Compressing the Channels in the Crystal Structure of Copper Squarate Metal-Organic Framework

Abstract

The crystal structure of a copper squarate metal-organic framework is fully determined using first principles methods based in density functional theory. The compressibility of this material is studied by optimizing the structure under different isotropic pressures and uniaxial stresses directed along the direction of minimum compressibility, [1 0 0]. Under isotropic compression, channels become wider along [1 0 0], leading to negative linear compressibility, NLC. Under compression along [1 0 0], the unit-cell volume increases leading to negative volumetric compressibility.

1. Introduction

The isotropic negative linear compressibility (INLC) phenomenon [1,2] is an important counterintuitive effect observed in materials that expand when compressed or shrink under tension. Under the effect of isotropic pressure (P), the total volume (V) of a thermodynamically and mechanically stable space-unconstrained [3] material should decrease. Therefore, the isotropic volumetric compressibility, $k_V=-1/V·{(\partial V/\partial P)}_P$, should always be positive. This law, however, does not hold for each dimension of the material considered, and one or two of its dimensions (negative area compressibility) can increase under isotropic pressure if the remaining dimensions and the total volume decrease. In this case, the linear compressibility along a certain direction $\ell$, $k_{\ell }=-1/\ell ·{\left(\partial \ell /\partial P\right)}_P$, can be negative. The anisotropic negative volumetric compressibility (ANVC) effect occurs when compression is not isotropic [4,5,6,7,8]. If a uniaxial stress is exerted along a certain direction $m$ $\left(P_m\right),$the total volume can increase and the volumetric compressibility $k_V^m=-1/V·{\left(\partial V/\partial P_m\right)}_P$ can be negative. Materials exhibiting negative compressibilities, frequently simultaneously displaying the negative Poisson’s ratio (NPR) phenomenon [9,10,11], have been proposed for an immense range of potential applications, such as the development of ultrasensitive pressure-sensing devices [1,12], pressure driven actuators [12,13], optical telecommunication cables [1], artificial muscles [14] and body armor [12,15], and for sound attenuation [16], superconductivity modulation [17], ferroelectric enhancement [1] and transmission stabilization [18]. The main difficulties encountered in the practical applications of the natural and man-made NLC materials that have been found to date, is the relatively small absolute value of their negative compressibilities and the short range of pressure for which NLC effects are displayed. Thus, even though the search for new NLC materials has been intense and fruitful [2], the investigation is still in a highly active state. One of the main lines of research concerns the underlying structural mechanisms of the NLC phenomenon because, in most cases, the presence of NLC effects is a direct consequence of the structural characteristics. The use of these structure-function relationships should simplify the search for NLC materials, for example by filtering the crystal structures displaying these structural motifs. Furthermore, since nanotechnology promises the construction of materials with almost any requested geometry, the artificial reproduction of the structural motifs leading to NLC with variable compositions should allow one to generate a wide range of metamaterials [19] with desirable mechanical properties. Some of the known NLC mechanisms are the presence of ferroelastic instabilities, phonon instabilities, ferroelastic phase transitions and the anomalous mechanical behavior found in materials displaying correlated polyhedral tilts and helical and wine-rack structural motifs [2]. NLC effects were also found in chain or sheet structures linked by means of van der Waals interactions [4,5] or in structures displaying rotating polygon structural motifs [20,21,22]. An additional NLC mechanism was recently found for the crystal structure of porous materials containing cylindrical channels [23,24]. The main objective of this work is to confirm that these types of highly flexible crystal structures are likely to exhibit NLC effects. Therefore, the structures of crystalline species containing a small number of chemical elements were filtered to find simple compounds displaying empty structural channels. Among the simplest structures found, one was that of copper squarate [25], ${\mathrm{CuC}}_4{\mathrm{O}}_4$, a simple metal-organic framework, which fitted perfectly with the search pattern. Then, a study of the behavior of this compound under pressure was performed. A significant NLC effect was found, and the computed crystal structures under the effect of pressure very clearly showed an NLC mechanism based on the structural modification of the channels, which is the only source of the NLC effect in copper squarate.

