Stretch-Induced Coil–Helix Transition in Isotactic Polypropylene: A

May 15, 2018 - The stretch-induced coil–helix transition (CHT) of isotactic polypropylene (iPP) was studied with full-atom molecular dynamics (MD) s...
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Stretch-Induced Coil−Helix Transition in Isotactic Polypropylene: A Molecular Dynamics Simulation Chun Xie,† Xiaoliang Tang,† Junsheng Yang,†,‡ Tingyu Xu,† Fucheng Tian,† and Liangbin Li*,† †

National Synchrotron Radiation Lab, CAS Key Laboratory of Soft Matter Chemistry, Anhui Provincial Engineering Laboratory of Advanced Functional Polymer Film, University of Science and Technology of China, Hefei, China ‡ Computational Physics Key Laboratory of Sichuan Province, Yibin University, Yibin, China S Supporting Information *

ABSTRACT: The stretch-induced coil−helix transition (CHT) of isotactic polypropylene (iPP) was studied with full-atom molecular dynamics (MD) simulations during the uniaxial stretch process. The results show that imposing stretch induces CHT, which increases both the content and the average length of helices. As strain exceeding a certain value, long helices initially not presented in melt start to emerge, which mainly follow a kinetic pathway of merging adjacent short helices, while overstretch at large strain leads to the helixextended coil transition. Based on statistics on the distribution of helical length and theoretical calculation, stretch is found to reduce free energy gap for CHT. At small strain, the single-chain model is sufficient to account stretch-induced CHT for the formation of short helices, but the gap reduction is mainly contributed by intrachain energy rather than entropy, which is different from current theories for stretch-induced CHT. While the formation of long helices at large strain requires interchain cooperative interactions, which is accompanied by the formation of helix-rich clusters. Additionally, we found that the content of helices with odd atoms in backbone is higher than their even counterparts, which exhibits an odd−even effect due to their corresponding helical lengths.

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INTRODUCTION Flow-induced crystallization (FIC) of polymer is an important research subject because of its inescapable occurrence in polymer processing.1−6 A great deal of effort has been devoted to studying FIC of polymers over the past few decades.7−19 Imposing flow can largely enhance nucleation rate and induce new crystal forms and oriented morphologies like shish kebab, during which density fluctuation or precursor is frequently reported prior to the occurrence of crystallization.12,20−25 These observations are more related to interchain orderings rather than intrachain conformational ordering like the gauche−trans26−28 or coil−helix transition (CHT).29−33 In polymer crystallization, these connective and flexible chains must be converted into conformational ordered rigid segments for packing into crystal,34−36 which is the most peculiar feature as compared to spherical atoms and small molecules. Unfortunately, how flexible chains are converted into conformational ordered rigid segments has not been explicitly addressed in current models for polymer crystallization at quiescent or flow conditions. Theoretically, imposing flow can induce intrachain conformational ordering like CHT. By incorporating elastic energy imposed externally into the free energy of polymer chain, Tamashiro and Pincus 37 and Buhot and Halperin 38,39 formulated a theoretical framework for stretch-induced CHT for single polymer chain, which was later demonstrated by Courty et al.40,41 on the study of gelatin. Flow-induced © XXXX American Chemical Society

conformational ordering has also been observed in synthetic polymers like isotactic polypropylene (iPP)42−45 and polyethylene (PE).46,47 With Fourier-transform infrared spectroscopy (FTIR), An et al.42 found that CHT can be induced prior to crystallization in iPP melt by shearing at temperatures below and above the melting point. Furthermore, Geng et al.43 observed that the contents of short and long helices decrease and increase after the cessation of shear, respectively. Nevertheless, spectroscopic methods like FTIR and Raman can only detect the contents of helices with special length or the content of all helices, while more detailed information about helical evolution at the molecular level is still unclear, which may be unveiled with the aid of computer simulation. Molecular dynamics (MD) simulation which shows detailed structure and interaction information is the favorable method to study the crystallization of polymer.28,48−55 With molecular dynamics simulation, Yamamoto studied strain-induced crystallization of iPP and observed strain-induced CHT.49 With Monte Carlo simulation (MC), Li et al.56 showed that the extended conformations of PE chains are preferred during deformation. Though the works with computer simulation demonstrate the presence of strain-induced conformational ordering in polymers, the detailed information related to the Received: February 13, 2018 Revised: April 15, 2018

