Stress Transmitters at the Molecular Level in the Deformation and

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Stress Transmitters at the Molecular Level in the Deformation and Fracture Processes of the Lamellar Structure of Polyethylene via Coarse-Grained Molecular Dynamics Simulations Yuji Higuchi*,†,‡ †

Institute for Solid State Physics, The University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-8577, Japan

‡

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S Supporting Information *

ABSTRACT: To improve the toughness of semicrystalline polymers against stretching, it is essential to understand the stress transmission processes at the molecular level. The deformation and fracture processes of the lamellar structure of polyethylene were studied using coarse-grained molecular dynamics simulations to investigate the influence of molecular structures such as tie chains and entanglements. First, two models with different numbers of tie chains and entanglements were successfully constructed and subjected to simulated stretching. The results revealed that tie chains and entanglements indeed transmit the stress upon stretching. The roles of these molecular structures were found to be similar at low strain, whereas the tie chains were more important at void generation owing to the rapid relaxation of the entanglements. Next, to unravel the behavior of the tie chains, a model containing defects was subjected to simulated stretching. In the model lacking defects, the tie chains functioned similarly in all four amorphous layers. Interestingly, in the model containing defects, the stresses of the tie chains in the amorphous layers containing defects were found to be higher than those in the amorphous layers lacking defects following void generation. Therefore, the nature of the stress transmitters in the lamellar structure of semicrystalline polymers has been successfully elucidated at the molecular level.

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INTRODUCTION The safe and resource-efficient use of polymers requires a comprehensive understanding of their deformation and fracture processes to optimize their mechanical properties. Semicrystalline polymers, such as polyethylene, polypropylene, poly(vinyl alcohol), and poly(oxymethylene), possess hierarchical structures, which results in complicated yet interesting deformation and fracture processes. Owing to their importance, these polymers have been actively studied.1−6 In general, lamellae consisting of crystalline and amorphous layers are a fundamental structural feature of semicrystalline polymers. Therefore, understanding the deformation and fracture processes of these lamellar structures is essential. At the molecular level, the crystalline layers are solid and act as blocks during the fracture process, which has been observed using electron microscopy.7 Strobl et al. reported the occurrence of the following processes: (i) the lamellar blocks slip; (ii) the lamellar blocks move together; (iii) under further stretching, fibrils form; and (iv) the polymer chains start to disentangle, which leads to memory loss.8,9 It has been suggested that tie chains connecting two neighboring crystalline layers play an important role during the deformation and fracture processes.3,4,6 Nitta et al. proposed the lamellar cluster model to describe the deformation and fracture processes of semicrystalline polymers.10−13 In this model, the lamellar clusters connected by tie chains are reallocated during © XXXX American Chemical Society

stretching. In addition to tie chains, entanglements in amorphous layers have also attracted attention. Fukuoka et al. investigated the influence of the number of entanglements in the amorphous layers on the mechanical properties.14 Humbert et al. suggested that the tie chains and entanglements in the amorphous layers connect the solid crystalline layers and therefore act as stress transmitters upon stretching.15,16 However, the experimental observation of polymer chain dynamics at the molecular scale is difficult. Furthermore, the dynamics are complicated as they are influenced by numerous parameters, such as the crystallinity and crystal thickness. Thus, the roles of the tie chains and entanglements in the deformation and fracture processes of semicrystalline polymers remain controversial. Molecular simulations are an effective strategy for elucidating the dynamics of polymer chains and their mechanisms at the molecular scale. Coarse-grained molecular dynamics simulations have been extensively applied to examine the deformation and fracture processes of a variety of polymers, such as polymer glasses,17−20 elastomers,21 rubbery and glassy structures,22,23 block copolymers,24,25 and polymer gels.26−29 On the other hand, the deformation and fracture Received: March 29, 2019 Revised: August 1, 2019

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DOI: 10.1021/acs.macromol.9b00636 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules processes of semicrystalline polymers are less well understood owing to the difficulty associated with constructing their structures. Rutledge et al. pioneered the investigation of the fracture process of semicrystalline polymers. They prepared an amorphous layer sandwiched by crystalline layers and stretched the resulting structure using the coarse-grained molecular dynamics method.30−34 Yamamoto has studied the crystallization of polymers35 and also performed stretching simulations of polyethylene.36 These studies revealed the fracture of the amorphous layers. We also investigated fracture processes in the block structure of semicrystalline polymers.37 However, the entire deformation and fracture processes of the lamellar structure could not be modeled owing to the small simulation sizes. Jabbari-Farouji et al. performed large-scale molecular dynamics simulations of poly(vinyl alcohol) using a coarse-grained model consisting of 4.3 × 106 beads.38−40 The polycrystalline structure was oriented along the stretching direction. These studies demonstrated the effectiveness of the simulation approach for elucidating the deformation and fracture processes at the molecular level. However, the structure used in the simulation was not the lamellar structure observed experimentally and the simulated mechanical properties were not in agreement with the experimental ones. Recently, we successfully constructed the lamellar structure of polyethylene, and the obtained stress−strain curve was in agreement with the experimental one.41,42 However, the nature of the stress transmitters and their roles at the molecular level have not yet been clarified. In this study, the deformation and fracture processes of semicrystalline polymers were investigated using coarse-grained molecular dynamics simulations. First, the details of the coarse-grained molecular dynamics simulations are introduced. Next, the crystallization process is discussed. The deformation and fracture processes of semicrystalline polymers are then presented, followed by a discussion of the implications of our work and a comparison with other studies.

