Retardation on Blending in the Entangled Binary Blends of Linear

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

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Retardation on Blending in the Entangled Binary Blends of Linear Polyethylene: A Molecular Dynamics Simulation Study Lukun Feng,†,‡ Peiyuan Gao,†,‡ and Hongxia Guo*,†,‡ †

Beijing National Laboratory for Molecular Sciences, Joint Laboratory of Polymer Sciences and Materials, State Key Laboratory of Polymer Physics and Chemistry, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China ‡ University of Chinese Academy of Sciences, Beijing 100049, China

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S Supporting Information *

ABSTRACT: We perform atomistic molecular dynamics simulations on the binary blends of entangled polyethylene wherein short probes are dilute in long-chain matrixes as well as on their monodisperse counterparts. Compared to monodisperse systems, the relaxations of the end-to-end vector of short probes are retarded for the suppression of constraint release (CR) on blending, and the suppression degree becomes weaker with increasing of probe length. A parameter reflecting the CR strength is introduced, and its critical value from no CR at all to CR taking effect ranges from 0.071 to 0.46. By subdividing the overall motion into the lateral and longitudinal motions, we find both motions are more restricted in blends, and the time scale for the onset of these differences or of CR effects is smaller than the overall relaxation time. Additionally, retardation on blending behavior is further confirmed by the probability distribution of the chain entanglement lifetime.

1. INTRODUCTION It is well-known that the dynamics of the linear polymers can be described by the Rouse model and reptation model.1−5 The former works well if the molecular weight of polymers is small enough to avoid the presence of any topological constraints. However, the description of the Rouse model breaks down when the molecular weight becomes larger than some critical value. This failure results from entanglements between uncrossable polymer chains and can be qualitatively explained by the reptation model or tube model, which simplifies the complicated many-body effects into the mean-field concept of a single chain via presenting the topological constraints of surrounding chains as a tube and assuming the motion of the probe chain to be confined in the tube. Therefore, within the reptation model, the probe chain takes only one-dimensional reptation motion when the time scale is larger than the entanglement relaxation time (τe) because since then the probe chain begins to feel the confinement from the tube. We note that in the framework of the reptation model, τe, the tube diameter (dT) and the entanglement length (Ne) or the entanglement molecular weight (Me) are related to each other, namely, dT ∼ Ne1/2 and τe ∼ Ne2. Consequently, the reptation time (τd) that needs for the probe chain to forget its initial tube would be increased more remarkably than the prediction from the Rouse model, namely, τd ∼ N3 rather than N2, where N is the chain length. However, both experiments and molecular dynamics (MD) simulations have indicated that the dependence of τd on N in the monodisperse melts is τd ∼ N3.5±0.2.2,3 In addition, in comparison to monodisperse polymers, the binary blend polymers like those either of the © XXXX American Chemical Society

same chemical composition but different molecular lengths or of different chemistry, which are often encountered in practical applications, the dynamic properties are more complicated due to the interplay between different components, and deviations from the reptation model predictions were also noted.6−8 These discrepancies are not entirely surprising, as the original reptation model is a conceptually simple single-chain approach. Further refinements in the reptation model have been made through consideration of extra relaxation mechanisms, i.e., contour length fluctuations (CLF) and constraint release (CR). Among these, CLF has been well incorporated into the present tube model9,10 because of its single-chain mechanism, whereas the inclusion of CR is still challenging due to its manybody effect. Nevertheless, several models such as CR Rouse motion,11,12 double reptation13,14 (or the improved version: dual constraint model15,16), and dynamic tube dilation (DTD)17 have been proposed to account for the CR effects. Although these efforts have resulted in significant improvements of the reptation model as far as comparison with some experimental observations is concerned, the underlying microscopic foundations of CR mechanism are yet to be achieved. In fact, the CR is activated by the reptation of the surrounding chains and is an important relaxation mechanism especially for the case of the polydisperse systems. As addressed above, the fundamental molecular mechanisms are Received: January 9, 2019 Revised: April 17, 2019

A

DOI: 10.1021/acs.macromol.9b00047 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

