Molecular Dynamics Simulation of Stretch-Induced Crystallization in

Feb 8, 2019 - In the last stage the crystals come to collide with each other to give well-aligned stacked lamellae. By dividing the system into mesh c...
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Molecular Dynamics Simulation of Stretch-Induced Crystallization in Polyethylene: Emergence of Fiber Structure and Molecular Network Takashi Yamamoto* Graduate School of Science and Engineering Yamaguchi University, Yamaguchi 753-8512, Japan

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ABSTRACT: Molecular processes of fiber formation and network genesis in polyethylene (PE) are studied by molecular dynamics simulation of crystallization from highly stretched melt. We consider a system comprising 150 molecules of 500 united atoms, which is much larger than that of our previous work on fiber structures and their deformation [Polymer 2013, 54, 3086], aiming to study crystal nucleation and growth separately. Rapid elongation of the isotropic melt gives rise to incipient crystal clusters or embryos much smaller than critical nuclei. Subsequent isothermal crystallization of the stretched melt leads to three distinct stages of structure formation. The initial stage is a period waiting for critical nucleation. In the second stage that follows, we observe independent growth of isolated lamellae and resulting polynomial increases in crystallinity. In the last stage the crystals come to collide with each other to give well-aligned stacked lamellae. By dividing the system into mesh cells (pixels) and using an algorithm for image processing, we analyze the growth of clusters in detail in terms of their sizes and shapes. Despite highly anisotropic chain conformation in the melt, we find rather isotropic growth of the clusters both along and perpendicular to the fiber axis in the early stage of crystallization, where crystallites show linear growth of similar rate for each direction. The exception is the crystallization under high tension, where lateral growth of lamellae seems to be hindered indicating characteristic diffusion-controlled growth. We also study genesis and development of polymer network in situ by properly defining fold, tie, and cilium segments connected to the growing crystallites. In the early stage of the network genesis, small crystal clusters are connected to very long cilia which are forming the dominant component of the system. In the following stage of network development, where the crystallites show marked growth in size, the long cilia are rapidly reeled into the crystalline region, and the folds and ties are continually tightened. Through statistical analyses of the folds and ties during crystallization, we find that both of them are initially slack and have broad length distributions, but they continually tighten and come to have characteristic asymptotic distributions. Of special interest is that the tie molecules have rather stretched conformation even after sufficiently long crystallization, which is an indication of memory of crystallization process from the highly stretched melt.

1. INTRODUCTION Major constituents of plastics are crystalline polymers, whose mechanical, thermal, and electrical properties are dominated by the amount of crystallites and their higher order structures. Polymer crystallization is the fundamental and governing process of structure formation ranging from atomistic to macroscopic scales. Besides innumerable experimental investigations over several decades,1−10 these 20 years have witnessed great enthusiasm for in situ observing crystallization by computer modeling. Even if we limit to the molecular scale modeling, various aspects of polymer crystallization have been studied. They range from the molecular mechanisms of crystal nucleation11−18 and growth19−29 to challenges to much larger scale phenomena;30−32 intensive efforts are also devoted to the structures of supercooled melt,14,33,34 the role of chain entanglement in crystallization,35 and polymers of special geometry or topology such as helical,36−38 branched,39,40 cyclic,41,42 and block copolymers.27 © XXXX American Chemical Society

In industrial processing, polymeric materials are crystallized or solidified under various fields of flow or deformation. Taking fiber spinning, for example, the crystallizing polymers are under large elongational flow, which exerts great influence on crystallization and gives rise to characteristic higher order structures. Understanding crystallization during flow or deformation is therefore of supreme industrial significance.43,44 It also imposes intriguing scientific problems since crystallization under flow or deformation is accompanied by simultaneous thermal and mechanical relaxations. Computer modeling of polymer flow is an active field of research due to great industrial significance but mostly based on continuous models.45 When crystallization must be taken into consideration, however, the continuum models are Received: December 2, 2018 Revised: January 29, 2019

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of different chains and between those of the same chain more than three bonds apart

confronted with a serious problem of molecular scale heterogeneity due to phase transitions governed by molecular characteristics of the polymers. Of great interest here are molecular level studies of flow-induced or stretch-induced crystallization. However, as far as the author is aware, molecular simulations of crystallization under large deformation or flow in polymers started late in the early 2000s46−59 since the first reports by Koyama et al.46 and Rutledge et al.47,48 where they observed the onset of rapid crystallization from the highly oriented melts of polyethylene (PE) and nalkanes. In studying very slow process of polymer crystallization in realistic molecular models, they took advantage of rapid crystallization from highly stretched melt, thereby enabling direct observation of crystallization without recourse to coarse-grained models. The present author also applied similar strategy to study crystallization in much more complex helical polymers such as isotactic polypropylene which exhibits intriguing chirality selection during crystallization.38 In our recent paper, we reexamined the stretch-induced crystallization of PE into fibers and studied large deformation of the obtained fibers both along and perpendicular to the fiber axis direction.49 We there confirmed a characteristic mode of chain folding through hairpin conformation, which was suggested to be operative in crystallization of the quiescent melt.22,29 We also studied large deformation of the fiber synthesized by MD simulation, where we could observe largescale reorganization by extreme transverse deformation perpendicular to the fiber axis. The present paper is an extension of this previous report49 with respect to stretchinduced crystallization. We here focus on important problems left untouched and study nucleation and growth of individual crystalline clusters in a larger system by employing a new cluster analysis algorithm. We also reveal the development of a network structure comprising crystals, folds, and ties as well as dangling cilia.

