Deformation and Fracture Processes of a Lamellar Structure in

Article pubs.acs.org/Macromolecules

Deformation and Fracture Processes of a Lamellar Structure in Polyethylene at the Molecular Level by a Coarse-Grained Molecular Dynamics Simulation Yuji Higuchi*,†,‡ and Momoji Kubo† †

Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-8577, Japan PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan

‡

ABSTRACT: Coarse-grained molecular dynamics simulations can model the deformation and fracture processes of the lamellar structure in polyethylene on a molecular scale; however, the simulations have not been performed due to the difficulty in building the structure and the limitations of small-scale simulations on the order of 104 beads. Thus, we propose a crystallization method for a large-scale lamellar structure on the order of 106 beads. The highly oriented lamellar structure is stretched in a coarse-grained molecular dynamics simulation. The stress and variation of crystallinity during stretching parallel and perpendicular to the crystal direction agree with experimental results, confirming the validity of our simulation results. During stretching parallel to the crystal direction, the amorphous layers crystallize and the crystalline layers fragment. We also find that the movement of the polymer chain ends from amorphous to crystalline layers, which is difficult to observe experimentally, increases the compression and generation of voids in the amorphous layers.

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INTRODUCTION The mechanical properties of semicrystalline polymers, such as polyethylene, are important for their applications as industrial materials, and their deformation and fracture processes have been investigated thoroughly.1−6 The lamellar structures in polyethylene consist of crystalline and amorphous layers, where the polymer chains fold. This is a fundamental part of the spherulite structure observed on a much larger scale (50−500 μm). During flow or deformation, polymers are represented as fibers and lamellar structures that are oriented in roughly the same direction. Therefore, the deformation and fracture processes of a lamellar structure are important for understanding the mechanical properties of the polymer. To reveal the origin of stress during stretching and to improve the toughness of semicrystalline polymers, it is essential to understand these processes on a molecular level. Many experiments have been performed, and many molecular models have been proposed. Nitta et al. suggested the lamellar cluster model and investigated the mechanical properties of semicrystalline polymers.7−10 In the model, the structure of the cluster consisting of lamellar layers remains, and the clusters are rearranged during stretching. Tie molecules, which connect lamellar clusters, play an important role in the mechanical properties. Indeed, the block structures of the crystal parts were observed in the deformation and fracture of oriented thin films of polyethylene by electron microscopy.11 Strobl et al. suggested the following process.12,13 During stretching, the lamellar blocks slip first, and then the lamellar blocks show collective motion. Under further stretching, fibrils form. Finally, disentanglement starts and the structure changes, leading to © XXXX American Chemical Society

memory loss. Mandelkern studied crystallization processes and mechanical properties and revealed the effect of the conditions on the crystallinity and crystal thickness. Crystallinity and crystal thickness are important factors in the mechanical properties,14 in addition to molecular mass.15 Hosoda et al. clarified the effect of the molecular weight on the stress against the strain.16 To simplify the problem, an oriented structure, where the lamellar directions were in roughly the same direction, was studied. Zhou et al. revealed the dependence of the stretching direction in a highly oriented structure on the mechanical properties and suggested a model for the changes in morphology at the molecular level.17 Fukuoka et al. investigated the effect of entanglement in the amorphous layer in an oriented structure.18 The deformation process has been well studied; however, important parameters, such as crystallinity, crystal thickness, and molecular weight, complicate the problem. Therefore, the deformation and fracture processes at the molecular level are still controversial. Coarse-grained molecular dynamics simulations are useful for elucidating the mechanisms of the deformation and fracture processes on the molecular scale because it is possible to model polymer chain dynamics. The mechanical properties of amorphous and glassy polymers have been well studied.19−21Furthermore, coarse-grained models are valuable in revealing the crystallization processes of polymers.22−25 For example, Meyer et al.22,23 studied the crystallization process in Received: December 5, 2016 Revised: April 11, 2017

A

DOI: 10.1021/acs.macromol.6b02613 Macromolecules XXXX, XXX, XXX−XXX

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where l is the bond length between neighboring monomers, θ is the bond angle, r is the distance between two monomers, and a is a diameter of the beads. The parameters are shown in Table 1. The equilibrium bond length, l0, is 0.154 nm. The bending

polyethylene and poly(vinyl alcohol) models. Ramos et al.24,25 examined the crystallization processes of polyethylene in atomistic and coarse-grained molecular dynamics simulations. However, there are few molecular dynamics simulation studies of the mechanical properties of semicrystalline polymers. Rutledge et al. modeled a lamellar structure and revealed the dynamical properties of the interface between amorphous and crystalline layers.26−28 Pioneering studies clarified the importance of entanglement in the amorphous layer. Yamamoto studied the crystallization process of semicrystalline polymers29−32 and performed a molecular dynamics simulation of the deformation process of semicrystalline polymers.33 We also studied a fracture process in the block structure of semicrystalline polymers.34 These simulations were still small scale, where the number of beads was less than 4.0 × 104. Thus, the mechanical properties of a lamellar structure have not been determined fully. A larger simulation on the order of 106 beads is needed to represent lamellar clusters. Jabbari-Faouji et al.35 recently performed a large-scale molecular dynamics simulation consisting of 4.3 × 106 beads to reveal the plastic deformation of semicrystalline polymers. Although they simulated recrystallization and fragmentation processes with a large-scale model, they used a polycrystalline structure. Therefore, the deformation and fracture processes of a lamellar structure have not been clarified with a molecular dynamics simulation. On the basis of this background, we study the fracture process of a lamellar structure by using a large-scale coarse-grained molecular dynamics simulation consisting of 3.0 × 106 beads. We focus on the effect of the polymer chain structures such as tie chains, loops, entanglement, and chain ends. First, the details of the coarse-grained molecular dynamics simulation are introduced. Next, the preparation method for the lamellar structure is described. The deformation and fracture processes of a highly oriented lamellar structure are presented, and the implications of our work and comparisons with other studies are discussed.

Table 1. List of Parameters and Their Values kabond l0 kabend kbbend kdbend θ0 ϵ a kbbond R0 R1 U0 m

J/(m2 mol) nm J/mol J/mol J/mol deg J/mol nm J/(m2 mol) nm nm J/mol kg

(4)

This potential was also used in ref 19 to investigate the fracture of glassy polymers during stretching. Here, we set the values so that bond is broken at 0.296 nm with an energy barrier of 3.854 × 102 kJ/mol, and the averaged bond length is approximately 0.154 nm. The length and energy of bond breaking in C9H20 are estimated by the results of Gaussian 03.36 The force, f, is calculated from eqs 1−4. The stress against the stretching, σ, is calculated from the forces and kinetic energy as σ = (−1/V)∑[rifi + (1/m)pipi] = σbond + σbend + σLJ + σkinetic, where V is the volume of the cell, m is the mass, and pi is the momentum. Pressure is controlled by a loose coupling method37 with a control parameter τp = 10 (Pτ/a), where P = ϵ/a3. The value of the control parameter has also been used in calculating the crystallization process of polypropylene with united atom models38 in a study with similarities to our work. Three angles of the sides in the simulation cell are set as π/2. As explained later, the vacuum phase is set in one direction of the lamellar structure to observe the partial deformation process. Therefore, we set the external pressures as 0 in the crystallization and stretching processes. The monomers obey the stochastic dynamics described by the Langevin equation

