Mechanical Characterization of Anhydrous Microporous Aluminophosphate Materials: Tridimensional Incompressibility, Ductility, Isotropy and NegativeLinear Compressibility

Here, a detailed mechanical characterization of five important anhydrous microporous aluminophosphate materials (VPI-5, ALPO-8, ALPO-5, ALPO-18, and ALPO-31) is performed using first principles methods based on periodic density functional theory. These materials are characterized by the presence of large empty structural channels expanding along several different crystallographic directions. The elasticity tensors, mechanical properties, and compressibility functions of these materials are determined and analyzed. All of these materials have a common elastic behavior and share many mechanical properties. They are largely incompressible at zero pressure, the compressibilities along the three crystallographic directions being frequently smaller than 5 TPa−1. Notably, the compressibilities of ALPO-5 and ALPO-31 along the three principal directions are smaller than this threshold. Likewise, the compressibilities of ALPO-18 along two directions are smaller than 5 TPa−1. All of the considered materials are shear resistant and ductile due to the large bulk to shear moduli ratio. Furthermore, all of these materials have very small mechanical anisotropies. ALPO-18 exhibits the negative linear compressibility (NLC) phenomenon for external pressures in the range P = 1.21 to P = 2.70 GPa. The minimum value of the compressibility along the [1 0 0] direction, ka = −30.9 TPa−1, is encountered for P = 2.04 GPa. The NLC effect in this material can be rationalized using the empty channel structural mechanism. The effect of water molecule adsorption in the channels of ALPO-18 is assessed by studying the hydrated ALPO-18 material (ALPO-18W). ALPO-18W is much more compressible and less ductile than ALPO-18 and does not present NLC effects. Finally, the effect of aging and pressure polymorphism in the mechanical properties of VPI-5 and ALPO-5 is studied. As hydration, aging leads to significant variations in the elastic properties of VPI-5 and increases substantially its compressibility. For ALPO-5, pressure polymorphism has a small impact in its elasticity at zero pressure but a large influence at high pressure.


Introduction
Aluminophosphate (ALPO) compounds are important synthetic microporous materials whose structure is characterized by the presence of large structural channels expanding along several different crystallographic directions [1]. Due to their high surface area and pore volume, ALPO materials have been employed in a wide range of important applications [1]. However, the full tensorial elasticity of these compounds, determining their behavior under stress and their mechanical performance in the applications, have surprisingly not been studied. A detailed mechanical characterization of a representative set of ALPO materials is performed in the present work. The set of materials considered
As shown in Figure 2A, the structure of ALPO-18 [10,11] exhibits 4-and 8-membered rings in (0 0 1) plane. The 8-membered nearly circular rings, have a diameter of d = 6.5 Å. A perspective view of the 8-MR channels is plotted in Figure 2B. The ALPO materials have commonly also channels expanding along several directions. While these channels have generally smaller apertures, the case of ALPO-18 is noticeable since also large channels are observed when the material is viewed from other directions. As can be observed in the second and third subfigures of Figure 2A, large 8-membered rings are also observed when the structure of ALPO-18 is seen from [1 0 0] and [1 1 0] directions. Figure 2C shows the crystal structure of hydrated ALPO-18 (ALPO-18W). Two water molecules per formula unit are adsorbed within the channels of ALPO-18. The structure of ALPO-18 changes significantly upon hydration. The space symmetry of this material changes from monoclinic (C2/c) to triclinic (P1). In fact, some of the aluminum atoms change their coordination environment from tetrahedral to octahedral to account for the presence of additional water molecules [11].
The computed lattice parameters for VPI-5, ALPO-8, ALPO-5, ALPO-18, ALPO-18W, and ALPO-31 along with the experimental parameters are reported in Table 1. The average difference of the computed and experimental unit cell volumes is quite good, 2.5 and 2.3%, for the PBE and PBEsol functionals. The impact of introduction of dispersion corrections in these materials is relatively small and the average difference of the computed and experimental unit cell volumes is reduced by only 0.3%. Since the improvement due to the inclusion of dispersion corrections was small, the PBEsol functional was used for all anhydrous materials to retain the ab initio character of the computations. Additional details about the impact of including dispersion interactions in the calculations will be given in Section 3.6.3. Dispersion corrections were only included for ALPO-18W since, as it is well-known [100,[288][289][290], they significantly improve the hydrogen bond geometries in the structures of hydrated materials. The X-ray diffraction patterns of VPI-5, ALPO-8, ALPO-5, ALPO-18, and ALPO-31, generated from the computed and experimental structures [4,6,9,10,12], are compared in Figure 3. The agreement is excellent. A detailed comparison of the positions of the main reflections in the X-ray diffraction patterns for these ALPO materials is provided in Tables S2-S6 of the SM. Similarly, Tables S7-S11 of the SM provide a comparison of the computed and experimental interatomic distances in the crystal structures of these materials. The computed PO and AlO average distances are 1.52 and 1.72 Å, respectively, which are in good agreement with the experimental values of 1.51 and 1.70 Å.   [4,6,9,10,12] crystal structu radiation (λ = 1.540598 Å).  [4,6,9,10,12] crystal structures using CuK α radiation (λ = 1.540598 Å).