l o oij σ yz12 ij σ yz6| o+U UvdW(r ) = 4εo m cutoff ojj r zz − jj r zz } o o o k { k { n ~

where ε and σ are assumed to have values 0.5 kJ/mol and 0.38 nm, respectively; the interactions are cutoff at a distance rc = 2.5σ, and no tail corrections are made. The numerical values of the parameters, and the related reduced units are reproduced in Table 1 for convenience. Table 1. Values of the Parameters Used in the Simulationa

a

parameters

values

units

m ε (reduced energy) σ (reduced length) r0 kb θ0 kθ k a0 a1 a2 a3 a4 a5

14 500 0.38 0.4 10000 70.5 1000 18 1.0 1.31 −1.414 −0.3297 2.828 −3.3943

g/mol J/mol nm σ ε/σ2 deg ε ε

Reduced pressure is 15.13 MPa.

We placed PE molecules within a rectangular parallelepiped MD cell having three sides of lengths (a, b, c), which is replicated by the periodic boundary condition (PBC).61 The temperature and pressure are controlled mostly by the loose coupling method of Clarke and Brown. All MD simulations are done using the program COGNAC (coarse-grained molecular dynamics program) in OCTA (the open computational tool for advanced materials technology).62 2.2. Analysis of Crystal Clusters. To capture emerging crystalline order, we must introduce several order parameters. We first define so-called chord vectors bi, which is defined as connecting the mid points of the two adjacent C−C bonds, corresponding to the ith united atoms, except the terminal atoms, located at ri. Then we define a local order parameter P2(r), which was also used in our previous paper,49 by dividing the MD cell into cubic mesh cells of side length about 2σ (volume 8σ3), and within each mesh cell we calculate the orientational order parameter

2. MODEL, SIMULATION, AND DATA ANALYSES 2.1. Model and Simulation Method. In this paper we study PE, as in our previous paper,49 using a conventional united atom model combining every hydrogen atom to its nearest carbon atom. It is well acknowledged that the united atom models do not reproduce the lowest energy structure of PE crystal, especially its space group symmetry Pnam. However, we are not here concerned with reproducing the accurate crystal structure but are interested in studying molecular process of crystallization and network formation. We consider 150 PE molecules each comprising 500 methylene groups. We adopt the conventional Rigby−Roe force field,60 which consists of the following intramolecular energy terms: C−C bond stretching (Ubond), C−C−C bond angle bending (Uangle), and dihedral angle rotation (Utorsion)

P2(r) = ⟨3 cos2 θi , j − 1⟩/2

Ubond(r ) = k b(r − r0)2 /2

(3)

where θij is the angle between chord vectors bi and bj within the same mesh cell located at the representative position r, and the average is taken over all pairs of chord vectors within the mesh cell. Then we estimate the crystallinity xmesh from the c number of mesh cells Nc that have local order parameter greater than a threshold 0.7, divided by a total number of mesh cells in the system Ntotal

Uangle(θ ) = kθ(cos θ − cos θ0)2 /2 5

Utorsion(τ ) = k ∑ an cosn τ n=0

(2)

(1)

where r0 is the equilibrium bond length 0.152 nm, θ0 is the equilibrium bond angle of 70.5°, and the torsion angle τ is measured from the trans position. The nonbonded interactions UvdW of the following form are assumed between united atoms

xcmesh = Nc(P2 > 0.7)/Ntotal B

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Macromolecules The choice of the threshold 0.7, which we have used in our previous simulations, might seem rather small, but a slightly stringent threshold did not much affect the results. It should be noted that such description of the crystalline regions is a sort of a space-averaging, with resolution of 2σ. However, taking advantage of the mesh-cell (lattice) structure, we can readily find and demarcate crystal clusters by adopting a 3D extension of the algorithm often used in 2D-image processing (the connected-component labeling). We divide the whole MD cell into mesh cells of volume about 8σ3 and consider each mesh cell as a pixel in the 3D image. For each pixel, we give binary values 1 or 0 according to mesh-cell order parameter P2: 1 for the crystalline pixel (P2 > 0.7) and 0 for the noncrystalline pixel (P2 < 0.7). Using the connectedcomponent labeling algorithm for the binary image in 3D space, we can readily identify crystalline clusters (continued clusters of crystalline mesh cells) and calculate their volumes and shapes. Though the spatial resolution is limited by the mesh-cell size, the image processing is very easy since the cluster criterion is based only on the nearest-neighbor pixels in the 3D lattice. The size of each crystalline cluster is readily estimated from the volume, that is, the number of mesh cells, of each crystallite, while the shape of each crystalline cluster is expressed through the diagonal components of the radius of gyration tensor (Rgx2, Rgy2, and Rgz2) calculated like R gx 2 =

