Microscopic Description of Entanglements in Polyethylene Networks and Melts: Strong, Weak, Pairwise,...

Microscopic Description of Entanglements in Polyethylene Networksand Melts: Strong, Weak, Pairwise, and Collective AttributesStefanos D. Anogiannakis,† Christos Tzoumanekas,*,†,‡ and Doros N. Theodorou†,‡

†School of Chemical Engineering, Zografou Campus, National Technical University of Athens, GR-15780 Athens, Greece‡Dutch Polymer Institute (DPI), P.O. Box 902, 5600 AX Eindhoven, The Netherlands

*S Supporting Information

ABSTRACT: We present atomistic molecular dynamics simu-lations of two Polyethylene systems where all entanglements aretrapped: a perfect network, and a melt with grafted chain ends.We examine microscopically at what level topological constraintscan be considered as a collective entanglement effect, as in tubemodel theories, or as certain pairwise uncrossability interactions,as in slip-link models. A pairwise parameter, which varies betweenthese limiting cases, shows that, for the systems studied, thecharacter of the entanglement environment is more pairwise thancollective. We employ a novel methodology, which analyzes entanglement constraints into a complete set of pairwiseinteractions, similar to slip links. Entanglement confinement is assembled by a plethora of links, with a spectrum of confinementstrengths, from strong to weak. The strength of interactions is quantified through a link “persistence”, which is the fraction oftime for which the links are active. By weighting links according to their strength, we show that confinement is imposed mainlyby the strong ones, and that the weak, trapped, uncrossability interactions cannot contribute to the low frequency modulus of anelastomer, or the plateau modulus of a melt. A self-consistent scheme for mapping topological constraints to specific, strongbinary links, according to a given entanglement density, is proposed and validated. Our results demonstrate that slip links can beviewed as the strongest pairwise interactions of a collective entanglement environment. The methodology developed provides abasis for bridging the gap between atomistic simulations and mesoscopic slip-link models.

1. INTRODUCTION

In the molecular description of structure−property relations ofpolymer melts and networks, one of the fundamental conceptsis chain entanglement. When macromolecules interpenetrate,the term entanglements intends to describe interactionsresulting from the uncrossability of chains. In flexible polymermelts and rubbers these interactions alter in a universal mannertheir dynamical, flow, and deformation properties.Molecular-based theories which aim to incorporate entangle-

ments have first to define them. This task poses certain difficulties.In mathematical terms, entanglement interactions originate fromtopological constraints (TCs). Thus, in a many-chain theory, andin order to derive entanglement effects from “first principles”, arigorous description1−3 invokes the use of topological invariantsand certain approximations to make the problem tractable. Insuch theories, however, the range of problems that can be studiedis limited by mathematical complexity.In order to deal with entangled matter, several conceptual

simplifications were introduced by Edwards,4,5 de Gennes,6,7

Doi and Edwards,8,9 and other pioneers in the field of polymerphysics. The crucial simplification was the description ofentanglement through a mean field and a single chain model.In tube model theories,4,7,9,10 it is postulated that the entangle-ments generate a confining, (quadratic) mean field potential,which restricts lateral monomer fluctuations to a tube-likeregion surrounding each chain. In a perfect network, the

potential attracts the monomers toward the centerline of thetube and chains are restricted to adopt conformations which arecompatible with the tube constraint. In polymer melts, thisconfinement is not permanent, but leads to reptation,6 a one-dimensional diffusion of the chain along the tube.The axis of the tube plays the role of a coarse-grained

representation of the real chain, and it is called the primitivepath5,8,9 (PP). It has a random-walk like conformation, like thereal chain, but with a larger step length, which corresponds tothe entanglement molar mass,9 Me. The latter is determinedfrom the plateau modulus of long chain polymer melts. Theaverage tube diameter is not so well-defined; it is considered tobe of the order9 of magnitude of the PP step length; it is oftenassumed that the two lengths are equal.An alternative, discrete, localized version of the tube

constraint is utilized in models employing slip-links.8,11−15

The tube is replaced by a set of slip links along the chain, whichrestrict lateral motion but permit chain sliding through them.The real chain is represented by the corresponding PP, whichis a series of strands of average molar mass Me connectingthe links. The tube model provides a basis for working outanalytically the linear and nonlinear viscoelasticty of polymer

Received: May 4, 2012Revised: October 11, 2012Published: November 19, 2012

Article

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© 2012 American Chemical Society 9475 dx.doi.org/10.1021/ma300912z | Macromolecules 2012, 45, 9475−9492

pubs.acs.org/Macromolecules

melts9,16−20 and networks10,15,21,22 at a molecular level. Slip linkmodels19,20,22 also allow for analytical8,11−13,15,23,24 or stochasticsimulation14,25−35 treatments with a molecular representation,at the single or many-chain level.The concepts conveyed by the tube and slip-link models

provide the prevailing microscopic view of entanglement. Instochastic simulations, the slip links are usually envisaged asbinary entanglements, a form of discrete pairwise contacts whichconfine (link) the motion of two neighboring chains. The tubeconstraint, on the other hand, acts along the whole chain;locally, it is thought that confinement is applied collectively bymany overlapping surrounding chains, and not by specificpairwise interactions. Any gain or loss in the local pairwiseinteraction density does not necessarily affect the tube diameteror length, as it does the local entanglement density of slip-linkmodels19 (see p 552 in ref 20).A microscopic picture of entanglement is also suggested by

scaling models relating the degree of entanglement to chemicalstructure. Notably, the most successful of them are based onsimilar36−39 ideas. A good correlation with polymer melt data isprovided by the packing length40 model, which is based on theconjecture of Lin,41 and Kavassalis and Noolandi.42,43 Theconjecture is in support of a collective38,44 argument; an entangle-ment results from a fixed number of polymer chains cohabitingthe same volume. On the other hand, the scaling model36,45 ofColby and Rubinstein, which is in agreement with data fromsolutions in a theta solvent, is in support of a pairwise38,44

argument; a fixed number of binary contacts between chains givesrise to an entanglement strand.Despite the success of tube and slip-link models in describing

the linear (especially) and nonlinear rheology of polymer melts,as well as rubber elasticity, the nature of entanglements remainselusive. It would be desirable, therefore, to probe entanglementsmicroscopically in order to correct, refine and support theperspective given above with microscopic information. Simu-lation is probably the method of choice, since we need access tosome kind of topological information which is not directlyaccessible with spectroscopic methods.The question, then, is how to define entanglements in a set

of densely packed chain molecules. A unique definition mightnot exist, but a certain microscopic description has to be invokedto address the above topics. In this direction, several attemptshave been made. Assuming that entanglements lead to persistentchain interactions, the detection of long-lived chain contacts46−49

was the target of many simulations. Although it was found thatsuch contacts do exist, it is rather difficult to evolve this picturefurther. Additionally, the topological origin of entanglements isoverlooked. It becomes prominent, however, in studies50−55

employing topological invariants. The latter can detect pairs ofentangled and unentangled chains, and the “amount” of windingbetween the chains. We have to note that, for open chains, trueinvariants do not exist. Moreover, the simplest invariant forclosed chains, the Gauss linking number, can misclassify2

topological constraints. In any case, the provided informationis at the global level of a chain. Though very useful, it is not clearhow to put it into the context of current tube and slip-link modelideas. The latter require a somewhat localized description ofentanglement, especially in connection with constraint release(CR) processes. In this respect, the use of a local topologicalmeasure may be more suitable.Microscopic investigations of tube model ideas have greatly

benefited from simulations which construct the PPs accordingto the Edwards definition,5 i.e., by holding chain ends fixed and

removing all chain slack from the system. The first implementa-tion56,57 was by Everaers et al., but soon thereafter othermethods for shrinking the chains in continuous58,59 and latticemodels60,61 appeared. There is strong evidence59,62−66 that thePP Kuhn segment coming out from these methods is in verygood agreement with Me estimations from plateau modulusmeasurements. The PP representation is also popular in slip-link simulations. Thus, it is hoped that these coarse grainingmethods could serve as a bridge67,68 between the differentlevels of description.The properties of PPs have been studied in several systems,

such as rubbers,69 melts, nanocomposites,70−74 grafted brushes,75

glasses,76 in equilibrium or under deformation73 and flow.77

Concepts and properties such as entanglement densityscaling,56,57,78 entanglement molecular weight,56,59,62−65,79 CLFpotential,60,79 tube potential,80 dilution exponent,61 onset ofentanglements,81,82 and even the literal picture of a tube83−87

enclosing a chain and the associated tube survival probability,have been investigated. Despite the available microscopicinformation, the many chain nature of entanglement remainsunclear. A possible reason is that most of the simulationsfollow tube model ideas and examine, with the help of PPs, singlechain properties. Nevertheless, it is the many chain nature ofentanglement which is elusive and is replaced by a mean field.Therefore, rather than examining single chain properties, whichare the result of tube confinement, we can try to analyze thechain environment that gives rise to confinement. This is thestrategy followed here.In order to make the problem tractable, we have only

examined systems with trapped entanglements; a polyethylene(PE) melt where all chain ends are grafted to other chains, anda perfect PE network formed by connecting cross-links initiallysituated at the sites of a diamond lattice. In this way, our resultsare free of complications related with CR processes. The lattercan be addressed after a suitable microscopic description ofentanglement has been given. In the overview of the paper givenbelow, the terms tube and tube constraint are used as genericterms for entanglement constraints, and are not specificallyrelated to tube model ideas.As probes of the entanglement environment, we have used

the PPs themselves. Exploiting the fact that they sample theuncrossability constraints of the surroundings, we show that,by mapping short molecular dynamics (MD) trajectories to PPtrajectories, it is possible to detect all the pairwise chaininteractions assembling the tube constraint. To verify that theseinteractions result from true topological constraints, we ensurethat they satisfy a local topological criterion. The latter is basedon the common idea of two concatenated rings that define abinary link. Therefore, entanglement constraints are decom-posed into a large set of local links, which is more or lesscomplete. Similar ideas have been explored, analytically, byIwata and Ewards.88,89

By construction (chain shrinking), all these links are sampledinside the pervaded volume of a chain. The mechanism bywhich link sampling takes place is clarified in detail. Whilesome of those links are continuously present, many of themare intermittent or “blinking” links. They are not sampledcontinuously over a specific observation time. Less constraining(weak) links are sampled less frequently. This leads to adistribution of linking times, and thus the links assembling thetube constraint have a spectrum of confinement strengths, withmaxima at the strong90 and weak part of the spectrum.