2. Methods

Copper squarate was studied using first-principles solid-state methods based in Density Functional Theory using plane wave basis sets and pseudopotentials [26]. The CASTEP program [27] was employed for all the computations. The specialized version of the Perdew–Burke–Ernzerhof (PBE) energy-density functional for solid materials, PBEsol [28], was used. The good behavior of this functional for describing anhydrous materials has been widely recognized [29,30,31,32,33,34,35,36]. The pseudopotentials utilized for C, O and Cu atoms were standard norm-conserving pseudopotentials [37] provided in CASTEP code (00PBE-OP type). The optimization of the atomic positions and unit cell lattice parameters was carried out using the Broyden–Fletcher–Goldfarb–Shanno technique [38]. All the optimizations, even those of the compound under pressure, were performed with stringent convergence thresholds in the variation of the total energy, maximum atomic force, maximum atomic displacement and maximum stress of $2.5\times {10}^{-6} \mathrm{eV}/\mathrm{atom}, 0.005 \mathrm{eV}/\AA ,2.5\times {10}^{-4} \AA \mathrm{and} 0.0025 \mathrm{GPa}$, respectively, with the purpose of obtaining smooth pressure-volume curves. In the calculations, a large plane-wave kinetic energy cutoff (ε = 1000 eV) and a dense $k$point mesh [39] (3 × 3 × 5, corresponding to a grid density of 0.04 ${\AA }^{-1}$), were employed. The elasticity matrix of copper squarate was determined for the optimized crystal structure using the method of finite deformations [40]. The methodology used in this paper has been employed successfully in many previous works [4,5,6,7,8,23,24,31,35,41,42] in order to describe the elastic response of solid materials. For example, the predicted anomalous mechanical behavior of silver oxalate [6] was recently verified experimentally [20].

3. Results

Copper squarate is monoclinic, space group $I2/a$ (No. 15) [25]. In this paper, the axes were reordered, and the standard setting $C2/c$ (No. 15) was adopted. The crystal structure of copper squarate was described in detail by Dinnebier et al. [25]. A view of the crystal structure from the [0 0 1] crystallographic direction, clearly showing the presence of empty channels in the structure, is given in Figure 1. A single channel is explicitly shown by means of the corresponding solvent-accessible surface in a 1 $\times$ 1 $\times$ 5 supercell in Figure 2. The calculated unit-cell parameters of copper squarate are given in

Table 1. Lattice parameters of copper squarate.

Parameter	$\mathit{\alpha }$$\left(\mathbf{\AA }\right)$	$\mathit{b}$$\left(\mathbf{\AA }\right)$	$\mathit{c}$$\left(\mathbf{\AA }\right)$	$\mathit{\alpha } \left(\mathbf{deg}\right)$	$\mathit{\beta } \left(\mathbf{deg}\right)$	$\mathit{\gamma } \left(\mathbf{deg}\right)$	$\mathbf{Vol}. \left({\mathbf{\AA }}^3\right)$	$\mathit{\rho }$$(\mathbf{gr}/\mathbf{c}{\mathbf{m}}^3)$
PBE	11.1926	9.1271	5.7595	90.0	116.07	90.0	528.5049	2.207
PBE + disp	11.1578	9.1640	5.5982	90.0	117.78	90.0	506.4448	2.303
B3LYP	11.1829	9.2835	5.9160	90.0	114.88	90.0	557.1953	2.093
PBEsol	11.0961	9.0074	5.6076	90.0	116.26	90.0	502.6355	2.320
Exp. [25]	10.6869(4)	8.9394(4)	5.6747(1)	90.0	117.60(2)	90.0	480.4546	2.427

