Dynamics and Thermodynamics of Crystalline Polymorphs. 2. Î²

Article pubs.acs.org/JPCA

Dynamics and Thermodynamics of Crystalline Polymorphs. 2. β‑Glycine, Analysis of Variable-Temperature Atomic Displacement Parameters Thammarat Aree,*,† Hans-Beat Bürgi,‡ Vasily S. Minkov,§,∥ Elena V. Boldyreva,§,∥ Dmitry Chernyshov,⊥ and Karl W. Törnroos# †

Department of Chemistry, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand Department of Chemistry and Biochemistry, University of Berne, CH-3012 Bern, and Institute of Organic Chemistry, University of Zürich, CH-8050 Zürich, Switzerland § Novosibirsk State University, Novosibirsk 630090, Russia ∥ Institute of Solid State Chemistry and Mechanochemistry, Russian Academy of Sciences, Novosibirsk 630128, Russia ⊥ Swiss-Norwegian Beamlines at the European Synchrotron Radiation Facility, 38043 Grenoble, France # Department of Chemistry, University of Bergen, N-5007 Bergen, Norway ‡

S Supporting Information *

ABSTRACT: The molecular dynamics in the crystal and the thermodynamic functions of the β-polymorph of glycine have been determined from a combination of molecular translation-libration frequencies reflecting the temperature dependence of atomic displacement parameters (ADPs), with frequencies derived from ONIOM(DFT:PM3) calculations on a 15-molecule β-glycine cluster. ADPs have been obtained from variable-temperature diffraction data to 0.5 Å resolution collected with X-ray synchrotron (10−300 K) and sealed tube radiation (50−298 K). At the higher temperatures, the ADPs of β-glycine from synchrotron are larger than those from sealed tube probably due to different experimental conditions. The lattice vibration frequencies from normal-mode analysis of ADPs and the internal vibration frequencies from ONIOM(B3LYP/6-311+G(2d,p):PM3) calculations agree with those from spectroscopy. Estimation of thermodynamic functions using the vibrational frequencies, the Einstein and Debye models of heat capacity, and the room-temperature compressibility provides Cp, Hvib, and Svib that agree with those from calorimetry. The β-phase with higher H and G is found to be less stable than the α-phase in the temperature range of the experiment. formation to the α-phase. Boldyreva and co-workers studied structural distortions of the crystal structure of β-glycine on cooling from sealed-tube X-ray data collected to 0.6 Å resolution at 294 and 150 K.10 Thus far, neutron diffraction data of β-glycine are not available. The goal of the present work is to analyze the atomic displacement parameters (ADPs), especially their temperature dependence, for information on the dynamics and thermodynamics of the β-polymorph with normal-mode analysis.11 X-ray data limited in resolution to 0.6 Å rather than 0.5 Å are marginal for obtaining accurate ADPs minimally contaminated by valence electron density. In addition, ADPs from sealed tube radiation have been reported to be systematically larger by ∼6 × 10−4 Å2 as compared to those from synchrotron radiation. The effect has been attributed to the higher dispersion of the beam from a sealed tube.12 Moreover, the data reported so far do not cover the temperature range from the quantum (low

1. INTRODUCTION Glycine crystallizes as the zwitterion +NH3CH2COO− in three polymorphic modifications. At ambient pressure, the relative stabilities are γ (trigonal, P31) > α (monoclinic, P21/n) > β (monoclinic, P21).1 Differences in the 3D arrangement of the zwitterions in the three glycine polymorphs result in differences in their physical and chemical properties. The β- and γ-phases exhibit piezoelectricity,2,3 whereas the α-phase shows pyroelectricity near room temperature.4 The metastable β-glycine is hygroscopic and readily converts to α-glycine in the presence of humidity. It is therefore the least studied among the three polymorphs. Iitaka reported the first βglycine crystal structure obtained by photographic techniques in 1959.2,5 Kozhin reported the unit cell constants at 77 and 293 K in a study of thermal expansion tensors.6 Using modern diffractometers, Drebushchak and co-workers determined the β-glycine crystal structure to 0.6 Å resolution at room temperature7 and followed the β- to α-phase transition in a crystal directly mounted on a diffractometer.8 Ferrari and coworkers9 published the β-glycine crystal structure to 0.77 Å resolution at 150 K in a study of solvent-mediated trans© 2013 American Chemical Society