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From this picture, we then develop a simple statistical modeland we introduce a pairwise parameter, that can quantify thepairwise versus collective character of mean-field type linksrepresenting many uncrossability interactions. With the help ofthese tools we show quantitatively that;

• the entanglement environment consists of a plethora ofpairwise interactions, so that it is inherently collective;

• there exist strong and weak pairwise uncrossability constraints;• the majority of pairwise interactions are weak and

confinement is mainly enforced by the strongest ones;• weak topological interactions can be fully relaxed by

stored length fluctuations, and they do not contribute tothe low frequency storage modulus of an elastomer, orthe plateau modulus of a melt;

• the average effect of the collective entanglement environ-ment is to confine a chain through a much smallernumber of mean-field type links, which have a dominantpairwise character;

• the pairwise parameter is chain length and entanglementdensity independent.

In order to be able to map the entanglement environment tospecific pairwise interactions, we present a procedure forselecting the strongest links according to a given link density.The procedure introduces an auxiliary time scale which is self-consistently determined to be approximately one τe. It is shownthat strong links have the property that they are sampled atleast once within this time interval. By following the space-timetrajectories of selected links, under equilibrium, deformation, orflow, we could obtain useful mesoscopic information for thedevelopment of hierarchical slip-link, or single particle91−93

simulation schemes. The mean square displacement of selectedlinks is presented in the appendices.The PPs employed in our study have been generated by

the CReTA algorithm,59 which applies a contour reduction(chain shrinking) scheme. Different schemes, based on timeaveraging,94,95 have been proposed. However, the presentedresults are not based on specific properties of the PPs, exceptthat they probe pairwise uncrossability constraints in thepervaded volume of a chain. Thus, our conclusions are fairlygeneral and independent of the particular scheme employed forobtaining the PPs.In order to make the article more readable and focus on the

analysis of entanglement constraints, many of our results arepresented in the Appendices. Our microscopic description isalso supported by several explanatory videos96 provided with theSupporting Information for this article.

2. SYSTEMS STUDIEDThe systems analyzed are a PE network “rubber” and an end-graftedPE melt (see next section). The rubber system contains 64tetrafunctional cross-links and 128 chains. The cross-links areinitially situated at the sites of a diamond lattice and all chainsare attached to cross-links at both ends. Thus, the network isperfect (absence of pendant chains, chain loops, etc.). Eachchain consists of 201 united atoms, not counting the cross-linksat its two ends.The melt system consists of 16 linear chains of 500 united

atoms each. In order to examine the chain length dependenceof some of our results, we also analyzed a melt with eight chainsand double the chain length, i.e., of 1000 united atoms. Theentanglement density of these melt configurations is approx-imately the same. Whenever required, these systems will be

denoted by (C500) and (C1000), but in general, our analysiswill focus on the (C500) system.The simulations were performed in the NPT ensemble, at

temperature 450 K, pressure 1 Atm, by employing the unitedatom TraPPE97,98 force field and 3D periodic boundaryconditions. United atom beads correspond to CH2 methyleneunits, except chain ends, trifunctional, and tetrafunctional cross-links, which correspond to CH3, CH units, and carbon atoms,respectively.The rubber was equilibrated by the gradual push-off99

method, discussed in Appendix I. The melt was equilibrated bystandard connectivity altering100,101 Monte Carlo algorithms.The shortest relaxation time related with the tube constraint, τe,is 1.3 ns for the rubber, and 1.7 ns for the melt (C500). It wasdetermined from the first crossover time in the mean squaredisplacement (msd) of midchain monomers (Appendix II). Thelongest relaxation time of each system, τR′, is determined fromthe msd crossover to an invariant regime (absence of long-rangediffusion), and is 9.8 ns for the rubber and 11.5 ns for the melt(C500). The corresponding crossover times for the (C1000)melt are 1.6 and 19.4 ns. Details about the dynamics and theequilibration of the systems will appear elsewhere.102,103

Both systems share, approximately, the same density,104,105

ρ ≃ 0.78 g/cm3, and characteristic ratio, C∞. The latter is8.5 for the rubber and 8 for the (C500) melt. Volumetric,structural,100,101 and entanglement59 properties of the meltsystem have been presented elsewhere. They are in very goodagreement with corresponding experimental estimates. Therubber system is structurally similar to the melt, except forentanglement properties. Its Me was estimated

59 from the Kuhnsegment of the PPs, and is about half that of the PE melt. Thelarge entanglement density of the rubber is also reflected102 in acomparatively high, entanglement-dominated, shear modulusfor an elastomer, of 5.9 MPa. As is discussed in Appendix I, it isdue to the short length of the precursor chains and the artificialpreparation conditions of the network.

3. METHODOLOGYFor both systems, we generated a molecular dynamics (MD)trajectory, which is of length 35τe for the rubber andapproximately 50τe for the melts, covering a few times thelongest relaxation time of each system. Our purpose is to use thetrajectory as an ensemble of states with invariant topology,representative of the entanglement constraints in the system.This means that under thermal motion all entanglements shouldremain trapped, a condition satisfied in the perfect network butnot in the melt, due to the free motion of chain ends.The melt topology can become invariant by introducing

additional topological constraints. Our choice is to graft (bond)all chain ends to their nearest united atom bead along adifferent nearby chain. Thus, the melt system will be referred toas end graf ted melt (EG-melt). Effectively, the EG-melt becomesa perfect trifunctional network with as many cross-links as chainends. The initial chains are grafted at both ends and have onaverage two randomly placed cross-links along their length(from the grafted ends of other chains).In a MD simulation of this network, the conformations

sampled by grafted chains are restricted by the same topologicalconstraints present in the parent melt state, plus the constraintsdue to grafting. From such a network trajectory we store chainconformations by excluding all bonds inserted to imposegrafting, i.e., the network trajectory is mapped to a melt system,with virtually free chain ends. Thus, starting from a melt



configuration with a specific topology, we can generate thermallyfluctuating chain conformations with approximately the sametopology, by eliminating CR through grafting.Our results for the EG-melt refer to this “perturbed” MD

trajectory. The perturbation preserves the average PP contourlength, but it alters contour length fluctuations. Because of thepresence of additional constraints, the grafted chains cannotsample certain Rouse modes with long length scales. However,the short and medium length scale modes, which are responsiblefor the sampling of the enclosing tube, are adequately sampled.In this respect, at short times, the dynamics of grafted chains aresimilar to those of melt chains with free chain ends. For example,at short times the mean square displacement (see Appendix II)scales as t0.5 and then displays a crossover to a regime scaling ast0.28, a crossover indicative of tube confinement.

4. REDUCTION TO NETWORKS OF LOCAL LINKS

The generated MD trajectories were subjected to topologicalanalysis, frame by frame, by using the CReTA59 (contourreduction topological analysis) algorithm. CReTA reduces adense system of polymer chains to the corresponding system ofprimitive paths (PPs), constructed as the shortest paths9 underthe same “topological constraints” (TCs) as the original chains.By fixing chain ends in space and by prohibiting chain crossing,the contour lengths of all chains are simultaneously minimized(shrunk), until they become piecewise linear objects comingtogether at the nodal points of a network. During contourreduction, chain thickness is progressively reduced. Finally, thePPs become very thin objects consisting of fused beads whichcan be also mapped to the initial chain monomers. Explanatoryvideos and other details can be found elsewhere.59,96

Network nodes are placed where PPs are mutually blocked(entangled) and contour reduction can no longer proceed (seenetwork pictures in Supporting Information). They representeffective spatial localization points of the TCs, that each chain issubjected to. They can also be envisaged as local links that havea physical resemblance to binary slip links. The details ofmapping a system of “tightened” entangled curves, consisting ofconsecutive beads, as our system of PPs here, to a network withpoint-like nodes and rectilinear edges, will be describedelsewhere.103 It will be shown that the mapping is very stablewith respect to changes in the free parameters involved. A shortaccount of the mapping is given in Figure 1, and in Appendix III.