, and the calculated interatomic distances and angles are given in Tables S1 and S2 of the Supplementary Material (SM), respectively. In these tables, the calculated values are compared with the corresponding experimental data [25]. It must be noted that a full structure determination of the crystal structure of copper squarate was not possible in the work by Dinnebier et al. [25]. These authors performed a rigid-body Rietveld refinement of the squarate anions without a determination of the corresponding internal degrees of freedom. Therefore, the fully optimized structure obtained in this work is reported in the SM as a file with a CIF (Crystallographic Information File) format. As can be observed in Table 1, the difference in the computed unit cell volume using the PBE functional with respect to the experimental value of Dinnebier et al. [25] is very large, 10.0%. The results were significantly improved when the PBEsol functional was used. The difference was reduced to 4.6%. The PBE functional supplemented with dispersion corrections [43] and several hybrid energy-density functionals, such as the PBE0 [44] and B3LYP [45] ones, was also used but led to larger differences in the computed unit cell volume and parameters with respect to the experiment. Therefore, the PBEsol functional was selected to determine the mechanical properties of copper squarate. The X-ray powder diffraction pattern of copper squarate was determined from the computed and experimental structures [46]. The results are compared in Figure S2 of the SM, and, as can be seen, they are in good agreement. The positions of the main reflections in the computed and experimental X-ray powder patterns are compared in Table S3 of the SM.

The rigid-body Rietveld refinement of the squarate anions [25] provided average bond lengths for the C-C (1.23 Å) and C-O (1.45 Å) bonds, which agreed reasonably with experimental values of these bond lengths reported in the literature. For comparison, the different C-C bond lengths in squaric acid lead to an average C-C distance of 1.46 Å, and the corresponding C-O bond lengths give an average C-O distance of 1.26 Å [4,47]. The computed C1-C2 and C1-C2′ bond distances (see Table S1) differ from the average value by about 0.04 Å. Similarly, the differences for the calculated C1-O1 and C2-O2 distances are smaller than 0.02 Å. Finally, the Cu-O1, Cu-O2 and Cu-O2′ distances agree with their experimental counterparts within about 0.1 Å. Due to the incomplete structure refinement, the computed C-C-C and O-C-C angles should not be compared with the corresponding experimental values. However, the theoretical O-Cu-O angles are in reasonable agreement with the experiment [25].

The computed elasticity matrix of copper squarate is:

$$C=\left(\begin{array}{cccccc}72.78& 44.11& 38.74& 0.00& -25.81& 0.00\\ 44.11& 44.30& 19.38& 0.00& -2.61& 0.00\\ 38.74& 19.38& 52.31& 0.00& -17.54& 0.00\\ 0.00& 0.00& 0.00& 14.24& 0.00& -4.70 \\ -25.81& -2.61& -17.54& 0.00& 30.60& 0.00\\ 0.00& 0.00& 0.00& -4.70& 0.00& 32.78 \end{array}\right)$$

(1)

In this equation all values are given in GPa. The mechanical stability of its structure was analyzed from the computed elasticity matrix. According to the necessary and sufficient Born stability condition [48], if a material is mechanically stable, the elasticity matrix should be positive definite, i.e., all its eigenvalues must be positive. Therefore, the elasticity matrix of copper squarate was diagonalized numerically. Since all the eigenvalues were found to be positive, copper squarate is mechanically stable. The mechanical properties of copper squarate were then evaluated as a function of the orientation of the applied stress, and tridimensional representations were performed using ElAM software [49]. The results are shown in Figure S3 of the SM. Bidimensional projections on the $xy$ and $xz$ planes are given in Figure 3 and Figure 4, respectively. As can be observed in Figure 4A, copper squarate exhibits negative compressibilities for a wide range of directions of the applied stress. The direction of minimum compressibility is close to [1 0 0]. Furthermore, Figure 4D shows that copper squarate displays the negative Poisson’s ratio phenomenon. The minimum value of the Poisson’s ratio is quite significant, ${\nu }_{min}=-$0.66, for an external stress applied along the minimum Poisson’s ratio direction, [−0.43, −0.36, +0.82].