Figure 1. Definitions of elements comprising crystal network: crystals, ties, folds, and cilia. The crystalline atoms (white) are first defined as composing straight parallel stems as described in the text. Then the tie and fold segments are considered as those connecting different crystalline stems; those entering into the crystalline region in parallel fashion (red) are here called ties, while those entering in opposite fashion (yellow) are called folds. The segments one end of which is floating in the melt phase (green) are called cilia.

according to the rule shown in Figure 1. Ties are segments connecting crystallites in a way that the chord vectors at the crystal exit and at the crystal entrance are parallel with positive inner product, while we consider folds those antiparallel with negative inner product. The definition of the folds might seem a little strange when the two crystallites are separated lumps (Figure 1), but when they come to be connected into a single lamella, the definition becomes consistent with the usual definition of folds. The remaining cilia segments are those exiting or entering the crystallites and ending in the melt phase.

∑ (rix − rjx)2 /2n2 (5)

i,j

where vector rix represents the x-coordinate of the ith pixel, and the sum with respect to i and j is taken over all n pixels within a single cluster. Then the averages of any quantity A, over Aα for the αth cluster, are taken as volume (weight) averaged such that A=

3. RESULTS OF MD SIMULATIONS 3.1. Uniaxial Stretching of Isotropic Melt. Highly oriented PE is generated by uniaxial drawing of isotropic melt as in our previous work.49 We first constructed amorphous PE within a cubic MD cell, which was annealed at 800 K for 2 ns under atmospheric pressure and then cooled and further annealed for 1 ns at 350 K. The equilibrated melt in the MD cell of the size a ≅ b ≅ c ≅ 33 was then stretched along the z direction by the rate c−1 dc/dt = 0.002 [1/unit time], which was very fast of about 2.72 time elongation within 1 ns (500 unit times) (Figure 2a). The stretching was conducted at 350 K slightly below the melting temperature and under constant volume condition to avoid cavity formation. The constant volume stretching can have delicate effects on many initial processes such as emergence of crystal embryos and their populations. The readers should have in mind that the present results of the initial processes can be slightly dependent on the detailed mechanical conditions of the initial drawing. Let us begin with examining the initial elongation of the isotropic melt. During the rapid elongation along the z-axis, the tension τzz rapidly grows up to about 4.5 (∼67 MPa) with concomitant small stress (τxx, τyy) in the transverse directions (x, y) going only up to 0.5 (Figure 2b). In the initial isotropic melt, we observed only a small structural fluctuation giving rise to a few small crystalline clusters, but when we stretched the melt a number of crystallites emerged (inset of Figure 2a), though the crystallinity is still very small of a few percent (Figure 2c). It was also found that comparable stretch-induced cluster formation is noticed even at a temperature Td = 400 K very close to the quiescent melting point. The sizes of the crystallites observed during the stretching (Figure 2d) are

∑ AαVα/∑ Vα α

α

(6)

where Vα is the volume of the αth cluster and the Vα/∑αVα is the volume (weight) fraction of the αth cluster. 2.3. Description of the Network. During the rapid formation of crystallites, we expect pronounced generation of fold and tie segments connecting the crystallites along with dangling tails or cilia. We here regard the crystallizing system as a network composed of the crystallites, ties, folds, and cilia. This may be similar to systems of partially crystalline rubber or silk. To describe the network structure, we here define basic network components as follows (Figure 1). Because the present discussion is based on molecular level details of the crystals, we redefine the crystal clusters a little bit in detail. The way we used as a criterion for the crystalline atoms is similar to what we have used before.22 We first pick up straight chain segments, which are made of at least nine chord vectors whose angle between neighboring chord vectors are less than 30° and consider these segments as crystal candidates. Of these candidates, we consider them crystalline those having more than two parallel and neighboring stems.22 This crystal criterion may be slightly weaker, and the resulting crystallinity xatomic is a little larger than that we defined before c using crystalline mesh cells, but the discussions are not affected by the change of the definition. Having defined the crystalline atoms in this way, we categorize the connecting segments as ties, folds, and cilia C

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Figure 2. Initial drawing of the isotropic melt along the z-axis at temperature Td = 350 K near the melting point, during which (a) the MD cell size c increases exponentially and (b) the stress component along the z-axis τzz increase rapidly while those along the x- and y-axes remain much smaller. During the initial drawing, there emerged appreciable crystalline clusters (blue clusters) in the background melt phase (red) (inset, part a). (c) Emergence of the embryonic crystallites is calculated versus time in terms of crystallinity, at two drawing temperatures Td = 350 K and Td = 400 K. (d) Distribution of the cluster size during the initial drawing Td = 350 K. The clusters are shown to be much smaller than the critical nucleus for isothermal nucleation described later.