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size at this time was 188.65 Å, 188.65 Å, and 60.83 Å in x, y, and z directions, respectively (Figure 1a). Then the uniaxial stretch was imposed with an engineering stain rate ε̇ = 109 s−1 and strain of 9 in the z direction. During the stretching, the box was deformed along the z direction (stretching direction), and two other directions varied with it. The Nosé−Hoover thermostat and barostat were used to control the temperature of the system as 360 K and the pressure as 1 atm in x and y directions while the pressure of stretching direction z was not controlled, and this external pressure would be used as a parameter of Nosé−Hoover barostat to control the position and velocity of particles. In all, the ensemble here can be called the NLzPxPyT ensemble.59−62 The force field in this work was COMPASS63 which has been used in many works.64−66 At last, the periodic boundary condition was applied. The sketch of simulation process is shown in Figure 1b.

evolution of helices with different length has not been fully elucidated yet. In this work, we employ MD simulation on all-atom model to study stretch-induced CHT of iPP, which adopts the 3/1 helical conformation in its crystal forms. By analyzing the evolution of helices with different lengths, we observe that merging short helices is the favorable kinetic pathway for the formation of long helices, while large strain induces overstretch of chain and leads to helix-extended coil transition. Interestingly, the content of helical segments with odd carbon atoms in backbone is obviously higher than that containing even atoms, which is attributed to the difference of helices length under strain.

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MODEL AND SIMULATION PROCESS A system containing of 50 iPP chains with 500 monomers per/ chain (see Figure 1a) was studied by all-atom molecule

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PARAMETER DEFINITION The conformational ordering of chains is represented by the dihedral angle along the backbone as shown in Figure 2a and calculated by using eq 1: φ = cos−1

(r1 × r2) ·(r2 × r3) |r1 × r2|·|r2 × r3|

(1)

where r is the vector along the bond. A dihedral angle is associated with four atoms or three bonds as shown in Figure 2a. So, we specified that every dihedral angle belongs to the second atom when we calculated the microconformation along the chain. Therefore, the length of sequence mentioned in the next, like several atoms, all actually represents the number of dihedral angles. Three kinds of microconformations67trans (T), right gauche (G−), and lef t gauche (G+)are defined based on the distribution of dihedral angles as shown in Figure 2b. Then, successive microconformations including TG−TG−TG−, TG+TG+TG+, and TTTTTT are defined as right 3/1 helix, left 3/1 helix, and extended coil, respectively, and each of them contains at least six carbon atoms along the backbone as shown in Figure 2c. For other sequences like TG−TG− (shorter than six atoms), TG−TG−TG+ (consisting of both TG+ and TG−) are regarded as the coil segments. Here, helical structures should have the lowest energy while the extended coil locates at the highest energy position.

Figure 1. (a) Initial system of this work consisting of 50 iPP chains with 500 monomers per/chain. (b) The whole simulation process. The initial model was relaxed at 500 K for 2 ns, and then it was quenched from 500 to 360 K. After a short relaxation, the system was stretched for 9 ns with an engineering strain rate of ε̇ = 109 s−1.

dynamics simulation performed on LAMMPS.57 The initial structure was built by Amorphous Cell module of Materials Studio software and then relaxed at NPT ensemble (T = 500 K, P = 1 atm) for 2 ns until averaged mean-square root radius of gyration Rg of chains was 4.36 nm, close to the value 4.64 nm calculated with the equation R g = C ∞ M nm, where M is the molecular weight and C∞ = 0.032.58 Then the system was quenched down to 360 K and relaxed for a short time. The box

Figure 2. (a) r1, r2, and r3 are the vectors along each backbone bond, and the dihedral angle φ is decided by those three vectors. (b) P(φ) is the distribution of dihedral angle φ. According to the distribution, three kinds of microconformationtrans T, right gauche G−, and lef t gauche G+are defined. (c) Then three kinds of conformationsright helix, left helix, and extended coil (serial trans)are defined based on the definitions of the microconformations above. B

