Analysis of Slow Modes in Ring Polymers: Threading of Rings

Jun 3, 2016 - For a linear chain with two free ends, the PP reptates backward and forward along the tube axis. In a branched polymer, each arm has one...
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Analysis of Slow Modes in Ring Polymers: Threading of Rings Controls Long-Time Relaxation Dimitrios G. Tsalikis,† Vlasis G. Mavrantzas,*,†,‡ and Dimitris Vlassopoulos§,∥ †

Department of Chemical Engineering, University of Patras & FORTH/ICE-HT, Patras GR 26504, Greece Particle Technology Laboratory, Department of Mechanical and Process Engineering, ETH-Z, CH-8092 Zurich, Switzerland § FORTH, Institute for Electronic Structure and Laser, Heraklion GR 71110, Greece ∥ Department of Materials Science & Technology, University of Crete, Heraklion GR 71003, Greece ‡

S Supporting Information *

ABSTRACT: Atomistic configurations of pure, precisely monodisperse ring poly(ethylene oxide) (PEO) melts accumulated in the course of very long molecular dynamics (MD) simulations at T = 413 K and P = 1 atm have been subjected to a detailed geometric analysis involving three steps (reduction to ensembles of coarse-grained paths, triangulation of the resulting three-dimensional polygons, and analysis of interpenetrations using vector calculus) in order to locate ring−ring threading events and quantify their strength and survival times. A variety of threading situations have been identified corresponding to single and multiple penetrations. The percentage of inter-ring threadings that correspond to full penetrations has also been quantified. By repeating the analysis for several PEO melts, the dependence of the degree of inter-ring threading on molecular weight (MW) has been obtained. Simulations with MWs up to 10 times the reported entanglement molecular weight (Me) of linear PEO have revealed several multiple threading events in all systems, with their relative number increasing with increasing MW. Our analysis indicates the existence of strong ring−ring topological interactions, which can last up to several times the corresponding average orientational ring polymer relaxation time. We show that these ring−ring interactions, together with the additional ring−linear threadings due to the remaining linear impurities, can explain the appearance of slow relaxation modes observed experimentally in entangled rings. 38k),11,12 together with best fits to the data using known theoretical models. For the rings, the data have been fitted with11

D

ue to their mutual uncrossability, topological constraints generated in high-molar-mass polymers dominate their dynamical and rheological properties. These topological interactions, known as entanglements, are well understood and successfully described by the tube model, an effective medium theory built on the concept of primitive path (PP) capable of addressing not only the simpler case of linear polymers but also the more complex one of branched chains.1−4 The tube model postulates that due to entanglements the lateral motion of the chain is restricted within a curvilinear tube-like region encompassing the chain. For a linear chain with two free ends, the PP reptates backward and forward along the tube axis. In a branched polymer, each arm has one free end, which retracts attempting to reach the other, immobile end. Reptation and retraction apply strictly to macromolecules with chain ends. Ring polymers lack chain ends; they lie therefore outside the conceptual framework of the tube model. As a result, understanding chain relaxation and dynamics in ring polymers requires a new mesoscopic view and new phenomenological models.5−10 A particularly puzzling issue is the appearance of long relaxation modes in the stress relaxation modulus, G(t). In Figure 1 we report the G(t) curves for monodisperse linear and pure ring polystyrene (PS) melts of molar mass 200 kg/mol (PS-200k) and of a ring 1,4polyisoprene (PI) melt of molar mass 38 kg/mol (PI© XXXX American Chemical Society

⎛ t ⎞ ⎛ t ⎞−2/5 ⎟⎟ G(t ) = GN0 ⎜ ⎟ exp⎜⎜ − ⎝ τe ⎠ ⎝ τring ⎠

(1)

while for the linear melt we used1 ⎛ τ ⎞1/2 G(t ) = GN0 ⎜ e ⎟ + GN0 ⎝t⎠

∑ p :odd

⎛ ⎛ 2 ⎞β⎞ 8 ⎜−⎜ p t ⎟ ⎟ exp ⎜ ⎝τ ⎠ ⎟ p2 π 2 D ⎠ ⎝

(2)

involving the sum of Rouse and reptation contributions, the latter in the form of a stretched exponential function (β is the stretch exponent), in order to account for polydispersity in the linear sample (for a perfectly monodisperse melt, β is strictly equal to 1, but for a polydisperse sample, β is less than 1). In eqs 1 and 2, G0N is the plateau modulus, τe the entanglement time, and τring the longest ring orientational relaxation time (τring typically corresponds to the time needed for the autocorrelation function of the unit vector directed along the Received: March 31, 2016 Accepted: May 31, 2016

