Acoustic Properties of Metal-Organic Frameworks

Metal-organic frameworks (MOFs) have attracted significant attention in the past two decades due to their diverse physical properties and associated functionalities. Although numerous advances have been made, the acoustic properties of MOFs have attracted very little attention. Here, we systematically investigate the acoustic velocities and impedances of 19 prototypical MOFs via first-principle calculations. Our results demonstrate that these MOFs exhibit a wider range of acoustic velocities, higher anisotropy, and lower acoustic impedances than their inorganic counterparts, which are ascribed to their structural diversity and anisotropy, as well as low densities. In addition, the piezoelectric properties, which are intimately related to the acoustic properties, were calculated for 3 MOFs via density functional perturbation theory, which reveals that MOFs can exhibit significant piezoelectricity due to the ionic contribution. Our work provides a comprehensive study of the fundamental acoustic properties of MOFs, which could stimulate further interest in this new exciting field.


Introduction
Metal-organic frameworks (MOFs), constructed by metal ions/clusters and organic ligands via directional coordination bonds, have attracted wide attention during the past two decades since the homogeneous hybrid nature gives rise to the integration of properties from both the inorganic and organic components [1]. On the one hand, porous MOFs have shown promising application potential in gas storage and separation, catalysis, and drug delivery [2]; on the other hand, this emergent class of materials can exhibit diverse physical properties, including fluorescence, ferromagnetism, ferroelectricity, and multiferroicity [3][4][5][6][7]. Although there are numerous studies on the diverse properties and functionalities of MOFs [8], little attention has been paid to exploring an important area, namely, the acoustic properties [9]. Very recently, the application of acoustics and energy-transfer mechanisms in MOFs has been experimentally validated [10,11], which prompts the necessity to systematically study the acoustic properties of MOFs. The most fundamental aspect of acoustics is to study the propagation of sound in materials, which includes the understanding of the acoustic velocity and acoustic impedance. Both of these parameters are closely related to the elastic properties of a given material and are determined by the stiffness constant and density [12]. Compared with purely inorganic (i.e., oxide) and organic (i.e., polymer) materials, MOFs have intermediate stiffness and density, which could enable them to show very different acoustic properties [13][14][15]. In this context, it is highly interesting to explore the fundamental acoustic velocity and acoustic impedance of MOFs and compare them with those of their purely inorganic and organic counterparts. In addition, it is well known that MOFs exhibit significant structural diversity and chemical variability; therefore, they are expected to show diverse acoustic properties, similar to their other properties [16]. Furthermore, the comprehensive study of the basic acoustic properties of MOFs is also of great importance for exploring their application potential in acoustics, such as acoustic switching devices and sensors.
In this work, we present a systematic study of the fundamental acoustic properties of 19 prototypical MOFs via firstprinciple calculations and compare them with those of traditional inorganic materials having similar structures. At the same time, potential acoustic applications of MOFs are proposed based on these properties. Moreover, the piezoelectric properties of several MOFs are calculated using density functional perturbation theory, and the advantageous properties of these MOFs over their inorganic counterparts are comprehensively discussed from the viewpoint of potential applications.