Received: May 4, 2013 Revised: July 18, 2013 Published: July 18, 2013 8001

dx.doi.org/10.1021/jp404408h | J. Phys. Chem. A 2013, 117, 8001−8009

The Journal of Physical Chemistry A

Article

beamline BM01A at the Swiss Norwegian Beamlines (SNBL) of the European Synchrotron Radiation Facility (ESRF) with a wavelength of 0.70128 Å, using an ONYX KUMA CCD, 6circle diffractometer. The other crystal was used for data collection at 10 K on beamline ID11 with a wavelength of 0.15927 Å, using a homemade CCD, 3-circle diffractometer. For comparison of the ADPs, one more crystal was selected for data collection with a sealed tube source to 0.5 Å resolution at 50, 100, 150, 200, 250, and 300 K, using an Oxford Diffraction Gemini R Ultra, 4-circle kappa CCD diffractometer (Mo Kα, λ = 0.71073 Å). Data from the SNBL and from the sealed tube source were processed and multiscan absorption corrections were applied with CryAlis RED;22 data from ID11 were treated with SAINT23 and SADABS.24 Completeness, redundancy, and Rint are 93−99%, 3.1−4.4, and 0.025−0.039 for the combined SNBL and ID11 synchrotron data and 98−100%, 7.0−9.7, and 0.029−0.075 for the sealed tube data (see Supporting Information, Tables 1S−3S). The unit cell dimensions from both sources agree to within 3 times the standard uncertainties; the volumes from synchrotron data are on average 0.5 Å3 larger than those from sealed tube data (Figure 1S, Supporting Information). Data were scaled and merged with XPREP.25 The structures were then refined using full-matrix least-squares on F2 with SHELXL97.26 All non-H atoms were refined anisotropically using conventional spherical scattering factors27 (Independent Atom Model, IAM). Hydrogen atoms found from difference Fourier maps were refined isotropically. The final R1 values are 0.024−0.041 for synchrotron and 0.024− 0.038 for sealed tube data. Refinement statistics for the data sets from both sources are summarized in Table 1. The temperature

temperature) to the classic (high temperature) regime needed for normal-mode analysis.11 We have therefore collected new synchrotron data in the temperature range 10−300 K. The work presented here is part of a systematic study on the dynamic and thermodynamic properties of the glycine polymorphs based on an analysis of their respective multitemperature ADPs. The ultimate aim is an insight into the relative stabilities of the glycine polymorphs. ADPs measured as a function of temperature are analyzed with the normal mode model developed by Bürgi and Capelli.11 It expresses the ADPs in terms of a molecular Einstein model with the relationship: ∑x (T ) = AgVδ(T )VTg TAT + ε x

(1)

where δ(T) is a diagonal matrix of temperature-dependent normal mode displacements with elements: δi(T ) =

⎛ hνi ⎞ h coth ⎜ ⎟ 8π 2νi ⎝ 2kBT ⎠

(2)

The matrices g and A transform the normal modes with frequencies νi and eigenvectors V into atomic displacements ∑x(T); ADPs are the 3 × 3 diagonal blocks of ∑x; εx is a temperature-independent term accounting for the highfrequency intramolecular vibrations, which are not significantly excited in the temperature range of the diffraction experiment; h is the Planck constant; and kB is the Boltzmann constant. The vibrational motion of molecules in the crystal is parametrized by the model of motion, that is, the frequencies νi, the independent elements of the orthonormal matrix V representing molecular displacement coordinates and the six independent elements of each of the 3 × 3 diagonal blocks of εx (one per atom). The vibration frequencies thus obtained are estimates of averages over the Brillouin zone of the lattice frequencies. Anharmonicity associated with the thermal expansion of the crystal is taken into account by a Grüneisen parameter13 (eq 3) derived from the ratio of the frequency change to the cellvolume change with temperature.14 γG, i

V (0) Δνi(T ) =− · ΔV (T ) νi(0)

Table 1. Refinement Statistics for β-Glycine from the Independent Atom Model (IAM) Using SHELXL9726 and from the Invariom Multipole Model (INV) Using XD29a temp (K)

R1(F)_IAM

10 80 90 100 130 190 250 300

0.024 0.032 0.030 0.041 0.032 0.035 0.033 0.034

50 100 150 200 250 298

0.031 0.024 0.025 0.032 0.035 0.038

(3)

The method has been applied for accurate determinations of heat capacities Cv, Cp of several molecular crystals using Einstein,15 Debye,16 and Nernst−Lindemann17 relations.18 Very recently, we have used it to determine vibration frequencies and thermodynamic functions of α-glycine.19,20 In this work, we report the results of a normal-mode analysis and estimates of thermodynamic functions based on β-glycine diffraction data from synchrotron and sealed tube sources.

