Computational Characterization of Nylon 4, a Biobased and Biodegradable Polyamide Superior to Nylon 6

This study is an attempt to develop a theoretical methodology to elucidate or predict the structural characteristics and the physical properties of an isolated polymeric chain and its crystalline state precisely and quantitatively. To be more specific, conformational characteristics of a biobased and biodegradable polyamide, nylon 4, in the free state have been revealed by not only ab initio molecular orbital calculations on its model compound but also nuclear magnetic resonance experiments for the model and nylon 4. Furthermore, the crystal structure and solid-state properties of nylon 4 have been elucidated by density functional theory calculations with a dispersion force correction under periodic boundary conditions. In the free state, the nylon 4 chain forms intramolecular N–H···O=C hydrogen bonds, which force the polymeric chain into distorted conformations including a number of gauche bonds, whereas nylon 4 crystallizes in the fully extended all-trans structure (α form) that is stabilized by intermolecular N–H···O=C hydrogen bonds. The intermolecular interaction energy (ΔECP) in the crystal was accurately calculated via a counterpoise (CP) method contrived here to correct the basis set superposition error, and the ultimate crystalline modulus (Eb) in the chain axis (b axis) direction at 0 K was also evaluated theoretically. The results were compared with those obtained from the α and γ crystalline forms of nylon 6, and, consequently, the superiority of nylon 4 to nylon 6 in thermal stability and mechanical properties was indicated: the ΔECP and Eb values are, respectively, −214 cal g–1 and 334 GPa (nylon 4), −191 cal g–1 and 316 GPa (α form of nylon 6), and −184 cal g–1 and 120 GPa (γ form of nylon 6). In conclusion, nylon 4 is expected to be put to practical use as a tough environmentally friendly polyamide.


■ INTRODUCTION
In order to suppress global warming, we are increasingly required to replace chemical production based on fossil resources with manufacture from carbon-neutral feedstock. Nylon 6 is produced by ring-opening polymerization of ϵcaprolactam, 1 and its monomeric unit includes six carbon atoms originating from the starting material, benzene, phenol, cyclohexane, or 1-cyclohexanol. If the carbon sources were produced from plants, nylon 6 would become a carbon-neutral material. Nylon 6 was suggested to be degraded by a white rot fungi strain IZU-154, Phanerochaete chrysosporium, and Trametes versicolor through oxidation processes; 2,3 however, the degradation is so slow that nylon 6 is far from biodegradable.
Nylon 4 is produced from 2-pyrrolidone, which has recently come to be prepared by fermentation of biomass using Escherichia coli from biobased γ-glutamic acid via γ-aminobutyric acid; 4 therefore, nylon 4 can be considered to be biobased and carbon-neutral. Furthermore, it was found that nylon 4 is hydrolyzed to γ-aminobutyric acid by Pseudomonas sp. ND-11 widely inhabiting activated sludge and finally decomposed into CO 2 , H 2 O, and NO 3 − . 5−10 Besides, this polyamide is degraded in vivo. 11 Nylon 6 is superior in mechanical strength, rigidity, thermal stability, and chemical residence. 12 These favorable properties are partly due to interchain N−H···OC hydrogen bonds. Nylon 4 also forms so strong interchain hydrogen bonds as to exhibit melting points of 260−265°C higher than that (225°C ) of nylon 6. 12−14 This means that, compared with nylon 6, nylon 4 has an advantage of superior thermal resistance and a disadvantage of higher energy costs in annealing and molding processes. Despite the drawback, nylon 4 is expected to become a biobased and biodegradable substitute for nylon 6. In expectation of future utilization of nylon 4, we have elucidated its structural characteristics and physical properties by computational chemistry and nuclear magnetic resonance (NMR) experiments as follows: (chain characteristics) conformational analysis via molecular orbital (MO) calculations and NMR experiments on its small model compound, N-acetyl-γ-aminobutyric acid N′-methylamide (designated herein as ABAMA, Figure 1) and nylon 4; (crystal structure and properties) density functional theory (DFT) calculations  The dihedral angle is defined here according to the IUPAC recommendation: 18 trans (t) ≈ 180 ± Δ°; gauche + (g + ) ≈ −60 ± Δ°; gauche − (g − ) ≈ +60 ± Δ°, where Δ stands for the allowance. b The dihedral angles of bond 6 are distributed too widely to be classified into a few RISs. c At the MP2/6-311+G(2d,p)//B3LYP/6-311+G(2d,p) level. Relative to ΔG k of the all-trans conformation. d For the bond numbers, see Figure 1. e Dimethyl sulfoxide. f The molecular geometry was optimized with the dihedral angles fixed at 180°so that the all-trans conformation would be the control for ΔG k and 13 C NMR chemical shifts.

