Volume48A, number5 PHYSICSLETTERS 15 July 1974

ON THE POLYMER PHASE TRANSITION

D.C. RAPAPORTPhysicsDepartment,Bar-han University,Ramat-Gan,Israel

Received16 May 1974

Theself-interacting,self-avoidinglattice walkmodelof a polymerchainhasbeenstudiedby exactenumerationof short chainconfigurations.Theapparentdevelopmentof a divergencein thespecific heatasthe chainlength isincreasedindicatesa phasetransitionin theinfinite chainlimit.

The randomwalk has longbeenregardedasthe we havedeterminedtheCnm exactlyfor n ~ 9 on thebasisfor developingmodelsof an isolatedpolymer FCClattice andn ~ 13 on theSC.chainin dilute solution [1]. However,any approach The reducedspecific heatC(O) = d2logc,~(0)/do2basedon the randomwalk neglectstwo important (the true specific heatis kBO2C)wasdeterminedasalong-rangeeffects: theexcludedvolume,and the function of 0 for eachn andits maximumvalueCmaxattractiveinteractionbetweennon-neighbouringchain computed.Cmax is plottedversuslog n in fig. 1, andunitswhich approachone anotherin particularspatial thevaluesof 0 at which the maximaoccur,0t~areconfigurations.TheseeffectsdestroytheMarkovian given in table 1.natureof themodel andarethuscapableof produc- Fig. 1 also containsestimatesof Cmaxextracteding a fundamentalchangein behaviour, from graphedFCC Monte Carlo dataon muchlonger

Thelattice versionof the chainwithexcludedvolume - the self-avoidingwalk - hasbeenextensivelystudied,[e.g.2], thewalk with attractiveforcesto a Ilesserextent [3—5]. The effect of the attractionis 8 - Ismallat hightemperatureandthechainbehavesasa /self-avoidingwalk,butat low temperatureenergy -

considerationsrequirethat the chaincondenseinto IF

anordered,tightly packed(coiled or folded) con- 6 - //

figuration.Theresultspresentedin this note pointto the existenceof aphasetransitionat someinter- - /

C SC(xIO) /mediatetemperature,anda specificheatwhich be- max //

comesinfinite at the transitionpoint. This tran- - \ / /sition presumablycoincideswith theonsetof long- j1 ,~

rangeorder. - / ,‘ FCC

Thepartitionfunctionof a chainof n + 1 units, 2 - ~ V ,‘

i.e. ann-stepwalk, embeddedin a regularlattice is

c~(0)= m~OCnm exp(mU) -

with0 = —J/kBT.J is theexcessenergydueto each 0 I b’ ~ 29 49 6999pairof non-adjacentunitswhich approachto nearest-neighbourseparation,andCnm is the numberof n-stepwalkswithm suchpairs.TheC,, (0) arefinite Fig. 1. Reducedspecificheatmaximumversuslog n. The

polynomialsin exp0 and,with the aid of a computer MonteCarloresults(n ~ 29) aretakenfrom ref. [3].

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Volume48A, number5 PHYSICSLETTERS 15 July 1974

Table 1 down,but the overalltrendappearssimilar to theFCC.Estimatesof for different chainlengthsn. Little canbe saidregardingthe trendsin 0~,except

that in the FCC caseapossiblelimiting valueis~

0t (FCC) ~ 8~(SC)_______ 0.2. Thismeansthat in then = limit the

5 0.405 9 1.020 specificheatis singularat finite temperature.(The6 0.405 10 0.924 irregularO~behaviouris thereasonwe choseto work7 0.379 11 1.005 with thereducedspecific heat.)8 0.356 12 0.808 Supportfor this type of analysisis providedby9 0.342 13 0.810

calculationsfor small Ising systems[6] which alsoyield specificheatmaximathatdivergesmoothly

chains [3]. Theerror barsin theMonte Carloestimates with increasingsystemsize. In this casethelimit iscorrespondto a 2% uncertainty;no estimatesof their knownexactly,andthe observedtrendscorrectlyvalueshavebeenpublished,butthey areunlikely to predict the limiting behaviour.besmaller thanthis. Furthermore,no attemptwasmadeto searchfor the locationof Cmax, sothatthe correctMonteCarlo Cmax maybeslightly larger. References

Evenwith thesereservationsthe resultsare striking.TheFCC exactdataestablisha smoothtrendwhich [l~ P.J.Flory, Statisticalmechanicsof chainmolecules

persistsinto theMonte Carlo results.The actualrate (Wiley N.Y. 1969).

of divergenceis fasterthanlogn, but noteasilydeter- [2] C.’Domb, Adv. Chem.Phys.15 (1969)229.[3] J. MazurandF.L. McCrackin,J. Chem. Phys.49 (1968)mined. A log-logplot of the samedatashowsdown- 648.

wardcurvature,indicatingthat the divergenceis not [4] F.L. McCrackin,J. MazurandC.M. Guttman,

a simplepowerlaw — at leastnot for the chainlengths Macromolecules6 (1973) 859.

considered,andtechniquessuchasNeville extrapolation [5] D.C. Rapaport,Macromolecules7 (1974)64.

fail to clarify matters.TheSC resultshaveyet to settle 16] A.E. FerdinandandM.E. Fisher,Phys.Rev. 185 (1969)

832.

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