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Equilibration of Long Chain Polymer Melts in Computer Simulations

Rolf Auhla, Ralf Everaersa∗, Gary S. Grestb, Kurt Kremera and Steven J. PlimptonbaMax-Planck-Institut fur Polymerforschung, Postfach 3148, D-55021 Mainz, Germany

bSandia National Laboratories, Albuquerque, New Mexico 87185 USA

Several methods for preparing well equilibrated melts of long chains polymers are studied. Weshow that the standard method in which one starts with an ensemble of chains with the correctend-to-end distance arranged randomly in the simulation cell and introduces the excluded volumerapidly, leads to deformation on short length scales. This deformation is strongest for long chainsand relaxes only after the chains have moved their own size. Two methods are shown to overcomethis local deformation of the chains. One method is to first pre-pack the Gaussian chains, whichreduces the density fluctuations in the system, followed by a gradual introduction of the excludedvolume. The second method is a double-pivot algorithm in which new bonds are formed across a pairof chains, creating two new chains each substantially different from the original. We demonstratethe effectiveness of these methods for a linear bead spring polymer model with both zero andnonzero bending stiffness, however the methods are applicable to more complex architectures suchas branched and star polymer.

PACS Numbers: 61.41+e

I. INTRODUCTION

A systematic investigation of the structure-propertyrelations for polymeric systems by computer simulationsrequires the preparation of equilibrated melts of long,entangled chains. For temperatures well above the glasstransition, this can, in principle, be achieved using suf-ficiently long molecular dynamics (MD) or Monte Carlo(MC) simulations. However since the longest relaxationof an entangled polymer melt of length N scales at leastas N3, giving at least N4 in cpu time, this method isonly feasible for relatively short chain lengths. Depend-ing on the detail of the model the longest chains whichcan presently be equilibrated by brute-force MD or MCsimulations are on the order of 2-7 entanglements lengthsNe. Using united atom potentials for polyethylene inwhich one treats the carbon and its bonded hydrogen as asingle united atom, it is possible to equilibrate high tem-perature (T & 450K) melts for chains of the order of ap-proximately 2Ne or about 200 monomers.1,2 For coarse-grained bead-spring models like the one employed hereit is possible to study longer chain lengths, on the orderof up to 500 monomers or approximately 7Ne.

3 Howeverthis reaches the very limits of present day fastest comput-ers. An increase of cpu speed by an order of magnitudedoes even not allow for a doubling of the chain length.While the present chain lengths are sufficient to followthe dynamics well into the reptation regime, they arenot long enough to study structure-property relations,such as the plateau modulus Go

N . This would requirechains of order 10 − 20Ne. The situation is even worsefor branched polymer melts or polymers at interfaces,

∗present address: Max-Planck-Institut fur Physik komplexerSysteme, Nothnitzer Str. 38, 01187 Dresden, Germany

where equilibration times even for relatively short chainsare prohibitively large to use brute force simulations toproduce equilibrated melts.Fortunately to produce equilibrated samples, there is

no need to follow the slow physical dynamics of entan-gled polymer melts. One possibility is to temporarilyturn off the excluded volume interactions and to allowthe chains to pass through each other. In order to ob-tain a well-defined Monte Carlo algorithm, it is useful tocombine this idea with parallel tempering. While it hasbeen demonstrated that this is feasible in principle,4 themethod has so far not proven particularly efficient, themain reason being the large amount of computer timespent on unphysical Hamiltonians. More conventionalMC algorithms which can be used to equilibrate poly-mer melts include reptation moves (generalized slitheringsnake algorithms),5 configuration bias algorithms,6–8 andconcerted rotation algorithms.8–10 Being relatively lo-cal, these methods work best for moderate chain lengthsand densities. The complete equilibration of very longchain melts still requires long runs. The fastest method,the generalized slithering snake algorithm, theoreticallyscales as N2. An alternative are algorithms, which areable to move large sections of a chain at once. Theprototype for such methods is the pivot algorithm11–15

in which a monomer is chosen at random and one ofthe two segments formed by that partitioning is piv-oted rigidly in a random direction about the unit. ABoltzmann weight is used to determine if the move is ac-cepted or not. This method is highly efficient for singlechains in a continuum, implicit solvent. Unfortunately,a direct application to dense melts is not feasible sinceany large scale conformational change is bound to vio-late the packing constraints imposed by the chain ex-cluded volume. However it is possible to introduce adouble bond crossing algorithm in a way that two newbonds or bridges are formed across a pair of chains, creat-

1

ing two new chains each substantially different from theoriginal.2,10,16–18 For the bead spring model concernedhere this can simply be done by cutting two bonds andintroducing two new bonds as discussed in more detailin Sec. VII. For atomistic models, the bond length ismuch shorter then the diameter of a monomer, and itis necessary to construct new bridges between the chainsegments. Karayiannis et al.17,18 have shown that byconstructing two trimer bridges between the two prop-erly chosen dimers along the backbone one can quicklyequilibrate long chain polyethylene melts. For the twochains involved this is essentially a double-pivot move,where each chain experiences such a pivot rotation.In addition to improved equilibration algorithms there

is an obvious interest in methods for generating ini-tial melt configurations which are as close to equilib-rium as possible.19–24 In our previous work on polymermelts,3,19,25,26 we prepared most melts by first generat-ing an ensemble of chains with the correct end-to-enddistance and randomly placing them in the simulationcell. We then introduced a weak, non-diverging excludedvolume potential in the form of e.g. a cosine potential,A(1 + cos(πr/21/6σ)), between non-bonded monomers,where r is the distance between two monomers and σ isthe bead diameter. The amplitude A was then increasedover a short time interval until the minimum distance be-tween monomers was sufficient to switch on the Lennard-Jones potential without creating numerical instabilities.In the first part of this paper we confirm observationsby Brown et al.21 that this method deforms the (longer)chains. As a consequence, long chain melts are not aswell equilibrated as originally believed. The deformationturns out to be strongest at short length scales and com-pletely relaxes away only after a time τmax(N), which isthe time for a chain to move its own size.21 This longtime is needed because the local chain-chain topologycan only relax by the slow physical dynamics. For lin-ear chains, τmax(N) ≈ τe(N/Ne)