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Article in the γ form, the optimum packing of the methylene units appears to dominate over the hydrogen bond. 17 Therefore, nylons 4 and 6 prefer the α form, whereas nylons n (n ≥ 8) show the γ preference. Figure 2 illustrates some conformations with low ΔG k values, showing the formation of N−H···OC close contacts, that is, intramolecular hydrogen bonds. Here, ΔG k represents the free energy difference of conformer k from that of the alltrans form. For the conformer number k, see Table 1. The ΔG k values including solvation of 2,2,2-trifluoroethanol (TFE) and H···O distances were evaluated to be, respectively, −3.92 kcal mol −1 and 2.00 Å (k = 15), −3.59 kcal mol −1 and 1.94 Å (k = 3), and −2.95 kcal mol −1 and 2.65 Å (k = 14). Therefore, it is obvious that the low ΔG k 's are due to the intramolecular hydrogen bonds.
The trans fractions (p t 's) of bonds 3−5 were calculated with the ΔG k data according to the conformational classification shown in Table 1. All the p t values (see Table 3) are so small as to indicate that ABAMA strongly prefers distorted shapes rich in gauche conformations ( Figure 2). Because the p t values tend to increase with medium polarity, the intramolecular hydrogen bonds would be weakened or cleaved by the polar solvents.
NMR Experiments. Vicinal Coupling Constants. Figure 3 shows the 1 H NMR spectra observed from three kinds of methylene protons, BB′, CC′, and DD′ (see Figure 1), of ABAMA and nylon 4. The gNMR simulations 19 for ABAMA exactly reproduced the observed spectra to yield the chemical shits and spin−spin coupling constants. Of the NMR parameters, only vicinal coupling constants, used in the conformational analysis, are listed in Table 2.
The vicinal 1 H− 1 H coupling constant ( 3 J XY ) between protons X and Y can be expressed as a function of trans (p t ) and gauche (p g ) fractions of the centrally intervening bond as follows: between protons A and B (B′) about bond 3 between protons C (C′) and D (D′) about bond 5 Figure 2. Stable conformers of ABAMA with the electrostatic potential surface: (a) k = 15, g + g − g − (∼g + ); (b) k = 3, tg + g − (∼t); (c) k = 14, g + g − g + (∼t). k is the conformer number (see Table 1), and approximate conformations of bond 6 are written within parentheses. The dotted lines represent intramolecular N−H···OC hydrogen bonds.  Figure S1 (Supporting Information).

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By definition, the following relation must always be fulfilled: The proton symbols and bond numbers are defined in Figure 1. The coefficients (J C 's J E ′, J G 's, and J T 's) of the individual equations, illustrated in Figure 4, were obtained as follows: (set A) from the Karplus equation for the same bond sequence as that of ABAMA; (sets B, D, and F) from MO calculations for ABAMA itself at the B3LYP/6-311++G-(3df,3pd)//B3LYP/6-311+G(2d,p) level; (sets C and E) from experimental coupling constants of a ring compound, 2methylpiperidine or cyclohexane, which has a bond sequence similar to that of ABAMA. The numerical values of the coefficients are written in the footnotes of Table 3. Substitution of the observed 3 J XY values into the above equations yields p t 's and p g 's of the individual bonds, and only the p t values are listed in Table 3 because of eq 6.
Compared with the p t 's derived from the MO calculations, those of sets D and F are somewhat larger, whereas those of sets A, B, C, and E show good agreement with the theoretical values (Table 3); in general, the agreement seems to be fully acceptable. As discussed above, the MO calculations suggest that the small p t values result from the intramolecular N−H··· OC hydrogen bonds. Therefore, the NMR data also indicate formation of the hydrogen bonds in the solutions.
Inasmuch as both MO and NMR data here suggest that ABAMA forms the strong intramolecular N−H···OC hydrogen bond, it is probable that the isolated nylon 4 chain also includes the hydrogen bonding. To prove this conjecture, we measured the 1 H NMR spectra of nylon 4 itself and attempted to reproduce the observed spectra using the same geminal and vicinal coupling constants as derived from ABAMA. In Figure 3c, as an example, the measured and calculated spectra of nylon 4 in TFE at 25°C are compared. It can be seen that both spectra are essentially identical. This fact indicates that nylon 4 has the 3 J XY and p t values (i.e., conformational distribution) close to those of ABAMA and hence supports the abovementioned conjecture. All spectra observed from nylon 4 dissolved in TFE at different temperatures were reproduced exactly. 13 C NMR Chemical Shifts. Ando et al. 23 measured solidstate 13 C NMR of nylon 4 via the cross-polarization/magicangle spinning (MAS), 13 C pulse saturation transfer/MAS, and 13 C low-power decoupling/MAS techniques, determined the 13 C chemical shifts of the crystalline and noncrystalline chains separately, and plotted the chemical shift differences (Δδ i 's, i, carbon species) between the crystalline and amorphous carbons against the carbon position as shown in Figure 5. The 13 C chemical shift differences of nylon 4 between in the crystallite and in a 1,1,1,3,3,3-hexafluoro-2-propanol (HFIP) solution were also investigated. As seen from Figure 5, both Δδ i values depend largely on the carbon position in the backbone, and the two plots (open triangle and square) oscillate similarly. It was suggested that the experimental data would be interpreted in terms of the so-called γ and δ substituent effects. The former and latter effects stem from the shielding or deshielding caused by chemical species (atoms) separated from the observed carbon atom by three and four bonds, respectively, thus depending on conformations of the in-between bonds. 24 On the basis of these effects, conformational analyses of polymers have been carried out: 25,26 the δ and γ effect parameters are appropriately assumed and weightaveraged over the possible conformations to yield 13 C chemical shifts, which are compared with the corresponding experimental values. 27,28 However, it is too difficult to carry out