2. DIFFRACTION EXPERIMENTS 2.1. Crystal Preparation. White powder of glycine purchased from Fluka was used as received. Plate-like, colorless single crystals of the β-glycine polymorph were prepared from a saturated acetic acid aqueous solution of glycine (volume ratio of water and acetic acid of 1:2) by slow evaporation, and not through addition of an antisolvent.21 The metastable β-phase crystals were dried before storing them in a desiccator to avoid transformation into the α-phase. 2.2. Data Collection and Processing. Two plate-like βglycine single crystals each were mounted on glass fibers using epoxy glue. Synchrotron data from one crystal were collected to 0.5 Å resolution at 80, 90, 100, 130, 190, 250, and 300 K on

a

R1(F)_INV

Δρ_IAM (e Å−3)

Synchrotron 0.019 0.44/−0.23 0.026 0.51/−0.26 0.023 0.47/−0.24 0.036 0.63/−0.45 0.025 0.42/−0.24 0.028 0.44/−0.24 0.025 0.38/−0.22 0.027 0.35/−0.17 Sealed Tube 0.029 0.36/−0.34 0.016 0.33/−0.33 0.018 0.27/−0.35 0.026 0.32/−0.46 0.028 0.33/−0.40 0.030 0.31/−0.37

Δρ_INV (e Å−3) 0.24/−0.18 0.26/−0.19 0.23/−0.17 0.41/−0.37 0.25/−0.17 0.20/−0.16 0.20/−0.13 0.22/−0.14 0.27/−0.35 0.16/−0.23 0.18/−0.25 0.28/−0.39 0.31/−0.37 0.27/−0.34

Data are from synchrotron and sealed tube experiments.

evolution of the β-glycine ADPs is depicted with PEANUT28 plots in Figure 1 and graphed in Figure 2a,b, which also shows some results from published sealed tube data measured to 0.6 Å resolution at 150 and 294 K.10 2.3. Separation of Thermal Motion from Valence Bonding Density. ADPs of the non-H atoms minimally contaminated by directional valence bonding density may be obtained by refining the diffraction data using a multipole model with program XD.29 We adopted the invariom model30 8002



Article

Å−3 (Table 1). At 300 K, the ADPs of atom O1 (particularly U33) decrease by as much as 1.0 × 10−3 and 1.5 × 10−3 Å2 for synchrotron and sealed tube data, respectively, corresponding to ∼5 and ∼8 times the standard errors (Figure 2c,d). By contrast, the elements U22 from synchrotron data increase by 0.8 × 10−3 Å2 or ∼4 times the standard errors (Figure 2c). Changes that are similar to those found for atom O1 are also observed for atoms O2 and C2, while all of the principal elements of atoms C1 and N1 decrease by ∼1 times the standard errors. Except for atom O1, the ADPs from the sealed tube data show a similar decrease upon multipole refinement, ∼4 times the standard errors (Figure 2d). The differences of mean square displacements along interatomic vectors of bonded non-H atoms are less than 6.0 × 10−4 Å2 for all temperatures, thus conforming to Hirshfeld’s rigid-bond criterion32 and documenting the quality of the ADPs obtained.

