integrity monitoring and framework diagnostics, and rotor dynamics for energy transduction. To summarise, our findings highlight that many of the mechanical mechanisms behind the anomalous elasticity of MOF materials are yet to be fully understood. In particular, we hope that further experimental studies will be performed on HKUST-1 to confirm the level of anisotropy, with regards to the DFT predicted directional dependant Young's modulus, shear modulus and Poisson's ratio.

Acknowledgements

M. R. R. would like to thank the UK Engineering and Physical Sciences Research Council (EPSRC) for a DTA postgraduate scholarship and also an additional scholarship from the Science and Technology Facilities Council (STFC) CMSD Award 13-05. We thank Dr. B. Van de Voorde, KU Leuven for provision of sample. We thank Diamond Light Source for access to the MIRIAM beamline B22 (Proposal SM9388). The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work (http://dx.doi.org/10.5281/zenodo.22558). We are also grateful to the SCARF cluster at the Rutherford Appleton Laboratory for provision of additional computing resources.

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Footnote

† Electronic supplementary information (ESI) available: Lattice parameters, basis set information, descriptions of all THz vibrational modes, and videos of selected collective vibrations. See DOI: 10.1039/c5ce02347e

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