5. RESULTS AND DISCUSSION

5.1. Fluctuations in the Number of Local Links andLinked Chains (LCs). Here, we examine the tube constraint atthe level of the surrounding chains that create it. To this end,we define as zα(t) the number of chains linked, instantaneously,to a specific chain α, at a specific time t. As shown in Figure 2,all chains in this set are dif ferent. That is, if at time t chain α islinked twice or more with chain β, at different places along theircontours, then β counts as one chain in zα(t), and vice versa.We will refer to this time-dependent set as the mate chains,or linked chains (LCs) of α. If, due to periodic boundaryconditions, α is also linked with different “images” of β or ofitself, then these images106 are considered as different chains inzα(t). Entanglements or knots of α with itself (same image) arenot considered. For the chain lengths studied here theiroccurrence is negligible.57,59 We also note that zα(t) character-izes a chain’s entanglement environment by keeping track of

the identities of linked chains at time t, in a form suitable forcomparisons at different times.In Appendix IV (Figure 16), we show that, when two chains

are linked, the probability that they are coupled with more thanone link is very small (less than 10%). Hence, the simplificationof dealing with linked chains instead of individual links isjustified. This choice has been made in order to facilitate thepresentation and the interpretation of our results. We could aswell work with individual links. Therefore, in the followingsections we will use interchangeably the terms local links andlinked chains for zα(t), though in our calculations zα(t) refers toLCs and not individual links.In Figure 3, we plot the distribution of zα(t) reduced by its

equilibrium time average ⟨zα(t)⟩. In contrast to what one wouldexpect for a system with solely trapped entanglements, zα(t)fluctuates around a time average value. In a corresponding slip-link system, zα(t) would be constant. This happens because

Figure 1. Network nodes, local links, and the topological criterionsatisfied by local links. (a) PPs are composed of consecutive beads.Parts ABC and abc, of chains α,β, respectively, constrain each otherand are sketched with a bead structure. All other parts are sketchedwith contour lines. The beads which “carry” a TC are colored gray.They mark the points along the PPs where a suitably definedcurvature103 shows a local maximum. Each TC bead is pairwiseassociated with a TC bead along the PP of a mutually constrainedchain. Thus, chain ends and TC bead pairs partition a chain intoconsecutive strands. In order to decide if a TC bead pair defines a locallink we apply a topological criterion at the bead level. For each pair,such as (B,b) between chains α,β, we construct the composite strandsABC and abc. The ends of these strands are virtually connected, andthen we check if the strand ABC crosses the area enclosed by thevirtual segment ac and the beads along abc. Similarly, we examine if thestrand abc crosses the area enclosed by the virtual segment AC and thebeads along ABC. In this way, we examine twice if the ABC and abcrings concatenate. If at least one of these checks is successful, the TCbead pair (B,b) is promoted to a network node (local link), otherwise itis discarded. Nodal coordinates are calculated from the vector averageof the coordinates of each TC pair. Hence, the configuration shown in(a) becomes the one shown in (b), and the underlying topology ismapped to a network. Dynamical videos of such networks, and realisticcloseups on PP networks of a PE melt can be found in the SupportingInformation.



each PP is a dynamically f luctuating object that samples itsenvironment by colliding with the contours of the surroundingPPs penetrating its pervaded volume.

Such “collisions” lead to the formation of local links whichare not permanent. Instead they are formed for certain times,i.e., they exhibit various linking times. However, there also existlinks which restrict continuously two fluctuating PPs andexhibit a permanent character. As a result, the instantaneousnumber of links sampled by a PP, zα(t), fluctuates around anaverage value. In Appendices V and VI, and in the videosprovided with the Supporting Information, this issue is discussedin more detail. In the inset to Figure 3a, we also see that theamplitude of these fluctuations seems to be smaller for longchains.The chain average of zα(t) is defined as

∑=α

α=

z tN

z t( )1

( )N

ch 1

ch

(1)

where Nch is the number of chains. In Figure 3b it is plotted as afunction of reduced time, t/τe. We see that for both systems it isapproximately constant. Its time average, ⟨z(t)⟩, is 9.5 for therubber, and 14.6, 25.3 for the EG-melt (C500) and (C1000),respectively. These values lead107 to a rubber entanglementdensity which is approximately twice that of the EG-melt.The fluctuations around ⟨z(t)⟩ are much smaller for the

rubber because chain averaging is more effective in smoothingindividual chain fluctuations. The rubber consists of eight andsixteen times more chains than the EG-melt (C500) and (C1000)systems, respectively. In a very large system, z(t) would appearas a thin horizontal line, i.e., it would remain constant overtime. Thus, despite the fluctuations in the number of links ofindividual chains, when viewing the system as a single chain(through ⟨z(t)⟩), we observe the behavior expected of a slip-link system.

5.2. Cumulative Number of Sampled Links: AComplete Set. Because of the absence of CR, after a certaintime the PP cannot sample any new links in its surroundings.This fact is exploited in order to gather all sampled links into acomplete set of pairwise uncrossability interactions characterizingthe entanglement environment.To elucidate these ideas further, we introduce Uα(t), the

cumulative number of different chains which have been linkedto a given chain, α, up to time t. All chains in Uα(t) are dif ferent.They were not necessarily linked to α at all times between0 and t. Some of them may have been linked for a long(discontinuous or continuous) subinterval, some of them forjust one instant, or for the whole time t. As shown in Figure 2,Uα(t + Δt) is larger than Uα(t) by the number of mate chainswhich link to α, for the f irst time, at any instant between t andt + Δt.An informative graph of Uα(t), together with individual “link

trajectories”, is shown in Figure 4. In this figure, the systems areobserved at equidistant time instants, spaced by 0.01 ns, over atotal observation time (OT), T, which covers a few times thelongest relaxation time of each system. Specifically, T ≃ 35τe forthe rubber, T ≃ 50τe for the EG-melts.We see that some links “blink”. They are not present during

the entire OT. From the number of points in a trajectory wecan estimate the corresponding linking time (LT), τ, i.e., thetotal time during which a particular link was active. In order toanalyze the plethora of links we introduce the link persistence,τ = τ/T, which is basically the reduced linking time of a link.As the observation time increases, τ becomes proportional toT, with a different prefactor for each link (see SupportingInformation). Thus, for adequately long OTs, as the ones

Figure 2. At times zero and Δt, chain α is linked to a group of five andsix different chains, respectively. Since four of these chains are the same,the overlap between these groups is zα(Δt)|zα(0) = 4. The cumulativenumber of different chains sampled by α within Δt is Uα(Δt) = 7. Bydefinition Uα(0) = zα(0). For the chain lengths studied, the probabilitythat two chains are entangled with more than one link, as chain α andchain 2 here, is less than 10% (see Appendix IV). Thus, throughout thedocument we will consider that, at any time t, the number of links ofchain α is equal to the number of linked chains zα(t).

Figure 3. (a) Distribution of zα(t) reduced by the corresponding timeaverage ⟨zα(t)⟩ (see text). Log−log coordinates in the inset. To obtainthese results, a separate distribution is constructed for each chain firstand then an average is taken over the distributions of all chains. This isdone because the entanglement environment (number of entangle-ments) and related fluctuations may be different for each chain. Thus,we consider each chain as a subsystem, we analyze the fluctuationswithin each subsystem, and then we average over the subsystems. This isnecessary in order to map the fluctuations to a single chain model. (b)Average number of links per chain at time t.



utilized here, the τ’s are equilibrium T-independent quantitiesthat characterize the “persistence” of link trajectories withvalues 0 < τ ≤ 1.A subset of the links appears to be permanent, with τ ≃ 1.

Most of the LCs sampled at t = 0 have this property.Instantaneous sampling at any t (a t-column in Figure 4), willhave a large overlap with this subset, which are basically stronglinks. This is evident in individual t-columns, where most of thepoints are located on horizontal lines originating at t = 0. Notethat, by construction, the number of points in a t-column is zα(t).A striking feature of Figure 4 is that the majority of the

Uα(T) links have small, even vanishing τ’s. Moreover, the

number of links sampled over the whole OT, Uα(T), is muchlarger than those sampled instantaneously, zα(t), at any time.Furthermore, after a certain time, t′, most of the links withappreciable τ have been sampled. The links sampled for the firsttime at later times, t > t′, appear to have a τ ≃ 0.The sampling mechanism that generates blinking links is

relatively simple, and is demonstrated in Figure 5. The driving

force for this mechanism are stored length f luctuations, asdescribed in detail in Appendix V. In order to avoid confusionwith constraint release (CR) mechanisms driven by chain endmotion, we call this mechanism lateral link sampling (LLS).However, upon nonlinear deformation or flow, LLS couldpossibly act as a CR mechanism. A similar mechanism wasimplemented93,108,109 by Padding and Briels to enforce chainuncrossability in a coarse grained model of polyethylene.The chain average of Uα(t) is defined as

∑=α=

U tN

U t( )1

( )N

ach 1

ch

(2)

and is shown in Figure 6. It is estimated by averaging overmultiple time origins, so that by construction U(0) = ⟨z(t)⟩(since Uα(0) = zα(0)). We see that U(t) increases rapidly andapproaches a plateau which is much larger than the averagenumber of links ⟨z(t)⟩ per chain (see Figure 3b). This is moreevident in the EG-melt (C500); the longer chains and the widertube of this system lead to a larger number of surroundingchains assembling the tube. In this system, U(t) does not seemto level off. It increases slowly, with a very small but nonzerodecreasing rate. A true plateau with a slightly larger value isexpected to exist (fixed tubes), at some longer t.However, it is not required to sample this regime, since the

links sampled for the first time at such long times have a vanishingτ (see Figure 4). Most probably, their sampling involves large,cooperative stored length fluctuations (Appendix V), which arenot frequent.110 While these links are part of the entanglementenvironment of a chain, their confinement strength is negligible,and they can be safely considered as weak.

Figure 4. Cumulative number of different links or linked chains(LCs), Uα(t), sampled by a given chain α within observation time, T,as a function of reduced time, t/τe. (a) EG-melt (C500), T ≃ 50τe. (b)Rubber, T ≃ 35τe. Uα(t) is a stepwise, increasing, integer valuedfunction, shown by a solid line. Here, it enumerates with uniqueinteger indices, i, cumulatively, the links sampled by chain α, up totime t. When a link, i, is found to be paired to α at time t, a point withcoordinates (i,t) is plotted. Therefore, horizontal lines, continuous orsegmented, correspond to individual “link trajectories”, as thosedenoted by m1, m2 and m3. The links sampled for the f irst time at atime t′ > 0 lack point entries for earlier times t < t′ (link m3). Forconvenience, the links sampled at t = 0 are also plotted as a separatecolumn to the left of t = 0. By construction, the number of points in at-column is equal to zα(t), and Uα(0) = zα(0). The linking time (LT),τ, of a link, is the total plotted length of the corresponding trajectory.The reduced linking time, τ = τ/T, is a T-independent quantitycharacterizing the “persistence” of each link, with values 0 < τ ≤ 1. Inthe videos provided as Supporting Information, intermittent linktrajectories with τ < 1, e.g., link m1, appear as “blinking” links and theyhave various, intermediate confinement strengths. Link trajectorieswith τ ≃ 1, e.g., link m2, correspond to permanent, strong links. Therealso exist links with τ ≃ 0, e.g., link m3, which are very weak links.Similar figures for other chains can be found in the SupportingInformation.

