Thermal and Mechanical Properties of Poly(methylene oxide

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Thermal and Mechanical Properties of Poly(methylene oxide) Polymorphs Unraveled by Periodic Density Functional Theory Yuichiro Fukuda and Yuji Sasanuma* Department of Applied Chemistry and Biotechnology, Graduate School and Faculty of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan

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S Supporting Information *

ABSTRACT: Density functional theory (DFT) calculations with a dispersion force correction under periodic boundary conditions have been applied to two kinds of crystalline forms of poly(methylene oxide) (PMO): trigonal lattice with a 9/5 helical chain (t-PMO) and orthorhombic lattice with two 2/1 helical chains (o-PMO). The following computational results were derived: optimized structure (lattice constants and atomic coordinates), interchain cohesive energy, infrared spectrum, thermodynamic functions, orthorhombic-to-trigonal transition temperature, and crystalline modulus. The DFT calculations reproduced the experimental polymorphic transition at 69 °C, and the evaluated elastic modulus of trigonal PMO lies within close range to the experimental values. This study has demonstrated that the advanced computational chemistry supplies reliable and quantitative information on polymer crystals.

■

INTRODUCTION Poly(methylene oxide) (PMO) has been widely used as an engineering thermoplastic and an alternative to metals because of its superior mechanical properties and chemical resistance.1 These characteristics are partly due to a strong gauche preference of the CO bond;2−4 the gauche conformation is more stable by −1.5 to −2.5 kcal mol−1 than the trans state, and the large energy difference yields a small configurational entropy (Sconf) of 3.0 cal K−1 mol−1.5 Inasmuch as Sconf corresponds to the entropic difference of a polymeric chain between in the crystalline and in the free (amorphous, melt, or solution) states, being close to the entropy of fusion (ΔSu, 3.5 cal K−1 mol−1),6 the small Sconf results in a high equilibrium melting point (479.2 K) and extremely poor solubility (i.e., chemical resistance). In the crystal, owing to the strong gauche preference, the PMO chain forms either of two all-gauche helices: trigonal (space group P32) 9/57−11 and orthorhombic (P212121) 2/1 helical structures.12 Herein, the former and latter crystals are designated as t-PMO and o-PMO, respectively. It has occasionally been discussed whether the true helical pitch of the former is 9/5 (1.80 monomeric units per turn) or 29/16 (1.8125).13−15 The helix notations, 9/5 and 29/16, give the impression that the two are significantly different from each other, but as it is, they are quite similar in both molecular conformation and interchain packing. Therefore, throughout this study the helix has been assumed to be 9/5, because it has so high a crystallographic symmetry as to be freer from arbitrariness in structure than the 29/16 helix. Ordinary melt or solution crystallization of PMO yields tPMO, whereas polymerization of formaldehyde in aqueous solution12 or cationic polymerization of trioxane16 results in o© XXXX American Chemical Society

PMO. The o-PMO modification will be transformed to t-PMO by heating above 70 °C or mechanical deformation,17 and hence o-PMO had been considered to be metastable. This is because the orthorhombic → trigonal (o → t) transition can be observed by differential scanning calorimetry (DSC) as an endothermic peak at 69 °C only on heating and hence appears to be monotropic (irreversible). However, it was found afterward that compression at 1.4−2.5 GPa induces the inverse trigonal → orthorhombic (t → o) conversion;18,19 therefore, the phase transition was proved to be enantiotropic (reversible), and the thermal equilibrium must be established between the two crystal modifications.20 The o → t transition has also been confirmed by recent variable-temperature X-ray diffraction measurements.21 The t-PMO crystal exhibits two large-scale morphologies depending on the sample preparation: one is the fully extended chain crystal (ECC) and the other is the folded chain crystal (FCC). The former was found in needle-like single crystals (the so-called polymer whisker) resulting from a heterogeneous cationic polymerization of trioxane,22 and the latter is the lamellar crystals grown from dilute solutions.19 The formation of ECC is a unique characteristic of PMO. It was found that ECC shows infrared (IR) spectra different markedly from those of FCC: the IR bands of ECC with transition moments parallel to the chain axis are selectively shifted toward the higher frequency by as much as 100 cm−1.17,19,23 This phenomenon was explained by the transition dipole− dipole coupling theory;23,24 the large dipole−dipole interaction Received: August 6, 2018 Revised: October 3, 2018