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Table 1. Summary of Parameters and Their Values kbond l0 kabend kbbend kdbend θ0 ϵ a m

METHODS

1 a k bond(l − l0)2 2

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

The equilibrium bond length, l0, was set to 0.154 nm. Polymer chains preferentially adopt a linear conformation owing to the energy minimum at θ = 180°, but they are flexible under the conditions of a local minimum of 2.51 kJ/mol at θ = 90° and an energy barrier of approximately 12.54 kJ/mol at θ = 130°. The unit length and energy are represented by a and ϵ, respectively. The Lennard-Jones potential, ULJ, includes the excluded volume and attractive effects. In the calculation of ULJ, a cutoff length of 2.0a was adopted. Interactions between monomers less than three bonds apart were excluded based on ref 45. Bond scission during the deformation and fracture processes was not considered because this was not observed in our previous study at strains from 0.0 to 1.0.41 The forces, f, were calculated from eqs 1−3. The stress during the stretching simulation, σ, was calculated from the forces and kinetic energy as σ = −1/V∑[rif i + 1/mpipi] = σbond + σbend + σLJ + σkinetic, where V is the volume of the cell, m is the mass, and pi is the momentum. A loose coupling method47 was adopted for pressure control with the control parameter τp = 10(Pτ/a), where P = ϵ/a3. This value of the control parameter was also used in the simulated crystallization of polypropylene with united atom models.48 Three angles of the sides of the simulation cell were fixed as π/2. A vacuum phase was introduced in one direction of the lamellar structure to allow the partial deformation process. Therefore, the external pressures were set to zero for the crystallization and stretching processes. To determine the roles of each of the molecular structures during stretching, the stress was calculated for the tie chains, entanglements, chain ends, and loops. The calculation of local stress is difficult because the volume cannot be estimated. Therefore, the stress was estimated via the following procedure based on the virial theorem and is referred to as monomer stress, σmon. Here, the stress was divided over each monomer equally for the two-body interactions shown in eqs 1 and 3. For the three-body interaction shown in eq 2, the force was reduced to two-body interactions using the central-force decomposition method.49 As the effective volume of each monomer is difficult to define, the value was finally divided by the total volume. Consequently, the stress of the cell is the summation of monomer stress, σ = ∑σmon. Since the fluctuations were too large to reveal the mechanism, the monomer stress was averaged every 4000 steps. The total number of simulation steps at a strain of 1.0 in model A was 106. The change of the strain for 4000 steps was 4.0 × 10−3, which was small enough to average the stress. The cell was drawn in the x- and zdirections with a velocity of v = 0.03a/τ. In the z-direction, the strain rate was calculated as v/Linit = 2.0 × 10−4/τ, where Linit is the initial cell length along the z-direction. The monomers obey the stochastic dynamics described by the Langevin equation

To investigate the stress transmitters in the deformation and fracture processes of the lamellar structure, coarse-grained molecular dynamics simulations were performed using a bead−spring model under a three-dimensional periodic boundary condition. The total number of beads was 4.0 × 106; the number of polymer chains was 4000; and the chain length was N = 1000. This simulation size was larger than that used in our previous study.41 These values of chain length and simulation size were considered sufficiently long and large, respectively, on the basis of previous coarse-grained molecular dynamics simulations,32,36,38 where the chain length ranged from 300 to 1078 and the size was on the order of 104−106. The potential energy was calculated using previously reported parameters for the crystallization of polyethylene.43−46 The potential energy of this system can be represented by the following bonding, bending, and attractive and repulsive terms Ubond =

values 3.5 × 1025 0.154 7.440 × 103 2.297 × 104 7.386 × 104 108.78 598.64 0.392 1.4 × 10−3

(1)

a b d Ubend = k bend − k bend (cos θ − cos θ0) + k bend (cos θ − cos θ0)3 (2) É ÅÄÅ Ñ Åi a y12 i a y6ÑÑ ULJ = 4ϵÅÅÅÅjjj zzz − jjj zzz ÑÑÑÑ ÅÅk r { k r { ÑÑÑÖ (3) ÅÇ

m

d2ri dt 2

= −γ

dri + fiU + ξi dt

(4)

where γ = 0.5 is the drag coefficient. The constant τ = a(m/ϵ) was chosen as the unit for the time scale, and the time step was set to dt = 5.0 × 10−3τ. 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. 1/2

where l is the bond length between neighboring monomers, θ is the bond angle, r is the distance between two monomers, and a is the bead diameter. The parameters used are summarized in Table 1. B


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Macromolecules To describe crystallinity, the local orientational order was calculated as η =1/2(3⟨cos2 ϕij⟩ − 1), where ϕ is the angle between the bond vectors i and j. The brackets, ⟨⟩, indicate averaging over the nonbonded pairs if the distance between them is less than 2.0a. In the snapshots, the crystallinity of the monomers is indicated using a color scale based on η. In the RGB color model, the values of blue and green are set as η and 1.0 − η, respectively, and when η is negative, the value is set as 0.0. The color changes continuously from green to blue over time with the order parameter of η.