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Article conformational analysis of nylon 4 and ABAMA in this manner, because, as discussed above, dihedral angles of bonds 3 and 6 are variable to a large extent; therefore, it is impossible to simulate the 13 C chemical shifts with a small number of δ and γ effect parameters.
Instead, we calculated the chemical shift differences (Δδ i 's) of ABAMA between in the all-trans conformation and in the free state. The chemical shift ⟨δ i ⟩ of carbon i of ABAMA in the free state can be calculated as the weight average over all the possible conformations where δ k i is the 13 C chemical shift of carbon i of conformer k, ΔG k is given in Table 1, R is the gas constant, and T is the

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Article absolute temperature. The calculated δ k i and ⟨δ i ⟩ values are listed in Table S1 (Supporting Information). The Δδ i value of ABAMA is defined as the difference from that (δ all-trans i ) of the all-trans state The Δδ i values, calculated from the δ k i and ΔG k data on the gas phase (filled circle) and TFE solution (filled triangle), are plotted in Figure 5. Because nylon 4 crystallizes in the all-trans conformation, it is meaningful to compare the Δδ i values between ABAMA and nylon 4. Figure 5 clearly shows that the Δδ i plots of ABAMA oscillate analogously to the data of Ando et al. 23 on nylon 4. This fact indicates that the model and polymer undergo similar γ and δ effects and hence have analogous conformational distributions. As shown above, ABAMA strongly prefers the bent conformations because of the intramolecular N−H···OC hydrogen bonds ( Figure 2); therefore, the nylon 4 chain also lies in such distorted conformations in the free state (viz., melt, amorphous phase, and solutions). Both vicinal 1 H− 1 H coupling constants and 13 C chemical shifts show that the free nylon 4 chain tends to form the intramolecular hydrogen bonds.
This conclusion is also based on Flory's assertion: 29−31 in the statistical thermodynamics of polymer systems, the partition function of polymer solutions can be factorable into intra-and intermolecular parts, and each is independent of the other. The latter factor depends on the composition, whereas the former is unaffected by dilution. It follows immediately that the configuration of the chain should be unperturbed (as if it were dissolved in a theta solvent) even in the amorphous and molten states with other chains at a high density. Configurational properties of polymers in the unperturbed state will be influenced only by short-range intramolecular interactions. 15 In fact, for example, the spatial configuration of nylon 6,6 with hydrogen bonds was predicted under the RIS scheme based on intramolecular interactions within the monomeric unit, 32 being in good agreement with the experiment. 33 Flory's assertion has been demonstrated by small-angle neutron-scattering experiments. 31,34 Structure Here, the experimental data are written in the parentheses. 16,35,36 The agreement between theory and experiment is satisfactory. Milani et al. 37 also optimized the α and γ structures of nylon 6 at the same level and derived slightly different lattice constants from ours; however, the discrepancies may stay within allowances of the structural optimization. Figure 6 shows the optimized crystal cell of the α form of nylon 4. For the optimized fractional coordinates, see to be 0.341 Å, where N at is the total number of atoms in the unit cell and, for example, x cal,i and x exp,i represent the calculated and experimental x positions of the i-th atom, respectively. In Table S4 (Supporting Information), the structure factors for X-ray diffraction are listed. The reliability factor, R . 23 i indicates the carbon species (CO, α CH 2 , β CH 2 , and ω CH 2 , see Figure 1a).