3, where the entangle-ment time τe = τRouse(Ne) is the Rouse time of a chainof length Ne (experimentally the measured exponent iscloser to 3.4 than 3 except for extremely long chains).This however was exactly what we tried to avoid by thatapproach. Not surprisingly, the longer the chain lengthN , the more severe the effect is. For relatively shortchains, such as those investigated in our previous stud-ies of the crossover from Rouse to reptation dynamics(N ≤ 350), the simulations were run long enough that theresults are independent of the starting state.3,19,25 How-ever for longer chains, which cannot be run long enoughcompared to the longest relaxation time, this simple pro-cedure for generating starting states is inadequate.In the present paper we report results from an ex-

tensive effort to prepare well-equilibrated melts of bead-spring polymers with chain lengths ranging from N =350 to N = 7000. For this purpose we use both ap-proaches outlined above. First we show how to avoidthe local stretching with only minimal computational ef-fort by a suitable modification of our standard method.

Then we demonstrate the capacity of the double-pivotalgorithm to dynamically equilibrate melts of mediumsized chains of length up to N = 700. The paper is or-ganized as follows: In Sec. II we define the model. InSec. III, we present some simple theoretical estimates ofthe structure of bead-spring chains with different intrinsicbending stiffness in dense melts. In Sec. IV, we discussways to characterize the single chain structure and ex-tract suitable target functions for long chain melts fromlong simulations of short chain melts. In Sec. V we ex-amine carefully the standard procedure used in the pastto prepare melt configurations. We show that for longchains this method deforms the chains at short lengthscales and that these deformations relax completely onlyafter the chains have moved their own size. In Sec. VI,we present the new pre-packing procedure, which signif-icantly reduces the density fluctuations particularly forlong chains. This combined with a gradual introduc-tion of the excluded volume, results in well equilibratedlong chain melts in a reasonable amount of cpu time. InSec. VI we describe the double-pivot algorithm and com-pare results for the internal dimensions of the chain withthose produced from pre-packing and a gradual introduc-tion of the excluded volume. In Sec. VIII we test the twomethods for chains with local bending rigidity. A briefsummary of our main conclusions are given in Sec. IX.

II. MODEL

We use a coarse grained model in which the polymeris treated as a string of beads of mass m connected bya spring. The beads interact with a purely repulsiveLennard-Jones excluded volume interaction,

ULJ(r) =

{

4ǫ{

(σ/r)12 − (σ/r)6 + 1

4

}

r ≤ rc0 r ≥ rc

(1)

cutoff at rc = 21/6σ. The beads are connected by a finiteextensible non-linear elastic potential (FENE),

UFENE(r) =

{

−0.5kR2o ln

(

1− (r/Ro)2)

r ≤ Ro

∞ r > Ro

in addition to the Lennard-Jones interaction. The modelparameters are the same as in ref. 19, namely k = 30ǫ/σ2

and Ro = 1.5σ unless otherwise noted. The temperatureT = ǫ/kB. The basic unit of time is τ = σ(m/ǫ)1/2. Weperformed constant volume simulations of monodispersemelts at a segment density ρ = 0.85σ−3. The temper-ature was kept constant by coupling the motion of eachbead weakly to a heat bath with a local friction Γ. Un-less otherwise specified Γ = 0.5m/τ . The equations ofmotion were integrated using a velocity Verlet algorithmwith a time step ∆t = 0.012τ . The average bond lengthis < b2 >1/2= 0.97σ. The polymer melts studied con-sisted of M chains of length N beads. The chain lengths

2

studied varied from N = 25 to 7000. The number ofchains are each system is specified in Table I.Equilibration algorithms for polymer melts which fol-

low the physical dynamics encounter a strong growth ofthe necessary equilibration times with chain length. Perunit of time τ , the simulation of a melt of M chains oflength N requires a computational effort of 102M × Nparticle updates. Since the longest relaxation time is ofthe order τmax ≈ τe(N/Ne)

3, equilibration of our largestsystem with M = 80 and N = 7000 would require102M × N(τe/τ)(N/Ne)

3 ≈ 8 × 1017 particle updates.Assuming a typical performance of 2.5× 105 particle up-dates per second, which is typical for this model on asingle processor, the required cpu time for the equilibra-tion is about 105 years. On a modern parallel cluster suchas the DEC alpha CPlant cluster at Sandia, the times aresomewhat less. On 64 processors, we get about 20 stepsper second for our largest system, which means that thetotal time for the chains to move on average their ownsize is approximately 1600 years. Increasing the numbersof processors helps somewhat but since the efficiency ofthe computation decreases as the number of processorsis increased for fixed system size, this type of brute forceapproach for long chain melts is clearly not feasible.In our previous studies, we considered only fully flex-

ible chains. However to demonstrate the effectiveness ofthe methodology, we also include a nonzero bending stiff-ness, which is modelled by

Ubend(θ) = kθ(

1− cos θ)

(2)

where cos θi = (ri − ri−1) · (ri+1 − ri), where (ri − ri−1)is the unit vector in the bond direction.

