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Single-chain folding of a quenched isotactic polypropylene chain through united atom molecular dynamics simulations
Polymer 183 (2019) 121861
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Single-chain folding of a quenched isotactic polypropylene chain through united atom molecular dynamics simulations Katsumi Hagita a, *, Susumu Fujiwara b a b
Department of Applied Physics, National Defense Academy, 1-10-20, Hashirimizu, Yokosuka, 239-8686, Japan Faculty of Materials Science and Engineering, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, 606-8585, Japan
A R T I C L E I N F O
A B S T R A C T
Keywords: Single isotactic polypropylene chain Structure formation United atom molecular dynamics simulation
The single-chain folding of an isotactic polypropylene (iPP) chain through united atom (UA) molecular dynamics simulations under quenching was investigated using force fields (FFs) based on TraPPE-UA. We estimated the degree of local rigidity of the folded chain along iPP undergoing single-chain folding. To maintain the tacticity of iPP, we introduced improper torsional angle potentials and/or explicit hydrogen atoms bonded to backbone carbon atoms. In the simulation using modified TraPPE-UA FFs with added hydrogen atoms, folded structures were observed. In the cases using modified TraPPE-UA FFs with only improper torsional potentials, folding of the quenched iPP chain was not observed. Moreover, to clarify the folding behavior of the iPP chain, we studied the chirality of a single iPP chain and subsequently observed spontaneous chirality selection during the quenched folding process. We also examined the effect of the angle and torsional potentials of the added hydrogen on the folding behavior of a single iPP chain into local crystals. Therefore, we confirmed that the strength of the angle and torsional potentials contributed to the acceleration of the folding behavior.
1. Introduction Crystallization is an important factor that considerably affects the material properties of polymers [1–8]. Crystallization can be regarded as a typical case of polymer self-organization [1]. As a fundamental aspect of crystallization, chain folding has been more extensively examined for homopolymers rather than heteropolymers such as proteins. In partic ular, molecular dynamics (MD) simulations of the folding behavior of a single polyethylene (PE) chain have been reported numerous times [9–15]. In polymer materials, lamellar structures related to chain folding play an important role in imparting physical and chemical properties. There have been many simulation studies on the lamellar crystal structure formation of linear PE chain melts [6,16–26] as well as on the crystallization of branched and cyclic PE chains [27–33]. In these crystallization processes, it is known that each PE chain folds. Thus, an understanding of the folding behavior of various polymer chains is a fundamental issue of polymeric materials that must be resolved. PE has been studied as the simplest polymer in which carbon atoms are linked in a straight line, whereas the simplest polymer with side chains is considered to be polypropylene (PP). Similar to PE, PP has been widely used in industry as the lowest-density polymer material among
thermoplastic resins. In PP, tacticity plays an important role in deter mining properties such as glass transition temperature Tg, elastic modulus, and crystallinity [34]. In the 1950s, Natta and Corradini established the synthesis of isotactic polypropylene (iPP) [35], wherein methyl groups are bonded in the same direction. The cases in which methyl groups are arranged in alternating and random directions are called syndiotactic PP and atactic PP, respectively, and each has differing properties. For example, Tg of iPP, sPP, and aPP are 255 K, 269 K, and 267 K, respectively [36]. To investigate the crystallization of iPP chains, various molecular simulations have been performed [4,37–41]. To reduce the number of atoms in the system and the computing costs, several united atom (UA) models of iPP chains have been utilized. Romanos and Theodorou [4,37] developed a force field with explicit carbon and hydrogen atoms on the main chain and methyl groups lumped into united atoms. As approaches based on a general UA model that omits hydrogen atoms, Yamamoto and Sawada [38,39] and Pütz et al. [40,41] proposed UA models of iPP chains to conserve their tacticity. Based on the molecular details of the static structure of iPP, the flip of a methyl group causes a change in tacticity. To avoid this methyl group flip, on the basis of a UA model of the transferable potentials for phase equilibria (TraPPE) force field [42],
* Corresponding author. E-mail address: [email protected] (K. Hagita). https://doi.org/10.1016/j.polymer.2019.121861 Received 22 July 2019; Received in revised form 27 September 2019; Accepted 2 October 2019 Available online 3 October 2019 0032-3861/© 2019 Elsevier Ltd. All rights reserved.
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Yamamoto and Sawada [38,39] introduced an explicit hydrogen atom opposite a methyl group, while Pütz et al. [40,41] introduced improper torsional angle potentials. Note that Yamamoto and Sawada [38] also proposed a method in which phantom bond-length constraints are imposed between the nearest CH3 and CH2 and between CH2 and CH2 instead of introducing explicit hydrogen atoms. We did not examine this extension of MD simulation methods. By using the method with explicit hydrogen atoms, Yamamoto confirmed the accelerated crystallization of iPP chains from a highly stretched amorphous state by using UAMD simulations [39]. Recently, Xie et al. confirmed the stretch-induced coil helix transition of iPP chains using full-atom MD simulations [43]. Pütz et al. [40] and Heine et al. [41] studied the chain statistics of iPP by using polymer interaction site model (PRISM) calculations and UAMD simulations. By using the Pütz parameters, Kuppa et al. per formed an atomistic Monte Carlo simulation to investigate the crystal lization behavior of the amorphous region between the crystalline lamellae of iPP [44]. Recently, by using the Pütz parameters, Tzounis et al. examined the tacticity effect on the conformational properties of PP polymers and PE-PP copolymers [34]. However, at present, there has been no comparison between the two approaches. Furthermore, studies on whether a single iPP chain exhibits folding behavior have not pro vided satisfactory results. Thus, the purpose of this study was to examine the folding behavior of a single iPP chain and to compare the two force fields (FFs) of the UA model. In studies on crystallization behavior, FFs that show chemically ac curate behavior are essential. However, because computational re sources that offer highly accurate results are critically insufficient in reproducing slow crystallization behavior similar to real phenomena, we considered that an accelerated FF for promoting crystallization would be useful as a basic aspect for determining the nature of crystallization. For PE chains, it was previously reported that the crystallization rate in MD simulations is sensitive to chain stiffness [16,17]. In our previous study on PE [33], we clarified that the DREIDING-UA FF [45] can be regarded as an accelerated FF to promote PE crystallization. Thus, this study also aimed to find an accelerated FF that promotes the crystallization and folding of iPP chains. First, we clarify our research stance. In order to obtain highly chemically accurate FF parameters, comparisons with experiments and first-principles DFT calculations are important. How ever, the direct observation of the folding of a single iPP chain remains difficult at this time, although AFM observations on a substrate are available [46]. Tracking the dynamics by first-principles DFT calcula tions of a thousand atoms is also impractical. In this paper, priority was given to searching for an FF parameter set that reflects the folding behavior rather than one with chemical precision. To explain the folding behavior of a single iPP chain, Varashey and Carri [47] as well as Yamamoto and Sawada [1,38] reported novel and groundbreaking results. Varashey and Carri examined the transitions between the folded states of helical subchains and coil states [46], for which they reported state diagrams for a single iPP chain with several chain lengths and interaction parameters. Yamamoto and Sawada examined the order-disorder transitions of a folded single iPP chain [1, 38] and investigated the mechanism of the chirality selection of a single iPP chain placed on an ordered iPP substrate. In this study, we investi gated the folding behavior of a single iPP chain and its chirality selection.
