How the H-Bond Layout Determines Mechanical Properties of

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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 Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.8b00067 • Publication Date (Web): 18 Apr 2018 Downloaded from http://pubs.acs.org on April 25, 2018

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HOW THE H-BOND LAYOUT DETERMINES MECHANICAL PROPERTIES OF CRYSTALLINE AMINO ACID HYDROGEN MALEATES Yury V. Matveychuk†∗, Ekaterina V. Bartashevich†, 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

Abstract The stiffness tensor and elastic anisotropy characteristics for the crystalline hydrogen maleates of L-isoleucinium, 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.

∗ Corresponding Author. E-mail: [email protected]; 76, Lenin pr., Chelyabinsk, 454080, Russia.



HOW THE H-BOND LAYOUT DETERMINES MECHANICAL PROPERTIES OF CRYSTALLINE AMINO ACID HYDROGEN MALEATES Yury V. Matveychuk†∗, Ekaterina V. Bartashevich†, 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

∗ Corresponding Author. E-mail: [email protected]; 76, Lenin pr., Chelyabinsk, 454080, Russia.


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ABSTRACT: The stiffness tensor and elastic anisotropy characteristics for the crystalline hydrogen maleates of L-isoleucinium, 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.



INTRODUCTION The practical usage of molecular crystals is based on 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 al2 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 stability4, the velocities of acoustic waves in crystals (via the Christoffel equation),5 the spatial anisotropy of elastic moduli,6 and various indexes of anisotropy based on different approaches: the Zener’s anisotropy factor, A,7 the general elastic anisotropy measure, A*,8 the last-proposed log-Euclidean anisotropy measure, AL,9 and others. Further we use the universal elastic anisotropy index, AU:10

GV KV A = 5 R + R − 6, G K U


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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 coworkers11-14 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 H-bonds, and stacks orthogonally linked only by weak van der Waals interactions, is the reason of easy crystal bending. Examples are 3,4dichlorophenol, 2-nitrobenzaldehyde, hexachlorobenzene crystals, etc. To the contrary, the crystals with intermolecular interactions of comparable force in three 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 towards 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 al17 have described the behavior of the L-valine and Dmethionine co-crystal, 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 acids20,21 and metal-organic frameworks.6,22,23 These studies paid the 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 value of 1092 GPa in the plane of layers, whereas it is equal to 39 GPa in 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


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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 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 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 has been done.20



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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 stress34. 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 non-covalent 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 non-covalent 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: Lisoleucinium hydrogen maleate hemihydrate VUKQEZ 1, space group P21 at 298 K,38 Lleucinium hydrogen maleate VUKQAV 2, space group C2 at 298 K,38 co-crystal L-norvalinium 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:


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– 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 molecular fragments and intermolecular non-covalent structure-forming interactions (H-bonds).

EXPERIMENTAL PART 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 valuation of elastic properties of various molecular organic crystals,22 and the full-electron basis set 6-31G(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 Towler.46 The experimental crystallographic data38,39 and the data from 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 has always equaled the experimental. The following convergence parameters have been used for all calculation: TOLDEG (root-mean-square on gradient) is less than 3⋅10-5 a.u., TOLDEX (rootmean-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



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k-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 Lalanine 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 directions. The sufficiency of such conditions for the absence of imaginary frequencies was also been checked.


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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

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, 2.32 for crystal 3. Comparing the spatial dependencies of elastic modules of 1 and 2 (Table 1 and Fig. 1-2, 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.

Table 1. Comparison of calculated data of the elastic moduli for crystals 1, 2, 3 Young's modulus, GPa

The modulus value

Linear

compressibility, TPa-1

Shear modulus, GPa

Poisson's ratio

Emin

Emax

βmin

βmax

Gmin

Gmax

νmin

νmax

1

6.23

18.15

11.08

59.53

1.96

6.36

0.14

0.61

2

4.26

21.64

1.34

72.24

1.24

7.33

-0.04

0.93

3

6.60

21.98

16.02

47.27

1.96

8.18

0.06

0.73



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Modulus

1

2.91

5.37

3.24

4.47

anisotropy

2

5.08

53.77

5.92

∞

ratio

3

3.33

2.95

4.17

13.15

1

0.67

0.48

-0.46

-0.89

0.66

0

0

0

-0.34

0.88

-0.89

0.46

Direction of minimum and maximum value (in

2

Cartesian

-0.73

0.11

0.29

-0.96

0.68

0.55

0

0

-0.01

0.83

-0.96

-0.29

coordinates)*

3

-0.65

0

0.67

0.74

0.69

1

0

0

-0.32

0

-0.74

0.67

-0.88

-0.37

-0.55

0.46

0

0.71

0.68

0.76

0.47

-0.60

0.48

-0.45

0

-0.37

-0.52

-0.70

-1

-0.71

0.17

0.63

0

-0.60

-0.84

0.34

-0.99

-0.86

0.31

0.44

0

0

0.60

0.47

0.11

-0.52

0.74

-0.77

0

0.52

-0.63

-0.82

-1

0

-0.45

0.56

0

-0.86

0.63

-0.12

0.92

-0.33

-0.99

0.91

0

0.71

0

0

0.39

0.62

0.12

-0.42

0

-0.33

0

-0.42

1

-0.71

-1

0

0

0.62

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


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a)