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Here, for example, E A{AB} is the energy of molecule A, calculated with basis sets {AB}; molecule B supplies only its basis set and hence is termed ghost. Therefore, the BSSE may be estimated from For three-dimensional crystals in which polymeric chains of innite length are packed densely, we have devised a suitable CP method (Figure 7). The BSSE per repeating unit may be dened as where E SC{Domain} (n g ) is the energy of a single chain (SC) located in a domain filled with the n g ghost chains, and E SC{SC} is the energy of the single isolated chain. Both E SC{Domain} (n g ) and E SC{SC} can be calculated under the one-dimensional periodic boundary condition (in the chain axis direction).
Therefore, the interaction energy per repeating unit, based on the CP method, can be calculated from where E Crystal is the total energy of the crystal including Z repeating unit, and E SC{SC}+D is the SC energy with the dispersion force correction. The E SC{Domain} (n g ), E SC{SC} , and E SC{SC}+D terms are evaluated per repeating unit. In principle, the ΔE CP (n g ) value must become more reliable (freer from the BSSE) with increasing domain size, that is, the number (n g ) of ghost chains (see Figure 7). Ultimately, the most reliable BSSE can be derived from From BSSE(∞), we can obtain the ideal interaction energy, ΔE CP (∞). In this study, the E SC{Domain} and E SC{SC} energies were calculated at the B3LYP/6-31G(d,p) and B3LYP/ pob_TZVP levels, and the E Crystal and E SC{SC}+D energies were evaluated at the same levels with the dispersion force correction (B3LYP-D/6-31G(d,p) and B3LYP-D/ pob_TZVP). Figure 8 shows the BSSE (n g ) versus n g plot for the α form of nylon 4, in which the magnitude of BSSE (n g ) increases with n g and finally converges on a horizontal line: BSSE(∞) = −6.06 kcal mol −1 (B3LYP/6-31G(d,p)) or −5.24 kcal mol −1 (B3LYP/pob_TZVP). The ΔE CP (∞) energy was derived from the BSSE(∞) with eq 15 as shown in Table 4. For more detailed numerical data, see Table S5 (Supporting Information). Similarly, the ΔE CP (∞) energies of nylon 6 were evaluated. Because nylon 4 is different from nylon 6 in monomer size, the ΔE CP (∞) value was recalculated in two units: kcal per mol of the skeletal bond; cal g −1 . The scaled values suggest that nylon 4 would be more thermally stable than nylon 6. Actually, the melting point of nylon 4 has been observed in the range of 260−265°C, 13 whereas the equilibrium melting point (T m 0 ) of nylon 6 was determined to be 225°C. 14 However, it should be noted that the
Crystal Elasticity. As for polymers, mechanical properties may be the most important characteristics especially from a practical point of view. Inasmuch as semicrystalline polymers comprise crystalline and amorphous regions, the crystalline modulus in the chain axis direction at 0 K is, in principle, the largest modulus that the polymer can exhibit. 43 Shown in Appendix A (Supporting Information) are the stiffness and compliance tensors calculated for the α form of nylon 4 and the α and γ forms of nylon 6. The compliance tensor yields Young's moduli (E a , E b , and E c , respectively) in the a, b (chain axis), and c axis directions of the crystal. In Table 5, the calculated crystalline moduli at 0 K are compared with other theoretical and experimental data. Most of the experimental values were determined from X-ray diffraction measurements at room temperature. Therefore, it is natural that the crystalline moduli calculated here are larger than the experimental values, because the crystalline modulus depends on the temperature (cf. E b 's (100 and 270 GPa) of the α form of nylon 6 at 18 and −150°C, respectively, Miyasaka, 1980). 44 Interestingly, other theoretical data are, in general, similar to our calculations. Among the three crystalline forms, the magnitude relation of E b can be expressed as α of nylon 4 > α of nylon 6 ≫ γ of nylon 6, and the modulus in the hydrogen bond direction is also in the order of α of nylon 4 (E a ) > α of nylon 6 (E a ) > γ of nylon 6 (∼E c ). In the α form, the hydrogen bond forms nearly in the a axis direction (see Figure 6), whereas, in the c axis direction, only van der Waals interactions exist; therefore, E a is expected to be larger than E c , as the present calculations indicate.