consists of a chain of n repeated units of C(C)C in the SMILES notation. The UAs interact via bonded potentials (bond-stretching, bond-bending, and dihedral and improper torsional potentials) and non-bonded po tentials. The total potential energy, Etot, consisted of four parts: (i) the 12-6 LJ potential between non-bonded atoms separated by more than three bonds, ELJ; (ii) the bond-stretching energies for C–C bonds, EBOND; (iii) the bond-bending energies, EANGLE; and (iv) the torsional and improper energies, ETORSION and EIMPROPER. The detailed equations are given in Refs. [38–41]. The large-scale atomic/molecular massively parallel simulator was used for all calculations [49]. The equations of motion for all atoms were solved numerically using the velocity version of the Verlet algorithm with an integration time step of Δt ¼ 0.5 fs. Here, we used a Δt shorter than that for a single PE chain in order to maintain the tacticity of iPP. We applied constant NVT conditions with a Nos�e–Hoover thermostat, and the cutoff distance for the LJ potential was 1.4 nm. The total mo mentum and total angular momentum were assumed to be zero in order to cancel the overall translation and rotation of the chain. The MD simulations were performed by the following procedure: First, a single iPP chain with n ¼ 120 or 240 units of the UA iPP model was placed in a large space. We used 2003 and 2503 nm3 cubic boxes under periodic boundary conditions for n ¼ 120 and 240, respectively. Note that the simulation box was not deformed during these simulations. In these simulations, we observed spontaneous folding in a vacuum space with no adsorption wall. Second, we obtained random configurations of the iPP chain at a high temperature (T ¼ 600 K) by means of an MD simulation lasting longer than 10 ns. Note that we confirmed that the longest Rouse relaxation times of the PE chain at T ¼ 600 K were approximately 0.5 ns and 2.0 ns for n ¼ 120 and 240, respectively. In this study, we used T ¼ 600 K in order to maintain the tacticity of iPP with Δt ¼ 0.5 fs. Third, a 1 ns run at T ¼ 500 K was performed. Finally, the configuration at T ¼ 500 K was cooled stepwise to T ¼ 100 K at a rate of 50 K/ns or 5 K/ns for quenching. 2.2. Modification of TraPPE-UA As explained in section 1, Yamamoto and Sawada [38,39] introduced the explicit hydrogen atoms, and Pütz et al. [40] and Heine et al. [41] introduced the improper torsional angle potentials to maintain the iPP tacticity. We expected both to be effective and that introducing both would improve the stereostructure and promote the crystallization and folding of iPP. The potential for the added hydrogen atoms was set as follows. Based on the original TraPPE-UA model, the non-bonded po tential for the added hydrogen atoms was set to zero. As for the bond potential of C–H, a loose bond was used to reduce the frequency. In practice, it was set to 52 kcal/mol, which is one-tenth of the potential of the C–C bond (520 kcal/mol). Here, we used a loose bond for C–H in order to lengthen the integration time step. For this problem, Yamamoto increased the mass of the explicit hydrogen atom while maintaining the C–H bond value the same as the C–C bond [50]. (Note that for simula tions of many iPP chains with computing costs much larger than that of a single-chain simulation, the SHAKE method [51] to fix the C–H bond length and the RESPA method [52] to lengthen the integration time step are essential.) Regarding angle potentials, we introduced hard angle potentials to accelerate the local crystallization. The angle potentials of C–C–H and R–C–H were set to 1400 kcal/mol, which is 11.2 times that of C–C–C (124.2 kcal/mol). Here, R denotes a united atom of the methyl group. Because the intention was to realize regular folding by balancing the torsional potentials, the torsional angle potentials of C–C–C–H and R–C–C–H had the same value as that of C–C–C–C. Note that the torsional angle potentials of C–C–C–H and R–C–C–H were zero in Yamamoto’s model with the explicit hydrogen atoms [38,39]. Based on the discussion by Pütz et al. [40] and Heine et al. [41], the coefficient of the improper torsional angle potentials was set to a value identical to the bond potential of C–C. To label the used potential model, the model with the improper torsional angle potentials is denoted as the
2. Methods 2.1. Simulation methods In this study, we performed UAMD simulations using the TraPPE-UA FF [42] with and without modification. Note that the FF parameters of TraPPE-UA were obtained by using enhanced Monte Carlo tools [48]. The used parameters were almost the same as those employed in pre vious studies by Yamamoto and Sawada [38,39], Pütz et al. [40], and Heine et al. [41]. We considered that the UA model of an iPP chain 2
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TraPPE-UA (Imp.) model hereafter. The TraPPE-UA (Imp. & H) model denotes the case with explicit hydrogen atoms and improper torsional angle potentials, and TraPPE-UA (H) denotes the model with explicit hydrogen atoms and without improper torsional angle potentials.
terms (Imp. and H): i) modified TraPPE-UA (Imp.) model; ii) modified TraPPE-UA (H) model; and ii) modified TraPPE-UA (Imp. & H) model. Fig. 1 presents the racemic diad ratios of a PP chain with n ¼ 120 after 1 ns runs at T ¼ 400 K, 600 K, 800 K, and 1000 K. The results at T ¼ 200 K, 300 K, 400 K, and 500 K are presented only for the original TraPPE-UA because the PP chain cannot conserve its tacticity. Here, the racemic diad ratio of a perfect iPP chain is zero. In Fig. 1(b), the racemic diad ratio was 0.0342 at t ¼ 1 ns and T ¼ 800 K. For the modified TraPPE-UA, the racemic diad ratio was 0 for T � 600 K. Thus, we found that the three modified versions of TraPPE-UA can maintain the tacticity of the iPP chain. In addition, we confirmed that Yamamoto’s explicit hydrogen atom model conserves the tacticity of the iPP chain despite the fact that torsional potentials associated with the explicit hydrogen atom were not introduced. On the other hand, the original TraPPE-UA failed at maintaining the tacticity of the iPP chain at T � 300 K. Thus, we concluded that the original TraPPE-UA is not suitable for studying the folding behavior of an iPP chain.
2.3. Indicator of a folded single iPP chain According to experimental knowledge [53–60], iPP exhibits a stable crystalline α-form consisting of chains in the 31 helical conformation. In this perfect helical structure, the two vectors riþ6 ri and riþ7 riþ1 are parallel, where ri is the position of the ith carbon atom in the main chain (backbone) [38]. Thus, we used the orientation order parameter P2;i ¼ ð3cos2 θi 1Þ=2, which was defined by the angle θi between vectors riþ6 ri and riþ7 riþ1 . To estimate the local rigidity of the folded single iPP chain along the chain, we computed the probability Pwhole of the angle jθj < π=20. For a single iPP chain with n units (2n carbon atoms in the main chain), we calculated Pwhole as follows: Pwhole ¼
2n X5
1 2n
7
P2;i :
3.2. Fast quenching at 50 K/ns
(1)
i¼3
Fig. 2 shows the temperature dependence of a fast-quenched single iPP chain with n ¼ 120 in the simulations using the modified TraPPEUA FFs. As a test, the quenching rate of 50 K/ns was considered on the basis of a previous study for a single PE chain [33]. From Fig. 2(a), for the TraPPE-UA (Imp.) model, folding of the single iPP chain was not observed. On the other hand, from Fig. 2(b) and (c), we can confirm that the single iPP chain with hydrogen atoms showed folding behavior. Moreover, we found that the folding of the iPP chain with the
In this sum, P2;i involving carbon atoms at both ends were removed. 3. Results 3.1. Verification of conservation of tacticity We examined three UA models as a combination of two additional
Fig. 1. Racemic diad ratios of a PP chain with n ¼ 120 after a 1 ns run. (a) Original TraPPE-UA model, (b) modified TraPPE-UA (Imp.) model, (c) modified TraPPE-UA (H) model, and (d) modified TraPPE-UA (Imp. & H) model. 3
K. Hagita and S. Fujiwara
Polymer 183 (2019) 121861
Fig. 2. Temperature dependence of a fast-quenched single iPP chain with n ¼ 120. (a) Modified TraPPE-UA (Imp.) model, (b) modified TraPPE-UA (H) model, and (c) modified TraPPE-UA (Imp. & H) model.
TraPPE-UA (Imp. & H) model was stronger than that with the TraPPE-UA (H) model. Thus, it was found that the two TraPPE-UA models with hydrogen atoms can be used as FFs to study the folding behavior of a single iPP chain. For T ¼ 100 K, we estimated the ratios of right- and left-handed helical monomer units (RR, RL) with the method reported by Yamamoto and Sawada [38]. (RR, RL) ¼ (0.38, 0.50), (0.52, 0.40), and (0.53, 0.35) for the modified TraPPE-UA (Imp.), (H), and (Imp. & H) models, respectively. The obtained values indicate that no chirality selection of the right- or left-handed helix occurred. Chirality
selection will be discussed in the next subsection. Furthermore, we estimated Pwhole of the three chains. For the modified TraPPE-UA (Imp.) model, Pwhole was 0.463, 0.416, 0.446, and 0.343 for T ¼ 100 K, 200 K, 300 K, and 400 K, respectively. For the modified TraPPE-UA (H) model, Pwhole was 0.652, 0.627, 0.571, and 0.524 for T ¼ 100 K, 200 K, 300 K, and 400 K, respectively. For the modified TraPPE-UA (Imp. & H) model, Pwhole was 0.678, 0.648, 0.614, and 0.601 for T ¼ 100 K, 200 K, 300 K, and 400 K, respectively. From the obtained results, Pwhole seems to function as an indicator of folding.