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RESULTS AND DISCUSSION Crystallization Process. To investigate the roles of the possible stress transmitters such as tie chains and entanglements during stretching, three different lamellar structures were prepared as models of the highly oriented structures observed experimentally.14,50 The molecular structures in the constructed lamellar structures were then examined. Recently, we successfully constructed a lamellar structure for which the stress−strain curve was in agreement with that obtained experimentally,41 and we applied the same method in this study. The procedure is briefly introduced in the following sections, and schematic images are presented in Figure 1. In the initial state of model A, all of the polymer chains possessed a straight alignment, which means that the bonding and bending energies were zero. In the initial state of model B, each of the polymer chains were bent once and the polymer chains were arranged in pairs to increase the number of entanglements. In the initial state of model C, the polymer chain ends were concentrated to generate defects. We previously reported that concentrations of chain ends serve as defects during the fracture process.41 After introducing 1000 polymer chains into the cell with dimensions of (Lx, Ly, Lz) = (30a, 30a, 420a), where Lx, Ly, and Lz are the lengths of the cell in the x-, y-, and z-directions, respectively, the cell was relaxed using the NPT ensemble method at 500 K. The NPT ensemble simulation was stopped when Lz was approximately 60 nm. Next, cyclic forces were applied in the z-direction to form the lamellar structure. If vector ro is the crystal direction, ri,i+1 is the bond vector, and ψ is the angle between ro and ri,i+1, the applied potential can be expressed as follows π l o o f (rz)koψ 2 , ψ< o o 2 o Uψ = m o π o o o f (rz)ko(ψ − π )2 , ψ > o o 2 n

Figure 1. (a) Schematic images showing the initial placements. In model A, the polymer chains were set randomly with a straight alignment. In model B, the polymer chains were bent once to increase the number of entanglements. In model C, the polymer chain ends were concentrated to introduce defects. (b) Schematic images showing the modeling method for the lamellar structure. (c) Threedimensional snapshot of the constructed lamellar structure.

(5)

both experimentally52 and in a simulation,36 and the angle was estimated as 40−60°. To mimic a gradual crystallization process, la was decreased, lc was increased, and the temperature was decreased from 500 to 300 K during the crystallization process. The NPT ensemble was used only in the y-direction to permit variation of the density. Next, a relaxation calculation using the NPT ensemble was performed without the force applied at 300 K. To investigate the mechanical properties and observe the partial deformation process, the simulation cell was copied and the vacuum space was introduced in the ydirection. After this enlargement of the simulation size, the relaxation calculation was performed via the following four steps: (i) NVT at 400 K, (ii) cooling to 300 K with NVT, (iii) NVT at 300 K, and (iv) NPT at 300 K. The constructed lamellar structure is depicted in Figure 1c. It was confirmed that the stress in the constructed structure was lower than atmosphere pressure and therefore negligible. To generate the model with defects, chain ends were concentrated in the initial state (Figure 1a), and then the

where f(rz) is a rectangular function of the z position, rz, and ko 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 z l o o 0, n(lc + la) < rz − L0 < n(lc + la) + la f (rz) = o m z o o n1, n(lc + la) + la < rz − L0 < (n + 1)(lc + la)

(6) z

where n is an arbitrary integer and L0 is the initial z coordinate of the cell in the z-direction. The lamellar thickness was set to approximately 15.0 nm based on the experimental estimation of 13.5−15.0 nm reported previously.51 Four lamellar layers were prepared in this simulation with ko = 5.0. The crystal direction, ro, was set to (rox, roy, roz) = (1.0, 0.0, 1.0) and (rox, roy, roz) = (−1.0, 0.0, 1.0), for odd and even values of n, respectively. This tilted crystal structure has been observed C


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approximately 0.1, indicating that these peaks corresponded to amorphous regions. Conversely, the four highly oriented regions corresponded to the crystalline layers. Here, the lamellar structure was divided into amorphous and crystalline layers using a threshold η value of 0.6. Next, the molecular structures in the amorphous layers were analyzed. Based on the types of molecular structures in amorphous layers identified by Nilsson et al.,54 the numbers of loops, chain ends, free chains, tie chains, and entanglements were counted. Entanglements consist of two loops constrained topologically. The constraints were evaluated using the method proposed by Klenin et al.55 The four amorphous layers observed in Figure 2 were numbered from the bottom. The numbers and number densities of each type of molecular structure in models A, B, and C are summarized in Tables 2, 3, and 4, respectively. In model A, the loops, chain ends, tie chains, and entanglements were homogeneously distributed. In model B, the numbers of tie chains and entanglements decreased and increased, respectively, compared with model A, demonstrating the successful construction of an entanglement-rich structure. In model C, a high number of chain ends was observed in the first and second layers. These results indicate that it is possible to control the characteristics of the lamellar structure during model construction. Rutledge et al. also analyzed the molecular structures in a lamellar structure consisting of an amorphous layer and a crystalline layer prepared via the Monte Carlo method.30−32 The number densities of the tie chains, loops, and loop−loop entanglements were 4.13 × 10−3, 2.76 × 10−1, and 1.280168 × 10−3 nm−3, respectively. The number densities of the loops and entanglements are similar to those obtained in this study. On the other hand, the number of tie chains observed in this study is much larger than that reported in the previous study. The lamellar structure in this study is a model structure of a highly oriented one. Difference between Tie Chains and Entanglements. To investigate the role of the stress transmitters in the deformation and fracture processes of the lamellar structure of polyethylene at the molecular level, stretching simulations were performed for models A and B, where the latter model contained a greater number of entanglements. The chain length of polymer chains was N = 1000 in these simulations. First, the roles of the tie chains and entanglements and the differences between the two structures are considered. Figure 3 shows the stress−strain curves for models A (control) and B (entanglement rich). Upon stretching perpendicular to the lamellae, the increases in the stress at strains from 0.0 to 0.3 were almost the same for both models. At strains from 0.3 to 0.5, the stress increased sharply and voids were then generated. The fracture stress was larger for model A than for model B. In contrast, upon stretching parallel to the lamellae, the stress increased gradually in almost the same manner for both