A

DOI: 10.1021/acs.macromol.8b01697 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 1. Summary of the Present Study and Comparison with Experiments trigonal (9/5 helix, space group P32) this studya a 4.288 (4.302)c

b 4.288 (4.302)c

C C C H H H H H H O O O

x/a 0.0035 0.0476 −0.1447 −0.2083 0.2057 0.3069 −0.0965 −0.4132 −0.1148 −0.1765 0.1238 −0.1096

y/b 0.1691 −0.1258 0.0205 0.2002 0.4171 −0.0675 −0.3993 −0.1451 0.2673 −0.1111 0.1307 −0.1960

C−O ∠ OCO ∠ COC C−O−C−O

1.416 112.2 114.7 80.2

BSSE (∞) ΔECP

−2.18 −3.17 Ea 17 (16)c

Eb 17 (16)c

experimentb Lattice Constant (Å) c a 17.765 4.373 (17.745)c ΔLC = 1.3%d Fractional Coordinates z/c x/a 0.0653 −0.0184 0.1754 0.0823 0.2892 −0.1388 0.0386 −0.193 0.0921 0.110 0.1522 0.271 0.1977 −0.070 0.2655 −0.370 0.3136 −0.162 0.1187 −0.1656 0.2329 0.1356 0.3450 −0.0936 ΔCHO = 0.046, ΔCO = 0.027e Geometrical Parameter (Å, °) 1.406 110.8 113.1 77.9 Energy Parameter (kcal mol−1)

b 7.831

C H H O

x/a 0.1377 0.3272 −0.4246 0.3802

y/b −0.0768 0.1968 0.0782 −0.0623

C−O ∠ OCO ∠ COC C−O−C−O φi

1.417 112.7 115.0 −65.3 50.2

BSSE (∞) ΔECP

y/b 0.1468 −0.0977 0.0318 0.182 0.363 −0.094 −0.338 −0.103 0.206 −0.1249 0.1599 −0.1746

Crystalline Modulus (GPa) Ec E⊥f (= Ea, Eb) 115 8.0 (114)c orthorhombic (2/1 helix, space group P212121)

this studya a 4.487

b 4.373

c 17.274

z/c 0.06536 0.17663 0.28741 0.040 0.090 0.150 0.209 0.263 0.310 0.11929 0.23268 0.34292

E∥g (= Ec) 100−105

experimenth Lattice Constant (Å) c a 3.601 4.714 ΔLC = 2.2%d Fractional Coordinates z/c x/a 0.2866 0.1342 −0.0664 0.3519 −0.3590 −0.4572 0.0376 0.3665 ΔCHO = 0.060, ΔCO = 0.056e Geometrical Parameter (Å, °) 1.411 113.9 112.9 −64.4 44.8 Energy Parameter (kcal mol−1)

b 7.505

c 3.554

y/b −0.0722 0.1823 0.0707 −0.0737

z/c 0.3426 −0.0155 −0.2967 0.0893

−2.03 −3.38 Ea 14

Eb 12

Crystalline Modulus (GPa) Ec 83

B

DOI: 10.1021/acs.macromol.8b01697 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 1. continued

a At 0 K. bAt−150 °C.14 cResults of our previous study,29 optimized and calculated under space group P31. dEquation 1. eEquation 2. fYoung’s modulus perpendicular to the chain axis. At room temperature.30 gYoung’s modulus parallel to the chain axis. At −150 °C.31−33 hAt 100 K.21 iThe set angle: the angle between the b-axis and O···O plane (see Figure 1).

| l o 1 o o ΔCHO o }= m o o o o N Δ o atom n CO o ~

vanishes in the ideal ECC and becomes maximum in the extremely thin FCC. It is also known that o-PMO forms a single crystal of ECC as a byproduct generated in the cationic polymerization of trioxane.16 Its vibrational spectra also differ from those of FCC.19,25 This study has aimed to elucidate the above-mentioned structures and properties characteristics of PMO by density functional theory (DFT) calculations26 with a dispersion force correction (DFT-D)27 under periodic boundary conditions. Herein, the computational results such as crystal structures (lattice constants and atomic coordinates), interchain cohesive energies that were evaluated via the counterpoise (CP) correction28 for the basis set superposition error (BSSE), IR spectra, the o → t transition temperature, and crystalline moduli are compared with experiments and discussed so as to investigate the applicability of the theoretical methodology to polymer crystals and comprehend structure−property relationships of the PMO crystals.

