Dynamical Monte Carlo study of equilibrium polymers: Static properties

Dynamical Monte Carlo study of equilibrium polymers: Static propertiesJ. P. Wittmer, A. Milchev, and M. E. Cates Citation: The Journal of Chemical Physics 109, 834 (1998); doi: 10.1063/1.476623 View online: http://dx.doi.org/10.1063/1.476623 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/109/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analysis of the configurational temperature of polymeric liquids under shear and elongational flows usingnonequilibrium molecular dynamics and Monte Carlo simulations J. Chem. Phys. 132, 184906 (2010); 10.1063/1.3415085 Static and dynamic properties of tethered chains at adsorbing surfaces: A Monte Carlo study J. Chem. Phys. 120, 8831 (2004); 10.1063/1.1691393 A Monte Carlo study of effects of chain stiffness and chain ends on dilute solution behavior of polymers. I.Gyration-radius expansion factor J. Chem. Phys. 118, 2911 (2003); 10.1063/1.1536619 Lattice polymers with structured monomers: A Monte Carlo study of thermodynamic properties of melts andsolutions J. Chem. Phys. 116, 10959 (2002); 10.1063/1.1478766 Dynamical Monte Carlo study of equilibrium polymers. II. The role of rings J. Chem. Phys. 113, 6992 (2000); 10.1063/1.1311622

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67

On: Thu, 18 Dec 2014 08:24:43

http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/327320036/x01/AIP-PT/JCP_ArticleDL_101514/PT_SubscriptionAd_1640x440.jpg/47344656396c504a5a37344142416b75?x

http://scitation.aip.org/search?value1=J.+P.+Wittmer&option1=author

http://scitation.aip.org/search?value1=A.+Milchev&option1=author

http://scitation.aip.org/search?value1=M.+E.+Cates&option1=author

http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcov

http://dx.doi.org/10.1063/1.476623

http://scitation.aip.org/content/aip/journal/jcp/109/2?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jcp/132/18/10.1063/1.3415085?ver=pdfcov








Dynamical Monte Carlo study of equilibrium polymers: Static propertiesJ. P. Wittmera)

Department of Physics and Astronomy, University of Edinburgh, JCMB King’s Buildings, Mayfield Road,Edinburgh EH9 3JZ, United Kingdom

A. MilchevInstitute for Physical Chemistry, Bulgarian Academy of Science, 1113 Sofia, Bulgaria

M. E. CatesDepartment of Physics and Astronomy, University of Edinburgh, JCMB King’s Buildings, Mayfield Road,Edinburgh EH9 3JZ, United Kingdom

~Received 26 January 1998; accepted 3 April 1998!

We report results of extensive dynamical Monte Carlo investigations on self-assembled equilibriumpolymers~EP! without loops in good solvent.~This is thought to provide a good model of giantsurfactant micelles.! Using a novel algorithm we are able to describe efficiently both static anddynamic properties of systems in which the mean chain length^L& is effectively comparable to thatof laboratory experiments~up to 5000 monomers, even at high polymer densities!. We sample up toscission energies ofE/kBT515 over nearly three orders of magnitude in monomer densityf, andpresent a detailed crossover study ranging from swollen EP chains in the dilute regime up to densemolten systems. Confirming recent theoretical predictions, the mean-chain length is found to scaleas ^L&}faexp(dE) where the exponents approachad5dd51/(11g)'0.46 andas51/2@11(g21)/(nd21)#'0.6,ds51/2 in the dilute and semidilute limits respectively. The chain lengthdistribution is qualitatively well described in the dilute limit by the Schulz-Zimm distributionp(s)'sg21 exp(2s) where the scaling variable iss5gL/^L&. The very large size of thesesimulations allows also an accurate determination of the self-avoiding walk susceptibility exponentg'1.16560.01. As chains overlap they enter the semidilute regime where the distribution becomesa pure exponentialp(s)5exp(2s) with the scaling variable nows5L/^L&. In addition to the aboveresults we measure the specific heat per monomercv . We show that the average size of the micelles,as measured by the end-to-end distance and the radius of gyration, follows a crossover scaling thatis, within numerical accuracy, identical to that of conventional monodisperse quenched polymers.Finite-size effects are discussed in detail. ©1998 American Institute of Physics.@S0021-9606~98!51526-7#

I. INTRODUCTION

Systems in which polymerization takes place under con-dition of chemical equilibrium between the polymers andtheir respective monomers are termed ‘‘equilibrium poly-mers’’ ~EP!. An important example is that of surfactant mol-ecules forming long flexible cylindrical aggregates, so-calledgiant micelles~GM!,1 which break and recombine constantlyat random points along the sequence~see Fig. 1!. Similarsystems of EP are formed by liquid sulfur,2,3 selenium4 andsome protein filaments.5 In the surfactant literature~e.g., Ref.1! giant micelles are often referred to as ‘‘living polymers’’although this is potentially confusing since they are distinctfrom systems that reversibly polymerize stepwise, in thepresence of fixed number of initiators, for which this termhas previously been reserved.6,7 As direct imaging methodsclearly demonstrate8 GM, which behave very much like con-ventional polymer chains,9 may become very long indeedwith contour lengths up to'1 mm. However, the constantreversible scission of the chains offers an additional stress

relaxation mechanism in comparison with conventional‘‘quenched’’ polymers whose identity is fixed for alltime.1,10–13

EP are intrinsically polydisperse and their molecularweight distribution~MWD! in equilibrium is expected1,14 tofollow an exponential decay with chain length. So far, we arenot aware of anydirect experimental measurements of theMWD in such systems. Of central interest is the mean-chainlength, and for GM there has been experimental and theoret-ical controversy1,15–18concerning its dependence on volumefraction, described by the growth exponent^L&}fa. A scal-ing theory~summarized below! gives a.0.6; although thisis consistent with some data on ionic micelles at intermediateor high salt levels,1 a much larger exponenta.1.2 is sug-gested by experiments on lecithin-in-oil reverse micelles andsome nonionic aqueous surfactants.18 Thus the experimentalevidence concerning the equilibrium growth law of GM re-mains controversial.

Given the shortcomings of any approximate analyticaltreatment and the difficulties with the laboratory measure-ments, numerical experiments, being exact within the frame-work of the respective model and able to account explicitlya!Author to whom correspondence should be addressed.

JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 2 8 JULY 1998

8340021-9606/98/109(2)/834/12/$15.00 © 1998 American Institute of Physics


On: Thu, 18 Dec 2014 08:24:43

for various factors that influence experiments, should helpmuch in understanding the thermodynamic behavior and theproperties, both static and dynamic, of EP. However, up tonow only a small number of simulational studies19–23 exist,in contrast to numerical experiments with conventional poly-mers. Indeed, while the connectivity of polymer chains andthe resulting slow dynamics render computer simulations ademanding task in its own terms, the scission-recombinationprocesses, which are constantly underway in EP, impose ad-ditional problems for computational algorithms, mainly interms of data organization and storage. Since chains con-stantly break while other fragments unite into new chains,objects can lose their identity or gain new ones at each stepof the simulation.

In earlier Monte Carlo~MC! simulations on EP19 thesystems of polydisperse polymer chains were mapped ontoan asymmetric Potts model, in which different spin valueswere taken to represent bonded and nonbonded monomers aswell as vacancies in a lattice. Such models are very efficientfor studying static properties of EP since at each update ofthe lattice all sites are assigned new spin values subject to aBoltzmann probability; dynamically however this violatesthe connectivityof the chains. Accordingly this approachfaithfully reproduces static properties, but since the kineticsof such models is fictitious these cannot be extended to studydynamics which is one of our goals.