not a full molecular model, which may affect its prediction and extension.27 Besides, Read et al.28,29 theoretically analyzed the influence of various CR environments (including monodisperse, binary, and ternary environments) on the relaxation of the end-to-end vector on the basis of the CR picture proposed by Viovy et al.12 and came to the same conclusion as Matsumiya et al.:24 CR does contribute to the relaxation of the end-to-end vector of the linear probe chains. Recently, Masubuchi et al.30 semiquantitatively reproduced the dielectric relaxation reported by Matsumiya et al.24 via performing the multichain slip-link simulations. The retardation of the relaxation of the end-to-end vector on blending for the suppression of CR is confirmed again. Although there is still some dispute over the effect of CR on the relaxation of the end-to-end vector, further systematic investigations on the relaxation mechanisms or the microscopic dynamics of linear chains in monodisperse bulks and binary blends will not only help to resolve the above issues but also be of critical importance to our understanding of the CR mechanism. However, a more thorough understanding requires detailed information at the molecular level; molecular simulations turn out to be an attractive alternative due to remarkable advances in computation power and algorithms throughout the past decades. As addressed above, within the context of the tube model, CR is activated by the motion of the surrounding chains and also activates local motion of the probe or primitive chain segments perpendicular to its contour or the initial primitive path. Direct examinations of these motions in monodisperse bulks and binary blends of linear chains are essential for a detailed understanding of the CR effects and developing improved microscopic theories for polymer dynamics. To explore the motion of the surrounding chains, Wang and Larson31 performed MD simulations on binary linear blends with strong CR events and proposed a method for the calculation of the lifetime of the surrounding chains staying in the tube region. Besides, a further refined method has been recently set up by Likhtman and Ponmurugan32 and Cao and Wang33 to record the lifetime of individual persistent contacts between mean paths, which also makes the investigation of CR process in different environment possible. With respect to the motion orthogonal to the chain or tube axis, relevant studies on the CR effect are rather limited. Nevertheless, some methods have been proposed to describe this motion. For example, Kremer and Grest34 extracted the lateral motion by subdividing the motion of the primitive chains (constructed in a continuously coarse-graining way) into two components along and perpendicular to the contour, respectively, and their results qualitatively support the reptation picture. Additionally, Zhou and Larson35 take the minimum distance from a bead (in instantaneous coordinates) to its corresponding primitive path (obtained from Z1-code and PPA36,37) as a perpendicular distance and perform direct tube sampling (DTS) to understand the lateral motion. As we have seen, the advances in computer technology and simulation algorithms considerably enhance our ability to access polymer dynamics by molecular simulations.18−21,31−35 Because the atom-level MD simulations can provide straightforward microscopic insights into structural, dynamical, and material properties of real systems, they have attracted more and more attention and been successfully implemented in the studies of microscopic dynamics of entangled polymers in recent years.18−21,38 Among various common commercial

still not well understood. There exist some unsettled questions, for example, whether or how CR influences the relaxation of the end-to-end vector, while the latter is usually used to determine the longest relaxation time. According to the Doi− Edwards theory,1 the CR gives a negligible impact on the longest relaxation time because CR represents the movement of the surrounding chains and results in conformational changes of the tube in the middle part. This notion has been employed in the works by Mavrantzas et al.,18−21 wherein the survival probability function of the primitive chain is calculated on the basis of topological state analysis and then compared with the relaxation of the end-to-end vector. Note that the latter is considered as a benchmark that is not affected by the CR.18−21 Nevertheless, in ref 19 where the long probe cis-1,4polybutadiene (PB) chains are blended with short PB matrix with the volume fraction being ∼40%, a further comparison between the relaxations of the end-to-end vector for the long probe chains in short matrixes with different chain length indicates that the shorter the matrix chain, the faster the probe chain. In addition, Glomann et al.22 proposed a unified description for the dielectric end-to-end vector and the viscoelastic stress relaxation of two type A homopolymer melts, i.e., polyisoprene (PI) and poly(butylene oxide) (PBO), in terms of the Likhtman−McLeish model10 and found that the relaxation of the end-to-end vector of these monodisperse melts is dominated by reptation and CLF and is not affected by CR. Later on, Pilyugina et al.23 conducted the simulations of the discrete slip-link model (DSM) for monodisperse and bidisperse PI and reached the same conclusions: an inappreciable CR effect on the relaxation of the end-to-end vector of the monodisperse PI. On the other hand, Matsumiya et al.24 measured the dielectric and viscoelastic data for the monodisperse linear PI with the number of entanglements ranging from 4 to 36 and for the binary PI/PI blends by diluting these PI in a much longer matrix background (i.e., the number of entanglements for the matrix is about 224). They discovered that both dielectric and viscoelastic relaxations are faster in the monodisperse bulks than in the blends because the CR mechanism is suppressed on blending. Particularly, they proposed a ratio of the viscoelastic relaxation time of the long component to the short one in the blend to describe the CR/ DTD effect; i.e., if the ratio is larger than 700, this CR/DTD effect is completely restrained, whereas if the ratio is smaller than 700, the CR/DTD effect enhances progressively with the decrement of the ratio. Additionally, this critical ratio is also true for the viscoelastic relaxation in the chemical similar PB/ PB blends reported by Liu et al.25 Furthermore, the retardation on blending was observed for the probe chain with smaller molecular weights26 (the number of entanglements is 1.6 and 2.8). In theory, Matsumiya et al.24 pointed out that the chain in the bulk shows reptation/CLF along the partially dilated tube that wriggles in the fully dilated tube, while in the blend the suppression of the partially dilated tube leads to the retardation on blending. Additionally, Watanabe et al.24,27 thought that the diameter of the partially dilated tube is determined by the process of CR-activated tension equilibration in both monodisperse and blend systems. On the basis of the molecular picture of CR-activated tension equilibration, Watanabe et al.27 formulated a blend model that can well describe the component relaxation times for both PI/PI blends and polystyrene blends with well and narrowly separated molecular weights. Nevertheless, the blend model employs some empirical equations during its formulation and thus is B