Stiffness Tensors and Mechanical Stability
The computed stiffness tensors of VPI-5, ALPO-8, ALPO-5, ALPO-18 and ALPO-31 are provided in Table 2. The number of non-vanishing and non-equivalent elements in the matrix representation of the symmetric stiffness tensor depends on the space symmetry of the corresponding crystal structure [192,325]. The P6 3 cm and P6cc structures of VPI-5 and ALPO-5 are hexagonal and have nine non-vanishing elastic constants in their stiffness matrices, five of which are non-equivalent (C 11 , C 33 , C 44 , C 12 , C 13 ). ALPO-8 is orthorhombic (Cmc2 1 ) and, therefore, its stiffness tensor has nine non-vanishing elements all of which are non-equivalent. The number of nonvanishing elastic constants for the monoclinic structure of ALPO-18 (C2/c), increases to thirteen due to its lower symmetry. For ALPO-31 (trigonal, R3h) there are fifteen non-vanishing elastic constants, seven of which are nonequivalent (C 11 , C 33 , C 44 , C 12 , C 13 , C 14 , C 15 ). A crystal structure is mechanically stable, if an only if, the Born mechanical stability conditions are fulfilled [189][190][191][192]. The generic Born mechanical stability condition can be written in mathematical form as an algebraic condition on the eigenvalues of the matrix representation of the stiffness tensor: the elastic matrix must be positive definite, that is, all its eigenvalues must be greater than zero [192]. A numerical diagonalization of the stiffness tensors of all of the ALPO materials was carried out. Since all of the elastic matrix eigenvalues for all materials were positive, they are mechanically stable.
The Cauchy pressure term, defined in terms of the elastic constants as CP = (C 11 − C 44 ), was proposed by Pettifor [199] as an indicator of the angular character of atomic bonding. The values of CP are positive and large, 22.49, 29.07, 41.56, 54.63, and 32.32 GPa, for VPI-5, ALPO-8, ALPO-5, ALPO-18, and ALPO-31, respectively, reflecting a largely angular bonding in these materials. The value of the Cauchy pressure term is particularly large for ALPO-18.
CP is also related with the brittle/ductile character of crystals [198,[200][201][202][203][204]. Large values of CP are associated with highly ductile materials. The ductility index, D = B/G, was proposed by Pugh [198], as a standard measure of the ductility of a material. A value of D = 1.75, separates the brittle and ductile materials [201,208]. All of the computed values of D, 2.66, 2.67, 2.92, 3.80, and 3.04, for VPI-5, ALPO-8, ALPO-5, ALPO-18, and ALPO-31, respectively, are much larger than 1.75. Therefore, the five materials are ductile. An improved ductility criterium has been provided recently by Niu et al. [200]. In their work, these authors noticed that the intrinsic ductility index, defined as the ratio of the Cauchi pressure term to the Young's modulus, D I = (C 11 − C 44 )/E, is strongly correlated hyperbolically with the Pugh's ratio. As shown in Table 3, the values obtained for the intrinsic ductility index for the ALPO materials considered ranges from 1.02 to 0.18 and are in the same range as that for common metals [200]. The value of D I for ALPO-18, 1.02, is close to that of Pt (0.98 ± 0.01) or Nb (1.00 ± 0.01). For ALPO-5, D I = 0.51, coincides with that of K (0.51 ± 0.01). The intrinsic ductility indices of ALPO-31 and ALPO-8, 0.45, and 0.43, respectively, are close to the ductility for Al (0.44 ± 0.04). Finally, the value of D I for VPI-5, 0.37, is near to that of Cu (0.38 ± 0.04).
The Vickers hardness (H) measures the resistance of a given material to indentation. A series of representative values of H for interesting materials may be obtained from several published papers [193][194][195][196][197]. As a reference, talc and halite (H = 0. 26

Mechanical Properties as a Function of the Orientation of the Applied Strain
In the previous subsection, a general view of the elasticity of the ALPO materials was achieved and average values of the elastic moduli, Poisson's ratios, and ductility, hardness, and elastic anisotropy indices were reported. A more detailed understanding of the elasticity of these materials is provided by the analysis of the variation of the mechanical properties with the strain orientation. Three dimensional representations of the dependence of the elastic moduli and Poisson's ratios for VPI-5, ALPO-8, ALPO-5, ALPO-18 and ALPO-31 with respect to the direction of the applied strain are displayed in Figures 4-8, respectively. These figures explain the low elastic anisotropy of these materials since all elastic moduli have a smooth variation with respect to the direction of the applied strain. The elastic properties of VPI-5 and ALPO-5 (with hexagonal space symmetries) show a nice orientational dependence which is axially symmetric around z axis.            figures show a smooth directional dependence of the shear modulus (E), without special directions associated with small values of this property (see the projections of the surfaces of minimum shear modulus). Therefore, there are not crystallographic planes along which shear failure can be predicted. The presence of shear slippages imposed serious limitations to the mechanical properties of some microporous materials including MOFs [72,75,[80][81][82] and carbon nanotube composites [215,216]. No signs of auxeticity (negative Poisson's ratios [334]) were found for any of the materials investigated since the Poisson's ratios are always positive for all strain directions. This is in contrast with the elasticity of other microporous materials (for example for zeolites), for which negative or zero Poisson's ratios were frequently encountered [51][52][53][54][64][65][66][67].