Figure 5. Lateral link sampling (LLS) mechanism by which PPsampling of new links takes place in a spatially fixed tube. (a) Theinstantaneously sampled PP segments are one above the other, at asmall distance h. (b) A small downward displacement of one nodeleads to a local link between the two PPs. The reverse process is alsopossible. In reality (see Appendix V), all nodes fluctuate due to thethermal motion of the real chains and the associated redistribution ofstored length. This leads to lateral PP fluctuations and to this kind ofsampling. Therefore, such events are frequent. Note that the strands inthe figure are topologically constrained, whether they form a link ornot. When h is of the order of lateral PP fluctuations they can becomelinked for certain time intervals. If the persistence, τ, of such anoccasionally formed link tends to zero, then the corresponding localpairwise interaction can be considered as weak.



Therefore, for the systems studied, and for all practicalpurposes, we consider that an observation time of T ≃ 40−50τe is adequate for sampling all pairwise interactions playing asignificant role in confinement. We also consider that the linkscontained in Uα(T) form a complete set. This set applies to α thereputed tube constraint. The state of entanglement of α, describedby zα(t), can be projected, at any time, to the Uα(T) links.Corresponding statements can also be made for U(T), whichviews the system as a single chain. In the next sections, we willrefer to these complete sets with the symbols Uα, U, defined as

= =α αU U T U U T( ), ( ) (3)

These are adequate representations of the full sets Uα(∞),U(∞), which may have contributions from additional weaklinks that can be neglected.5.3. Number and Weight Distribution of Link

Persistences. We have seen that for the average singlechain, the entanglement environment consists of a complete setof U dif ferent links. Here, we characterize this set through thedistribution of the persistence of links. In Figure 7a, we presentthe normalized probability distribution, p(τ), discretized inintervals of Δτ = 0.1. The fraction of the U links falling in eachinterval is p(τ)Δτ. We observe that p(τ) has an absolutemaximum at τ ≃ 0, and a second maximum at τ ≃ 1. Hence, themajority of the U links are weak, and the average linkpersistence, τn, is quite small, namely, 0.30 for the rubber, and0.24 for the EG-melt.In the Supporting Information, we present p(τ) for several

increasing T, starting from T = τe. It is shown that for theobservation times discussed here the distribution becomesinvariant. That is, had we increased T, we would have obtained,practically, the same distribution. This is a necessary condi-tion in order to consider U = U(T) as a complete set andT-independent. It is satisfied when sampling the entanglementenvironment for long OTs, where the linking time of any linkbecomes proportional to T, so that the correspondingpersistence, τ = τ/T, acquires an equilibrium value. Addition-ally, at long T’s the weak links of the environment have beenadequately sampled.In order to associate the persistence of a link with a

confinement strength, we have to note that in p(τ) all links are

equally weighted. Therefore, τn corresponds to the number-average link persistence. However, equal weighting hides thefact that the tube constraint is applied mainly by links withτ ≃ 1. In order to give greater importance to these links weneed to estimate the weight-average distribution of τ’s, w(τ),which is defined as

τ ττ

τ =

w p( ) ( )n (4)

In this way, the link of a chain is weighted according to the ratioof its persistence, over the sum of the persistencies of all linksto this chain. For a link between chains α, β, with persistenceταβ, the corresponding weight is

ττ

τ

τ =

∑ =

⟨ ⟩αβ αβ

αβ

β αβ

αβ

α=α

wz t

( )( )U

1 (5)

and corresponds to the conf inement strength of this link. Thecloser τ is to unity, the stronger the confinement provided by alink. Similarly, w(τ) dτ is the fraction of confinement providedby the U links having a persistence between τ and τ + dτ. Notethat p(τ) and w(τ) view the system as a single chain.The normalized weight distribution w(τ) is shown in

Figure 7b. We see that confinement comes from, mainly,links with τ ≃ 1. However, the form of w(τ) does not allow fora separation of links into strong and weak. This issue is discussedin detail in section 5.7. The weight-average link persistence, τw,is 0.76, for both systems. It is much larger than τn, and itcorresponds to the strength weighted average persistence of links.In order to clarify this section further, it is useful to make an

analogy with a corresponding slip-link system where all entangle-ments are trapped. In this case the links do not blink and all linktrajectories are continuous. Then, assuming Z links per chain, wehave z(t) = Z, at any t, and therefore U = ⟨z(t)⟩ = Z, for anyobservation time. For each link τ = 1 (permanent links), whichleads to τn = 1. Regarding p(τ)dτ, it would be zero for τ ≠ 1,

Figure 6. Average number of dif ferent linked chains or links whichmake the tube constraint up to time t. τe is 1.3 ns for the rubber and1.7 ns for the EG-melt (C500).

Figure 7. (a) Normalized distribution of link persistences. Since alllinks are equally weighted, this distribution corresponds to the numberdistribution of τ’s. (b) Normalized weight distribution of linkperistences. Log−log coordinates in the insets. The observationtime, T, is 35τe for the rubber, and 50τe for the EG-melt (C500).



and unity for τ = 1 (delta function at τ = 1). Regarding w(τ),from eq 4 we have w(τ) = p(τ), and thus, τw = τn = 1.Concerning individual links, from eq 5, we find that all links havean equal confinement strength, w = 1/Z.Finally, we note that the strength of a link could be associ-

ated with additional quantities. For example, a time average ofsuitably defined elastic interactions over the four strands makingthe link, the local friction between the mated chains, the realspace fluctuations of the link, the monomer diffusion through alink, etc. At a first approximation, here, we consider that all thesequantities are expressed indirectly through the link “persistence”.They could be extracted from atomistic simulations as long asindividual link trajectories can be defined, as discussed in section5.6 and in Appendix II).5.4. Collective Entanglement Environment and Mean-

Field Type Links. A useful relation can be obtained byexamining Figure 4, again. Denoting by ταβ the LT of a linkbetween chains α, β, the following equation holds,

∫∑ τ =β

αβ=

α

z t t( ) dU T

a1 0 (6)

where Uα = Uα(T). With reference to Figure 4, eq 6 states thatin order to add up the LTs of the links sampled by α, we cansum either segmented horizontal lines (left-hand side), or thenumber of points in vertical t-columns (right-hand side).In either case, the sums correspond to the total plotted lengthof the link trajectories, up to time T.To proceed further we note that the average LT of a link of

chain α, τn,α, and ⟨zα(t)⟩, can be written as

∑τ τ τ= = αα β

αβ α=

α

UT

1n

U

n,1

,(7)

∫⟨ ⟩ =αz tT

z t t( )1

( ) dT

a0 (8)

Then, by dividing eq 6 with UαT, we find

τ = ⟨ ⟩α α αU z t T( )n, (9)

Since Uα and τn,α are independently distributed variables111

over the chains of the system (numerically checked), the chainaverage of eq 9 reads

τ = ⟨ ⟩U z t T( )n (10)

that can also be written in the T-independent form

τ = ⟨ ⟩U z t( )n (11)

which shows that the average link persistence is τn = ⟨z(t)⟩/U.Equation 10, which is exact, provides a concise microscopicdescription of entanglements. It can be understood as follows.By observing the system over a time T, which covers a few

times its longest relaxation time, we found that for the averagesingle chain the entanglement environment is made by Udif ferent chains, with which the constrained fluctuating PPcollides and forms binary links. Each link represents a pairwiseuncrossability interaction, with a specif ic mate chain. Theaverage linking time, τn, is much smaller than T, because themajority of links are active only for very short times (blinkinglinks). The large number of these interactions shows that theentanglement environment is inherently collective. This descrip-tion corresponds to the left-hand side of eq 10.

The right-hand side is basically a mean-field type average ofthe left-hand side, where the identities of the mate chains aremixed. It is understood as follows. For the average single chain,the entanglement environment can be viewed as ⟨z(t)⟩ linksper chain which are active over the entire observation time, T.These links do not blink, and their linking time is τ = T(permanent links). However, the identities of their mate chains,and therefore their coordinates in real space and along thechain, are not specific. Thus, they are of mean-field type. Eachone represents many pairwise interactions, and all togetherrepresent the complete set of the U links on the left-hand side.This view leads to the foundations of the mean field pictureemployed in slip-link models.The above discussion is also valid in the limit T → ∞. In this

case, the appropriate equation for the mapping of the collectiveentanglement environment to a small number of ⟨z(t)⟩ mean-field type interactions is eq 11. In stochastic slip-link models,these mean-field interactions are usually realized as binary sliplinks, though they are not necessarily pairwise. For example, itis possible to consider that slip links represent multichaininteractions, as has been discussed by Nair and Schieber.31 Thedegree of multichain or pairwise character of these mean fieldinteractions is examined in the next section.