III. CHAIN CHARACTERISTIC IN THE MELT

In dense polymer melts excluded volume interactionsbetween different parts of a given polymer chain arescreened. Since in our model there are no explicit in-trachain interactions beyond those between neighboringbonds, the single chain structure should be well charac-terized by the expectation value 〈cos θ〉 of the bond angleθ. For example, the knowledge of 〈cos θ〉 is sufficient tocalculate an estimate for the mean-square end-to-end ex-tension 〈R2〉 of chain segments of length N monomersand n = N − 1 bonds,

〈R2〉(n) = nb2

(

1 + 〈cos θ〉

1− 〈cos θ〉−

1

n

2〈cos θ〉(

1− 〈cos θ〉n)

(

1− 〈cos θ〉)2

)

(3)where the asymptotic prefactor is referred to as

c∞ =1 + 〈cos θ〉

1− 〈cos θ〉. (4)

0 1 2 3 4

0

2

4

6

8

β kθ

c∞

FIG. 1. Estimates of the effective stiffness c∞ of our chainsas a function of the strength kθ of the bending potentialEq. (2). Eq. (5) (solid line) only accounts for the bendingenergy, while Eq. (6) (•) also considers excluded volume in-teractions between next-nearest neighbor monomers along thechain. The gray dots indicate estimates based on the analy-sis of simulation data for equilibrated melts of chain lengthN = 200− 350 for kθ = 0 and 50− 100 for kθ > 0.

The simplest estimate of 〈cos(θ)〉 only takes the barebending energy Eq. (2) into account. Neglecting smallvariations in the bond length around the mean value b =0.97σ, one finds:

〈cos θ〉 =

∫ π

0sin θ cos θ exp (−βUbend(θ)) dθ∫ π

0sin θ exp (−βUbend(θ)) dθ

=1 + e2βkθ (βkθ − 1) + βkθ

(e2βkθ − 1)βkθ, (5)

where β = 1/kBT . Eq. (5) is represented as a solidline in Fig. 1. However this result underestimates theeffective chain stiffness, since the chain cannot fold back.This effect can be accounted for approximately by mod-elling the chains as non-reversal-random-walks (NRRWs)with excluded volume interactions between next-nearestneighbors along the backbone. As a function of the bondangle θ their distance is given by lθ = |~ri+1 − ~ri−1| =b[2(1 + cos θi)]

1/2 so that

〈cos θ〉 =

∫ π

0sin θ cos θ exp (−β(Ubend(θ) + ULJ(lθ))) dθ∫ π

0sin θ exp (−β(Ubend(θ) + ULJ(lθ))) dθ

(6)

which has to be evaluated numerically (see the black dotsin Fig. 1). For kθ = 0 this gives c∞ = 1.76 compared toc∞ = 1 obtained from Eq.( III) and in good agreementwith simulation results for chains of length 20 ≤ N ≤350. This value is slightly smaller than the value c∞ =2.06 obtained at the Θ-point for the same Lennard-Jonesinteraction truncated at 2.5σ in an implicit, continuumsolvent.15

Ensembles of single chains which obey Eq. (3) can begenerated by simple sampling. This is particularly sim-

3

ple, if the allowed bond vectors are distributed evenlyover a solid angle with θ ≤ θmax. In this case

c∞ − 1

c∞ + 1= 〈cos θ〉θmax

= (cos θmax/2)2

(7)

so that the appropriate value for θmax can be easily de-termined for any desired effective stiffness.

IV. TARGET FUNCTIONS AND SUITABLE

WAYS OF CHARACTERIZING SINGLE CHAIN

STRUCTURE

The estimates of the chain structure presented in theprevious section were obtained within an effective sin-gle chain picture. In reality, the formal characterizationof a polymer melt is a complicated many-body problem.For example, we have completely ignored the influence ofpacking effects on the chain conformations. As was re-cently shown27 the local melt structure very sensitivelydepends on the ratio of bond length to effective excludedvolume of the beads. This effect is not well represented bythe value of the overall chain extension. Here we use ourstandard bond length/diameter ratio and focus on thesingle chain properties, however the methods discussedbelow apply to systems with different parameters as well.In the present case, sufficiently long MD or MC simula-tions represent the only “ab initio” method which allowsto take into account all interactions.In order to characterize the chain conformations in

the melt we mainly rely on mean-square internal dis-tances

⟨

R2⟩

(|i− j|, N) averaged over all segments of sizen = |i − j| along the chains, where i < j ∈ [1, N ] aremonomer indices. In addition we characterize deviationsfrom Gaussian statistics on short length scales using the

ratio⟨

R4⟩

/⟨

R2⟩2. Figures 1 and 2 show a compari-

son of simulation results for equilibrated melts and ournaive estimates. The results from the melts have beenrun sufficiently long that the chains have moved severaltimes their own size. For N = 350, kθ = 0, the run timeTt = 2.6 × 106τ .3 Clearly, the description of chains in amelt as simple freely jointed chains is too naive. Nev-ertheless, the deviations are not large and the measuredand estimated values for c∞ agree quite well. In particu-lar, the simulation data show neither unexpected featuresnor significant finite chain length effects.The rest of the present paper deals with algorithms

which try to circumvent the slow physical equilibrationpath. These algorithms will be judged according to twocriteria: their capacity to reproduce the target functionsand their performance. We mostly concentrate on thefully flexible case. In the end, we come back to the stiffersystems as a kind of “blind test”.