Compressibility Functions
The crystal structures of VPI-5, ALPO-8, ALPO-5, ALPO-18, and ALPO-31 were fully optimized under different external isotropic pressures in the pressure range from −0.5 to 5.0 GPa. The computed unit cell volumes and lattice parameters for VPI-5, ALPO-8, ALPO-5, ALPO-31 and ALPO-18 at different pressures are plotted in Figures 9-13, respectively. The calculated volumetric compressibilities, k V = −1/V·(∂V/∂P) P , and the linear compressibilities, k l = −1/l·(∂l/∂P) P (l = a, b, c) along the three crystallographic directions between P = 0.0 and P = 4.0 GPa are also displayed in these figures. The values of the calculated unit cell volumes, lattice parameters and compressibilities are given in Tables S12-S21. The structure of ALPO-18 was also optimized under the effect of different uniaxial pressures (see Figure 14 and Tables S22 and S23). The calculated compressibilities of these materials at zero pressure are collected in Table 4. From this table, it follows that the volumetric compressibilities at zero pressure are very small, the most compressible material being VPI-5 (k V = 16.46 TPa −1 ) and the less compressible one being ALPO-31 (k V = 11.06 TPa −1 ). Solids 2022, 3, FOR PEER REVIEW 16 special directions associated with small values of this property (see the projections of the surfaces of minimum shear modulus). Therefore, there are not crystallographic planes along which shear failure can be predicted. The presence of shear slippages imposed serious limitations to the mechanical properties of some microporous materials including MOFs [72,75,[80][81][82] and carbon nanotube composites [215,216]. No signs of auxeticity (negative Poisson's ratios [334]) were found for any of the materials investigated since the Poisson's ratios are always positive for all strain directions. This is in contrast with the elasticity of other microporous materials (for example for zeolites), for which negative or zero Poisson's ratios were frequently encountered [51][52][53][54][64][65][66][67].

Compressibility Functions
The  Figure 14 and Tables S22 and S23). The calculated compressibilities of these materials at zero pressure are collected in Table 4. From this table, it follows that the volumetric compressibilities at zero pressure are very small, the most compressible material being VPI-5 ( = 16.46 TPa −1 ) and the less compressible one being ALPO-31 ( = 11.06 TPa −1 ).          The linear compressibilities along the different directions are frequently smaller than 5 TPa −1 . For ALPO-5 and ALPO-31, the three linear compressibilities are smaller than this threshold. The same occurs for two linear compressibilities of ALPO-18, although the compressibility along direction is also very near to this limit ( = 5.17 TPa −1 ). For VPI-5 and ALPO-8, only the compressibilities along direction satisfy < 5 TPa −1 . However, the value of for ALPO-8 is the lowest linear compressibility found for all of the ALPO materials considered, ( = 1.81 TPa −1 ). The criterium usually used for zero linear compressibility (ZLC) [102,103,[132][133][134][135] is that the absolute value of the linear com-  Material The linear compressibilities along the different directions are frequently smaller than 5 TPa −1 . For ALPO-5 and ALPO-31, the three linear compressibilities are smaller than this threshold. The same occurs for two linear compressibilities of ALPO-18, although the compressibility along b direction is also very near to this limit (k b = 5.17 TPa −1 ). For VPI-5 and ALPO-8, only the compressibilities along c direction satisfy k c < 5 TPa −1 . However, the value of k c for ALPO-8 is the lowest linear compressibility found for all of the ALPO materials considered, (k c = 1.81 TPa −1 ). The criterium usually used for zero linear compressibility (ZLC) [102,103,[132][133][134][135] is that the absolute value of the linear compressibility along a certain direction is smaller than 1.0 TPa −1 , | k l | ≤ 1.0 TPa −1 [133]. While this criterium is not met for ALPOs, the presence of three simultaneously small linear compressibilities is very infrequent. The term near zero tridimensional linear compressibility (NZTLC) is proposed for materials satisfying, 1 ≤ | k l | ≤ 5.0 for l = a, b, c. ALPO-5, ALPO-31 and, in practical terms, ALPO-18 are NZTLC materials at zero pressure.
As shown in Figure 9, the compressibilities of VPI-5 along the three crystallographic directions are lower than 10 TPa −1 for the full range of pressure considered. The compressibility along the c direction is smaller than 5 TPa −1 from 0 to 4 GPa except for applied pressures near 4.0 GPa. Therefore, as shown in Figure S1, the isotropic compression of VPI-5, leads to very small changes of its structure. For ALPO-8 ( Figure 10), although the linear compressibility along a direction remains small from 0 to 4 GPa and attains a minimum near P = 2.75 GPa (k a = 3.4 TPa −1 ), k b and k c increase rapidly and reach maxima near P = 2.5 GPa. Consequently, the volumetric compressibility increases from 0 to 2.5 GPa and then decreases up to 4 GPa. Since the compressibility along c direction is very small at zero pressure, and it is a strongly decreasing function as the pressure diminishes, the presence of negative values of k a under tension (negative pressure) is highly probable. For ALPO-5 ( Figure 11), as for VPI-5, the linear compressibilities remain small in the range from 0 to 4.0 GPa. However, for ALPO-31 ( Figure 12), as for ALPO-8, the compressibilities increase largely as the pressure increases and the volumetric compressibility reach a maximum near P = 2.0 GPa. The behavior of ALPO-18 under pressure is extremely anomalous and is studied in the next Subsection.