In a recent work by Rouault and one of us, a dynamicalMonte Carlo algorithm~DMC! was proposed,20 based on thehighly efficient bond fluctuation model~BFM!24–27 which isknown to be very accurate in reproducing both static anddynamic properties~Rouse behavior! of polymer chains inmelts and solutions.25 However, the data structure used wasbased on a quenched polymer algorithm and was thereforerather slow and memory consuming; a radical new approachis required. Below we present a method to deal with theseproblems efficiently and are thereby able to study muchlarger systems.

In several EP systems, the behavior is strongly affectedby the presence of polymeric rings.28 For reasons that are notentirely clear however, ring-formation seems to be negligible

in other cases, included that of GM.~This does not necessar-ily exclude a small number of closed loops in GM systems asmay sometimes be seen using direct imaging methods.8!Since the latter are among the most widely studied examplesof EP, our results on the ring-free case are presented here.~This elaborates a brief previous discussion.29! Correspond-ing results on EP systems containing rings will be presentedelsewhere.30 Under some conditions GM systems can alsocontain branch points;31 we forbid these in the present work.

After recalling some analytical predictions in Sec. II wediscuss our new approach in detail in Sec. III. With thisalgorithm we are able to vary the volume fraction overnearly three orders of magnitude and we obtain equilibratedsystems with average chain length up to about^L&'5000~see Table I!. We present our computational results on staticproperties of EP without rings in Sec. IV. A complete cross-over scaling analysis ranging from the dilute regime of swol-len EP up to the dense Gaussian limit is performed; for sys-tems with large enough chains (^L&@5) we obtain closeagreement with recent analytical predictions.1,15,16A delicateissue concerns finite-size effects which arise if one works attoo low a temperature for any given system size. These forcethe breakdown of the~essentially! exponential MWD as oneenters a state where a large fraction of the monomers residein a single chain. Particular care is taken in this regard inSec. V. In the final Sec. VI we summarize our findings. Ex-tension and development of these investigations to dynamic

FIG. 1. Sketch of model: Bonds of EP chains break and recombine con-stantly with ratesks5exp(2(E1B)/kBT) and kr5exp(2B/kBT) dependingon the scission energyE and the activation energy barrierB, both supposedto be independent of monomer position and density. Closed rings andbranching of chains are not allowed within our model.

TABLE I. Summary of measured static quantities for configurations with^L&.50. Three decades of volume fractions betweenf50.001 andf50.6 have been sampled with bond energies up toE515. Note thatSB(f,0.1)5200a and LBox(f>0.1)5100a. Quantities tabulated: the meannumber of chains M &, the mean-chain lengthL&, the specific heat permonomercv , the mean end-to-end distanceRe , the mean gyration radiusRg and the average distance between chainsH. All length scales are given inunits of the lattice constanta.

f E ^M & ^L& cv Re Rg H

0.001 13 16 62 0.7 32 13 7915 8.5 124 0.7 50 20 98

0.005 13 41 125 0.5 51 2 5615 17 311 0.3 86 34 78

0.01 13 60 168 0.5 61 24 5115 24 424 0.26 105 41 69

0.05 10 498 101 0.45 42 17 2515 46 1110 0.13 165 64 56

0.1 8 218 57 0.58 29 12 1710 81 155 0.31 50 20 2313 20 641 0.1 112 44 3715 7.5 1777 0.06 163 68 51

0.4 6 970 52 0.36 23 10 108 360 139 0.23 39 16 1410 133 375 0.128 65 26 2013 30 1898 0.05 132 54 3315 13 4017 0.02 196 81 43

0.5 6 1058 59 0.29 24 10 107 642 98 0.24 31 13 128 391 160 0.22 40 16 149 237 263 0.16 52 21 1610 145 432 0.1 67 27 1913 32 1968 0.05 140 58 3115 13 5113 0.02 231 92 43

0.6 10 157 480 0.11 68 28 19

835J. Chem. Phys., Vol. 109, No. 2, 8 July 1998 Wittmer, Milchev, and Cates


On: Thu, 18 Dec 2014 08:24:43

properties of EP~without rings! will be reported in a com-panion paper.32

II. MODEL AND ANALYTICAL PREDICTIONS

Before describing our computational investigation wedefine the physical model, and recall some essential analyti-cal predictions concerning various properties of equilibriumconformations of EP.

On a coarse-grained level~see Fig. 1! systems of EP arecharacterized by the monomer volume fractionf, the energydifferenceE between saturated and unsaturated bond states,the height of the activation energy barrierB, the persistencelength l p and the excluded volume sizeb of the monomer~related to the cross sectional diameter of a giant micelle!.Additionally, parameters for the non-bonded interactionsmay be defined.

As appropriate to the GM case we do not allow~i!closed rings or~ii ! branching points and suppose that~iii ! allscission energies are independent off and position of themonomer within the chain. Hence, the rates for scission andrecombinationks5exp(2(E1B)/kBT) and kr}exp(2B/kBT)are taken to be constant. For GM simplification~iii ! is ex-pected to become accurate at high salinity where electrostaticeffects may be neglected.1

For EP that are long compared to the persistence lengthl p , the main departure from conventional theory of polymersolutions33 is that in micellar systems, the reversibility of theself-assembly process ensures that the MWDc(L) of theworm-like polymeric species is in thermal equilibrium. Thiscontrasts with ordinary quenched polymers for which theMWD is fixed a priori, and equilibrium only applies to theremaining~configurational! degrees of freedom. For EP, onlythe total volume fraction of the system

f5 (L

Lmax

Lc~L ! ~1!

is a conserved quantity, rather than the entire distributionc(L). It is useful to introduce the normalized probabilitydistribution p(L)5(^L&/f)c(L), so that (Lp(L)51. Themaximum chain lengthLmax, given by the total number ofmonomers in the system, becomes of relevance in the con-text of finite-size effects~Sec. V!.

At the level of a Flory-Huggins mean-field approxima-tion ~MFA! the grand potential densityV for ad-dimensional system of EP may be written as

V5(L

c~L !@ ln~c~L !bd!1E1Lm#, ~2!

where we choose energy units so thatkBT51. The first termis the entropy of mixing, the second the scission energyE ofa bond and the last term entails the usual Lagrange multiplierfor the conserved monomer density Eq.~1!. Without loss ofgenerality, we have suppressed in Eq.~4! the part of the freeenergy linear in chain length. Minimizing with respect to theMWD, paying attention to the constraint of Eq.~1!, yieldsthe exponential size distributionp(s)ds5exp(2s)ds wherethe chemical potential defines a scaling variables5mL. Thisdistribution has L&m51, i.e., the scaling variables is given

in the MFA by the reduced chain lengthx[L/^L&5s. Usingagain Eq.~1! we then obtain the mean-chain length

^L&5Afa exp~dE! ~3!

with a ~nonuniversal! amplitudeA and the MFA-exponentsaMF5dMF51/2. This result is expected to be a good ap-proximation near theu-temperature and in the melt limitwhere chains become free random walks uncorrelated withthemselves and their neighbors.20

Note that the chain stiffness may in principle be incor-porated in the above description of flexible chains by addingto Eq. ~2! a free energy term for the higher probability oftrans-states. It is straightforward to work out, that for reason-able bending energies this renormalizes the scission energyE by a constant of order unity~that is, of orderkBT). Herewe do not pursue stiffness effects and instead choose a per-sistence lengthl p comparable to the monomer sizeb.