5.5. Pairwise Parameter. To examine the mean fielduncrossability interactions in more detail, we define zα(t)|zα(0)as the number of links at t = 0, which also contribute to zα(t) attime t, i.e., the “overlap” between the sets contributing to zα(0)and zα(t). By definition, this number is smaller than or equalto zα(0) (see Figure 2). By averaging zα(t)|zα(0) over multipletime origins along an equilibrium trajectory, we obtain theautocorrelation function ⟨zα(t)|zα(0)⟩. Note that the linkswhich contribute to zα(t)|zα(0), for a different time origin, neednot always be the same. The ratio

=⟨ | ⟩

⟨ ⟩αα α

αq t

z t zz t

( )( ) (0)

( ) (12)

is the percentage of links at time t = 0 which are also present attime t, irrespective of their state of entanglement with α attimes between 0 and t. The corresponding average over allchains, q(t), is plotted in Figure 8. It is defined as

=∑ ⟨ | ⟩

∑ ⟨ ⟩= ⟨ | ⟩

⟨ ⟩α α α

α α

=

=

q tz t z

z tz t z

z t( )

( ) (0)

( )( ) (0)

( )

N

N1

1

ch

ch(13)

We observe that q(t) starts from unity (t = 0), the largestpossible value, then decreases, and after a certain time reaches aplateau. For all systems, this time is around t = 10τe. The decayof q(t) is due to the mean-field nature of the z(t) links. Thepairwise character of these links is quantified by the pairwiseparameter, q, which is defined as the asymptotic value q = q(t→∞). We find that q is 0.76 for the rubber, 0.74 for the EG-melt(C500), and 0.81 for the EG-melt (C1000).In principle, 0 < q ≤ 1, though it is obvious that it cannot

become arbitrarily112 small. According to eq 13, when q = 0, thechains linked to a reference chain at times zero and t would allbe different to each other, something impossible in a systemwith fixed topology. q = 1 corresponds to a strictly pairwisecharacter. The larger the deviation of q from unity, the higherthe multichain or collective character of the environment. Here,the mean field links have a strong pairwise character, supportingthe employment of binary links in slip-link models. Moreover, itseems that the pairwise parameter is practically chain length



and entanglement density independent, a very interestingresult.A simple model which relates q to the weight-average link

persistence, and therefore to the confinement strength ofpairwise interactions, is presented below. The linking proba-bility, Pαβ, that chain α is linked to chain β, can be estimated asthe ratio of the linking time ταβ, over the observation time T, i.e.,Pαβ = τaβ. The probability that these chains are not linked is1 − Pαβ. Obviously, Pαβ = 0 for chains not included in Uα. Theanalysis below concerns long observation times, T → ∞, wherethese probabilities become T-independent, with the conventiont T.The average number of different chains linked to chain α is

∑ ∑ τ⟨ ⟩ = = αβ

αββ

αβ= =

α α

z t P( )U U

1 1 (14)

On the same footing, it is meaningful to assume that the timecorrelation function defined in eq 12 can be written as

∑⟨ | ⟩ =α αβ

αβ αβ=

α

z t z P P( ) (0)U

1 (15)

Then, the weight-average persistence of the links of chain αreads

∑ ∑τ τ = =⟨ ⟩α

βαβ αβ

α βαβ αβ

= =

α α

wz t

P P1( )w

U U

,1 1 (16)

which leads to

τ⟨ ⟩ = ⟨ | ⟩α α α αz t z t z( ) ( ) (0)w , (17)

As in eq 9, ⟨zα(t)⟩ and τw,α are independently distributedvariables over the chains of the system (numerically checked).Thus, the chain average of eq 17 reads

τ⟨ ⟩ = ⟨ | ⟩z t z t z( ) ( ) (0)w (18)

where τw is the strength-weighted average persistence of links.From eqs 13 and 18, and by taking into account that T → ∞,we then find

τ= q w (19)

Moreover, eq 18 can now be written as

τ τ = ⟨ ⟩ qU z t( )n w (20)

which is the “weighted” form of eq 11.Equation 19 is in very good agreement with the data of

Figure 8. Even for short observation times T, where τw isT-dependent (see Supporting Information), the relation q(T) =τw(T) is able to follow the data, while in the asymptotic regimethe agreement is very good. The small deviations are due toeq 15, which holds only approximately.The above eqs when τn = τw = 1, i.e., when the links do not

“blink”, lead to ⟨z(t)⟩ = ⟨z(t)|z(0)⟩ = U, and q = 1. This is thecase of a corresponding slip-link model, i.e., of a strictly pairwiseentanglement environment (see section 5.3). In practical terms,the meaning of the pairwise parameter, and of eq 13, is that themean-field links can be realized as q⟨z(t)⟩ ef fective binary links,with specific mate chain identities and coordinates.

5.6. Selecting the Strongest Pairwise Interactions. Insection 5.3, we have seen that the links have various strengths,which were categorized according to their link persistence, τ.However, a specific persistence scale separating the links intostrong and weak does not seem to exist. The reason is that, forintermittent links, it is not only the linking time which qualifiesa link, but also how this time is distributed over the observationtime. The more uniformly it is distributed, the less weak is thelink, as long as its persistence is appreciable. Therefore, amethodology for selecting the strongest links has to considerthe idle time intervals, or time gaps, τg, of the blinking links.These are the gaps in the “link trajectories” of Figure 4.In Figure 9 we present the normalized distribution of τg

reduced by τe. The time gaps before the first, or after the last

appearance of a link contribute to the distribution. We see thatit follows a power-law decay (inset), and thus it is scale-free.The exponent is somewhat higher than −2. Almost all valuesare concentrated at τg ≪ τe, and large time gaps are rare.In order to select the strongest links, a possible strategy is to

consider that, for idle time intervals less than or equal to amaximum τg,max, the links are instead active. This way we actually“fill in” certain time gaps of intermittent link trajectories, such asthose in Figure 4. When all the time gaps of a trajectory are lessthan or equal to the specified τg,max then this link spans the wholeobservation time, and it can be selected as a link which issampled at least once within this time interval. The idle times at

Figure 8. Fraction of links at time t = 0, which are also present at timet. The plateau indicates the pairwise character of the mean-fieldinteractions that confine the average single chain. The squares denotecorresponding predictions from the weight-average link persistenceτw(T), for observation times (T/τe) = 0, 1, 2, 5, 10, 15, ..., 50 (see text).

Figure 9. Distribution of idle time intervals (time gaps), τg, ofintermittent link “trajectories” such as those presented in Figure 4.Log−log coordinates in the inset.



the beginning and end of the observation time, if present, can betreated with a different τg,max. Here, for simplicity, a commonτg,max has been used.The ratio of the average number of selected links per chain,

zsel, over ⟨z(t)⟩, for various values of τg,max, is shown in Figure 10.

In the regime of interest the ratio scales as (τg,max/τe)0.42. From

this plot we can pick out an appropriate time scale for filling thegaps of link trajectories. Below we examine two possible choices,and a third one is examined in the next section.Following the outcome of weighted averaging, τg,max can be

adjusted so that we select as many links as the ef fective binarylinks, i.e., zsel = q⟨z(t)⟩, where q is the pairwise parameter. InFigure 10, this case is shown with filled symbols, and the cor-responding τg,max is approximately one τe, for both systems.The other choice is to select as many links as the average

number of links per chain, which are mean-field type links (seesection 5.4). That is, zsel = ⟨z(t)⟩. This scenario corresponds toa “realization” of the mean f ield in terms of specific pairwiseinteractions and link coordinates. In Figure 10, the correspond-ing τg,max is estimated from the intersections of the dashed linewith the data. It is 1.5τe for the rubber and 2τe for the EG-melt(C500).The time scale of the maximum time gaps is very reasonable.

The distribution of the persistence of selected links is shown inFigure 11. The peak at weak links has now disappeared. Wealso see that a smaller τg,max leads to fewer selected links, withlarger persistences. It also seems that the distributions aresuperposable. By adjusting τg,max to fit the effective binary links,contributions from links with τ < 0.45 vanish.To examine the effectiveness of this strategy, in Figure 12 we

plot the trajectories of all the links selected under the conditionzsel = q⟨z(t)⟩, with τ ≃ 0.45. There are few of them. It is seenthat they blink in such a way that they cover the wholeobservation time.The above results show that the entanglement environment

can be mapped to specific pairwise interactions, according to agiven link density. The selection procedure sorts out the linksaccording to their strength, and then selects interactions fromthe strong part of the confinement strength spectrum. A certaintime scale (interval), τg,max, is introduced, which is self-consistentlydetermined to be of the order of one τe. The selected links havethe property that they are sampled at least once within this timeinterval.

The use of this time scale permits a controlled study withrespect to a slip-link mapping of the system, and the developmentof simulation schemes which track the motion of entanglementsat equilibrium, or under deformation and flow. For trappedentanglements, this is a straightforward procedure. First, thestrong links can be selected by analyzing a short equilibrium MDtrajectory. Then, it is possible103 to follow these links upon anykind of deformation. The whole method is relatively fast, as longas the necessary software exists. When CR is active, an additionalscheme103 for dealing with the update of links would be required.

5.7. Link Density, Storage Modulus, and Slip-LinkMapping. Rheological estimates of Me from plateaumodulus measurements are in very good agreement with Me

Figure 10. Ratio between selected links, zsel, and average number oflinks per chain ⟨z(t)⟩ for various time gaps τg,max. Filled symbolscorrespond to ratios dictated by the pairwise parameter, q.

Figure 11. Distribution of the persistence of selected links. Themaximum time gap, τg,max, is adjusted so that zsel = q⟨z(t)⟩, (filledsymbols in Figure 10). In the inset the maximum time gap is adjustedso that zsel = ⟨z(t)⟩. The complementary distribution of discarded linkscan be found in the Supporting Information.