Figure 3. (a) Increases in the crystallinity during crystallization of the stretched melt at 350 K under different stress conditions: τzz = 3 (black), τzz = 1 (blue), and τzz = 0 (red); averages are made over three runs. Changes in the MD cell size c are also plotted vs time in the inset. (b) The same crystallinity changes are plotted in log−log scales. After short waiting time (prenucleus stage), we observe time periods where clear polynomial increases in the crystallinity are noticed, indicating the independent growth of lamellae. Then followed saturation mainly due to limited lateral sizes of the MD cell, where we observe marked stacking of lamellae. Ends of the postnucleus stages are marked by large colored circles, which are located around 5, 10, and 20 ns for τzz = 3, 1, and 0, respectively.

much smaller than the critical nucleus size of about 80 σ3 which will be explained later. 3.2. Nucleation and Growth in the Stretched Melt. The highly stretched melt obtained at Td = 350 K and under resulting stress (τxx, τyy, τzz) = (0.5, 0.5, 4.5) (Figure 2b) was used as an initial state for the following crystallization

simulations under constant temperature Tc and constant stress {τ}. Though we performed simulations at several temperatures, we here discuss cases for Tc = 350 K only where we could observe well-developed lamellae. The stress conditions for {τ} was changed from the initial value of τzz = 4.5 to selected values of 3.0, 1.0, and 0, while the lateral stress components are D

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Macromolecules nearly fixed near the initial value of 0.5, with the exception for τzz = 0 where τxx = τyy are also set 0. Generally, the system rapidly transformed into fiber structures made of alternating layers of crystalline and amorphous phases within a few tens of nanoseconds. (eq 4) Figure 3a shows the increases in the crystallinity xmesh c during crystallization under various tension conditions τzz = 3.0, 1.0, and 0; three independent runs were made to average. As is readily noticed, larger tension τ zz gives faster crystallization. Though the present system is restricted severely in the lateral x- and y-directions, the increase in the crystallinity is expected to follow the Avrami equation in the early stage. The Avrami equation for the crystallinity xc is written for the steady-state growth of nonoverlapping lamellae as xc = 1 − exp(−Ktn) ≅ Ktn (t ≪ 1), and then ln(xc) ∼ n ln(t) + ln(K). Figure 3b show the log−log plot of the crystallinity versus time given in Figure 3a. After some waiting times for the onset of steady-state growth of lamellae, the crystallinity shows simple power low increases. These time regions are considered to show the stages of independent growth of lamellae after nucleation. The waiting times for the prenucleus stages are around 1−5 ns according to the value of tension τzz, and they are considered the times needed for the emergence of the primary nucleus described later. On the other hand, marked deviations from the power law at later times are evidently due to overlap of the growing lamellae due to the limited lateral sizes of the MD cell; the stages are here call lamellar stacking stage since well-developed lamellae start stacking in these time regions. Also shown in the inset of Figure 3a is the changes in the MD cell dimension c along the stretch direction. During the initial few nanoseconds, where the crystallinity increases rapidly, the c dimension of the MD cell, the length of the fiber, increases markedly for τzz = 3.0 and 1.0. These initial fiber elongation will be the mechanical deformation by the applied tension τzz; the elongation continues until considerable crystallinity of about xmesh = 0.2−0.3 is attained (filled colored c circles). It is interesting to note that crystallinities of about = 0.2−0.3 are nearly half of the limiting crystallinities xmesh c obtained after sufficiently long crystallization. Appreciable elongation is also noted even at τzz = 0 after the initial contraction, which is probably due to large reduction in tension from τzz = 4.5 to τzz = 0. This elongation will be due to the equilibrium thermodynamic origin by chain straightening and alignment during crystallization. 3.2.1. Identifying Nucleation and Growth by Image Processing. We want to mark out crystalline regions to study the number of crystallites and their sizes and shapes. For the convenient delineation of the crystalline region, we divided the whole system into a 3D mesh; for each mesh cell we defined the local order parameter P2(r) and the binary index according to whether the mesh cell is crystalline or noncrystalline, as explained in section 2.2. Considering each mesh cell as a pixel of the 3D image, we applied the connected component labeling algorithm to delineate crystalline clusters. Figure 4 shows, for example, crystalline clusters growing at 350 K and τzz = 1; different colors indicate different clusters, and the rodlike entities in the clusters represent crystalline stems of the polymer chains. It is readily noticed that each crystal grows both in the fiber direction z (horizontal) and in the lateral directions x and y (vertical), indicating thickening and lateral growth, respectively, of the crystalline lamellae.

Figure 4. Development of crystalline clusters, in the postcritical nucleus stage, at 350 K and τzz = 1. Clusters are delineated using a 3D extension of the connected-component labeling algorithm, and separated clusters are painted in different colors. Marked growth in the lateral directions x−y (vertical) is readily seen along with appreciable lamella thickening in the fiber axis direction z (horizontal). Typical dimensions of the figures about 100σ (horizontal) and 20σ (vertical).