■ CONCLUSIONS
The MO calculations on the model compound, ABAMA, well reproduced not only the bond conformations determined from Figure 8. BSSE of the α form of nylon 4 as a function of the number (n g ) of ghost chains. As n g increase, the BSSEs at the B3LYP/6-31G(d,p) (filled circle) and B3LYP/pob_TZVP (filled square) levels approach −6.06 and −5.24 kcal mol −1 (horizontal dotted lines), respectively. A function of A + B exp(−n g /C) (dotted curve) was fitted to the calculated data, where A, B, and C are adjustable parameters and A corresponds to BSSE(∞).  a E a , E b , and E c are the crystalline moduli in the a, b (chain axis), and c directions of the crystal, respectively. b Only when the measurement temperature was explicitly written in the literature.

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Article NMR vicinal 1 H− 1 H coupling constants but also the conformational dependence of 13 C NMR chemical shifts of nylon 4 existing in the amorphous and solution environments. These facts show that the nylon 4 chain in itself tends to lie in distorted conformations rich in gauche states because of intramolecular N−H···OC hydrogen bonds. The DFT calculations at the B3LYP/6-31G(d,p) level including the dispersion force correction under the three-dimensional periodic boundary condition reproduced the experimental crystal structures of nylons 4 and 6. A CP method for correcting the BSSE of polymer crystals has been devised and applied to the three crystalline forms to yield the following interaction energies: −214 cal g −1 (α form of nylon 4); −191 cal g −1 (α form of nylon 6); and −184 cal g −1 (γ form of nylon 6). Therefore, nylon 4 is suggested to be more thermally stable than nylon 6. The B3LYP-D calculations also yielded the crystalline moduli parallel to the crystalline a, b, and c axes. As the maximum stiffness, Young's moduli in the chain axis (b axis) direction of the crystals at 0 K were predicted: 334 GPa (α form of nylon 4); 316 GPa (α form of nylon 6); and 120 GPa (γ form of nylon 6). These data suggest that nylon 4 is also superior to nylon 6 in stiffness. Because of the potential carbon neutrality and biodegradability, nylon 4 will be used for strong environmentally friendly materials and, probably, also biomedical materials left in human bodies, such as sutures for surgery, whereas its biodegradability would be disadvantageous for the use of durable materials kept long in the environment, for example, fishing lines and nets. In conclusion, nylon 4 will be partly substituted for nylon 6 and open up the possibility of new functional materials. The computational characterization developed here can also be applied to other polymers, furthermore, for molecular design of new polymers.