model was crystallized. The distributions of chain ends before and after the crystallization are shown in Figure S1. Before the crystallization, chain ends were located in half of the cell. There were no chain ends around a normalized distance of 0.7. After crystallization, the chain ends were concentrated and there were two large peaks at normalized distances of 0.1 and 0.3. For comparison, the control (model A) is shown. There were four peaks, meaning that the chain ends were distributed evenly. Chain ends tend to concentrate in the amorphous layer to increase crystallinity and gain attractive energy, which is consistent with the experimental results53 and is discussed in our previous study.41 The constructed structure was next analyzed. The methods were described in detail in the previous study.41 The probability densities of the orientational order, η, in the melt state and the lamellar structure are used to evaluate crystallinity (Figure S2). There is a large peak at η = 0.0 in the melt state. The probability becomes almost 0 at η = 0.4. In the lamellar structure, there is a sharp peak at η = 0.9, indicating a crystalline state. There is a small peak at η = 0.1 and the distribution is broad, indicating an amorphous state. The crystalline and amorphous states coexist. According to these results, the threshold value is set as 0.4. Then, the crystallinity is decided by the ratio of the monomers above η = 0.4. In previous simulations studying semicrystalline polymers,36,38 the crystallinity was determined by a similar calculation. First, the orientational order in a small space is averaged. Then, the crystallinity is determined by the ratio of space above the threshold value. In the molecular dynamics simulations, the threshold value is essential for determining the crystallinity. In this article, the value is defined based on the bimodal distribution (Figure S2). The estimated crystallinities of the constructed structures were 0.79, 0.74, and 0.78 for models A, B, and C, respectively. Next, the amorphous and crystalline layers were determined based on the average orientational order in the xy plane along the z axis, as plotted in Figure 2b.

Figure 2. (a) Cross-sectional snapshot of the constructed lamellar structure. (b) Average orientational order, η, along the z axis.

For comparison, the cross-sectional snapshot is shown in Figure 2a. The minimum average η value at the four peaks was

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

3473 740 3089 9

second layer

number density (nm−3) 5.84 1.24 5.19 1.51

× × × ×

10−1 10−1 10−1 10−3

no. 3270 1038 3013 8

third layer


× × × ×

10−1 10−1 10−1 10−3 D

no. 3034 975 3125 6

fourth layer


× × × ×

10−1 10−1 10−1 10−4

no. 3240 1063 3045 6


× × × ×

10−1 10−1 10−1 10−3


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Table 3. Number and Number Density of Loops, Chain Ends, Tie Chains, and Trapped Entanglements in the Amorphous Layers of Model B first layer no. loops chain ends tie chains entanglements

3926 893 2618 56

second layer −3

number density (nm ) 5.07 1.15 3.38 7.24

× × × ×

−1

10 10−1 10−1 10−3

no. 3745 1078 2455 50

third layer −3


× × × ×

−1

10 10−1 10−1 10−3

no. 3762 998 2450 92

fourth layer −3


× × × ×

−1

10 10−1 10−1 10−2

no. 1088 1088 2404 71


× × × ×

10−1 10−1 10−1 10−3

Table 4. Number and Number Density of Loops, Chain Ends, Tie Chains, and Trapped Entanglements in the Amorphous Layers of Model C first layer no. loops chain ends tie chains entanglements

3835 1651 2197 4

second layer


× × × ×

10−1 10−1 10−1 10−4

no. 3503 1816 2164 9

third layer


× × × ×

10−1 10−1 10−1 10−3

fourth layer

no.

number density (nm−3)

no.

2591 216 3942 0

5.31 × 10−1 4.43 × 10−2 8.08 × 10−1 0.00

2785 291 3948 1


× × × ×

10−1 10−2 10−1 10−4

models. Since there was a clear difference between models A and B, the results for stretching perpendicular to the lamellae are focused upon. Figure 4a,c shows the deformation and fracture processes for models A and B upon stretching perpendicular to the lamellae, where the color indicates the orientation of the polymer chains. At a strain of 0.00, the crystallinity in model B was less than that in model A owing to the greater number of entanglements in the former. In both models, the sample deformed upon stretching at a strain of 0.50. Void generation was then observed at a strain of 0.75 followed by void growth at a strain of 1.0. During stretching, the polymer chains oriented gradually. The polymer chains around the voids became oriented and formed fibrils. The processes were similar for the two models. Figure 4b,d shows the monomer stress during stretching for models A and B,

Figure 3. Stress−strain curves for the lamellar structures of models A (control) and B (entanglement rich) during stretching perpendicular (z-direction) and parallel (x-direction) to the lamellae. The strain rate is approximately 2.0 × 10−4a/τ.