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FIG. 2. Mean square internal distances from long MD runsfor chain length N = 100 and 350 respectively for four valuesof the intrinsic bending stiffness: kθ = 0 (�), kθ = 0.75 (×),kθ = 1.5 (∗), and kθ = 2.0 (+). We refer to these data sets as“target functions”. For comparison, they are included as thickblack lines in subsequent, corresponding plots. The dashedlines show to a fit of Eq. (3) to the asymptotic behavior. Thecorresponding values for c∞ are included in Fig. 1.

V. STANDARD PROCEDURE FOR PREPARING

POLYMER MELTS

Our standard method for preparing melt conforma-tions is as follows:

1. Start from an ensemble of chains with the correctend-to-end distance R2(N) = b2c∞(N−1) on largelength scales. For a known c∞ this step is trivial.

2. Arrange the chains randomly in the simulation box.

3. Introduce a weak, non-diverging excluded volumepotential. A convenient form for the soft potentialis a cosine potential,

Usoft(r) =

{

A(1 + cos(πr/rc)) r ≤ rc0 r ≥ rc

between non-connected monomers with rc = 21/6σ.The initial amplitude of A = 4ǫ was linearly in-creased to a final values of A ≥ 100ǫ (“push-off”)over a short time interval of 10-20 τ . In the finalstate the inter monomer distances are large enoughto allow one to switch to the LJ potential withoutcreating numerical instabilities. During this phase,we often use a stronger coupling to the thermostatby increasing the friction constant to Γ = 2.0m/τ .In addition, the velocities of all monomers are set tozero every 50 time steps. This facilitates monomersthat are strongly overlapping to separate. We re-fer to this method in which the excluded volumeinteractions are introduced fairly rapidly as “fastpush-off.”

4

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1.2

1.4

1.6

1.8

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1 10 100 1000

<R

2 (n)>

/n

n

FIG. 3. Mean square internal distances after a fast push-offfor randomly packed phantom chains of length N = 25 (+),50 (×), 350 (∗) and 7000(�). The thick line is our targetfunction from Fig. 2.

4. Relax the system with the full LJ-potential by ashort MD run.

Figure 3 shows that this procedure actually deformsthe chains so that the melt is not equilibrated after step3. The deformations are strongest on short lengths.With increasing N only the amplitude but not the po-sition of the deformation along the chain changes. Themonomer displacements during step 3 are too small toaffect the conformations on large scales, allowing us totune them to preselected values. Since the largest lengthscales also take the longest time to relax, one might hopethat the MD relaxation in step 4 can be significantlyshorter than for a starting conformation of, say, com-pletely stretched chains. To be more specific, a naiveexpectation is that chain segments of length n ≤ N equi-librate on time scales comparable to the Rouse/reptationtime of chains of length n independent of N . Fig-ure 4 compares the result of equilibration runs of lengthTt = 104τ ≈ τmax(N = 90) to the target function inFig. 2. As expected, the internal distances measured forchains of length N = 25 and N = 50 coincide. Howeversegments with n . 50 of longer chains for N = 350 arestill far from being equilibrated after 104τ . Even aftera time t ≈ τRouse(N) ∼ τe(N/Ne)

2 ∼ 105τ (not shown)the chains of length N = 350 were not fully equilibrated.Rather the local equilibration is completed only after thechains have moved their own size, t > τmax(N). In lat-tice simulations it is sometimes possible to avoid the slowreptation dynamics by allowing the chains to cut througheach other.28,29 In this case τmax is given by a Rouse-time and scales as N2. In off-lattice bead-spring modelstopology conservation is the result of energetic barrierswhich prevent chain crossing. Since these barriers are aresult of the microscopic interaction potentials, they aredifficult to manipulate without affecting the local chainstructure. Hence Rouse-like relaxation mechanisms due

1

1.2

1.4

1.6

1.8

2

1 10 100 1000

<R

2 (n)>

/n

n

FIG. 4. Mean square internal distances for the systemsshown in Fig. 3 after an additional MD relation of 104τ whichcorresponds to τRouse(N = 90). Symbols as in Fig. 3. Thethick line is our target function from Fig. 2.

M N

500 100250 200

120,200 350500 500200 700200 140080 350080 7000

TABLE I. System size studied: Number of chains M andchain length N .

to barrier crossing dominate reptation only for extremelylong (and therefore inaccessible) chain lengths. Attemptsto circumvent these barriers using parallel tempering4

have met with limited success at least as far as efficiencygains are concerned.

VI. OPTIMIZED PROCEDURE FOR POLYMER

MELT CONFORMATIONS

The methods presented in this section aim at a morecareful implementation of the idea underlying our stan-dard procedure: the local build-up of the characteristicmelt packing for chains with the correct large length scalestatistics. In this section we show that we can achievethis goal by first pre-packing the phantom chains and asubsequent slow and improved push-off scheme for theexcluded volume. Both steps are needed to produce wellequilibrated melts in a reasonable amount of cpu time.Applying only one does not achieve this objective. Allresults in this section are averaged over five independentrealizations of the packing and push-off procedures. Thesystem sizes studied are listed in Table I.

5

A. Prepacking of phantom chains

The chain deformations created by the standard pro-cedure are due in part to the large local density fluctua-tions in a “melt” of randomly overlapping NRRW chains.As the chain length increases, these fluctuations increasein size as shown in Fig. 5. In this section we describea dynamic Monte-Carlo-like pre-packing procedure forphantom chains, which significantly reduces these den-sity fluctuations.As a merit function we have chosen the fluctuations

E = 〈n2〉 − 〈n〉2 in the number ni of particles found ina sphere of radius d around particle i. In a homoge-neously dense system E = 0. We perform a zero temper-ature Monte Carlo (MC) optimization in which all moveswhich decrease the density fluctuations are accepted andall moves which increase the density fluctuations are re-jected. In the course of the packing run d is decreasedfrom d = 4σ to 2σ. We use a linked cell structure in orderto calculate the ni efficiently. Choosing d > σ effectivelyincreases the range of the excluded volume correlations.All MC moves change the positions or orientations of

entire chains which are treated as rigid. This pre-packingprocedure therefore does not affect the single chain statis-tics, which by construction already have the correct end-to-end distance R as well as the targeted internal distancedistributions. We consider five types of MC moves:

Translation of individual chains in a random directions.