Fig. 3. Snapshots of a quenched single iPP chain for n ¼ 120 and 240 at T ¼ 100 K. (a) Modified TraPPE-UA (Imp.) model, (b) modified TraPPE-UA (H) model, and (c) modified TraPPE-UA (Imp. & H) model. Here, the quenching rate from T ¼ 500 K to 100 K was 5 K/ns. These 3D visualizations are given in the SI in addition to the snapshots of the other three ensembles. 4
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Polymer 183 (2019) 121861
Discussions for Pwhole are provided in section 3.3 with the results for slow quenching at 5 K/ns.
behavior is high. However, Pwhole is clearly not a sensitive quantity. For the cases with hydrogen atoms, we found that the difference in Pwhole values was small for determining the local crystallinity, although the snapshot differs considerably depending on the presence or absence of improper torsional potentials. Actually, Pwhole is an indicator of the local rigidity along the chain. For an appropriate crystallinity because crys tallization should be a first-order phase transition, it is necessary to generate the local crystallinity from the stem length (local structure continuity) and the amount related to the three-dimensional packing of local crystal structures. We consider that machine learning could play an active role in the generation of simulated crystallinity, as suggested by Welch [61]. Studies in this direction will be performed in the future.
3.3. Slow quenching at 5 K/ns We verified the structural formation of a single iPP chain under quenching from T ¼ 500 K to 100 K at a rate of 5 K/ns. Here, we exam ined the simulations with the three modified TraPPE-UA FFs. Fig. 3 shows snapshots of the quenched single iPP chain for n ¼ 120 and 240 at T ¼ 100 K. As shown in Fig. 3(a), folded structures were not observed for the modified TraPPE-UA (Imp.) model even at a quenching rate of 5 K/ns. On the other hand, from Fig. 3(b) and (c), for the quenched iPP chain using the modified TraPPE-UA with added hydrogen atoms, we found that the case with improper torsional potentials showed stronger folded structures than that without improper torsional potentials. Although the local order of crystallization was high in both cases, the regularity of the entire structure with improper torsional potentials was higher than that without improper torsional potentials. Table 1 shows the (RR, RL) values from four independent ensembles for n ¼ 120 and 240 for the modified TraPPE-UA (Imp.), (H), and (Imp. & H) models at T ¼ 500 K, 400 K, and 100 K. The results of ensemble 1 are shown in Fig. 3. With the modified TraPPE-UA (Imp. & H) model, it was found that chirality was spontaneously selected before folding. In addition, we concluded that the selection of chirality pro gressed slowly even after folding. For the modified TraPPE-UA (H) model, although spontaneous chirality selection before folding did not clearly occur, it slowly progressed after the occurrence of folding. Fig. 4 shows snapshots of a quenched single iPP chain colored with right- and left-handed helical monomer units. We found that chirality selection began above T ¼ 400 K, before folding and after the formation of local rigid structures. Below T ¼ 500 K, for the TraPPE-UA (H) and (Imp. & H) models, we found that the length of helical monomer unit chains with the same chirality became longer for smaller T. From the behaviors between T ¼ 500 K and 400 K as presented in Appendix B, helix inversion occurred with a large deformation of the iPP chain to ward folding. To understand the chirality selection dynamics, it will be necessary to perform further simulations between T ¼ 500 K and 400 K. In particular, whether the helix inversion has been swept out into the folding region or has been eliminated near the chain ends is of interest. Detailed analyses of the helix inversion will be performed in future work. Fig. 5 shows the temperature dependence of the indicator Pwhole for quenching at 5 K/ns. The values of Pwhole for the case of the TraPPE-UA (Imp.) model, which is without hydrogen atoms, were smaller than those for the other cases. Considering the results discussed in section 3.2, it can be concluded that the correlation between Pwhole and the folding
3.4. Effects of strength of angle and torsional angle potential of added hydrogen From the comparison between the presence of hydrogen atoms and/ or improper torsional angle potential, it was considered that the addi tion of hydrogen atoms was essential. We expected that the potentials of angles and/or torsional angles related to the added hydrogen atoms have important roles in determining the folding behavior. To confirm the effect of the angle and torsional angle potentials, we estimated Pwhole of a single iPP chain with n ¼ 120 for several angle and torsional angle potentials strengths. As noted in section 2.2, the coefficient of angle potential related to the added hydrogen atom was set to a value 11.2 times that of the other angle potential. Thus, we examined the cases with magnification factors Cangle ¼ 0.0, 0.1, 0.5, 1.0, 2.0, 3.0, 5.0, and 11.2 with Ctorsion ¼ 1.0. On the other hand, to investigate the magnifi cation factor Ctorsion of the coefficient of torsional angle potential against that for atom types aside from the added hydrogen atom, we examined the cases with Ctorsion ¼ 0.0, 0.1, 0.5, and 1.0 with Cangle ¼ 11.2. Fig. 6 shows the effects of Cangle and Ctorsion on the single iPP chain of n ¼ 120 with the modified TraPPE-UA (Imp. & H) model. From Fig. 6 (a), we found that Pwhole decreased for smaller Cangle < 1.0. On the other hand, for Cangle � 1.0, the difference in the value of Pwhole was expected to be within reasonable error. In fact, no significant difference was found in the results calculated with four independent ensembles. The detailed results are presented in the SI. From Fig. 6(b), it was found that Pwhole decreased for smaller Ctorsion. Based on these results, it was revealed that both Cangle and Ctorsion affect the local crystallization of a single iPP chain, where the effect of Ctorsion appears to be stronger than that of Cangle. The effects of Cangle and Ctorsion on the entire chain structure are examined as follows. Fig. 7 shows the entire chain structures of the quenched single iPP chain of n ¼ 120 for various Cangle and Ctorsion using the modified TraPPE-UA (Imp. & H) model. For Cangle ¼ 0.0, in Fig. 7(a), we found a
Table 1 (RR, RL) for n ¼ 120 and 240 obtained using the modified TraPPE-UA (Imp.), (H), and (Imp. & H) models at T ¼ 500 K, 400 K, and 100 K. n
Modification
Temp.
ensemble 1
ensemble 2
ensemble 3
ensemble 4
120
Imp.
120
H
120
Imp. & H
240
Imp.
240
H
240
Imp. & H
T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K
(0.32, (0.52, (0.47, (0.51, (0.47, (0.73, (0.50, (0.80, (0.83, (0.43, (0.41, (0.54, (0.50, (0.49, (0.59, (0.56, (0.68, (0.70,
(0.35, (0.39, (0.52, (0.62, (0.56, (0.68, (0.72, (0.85, (0.94, (0.32, (0.36, (0.58, (0.40, (0.42, (0.48, (0.67, (0.74, (0.81,
(0.38, (0.42, (0.44, (0.59, (0.62. (0.60, (0.64, (0.85, (0.95, (0.39, (0.35, (0.50, (0.46, (0.50, (0.55, (0.68, (0.82, (0.86,
(0.41, 0.34) (0.36, 0.42) (0.48, 0.40) (0.51, 0.32) (0.47, 0.38) (0.44, 0.49) (0.63, 0.18) (0.72, 0.12) (0.88, 0.09) (0.34, 0.39) (0.45, 0.34) (0.58, 0.30) (0.47, 0.30) (0.50, 0.35) (0.53, 0.41) (0.60, 0.21) (0.71, 0.16) (0.80, 0.16)
0.32) 0.29) 0.43) 0.33) 0.41) 0.23) 0.27) 0.09) 0.13) 0.35) 0.37) 0.37) 0.33) 0.36) 0.35) 0.24) 0.22) 0.22)
5
0.42) 0.35) 0.38) 0.24) 0.28) 0.27) 0.14) 0.06) 0.04) 0.39) 0.43) 0.31) 0.41) 0.47) 0.45) 0.19) 0.11) 0.16)
0.33) 0.41) 0.44) 0.20) 0.22) 0.35) 0.18) 0.03) 0.03) 0.38) 0.43) 0.38) 0.33) 0.31) 0.40) 0.17) 0.09) 0.11)
K. Hagita and S. Fujiwara
Polymer 183 (2019) 121861
Fig. 4. Snapshots of a quenched single iPP chain colored with right- (red) and left-handed (blue) helical monomer units. Green means other. Here, the quenching rate from T ¼ 500 K–100 K was 5 K/ns. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
Fig. 5. Temperature dependence of indicator Pwhole under quenching at 5 K/ns.