Acoustic
Velocity. Generally, there are two types of acoustic waves: pressure waves (p-waves, longitudinal waves) and shear waves (s-waves, transverse waves). Here, we divide the velocity of shear waves into the velocities of pure-s-waves (v 1 ) and quasi-s-waves (v 2 ). The quasi-swaves have a small longitudinal wave component, which means that they are not purely transverse waves. The velocities of p-waves (v 3 ) and s-waves are determined by the Young's moduli and shear moduli of the materials, respectively. As the Young's moduli of most materials are greater than their shear moduli, the magnitudes of these wave velocities have the following trend: v 1 The calculated independent elastic constants of 15 materials by us and 7 materials by others all conform to the elastic stability criterion and are tabulated in Table S1 along with some experimentally measured densities [32][33][34]. Meanwhile, the acoustic velocities were obtained by solving the Christoffel equations based on the elastic constants and densities. The 3D surfaces of the acoustic velocities for these 22 materials are presented in Figs. S1-S21. Figure 2 shows the maximum and minimum values of v 1 , v 2 , and v 3 in these 22 materials, which demonstrate that acoustic waves propagate in these MOF crystals with a wide range of velocities. For the maximum values of all three velocities, there are no obvious boundaries between the 19 MOFs and 3 inorganic materials. These results are reasonable considering the medium maximum moduli and low densities of MOFs. A similar trend is also found between the porous and dense MOFs. Notably, the maximum velocity values of some MOFs are even higher than those of quartz, BaTiO 3 , , and sodalite. For example, the v 3 value of MIL-47 is 9.94 km/s, which is 138% larger than that of BaTiO 3 . In terms of the minimum values of the three velocities, v 3min and v 2min are comparable with those of inorganic materials. However, the v 1min values of dense MOFs are comparable with those of inorganic materials and generally higher than those of porous MOFs. This is because the minimum shear moduli of some porous MOFs studied here are significantly lower than those of the dense systems and inorganic materials. For instance, MIL-53(Ga)lp has the lowest v 1min of 0.219 km/s among all 19 MOFs.
The ratios of the maximum and minimum velocities of each type of wave for these 22 materials were calculated as indicators to evaluate their anisotropy (A v = v max /v min ), and the results are shown in Figure 3. Notably, the A v values of DMOF-1loz, DMOF-1sq, MIL-47, MIL-53(Al)lp, and MIL-53(Ga)lp are up to an order of magnitude higher than those of the 3 inorganic materials, which could give rise to some potential applications unavailable in conventional materials, as discussed below. In particular, MIL-53(Ga)lp has the highest acoustic velocity anisotropy, with anisotropy indices of the longitudinal wave velocity (A v3 ) and pure shear wave velocity (A v1 ) of 2.35 and 15.17, respectively.
The 3D surface contours and projected 2D plots of the acoustic velocities for MOF-5 are displayed in Fig. S1, which demonstrates that its v 1 , v 2 , and v 3 are in the ranges of 1.67-3.13, 1.67-3.67, and 5.50-6.83 km/s, respectively. The v 1min and v 2min of MOF-5 are approximately half of those of BaTiO 3 , which also has a ReO 3 -related structure, while the v 1max and v 2max are close to those of BaTiO 3 . In addition, both the v 3max and v 3min values of MOF-5 approximate those of BaTiO 3 . These findings disclose that acoustic waves can propagate similarly in porous MOFs and dense oxides. Furthermore, the anisotropy of the 3 types of wave velocities in MOF-5 is also distinct, and A v3 (1.19) is significantly smaller than A v1 (1.88) and A v2 (2.19). As the Young's modulus of MOF-5 is determined by the connectivity of organic ligands, the isotropic periodic extension of the organic linkers in the structure leads to an isotropic Young's modulus corresponding to a small A v3 . However, unlike the p-wave velocity, the swave velocity in MOF-5 is affected by the shear modulus. For the square-shaped building block in the structure, it is much more easily sheared along the organic linker direction (edge direction) than along the diagonal direction, hence resulting in dramatic anisotropy in v 1 and v 2 [32].
The 3D surface contours and 2D plots of the acoustic velocities for ZIF-8 and MAF-7 are displayed in Figs. S2 and S3. Although they are isomorphic with the sodalite structure, the v 1 , v 2 , and v 3 of ZIF-8 are all slightly smaller than those of MAF-7. According to formulas (1)-(3), the difference is attributed to the enhanced Young's modulus of MAF-7 due to the electron-donating effect and its lower density (ZIF-8, 1.14 g/cm 3 ; MAF-7, 1.08 g/cm 3 ). In addition, the v 1 and v 2 values of ZIF-8 and MAF-7 are only approximately 26-31% of those of sodalite, while the v 3 values are approximately 53-62% of the value of sodalite. These phenomena arise from the more obvious difference in the Young's moduli between ZIF-8/MAF-7 and sodalite than that in the shear moduli. Moreover, both ZIF-8 and MAF-7 exhibit very small anisotropy in sound velocities, which is reminiscent of that of sodalite.
DMOFs have unusual guest-dependent dynamic behaviour: the framework shrinks to a lozenge (DMOF-1loz) when the guest is present and expands to a square (DMOF-1sq) when the guest is released. Along with the change in unit cell volume from 1147 (DMOF-1sq) to 1114 Å 3 (DMOF-1loz), the density increases from 0.826 to 0.850 g/cm 3 . This drastic structural change also leads to a significant difference in framework stiffness and corresponding acoustic velocities (Figs. S4 and S5). The v 1max , v 2max , v 3max , and v 3min values of DMOF-1lsq are all larger than those of DMOF-1loz, while the v 1min and v 2min exhibit an inverse trend. Notably, the In particular, the anisotropy in v 1 of MIL-53Ga is the highest among all 22 materials, and such a large anisotropy is even larger than those of many 2D crystals. In terms of potential applications, MIL-53(Ga)lp could be utilized as a new type of acoustic switch through which acoustic waves can travel approximately 14 times faster in one direction than in the other, as shown in Figure 4(c).
2.3. Acoustic Impedance. Acoustic impedance (Z) describes the ratio of acoustic pressure to sound flow propagating in a material, which can be defined as the product of the acoustic velocity and the density of the material: Z = v * ρ, where v and ρ are the acoustic velocity and density, respectively. The acoustic impedance can also be divided into transverse acoustic impedance and longitudinal acoustic impedance according to the different acoustic velocities. The longitudinal acoustic impedance is widely used in practice, so we calculated the maximum and minimum longitudinal acoustic impedances of these 22 materials.
As seen from Figure 5, the longitudinal acoustic impedances of MOF-5, ZIF-8, and MAF-7 have very similar values Taking the DMAMg(HCOO) 3 perovskite as an example, its Z min and Z max are approximately 45% and 40% of those of the perovskite oxide BaTiO 3 . By comparison, it can be seen that the acoustic impedances of MOFs are generally lower than those of traditional inorganic materials.