3. THEORETICAL CALCULATIONS The zwitterionic glycine molecule is unstable in the gas phase, but is stabilized by the crystal field in the solid phase. The crystal field was mimicked by a 15-molecule cluster whose central molecule sits in the hydrogen-bonding network of the 14 surrounding neighbors. The cluster size is consistent with Kitaigorodski’s rule of molecular close packing.33 The cluster was built using the atomic coordinates of the β-glycine structure at 10 K determined from the ESRF-ID11 data. A two-layer ONIOM(B3LYP/6-311+G(2d,p):PM3) method was employed to estimate the vibration frequencies of the glycine molecule in its crystalline environment with program Gaussian 03.34 The central molecule was treated at the B3LYP/6-311+G(2d,p) level of theory, the 14 surrounding molecules with the semiempirical PM3 method. With this approach, the structural optimization converged quite rapidly. The same procedure has

Figure 1. PEANUT plots (rmsd scale 2.50) of β-glycine ADPs derived from synchrotron diffraction data (IAM refinement). The Hatoms are omitted for clarity. 28

based on the approximation that a nonspherical electron density fragment remains intact in related molecules and that the multipole population parameters are therefore transferable. This treatment has been successfully applied to the α-glycine synchrotron data,19,20 providing well-defined deformation electron density maps and improving R-values by ∼1% as originally observed for Mo Kα data sets of organic molecules.31 For the β-glycine synchrotron and sealed tube data, the invariom refinement decreases the R-values by up to 0.8%, primarily reduces the ADPs (Figure 2c,d), and flattens the residual electron density Δρ to values mostly below ∼0.25 e

Figure 2. The variable-temperature ADPs of β-glycine (a) C1 and (b) O1 atoms from IAM refinement of synchrotron data and sealed tube data at 0.5 and 0.6 Å (150 and 294 K).10 Comparison between ADPs of O1 from IAM and INV refinements of (c) synchrotron and (d) sealed tube data. The standard errors are 2 × 10−4 Å2, or ca. the line thickness. 8003



Article

Table 2. Normal Mode Analysis of Multitemperature ADPs of β-Glycine frequency (ν̅, cm−1), Grüneisen, and eigenvector ADP Set 1: Sealed Tube + IAM; Model rbegb 65.8(0.7) 77.1(1.5) 83.1(1.0) 2.19(0.41) 2.19(0.41) 2.19(0.41) Lx −0.505(18) −0.078(114) −0.483(27) Ly 0.335(29) 0.117(71) −0.324(14) Lz 0.060(18) −0.491(20) 0.004(113) Tx −0.570(17) 0.027(59) −0.421(44) Ty 0.508(15) 0.260(127) −0.657(78) Tz 0.207(33) −0.817(49) −0.221(144) U1 0.054(17) 0.054(29) 0.069(20)

ADP Set 2: Synchrotron + IAM; Model rbegb 63.9(0.6) 72.0(0.9) 2.34(0.25) 2.34(0.25) Lx −0.450(12) −0.002(56) Ly 0.324(28) 0.211(41) Lz 0.104(21) −0.462(13) Tx −0.561(15) −0.023(39) Ty 0.513(16) 0.403(60) Tz 0.320(43) −0.759(40) U1 0.028(15) 0.062(26)

80.7(1.0) 2.34(0.25) −0.515(23) −0.312(17) −0.076(53) −0.469(42) −0.557(47) −0.317(60) 0.038(19)

ADP Set 3: Sealed Tube + INV; Model rbegbc 70.5(0.6) 79.5(1.4) 93.9(0.9) 2.41(0.37) 2.41(0.37) 2.41(0.37) ADP Set 4: Synchrotron + INV; Model rbegbc 62.0(0.7) 68.0(0.7) 74.2(1.8) 1.13(0.31) 1.13(0.31) 1.13(0.31)

ϵ (×10−4)a,b 138.2(5.6) 2.19(0.41) −0.537(46) 0.436(76) 0.349(73) 0.564(42) −0.153(91) −0.140(21) −0.197(103)

148.8(5.8) 2.34(0.25) −0.597(30) 0.348(92) 0.212(82) 0.547(54) −0.088(102) −0.126(23) −0.394(108)

N1 C1 C2 O1 O2 29(1) 0(1) 33(1) H1 H2 63

H3 H4 H5 63

0 134

0 0 152

0 210

0 0 154

N1 C1 C2 O1 O2 29(1) −1(1) 36(1) H1 H2 63

H3 H4 H5 63

1(1) 3(1) 35(2)

GOF

R (%)

2.76

3.59

observations: 210 restraints: 25 parameters: 54

2.74 0(1) 1(1) 30(2)

0 134

0 0 152

0 210

0 0 154

4.38

observations: 280 restraints: 25 parameters: 54

101.1(3.6) 2.41(0.37)