Fig. 6. Temperature dependence of indicator Pwhole under quenching at 5 K/ns.
failure in the folding, although folded chains were observed for Cangle � 0.1. Here, for Cangle ¼ 0.0 and 0.1, the tacticity was not conserved, and the racemic diad ratios of the PP chain were 0.428 and 0.034, respec tively. For larger Cangle, the structure was straighter and more regular; this behavior appears to correspond to indicator Pwhole. For Cangle ¼ 5.0 and 11.2, the folded structures were similar to those for Cangle ¼ 2.0 and 3.0, although they are not presented here. From these behaviors, it became clear that the stiffness of the iPP chain originating from the C–C–H and R–C–H angle potentials is related to the ease of chain folding. For Ctorsion ¼ 0.0, in Fig. 7(g), we found a failure in the folding, although folded chains were observed for Ctorsion � 0.1. From this behavior, we considered that the stiffness of the iPP chain originating from the C–C–C–H and R–C–C–H torsional potentials also plays an important role
in the ease of chain folding. Note that the case with Ctorsion ¼ 0.0 was slightly different from Yamamoto’s model [38,39] shown in Appendix A since the improper potential was considered. From these results, we can conclude that both the angle and torsional angle potentials related to the added hydrogen atoms are essential for reproducing the folded structure of a single iPP chain. 3.5. Very slow quenching at 5 K/5 ns for iPP model without hydrogen atoms To confirm the folding behavior using the modified TraPPE-UA (Imp.) model, which did not show any folded structures in Figs. 2 and 3, we verified the structural formation of a single iPP chain under 6
K. Hagita and S. Fujiwara
Polymer 183 (2019) 121861
Fig. 7. Snapshots of a quenched single iPP chain of n ¼ 120 with the modified TraPPE-UA (Imp. & H) model for various Cangle and Ctorsion.
quenching at 5 K/5 ns, which means 5 K decreases every 5 ns, from T ¼ 500 K to 200 K. Fig. 8 shows the snapshots at T ¼ 100 K, 200 K, 300 K, and 400 K. Compared with quenching at 5 K/ns, the regularity of the entire chain appears to be improved. Because the change at the three temperatures was small, the regularity is expected to further improve for long-time runs. This confirmation will be a topic of future research.
right- and left-handed helical monomer units (RR, RL) during the quenched folding processes. For the modified TraPPE-UA (Imp. & H) model, we found that chirality was spontaneously selected before folding and that the selection of chirality progressed slowly even after folding. From the snapshots colored with right- and left-handed helical monomer units, we found that the length of helical monomer unit chains with the same chirality became longer at smaller T with the TraPPE-UA (H) and (Imp. & H) models. Moreover, we evaluated the indicator Pwhole to determine the tem perature dependence of the folding behaviors. The indicator Pwhole of the unfolded structures was smaller than that of the folded structures. Although the indicator Pwhole describes the local rigidity along the chain, it may not be appropriate for describing the “crystallinity” of the folded iPP chain because crystallization should be a first-order phase transition. We consider the development of order parameters to describe the entire chain crystallization of the folded iPP chain as a topic for future studies. Furthermore, we investigated the effects of angle and torsional po tentials related to the added hydrogen atoms on the folding behavior of the quenched single iPP chain. For Cangle � 0.1 and Ctorsion � 0.1, the folding behavior of the single iPP chain was observed, but folding was not observed for Cangle ¼ 0.0 and Ctorsion ¼ 0.0. Based on these behaviors for large Cangle, we found that the stiffness of the iPP chain originating from the angle potentials associated with the added hydrogen atoms plays an important role in the ease of chain folding. At the same time, we considered that the stiffness derived from torsion potentials associated with the added hydrogen atoms is also important. Thus, we concluded that both angle and torsional angle potentials associated with the added hydrogen atoms are essential to reproduce the folded structure of a single iPP chain. Moreover, the balance of FF parameters revealed that the folding behavior was significantly altered. Thus, the determination of FF parameters that optimize folding and crystallization acceleration should be included in future research. In this study, we clarified the different dynamics of the folding behavior of a single iPP chain depending on the details of the FFs. It is an interesting topic, and future tasks include investigations into the effect of difference in the FFs on the crystallization and glass transition tem perature Tg of many-chain systems.
4. Summary and conclusion In order to examine the single-chain folding of a single iPP chain, we performed UAMD simulations with modified TraPPE-UA FFs. By quenching the system, we observed the folding behavior of a single iPP chain and estimated the indicator Pwhole of a single-chain folded iPP. We observed highly folded structures through the simulations with modified TraPPE-UA FFs with the addition of hydrogen atoms even at a high quenching rate of 50 K/ns. Thus, we clarified that the folding of a single iPP chain into local crystals could be easily achieved with the modified TraPPE-UA FFs with the addition of hydrogen atoms. Through slow quenching at 5 K/ns, we confirmed the folded structures of a single iPP chain with n ¼ 120 and 240. For the modified TraPPE-UA FF with improper potentials and without the addition of hydrogen atoms, no folding behavior was observed even under very slow quenching at 5 K/5 ns. Thus, we concluded that the addition of hydrogen atoms causes folding. Regarding spontaneous chirality selection, we evaluated the ratios of
Author contributions The manuscript was written with contributions from all the authors. The computations and analysis were mainly performed by K.H. All au thors have given their approval of the final version of the manuscript.
Fig. 8. Snapshots of the folded structures of an iPP chain with the modified TraPPE-UA (Imp.) model at T ¼ 100 K, 200 K, 300 K, and 400 K under quenching at 5 K/5 ns. 7
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Acknowledgment
plinary Large-scale Information Infrastructures (JHPCN) and the HighPerformance Computing Infrastructure (HPCI) in Japan (jh160036NAH, jh180020, hp180016, and hp180116). This work was partially supported by JSPS KAKENHI, Japan, grant nos.: JP18H04494 and JP19H00905, and JST CREST, Japan, grant nos.: JPMJCR1993 and JPMJCR19T4.
The authors thank Prof. T. Yamamoto and Dr. T. Konishi for useful discussions. We also thank K. Yogome, Dr. T. Oshiyama, and T. Naka mura for preliminary works related to this paper. The authors were partially supported by the Joint Usage/Research Center for Interdisci
Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.polymer.2019.121861. Appendix A. Folding Structure of a Single iPP Chain based on Yamamoto’s Model For comparison, we also performed UAMD simulations with Yamamoto’s model [38,39] at a quenching rate of 5 K/ns from T ¼ 500 K to 100 K. In this model, the C–C–H and R–C–H angle potentials were considered, although no torsional potential related to the additional explicit hydrogen atoms and no improper potential related to the methyl group were introduced. Here, we investigated two cases with magnification factors Cangle ¼ 11.2 and 1.0. A fine folded structure was only observed for n ¼ 120 and Cangle ¼ 11.2. From these results, we concluded that the folding in Yamamoto’s model is not faster than that of the modified TraPPE-UA (Imp. & H) model.
Fig. A1. Snapshots of a quenched single iPP chain with Yamamoto’s model at T ¼ 100 K. (a) n ¼ 120 and Cangle ¼ 11.2, (b) n ¼ 120 and Cangle ¼ 1.0, (c) n ¼ 240 and Cangle ¼ 11.2, and (d) n ¼ 240 and Cangle ¼ 1.0. Here, the quenching rate from T ¼ 500 K–100 K was 5 K/ns.
Appendix B. Snapshots of a Single iPP Chain at Temperatures from T ¼ 500 K to 400 K To observe the helix inversion behaviors with large deformations during quenching from T ¼ 500 K to 400 K, Fig. B1 shows snapshots of a quenched single iPP chain with n ¼ 120 at temperatures from T ¼ 495 K to 405 K. Here, we considered the modified TraPPE-UA (H) and (Imp. & H) models. In Fig. B1(b), elimination of the helix inversion near the chain end occurred at temperatures from T ¼ 485 K to 465 K. Detailed investigations to capture the dynamics of the helix inversions are problems for future works.
8
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Fig. B1. Snapshots of a quenched single iPP chain colored with right- (red) and left-handed (blue) helical monomer units at temperatures from T ¼ 495 K to 405 K. Green means other. Here, the quenching rate was 5 K/ns.