Isotropic Negative Linear Compressibility (INLC)
The three lattice parameters of VPI-5, ALPO-8, ALPO-5, and ALPO-31 decrease invariably under isotropic compression. However, as can be observed in Figure 13B, the a lattice parameter of ALPO-18 increases sharply from P = 1.21 to P = 2.70 GPa. Therefore ALPO-18 exhibits the isotropic negative linear compressibility (INLC) phenomenon [129][130][131] in this pressure range. The minimum value of the compressibility along the a direction is encountered at P = 2.04 GPa, k a = −30.9 TPa −1 .
The INLC effect in ALPO-18 can be rationalized in terms of the empty channel structural mechanism [100,335]. The deformation of the crystal structure of ALPO-18 induced by the application of increasing isotropic pressures is illustrated in Figure 15 (Figure 2) in the dimensions of the crystal is much smaller, as shown in Figure S2. Therefore, the dominance of the deformation of the 8-MR channels expanding along [0 0 1] makes observable the NLC effect based in the empty structural mechanism in the multichannel ALPO-18 material.

Anisotropic Negative Volumetric Compressibility (ANVC) Effect
In previous works, the INLC effect due to the empty channel structural mechanism was observed to be accompanied by the anisotropic volumetric NLC effect (ANLC)

Anisotropic Negative Volumetric Compressibility (ANVC) Effect
In previous works, the INLC effect due to the empty channel structural mechanism was observed to be accompanied by the anisotropic volumetric NLC effect (ANLC) [100,336], i.e., the increase of the volume of a material when an external anisotropic pressure is applied to it. This effect was discovered in 2015 by Baughman and Fonseca [336] in porous materials and, independently, by Colmenero in 2019 for non-porous materials as the cyclic oxocarbon acids [307], oxalic acid [308] and uranyl squarate monohydrate [311]. The unit cell volumes, lattice parameters and compressibilities of ALPO-18 under the effect of increasing uniaxial pressures along the direction of minimum compressibility, [1 0 0], are shown in Figure 14 and provided in Tables S22 and S23 of the SM. As can be appreciated, the unit cell volume increases under tension from P = −1.0 up to P = −0.20 GPa. Therefore, ALPO-18 exhibits the ANVC effect in this pressure range. The minimum value of the compressibility is found at P = −0.76 GPa, k V = −6.0 TPa −1 .

Effect of Dispersion Interactions in the NLC Effect of ALPO-18
In Section 3.1, the influence of dispersion interactions in the crystal structures of the considered ALPO materials was shown to be small. However, due to the relevance of the NLC phenomenon, the crystal structure of ALPO-18 was also completely optimized under the effect of different isotropic pressures using the PBE functional supplemented with Grimme's dispersion corrections [283] The computed values of the a lattice parameter are compared with those obtained using the PBEsol functional in Figure 16. The results are quite simitar, thus confirming the NLC effect in ALPO-18 and the good performance of the PBEsol functional for the description of anhydrous materials . Using the dispersion corrected treatment, ALPO-18 displays an even larger isotropic NLC effect from P = 0.61 GPa to P = 2.50 GPa. The minimum value of the compressibility along the [1 0 0] direction is k a = −38.7 TPa −1 at P = 1.74 GPa.
[1 0 0], are shown in Figure 14 and provided in Tables S22 and S23 of the SM. As can be appreciated, the unit cell volume increases under tension from P = −1.0 up to P = −0.20 GPa. Therefore, ALPO-18 exhibits the ANVC effect in this pressure range. The minimum value of the compressibility is found at P = −0.76 GPa, = −6.0 TPa −1 .

Effect of Dispersion Interactions in the NLC Effect of ALPO-18
In Section 3.1, the influence of dispersion interactions in the crystal structures of the considered ALPO materials was shown to be small. However, due to the relevance of the NLC phenomenon, the crystal structure of ALPO-18 was also completely optimized under the effect of different isotropic pressures using the PBE functional supplemented with Grimme's dispersion corrections [283] The computed values of the lattice parameter are compared with those obtained using the PBEsol functional in Figure 16. The results are quite simitar, thus confirming the NLC effect in ALPO-18 and the good performance of the PBEsol functional for the description of anhydrous materials . Using the dispersion corrected treatment, ALPO-18 displays an even larger isotropic NLC effect from P = 0.61 GPa to P = 2.50 GPa. The minimum value of the compressibility along the [1 0 0] direction is = −38.7 TPa −1 at P = 1.74 GPa.

