Molecular Dynamics Simulation of the Effects of Layer Thickness and

Article pubs.acs.org/Macromolecules

Molecular Dynamics Simulation of the Effects of Layer Thickness and Chain Tilt on Tensile Deformation Mechanisms of Semicrystalline Polyethylene In-Chul Yeh,*,† Joseph L. Lenhart,† Gregory C. Rutledge,‡ and Jan W. Andzelm*,† †

Macromolecular Science & Technology Branch, Materials & Manufacturing Science Division, U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005, United States ‡ Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: We performed molecular dynamics simulations to investigate the effects of layer thicknesses of both crystalline and noncrystalline domains and chain tilt within the crystalline lamellae on tensile deformation mechanisms of the lamellar stack model of semicrystalline polyethylene. For equal thicknesses of crystalline and noncrystalline regions, similar stress−strain profiles were obtained with two different initial orientations of the crystal stem relative to the tensile direction. Repeated melting/recrystallization transitions were observed, at the slower strain rate of 5 × 106 s−1, characterized by oscillating stress− strain profiles. With increasing thickness of the crystalline regions, these oscillations occurred less frequently. For systems with initially tilted chain stems in the crystalline domain, decreasing the thickness of the noncrystalline region increased the number of short bridge segments in the noncrystalline region connecting the two crystalline regions and induced significant shear stresses, rearrangements in the crystalline region, and the strain hardening during the tensile deformation.

■

INTRODUCTION Semicrystalline polyethylene (PE) with diverse thermal and mechanical properties can be prepared by choice of molecular weights and adjusting the crystallinities through various processing conditions. The complex interactions between polymer chains spanning both crystalline and amorphous phases affect the structural and mechanical properties of semicrystalline PE. There have been significant experimental efforts to characterize the plastic deformations of semicrystalline PE and to optimize its structural and mechanical properties.1−8 Molecular dynamics (MD) simulation can be a valuable tool to complement the experimental efforts by revealing the structural and dynamical details of semicrystalline PE under varying conditions. Tensile deformations of semicrystalline PE have been investigated with MD simulations.9−14 Lee and Rutledge11 introduced a molecular model of semicrystalline PE based on the lamellar stack, comprising alternating layers of fully thermalized crystalline and noncrystalline domains, and subjected it to tensile deformations at constant lateral dimension and at constant volume. Subsequently, Kim et al.12 expanded this approach to plastic deformation in a variety of modes, including extension, compression, and shear. Mechanisms of fine crystallographic slip in the (100)[001] system and melting/recrystallization of © XXXX American Chemical Society

the crystalline domain, as well as network stretch and cavitation in the noncrystalline domain, were identified to explain their observations. Recently, Yeh et al.13 studied tensile deformations of larger semicrystalline PE systems having increased lateral dimensions but equal thicknesses in crystalline and noncrystalline regions at 350 K with MD simulations, using two different strain rates of 5 × 107 and 5 × 106 s−1 and achieving larger strains. No significant system-size effects were observed, but in accord with the earlier works of Lee and Rutledge11 and of Kim et al.,12 cavity formation was the dominant deformation mechanism during tensile deformations at the faster strain rate of 5 × 107 s−1, characterized by monotonically declining stress−strain profiles caused by the expansion of free volume during cavitation after the point of yield. However, at the slower strain rate of 5 × 106 s−1, they observed oscillating stress−strain profiles and strain hardening associated with repeated melting and recrystallization phenomena within the semicrystalline PE, observable only with sufficiently large systems and strains.13 This oscillatory behavior was traced to the extraction of chain stems from crystalline to noncrystalline Received: August 10, 2016 Revised: January 26, 2017

A

DOI: 10.1021/acs.macromol.6b01748 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

with crystal stems aligned along the z-axis, the parallelpiped cells with tilted crystal stems have longer x and shorter z dimensions and a smaller areal number density of chains passing through the x−y plane. Then, lamellar semicrystalline PE configurations were prepared by designating a subregion of specified relative z thickness in the respective crystalline PE configurations as the noncrystalline region and amorphizing this region under periodic boundary conditions using the interphase Monte Carlo (IMC) method,22,23 wherein the unitedatom (UA) sites within this noncrystalline region are subjected to displacement, end-rotation, internal rebridging, reptation, and endbridging moves. Each UA site represented a carbon atom and the hydrogen atoms bonded to it. The number of the UA sites in the noncrystalline region was adjusted in proportion to the thickness of the noncrystalline region in order to match the lower density of the amorphous PE (0.85 g/cm3)13,24,25 prior to any MC moves. A total of 48 chain ends (Ne) was introduced in this process. The number of UA sites (N) and the number of linear chains (Ne/2) were kept constant during IMC modeling. The IMC equilibration started with 10 000 MC cycles at 10 000 K, followed by cooling to 350 K during 10 000 MC cycles, and an additional 240 000 MC cycles at 350 K. One hundred initial configurations were prepared for each of eight different lamellar semicrystalline PE systems, distinguished by different initial crystallinities and crystal stem orientations, as illustrated in Figure 1. In Figure 1, each semicrystalline PE system is labeled by the numbers of c replications of the unit cell in crystalline and noncrystalline regions and whether crystal stems are initially aligned or tilted with respect to the z-axis. For example, “c100n60a” designates a PE system with 100 and 60 c replications in crystalline and noncrystalline regions, respectively, in the rectangular prism simulation cell with crystal

regions and the subsequent recrystallization that occurred. When tensile deformations were performed at a lower temperature of 250 K, the cavity mechanism was dominant even at the slower strain rate. Jabbari-Farouji et al.15 used a different coarse-grained model of semicrystalline polymer, represented by prolifically nucleated, randomly oriented small crystallites connected by numerous thin noncrytalline domains instead of a single lamellar stack, in a recent simulation of plastic deformation of semicrystalline poly(vinyl alcohol) and observed strain hardenings associated with strain-induced chain alignment. Effects of different crystallinities and tensile directions on deformation mechanisms of semicrystalline PE have been investigated experimentally.5−7,16−18 Pawlak and Galeski5−7 observed that the cavitation process in the noncrystalline region during tensile deformations of high-density PE is initiated if the crystal region is thick and the crystallinity is high. Fu et al.18 studied the stretching direction dependency of uniaxial deformation in overstretched PE and suggested that the stretching direction may influence the performance of semicrystalline PE. In earlier simulations,11−13 the crystallographic plane {201} in the lamellar domain of the semicrytalline PE was parallel to the interface, resulting in the crystal stem orientation tilted with respect to the interface normal; this tilt was in agreement with the experimental observations in melt-crystallized PE spherulites.19,20 The effect of the crystalline chain tilt on topological properties and interfacial energy of semicrystalline PE has been studied computationally by Gautam et al.,21 using the interphase Monte Carlo model,22 and the {201} oriented interface was found to be preferable. In this study, using MD simulations, we investigated the effects of both the crystalline and noncrystalline layer thicknesses, as well as different chain orientations in the crystalline region, on tensile deformation mechanisms of semicrystalline PE. Higher initial crystallinities were achieved using thicker crystalline regions as well as thinner noncrystalline regions. In addition to configurations with crystallographic plane {201} normal to the elongation direction, which were used in previous simulations,11−13 those with {001} plane normal to the elongation direction were prepared as well to study effects of different initial orientations of crystal stems on tensile deformations of semicrystalline PE.