■ METHODS
Materials. Synthesis of Model Compound, ABAMA. The model compound, ABAMA, was, in principle, prepared by reference to previous studies. 52,53 Aqueous solution of methylamine (40%, 40 mL) was added to ethyl 4-aminobutyrate hydrochloride (2.5 g, 15 mmol), stirred at 0°C for 15 min, and then condensed on a rotary evaporator. The residue was dissolved in a small amount of methanol and added dropwise to petroleum ether to yield a white precipitate. The precipitate was collected by filtration, dissolved in methanol, and dropped into tetrahydrofuran including 7% methylamine to separate out a white solid. After removal of the precipitate, the filtrate was condensed and dried under reduced pressure to yield a yellow oily product, which was identified as γaminobutyric acid N-methylamide by 1 H and 13  Dry pyridine (0.87 mL, 11 mmol) was added to dry chloroform (10 mL) including the γ-aminobutyric acid Nmethylamide (0.44 g, 3.8 mmol) prepared as above. With the solution stirred at 0°C under nitrogen atmosphere, acetyl chloride (0.57 mL, 8.0 mmol) was added slowly. The mixture was allowed to warm to room temperature and stirred overnight. The reaction mixture underwent extraction with 0.1 M hydrochloric acid and 5 wt % NaHCO 3 aqueous solution. After the water layer was condensed, the residue was dissolved in a small amount of methanol and added dropwise to tetrahydrofuran including 7% methylamine to yield a white precipitate. After filtration, the filtrate was evaporated and dried in vacuo to yield a yellow solid, which was identified as ABAMA by 1 H and 13 C NMR and electrospray ionization mass spectrometry (ESI MS). 1 4,8,9,11 The molecular weight distribution was measured by TOSOH Analysis and Research Center Co., Ltd. on a TOSOH HLC-8220GPC system equipped with two TOSOH TSKgel Super AWM-H columns and a refractive index detector under the following conditions: eluting solvent, HFIP including sodium trifluoroacetate (10 mM); polymer concentration, 1 mg mL −1 ; column temperature, 40°C; flow rate, 0.3 mL min −1 ; injection volume, 20 μL; molecular weight calibration, poly(methyl methacrylate) standards. The number-average and weight-average molecular weights were evaluated to be 10.3 and 67.7 kDa, respectively. The size-exclusion chromatographic (SEC) differential curve is shown in Figure S1 (Supporting Information). 1 H NMR Measurements. 1 H NMR spectra of ABAMA were recorded at 400 MHz on a JEOL JNM-ECX400 spectrometer equipped with a variable temperature control unit in the Center for Analytical Instrumentation of Chiba University. The typical conditions were as follows: scan, 32; 90°pulse width, 12 μs; acquisition time, 6.6 s; recycle delay, 5.0 s; temperature range, 15 or 25°C to 45 or 55°C; temperature interval, 10°C. The free induction decay was zero-filled before the Fourier transform so as to yield enough digital resolution for the subsequent spectrum simulation.
The model compound was dissolved at a concentration of 50 mM in chloroform-d, methanol-d 4 , or dimethyl-d 6 sulfoxide, injected into a 5 mm NMR sample tube, and underwent the measurement. However, the TFE solution was prepared differently: ABAMA was dissolved in nondeuterated TFE and injected into a 3 mm glass tube (coaxial insert), the tube was inserted into a normal 5 mm one, and the space between the inner and outer tubes was filled with chloroform-d. The spectra thus obtained were simulated with the gNMR program 19 to yield 1 H chemical shifts and 1 H− 1 H coupling constants. 1 H NMR of nylon 4 was measured and analyzed similarly.
MO Calculations of the Model Compound. MO calculations of ABAMA were carried out with the Gaussian 09 program 54 installed on an HPC Silent-SCC T2 workstation. For each conformer, the molecular geometry was fully optimized at the B3LYP/6-311+G(2d,p) level under tight convergence, thermochemical energies at 25°C and 1 atm were computed by the frequency calculation at the same level, and an accurate electronic energy was calculated at the MP2/6-311+G(2d,p) level. The Gibbs-free energy (ΔG k ) of conformer k was evaluated from the MP2 electronic and B3LYP thermochemical energies, being expressed herein as the

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Article difference from that of the all-trans conformer. The solvation effects on the MP2 electronic energy were also evaluated by the polarizable continuum model using the integral equation formalism variant (IEF-PCM). 55 DFT Calculations on Crystal Structures under Periodic Boundary Conditions. DFT calculations under the periodic boundary condition were carried out with the CRYSTAL14 56,57 program installed in an HPC 5000-XBW216TS-Silent workstation. The initial data (i.e., lattice constants and atomic positions) for the structural optimization of the α form 35 of nylon 4 and the α 36,58 and γ 16 forms of nylon 6 were taken from the crystal structures determined by X-ray diffraction, and the total energy was minimized under the following conditions: DFT, B3LYP with a dispersion force correction (B3LYP-D) described below; basis set, 6-31G(d,p) or pob_TZVP; space group, P2 1 (α form) or P2 1 /a (γ form); self-consistent field (SCF) convergence threshold, 10 −7 ; 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); 57 shrinking factor, 4.
Dispersion Force Correction. As often pointed out, the DFT calculations tend to underestimate the dispersion force, thus being combined here with the empirical function (E disp ) formulated by Grimme 59,60 where E DFT-D is the corrected energy and E DFT is the energy as derived from the DFT calculation. The E disp energy is expressed as = is the dispersion coefficient for atom pair ij, and R ij is the distance between atoms i and j. Here, the damping factor is given as where d represents the steepness of the damping function and R r is the sum of van der Waals radii (R vdW 's) of atoms i and j. For nylons 4 and 6, the dispersion force parameters optimized by Milani et al. 37 were used: s 6 = 1.00, C 6 H = 0.14, C 6 C = 1.75, The C and S tensors were calculated with the ELASTCON routine 63,64 built 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 and the strain-step size along a given deformation were set equal to 5 and 0.005, respectively.