Piezoelectric Properties.
The acoustic and piezoelectric properties of materials are closely related since acoustic waves can be detected by piezoelectric materials or vice versa. According to the type of energy conversion, the piezoelectric effects can be divided into direct and indirect effects. The direct piezoelectric effect occurs when a piezoelectric material generates a dipole moment under external stress, while the indirect piezoelectric effect refers to the strain generated in piezoelectric materials when they are placed in an electric field [35].
Among the 19 crystals, QMOF-1, QMOF-2, and DMAMg(HCOO) 3 have piezoelectric properties owing to their 3, 622, and m noncentrosymmetric point groups, respectively. The voltage signal harvested by our experiment also confirmed the piezoelectric property of QMOF-1 (Fig. S22). Because of the structural similarity, the origin of the piezoelectric effect of QMOF-1 is reminiscent of that of quartz. As shown in Figure 6, Zn-O polyhedra and isonicotinate linkers are considered negatively and positively charged components, respectively. In the absence of stress, the positive and negative charge centres overlap, which leads to no macroscopic spontaneous polarization. However, negative and positive charges are generated on the top and bottom surfaces of the crystal, respectively, when the a-axis is stressed.
To further analyse the piezoelectric properties of QMOF-1, QMOF-2, and DMAMg(HCOO) 3 , the density functional perturbation theory (DFPT) calculation method is used to obtain their piezoelectric tensors. It is well known that the piezoelectric stress tensor ½e, elastic compliance matrix ½s, and piezoelectric strain tensor ½d can be related via the following formula: ½d = ½e½s [36,37]. The full piezoelectric stress and strain tensors of all three MOFs are listed in Tables S2, S4, and S5. In all 3 MOFs, the ionic contribution is much larger than the electronic contribution to the piezoelectric stress tensors. For QMOF-1, the piezoelectric strain tensor d 14 (23.64 pC/N) is 31.3 times larger than that of quartz (d 14 = 0:73 pC/N), and its d 11 is 4.60 pC/N. QMOF-2 only has a d 14 tensor (2.52 pC/N) due to its 622 symmetry, which is approximately 3.5 times higher than that of quartz. The  [38,39].
The piezoelectric strain tensor d in these 3 MOFs is mainly affected by two factors: the magnitude of the atomic displacement caused by the strain and the charge change caused by the atomic displacement. The Born effective charge (BEC) is defined as the change in electron polarization caused by ion displacement and is the second derivative of the energy with respect to the displacement and electric field. In this regard, the BEC (Z mα ) can directly reflect the contribution of the second factor to the piezoelectric strain tensor. The BEC can be expressed as: where μ, σ, and η represent the atomic displacement, electric field, and strain, respectively. E denotes the total energy of the material in the ground state, and 0 is the unit cell volume of the material.
To explore the direction-dependent piezoelectric effects, the BEC of the zinc ions along different directions in QMOF-1 was calculated, and the results are listed in Table S3. There are three nonequivalent zinc ions in the unit cell of QMOF-1, and the sum of their BECs along the c -axis is approximately half of those along the a/b-axis. As the zinc ion normally shows a positive divalence, it is prone to exhibit charge transfer if its BEC is larger than 2. The BECs of the three zinc elements are 1.59, 1.59, and 1.58 along the c-axis, while they are 2.11, 3.79, and 2.33 along the a-axis and 3.38, 1.69, and 3.16 along the b-axis. Such a significant difference indicates that transfer of their charges is more difficult along the c-axis than along the a/b-axis, hence giving rise to smaller d 33 but larger d 11 and d 22 .