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DOI: 10.1021/acsmacrolett.6b00259 ACS Macro Lett. 2016, 5, 755−760

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down ring dynamics has remained unexplored. Yet, it is of great importance because of its significant implications for the dynamics of entangled polymers but also for biophysics problems such as organization of chromatin in the cell nucleus or for applications ranging from DNA separation to enzymology and from protein structure stabilization to drug delivery.16−20 This is exactly the challenge we address in this Letter. We present a detailed, microscopic analysis of topological interactions in melts of pure ring polymers through a geometric algorithm, which provides direct evidence for significant ring−ring threading that can survive for times that extend up to several times the ring orientational relaxation time τring. We find that a non-negligible fraction of these threadings participate in multiple interpenetrations, which implies a large impact on ring dynamics. Our algorithm generalizes a previous study on a simpler problem (ring threading by linear chains13) and involves three steps. First, atomistic MD simulations are performed in the isothermal−isobaric (NchnPT) statistical ensemble (Nch denotes the total number of chains, n the total number of atoms, P the pressure, and T the temperature) in a 3d-periodic box to generate a large ensemble of fully equilibrated melt configurations of the ring polymer under study, which is PEO. Second, accumulated trajectories from the MD simulations are reduced to ensembles of shortest disconnected multiple paths, known as primitive paths (PPs) for linear polymers, by applying the CReTA21 algorithm (Contour Reduction Topological Analysis). At the end of the topological analysis with CReTA, the ensemble of atomistic chains has been mapped to an ensemble of closed geometric paths having the shape of 3d polygons. In the third and final step of our methodology, we use geometric operations on the generated ensemble of 3d polygons based on vector calculus to identify and quantify situations where a ring threads another ring. A key element of our analysis is the process of triangulation:22 each ring molecule (e.g., the yellow and green rings in Figure 2) is regarded as a 3d object having the shape of an irregular, solid polygon which spans a surface in 3d space that can be approximated by the sum of surfaces of all successive triangular edges making up the polygon. In this respect, our approach,

Figure 1. Stress relaxation modulus, G(t), for a linear PS melt, a ring PS melt, and a ring 1,4-PI melt. Linear PS-200k (black square); Ring PS-200k (black circle); Ring PI-38k (black triangle). Red and blue lines are best fits (see text). In order for the data to collapse at short times, they are reported at isofrictional conditions and shifted vertically by the ratio of Me of PI and PS.

ring diameter to drop to zero). In fitting the experimental data, we fixed the value of G0N and τe for PS to known values from the literature, and the same for PI.11,12 Their values as well as the best-fit ones for the rest of the model parameters are reported in Table 1. As far as the power-law exponent of −2/5 appearing Table 1. Numerical Values (Experimental or Best-Fit) of All Parameters Appearing in Equations 1−3 for the Stress Relaxation Modulus of the Three Experimental Samples ring PI-38k G0N (Pa) τe (s) τring (s) β GN,RR (Pa) GN,RL (Pa) τd (s) τd,RR (s) τd,RL (s)

5.8 3.4 1.6 1 3.6 9.4 8.4 4.2

× 105 × 10−5 × 10−3 × 104 × 103 × 10−3 × 10−2

ring PS-200k 2.0 2.5 2.6 1 6.0 9.0 8.2 3.0

× 105 × 10−4 × 10−1 × 103 × 102 × 10−1

rinear PS-200k 2.0 × 105 1.6 × 10−4 0.65 1.0 -

in eq 1 is concerned, this arises from the lattice animal model fractal exponent (df = 4) which suggests that it is constant and independent of chain length.5,11 A more recent scaling model of self-similar conformations and dynamics for nonconcatenated entangled ring polymers10 results in a slightly different exponent of −3/7. Hence, the use of the value −2/5 in eq 1 here can be considered as a convenient ad hoc assumption, which happens to provide a satisfactory description of the experimental data. This however does not affect the results and message of this work. For the two ring polymers, G(t) exhibits slow relaxation modes at very long times which cannot be captured by the theoretical model (eq 1). Remaining impurities not resolved by advanced chromatography (estimated to be at fractions below 0.1%)11,12 are one factor that is expected to contribute to these slow modes. A second one could be topological constraints in the ring polymer melt resulting from ring−ring interpenetration. Whereas the threading of rings by linear chains has been established,11−14 the mutual interpenetration of rings has been observed in simulations,15 but its potential role in slowing