Rotation of individual chains by random angles aroundrandom axes through their centers of mass.

Reflection of individual chains at random mirror planesgoing through the center of mass.

Inversion of individual chains at their centers of mass.

Exchange of two chains preserving the center of masspositions.

Typical run times were of the order of days on individualworkstations and therefore negligible.The amplitude of the density fluctuations is drastically

reduced as the packing proceeds. The final result is bestassessed by the q → 0 limit of the system structure func-tion S(q). Figure 5 shows that the pre-packing reducesthe density fluctuations to one percent of the initial value.Applying the fast push-off procedure outlined in Sec. V tothese pre-packed states, we see by comparing Figs. 3 and6a that the local deformations are eliminated for shortchains (N . 50) but not for long chains. For long chains,there is a significant reduction in the local stretching butnot enough to avoid the necessity of subsequent long MDruns, which again would need to be of the order of thedisentanglement time τmax(N).

1

10

100

1000

10000

100 1000 10000

S(q

= 0

)

N

FIG. 5. Density fluctuation of randomly distributed (�)and pre-packed Gaussian chains ( ) for various chain lengths.By pre-packing the density fluctuations are about 1% forN = 350 and 0.5% for N = 7000 of that for randomly dis-tributed chains.

B. Slow push-off

In order to further reduce the chain deformations wehave modified the push-off procedure in three ways:

1. Replace the cos-potential Eq. (3) with a force-capped-Lennard-Jones-potential,21.

2. Keep the full excluded volume between next-nearest neighbor monomers to preserve to non-reversal-random-walk structure (see sec. III).

3. Increase the push-off time.

Force-capping21 can either be defined directly via amaximum force Fmax or implicitly via a distance rfcwhere ULJ

′(rfc) ≡ Fmax. At larger separations, the in-teraction is defined by Eq. (1), at smaller distances theforce becomes constant so that the interaction potentialgrows only linearly with (r − rfc):

UFCLJ(r) =

{

(r − rfc) ∗ULJ′(rfc) + ULJ(rfc) r < rfc

ULJ(r) r ≥ rfc

In the present case, rfc is gradually reduced from the

Lennard-Jones cut-off radius rc = 21/6σ to 0.8σ which issignificantly smaller than the relevant interparticle dis-tances.Force-capping has the advantage that the soft potential

systematically approaches the true potential. In contrast,the cos-potential overcompensates the missing singular-ity at the origin by a fast rise of the repulsive interactionsclose to the cut-off distance. This leads to slight differ-ences in the monomer packing. As a result, the chainconformations are slightly perturbed after switching tothe full LJ-potential.

6

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<R

2 (n)>

/n

n

a)

1

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1.6

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2

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<R

2 (n)>

/n

n

b)

FIG. 6. Mean square internal distances for pre-packed sys-tems after a (a) fast push-off and (b) slow push-off for chainswith kθ = 0. Symbols as in Fig. 3. The thick line is ourtarget function from Fig. 2.

We have increased the push-off time in order to intro-duce the excluded volume interactions quasi-statically.In practice, one has in this context to worry about twoissues: (i) How slow is slow enough? (ii) What is theequilibrium statistics for polymers with force-capped in-teractions? Concerning the first point, we have chosen apush-off time of 5000τ , which is of the order τRouse(50).This covers the typical subchain regime, up to which theintrachain distances still were disturbed by a fast push-off. The second question is more difficult to answer. Ifwe simply switch off the repulsive monomer-monomer in-teraction, we find

⟨

R2⟩

= 0.94σ2N , while the effectivestiffness c∞ ≈ 1.7 of our chains in the melt is due tolocal bead packing effects. Thus a slow push off start-ing from a potential for which the end-to-end distanceis an ideal random walk is also dangerous. For bead-spring chains these effects can easily be accounted forby treating the chains as NRRWs by always keeping thefull LJ excluded volume between next-nearest neighborsalong the chain. For atomistic polymer models, it is nec-essary to extend this to larger distances along the chain(“pentane effect”).21 Once this adjustement is made the

push off procedure cannot be “too slow”and the chainsize varies only very weakly with the maximum excludedvolume force.Figure 6b shows that the new procedure works quite

well. Independent of the total chain length N the chainsare no longer stretched on intermediate scales and thelarge scales remain unaffected by the introduction of ex-cluded volume interactions. As an additional check, wehave varied the stiffness of the initially generated randomwalks over a range of about 20% (Fig. 7a). Fig. 7b com-pares the internal distances after pre-packing and slowpush-off and confirms our previous observations. Thechains are fully equilibrated on short scales, where theresults are independent of the initial conditions. In con-trast, internal distances beyond n = 100 remain prac-tically unaffected. We therefore conclude that our newmethod allows the preparation of well-equilibrated meltsof extremely long chains within reasonable computationaleffort, provided the correct asymptotic chain stiffness c∞is known from a careful extrapolation of simulation re-sults for well-equilibrated short and medium chain lengthmelts. For our largest system of M = 80 chains of lengthN = 7000, an equilibration time of 5000τ takes aboutone month on a single processor or about 8 hours onthe 100 processor DEC alpha cluster. Thus even withmoderate computational means it is possible to equili-brate very large systems using a combined pre-packingand slow push-off procedure.