(1)

a b d Ubend = k bend − k bend (cos θ − cos θ0) + k bend (cos θ − cos θ0)3 (2)

⎡⎛ a ⎞12 ⎛ a ⎞6 ⎤ ULJ = 4ϵ⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r ⎠ ⎦ ⎣⎝ r ⎠

3.5 × 10 0.154 7.440 × 103 2.297 × 104 7.386 × 104 108.78 598.64 0.392 4.401 × 1026 0.296 0.107 3.854 × 105 1.4 × 10−3

b b U bond = k bond (l − R 0)3 (l − R1) + U0

METHOD To investigate the deformation and fracture processes of a lamellar structure, we perform coarse-grained molecular dynamics simulations by using a bead−spring model with a three-dimensional periodic boundary condition. The number of polymer chains is 3000, and the chain length is N = 1000. The number of total beads is 3.0 × 106. In previous coarse-grained molecular dynamics simulation studies,28,33,35 the chain lengths and the number of total beads were N = 1078 and approximately 2.7 × 104, N = 513 and approximately 1.1 × 104, and N = 300 and 4.3 × 106, respectively. Our chain length is longer and simulation size is larger than the previous simulations. To reproduce the lamellar structures of polyethylene, we adopt the potential energies and parameters that have been widely used for the crystallization process of polyethyelene.29−32 The potential energy of the system is represented by the bonding, bending, and attractive and repulsive terms 1 a k bond(l − l0)2 2

units

25

potential gives the lowest energy at θ = 180° and a local minimum of 2.51 kJ/mol at θ = 90°. The energy barrier is about 12.54 kJ/mol at θ = 130°. Polymer chains are flexible and prefer to adopt a straight conformation. The length and energy are measured in units of a and ϵ, respectively. The excluded volume and attractive effects are included in the Lennard-Jones potential, ULJ. On the basis of ref 31, we exclude the interaction of ULJ between monomers less than three bonds apart, and the cutoff length of ULJ is 2.0a. During stretching, we should consider bond dissociation. The following potential is used to allow the bond breaking.

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a U bond =

values

m

d2ri dt

2

= −γ

dri + f Ui + ξi dt

(5)

where γ = 0.5 is the drag coefficient. The constant τ = a(m/ ϵ)1/2 is chosen as the unit for the time scale. We set the time

(3) B


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Macromolecules step as dt = 5.0 × 10−3τ for eq 1 in the crystallization process and dt = 3.0 × 10−3τ for eq 4 in the stretching process. The Brownian force, ξi, satisfies the fluctuation dissipation theorem ⟨ξi(t)ξi(t′)⟩ = 6kBTγδijδ(t − t′), where kB is the Boltzmann constant and T is the absolute temperature. We introduce local orientational order η to describe crystallinity as η = 1/2(3⟨cos2 ϕij⟩ − 1), where ϕ is the angle between bond vectors i and j. The brackets, ⟨ ⟩, indicate averaging over the bond pairs if the distance between them is less than 2.0a, and they are not bonded. The crystallinity of the polymers is indicated by a color scale based on η: blue and green show the crystalline and amorphous states, respectively. The color changes continuously from green to blue with the order parameter of η.

⎧ 0, n(lc + la) < rz − L0z < n(lc + la) + la ⎪ ⎪ f (rz) = ⎨1, n(lc + la) + la ⎪ < rz − L0z ⎪ < (n + 1)(lc + la) ⎩

where n is an arbitrary integer number and Lz0 is the start zcoordinate of the cell in the z-direction. We examine the crystallization process based on this method. Crystallite thickness depends on the polymer chain length and crystallization process and has been determined experimentally as 13.5−15.0 nm.14 Here, the lamellar thickness is set as approximately 15.0 nm. Initially, 500 polymer chains are located in the cell size of (Lx, Ly, Lz) = (30a, 30a, 420a), where Lx, Ly, and Lz are the lengths of the cell in the x-, y-, and zdirections, respectively. When the cell length in the z-direction is approximately 60 nm, relaxation by the NPT ensemble at 500 K is stopped, where the cell size is (Lx, Ly, Lz) = (43.0a, 42.8a, 152.4a). Four lamellar layers are prepared in this simulation. The forces are applied with k0 = 5.0. Experimentally39 and in a simulation,33 tilt crystal structures have been observed in lamellae. The angle was estimated as 40°−60°. Therefore, we set the crystal direction, r0, as (r0x, r0y, r0z) = (1.0, 0.0, 1.0) and (r0x, r0y, r0z) = (−1.0, 0.0, 1.0), for odd and even n, respectively. To mimic the gradual crystallization process, la decreases from 38.1a to 2.0a and lc increases from 0.0a to 37.1a in t = 2.5 × 103τ. The temperature decreases from 500 to 300 K during the crystallization process. Because the density increases with an increase in crystallinity, the NPT ensemble is used only in the y-direction. Figure 1 shows cross-sectional snapshots of the yz-

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RESULTS AND DISCUSSION Crystallization Process. In previous studies, simulation sizes on the order of 104 beads were used to reproduce lamellar structures.26−28,33 In a large-scale simulation on the order of 106 beads, a polycrystalline structure was used.35 Therefore, a lamellar structure has not been reproduced in a large-scale simulation although this is essential for revealing the mechanical properties. Thus, we develop a crystallization method for the large-scale simulation to construct a lamellar structure. First, we prepare a lamellar structure, which is a model of the highly oriented structure observed experimentally.17,18 Fluids can also exhibit highly oriented or fiber crystal structures. Yamamoto33 studied the crystallization process in fluids and found a lamellar structure. We also model the crystal structure in fluid. Because the relaxation time is too long to calculate the whole crystallization process in a large-scale simulation consisting of 3.0 × 106 beads, an acceleration method is needed. For example, a previous simulation used the Monte Carlo method.26−28 In a molecular dynamics method, the walls are placed to accelerate the crystallization process.29−32 In previous work, to prepare a large-scale structure consisting of 4.3 × 106 beads, small crystal grains were copied and positioned.35 In this study, we apply forces to promote the crystallization process and then enlarge the system by copying a basic cell. Because entanglements are conserved during the crystallization process,1 the number of entanglements should be small in the initial conditions to achieve high crystallinity. Therefore, to decrease entanglements, the melted state is not used here. In the initial state, all polymer chains are completely straight, which means that the bonding and bending energies are zero. Then, the cell is relaxed by the NPT ensemble method. We assume that the procedure corresponds to the reverse of the fluid process. Next, cyclic forces are applied in the z-direction to form a lamellar structure. Vector r0 is the crystal direction, ri, i+1 is the bond vector, and ψ is the angle between r0 and ri,i+1. The applied potential is π ⎧ 2 ψ< ⎪ f (rz)k 0ψ , 2 Uψ = ⎨ π ⎪ f (r )k (ψ − π )2 , ψ > ⎩ z 0 2