We thus identified the crystalline clusters whose possible minimum volume is about 8σ3 (one mesh cell). Figure 5 shows

Figure 5. Number of crystalline clusters, averaged over ten consecutive snapshots, plotted versus time during crystallization at 350 K under tension (a) τzz = 1 and (b) τzz = 3. Black dots are the total number of clusters smaller than 50 mesh cells which are considered well-separated. Red dots are, on the other hand, the total numbers of the precritical sizes smaller than 11 mesh cells (88σ3).

total numbers of crystallites versus time during nucleation and growth at τzz = 1 and τzz = 3. Initial increases in the number are noticeable during around 2 ns at τzz = 1 (Figure 5a) and around 1 ns at τzz = 3 (Figure 5b); the time periods roughly correspond to the prenucleus stage in Figure 3b. The crystallites then show rapid decreases in number after around 5 ns (τzz = 1) and 2 ns (τzz = 3), where the crystals grow rapidly by eliminating small clusters or gathering into larger ones. It should be remembered that the constant supply of new embryonic crystallites is not possible due to finite system size. It shows that crystallization in the present system originate from the crystallites implanted during the initial drawing process. In this sense, crystallization is started heterogeneously from limited number of initiators. E

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independent growth (Figure 3b). The method of characterizing the size and the shape of the crystallites was already explained in the previous section 2.2. Figure 7a shows time evolutions of the average volume of the crystalline clusters, weight-averaged in each snapshot and further averaged over successive ten snapshots for each time. In early stages up to around 1−3 ns for τzz = 3−0, only crystalline clusters smaller than 100σ3 are observed. After the times of 1− 3 ns waiting for critical nucleation, the volume of the clusters increases in polynomial order of time t: t2, t3, and t3 according to the value of τzz = 3, 1, 0, until the lateral sizes of the clusters cross the periodic boundary around 6−20 ns for τzz = 3−0. On the other hand, the average shapes of the clusters in terms of (Rgx2)0.5, (Rgy2)0.5, and (Rgz2)0.5 are shown in Figures 7b, 7c, and 7d for τzz = 3, 1, and 0, respectively. For the low tension conditions of τzz = 1 and 0 (Figures 7c and 7d), the postcritical clusters grow linearly with time in similar rates along all directions x, y, and z, until the lateral size of the crystals get large enough to cross the lateral periodic boundaries. The independent growth of the crystalline clusters is thus found unexpectedly isotropic as long as the present time and space region are concerned. This observation also conforms to the volume increases with t3 in Figure 7a. The exception is the case under large tension τzz = 3 (Figure 7b), where the crystal dimension along the fiber direction (Rgz2)0.5 grows linearly with time, but the lateral growth in (Rgx2)0.5 and (Rgy2)0.5 follows t0.5, and the resulting volume increase is proportional to t2 (Figure 7a). The reduced power law for the lateral growth is suggestive of the diffusion-controlled growth. Indeed recent experiments showed that the lateral growth of the clusters can be diffusion limited under strong tension.44 When the lateral sizes of the clusters grow sufficiently large to cross the periodic boundary, present calculations underestimate the lateral cluster sizes. Indeed, they show pronounced saturation at (Rgx2)0.5 ∼ (Rgy2)0.5 ∼ from 3σ to 4σ depending on the lateral sizes of the MD cell. On the other hand, the cluster size in the z-direction (R2gz)0.5 saturates due to the lamella thickness. 3.3. Network Structure. Rapid crystallization observed in the oriented melt yielded stacked crystalline lamellae, which are presumably connected by many ties and folds, thereby forming an interlinked network. To visualize and investigate the network formation, we defined the ties, the folds, and the cilia as described in section 2.3 and in Figure 1. In this paper we only discuss the case of crystallization at 350 K and τzz = 1, while other cases of crystallization under different tension conditions are qualitatively similar. 3.3.1. Evolution of the Network. Figure 8 shows the network evolution by coloring constituent atoms being colored according to their attributes: crystals in white, folds in yellow, ties in red, and cilia in green (Figure 1). In early stage of crystallization (2 ns), the system is dominantly composed of long cilia (green) connected to a few small crystallites. From the sea of dominant cilia, the crystallites continuously grow in sizes with concomitant contraction of the cilia. It is also noteworthy that the crystallites keep rather rounded or undulated shapes with tapered growth fronts, until they collide each other and form stacked lamellar structure. To describe structural changes in the network, we counted the number of C−C bonds comprising crystallites, folds, ties, and cilia defined in Figure 1. Figure 9a shows typical changes in the numbers of bonds for crystals, folds, ties, and cilia during crystallization at 350 K and τzz = 1.0. In the early stage up to

The birth and growth of crystalline clusters of various sizes are described by the relative abundance of crystallites versus their sizes. When we define the number ρ(υ) of crystal clusters of size υ, the volume occupied by these crystals is υ·ρ(υ). Figure 6 shows the occupied volume υ·ρ(υ) versus cluster size

Figure 6. Histograms of the volume υ·ρ(υ) occupied by clusters of given size υ, during crystallization at 350 K under τzz = 1. Red, green, and black bars are those after 1, 3, and 4 ns of crystallization, respectively. In the very early stage (red), the appreciable volume is only for clusters smaller than 80σ3, but after 3 and 4 ns the volume of larger clusters turn to increase, and the histogram shows a minimum around the volume 80σ3 (red arrow), suggesting a critical nucleus of size around 80σ3. Inset shows a typical free energy diagram of the CNT.