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RESULTS The atomic snapshots of the system under stretching are presented in Figure 3 with the conformations including coil,

sharp increase corresponding to a fast generation of Nt and a weak reduction of Nh after the strain exceeds 4.4. This phenomenon suggests that overstretch induces the transition from the low energy helical conformation to high energy extended coil, in line with early theoretical prediction and other simulations in biopolymers.39,69 To directly view the evolution of different conformations during stretch process, conformations belonging to two representative chains are plotted as the function of ε in Figure 5.70 Here, the colors from blue to red correspond to an increase

Figure 3. Atomic snapshots of stretch process of the system at several representative strains ε. The green sections are atoms in coil state while the red and blue ones represent right and left helices, respectively. Figure 5. (a) Conformations extracted from one representative chain. Vertical axis ε is strain, and the horizontal axis Sa is the serial number of atoms along the chain backbone. The color from blue to red corresponds to increase of helical length as shown in color label, while dark blue and dark red represent coil and extended coil, respectively. Different pathways of helical formation are highlighted with the zooming-in pictures. (a1) Merging of two short helices to form a longer helix, (a2) disappearance of a short helix, (a3) formation of a helix from melt, (a4) and (a6) short helices gradually merge to form a long helix, (a5) a short helix growing to a longer one, and (a7) destruction of a helix and generation of an extended coil segment in over stretch condition. (b) Conformations belonging to another representative chain, which shows the destruction of the helices and the generation of extended coil after overstretch.

right, and left helical sections which are colored by green, red, and blue, respectively. A few helical (red and blue) states can be found in the system at 0 strain while imposing stretch induces continuous increase of helical content until strain of about 4, which indicates that stretch can indeed promote the formation of helices. Further increasing strain does not lead to obvious increase of helical content until strain of 9. To quantitatively analyze CHT, the evolution of the total number (Nhs) and the average length (Lhs) of helical segments as a function of strain ε is shown in Figure 4a. Nhs increases

of helical length, while dark blue and dark red represent coil and extended coil conformations, respectively. Various classic possible kinetic pathways of helix evolution are observed in Figure 5.42 Figures 5(a1), 5(a4), and 5(a6) show that two or more short helices gradually merge to form a long one, which represent the merging mode of helix.68 Here, the merger only happens in same-handed short helices, and the opposite-handed helical helices cannot merge even if they come closer because of the existence of helix reversal defects. Besides, short helix can disappear or appear abruptly (Figures 5(a2) and 5(a3)), which can also grow to form long one (Figure 5(a5)). The transition from helix to extended coil conformations is also found under overstretch as shown in Figure 5(a7), which is demonstrated more obviously with another representative chain in Figure 6b. At strain exceeding 4.4, a large portion of helices breaks and then form extended coil segments gradually. These snapshots in Figure 5 demonstrate that there are two kinetic pathways for the formation of long helices, namely (i) growth of short helix and (ii) merging of adjacent short helices. Combing results from Figures 4a and 5a, we can conclude that merging of short helices is the dominant kinetic pathway for the formation of long helices. Helix merging consumes short helices and leads the decrease of the number of helices Nhs (Figure 4a). As shown in Figure 5a, most atoms belonging to long helices are in short helix state before becoming the long helices, which also supports that merging short helices plays the dominant role to generate long helices. Besides the figure of conformation evolution, we also tried to quantitatively study the source of

Figure 4. (a) Evolution of the number of helices Nhs (black) and the average length of helices Lhs (red). (b) Evolution of interaction energy of intrachain Eintra (black) and the numbers of atoms in helices Nh (red) and in extended coil Nt (blue). The turning points appear at the same strain of 4.4 for the Eintra, Nh, and Nt curves.

with strain first and then turns to decrease at strain of about 3, while Lhs increases monotonically with strain and reaches an average monomer number of 6.5 at strain of 9. The unsynchronized trends of Nhs and Lhs imply that the formation of long helices (the increase of Lhs) takes a kinetic process of merging neighboring short helices (the decrease of Nhs),42,68 which can reduce the number of interfaces between coil and helix.39 CHT can be reflected directly by intrachain energy (Eintra) as shown in Figure 4b (black), in which the numbers of atoms in helix state (Nh, red) and extended coil state (Nt, blue) are also presented for comparison. At strain below 4.4, a fast increase of Nh and a relative weak increase of Nt are observed to accompany with a decrease of Eintra, which indicates that CHT leads to a lower energy state of chain. However, Eintra shows a C