■

METHODS

Lamellar stack configurations of semicrystalline PE were prepared in a similar manner as in the previous study by Yeh et al.13 For one set of simulations, the chain stems within the crystalline domains of the semicrystalline PE simulations were aligned with and parallel to the zaxis of the simulation cell, such that the crystallographic plane {001} was normal to the z-axis. In these configurations, the orthorhombic unit cell of crystalline PE was replicated 6 and 10 times in a and b (equivalently, x and y) directions, respectively, and 120 or 160 times in the c direction, which coincides with the z-axis. These replications resulted in rectangular prism supercells with x and y box lengths of 4.630 and 4.445 nm, respectively, and z box lengths of 30.321 or 40.428 nm for 120 or 160 c replications, respectively. For a second set of simulations, the chain stems within the crystalline domains were tilted with respect to the z-axis, such that the crystallographic plane {201} was normal to the z-axis, as in previous studies.11−13 These reoriented crystalline PE configurations were described by parallelepiped cells with two rectangular faces, with x and y box lengths of 5.535 and 4.445 nm, respectively, one lying within the x−y plane at z = 0 and the other at a z height of 25.366 or 33.822 nm, displaced by 0.007 or 0.010 nm along the x-axis, for 120 or 160 c replications, respectively. Compared to the corresponding rectangular prism cells

Figure 1. Illustrations of semicrystalline PE starting configurations used in this study, with different thicknesses of crystalline and noncrystalline regions and chain orientations in the crystalline region. Each snapshot illustrates a primary simulation cell, which is replicated in all three directions under periodic boundary conditions to eliminate surface effects and model a bulk semicrystalline PE lamellar stack. The numbers after “c” and “n” refer to the number of replications of the crystalline unit cell along the c-axis in crystalline and noncrystalline regions, respectively, while the suffixes “t” and “a” signify crystal stems tilted and aligned with respect to the z-axis, respectively. B


Article

Macromolecules Table 1. Summary of Simulated Systems systema

Lcb (nm)

L0c (nm)

Crvd (%)

Crwe (%)

cryst planef

c60n60t c100n60t c130n30t c145n15t c60n60a c100n60a c130n30a c145n15a

12.683 21.139 27.480 30.651 15.161 25.268 32.848 36.638

12.683 12.683 6.342 3.171 15.161 15.161 7.580 3.790

50 63 81 91 50 63 81 91

54 66 84 92 54 66 84 92

{201} {201} {201} {201} {001} {001} {001} {001}

av chain lengthg 1078 1460 1502 1529 1069 1445 1481 1510

strain rateh

(782) (1091) (1242) (1326) (802) (1066) (1243) (1287)

f, f, f, f, f, f f f

s s s s s

The numbers after “c” and “n” signify the number of replications of the unit cell of crystalline PE along the c-axis in crystalline and noncrystalline regions, respectively, while the suffixes “t” and “a” signify crystal stems tilted or aligned with respect to the z-axis, respectively. bInitial thickness of crystalline region. cInitial thickness of noncrystalline region. dThe initial crystallinity by volume. eThe initial crystallinity by weight was estimated by the numbers of UA sites in crystalline and noncrystalline regions. fThe crystallographic plane perpendicular to the z-axis is indicated. gThe average chain length is estimated for the linear chain with the distinct end groups and is smaller than the number estimated by the total number of UA sites because of the existence of cyclic chains without the end groups, which are allowed in the IMC method. The numbers in parentheses indicate the standard deviation of polydisperse chain lengths. hThe characters “f” and “s” signify “faster” and “slower” strain rates of 5 × 107 and 5 × 106 s−1, respectively, used in tensile deformation simulations. a

Table 2. Topological Properties in the Noncrystalline Regiona system c60n60t c100n60t c130n30t c145n15t c60n60a c100n60a c130n30a c145n15a

nsites,tail 161 159 43 11.6 181 189 43 9.3

(5) (4) (2) (0.3) (6) (5) (2) (0.4)

⟨nloop⟩ 94.7 94.7 88.6 72.6 95.5 95.7 91.2 81.4

⟨nbridge⟩

nsites,loop

(0.2) (0.2) (0.5) (0.8) (0.2) (0.1) (0.4) (0.6)

40 41 33 20.6 33 31 33 20.6

(2) (2) (1) (0.3) (2) (2) (1) (0.4)

1.3 1.3 7.4 23.4 0.5 0.3 4.8 14.6

(0.2) (0.2) (0.5) (0.8) (0.2) (0.1) (0.4) (0.6)

nsites,bridge 572 594 161 43.2 830 732 233 65

(49) (49) (8) (0.7) (119) (170) (10) (1)

⟨nloop⟩ and ⟨nbridge⟩ refer to average numbers of loop and bridge segments, respectively, in each PE configuration. The number of tail segments was 48. nsites,tail, nsites,loop, and, nsites,bridge refer to average numbers of UA sites in each of tail, loop, and bridge segments, respectively, for each system. Error bars in parentheses were estimated as 1.96 times the standard error, based on the 95% confidence interval. a

stems initially aligned with the z-axis, while “c100n60t” designates a PE system with the same numbers of c replications in crystalline and noncrystalline regions, respectively, but in the parallelepiped simulation cell with initially tilted crystal stems. Semicrystalline PE configurations prepared in this manner consisted of chains with polydisperse molecular weight distributions. The averages and standard deviations of chain lengths of all the semicrystalline PE systems are summarized in Table 1, along with initial thicknesses of crystalline and noncrystalline regions, initial crystallinities by volume and weight, the crystallographic plane normal to the z-axis initially, and strain rates used for each system during tensile deformation simulations. MD simulations were performed using the LAMMPS program26 starting with the configurations of the lamellar stack model prepared by the IMC method. Interactions within and between PE chains were described by the united-atom force field27,28 used in the previous study13 and are summarized in Table S1. Chemical bond breakings can be modeled by using a reactive all-atom force field.29 However, the bond breakings are not expected to play an important role in the tensile deformation simulation conditions in this study and were not considered. The nonbonded interactions within the cutoff distance rc = 2.5σ, where σ is a Lennard-Jones potential parameter specified in Table S1, were calculated explicitly. The nonbonded interactions beyond rc were taken into account with a long-range van der Waals tail correction to the energy and pressure.30 The time step of 2 fs was used unless specified otherwise. The initial configurations were equilibrated with a brief energy minimization followed by successive 10 ps and 8 ns simulations with time steps of 1 and 2 fs, respectively, at the temperature of 350 K and the pressure of 1 bar. Temperature and pressure were maintained using the Nosé−Hoover thermostat and barostat with relaxation time constants of 10 fs and 1 ps, respectively. The thermostat imposes an isothermal condition during tensile

deformation, which precludes the softening due to adiabatic heating sometimes observed experimentally during deformation at high strain rates,31 but provides stress−strain relationships more comparable to those obtained at low strain rates through an empirical temperature/ strain rate equivalence relation.32 Tensile deformations from the equilibrated configurations were performed along the z-direction at two different strain rates of 5 × 106 and 5 × 107 s−1, up to strain of 123%, at the temperature of 350 K. This combination of strain rates and temperature was chosen based on previous studies,11−13 where two distinct deformation mechanisms, cavitation and melting/ recrystallization, were observed. During tensile deformations, normal stresses in x- and y-directions were maintained at 1 bar independently. In earlier simulations of semicrystalline PE,11−13 each UA site i was designated as either crystalline or noncrystalline during equilibrations and tensile deformations according to its local orientational order parameter P2,i = (3⟨cos2 θij⟩ − 1)/2,33 averaged over all UA sites j within the cutoff distance rc from site i, where θij is the angle formed by the chords that join the UA sites bonded to i and j, respectively. Similarly, to estimate the overall alignment of PE chains along the tensile direction, we calculated a global orientational order parameter defined as

P2, z = (3⟨cos2 θi , z⟩− 1)/2

(1)

averaged over all UA sites i, where θi,z is the angle between the z-axis and the chord that join the UA sites bonded to site i.