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Polymer journal homepage: http://www.elsevier.com/locate/polymer
Single-chain folding of a quenched isotactic polypropylene chain through united atom molecular dynamics simulations Katsumi Hagita a, *, Susumu Fujiwara b a b
Department of Applied Physics, National Defense Academy, 1-10-20, Hashirimizu, Yokosuka, 239-8686, Japan Faculty of Materials Science and Engineering, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, 606-8585, Japan
A R T I C L E I N F O
A B S T R A C T
Keywords: Single isotactic polypropylene chain Structure formation United atom molecular dynamics simulation
The single-chain folding of an isotactic polypropylene (iPP) chain through united atom (UA) molecular dynamics simulations under quenching was investigated using force fields (FFs) based on TraPPE-UA. We estimated the degree of local rigidity of the folded chain along iPP undergoing single-chain folding. To maintain the tacticity of iPP, we introduced improper torsional angle potentials and/or explicit hydrogen atoms bonded to backbone carbon atoms. In the simulation using modified TraPPE-UA FFs with added hydrogen atoms, folded structures were observed. In the cases using modified TraPPE-UA FFs with only improper torsional potentials, folding of the quenched iPP chain was not observed. Moreover, to clarify the folding behavior of the iPP chain, we studied the chirality of a single iPP chain and subsequently observed spontaneous chirality selection during the quenched folding process. We also examined the effect of the angle and torsional potentials of the added hydrogen on the folding behavior of a single iPP chain into local crystals. Therefore, we confirmed that the strength of the angle and torsional potentials contributed to the acceleration of the folding behavior.
1. Introduction Crystallization is an important factor that considerably affects the material properties of polymers [1–8]. Crystallization can be regarded as a typical case of polymer self-organization [1]. As a fundamental aspect of crystallization, chain folding has been more extensively examined for homopolymers rather than heteropolymers such as proteins. In partic ular, molecular dynamics (MD) simulations of the folding behavior of a single polyethylene (PE) chain have been reported numerous times [9–15]. In polymer materials, lamellar structures related to chain folding play an important role in imparting physical and chemical properties. There have been many simulation studies on the lamellar crystal structure formation of linear PE chain melts [6,16–26] as well as on the crystallization of branched and cyclic PE chains [27–33]. In these crystallization processes, it is known that each PE chain folds. Thus, an understanding of the folding behavior of various polymer chains is a fundamental issue of polymeric materials that must be resolved. PE has been studied as the simplest polymer in which carbon atoms are linked in a straight line, whereas the simplest polymer with side chains is considered to be polypropylene (PP). Similar to PE, PP has been widely used in industry as the lowest-density polymer material among
thermoplastic resins. In PP, tacticity plays an important role in deter mining properties such as glass transition temperature Tg, elastic modulus, and crystallinity [34]. In the 1950s, Natta and Corradini established the synthesis of isotactic polypropylene (iPP) [35], wherein methyl groups are bonded in the same direction. The cases in which methyl groups are arranged in alternating and random directions are called syndiotactic PP and atactic PP, respectively, and each has differing properties. For example, Tg of iPP, sPP, and aPP are 255 K, 269 K, and 267 K, respectively [36]. To investigate the crystallization of iPP chains, various molecular simulations have been performed [4,37–41]. To reduce the number of atoms in the system and the computing costs, several united atom (UA) models of iPP chains have been utilized. Romanos and Theodorou [4,37] developed a force field with explicit carbon and hydrogen atoms on the main chain and methyl groups lumped into united atoms. As approaches based on a general UA model that omits hydrogen atoms, Yamamoto and Sawada [38,39] and Pütz et al. [40,41] proposed UA models of iPP chains to conserve their tacticity. Based on the molecular details of the static structure of iPP, the flip of a methyl group causes a change in tacticity. To avoid this methyl group flip, on the basis of a UA model of the transferable potentials for phase equilibria (TraPPE) force field [42],
* Corresponding author. E-mail address: [email protected] (K. Hagita). https://doi.org/10.1016/j.polymer.2019.121861 Received 22 July 2019; Received in revised form 27 September 2019; Accepted 2 October 2019 Available online 3 October 2019 0032-3861/© 2019 Elsevier Ltd. All rights reserved.
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Yamamoto and Sawada [38,39] introduced an explicit hydrogen atom opposite a methyl group, while Pütz et al. [40,41] introduced improper torsional angle potentials. Note that Yamamoto and Sawada [38] also proposed a method in which phantom bond-length constraints are imposed between the nearest CH3 and CH2 and between CH2 and CH2 instead of introducing explicit hydrogen atoms. We did not examine this extension of MD simulation methods. By using the method with explicit hydrogen atoms, Yamamoto confirmed the accelerated crystallization of iPP chains from a highly stretched amorphous state by using UAMD simulations [39]. Recently, Xie et al. confirmed the stretch-induced coil helix transition of iPP chains using full-atom MD simulations [43]. Pütz et al. [40] and Heine et al. [41] studied the chain statistics of iPP by using polymer interaction site model (PRISM) calculations and UAMD simulations. By using the Pütz parameters, Kuppa et al. per formed an atomistic Monte Carlo simulation to investigate the crystal lization behavior of the amorphous region between the crystalline lamellae of iPP [44]. Recently, by using the Pütz parameters, Tzounis et al. examined the tacticity effect on the conformational properties of PP polymers and PE-PP copolymers [34]. However, at present, there has been no comparison between the two approaches. Furthermore, studies on whether a single iPP chain exhibits folding behavior have not pro vided satisfactory results. Thus, the purpose of this study was to examine the folding behavior of a single iPP chain and to compare the two force fields (FFs) of the UA model. In studies on crystallization behavior, FFs that show chemically ac curate behavior are essential. However, because computational re sources that offer highly accurate results are critically insufficient in reproducing slow crystallization behavior similar to real phenomena, we considered that an accelerated FF for promoting crystallization would be useful as a basic aspect for determining the nature of crystallization. For PE chains, it was previously reported that the crystallization rate in MD simulations is sensitive to chain stiffness [16,17]. In our previous study on PE [33], we clarified that the DREIDING-UA FF [45] can be regarded as an accelerated FF to promote PE crystallization. Thus, this study also aimed to find an accelerated FF that promotes the crystallization and folding of iPP chains. First, we clarify our research stance. In order to obtain highly chemically accurate FF parameters, comparisons with experiments and first-principles DFT calculations are important. How ever, the direct observation of the folding of a single iPP chain remains difficult at this time, although AFM observations on a substrate are available [46]. Tracking the dynamics by first-principles DFT calcula tions of a thousand atoms is also impractical. In this paper, priority was given to searching for an FF parameter set that reflects the folding behavior rather than one with chemical precision. To explain the folding behavior of a single iPP chain, Varashey and Carri [47] as well as Yamamoto and Sawada [1,38] reported novel and groundbreaking results. Varashey and Carri examined the transitions between the folded states of helical subchains and coil states [46], for which they reported state diagrams for a single iPP chain with several chain lengths and interaction parameters. Yamamoto and Sawada examined the order-disorder transitions of a folded single iPP chain [1, 38] and investigated the mechanism of the chirality selection of a single iPP chain placed on an ordered iPP substrate. In this study, we investi gated the folding behavior of a single iPP chain and its chirality selection.