Figure 2. Indicative snapshot from the geometric analysis of the PEO10k melt, demonstrating the threading of a ring PEO molecule (represented by the sequence of yellow beads and strands) by another ring molecule (represented by the sequence of green beads and strands), after the reduction of the corresponding atomistic chains to PPs. 756

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the procedure of dictating binary intersections in order to account for all possible sequences of successive triangles that can be defined on such a polygon. This exhaustive procedure of triangulating rings proved sufficient to detect all intersections of a given ring with all other rings in the melt and to define the corresponding binary intersections (taking of course due care of the periodic boundary conditions that the simulation cell is subjected to). To provide information about the degree of ring penetration, in particular in order to decide whether or not complete penetration occurs, we do the following (see Figure 3): at the

which traces back to the early theoretical work of Grosberg et al.23 in the 80s and to the numerical studies by Müller et al.24 in the 90s, bears many similarities with the more recent study by Lang25 based on the interpenetration of minimal surfaces. In our study, here, we go a step further and calculate the dynamics of threading events, their survival times, and their effect on the stress relaxation modulus of the ring melt. We investigate model ring PEO chains of the chemical structure −CH2−O−(CH2−CH2−O)X−CH2− where X denotes the number of monomers per chain. Four different systems were considered with values X = 40, X = 113, X = 227, and X = 455 chosen to match experimental samples.12 The corresponding molar masses are 1806, 5022, 10044, and 20088 g/mol; we refer to them as PEO-2k, PEO-5k, PEO-10k, and PEO-20k, respectively. With the entanglement molar mass of (linear) PEO equal to Me ≈ 2020 g/mol, the four PEO melts have Z = 0.9, 2.5, 5, and 10 entanglements per chain, respectively.26 In terms of Kuhn segments, the simulated PEO melts are rather short, so asymptotic chain length behavior is probably not described. However, they do serve well the purpose of analyzing threading events in detail and their implications for the dynamics of melts of nonconcatenated ring polymers. CReTA identifies all possible interchain topological interactions in the system, including the very weak ones where a ring “wraps” or “folds” around another ring. To locate which of them correspond to threadings, we adopt the following procedure (see Figure 2): We represent the threading ring (sketched by the sequence of green strands and beads in Figure 2) by the topological sum of two sequences of segments making up two linear subchains sharing the same end points (A1, AN) with it. In Figure 2, the first linear subchain is spanned by the four segments denoted as A1A2, A2A3, A3A4, and A4AN and the second by the five segments denoted as A1A′2, A′2A′3, A′3A′4, A′4A′5, and A′5AN, where N stands for the number of kinks detected by CReTA along the ring. Similarly, the threaded ring (sketched by the sequence of yellow beads and strands in Figure 2) is represented by the topological sum of the two subchains composed of the straight segments marked as B1B2, B2B3, B3B4, B4B5, and B5BM and B1B′2, B′2B′3, B′3B′4, B′4B′5, and B′6BM, respectively, where M denotes the number of kinks detected by CReTA along this molecule. The problem then of determining whether or not the ring composed by the sequence of green strands in Figure 2 threads the ring composed by the sequence of yellow strands reduces to the problem of determining whether or not the sequence of straight segments along the two linear subparts A1A2, A2A3, ..., AN−1 AN and A1A′2, A′2A′3, ..., A′N−1 AN of the former intersects the corresponding sequence of planar triangles B1B2B′2, B2B′2B3, ..., BM−1B′M−1BM of the latter. This is addressed by using vector calculus to identify all intersections of the sequence of straight strands A1A2, A2A3, ..., AN−1AN and A1A′2, A′2A′3, ..., A′N−1AN along the two linear subchains of the green ring with the surface spanned by the yellow ring. Since the green molecule is actually a ring chain, and given that our rings in their molten state are nonconcatenated, these intersections correspond to binary interactions signifying cases where the same ring crosses the surface spanned by the other ring twice (see, e.g., points C1 and C′1 in Figure 2). In practice the situation is more complicated because of the occurrence of multiple threading events (see Figure S1). To be able to locate binary interactions corresponding to such situations, we generalized the triangulation process by repeating

Figure 3. Binary crossing and time evolution of a threading corresponding to complete penetration. Snapshot taken from the topological analysis of the PEO-10k ring melt.