ÉÑ2 ÄÅ ÉÑ2 lÄÅÅ o Ñ ÅÅ y Ñ o oÅÅÅij x yz ÅÅij yz ij x yz ÑÑÑ ij y yz ÑÑÑ Å Ñ Å ∑ omoÅÅÅjj zz − jj zz ÑÑÑ + ÅÅÅjj zz − jj zz ÑÑÑÑ Åk a {calc k a {expt ÑÑÖ ÅÅÇk b {calc k b {expt ÑÑÖ o atom o nÅÇ ÄÅ ÉÑ2 |1/2 Ño ÅÅ o ÅÅij z yz ij z yz ÑÑÑ o Å j z j z + ÅÅj z − j z ÑÑÑ o } ÅÅk c {calc k c {expt ÑÑ o o o ÅÇ ÑÖ ~

(2)

where ΔCHO includes and ΔCO excludes the hydrogen atoms, and Natom is the number of atoms in the asymmetric unit. From the data shown in Table 1, we obtained ΔLC = 1.3%, ΔCHO = 0.046, and ΔCO = 0.027 for t-PMO and ΔLC = 2.2%, ΔCHO = 0.060, and ΔCO = 0.056 for o-PMO. Inasmuch as the DFT-D data represent the crystals at 0 K, the optimized structures of tPMO and o-PMO are, respectively, compared with the X-ray experiments at −150 °C14 and 100 K21 in Table 1. The bond lengths, bond angles, and dihedral angles evaluated as a result of the DFT-D optimization are also close to the experimental data. For o-PMO, an apparent difference between the DFT-D and X-ray structures can be found. In the X-ray experimental lattice (not shown),12,21 the sides of the 2/1 helix projected on the ab plane are almost parallel or perpendicular to the a and b axes, whereas in the optimized crystal the squares are somewhat slanted. The set angle (the angle between the b-axis and O···O plane, see Figure 1b) was calculated to be 50.2° (optimized here) or 44.8° (X-ray experiment at 100 K).21 However, the actual crystal at 0 K would have a set angle different from ∼45.0°. For example, a homologue of PMO, poly(methylene selenide) ([−CH2Se−]n), also forms an orthorhombic (P212121) 2/1 helical structure,34 and its set angle was determined at room temperature to be 66.3°. Nonetheless, the fully small ΔLC, ΔCHO, and ΔCO values and the acceptable reproduction of the geometrical parameters clearly indicate that both optimized crystal structures are sufficient for us to evaluate physical properties of t-PMO and o-PMO therefrom. Interchain Cohesive Energy and the Basis Set Superposition Error (BSSE). In a previous study,35 we have developed a counterpoise (CP) method28,36 to correct for the basis set superposition error (BSSE) of polymer crystals. In this study, the BSSEs of t-PMO and o-PMO were evaluated by the method. The BSSE of a single polymeric chain surrounded by ng ghost chains can be calculated from

■

RESULTS AND DISCUSSION Structural Optimization. The computational results are summarized in Table 1, and the optimized crystal structures are illustrated in Figure 1. The reproducibility of the

Figure 1. Optimized crystal structures (a) trigonal lattice with a 9/5 helical chain (abbreviated as t-PMO) and (b) orthorhombic lattice with two 2/1 helical chains (o-PMO). ϕ is the set angle (the angle between the b-axis and O···O plane). For the lattice constants and fractional coordinates, see Table 1.