Effect of Hydration in the Mechanical Properties of ALPO-18
In this Section, the effect of the presence of water molecules adsorbed in the structural channels of ALPO-18 on the mechanical properties of this material is studied. This is relevant from the point of view of applications since if one desires to take advantage of the mechanical properties of ALPOs, such as their large incompressibility and ductility, the influence of water adsorption should be considered. If the impact in the elastic properties

Effect of Hydration in the Mechanical Properties of ALPO-18
In this Section, the effect of the presence of water molecules adsorbed in the structural channels of ALPO-18 on the mechanical properties of this material is studied. This is relevant from the point of view of applications since if one desires to take advantage of the mechanical properties of ALPOs, such as their large incompressibility and ductility, the influence of water adsorption should be considered. If the impact in the elastic properties is large, hydration should be avoided as much as possible. The calculated lattice parameters of ALPO-18W are in given in Table 1. The computed unit cell volume differs from the experimental value [11] by only 0.5%. The computed stiffness tensor and mechanical properties of ALPO-18W are reported in Tables 5 and 6, respectively, and the dependence of its mechanical properties on the orientation of the applied strain is shown in Figure S3. The unit cell volumes, lattice parameters and compressibilities of ALPO-18W under different isotropic pressures are shown in Figure S4 and given in Tables S24 and S25.   Since ALPO-18 is triclinic, all of the elements of the matrix representation of its elastic tensor are non-vanishing and non-equivalent. As with ALPO-18, ALPO-18W is characterized by large bulk, Young's, and shear moduli. However, due to the adsorption of water molecules, the bulk and shear moduli of ALPO-18W become much smaller and larger, respectively, than those of ALPO-18. Consequently, although ALPO-18W is also ductile, the ductility index is smaller. Since the intrinsic ductility index (D I ) is strongly correlated with Pugh's ratio [190], its value is reduced from 1.02 to 0.18. Therefore, hydration makes this material more compressible and less ductile. The universal anisotropy index is very small, A U = 0.24, as with the other ALPO materials. As expected from the small elastic anisotropy index, the dependence of the mechanical properties on the direction of the applied strain is smooth. No preferred directions for fracture or shear failures nor negative Poisson's ratios are observed.
In a recent work [100], a strong reduction of the NLC effect in titanium oxalate dihydrate was also found as a result of water molecule adsorption, although the NLC effect in this microporous metal organic framework does not disappear completely. In contrast, for some microporous zeolites and MOFs [56,57,98,99], water intrusion leads to a strong increase of the unit-cell volume and NLC effects. The strong influence of the presence of guest molecules in the structural channels of porous materials in their mechanical properties has been found by several research groups in previous works [87,88,337]. In fact, Terracina et al. [88], showed that the main source of structural instability in the MOF HKUST-1 during compaction was the presence water molecules adsorbed by the powdered samples and a new tableting method preserving the crystal structure and porosity of the pristine powders was reported. The influence of guest molecule adsorption in the elastic properties of microporous materials is highly dependent on the material under study and the type of interaction between the molecules with the walls of the channels and between the molecules themselves. The presence of water in contact with the material may increase the internal tensions, lead to phase transformations or even be the origin of crack propagation and fracture [238]. The important NLC effect induced by water or guest molecule intrusion should be distinguished from the conventional NLC phenomenon, encountered for ALPO-18 in this paper, due to the need of specifying the origin of pressure and the requirement of the description of the interaction of a variable number of guess molecules with the material for his theoretical study. It is a common belief that the absorption of water molecules in the channels of a microporous material should reduce its compressibility due to the stiffening of the structure due to increased density [338]. However, the opposite its true in ALPO-18. The counterintuitive softening upon adsorption of guest molecules in microporous materials was first observed by Mouhat et al. [339] and Canepa et al. [340]. It is difficult to find an explanation for the softening in ALPO-18 based on the changes in the chemical bonding due to the drastic geometric rearrangement occurring upon water absorption. This requires a further study which is out of the scope of the present work.
The great influence of water adsorption in the structure of ALPO-18, underlines the need of using non-hydrous pressure transmitting media (PTM) to measure their compressibilities or experimental methods which are not based in the use of DAC in order to study its full tensorial elasticity. The compressibility of hydrated ALPO-5 [165][166][167] material was measured experimentally using DACs and different pressure transmitting media. The measured compressibilities were highly dependent of the PTM used. Furthermore, the compressibilities measured using the same PTM vary significantly from one study to another. For example, the volumetric compressibilities of ALPO-5W using a 16:3:1 methanol-ethanol-water (MEW) mixture as PTM measured by Kim et al. [166] and Lotti et al. [167] were 19.8 and 45.0 TPa −1 (corresponding to measured bulk moduli of 50.5(7) and 22.2(9) GPa), respectively. For VPI-5W, Alabarse et al. [168], using silicone oil as PTM, obtained of volumetric compressibility of 41.2 Tpa −1 (B = 24.3(5) Gpa). The present results for anhydrous VPI-5 and ALPO-5 and the experimental results for VPI-5W and ALPO-5W show that, as for ALPO-18, the influence of the presence of water in the channels of these materials in the elastic properties are substantial. Again, the use of complementary experimental methods not based in the use of DAC with a PTM for the study of the full tensorial elasticity of these materials is suggested.

