Marco Baiesi,1 Enrico Carlon,1 Enzo Orlandini,1 and Attilio L. Stella1,2
1INFM–Dipartimento di Fisica, Universita` di Padova, I-35131 Padova, Italy2Sezione INFN, Universita` di Padova, I-35131 Padova, Italy
~Received 20 July 2000; published 21 March 2001!
Using exact enumeration methods and Monte Carlo simulations, we study the phase diagram relative to theconformational transitions of a diblock copolymer in two dimensions. The polymer is made of two homoge-neous strands of monomers of different species which are joined to each other at one end. We find that,depending on the values of the energy parameters in the model, there is either a first order collapse from aswollen phase to a compact phase of spiral type, or a continuous transition to an intermediate zipped phasefollowed by a first order collapse at lower temperatures. Critical exponents of the zipping transition arecomputed, and their exact values are conjectured on the basis of a mapping onto percolation geometry, thanksto recent results on path-crossing probabilities.
Polymers in solution typically undergo a coil-globutransition from a high temperature~T! swollen phase to a lowT phase where the polymer assumes compact conformatIn the case of homopolymers, for which all the monomare identical, this transition is by now well understood@1,2#.It is known asQ collapse, and has been extensively invesgated in the past years using various methods such as mfield approximations@3#, exact enumerations of interactinself-avoiding walks on lattices@4#, Monte Carlo calculations@5#, transfer matrix calculations@6#, and field theoretical cal-culations@7#. In two dimensions the exponents of theQ col-lapse have been related to the fractal properties of the pelation cluster, and are believed to be known exactly@8#.
The study of the conformational properties and phtransitions of macromolecules with inhomogeneous or rdom sequences of monomers is an interesting frontier inrent polymer statistics@9#. These systems pose new theorical and numerical challenges, compared to their mstandard, homogeneous counterparts. Particularly intereis the possibility that inhomogeneities along the chain colead to transitions and universality classes of scaling behior, which are not realized for homopolymers@10,11#. More-over, the most complex versions of models of this classalso expected to be useful for descriptions of phenomenaprotein folding@12#, DNA denaturation@13# and RNA sec-ondary structure formation@14#. Thus, an investigation of theuniversal properties of the simplest among these systemsoffer an important gauge of the relevant model ingredienecessary in order to reproduce the basic conformatiomechanisms in more sophisticated descriptions.
One of the most elementary conformational transitio~not realized in homopolymers! one can try to describe inrelatively simple terms is what we call here azippingtransi-tion. By zipping we mean a process in which two strancomposing the polymer come in contact in such a way aform a bound double structure, which remains swollen adoes not assume compact configurations. In order to indutransition from an unzipped state to a zipped state, the mmal inhomogeneity required implies a distinction betwe
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the two strands: if the polymer is made of two blocks coposed of monomers of different species~diblock copolymer!,and there is a dominant attractive interaction acting betwthese different monomers, one would expect such a transto be possible. Of course, the zipping occurring in biomecules results in general from higher degrees of inhomoneity than those of a simple diblock copolymer.
From a physical point of view one can think of dibloccopolymers with oppositely charged monomers in the tblocks; in the model discussed here the interactions areshort range, and this would correspond to the casescreened Coulomb forces. Another possibility is that attrtive interactions between monomers of the two blocksestablished through a preferential formation of hydrogbonds.
Attractive interactions between the two blocks, besidzipping, also tend to produce collapse into a globular copact state, unless some contrasting effect limits the capabof a given monomer to attract monomers of the other bloIn a recent paper@15#, a model of a diblock copolymer withsome of the features discussed above was studied in bothand three dimensions. In this model the two blocks wrepresented by two halves of a self avoiding walk~SAW! ona hypercubic lattice with attractive interactions betwenearest neighbor sites~monomers! visited by the two blocks.So, apart from the steric constraints, there was no interacmechanism possibly opposing the tendency of a given momer to be surrounded by as many monomers of the oblock as possible. The transition of the diblock copolymfrom a highT swollen phase to a lowT compact phase, hadanalogies with both polymer adsorption on a wall, andQcollapse, but turned out to belong to a universality classferent from both@15#. An intriguing question remained opeconcerning the very nature of this transition: indeed, the psibility that a zipped, swollen phase could exist for tempetures just below the transition could not be excluded. If twere the case, the adsorptionlike collapse found in Ref.@15#would correspond to a zipping, and a further transition tocompact globular phase should be expected to take placelower T.