(7)

Figure 1. Cross-sectional snapshots from the crystallization process.

plane during the crystallization process under applied forces. At t = 0.00τ, the polymers are melted. After the forces are applied, the number of crystalline regions increases gradually. At t = 2.5 × 103τ, four layers are observed. Next, a relaxation calculation is performed without the force applied by the NPT ensemble at 300 K in t = 2.5 × 103τ. Figure 2a shows cross-sectional snapshots of the yz-plane at t = 2.5 × 103τ. Four crystalline layers remain after the relaxation calculation. Figures 2b and 2c show typical snapshots of a polymer chain along the x- and yaxes, respectively. The polymer chain folds and tilts in the xzplane. Figure 2d shows the stress tensor of the relaxation calculation. The stress is almost relaxed, indicating that the structure is stable. We note that σzx is not relaxed fully due to the tilt. However, we continue the preparation of the lamellar structure because the stress is smaller than atmosphere pressure

(6)

where f(rz) is a rectangular function of the z position, rz, and k0 is the strength of the applied potential. The lengths of the crystal and amorphous layers are important parameters. To obtain any combination of lengths, the crystal length (lc) and amorphous length (la) are given by f(rz) as C


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Figure 3. Cross-sectional snapshots of the yz-plane (a) after the enlargement of the cell with vacuum spaces in both sides, (b) after the annealing process at 400 K, (c) after the cooling process from 400 to 300 K, and (d) after the relaxation process at 300 K. (e) Threedimensional snapshot of the lamellar structure.

Figure 2. (a) Cross-sectional snapshots after the crystallization process. A typical polymer chain viewed along the (b) x- and (c) yaxes. (d) Time variation of stress in the relaxation process. (e) Probability density of orientational order, η, in the melted state at 500 K (left snapshot in Figure 1) and a lamellar structure at 300 K (a).

spread over the surface. After the enlargement of the cell, the relaxation calculation is performed by the NVT ensemble at 400 K in t = 5.0 × 102τ (Figure 3b). The surfaces become flat, and the size of the amorphous region increases at high temperature. Then, the temperature is decreased from 400 to 300 K in t = 5.0 × 102τ by the NVT ensemble (Figure 3c). The crystalline layer increases again after cooling. The relaxation calculations by the NVT and NPT ensembles are performed in t = 5.0 × 102τ. After then, the structure in Figure 3d is similar to that in Figure 3c. The amorphous layers spread slightly near the surfaces. Finally, the relaxation calculation by the NPT ensemble at 300 K is performed with the bond potential in eq 4 in t = 2.5 × 102τ. Figure 3e shows a three-dimensional snapshot. Four crystalline and amorphous layers are observed. Thus, a lamellar structure with surfaces is constructed. The crystallinity in the structure with surfaces in Figure 3e is 0.76, and that in the bulk state in Figure 2a is 0.8. Therefore, crystallinity of the surfaces decreases. The experimental crystallinity, which we aim to replicate here, is about 0.8. In a previous experiment,17 the crystallinity was not described. However, another study showed that the range of crystallinity

and is negligible. To evaluate the crystallinity, the probability density of the orientational order, η, in the melted state in Figure 1 at t = 0.00τ and the crystalline state in Figure 2a are calculated, and they are shown in Figure 2e. In the melted state, the peak is around 0.0 and the probability is almost zero at η = 0.4. In the crystalline state, a bimodal distribution is observed with two peaks at around 0.1 and 0.9. This indicates that the crystalline and amorphous states exist, as shown in Figure 2a. We set the threshold value of the crystalline state as 0.4. The crystallinity is calculated as 0.80. In this work, we estimate the crystallinity using this procedure. To reveal the mechanical properties, the simulation size must be increased. Furthermore, the vacuum phase is set to observe the partial deformation process because the periodic boundary condition and NPT ensemble result in homogeneous deformation. Figures 3a−d show cross-sectional snapshots of the yz-plane in the enlargement and relaxation process. The cell is copied twice in the x-direction and three times in the ydirection (Figure 3a). The vacuum space is set in the ydirection. To prevent bond dissociation, the polymer chains are D


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ignored. A statistical loop is defined as when a polymer chain enters an amorphous or crystalline layer from a crystal or amorphous layer and then returns to the original crystalline or amorphous layer. If a loop is shorter than five monomers, it is classified as short loop. The number of entanglements are counted when two loops are constrained topologically. The topology of the two loops are evaluated by the method proposed by Klenin et al.42 Amorphous and crystalline layers are divided by a threshold value of 0.4 in Figure 4. There are four amorphous and crystalline layers, and they are numbered from the bottom. In particular, the bottom of the snapshot in Figure 4a is defined as the first crystalline layer. The amorphous layer above the first crystalline layer is defined as the first amorphous layer. Under periodic boundary conditions, the top crystal part is a continuous layer of the bottom crystal part; therefore, it is defined as the first crystalline layer. Tables 2 and 3 show the numbers of amorphous and crystalline layers, respectively. The numbers of short loops and statistical loops in the amorphous layer are greater than in the crystalline layer. This indicates that the polymer chains fold in the amorphous layer. The numbers of chain ends are higher in the second and fourth amorphous layers and in the third and fourth crystalline layers. The distribution of chain ends is calculated. Figure 5 shows the probability density of chain ends along the z-axis in the initial state (Figure 1a) and lamellar structure (Figure 3e). For comparison, Lz is normalized. In the initial state, the chain ends are distributed randomly. After the crystallization process, there are four peaks in the amorphous layers (see Figure 4), indicating that the chain ends gather in the amorphous layers. The number of tie chains is similar in all the layers, suggesting that the effect of tie chains on the mechanical properties is similar in each layer. The number of entanglements in the amorphous layer is greater than that in the crystalline layer. There are few entanglement defects in the crystalline layers. In this analysis, the number of entanglements is calculated only from two loops belonging to the same layer, and more complicated topological structures are ignored. Rutledge et al.26−28 prepared a lamellar structure consisting of an amorphous layer and a crystalline layer by the Monte Carlo method. They analyzed the structure of polymer chains in the lamellar layers. There were 1.29 tie chains, 86.28 loops, and 0.4 loop−loop entanglements on average in the amorphous layer. The amorphous and crystalline lengths were 12.7 and 12.666 nm, respectively. The cell lengths perpendicular to the crystal direction were 5.535 and 4.445 nm. Based on these numbers, the number densities of tie chains, loops, and loop− loop entanglements were 4.13 × 10−3, 2.76 × 10−1, and 1.280168 × 10−3 nm−3, respectively. We also analyze the number densities and compare them with those of the previous study. The amorphous and crystalline thicknesses are estimated with a threshold of 0.4. The thicknesses of the first, second,