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Figure 7. (a, b) Number of helices Nha with a low and high atom numbers as a function of strain ε, respectively. To more clearly show the tendency, the curves shown here are the smooth curves of all data points, which are shown in the Supporting Information (SI 2).

follows our general expectation that the number Nha of the shorter helical segments is always larger than that of the longer ones (see Figure 7a). However, imposing strain unexpectedly results in that the number of helical segments with odd atoms gradually exceeds the number of helical segments with even atoms even if the former is longer, which we name the odd− even effect. For example, with strain above 3 the Nha of helical segment with 9 atoms is larger than the one of helical segments with either 8 or 10 atoms. A similar phenomenon is also observed in Figure 7b, which presents the evolution of long helical segments. The number Nha of helices with 17 atoms is even larger than the ones with 12, 14, or 16 atoms at the end of stretch. Evidently, the helix with odd number atoms is energetically favorable as compared with their adjacent evennumbered counterparts during stretch.

Figure 6. (a) Numbers of helices Nhm with 4, 5, 6, 7, and 8 monomers as a function of strain ε. (b) Numbers of helices Nhm with 9, 10, 11, 12, and 13 monomers as a function of strain ε.

long helices while merger and growth are our main goals. Based on the subjective but reasonable definitions of merger and growth, the proportions of merged long helices and growing ones were obtained. The result that merged ones occupy nearly 60% and growing ones occupy nearly 40% further supports the conclusion that merging of short helices is the dominant mechanism for the formation of long helices, and more details can be seen in the Supporting Information (SI 1). Figure 6 presents the numbers of helical segments (Nhm) with different lengths (represented with monomer number n) during stretch. Short helices with monomer number n below 10 already present in the melt before imposing stretch, which is consistent with experimental observations.43 Imposing stretch increases Nhm of short helices immediately, which leads to a maximum and then following a decrease with further increasing strain (Figure 6a). Here, the decrease of short helices number further proves that the merging of short helices is the main pathway to form long helices. The strain at Nhm maximum increases with the increase of helical length n, and no Nhm maximum appears for n > 8 within the strain range we studied here. For n > 10, an incubation ε exists for the appearance of long helices (Figure 6b), which is consistent with the experimental observation,43 and the similar onset of flowinduced crystallization has also been observed.71 For example, the appearance of helices with monomer number n of 13 requires an onset strain of about 1.2. This demonstrates that the formation of long helices with n > 10 is mainly driven by external stretch, namely stretch-induced CHT. The Nhm of long helices (n > 10) increases slowly and enters nearly a plateau after strain exceeding 4.4, suggesting that overstretch does not promote further growth of helices but induces helix-extended coil transition as shown in Figure 5b. In Figure 7, the unit of helical length is replaced with the number of carbon atoms in the backbone instead of monomer number as presented in Figure 6. To avoid confusion, we use Nha to represent the number of helical segments with different length (represented with atom number). Compared to the Figure 6, our original expectation was that the number of helices with N monomers should be the sum of helices with 2N atoms and 2N + 1 atoms, and the number of helices with 2N atoms should be more than the one with 2N + 1 atoms in general as the former is shorter. At zero strain before stretch, it