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Macromolecules

computational convenience. The angle bending interaction is described by a harmonic potential of the form

polymers, polyethylene (PE) is by far the most utilized and intensively studied one in industrial applications, experimental studies, and MD simulations. Investigations on the relaxational properties and microscopic dynamics of entangled PE in different CR environments could be instructive for both industrial processing and theoretical refinement. Thus, in present work, extensive atomistic MD simulations of monodisperse and bidisperse linear PE melts are performed, and a more detailed comparison between them in terms of the end-to-end vector relaxation, the primitive or probe chain lateral (perpendicular) and longitudinal motion, and the chain entanglement lifetime is made. To gain new insight into the microscopic mechanism of CR, we focus on the weak CR condition under which the short probe chains are diluted in a matrix of longer chains instead of strong CR environments such as diluted long probe chains in a matrix of short chains reported in recent studies.31,39 In addition to providing benchmark tests of the CR effect, the bidisperse melts of this kind possess potential utilization in the probe rheology, which serves as a simple method to probe the tube motion and has drawn a considerable amount of interest recently.25 In addition, the ratios of the component chain length in the blends, i.e., NL/NS where NL and NS are the number of carbon atoms per long and short chain, respectively, cover a relatively wide range from 10 to 3.3 to systemically investigate different degrees of the suppression of CR. Lastly, we note that the movement of topological constraints on a probe chain can lead to a dynamical variation of the tube. Recent atomic MD simulations show that in the binary mixtures of the entangled PB the entanglement statistics remains unaffected by the environment.19 To obtain a more comprehensive understanding of CR effects, it is interesting to analyze the static properties of the entanglement network in our studied monodisperse and bidisperse linear PE melts and check whether the same conclusion as above can be obtained in these weak CR environments. More interestingly, the perpendicular displacement of the chain is thought to be confined by the surrounding chains to an average distance, while recent coarsegrained MD simulations indicated that this distance does not correspond to the tube diameter.40 By extracting the entanglement network parameters like the length of an entanglement strand and the atom number in an entanglement strand from the static property analysis, localizing the value of perpendicular displacements in reptation regime and the value of longitudinal displacements at the entanglement time which signifies the onset of the tube constraints on their movement, and then comparing them with the tube diameter which is derived indirectly by fitting mean-squared displacement (MSD) data in different regimes to the scaling laws based on various assumptions of the tube model, we can reexamine the above issue with the real polymers. Relevant information would be very beneficial for the development of improved microscopic theories for the dynamic properties of entangled polymeric systems.

Ubending(θ ) =

1 kθ(θ − θ0)2 2

(1)

where kθ and θ0 are the force constant and equilibrium angle. Their values are set to 519.625 kJ/(mol rad2) and 114°, respectively. The torsional interaction is described by Utorsion(ϕ) = c1[1 + cos ϕ] + c 2[1 − cos(2ϕ)] + c3[1 + cos(3ϕ)]

(2)