υ at times 1, 3, and 4 ns during crystallization at 350 K and τzz = 1. At the very early stage around 1 ns, all clusters are smaller than 80σ3. With laps of time, the clusters grow in size, and the distribution υ·ρ(υ) comes to show clear minimum around υ ≅ 80σ3. Simple calculation by the classical nucleation theory (CNT) suggests that this minimum corresponds to the critical nucleus volume (Supporting Information), and the size of about 80σ3 indicates that the critical nucleus is made of about 180 CH2 units. Assuming CNT and the usual enthalpy of fusion ΔH ∼ 2.8 × 109 erg/cm3 and surface free energies σe ∼ 90 erg/cm2 and σs ∼ 12 erg/cm2, given by Hoffman and Miller,6 actually if we assume σeσs2 ∼ 1.3 × 104, the critical nucleus size v* satisfies 3 16π 2σs 2σe ij Tm yz3 ij Tm yz v* j z j z = ∼ 1.7 j z j z σ3 (ΔH v)3 σ 3 k ΔT { k ΔT {

Because crystallization temperature is now Tc = 350 K, Tm is estimated to be around 480 K, which is about 80 K above the quiescent melting temperature of 400 K. It is well acknowledged that large extension results in effective melting temperature much higher than that under quiescent condition. However, a reliable experimental melting temperature under such large tension is not available. 3.2.2. Evolution in Size and Shape of the Growing Clusters. Molecular processes of nucleation and growth in the quiescent melt have been extensively investigated by molecular simulations, where pointlike nuclei of nearly isotropic shape first appear and then they grow predominantly in directions perpendicular to the chain axis leading to thin platelike lamellae.14,16 Our present interest is the crystallization in highly stretched melt, and we here study evolution of the size and shape of the crystallites over the stages of prenucleus and F

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Figure 7. (a) Growth in the weight-averaged volume of clusters, which was further averaged over ten consecutive snapshots for each time, under τzz = 3 (black) τzz = 1 (blue), and τzz = 0 (red). Parts b, c, and d are changes in cluster size and shape expressed by radius of gyration in the x, y, and z directions, ⟨Rgx2⟩0.5 (●), ⟨Rgy2⟩0.5 (○), and ⟨Rgz2⟩0.5 (△), under τzz = 3 (b), τzz = 1 (c), and τzz = 0 (d). In the postnucleation stage, clusters show clear polynomial growth of time (dashed lines). Each end of the polynomial growth of clusters is marked by large colored circle, which is located around 5, 7, and 15 ns for τzz = 3, 1, and 0, respectively.

bonds, or crystallinity xatomic , increased fast along with a c remarkable decrease in the number of cilia bonds; this is consistent with the rapid decrease in the green atoms in Figure 8. On the other hand, the number of fold and tie bonds increased approximately twice as many. The crystallinity increase is thus considered mainly due to the cilia segments being taken into the crystalline regions. In the later stage after 8 ns, the lamellar stacking stage (Figure 3b), crystallinity still shows a gradual but steady increase, while the numbers of fold and tie bonds slightly decrease, suggesting tightening of initial loose folds and ties. The fold, tie, and cilia segments are very important elements of the network playing major mechanical roles. Figure 9b shows the numbers and average lengths of the folds, ties, and cilia versus crystallization time. In the initial stage up to 8 ns, the numbers of fold and tie segments rapidly increase indicating the “network development”, while the number of cilia segments is much smaller and nearly constant. It is interesting to notice that the time period of we call “network development” roughly corresponds to the stage of the independent growth of the lamellae in Figure 3b. The continued tightening of the network elements is clearly noticed from the average lengths of folds, ties, and cilia segments (open circles connected by dashed lines in Figure 9b). Besides the pronounced decrease in the cilia length, the average lengths of folds and ties are seen to shrink considerably until the end of crystallization. Of considerable interest is that the average segment lengths of folds and ties tend to a

Figure 8. Development of the network during crystallization at 350 K and τzz = 1: crystallites (white) and connecting segments of folds (yellow), and ties (red), together with cilia (green); the colors are the same as those used in Figure 1. In the initial state of network genesis (2 ns), dominant components are cilia connected to small crystallites. In the following stages of network development and completion, the crystals are growing rapidly taking most of the cilia segments into crystals.

around 8 ns, which corresponds to the stage of independent growth of lamellae (Figure 3b), the number of crystalline G

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Figure 9. (a) Changes in the numbers of atoms (actually C−C bonds) forming crystals (black), folds (red), ties (green), and cilia (blue) plotted versus time for crystallization at 350 K and τzz = 1. (b) Detailed structures of folds, ties, and cilia in terms of the numbers of corresponding segments (filled circles with left axis scale) and their averaged lengths (open circles with right axis scale). The time period below around 8 ns (yellow shading) we call the “network development stage”.