2.33

3.22

168.5(13.7) 1.13(0.31)

3.44

5.51

a

The H-atom epsilons are restrained to the values from ONIOM calculations. bThe diagonal elements of epsilon for non-H from ONIOM calculations are 23 × 10−4, 32 × 10−4, and 29 × 10−4 Å2. cThe INV results are discrepant for unknown reasons and are not discussed in the text.

been applied to estimate the vibration frequencies of αglycine19,20 and to model the structures of the three glycine polymorphs.35

the principal elements U11 and U33 of the latter are larger by 1 × 10−3 Å2, whereas the elements U22 differ only slightly (Figure 2a,b). The H-atom ADPs increase with increasing temperature, similarly to the non-H atoms they are attached to (for an example, see Uiso of the methylene H-atoms in Supporting Information, Table 2S). 4.2. Dynamics in the Crystal from Normal Mode Analysis. Four simple normal mode models of motion were fitted to four sets of multitemperature ADPs with program NKA:38 model 1 “rigid body” (rb), the simplest model of motion takes only molecular librations and translations into account;39,40 model 2 (rbe) includes internal vibration effects (in terms of 3 ε tensors: the tensor for non-H atoms is refined, the tensors for the methylene H-atoms H1, H2 and for the ammonium H-atoms H3, H4, H5 atoms are restrained to the values obtained by ONIOM calculations; the local coordinate systems of these tensors are given in the Supporting Information); model 3 (rbeg) includes anharmonic effects accounted for by a single Grüneisen constant; and model 4 (rbegb) includes torsional oscillation of the CO2− group (U1 in Table 2). The molecular coordinate system for normal-mode analysis is defined with the x-axis along the vector O1→N1, the z-axis orthogonal to the plane defined by the vectors along O1→N1 and C2→C1, the y-axis orthogonal to the x- and zaxes and completing a right-handed system (Figure 3). The C−

4. RESULTS AND DISCUSSION 4.1. Multitemperature ADPs. The dependence of the βglycine ADPs on temperature from both synchrotron and sealed tube sources is smooth and continuous from 10 to 300 K. Neither the ADPs nor the normal-mode analysis indicate the second-order phase transition seen at 252 K by calorimetric36 and inelastic neutron scattering37 measurements. The steeperthan-linear increase of the ADPs at elevated temperatures indicates anharmonic motion due to thermal expansion (Figure 2a,b). In the temperature range below 100 K, the ADPs from both sources are similar. As the temperature increases, the ADPs from synchrotron data become larger than those from sealed tube data by as much as 3 × 10−3 Å2 at 300 K (Figure 2a,b). This observation contradicts the results from Rousseau and co-workers.12 The differences are probably due to the different crystals, different experimental conditions, and different data processing software, as previously noted for αglycine where the ADPs from two different synchrotron beamlines (KEK and SPring-8) differed to a similar degree at 298 K.19,20 Comparison between the ADPs in the classic regime from sealed tube data at 0.5 and 0.6 Å resolution10 shows that 8004



Article

covered by the diffraction data and to be reasonably determined from the temperature dependence of the ADPs. The four lattice frequencies obtained with the ADPs from synchrotron data (Table 2, ADP set 2) cover the same range as those from Raman spectroscopy42 (Table 3). A frequency-to-frequency comparison has to be taken with a grain of salt though: the spectroscopic frequencies refer to the origin of the Brillouin zone, and an assignment in terms of eigenvectors is not available; in contrast, the normal modes from ADPs represent averages over the Brillouin zone, but are characterized by both frequencies and eigenvectors. The above frequencies are used for estimating thermodynamic functions in section 4.4. 4.3. Internal Vibrations from ONIOM Calculation. The ONIOM(B3LYP/6-311+G(2d,p):PM3) method applied to the 15-molecule cluster provides 30 vibrational frequencies (58− 3419 cm−1), mostly internal modes. Averaging the ratios of observed to calculated vibrational frequencies from the 20 highest energy modes gives a scaling factor of 0.974 that is similar to the values proposed by Scott and Radom.43 Although the calculated frequencies agree reasonably well with those from spectroscopy44 (see Supporting Information, Table 4S), the caveat mentioned at the end of section 4.2 applies here as well. The ONIOM results served not only for calculating thermodynamic functions (section 4.4), but also to estimate the anisotropic, temperature-independent contributions ε to the calculated ADPs. They compare well with the refined values for C, N, and O atoms (Table 2, footnote b). For H atoms, they were kept at calculated values. 4.4. Thermodynamic Functions. The four temperaturedependent vibration frequencies from the normal-mode analysis of synchrotron data and the 26 highest (mostly internal) vibration frequencies from the ONIOM calculation are used together with the Einstein15 and Debye16 models of heat capacity to estimate the specific heat Cv. β-Glycine does not have a well-defined melting point because it transforms into either the α- or the γ-polymorph on heating above ambient temperature; hence, Cv is converted to Cp using the experimentally available compressibility κ(T) (eq 4) instead of the Nernst−Lindemann relation17 used previously for the molecular crystals of α-glycine,19,20 HMT, naphthalene, and anthracene.18 The molecular heat capacity Cp(T) can be expressed as