Discussion
Our above analyses demonstrate the tight structural-acoustic property relationships of MOFs and suggest some tuning approaches via crystal engineering. First, variation of the organic ligands can result in prominent differences in the elastic moduli of MOFs [14] and the corresponding acoustic velocities and impedance for the same structural topology and similar densities. As this strategy has been well demonstrated by the difference in acoustic properties between the prototypical ZIF-8 and MAF-7 by simply varying the ligand, it could be widely applied in other MOF systems. Second, metal nodes play an important role in determining the framework rigidity and therefore substantially influence the corresponding acoustic properties. For the MIL-53 family, the Al analogue is stiffer than the Ga counterpart due to the shorter coordination bonds [32], which leads to its higher sound velocities but lower acoustic anisotropy. Similarly, the Jahn-Teller effect and ligand splitting stabilization energy could be employed to modulate the elasticity and corresponding acoustic properties of MOFs [40]. Furthermore, the coordination number of metal nodes would be an alternative way to tune the acoustic properties of MOFs. For instance, MOFs with highly coordinated metal nodes show higher resistance towards shear stress than those with lowcoordinated structures and therefore exhibit higher transverse wave velocities.  3 ] (AZE = azetidinium), which naturally results in its much larger acoustic velocities and impedance [41]. On the other hand, gas molecule-filled MOFs exhibit lower acoustic velocities owing to unaltered elastic moduli but significantly increased density compared to their parent phases [42].
In terms of the piezoelectricity of MOFs, very few studies have been reported, and more research efforts need to be devoted to this field, especially considering the availability of vast numbers of noncentrosymmetric MOFs with diverse structures and topologies. When sufficient piezoelectric MOFs are identified and their structure-property relationship is corroborated, the next goal would be to optimize their performance to catch up with those of their conventional ceramic counterparts. In this regard, screening more noncentrosymmetric MOFs using DFPT calculations with the aid of artificial intelligence would be a promising method if combined with complementary experimental confirmation.

Conclusions
In conclusion, the acoustic properties of 19 MOFs were systematically investigated via density functional theory (DFT) calculations and by solving the Christoffel equations. Our results show that the MOFs have a very wide range of acoustic velocities due to their structural diversity and abundant chemical compositions. Meanwhile, some MOFs can exhibit very high anisotropy in acoustic velocities, which could be applicable in acoustic switches. In addition, the low densities and moduli of MOFs are responsible for their small acoustic impedances compared with their inorganic counterparts. Furthermore, the piezoelectric properties, which are closely related to the acoustic properties, were calculated for two quartz-like and one perovskite-like MOFs. Our results reveal that these MOFs can exhibit significant piezoelectricity, which mainly originates from the ionic contribution. Finally, we propose some strategies for engineering the acoustic properties of MOFs and discuss future directions for studying the piezoelectricity of MOFs. This work offers an original study of the fundamental acoustic properties of MOFs, which we believe could inspire further attention in this largely unexplored area.