The crystal structure of copper squarate was optimized under the effect of twenty-one different isotropic pressures in the range of −1.50 to 6.00 GPa. The optimized unit cell volumes and lattice parameters are plotted in Figure 5 and given in Table S4 of the SM. As can be seen in Figure 5B, the $a$ lattice parameter increases significantly from the lowest isotropic pressure considered, p = −1.50 GPa, up to p = 4.14 GPa. Therefore, copper squarate exhibits the INLC phenomenon in this pressure range. The $b$ and $c$ lattice parameters and the total unit-cell volume decrease. The computed compressibilities are shown in Figure 5E and given in Table S5 of the SM. The minimum value of the compressibility is found for P = 1.35 GPa, $k_a=-8.14$ ${\mathrm{TPa}}^{-1}$. Although a large variation of the unit cell volume and parameters is observed between 1.0 and 2.0 GPa, no topological changes on the bond structure were observed, and, therefore, copper squarate does not undergo a pressure-induced phase transition in this region. Furthermore, the enthalpy pressure equation of state (Figure S4 of the SM) does not show any slope change, ruling out phase transitions within the complete range of pressure considered.

The structure of ${\mathrm{CuC}}_4{\mathrm{O}}_4$ was also optimized under the effect of eleven different stresses applied in the minimum compressibility direction, [1 0 0], in the range going from P = −0.33 to P = 0.25 GPa. The corresponding optimized unit cell volumes and lattice parameters are plotted in Figure 6 and given in Table S6 of the SM. Figure 6A shows that the unit-cell volume increases from P = $-$0.33 GPa to P = 0.17 GPa. Hence, copper squarate displays the ANVC phenomenon in this pressure range. The calculated compressibilities are plotted in Figure 6E and given in Table S7 of the SM. The minimum value of the compressibility, $k_V=-1/V·{\left(\partial V/\partial P\right)}_P=-$81.3 ${\mathrm{TPa}}^{-1}$, is obtained for P = 0.12 GPa.

4. Discussion

The structural mechanism of INLC was studied by analyzing the variation of the optimized structures under isotropic pressure. As shown in Figure 7, the width and height of the channels become increasingly larger and smaller, respectively, under pressure. As a result, the $a$ lattice parameter increases strongly, and the compressibility is negative for a wide pressure range. The more important structural change that is observed is the variation of the relative angle between the copper octahedra and the squarate rings. The variation of the $\mathrm{Cu}-\mathrm{O}-\mathrm{C}$ angle, mainly responsible for the largest peak in the compressibility found in Figure 5E, is reduced from 127.7° to 123.3° between P = 1.0 GPa and 2.0 GPa.

Similarly, the mechanism of ANVC was revealed by analyzing the modification of the optimized structures under external stresses applied along the [1 0 0] direction. As shown in Figure 8, contrary to the behavior of the crystal structure under isotropic pressure, the width and height of the channels become smaller and larger, respectively, as the stress increases and the total unit-cell volume increases. Therefore, the volumetric compressibility is negative.

5. Conclusions

The isotropic compression of structures containing empty channels shows a clear trend for producing large NLC effects due to the elongation of the elliptic section of the channels along its major semiaxis accompanied by the reduction of its minor semiaxis. Under compression along the direction of the major semiaxis, however, the corresponding dimension of the unit cell is reduced and the minor semiaxis and total volume increase, leading to a negative volumetric compressibility. Similar results were obtained in previous studies [23,24]. Therefore, additional studies on the mechanical properties of an extended set of porous solid compounds containing the empty channel structural motif, such as other MOFS, zeolites and aluminophosphate materials, are now being carried out. The results obtained so far have shown that, essentially, the same behavior is obtained for a large fraction of the materials considered. The minimum isotropic compressibility of copper squarate was found to be significant, $k_a$ = −81.4 ${\mathrm{TPa}}^{-1}$ at P = 1.35 GPa. The computed minimum anisotropic volumetric compressibility is $k_V$ = −81.3 ${\mathrm{TPa}}^{-1}$ at P = 0.12 GPa. The compressibility functions of copper squarate must be evaluated experimentally to confirm the present theoretical results. Although the results of accurate non-empirical calculations are generally in agreement with the experiment [20], the study of the effect of temperature, pressure-transmitting media and a partial occupation of the channels with water could be very interesting.