Figure 4. Cross-sectional snapshots showing the deformation and fracture processes of the lamellar structures during stretching perpendicular to the lamellae for (a) and (b) model A (control) and (c) and (d) model B (entanglement rich). The color indicates the orientational order in panels (a) and (c) and the monomer stress in panels (b) and (d). E


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Table 5. Average Monomer Stress in Simulation Units of Tie Chains, Entanglements, Loops, Chain Ends, and Loose Chain Ends for Model A (Control) at Strains of 0.2 and 0.5a tie chains at a strain of 0.20 at a strain of 0.50

0.426 14.3 2.72 12.7

(7.03) (236) (44.9) (210)

entanglements 0.454 16.0 2.46 15.2

loops

(7.49) (264) (40.6) (251)

0.0867 19.9 2.09 18.5

(1.43) (328) (34.5) (305)

chain ends −52.6 21.7 −59.7 26.2

(−868) (358) (−985) (432)

loose chain ends 2.37 17.8 5.33 18.6

(39.1) (294) (87.9) (307)

a

The values in the unit of MPa are indicated in the parentheses. Standard deviation values are indicated below.

Table 6. Average Monomer Stress in Simulation Units of Tie Chains, Entanglements, Loops, Chain Ends, and Loose Chain Ends for Model B (Entanglement Rich) at Strains of 0.2 and 0.5a tie chains at a strain of 0.20 at a strain of 0.50

0.397 14.6 2.23 13.5

(6.55) (241) (36.8) (223)

entanglements 0.231 16.4 2.13 15.1

loops

(3.81) (271) (35.1) (249)

0.188 18.6 1.82 17.6

(3.10) (307) (30.0) (290)

chain ends −50.2 21.6 −57.1 26.9

(−828) (356) (−942) (444)

loose chain ends 1.68 17.7 3.95 18.2

(27.7) (292) (65.2) (300)

a

The values in the unit of MPa are indicated in the parentheses. Standard deviation values are indicated below.

monomer stresses of tie chains and entanglements are higher than those of loops. Thus, it was quantitatively clarified that the tie chains and entanglements transmit the stress against the stretching and play a greater role. To reveal the difference between the roles of tie chains and entanglements during the stretching process, the average monomer stresses for the tie chains and entanglements are plotted in Figure 5. The layers were distinguished by the

respectively. At the low strain of 0.10, the monomer stress was almost uniform in the crystalline layers because almost all of the polymer chains were aligned and the molecular structures were the same. As the densities and structures were uniform, so too was the monomer stress. In contrast, in the amorphous layers, the monomer stress varied widely owing to the presence of numerous molecular structures, such as tie chains and entanglements. It was noted that the local stress along the stretching direction (σzz) is equal for any xy plane based on the Irving−Kirkwood procedure.56 Therefore, the monomer stress was slightly higher in the amorphous layers than in the crystalline layers owing to the lower density of the amorphous layers compared with the crystalline layers. According to the same mechanism, following deformation and void generation, the monomer stress in the oriented regions became uniform, whereas the monomer stress in the disordered regions varied widely owing to the numerous molecular structures and lower density. As model B (Figure 4d) contained more disordered regions than model A (Figure 4b), low and high monomer stresses were observed. The monomer stress during the deformation and fracture processes of the lamellar structure was therefore successfully visualized. To quantitatively evaluate the monomer stress for each molecular structure, the monomers in the amorphous layers were distinguished according to their molecular structures. Average monomer stresses for models A and B are summarized in Tables 5 and 6, respectively. The probability density distributions of the monomer stresses at strains of 0.2 and 0.5 were examined for models A and B, as shown in Figure S3. The distributions for the tie chains, entanglements, and loops were normal and almost identical. At the strain of 0.2, the monomer stress of the chain ends was lowest. This is because chain ends can move freely, which results in stresses in the expansion direction even under stretching. Loose chain ends connected with chain ends tend to contract against the expansion stress of the chain ends. At the strain of 0.5, the monomer stress for the chain ends decreased, which indicates that the chain ends moved from the amorphous layers to the crystalline layers. This result is consistent with our previous report.41 Average monomer stresses of tie chains, entanglements, and loops increase from those at a strain of 0.2. Those in model A are higher than those in model B. The results are consistent with the stress shown in Figure 3. At the strain of 0.5, average

Figure 5. Stress−strain curves for the tie chains and entanglements of model A (control) and model B (entanglement rich) during the deformation and fracture processes of the lamellar structure. The strain rate is approximately 2.0 × 10−4a/τ.

crystallinity along the cell for strains from 0.0 to approximately 0.6; at higher strains, the layers could not be determined owing to the crystallization of the amorphous layers and breakage of the crystalline layers. Therefore, the monomer stress at strains from 0.0 to 0.6 is discussed. At low strains from 0.0 to 0.2, the monomer stress of the tie chains was slightly higher than that of the entanglements. No obvious differences between models A and B were observed. At strains from 0.2 to 0.4, the monomer stresses of the tie chains and entanglements were almost the same. However, at a strain of approximately 0.4, the stresses in model A became larger than those in model B. Then, remarkably, differences between the stresses of the tie chains and entanglements were observed at a strain of 0.5. These results indicate that tie chains and entanglements serve to transmit stress similarly at low strain. In contrast, at fracture, the stress of the tie chains is larger than that of the entanglements. It is interesting to note that increasing the number of tie chains is essential for increasing the fracture stress but inconsequential for the mechanical properties at low strain. F


