Can Polymer Chains Cross Each Other and Still Be Entangled

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Can Polymer Chains Cross Each Other and Still Be Entangled? Rakwoo Chang*,† and Arun Yethiraj‡ †

Department of Chemistry, Kwangwoon University, Seoul 01897, Republic of Korea Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, United States

‡

Macromolecules Downloaded from pubs.acs.org by WEBSTER UNIV on 02/26/19. For personal use only.

S Supporting Information *

ABSTRACT: The effect of chain crossing on polymer entanglement behavior is studied using molecular dynamics simulations of linear hard-sphere chains. The degree of chain crossing is controlled by changing the amplitude of bond vibrations while keeping the average bond length fixed. When the vibration amplitude is small, chain crossing is strictly prohibited, but for larger amplitudes the chains can cross. When chain crossing is strictly prohibited, the apparent scaling of the self-diffusion coefficient, D, and rotational relaxation time, τR, with degree of polymerization, N, is consistent with entangled behavior, and when chains cross freely Rouse dynamics is recovered. There is an intermediate regime, however, when chains are allowed to cross, but crossing events are rare. Under these conditions entanglement behavior is recovered. We therefore conclude that polymer chains can cross and still be entangled for finite length chains. There is no discernible change in the intermolecular static structure factor between the unentangled and entangled systems.

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INTRODUCTION Concentrated polymer solutions and melts display a fascinating combination of viscous and elastic behavior. For example, silly putty can be stretched without breaking but when rolled into a sphere can bounce like a rubber ball. This viscoelastic behavior is attributed to the entanglement of the polymers at a molecular level.1,2 Polymer molecules cannot cross each other, and this places restrictions on how they can move in concentrated solutions and melts. An appealing picture is that the topological constraints imposed by surrounding chains confine a given polymer into a tube inside which the polymer moves. In this paper we investigate, using computer simulations, the relationship between noncrossing topological constraints and the transport properties. The focus of this study is the following question: Is the constraint that two chains cannot cross necessary for polymer entanglement? In the reptation theory topological noncrossing constraints are replaced with a constraining tube, thus resulting in an anisotropic motion of the polymer.1−5 In an alternative mode-coupling theory (MCT) approach, however, these topological constraints are not relevant. In the polymeric MCT,6−11 the surrounding polymers are replaced with a matrix, and entanglements arise from temporal and spatial intramolecular correlations induced by the matrix. Both MCT and tube model make identical predictions for the scaling of transport properties such as the self-diffusion coefficient, D, or viscosity, η, with the degree of polymerization, N. It has been demonstrated that chain noncrossing is suf f icient for polymer entanglement.12,13 Shaffer investigated the effect of the bond crossing on the polymer entanglement using Monte Carlo simulations of a lattice polymer model. He used two classes of Monte Carlo moves, with chain crossing allowed in one but not in the other, and showed that when chain crossing © XXXX American Chemical Society

was allowed the entanglement regime disappeared. He therefore concluded that forbidding bond crossing is sufficient for entanglement behavior. It is usually assumed that chain noncrossing is the cause of entanglements14−16 although this has not been definitively established. Imposing the noncrossing of bonds in a microscopic theory is difficult because to dynamically describe the crossing (or noncrossing) of two bonds on two different segments, one requires the relative positions of the two segments before and after a possible crossing event. Because each segment is described by two particles, one would require (at least) a fourpoint dynamic correlation function.11 Fashioning approximations for such a correlation function is a daunting task. In fact, MCT decouples such correlations into products of two-point correlation functions. Understanding whether chain noncrossing is a necessary condition for entanglements is therefore of fundamental interest. Szamel developed a dynamical mean-field theory for a model of infinitely thin nonrotating rods and showed that the dynamics are solely determined by topological constraints.17,18 Being infinitely thin, the thermodynamic properties are in the ideal gas limit. His theory enforces both polymer connectivity and noncrossing exactly at the two-rod collision level and gives reptation scaling behavior. It was later shown by Miyazaki and Yethiraj11 that the theory can be related to MCT with a particular approximation to the four-point correlation function, which goes beyond the Gaussian approximation in the conventional MCT. The theory has been extended to a selfconsistent theory for both rigid and flexible polymer chain Received: October 25, 2018 Revised: February 15, 2019