BSSE(ng ) = E SC{Domain}(ng ) − E SC{SC}

where ESC{Domain}(ng) is the energy of a single chain (SC) located at the center of the domain filled with ng ghosts, and ESC{SC} is the energy of the single isolated chain. Here, all the energy parameters are in principle defined per repeating unit (−CH2O−). Both ESC{Domain}(ng) and ESC{SC} can be calculated under the one-dimensional periodic boundary condition (in the chain axis direction). As ng increases, the BSSE (ng) will approach a certain value, which may be considered as BSSE (∞). The interchain cohesive energy (ΔECP), corrected for the BSSE via the CP method, may be expressed as

experimental structures was examined by two criteria. For the lattice constants, a, b, and c, we defined ΔLC

(3)

É ÄÅ 2 2 2 Ñ1/2 ÅÅ ij ccalc − cexpt yz ÑÑÑÑ ij bcalc − bexpt yz 1 ÅÅÅijjj acalc − aexpt yzzz z j z j zz ÑÑ × 100 zz + jj = ÅÅÅjj zz + jjj zz jj cexpt zz ÑÑÑÑ j bexpt z 3 ÅÅÅj aexpt { ÑÑÖ k { k { ÅÇk

(1)

where the subscripts, calc and expt, represent the optimized and experimental data, respectively. For the atomic positions, x/a, y/b, and z/c, we evaluated C

DOI: 10.1021/acs.macromol.8b01697 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

ij E Crystal yz ΔE CP = jjj − E SC{SC}+ Dzzz − BSSE(∞) j Z z (4) k { Crystal where E is the total energy of the crystal cell including Z repeating units and ESC{SC}+D is the SC energy with the dispersion force correction. Figure 2 shows the BSSE (ng) versus ng plots, which are seen to converge rapidly. Even at ng = 6, the BSSE (6) value is fully

Figure 2. Basis set superposition errors (BSSEs) of t-PMO (●) and o-PMO (■) as a function of the number (ng) of ghost chains. The dotted curve passing through the data points can be expressed as A + B exp(−ng/C), where A, B, and C are adjustable parameters, and A is represented by the horizontal dotted line and corresponds to the BSSE (∞) value of −2.18 kcal mol−1 (t-PMO) and −2.03 kcal mol−1 (o-PMO).

lowered and almost equal to BSSE (18). In Figure 2, the dotted lines expressed as A + B exp(−ng/C) (A, B, and C, adjustable parameters) were fitted to the plotted data, and A gives the BSSE (∞) value: − 2.18 kcal mol−1 (t-PMO); − 2.03 kcal mol−1 (o-PMO). With the BSSE (∞) value, ΔECP was calculated from eq 4 to be −3.17 kcal mol−1 (t-PMO) or −3.38 kcal mol−1 (o-PMO). Therefore, the o-PMO chain is suggested to interact with neighbors more firmly than t-PMO. Infrared (IR) Spectra. The polymer crystal is characterized by the chain conformation and interchain bindings, and the two factors are reflected in its molecular vibrations. Figure 3 shows the calculated IR spectra of t-PMO and o-PMO. As mentioned in the Introduction, the IR spectra of PMO depend on the large-scale morphology, that is, whether ECC or FCC. The DFT-D calculations were carried out under threedimensional periodic boundary conditions; the crystal is assumed to be infinite in size. Preferably, the calculated IR spectra should be compared with those observed from the ECCs, whose spectra are overlaid in Figure 3a.37 As has often been pointed out, molecular orbital calculations overestimate vibrational frequencies, and hence a scale factor for the frequency calibration has been proposed for each Hamiltonian and basis set. For example, for the B3LYP/6-31G(d) combination similar to the B3LYP/6-31G(d,p) level employed here, the computed frequencies were recommended to be multiplied by 0.9614, which was derived by comparison with experiments on 122 molecules and 1066 frequencies.38 However, this value seemed to be slightly too small to match the calculated and observed frequencies of t-PMO. Therefore, with respect to the most intense peak at 895 cm−1 the scale factor was adjusted so as to achieve the good agreement, and a value of 0.9846 was obtained. The as-calculated frequencies (see Table S1, Supporting Information) were multiplied by

Figure 3. Infrared spectra of (a) t-PMO and (b) o-PMO. Each calculated spectral line (Table S1, Supporting Information) was converted to a Lorentzian function whose position and integrated area correspond to the scaled frequency and intensity, respectively. The experimental spectra observed from fully extended chain crystals (ECCs) in (a) needle-like single crystals of t-PMO and (b) a micronsized single crystal of o-PMO were adapted with permission from (a) Shimomura, M.; Iguchi, M.; Kobayashi, M. Vibrational spectroscopic study on trigonal polyoxymethylene and polyoxymethylene-d2 crystals. Polymer 1988, 29, 351 (copyright 1988 Elsevier) and (b) Kobayashi, M.; Adachi, T.; Matsumoto, Y.; Morishita, H.; Takahashi, T.; Ute, K.; Hatada, K. Polarized Raman and infrared studies of single crystals of orthorhombic modification of polyoxymethylene and its linear oligomer: crystal structures and vibrational assignments. J. Raman Spectrosc. 1993, 24, 533 (copyright 1993 Wiley).