in the lower oriented structure was 0.56−0.66, which was estimated from the enthalpy of fusion.40 The ranges of the crystallinity estimated from the density and calculated from the enthalpy of fusion in various samples are 0.53−0.83 and 0.39− 0.77, respectively.15 The crystallinity estimated by the enthalpy of fusion was lower than that based on density. Considering the decrease in crystallinity near the surfaces, our result agrees well with the experimental results.17 In this study, the monomers are divided into amorphous or crystalline states based on the orientational order, which is similar to deciding the crystallinity by fitting the crystalline and amorphous areas in X-ray diffraction patterns. Thus, the crystallinity is calculated from the monomer fraction, which shows a similar result to the mass fraction of the crystalline region. In previous simulations investigating the deformation process of semicrystalline polymers, the crystallinity was around 0.43 and 0.5235 and 0.4.33 The definition was different from our method: they averaged the orientational order in a small space and determined the crystallinity, and their crystallinity was much lower than the experimental value. We think that their threshold was stringent and that our method is appropriate for comparing the calculated results with experimental values. The lamellar structure and the structure of polymer chains in Figure 3e are analyzed in detail. First, the orientational order, η, of monomers is averaged in the xy-plane (Figure 4b). For

Figure 4. (a) Cross-sectional snapshot and (b) averaged orientational order, η, along the z-axis.

comparison, we show a cross-sectional snapshot of the yz-plane (Figure 4a). Minimum averaged η values of approximately 0.1 are at around positions 6, 44, 81, and 117 along the z-axis. The lamellar thickness is estimated as 37a ≃ 14.5 nm. The threshold of crystallinity is 0.4, which is decided by Figure 2e. These results confirm that there are four amorphous and crystalline layers. Nilsson et al.41 investigated the structure of polymer chains in an amorphous layer. The polymer chains can form tight folds (short loops), statistical loops, chain ends, free chains, tie chains, and entanglement of two loops. In our model, the polymer chains are long enough for free chains to be

Table 2. Number and Number Density of Short Loops, Statistical Loops, Loose Chain Ends, Tie Chains, and Trapped Entanglements in the Amorphous Layers first layer no. short loops statistical loops chain ends tie chains entanglements

245 2167 479 2382 7

second layer −3

number density (nm ) 6.93 6.13 1.35 6.74 1.98

× × × × ×

−2

10 10−1 10−1 10−1 10−3

no. 220 2017 611 2378 2

third layer −3


× × × × ×

−2

10 10−1 10−1 10−1 10−4 E

no. 235 1885 540 2461 7

fourth layer −3


× × × × ×

−2

10 10−1 10−1 10−1 10−3

no. 238 1967 612 2360 7

number density (nm−3) 7.16 7.16 1.84 7.10 2.10

× × × × ×

10−2 10−1 10−1 10−1 10−3


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Table 3. Number and Number Density of Short Loops, Statistical Loops, Loose Chain Ends, and Trapped Entanglements in the Crystalline Layers first layer no. short loops statistical loops chain ends entanglements

152 1922 913 0

second layer −3

number density (nm ) −3

9.35 × 10 1.18 × 10−1 5.62 × 10−2 0.00

no. 139 2013 860 3

third layer −3

number density (nm ) 8.42 1.22 5.21 1.82

× × × ×

−3

10 10−1 10−2 10−4

no. 151 1888 1034 4

fourth layer −3

number density (nm ) 9.48 1.18 6.49 2.51

× × × ×

−3

10 10−1 10−2 10−4

no. 187 1816 951 2

number density (nm−3) 1.22 1.19 6.21 1.31

× × × ×

10−2 10−1 10−2 10−4

because the amorphous length in our work is thinner than that in the previous study. The crystal structure of polyethylene depends on the preparation path. This leads to the different thicknesses of the amorphous and crystalline layers. In this study, we model the highly oriented lamellar structure and compare the mechanical properties with experimental measurements.17 The lamellar thickness and the crystallinity are set as 15 nm and 0.8, respectively, which are typical experimental values.14,15 The thicknesses of the amorphous and crystalline layers are set based on these numbers. Therefore, we think that in this study the experimental structure is reproduced well by the thin amorphous layer compared with previous studies. We also discuss the crystallization procedure. There are still many parameters for controlling the lamellar structure and folded structure of polymer chains that should be clarified in the future. In this work, our aim is to reveal the mechanical properties of a lamellar structure. For example, we can change the length of the amorphous and crystalline layers to control the length of the applied force area described in eqs 6 and 7. Crystallinity also changes with the length fraction of the amorphous and crystalline layers. From the beginning of the NPT relaxation process, the structure of the polymer chains is also controlled. That is, when the relaxation calculation is short and the cell length along the oriented direction is long, the number of extended chains is large. Conversely, the number of folded chains increases with an increase in the relaxation time. Entanglements are controlled and increase as the NVT ensemble simulation time increases. Thus, our method can

Figure 5. Probability density of chain ends versus normalized distance of the cell length in the z-direction. An initial state and a lamellar structure are shown in Figures 1a and 3e, respectively.

third, and fourth amorphous layers are 6.70a, 7.10a, 7.00a, and 6.30a, respectively. The thicknesses of the first, second, third, and fourth crystalline layers are 30.8a, 31.3a, 30.2a, and 29.0a, respectively. The volume is estimated as 1.30 × 106a3. The cell length along the z-axis is 148.4a. Therefore, the area of the xyplane is 8.76 × 103a2. Based on these numbers, the number densities are calculated (Tables 2 and 3). The number densities of the loops and entanglements are 6.0 × 10−1 and 2.0 × 10−3 nm−3, respectively, which are similar to those in previous studies. In contrast, the number densities of tie chains are on the order of 10−1 nm−3 and are much larger than that of the previous study, which was on the order of 10−3 nm−3. This is

Figure 6. Cross-sectional snapshots of the deformation and fracture processes of the lamellar structure during stretching (a) parallel and (b) perpendicular to the crystal direction. F


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used a larger, more coarse-grained model. In contrast, the lamellar structure in our study consists of crystalline and amorphous layers and we use a coarse-grained model representing the properties of polyethylene. The buckling is strongly related to the stability of crystalline layers, which is interesting because the block structure has been observed by electron microscopy.11 Similarly, the slip of the crystalline layers is also interesting. These processes are still not fully understood at the molecular level and may be revealed by larger scale simulations in the future. Next, we evaluate the mechanical properties and the crystallinity. These values have also been determined experimentally;17 therefore, we compare the experimental values with our results. Figures 7a and 7b show the stress−