In the present paper we extend the model of Ref.@15# in
BAIESI, CARLON, ORLANDINI, AND STELLA PHYSICAL REVIEW E 63 041801
two dimensions to include an interaction among alternattriplets of different monomers. Depending on its sign, tadditional interaction can enhance the tendency of thetem either to form compact structures or to take zipped cformations. We draw an accurate phase diagram for thetem, in which a zipping transition line is well identified ancharacterized. Our analysis seems to indicate that the adstionlike collapse of Ref.@15# belongs to the zipping universality class as well. Specifically, we find that, dependingthe triplet interaction energy, one has either a continuswollen-zipped transition followed by a first order collapinto compact conformations, or a direct first order swollecompact transition. Although we mainly focus on the zping, it turns out that the first order collapse has interestfeatures as well, since it shows remarkable analogies wthat found in homopolymers with orientation dependentteractions, which attracted some attention recently@16#.
We will argue that the exact exponents for the zippitransition in two dimensions can be found through an idtification between the stochastic geometry of the blocksthat of a percolation cluster backbone@17#. A preliminaryform of this argument was presented in Ref.@15#. The rel-evant dimensions of the percolation cluster can be identithanks to some recent results for path-crossing probabil@18#. Our numerical estimates for the zipping exponentsin very good agreement with the conjectured values. Toknowledge, the mapping onto percolative stochastic geetry we discuss here is the first example of exact resderived for a genuinely inhomogeneous polymer problin two dimensions. The connection with percolation geoetry also shows that the physics of zipping is closely cnected with that of theQ-point transition, for which a rep-resentation in terms of percolation geometry was establislong ago@8#.
Besides the prototypical importance that zipping acquhere also on the basis of our exact results, one should rethat many of the conformational transitions occurring in bmolecules show aspects which, to some extent, are remcent of zipping. This is certainly the case for DNA, in whicupon lowering the temperature below the denaturation othe conjugate bases form pairs, so that the molecule arraitself in a double stranded helical structure@13#. Doublestranded, zipped structures also appear in the foldingb-hairpin peptides@19#. In the philosophy mentioned at thbeginning of this section, one can hope that studies likeone presented here can teach something about how to mthese more complicated systems properly.
This paper is organized as follows: In Sec. II we presthe model and the main features of the phase diagramSecs. III and IV we discuss the numerical results obtainedexact enumerations and Monte Carlo simulations, resptively. In Sec. V, using recent results for crossing probabties of percolation paths in two dimensions, we conjectthe exact values of the exponents of the zipping transitSection VI concludes the paper, with a summary of thesults and a general discussion.
II. MODEL AND PHASE DIAGRAM
We model the two-dimensional~2D! diblock copolymerby an interacting SAW on the square lattice. In the config
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ration w the SAW hasuwu5N vertices~monomers!, with Neven, and consists ofN/2 consecutive monomers of typeA(wA) followed by N/2 monomers of typeB (wB). A pair ofvertices (A,B) form a contact if they are a unit lattice distance apart. The interaction between the two blockswA andwB is taken into account by assigning an energy« ~«,0! toeachA-B contact. In addition we introduce a second enerparameterd, associated with contacts formed by a sequeA-B-A or B-A-B of neighboring monomers on a line. Wrefer to these sequences as totriple contacts. The Hamil-tonian of the system in configurationw is given by
H~w!5NAB~w!«1N3~w!d ~1!
whereNAB(w) andN3(w) are the number ofA-B and triplecontacts, respectively.
For d50 we recover the model introduced in Ref.@15#; inthe present work we consider both signs ofd: a positivevalue ofd must prevent the polymer from collapse into compact conformations and favor an intermediate zipped phwhile for a negatived the tendency to collapse is enhance
Letting cN(NAB ,N3) be the number of copolymer configurations withN edges,NAB contacts of typeA-B andN3triple contacts, we define the finite-N free energy per monomer
FN~b,d!5N21logZN~b,d!, ~2!
where
ZN~b,d!5 (NAB ,N3
cN~NAB ,N3!e2b(NAB«1N3d) ~3!
is the partition function andb51/T @20#. Throughout therest of the paper we set«521. By varyingT andd we haveexplored the phase diagram of the model~see Fig. 1! on the
FIG. 1. Phase diagram in thed-T plane. Line~1! is the continu-ous zipping transition line separating the high temperature swophase from the zipped phase. Lines~2! and ~3! are of first ordertype.