It is relatively simple to extend the above analysis todilute and semi-dilute solutions of EP. We recall33 from stan-dard polymer theory that the correlation lengthJ for chainsof lengthL in the dilute limit is given by the size of the chainJ5R}Ln. When the chains~at given number density! be-come so long that they start to overlap atL'L*'f21/(nd21) the correlation length of the chain levels offand becomes the~chain-length independent! size of a‘‘blob’’ J5j}L* n. Here d53 is the dimension of space,and n'0.588 is the swollen chain~self-avoiding walk! ex-ponent.

The mean-field approach remains valid10 so long as thebasic ‘‘monomer’’ is replaced by a blob of lengthJ. Acrossthe entire concentration range we may write the grand poten-tial density as

V5(L

c~L !@ ln~c~L !Jd!2 ln~~J/b!d1u!1E1Lm#.

~4!

We have taken into account here of the free energy changeresulting from thegain in entropywhen a chain breaks sothat the two new ends can explore a volumeJd. This gain isenhanced by the fact that the excluded volume repulsion onscales smaller thanJ is reduced by breaking the chain; thisis accounted for by the additional exponentu. Note thatu5(g21)/n'0.3 with g'1.165~as we will confirm in Sec.IV B !. In MFA g51,u50, and Eq.~4! simplifies to Eq.~2!,but this ignores correlations arising from excluded volumeeffects. In the dilute regime Eq.~4! can be rewritten as

V5(L

c~L !@ ln~c~L !bd!1~g21!ln~L !1E1Lm# ~5!

so that the relation to the well-known partition function ofself-avoiding walks~with an effective coordination numberz), QL} zLLg21, is recovered.

Note that in concentrated and semidilute systems thereare minor corrections to Eq.~2! from the small fraction ofchains that are too long for their excluded volume interac-tions to be screened by the surrounding chains~under meltconditions, this applies33 for those chains whose length ex-ceeds^L&2). This contribution is exponentially small andwill be neglected. Note, however, that the contribution of

836 J. Chem. Phys., Vol. 109, No. 2, 8 July 1998 Wittmer, Milchev, and Cates


On: Thu, 18 Dec 2014 08:24:43

short chains~smaller than or comparable to the blob size! inthe semidilute regime is properly included in Eq.~4!, so longas theL-dependence ofJ is taken into account. In general,this dependence is contained in a scaling function

J~s!5J~s/s* !, ~6!

where the scaling variables5Lm was already introducedabove, ands* 5L* m. This function asymptotes to the radiusof gyration and the blob size at large and smalls, respec-tively.

Minimization of Eq.~4! at fixedf yields the MWD

p~L !dL5p~s!ds}J~s!exp~2s!ds, ~7!

which depends viaJ on s* . We see that the effective expo-nent ge(s* )[s/x5^L&m5^s&5*sp(s)ds is not in generalequal to unity. Hence, the scaling variables is not necessar-ily given by the reduced chain lengthx as in MFA and assupposed in Ref. 34, but bys5gex. In the two limits faraway from the crossover line one can readily calculate theintegrals. In the semi-dilute limit (s* !1) we obtain^L&m5ge51, i.e., as in the MFAs5x, but in the dilute limit(s* @1) we get ^L&m5ge5g, i.e., s5gx. Substitutingeverything we obtain finally the distributions

p~x!dx5H exp~2x!dx ~^L&@L* !,

gg

G~g!xg21exp~2gx!dx ~^L&!L* !.

~8!

We remark that any observed breaking of the scalingp(L)dL5p(x)dx is an indication of either crossover be-tween both density regimes (^L&'L* ) or finite-size effects(^L&'1 or ^L&'Lmax). One expects for semi-dilute configu-rations close to the crossover line some reminiscence of thedilute behavior, which should show up in a slightly higherge .

As in the MFA-case we obtain a mean chain length

^L&5L* ~f/f* !a ~9!

which may be cast in the generic form Eq.~3!, but with theexponentsad5dd51/(11g)'0.46 in the dilute andas

51/2(11(g21)/(nd21)'0.6, dd51/2 in the semi-dilutelimit. Thus the concentration dependencies of the mean mo-lecular weight^L& in the dilute and semidilute limits differslightly form the one predicted by simple mean-field theory.Here finally wedefine(L* ,f* ) quantitatively as the coordi-nates of the intercept of the dilute and semi-dilute asymp-totes on a plot of L& vs. f ~see Fig. 4 below!. Accordinglywe may write

f* 5P exp~2E/w!,~10!

L* 5Q exp~E/k!,

with exponents w5(as2ad)/(ds2dd)511(g11)/(nd21)'3.8 andk5(nd21)w'2.93. The amplitudesP andQ are similarly related to the prefactorsAd and As of themean-chain length, as defined in Eq.~3! in each density re-gime @see also Eq.~12! below#.

We consider finally the specific heat capacity of the sys-tem. This offers a possible experimental measure of the typi-cal scission energyE. Assuming this to be purely enthalpic

and independent of temperature, the internal energy densityU given by the density of end monomersU5E( lc( l )5Ef/^L&. From this we get the specific heat per monomer

cv51

f

]U

]T5d

E2

^L&5

d

AfaE2 exp~2dE!. ~11!

One verifies thatcv has a maximum atE52/d which shiftsslightly to higher values at lower densities~from 4 aboveL*to 4.4 below!.

III. THE ALGORITHM AND CONFIGURATIONS

In EP systems bonds between monomers break and re-combine constantly and the chains are only transient objects~Fig. 1!. Therefore it is relatively inefficient to base the datastructure on thechains~such an approach penalized either bysorting times or waste of memory!. Rather, one has to base iton the monomers, or even better on the two saturated orunsaturatedbondsof each monomer. This brings the algo-rithm as close as possible to what actually happens in sys-tems of EP and makes it possible via pointers between bondsto avoid all sorting procedures, time consuming nested loopsand arrays, at the expense of only one additional list re-quired. Our chosen data structure is explained for three ini-tial chains in Fig. 2, one of which is actually a free monomerwith two unsaturated bonds.

Using the assumption that no branching of chains is al-lowed, the two~possible! bonds of each monomerimon arecalled ibond5 imon and ibond52 imon. No specificmeaning~or direction! is attached to the sign: this is merelya convenience for finding the monomer from the bond list( imon5u ibondu). Pointers are taken to couple indepen-dently of sign; see Fig. 2. In the proposed algorithm thebonds are coupled by means of a pointer list in a completelytransitive fashion @ jbond5pointer( ibond)⇔ ipoint

FIG. 2. Sketch of algorithm:~a! Each monomer has two~saturated or un-saturated! bonds. Chains consists of symmetrically connected lists of bonds:jbond5 pointer(ibond)⇔ ipoint5 pointer(jpoint). The pointers of all end-bonds point to NIL.~b! Recombination of two initially unsaturated bondsibond52 andjbond55 connects the respective monomersimon52 andjmon55. Note that only two pointers have to be changed and that the remainingchains~bold lines! behind both monomers are not involved.~c! Breaking asaturated bondibond requires resetting the pointers of the two connectedbondsibond and jbond 5 pointer(ibond)to NIL.



