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How the H‑Bond Layout Determines Mechanical Properties of Crystalline Amino Acid Hydrogen Maleates Yury V. Matveychuk,*,† Ekaterina V. Bartashevich,† and Vladimir G. Tsirelson†,‡ †

Research and Education Center “Nanotechnology”, South Ural State University, 76, Lenin av., Chelyabinsk, 454080, Russia D.I. Mendeleev University of Chemical Technology, Miusskaya Sq. 9, Moscow, 125047, Russia

‡

S Supporting Information *

ABSTRACT: The stiffness tensor and elastic anisotropy characteristics for the crystalline hydrogen maleates of Lisoleucinium, L-leucinium, and L-norvalinium with L-norvaline have been calculated using the periodic DFT calculations and atom-centered basis sets. The H-bond orientations have been compared with spatial directions of the minimum and maximum values of Young’s modulus, shear modulus, and linear compressibility. In spite of the similar layered structures, L-isoleucinium and L-leucinium hydrogen maleates show significant difference in elastic moduli anisotropy. The flexibility of L-leucinium hydrogen maleate is explained by the relatively high universal elastic anisotropy index and the large anisotropy ratios of elastic moduli. In its turn, this index is determined by the almost coincidental Young’s modulus maximum direction and the orientation of the strongest H-bonds.

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Euclidean anisotropy measure, AL,9 and others. Further, we use the universal elastic anisotropy index, AU10

INTRODUCTION

The practical usage of molecular crystals is based on the combination of their useful properties and other essential characteristics, such as synthesis processability, low defectiveness, good crystal habit, thermodynamic and chemical stability in the required temperature and pressure ranges, and, last but not least, necessary mechanical characteristics, for example, flexibility and lack of brittleness. Elastic constants are the major characteristics describing the behavior of a crystal under mechanical impact.1 These constants show what the material is brittle, ductile, or flexible, and how these properties depend on the applied stress. Within validity of Hooke’s law, there are two types of the fourth rank tensors that are transformed one into another by inversion:1 the stiffness tensor (elastic constants of the second order) and the compliance tensor (elastic moduli of the second order). The algorithm of the stiffness tensor calculation suggested by Perger et al.2 uses the dependence “stress-strain”, which is derived from the total energy of a crystal. Then the values of bulk (polycrystalline) elastic Young’s modulus, E, Poisson’s ratio, ν, shear modulus, G, and bulk modulus, K, might be determined. The approximations described by Voigt, Reuss, and Hill3 are used; therefore, the specified bulk moduli differ from the elastic moduli as compliance tensor components. The stiffness tensor allows us to find the criteria of mechanical stability,4 the velocities of acoustic waves in crystals (via the Christoffel equation),5 the spatial anisotropy of elastic moduli,6 and various indices of anisotropy based on different approaches: the Zener’s anisotropy factor, A,7 the general elastic anisotropy measure, A*,8 the last-proposed log© XXXX American Chemical Society

AU = 5

GV KV + −6 GR KR

where GV, GR are shear modulus polycrystalline approximations by Voigt and Reuss; KV, KR are bulk modulus polycrystalline approximations by Voigt and Reuss.3 This index quantifies the directional variability of the shear and bulk moduli. In recent years, many experimental and theoretical works aimed to establish the relationship between features of crystal packing, such as mutual orientation of molecular chains and arrangement of molecular layers, and spatial anisotropy of crystalline mechanical characteristics associated with the stiffness tensor.11−23 In particular, Desiraju and co-workers11−14 have considered the qualitative relationships of the substituent effect in some series of crystals (halogen- and nitro-substituted benzene, phenol, benzoic acid, pyridine, and bipyridine), as well as of the features of intermolecular interactions formed by a substituent, on the crystal packing and the manifestation of ductile and flexible behavior. They have concluded that the packing of molecules into layers due to relatively strong Hbonds, and stacks orthogonally linked only by weak van der Waals interactions, is the reason for easy crystal bending. Examples are 3,4-dichlorophenol, 2-nitrobenzaldehyde, hexachlorobenzene crystals, etc. To the contrary, the crystals with intermolecular interactions of comparable force in three Received: January 12, 2018 Revised: April 4, 2018 Published: April 18, 2018 A

DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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mechanical characteristics of newly synthesized crystals at simultaneous monitoring of their nonlinear optical properties. The experimental data about the stiffness tensor components and elastic moduli are limited and fragmented, even for such a well-studied crystal as L-alanine. The experimental microhardness data for different crystal faces have been given,29 the components of the stiffness tensor have been measured,21 and the velocities of longitudinal and transverse acoustic waves have also been measured.30 The Young’s modulus measurements on some crystal faces have been given,31 while the measurement by the microhardness method and the calculation of the c11 component by Wooster’s empirical equation have been provided.32 The latter results seem to be somewhat overestimated, though comparable with other data for β-alanine.33 The comparison of the experimental and calculated values of Young’s modulus on certain crystal faces and the analysis of spatial anisotropy of Young’s modulus from the calculated stiffness tensor have been done.20 Recently, modeling of solid mechanical characteristics has come to the new level: we can predict the new mechanical properties of new artificial materials, such as the overall twist of metamaterial under the uniaxial stress.34 So, revealing the relationship between the anisotropy of elastic moduli and such crystal mechanical characteristics as brittleness, plasticity, and elasticity demands the complex consideration of the stiffness tensor and structural crystal features: certain orientations of the covalent bonds and rather strong structure-forming noncovalent interactions with regard to the crystallographic and morphological axes of a crystal, as well as chains and motifs of the molecules forming this crystal. In addition, we consider that it is important to combine the knowledge of the influence of noncovalent interactions on mechanical characteristics of solid compounds with the role of these interactions in biological processes with the participation of these substances. This comparison is especially important at changing external conditions, in particular, environment hydrostatic pressure.35 The modern quantum chemical methods and software (VASP,36 CASTEP,37 CRYSTAL14,5 etc.) allow us to get the components of the stiffness tensor and the related mechanical characteristics by calculation, without the necessity for synthesis of large perfect crystals of all studied compounds and acoustic experiments. From this point of view, the following crystalline amino acids and their salts have been considered in the present study: L-isoleucinium hydrogen maleate hemihydrate VUKQEZ 1, space group P21 at 298 K,38 L-leucinium hydrogen maleate VUKQAV 2, space group C2 at 298 K,38 co-crystal Lnorvalinium hydrogen maleate with L-norvaline VUKQID 3, space group P21 at 298 K,38 and L-alanine LALNIN 4, space group P212121 at 283−303 K.39 The latter crystal has been considered as the reference crystal for the calculation technique validation. We focused on the following objectives in this study: • Theoretical estimation of the stiffness tensor components and the spatial anisotropy of Young’s and shear moduli and linear compressibility for the specified crystals. • Analysis of the quantitative relationship of the anisotropy of elastic constants and the mechanical (elastic) characteristics of the crystals marked in the experiment earlier. • Calculation of the minimum and maximum values of Young’s and shear moduli and linear compressibility in order to relate them to the spatial orientation of