A

DOI: 10.1021/acs.macromol.8b02299 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules systems by Sussman and Schweizer.19−22 Recently, Dell and Schweizer developed another segment-scale, force-based theory for the breakdown of the unentangled Rouse model, where they showed that the chain uncrossability condition is essential for the entanglement dynamics.23 Although our focus is on the fundamental concepts of polymer entanglement, there is the potential of direct comparison to experiment. Kilfoil recently performed brightfield microscopy experiments to directly track micrometersized particles, which couple to the DNA fluctuating movement, and reported their results as a poster in a conference.24 In their report, they observed the dynamic transition from entangled to Rouse behavior of λ-DNA networks in the presence of the topoisomerase II motor and ATP. A detailed experimental analysis with which we can compare our simulations has not, however, appeared yet. There have been many computer simulation studies for polymer entanglement by implementing various bond-noncrossing constraints including bead−spring, 25−29 slipspring,30−36 and slip-link models.37−44 However, to the best of our knowledge, the correlation between the polymer dynamics and the bond-noncrossibility condition has never been investigated systematically. We investigate the effect of noncrossing on polymer dynamics using a hard-sphere chain model of polymers and find that even when chains can (and do) cross, one recovers apparent entangled behavior if the crossing of chains is sufficiently rare. We therefore conclude that strict chain noncrossing is not necessary for the polymer melt to display entanglement behavior.

growth and equilibration method47 and equilibrated using DMD. We study δB = 0.05, 0.4, 0.45, 0.5, 0.6, 0.7, 0.8, and 0.9. We define the system length, mass, and time by setting σ = 1, m = 1, and τ = mσ 2/kBT = 1. Details regarding the simulation method can be found elsewhere.48,49 The intermolecular pair correlation function g(r) and the static structure factor S(k) are defined as50 Np

g (r ) =

Np

N

1 ⟨∑ ∑ ∑ 2πr 2ρNNp k < l l = 1 α = 1

N

∑ δ(r − rkα ,lβ)⟩ β=1

(1)

and S( k ) =

1 NNp

Np

Np

N

N

∑∑∑ ∑ k=1 l=1 α=1 β=1

sin(krkα , lβ) krkα , lβ

(2)

where k is the wave vector, ρ = NNp/V, and rkα,lβ is the distance between the αth monomer in the kth chain and the βth monomer in the lth chain. The translational self-diffusion coefficient, D, and the rotational relaxation time, τR, are the key dynamic properties of polymers in melts. The translation self-diffusion coefficient is given by the Einstein relation: D = lim

t →∞

⟨(R com(t ) − R com(0))2 ⟩ 6t

(3)

where Rcom(t) is the center-of-mass coordinate of each polymer chain at time t, and the rotational relaxation time is obtained by fitting the time autocorrelation function U(t) of the normalized end-to-end distance vector u(t) using an exponential function y0 exp(−t/τR):

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METHODS In this study, we have employed a primitive polymer model, where the polymer molecules are modeled as chains of N hard spheres of diameter σ, where N is the number of monomers (or degree of polymerization) in a chain. The bond length between adjacent beads is allowed to fluctuate freely between (1 − δB)σ and (1 + δB)σ.45 In this primitive model the minimum bond length with which the bond crossing is possible is √2σ or δB = 0.414 (see Figure 1). The detection scheme for bond-crossing events is also provided in the Supporting Information.

U (t ) = ⟨u(t ) ·u(0)⟩

(4)

where u (t ) =

■

r1(t ) − rN (t ) |r1(t ) − rN (t )|

RESULTS AND DISCUSSION The effect of δB on the chain conformation can be examined from the average bond length lB and the radius of gyration R g ≡ ⟨R g 2⟩ , which are seen in Figure 2. The bond length

Figure 1. Cartoon where the bond crossing event can occur in the primitive model. The minimum bond length with which the bond crossing is possible is √2σ or δB = 0.414.

Simulations are performed using discontinuous molecular dynamics (DMD)46 where trajectories are evolved as a sequence of elastic collisions. The simulation cell is a cube of side length L which contains Np chains, with Np ≥ 32 for most cases (Np = 16 for N = 512). We investigate two volume fractions, ϕ ≡ NNpπσ3/6L3, of ϕ = 0.3 and 0.4 for N ranging from 8 to 512. Initial configurations are generated using the

Figure 2. Average bond length lB (left, blue color) and the radius of gyration R g ≡ ⟨R g 2⟩ (right, red color) as a function of δB for N = 256 and ϕ = 0.4. B

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Figure 3. (a) Pair correlation function g(r) and (b) static structure factor, S(k), divided by the single chain structure factor, ω(k), for N = 256, ϕ = 0.4, and various values of δB.