consists of a chain of n repeated units of C(C)C in the SMILES notation. The UAs interact via bonded potentials (bond-stretching, bond-bending, and dihedral and improper torsional potentials) and non-bonded po tentials. The total potential energy, Etot, consisted of four parts: (i) the 12-6 LJ potential between non-bonded atoms separated by more than three bonds, ELJ; (ii) the bond-stretching energies for C–C bonds, EBOND; (iii) the bond-bending energies, EANGLE; and (iv) the torsional and improper energies, ETORSION and EIMPROPER. The detailed equations are given in Refs. [38–41]. The large-scale atomic/molecular massively parallel simulator was used for all calculations [49]. The equations of motion for all atoms were solved numerically using the velocity version of the Verlet algorithm with an integration time step of Δt ¼ 0.5 fs. Here, we used a Δt shorter than that for a single PE chain in order to maintain the tacticity of iPP. We applied constant NVT conditions with a Nos�e–Hoover thermostat, and the cutoff distance for the LJ potential was 1.4 nm. The total mo mentum and total angular momentum were assumed to be zero in order to cancel the overall translation and rotation of the chain. The MD simulations were performed by the following procedure: First, a single iPP chain with n ¼ 120 or 240 units of the UA iPP model was placed in a large space. We used 2003 and 2503 nm3 cubic boxes under periodic boundary conditions for n ¼ 120 and 240, respectively. Note that the simulation box was not deformed during these simulations. In these simulations, we observed spontaneous folding in a vacuum space with no adsorption wall. Second, we obtained random configurations of the iPP chain at a high temperature (T ¼ 600 K) by means of an MD simulation lasting longer than 10 ns. Note that we confirmed that the longest Rouse relaxation times of the PE chain at T ¼ 600 K were approximately 0.5 ns and 2.0 ns for n ¼ 120 and 240, respectively. In this study, we used T ¼ 600 K in order to maintain the tacticity of iPP with Δt ¼ 0.5 fs. Third, a 1 ns run at T ¼ 500 K was performed. Finally, the configuration at T ¼ 500 K was cooled stepwise to T ¼ 100 K at a rate of 50 K/ns or 5 K/ns for quenching. 2.2. Modification of TraPPE-UA As explained in section 1, Yamamoto and Sawada [38,39] introduced the explicit hydrogen atoms, and Pütz et al. [40] and Heine et al. [41] introduced the improper torsional angle potentials to maintain the iPP tacticity. We expected both to be effective and that introducing both would improve the stereostructure and promote the crystallization and folding of iPP. The potential for the added hydrogen atoms was set as follows. Based on the original TraPPE-UA model, the non-bonded po tential for the added hydrogen atoms was set to zero. As for the bond potential of C–H, a loose bond was used to reduce the frequency. In practice, it was set to 52 kcal/mol, which is one-tenth of the potential of the C–C bond (520 kcal/mol). Here, we used a loose bond for C–H in order to lengthen the integration time step. For this problem, Yamamoto increased the mass of the explicit hydrogen atom while maintaining the C–H bond value the same as the C–C bond [50]. (Note that for simula tions of many iPP chains with computing costs much larger than that of a single-chain simulation, the SHAKE method [51] to fix the C–H bond length and the RESPA method [52] to lengthen the integration time step are essential.) Regarding angle potentials, we introduced hard angle potentials to accelerate the local crystallization. The angle potentials of C–C–H and R–C–H were set to 1400 kcal/mol, which is 11.2 times that of C–C–C (124.2 kcal/mol). Here, R denotes a united atom of the methyl group. Because the intention was to realize regular folding by balancing the torsional potentials, the torsional angle potentials of C–C–C–H and R–C–C–H had the same value as that of C–C–C–C. Note that the torsional angle potentials of C–C–C–H and R–C–C–H were zero in Yamamoto’s model with the explicit hydrogen atoms [38,39]. Based on the discussion by Pütz et al. [40] and Heine et al. [41], the coefficient of the improper torsional angle potentials was set to a value identical to the bond potential of C–C. To label the used potential model, the model with the improper torsional angle potentials is denoted as the
2. Methods 2.1. Simulation methods In this study, we performed UAMD simulations using the TraPPE-UA FF [42] with and without modification. Note that the FF parameters of TraPPE-UA were obtained by using enhanced Monte Carlo tools [48]. The used parameters were almost the same as those employed in pre vious studies by Yamamoto and Sawada [38,39], Pütz et al. [40], and Heine et al. [41]. We considered that the UA model of an iPP chain 2
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TraPPE-UA (Imp.) model hereafter. The TraPPE-UA (Imp. & H) model denotes the case with explicit hydrogen atoms and improper torsional angle potentials, and TraPPE-UA (H) denotes the model with explicit hydrogen atoms and without improper torsional angle potentials.
terms (Imp. and H): i) modified TraPPE-UA (Imp.) model; ii) modified TraPPE-UA (H) model; and ii) modified TraPPE-UA (Imp. & H) model. Fig. 1 presents the racemic diad ratios of a PP chain with n ¼ 120 after 1 ns runs at T ¼ 400 K, 600 K, 800 K, and 1000 K. The results at T ¼ 200 K, 300 K, 400 K, and 500 K are presented only for the original TraPPE-UA because the PP chain cannot conserve its tacticity. Here, the racemic diad ratio of a perfect iPP chain is zero. In Fig. 1(b), the racemic diad ratio was 0.0342 at t ¼ 1 ns and T ¼ 800 K. For the modified TraPPE-UA, the racemic diad ratio was 0 for T � 600 K. Thus, we found that the three modified versions of TraPPE-UA can maintain the tacticity of the iPP chain. In addition, we confirmed that Yamamoto’s explicit hydrogen atom model conserves the tacticity of the iPP chain despite the fact that torsional potentials associated with the explicit hydrogen atom were not introduced. On the other hand, the original TraPPE-UA failed at maintaining the tacticity of the iPP chain at T � 300 K. Thus, we concluded that the original TraPPE-UA is not suitable for studying the folding behavior of an iPP chain.
2.3. Indicator of a folded single iPP chain According to experimental knowledge [53–60], iPP exhibits a stable crystalline α-form consisting of chains in the 31 helical conformation. In this perfect helical structure, the two vectors riþ6 ri and riþ7 riþ1 are parallel, where ri is the position of the ith carbon atom in the main chain (backbone) [38]. Thus, we used the orientation order parameter P2;i ¼ ð3cos2 θi 1Þ=2, which was defined by the angle θi between vectors riþ6 ri and riþ7 riþ1 . To estimate the local rigidity of the folded single iPP chain along the chain, we computed the probability Pwhole of the angle jθj < π=20. For a single iPP chain with n units (2n carbon atoms in the main chain), we calculated Pwhole as follows: Pwhole ¼
2n X5
1 2n
7
P2;i :
3.2. Fast quenching at 50 K/ns
(1)
i¼3
Fig. 2 shows the temperature dependence of a fast-quenched single iPP chain with n ¼ 120 in the simulations using the modified TraPPEUA FFs. As a test, the quenching rate of 50 K/ns was considered on the basis of a previous study for a single PE chain [33]. From Fig. 2(a), for the TraPPE-UA (Imp.) model, folding of the single iPP chain was not observed. On the other hand, from Fig. 2(b) and (c), we can confirm that the single iPP chain with hydrogen atoms showed folding behavior. Moreover, we found that the folding of the iPP chain with the
In this sum, P2;i involving carbon atoms at both ends were removed. 3. Results 3.1. Verification of conservation of tacticity We examined three UA models as a combination of two additional
Fig. 1. Racemic diad ratios of a PP chain with n ¼ 120 after a 1 ns run. (a) Original TraPPE-UA model, (b) modified TraPPE-UA (Imp.) model, (c) modified TraPPE-UA (H) model, and (d) modified TraPPE-UA (Imp. & H) model. 3
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Fig. 2. Temperature dependence of a fast-quenched single iPP chain with n ¼ 120. (a) Modified TraPPE-UA (Imp.) model, (b) modified TraPPE-UA (H) model, and (c) modified TraPPE-UA (Imp. & H) model.
TraPPE-UA (Imp. & H) model was stronger than that with the TraPPE-UA (H) model. Thus, it was found that the two TraPPE-UA models with hydrogen atoms can be used as FFs to study the folding behavior of a single iPP chain. For T ¼ 100 K, we estimated the ratios of right- and left-handed helical monomer units (RR, RL) with the method reported by Yamamoto and Sawada [38]. (RR, RL) ¼ (0.38, 0.50), (0.52, 0.40), and (0.53, 0.35) for the modified TraPPE-UA (Imp.), (H), and (Imp. & H) models, respectively. The obtained values indicate that no chirality selection of the right- or left-handed helix occurred. Chirality
selection will be discussed in the next subsection. Furthermore, we estimated Pwhole of the three chains. For the modified TraPPE-UA (Imp.) model, Pwhole was 0.463, 0.416, 0.446, and 0.343 for T ¼ 100 K, 200 K, 300 K, and 400 K, respectively. For the modified TraPPE-UA (H) model, Pwhole was 0.652, 0.627, 0.571, and 0.524 for T ¼ 100 K, 200 K, 300 K, and 400 K, respectively. For the modified TraPPE-UA (Imp. & H) model, Pwhole was 0.678, 0.648, 0.614, and 0.601 for T ¼ 100 K, 200 K, 300 K, and 400 K, respectively. From the obtained results, Pwhole seems to function as an indicator of folding.