0.9846, and the calibrated IR spectrum is shown in Figure 3a, where all the peaks exactly correspond to the observed bands. Surprisingly, even very small peaks around 1030−1070 and 1180 cm−1 were reproduced well. The IR spectrum of o-PMO was also calculated, the scale factor was similarly determined with respect to the most intense peak at 897 cm−1, and the vibrational frequencies were calibrated with a scale factor of 0.9793. In Figure 3b, the scaled spectrum is compared with polarized microfocus Fourier transform IR spectra observed from ECC in a micron-sized single crystal of o-PMO.25 Because mechanical deformations will induce changes from o-PMO to t-PMO and from ECC to FCC, the comparatively large single crystal as it was would undergo the IR measurements; therefore, intensities of the spectral bands were not assigned and no spectrum of isotropic D

DOI: 10.1021/acs.macromol.8b01697 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 2. Comparison between Observed and Calculated IR Data on PMO Crystals orthorhombic

trigonal experimentala mode A2

experimentalb

this study

frequency (cm−1)

intensity

2985 1385

m vw

1093

s

895 220

vs

E1 2999 2928

w m

1470 1435 1240

vw vw m

1093

s

938

m

633 457

w

frequency (cm−1)

intensity (km mol−1)

3127 1404 1100 1094 928 895 204 105 3132 3127 3061 3055 1496 1469 1429 1242 1182 1103 1068 1033 938 924 922 624 454

3.3 58.5 314.8 473.4 57.4 4332.5 133.4 7.4 37.1 22.2 12.5 148.9 17.1 10.9 0.7 233.9 27.2 588.2 12.4 15.2 707.0 156.2 18.7 157.1 116.3

mode B1

frequency (cm−1)

this study

intensityc

1392 1090 897 636 304

B2 1487 1430 1291 1221 1110 936 428 B3

1238

594 434

frequency (cm−1)

intensity (km mol−1)

1402 1086 897

11.1 249.0 1504.5

282 150 3043 1481 1452 1296 1209 1131 934 589 424 3108 3038 1500 1455 1304 1212 1112 930 590 428

174.0 11.0 19.5 1.7 16.5 0.3 43.8 188.1 226.4 19.1 16.2 25.5 156.3 8.1 9.6 17.5 28.9 282.6 126.7 34.3 9.6

a From needle-like single crystals.37 bFrom a micron-sized single crystal.19,25 cThe intensities of the bands were not assigned.

ECC of o-PMO was reported.25 In this case also, the agreement between theory and experiment is generally satisfactory although some discrepancies are found above 1400 cm−1. In Table 2, the major observed and calculated IR peaks are compared. In the IR experiments, the following differences between t-PMO and o-PMO were used to trace the o → t transition and estimate the t-PMO/o-PMO ratio: (a) 633 (624) cm−1 (t-PMO) and 594 (590) cm−1 (o-PMO) (see Figure 3);19 (b) 101 (105) cm−1 (t-PMO at 102 K) and 142 (150) cm−1 (o-PMO at 93 K);17 and (c) 220 (204) cm−1 (tPMO) and 297−304 (282) cm−1 (o-PMO),19 where the observed and calculate frequencies are given outside and inside the parentheses, respectively. In addition, the so-called Davydov split pair characteristic of o-PMO was observed at 428 (424) cm−1 and 434 (428) cm−1 (see Figure 3).19 In general, the agreement between theory and experiment seems to be satisfactory. The acceptable reproduction of the IR spectra as well as the crystal structures shows the reliability of the present B3LYP-D computations. Transitions between Orthorhombic and Trigonal Modifications. The calculated thermodynamic functions of the t-PMO and o-PMO crystals are listed as a function of temperature in Table S2 (Supporting Information). The Gibbs free energy, G, may be calculated from G = Eel + E0 + E T + PV − TS = H − TS

energy; P, pressure; V, volume; T, absolute temperature; S, entropy; and H, enthalpy. Here, all the thermodynamic quantities are defined per monomeric unit, −CH2O−. It has been pointed out that thermochemical data derived from periodic DFT calculations using atom-centered basis sets such as 6-31G(d) may suffer from the BSSE.39,40 As described above, the BSSE values of t-PMO and o-PMO were determined by the CP method. With the BSSE term, the BSSE-corrected free energy, GCP, may be evaluated from GCP = (Eel − BSSE) + E0 + E T + PV − TS = H CP − TS = G − BSSE