b)

c)

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

a)

b)

c)

Figure 2. The 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 are shown by green color, the surface of the maximum shear modulus value are indicated by transparent blue color

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 6S). Values



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of linear compressibility are an exception. If 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 Hbonds 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.

a)

b)


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c) Figure 3. The 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, the arrows Emax and Emin are specified irrespectively of the modulus values

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 (Fig. 3). The layer structure is caused, among other things, by rigidity of the planar hydrogen maleate anion having the strong intramolecular H-bond (~ 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 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 L-isoleucinium 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 zwitter-ions and L-norvalinium cations. Second, let us analyze how H-bonds in 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 fourteen intermolecular H-bonds, from which nine are oriented at a larger angle ( > 20º ) 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 forty H-bonds, from which only twelve 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 four of ten 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.


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Figure 4. The 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

In crystal 1 the most of 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 (Fig. 4). However, in crystal 2 another picture has been found. The most of 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 direction. The orientation of all medium Hbonds 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 (Fig. 3b). In crystal 1 the



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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).

a)

b)

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

The Emax direction along the structural channel, which is formed by the rings of Lnorvaline zwitter-ions and L-norvalinium cations (Fig. 3c), mentioned above for crystal 3, is a feature of this crystal only. We associate it with existence of 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 (Fig. 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. “extinguish” each other. We consider the relative arrangement of such bonds to be the main


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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 L-leucinium hydrogen maleate crystal. On the contrary, the participation of water molecules in several adjacent Hbonds, “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 precise prediction it is necessary to consider other elastic moduli for the refining of mechanical behaviour of the crystal.

Anisotropy of linear compressibility Let us compare the spatial dependences of linear compressibility β for three considered hydrogen maleate crystals of amino acids, which describes the crystal behavior at isotropic hydrostatic compression (Fig. 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 anisotropy ratio of this module is here the relation of the maximum value to



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minimum value, it increases in the crystal series 3 < 1 60º ). Any dependence of these angles on the type and length of a bond has not been detected. It should be noted that similar distribution of angles both with 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


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crystal are oriented almost orthogonally to the βmax direction (Fig. 6b). At the same time, the angles of H-bonds with the βmin direction do not exceed 60º. From this point of view, the experimentally observed brittleness of crystal 1 and elastic behaviour 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 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 (< 1 GPa) or high (> 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.



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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 (Fig. 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 existence of the channel structure.

a)

b)


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c) Figure 7. The 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

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 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 abovementioned 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 (Fig. 7c, Table 9S). In crystal 2 the particular influence on the Gmax value of shear modulus should be expected from medium Hbonds (Fig. 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 Hbonds 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.

CONCLUSION In this research we have studied three hydrogen maleate crystals of amino acids: Lisoleucinium 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 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


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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 Hbonds, or at small angle to them. It has 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 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 reflects 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 Lisoleucinium 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.

Conflicts of interest. The authors declare no competing financial interest. Acknowledgements. This work was supported by the Ministry of Education and Science of the Russian Federation, grant 4.1157.2017/4.6 and by Government of the Russian Federation, Act 211, contract № 02.A03.21.0011.

Supporting Information. The description of validation of the method and level of calculations with periodically 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 H-bonds 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.


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For Table of Contents Use Only HOW THE H-BOND LAYOUT DETERMINES MECHANICAL PROPERTIES OF CRYSTALLINE AMINO ACID HYDROGEN MALEATES

Yury V. Matveychuk Research and Education Center “Nanotechnology”, South Ural State University, 76, Lenin av., Chelyabinsk, 454080, Russia Ekaterina V. Bartashevich Research and Education Center “Nanotechnology”, South Ural State University, 76, Lenin av., Chelyabinsk, 454080, Russia 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

Synopsis: The flexibility of amino acid hydrogen maleate crystals depends on anisotropy of elastic moduli defined by the spatial arrangement of H-bonds.


How the H-Bond Layout Determines Mechanical Properties of

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