■

RESULTS AND DISCUSSION Topological Properties. We identified and analyzed topological segments such as tails, loops, and bridges in the noncrystalline region of the lamellar stack model of semicrystalline PE configurations prepared by the IMC method. A

C


Article

Macromolecules

properties of bridge segments. Overall, as L0 decreased, the number of bridge segments increased significantly, at the expense of the number of loop segments, while the average length of bridge segments relative to the thickness of the noncrystalline region, Lbridge/L0, decreased, as shown in Table 2 and Figure 2. The near-linear dependence of Lbridge/L0 on L0 is consistent with random walk chain statistics,33 where the chain length is proportional to the mean square end-to-end distance, and is in qualitative agreement with a thermodynamic prediction of the bridge length in semicrystalline polymers34 and earlier interphase MC simulations by Balijepalli and Rutledge35 using freely rotating chains. The thickness of the noncrystalline region was smaller when the crystal stems were tilted with respect to the z-axis than when they were aligned with the z-axis as shown in Figure 1 and Table 1. As a result, the configurations with the tilted crystal stems had more and shorter bridges compared to the corresponding values with the aligned crystal stems, as shown in Figure 2 and Table 2. Figure 3 shows distributions of lengths of tail, loop, and bridge

tail segment is connected to a crystal stem at only one end. A loop segment connects crystal stems within the same crystal lamella. A bridge segment connects crystal stems across the noncrystalline region. The number of tail segments was fixed at Ne = 48, the number of chain end sites introduced during system construction; the total number of loop and bridge segments in any one configuration was similarly fixed at 96, by construction. Table 2 summarizes the average numbers of loop and bridge segments in each system. The length of each topological segment, expressed as the number of UA sites in each segment, is also summarized in Table 2. With c100n60t and c100n60a systems, the z thicknesses of the noncrystalline regions were the same as in the c60n60t and c60n60a systems, respectively, while the crystalline region was increased from 60 to 100 c replications. As a consequence, c100n60t and c100n60a systems displayed essentially the same topological properties in the noncrystalline region as c60n60t and c60n60a systems, respectively, as shown in Table 2. Figures 2a and 2b show the average length of bridge segments, Lbridge, and the average number of bridge segments, respectively, as functions of the thickness of the noncrystalline region, L0. Figure 2 shows that L0 is the main determining factor for the topological

Figure 3. Probability density distributions of lengths of tail, loop, and bridge segments in the noncrystalline region of semicrystalline PE configurations with initially tilted crystal stems. The length was expressed as the number of UA sites in each segment. The distributions for tail, loop, and bridge segments are shown in (a), (b), and (c). An inset in (c) displays the length distribution of bridge segments for the c145n15t system in more detail. Solid, dashed, and dot-dashed lines represent the distributions with c100n60t, c130n30t, and c145n15t systems, respectively.

Figure 2. Average length and number of bridge segments as a function of the thickness of the noncrystalline region. (a) The average length of the bridge segment, Lbridge, relative to the thickness of noncrystalline region, L0, which corresponds to the minimum length of the bridge segment. Lbridge/L0 was estimated by the ratio of nsites,bridge, the average number of UA sites in the bridge segment, summarized in Table 2, and the number of UA sites in the shortest path across the noncrystalline region in crystalline PE chain conformation. Lbridge/L0 = 1 signifies the highly strained shortest bridge segment spanning noncrystalline region. (b) The average number of bridge segments in each system, nbridge, normalized by the cross-sectional area. Error bars shown in this and following figures were estimated as 1.96 times the standard error, based on the 95% confidence interval.

segments in the configurations with different initial crystallinities when the crystal stems were initially tilted with respect to the z-axis. With the thinner noncrystalline region in the c130n30t system, the lengths of tail and bridge segments were significantly smaller compared to those with the c100n60t system. We observed similar trends in the c145n15t system, where the average number of bridge segments increased to 23.4 D


Article

Macromolecules ± 0.8, while the average length of the bridge segment decreased to 43.2 ± 0.8 UA sites, as shown in Table 2. The inset in Figure 3c shows that there is a significant number of short bridge segments in the c145n15t system. It has been speculated that such short bridge segments (also known as “taut tie chains”) significantly influence mechanical properties of semicrystalline PE during tensile deformations.1,36−38 As shown in Figure 2a, for the c145n15t system, the initial ratio of the length of a bridge segment and the thickness of the noncrystalline region, Lbridge/L0, is estimated to be 1.72 on average. This means that these short bridge segments can influence stress−strain profiles during tensile deformation of the c145n15t system with significant tightening of bridge segments starting at about a 0.72 (or 72%) strain on average. In a subset of semicrystalline PE configurations, we observed configurations having bridge segments connecting the same crystal stem across the noncrystalline region. This type of connection results in a special form of cycle that spans both crystalline and noncrystalline regions, without chain ends, due to periodic boundary conditions. The configurations containing such cycles displayed artificially large stresses in stress−strain profiles and were excluded from the subsequent analysis of the stress−strain profiles. z-Direction Density Distributions. Figure 4 shows mass density distributions along the z-direction for semicrystalline PE with different crystallinities and relative orientations of crystal stems. They were calculated from the last 4 ns of the 8 ns equilibration period and averaged over 100 runs with

different starting configurations for each crystallinity and orientation. In agreement with the results in the previous simulation study13 with the same force field and the same temperature of 350 K, all z-direction density distributions displayed well-defined crystalline and noncrystalline regions, characterized by mass densities near 1 and 0.85 g/cm3, respectively. Figure 4a compares density distributions of semicrystalline PE systems with initially tilted crystal stems. The density distributions across the interface between crystalline and noncrystalline regions in Figure 4a were similar. However, the thicknesses of the noncrystalline regions were significantly smaller for the c130n30t and c145n15t systems. For instance, with the c145n15t system, the noncrystalline region, as measured by density (i.e., excluding the interfacial region), spans about 1.6 nm along the z-direction, as shown in Figure 4a. Figure 4b shows density distributions of semicrystalline PE systems with crystal stems initially aligned with the zaxis. Compared to the z-direction density profiles of systems with tilted crystal stems shown in Figure 4a, the density first increased slightly near the interface and then decreased more sharply from crystalline to noncrystalline regions. Similar observations were reported previously by Gautam et al.21 In addition, densities in the crystalline region of semicrystalline PE with higher crystallinities were slightly higher than those with lower crystallinities, even though the values were all close to 1 g/cm3. These results are consistent with the presence of expansive interfacial stresses at the boundaries between the crystalline and noncrystalline domains, as reported previously by Hütter et al.,24 which has in turn a stronger influence on the volumetric properties of semicrystalline PE comprising thinner crystalline lamellae.39 Effects of Crystal Stem Orientations. Tensile deformations were performed previously13 on semicrystalline PE configurations with crystal stems initially tilted with respect to the tensile or z-direction, which are designated as the c60n60t system in Figure 1 and Table 1. To study the effects of initial crystalline stem orientations on the mechanical properties, we performed tensile deformations of the semicrystalline PE configurations having crystal stems initially aligned along the z-direction, which are designated as the c60n60a system in Figure 1 and Table 1. Figure 5a shows the average stress−strain profile of the c60n60a system at the faster strain rate of 5 × 107 s−1. The stress declined monotonically after the peak, as shown in Figure 5a, which is consistent with the cavitation mechanism observed previously with the c60n60t system at the same strain rate. However, there were 13 cases in the c60n60a system that display stress−strain curves that deviated from the typical behavior, whose average stress−strain profile is also shown in Figure 5a. Figure 5b compares average stress−strain curves of c60n60t and c60n60a systems at the strain rate of 5 × 107 s−1. The average stress−strain profile for c60n60a is similar to that for c60n60t. However, a gradual increase in stress at strains up to 0.1, associated with the mechanism of fine crystallographic slip and the rotation of the initially tilted crystal stems in c60n60t toward alignment with the tensile stress, is replaced by a more immediate increase to the peak stress and shift of the yield point to smaller strain for the c60n60a system, as summarized in Table 3. Figure 5c compares the average stress− strain curves obtained from tensile deformations of c60n60t and c60n60a systems at the slower strain rate of 5 × 106 s−1. The yield strain was again shifted to a smaller value within the c60n60a system, as summarized in Table 3. In individual stress−strain curves (data not shown) for the c60n60a system,