control complicated parameters, such as crystallinity, lamellar thickness, and entanglements, and reveal the various mechanical properties of the semicrystalline polymers. Deformation and Fracture Processes. To reveal the deformation and fracture processes of the lamellar structure in polyethylene at the molecular level, which has not been achieved by molecular dynamics simulations, we perform stretching simulations. The cell is stretched parallel and perpendicular to the crystal directions. This has also been performed experimentally,17 and we compare our results with the experimental results. In the stretching simulation, the temperature is set at 300 K. The cell is drawn in the x- and zdirections with a velocity of v = 0.03a/τ. This value was also used in previous simulations.19,34 The cell length in the ydirection is fixed; however, the vacuum space is set in this direction, and there are surfaces. Therefore, the polymers can deform and contract freely. The other direction is calculated with the NPT ensemble method. The stress is calculated by σ = (−1/V)∑[rifi + (1/m)pipi]. We use the initial volume as a constant value for volume V during the stretching simulation for calculating the stress. To observe the partial deformation, the vacuum spaces are set in the y-direction. During the partial deformation process, it is difficult to define the surface and calculate the deformed volume. Furthermore, when voids are generated and grow, an accurate volume is difficult to calculate. Thus, the initial volume is used for calculating stress. The drawing method to calculate stress from the initial volume was also used in a previous study to determine the effect of surfaces on the mechanical properties.43 The strain is calculated by ΔL/ L0, where L is the cell length in the stretching direction and ΔL is the drawing length. Figures 6a and 6b show cross-sectional snapshots during stretching parallel and perpendicular to the crystal direction, respectively. During stretching parallel to the crystal direction (Figure 6a), the sample becomes thinner in the y-direction for strains from 0.00 to 0.49. For strains from 0.49 to 0.73, the amorphous layers contract and a void is generated. Finally, at a strain of 0.97, the void grows larger. Simultaneously, the amorphous layer around the void changes to a crystalline layer. During stretching perpendicular to the crystal direction (Figure 6b), the sample becomes thinner in the z-direction for strains from 0.00 to 1.00. The four amorphous and crystalline layers are conserved in the process. Makke et al.44,45 performed stretching simulations of a lamellar structure consisting of rubbery and glassy layers. Although the lamellar structure did not contain crystalline layers, the results were interesting. They found that cavitation occurred in the amorphous layers, similar to our results. They also observed void generation in the rubbery layers, regardless of crystallinity. Buckling instability accompanied by cavitation and waves in the lamellar structure was also observed in the previous study, where a more coarse-grained model than that in this article was used.44 The noncrystalline polymers were located in a cell size of (32.4a′, 200−800a′, 74.2a′), where a′ is the length unit of the simulation. The cell was stretched perpendicular to the lamellar direction (along the z-axis). The buckling wave was observed along the lamellar direction (along the y-axis). The buckling always occurred when the samples were larger than about 400a′ in the y-direction. This indicated that the buckling was cyclic, and a large simulation size is required to observe buckling. Buckling is not observed in the present work. This difference may be caused by the simulation size and/or the crystallinity. The lamellar structure in the previous study44 consisted of noncrystalline layers, and they

Figure 7. (a) Stress−strain curve of the lamellar structure during stretching parallel (z-direction) and perpendicular (x-direction) to the crystal direction. (b) Variation of crystallinity versus strain during stretching parallel (z-direction) and perpendicular (x-direction) to the crystal direction.

strain curves and time variation of the crystallinity, respectively. During stretching parallel to the crystal direction, the stress increases for strains from 0.0 to 0.3. Then, the slope becomes large around a strain of 0.3, and the stress increases. At a strain of 0.6, the stress decreases. Therefore, the fracture point is at a strain of 0.6, and the fracture stress is approximately 140.25 MPa. In Figure 7b, the crystallinity increases with the stretching. The inflection points are similar to those of the stress−strain curve. During stretching perpendicular to the crystal direction, the stress increases for strains from 0.0 to 1.0 (Figure 7a). The slope gradually decreases during the process. A fracture point is not observed. To estimate Young’s modulus, the stress, σ, is fitted by σ = Y (ΔL /L0) + A(ΔL /L0)2

(8)

where Y is the Young’s modulus and A is the coefficient of the second-order term. The fitted range of the strain is varied from G


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elongation of bonds, even though the elongation is disfavored. Generally, the relaxation of a polymer chain is fast compared with that of the assembly structure. Thus, it is difficult for a polymer chain to relax against the stretching for strains from 0.4 to 0.6, and the bonds are elongated. However, bond dissociation is not observed in the simulation. The difficulty in relaxation is also indicated by the bending stress. The bending stress decreases for strains from 0.0 to 0.45, indicating that the polymer chains are aligned in the stretching direction or they unfold. Subsequently, the bending stress becomes constant, and the polymer chains cannot align and unfold any more. According to these results, the process can be described as follows. During stretching, the sample is compressed perpendicular to the stretching direction as the polymer chains become aligned to the stretching direction and unfold. After the polymer chains cannot align and unfold any more, the bonds are elongated. Finally, a void is generated and the sample is fractured. We also analyze the change in the crystallinity along the stretching direction to evaluate the crystallization process, the solidity, and the fragmentation, quantitatively. Figure 9a shows

0.0 to 0.05, 0.10, 0.20, and 0.30 because the estimation of the yield point is difficult in Figure 7a. For the stretching parallel (z-direction) and perpendicular (x-direction) to the crystal direction, the Young’s moduli are 501.68745, 357.1029, 277.992, and 227.00865 MPa, and 544.24425, 333.498, 231.759, and 194.2182 MPa in strain ranges of 0.05, 0.10, 0.20, and 0.30, respectively. In a previous study,17 where highly oriented polyethylene was stretched, the Young’s moduli were 530−1500 and 750−1600 MPa for the stretching parallel and perpendicular to the crystal direction, respectively. The Young’s modulus estimated by our simulation results are in the same order of magnitude as the previous experimental result. The Young’s moduli are approximately 0.1, 170, and 340 GPa for isotropic, single-crystal, and perfectly oriented polyethylene.46 The highly oriented lamellar structure calculated in this article has a higher Young’s modulus than that of the isotropic polyethylene but much lower than those of single-crystal and perfectly oriented polyethylene. In Figure 7b, the crystallinity increases slightly during stretching. The increase in the crystallinity during stretching perpendicular to the crystal direction is smaller than that during stretching parallel to the crystal direction. In a previous experiment,17 the stress during stretching parallel to the crystal direction was larger than that during stretching perpendicular to the crystal direction, although the initial stresses were similar. During stretching parallel to the crystal direction, there was a fracture point after a sharp increase in the stress, whereas the stress became constant and no fracture point was observed during stretching perpendicular to the crystal direction. They also investigated the crystallinity by X-ray scattering and found that the crystallinity increased during stretching parallel to the crystal direction, whereas it was constant during perpendicular stretching. These trends agree with our results. Therefore, our method for representing a lamellar structure and the stretching process is validated. Stretching Parallel to the Crystal Direction. We investigate the origin of the stress during stretching parallel to the crystal direction. Figure 8 shows the stresses along the stretching

Figure 9. (a) Variation of averaged orientational order, η, versus normalized distance of the cell length in the z-direction during stretching parallel to the crystal direction. (b) Increase in thicknesses in the amorphous and crystalline layers versus strain during stretching parallel to the crystal direction.