Effect of Aging in the Elastic Properties of VPI-5
De Oñate Martinez et al. [5], noted that the space symmetry of VPI-5 depends on the method of preparation of this material and that aging also leads to space symmetry variations. Clearly, the origin of this effect is the small differences in the relative thermodynamic stabilities of the different crystal structures of ALPO materials [341,342]. Therefore, to assess the influence of aging, the monoclinic C1m1 crystal structure reported by De Oñate Martinez et al. [5] obtained from an aged sample of anhydrous VPI-5 was employed in order to compute its mechanical properties. The computed lattice parameters are given in Table S26 of the SM and the calculated stiffness tensor and mechanical properties are given in Tables 5 and 6, respectively. The dependence of its mechanical properties on the direction of the applied strain is shown in Figure S5. The bulk modulus for the monoclinic structure of VPI-5 diminishes significantly with respect to that for hexagonal VPI-5 (from 60.5 to 37.1 GPa) and, consequently, the ductility index is largely reduced (from 2.66 to 1.77). The elastic anisotropy in monoclinic VPI-5 (A U = 0.24) is smaller than in hexagonal VPI-5 (A U = 0.66). Although the directional dependence of the elastic properties for the monoclinic structure are significantly modified, the axial symmetry around z axis is conserved ( Figure S5B) and no preferred directions for fracture and shear failure are observed. Therefore, aging in VPI-5, as hydration in ALPO-18, reduces its incompressibility and ductility.
The computed unit-cell volume, lattice parameters and compressibilities of monoclinic VPI-5 as a function of the external isotropic pressure are shown in Figure S6 and given in Tables S27 and S28 of the SM. While the linear compressibilities of hexagonal VPI-5 remains small as pressure increases, the same is not true for the monoclinic structure.
3.9. Effect of Pressure Polymorphism 3.9.1. VPI-5 Since in this paper we are interested in studying the behavior of ALPO materials under the effect of pressure, the relative thermodynamic stability of the P6 3 cm4 and C1m1 [5] structures of VPI-5 under pressure was investigated. Both crystal structures were fully optimized under the effect of different external isotropic pressures and the corresponding enthalpies were determined. As shown in Figure S7 of the SM, the X-ray diffraction patterns of both structures at zero pressure are remarkably similar. The positions of the main reflections in the X-ray diffraction patterns of both structures are reported in Tables S2 and S29, respectively. The computed unit cell volumes and enthalpies are compared in Figure 17. In this figure the volumes and enthalpies of the P6 3 cm structure have been doubled since the unit-cell of the C1m1 structure is twice as large as the hexagonal unit cell. As can be seen, while the enthalpies of both structures are very close at zero pressure, the monoclinic structure is increasingly more stable as pressure increases. The difference of the enthalpies of both polymorphs, 0.3 kJ per formula unit at zero pressure, becomes 28.6 kJ at P = 5.0 GPa. Therefore, the VPI-5 monoclinic polymorph appears to be significantly more stable than the hexagonal one at high pressure conditions. The transition pressure between these structures is conditioned by thermodynamic and kinetic considerations [343]. In the initial studies concerning the structures of ALPO materials [344], great effort was devoted to the identification of the symmetry of their structures. Present results show that the difficulty in the identification, is further complicated by pressure polymorphism. At the same time, the results point to a form of obtaining monoclinic VPI-5 by submitting hexagonal VPI-5 to high isotropic pressures. As was shown in Section 3.8, the hexagonal-monoclinic polymorphic transformation reduces the incompressibility and ductility of VPI-5 substantially. It should be noted that an additional monoclinic C1m1 structure for VPI-5 under pressure was also recently obtained by Fabbiani et al. [345] with a volume four times that of the hexagonal structure and two times that of the monoclinic structure reported by De Oñate Martinez et al. [5]. As shown in Table S30, the X-ray diffraction pattern derived from the structure of De Oñate Martinez et al. [5] is nearly the same as that derived from this structure, the difference in the positions of the main reflections being lower than 0.1 • . This structure was obtained under the effect of isotropic pressure, in agreement with the present results favoring the monoclinic structures under pressure. fraction pattern derived from the structure of De Oñate Martinez et al. [5] is ne same as that derived from this structure, the difference in the positions of the mai tions being lower than 0.1°. This structure was obtained under the effect of isotrop sure, in agreement with the present results favoring the monoclinic structures und sure.

ALPO-5
For ALPO-5, the relative thermodynamic stability of the cc2 [8] and structures under pressure was investigated. Figure S8 of the SM shows the great si of X-ray diffraction patterns of both structures at zero pressure. The positions of t reflections in the X-ray diffraction patterns of both structures are given in Tables S31, respectively. The unit cell volumes and enthalpies associated with both str under the effect different external pressure are compared in Figure 18. The volum enthalpies of the 6 structure were doubled in Figure 18 (the orthorhombic un twice as large as the hexagonal one). Both structures are nearly degenerate at ze sure. However, the orthorhombic structure is increasingly more stable as the pres creases. The enthalpy difference becomes 12.1 kJ per formula unit (AlPO 4 ) at P = The hexagonal-orthorhombic polymorphism in the anhydrous and hydrated f ALPO-5 is a long-standing problem [8,9,[346][347][348][349][350][351][352][353][354][355][356]. The presence of one or anoth morph is not only dependent on the temperature but also on the method used synthesis of this compound [356]. The results obtained here show that the orthor