Figure 3. PEANUT28 plots showing difference displacement parameters 10 × (Uobs − Ucalc)1/2 of β-glycine from synchrotron data (IAM refinement, model rbegb). Positive differences are shown with solid lines, and negative differences are shown with dashed lines.

H and N−H distances from X-ray structures are systematically too short; for the ADP analysis, they were placed to the more appropriate positions found in the neutron structures of αglycine19,20,41 to ensure optimal moments of inertia for the librational motion. Although included in the normal-mode analysis with their isotropic ADPs, the hydrogen atoms influence the models only slightly because the standard uncertainties of their ADPs are large (see Supporting Information, Table 2S). The four sets of ADPs tested were obtained from the IAM and invariom refinements of the synchrotron and sealed tube data. Generally, each of the four models of motion explains the four ADP sets similarly well with R-factors in the range 0.032− 0.055 (Table 2). The IAM ADPs from both sealed tube and synchrotron data lead to fairly comparable dynamic models (Table 2, ADP sets 1 and 2). In contrast, the frequencies and Grüneisen parameters obtained from the ADPs based on sealed tube and synchrotron data via invariom refinements differ significantly (Table 2, ADP sets 3 and 4). The reasons for this behavior are unclear (see also section 4.1). Therefore, we only discuss the model rbegb fitted to the ADPs from IAM refined synchrotron data, which sample a wider temperature range with more temperature points than do the sealed tube data (Table 1). The PEANUT plots in Figure 3 show that the differences between the calculated and observed displacement parameters are small and randomly distributed. Glycine is a relatively small molecule with substantial hydrogen bonding and consequently limited mobility in the crystal lattice. Thus, only four normal modes have frequencies low enough to be significantly excited in the temperature range

Cp(T ) = Cv(T ) + Tχ 2 (T )V (T )/κ(T )

(4)

where χ(T) is the thermal expansivity and V(T) is the molar volume. The thermal expansion and the molar volume are available from the multitemperature diffraction experiments. The compressibility of 5.78 × 10−5 (MPa)−1 is derived from the pressure-dependent unit cell volume obtained from X-ray diffraction45,46 at room temperature of the reversible β ↔ β′ phase transition at 0.8 GPa. The calculated values of Cp, Svib,

Table 3. Lattice Vibration Frequencies of β-Glycine in Comparison with α-Glyine β

α

a

lattice vib freq (cm−1)

technique

T (K)

sample

ref

63.9, 72.0, 80.7, 149 56.2, 96.4, 160, 169a 64, 88, 117, 131, 150, 160 71.3, 77.9, 86.7, 121 144, 154, 166a 56, 75, 118, 150

ADP-analysis on X-ray (ID11+BM01A) ONIOM(B3LYP/6-311+G(2d,p):PM3) Raman spectroscopy ADP-analysis on X-ray (SPring-8) and neutron ONIOM(B3LYP/6-311+G(2d,p):PM3) Raman spectroscopy

10−300

single crystal cluster model single crystal single crystal cluster model single crystal

this work this work 42 19, 20 19, 20 42

40 18−323 60

After scaling with 0.974 and 0.982 for the β- and α-phases, respectively. 8005



Article

and Hvib are found to agree with those from calorimetry36 (Figure 4a,b). The largest Cp differences are

Dynamics and Thermodynamics of Crystalline Polymorphs. 2. Î²

Recommend Documents