Figure 8. (a) Total stress (σzz), bonding stress (σbond zz ), bending stress LJ (σbend zz ), attractive and repulsive stresses (σzz), and kinetic stress ) in the stretching direction (z-direction) versus strain during (σkinetic zz stretching parallel to the crystal direction.

the averaged crystallinity along the stretching direction. The cell length along the stretching direction is normalized, and η is averaged in the xy-plane. At a strain of 0.00, there are four minima, indicating four amorphous layers. Four crystalline layers are observed, and the crystallinity is around 0.8. With the stretching, the crystallinity of amorphous layers gradually increases for strains from 0.00 to 0.49; however, the layers are still amorphous. The crystallinity of the crystalline layers does not change. The crystalline layers are solid and behave as blocks. After the fracture, the amorphous layers are crystallized at a strain of 0.73. Finally, the amorphous layers are crystallized

direction. The stress caused by the Lennard-Jones potential, which indicates attractive interactions, increases with stretching for strains from 0.0 to 0.5. The sample is compressed by the stretching, decreasing the loss of attractive interactions. Then, the stress decreases and becomes constant at a strain of 0.5. This indicates the generation of the void. The stress caused by bonds is constant for strains from 0.0 to 0.4, although it increases for strains from 0.4 to 0.6. This indicates the H


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Macromolecules and the crystallinity is from 0.5 to 0.7. Remarkably, the crystallinity of crystalline layers decreases, which corresponds to their fragmentation. We estimate the growth of the amorphous and crystalline layer thicknesses for strains from 0.0 to 0.5 (Figure 9b). Because all the amorphous layers are crystallized and the crystallinity is over the threshold of 0.4, the thickness cannot be defined for strains from 0.5 to 1.0. The thicknesses of the crystalline layers increase, whereas the thicknesses of the amorphous layers increase only slightly. The amorphous layers crystallize and transform into crystalline layers. Therefore, the crystallization of the amorphous layers increases the thicknesses of the crystalline layers. The crystallization of the amorphous layers was also observed in a previous simulation of a crystalline and an amorphous layer,26−28 and the growth of the crystalline layers was on the order of 104 beads.33 We also show the crystallization of the amorphous layers on a larger scale of 3.0 × 106 beads, where there are four crystalline and amorphous layers. The solidity of the crystalline layers was also observed experimentally by an electron microscope.11,17 The crystalline layers also behaved as a block, and the crystal blocks were separate and were connected by tie chains during the fracture process. The lamellar cluster model suggested by Nitta et al.7−10 also indicated that the crystalline layers are solid and behave as blocks. Our simulation results are consistent with their results. The solidity of the crystalline layers and the fragmentation cannot be revealed by small molecular dynamics simulations on the order of 104 beads. Thus, we reveal the time evolution of the crystal structure on a molecular scale by using a large-scale coarse-grained molecular dynamics simulation consisting of 3.0 × 106 beads. To reveal the effect of the molecular structure of the polymer chains on the mechanical properties, we investigate the numbers of chain shapes in each crystalline and amorphous layer for strains from 0.0 to 0.5. The effect of polymer chain ends is critical, and the change in the number of chain ends is shown in Figure 10a. In the crystalline layers, the number of chain ends increases, whereas the number decreases in the amorphous layers. Around a strain of 0.35, the slopes of the numbers in the crystalline layers increase and those in the amorphous layers decrease. Figure 10b shows the number of chain ends that move from the amorphous to the crystalline layers. Slopes for ΔL1.0 and ΔL2.0 in the double-logarithmic axis are shown as visual guides, where ΔL is the drawing length. The negative value disappears in the double-logarithmic plot. For strains from 0.00 to 0.35, the summation number increases with a slope of 1.0 (Figure 10b), and the thickness of crystalline layers also increases linearly. Figure 10b shows that the movement of chain ends from the amorphous layers to the crystalline layers increases at a strain of 0.35 because the slope is 2.0 for strains from 0.35 to 0.5. These results mean that chain ends move from the amorphous layers to the crystalline layers before the voids are generated. After the movement of chain ends, space is generated in the amorphous layers, leading to the thinning and generation of voids in the amorphous layers. The slopes for ΔL1.0 may indicate the crystal growth from the amorphous layers to the crystalline layers with stretching in the z-direction linearly. The thicknesses of the crystalline layers increase linearly (Figure 9b). We assume that the slopes for ΔL2.0 indicate the contraction of the amorphous layers in addition to the crystal growth from the amorphous layers to the crystalline layers. The contraction decreases the space in the amorphous layers, increasing the movement of the chain ends from the amorphous layers to the crystalline layers.

Figure 10. (a) Change in number of polymer chain ends in the amorphous and crystalline layers versus stress during stretching parallel to the crystal direction. Differences are calculated from the number at a strain of 0.0. (b) Number of chain ends that moved from the amorphous to crystalline layers. Slopes for ΔL1.0 and ΔL2.0 in the double-logarithmic axis are shown as visual guides, where ΔL is the drawing length.

Stretching Perpendicular to the Crystal Direction. We also investigate the deformation process during stretching perpendicular to the crystal direction. Figure 11a shows the stresses along the stretching direction. The stress caused by the

Figure 11. (a) Total stress (σzz), bonding stress (σbond zz ), bending stress LJ kinetic ) (σbend zz ), attractive and repulsive stress (σzz), and kinetic stress (σzz in the stretching direction (x-direction) versus strain during stretching perpendicular to the crystal direction. (b) Dynamics of a typical polymer chain during stretching perpendicular to the crystal direction. I


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desirable; however, this is difficult because the factors are interrelated. Here, we discuss the factors on the molecular scale. The effect of tie chains is important,3,4,6 which is consistent with the effect of the chain ends revealed in this article. When the number of chain ends in the amorphous layers increases, the number of tie chains decreases because the space in the amorphous region is limited. A coarse-grained molecular dynamics simulation showed the effect of tie chains.45 Makke et al. stretched a lamellar structure that contained rubbery and glassy layers and revealed an increase in resistance to cavitation with an increase in tie chains. Although the material in their study did not contain crystalline layers, our results showing that the effect of chain ends increases the void generation are similar to their results revealing the relationship between cavitation and tie chains. The chain ends prefer to be in the amorphous layers (Figure 5); therefore, this effect is crucial to the strength. Chain length (molecular mass) also affects the mechanical properties.16 When the molecular mass is low, the stress at the yield and rupture points decreases. In particular, the stress at a rupture point decreases greatly with a decrease in molecular mass. In our results, the movement of chain ends from the amorphous layers to the crystalline layers generates voids, which leads to fracturing. Low-molecular-mass polymers contain many more chain ends than high-molecular-mass polymers. Therefore, we propose that the effect of chain ends is a factor in the decrease of the stress at the rupture point. In polyacetal, which is a crystalline polymer, mixing of lowmolecular-weight molecules is a severe problem that decreases dart impact strength.47 In this case, although other factors, such as crystallinity and lamellar thickness, were different, the mechanism of void generation may be the same as that in the present work. Thus, the effect of polymer chain ends is important for fracturing in crystalline polymers. We discuss the effect of a force field on the deformation and fracture processes of semicrystalline polymers. The united atom model used in previous simulation studies26−28,33 showed a zigzag structure consisting of CH2 groups. In contrast, the coarse-grained model used in this article shows a straight structure. This coarse-grained model has been used to reveal the crystallization process of polyethylene, and its validity was confirmed in previous studies.29−32 Therefore, the model used in the present study can reproduce the correct crystal structure. The present model is more coarse-grained than the model used in previous simulation studies,26−28,33 decreasing the simulation time. Thus, we use a force field to perform larger-scale simulations. A similar force field was used to perform a largescale simulation to reveal the deformation process of semicrystalline polymers.35 Slip is an important problem in the deformation process of polyethylene. For the slip of the polymer chains, an all-atomic molecular dynamics simulation is more accurate than a coarse-grained model. O’Connor et al. observed the slip of polyethylene with a molecular dynamics simulation using an all-atomic model.48 In contrast, the slip of crystal layers, leading to void generation, is only observed in larger scale simulations on the order of 106 monomers in this work and in a previous study.35 The potential eq 4 is chosen to allow bond breaking. This type of the potential was also used in a coarse-grained molecular dynamics simulation of the fracture process of glass polymers.19 We also used the potential in a previous study of the fracture process of semicrystalline polymers and observed bond breaking.34 On the basis of these previous studies, we used the potential to allow bond