ALPO-5
For ALPO-5, the relative thermodynamic stability of the Pcc2 [8] and P6cc [9] structures under pressure was investigated. Figure S8 of the SM shows the great similarity of Xray diffraction patterns of both structures at zero pressure. The positions of the main reflections in the X-ray diffraction patterns of both structures are given in Tables S4 and  S31, respectively. The unit cell volumes and enthalpies associated with both structures under the effect different external pressure are compared in Figure 18. The volumes and enthalpies of the P6cc structure were doubled in Figure 18 (the orthorhombic unit cell is twice as large as the hexagonal one). Both structures are nearly degenerate at zero pressure. However, the orthorhombic structure is increasingly more stable as the pressure increases. The enthalpy difference becomes 12.1 kJ per formula unit (AlPO 4 ) at P = 5.0 GPa. The hexagonal-orthorhombic polymorphism in the anhydrous and hydrated forms of ALPO-5 is a long-standing problem [8,9,[346][347][348][349][350][351][352][353][354][355][356]. The presence of one or another polymorph is not only dependent on the temperature but also on the method used for the synthesis of this compound [356]. The results obtained here show that the orthorhombic structure is the high-pressure polymorph. The orthorhombic structure should be obtainable submitting the hexagonal polymorph to high pressure. Similarly, the synthesis of ALPO-5 at sufficiently high pressure should favor the production of the orthorhombic form, independently of the synthetic method employed. structure is the high-pressure polymorph. The orthorhombic structure should be obtainable submitting the hexagonal polymorph to high pressure. Similarly, the synthesis of ALPO-5 at sufficiently high pressure should favor the production of the orthorhombic form, independently of the synthetic method employed. The computed lattice parameters of the cc2 structure [8] at zero pressure are given in Table S26 of the SM. The calculated stiffness tensor and mechanical properties are reported in Tables 5 and 6, respectively. The dependence of its mechanical properties on the direction of the applied strain is shown in Figure S9. In contrast with the result obtained for VPI-5 in previous Section, the variation of the elastic properties for the orthorhombic structure with respect to those of the hexagonal one is exceedingly small. For example, the bulk modulus for the hexagonal polymorph, = 88.2 GPa, becomes 88.6 GPa. The intrinsic ductility index, = 0.51, is unchanged. The directional dependence of the elastic properties for the two polymorphs is quite similar. The axial symmetry around the z axis for the hexagonal structure is slightly distorted in the orthorhombic polymorph ( Figure  S9B). Consequently, the impact of pressure polymorphism in the mechanical properties of ALPO-5 at zero pressure is very small. The computed unit-cell volumes, lattice parameters, and compressibilities of orthorhombic ALPO-5 as a function of the applied isotropic pressure are displayed in Figure S10 and given in Tables S32 and S33 of the SM. Although the influence of the pressure polymorphism in the elastic properties of ALPO-5 is small, The computed lattice parameters of the Pcc2 structure [8] at zero pressure are given in Table S26 of the SM. The calculated stiffness tensor and mechanical properties are reported in Tables 5 and 6, respectively. The dependence of its mechanical properties on the direction of the applied strain is shown in Figure S9. In contrast with the result obtained for VPI-5 in previous Section, the variation of the elastic properties for the orthorhombic structure with respect to those of the hexagonal one is exceedingly small. For example, the bulk modulus for the hexagonal polymorph, B = 88.2 GPa, becomes 88.6 GPa. The intrinsic ductility index, D I = 0.51, is unchanged. The directional dependence of the elastic properties for the two polymorphs is quite similar. The axial symmetry around the z axis for the hexagonal structure is slightly distorted in the orthorhombic polymorph ( Figure S9B). Consequently, the impact of pressure polymorphism in the mechanical properties of ALPO-5 at zero pressure is very small. The computed unit-cell volumes, lattice parameters, and compressibilities of orthorhombic ALPO-5 as a function of the applied isotropic pressure are displayed in Figure S10 and given in Tables S32 and S33 of the SM. Although the influence of the pressure polymorphism in the elastic properties of ALPO-5 is small, the linear compressibilities along a and c directions and the volumetric compressibility of the orthorhombic polymorph are strongly affected by the increase of pressure. Only the linear compressibility b direction remains nearly constant under pressure with a value close to k b = 5 TPa −1 .