Figure 4. Mass density profiles of semicrystalline PE configurations along the z-direction. z = 0 was defined as the center of the noncrystalline region. The distributions were calculated from the last 4 ns of 8 ns simulations before the tensile deformations. Solid, dashed, dot-dashed, and double-dot-dashed lines represent the results with c60n60t, c100n60t, c130n30t, and c145n15t systems, respectively, in (a) and c60n60a, c100n60a, c130n30a, and c145n15a systems, respectively, in (b). E


Article

Macromolecules

Table 3. Elastic Moduli, Yield Stresses, and Yield Strains systema

strain ratea

Eb (MPa)

c60n60t

f s f s f s f s f s f f f

185 61 113 40 332 143 581 314 337 154 1349 5159 9127

c100n60t c130n30t c145n15t c60n60a c100n60a c130n30a c145n15a

yield stressc (MPa) 108 91 113 108 123 112 144 117 105 88 108 129 141

(2) (3) (2) (1) (3) (3) (3) (2) (2) (2) (3) (1) (2)

yield straind 0.2411 0.2245 0.2392 0.2461 0.2129 0.2436 0.2605 0.2892 0.1659 0.1428 0.0904 0.0413 0.0228

a

See Table 1 for explanations. bElastic modulus was estimated by the slope of the linear fit of the average stress−strain profiles up to 1% strain. cYield stress was estimated as the first peak stress in the average stress−strain profiles. dYield strain was estimated as the strain corresponding to the first peak in the average stress−strain profiles.

There is a competition between cavitation in the noncrystalline region and plastic deformation of the crystalline region during tensile deformations of semicrystalline PE.3 As mentioned earlier, the chain extraction from the crystalline region is a critical step in the plastic deformation of the crystalline region. The cavitation was observed near the yield point in the stress−strain profiles,11−13 and critical yield stresses for cavitation were found to be 108 ± 2 and 105 ± 2 MPa for c60n60t and c60n60a systems, respectively, which were estimated from yield stresses in stress−strain profiles undergoing cavitation at the faster strain rate as summarized in Table 3. By contrast, at the slower strain rate we estimated much smaller critical yield stresses of 91 ± 3 and 88 ± 2 MPa for the c60n60t and c60n60a systems, respectively, which also occurred at smaller yield strains than at the faster strain rate. This indicates that at the slower strain rate the tensile deformation proceeded with the plastic deformation of the crystalline region, which has a lower critical yield stress at an earlier strain. Effects of Thicker Crystalline Regions. We investigated the effects of a thicker crystalline layer on mechanical properties using the c100n60t system, which had a crystalline region about 1.33 times thicker than the c60n60t system. The thickness of the noncrystalline region remained the same as that in the c60n60t system. Figures 6a and 6b show average stress−strain curves from tensile deformations of the c100n60t system at two different strain rates of 5 × 107 and 5 × 106 s−1, respectively. As shown in Figure 6a, at the faster strain rate, the stress−strain profile is qualitatively similar to the one obtained with the c60n60t system shown in Figure 5b. This indicates that cavitation is still the dominant mechanism underlying the yield response in the c100n60t system at the faster strain rate, as in the c60n60t system.13 However, as shown in Figure 6b, at the slower strain rate, we observed both oscillatory and monotonically declining stress−strain profiles, which are signatures of the melting/recrystallization and cavitation mechanisms, respectively. Figure 6b shows the average stress−strain profiles of 68 monotonically declining and 29 oscillatory stress−strain profiles as well as the overall average. The average stress− strain profile for the monotonically declining cases resembles the one obtained for the same system at the faster strain rate. The average stress−strain profile for the oscillating cases is

Figure 5. Effect of initial orientations of crystal stems on stress−strain profiles. (a) Average stress−strain profile from the tensile deformation of the c60n60a system, where the crystal stems were initially aligned along the z-axis, at the faster strain rate of 5 × 107 s−1. A solid line shows the overall average while the dashed line shows the average from 13 cases that deviate from the majority pattern. Stress and strain values in stress−strain profiles in this and subsequent figures show true stresses, estimated from the pressure component along the tensile direction, and engineering strains, respectively. (b) The average stress−strain profile from tensile deformation of the c60n60t system with the initially tilted crystal stems at the strain rate of 5 × 107 s−1, shown with a dashed line, is compared to that of the c60n60a system, shown with a solid line. The dot-dashed line represents the stress− strain profile averaged over all the configurations from both c60n60t and c60n60a systems, having different initial crystal stem orientations. (c) The similar average profiles as in (b) but obtained at the slower strain rate of 5 × 106 s−1.

we observed rapid oscillations of stress as for the c60n60t system.13 These oscillations were reduced upon averaging over 100 different starting configurations, shown in Figure 5c, but small residual oscillations are still evident. Yeh et al.13 opined that the uniformity of crystalline chain orientations might be partly responsible for these persistent residual oscillations and that such oscillations would not be apparent when averaged over all the sizes and orientations of lamellar stacks in spherulitic experimental samples. Similar compensation might be expected if lamellar stacks with different crystal stem orientations are present. To confirm this, Figures 5b and 5c show the stress−strain curves averaged over all 200 configurations from both c60n60t and c60n60a systems, having the two different initial crystal stem orientations, at faster and slower strain rates, respectively. As anticipated, the residual oscillations after the first peak in Figure 5c for the slower strain rate were significantly diminished when the average was taken over different crystal stem orientations, in closer agreement with the expected responses of realistic experimental samples with diverse crystal stem orientations. F


Article

Macromolecules

semicrystalline PE system, we fixed the overall number of c replications but changed the ratio of c replications corresponding to crystalline and noncrystalline regions, viz. c130n30t and c145n15t. Figure 7 compares the stress−strain profiles during

Figure 6. Effect of the crystalline thickness on the stress−strain profile. (a) The solid line shows the average stress−strain profile from tensile deformations of the configurations with the c100n60t system, which has a thicker crystalline layer than the c60n60t system (cf. Figure 5), at the faster strain rate of 5 × 107 s−1. (b) The solid line shows the overall average stress−strain profile of the c100n60t system during the tensile deformation at the slower strain rate of 5 × 106 s−1. Dashed and dotdashed lines represent the profiles averaged from 68 nonoscillating and 29 oscillating cases, respectively.