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ZIPPING AND COLLAPSE OF DIBLOCK COPOLYMERS PHYSICAL REVIEW E63 041801
basis of exact enumeration and Monte Carlo simulationsults.
As expected, we find a zipped phase in the positivedregion, while for negatived there is a direct transition fromthe swollen to the compact phase. The numerical resshow that the line~1! separating the swollen and zippephases is continuous, while line~2! and ~3! are first order.Unfortunately, the numerical methods at our disposal aresufficiently accurate to determine the precise location ofintersection point between the lines, nor the character ofphase transition in this point, which could be of special tyThe location of this intersection point seems to fall at slighnegative values ofd.
It is rather instructive to show some typical equilibriuconformations of the copolymer for various values ofT andd, in the three different phases~see Fig. 2!. The configura-tions are snapshots obtained by Monte Carlo simulations~a!and ~b! are conformations in the swollen highT phase, with~b! close to the zipping transition.~c! and ~d! are insteadzipped configurations. Note that the pairing of the twstrands in~d! follows an opposite orientation with respect~c!. Finally, ~e! and ~f! are both compact, but of differennature: the latter occurs atd,0 where triple contacts arenergetically favored. Therefore, the polymer assumespiral-like shape with straight segments turning aroundcenter in order to maximize the number of triple contactscase~e!, d is positive: the configuration is still of spiral typebut in this case the arms of the spiral are oriented prefetially at 45° with respect to the axes of the square latticeorder to avoid the formation of triple contacts.
Now one can easily understand why lines~1! and ~3! forlarged run practically horizontal in the phase diagram of F1. In the whole zipped phase triple contacts, which costenergyd, seldom occur@see Fig. 2~c!#; therefore, the zippingtemperature should depend rather weakly ond. In addition
FIG. 2. Typical Monte Carlo equilibrium configurations for aN560 diblock copolymer:~a! and ~b! swollen chains;~c! and ~d!zipped chains;~e! and ~f! compact chains withd51.5 ~e! andd521.5 ~f!.
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the polymer can form compact conformations such asshown in Fig. 2~e!, which also avoids this type of contactHence the zipped-compact transition temperature also shnot depend ond, whend is positive and large enough. Botlines~1! and~3! in Fig. 1 should be asymptotically horizontafor larged.
III. EXACT ENUMERATIONS
Exact enumerations of interacting SAW’s are standtechniques for the study of the homopolymerQ-collapsetransition@4#. In the present calculation we generated all posible configurations for copolymers up toN530 monomers,a length which is already sufficient to characterize ratwell the critical behavior of the zipping transition.
The occurrence of phase transitions in interacting polymsystems can be detected by studying the largeN behavior ofthe canonical average squared radius of gyration,
Rg25
(w
exp@2bH~w!#R2~w!
(w
exp@2bH~w!#
, ~4!
where the sums extend to allN-step configurationsw of thecopolymer, with radiusR(w) relative to the center of massIndeed, in the proximity of a conformational transition temperatureTc we expect that
Rg~N,T!;NncR@~T2Tc!Nf#, ~5!
wherenc andf are the exponents characterizing the trantion, andR is a scaling function that is assumed to approaa positive constant if its argument approaches zero.
Another important quantity is the specific heatC(N,T)5(1/N)]^H&/]T, which for N large andT close toTc isexpected to obey the scaling
C~N,T!;N2f21C@~T2Tc!Nf#, ~6!
whereC is again a suitable scaling function.Figures 3 and 4 showdRg
2/db andC as functions ofb ford50 ~a! andd521.5 ~b!. As the radius of gyration drops aa transition, its derivativedRg
2/db shows a peak in correspondence to the transition point.