time a threading event is recorded, we compute and store the two contour lengths from the common end point A1 up to the intersection points C1 and C′1 for each of the two linear subchains (the two strands from A1 to AN in Figure 3) linking the threading molecule with the surface of the threaded molecule. In the course of the geometric analysis, these two contour lengths are continuously monitored and updated inasmuch as the topological constraint between the two rings remains active, and when the threading event becomes inactive 757

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ACS Macro Letters we compare final and initial values. Then, if C1 and C′1 are found to be located closer to A1 at the beginning of the interaction and closer to AN at the end of the interaction (and vice versa), one of the rings is considered to have completely penetrated the other. An enlightening example is shown in Figure 3 where the green ring fully penetrates the yellow ring. Our geometric analysis showed (see Figure S2) that the number of threadings increases linearly with MW, in agreement with a recent computational study.27 We also found that as the chain length increases the probability to observe multithreading events increases (see Figure S3). Furthermore, the number of threading events per ring molecule is significantly smaller than the number of uncrossability constraints initially computed by CReTA (equal approximately to 6, 11, and 17 constraints/chain for the PEO-5k, PEO-10k, and PEO-20k melts, respectively). Only a small percentage (∼20%) of these corresponds to threading events, suggesting that, although the main mechanism of stress relaxation is the self-similar response based on cooperative local rearrangements,5−11 there is still an important contribution of a different origin associated with ring−ring disengagement. By analyzing the statistics of persistence and exchange times of polymer contacts, Lee et al.28 have reported recently that threading and unthreading events are important factors in determining the dynamics of ring polymers. We have also computed the average number K1(r) of neighboring rings whose center-of-mass is within a distance r equal to the ring mean radius-of-gyration Rg (Rg ≡ ⟨R2g⟩1/2) from the center-of-mass of a reference ring. The variation of K1(r) with the ring chain length X (number of monomers per chain) is shown in Figure S4. K1(r) attains values between 1.75 and 2.75 for the PEO ring melts addressed here, but these increase as X increases. A similar calculation has been reported by Halverson et al.15 using a generic bead−spring chain model yielding similar numerical results. How the threading time compares with τring is shown in Figure 4. To compute ring−ring survival times, we monitored a

however, there also exists a non-negligible number of threadings that live for times significantly longer than τring (e.g., up to six times the value of τring in PEO-20k). These longlived ring−ring engagements correspond to strong threadings which along with the already mentioned threading by remaining linear chain contaminants11−13 should be at the origin of the slow relaxation. Nevertheless, the vast majority of threading events survive for times on the order of τring and smaller and can be classified as weak threadings. Overall, the percentage of threadings that correspond to full penetrations is ∼6.4 ± 0.2% for PEO-2k, ∼5.8 ± 0.3% for PEO-5k, ∼6.2 ± 0.3 for PEO-10k, and ∼7.3 ± 0.3% for PEO-20k. Compared to ring-linear threading events,13 ring−ring topological constraints live for shorter times, implying a stronger effect of linear impurities on the long time decay of the stress relaxation modulus G(t) than that of ring−ring crossings. Together, they can explain the slow relaxation behavior of G(t) observed experimentally even for highly purified ring melts since they imply the existence of extra relaxation modes associated with the release of deep ring−ring and ring−linear entanglements. We confirm this by fitting the G(t) curves shown in Figure 1 for the experimental, nearly monodisperse ring melts (β = 1) with the function G (t ) =

⎛ ⎛ t ⎞−2/5 t ⎞⎟ exp⎜⎜ − ⎟ ⎟+ ⎝ τe ⎠ ⎝ τring ⎠

GN0 ⎜

GN ,RR

∑ p :odd

GN ,RL

∑ p :odd

⎛ p2 t ⎞ 8 ⎟⎟ + exp⎜⎜ − p π ⎝ τd,RR ⎠ 2 2

⎛ p2 t ⎞ 8 ⎜⎜ − ⎟⎟ exp p2 π 2 ⎝ τd,RL ⎠

(3)