Figure 4. (a) Self-diffusion coefficient (multiplied by N), D, and (b) rotational relaxation time (divided by N2), τR, as a function of N for ϕ = 0.4.

0.4, where the bond crossing is strictly prohibited, N × D is almost flat up to N = 32 but starts decreasing from N = 64 with the slope approaching 1, the dynamic scaling of which corresponds to D ∼ N−2. However, for systems with δB ≥ 0.6, N × D stays constant up to N = 512, which implies the dynamic scaling is D ∼ N−1. Interestingly, for δB = 0.45, where bond crossing is possible, the N-dependence of of D is similar to that with δB = 0.4 (no bond crossing is allowed). As will be discussed below, in this system most of the bond-crossing events occur between intrachain bonds, but the interchain bond-crossing event is extremely rare. On the other hand, the system with δB = 0.5 shows an intermediate dynamic scaling behavior: D ∼ N−1−α, where 0 < α < 1. The crossover between entangled and Rouse dynamics is also seen in the rotational dynamics as shown in Figure 4b. Consistent with the translational dynamics, the system with δB = 0.4 shows entanglement behavior (τR ∼ N3) and those with δB ≥ 0.6 show Rouse behavior (τR ∼ N2). The system with δB = 0.45 follows entanglement behavior. What is interesting, though, is that its rotational dynamics is somewhat faster than the system with δB = 0.40. It implies that the intrachain bond crossing event has a larger impact on the rotational dynamics than on the translational dynamics. For δB = 0.5 an intermediate regime is observed for τR, similar to what was seen for D. It may be argued that the entanglement length Ne is dependent on δB, so that Ne of chains with δB ≥ 0.5 is beyond the degree of polymerization investigated in this study. Therefore, we plotted results from both us and Smith et al.53

remains constant (∼1) up to δB = 0.4 but starts increasing as δB gets larger. This results from the fact that the bond crossing causes the bonded monomers to be pushed away once the event is successful. The bond-length increase is reflected in the chain size as Figure 2 (right) shows about 6% increase in Rg going from δB = 0.4 to δB = 0.9. However, this increase is less drastic than that of the bond length itself (∼22% increase), and therefore, the overall chain conformation is less sensitive to δB. The static structure of the melt is not sensitive to δB. Figures 3a and 3b depict the intermolecular pair correlation function, g(r), and the ratio of the static structure factor to the single chain structure factor for N = 256, ϕ = 0.4, and various values of δB. The overall structure is not affected significantly by the change of δB. The reason for the slightly enhanced peak at r = σ with larger δB is that the accessible area of each polymer chain by other chains becomes larger as δB is increased or the bond length gets longer. The peak and the position of S(k)/ ω(k) are insensitive to δB. The dynamics of polymer melts show two distinct scaling regimes depending on the degree of polymerization N. At short polymer chains the dynamics is suitably described by the Rouse model: D ∼ N−1 and τR ∼ N2. When the chain length is greater than the entanglement length, the scaling exponents change: D ∼ N−2.3±0.1 and τR ∼ N3.4.1,2,51,52 The simulations show a transition from entangled to Rouse behavior as δB is increased. Figures 4a and 4b show the translational diffusion coefficient, D, and the rotational relaxation time, τR, as a function of the degree of polymerization, N, for a volume fraction 0.4. (We plot N × D and τR/ N2 to emphasize deviations from Rouse behavior.) For δB = C

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Macromolecules for various conditions with different volume fractions and bond fluctuations in Figure 5. The model and simulation setup of

Figure 6. Mean-square displacement of central monomers and center of mass for N = 256, ϕ = 0.4, and δB = 0.4, 0.45, 0.5, and 0.6. Figure 5. Data were redrawn from Figure 4, where each data set has been shifted to match a point at N = 64 to 1.0. Data from Smith et al. were taken from ref 53.