where the force parameters are c1 = 2.952 kJ/mol, c2 = −0.567 kJ/mol, and c3 = 6.579 kJ/mol. The nonbonded interactions, including intermolecular interaction sites and intramolecular sites that are separated by more than three bonds, are described by the Lennard-Jones (LJ) 12−6 potential of the form ÄÅ ÉÑ ÅÅÅij σ yz12 ij σ yz6ÑÑÑ ij ij Å Ñ ULJ(rij) = 4ϵijÅÅÅÅjjjj zzzz − jjjj zzzz ÑÑÑÑ j rij z ÑÑ ÅÅj rij z k { ÑÑÖ ÅÅÇk { (3) where rij, εij, and σij are the distance, interaction strength, and collision diameter between interaction sites i and j. The values of ε and σ are 0.382 kJ/mol and 0.395 nm, respectively. A cutoff distance of 1.4 nm for the nonbonded interactions and a van der Waals tail correction are employed. All MD simulations are performed using the molecular dynamics package GROMACS42 with the time step fixed to 2 fs. All the initial configurations of the PE melts in present work are generated using the Monte Carlo algorithm, and then the melts are pre-equilibrated at 450 K and 1 atm using the Nosé− Hoover thermostat and Parrinello−Rahman barostat of a coupling time of 0.1 and 0.5 ps, respectively, for a time of 500 ns to fully equilibrate the melts. In addition to the end-to-end distance, we calculate the mean-squared internal distances, ⟨R2(n)⟩, which are defined as the mean-squared end-to-end distance over all segments of length n43 to guarantee the chains are equilibrated at all length scales. A typical result is shown in Figure S1 of the Supporting Information. Then the long-time MD simulations in the NVT ensemble at temperature of T = 450 K are performed on these equilibrated PE samples. The temperature is kept constant through the Nosé−Hoover thermostat with a coupling time of 0.1 ps. Note that sufficiently long trajectories, i.e., 1 μs, are accumulated for MD analyses, which cost more than 1 million core hours. To ensure good statistics, in our MD simulations each run is repeated independently at least two times, and then the results are averaged. Actually, because of the relatively large number of chains and long production run times when compared to the correlation time, results from these parallel runs are close to each other within hardly noticeable error bars, as typically illustrated in Figure S2. As mentioned before, we concentrate on the monodisperse PE melt and bidisperse PE mixtures with the short probe chains diluted in a matrix of longer chains. The volume fraction of the short chain is fixed at a low value of about 0.15 in the bidisperse PE mixtures. Following current studies on the dynamics of the entangled PE melts,38,44,45 the entanglement length and entanglement relaxation time obtained from fitting MSD data in different regimes to the scaling laws are chosen as a reference, namely, the entanglement length Ne = 56 carbon

2. SIMULATION DETAILS 2.1. MD Simulation. In this work, the widely used TraPPE united-atom force field41 is adopted. In the description of TraPPE-UA force field (FF), each methyl (CH3) and methylene (CH2) group is treated as a single interaction site. The FF includes the bonded (including bond stretching, bond angle bending, and torsional rotation) and nonbonded interactions. All the bond lengths are fixed at 0.154 nm for C


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Macromolecules atoms, the entanglement relaxation time τe = 2.3 ns, and the tube diameter dT = 3.3 nm. To conserve the weak CR condition, the long matrix chain length NL is taken to be 1200 carbon atoms (the number of the entanglement segments, Z, is ∼21) and the short probe chain length NS is increased from 120 to 360 carbon atoms (Z ranges from 2.1 to 6.4). Meanwhile, a series of monodisperse systems with the chain length ranging from 120 to 600 carbon atoms (Z ranges from 2.1 to 10.7) are conducted for comparisons as well. Note that the maximum chain length for the probe chains in bidisperse blends is shorter than that in monodisperse systems, and the length of the matrix chains is quite shorter when compared to the length in experiments due to the limited simulation time. In what follows, we refer to these simulated systems as b120, m120, b180, m180, and so on for conciseness. Here, the letter b or m represents bidisperse or monodisperse, respectively, and the numerals behind b or m denote the number of carbon atoms for the short PE chain in the bidisperse blend or for the PE chain in the monodisperse melt. All systems in this work are listed in Table 1. Because of the effect of free ends, the bulk

motion of the (primitive) chains or the (primitive) chain segments. For the former, the time autocorrelation function (ACF) for the end-to-end vector is calculated, which is defined as ACF(t ) =

monodisperse systems

code

NL

ML

NS

MS

ns

bl20 bl80 b240 b300 b360 ml20 ml80 m240 m300 m360 m480 m600

1200 1200 1200 1200 1200

21 25 34 40 51

120 180 240 300 360 120 180 240 300 360 480 600

40 30 30 30 30 100 90 90 90 90 90 90

0.160 0.153 0.150 0.158 0.150 1.00 1.00 1.00 1.00 1.00 1.00 1.00

βy ij i j j t yzz zzz ACF(t ) = expjjj−jjj z zz zzz jj j τ KWW { z k k {

ρ∞(T ) 1 + a0(T ) / N

(5)

where the τKWW is the characteristic time and β is the stretching exponent. The value of the correlation time, τc, is then calculated through τc = τKWW(Γ(1/β)/β)

(6)

For the microscopic motion, we calculate the chain contact lifetime, tlife, which could be regarded as the time that the probe chains stay closely enough to the matrix chains based on the microscopic definition of polymer entanglements proposed by Likhtman and Ponmurugan.32 In ref 32, the entanglement is defined as a persistent contact between mean paths of the chains, and the resulting entanglement survival probability of an entangled linear melt not only follows the stress relaxation rather well but also fits well with reptation theory. Here, the mean paths are calculated with the averaging time τave ≅ 2τe, and the cutoff distance determining a contact is chosen to be triple bond length. Note that similar methodology has been performed in the bead−spring model by Wang and Larson31 where the primitive chain is constructed by space averaging. Besides this, to further explore the microscopic motion of the probe or primitive chain segments, we subdivide the chain motion into the longitudinal and lateral motions based on the chain or tube axis by following the analysis of Kremer and Grest34 and define the lateral motion as

density is not fixed at a constant value for our studied monodisperse systems. It first increases relatively strongly with the increasing of chain length, for example, the mass density of m120 is 0.738 g/cm3 and m360 is 0.753 g/cm3, and then becomes saturated for larger chain length. This dependency can be described by a hyperbolic expression: 4 6 ρ (N , T ) =