For d521.5 both quantities have a single isolated peindicating that there is a single transition from a swollphase to a compact phase. Ford50, instead, the derivative othe radius of gyration has two distinct peaks@Fig. 3~a!#,while the picture emerging from the specific heat@Fig. 3~b!#is somewhat more confusing, since theN dependence of thepeak positions and heights is rather irregular, and theirtrapolation toN→` becomes impossible. Therefore we fcus on the peaks ofdRg
2/db. Extrapolating their positions ad50, we find the following two estimates of critical temperaturesTc51.40(15) andT2c51.1(20), which overlapsomewhat within error bars. For this reason it is difficultdiscern between two separate, but close, transitions, asingle transition. For positive values ofd the two sets of
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BAIESI, CARLON, ORLANDINI, AND STELLA PHYSICAL REVIEW E 63 041801
peaks are clearly separated and extrapolations yield twotinct transition temperatures. Here we focus on the chaterization of the highT transition from the swollen phase tthe zipped phase. The scaling form of Eq.~5! implies thatTc(N), the temperature wheredRg
2/db has a maximum,scales for largeN asTc(N)2Tc;x0N2f, with x0 a suitableconstant. We calculated both the radius of gyration and scific heat atTc(N); from Eq. ~5! and ~6!, one has
Rg@N,T5Tc~N!#;NncR~x0! ~7!
and
C@N,T5Tc~N!#;N2f21C~x0!. ~8!
FIG. 3. Derivatives of the squared radius of gyration withspect to the inverse temperature ford50 ~a! and d521.5 ~b! andfor N512,14, . . . 30. Thedouble peak structure ford50 suggests asequence of two transitions: swollen zipped and zipped collaps
FIG. 4. Solid lines: specific heat from exact enumerationsN512, 14, . . . ,30 ford50 ~a! and d521.5 ~b!. Circles: MonteCarlo results forN530.
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For the calculation of the critical exponents we first formthe finiteN approximants, for instance
nc~N![ln~Rg~N12!/Rg~N!!
ln~~N12!/N!~9!
@hereRg(N) is a shorthand notation forRg„N,Tc(N)…], andthen extrapolatednc(N) to N→`. The same procedure wafollowed for f.
The extrapolated values are reported in Table I, togetwith the exponentnc8 , which is that associated with the radius of the half-chain, or single block, which at the critictemperature should scale as
Rg8~N,T5Tc!;S N
2 D nc8
. ~10!
The values of the exponentsf andnc vary slightly alongline ~1! of Fig. 1 whend is increased, whilenc8 is ratherstable. We believe that the variation ofnc andf is a spuriouseffect, due to the vicinity of an additional transition in thneighborhood ofd50. It is much more plausible that thexponents are constant along line~1!; the most reliable esti-mates fornc andf should be those for larged, where lines~1! and ~3! of Fig. 1 are clearly separated. The values ofnc
and nc8 are consistent~the former only at larged! with thescaling behavior of a SAW, namely,Rg;N3/4. The value off is instead consistent withf59/1650.5625, which wasconjectured in Ref.@15# for the transition atd50, and will bederived in detail in Sec. V.
As for transition lines~2! and ~3!, the exact enumerationanalysis is not at all conclusive since the scaling behaviothe peaks with the chain lengthN is rather irregular, andprecise extrapolations turn out to be impossible. This iswill be clarified with the use of Monte Carlo simulationwhich allow one to achieve much larger copolymer lengt
IV. MONTE CARLO SIMULATIONS
In order to sharpen and extend the results of exact emerations, in particular concerning the properties of zippand collapsed phases, we performed Monte Carlo simtions for variousN andd and for a wide range of temperatures. Since the simulations considered involve samplingpoints which include low values ofT, a standard Markovchain Monte Carlo approach is unlikely to be successfubeing difficult to construct a Markov chain sufficiently ‘‘mo
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TABLE I. Extrapolated values ofnc andf from the exact enu-meration data relative to diblock copolymers up to lengthN530.The exponentnc8 is obtained from the scaling behavior of the radiof one of the blocks.
d nc f nc8
0.0 0.72~1! 0.60~5! 0.74~1!
0.5 0.73~1! 0.58~3! 0.750~5!
1.0 0.73~1! 0.57~3! 0.750~5!
3.0 0.74~1! 0.56~3! 0.750~5!