On: Thu, 18 Dec 2014 08:24:43

5pointer(jpoint)# to make recombinations and scissions asfast as possible; only two simple vector operations are re-quired for breaking bonds or recombination as shown in Fig.2. Note that this would be impossible in any algorithm in-volving only one bond per monomer; the latter requires animplicit sequential order of segments in the chains, forcingsorting operations of the order of the mean-chain length forevery recombination. Unsaturated bonds at chain ends pointto NIL. Only these bonds may recombine. With this algo-rithm, no explicit distinction between end-monomers, freemonomers or middle monomers is required.

As a work-horse, harnessed to this new data organiza-tion, we exploit thebond-fluctuation model~BFM!24 for theDMC simulation of polymers. This choice was mainly moti-vated by its efficiency, and the large amount of existing dataavailable on monodisperse conventional polymers which canserve as a reference against which to compare the presentwork. However, we emphasize that a different choice, suchas an off-lattice MC algorithm,35 could equally well be com-bined with our data structure.

In the BFM an ‘‘effective monomer’’ consists of an el-ementary cube whose eight sites on the hypothetical cubiclattice are blocked for further occupation.24 We consider theformulation of the BFM on a dual lattice, i.e., the center ofan effective monomer is represented byonesite on a simplecubic lattice as introduced by Mu¨ller.26 Excluding all 26neighboring sites from occupancy renders the model equiva-lent to the original model proposed by Carmesin andKremer.24 The volume fractionf is the fraction of latticesites blocked by the monomers. The monomers of a polymerchain are connected via bond vectorsb, which are takenfrom the allowed setP(2,0,0),P(2,1,0),P(2,1,1),P(3,0,0),and P(3,1,0), whereP stands for all permutations and signcombinations of coordinates. A bond corresponds physicallyto the end-to-end distance of a group of 3–5 successivemonomers and can thereforefluctuatewithin some range oflengths. All length are measured in units of the lattice spac-ing a. The algorithm combines typical advantages of the lat-tice MC methods with those from the continuous Browniandynamics algorithm. As defined above, it corresponds togood solvent conditions without hydrodynamics~but respect-ing entanglement constraints!.24,25

As explained above~Sec. II! the ends of a given EP arenot allowed to bind together in the presented study; beforeevery recombination we have therefore to check that bothmonomers do not belong to the same chain. Because there isno direct chain information in the data structure this has to bedone by working up the list of links~which adds only fourlines to the source code!. In physical time units the simula-tion becomesfaster for higherE: the number of recombina-tions per unit time goes down like exp(2E), but the chainlength only up as exp(E/2).

The barrier energyB.0 is taken into account by settingan attempt frequencyvB for scissions and recombinations.This is a convenient tool for testing dynamic behavior of thesystem at different lifetimes of the chains,32 although forstatic properties as studied in this paper, the choice ofB isimmaterial. Therefore in almost all runs reported here we setB50. Those sites of the lattice that are not occupied by

monomers are considered empty~vacancies! and contributeto the free volume of the system. We may in principle assignan energy2w (w.0) for the nonbonded interaction be-tween monomers in the system,20 and a bendingenergys sin(uij) with u i j being the angle between consecu-tive bonds.27 In the present investigation, however, we focusexclusively on the process of equilibrium polymerization ofentirely flexible chains in an athermal solvent settingw5s50.

Time is measured, as usual, in Monte Carlo steps~MCS!per monomer. Each MCS is organized as follows:

~i! A monomer is chosen at random and allowed to per-form a move according to the BFM algorithm.24

~ii ! With a frequencyvB[exp(2B), i.e., every 1/vB

MCS, one of the bonds is chosen at random~remem-ber that there are twice as many bonds as monomers!.If one of the bonds happens to be a saturatedP(2,0,0)-bond an attempt is made to break it, other-wise if it is unsaturated, i.e., the monomer is at theend of a chain or a free monomer, an attempt is madeto create a bond with another monomer that might bepresent onany of the six neighboringP(2,0,0) sites.Applying the Metropolis algorithm36 a scission is per-formed whenever the value of a random number be-tween 0 and 1 is smaller than exp(2E). On the otherhand, with the bond energy being positive (E.0),recombination is always accepted so long as a ring isnot thereby created.

Note that P(2,0,0)-bonds are broken irrespective ofwhich particular bond vector, i.e., which of the six possibili-ties, they stand for, and that, therefore, detailed balance re-quires that for recombinationall ~and not justoneas in Ref.20! of these sites have to be checked for possible unsaturatedbonds. These neighboring sites have to be checkedran-domly; a typewriterlike search along the list of possible bondvectors creates correlations in violation of the detailed bal-ance requirement.~This has subtle but measurable conse-quences.! Note finally that our decision to restrict the break-ing and recombination to the six shortest@P(2,0,0)# bondsavoids ergodicity problems arising from crossed pairs ofbonds that can result in immobile monomers.37

In the presented study the volume fraction was variedover nearly three orders of magnitude fromf50.001 (Lmax

51000 monomers per box! to f50.6 (Lmax575 000 mono-mers per box! as shown in Fig. 3. For densities smaller thanf50.1 cubic lattices with linear dimensionSB5200 wereused, for higher densities a smaller box withSB5100. Thisshould be compared to Ref. 20 where aSB530-box withLmax51300 particles (f50.4) was the largest achievable.One should bear in mind that densities around 0.5 corre-spond to extremely dense systems~melt conditions! in theBFM, since at higher densities the blocking of neighboringsites by other monomers leaves no room for movement andthe system goes effectively into a glassy state.24,25

The starting configuration consists of randomly distrib-uted and nonbonded monomers which we cool down step bystep~a sequence of so-called ‘T-Jumps’! each step samplinga higher scission energy up to the maximumE515. As men-



On: Thu, 18 Dec 2014 08:24:43

tioned above, for the static results we usually setB50, i.e.,vB51. Due to the constant breaking and recombining of thebonds the equilibration is then much faster than in systems ofquenched polymers. Following Rouaultet al.21 we havechecked this explicitly by monitoring the evolution of mean-chain length L(t)& and radius of gyrationRg

2(t)& after ev-ery T-Jump.

Measurements of static properties such as the mean-square end-to-end distance^Re

2&, the radius of gyrationRg2&

or the specific heatcv~see Table I! were performed in inter-vals of roughlyt t/10, wheret t is the terminal relaxation timeof the system for the parameters (f,E) considered.32 Typicalruns~with B50) covered easily up to 102100t t . The meltdensityf50.5 was a particular focus for our study of dy-namical properties,32 requiring better statistics. Hence, forthis f we have sampled over 128 independent configura-tions. However, for all static properties other than the spe-cific heatcv and, for the largestE values some MWD data, along run from a single starting configuration was quite suf-ficient. Indeed, statistical errors are generally within the sym-bol size in the data presented below.

There are two different kinds of ‘‘finite-size effects’’ inthis simulation. The first, and most important, arises fromchains that are too small. Not surprisingly, we found in con-figurations with ^L&,5 ~below the dotted line in Fig. 3!non-universal behavior. For clarity these data points areomitted in most of our plots. Second, for systems with~es-sentially! exponential MWD one has to worry about the sys-tem size wheneverL& becomes of the order of the totalnumber of monomers in the boxLmax. This important issuewill be addressed in Sec. V where we present results of asystematic finite-size study summarized in Table II confirm-

ing unambiguously that the systems reported are indeed suf-ficiently large.