Figure 7. Comparison of the stress−strain profiles with different relative thicknesses of crystalline and noncrystalline regions. (a) The average stress−strain profiles obtained from tensile deformations at the faster strain rate of 5 × 107 s−1 with c100n60t, c130n30t, and c145n15t systems are shown with solid, dashed, and dot-dashed lines, respectively. (b) The similar stress−strain profiles as in (a) but at the slower strain rate of 5 × 106 s−1. Excluding runs having the bridge segments connected to the same crystal stem, 97, 83, and 82 runs for c100n60t, c130n30t, and c145n15t systems, respectively, were used to calculate the average stress−strain profiles.

similar to the one for c60n60t at the slower strain rate but displayed higher stress values and a longer period between oscillations. Similar yield stresses for these two stress−strain profiles, shown in Figure 6b and summarized in Table 3, suggest that the critical yield stress for plastic deformation of the crystalline region was increased with thickness of the crystalline region. The overall average stress−strain profile is similar to the average for the majority of cases for which the stress declines monotonically but displayed slightly higher stresses and small oscillatory structures due to contributions from the minority population of oscillating cases. These results indicate that the lamellar thickness in the crystalline region influences deformation mechanisms of semicrystalline PE. Evidently, a thicker crystalline region hinders extraction of chain stems from the crystalline region and suppresses the melting/recrystallization mechanism, so that cavitation becomes more prevalent; this is true even at the slower strain rate. Our results are in qualitative agreement with experimental study of cavitation during tensile deformation of high-density PE with different thicknesses of crystalline regions.7 Effects of Thinner Noncrystalline Regions. The long period of the lamellar stack in semicrystalline PE, spanning both crystalline and noncrystalline regions, has been estimated to be about 20−40 nm.18,40,41 With 160 c replications in the c100n60t and c100n60a systems, the long period was about 34 and 40 nm for systems with tilted and aligned crystal stems, respectively. To study semicrystalline PE with higher crystallinity but with its long period matching that of a realistic

tensile deformations of the c100n60t, c130n30t, and c145n15t systems, with initially tilted crystal stems, at two different strain rates. These systems have the same overall thickness in the zdirection but have different relative thicknesses of crystalline and noncrystalline regions. Elastic moduli and yield stresses correlated positively with initial crystallinities, as shown in Figure 7 and Table 3. However, yield strains, the strain values corresponding to the peak stresses, displayed a different pattern because of two competing structural features affecting them, namely, thicker crystalline regions and thinner noncrystalline regions. In response to increased strains before reaching the peak stress during the tensile deformation of semicrystalline PE with initially tilted crystal stems, the crystal stems rotated toward the direction of the tensile deformation through the mechanism of fine crystallographic slip, while the material in the noncrystalline region was stretched to a lower density. Therefore, for systems with tilted crystal stems, a thicker crystalline region results in a larger strain over which fine crystallographic slip is operative, while a thinner noncrystalline region results in a smaller strain subsequently to reach the peak stress. The observed yield strain values were expected to be a compromise of these two offsetting trends. However, we note G


Article

Macromolecules

with the corresponding tensile stress−strain profile. These snapshots indicate that the crystal stems did not become fully aligned along the tensile direction; we attribute this behavior partly to topological constraints in the noncrystalline region, such as shorter bridge segments. A significant buildup of shear stresses was observed in Figure 9a, which compares average

that the crystal stems were not fully aligned, and significant structural changes took place at the interface, as will be described later. The yield strain was the largest for the c145n15t, which has the thickest crystalline region but the thinnest noncrystalline region. The c100n60t system, which has thinner crystalline but thicker noncrystalline regions, showed a slightly larger yield strain than the c130n30t system. The stress declined monotonically after the peak stress for the c100n60t and c130n30t systems, as shown in stress−strain profiles in Figure 7a, indicating that the cavitation mechanism was dominant at the faster strain rate. However, surprisingly, with the c145n15t system, which had the thinnest noncrystalline region, the stress decreased to 90 MPa at strain of 0.6 after the peak and then gradually increased to 113 MPa at strain of 1.23. This is consistent with an expected tightening of bridge segments at about 0.72 strain for the c145n15t system, discussed earlier based on the analysis of the length of bridge relative to the thickness of noncrystalline region shown in Figure 2a. Structural analysis revealed that complex deformation mechanisms involving combinations of cavitation and chain displacements took place after the peak stress in the c145n15t system. Figure 8 shows representative snapshots during the tensile deformation of the c145n15t system along

Figure 9. Profiles of average shear stress and structural properties of semicrytalline PE with initially tilted crystal stems during tensile deformations. Profiles of stress component σxz, overall mass density, cross-section area, and global orientational order parameter P2,z at the strain rate of 5 × 107 s−1 are shown in (a), (b), (c), and (d), respectively. P2,z was calculated by eq 1 to estimate the overall alignment of PE chains along the tensile direction. Respective profiles at the slower strain rate of 5 × 106 s−1 are shown in (e), (f), (g), and (h). Solid, dashed, and dot-dashed lines represent the results with c100n60t, c130n30t, and c145n15t systems, respectively.

shear stress profiles during tensile deformations of semicrystalline PE with initially tilted crystal stems but having different crystallinities. Bartczak et al.8 showed that the chain slip (100) [001], which is relevant to the crystal stem alignment in our study, is the easiest with the critical shear stress of 7.2 MPa. The higher critical shear stresses with increasing crystallinities and also compared to experiments may be partly due to topological constraints such as short bridge segments in thinner noncrystalline regions. In Figure 9b, we also show the change of the overall mass density, which can provide information about the degree of the cavitation during the tensile deformation. In Figure 9c, we show the change in the cross-sectional area along the x−y direction, which can indicate the degree of the chain segment removal from crystalline to noncrystalline regions. Profiles of the global orientational order parameter P2,z, which

Figure 8. Snapshots and the corresponding tensile stress profile during tensile deformation of a representative semicrystalline PE configuration for the c145n15t sytem at the strain rate of 5 × 107 s−1. Blue lines represent boundaries of primary simulation cells. Periodic images of the simulation cell along the x-direction are shown to depict the topology of chains across the periodic boundaries. One of PE chains is randomly chosen and shown in red to illustrate the chain orientations during the tensile deformation. The rest of chains are shown with black lines. All snapshots are viewed along the y-axis except for the rightmost snapshot, which is viewed from the x-axis to illustrate the cavity formed during deformation. H


Article

Macromolecules was calculated by eq 1 to estimate the overall alignment of polymer chains along the tensile direction, are compared in Figure 9d. Tilted crystal stems orient toward the tensile direction before the initial peak, which contributes to the increased P2,z values at small strains in Figure 9d. At larger strains, alignment of chains in the noncrystalline region is expected to contribute to P2,z. However, changes of P2,z at larger strains are more modest because of relatively small numbers of bridge segments in semicrystalline PE configurations, as summarized in Table 2. A smaller decrease in mass density, a larger decrease in areal density, and an increasing global orientational order parameter observed at large strains in the c145n15t system indicate deformation mechanisms involving a smaller degree of cavity formation, more frequent chain segment removal (despite the larger crystal lamellar thickness), and increasing overall chain alignment compared to c100n60t and c130n30t systems. These structural changes could explain the apparent strain hardening observed at the later stage of deformation of the c145n15t system shown in Figure 7a. Figure 7b compares the stress−strain curves of c100n60t, c130n30t, and c145n15t systems at the slower strain rate. Compared to the corresponding stress−strain curves in Figure 7a for the faster strain rate, more oscillatory stress−strain curves are shown in Figure 7b for the slower strain rate. This behavior indicates significant chain segment removal from the crystal lamella and rearrangements. However, overall trends in stress− strain curves at the two different strain rates shown in Figure 7a,b are remarkably similar. Profiles of shear stresses, mass densities, cross-sectional areas, and global orientational order parameters at the slower strain rate shown in Figures 9e−h, respectively, are also similar to the corresponding profiles at the faster strain rate shown in Figures 9a−d. This is in contrast to the significant effects of different strain rates on the stress− strain profiles observed for c60n60t. This indicates that the stress−strain behaviors of semicrytalline PE are affected significantly by the topological constraints in the noncrystalline region and the thicknesses of the crystalline region, in addition to the strain rates. Figure 10 compares stress−strain curves of c100n60a, c130n30a, and c145n15a having crystal stems initially aligned