Lennard-Jones potential gradually increases with stretching for strains from 0.0 to 1.0. The trend is similar to that for the total stress. Therefore, the stress caused by the Lennard-Jones potential is the main factor in the process. The bonding and kinetic stresses are constant for strains from 0.0 to 1.0 and do not affect the stress during stretching substantially, whereas the bonding stress increases during stretching parallel to the crystal direction. The bending stress decreases for strains from 0.0 to 1.0 gradually. This means that polymer chains gradually align with the stretching direction. Figure 11b shows typical snapshots of a polymer chain. During the stretching process, the polymer chain gradually tilts in the stretching direction, and the crystal direction changes to the stretching direction. To reveal the effect of chain ends, we investigate the change in numbers of chains ends in each crystal and amorphous layer for strains from 0.0 to 1.0 (Figure 12a). In the crystalline layers,

Figure 12. (a) Change in the number of chain ends in amorphous and crystalline layers versus stress during stretching perpendicular to the crystal direction. Differences are calculated from the number of chain ends at a strain of 0.0. (b) Number of chain ends that moved from the amorphous to the crystalline layers. The slope for ΔL1.0 in the doublelogarithmic axis is shown as a visual guide, where ΔL is the drawing length.

the number of chain ends starts to increase at a strain of 0.50. Figure 12b shows the number of chain ends that move from the amorphous to the crystalline layers. The negative values disappear in the double-logarithmic plot. For strains from 0.00 to 0.20, the value is negative. The chain ends move randomly between the crystalline and amorphous layers. The total numbers are smaller than those during stretching parallel to the crystal direction. Furthermore, the slope maintains a value of 1.0 for strains from 0.60 to 1.00. Therefore, the movement of chain ends does not affect the process much. Deformation and Fracture Process on a Molecular Scale. There are still many factors that affect the mechanical properties of polyethylene. To separate the factors and reveal their effects on the mechanical properties individually is J


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Macromolecules breaking so that bond dissociation was possible. However, we did not observe bond breaking in the present work. Physical aging of polymers is a severe problem and has been studied extensively because of its importance.49 Recycled polypropylene is brittle and the fracture surface changes, although molecular mass does not,50 and it was suggested that the structure changes at the molecular level. In this study, we examine the fracturing caused by the movement of polymer chains. Therefore, we predict that the concentration of polymer chain ends leads to the easy generation of voids and fracturing. The concentration of polymer chain ends requires long-range chain diffusion at room temperature and is one possible mechanism of physical aging. To confirm the effect of the concentration, polymer chain ends are located in the lower part of the cell in the z-direction. Then, the lamellar structure is prepared with a crystallinity of 0.75, which is similar to the structure in Figure 3e. The probability density of the chain ends is shown in Figure 13a. The probability density of the lamellar structure in Figure 3e is also shown as the control. Polymer chains are located in the lower amorphous layers. Figure 13b shows the stress−strain curve. During stretching perpendicular to the crystal direction, the stress does not change much, indicating that the concentration of polymer chain ends does not affect the mechanical properties. Remarkably, the maximum stress at a strain of 0.6 decreases during stretching parallel to the crystal direction; however, the stress does not change greatly for strains from 0.0 to 0.3. The concentration of polymer chain ends only affects the generation of voids. This weakening of the maximum stress is clearly caused by localized polymer chains. This is confirmed by the cross-sectional snapshots in Figure 13c. Polymer chain ends are concentrated in the two lower amorphous layers, in which the compression and generation of voids are observed. In contrast, the structure of the upper amorphous layers is almost conserved during the stretching process. In Figure 6a, all amorphous layers are compressed by stretching. Therefore, the concentration of polymer chain ends leads to the easy fracturing, which is a possible reason for the physical aging of crystalline polymers. The generation of voids and their growth are observed in this work. This cannot be calculated by a small-scale molecular dynamics simulation on the order of 104 beads. Therefore, the large-scale coarse-grained molecular dynamics simulation on the order of 106 beads is a powerful method for revealing the deformation and fracture processes at the molecular level. Necking and slip in lamella are larger-scale phenomena and are still controversial. Farge et al. discussed the generation of voids and the crystallization in necking regions.51 To address these problems, the phenomena need to be understood from molecular to macro scales. Humbert et al. also discussed crystallite shearing and the generation of voids at yielding.52,53 They suggested that an increase in stress transmitters, which are molecular networks, such as tie chains and entanglements, increases the yielding stress and decreases the generation of voids. This indicates that the molecular structure determines the generation of voids and the yielding stress. They also proposed that the physical explanation should relate the mechanical properties to the molecular structure such as crystallinity and crystallite thickness. Therefore, further understanding at a molecular level is needed and large-scale simulations by a coarse-grained molecular dynamics method are desirable for revealing the complicated deformation and fracturing processes in crystalline polymers.

Figure 13. (a) Probability density of chain ends versus normalized distance of the cell length in the z-direction. Polymer chain ends are locally concentrated in the biased model. The probability density of the control model in Figure 3e is also shown for comparison. (b) Stress−strain curve of the lamellar structure during stretching parallel (z-direction) and perpendicular (x-direction) to the crystal direction. (c) Cross-sectional snapshots of the deformation and fracture processes in the biased model during stretching parallel to the crystal direction.