orthogonal directions, as naphthalene, 4-hydroxybenzoic acid, pentafluorobenzamide, etc., are rigid or brittle, as a rule. The character of intermolecular interactions does not play an essential role; only their isotropic orientation is important.11 Also, on the same crystals example, the other approach to the analysis of interaction anisotropy and its influence on mechanical properties, namely, the “energy frameworks” method, has been considered.15 The accumulation of experimental data and their generalization enable us to develop the crystal design toward the improvement of mechanical characteristics of potentially useful crystalline materials. In particular, to increase the spatial anisotropy of elastic moduli of the brittle 4,4-dipyridine and salicylic acid co-crystal, the authors13 have introduced halogens into a molecule for the formation of the one-dimensional or two-dimensional network of halogen bonds. It has allowed them to get more ductile crystals that are steady enough under external impacts. The qualitative relationship of a structure and special mechanical characteristics of the molecular crystals depending on the direction of external mechanical impact has also been noted by Rupasinghe et al.16 Zolotarev et al.17 have described the behavior of the L-valine and D-methionine cocrystal; it is similar to the behavior of a pack of paper sheets at application of the bending stress. In these co-crystals, the molecules of amino acids, which are hydrogen-bonded with each other, form layers connected only by weak van der Waals forces. Such layers are easily shifted in relation to each other at the external bending stress. Unfortunately, the studies of the quantitative relationships between the stiffness tensor, calculated or experimental, and mechanical characteristics of molecular crystals are rather rare now. There are a few researches on studying the crystal properties of urea and pentaerythritol,18,19 amino acids,20,21 and metal−organic frameworks.6,22,23 These studies paid attention to the spatial anisotropy of elastic Young’s and shear moduli depending on the mainly linear and two-dimensional arrangement of H-bonds or other specific interactions. The strong anisotropy of elastic moduli indicates either mechanical fragility or elasticity of crystals in certain directions. Thus, on the one hand, the layers of the covalent-bonded carbon atoms in crystalline graphite are connected with each other by weak van der Waals interactions.24 Such structure has high anisotropy of elastic constants: Young’s modulus shows a value of 1092 GPa in the plane of layers, whereas it is equal to 39 GPa in the orthogonal direction to layers. On the other hand, it has been shown for MOF MIL-53 that the high anisotropy of Young’s modulus is the sign of crystal flexibility.6 Similarly, the high anisotropy of the shear modulus of a layered crystal structure suggests the possibility of the molecular layer shift in relation to each other and the elasticity of a crystal. The analysis of eigenvalues of the stiffness tensor can also confirm such property. The authors6 have offered the criterion of existence of “soft deformation modes”, meaning that, among eigenvalues of the stiffness tensor of such crystals, there are the values small enough. The consideration of the crystals of amino acids and their salts in this aspect is also interesting because of their nonlinear optical properties.25−28 At that, an important role is acquired by the special mechanical characteristics of these crystals: the brittleness or elasticity, the possibility of a split in a certain direction, the stiffness anisotropy, etc. Knowledge of the relationship between features of a crystal structure and the stiffness tensor leads to the unique opportunity to operate the B

DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Table 1. Comparison of Calculated Data of the Elastic Moduli for Crystals 1−3 Young’s modulus, GPa Emin the modulus value

modulus anisotropy ratio

direction of minimum and maximum value (in Cartesian coordinates)a

Emax

6.23 18.15 4.26 21.64 6.60 21.98 2.91 5.08 3.33

1 2 3 1 2 3 1

linear compressibility, TPa−1

shear modulus, GPa

Poisson’s ratio

βmin

βmax

Gmin

νmin

11.08 1.34 16.02

59.53 72.24 47.27

1.96 1.24 1.96

5.37 53.77 2.95

6.36 7.33 8.18 3.24 5.92 4.17

0.67 0.66 −0.34

0.48 0 0.88

−0.46 0 −0.89

−0.89 0 0.46

−0.73 0.68 −0.01

0.11 0.55 0.83

0.29 0 −0.96

−0.96 0 −0.29

−0.65 0.69 −0.32

0 1 0

0.67 0 −0.74

0.74 0 0.67

2

3

a

Gmax

−0.88 0 0.47 0 −1 0 −0.99 0 0.11 0 −1 0 0.92 0 0.39 0 1 0

−0.37 0.71 −0.60 −0.37 −0.71 −0.60 −0.86 0 −0.52 0.52 0 −0.86 −0.33 0.71 0.62 −0.33 −0.71 0.62

0.14 −0.04 0.06

νmax 0.61 0.93 0.73

4.47 ∞ 13.15 −0.55 0.46 0.68 0.76 0.48 −0.45 −0.52 −0.70 0.17 0.63 −0.84 0.34 0.31 0.44 0.60 0.47 0.74 −0.77 −0.63 −0.82 −0.45 0.56 0.63 −0.12 −0.99 0.91 0 0 0.12 −0.42 0 −0.42 −1 0 0 −0.91

In this table for G and ν: the direction of the applied stress is in the top line; the direction of measurement of the modulus is in the bottom line.50

Figure 1. 3D spatial dependences of Young’s moduli for crystals: (a) 1, (b) 2, (c) 3. Values on Cartesian axes are in GPa.