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ZIPPING AND COLLAPSE OF DIBLOCK COPOLYMERS PHYSICAL REVIEW E63 041801
bile’’ at low T where the interaction energies become revant. Instead we use a multiple Markov chain techniquewhich one samples simultaneously at various values oT,including T5` where convergence is rapid. Most recenthis method was used successfully to investigate collatransitions in homopolymers@21#, hetheropolymers@11#, andadsorption inQ solvent@22#.
First one defines a Metropolis based Markov chain fotemperatureT. This procedure makes use of a hybrid algrithm based on pivot@23# as well as on local moves@24#.Pivot moves are of global type, and operate well in the swlen regime, whereas local moves turn out to be essentiaspeeding up Monte Carlo convergence at low temperat@21#. In our calculations, each Monte Carlo step consistsO(1) pivot moves andO(N) local moves. In this modelhowever, we have to deal also with a zipped phase wheremost probable configurations are characterized by havingtwo blocks A and B paired together, but still not compac@see Figs. 1~b!–1~d!#. To increase the mobility of the Markochain in this region we added a set of bilocal moves, suchend-end reptation and kink-end~and end-kink! moves@25#.These moves are particularly effective for dense chains,even more effective for zipped chains, where typically oside of each half-chain is free and can hold a new kink. Tresulting algorithm is slightly heavier, but enables the recrocal sliding of the half-chains and a more efficient explotion of the configuration space. One may then run in paraa numberm ~typically 20–40) of these Markov chains adifferent temperatures. The sampling at lowT is then con-siderably enriched by swapping configurations between Mkov chains contiguous inT. The whole process is itself~composite! Markov chain, obeys detailed balance, andergodic@21#.
Monte Carlo simulations were performed for three dtinct values ofd: d51.5,d521.5, andd50. As a test of theperformance of the multiple Markov chain algorithm wcompared the Monte Carlo results with those obtained frthe exact enumeration for chains up toN530 monomers. Inall cases analyzed the agreement turned out to be extregood @see, for instance, Fig. 4~a!#.
A. dÄ1.5
In the cased51.5 we considered diblocks of lengths upN5400, and sampled at a set ofm.40 temperatures typically in the [email protected], #. In Fig. 5 we plot the specificheat as a function ofb for different N values. Clearly eachcurve displays a double peak structure indicating two subquent transitions. We can rule out the possibility that sudouble peaked structure is a finite size effect by noting tthe peaks sharpen and grow withN. Let us focus first on theset of peaks at higher temperatures, i.e., on the transfrom a swollen phase to a zipped phase. The corresponTc and f could be deduced from theN dependence of theheight,h(N), and position,Tc(N), of the peak maxima. In-deed, from the scaling behavior@Eq. ~6!#, we expect, asNincreases,
h~N!;N2f21 and Tc~N!2Tc;N2f. ~11!
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Since a linear least squares fit of logh vs logN gives a verylarge x2 statistical error, we fit the data with a functioA N2f21(11B/N) where a scaling correction 1/N is in-cluded. The least squares fit in this case givesf50.5760.02,in agreement with the valuef59/1650.5625 conjectured inRef. @15#, and also with the estimates obtained by exact emeration. This procedure yields results consistent with arect extrapolation of effective finiteN exponents. We havealso tried to fit the data by using the more general scacorrection 1/ND, and found that the best fits are those w0.5<D<1 yielding a stable value forf. The estimated valueof f allowed us to extrapolateTc by plottingTc(N) vs 1/Nf.This givesTc51.51(4) @bc50.66(2)#.
The inset of Fig. 5 shows a plot ofC/N2f21 vs (b2bc)N
f, where we have usedf59/16 and the estimatebc50.66. As expected, the high temperature peaks collaonto a single curve quite nicely. Conversely, for a setpeaks at lower temperatures, the same rescaling proceturns out to be inappropriate. In particular, by using the rcaled variables adequate for the former set of peaks, thesitions of the latter set tend to move away from zero whtheir heights still increase withN: this is only consistent witha scenario in which a new transition, at a lowerT5T2c ,exists. This transition should be also characterized bcrossover exponent greater thanf59/16. We have tried toverify this by looking for two new valuesb2c and f2 thatallow a reasonable fit of the scaling behavior of the secoset of peaks. In this way we obtained the rough estimab2c'1.0 andf2'0.7.9/16. Unfortunately the sampling alow temperatures is not good enough to make such estimsufficiently sharp. Moreover, in the zipped phase the efftive size of anN monomers system drops toN/2, making thefinite size corrections to scaling more pronounced.