Most of the simulations on static properties were per-formed on two single DEC Alpha workstation over a periodof year. A parallel version of the algorithm, however, wasalso developed and some of the computations at melt densitymentioned above have been carried out on the facilities ofEPCC, Edinburgh. The latter computations focused mainlyon the dynamics of equilibrium polymers as we reportelsewhere.32

IV. COMPUTATIONAL CONFIRMATION OF SCALINGPREDICTIONS

In what follows we examine the influence of densityfand scission energyE on mean-chain lengthL&, MWDc(L), specific heat per monomercv and the size of thechains, as measured, e.g., by the radius of gyration^Rg

2&.

A. Mean-chain length

In Fig. 4~a! we show in a semi-log plot the measuredvariation of the mean-chain lengthL& versus the bond en-ergyE. Note that we have been able with this new algorithmto obtain mean chain length of up to^L&'5100~see Table I!which is comparable with~at least some! experimental sys-tems of EP. Giant micelles, for example, are somewhat rigidand the persistence length relatively large,l p'16 nmwhereasRg'100 nm, resulting in around 100 statisticalsegments,17 although much longer chains could arise in somesemidilute systems.38 As mentioned in the Sec. III and elabo-rated further in Sec. V below, it is the finite size of thesystems, rather than the equilibration time, that prevents usstudying higherE ~lower temperatures! since the largestchain would then comprise too high a proportion of the totalavailable monomers in the system.

For long enough chains~above the dotted line in Fig. 3!the chain length increases exponentially with scission energyE, as predicted by Eq.~3!. The data confirm with high pre-

FIG. 3. Simulation parameters (E,f) in relation to the crossover densityf* . Above and to the right of the dashed line is the semi-dilute regime,below and to the left the dilute regime. The crossover density follows anexponential decay withf* 'P exp(2E/w) where w53.82 andP50.26consistent with Eqs.~10! and~12!. At volume fractions of aboutf50.5 thecorrelation length becomes of order of the monomer size~the melt regime!.The dotted line corresponds to^L&55; to the left of this line chains are tooshort for good scaling. For some systems~filled symbols! we have in addi-tion systematically varied the frequency of scission/recombinationvB byvarying the activation energy Ref. 32. For certain parameter choices~circles! we have checked carefully for finite-size effects.

TABLE II. Variation of system sizeSB for f50.5 for two high scissionenergies. We give the average number of chains^M &, the mean-chain length^L&, the specific heatcv , the end-to-end distanceRe and the radius ofgyrationRg .

E SB/Lmax ^M & ^L& cv Re Rg

10 10/62 1.0 62 0.01 23 1015/210 1.1 200 0.05 42 1720/500 1.5 384 0.09 57 2325/976 2.6 468 0.12 62 2530/1687 4.1 466 0.11 64 2635/2679 6.4 448 0.11 66 2740/4000 9.3 458 0.13 67 2750/7812 18 436 0.10 65 2770/21437 50 435 0.11 67 27100/62500 145 432 0.10 67 27

15 20/500 1.0 499 0.001 67 2730/1687 1.1 1627 0.01 136 5550/7812 2.0 4706 0.02 208 8060/13500 2.5 6512 0.02 277 10470/21437 4.4 6180 0.03 240 95100/62500 12.7 5113 0.02 231 92



On: Thu, 18 Dec 2014 08:24:43

cision the predicted exponentsdd'0.4660.01 andds50.560.005 in the dilute and semidilute regimes respectively.The growth exponentsa are most readily confirmed@follow-ing the scaling prediction Eq.~9!# by directly plotting the‘‘number of blobs’’ l 5^L&/L* against the reduced densityf/f* . The data collapse onto a single master curve, seen inFig. 4~b!, is indeed remarkable and is one of the main resultsof this work. The two indicated slopes are comparisons withthe two asymptotically predicted growth exponents (ad

50.46, as'0.6). @Note that in the dilute limit finite-sizeeffects are visible for lowE values where chains are ex-tremely short (L&,5).# This finding is at variance with Ref.23 where a much stronger growth with density is reported forsystems containing only 8400 monomers with mean-chainlengths up to 210.

By plotting A5^L&/fa/exp(dE) versus L& or E one fitsfor each density regime an amplitudeAd'3.660.2 andAs

'4.460.1 respectively. From these prefactors the ampli-tudes governingf* and L* defined in Eq.~10! can be ob-tained

P5~Ad /As!1/~as2ad!'0.360.1,

~12!Q5AsP

as'1.960.5.

@The high error bars are caused by the small difference in thetwo growth exponents resulting in the large exponent 1/(as

2ad).# Although these are defined by the crossing point ofthe two asymptotes in Fig. 4~b!, it is notable that the asymp-totes are followed, within numerical accuracy, all the way tothe crossing point.~A large deviation is of course not pos-sible since the two asymptotic slopes are not very different.!Thus the crossover between the two regimes occurs rathersharply at L&/L* 5f/f* 51, and this can be used to definea crossover line on thef, E or ^L&, E plot. Indeed dashedlines in Fig. 3 and Fig. 4~a! indicate the position of the cross-over linesf* and L* using the exponentsw53.84 andk52.93 together with the above amplitudesP andQ.

B. Molecular weight distribution

We consider first the MWD in the dilute (l !1) andsemidilute (l @1) limits, far away from the crossover. There-after we discuss the crossover effects nearl'1 where a sig-nificant fraction of the chains are still smaller than the blobsize of the semidilute network.

The distribution of chain lengths at equilibrium,c(L), ispresented in Fig. 5~a! on semi-log axes for variousE at highdensityf50.5. The fluctuations in the sampled lengths in-crease considerably for very long chains where correlationsbetween successive configurations deteriorate the statistics.To the available accuracy, the normalized distributionp(x)plotted versus the reduced chain lengthx5L/^L& collapseperfectly on single ‘‘master’’ curve as shown in the insert ofFig. 5~a!; thus the mean chain-length^L& contains all energyinformation. The exponential decay confirms Eq.~8a!. Forcomparison we have indicated the exp(2gx) behavior whichis clearly not compatible with the data. This finding is inagreement with Ref. 22, but in clear contrast to Gujrati34

according to whom the Schulz distribution holds indepen-dently of the overlap.

While at high densities we observe perfect exponentialscaling of p(x), at lower dilute densities~with sufficientlylong chains! our results are qualitatively consistent with theSchulz distribution Eq.~5b!. We compare in Fig. 5~b! thisprediction~bold line! with data sets for configurations in thedilute limit. To stress the systematic difference, we have in-cluded the high density prediction.

As shown in the insert of Fig. 5~b! our MWD at dilutedensities are also qualitatively consistent with the additionalpower-law dependencep}xg21 in the limit of smallx. Notethat the maximum of the distribution is atxM5(g21)/g'0.1 corresponding to a chain lengthL50.1 L&'3.1 forf50.005 and'4.2 for f50.01. Hence, we could not ex-pect to reproduce accurately the power-law regime forx!xM . This would require configurations of at least^L&'1000 in the dilute regime; due to the finite-size effects inthe range of scission energies used~discussed in Sec. V be-low! this is at present not feasible.~Since in 2d one hasg543/32 which is much farer from the MFA valueg51 thispart of the distribution can be more efficiently probed in 2d,

FIG. 4. ~a! The average chain length^L& for a wide range of densitiesf andenergiesE. ~b! Rescaled average chain lengthl 5^L&/L* versus reduceddensityf/f* confirming the scaling Eq.~9! and the amplitudes Eq.~12!.The two slopes are comparisons with the predicted growth exponentsad

'0.46 andas'0.6.