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Towler.46 The experimental crystallographic data38,39 and the data from the Cambridge Structural Database47 have been taken as initial structural data for our calculations. The structure optimization (relaxation) has been performed for atomic coordinates only; both the volume of the unit cell and the crystal density have always equaled the experimental. The following convergence parameters have been used for all calculations: TOLDEG (root-mean-square on gradient) is less than 3 × 10−5 a.u., TOLDEX (root-mean-square on estimated displacements) is less than 5 × 10−5 a.u., TOLDEE (energy change between optimization steps threshold) is less than 10−12 a.u., TOLINTEG (truncation criteria for bielectronic integrals) is 10 10 10 10 16. The factor SHRINK, which determines the number of k-

molecular fragments and intermolecular noncovalent structure-forming interactions (H-bonds).

EXPERIMENTAL SECTION

The Kohn−Sham calculations40 with periodic boundary conditions have been carried out for all structures using a software package CRYSTAL14.5 We have used the B3LYP functional,41,42 which provides significantly precise evaluation of elastic properties of various molecular organic crystals,22 and the full-electron basis set 631G(d,p),43 offered for periodic calculations on the basis of the previously published basis set for molecular calculations.44,45 The basis set has been corrected for using in the CRYSTAL14 according to C

DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Figure 2. 3D spatial dependences of shear modulus for crystals: (a) 1, (b) 2, (c) 3. Values on Cartesian axes are in GPa. The surface of the minimum shear modulus value is shown by green color; the surface of the maximum shear modulus value is indicated by transparent blue color.

Figure 3. Calculated directions of the maximum Emax and minimum Emin Young’s modulus values for crystals: (a) 1, (b) 2, (c) 3. Hereinafter, a, b, and c are crystallographic axes, and the arrows Emax and Emin are specified irrespectively of the modulus values. points in reciprocal space in the Pack−Monkhorst scheme,48 at which the KS matrix is diagonalized, has been set to 8. For the relaxed structures, the components of the stiffness tensor have been calculated according to Perger et al. (Table 1S).2 For the rational use of supercomputer resources, the convergence parameters, more weak but sufficiently accurate, have been applied in these calculations: TOLDEG < 5 × 10−5 a.u., TOLDEX < 1 × 10−4 a.u., TOLDEE < 10−10 a.u., TOLINTEG is 8 8 8 8 16. The values of bulk moduli according to the polycrystalline averaging by Voigt, Reuss, and Hill3 (Table 5S) and acoustic wave velocities in the considered compounds have been obtained at the same time. The universal elastic anisotropy indices AU have also been calculated according to Ranganathan et al. (Table 6S).10

Further, the ELATE online tool,49,50 created on the basis of ElAM,51 has been used. This tool provides the following output: all the values of bulk moduli, the tensor eigenvalues, the spatial dependences of elastic moduli, the minimum and maximum values of Young’s and shear moduli, linear compressibility and Poisson’s ratio, as well as their Cartesian directions. Mechanical stability of the relaxed structures has been estimated from the eigenvalues of the stiffness tensor. All the considered structures have shown positive eigenvalues of their tensors and corresponded to the generalized stability criterion.4 The preliminary validation of the above-mentioned methods was carried out for the L-alanine crystal (4), and these calculations are performed in the Supporting Information. The results allowed us to consider the chosen calculation technique as suitable for both the stiffness tensor components and the elastic moduli measured in certain D

DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Figure 4. Distribution of angles α between the H-bonds, ranged in length (Å), and the calculated directions of minimal Emin and maximal Emax resistance to external uniaxial stress, for crystals: (a) 1, (b) 3, (c) 2. The numbers of H-bonds correspond to Tables 7S−9S.

we rely only on the AU and ratios, then we can expect that crystal 3 will show a certain elasticity or plasticity. Anisotropy of Young’s Modulus. The bonding of molecular fragments into 2D layers by means of relatively strong H-bonds and formation of the 3D structure via more weak van der Waals interactions between layers actually specify the mechanical characteristics of many molecular crystals.14,17 Let us consider how differences in mechanical characteristics of the 1, 2, and 3 crystals are determined by the H-bonds orientation in relation to the calculated directions of the minimum (Emin) and maximum (Emax) values of Young’s modulus. According to the origin paper,38 crystals 1, 2, and 3 consist of the layers formed by the H-bonded chains of hydrogen maleate anions with amino acids cations. At first, let us discuss how the layered organization of the considered crystals correlates with the directions of the Emin and Emax values of Young’s modulus. Undoubtedly, the common feature for all three crystals is the orientation of directions of the maximum values of Young’s modulus, Emax, almost parallel to layers (Figure 3). The layer structure is caused, among other things, by rigidity of the planar hydrogen maleate anion having the strong intramolecular Hbond (∼1.33 Å). The quantitative estimation shows that the Young’s modulus values along the directions parallel to layers are greater than in all other directionsabout 20 GPafor all three crystals. The minimum values of Young’s modulus, Emin, for all considered crystals are oriented at an angle to the planes of molecular layers and have low values in the range 4.00−6.00 GPa. It is assumed that the shift of molecular fragments under