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entanglement is strongest against stretching because the tail (chain end) can relax rapidly.41 In this article, because we evaluated stress transmission at the molecular level, we focused on the monomer stress of loop−loop entanglement. To reveal the work of the tie chains, the monomer stress of the tie chains was divided into 10 types. The side parts of the tie chains were classified as loose chain ends, loops, entanglements, and sustaining chains, which passed through the crystalline layer and to the other side of the amorphous layer. Then, the average monomer stress was calculated for all 10 combinations. Figure 8 shows the average monomer stress

To elucidate the fracture process in detail, the processes of void generation and growth were also examined. Here, spaces were identified as voids when no monomers were present in an area of approximately 2.0a3. To determine the void size, the cell was divided into small grids. The length of the grid was approximately 2.0a, which was the cutoff length of the Lennard-Jones potential. The cell length cannot be divisible by 2.0 during the stretching; therefore, a length of approximately 2.0a was used here. The length was always slightly longer than 2.0a. Then, a small cell containing no monomer was classified as void. The total volume of voids was calculated during the stretching. The void generation and growth processes are shown in Figure 6. For model A, voids

Figure 8. Monomer stresses for the tie chains of model A. Both sides of the tie chains are classified as sustaining chains, loops, and chain ends. The strain rate is approximately 2.0 × 10−4a/τ.

Figure 6. Void generation and growth during the deformation and fracture processes for models A (control) and B (entanglement rich). The strain rate is approximately 2.0 × 10−4a/τ.

in model A. For low strains of 0.0−0.45, the monomer stresses with the chain end were slightly higher because chain ends could move freely. For high strains of 0.45−0.60, the monomer stresses for the sustaining chain were higher than the others. The results revealed qualitatively that the sustaining chains work against the stretching, whereas the loops relax more rapidly at high strain. Entanglements are rare in crystalline layers because it decreases crystallinity and leads to loss of energy. The numbers of monomers in the combination are summarized in Table S1. Because the chain length was sufficiently long, there were no tie chains with both sides that were chain ends. The number of entanglements was small because entanglements in the crystalline layer decrease crystallinity and loss of energy. Therefore, the monomer stress fluctuated (Figure S4) and it is difficult to discuss the difference among the monomers. Crystallinity is one of the most important factors in the fracture process. A low-crystallinity, low-density entanglement model was prepared (model D) and stretched. To decrease the crystallinity, the crystallization parameters, ko and lc, were decreased from 5.0 to 2.0 and from 37.0a to 28.0a, respectively. The other conditions were the same as those in model A. Figure 9a shows the stress−strain curves of models A, B, and D. There was almost no difference at low strains of 0.0− 0.2. A difference appears at a strain of 0.2. The stress of model D was lowest at high strains of 0.4−0.7. To clarify the difference, the monomer stress for tie chains and entanglements is shown in Figure 9b. In the low-crystallinity model (model D), there was no difference between monomer stresses for the tie chains and entanglements. A decrease in the crystallinity decreased the stress against stretching. In particular, the monomer stress for tie chains decreased drastically. Next, the effect of the decrease in crystallinity on the void generation is examined. Figure S5 shows the void generation and growth processes of models A, B, and D. In model D, no voids were generated. Therefore, the decrease in the crystallinity inhibited void generation. Figure 10 shows the

occurred at a strain of 0.5. This is almost the same as the strain at maximum stress, which confirms that the stress peak corresponds to the fracture stress. The voids then grew larger. Interestingly, a delay in void generation and slower void growth were observed for model B. Ge et al.57 studied the change in the entanglement of glassy polymer crazing. Interestingly, the molecular structure of the entanglements changed during the stretching, whereas the number of entanglements remained similar. The number of entanglements and tie chains in model B in the semicrystalline polymers in this article are shown in Figure 7. The change in

Figure 7. Change in numbers of entanglements and tie chains during stretching in model A.

the number of tie chains during the stretching was 1%. The number of entanglements decreased slightly; however, the fluctuations were too large and the change was less than 5%. Therefore, the molecular structures of the tie chains and entanglements were unchanged during the stretching for strains of 0.0−0.6 in the semicrystalline polymers. Entanglements of polymers are efficiently analyzed by the following methods.58,59 Yeh et al.32 used Z-code and analyzed entanglements at the initial state. They also discussed the change in the number of entanglements. They categorized them as loop−loop, tail−loop, and tail−tail. The loop−loop G


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Figure 11. (a) Stress−strain curves for the lamellar structure of rigid and flexible models. (b) Stress−strain curves for the tie chains and entanglements of rigid and flexible models during the deformation and fracture processes of the lamellar structure.