Fig. 3. Snapshots of a quenched single iPP chain for n ¼ 120 and 240 at T ¼ 100 K. (a) Modified TraPPE-UA (Imp.) model, (b) modified TraPPE-UA (H) model, and (c) modified TraPPE-UA (Imp. & H) model. Here, the quenching rate from T ¼ 500 K to 100 K was 5 K/ns. These 3D visualizations are given in the SI in addition to the snapshots of the other three ensembles. 4
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Discussions for Pwhole are provided in section 3.3 with the results for slow quenching at 5 K/ns.
behavior is high. However, Pwhole is clearly not a sensitive quantity. For the cases with hydrogen atoms, we found that the difference in Pwhole values was small for determining the local crystallinity, although the snapshot differs considerably depending on the presence or absence of improper torsional potentials. Actually, Pwhole is an indicator of the local rigidity along the chain. For an appropriate crystallinity because crys tallization should be a first-order phase transition, it is necessary to generate the local crystallinity from the stem length (local structure continuity) and the amount related to the three-dimensional packing of local crystal structures. We consider that machine learning could play an active role in the generation of simulated crystallinity, as suggested by Welch [61]. Studies in this direction will be performed in the future.
3.3. Slow quenching at 5 K/ns We verified the structural formation of a single iPP chain under quenching from T ¼ 500 K to 100 K at a rate of 5 K/ns. Here, we exam ined the simulations with the three modified TraPPE-UA FFs. Fig. 3 shows snapshots of the quenched single iPP chain for n ¼ 120 and 240 at T ¼ 100 K. As shown in Fig. 3(a), folded structures were not observed for the modified TraPPE-UA (Imp.) model even at a quenching rate of 5 K/ns. On the other hand, from Fig. 3(b) and (c), for the quenched iPP chain using the modified TraPPE-UA with added hydrogen atoms, we found that the case with improper torsional potentials showed stronger folded structures than that without improper torsional potentials. Although the local order of crystallization was high in both cases, the regularity of the entire structure with improper torsional potentials was higher than that without improper torsional potentials. Table 1 shows the (RR, RL) values from four independent ensembles for n ¼ 120 and 240 for the modified TraPPE-UA (Imp.), (H), and (Imp. & H) models at T ¼ 500 K, 400 K, and 100 K. The results of ensemble 1 are shown in Fig. 3. With the modified TraPPE-UA (Imp. & H) model, it was found that chirality was spontaneously selected before folding. In addition, we concluded that the selection of chirality pro gressed slowly even after folding. For the modified TraPPE-UA (H) model, although spontaneous chirality selection before folding did not clearly occur, it slowly progressed after the occurrence of folding. Fig. 4 shows snapshots of a quenched single iPP chain colored with right- and left-handed helical monomer units. We found that chirality selection began above T ¼ 400 K, before folding and after the formation of local rigid structures. Below T ¼ 500 K, for the TraPPE-UA (H) and (Imp. & H) models, we found that the length of helical monomer unit chains with the same chirality became longer for smaller T. From the behaviors between T ¼ 500 K and 400 K as presented in Appendix B, helix inversion occurred with a large deformation of the iPP chain to ward folding. To understand the chirality selection dynamics, it will be necessary to perform further simulations between T ¼ 500 K and 400 K. In particular, whether the helix inversion has been swept out into the folding region or has been eliminated near the chain ends is of interest. Detailed analyses of the helix inversion will be performed in future work. Fig. 5 shows the temperature dependence of the indicator Pwhole for quenching at 5 K/ns. The values of Pwhole for the case of the TraPPE-UA (Imp.) model, which is without hydrogen atoms, were smaller than those for the other cases. Considering the results discussed in section 3.2, it can be concluded that the correlation between Pwhole and the folding
3.4. Effects of strength of angle and torsional angle potential of added hydrogen From the comparison between the presence of hydrogen atoms and/ or improper torsional angle potential, it was considered that the addi tion of hydrogen atoms was essential. We expected that the potentials of angles and/or torsional angles related to the added hydrogen atoms have important roles in determining the folding behavior. To confirm the effect of the angle and torsional angle potentials, we estimated Pwhole of a single iPP chain with n ¼ 120 for several angle and torsional angle potentials strengths. As noted in section 2.2, the coefficient of angle potential related to the added hydrogen atom was set to a value 11.2 times that of the other angle potential. Thus, we examined the cases with magnification factors Cangle ¼ 0.0, 0.1, 0.5, 1.0, 2.0, 3.0, 5.0, and 11.2 with Ctorsion ¼ 1.0. On the other hand, to investigate the magnifi cation factor Ctorsion of the coefficient of torsional angle potential against that for atom types aside from the added hydrogen atom, we examined the cases with Ctorsion ¼ 0.0, 0.1, 0.5, and 1.0 with Cangle ¼ 11.2. Fig. 6 shows the effects of Cangle and Ctorsion on the single iPP chain of n ¼ 120 with the modified TraPPE-UA (Imp. & H) model. From Fig. 6 (a), we found that Pwhole decreased for smaller Cangle < 1.0. On the other hand, for Cangle � 1.0, the difference in the value of Pwhole was expected to be within reasonable error. In fact, no significant difference was found in the results calculated with four independent ensembles. The detailed results are presented in the SI. From Fig. 6(b), it was found that Pwhole decreased for smaller Ctorsion. Based on these results, it was revealed that both Cangle and Ctorsion affect the local crystallization of a single iPP chain, where the effect of Ctorsion appears to be stronger than that of Cangle. The effects of Cangle and Ctorsion on the entire chain structure are examined as follows. Fig. 7 shows the entire chain structures of the quenched single iPP chain of n ¼ 120 for various Cangle and Ctorsion using the modified TraPPE-UA (Imp. & H) model. For Cangle ¼ 0.0, in Fig. 7(a), we found a
Table 1 (RR, RL) for n ¼ 120 and 240 obtained using the modified TraPPE-UA (Imp.), (H), and (Imp. & H) models at T ¼ 500 K, 400 K, and 100 K. n
Modification
Temp.
ensemble 1
ensemble 2
ensemble 3
ensemble 4
120
Imp.
120
H
120
Imp. & H
240
Imp.
240
H
240
Imp. & H
T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K T ¼ 500 K T ¼ 400 K T ¼ 100 K
(0.32, (0.52, (0.47, (0.51, (0.47, (0.73, (0.50, (0.80, (0.83, (0.43, (0.41, (0.54, (0.50, (0.49, (0.59, (0.56, (0.68, (0.70,
(0.35, (0.39, (0.52, (0.62, (0.56, (0.68, (0.72, (0.85, (0.94, (0.32, (0.36, (0.58, (0.40, (0.42, (0.48, (0.67, (0.74, (0.81,
(0.38, (0.42, (0.44, (0.59, (0.62. (0.60, (0.64, (0.85, (0.95, (0.39, (0.35, (0.50, (0.46, (0.50, (0.55, (0.68, (0.82, (0.86,
(0.41, 0.34) (0.36, 0.42) (0.48, 0.40) (0.51, 0.32) (0.47, 0.38) (0.44, 0.49) (0.63, 0.18) (0.72, 0.12) (0.88, 0.09) (0.34, 0.39) (0.45, 0.34) (0.58, 0.30) (0.47, 0.30) (0.50, 0.35) (0.53, 0.41) (0.60, 0.21) (0.71, 0.16) (0.80, 0.16)
0.32) 0.29) 0.43) 0.33) 0.41) 0.23) 0.27) 0.09) 0.13) 0.35) 0.37) 0.37) 0.33) 0.36) 0.35) 0.24) 0.22) 0.22)
5
0.42) 0.35) 0.38) 0.24) 0.28) 0.27) 0.14) 0.06) 0.04) 0.39) 0.43) 0.31) 0.41) 0.47) 0.45) 0.19) 0.11) 0.16)
0.33) 0.41) 0.44) 0.20) 0.22) 0.35) 0.18) 0.03) 0.03) 0.38) 0.43) 0.38) 0.33) 0.31) 0.40) 0.17) 0.09) 0.11)
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Polymer 183 (2019) 121861
Fig. 4. Snapshots of a quenched single iPP chain colored with right- (red) and left-handed (blue) helical monomer units. Green means other. Here, the quenching rate from T ¼ 500 K–100 K was 5 K/ns. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
Fig. 5. Temperature dependence of indicator Pwhole under quenching at 5 K/ns.