directions. The sufficiency of such conditions for the absence of imaginary frequencies has also been checked. Visual representation of the fragments of crystal structures and the selected directions in crystals has been performed by means of the software package ChemCraft.52 The present study has utilized the supercomputer resources.53

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RESULTS AND DISCUSSION

Estimation of Elastic Anisotropy. The values of the universal elastic anisotropy index AU obtained from the calculated bulk moduli equal 1.38 for crystal 1, 3.76 for crystal 2, and 2.32 for crystal 3. Comparing the spatial dependencies of elastic moduli of 1 and 2 (Table 1 and Figures 1 and 2, Figures 1S−3S), we see that the anisotropy ratios for crystal 2 exceed those for crystal 1 by 2−3 times. Besides, the maximum values of Young’s and shear moduli for crystal 2 are higher, and minimum values are lower, than for crystal 1. These results verify the larger flexibility of crystal 2, in agreement with experiment:38 the needles of 2 show the elastic bending first, then the plastic strain at increasing of the applied stress. Unlike them, the needles of 1 are extremely brittle and fragile. Thus, the calculated bulk elastic moduli of the considered compounds correspond to the observed mechanical characteristics of these crystals. The universal elastic anisotropy index, AU, and spatial anisotropy ratios of elastic moduli for compound 3 show intermediate values between compounds 1 and 2 (Table 1 and Table 6S). Values of linear compressibility are an exception. If E

DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Figure 5. Strongest H-bonds forming the structural channel in crystal 3: (a) along the Emax direction, (b) across the Emin direction.

found. The most of the weak H-bonds with the length 1.70− 2.00 Å in crystal 2 are oriented at large angles to both the Emax and Emin directions. The orientation of all medium H-bonds with the length up to 1.80 Å is practically parallel to a molecular layer. Such arrangement of H-bonds is in good agreement with the larger Emax value for crystal 2 and with the rather large angle between a molecular layer and the Emin direction in this crystal (Figure 3b). In crystal 1, the less anisotropic spatial orientation of the strong and medium H-bonds is observed. This is reflected in the larger Emin value and the smaller Emax value in relation to the corresponding values for crystal 2 (Table 1). The Emax direction along the structural channel, which is formed by the rings of L-norvaline zwitterions and Lnorvalinium cations (Figure 3c), mentioned above for crystal 3, is a feature of this crystal only. We associate it with the existence of a short H-bond (1.29 Å) between carboxyl groups of L-norvaline and L-norvalinium. The “zigzag” created by these bonds is oriented along the structural channel (Figure 5). The angles between such H-bonds and the direction of the maximal resistance to stress are equal in magnitude, but opposite in signs; i.e., they “extinguish” each other. We consider the relative arrangement of such bonds to be the main reason for the largest value of Emax (Table 1) from all three crystals, which is also true for the largest value of bulk Young’s modulus EH (Table 5S). This direction is also parallel to a layer in general. The minimum value of Young’s modulus for crystal 3, Emin, is also the largest from all crystals (Table 1). In general, we can claim that the influence of the relative arrangement of H-bonds on Young’s modulus can be manifested in macroscopic mechanical characteristics. Thus, the smaller value of Young’s modulus in the direction, orthogonal to layers, means the larger “softness” and, respectively, the high probability of elastic properties of Lleucinium hydrogen maleate crystal. On the contrary, the participation of water molecules in several adjacent H-bonds, “pinning” molecular layers, can be shown either in increasing brittleness or in increasing flexibility of crystals. However, the more uniform distribution of the H-bonds relative orientations, which is shown in smaller spatial variability of Young’s modulus values, can have an opposite effect: increasing crystal mechanical durability in relation to external influences and decreasing probability of grown crystal splitting. For more