where the second and third terms account for the additional small elasticity due to ring−ring and ring−linear interpenetration. In proposing eq 3, we envisage that ring−ring and ring−linear disengagement processes (which signify the two slow modes) are akin to a chain escaping a network; hence to a first, rough approach they can be approximated by the mathematical expressions of reptation with different effective “plateau” moduli GN,RR and GN,RL and different disentanglement times τd,RR and τd,RL. Equation 3 contains seven parameters (G0N, τe, τring, GN,RR, τd,RR, GN,RL, τd,RL), but for a given polymer three of them (G0N, τe, τring) are known (thus fixed) or can be easily estimated by atomistic MD simulations. G0N, e.g., is given in the literature for any polymer. The same is true for τe, whereas τring is computed directly from the simulations. On the other hand, τd,RR and τd,RL are precisely determined by our MD simulations by monitoring threadings from birth to death (and remarkably the relative ratios τd,RR/τring and τd,RL/τd,RR as computed directly from the MD simulations for the simulated PEO samples and as extracted indirectly (through fittings) from the rheological measurements for the two ring samples come out to be the same, between 3 and 5). Hence, only two adjustable parameters remain, GN,RR and GN,RL. Very nicely, our approach reduces the problem of computing the modulus of relaxation of a pure ring to the problem of computing the elastic moduli associated with ring−ring and ring−linear threading. The new fits of the experimentally measured G(t) curves (upon keeping the values of G0N, τe, and τring unaltered; see Table 1) are reported in Figure 5 (the dashed lines in the figure

Figure 4. Distribution of ring−ring disengagement times.

large number of threadings from birth to death, and in addition, we made sure that the MD simulation time (∼2 μs) was significantly longer than the longest ring−ring threading time recorded (∼1.2 μs). The results indicated a broad distribution of survival times. The vast majority of threadings (more than 80%) lives for rather short times (shorter than 0.5τring); 758

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Additional analytical details and computational results (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.G.T and V.G.M. acknowledge financial support from the Limmat Foundation, Zurich, Switzerland through the project “Multiscale Simulations of Complex Polymer Systems” (MuSiComPS). D.V. acknowledges support from the Greek General Secretariat for Research and Technology (ESPA, ARISTEIA-RINGS). This work was supported by computational time granted from the Greek Research & Technology Network (GRNET) in the National HPC facility - ARIS under project pr001030-NanoComp.

Figure 5. G(t) fittings for the PS-200k and PI-38k rings with eq 3.

show the separate contributions from the three different terms in eq 3) and, admittedly, provide an excellent description of the data. In Figure S5, the same data are reported in the form of G(t)t1/2 versus t. From a modeling point of view (network theory), we can write GN,RR = νRRkBT and GN,RL = νRLkBT where νRR and νRL denote the number of active or stress carrying entanglement strands per unit volume due to ring−ring and ring−linear interpenetration, and kB is the Boltzmann constant. As stress carrying strands we consider the strands that survive for times longer than τring, i.e., those involved in strong threadings. Application of the above expression to the PEO melts addressed here yields GN,RR and GN,RL values around 0.04 MPa, i.e., of the same order of magnitude as the best-fit ones in Table 1. In summary, we have presented a novel analysis of ring−ring interpenetrations in melts of entangled PEO rings, which has revealed a significant number of threading events (most of which correspond to multiple interpenetrations). The population of mutually penetrating rings increases linearly with polymer MW. Threading events can survive for times up to several times the orientational relaxation time τring of ring molecules. We have also calculated the percentage of full penetrations, and their number came out to be between 5.8% and 7.3% exhibiting practically no dependence on ring MW. Motivated by the results of the geometric analysis, we have proposed a network-type correction to the slow relaxation for ring melts (with a very small elasticity modulus corresponding to ring interpenetration) which was found to provide an accurate description of experimentally measured G(t) data for two ring polymers, PS-200k and PI-38k. Our findings provide direct evidence for the intimate connection of ring−ring and ring−linear interpenetration with the slow relaxation modes observed in experimental measurements of the long time ring dynamics, with important implications for a wide range of problems in soft matter physics (from entanglement dynamics to spatiotemporal organization of chromatin and DNA motion).

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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/acsmacrolett.6b00259. 759

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ACS Macro Letters (23) Grosberg, A. Y.; Nechaev, S. K.; Shakhnovich, E. I. J. Phys. (Paris) 1988, 49, 2095−2100. (24) Müller, M.; Wittmer, J. P.; Cates, M. E. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1996, 53, 5063−5074. (25) Lang, M. Macromolecules 2013, 46, 1158−1166. (26) Tsalikis, C.; Koukoulas, T.; Mavrantzas, V. G. React. Funct. Polym. 2014, 80, 61−70. (27) Michieletto, D.; Marenduzzo, D.; Orlandini, E.; Alexander, G. P.; Turner, M. S. ACS Macro Lett. 2014, 3, 255−259. (28) Lee, E.; Kim, S.; Jung, Y. Macromol. Rapid Commun. 2015, 36, 1115−1121.

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