= 0.5. It implies that the anisotropic motion of the chains parallel and perpendicular to the primitive path is partially released by the bond-crossing event. The g(t) for δB = 0.45 is indistinguishable from that for δB = 0.40. This is further evidence that entangled behavior can be seen even when chain crossing is allowed. It is also seen in the figure that the meansquare displacement of the center of masses of a polymer chain shows different scaling behavior depending on δB, consistent with g(t). The entanglement behavior can be understood by considering the bond crossing time, τBC, which is monitored in the simulations. Figure 7a depicts the average intermolecular and intramolecular bond crossing times (time of simulation divided by the number of bond-crossing events per chain) as a function of δB. As expected, τBC is very short for large values of δB and increases sharply as δB is decreased; note that τBC → ∞ as δB → √2 − 1. For δB = 0.5, τBC ≈ 600τ, and the intermolecular bond-crossing events occur on the time scale comparable to the tube confinement time, which is the order of 103τ. Clearly, chains with the bond-crossing time sufficiently longer than the tube confinement time show entanglement behavior, even though the bond crossing is allowed. For δB = 0.45 intermolecular bond crossing events are not observed over the time scale of the simulations, but intramolecular bond crossing events are seen, which explains the difference in behavior of the translational diffusion and the rotational relaxation time. Also depicted in Figure 7b is the trend of bonding crossing events for ϕ = 0.3 and δB = 0.9. As the degree of polymerization, N, increases, the ratio of the intramolecular (Nintra) to total (Ncross) bond crossing events also increases at first up to N = 128 but saturates to around 80% for N ≥ 128. It is also noted that most of the intramolecular bond crossing events come from those between neighboring bonds, which are defined as two bonds separated by one bond in the same chain. For example, in the case of N = 512, Nintra/Ncross and Nneighbor/ Nintra are 0.74 and 0.81, respectively. In other words, only 14% of the total bond crossing events are attributed to the nonneighboring intramolecular bond crossing events, which is even less than the intermolecular bond crossing events (26%). Hence, the shorter bond crossing time of intramolecular bond crossing events compared to that of intermolecular events is mainly due to the bond crossing events of neighboring bonds. The reason why the neighboring bond pairs are dominant in bond crossing can be explained by their preferential relative orientation, rather than the contact probability. As shown in the Supporting Information, the neighboring bond pairs prefer to have the relative orientation of θ = 96°−117°, which is close

Smith et al. is the same as our study except that they fixed δB = 0.1 in their study. In the figure, each result is shifted to set the value at N = 64 to 1.0. As shown in the figure, regardless of the volume fraction (ϕ) and the amplitude of bond vibration (δB), the curves show the same curves show the same dynamic behavior as a function behavior as a function of the degree of polymerization. It implies that the entanglement length is somewhat insensitive to ϕ. According to Smith et al., the entanglement length Ne for their systems is around 35 for ϕ = 0.4. However, our data show no signature of entanglement for chains with δB ≥ 0.6 almost up to 10 times as large as the entanglement length for chains with small δB. Although we cannot completely exclude the possibility that the entanglement length for chains with high δB is larger than degree of polymerization investigated in this study, it is clear that the absence of the entanglement for high δB is not directly related to the volume fraction of the system at least in the range of ϕ = 0.3−0.4. Entanglement effects are manifested in the different regimes in the atomic mean-square displacement, g(t), which is defined as g (t ) = ⟨(rcm(t ) − rcm(0))2 ⟩

(5)

where rcm(t) is the coordinate of the central monomers in a polymer at time t. Within the framework of the tube model,3,4,54−56 at short times the atomic motion is independent of the constraints of the tube and is therefore governed by Rouse dynamics with g(t) ∼ t1/2. Once the atomic displacement is greater than the tube diameter, their motion perpendicular to the primitive path is restricted but the motion along the primitive path is not, and on this time scale g(t) ∼ t1/4. At longer times, the chain diffusion along the contour of a rigid tube follows g(t) ∼ t1/2. Finally, at long times g(t) ∼ t. These scaling regimes have been confirmed in previous simulations27,53 and are also observed in this study for δB ≤ 0.4. Figure 6 depicts g(t) for various values of δB. For δB = 0.4 all of the above regimes are observed, but for δB ≥ 0.6, the t1/4 scaling regime is absent, which indicates that the motion of individual chains is isotropic on these intermediate intermediate time scales. For δB = 0.5 the behavior of g(t) is similar to that of δB = 0.4 for times up to t ∼ 2000τ, and there are differences at longer times. This is consistent with the intermediate dynamic scaling observed in both translational and rotational diffusions for the system of δB D