Figure 9. (a) Stress−strain curves for the lamellar structure of models A (control), B (entanglement rich), and D (low-crystallinity, lowdensity entanglement). (b) Stress−strain curves for the tie chains and entanglements in models A, B, and D during the deformation and fracture processes of the lamellar structure. The strain rate is approximately 2.0 × 10−4a/τ.

the average monomer stresses were calculated. Figure 12a,b shows the monomer stresses in rigid and flexible chains,

Figure 10. Change in the crystallinity of models A (control), B (entanglement rich), and D (low-crystallinity, low-density entanglement).

evolution of the crystallinity of models A, B, and D. The crystallinity increases during the stretching. The initial differences decrease slightly in the deformation and fracture processes. The trends are similar in these three models. In a previous experiment,50 the crystallinity measured by X-ray scattering increased during stretching parallel to the crystal direction, which is consistent with the results in this article. The results revealed the stress transmission process of entanglement qualitatively. The entanglement was strongly affected by the bending potentials. Here, to determine the effect of bending potentials, more rigid or flexible polymers were added to the lamellar structure. The structure of model A was used as the initial structure. Then, the values of kabend, kbbend, and kdbend were set as twice and half of the original values. After the relaxation calculation, the stretching simulations were performed. Figure 11a shows the stress−strain curve of rigid and flexible models and model A (control). The rigidity of the polymer chains increased the Young modulus and fracture stress. Figure 11b shows the average monomer stress for tie chains and entanglements in rigid and flexible chains. The rigidity increased the work of the tie chains considerably. To reveal the mechanism, the monomer stress for tie chains was categorized. Both sides of the tie chains were classified as sustaining chains, loops, chain ends, and entanglements. Then,

Figure 12. Monomer stresses for the tie chains of (a) rigid and (b) flexible models. Both sides of the tie chains are classified as sustaining chains, loops, and chain ends.

respectively. The number of entanglements was too small to evaluate and thus is not shown. A clear difference appeared gradually above a strain of 0.45 in the rigid chains. The monomer stress was highest when both sides of the tie chains were sustaining chains. Loops were weaker than the sustaining chains. In flexible chains, the same trends were observed; however, the difference was smaller. Therefore, the rigidity increased the work of the sustaining chains in crystalline layers rather than that of the loops. Behavior of Tie Chains in Defect Model. Tie chains are the most important stress transmitters during the deformation and fracture processes of semicrystalline polymers. ExperH


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Macromolecules imental results suggest that tie chains can be divided into active and inactive states.60,61 In this subsection, the behavior of the tie chains in each amorphous layer is discussed. The amorphous layers were numbered from the bottom to the top (see the snapshot at a strain of 0.00 in Figure 13a). To

Figure 14. Void generation and growth during the deformation and fracture processes in the presence and absence of defects. The strain rate is approximately 2.0 × 10−4a/τ.

containing the defects grew larger and more rapidly than in the case of the structure without defects. To investigate the behavior of the tie chains in each layer, the average monomer stresses were calculated and the results are plotted in Figure 15. Since the layers could not be

Figure 13. Deformation and fracture processes of semicrystalline polymers containing defects. The cross-sectional snapshots are colored according to (a) orientational order and (b) monomer stress. Figure 15. Monomer stresses for the tie chains of layers 1, 2, 3, and 4 for (a) model A (without defects) and (b) model C (with defects). The strain rate is approximately 2.0 × 10−4a/τ.

determine the behavior of the tie chains, defects were introduced into layers 1 and 2, where the chain ends were concentrated. We previously reported that concentrated chain ends act as defects and decrease the fracture stress.41 Figure 13a shows the deformation and fracture processes of the lamellar structure containing defects. Voids were generated in the amorphous layers 1 and 2. Figure 13b shows the monomer stresses during stretching. The monomer stress in the amorphous layers varied greatly at a strain of 0.50, whereas the stress in the crystalline layers was almost uniform. Wide variation of the monomer stress was also observed in the regions of the crystalline layer where the orientational order was low (see the snapshot at a strain of 0.50 in Figure 13a). No large differences were observed between the amorphous layers. Then, at a strain of 0.75, the stress varied greatly around the voids. Finally, at a strain of 1.00, the stress was high at the sides of the voids. The amorphous layers 3 and 4 were not deformed, and the monomer stresses were similar. The surrounding crystalline layers were also conserved, and the monomer stresses did not change. Obvious changes were observed only in the amorphous layers 1 and 2, where the defects had been introduced. To elucidate the fracture process, the void generation and growth processes were examined, as shown in Figure 14. Void generation occurred at a strain of approximately 0.52. The voids in the lamellar structure

distinguished after fracture, the results are shown for strains from 0.0 to 0.6. For model A (control) without defects (Figure 15a), the monomer stresses of the tie chains of each layer remained almost the same during stretching. This indicates that all of the tie chains were functional and active. This result is consistent with those shown in Figure S3a, where the distribution of the monomer stresses of the tie chains was normal rather than bimodal. For model C with defects (Figure 15b), the monomer stresses of the tie chains in each layer were almost the same at strains from 0.0 to 0.5. Remarkably, small differences between the layers were clearly observed at strains from 0.5 to 0.6, despite the fluctuations that occurred during stretching. This difference indicates that the behavior of the tie chains changed just before and after void generation. Following void generation, vacant spaces are present, resulting in a lower density. Consequently, the stress of the remaining tie chains increases because the stress along the stretching direction on the xy plane remains equal. Stress transmitters such as tie chains and entanglements have been studied experimentally owing to their importance for increasing the toughness of semicrystalline polymers against stretching.3,4,6,14−16 However, the influence of tie chains and I