(4)

where R(t) is the end-to-end vector of a chain at time t. To quantify the relaxation in the bidisperse and monodisperse systems, the ACF curves are fitted with the stretched exponential functions or KWW functions of

Table 1. System Parameters: Number of Carbon Atoms per Chain N, Number of Chains M, and Volume Fraction of the Short Chains in the Binary Blend ns (Subscripts L and S Denote Long and Short Chains, Respectively) binary blends

⟨R(t )R(0)⟩ ⟨R2(0)⟩

g⊥(i , t ) = ⟨min |ri(t + τ ) − rj(τ )|2 ⟩ j

(7)

where the ri(t) is the position of atom i at time t. The longitudinal motion is also measured by the difference between atomistic indexes i and j: g ( i , t ) = | i − j|

, where ρ∞(T) is the density of infinite

molecular weight at a given temperature T. The fitting parameter ρ∞(450 K) here is 0.760 g/cm3. We should note that the value of ρ∞ is close to 0.767 g/cm3 reported by Jeong et al.47 and 0.766 g/cm3 by Pearson et al.48 at practically the same conditions. In contrast, the density in the bidisperse systems hardly changes with the increase of short chain length by adding a small amount of the short chains and remains around 0.760 g/cm3, which is quite close to the fitting parameter of ρ∞(450 K) for monodisperse systems. 2.2. Observables. In this work, we study the relative effects of CR on the static and dynamic properties of the entangled PE monodisperse and bidisperse melts. For the static properties, we calculate the topological state of these melts by utilizing the Z1-code.49−52 As for the dynamics, we calculate the overall relaxation behavior as well as the microscopic

(8)

Note that we use both instantaneously atomistic and meanpath positions when evaluating g⊥(i,t) and g∥(i,t). g⊥(i,t) could serve as a measure of the perpendicular displacements of the primitive chain or tube constraints.

3. RESULTS AND DISCUSSION For a better understanding of the CR effects, the overall relaxation, namely, the relaxation of the end-to-end vector, the microscopic motions like the primitive or probe chain lateral and longitudinal displacement, and the chain entanglement lifetime as well as the static entanglement network structures in both monodisperse and bidisperse linear PE melts are calculated, and a detailed comparison between our results, other simulations, and experimental and theoretical studies in the literature is made. D


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Macromolecules 3.1. Chain Relaxation. Figure 1 presents the decay of ACF as a function of time in semilogarithmic coordinates. We see all

Figure 2. Dependence of the correlation time, τc, on the length of the probe chain, N, in the monodisperse systems (m) and the binary blends (b) denoted as black squares and red circles, respectively.

τKWW and τc, the values of τc of the monodisperse PE systems calculated here are close to those reported by Karayiannis and Mavrantzas54 (the consistency is shown in Figures S3 and S4) and are rather smaller than in the bidisperse blends, quantitatively indicating again the retardation on blending. Furthermore, from Figure 2 we can observe that for the chain length of N ≥ 180 the τc-vs-N curves of monodisperse systems could be described with a power law of the form τc ∝ N∝ with the exponent α ≈ 3.3 (= 3.25 ± 0.06). This dependence is close to the predication of reptation models.2 Note that the τc value of m120 is excluded during fitting with the power law because for the m120 the ACF curve can be well represented by both the Rouse model expression and the KWW function. The same also holds true for the b120 system. Additionally, recent work suggests the Rouse model even works for the PE chain length up to 156.55 On the other hand, for the bidisperse blends, because of the suppression of CR, besides the τc of the short probe chains longer than these in monodisperse ones, the resulting exponent of power law τc ∝ N∝ is α ≈ 3.1 (= 3.05 ± 0.18), obviously smaller than 3.3 of the monodisperse systems. This result strongly indicates that the relaxation time of the short probes in a long chain matrix approaches gradually that in monodisperse systems as the probe chain becomes longer. We hence conclude that the CR suppression from the matrix in bidisperse blends becomes weaker with decreasing the ratio of the chain length of the probe to the matrix. Experimentally, the multimode Maxwell functions are widely used to calculate the relaxation time of polymers.56 Thus, we also apply the multimode Maxwell functions to fit the ACF curves. As shown in Figure S7, the multimode Maxwell

Figure 1. Decay of ACF for the (probe) chain end-to-end vector in both monodisperse and bidisperse systems in semilogarithmic coordinates.