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Figure 7. (a) Intra- and intermolecular molecular bond-crossing times per polymer as a function of δB for ϕ = 0.4 and N = 256 and (b) the ratios of intramolecular (Nintra) to total (Ncross) (left) and neighboring (Nneighbor) to intramolecular bond-crossing events (right) as a function of N for ϕ = 0.3 and δB = 0.9. Neighboring bonds are defined as two bonds separated by one bond in the same chain.

to 90°, the best orientation for bond crossing. On the other hand, other bond pairs including intermolecular bond pairs have no preferential orientation. Figure 8 also displays the intermolecular bond crossing rate per bond, v, for ϕ = 0.4 and δB = 0.5 and 0.9. For both cases, v

Figure 9. Comparison between the intermolecular bond crossing rate, v, and the bond crossing probability, PBC, for ϕ = 0.4 and N = 256. The dotted line corresponds to the power-law fit of v.

course, that static correlations are not important because these must be the origin of the anisotropic tube in the first place.

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Figure 8. Intermolecular bond-crossing rate per bond, v, as a function of N for ϕ = 0.4 and δB = 0.5 and 0.9.

SUMMARY AND CONCLUSIONS In summary, we have investigated the effect of bond crossing on the entanglement behavior of polymer melts using an efficient discrete molecular dynamics simulation technique where bond crossing is controlled by the nature of strings between neighboring beads. The system shows entangled behavior even when bond crossing is allowed as long as the intermolecular bond crossing events occur on a time scale longer than the tube confinement time. We can therefore conclude that the strict noncrossing of bonds is a sufficient but not a necessary condition for the entanglement of polymer melts for chains of finite length. The simulations also suggest that structure does not play as a significant role in polymer dynamics as is often thought, for example, in the application of mode-coupling techniques. As a caveat, we have studied a very primitive model of polymers, and it would be of interest to see if the conclusions continue to be valid for more realistic polymer models. As a final remark, although the purpose of this study is mainly the validation of concepts in polymer entanglement, our observation can be examined by some experiments. For example, as already mentioned in the Introduction, Kilfoil found the ATP-activated topoisomerase II motor can release the topological constraint of the lambdaDNA network and observed the dynamic transition from entanglement to Rouse dynamics by varying the ATP concentration. By use of this technique, the correlation between the ATP concentration (or the bond-crossing rate)

is independent of the degree of polymerization. This implies that the intermolecular bond crossing is mainly affected by local segmental dynamics rather than global polymer conformation and dynamics. We have also developed a simple theory for bond-crossing probability, PBC, the details of which are given in the Supporting Information. The comparison between the intermolecular bond crossing rate, v, and the bond crossing probability, PBC, is displayed in Figure 9 for ϕ = 0.4 and N = 256. PBC when multiplied by 0.32 shows a good agreement with v except for δB = 0.5. As the power fit indicates, the intermolecular bond crossing rate steeply decreases as δB → √2 − 1. The simulations support the classical picture of entanglements originating from topological constraints. Although a strict noncrossing of bonds is not necessary for entangled behavior, signatures of entanglement are observed when the bond-crossing time is larger than the tube confinement time. On the other hand, the transition from entangled to unentangled behavior occurs for δB going from 0.4 to 0.6, a range over which the structure factor is essentially unchanged. This questions the notion that entanglement behavior arises from radially averaged static correlations between the polymer chain and the surrounding matrix. This does not mean, of E

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and the entanglement behavior can be experimentally determined.

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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.8b02299. Detection scheme for bond crossing, density and angle distribution of bond pairs, and prediction of bond crossing probability (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(R.C.) E-mail: [email protected]; Ph +82 2 940-5243; Fax +82 2 942-0108. ORCID

Rakwoo Chang: 0000-0002-5900-3121 Arun Yethiraj: 0000-0002-8579-449X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon the work supported by the National Research Foundation of Korea (NRF) (Nos. 2015R1D1A1A01058045 and 2018R1D1A1B05050938), Korea Institute of Science and Technology Information (No. KSC-2017-C3-0003), and Kwangwoon research grant (2017).

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REFERENCES

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DOI: 10.1021/acs.macromol.8b02299 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.8b02299 Macromolecules XXXX, XXX, XXX−XXX