Figure 12. Intermittent link trajectories that correspond to ef fectivebinary links, discussed in section 5.5. They have been selected by usinga maximum time gap such that zsel = q⟨z(t)⟩. (a) EG-melt (C500);(b) rubber. All the trajectories have a link persistence τ ≃ 0.45. τg,max is0.7τe for the rubber and one τe for the EG-melt (C500). Thetrajectories are plotted with different colors and parallel to each other.The vertical axis does not provide any kind of information.



estimates56,59,62−65,79 from the Kuhn segment of the PPs, asobtained from atomistic systems. However, this length scale islarger59,67 than the mesh length,113 MTC, of the networkspresented here, i.e., Me ≃ 2 to 2.5MTC, leading to a link densitywhich is larger114 than the “rheological” entanglement density.In the language of this paper, MTC is the average molar mass

between the successive, ⟨z(t)⟩ on average, links per chain. Thestatistical properties of these networks have been presentedelsewhere.59 Conformationally speaking, the larger link densityis due to exponentially decaying orientational correlations59

along successive steps of the PP. A slowing down of Rousemodes conforming to the mesh length MTC of the networks hasalso been observed.82,109

Hence, a one-to-one mapping of our networks to slip-linkmodels employing the rheological entanglement density, isnot viable. However, given that the links here have variousstrengths, a way to resolve this issue is to consider the PPlength as obtained from the networks, but for the slip-linkmotion under deformation, flow, or in equilibrium, to consideronly a subset of very strong links. For example, in Appendix IIwe present the mean square displacement of the selected linkszsel = q⟨z(t)⟩, which was constructed by averaging over theirindividual real space trajectories.Then, a practical approach is to adjust τg,max of the previous

section, so that the selected link density matches the rheologicalentanglement density. Applying this scheme here, we foundthat only the very strong links, with a persistence τ > 0.96, wereselected (see below). The required τg,max is very small,approximately 0.1τe, for both systems. It remains to be shownthat the spatial statistics of these links provide a meaningfulmesoscopic picture. This approach seems promising as a simplecoarse graining scheme for bridging the gap between atomisticand slip-link models. We note that this scheme is self-consistent,since τg,max is adjusted by the Me obtained from the Kuhnlength of PPs. Alternatively, τg,max can be adjusted so that theselected link density matches a relaxation modulus obtainedfrom simulation.Evidently, the most important question is how the trapped

links revealed here contribute to the low frequency storagemodulus of an elastomer. For example, we have seen that themajority of the links blink instead of being continuously present(section 5.2). The blinking is due to stored length fluctuations(Appendix V), which at the network level lead to thermal PPfluctuations and considerable idle times in certain linktrajectories (Figure 4). Stored length fluctuations can “relax”the constraining effect of these links, either fully or partially.Most probably, weak links can be fully relaxed so that they donot contribute to the modulus, while strong links can onlypartially relax. In this respect, the persistence of each link isindicative of its contribution to the low frequency modulus.To investigate this issue further, we define the cumulative

distribution function (CDF) of the persistence of links

∫τ τ τ = ′ ′τ

p p( ) ( ) dfcd

1

(21)

where p(τ) is integrated backward, from the right to the left.pcdf(τ) is the fraction of U links with a persistence τ ≤ τ′≤ 1.We also introduce wcdf(τ), the CDF corresponding to w(τ),which is defined similarly to pcdf(τ). wcdf(τ) is the fraction ofconfinement strength provided by links with a persistence τ ≤τ′≤ 1. Both distributions are shown in Figure 13. From theform of the CDFs, we see that a τ-scale separating the links intostrong and weak is not obvious. However, we observe that the

two wcdf(τ) superpose, and that there are two regimes. Fromτ ≃ 1 to τ ≃ 0.95, the confinement strength provided by theselinks increases abruptly, while below τ ≃ 0.95 it follows asmooth increase.A way to examine Figure 13 with respect to relaxation effects

is the following. In the inset we present a table with therequired maximum time gap, τg,max, so that the number ofselected links, zsel, matches certain link densities discussed inthe article. Z = (⟨Lpp⟩/d − 1), where ⟨Lpp⟩ is the average PPcontour length, and d is the PP Kuhn length obtained bymapping the PPs to random walks59 (see Appendix II). Zcorresponds to the number of “rheological” entanglements, i.e.,Z = (M/Me) − 1, and is much smaller than ⟨z(t)⟩.It is meaningful to assume that links with a large τg,max can

relax to a larger degree than links with a small τg,max. Forexample, we can safely consider that the links with τg,max ≥ τR′,where τR′ is the longest monomer relaxation time in the sys-tem (Appendix II), do not contribute to the low frequencymodulus. On the other hand, links with τg,max ≤ τe can con-tribute to the modulus, but they can also partially relax,possibly to a different degree. Grossly speaking, we can assumethat at low frequencies links with τg,max > τe become fullyrelaxed. We envisage that links which disappear and reappear,with idle times longer than τe, have been relaxed by storedlength fluctuations.To proceed further we need to associate a pair of (zsel, τg,max),

to a link persistence. This can be done by associating zsel to aspecific τ, such that the fraction of the U links with τ ≤ τ′≤ 1is zsel. That is, Upcdf(τ) = zsel. For example, each symbol ofFigure 13 is plotted at a τ associated with a specific pair(zsel,τg,max) of the table in the inset. We see that the fraction oflinks with τg,max < τe is approximately 20%, with τ > 0.65, for bothsystems (circle symbols). These links provide approximately 70%of the confinement strength (see wcdf(τ)), they are q⟨z(t)⟩ innumber, and they can potentially contribute to the low frequencymodulus. In Figure 13 they are enclosed in a gray shaded area.The rheological entanglement density corresponds to Z very

strong links with τg,max ≃ 0.1τe and τ > 0.96 (square symbols).These links provide only 30% to 40% of the confinement

Figure 13. Cumulative distribution function of p(τ) and w(τ) (inset),which were presented in Figure 7. The symbols are put at the specificτ’s for which Upcdf(τ) = zsel, according to the table of the inset (seetext). The curves should be examined from the right to the left. Theshaded area encloses the spectrum of τ’s corresponding to links whichcan potentially contribute to the low frequency modulus of anelastomer or to the plateau modulus of a melt.



strength. Interestingly, the full confinement strength is providedby links with τg,max < τR′ (diamond symbols). The above resultsdemonstrate that weak links, though essentially trapped, do notcontribute to the low frequency modulus, in contrast tocommon belief. It seems that stored length fluctuations canfully relax weak topological interactions. The role of thesefluctuations in the viscoelasticity115 of elastomers and melts iscentral. Regarding the contribution of weak links to the plateaumodulus of a melt, the above discussion is still valid.5.8. Discussion. We have seen that the pairwise

uncrossability constraints detected as kinks by chain shrinkingmethods, such as the Z1,58,79 CReTA59 algorithms, or theprimitive path analysis method,56,116 can be either strong orweak interactions, and not detected at all times (link blinking),even in the absence of CR. The entanglement environment iscollective, and certain binary interactions are more importantthan others. Thus, under nonlinear deformation or flow, wheresome kind of entanglement network evolution takes place (dueto CR), it is possible that changes in the number of detectedkinks73,116,117 do not reflect the actual microscopic changes ofthe entangled state of the system.

6. SUMMARY AND CONCLUSIONSIn molecular theories of rubber elasticity and polymer rheology,the microscopic entanglement constraints are replaced with aconfining mean field, that restricts laterally chain dynamics andfluctuations. Here, we examine in detail at what level this meanfield is a collective entanglement effect, as in tube model theories,or can be described through certain pairwise uncrossabilityinteractions, as in slip-link models (see p. 552 in ref 20).To this end, for the average single chain, the entanglement

environment was expanded to a set of U local links. Each linkrepresents a pairwise uncrossability interaction with a dif ferentneighboring chain, resembling a familiar slip-link. Since in oursystems all entanglements are trapped, we claim that the Upairwise interactions form a complete set, and that the entangledstate of the system, at any time, can be projected to them. Evenif some interactions have been left out of the analysis, it isshown that they are too weak to be considered.The number of the U interactions, or links, is very large,

which means that the entanglement environment is inherentlycollective. However, their confinement strength varies fromstrong to weak, and the majority of them are weak. The plethoraof these links is due to the fact that primitive paths (PPs) arefluctuating objects. Thus, as a PP fluctuates, it collides with thecontours of other PPs and forms local links with them. Thissampling mechanism reveals that weak links appear asintermittent, blinking links, because they are not continuouslysampled. However, there also exist strong links which are sampledcontinuously, and appear as permanent constraints. The weaklinks can “disappear” for time intervals longer than τe, thus theydo not contribute to the low frequency modulus of an elastomer,or the plateau modulus of a melt. Overall, the average effect ofthese interactions is to confine a chain through a much smallernumber of mean-field type, permanent links, ⟨z(t)⟩, which inprinciple represent many other pairwise interactions.The multichain, or pairwise character of these mean-field links

was quantified by defining a pairwise parameter, and it was shownthat the pairwise character dominates. Moreover, for the systemsstudied the pairwise parameter is entanglement density and chainlength independent. These results imply that the realization of aconfining mean field with binary interactions, as in slip-link models,is not in contrast with a collective entanglement environment.

On this basis, we proposed a self-consistent scheme formapping the entanglement environment to a slip-link model.The scheme discards weak links and selects the strongestbinary interactions conforming to a given entanglement density.It is directly applicable to trapped entanglements, and to theanalysis of MD deformation trajectories of bead−spring andatomistic polymer networks. From such studies, available slip-link network models can be refined, corrected, or extended, in ahierarchical manner. With suitable extensions,103 the tracking ofconstraint release events in polymer melt simulations, seemsalso possible. A list of symbols and abbreviations for this paperis given in Table 1.

■ APPENDIX I: CONSTRUCTION OF A DEFECT-FREEENTANGLED PE NETWORK

In flexible polymers, the degree of chain overlap is controlled118

by density and stiffness. Therefore, it seems unusual that the twosystems examined here (rubber and end-grafted melt) have adifferent degree of entanglement, though the slightly larger C∞ ofthe rubber would decrease118 Me, in comparison to the melt.Since both systems are well equilibrated, the difference must bedue to the short subchains and the artificial preparationconditions of the network. The latter is constructed as follows.