external stress is hindered within the layers and is facilitated at an angle to them. In the last case, mainly weak van der Waals interactions between amino acid alkyl fragments are most probable. However, the following significant differences are revealed by detailing of the Emin and Emax orientation features in three considered crystals. For crystal 1, the maximal resistance to uniaxial stress Emax is shown along the chains consisting of Lisoleucinium cations. For crystal 2, the direction of the maximal resistance to stress Emax is at an angle to all chains forming a layer. For crystal 3, such direction is oriented along the structural channel created by L-norvaline zwitterions and Lnorvalinium cations. Second, let us analyze how H-bonds in the three crystals are oriented in relation to the layers and directions, in which the maximum and minimum Young’s modulus values, Emin and Emax, have been determined. At the same time, we do not consider the intramolecular H-bonds in hydrogen maleate anion, as its conformational flexibility is limited. We do not additionally accentuate the bond kinds defined by the H-bond donor and acceptor atoms. In the unit cell of 1 with 142 atoms, there are 14 intermolecular H-bonds, from which nine are oriented at a larger angle (>20°) to the plane of a layer (Table 7S). It is necessary to emphasize that the majority of such bonds is represented by the bonds with average length 1.50−1.80 Å. The hydrate water participates in forming seven such bonds. In crystal 2, the larger quantity of H-bonds is located parallel to the molecular layer plane. In the unit cell with 204 atoms, there are 40 H-bonds, from which only 12 are oriented at a larger angle to the plane of a layer (Table 8S). The bonds thus located are rather weak; their length is in the range 1.80−2.00 Å. That is, the role of H-bonds in the resistance to external stress, applied orthogonally to a layer, has to be much less than in crystal 1. In the unit cell of crystal 3, consisting of 100 atoms, only 4 of 10 H-bonds are oriented at a larger angle to the plane of a layer (Table 9S). These, rather weak, bonds are formed by amino groups. In crystal 1, the most of the H-bonds up to the length 1.80 Å form rather large angles with the Emax direction. The mainly weak H-bonds with the length 1.70−2.00 Å are oriented along the Emax direction. The same trend is observed for crystal 3 (Figure 4). However, in crystal 2, another picture has been F

DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Figure 6. Numerical distribution of the angles (deg) between the H-bonds and the calculated directions of the minimum βmin and maximum βmax linear compressibility, for crystals: (a) 1, (b) 2, (c) 3.

Figure 7. Calculated directions of measurement of the maximum shear modulus value Gm and of the corresponding applied shear stress Ga for crystals: (a) 1, (b) 2, (c) 3.

anisotropy ratio of this module is here the relation of the maximum value to minimum value; it increases in the crystal series 3 < 1 ≪ 2 (Table 1). At the same time, the angle between the βmin direction and molecular layer increases from ∼0° for crystal 1 to ∼30° for crystal 2 (Figure 5S). Comparing the orientation of H-bonds in crystals with the βmin and βmax directions, we can note the following. The minimum anisotropy of resistance to hydrostatic compression is shown for crystal 3 (Table 1). We see (Figure 6c) that the most H-bonds are oriented in relation to both directions βmin and βmax at larger angles (>60°). Any dependence of these angles on

precise prediction, it is necessary to consider other elastic moduli for the refining of mechanical behavior of the crystal. Anisotropy of Linear Compressibility. Let us compare the spatial dependences of linear compressibility β for the three considered hydrogen maleate crystals of amino acids, which describes the crystal behavior at isotropic hydrostatic compression (Figures 1S−3S and Table 1). The minimum values βmin of this modulus for all crystals have been found in the direction of the same molecular chain. However, the direction of the maximum value of linear compressibility βmax is orthogonal to a molecular layer only for crystal 1. The G

DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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arrangement in relation to the plane Gmax‑m−Gmax‑a is important, too. This plane is determined by crossing two directions: the direction of measurement of shear modulus value, Gmax‑m, and the direction of corresponding applied shear stress, Gmax‑a. Thus, at shear stress, the largest distortion occurs for H-bonds, which are orthogonal to the Gmax‑m direction and oriented at small angles to the above-mentioned plane. Therefore, the bonds, located thus, significantly influence the shear modulus. We have determined that the largest calculated maximum value of shear modulus, Gmax, belongs to crystal 3 (Table 1). Most likely, the strongest H-bond, forming the structural channel, renders the main influence on the value of this modulus (Figure 7c, Table 9S). In crystal 2, the particular influence on the Gmax value of shear modulus should be expected from medium H-bonds (Figure 6S). In crystal 1, we can consider only weak H-bonds affecting the maximum value, Gmax, of shear modulus. The half of these bonds is the H-bonds with participation of hydrate water. The marked differences are shown in small increasing of the maximal value, Gmax, of shear modulus from crystal 1 to crystal 3. Thus, the analysis of the stiffness tensor of the considered crystals and the revealed relations with the arrangement of intermolecular H-bonds shows the high degree of anisotropy and the smaller minimum values of three elastic moduli for crystal 2 in comparison with crystals 1 and 3. This fact, in general, is the consequence of mainly two-dimensional arrangement of H-bonds in molecular layers, and arrangement of weak van der Waals interactions orthogonally to molecular layers. Besides, for crystal 2, the calculations have revealed the minimum value of shear modulus, which is close to the “specific” value, namely, 1.24 GPa. These data well explain the experimentally observed flexibility and elasticity of crystal 2. Despite the intermediate values of the elastic moduli anisotropy in crystal 3, the detailed comparison of the calculated minimum and maximum values of all considered moduli with the features of H-bond arrangement shows that mechanical characteristics of crystal 3 may be expected closer to observed ones for crystal 1.

the type and length of a bond has not been detected. It should be noted that similar distribution of angles with both the direction βmin and direction βmax remains. For crystal 1, the great variability of angles between H-bonds and both the βmin and βmax directions is observed. Compared to other crystals, the largest βmax value is observed for crystal 2 (Table 1). The most H-bonds in this crystal are oriented almost orthogonally to the βmax direction (Figure 6b). At the same time, the angles of Hbonds with the βmin direction do not exceed 60°. From this point of view, the experimentally observed brittleness of crystal 1 and elastic behavior of crystal 2 are well explainable. Thus, on the one hand, the anisotropy of this modulus can indicate mainly elastic or brittle properties, which the crystal shows at hydrostatic compression. On the other hand, because of the isotropic character of external stress, the linear compressibility is the cumulative characteristic of resistance of the crystalline structure as a whole to this stress. The influence of separate, even the strong, H-bonds on this resistance is difficult to reveal. Anisotropy of Shear Modulus. The unusually low (50 GPa) values of Young’s and shear moduli,20,22 and the negative Poisson’s ratio24 can be used as criteria of the special features of a crystal. The range of the elastic moduli values of the considered hydrogen maleate crystals is ordinary. Therefore, all considered crystals under unidirectional mechanical stress, as at measurement of Young’s modulus, would not show noticeable plasticity. The demonstrated anisotropy of Young’s modulus also influences elastic and plastic crystal properties at bending in small degree. Such properties of crystals much stronger depend on shear modulus and its anisotropy, determined by the stiffness tensor components c44, c55, and c66, primarily on the minimum modulus value and its direction. Shear modulus G depends on two orthogonal vectors: the direction of the applied shear stress, Ga, and the direction of measurement of the response shear strain, Gm.50 The calculated directions of measurement of the minimum shear modulus value, Gmin‑m, are almost parallel for all considered crystals, in view of that the sign of this direction is not considered in our analysis. At the same time, the direction of the applied stress, Gmin‑a, must be oriented orthogonally to molecular layers. These directions correspond to the shift of molecular layers one from another, quite expected. The resistance to such shift is rendered by only van der Waals interactions formed by the alkyl groups of amino acids located between layers. Despite the similarity of chains and layers arrangement, the calculated minimum value of shear modulus, Gmin, in crystal 2 significantly differs from those in crystals 1 and 3 (Table 1). It is probably caused mainly by steric factors, though it is rather difficult to estimate their influence on the interaction of layers. The calculated directions of measurement of the maximum shear modulus values that are parallel to layers and at an angle to all chains are quite explainable for crystals 1 and 3 (Figure 7), but their orientation is less obvious and explainable for crystal 2. The shear strain in a molecular layer for crystal 1 follows distortion of the chains, bonded by medium H-bonds, and imminent rupture of some bonds. The larger value of shear modulus for crystal 3 in comparison with crystal 1 is caused by the existence of the channel structure. Considering the influence of the H-bonds arrangement on the directions defining the maximal resistance to shear stress, it is important to note the multifactorial character of such influence. Besides the length of bond and its angle with the direction of measurement of the shear modulus value, the bond