On: Thu, 18 Dec 2014 08:24:43

as done recently by Rouault and Milchev.22! Qualitatively,however, we believe that this result is unambiguous. Notethat we see no evidence for a possiblenegativeexponent inthe power law in Eq.~8!, as postulated in some treatments ofthe unusual diffusive behavior in GM.12

At intermediate densities, slightly above the crossoverline, a non-negligible fraction of chains are smaller than theblob size, and are thus fully swollen~these chains may fitamong the network of chains of average size^L& withoutbeing seriously perturbed by the interchain interaction!. Thedistribution therefore crosses over smoothly from the dilutelimit as depicted in Fig. 5~b! to the semi-dilute limit of Fig.5~b! ~data not shown!. Throughout the parameter range wehave fitted systematically the effective exponentge from theexp(2gex)-tail of the MWD. The values obtained are plottedin Fig. 6 versus the number of blobsl . We confirmge→g51.165 andge→g51 in the dilute and semi-dilute limit,respectively. In between we observe a crossover for the ef-fective exponent. Note that the error bars are mostly around60.01, however, a much higher accuracy is in principle fea-sible by this method. Note that the value obtained in the

dilute limit compares well with the best renormalizationgroup estimateg51.161560.0011.39 We believe this is themost reliable simulation determination so far of this well-known polymer exponent.

In passing, we recall that the polydispersity indexI5^L2&/^L&2, i.e., the ratio of weight average and numberaverage molecular weights, becomesI 5111/g and I 52 inthe dilute and semidilute limits, respectively. This offers~inprinciple! an additional method to check theg-exponent.However, due to the difficulty to measure accurately the dis-tribution for small x ~which contributes strongly toI ), weobtain with this method values that are slightly larger thanthat quoted above for the dilute limit.

To summarize, the behavior found in a large range ofl isin support of recent treatments of the problem by means ofrenormalization group and scaling analyses10,15,16and in con-trast to earlier claims34 that the Schulz distribution, Eq.~8!,will hold independent of the degree of overlap between thechains.

C. Specific heat

A quantity that may be accessible experimentally is thespecific heatcv~see also Table I!. Due to its variable numberof broken bonds~chain ends! EP should absorb or releaseenergy as the temperature, i.e.,E, is varied. In Fig. 7 we plotthe specific heat per monomercvfas versus scission energyE for two densities within the semidilute regime. We com-pare with the prediction of Eq.~11! where we have used theexponentsas50.6 andds50.5 and the amplitudeAs54.4estimated in Sec. IV A. We find a qualitatively good agree-ment, especially, as expected, for larger chains (E.5).

In the insert we check the predictioncv^L&/E25d ~plot-ted versus the numbers of blobsl ) for all configurations with^L&.5. Although the statistics is too poor to separate theextremely small difference betweendd andds the data ‘‘col-lapse’’ aroundd50.560.1 is qualitatively satisfactory forthis ambitious test.

Note that although this heat capacity has some resem-blance to that predicted for stepwise living polymers in the

FIG. 5. Molecular weight distribution.~a! c(L) at high ~melt! density f50.5 for energiesE as indicated in the figure. Insert: Data collapse of thenormalized distributionp(L)dL5p(x)dx5exp(2x)dx. ~b! Dilute limit con-firming p(x)}exp(2gx). Insert: MWD for smallx for two configurationswith f50.005 ~squares! and f50.01 ~diamonds! both at E510 in thedilute regime. We compare~bold line! with the prediction of Eq.~8b!.

FIG. 6. Effective exponentge versus the number of blobs per chainl5^L&/L* .



On: Thu, 18 Dec 2014 08:24:43

present of initiators, the latter systems are fundamentally dif-ferent in that they can show a~possibly rounded! phase tran-sition at nonzero temperature.3,7 In the present system nocritical phenomena are observed at finite temperature; this isconfirmed by the fact that the measured heat capacity be-comes system-size independent in the limit of large systems~compare Table II!.

D. Conformational properties

Following Ref. 25 the average over all chains of themean-square end-to-end distance^Re

2& and the radius of gy-ration ^Rg

2& have been measured and are listed in Table I. Toobtain meaningful results for chain size, when large chainsare present, we must, however, ‘‘unwrap’’ the periodic boxstarting from the position of one chain end, using the bondvectors between consequent monomers of the chain. Not sur-prisingly the mean bond lengthb2& is nearly identical to theones obtained in the study by Paulet al.25 and is likewiseslightly decreasing with density. Also we observe~for suffi-ciently long chains! a persistence lengthl p'1.1b. We in-clude in the table the mean interchain distanceH5(8^L&/f)1/3 which is of relevance in the context of diffu-sion controlled scission and recombination events.32

In Fig. 8 we demonstrate that the mean chain sizes forEP follow the same universal function as conventionalquenched polymers. We want to compare sizes of chains ofgiven mean-chain length with the size of swollen dilutechains of the same length; therefore we defineR05b^L&n.We plot the reduced average chain sizeue5^Re

2&/R02 and

ug5^Rg2&/R0

2 for the end-to-end distance and the radius ofgyration respectively as function of the scaling variablev5(R0 /H).3 This choice of variable, rather than the alterna-tive f/f* ~to whichv is proportional! enables a comparisonbetween EP and conventional monodisperse polymers~aster-isks in Fig. 8!. Data for the latter is taken from Paulet al.25

In the dilute regime we haveRe'Rg'R0 and the scalingfunction u approaches a constant as can be seen in Fig. 8.

Note that the plateau forue is slightly above 1 in agreementwith the persistence length given above. In the semi-dilutelimit the chains are Gaussian on length scales larger than theblob sizej ~i.e., Rg}^L&1/2) implying the scalingu}v2b

with exponentb5(2n21)/(3n21)'0.23. It is remarkablethat, on this scaling plot, EP and conventional monodispersepolymers are nearly indistinguishable; the two universalfunctions are virtually identical to within numerical accu-racy. Note the location of the crossover density atv* '361, i.e.,Re'R0'1.4H. A consistency check with the esti-mates of the amplitudesP and Q from Eq. ~12! gives aslightly lower valuev* 5PQ3n21(b/a)3/8'1. However, inview of the error in locating the crossover values and thelarge exponents involved in the estimation ofP andQ this isreasonably consistent.

This conclusion is corroborated in Fig. 9 where we showthe distribution of chain sizes,Rg

2&L and ^Re2&L , averaged

not over all chains present~as considered above! but over allchains of lengthL, plotted againstL ~rather ^L&). For themelt densityf50.5 we find ^Re

2&L'6^Rg2&L}L2n with a

Flory exponentn'1/2. For the much smaller semi-dilutedensityf50.01 we obtain swollen chains with an excludedvolume exponentn'0.6 ~or perhaps slightly larger!.