the ACF curves in both monodisperse and bidisperse systems eventually decay to zero, and the time taken increases with the (probe) chain length. It is also observed that the ACF curves of the probe chains in the bidisperse blends relax more slowly than those in the monodisperse systems, denoted as solid and dashed lines, respectively. The slower relaxation clearly indicates that the relaxation is retarded on blending in the weak CR environments. Similar results have been observed in experiments24,25,29,53 and simulations.29,31,53 The resulting characteristic relaxation time, τKWW, stretching exponent, β, and correlation time, τc, are listed in Table 2. The dependence of τc on the chain length in both monodisperse and bidisperse systems is shown in Figure 2. From Table 2, we first note that the values of β in monodisperse systems are in the range from 0.49 to 0.72, very close to the values characterizing the second relaxation process of the linear PE, wherein the ACF are fitted in a piecewise way.54 Additionally, in monodisperse and bidisperse samples the values of β show a definite dependence on the chain length: β is decreasing with the increase of (probe) chain length, indicating that the longer the (probe) chain length, the wider the relaxation process. Moreover, β is slightly larger in the monodisperse systems than in the binary blends, which further indicates that the longer matrix chain length, the wider the relaxation process. As for

Table 2. Values of the Characteristic Relaxation Time, τKWW, the Stretching Exponent, β, and the Corresponding Correlation Time, τc, for Both Monodisperse and Bidisperse PE Systems mono systems N 120 180 240 300 360 480 600

τKWW (ns) 4.026 11.51 25.44 52.39 84.88 181.7 368.8

± ± ± ± ± ± ±

0.040 0.14 0.28 0.75 1.27 2.6 5.9

β 0.725 0.677 0.636 0.578 0.583 0.510 0.491

± ± ± ± ± ± ±

blend systems τc (ns)

0.006 0.006 0.005 0.006 0.006 0.004 0.005

4.936 15.06 35.63 82.89 131.5 350.1 762.2

± ± ± ± ± ± ±

τKWW (ns)

0.063 0.23 0.52 1.64 2.7 7.4 18.1 E

5.407 14.92 29.47 64.77 95.18

± ± ± ± ±

0.080 0.23 0.47 1.26 1.71

β 0.706 0.650 0.622 0.582 0.542

± ± ± ± ±

τc (ns) 0.009 0.008 0.007 0.008 0.006

6.792 20.38 42.40 101.5 166.0

± ± ± ± ±

0.129 0.42 0.91 2.8 4.3


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Macromolecules function fitting has no qualitative effect on our discussions and conclusions from the KWW fitting. The dielectric relaxation which reflects the orientational memory of the end-to-end vector and the viscoelastic relaxation which reflects the orientation anisotropy of the subchains are often experimentally combined to get insight into the CR mechanism.22−24,26 However, a reliable estimate of the viscoelastic relaxation or the stress relaxation G(t) is more difficult than that of the endto-end relaxation due to the poor signal-to-noise ratio in the calculation of G(t)57 as stated in detail in the Supporting Information. Despite the infeasibility of calculating the stress relaxation directly, an indirect way resorting to the use of the stress-optical law is included in the Supporting Information, where based on Figure S14 qualitative agreements with results of the end-to-end vector relaxation are obtained. As mentioned above, Watanabe et al.24 used the viscoelastic relaxation time ratio of the long and short components in the blends of PI/PI to check whether the CR/DTD mechanism is ignorable or not; namely, the CR/DTD-free criterion for the [G] short component would meet as τ[G] long/τshort > 700. Interestingly, the criterion obtained from PI/PI works for the binary blends of PB/PB reported by Liu et al.25 as well. In our PE blends, the relaxation time of the long components is inaccessible for the limited simulation time, and thus a direct calculation of the relaxation time ratio to determine whether the above criterion is met or not is impossible. In the probe rheology where the short probe chains are diluted in the matrix of long chains of the same chemical species, it is not unreasonable to assume that the dynamics of the long matrix is only slightly affected by the short probes. Besides, if the reptation motion of the short probe chains is faster than the Rouse relaxation of the long matrix, the short chains are confined in the tube whose coherent moves are still not allowed. Thus, in this case, we believe that CR is completely suppressed. Oppositely, CR starts working and is partially suppressed. In consequence, we can introduce a new dimensionless parameter, rpr, as an alternative to characterize the relaxation mechanism of the short probe chains and rpr =

τd,S τR,L

= 3NS3/(NL 2Ne)

Table 3. Values of rpr in PI, PB, and PE Blends systems

ZL

ZS

rpr

PI

224 224 224 224 792 792 792 792 792 792 21.4 21.4 21.4 21.4 21.4

35.8 19.7 8.6 4.3 206 104 63 24.6 14.5 8.3 6.4 5.4 4.3 3.2 2.1

2.74 0.46 0.039 0.0047 41.8 5.38 1.20 0.071 0.015 0.0027 1.74 1.00 0.51 0.22 0.064