VII. DOUBLE-PIVOT-MD HYBRID

ALGORITHM

The present section deals with the problem of acce-larating the dynamic equilibration of a long chain poly-mer melt. We describe a double-pivot-MD hybrid algo-rithm which performs this task significantly faster thansimple MD relaxation. In contrast to the methods dis-cussed in the previous section, the dynamic equilibrationof a dense polymer melt does not require independentknowledge of the effective c∞. However, the preparationof locally equilibrated samples can significantly reducethe required relaxation times.In the original pivot algorithm11–14 one choses a

monomer at random and one of the two segments formedby that partitioning is pivoted rigidly in a random direc-tion about the selected monomer. The pivot algorithmis extremely efficient when applied to isolated chains inan implicit solvent. However, in a melt or even in a solu-tion at moderate density, pivot moves applied to a single

chain are bound to be rejected, since they lead to strongexcluded volume interactions with other chains.An attractive alternative are Monte Carlo moves in-

volving at least two chains, which change the connectivitywithin the melt in a way that the overall packing of beadsremains (almost) unaffected. Such algorithms were firstintroduced for lattice models30,31 and recently extended

7

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a)

1

1.2

1.4

1.6

1.8

2

1 10 100 1000

<R

2 (n)>

/n

n

b)

FIG. 7. Influence of the effective stiffness used to generatethe initial sets of random walks on the final configurations forsystems with kθ = 0. The figure shows mean square inter-nal distances (a) before and (b) after pre-packing and slowpush-off. From top to bottom the curves represent results forchains with lengthN = 3500 which were set up with a stiffnessvarying by −4%, −2%, 0%, +2%, +4%, +6%, +8%, +10%and +12% relative to our initial guess of c∞ = 1.7. The thickline is our target function from Fig. 2.

to off-lattice models.2,10,16–18 The efficient equilibrationusually comes at the expense of a certain amount of poly-dispersity which is introduced into the samples, thoughrecently Karayiannis17,18 have overcome this limitationusing double-bridging Monte Carlo moves. For the bead-spring polymers studied here, the algorithm is straight-forward to implement since the bond length l is compara-ble to the excluded volume parameter σ. This certainlyis a unique situation, as for many bead spring coarsegrained models of specific chemical species, the above ra-tio is not as close to one. For example in the case ofa united atom model for polyethylene where the bondlength l = 1.54A is significantly smaller then σ ≈ 4.0A,it is necessary to construct two trimer bridges betweenthe two properly chosen dimers along the backbone.17,18

For the bead spring model studied here, complex bridg-ing moves are not necessary. For this reason we refer

to the method as the double-pivot algorithm since it ismuch closer to the pivot algorithm used for single chainsin an implict, continuum solvent.In order to maintain a monodisperse melt we search

for a pair of spatial neighbor monomers i and j on differ-ent chains or on the same chain, which happen to also bethe same distance from one end of their respective chain.Then one can replace two bonds along the original chainswith two new bonds or bridges across the pair of chains.The total change in energy ∆E is local and consists ofthe sum of the energies of the two new bonds minus theenergies of the two previous bonds as well as the dif-ference in the bending energy, which involves the sumof the energies of four new angles minus the energies ofthe four previous angles for the case kθ 6= 0. The moveis accepted by a standard Metropolis criterion, namelywith a Boltzmann weight exp (−∆E/kBT ) if ∆E > 0and 1 if ∆E ≤ 0. As in the original pivot algorithm, thedouble-pivot algorithm generates new chains which aresubstantially different from the original two chains sinceas many as N/2 monomers of a chain are replaced bymonomers from the other chain. This results in immedi-ate large changes in the end-to-end distance and radiusof gyration.We have implemented the double-pivot algorithm into

both our shared memory and distributed memory MDcodes. The details of the implementation and a discus-sion of its computational efficiency will be published else-where. Briefly in our shared memory code, every 3 − 5time steps, we randomly chose a monomer i and checkits non-bonded nearest neighbors to determine if anysatisfy the condition that they are equidistant from theend of their chain. If so, the energy change ∆E of thedouble-pivot move is determined and the move acceptedor rejected on the basis of the Metropolis criterion. Ifthe move is accepted, monomers and/or the connectiv-ity table are re-labeled depending on which of the twocodes is used and the MD simulation is continued. If themove is rejected or no suitable pair is identified, a newmonomer is chosen at random and the process repeated.If no suitable pair is generated after searching a specifiedfraction of monomers (usually 2 − 5% depending on thechain length N), we continue with the MD simulationsfor another 3 − 5 steps and repeat the procedure. Onthe distributed memory code, the procedure is similarexcept that each processor randomly choses a monomeri and checks its non-bonded nearest neighbors on thesame processor to see if they satisfy the condition thatthey are equidistant from the end of their chain. If nosuitable non-bonded neighbor is found, another monomeri is randomly selected. The process is continued until aspecified fraction of the momomers, typically 50%, aretested. To facilitate determining the distance from thefree end, each chain carries an extra label starting from1 at either end to N/2 at the center. Because each pro-cessor searches a unique region of space independently,it is possible to have multiple successful moves each timethe procedure is applied. The search is restricted to sets