Figure 9~b! shows the distribution ofRg2&L at an inter-

mediate density which can be used to determine the averagesize,j, of a blob containingL* monomers. ForL!L* and^Rg

2&L!j2, i.e., within the blob, the excluded volume inter-actions dominate andRg

2&L}L1.2. For largerL ~i.e., alsolarger distances! the chains become Gaussian^Rg

2&L}L. Wemay measure the values ofL* andj directly from the cross-over between both regimes and compare them with the val-ues obtained above. From the intersection point of the twoslopes in Fig. 9~b! one hasj2'420 a2 and L* '160. Re-membering that every monomer occupies eight sites of thelattice one obtains as a consistency check roughlyf58L* /j3'0.13 which nearly matches the actual density ofthe system. Figure 9~b! confirms~within numerical accuracy!the sharp crossover between asymptotic forms as discussed

FIG. 7. The specific heat per monomercv . We plot cvfas versus scissionenergyE which is here equivalent to plot against the inverse temperature.The line is the prediction in the semi-dilute regime. Insert: Scaling-plot ofcv^L&/E25d versus number of blobsl in agreement with Eq.~11!. Symbolsfor the various densities as in Fig. 4~a!.

FIG. 8. Crossover scaling plot of the reduced mean-square end-to-end dis-tanceue5^Re

2&/R02 and the reduced radius of gyrationug5^Rg

2&/R02 vs the

scaling variablev5(R0 /H)3. Same symbols as in Fig. 4~a!, asterisks denotemonodisperse chains.



On: Thu, 18 Dec 2014 08:24:43

in connection with Fig. 4~b! and is another of our main re-sults.

V. FINITE-SIZE EFFECTS

Since no chain can be larger than the total number ofmonomers present in the system, the exponential MWD mustbreak down whenever the mean chain length becomes toolarge, i.e., when the average number of chains per box^M &'Lmax/^L& of order one.~For the same reason it must alwaysbreak down in the high molecular weight tail, in any case.!Note that for the highest energyE515 used we find typicallyan average chain number^M &'10. In fact, the~rather noisy!MWD show even then qualitatively an exponential decay.

In order to understand finite-size effects quantitativelywe have explicitly performed a systematic finite-size studyfor E510 andE515 at densityf50.5 which is summarizedin Table II. In Fig. 10~a! the mean-chain lengthL& and theend-to-end distanceRe reduced by their asymptotic valuesfor infinite systems~taken from the predictions in Sec. IItogether with the amplitudes obtained above!, are plotted

against an obvious finite-size scaling variable, which is thenumber of monomers in the boxLmax divided by the averagechain length of an infinite system,^L&. We see that the sys-tems for which we have presented results above are indeedlying in the asymptotic plateau region.~Additionally, thisconfirms the amplitudes and exponents presented alreadyabove.! The scaling curves for the mean chain length and thespecific heat~not shown! are—not surprisingly—similar, thespecific heat being the noisier quantity. Both decay in thesmall system limit with the system size~slope one!, implyingthe formation of a single long chain of length'Lmax}f.Hence, the effective growth exponenta is likely to be over-estimated in computational studies on relatively small sys-tems. FromLmax/^L&}f12a we see that increasing the den-sity at fixed scission energy should decrease the finite-sizeeffects shifting the system to the right in Fig. 10. Note that atintermediate system sizes aboveLmax/^L&'1 the mean chainlength is slightlylarger than the asymptotic value~see TableII !.

For the chain size in the limit of strong finite size effectsour results are qualitatively consistent with a square root de-pendency,R}Lmax

1/2, rather than a 1/3 exponent for a single

FIG. 9. Variation of chain size with chain lengthL. ~a! At very low [email protected].,f50.05 ~dashed lines!# the chain are swollenR}Ln with n'0.6. Atdensityf50.5 one has a melt of Gaussian chains (n50.5). ~b! For a semi-dilute system, as forf50.1 at E510 andE515, one can identify tworegimes. Up to sizesj the chains are swollen, on sizes larger thanj they areGaussian.

FIG. 10. Systematic variation of the system dimensionSB . ~a! Reducedchain length^L&, end-to-end sizeRe versusLmax reduced by the averagechain length of an infinite system. Solid symbolsE515, open symbols,E510. ~b! MWD for f50.5 at scission energyE510 for various systemsizes as indicated in the figure.



On: Thu, 18 Dec 2014 08:24:43

compact chain; this is because the chain size is defined by‘‘unwrapping’’ the periodic box starting from one chain end~as mentioned above! and nothing prevents a single Gaussianchain from wrapping repeatedly to fill the periodic box. InFig. 10~b! we see directly from the MWD how the transitionto the single chain ‘‘phase’’ occurs. For small systems (SB

<30) we find clear peaks in the distribution atL5Lmax

which disappear as we further increase the system size. Thisfinite-sizes study confirms unambiguously that the configu-rations presented above are indeed large enough—at least forthe static properties. It is however not so clear that this is stillholds for dynamic properties in the limit of large barrierB,and we will address this issue in the second part of thiswork.32

VI. DISCUSSION

In the present Monte Carlo simulation of static~anddynamic32! bulk properties of EP we have used a greatlyimproved version of a recently proposed algorithm.20 Theoriginal data structure based on the BFM polymer chainmodel was completely altered, recognizing that the funda-mental entities in EP systems are not chains but monomers,or ~in our algorithm! the bonds connecting these. This makesthe algorithm much more powerful, and allows us to studysystems across three decades in monomer density with up to75 000 monomers per configuration~for f50.6 on a DECAlpha workstation!. With this algorithm we can equilibratesystems with mean-chain length up to^L&'5000 in the melt,which is at least one orders of magnitude larger than anyother previous study.20,23 This was achieved with negligiblefinite-size effects as we have explicitly checked.

Therefore, we are able to compare simulational results todata extracted from laboratory observations and to the scal-ing predictions of theoretical treatments, known to hold inthe asymptotic limit of sufficiently long chains. Some earliernumerical observations which showed discrepancies withanalytic results could thereby be traced to finite-size effectsarising from small systems and/or short chains.

In the present investigation we simulated EP where theformation of rings is not allowed.~A study for systems con-taining rings is underway.30! The non-bonded monomer in-teractions have been set to zero~athermal conditions! so thatthe overall picture was not complicated by phase separationat low temperature. For simplicity the chains are modelled astotally flexible.

At variance to some recent computational studies23 wefind that the static properties in the asymptotic limit of large^L& agree well with analytical predictions of the scalingtheory1,10 and renormalization group studies:15,16

~1! The mean-chain length varies asL&5L* (f/f* )a

}fa exp(dE) with densityf and end-cap energyE. Aspredicted analytically,10,16 we find the exponentsad

5dd51/(11g)50.46 in the dilute regime andad

50.6,dd50.5 in the semi-dilute limit. We confirmed thescaling behavior of the mean number of blobsl5^L&/L* versus the reduced densityf/f* . This showsa rather sharp crossover which enables clean crossoverlines f* (E) andL* (E) have been located.

~2! In the dilute limit the MWD scales consistently with aSchulz distributionp(s)}sg21 exp(2s) with a scalingvariables5gL/^L&. In the semi-dilute regime for largeenough^L& the MWD decays exponentially with chain-length p(s)5exp(2s) where the scaling variable be-comes the reduced chain lengths5L/^L&. Between bothlimiting regimes we observe a relatively gradual cross-over atl'1, the overlap threshold of the polymers. Theextremely large size of our chains, allied to a carefulanalysis of the size distribution in the dilute limit, allowsus to extract an accurate estimate of the self-avoidingwalk susceptibility exponentg51.16560.01.

~3! A satisfactory scaling collapse of the specific heat inboth density regimes was obtained by plottingcv^L&/E25d'0.5 versusl . As suggested earlier,20 themaximum of the specific heat per monomercv occurs atE52/d54 for high concentrationsf.

~4! Chain conformations are described within numerical ac-curacy by the same universal functions as for conven-tional polymers.