(6) CP

Here, the BSSE-corrected enthalpy, H , is given by H−BSSE. Let the difference in the thermodynamic function (A) between t-PMO and o-PMO be defined as ΔA = A t − Ao

(7) CP

CP

where A = G, H, TS, G , or H , and the subscripts, t and o, represent t-PMO and o-PMO, respectively. In Figure 4a, the calculated ΔGCP values are plotted against temperature, together with ΔHCP and ΔTS. In Figure 4a, the solid line fitted to the data points at intervals of 20 K is expressed by a cubic functions ΔGCP(t) = at3 + bt2 + ct + d, where t is temperature in °C and a, b, c, and d are adjustable parameters. The cubic equation ΔGCP(t) = 0 yields t = 69 °C, at which the two crystalline phases are suggested to reach thermal equilibrium. Below 69 °C, GCP t > GCP o ; this means that o-PMO is thermodynamically more stable CP than t-PMO. Above 69 °C, GCP t < Go ; t-PMO is preferred to

(5)

where the symbols are as follows: Eel, electronic energy; E0, zero-point energy; ET, thermal contribution to the vibrational E

DOI: 10.1021/acs.macromol.8b01697 Macromolecules XXXX, XXX, XXX−XXX

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boundary condition in the chain-axis direction; the t-PMO and o-PMO chains keep the conformation as in the crystalline cell. In the isolated state, the ΔTS term markedly increases with temperature, hence the 9/5-helical t-PMO chain becomes more stable than 2/1-helical o-PMO even at much lower temperatures, and as a consequence the o → t transition is suggested to occur at −108 °C. In Figure 4c, it is implied that the single isolated t-PMO free from two-dimensional constraints by its neighbors would be superior in entropy to o-PMO by as much as 7.6 cal K−1 mol−1 at the transition temperature (69 °C) predicted in Figure 4a. By comparison between Figure 4, panels a and c, it is revealed that interchain interactions of the o-PMO crystal enthalpically stabilize the crystalline structure and simultaneously suppress the entropic movement to change to t-PMO. As stated above, the DFT calculations exactly reproduced the transition temperature; however, the ΔHCP value at 69 °C is evaluated to be 0.82 kcal mol−1, being much larger than the DSC value of 0.6 kJ mol−1.17 Compared with the entropy of fusion (3.5 cal K−1 mol−1) of t-PMO at the equilibrium melting temperature of 206 °C,6 the o → t transition entropy (820/ 342 = 2.4 cal K−1 mol−1) might be rather overestimated. The enthalpy and entropy differences may be partly due to the crystal structures at 0 K; in the thermodynamical computations, the crystal structures of t-PMO and o-PMO were always kept as optimized at 0 K. Preferably, the crystal structure is so optimized at each temperature as to minimize the Gibbs free energy and reach the equilibrium structure, for which the thermodynamic functions should be calculated. Such algorithmic improvement in the computation is expected to be introduced in the near future. By the DFT method using the PBE functional and DNP basis set, Tashiro et al.21 have calculated the o → t transition temperature, enthalpy, and entropy to be 225 K, 0.297 kcal mol−1, and 1.32 cal K−1 mol−1, respectively. The enthalpy and entropy are, respectively, twice and three times larger than the DSC values. The DSC experiment was conducted under specific conditions because the normal sample preparation caused diffuse multiple peaks on the DSC curve. Although the DSC measurement exhibited a sharp peak at 69 °C, it had a long tailing.17 The variable-temperature X-ray diffraction experiment detected a wide transitional region of 330−350 K, where o-PMO and t-PMO coexist.21 Therefore, the experimental ΔH and ΔS values may include significant allowances. Crystalline Moduli. Mechanical properties of a polymer are most important factors for its practical applications, and the crystalline modulus at 0 K corresponds to its ultimate hardness. In a previous paper,29 we assigned the space group of t-PMO to P31, optimized the crystal structure, and derived its stiffness and compliance tensors and related mechanical properties. The results, lattice constants, and Young’s moduli in the a-, b-, and c-axis directions are written in the parentheses of Table 1. In the present study, the structure was optimized under the space group of P32, and mechanical properties were calculated therefrom. As seen from Table 1, the differences between the previous and present studies are negligible. For details of mechanical properties of t-PMO, its stiffness (C) and compliance (S) tensors, and the related discussion, see the previous paper.29 The C and S tensors of o-PMO were obtained as follows