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CONCLUSION In this research, we have studied three hydrogen maleate crystals of amino acids: L-isoleucinium hydrogen maleate hemihydrate, L-leucinium hydrogen maleate, and the co-crystal of L-norvalinium hydrogen maleate with L-norvaline. The mechanical characteristics of these crystals have been quantitatively estimated based on the stiffness tensor, which has been calculated by the Kohn−Sham method with periodic boundary conditions. The directions determining the maximal and minimal resistance to uniaxial and hydrostatic compression and shear stress have been revealed by means of spatial dependences of Young’s modulus, shift modulus, and linear compressibility. The relative orientation of these directions and intermolecular H-bonds has been analyzed. We have found for all considered crystals that the directions of the Young’s modulus maximum values, corresponding to the maximal resistance to uniaxial stress, are oriented parallel to molecular layers: either along chains in which the molecular fragments are connected by H-bonds or at a small angle to them. It has been shown that the anisotropy of Young’s modulus and the corresponding minimum and maximum values are defined by both force of H-bonds and their orientation in the unit cell. The significantly larger range of the Young’s modulus values for the elastic L-leucinium H

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ACKNOWLEDGMENTS This work was supported by the Ministry of Education and Science of the Russian Federation, grant 4.1157.2017/4.6, and by the Government of the Russian Federation, Act 211, contract No. 02.A03.21.0011.

hydrogen maleate crystal corresponds to the apparent anisotropy in orientation of H-bonds, which are mainly oriented parallel to the plane of a molecular layer. The Young’s modulus anisotropy ratios quantitatively reflect the mechanical characteristics of the considered crystals. We have shown that it is difficult to establish the influence of the H-bonds orientation on the linear compressibility of crystals, as hydrostatic pressure has the isotropic character and equally influences the variously oriented bonds. However, the consideration of anisotropy of this elastic modulus is useful for the analysis of macroscopic mechanical characteristics of a crystal in general. We have noted the conformity between the higher anisotropy of linear compressibility and elastic properties of L-leucinium hydrogen maleate observed in experiment. The directions of measurement of the maximum values of shear modulus are oriented along molecular layers in the considered crystals, on condition of the corresponding stress being applied orthogonally to layers. Not H-bonds, but interlayered van der Waals interactions are here the main obstacles to layer shift. The maximum values of the shear modulus increase from L-isoleucinium hydrogen maleate hemihydrate to the co-crystal of L-norvalinium hydrogen maleate with L-norvaline. We consider that, in the latter case, it is caused by influence of the very strong H-bond forming the structural channel. It has been shown that the shear modulus, as well as other elastic moduli, is characterized by the largest anisotropy for the crystal with elastic properties. We have revealed that the ratio of the maximum and minimum values of different elastic moduli must be more than 5 for the elastic properties of crystals to be obvious. The obtained results visually illustrate the fact that the quantitative estimation of elastic anisotropy is useful for the search and design of elastic crystals. The suggested approach is recommended for design of new crystals with the preset properties of elasticity.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.8b00067. The description of validation of the method and level of calculations with periodic boundary conditions, the elastic tensors of considered crystals 1−4 and derivative data: bulk moduli and AU for 1−4, seismic wave velocities and Young’s moduli for 4, projections of the spatial dependences of elastic moduli to the Cartesian planes (XY), (XZ), and (YZ) for 1−4, the spatial data of elastic moduli of 4. In addition, there are the data for Hbonds in the unit cell of 1−3, the calculated directions of the maximum and minimum linear compressibility values for crystals 1−3, the distribution of angles between the ranged H-bonds, and the calculated directions of shear modulus values (PDF)

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Yury V. Matveychuk: 0000-0001-5255-3262 Notes

The authors declare no competing financial interest. I

DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

Article

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DOI: 10.1021/acs.cgd.8b00067 Cryst. Growth Des. XXXX, XXX, XXX−XXX