Figure 10. Number of folds versus their contour length during crystallization at 350 K and τz = 1: at (a) 4 ns, (b) 7 ns, (c) 23 ns, and (d) 39 ns. Emergence and following compaction of the folds are clearly noticed. The histograms are linear scaled numbers (left scale), while the open circles represent log-scaled numbers (right scale). Shorter folds show faster equilibration to the exponential distribution (red dashed lines).

3.3.2. Statistical Structures of Folds and Ties. In the above discussions we found that the folds and ties were tightened considerably during development and completion of the network. We here study their statistical structures in a little more detail during crystallization at 350 K and τzz = 1.0. Figure 10 shows the population of folds pfold(lfold) versus their lengths lfold; the numbers of folds is plotted both in linear scale (histogram with left scale) and in log scale (open circles with right scale). Initially (4 ns) the folds are small in number as is also shown in Figure 9b, and they distribute rather uniformly over a wide range of length lfold. As time elapses, shorter folds markedly increase in numbers. The limiting distribution pfold(lfold) tends to an exponential form pfold ∼ exp(−lfold/l̅fold) when we neglect longer folds. The expected average length fold l̅fold is about 17 bonds, close to that shown in Figure 9b. As

common length around 20−30 bonds; it seems that the entropic drive to take longer folds and ties are counterbalanced by the same degree due to the energetic drive to crystallize. We can notice a marked trend of the cilia to decrease in length further even at the end of the simulation, which is consistent with the well-known fact that chain ends are exclusively located near the fold surfaces. As a stress transmitter between lamellae, the role of tie molecules is of prime importance in the discussion of mechanical properties. We will show, in the Supporting Information, how the number of tie molecules changes in the crystallization under larger tension τzz = 3. We will see clear increase in the number of tie molecules relative to that of folds in crystallization under larger tension. H

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Figure 11. Number of ties versus their contour length during crystallization at 350 K and τz = 1. The emergence and compaction (shortening) of the tie molecules are clearly seen from the distributions at (a) 4 ns, (b) 7 ns, (c) 23 ns, and (d) 39 ns. The distribution, which is rather uniform distribution in the early stage of crystallization, comes to show a sharp peak around 20 bonds. The black line in (d) is a curve fitted by an analytic function ln exp(−bl).

often reported previously,22,49 the folds were predominantly compact connecting the nearest- or the next-nearest-neighbor crystalline stems. In a similar way, the populations of tie molecules ptie(ltie) are shown in Figure 11 versus their lengths ltie. Initially the length of ties distribute over wide range, but their tightening with time is clearly seen from the distributions. Marked preference of shorter ties are readily noticed, and the distribution ptie(ltie) tends to have an asymptotic form ptie(ltie) ∼ lntie exp(−bltie), where n ∼ 1.33 and b ∼ 0.08 (Figure 11d). The asymptotic form has a peak around 20 bonds, and the average tie length is calculated to be about 29 bonds, which is again in good agreement with Figure 9b. The number and structure of the tie molecules (or tie segments) are expected to dominate mechanical properties of crystalline polymers, which on the other hand are known to depend on thermal and mechanical history of crystallization. The structure of the tie molecules is therefore considered to have clear memory of crystallization. However, present knowledge of tie molecules seems rather scarce except that based on equilibrium considerations.63 We here studied the structure of tie molecules in terms of their squared end-to-end distances Rnm2 ≡ (rm − rm+n)2 between both ends of the tie segments rm and rn+m n-bonds apart along the chain contour (inset of Figure 12). If a tie segment has a random coil conformation, the average Rn2 = ⟨Rnm2⟩ should follow the Gaussian statistics Rn2/n ∼ constant. On the other hand, if they are rather stretched and not fully relaxed, Rn2/n is expected to be a linear function of n, Rn2/n ∼ n. Figure 12 shows plots of Rn2/n vs n for all tie segments in the system crystallized at 350 K and τzz = 1 for 7 ns (red), 19 ns (green), and 39 ns (blue)

Figure 12. Conformational statistics of the tie segments in terms of their squared end-to-end distance Rn2 divided by the contour length n (inset) plotted versus n during crystallization at 7 ns (red), 19 ns (green), and 39 ns (blue) and after long equilibration for 4 ns under stress-free conditions, all at 350 K. The solid line is for comparison, which is for the segments in the isotropic melt (random coil) of the same polyethylene at 345 K; ideal Gaussian chains indeed follow Rn2/ n = constant for segments longer than around 10−20 monomers (i.e., bonds) corresponding to the Kuhn length. For convenience, the data at each time is shifted upward by five divisions. Shorter tie molecules follow the solid line for the short random coils, while longer ties deviate appreciably from the random coils in the isotropic melt.