Figure 4. Differences in Gibbs free energy (filled circle), enthalpy (open square), and entropy (open triangle) terms between t-PMO and o-PMO as a function of temperature: (a) for the crystalline cell with the CP correction (ΔGCP, ΔHCP, and ΔTS); (b) for the crystalline cell without the BSSE correction (ΔG, ΔH, and ΔTS); and (c) for the single chain of infinite length (ΔG, ΔH, and ΔTS). The solid curves passing through the data points are expressed by cubic functions of temperature t. The cubic equation ΔGCP(t) or ΔG(t) = 0 (horizontal line) yields t = (a) 69 °C, (b) 6 °C, or (c) −108 °C.

o-PMO in this temperature range. Actually, Kobayashi et al.17 observed the o → t transition at 69 °C as a sharp endothermic peak by DSC. Therefore, the DFT-D computations exactly reproduced the phase transition temperature. Figure 4a also indicates that o-PMO is enthalpically more stable and entropically less favored than t-PMO. This can be confirmed from the consequence that o-PMO is lower in ΔECP by 0.21 kcal mol−1 than t-PMO. As temperature is raised, the ΔTS term increases and, above 69 °C, t-PMO becomes preferred to o-PMO. Figure 4b shows plots of ΔG, ΔH, and ΔTS versus temperature. The thermodynamic functions are subjected to the BSSE, and the o → t transition temperature was estimated to be 6 °C. This result shows that the BSSE considerably affects the relative stabilities of the two polymorphs and, consequently, shifts the transition temperature. Figure 4c represents the thermodynamic stabilities of t-PMO and oPMO as a single isolated chain in vacuo. The thermodynamic functions were calculated under one-dimensional periodic F

DOI: 10.1021/acs.macromol.8b01697 Macromolecules XXXX, XXX, XXX−XXX

ÅÄÅ 26.307 16.163 13.260 ÅÅ ÅÅ ÅÅ 21.905 7.396 ÅÅ ÅÅ ÅÅ 89.585 C = ÅÅÅÅ ÅÅ ÅÅ ÅÅ ÅÅ ÅÅ ÅÅ ÅÇ

Macromolecules

0.000 0.000 0.000 5.450

ÄÅ ÅÅ 73.0944 − 51.7252 − 6.5483 ÅÅ ÅÅ ÅÅ 83.5651 0.7567 ÅÅ ÅÅ ÅÅ 12.0694 S = ÅÅÅ ÅÅ ÅÅ ÅÅ ÅÅ ÅÅ ÅÅ ÅÅÇ

0.000 0.000 0.000 0.000 7.599

ÑÉÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ(GPa) ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ 12.763ÑÑÑÖ

Article

0.000 0.000 0.000 0.000 0.000

É 0.0000 0.0000 0.0000 ÑÑÑÑ ÑÑ 0.0000 0.0000 0.0000 ÑÑÑÑ ÑÑ 0.0000 0.0000 0.0000 ÑÑÑÑ Ñ 183.4991 0.0000 0.0000 ÑÑÑÑ ÑÑ 131.6020 0.0000 ÑÑÑÑ ÑÑ 78.3489 ÑÑÑÖ

(TPa−1)

(8)

(9)