Table 1. List of Symbols and Abbreviations

Nch number of chains in the systemα chain index, 1 ≤ α ≤ Nch

τe entanglement timeτR Rouse timeMe entanglement molecular weightT observation time (OT)τ linking time (LT)τ = τ/T link persistencezα(t) number of different links or linked chains (LCs) constraining

α at time tUα(t) cumulative number of the different links sampled by α up to

time tzα(t)|zα(0) subset of links constraining α at both times, 0 and t⟨zα(t)|zα(0)⟩ zα(t)|zα(0) averaged over multiple time originsqα(t) ratio of ⟨zα(t)|zα(0)⟩ over ⟨zα(t)⟩z(t), U(t), q(t) chain averages of the corresponding quantities aboveq pairwise parameter, limt→∞q(t)p(τ) number distribution of link persistencew(τ) weight distribution of link persistenceτn,α average persistence of links of chain α

τn average link persistenceτw,α weight-average persistence of links of chain α

τw weight-average link persistenceτα,β persistence of a link between chains α,βwαβ(ταβ) confinement strength of a link with persistence ταβPαβ probability that chain α is linked to chain β

τg,max maximum time gap of link trajectorieszsel number of selected links per chainPP primitive pathTC topological constraintLC linked chainLLS lateral link samplingCDF cumulative distribution functionEG-melt end-grafted meltEC-melt end-constrained meltEL-CR end-looping constraint-release



At the nodes of the diamond lattice we place carbon atoms. Thelattice constant is then adjusted so that the carbon atoms arebonded with PE chains at full extension (all-trans state of dihedralangles). This is an unentangled state. Then, the network is let torelax at 1 Atm. Because of entropic elasticity and cohesiveinteractions it contracts to the melt density. At this stage thesystem is not equilibrated conformationally. To imposeequilibration a phantom chain simulation at melt density (NVTensemble) is performed. Bead overlaps are then eliminated by thegradual push-off99 method.Entanglements are introduced dynamically after the chains

have been linked to a network, by letting them interpenetrate atthe phantom chain simulation stage. It seems that thisprocedure, together with the short length of the chains, leadsto a larger degree of chain overlap in comparison to the meltstate. We expect that in networks of very long subchains, wherethe cross-link density vanishes, this effect would subside. It canalso be avoided through random linking of a melt system, at thecost of introducing network defects. On the other hand, the factthat our results have been obtained with different entanglementdensities makes our conclusions fairly general.A different path to defect-free networks has been followed by

Everaers.50,119 It invokes constructing identical interpenetratingdiamond networks, where the lattice constant is set to the rootmean squared end-to-end distance of melt chains. The systemconsists of as many networks as are required to reach the meltdensity, which is held fixed. As in our case, equilibration isachieved by letting the networks interpenetrate at a phantomchain simulation stage. Chain overlaps are then eliminatedthrough a gradual99 “push-off” stage. In comparison to ournetworks the chains can also entangle with chains from otheridentical networks.

■ APPENDIX II: MEAN SQUARE DISPLACEMENT ANDCHARACTERISTIC RELAXATION TIMES

In Figure 14, we present the mean square displacement (msd),g1(t), of middle chain monomers (10% of each chain), reduced

by t0.5. Four regimes are discernible. The first one isunimportant. It is usually called ballistic. A second regime,scaling as t0.5, corresponds to free Rouse motion within the tube.The next regime is indicative of tube confinement (restrictedRouse motion along a random-walk like tube), and it scales,

approximately, as t0.28, for both systems. The corresponding tubemodel prediction7 is t0.25. The last regime corresponds to aninvariant msd. From the first crossover (between second andthird regime) we estimate τe, the time that a monomer “hits” thetube for the first time. It is 1.3 ns for the rubber and 1.7 ns forthe EG-melt. From the second crossover we estimate τR′, thelongest monomer relaxation time of each system, which is 9.8 nsfor the rubber and 11.5 ns for the EG-melt. This timecorresponds to the Rouse time, τR. Here, it is denoted with aprime, since the boundary conditions for a Rouse chain aredifferent (free chain ends). For the (C1000) EG-melt, thecorresponding crossover times are 1.6 and 19.4 ns (not shown).In Figure 15, we present the msd of middle chain monomers

together with the msd of links selected according to the

pairwise parameter, i.e., zsel = q⟨z(t)⟩. The plateaus indicate theabsence of reptation and long chain diffusion. We see that theselected links span much shorter distances than real chainmonomers. Thus, we can say that the enclosing tube of PPs is ashrunk version of the alleged tube that restricts lateral motionof real chain monomers.The bars plotted parallel to the plateaus correspond to

estimations of certain length scales related to entanglementconstraints. d is the tube diameter estimated from the Kuhnsegment of the PP as d = ⟨R2⟩/⟨Lpp⟩, where ⟨R2⟩ is the meansquare end-to-end distance. ⟨dTC

2⟩ is the mean square edgelength of the link network.59,81 wpp

2 is an estimation of themean square distance between PP contours59,81 from the in-verse of the PP contour length density. It is defined as wpp

2 =⟨V⟩/(Nch⟨Lpp⟩), where ⟨V⟩ is the average volume. Note thatthe average distance between PP contours is smaller than theaverage distance between connected links along the PPs, ⟨dTC⟩.For example, in Figure 5, the distance between PP contours, h,is much smaller than the PP segments with average length⟨dTC⟩.For both systems, we see that the msd plateau of the

monomers shows a better correlation with d2/4, whichcorresponds to the squared radius of the tube, but is actuallylarger than that. This is not surprising, since real chain

Figure 14. Mean square displacement of middle chain monomers(10%), reduced by t0.5, for the rubber and the (C500) EG-melt. Fromthe crossovers we estimate the shortest and longest relaxation time ofeach system.

Figure 15. Mean square displacement of middle chain monomers, andselected links. Key: (a) EG-melt (C500); (b) rubber. The links wereselected according to the condition zsel = q⟨z(t)⟩. The bars parallel tothe plateaus show certain length scales related to entanglementconstraints (see text).



monomers can penetrate the tube walls, as parts of unentangledloops for example. In both systems we also observe acorrelation between the squared tube radius and the meanedge length of the link network. The msd plateaus of selectedlinks are much smaller than these quantities, which means thatlink fluctuations are restricted to the inner tube region alongthe tube axis. These plateaus are also larger than wpp

2, especiallyin the EG-melt, which means that the fluctuating PPs cancollide to each other, in accordance with the lateral linksampling mechanism.

■ APPENDIX III: BINARY LINKS, LOCAL KNOTS, ANDTERNARY NODES

As shown in Figure 1, each node links locally two chains, exceptfrom few ternary nodes (see below). Within this terminology,each node resembles a binary link that can be resolved into twoTCs along two “mate chains”. Iwata and Edwards havediscussed88,89 melt dynamics and rheology in terms of localknots which are conceptually similar to the ones presentedhere. The knots were defined at the real chain level by using astopological criterion a local form of the Gauss linking number.For open chains this criterion is not more rigorous53,52,120 thanthe one defined here, while it is computationally more complex.In our networks about 5% of the nodes are ternary. That is,

there exist situations where three chains, let us say α, β, and γ,are mutually constrained and form local links with each other(see videos). Detection of these cases is hard and needsspecial103 attention. They are mapped to nodes with six edges,which are further resolved103 into three links, αβ, αγ, βγ, so thatall three chains are mutually coupled in pairs. With this choice,the mapping to a network leads exclusively to binary links, as inslip-link models. Information about ternary nodes is retrievable,though.Upon dynamical evolution of the network, most of the

ternary nodes tend to break up into consecutive binary nodesalong the same chain, e.g., αβ, αγ, in the former example. Thereverse procedure is also possible (see videos). A statisticalanalysis59 has shown that between successive links of a chainthere exists an effective repulsion. Thus, we expect that, whileternary links certainly appear, they are transient. Interestingly, arepulsive interaction also emerges when enforcing121 detailedbalance in the dynamical equations of the slip-link basedprimitive chain network model.

■ APPENDIX IV: MULTIPLY LINKED CHAINSIn Figure 16, we show the normalized probability, P(n), that achain α forms n local links with a mate chain in zα(t),instantaneously, at time t, (see also Figure 2). For the rubberand the short chain EG-melt P(n) is highly peaked at unity.Therefore, the simplification of dealing with LCs instead ofindividual links is justified. However, as chain length increasesP(1) decreases. Thus, in systems with long chains the analysisshould be made against individual links.

■ APPENDIX V: LATERAL LINK SAMPLING ANDCReTA MAPPING

The large set of different chains sampled by the PP is rathersurprising. Since the PPs are determined by contour reduction,then, by construction, the sampled chains have portions insideor at the boundaries of the pervaded volume of a referencechain. Moreover, they are not just simple chain contacts, sincelinked chains satisfy the local topological criterion of Figure 1,

(although some very weak links could be considered asfluctuating chain contacts). The lateral link sampling (LLS)mechanism that takes place is simple and was demonstrated inFigure 5.In order to provide more evidence for this mechanism, we

first define the reduced curvilinear coordinates of a link, sα(t),sβ(t) between chains α,β. These are defined such that 0 ≤ s ≤ 1.That is, sα(t) = xα(t)/Lα(t), where xα(t) is the curvilinearcoordinate of the link along chain α, and Lα(t) is the PPcontour length. Likewise for sβ(t). Note that a local link canslide along PP contours, hence the time dependence of sα, sβ.In Figure 17, we plot sα(t), sβ(t), for all the links that were

first sampled at a time t > 0. In Figure 4, these links correspondto the trajectories with Uα(t) > Uα(0), (continuous or

segmented trajectories above the column to the left of t = 0in Figure 4). For each one of these links, we plot a single pointthat denotes sα(t), sβ(t), at the time the link was first sampled.Beyond some explainable heterogeneities, the homogeneouspatterns support the LLS mechanism of Figure 5. The EG-meltdata are less dense due to the smaller number of simulatedchains.

Figure 16. Normalized probability that a chain forms n local links withanother mate chain.

Figure 17. Reduced curvilinear coordinates of the links sampled by themechanism of Figure 5: (a) EG-melt (C500); (b) rubber. In order toincrease statistical accuracy we exploit head to tail symmetry.Coordinates with s > 0.5 are plotted at |1 − s|.