Fig. 6. Temperature dependence of indicator Pwhole under quenching at 5 K/ns.
failure in the folding, although folded chains were observed for Cangle � 0.1. Here, for Cangle ¼ 0.0 and 0.1, the tacticity was not conserved, and the racemic diad ratios of the PP chain were 0.428 and 0.034, respec tively. For larger Cangle, the structure was straighter and more regular; this behavior appears to correspond to indicator Pwhole. For Cangle ¼ 5.0 and 11.2, the folded structures were similar to those for Cangle ¼ 2.0 and 3.0, although they are not presented here. From these behaviors, it became clear that the stiffness of the iPP chain originating from the C–C–H and R–C–H angle potentials is related to the ease of chain folding. For Ctorsion ¼ 0.0, in Fig. 7(g), we found a failure in the folding, although folded chains were observed for Ctorsion � 0.1. From this behavior, we considered that the stiffness of the iPP chain originating from the C–C–C–H and R–C–C–H torsional potentials also plays an important role
in the ease of chain folding. Note that the case with Ctorsion ¼ 0.0 was slightly different from Yamamoto’s model [38,39] shown in Appendix A since the improper potential was considered. From these results, we can conclude that both the angle and torsional angle potentials related to the added hydrogen atoms are essential for reproducing the folded structure of a single iPP chain. 3.5. Very slow quenching at 5 K/5 ns for iPP model without hydrogen atoms To confirm the folding behavior using the modified TraPPE-UA (Imp.) model, which did not show any folded structures in Figs. 2 and 3, we verified the structural formation of a single iPP chain under 6
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Polymer 183 (2019) 121861
Fig. 7. Snapshots of a quenched single iPP chain of n ¼ 120 with the modified TraPPE-UA (Imp. & H) model for various Cangle and Ctorsion.
quenching at 5 K/5 ns, which means 5 K decreases every 5 ns, from T ¼ 500 K to 200 K. Fig. 8 shows the snapshots at T ¼ 100 K, 200 K, 300 K, and 400 K. Compared with quenching at 5 K/ns, the regularity of the entire chain appears to be improved. Because the change at the three temperatures was small, the regularity is expected to further improve for long-time runs. This confirmation will be a topic of future research.
right- and left-handed helical monomer units (RR, RL) during the quenched folding processes. For the modified TraPPE-UA (Imp. & H) model, we found that chirality was spontaneously selected before folding and that the selection of chirality progressed slowly even after folding. From the snapshots colored with right- and left-handed helical monomer units, we found that the length of helical monomer unit chains with the same chirality became longer at smaller T with the TraPPE-UA (H) and (Imp. & H) models. Moreover, we evaluated the indicator Pwhole to determine the tem perature dependence of the folding behaviors. The indicator Pwhole of the unfolded structures was smaller than that of the folded structures. Although the indicator Pwhole describes the local rigidity along the chain, it may not be appropriate for describing the “crystallinity” of the folded iPP chain because crystallization should be a first-order phase transition. We consider the development of order parameters to describe the entire chain crystallization of the folded iPP chain as a topic for future studies. Furthermore, we investigated the effects of angle and torsional po tentials related to the added hydrogen atoms on the folding behavior of the quenched single iPP chain. For Cangle � 0.1 and Ctorsion � 0.1, the folding behavior of the single iPP chain was observed, but folding was not observed for Cangle ¼ 0.0 and Ctorsion ¼ 0.0. Based on these behaviors for large Cangle, we found that the stiffness of the iPP chain originating from the angle potentials associated with the added hydrogen atoms plays an important role in the ease of chain folding. At the same time, we considered that the stiffness derived from torsion potentials associated with the added hydrogen atoms is also important. Thus, we concluded that both angle and torsional angle potentials associated with the added hydrogen atoms are essential to reproduce the folded structure of a single iPP chain. Moreover, the balance of FF parameters revealed that the folding behavior was significantly altered. Thus, the determination of FF parameters that optimize folding and crystallization acceleration should be included in future research. In this study, we clarified the different dynamics of the folding behavior of a single iPP chain depending on the details of the FFs. It is an interesting topic, and future tasks include investigations into the effect of difference in the FFs on the crystallization and glass transition tem perature Tg of many-chain systems.
4. Summary and conclusion In order to examine the single-chain folding of a single iPP chain, we performed UAMD simulations with modified TraPPE-UA FFs. By quenching the system, we observed the folding behavior of a single iPP chain and estimated the indicator Pwhole of a single-chain folded iPP. We observed highly folded structures through the simulations with modified TraPPE-UA FFs with the addition of hydrogen atoms even at a high quenching rate of 50 K/ns. Thus, we clarified that the folding of a single iPP chain into local crystals could be easily achieved with the modified TraPPE-UA FFs with the addition of hydrogen atoms. Through slow quenching at 5 K/ns, we confirmed the folded structures of a single iPP chain with n ¼ 120 and 240. For the modified TraPPE-UA FF with improper potentials and without the addition of hydrogen atoms, no folding behavior was observed even under very slow quenching at 5 K/5 ns. Thus, we concluded that the addition of hydrogen atoms causes folding. Regarding spontaneous chirality selection, we evaluated the ratios of
Author contributions The manuscript was written with contributions from all the authors. The computations and analysis were mainly performed by K.H. All au thors have given their approval of the final version of the manuscript.
Fig. 8. Snapshots of the folded structures of an iPP chain with the modified TraPPE-UA (Imp.) model at T ¼ 100 K, 200 K, 300 K, and 400 K under quenching at 5 K/5 ns. 7
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Acknowledgment
plinary Large-scale Information Infrastructures (JHPCN) and the HighPerformance Computing Infrastructure (HPCI) in Japan (jh160036NAH, jh180020, hp180016, and hp180116). This work was partially supported by JSPS KAKENHI, Japan, grant nos.: JP18H04494 and JP19H00905, and JST CREST, Japan, grant nos.: JPMJCR1993 and JPMJCR19T4.
The authors thank Prof. T. Yamamoto and Dr. T. Konishi for useful discussions. We also thank K. Yogome, Dr. T. Oshiyama, and T. Naka mura for preliminary works related to this paper. The authors were partially supported by the Joint Usage/Research Center for Interdisci
Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.polymer.2019.121861. Appendix A. Folding Structure of a Single iPP Chain based on Yamamoto’s Model For comparison, we also performed UAMD simulations with Yamamoto’s model [38,39] at a quenching rate of 5 K/ns from T ¼ 500 K to 100 K. In this model, the C–C–H and R–C–H angle potentials were considered, although no torsional potential related to the additional explicit hydrogen atoms and no improper potential related to the methyl group were introduced. Here, we investigated two cases with magnification factors Cangle ¼ 11.2 and 1.0. A fine folded structure was only observed for n ¼ 120 and Cangle ¼ 11.2. From these results, we concluded that the folding in Yamamoto’s model is not faster than that of the modified TraPPE-UA (Imp. & H) model.
Fig. A1. Snapshots of a quenched single iPP chain with Yamamoto’s model at T ¼ 100 K. (a) n ¼ 120 and Cangle ¼ 11.2, (b) n ¼ 120 and Cangle ¼ 1.0, (c) n ¼ 240 and Cangle ¼ 11.2, and (d) n ¼ 240 and Cangle ¼ 1.0. Here, the quenching rate from T ¼ 500 K–100 K was 5 K/ns.
Appendix B. Snapshots of a Single iPP Chain at Temperatures from T ¼ 500 K to 400 K To observe the helix inversion behaviors with large deformations during quenching from T ¼ 500 K to 400 K, Fig. B1 shows snapshots of a quenched single iPP chain with n ¼ 120 at temperatures from T ¼ 495 K to 405 K. Here, we considered the modified TraPPE-UA (H) and (Imp. & H) models. In Fig. B1(b), elimination of the helix inversion near the chain end occurred at temperatures from T ¼ 485 K to 465 K. Detailed investigations to capture the dynamics of the helix inversions are problems for future works.
8
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Fig. B1. Snapshots of a quenched single iPP chain colored with right- (red) and left-handed (blue) helical monomer units at temperatures from T ¼ 495 K to 405 K. Green means other. Here, the quenching rate was 5 K/ns.
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