From the S tensor, Young’s moduli in the a, b, and c directions, Ea, Eb, and Ec, of o-PMO can be evaluated to be 14, 12, and 83 GPa, respectively. For o-PMO, however, no experimental crystalline moduli have been reported. This is because a mechanical stress on o-PMO gives rise to the o → t conversion. This study has indicated that t-PMO is stiffer especially in the c axis (chain axis) direction than o-PMO. Young’s modulus E(l1, l2, l3) in an arbitrary direction indicated by a unit vector (l1, l2, l3) of the orthorhombic cell can be calculated from E−1(l1, l 2 , l3) = l o l14s11 + 2l12l 22s12 + 2l12l32s13 o o o o o o o o + l 24s22 + 2l 22l32s23 o o o o o o o + l34s33 o o m o o o + l 22l32s44 o o o o o o o o + l12l32s55 o o o o o o o + l12l 22s66 n

Figure 5. Young’s modulus distribution on the ab plane (a) t-PMO29 and (b) o-PMO. The spacing between the neighboring dotted lines corresponds to 10 GPa.

respectively, evaluated to be 17, 17, and 115 GPa (t-PMO) and 14, 12, and 83 GPa (o-PMO); t-PMO is stiffer than o-PMO. In conclusion, this study has given a demonstration that the computational chemistry scheme enables us to comprehensively elucidate structure−property relationships of polymer crystals.

(10)

where sij is the (i, j) element of the S tensor. The ab-plane distributions of the crystalline modulus of t-PMO and o-PMO are depicted in Figure 5. The distribution of t-PMO is perfectly circular, whereas that of o-PMO is symmetric but anisotropic, that is, large in the diagonal direction and small in the a- and baxis direction.

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METHODS

Density Functional Theory Calculations under Periodic Boundary Conditions. DFT calculations were performed under the periodic boundary condition with the CRYSTAL14 program26,41 installed in an HPC 5000-XBW216TS-Silent computer. The structural optimization was carried out with the experimental lattice constants and fractional coordinates set as the initial data. The computational conditions were as follows: DFT, B3LYP with Grimme’s dispersion force expression (B3LYP-D)27,42 and Milani’s parameters;43 basis set, 6-31G(d,p); space group, P32 (trigonal 9/5 helix crystal) or P212121 (orthorhombic 2/1 helix crystal); selfconsistent field (SCF) convergence threshold, 10−8; truncation criteria for bielectronic integrals, 10−7, 10−7, 10−7, 10−7, and 10−14; integration grid, 75 radial and 974 angular points; Fock/Kohn−Sham matrix mixing, 80% (with the modified Broyden method); and shrinking factor, 4. For the optimized crystal structures, infrared spectroscopic frequencies and intensities, and thermochemical quantities were evaluated.44,45 The stiffness and compliance tensors were calculated with the ELASTCON routine46,47 in the CRYSTAL14 program under a rigid threshold (10−8) for the SCF convergence. The number of points for calculation of the numerical second derivative was 5, and the strain-step size was 0.005.

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CONCLUSIONS Density functional theory calculations with a dispersion force correction under the periodic boundary condition were carried out for t-PMO and o-PMO. The optimized crystal structures are consistent with those determined by X-ray diffraction. The interchain cohesive energies corrected for the BSSE via the counterpoise method were evaluated to be −3.17 kcal mol−1 (t-PMO) and −3.38 kcal mol−1 (o-PMO). The computed IR frequencies, calibrated with scale factors of 0.9846 (t-PMO) and 0.9793 (o-PMO), agree closely with those observed from fully extended chain crystals. The thermodynamic functions were calculated, and the difference in Gibbs free energy between t-PMO and o-PMO was expressed as a function of temperature and found to become null at 69 °C, where t-PMO and o-PMO are suggested to reach thermal equilibrium and above which the o → t transition can occur. The predicted transition temperature exactly agrees with experiment (69 °C). Young’s moduli in the a-, b-, and c-axis directions were, G

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Macromolecules

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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.macromol.8b01697.

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Table S1: as-calculated IR frequencies and intensities. Table S2: calculated thermodynamic parameters without the BSSE correction (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81 (0)43 290 3394. Fax: +81 (0)43 290 3394. ORCID

Yuji Sasanuma: 0000-0003-0374-0621 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the late Professor Roberto Orlando of University of Torino for helpful advice on computations with the CRYSTAL14 program. This study was partially supported by the Grants-in-Aid for Scientific Research (C) (16K05906) from the Japan Society for the Promotion of Science.

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DOI: 10.1021/acs.macromol.8b01697 Macromolecules XXXX, XXX, XXX−XXX