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DISCUSSION The above MD computer simulation results demonstrate that stretch can indeed induce the formation of long helix, which mainly follows the kinetic pathway of merging adjacent two or more short helices, while overstretch drives the transition from helix to extended coil at large strain. All these results are consistent with experimental observation and in line with the theory of stretch-induced CHT. With the quantitative distribution of helical length in terms of monomer number (Figure 6) and atom number (Figure 7), we are going to discuss their thermodynamic origins in the following. A. Effect of External Force on CHT. As shown in Figure 6, imposing stretch results in the increase of helical content and the formation of long helices which do not present at quiescent melt. CHT is a competition between the potential energy gain Δh and the averaged entropy loss Δs per monomer, which can be expressed as free energy gap (or chemical potential difference) per monomer Δf = Δh − TΔs.37−39 By changing entropy loss Δs and potential energy gain Δh, and consequently lowering Δf = Δh − TΔs, the external force can stimulate the formation of helices. In order to estimate the average free energy gap Δf between helix and coil, two assumptions are proposed. First, it is assumed that the free energy gap Δf per monomer between helix and coil is constant at the same strain, meaning that free energy gap per monomer will not change with the increase of length of helix. Based on this assumption, the whole energy gap of the helix with n monomers can be written as ΔE = nΔf. Here, we should indicate that this free energy gap Δf represents the difficulty of coil−helix transition in the whole, and the boundary energy has been attributed to that. The other assumption is that the content of the helix follows the D

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Figure 8. (a, b) Linear fitting of relationship between ln Nhm and n under different strain. (c) Free energy gap Δf of coil−helix transition based on slope of (a) and (b) as a function of strain ε.

calculated based on force field we used, and more details are shown in the Supporting Information (SI 4). Note here we only consider intrachain energy, and the cooperative effect of chains through interchain interactions will be discussed later. Figure 9a

Boltzmann distribution, indicating that its content will be

(

decided by its energy, so the formula Nhm(n) = N0 exp

ΔE − kT

)

can be obtained. On the basis of two assumptions, we can get the equation ⎛ nΔf ⎞ ⎟ Nhm(n) = N0 exp⎜ − ⎝ kT ⎠

(2)

where k is the Boltzmann constant and N0 is a constant. Then eq 2 can be transformed to ⎛ Δf ⎞ ln Nhm(n) = ln N0 + n⎜ − ⎟ ⎝ kT ⎠

(3)

With eq 3, we fit the linear function of ln Nhm(n) vs n under different strains, as shown in Figure 8a,b. From the slope of linear fitting, the free energy gap for CHT Δf per monomer under different strains can be obtained, which is plotted vs strain ε in Figure 8c. From Figure 8c, it can be seen that the sign of Δf is positive, which means that the temperature of system is above the coil−helix transition temperature. The result shows that the free energy gap Δf decreases with the increase of strain, which is close to a linear relation with strain below 4. Further increasing strain leads to a weak reduction of Δf, which is in line with occurrence of overstretch. Following the theory of stretch-induced CHT for single chain37−39 and also considering the change of intrachain potential energy gain Δh induced by stretch, we express the free energy gap Δf between helix and coil states as

Figure 9. (a) Evolutions of elastic energy Fel, energy difference ΔEintra within the chain between helix and coil state, and their sum as functions of strain. (b) Comparison between the free energy gap Δfcal calculated by eq 4 and the one Δf fit fit from the Figure 8 as a function of strain. (c) Variation of interchains energy difference between helix and coil state ΔEinter as a function of strain. (d) The left section is a snapshot of system at strain 4.4, where red and blue sections respectively represent helix and coil. Then the right section is about helical clusters and some enlarged ones.

Δf = Δf0 + T ΔSε + ΔHε = Δf0 + Fel + ΔE intra

(4)

Here Δf 0 is the free energy gap at quiescent, which is about 0.87kT. Fel is the elastic energy of segment between entanglement points, which represents the change of entropy loss,39,72−75 and can be expressed as 2 3kT ⎛ R ete ⎞ Fel = − ⎜ ⎟ 2N0 ⎝ R 0 ⎠

shows the variation of Fel, ΔEintra, and Fel + ΔEintra with strain up to 4.4 as we only focus on stretch-induced CHT rather than helix-extended coil transition. At strain of 4.4, Fel reduces about 0.074kT, and ΔEintra decreases about 0.32kT, which is about 4 times of the former. This indicates that the change of potential energy gain ΔEintra plays a more important role than Fel does on stretch-induced CHT, which is in contrast with the theories of stretch-induced CHT.37−39 The early theories only consider the effect of Fel while ΔEintra is not introduced in. The free energy gap Δfcal calculated with eq 4 and the Δf fit fit from Figure 8 are compared in Figure 9b. Δfcal gives nearly the same value of Δf fit with strain below 1.3, suggesting that stretch-induced CHT can perfectly be attributed to the reduction of free energy gap, which, however, is mainly contributed by the decrease of intrachain energy gap ΔEintra between helix and coil rather than entropic reduction Fel.