with the tensile direction at the faster strain rate. With crystal stems already aligned along the tensile direction, the stretching in the noncrystalline region is expected to begin immediately at low strain, resulting in the yield strain increasing with the thickness of the noncrystalline region. Indeed, Figure 10 shows that the c145n15a system, with the thinnest noncrystalline region, reached the yield stress at the smallest yield strain, in contrast to the trend shown in Figure 7a for configurations with initially tilted crystal stems. In addition, the stress−strain profile for c145n15a in Figure 10 follows the pattern corresponding to the cavitation mechanism except at the larger strains, in contrast to c145n15t in Figure 7a with initially tilted crystal stems. Elastic moduli of the semicrystalline lamellar stack estimated from the slopes of linear fits to the stress−strain curves at small strains also correlated with the crystallinity, with the highest elastic modulus of 9.1 GPa estimated for c145n15a, as summarized in Table 3. We also estimated the elastic modulus from a c145n15a system containing only four chain ends, and thus significantly longer chains, but observed no significant difference (data not shown). Figure 11 shows typical

Figure 10. Effect of different relative thicknesses of crystalline and noncrystalline regions on stress−strain profiles with crystal stems initially aligned along the tensile direction. Solid, dashed, and dotdashed lines represent the average stress−strain profiles with c100n60a, c130n30a, and c145n15a systems, respectively, during the tensile deformations at the faster strain rate of 5 × 107 s−1. Excluding runs having the bridge segments connected to the same crystal stem, we used 100, 92, and 70 runs for c100n60a, c130n30a, and c145n15a systems, respectively, to calculate the average stress−strain profiles.

Figure 11. Snapshots and the corresponding tensile stress profile during tensile deformation of a representative semicrystalline PE configuration for the c145n15a system at the strain rate of 5 × 107 s−1. All snapshots at the top are viewed along the y-axis. Blue lines represent boundaries of primary simulation cells. As in Figure 8, one of PE chains is randomly chosen and shown in red, with periodic images of the simulation cell repeated along the x-direction, while the rest of chains are represented with black lines. I


Article

Macromolecules

c145n15a at the later stage of deformation as we observed in the c145n15t system. These differences can be partly explained by the presence of the thicker noncrystalline region, and thus fewer and longer bridges, in the c145n15a system; the presence of fewer and longer bridges reduces the degree of topological constraint at the early stage of deformation compared to c145n15t with tilted crystal stems. This could induce significantly less chain removal from the crystalline region and smaller degree of chain alignment in the c145n15a system compared to the c145n15t system. However, at larger strains with more extended tensile deformations, the topological constraint in the noncrystalline region may induce the structural change leading to the strain hardening effect even with initially aligned crystal stems, as we observed with tilted crystal stems. Comparison with Experimental Observations. A direct comparison of the results from our simulations and the experiments is challenging because of the differences in temperatures, strain rates, and morphologies between the samples examined in simulations and experiments. According to an empirical temperature/strain rate equivalence relation reported for polyethylene, a 1 decade increase in strain rate is approximately equivalent to a 10 K drop in temperature.31,32 By this measure, strain rates of 5 × 107 and 5 × 106 s−1 at 350 K would be equivalent to 5 × 102 and 5 × 101 s−1 at room temperature (300 K). These are still significantly higher strain rates that those typically used in experiments, at about 1 × 10−3 s−1.7,42 However, in agreement with experimental findings,3,5−7,43 the simulations show that cavitations were favored with thicker crystalline regions and higher strain rates. Elastic moduli, yield stresses, and yield strains of semicrystalline PE systems estimated from our simulations are summarized in Table 3. Experimental values of elastic moduli of semicrystalline PE samples at the room temperature range from 0.41 to 1.1 GPa, depending on crystallinities and preparation method.7,42 Elastic moduli of the lamellar stack obtained by our simulations at the two different strain rates were 40 and 581 MPa, when the crystal stems were initially tilted, and 154 MPa and 9.1 GPa when the crystal stems were aligned with the tensile direction. These values are all larger than the elastic modulus of amorphous PE, estimated to be about 30 MPa13 at 350 K and a strain rate of 5 × 107 s−1. Yield strains also varied significantly depending on the crystal stem orientation. Experimental values of yield stress range from 13.8 to 27.4 MPa.7,42,44 Yield stress values estimated from our simulations range from 88 to 141 MPa, depending sensitively on the length

snapshots during the tensile deformation of the c145n15a system along with the corresponding tensile stress profile. The average profiles of shear stress, density, cross-sectional area, and global orientational order parameter of c100n60a, c130n30a, and c145n15a are compared in Figure 12. The snaphots in

Figure 12. Average shear stress−strain profiles and structural properties of semicrytalline PE with crystal stems initially aligned along the z-axis during tensile deformations at the strain rate of 5 × 107 s−1. Profiles of stress component σxz, overall mass density, crosssection area, and global orientational order parameter P2,z described by eq 1 are shown in (a), (b), (c), and (d), respectively. The solid, dashed, and dot-dashed lines represent the results with c100n60a, c130n30a, and c145n15a systems, respectively.

Figure 11 illustrate the stretching of the noncrystalline region and the caviation observed at early and later stages of deformation, respectively. In contrast to the shear stress profiles shown in Figure 9a for semicrystalline PE with tilted crystal stems, no significant shear stresses developed in systems with aligned crystal stems, as shown in Figure 12a. The extent of cavitation increased steadily, as shown by the steady decrease of the overall density in Figure 12b. The cross-sectional area in Figure 12c, which correlates with the number of crystal stems, started to decrease at later strains and at a smaller rate with c145n15a, compared to the corresponding plot with c145n15t in Figure 9c. This indicates that the chain removal from crystalline to noncrystalline regions happened less frequently and at later stages of deformation in c145n15a than in c145n15t. In addition, as shown in Figure 12d, we did not observe an increase of global orientational order parameter in

Table 4. Effects of Number of Bridges on Yield and Final Stresses final stressc (MPa)

yield stressc (MPa) systema

strain ratea

c130n30a c130n30t c130n30t c145n15a c145n15a c145n15t c145n15t

f f s f s f s

nsites,bridgeb 233 161 161 65 65 43.2 43.2

(10) (8) (8) (1) (1) (0.7) (0.7)

nbridege < ⟨nbridge⟩ 138 142 127 152 135 159 127

(1) (1) (1) (2) (1) (2) (1)

nbridge > ⟨nbridge⟩ 139 145 127 155 138 170 128

(1) (1) (1) (2) (1) (3) (2)

nbridge < ⟨nbridge⟩ 37 48 39 58 41 89 81

(3) (2) (2) (3) (2) (5) (7)

nbridge > ⟨nbridge⟩ 45 58 50 72 52 139 111

(2) (5) (6) (5) (3) (11) (9)

See Table 1 for explanations. bSee Table 2 for explanations. cAverage yield and final stresses were calculated from stresses at the yield point and 123% strain, respectively, of two groups of configurations: one consisting of configurations having nbridge, the number of bridge segments in each configuration, less than ⟨nbridge⟩, the average number of bridge segments in each configuration, and the other consisting of those having nbridge more than ⟨nbridge⟩. a