PB

PE

[G] namely τ[G] long/τshort > 700, indicating again that the chain length of the probe should be much shorter than that of the matrix for the full suppression of CR. Even more exciting is that the results for our PE blends as listed in Table 3 come to support the validity of rpr,c, although the chain length of the matrix (ZL = 21.4) is smaller than that used in experiments (ZL = 224 and 792 for PI and PB, respectively) due to the limited computing resource and the length differences between the probe and matrix chains are some limited. For example, the resulting values of rpr for b240 (ZS = 4.3), b300 (ZS = 5.4), and b360 (ZS = 6.4) are 0.51, 1.00, and 1.74, respectively, namely rpr > rpr,c. This is in reasonable accordance with the results of τc-vs-N wherein the slope of corresponding τc-vs-N curve of bidisperse systems is smaller than that of monodisperse systems, and thus the relaxation time of the short probes approaches gradually to that in monodisperse systems as the probe chain becomes longer. Again, we come to the same conclusion that CR effect is nonnegligible for our studied b240, b300, and b360, and the degree of CR suppression becomes weaker with the decreasing of differences between chain length of the probes and matrixes. Furthermore, for the short-chain blend of b180 (ZS = 3.2), rpr is 0.22, which is in the range of rpr,c. In particular, the smallest value of rpr is 0.064 for b120 (ZS = 2.1), clearly smaller than rpr,c. Altogether, our results provide further corroboration of the existence of rpr,c. However, it should be noted that the parameter rpr for b120 may be invalid because its number of entanglements is only 2, which corresponds to a critical number for the onset of entanglement, and the Rouse model may work better.26,55 Therefore, it is necessary to elucidate this issue of the correspondence between rpr < rpr,c and complete CR suppression further using coarse-grained models, wherein the limited chain length and time scale accessible to current atomistic MD simulations can be overcome to a major extent. As such, we can accurately locate rpr,c or ascertain its actual value. Besides, to make a further analysis according to the tube theory and to give a preliminary insight into the CR function, analogous to the analysis of Stephanou et al.18 and Likhtman and McLeish,10 it seems not unreasonable to assume the tube relaxation function, μ(t), would be equal to the normalized time autocorrelation function (ACF) for the end-to-end vector, namely, μ(t) = ACF(t), in the binary blends, since constraint release effects are weakened in our binary blends. While in the monodisperse systems, the CR function, R(t), is introduced to

(9)

which is the ratio of the reptation time of the short probe chains, τd,S ≅ 3τe(NS/Ne)3, to the Rouse time of the long matrixes, τR,L ≅ τe(NL/Ne)2. The resulting values of rpr for the PE blends presented here and for PI and PB in ref 24 are listed in Table 3. As shown in Table 3, for the CR/DTD-free cases in ref 24, the rpr values are 0.0047 and 0.039 for the PI blends with the molecular weight of short probe chains of MS = 21K (the corresponding entanglement number ZS = 4.3) and MS = 43K (ZS = 8.6) and 0.0027, 0.015, and 0.071 for the PB blends with MS = 14K (ZS = 8.3), 22K (ZS = 14.5), and 39K (ZS = 24.6), respectively. For others that dissatisfy the above CR/DTD-free criterion, rpr > 0.46. Thus, based on the CR/DTD-free criterion, a critical value, rpr,c, ranging from 0.071 to 0.46, is found. Apparently, rpr reflects the strength of the CR mechanism, i.e., rpr < rpr,c no CR at all and rpr > rpr,c CR taking effect. The critical value in this case is 0.35.34 3.3. Chain Entanglement Lifetime. Figure 6 presents the probability distribution functions, P(tife), of the chain entanglement lifetime, tlife, for both monodisperse and bidisperse systems at four typical (probe) chain lengths of N = 180, 240, 300, and 360. Broad tlife distributions are observed, indicating that the CR events have a wide time spectrum, which is basically in line with the spectrum reported by Wang and Larson.31 Generally, the CR events with short lifetime correspond to the restrictions appearing at the chain ends, and those with long lifetime are the restrictions appearing at the inner parts of the chains.5 For the lifetime shorter than τe (τe = 2.3 ns),38,44,45 their probability distributions in most of our studied samples (except for the shortest systems, namely, m120 and b120) show no clear differences between the J