Comparison with Experimental Data
There are very few data to compare the results of the present paper with experimental data [165][166][167][168][169][170][171][172]. They are mostly for hydrated ALPO materials and limited to compressibility values measured using the DAC technique using a given PTM. There appears to exist a large difference between the experimental values of the compressibility measured using this technique and the theoretical results for empty porous structures. The same is true for the experimental results obtained with and without a PTM [150], using two different PTMs or from two different studies using the same PTM [166][167][168]. For different PTMs, involving different molecules, the collisions of molecules with the surfaces of the material considered produce different effects. In many cases, pressure induced transitions and pressure induced amorphizations (PIA) appear at very different pressures for different PTMs [166][167][168]. For instance, for ALPO-5W [166], a PIA was observed at 15.9 GPa using liquid nitrogen as PTM and at 8.5 GPa using silicone oil (non-pore-penetrating PTM). For dense crystal structures accurate values of the compressibilities are generally obtained [314]. The measurements performed for some ALPO materials indicate large compressibilities. For example, for a crystalline sample of dehydrated VPI-5, Alabarse et al. [169] using a DAC with silicone oil observed a pressure induced phase transition to ALPO-8 beginning at 0.8 GPa, which do not appear from the theoretical calculations, obtained a compressibility of k = 80.6 TPa −1 (bulk modulus B = 12.4(2) GPa) for VPI-5 from a fit to a to a second order Birch−Murnaghan (2-BM) equation of state (EOS) from pressure−volume data below 1.6 GPa and the same value for ALPO-8 from data below 3.4 GPa. For dehydrated ALPO-17, Alabarse et al. [171] using a DAC with silicone oil observed a pressure induced amorphization beginning at 1 GPa and obtained a compressibility of k = 32 TPa −1 (bulk modulus B = 31.2(5) GPa) from a fit to a to a 3-BM EOS from pressure−volume data below 3.1 GPa. The results obtained in this work showed that the compressibilities obtained from fits to a BM EOS are highly dependent on the pressure range used in the fits and that many pressure-volume data points should be used to obtain reliable compressibilities. Furthermore, the emergence of pressure induced amorphizations should influence substantially in the measured compressibilities due to the reduction of the volume involved in the pore collapse. NLC phenomena were also observed for ALPO materials in previous works. However, the NLC effects observed were much less significant than that found for ALPO-18 in this work. For ALPO-17, Alabarse et al. [171], found a small increase of the a lattice parameter near the PIA associated to the pore collapse. Likewise, for ALPO-5W, Kim et al. [166] found that a lattice expands at small pressures before it starts to contract using a MEW mixture as a PTM (probably due to molecule pore intrusion).
The results obtained using theoretical techniques are unique. The underlying reason for the different theoretical and experimental results based in DAC must be the different origin of pressure in these experimental techniques (collisional mechanism) and in the theoretical calculations. In the theoretical calculations, the pressure is defined in terms of the stress tensor and the elastic properties are determined from the stress tensor resulting from an applied strain and the action of molecules over the material is not invoked. The data presented in this paper are rigorous quantum mechanical results and were fully revised and carried out twice to check their reproducibility. Furthermore, the compressibilities obtained from the computed elastic tensors and fits of computed pressure-volume data are in excellent agreement. Thus, it is concluded that the data obtained from the theoretical calculations and experimental measurements using DAC correspond to different physical quantities for highly porous materials. To obtain a better agreement between theoretical and experimental results, either experimental measurements not based in DAC or quantum molecular dynamic calculations with specific solvents simulating the effect of the different PTMs should be carried out. The effect of temperature should also be investigated since the present theoretical results correspond to zero temperature.

Conclusions
In this work, the crystal structures, mechanical properties, and compressibility functions of five important anhydrous microporous aluminophosphate materials have been determined using first principles methods based on density functional theory. The calculated crystal structures and associated X-ray diffraction patterns are in good agreement with their experimental counterparts. The elastic tensors of all of these materials have been reported and the mechanical stability of their structures has been confirmed. A detailed mechanical characterization is performed, and a rich set of mechanical properties was derived. This set includes the bulk, shear and Young's moduli, as well as ductility, hardness, and elastic anisotropy indices. The elastic behavior of the five materials shares many common mechanical properties such as high incompressibility, ductility, and low elastic anisotropy. Their intrinsic ductility indices are in the same range as that for common metals. The analysis of the dependence of the mechanical properties of ALPOs in the orientation of the applied strain, show that they are resistant with respect to the application of external isotropic and uniaxial pressures and shear stresses. A smooth directional dependence is found in all of the cases and no special directions for material fracture or shear instability are encountered. The only previous study from which some clue about the incompressibility of ALPO materials was found is the work by Polisi et al. [210] where the dehydration mechanism of hydrated ALPO-5 was studied. The small volume change of this material under the effect of temperature led these authors to state that ALPO-5 was one of the most rigid zeolite frameworks. While this finding concerns only the thermal behavior of hydrated ALPO-5, the present results show that high incompressibility is a general property of anhydrous ALPO materials under the effect of pressure.
The crystal structures of all of the materials were completely optimized under the effect of different isotropic pressures and the linear and volumetric compressibilities were determined. At zero pressure, the ALPO materials have small linear compressibilities along the three crystallographic directions. The tridimensional incompressibility of ALPO-5, ALPO-18 and ALPO-31 is notable since the compressibilities along the three principal directions are lower or close to 5 TPa −1 . The incompressibility ALPO-8 and ALPO-31 materials is lost at high pressures. ALPO-18 displays an extremely anomalous mechanical behavior at relatively low external pressures. It exhibits a large negative linear compressibility effect between P = 1.21 and P = 2.70 GPa. The minimum value of the compressibility along [1 0 0] direction, k a = −30.9 TPa −1 , is encountered at P = 2.04 GPa. The NLC phenomenon in this material can be rationalized using the empty channel structural mechanism. The width and height of main 8-MR channels expanding along [0 0 1] increase and decrease substantially under increasing pressure. The widening of the channels along [1 0 0], coinciding with the direction of minimum compressibility in ALPO-18, leads to the increase of the a lattice parameter and to the NLC effect in this material. Furthermore, ALPO-18 exhibits the anisotropic negative volumetric compressibility effect for uniaxial pressures applied along the [1 0 0] direction.
The effect of water molecule adsorption in the channels of ALPO-18 in its elastic properties is assessed by studying the mechanical behavior of the hydrated ALPO-18 material (ALPO-18W). ALPO-18W is much more compressible and less ductile than ALPO-18 and does not present the NLC effect found in ALPO-18. The effect of aging and pressure polymorphism in the mechanical properties of VPI-5 and ALPO-5 is also studied. As hydration, aging and pressure polymorphism leads to significant variations in the elastic properties of VPI-5 and reduces its incompressibility and ductility. For ALPO-5, pressure polymorphism has only a small impact in its elasticity at zero pressure but a large influence at high pressure.