and that after further relaxation under stress-free conditions (τzz = 0) (black open circles); for convenience, each data at different times are shifted upward by five divisions. Attached to I

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Macromolecules each data is the black solid line which represents Rn2/n calculated for the random coils in fully relaxed isotropic melt of PE at 345 K; segments shorter than the Kuhn length of about ten bonds are rather stretched while longer segments are Gaussian. Figure 12 shows that the numbers of longer ties are decreasing with progressive crystallization. Shorter ties are stretched following Rn2/n ∼ n, while longer ties tend to gather around the solid line of Gaussian but show appreciably deviations. Though the deviations seem to decrease with time, appreciable disparity remains between the observed tie conformations (black circles) and ideal Gaussian (solid line), especially for ties longer than around 20 bonds. The tie molecules in the fiber are considered to be appreciably stretched retaining the memory of their birth from the highly stretched melt, as far as the present time scale of simulation is concerned. We recently found a work reporting that similar tightening of tie molecules was noticed even in crystallization from the isotropic melt; deviation from the Gaussian also started around 20 bonds.29

Figure 13. Schematic image of the network formation during crystallization of the highly stretched melt. The initial network comprises subcritical crystallites (embryos) connected to very long cilia and loose folds and ties. Dominant processes of the network development are independent growth of lamellar crystals and large shrink of the cilia segments; long cilia are continuously taken into the crystalline regions along with the folds and ties being tightened into compact forms. The stage of the network completion is the formation of final stacked lamellae. The compacted folds and ties come to have similar average lengths, probably due to similar competition of the enthapic and entropic drive for crystallization.

4. CONCLUSIONS AND DISCUSSION We could directly observe primary nucleation and subsequent growth of crystals into stacked lamellae from the highly stretched melt. Contrary to the usual expectations, the primary nuclei have nearly isotropic shape having similar dimension in both directions along and perpendicular to the fiber axis. In early stage of crystal growth, the crystallites grow in similar rates in all three directions. The exception was crystallization under large tension τzz = 3, where crystallites showed usual linear growth along the fiber axis, but in the perpendicular directions they showed slower diffusion controlled growth. Crystallization started from embryonic small crystallite, many of which were built in during initial drawing of the melt. During the initiation of the fiber, we observed the genesis and development of a network comprising crystallites and interconnecting folds and ties together with dangling cilia. The crystal growth proceeded by reeling in long cilia along with tightening of folds and ties. The final stage of the fiber formation was the fusion of crystallites, leading to the stacked lamellae, where further tightening of folds and ties went on. These processes are schematically depicted in Figure 13. The final fiber structure was found to have compact folds and neat ties, and the tie molecules were found to be appreciably stretched with considerable deviation from the Gaussian reflecting the history of their birth. The development of the network during crystallization is expected to result in rapid changes in mechanical properties of the system, the detailed clarification of which needs separate investigations on chain dynamics during crystallization, but it is beyond the reach of the present work. One thing we want to comment here is that the changes in the MD cell length along the draw direction observed in inset of Figure 3a can be the reflection of the mechanical property of the crystallizing fibers. The pronounced elongation along the stretch direction in the initial stage came to stop around the halfway of crystallization, when the crystallinity attained about the half of the final crystallinity. The process also corresponds to the end of independent growth (Figure 3b) or around the end of the network development (Figure 9b). Because of the development of the tie chains, the elongation along the stretch direction is considered to be markedly hindered. A similar mechanical response is also observed during melting by constant heating of the fiber under tension. Figure 14 shows

Figure 14. Nonbonded energy (open circles with left scale) and the MD cell size c (filled circles with right scale) during melting by rapid heating 10 K/ns under constant tension τzz = 1. Rapid elongation is seen to set in around 440 K, which is about the halfway of melting between 410 and 460 K.

the changes in the nonbonded energy (open circles) and the MD cell size c (filled circles) by rapid heating (10 K/ns). The melting occurred between 410 and 460 K, while the macroscopic deformation of the fiber also started around 440 K, midway between the onset and final melting. Disruption of the network linking the crystallites during melting is considered to cause the rapid elongation by the tensile force.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b02569. Our method of estimating the critical nucleus size from the volume distribution of crystalline clusters; the J

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lamella morphology and the statistics of tie molecules in crystallization under larger tension τzz = 3 (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Takashi Yamamoto: 0000-0002-0318-0823 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS I am very grateful to the Ministry of Education, Science, and Culture, Japan, for the financial support by JSPS KAKENHI Grant 15K05631 and to the members of the Doi-Project for free use of the simulation platform OCTA and for their generous support during the present work. Discussions with Prof. Rutledge and members of his laboratory during my short visit to MIT were very helpful in reconsidering simulation results, especially on the mechanical response during crystallization and melting of the fiber.

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