8

of monomers on the same processor to avoid the need tocommunicate changes in chain topologies between proces-sors. This restriction also prevents two or more proces-sors from performing simultaneous swaps that could en-ergetically conflict with each other. While this restrictionmeans (slightly) less swaps are considered, this is morethan outweighed by the parallelism, e.g. up to P swapstake place in one time step, where P is the number ofprocessors. With the standard parameters of the FENEpotential, k = 30ǫ/σ2, the acceptance rate is quite low.A way to improve the acceptance rate is to start with asomewhat lower value, k = 10−15ǫ/σ2 and gradually in-crease k during the course of the simulation. This changehas little effect on the overall structure of the chains butincreases significantly the acceptance rate. Using our dis-tributed memory code on 27 DEC alpha processors for asystem of M = 500 chains of length N = 500, the num-ber of successful exchanges for 105∆t with kθ = 0 wasapproximately 1750 for k = 30ǫ/σ2 compared to 8500 fork = 20ǫ/σ2 and 49500 for k = 10ǫ/σ2.To demonstrate the effectiveness of the double-pivot al-

gorithm, we have applied it to a melt of M = 500 chainsof length N = 500 generated using the standard fastpush-off procedure outlined in Sec. V. Figure 8 showsthe internal distances as a function of time during adouble-pivot/MD relaxation. Note that the character-istic “bump” in the internal distance function does notsimply decay. Rather, Figure 8 shows that the (seem-ingly equilibrated) largest scales are first swollen, beforethe chain dimensions decay homogeneously on all scales.Equilibration for the reduced spring constant k = 10ǫ/σ2

was achieved after about 6 × 104τ or 10 accepted pivotmoves per monomer. The penalty for reducing the springconstant to k = 10 is an increase of the mean-square bondlength by 7%, while the chain stiffness c∞ remains un-affected. In subsequent runs, the spring constant k isincreased to slowly “reel in” in the chains. This requiredan additional 6×104τ . We made no attempt to optimizethe number of steps for each value of k. Nevertheless,the required equilibration times are considerably shorterthan the time τmax(500) ≈ 4 × 106τ for the chains tomove their own size in a pure MD simulation.These observations illustrate the close analogy between

the pivot algorithm for single chains and the double-pivotalgorithm for dense polymer melts. As pointed out byMadras and Sokal13, it is useful to distinguish betweendecorrelation and equilibration in discussing the perfor-mance of such algorithms. Pivoting is extremely efficientin decorrelating large length scales, provided the initialconformation is properly equilibrated. Up to small cor-rections, the number of pivot moves needed to decorrelatethe end-to-end distance of isolated chains in an implicitsolvent is O(10) independent of chain length. Since thesame holds for subchains of arbitrary length, correspond-ingly more moves are required to decorrelate higher or-der (Rouse) modes. In other words, the decorrelationproceeds from large to small scales. However, the sameis not correct for the equilibration of a perturbed ini-

tial configuration. In this case it is necessary to run thesystem up to the decorrelation time of the shortest per-turbed length scale in order to equilibrate the chains onall length scales. For single chain studies aiming at globalproperties Madras and Sokal13 therefore recommend toapply the pivot algorithm to equilibrated initial config-urations generated by other techniques. In the presentcase, the combination of pre-packing with a slow push-off at least partially fulfills these requirements, since thechains are locally equilibrated.Since equilibration requires that each monomer comes

close enough to suitable exchange partners, one may askwhether of O(10) exchanges per monomer independent ofN is sufficient or whether the exchanges are simply beingmade back and forth between a limited number of states.To estimate whether enough independent configurationsare being sampled, consider the time for a monomerto diffusive a typical distance between monomers withthe same index (ρ/(N/2))−1/3 between successful pivotmoves. If we assume that monomers diffuse with thetypical 〈δr2〉 =

√

kBTb2t/ζ Rouse dynamics, this corre-sponds to

tpiv =

(

(N/2)

ρ

)4/3ζ

kBTb2(8)

which is equivalent to the Rouse time of a chain of lengthNeff = (N/2)2/3. For our longest chains N = 7000,Neff ≈ 236. The Rouse time for a chain of lengthN = 236 is approximately 7 × 104τ , which is still muchless than the time needed to make O(10) exchanges permonomer even if we used the reduced spring constantk = 10. Thus only for much longer chains or more com-plex architectures does one have to worry about sufficientindependent configurations being sampled by this hybriddouble-pivot-MD algorithm.

VIII. LOCAL BENDING RIGIDITY

As a last point we present results for chains with in-trinsic stiffness Eq. (2). In section IV we referred to thiscase as a “blind test,” because the starting states weregenerated based on our simple estimate Eq. (6) for theeffective chain stiffness in the melt and before the brute-force MD runs for the target functions shown in Fig. 2were completed. As can be seen in Fig. 1, our originalestimates were slightly too large. A comparison with thetarget functions after the pre-packing and slow push-offphase (Fig. 9a) shows that as for fully flexible chainsthe correct chain statistics is reproduced on short lengthscales. However, we are now in a situation comparable tothe situtaion presented in Fig. 7 where the large lengthscales are not fully equilibrated, because the initial chainswere generated with a slightly incorrect effective stiffness.Subsequently, we ran the combined double-pivot-MD

simulation for 1.08 × 105τ with k = 10 for the first 4

9

1

1.2

1.4

1.6

1.8

2

1 10 100 1000

<R

2 (n)>

/n

n

FIG. 8. Equilibration of mean square internal distancesusing combined double-pivot-MD relaxation following a fastpush-off for 500 chains of length N = 500. Results fort = 0 (+) just after the push-off, 6× 103τ (×), 1.2× 103τ (∗),6 × 104τ (�) and 1.2 × 105τ ( ). In order to increase theacceptance rate of the pivot moves we used a reduced springconstant k = 10ǫ/σ2 for 0 < t < 6× 104τ and k = 20ǫ/σ2 for6× 104τ < t < 9.6× 104τ before switching to k = 30ǫ/σ2 for9.6× 104τ < t < 1.2× 105τ .