(5)

where N0 = 100 monomers is regarded as the length of entanglement segment,76 and R0 = 5.1 nm is the end-to-end distance of the entanglement segment in a random state58 while Rete is the one in the stretching state. Here, eq 5 is based on the entropy elastic energy formula of Gaussian chain, and more details can be seen in the Supporting Information (SI 3). The energy difference ΔEintra between helix and coil states, which represents the change of intrachain potential energy gain, is E

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Macromolecules With strain above 1.3, Δfcal becomes obviously deviating from Δf fit with larger value, suggesting that the above singlechain model without considering interchain interactions is not sufficient to account the increases of helical content and length. Note as shown in Figure 6b long helices initially not existing in quiescent melt start to emerge around this strain. To explore other possible contributions for the formation of long helices, we estimate the variation of interchain energy gap ΔEinter between helix and coil states, which is plotted vs strain in Figure 9c (more details for the calculation are provided in the Supporting Information, SI 4). A sharp transition of ΔEinter occurs at strain of about 1.3. Below strain of 1.3, ΔEinter keeps nearly constant while it turns to a nearly linear decrease with further increasing strain. Two straight lines are plotted in Figure 9c to guide the eye. Evidently, at strain below 1.3 the content increase of short helices does not involve in the cooperative effect of interchain energy ΔEinter. In other words, the above single-chain model is sufficient to account stretch-induced CHT for the formation of short helices, which is already demonstrated by the results shown in Figure 9b. Furthermore, the onset strain for (i) the formation of long helices (Figure 6b), (ii) the deviation between Δfcal and Δf fit (Figure 9b), and (iii) the reduction of interchain energy gap ΔEinter between coil and helix (Figure 9C) are all coincident together, which strongly supports that the formation and growth of long helices require not only intrachain free energy change but also the assistance of interchain interactions. The left section of Figure 9d presents the distributions of helices (red) and coil (blue) of the system at strain of 4.4, which shows helices indeed aggregate into small clusters. And in the right section, the helical clusters and their zooming-in pictures are presented to illustrate the parallel packing of helices, which may be responsible for the reduction of ΔEinter (there is a more detailed movie in the Supporting Information). It is demonstrated that the cooperative effect of helices with interchain interactions indeed contributes to the formation of long helices. With the above analysis, we can conclude that stretch-induced CHT for the formation of short helices proceeds through reducing Fel and ΔEintra, where the latter plays the major role, while the formation of long helices is assisted by the coupling between intrachain conformational and interchain orderings, which is in line with early experimental and simulation observations.43,44,75,77,78 B. Cause of Stretch-Induced Odd−Even Effect. For the sake of simplicity, we used Hodd and Heven to represent helices with odd-number of atoms and helices with even-number of atoms, respectively. As shown in Figure 7, the content of Hodd gradually exceeds that of the adjacent Heven during stretch, which suggests that Hodd has lower free energy than the adjacent Heven. Investigating the Hodd TG−TG−TG−T and the adjacent Heven TG−TG−TG−G, which adds a random microconformation G− or G+, we found that the Hodd is longer than the adjacent Heven adding a random microconformation, which are detailed shown in the Supporting Information (SI 5), and the Hodd with longer length may be preferred in stretching condition. For a segment of polymer chain consisting of total m atoms in which mh atoms are in helical state, its total free energy is given by Buhot and Halperin38,39 as follows: Fch = mh Δfa +