The different nature of the two transitions can be bet
FIG. 5. d51.5: Plot of the specific heat vsb, for chains ofvarious lengths. ForN5200, 300, and 400 the low temperatupeak was not reached because of the considerable autocorretime caused by low mobility of long collapsed chains. Inset: clapsed specific heatC/N2f21 vs (b2bc)N
f, with f59/1650.5625 andbc50.66.
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BAIESI, CARLON, ORLANDINI, AND STELLA PHYSICAL REVIEW E 63 041801
detected from the behavior ofP(E,N), the probability dis-tribution of the energyE, for a chain of lengthN. Figure 6shows a plot ofP(E,N) as a function ofE/N for N5200. Atsufficiently high temperatures this quantity has a maximin E50, and decreases rapidly withE/N @Fig. 6~a!#. As thetemperature is lowered the maximum shifts continuouslylarger values ofE/N @Figs. 6~b! and 6~c!#. For lower tem-peraturesP develops a double peak structure@Figs. 6~a!–6~f!#. In the case of Fig. 6~e! the peaks have equal heighwhile at temperatures below or above it one of the two pedominates over the other. This behavior, which persistsbecomes more pronounced upon increasingN, is an indica-tion of phase coexistence; hence the transition at lowerT isof first order type. In terms of specific heat this would meC(T);N as T5T2c , i.e. f251. By extrapolatingb2c(N)vs 1/N, we findb2c51.260.2.
Another way to characterize the different phases ofmodel consists in looking at the scaling behavior of mequantities such asRg
2 defined in Eq.~4!, and the meansquared end-to-end distanceRe
2(N)5^(r N2r 0)2&, wherer 0
and r N are the two end monomers of the copolymer. FlargeN we expect
Re2~N!;reN
2n, ~12!
Rg2~N!;rgN2n, ~13!
and an interesting quantity to be computed is the rare /rg , which is expected to be universal@26#. For noninter-acting SAW’s on a square lattice, exact enumerationsMonte Carlo simulations givere /rg;7.13 ~see Ref.@27#,and References in Ref.@26#!. Figure 7 shows the ratioRe
2/Rg2
as a function ofb for severalN. Note that in the range o0,b,0.66 the curves tend to assume a constant vare /rg57.15(5), in agreement with the value expected fnon interacting SAW’s. In the proximity of the zipping transition the curves start to bend downward, and, atbc'0.66,they cross each other almost in a unique point~see the inset!.At the crossing point our estimate of the universal amplitu
FIG. 6. Plot ofP(E,N) with N5200 and for various temperatures:~a! b50.3, ~b! b50.75,~c! b51.4, ~d! b51.58,~e! b51.66,and ~f! b51.8.
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is re /rg56.3560.20, which is definitively different fromthe amplitude ratio of the SAW universality class. The zping transition that cannot be distinguished from the swolphase in terms of then exponent, is, however, characterizeby a different value of the universal ratiore /rg @28#.
If the temperature is further lowered theRe2/Rg
2 curvesreach a minimum value that decreases asN increases. This isan indication that the end-to-end distance in the zipped phno longer scales like the radius of gyration, as assumeEqs.~12! and~13!. For sufficiently low temperatures,re /rgstarts to grow back, indicating that the compact phasecharacterized by an end-to-end distance and mean radiugyration that scale in the same way withN. As the typicallow T configurations are of spiral type@see Figs. 2~e! and2~f!# with end points at opposite sides of the spiral, itnatural to expect thatRe;Rg;N1/2.
B. dÄÀ1.5
For d,0 triple contacts are favored and we expect~as theexact enumerations already indicate! a single transition fromswollen directly to compact phase. To investigate the natof such transition we have performed runs withd521.5, forseveral values ofN, sampling atm.30 different tempera-tures in the [email protected], #. As in the cased51.5, wehave examined the probability of finding the copolymer inconfiguration with energyE, as a function of the temperature. A plot ofP for N5200 and three different temperatureis shown in Fig. 8. Close to the transition temperatureP hastwo maxima@see Fig. 8~b!#, one atE50, and the other atE/N'0.7. This is a clear indication of a first order transitioThe evidence of such behavior is stronger than in the cd51.5, since here the coexistence is between two phases~theswollen and compact phases! with a rather large difference inenergy; therefore, the double peak structure ofP can alreadybe seen for smallN.