Methods
Structural optimizations and calculations of elastic constants based on DFT were performed using a plane-wave basis set as implemented in the Vienna ab initio simulation package (VASP) [43][44][45]. Projector augmented wave (PAW) pseudopotentials were employed to describe ion-electron interactions [46]. The considered PAW pseudopotentials and numbers of valence electrons were H (1), C (4), N (5), O (6), Na_pv (7), Mg (2), Al (3), Si (4), Cl (7), K_sv (9), Ca_ sv (10), Ti_sv (12), V_sv (13), Mn_pv (13), Fe (8), Co (9), Cu (11), Zn (12), Ga_d (13), and Ba_sv (10). The GGA (PBEsol) functional was used to further accurately describe the interactions within each MOF structure. Considering the abundant van der Waals interactions existed in MOF systems, dispersion corrected calculations are known to be crucial in the correct optimization of their crystal structures. Specifically, GGA (PBEsol) underestimates the binding energy, hence resulting in larger lattice parameters for these molecular systems. By applying the dispersion corrections, more reasonable binding energies could be obtained, which gives rise to more reliable lattice parameters [47,48]. As elastic constants are obtained by the second derivative of the total energy with respective to specific strain and strongly correlated to the relative change of the crystal structure, obtaining accurate cell parameters are of vital importance in the calculations. In this regard, the van der Waals corrected zero damping DFT-D3 method was chosen in our calculations.
Considering that the calculation of elastic constants requires a high standard for the structures of materials, the atomic positions and cell parameters were completely relaxed when the structure was optimized. The elastic constants were calculated by setting six finite distortions of the lattice and the step size to 0.015 Å. The total energy converged to within 10 -7 eV, and the residual forces on each atom were less than 0.005 eV/Å. The energy cutoff for the plane-wave basis and the smallest allowed spacing between k-points of these compounds were set to 500 eV and 0.5 Å -1 , respectively. The acoustic velocity was obtained by solving the Christoffel equation with the material density and the elastic constants calculated by DFT [49]. The Christoffel equation can be expressed as: where u is the elastic displacement field and Γ, v, ρ, and α are the Christoffel matrix, acoustic velocity in the crystal, density, and direction vector, respectively.
The second derivative of the total energy reflects many properties of the calculated structure, such as the Born effective charge (BEC) and piezoelectric and dielectric tensors. The density functional perturbation theory proposed by Baroni et al. in 1987 can effectively calculate the derivative of the total energy with respect to strain, atomic displacement, and electric field [50][51][52][53]. The expression for the total energy E (μ, σ, η) under perturbation is as follows: where μ, σ, and η represent the atomic displacement, electric field, and strain, respectively. E (μ, σ, η) represents the total energy of the material in the ground state, 0 is the unit cell volume of the material and is the unit cell volume of the material under perturbation, and P is the electric field polarization.
The piezoelectric effect reflects the deformation degree of a material under an electric field, which can be evaluated by the piezoelectric stress tensor (e αj ). The piezoelectric tensors calculated by DFPT are divided into the electronic contribution (e αj ) and ionic contribution (ê αj ). The former is an ionclamped piezoelectric tensor that ignores the effects of atomic relaxation; in contrast, the latter takes into account the effect of the displacement of atoms under strain on the piezoelectric tensor. Therefore, the piezoelectric tensor can be expressed as: In this part of the DFPT calculation, we used the previously highly optimized crystal structure as the calculation model and improved the self-consistent field convergence standard to 10 -8 eV.

Data Availability
All data needed in the paper are present in the paper and in the supplementary section. Additional data related to this paper may be requested from the authors.

Conflicts of Interest
The authors declare no conflict of interest.