Here, Δfa is the free energy change per atom of CHT and λ is the stretch ratio of end-to-end distance.R = 0.036 m (unit nm) is the end-to-end distance of the Gaussian iPP chain with m atoms,58 and Lh(mh) is the length of helix with mh atoms. For eq 6, the first term is the free energy change of the helix part, and the second term is the elastic energy of the rest random segment with (m − mh) atoms. Current simulation system consists of 50 000 backbone carbon atoms, in which approximately 2200 segments are in the helical state. Thus, the length of each segment m is about 22 (50 000/2200) atoms containing a helical section and a random coil section. In this small research scale, the influence of elastic energy may be important in contrast to the small influence of that in large research scale like entanglement segment. Δfa is 0.435kT per atom at the strain of 0, half of the free energy Δf 0 = 0.87kT per monomer obtained from Figure 8c. Considering the change of intrachain potential energy gain ΔEintra (Figure 9a), Δfa will decrease with the strain, and its values at strain of 1, 2, and 3 are 0.87 − 0.1378 0.87 − 0.2281 0.366kT ( = 0.366), 0.321kT ( = 0.321), 2 2 0.87 − 0.2752

and 0.297kT ( = 0.297 ). The helix length Lh(mh) is 2 approximately given by the equation For Hodd,

L h(mh ) = 0.108mh while for Heven L h(mh ) = 0.108(mh + 1) − 0.16

(8)

where 0.108 nm is the average length between adjacent two helical atoms and 0.16 nm is the length gap between the oddnumbered helix with mh atoms and even-numbered helix with (mh − 1) atoms adding a random micro conformation G− or G+, which is obtained by the model in Material Studio (see Supporting Information SI 5). Based on the above parameters, the energy of the whole segment with different helical length is calculated. Figure 10a−d presents the results at strains of 0, 1, 2, and 3, respectively, in which the black one is the calculated results based on eqs 6−8 while the blue one is obtained by the

Figure 10. Calculated energy Fcal (black) of whole segment based on eqs 6−8 and the energy Fdata (blue) obtained from distribution as a function of helix length mh contained in the segment. (a)−(d) are at strains of 0, 1, 2 and 3, respectively.

3kT (λR − L h(mh ))2 2R2(m − mh )

(7)

(6) F

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Macromolecules real distribution P(mh) of helices with different lengths and Boltzmann equation as follows: Fdata = −kT ln

P(mh ) P(mh = 6)

(9)

ACKNOWLEDGMENTS

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REFERENCES

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CONCLUSION Our molecular dynamics simulations reveal that stretch can indeed induce CHT in iPP and generate long helices which do not exist in quiescent melt while overstretch destroys conformational ordering, leading to helix-extended coil transition. Although both growth and merger of short helices are observed, the latter plays the major role in the formation of long helices. By means of mathematical statistics and fitting methods, we find that the external force reduces the free energy gap of CHT and promotes conformational ordering. At small strain (below about 1.3), the single-chain model of stressinduced CHT is sufficient to account the increase in short helical content, which, however, is mainly contributed by the stretch-induced reduction of intrachain energy gap ΔEintra between helix and coil rather than the reduction of entropy or elastic energy Fel proposed by current theories. While the single-chain model is insufficient to explain the formation of long helices, which is promoted by the coupling between intrachain and interchain interactions. Moreover, the statistical results show that the content of helical segments with odd atoms gradually exceeds that with even atoms during stretch, which can be explained with the free energy differences between the odd and even helices under stretching condition. ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00325. Quantitative analysis of the source of long helices; the numbers of helical segments (Nha) with different lengths (represented with atom number) during stretch; entropy elastic energy of Gaussian chain; calculation of intrachain energy and interchains energy; the length difference between odd and even helix (PDF) Rotating movie of system which shows the aggregation and orientation of helices (MPG)

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This work is supported by the National Natural Science Foundation of China (51633009 and 21704096) and the Key research and development tasks of MOST (2016YFB0302501). The work is carried out at National Supercomputer Center in Tianjin, and the calculations are performed on TianHe-1(A).

where Fdata has a similar meaning as ΔE = nΔf in subsection A. The two results displayed in Figure 10 are similar, and the odd−even effects appear after strain larger than 2, which show an oscillation with the switch of odd and even atom numbers. Thus, we propose that the odd−even effect is due to the effect of different lengths of Hodd and Heven on the overall free energy of the segment, which is gradually enlarged by stretch.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (L.L.). ORCID

Liangbin Li: 0000-0002-1887-9856 Notes

The authors declare no competing financial interest. G

DOI: 10.1021/acs.macromol.8b00325 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.8b00325 Macromolecules XXXX, XXX, XXX−XXX