Data concentration at the lower left corner of Figure 17bimplies that many of the new links are formed between chainends. This is explained by the higher probability ofinstantaneous link formation between PPs attached to thesame tetrafunctional cross-link of a network. In the EG-melt,local links with sα ≃ 0 and sβ ≃ 0 are favored. This is an artificialeffect, due to the less mobile chain ends because of grafting.The fluctuating PPs have higher probability to collide withcertain PP segments which are relatively immobile, or attachedto fixed points. In systems with very long chains theseheterogeneites would disappear.The physical origin of the LLS mechanism is the thermal

motion of real chains and, more specifically, the thermal motionof stored length,7 or chain slack. The spatial distribution of thislength is different at different time-snapshots of our systems. Inslip-link models this “redistribution” is taken into account bythe thermal motion of links and by a random transfer of Kuhnsegments between neighboring strands of the PP. Similarmechanisms apply in our simulations. Since CReTA operatesby consuming chain slack, the spatial redistribution of storedlength leads to associated thermal fluctuations of PP conforma-tions (in real space), and transfer of monomers between thestrands of the PP. As a result, there exist small displacements ofnetwork nodes which lead to the LLS mechanism.As regards the mapping of a specific spatial distribution of

stored length to a network of links by CReTA, this is notunique. CReTA minimizes the total contour length of thesystem. Assuming that there is a unique, “global” minimum,the value of the objective function will hardly be changed if weremove some small length from a PP and add it to another. Thestochastic nature of the reduction performed by CReTA issubject to this effect. At the network level, this will lead to smalldisplacement of nodes, and therefore, possibly, to link samplingby the mechanism of Figure 5.Thus, individual CReTA reductions of the same config-

uration (same spatial distribution of stored length), performedwith different pseudorandom numbers, may sample, each time,a slightly different set of links, with practically the same shortestpaths (PP conformations and length). This uncertainty isinherent to the reduction problem. Equivalently, one could saythat the stored length can only be mapped to a small irregularvolume around the centerline of the corresponding tube (seeFigure 18), and not to a specific path. This volume encloses asubset of the links which define the tube constraint and haslarge overlap with the tube core. An individual reduction byCReTA, then, will sample a path line in this volume. The linksof the subset which are inside the tube core will be sampledwith higher probability. This kind of inherent “noise” in thenetwork mapping does not affect any of our conclusions. It isvery small103 relative to the thermal noise driving the samplingof the links that assemble the tube constraint.

■ APPENDIX VI: ENTANGLEMENT CONSTRAINTS ATTHE LEVEL OF SHORTEST PATHS AND LOCALLINKS

A full view of what happens at the link level requires theknowledge of the distances spanned by fluctuating PPs. A shortaccount is given in Appendix II. A more detailed presentationwill be given elsewhere.103 For the systems studied here, itsuffices to say that, as time proceeds, the PP segments cover anincreasing distance (through lateral fluctuations), until theyhave fully sampled their corresponding tubes. Because of thecoarse graining procedure they span shorter distances than the

real chain monomers. Thereby, their enclosing tubes are shrunkversions of the real chain tubes, and while they restrict lateralPP fluctuations they do not restrict lateral monomerfluctuations. We could say that a PP samples the domainaround the centerline of the real chain tube. The picture thatemerges is described in Figure 18.

■ APPENDIX VII: CONSTRAINT RELEASE EFFECTSON q(t)

In Figure 19 we present q(t) for the (C500) melt, but this timeanalyzed as an end-constrained melt (EC-melt). That is, insteadof grafting chain ends to nearby chains, the MD trajectory was

Figure 18. Schematic representation of entanglement constraints,at the level of local links. A chain α in a system with fixed tubes isconsidered. The PP is a fluctuating object outlined with a black line.The dashed lines denote parts of the PP which are not shown. Theopen and filled circles denote the fluctuating PPs of other chains,which cross the figure plane at various directions. Because of lateralfluctuations of all PP conformations, the PP of α can collide withthe other PPs denoted by circles, and form local links with them.Therefore, the PP of α samples its environment by tracing afluctuating path through these links. The sampling of links inside theshaded tube, which are Uα in number, does not violate uncrossabilityconstraints. Links outside the tube cannot be sampled, or are tooweak to be considered. After a certain time, which for the systemsstudied is T ≃ 40−50τe, (and depends on entanglement density),the PP has sampled all the Uα links within its tube. Instantaneously,the PP is linked to zα(t) links, a subset of Uα. The linking time (LT)of individual links, τ, varies significantly. In the figure, the tubeconstraint is colored blue along the tube core and light blue on theexterior, conveying the fact that the inner links (filled circles) aresampled more frequently than outer ones. Thus, there exist strongand weak links, and the tube constraint is applied collectively by all ofthem. In the provided videos weak links appear as intermittent,blinking nodes. The strong links, making the tube core, appear asnonblinking, continuously present nodes. The diameter of thedepicted tube is much smaller for PP segments,103 than for real chainmonomers (Appendix II), which are not restricted by the shadedtube. The mean distance between the circles is proportional to theaverage distance between PP contours. Note that this distance issmaller than the average mesh length of the link network. The formerdistance is based on the density of sampled links Uα, which is largerthan the link density defined by ⟨zα(t)⟩ (section 5.2). Effectively, thismeans that the average distance between the circles is smaller thanthe average distance between the filled circles along the tube axis. Forexample, in Figure 5, the distance between PP contours, h, is muchsmaller than the PP segments with average length ⟨dTC⟩, which onthe average make the tube axis. The sampling of links is described inFigure 5, and in Appendix IV.



generated by keeping all chain ends fixed. This difference has astrong effect on q(t) due to the presence of a CR mechanism,which can be called end-looping constraint release (EL-CR). Apolymer chain, say chain A, can disentangle from a mate chain,say chain B, by developing a lateral unentangled hernia thatpasses over a chain end of B. That is, by hopping of an internalportion of A over one end of B, through Rouse motion (seeFigure 7 in the work of Zhou80 and Larson). Here, we see thatEL-CR reduces q(t) to a lower plateau value, which becomesdiscernible at t/τe ≃ 100. However, a safe estimation of thisplateau would require a longer observation time.Grossly speaking, the EC-melt can be viewed as the

EG-melt with some links not being trapped, but subject toupdating by EL-CR. This is similar to a network with danglingchains, i.e., with both trapped and temporary links. In such asystem q(t) will decay toward a certain plateau, characteristic ofthe density of trapped entanglements. In a melt with free chainends (exclusively temporary links), q(t) will eventually vanish.The plateau value for the EC-melt will be smaller than that ofthe EG-melt (smaller trapped entanglement density). However,the pairwise parameter is defined as the plateau of q(t) when asystem is examined with all of its entanglements being trapped(fixed tubes). For this purpose, as explained in section 3, CR iseliminated by grafting all free chain ends. Therefore, a plateauin q(t), in the presence of CR, does not correspond to apairwise parameter for the entanglement environment. Thedecay of q(t), or the difference between the EG-melt and EC-melt plateau, can possibly provide some kind of CRinformation, but this issue was not examined here.

■ ASSOCIATED CONTENT*S Supporting InformationEntanglement network views of a PE melt (Figure S1), linktrajectories for representative chains (Figure S2), number andweight distribution of link persistences, for increasingobservation time T (Figure S3), number-average and weight-average link persistence as a function of increasing T (FigureS4), distribution of unselected link persistences (Figure S5),magnified section of Figure 13 (Figure S6), and 15 explanatoryvideos for the material discussed in the document, with anassociated description. This material is available free of chargevia the Internet at http://pubs.acs.org/.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected].

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The authors would like to acknowledge collaborative work withMark Robbins, Ting Ge, and Robert Hoy on the problem ofquantifying entanglement loss in crazed polymer glasses, whichhelped in posing some of the problems examined here. We alsowould like to thank Athanasios Morozinis for sharing the MDtrajectories used in this study. The work of C.T. and D.N.T., inthe context of this paper, is part of the Research Programme ofthe Dutch Polymer Institute (DPI), Eindhoven, The Nether-lands, Project No. 650. S.D. Anogiannakis is thankful to theNational Technical University of Athens for a PhD fellowship.

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Figure 19. Fraction of links at time t = 0, which are also present attime t, for the end-constrained melt (C500) generated from the samemelt configuration as the EG-melt (C500). The squares denote q(t)estimations from the weight-average link persistence, as described insection 5.6.



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(107) By estimating the ratio of chain length over (⟨z(t)⟩ + 1), whichis proportional to the entanglement molar mass, we can compare thelink density between the two systems. In the rubber it is about twicethat of the EC-melt. Estimating Me from the Kuhn segment of the PPswe are led to the same conclusion. Note that the link density of oursystems is larger than the rheological entanglement density, asdiscussed in section 5.8.(108) Padding, J. T.; Briels, W. J. J. Chem. Phys. 2001, 115, 2846−2859.(109) Padding, J. T.; Briels, W. J. J. Chem. Phys. 2002, 117, 925−943.(110) On this point, strong evidence is provided by q(t), introducedin section 5.5, which reaches a plateau at times where U(t) is stillincreasing.(111) A chain average over these quantities satisfies the relation⟨Uτn⟩ = ⟨U⟩⟨τn⟩.(112) Note that, as explained in Appendix VII, the pairwiseparameter can only be defined in a system where all entanglementsare trapped.(113) In ref 59, MTC is denoted as MES.(114) Everaers, R. Phys. Rev. E 2012, 86, 022801.(115) Edwards, S. F.; Takano, H.; Terentjev, E. M. J. Chem. Phys.2000, 113, 5531−5538.(116) Leonforte, F. Phys. Rev. E 2010, 82, 041802.(117) Mahajan, D. K.; Singh, B.; Basu, S. Phys. Rev. E 2010, 82,011803.(118) Graessley, W. W.; Edwards, S. F. Polymer 1981, 2, 1329−1334.(119) Everaers, R. New J. Phys. 1999, 1, 12.1−12.54.(120) Millet, K.; Dobay, A.; Stasiak, A. Macromolecules 2005, 38,601−606.(121) Uneyama, T.; Masubuchi, Y. J. Chem. Phys. 2011, 135, 184904.



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