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effectiveness of large-scale simulation on the order of 106 monomers. Luo et al. revealed the effect of entanglements on the crystallization dynamics at the molecular level by coarsegrained molecular dynamics simulation.63 They showed that disentangled sequences are important for homogeneous nucleation, which also relates to the thermal history in the crystallization process. The relevance of entanglements and disentanglements at the molecular level is important in the crystallization process. In the fracture process of semicrystalline polymers, there is little change in the number of entanglements (see Figure 7). In future work, we intend to examine the effect of disentanglement on the fracture process. In our previous study, we proposed a method for constructing the lamellar structure and successfully elucidated its deformation and fracture processes.41 However, the influence of stress transmitters and void growth processes was not clarified. In this work, the number of entanglements in the model was successfully increased, the nature of the stress transmitters was determined based on the monomer stress, and the void generation and growth processes were elucidated.

entanglements on mechanical properties at the molecular level and the differences between them remain unclear. It was successfully confirmed that the tie chains and entanglements transmit the stress against the stretching as suggested by Humbert et al.15,16 Their difference appears during the fracture process but not in the deformation process. In the experiment,60 the yield stress decreased with a decrease in the tie molecule fraction. Then, the extrapolated yield stress became zero at the tie molecule fraction of 0.2. Therefore, the existence of inactive tie molecules was suggested. In this study, the distribution of monomer stress for the tie chains is normal in Figure S3 and the tie chains transmit the stress even if the defect exists, as shown in Figure 15b. Thus, inactive tie chains were not observed in the highly oriented lamellar structure adopted in this simulation. Since spherulites are stretched in many experiments and many different conditions are adopted in the experiments, it is difficult to compare them directly. However, the results in this study clearly suggest that activity of tie chains is still controversial. Three different models were stretched in this study. Remarkably, there are no obvious differences among them in the deformation process (see Figure 3 and the previous study41). The difference was observed only in the fracture process. The roles of the molecular structures such as tie chains, entanglements, chain ends, and so on become important in the fracture process. The crystallinity, crystal thickness, and amorphous thickness are almost same in these models. Therefore, it is thought that the parameters are important factors in the deformation processes. Comparison with Simulation Studies. Molecular simulation is a valuable technique for elucidating the role of stress transmitters in mechanical properties at the molecular level. Rutledge et al. used coarse-grained molecular dynamics simulations to study the influence of molecular structures such as tie chains and entanglements on mechanical properties.31−34 They showed snapshots of molecular structures during the stretching. It was found that tie chains and entanglements in an amorphous layer bridge crystalline layers, whereas loops and chain ends disengage. Then, entanglements are released earlier than tie chains at a higher strain. The results are consistent with the results shown in Figure 5 and Tables 5 and 6. In this study, qualitative relaxation processes were revealed by comparing with monomer stresses. Monasse et al. found that tie chains are pulled from the crystalline layer during stretching and demonstrated an increase in the stress upon stretching with an increasing number of tie chains via all-atomic molecular dynamics simulations.62 In this study, the difference between tie chains and entanglements was clearly revealed, where entanglements relax rapidly and play less of a role in void generation. Makke et al. prepared a lamellar structure consisting of rubbery and glassy states and studied its fracture process by coarse-grained molecular dynamics simulations.22 They reported that an increase in the numbers of tie chains in rubbery layers, which connect solid glassy layers, improves the resistance to void generation and damage in the glassy layers. The stress transmission between hard glassy layers was also revealed. Our work also reveals the importance of tie chains as stress transmitters. It is intriguing that the importance of tie chains is universal in the lamellar structure. Furthermore, we present a qualitative comparison of tie chains and entanglements. Sufficient chain length and simulation size are required to reveal the fracture process of semicrystalline polymers and the effect of entanglement, indicating the importance and

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CONCLUSIONS Determination of the nature of the stress transmitters at the molecular scale is essential for improving the toughness of semicrystalline polymers against stretching. The deformation and fracture processes of the lamellar structure of semicrystalline polymers were examined by focusing on the stresses at the molecular level using coarse-grained molecular dynamics simulations. First, an entanglement-rich model was successfully constructed. Then, the fracture processes occurring in the normal model, an entanglement-rich model, and a model containing defects were compared. It was found that tie chains and entanglements transmit the stress similarly upon stretching at low strain and then the tie chains play a greater role at void generation owing to the rapid relaxation of the entanglements. An increased number of entanglements leads to decreased crystallinity and fracture stress and delays void generation and growth. Interestingly, these processes are similar at low strain for the normal and entanglement-rich models and change only upon fracture. Next, the behavior of the tie chains was investigated using the defect model. In the normal model, the stresses of the tie chains were the same for all four amorphous layers. Remarkably, in the model containing defects, the stresses of the tie chains in the amorphous layers containing defects were found to be higher than those in the amorphous layers lacking defects after void generation. Therefore, this study has successfully elucidated the stress transmission processes of tie chains and entanglements and their differences at the molecular level during the deformation and fracture processes of semicrystalline polymers.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.9b00636. Figures of probability density of chain ends, probability density of orientational order, probability densities of monomer stresses, monomer stresses for the tie chains of model A, void generation for model D, and a table of number of monomers for tie chains with different chain types (PDF) J


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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Yuji Higuchi: 0000-0001-8759-3168 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by JSPS KAKENHI (Grant Number JP17K14534) and MEXT as “Exploratory Challenge on Post-K Computer” (Challenge of Basic Science−Exploring Extremes through Multi-Physics and Multi-Scale Simulations). The author thanks the Supercomputer Center at the Institute for Solid State Physics of the University of Tokyo for permitting the use of its facilities.

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Stress Transmitters at the Molecular Level in the Deformation and

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