~5! As in the case of conventional polymers, the chains areswollen within the excluded volume blobs and Gaussianat larger distances; this has been made directly evidentfrom the scaling of coil size against chain length withina single system.

We believe that the present work unambiguously con-firms the scaling results for an idealized model of EP basedon the classical behavior of conventional quenchedpolymers.10 Accordingly, it leaves completely unansweredthe question of how, in some experimental systems to whichthe model appears closely applicable, a growth exponenta.1.2 is convincingly argued to arise.18 This question re-mains open, but clearly must involve physics not in thepresent model, which is based on assuming a fixed scissionenergy, athermal excluded volume interactions, and the ab-sence of rings.

A parallel version of the algorithm has also been imple-mented and used for some of the results presented above. Itforms the basis of the natural extension of this work to thequestion of EP dynamics, which will be publishedelsewhere.32

ACKNOWLEDGMENTS

The authors are indebted to J. P. Desplat, Y. Rouault, M.Muller, P. van der Schoot and F. Lequeux for valuable dis-cussions and assistance during the present investigation.JPW acknowledges support by EPSRC under Grant No. GR/K56223 and is indebted to D. P. Landau for hospitality in theCenter for Simulational Physics at the University of Georgia.AM acknowledges the hospitality of the EPCC in Edinburgh~TRACS program!, and the support by theDeutsche Fors-chungsgemeinschaft~DFG! under Grant No. 436-BUL 113/45, and Grant No. 301/93 given by theBulgarian Ministryfor Science and Education.

1M. E. Cates and S. J. Candau, J. Phys.: Condens. Matter2, 6869~1990!.2R. L. Scott, J. Phys. Chem.69, 261 ~1965!.3J. C. Wheeler, S. J. Kennedy, and P. Pfeuty, Phys. Rev. Lett.45, 1748~1980!; S. J. Kennedy and J. C. Wheeler, J. Phys. Chem.78, 953 ~1984!.



On: Thu, 18 Dec 2014 08:24:43

4G. Faivre and J. L. Gardissat, Macromolecules19, 1988~1986!.5F. Oozawa and S. Asakura,Thermodynamics in the Polymerization ofProteins~Academic, New York, 1975!.

6M. Szwarc, Nature~London! 178, 1168~1956!.7LP polymerize~above or below some critical temperature! starting froman imposed density of initiators by reversible addition of monomers onactive chain ends. They are held together by~strong! covalent carbon-to-carbon bonds, hence, they do not break in the middle of the polymerchain. Neither do LP combine together to make larger LP or to form rings.For a recent review see S. C. Greer, Adv. Chem. Phys.96, 261 ~1996!.

8T. M. Clausen, P. K. Vinson, J. R. Minter, H. T. Davis, Y. Talmon, andW. G. Miller, J. Phys. Chem.96, 474 ~1992!.

9R. Messager, A. Ott, D. Chatenay, W. Urbach, D. Langevin, Phys. Rev. E60, 1410~1988!; J. Appel and G. Porte, Europhys. Lett.12, 185 ~1990!.

10M. E. Cates, J. Phys.49, 1593~1988!.11E. Facetibold and G. Waton, Langmuir11, 1972~1995!.12A. Ott, J. P. Bouchaud, D. Langevin, and W. Urbach, Phys. Rev. Lett.65,

2201 ~1990!; J. P. Bouchaud, A. Ott, D. Langevin, and W. Urbach, J.Phys. II1, 1465~1991!.

13B. O’Shaughnessy and J. Yu, Phys. Rev. Lett.74, 4329~1995!.14P. J. Flory,Principles of Polymer Chemistry~Cornell University Press,

Ithaca, NY, 1953!.15L. Schafer, Phys. Rev. B46, 6061~1992!.16P. van der Schoot, Europhys. Lett.39, 25 ~1997!.17J. F. Berret, J. Appell, and G. Porte, Langmuir9, 2851~1993!.18G. Jerke, J. S. Pedersen, S. U. Egelhaaf, and P. Schurtenberger, Phys. Rev.

E 56, 5772~1997!; P. Schurtenberger, C. Cavaco, F. Tiberg, and O. Re-gev, Langmuir12, 2894~1996!.

19A. Milchev, Polymer34, 362~1993!; A. Milchev and D. P. Landau, Phys.Rev. E52, 6431~1995!; J. Chem. Phys.104, 9161~1996!.

20Y. Rouault and A. Milchev, Phys. Rev. E51, 5905~1995!.21Y. Rouault and A. Milchev, J. Phys. II5, 343 ~1995!.22Y. Rouault and A. Milchev, Phys. Rev. E55, 2020~1997!.23M. Kroger and R. Makhloufi, Phys. Rev. E53, 2531~1996!; W. Carl, R.

Makhloufi, and M. Kroger,7, 931 ~1997!.24I. Carmesin and K. Kremer, Macromolecules21, 2819~1988!.

25W. Paul, K. Binder, D. Herrmann, and K. Kremer, J. Chem. Phys.95,7726 ~1991!.

26M. Muller and K. Binder, Comput. Phys. Commun.84, 173~1994!; Mac-romolecules28, 1825~1995!.

27J. Wittmer, W. Paul, and K. Binder, Macromolecules25, 7211~1992!.28R. Petschek, P. Pfeuty, and J. C. Wheeler, Phys. Rev. A34, 2391~1986!.

See also, M. E. Cates, J. Phys.~France! Lett. 46, L837 ~1985!.29J. P. Wittmer, A. Milchev, and M. E. Cates, Europhys. Lett.41, 291

~1998!.30J. P. Wittmer, P. van der Schoot, A. Milchev, and M. E. Cates~unpub-

lished!.31F. Lequeux and S.J. Candau, inStructure and Flow in Surfactant Solu-

tions, edited by C. A. Herb and R. K. Prud’homme~ACS Books, Wash-ington, 1994!.

32J. P. Wittmer, A. Milchev, and M. E. Cates~unpublished!.33P. G. de Gennes, inScaling Concepts in Polymer Physics~Cornell Uni-

versity Press, Ithaca, NY, 1979!, Chap. 1.34P. D. Gujrati, Phys. Rev. B40, 5140~1989!.35I. Gerroff, A. Milchev, K. Binder, and W. Paul, J. Chem. Phys.98, 6526

~1993!; A. Milchev, W. Paul, and K. Binder,ibid. 99, 4786~1993!.36See, e.g.,Monte Carlo Methods in Statistical Physics, edited by K. Binder

~Springer Verlag, Berlin, 1979!.37This is rather unfortunate the relaxation times being much faster using the

full 108 bonds available and no difference having been observed in thestatic properties where the~extremely unlikely! crossed bonds do not con-tribute. However, the rare event of fixed monomers~in the limit of highbarrierB where crossed bonds cannot be broken sufficiently fast! changescompletely the dynamic properties from Rouse to Reptation behavior~dueto the view artificial fixed obstacles! as obvious from the mean-squaredisplacement~Ref. 32!. We are indebted to Y. Rouault and M. Mu¨ller forpointing our attention on this technical difficulty related to the latticenature of the underlying BFM.

38R. Granek and M. E. Cates, J. Chem. Phys.96, 4758~1992!.39J. des Cloizeaux and G. Jannink,Polymers in Solution~Clarendon, Ox-

ford, 1990!.



On: Thu, 18 Dec 2014 08:24:43

Documents

Dynamical Monte Carlo study of equilibrium polymers: Static properties