FIG. 7. Re2/Rg
2 vs b for d51.5 and for various chain lengths. ThN530 curve is calculated from exact enumerations, while the oers are obtained from Monte Carlo simulations. Inset: blowup ofcrossing region.
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ZIPPING AND COLLAPSE OF DIBLOCK COPOLYMERS PHYSICAL REVIEW E63 041801
From the analysis of the specific heat peaks we findthey become sharper asN increases and their height appeato grow with a power ofN slightly exceeding one~the physi-cal upper limit!. At the same time, in a plot of energy vtemperature we see curves that seem to approach steptions. These data support the idea that the correspontransition should be of first order.
C. dÄ0
The more delicate region to be explored in the phasegram is the neighborhood ofd50, where three transitionlines meet each other. We have chosen in particular thed50, since it was considered in Ref.@15#.
In Fig. 9 we plot the specific heat as a function ofb forseveralN values. For the smallest chains (N560 and 80),one observes a peak in the specific heat with a shoulde
FIG. 8. P(E,N) vs E/N for d521.5 andN5200 for threedifferent temperatures~a! b50, ~b! b50.545, and~c! b50.6. Inset:Plots ofP(E,N) vsE/N near phase coexistence forN560, b50.66~solid line!, N5100, b50.56 ~dotted line!, andN5140, b50.545~dashed line!.
FIG. 9. d50: Plot of the specific heat vsb, for various chainlengths.
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smaller b. When the copolymer length is increased tshoulder becomes hardly noticeable. From the specific hplot one cannot rule out the possibility that the shouldeventually vanishes leaving out a single transition fromswollen phase to a compact phase. The other possibilitthat there are two separate transitions, but very close in tperature.
The presence of two distinct transitions is suggested bplot of the temperature derivative of the total radius of gytion, shown forN5100 and 200 as thick lines in Fig. 10. Ithis case one clearly detects two peaks, which although cing closer to each other asN increases are still noticeable ansharp forN rather large. The thin lines in Fig. 10 are thtemperature derivatives of the radius of gyration of a sinblock, which show only one peak in correspondence tolow temperature peak of the derivative of the total radiusgyration. This behavior is consistent with the following piture: coming from the swollen phase~smallb), one first hasa zipping transition characterized by a drop of the totaldius of gyration, while the radius of gyration of a singblock still behaves as a SAW and is not sensitive tozipping transition. However, at lower temperatures, in corspondence to the transition from zipped to compact phboth quantities drop and their derivatives show a peak.
Another quantity which we investigated is the universamplitude ratio between the end-to-end distance and raof gyration squared, which is plotted in Fig. 11. Here, asthe d51.5 case, this universal quantity takes the SAW va;7.13 at highT and drops in correspondence with the trasition. The fact that we find intersections withRe
2/Rg2
'0.635~the same value as ford51.5! strongly suggests thepresence of a zipping transition with the same univerproperties as that atd51.5. Unlike Fig. 7, hereRe
2/Rg2 drops
and increases again in a narrow range ofb values, indicatingthe the zipped phase is restricted to a small temperatureterval.
FIG. 10. d50. Thin lines: plot of the derivative of the half-chaisquared gyration radius, as a function ofb. Thick lines: two ex-amples of derivative of the total-chain squared gyration radiusN5120 and 200!.
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BAIESI, CARLON, ORLANDINI, AND STELLA PHYSICAL REVIEW E 63 041801
In summary, although the numerical evidence is not fuconclusive, our data seem to favor the existence of two serate transitions ford50. As in the cased51.5, it is natural toexpect that the lowT one~zipped-collapsed! is of first ordertype.
V. PERCOLATION PATHS AND EXACT EXPONENTS OFTHE ZIPPING TRANSITION
In this section we present a conjecture on the relatbetween the statistics of some percolation paths at thresand the diblock copolymer zipping transition. This conjeture leads to a prediction of exact values of the exponentpreliminary, less precise, version of the arguments bewas given in Ref.@15#.