Article

Macromolecules weaker for longer chain length from the view of microscopic motion. 3.4. Entanglement Structures. To have an insight into the CR effects on the static properties of the entanglement network, the probability distribution function of the number of entanglement segments, Z, and the primitive path contour length, LPP, of the (probe) chains in the binary blends and the monodisperse systems are first calculated, and the corresponding results are shown in Figure 7. We see that these distribution functions for the same chain length in different matrix environments almost fall on top of each other, though a slight discrepancy exists. Thus, the matrixes do not strongly affect the entangled topological structures. Meanwhile, we should note that Baig et al. have reported similar results in the binary mixtures of the entangled PB.19 Other local topological properties, including the length of an entanglement strand, des, and the number of atoms in an entanglement strand, Nes, are also calculated and shown in Figure 8. Both des and Nes in the binary blends are just slightly larger than those in the monodisperse counterparts. However, the differences are relatively small and decrease with the increasing of the length of chains. The local topological structures are not strongly influenced by the matrixes, too. We note that the values of des are slightly smaller than 2 nm, and thus (des/2)2 would be approximately equal to 1 nm2. (des/2)2 seems comparable to the value of g⊥(i,t) in the plateau-like region, which as mentioned above is generally smaller than 1 nm2 and is smaller than (dT/2)2. Besides, Nes also appears comparable to the values of g∥(i,t) at the crossover at τave which are in the range from 20 to 30 as indicated by Figure 4a. Additionally, the deviation of (des/2)2 from (dT/2)2 is in good agreement with recent simulation result that the tube persistence length estimated in the indirect ways which are based on the Gaussian assumption is larger than that measured directly from the actually topological constraints.47 Indeed, such deviation can be rationalized considering the nonGaussian features for PE model on the local scale as mentioned above. Lastly, as for Ne, we should note that several estimators have been proposed to improve the calculation of this basic parameter in tube model from the static PPA.51 Here, we follow the methods proposed by Hoy et al.51 and present the results in the Supporting Information. Similar to the above results of Z and LPP, Ne hardly changes with the matrix as shown in Figure S18. Besides, note that the value of Ne is different for different analysis methods for not clear reasons.38

blends than in the monodisperse systems at the time scale corresponding to the reptation regime. For g⊥(i,t), the times leaving the “expected plateau”, τl, in the blends are longer than those in the monodisperse systems and approach the latter with the increase of the chain length, which is similar to the behavior of ACF, indicating that CR starts to work at the time scale smaller than τc. The values of g⊥(i,t) at the “plateau” region are comparable with the squared half of the length of an entanglement strand directly measured via PPA. From the view of microscopic motion, the probability distribution functions of the chain entanglement lifetime show rather broad spectra and decrease quickly with the increase of the lifetime at the time scale beyond the entanglement time. The distribution functions are also retarded on blending, in qualitatively consistency with ACF and g⊥(i,t). Additionally, the CR function is extracted analogous to the analysis of Stephanou et al.18 and Likhtman and McLeish.10 The resulting R̂(t ) is larger for shorter chains, indicating that the shorter chain length, the stronger the CR effect, in consistence with results from ACF of the end-to-end vector and lifetime of entanglement. Lastly, to gain preliminary insight into the CR effects on the stress relaxation of our monodisperse and bidisperse PE systems, we resort to the use of the stress-optical law and calculate the normalized ACF of orientation relaxation within the same chain. As expected, the CR effects on the stress relaxation are qualitatively similar to the relaxation of the end-to-end vector.

4. CONCLUSION In this work, we conduct massive atomistic molecular dynamics simulations on the monodisperse melts and the binary blends of the linear entangled PE systems. Compared with the monodisperse systems, the relaxation of the end-toend vector of the short probes is retarded due to the suppression of the CR on blending and the degree of the retardation becomes weaker with the increase of the chain length of probes. We define a parameter, rpr, to characterize the relaxation mechanism of the short probes in the blends. On the basis of the CR/DTD-free criterion of PI/PI and PB/PB,24 we find a critical value, rpr,c (i.e., rpr < rpr,c no CR at all and rpr > rpr,c CR taking effect), ranging from 0.071 to 0.46, which is in general applicable for the PE/PE blends in present work. Furthermore, the lateral motion, g⊥(i,t), and longitudinal motion, g∥(i,t), of the short probes are more restricted in the

*Tel +86 10 82618124; fax +86 10 62559373; e-mail hxguo@ iccas.ac.cn.

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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.9b00047. Mean-squared internal distances, time decay of ACF for the independent runs of m120 and b120, consistent with the earlier PE simulations, multimode Maxwell functions’ fitting on the ACF curves, fluctuations in the viscoelastic relaxation functions and results of the orientation relaxation functions, MSD for mean path, Kremer−Grest model simulation, self-looping, and estimators of the entanglement length (PDF)

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AUTHOR INFORMATION

Corresponding Author

ORCID

Peiyuan Gao: 0000-0002-2906-6551 Hongxia Guo: 0000-0003-3655-6441 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the NSF of China (21790343, 21574142, and 21174154) and the National Key Research and Development Program of China (2016YFB1100800) and thank Prof. M. Kröger for providing the Z1-code and Prof. Z. Wang for helpful discussions. K


Article

Macromolecules

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Retardation on Blending in the Entangled Binary Blends of Linear

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