k = 10 k = 20 k = 30

kθ = 0.0 24800 3900 840kθ = 0.75 27000 4200 894kθ = 1.5 21200 3500 770kθ = 2.0 15500 2700 600

TABLE II. Influence of the strength of the FENE springconstant k on the number of successful double-pivot exchangesper 105 time steps for a system of 200 chains of lengthN = 350 on 27 DEC alpha processors.

millions steps, k = 20 for the second 4 million steps andk = 30 for the last million steps for a system of 200chains of N = 350 for kθ = 1.5 and 2.0 and N = 700 forkθ = 0.75. The resulting mean squared end-to-end dis-tance were in excellent agreement with the target func-tions generated by brute force MD simulations for shorterchains. Table II shows an example of the number ofsuccessful exchanges per 105 steps for the system of 200chains of length N = 350 monomers for three values ofthe spring constant k for four values of kθ for our dis-tributed memory code run on 27 DEC alpha processors.As seen from this table, the procedure used gives approx-imately 9−12 successful exchanges for k = 10, 2 success-ful exhanges for k = 10 and 0.1 for k = 30. Fortunatelythere is little difference in the local structure of the chainfor k = 20 compared to k = 30, though as seen fromFig. 8, the chain is expanded for k = 10. Thus there is adefinite tradeoff between reducing the bond spring con-stant k, which increases the number of exchanges while atthe same time increasing the mean squared bond length.Note that for longer chains lengths, the length of the

1

1.5

2

2.5

3

3.5

1 10 100 1000 10000

<R

2 (n)>

/n

n

a)

1

1.5

2

2.5

3

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1 10 100 1000 10000

<R

2 (n)>

/n

n

b)

FIG. 9. Mean square internal distances for chains with lo-cal bending rigidity (a) after pre-packing and slow push-offand (b) after an additional double-pivot-MD relaxation oflength 1.08 × 105τ . Results for kθ = 0 (∗), 0.75ǫ (+),1.5ǫ (×), and 2.0ǫ (�). We did not apply the additional dou-ble-pivot-MD relaxation to our fully flexible systems, whichare well-equilibrated after pre-packing and slow push-off. Thethick lines are our target functions from Fig. 2 which weregenerated a posteriori for chains with intrinsic stiffness.

run and/or the number of processors would have to beincreased so that the number of successful exchanges ison the order of O(10). This limits the double-pivot/MDhydrid method to moderate chain lengths.

IX. CONCLUSIONS

In this paper, we have discussed the preparation ofequilibrated melts of long chain polymers in computersimulations. The interest in reliable algorithms for thistask is due to two problems: (i) the prohibitively longrelaxation times encountered in brute force MD simula-tions and (ii) the difficulty to predict the chain statisticson large scales (i.e. the effective chain stiffness c∞) on thebasis of microscopic intra- and interchain interactions.Our results can be summarized as follows:

10

• It is insufficient to judge the quality of the equi-libration from the statistics of the chain end-to-end distances or radius of gyration. The first justmeasures one length, while the latter gives a nonspecific average over all internal distances. As amuch more sensitive criterion we measure meansquare internal distances on all length scales fromthe monomer size up to the contour length of thechains.

• Our standard method to rapidly introduce ex-cluded volume interactions between randomly as-sembled Gaussian chains with the correct overallstatistics (“fast push-off”) fails, when judged ac-cording to this improved criterion. The fast push-off introduces significant chain distortions on lengthscales of the order of the tube diameter. They canonly vanish when the local melt topology is equili-brated. In MD simulations this is only possible viareptation dynamics so that the proper equilibrationrequires runs whose length exceeds the disentangle-ment time of the chains. To overcome this problemwe have introduced two different methods.

• We have modified our standard approach by re-ducing the density fluctuations in the assembly ofGaussian chains (“pre-packing”) and by introduc-ing the excluded volume interactions in a quasi-static manner (“slow push-off”). As before, thismethod requires prior knowledge of c∞. Testsfor bead-spring models with chain lengths up toN = 7000 (N/Ne = O(100)) show that suitablepush-off times are the Rouse times of the character-istic chain lengths of the overshoot. In the presentcase this time is of the order of a few entanglementtimes, independent of N .

• We have applied a “double-pivot” algorithm whichis inspired by the highly efficient pivot algorithmfor single chains and the double-bridging methodfor dense systems. For intermediate chain lengths,this is a viable method to equilibrate melts in areasonable amount of cpu time, particularly whenone has no prior knowledge of c∞.

The combination of MD relaxation, double-pivot andslow push-off allows the efficient and controlled prepa-ration of equilibrated melts of short, medium and longchains respectively. While our results were obtained foran off-lattice bead spring model with chain lengths upto N = 7000 beads, the methods should work as ef-ficiently for lattice polymer models such as the bondfluctuation model.32,33 The pre-packing and gradual in-troduction of the excluded volume are also applicableto united and explicit atom simulations. Furthermore,the methods are applicable to polydisperse melts andto branched polymers, including long chain branching,comb and star polymers. The double-pivot and doublebridging algorithms has also been used to equilibrate long

end-tethered, polymer brushes in contact with a melt oflong chains.18,34 For these more complex chain architec-tures, in which the chains are not necessarily Gaussianand there is no a priori way to know the conformation ofthe chains, algorithms like the double-pivot and double-bridging algorithms are presently the only possible wayto equilibrate systems faster than extremely long MD orMC runs.

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