J


Article

Macromolecules of crystal stem regardless of the crystal stem orientation. These higher yield stress values compared to experimental values may be due to significantly higher strain rates used in our simulations than those in experiments. They may also be indicative of softer modes of yield in spherulitic morphologies, wherein lamellar stacks are more or less randomly oriented with respect to the direction of deformation, than in the elongation of a single lamellar stack, considered here. Effects of Topological Constraints on Mechanical Properties. Finally, we investigated whether there are correlations between topological constraints, such as number and length of bridge segments, and the observed mechanical properties such as elastic moduli, yield stresses, and strain hardening. Elastic moduli and yield stresses increased with decreasing thicknesses of noncrystalline region, hence with increasing numbers and decreasing lengths of bridge segments, as shown in Table 3. In order to examine the dependence of mechanical properties on topology, we divided the lamellar stack configurations in each system into two groups, based on the number of bridge segments. One consisted of configurations having the number of bridge segments more than the average, and the other group consisted of those having less than the average. We did not observe significant differences in elastic moduli, which reflect strain−stress relationship at the very early stage of deformation, between the two groups (data not shown). Average yield stresses shown in Table 4 were estimated from the average of the yield stresses observed in individual stress−strain curves, not from the averaged stress strain curve. A small difference in yield stresses between the two groups was observed for the c145n15t system at the faster strain rate. However, we observed significant differences in final stresses at 123% strain, which can be related with the strain hardening at later stages of deformation, as shown in Table 4. We conducted a similar analysis for each system based on the length of bridge segments instead of the number of bridge segments and found a similar but smaller degree of difference (data not shown). In our simulations, number and length of bridge segments connecting crystalline regions depend critically on the thickness of the noncrystalline region, and a strain hardening due to alignment of bridge segments was evident only when there was a significant number of short bridge segments, as in c145n15t. As discussed earlier, a significant influence of short bridge segments on stress−strain profiles of the c145n15t system was expected, beginning at about 72% strain. Figure 13 compares average stress−strain profiles of the two groups of configurations for c145n15t at the faster strain rate. It shows higher stress values at larger strains for configurations having larger number of bridge segments. The smaller yield stresses in Figure 13 than those shown in Table 4 were caused by the averaging of individual stress−strain profiles with different yield stress and strain. These results show that topological constraints such as short bridge segments can be correlated with strain hardening observed at the later stage of deformation.

Figure 13. Effects of number of bridge segments on mechanical properties. Average stress−strain profiles during tensile deformation of the c145n15t system at the strain rate of 5 × 107 s−1 for two groups of configurations, one having number of bridge segments less than the average of 23.4 shown with a solid line with circles and the other having more than the average shown with a dashed line with diamonds.

was reached earlier when the crystal stems were prealigned with the tensile direction. This behavior is readily traced to the removal of fine crystallographic slip as a mechanism for deformation in the prealigned systems. Residual oscillations in the stress−strain profiles at the slower strain rate of 5 × 106 s−1 were observed for both crystal stem orientations, but the observed oscillations were reduced when averaged over both initial crystal stem orientations. Similar behavior is expected when averaged over different sizes and orientations of the lamellar stack, which could explain why such oscillations are not observed experimentally. Occurrences of repeated melting/ recrystallization transitions observed with the lamellar stack with equal thicknesses in crystalline and noncrystalline regions during tensile deformations at the strain rate of 5 × 106 s−1 were less frequent when the crystalline thickness was increased, indicating reduced propensity for segment removal from crystalline to noncrystalline regions with increasing crystalline lamellar thickness. The crystallinity was increased further by decreasing the thickness of the noncrystalline region while maintaining the overall long period. The thinner noncrystalline region resulted in increased number and decreased length of bridge segments in the noncrystalline region. These topological constraints in the noncrystalline region induced significant shear stresses, rearrangement of crystal stems near the noncrystalline region, and strain hardening during the tensile deformation of semicrystalline PE systems with initially tilted crystal stems. Insights gained in this study about the differences in deformation mechanisms depending on thicknesses of crystalline and noncrystalline regions and crystal stem orientations relative to the tensile direction could be used to understand and improve the mechanical properties of semicrystalline PE.18 Methods to account for lower strain rates than are readily accessible by MD simulations at present and for the more complex morphology of real semicrystalline PE samples will facilitate closer reconciliation of experimental results and the simulations reported here.

■

SUMMARY AND CONCLUSIONS We have studied the effects of crystal stem orientation and thicknesses of the crystalline and noncrystalline layers on tensile deformation of semicrystalline PE using MD simulations. For semicrystalline PE configurations with equal thicknesses of crystalline and noncrystalline layers, stress− strain profiles obtained with two different initial crystal stem orientations were qualitatively similar, although the peak stress K


Article

Macromolecules

■

(13) Yeh, I. C.; Andzelm, J. W.; Rutledge, G. C. Mechanical and Structural Characterization of Semicrystalline Polyethylene under Tensile Deformation by Molecular Dynamics Simulations. Macromolecules 2015, 48, 4228−4239. (14) Schürmann, B. L.; Niebergall, U.; Severin, N.; Burger, C.; Stocker, W.; Rabe, J. P. Polyethylene (Pehd)/Polypropylene (Ipp) Blends: Mechanical Properties, Structure and Morphology. Polymer 1998, 39, 5283−5291. (15) Jabbari-Farouji, S.; Rottler, J.; Lame, O.; Makke, A.; Perez, M.; Barrat, J.-L. Plastic Deformation Mechanisms of Semicrystalline and Amorphous Polymers. ACS Macro Lett. 2015, 4, 147−150. (16) Wang, Y. T.; Jiang, Z. Y.; Wu, Z. H.; Men, Y. F. Tensile Deformation of Polybutene-1 with Stable Form I at Elevated Temperature. Macromolecules 2013, 46, 518−522. (17) Wang, Y.; Jiang, Z.; Fu, L.; Lu, Y.; Men, Y. Lamellar Thickness and Stretching Temperature Dependency of Cavitation in Semicrystalline Polymers. PLoS One 2014, 9, e97234. (18) Fu, L. L.; Jiang, Z. Y.; Enderle, H. F.; Lilge, D.; Wu, Z. H.; Funari, S. S.; Men, Y. F. Stretching Temperature and Direction Dependency of Uniaxial Deformation Mechanism in Overstretched Polyethylene. J. Polym. Sci., Part B: Polym. Phys. 2014, 52, 716−726. (19) Bassett, D. C.; Hodge, A. M. On the Morphology of MeltCrystallized Polyethylene I. Lamellar Profiles. Proc. R. Soc. London, Ser. A 1981, 377, 25−37. (20) Patel, D.; Bassett, D. C. On the Formation of S-Profiled Lamellae in Polyethylene and the Genesis of Banded Spherulites. Polymer 2002, 43, 3795−3802. (21) Gautam, S.; Balijepalli, S.; Rutledge, G. C. Molecular Simulations of the Interlamellar Phase in Polymers: Effect of Chain Tilt. Macromolecules 2000, 33, 9136−9145. (22) Balijepalli, S.; Rutledge, G. C. Molecular Simulation of the Intercrystalline Phase of Chain Molecules. J. Chem. Phys. 1998, 109, 6523−6526. (23) in ‘t Veld, P. J.; Hütter, M.; Rutledge, G. C. TemperatureDependent Thermal and Elastic Properties of the Interlamellar Phase of Semicrystalline Polyethylene by Molecular Simulation. Macromolecules 2006, 39, 439−447. (24) Hütter, M.; in ’t Veld, P. J.; Rutledge, G. C. Polyethylene {201} Crystal Surface: Interface Stresses and Thermodynamics. Polymer 2006, 47, 5494−5504. (25) Yi, P.; Locker, C. R.; Rutledge, G. C. Molecular Dynamics Simulation of Homogeneous Crystal Nucleation in Polyethylene. Macromolecules 2013, 46, 4723−4733. (26) Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular-Dynamics. J. Comput. Phys. 1995, 117, 1−19. (27) Paul, W.; Yoon, D. Y.; Smith, G. D. An Optimized United Atom Model for Simulations of Polymethylene Melts. J. Chem. Phys. 1995, 103, 1702−1709. (28) Chang, J.; Han, J.; Yang, L.; Jaffe, R. L.; Yoon, D. Y. Structure and Properties of Polymethylene Melt Surfaces from Molecular Dynamics Simulations. J. Chem. Phys. 2001, 115, 2831−2840. (29) O’Connor, T. C.; Andzelm, J.; Robbins, M. O. Airebo-M: A Reactive Model for Hydrocarbons at Extreme Pressures. J. Chem. Phys. 2015, 142, 024903. (30) Sun, H. Compass: An Ab Initio Force-Field Optimized for Condensed-Phase Applications - Overview with Details on Alkane and Benzene Compounds. J. Phys. Chem. B 1998, 102, 7338−7364. (31) Furmanski, J.; Cady, C. M.; Brown, E. N. Time−Temperature Equivalence and Adiabatic Heating at Large Strains in High Density Polyethylene and Ultrahigh Molecular Weight Polyethylene. Polymer 2013, 54, 381−390. (32) Brown, E. N.; Willms, R. B.; Gray, G. T., III; Rae, P. J.; Cady, C. M.; Vecchio, K. S.; Flowers, J.; Martinez, M. Y. Influence of Molecular Conformation on the Constitutive Response of Polyethylene: A Comparison of Hdpe, Uhmwpe, and Pex. Exp. Mech. 2007, 47, 381− 393. (33) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01748. Summary of functional forms and parameters of the united-atom force field used in this study (PDF)