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CONCLUSION We studied the deformation and fracture processes of a highly oriented lamellar structure in polyethylene by using a coarsegrained molecular dynamics simulation consisting of 3.0 × 106 beads. First, a lamellar structure was prepared by applying cyclic forces to promote crystallization. Four amorphous and crystalline layers were used and the lamellar thickness was approximately 14.5 nm. The crystallinity was estimated as 0.76, which was similar to the experimental value and higher than those estimated by previous simulations. Our proposed method can control factors such as the crystalline, amorphous, and lamellar thicknesses, crystallinity, and entanglements in the amorphous layers. Therefore, our proposed method is suitable for revealing the deformation and fracture processes of a lamellar structure at the molecular level. The lamellar structure was stretched parallel and perpendicular to the crystal direction. During stretching parallel to the K


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(5) Galeski, A. Strength and Toughness of Crystalline Polymer Systems. Prog. Polym. Sci. 2003, 28, 1643−1699. (6) Patlazhan, S.; Remond, Y. Structural Mechanics of Semicrystalline Polymers Prior to the Yield Point: A Review. J. Mater. Sci. 2012, 47, 6749−6767. (7) Nitta, K.; Takayanagi, M. Application of Catastrophe Theory to the Neck-initiation of Semicrystalline Polymers Induced by the Intercluster Links. Polym. J. 2006, 38, 757−766. (8) Kuriyagawa, M.; Nitta, K. Structural Explanation on Natural Draw Ratio of Metallocene-catalyzed High Density Polyethylene. Polymer 2011, 52, 3469−3477. (9) Nitta, K.; Kuriyagawa, M. Application of Catastrophe Theory to Neck Initiation of Metallocene-catalyzed High-density Polyethylene. Polym. J. 2012, 44, 245−251. (10) Mizushima, M.; Kawamura, T.; Takahashi, K.; Nitta, K. In Situ Near-infrared Spectroscopic Studies of the Structural Changes in Polyethylene during Tensile Deformation. Polym. Test. 2014, 38, 81− 86. (11) Adams, W.; Yang, D.; Thomas, E. Direct Visualization of Microstructural Deformation Processes in Polyethylene. J. Mater. Sci. 1986, 21, 2239−2253. (12) Men, Y.; Rieger, J.; Strobl, G. Role of the Entangled Amorphous Network in Tensile Deformation of Semicrystalline Polymers. Phys. Rev. Lett. 2003, 91, 095502. (13) Hong, K.; Rastogi, A.; Strobl, G. A Model Treating Tensile Deformation of Semicrystalline Polymers: Quasi-Static Stress-Strain Relationship and Viscous Stress Determined for a Sample of Polyethylene. Macromolecules 2004, 37, 10165−10173. (14) Mandelkern, L. The Relation between Structure and Properties of Crystalline Polymers. Polym. J. 1985, 17, 337−350. (15) Kennedy, M. A.; Peacock, A. J.; Mandelkern, L. Tensile Properties of Crystalline Polymers: Linear Polyethylene. Macromolecules 1994, 27, 5297−5310. (16) Hosoda, S.; Uemura, A. Effect of the Structural Distribution on the Mechanical Properties of Linear Low-Density Polyethylenes. Polym. J. 1992, 24, 939−949. (17) Zhou, H.; Wilkes, G. L. Orientation-dependent Mechanical Properties and Deformation Morphologies for Uniaxially Meltextruded High-density Polyethylene Films Having an Initial Stacked Lamellar Texture. J. Mater. Sci. 1998, 33, 287−303. (18) Fukuoka, M.; Aya, T.; Saito, H.; Ichihara, S.; Sano, H. Role of Amorphous Region on the Deformation Behavior of Crystalline Polymers. Polym. J. 2006, 38, 542−547. (19) Rottler, J.; Robbins, M. O. Growth, Microstructure, and Failure of Crazes in Glassy Polymers. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 011801. (20) Rottler, J. Fracture in Glassy Polymers: A Molecular Modeling Perspective. J. Phys.: Condens. Matter 2009, 21, 463101. (21) Makke, A.; Perez, M.; Lame, O.; Barrat, J.-L. Mechanical Testing of Glassy and Rubbery Polymers in Numerical Simulations: Role of Boundary Conditions in Tensile Stress Experiments. J. Chem. Phys. 2009, 131, 014904. (22) Meyer, H.; Müller-Plathe, F. Formation of chain-folded structures in supercooled polymer melts. J. Chem. Phys. 2001, 115, 7807−7810. (23) Vettorel, T.; Meyer, H. Coarse Graining of Short Polyethylene Chains for Studying Polymer Crystallization. J. Chem. Theory Comput. 2006, 2, 616−629. (24) Ramos, J.; Vega, J. F.; Martínez-Salazar, J. Molecular Dynamics Simulations for the Description of Experimental Molecular Conformation, Melt Dynamics, and Phase Transitions in Polyethylene. Macromolecules 2015, 48, 5016−5027. (25) Ramos, J.; Vega, J.; Sanmartín, S.; Martínez-Salazar, J. Coarsegrained simulations on the crystallization, melting and annealing processes of short chain branched polyolefins. Eur. Polym. J. 2016, 85, 478−488. (26) Lee, S.; Rutledge, G. C. Plastic Deformation of Semicrystalline Polyethylene by Molecular Simulation. Macromolecules 2011, 44, 3096−3108.

crystal direction, there was a fracture point after a sharp increase in the stress, whereas during stretching perpendicular to the crystal direction, the stress gradually increased and became constant, and no fracture point was observed. The stress during stretching parallel to the crystal direction was larger than that during stretching perpendicular to the crystal direction. These trends agree with a previous experiment, where a highly oriented lamellar structure was stretched. Therefore, we represented the lamellar structure and stretching process accurately. During stretching parallel to the crystal direction, the amorphous layers crystallized and the thickness of the crystalline layers increased. After the fracture point, voids were generated in the amorphous layers and the crystalline layers fragmented, which could not be revealed by previous small-scale molecular dynamics simulations on the order of 104 beads. During stretching perpendicular to the crystal direction, the crystal direction gradually tilted toward the stretching direction. The amorphous and crystalline layers were conserved and the crystallinity did not increase much. Finally, we analyzed the molecular structure during the deformation and fracture processes. Before the fracture point during stretching parallel to the crystal direction, the polymer chain ends moved from the amorphous to the crystalline layers. This led to the constriction of the amorphous layers and void generation. When the polymer chain ends were concentrated, the maximum stress decreased, demonstrating that the concentration of polymer chain ends leads to easy fracturing. These findings at a molecular level could not be obtained experimentally or by small-scale molecular dynamics simulations. Thus, the effect of polymer chain ends on the fracturing was revealed by large-scale coarse-grained molecular dynamics simulation on the order of 106 beads. We elucidated the deformation and fracture processes of the lamellar structure at the molecular level, which will contribute to the development of tough crystalline polymers.

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AUTHOR INFORMATION

Corresponding Author

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

Yuji Higuchi: 0000-0001-8759-3168 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by JST, PRESTO “Molecular technology and creation of new functions” (Project No. JPMJPR13KF) and by MEXT as “Exploratory Challenge on Post-K computer” (Challenge of Basic Science−Exploring Extremes through Multi-Physics and Multi-Scale Simulations).

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REFERENCES

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