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (I.-C.Y.). *E-mail: [email protected] (J.W.A.). ORCID

In-Chul Yeh: 0000-0001-9637-1700 Gregory C. Rutledge: 0000-0001-8137-1732 Jan W. Andzelm: 0000-0002-2451-3770 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported in part by an appointment to the Postgraduate Research Participation Program at the U.S. Army Research Laboratory (ARL) administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and ARL. The DoD HPC Modernization Office supported this project by supplying supercomputer time. We thank Drs. Kevin Masser and Timothy Sirk for helpful discussions and comments.

■

REFERENCES

(1) Lin, L.; Argon, A. S. Structure and Plastic Deformation of Polyethylene. J. Mater. Sci. 1994, 29, 294−323. (2) Galeski, A. Strength and Toughness of Crystalline Polymer Systems. Prog. Polym. Sci. 2003, 28, 1643−1699. (3) Pawlak, A.; Galeski, A.; Rozanski, A. Cavitation During Deformation of Semicrystalline Polymers. Prog. Polym. Sci. 2014, 39, 921−958. (4) Butler, M. F.; Donald, A. M.; Ryan, A. J. Time Resolved Simultaneous Small- and Wide-Angle X-Ray Scattering During Polyethylene DeformationIi. Cold Drawing of Linear Polyethylene. Polymer 1998, 39, 39−52. (5) Pawlak, A. Cavitation During Tensile Deformation of HighDensity Polyethylene. Polymer 2007, 48, 1397−1409. (6) Pawlak, A.; Galeski, A. Cavitation During Tensile Drawing of Annealed High Density Polyethylene. Polymer 2010, 51, 5771−5779. (7) Pawlak, A. Cavitation During Tensile Deformation of Isothermally Crystallized Polypropylene and High-Density Polyethylene. Colloid Polym. Sci. 2013, 291, 773−787. (8) Bartczak, Z.; Argon, A. S.; Cohen, R. E. Deformation Mechanisms and Plastic Resistance in Single-Crystal-Textured High-Density Polyethylene. Macromolecules 1992, 25, 5036−5053. (9) Queyroy, S.; Monasse, B. Molecular Dynamics Prediction of Elastic and Plastic Deformation of Semicrystalline Polyethylene. Int. J. Multiscale Comput. Eng. 2011, 9, 119−136. (10) Queyroy, S.; Monasse, B. Effect of the Molecular Structure of Semicrystalline Polyethylene on Mechanical Properties Studied by Molecular Dynamics. J. Appl. Polym. Sci. 2012, 125, 4358−4367. (11) Lee, S.; Rutledge, G. C. Plastic Deformation of Semicrystalline Polyethylene by Molecular Simulation. Macromolecules 2011, 44, 3096−3108. (12) Kim, J. M.; Locker, R.; Rutledge, G. C. Plastic Deformation of Semicrystalline Polyethylene under Extension, Compression, and Shear Using Molecular Dynamics Simulation. Macromolecules 2014, 47, 2515−2528. L


Article

Macromolecules (34) Popli, R.; Roylance, D. A Thermodynamic Prediction of Tie Chain-Length Distribution in Drawn Semicrystalline Polymers. Polym. Eng. Sci. 1985, 25, 828−833. (35) Balijepalli, S.; Rutledge, G. C. Conformational Statistics of Polymer Chains in the Interphase of Semi-Crystalline Polymers. Comput. Theor. Polym. Sci. 2000, 10, 103−113. (36) Keith, H. D.; Fadden, F. J. Deformation Mechanisms in Crystalline Polymers. J. Polym. Sci. 1959, 41, 525−528. (37) Peterlin, A. Bond Rupture in Highly Oriented Crystalline Polymers. Journal of Polymer Science Part A-2: Polymer Physics 1969, 7, 1151−1163. (38) Seguela, R. Critical Review of the Molecular Topology of Semicrystalline Polymers: The Origin and Assessment of Intercrystalline Tie Molecules and Chain Entanglements. J. Polym. Sci., Part B: Polym. Phys. 2005, 43, 1729−1748. (39) Cammarata, R. C.; Eby, R. K. Effects and Measurement of Internal Surface Stresses in Materials with Ultrafine Microstructures. J. Mater. Res. 1991, 6, 888−890. (40) Litvinov, V. M.; Xu, J.; Melian, C.; Demco, D. E.; Möller, M.; Simmelink, J. Morphology, Chain Dynamics, and Domain Sizes in Highly Drawn Gel-Spun Ultrahigh Molecular Weight Polyethylene Fibers at the Final Stages of Drawing by Saxs, Waxs, and 1h Solid-State Nmr. Macromolecules 2011, 44, 9254−9266. (41) Che, J.; Locker, C. R.; Lee, S.; Rutledge, G. C.; Hsiao, B. S.; Tsou, A. H. Plastic Deformation of Semicrystalline Polyethylene by XRay Scattering: Comparison with Atomistic Simulations. Macromolecules 2013, 46, 5279−5289. (42) Crist, B.; Fisher, C. J.; Howard, P. R. Mechanical Properties of Model Polyethylenes: Tensile Elastic Modulus and Yield Stress. Macromolecules 1989, 22, 1709−1718. (43) Pawlak, A.; Galeski, A. Plastic Deformation of Crystalline Polymers: The Role of Cavitation and Crystal Plasticity. Macromolecules 2005, 38, 9688−9697. (44) Brooks, N. W. J.; Mukhtar, M. Temperature and Stem Length Dependence of the Yield Stress of Polyethylene. Polymer 2000, 41, 1475−1480.

M


Molecular Dynamics Simulation of the Effects of Layer Thickness and

Recommend Documents