It is well known that, in two dimensions, the statistics oring polymer at theQ transition is identical to that of theexternal perimeter, or hull, of a percolation cluster. Throuthis identification the exact exponents of theQ transition,nQ54/7 andfQ53/7, were derived@8#. Here we show howsimilar arguments can be invoked for the zipping transitiThe differences are mainly associated with the fact thatrelevant percolative set appropriate for the zipping is nothull, as for the homopolymerQ point, but the backbone othe percolation cluster.
As in the Q-point case, here it is convenient to considsite percolation on a triangular lattice. For this problemrelevant percolation contours, like the hull of a cluster, arefact strictly self-avoiding paths on the dual, hexagonal ltice. Thus the equivalent diblock copolymer problem reized here by percolation paths will also be on a hexagorather than square lattice. On the basis of universality,expect our results to extend also to the square lattice ca
Let us consider a percolation cluster as sketched in12. Its external perimeter is a self-avoiding ring. The esemble of all possible conformations of an external hullthe lattice can be regarded as a problem of ring polym~grand canonical! statistics, as discussed in Ref.@8#. Onefurther realizes that this effective ring polymer problem
FIG. 11. d50: Square end-to-end distance over the squareration radius as a function ofb.
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characterized by attractive interactions. These originate frthe fact that, at threshold (p5pc51/2), multiple visitations@29# by the hull of the same occupied, or vacant, hexaggive a higher probability to the realization of a ring configration. Indeed, when the contour proceeds essentiallystraight direction, to each new step then corresponds ahexagon whose state~occupied or vacant! has to be deter-mined. This each step implies a factorpc51/2 in the prob-ability weight of the whole configuration. When the contofolds on itself and revisits, after some steps, the perimetethe same hexagon, the factor 1/2 does not apply, resultina higher global probability. This is equivalent to an attractiinteraction favoring the multiple visitations of the samhexagon.
Here it is convenient to summarize some very recentact results concerning the fractal dimensions of various pcolative sets. Following Ref.@18# we consider an annularegion of the hexagonal lattice delimited by an inner circlesmall radiusr, and an external one of radiusR@r . Two typesof paths connecting the two circles are also considerThese paths are formed by connected and self-avoidingquences of either occupied or empty hexagons. The so-capath-crossingprobability, namely, the probability thatl non-overlapping paths connect inner to outer circles, was fouto behave asymptotically as
Pl~r ,R!'~r /R!xl, ~14!
where
xl5l 221
12. ~15!
The formula is valid if there is at least a path of each typand the probability depends only on the total number of thpaths, not on their type@18#.
Figure 13 shows examples of crossing paths; withoutsolving the underlying lattice structure, we draw themsolid lines if they connect filled hexagons, while dashed linare used for paths connecting empty hexagons. As aexample@see Fig. 13~a!# we consider the crossing probabilitfor a continuous and a dashed path, which according to E
y-
FIG. 12. Percolation cluster of connected occupied hexag~dashed!. Each hexagon is centered on a site of the dual, trianglattice.
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ZIPPING AND COLLAPSE OF DIBLOCK COPOLYMERS PHYSICAL REVIEW E63 041801
~14! and~15! decays asPl 52'(r /R)1/4. One recognizes im-mediately that the set of points for which two of such seavoiding paths can be drawn are those of the external peeter or hull of the percolative cluster@see Fig. 13~a!#. Thisidentification allows one to derive the fractal dimensionthe hull. Since the area enclosed by the annulus is protional toR2, the perimeter of the hull enclosed in the annumust scale asLeh;R2Pl 52;R22x25R7/4. Identifying theexternal hull as a polymer ring at theQ point, one thenderives that the latter has a fractal dimensionDQ5Dl 52[22xl 5257/4.
In order to make contact with the diblock copolymer ziping, let us now imagine to identify two points 1 anddividing the cluster hull into two equally long parts~see Fig.14!. By fixing these two points on the cluster perimeter, oautomatically defines a backbone as a subset of the wcluster. The backbone is the union of all connected pathoccupied hexagons, which are strictly self-avoiding~i.e., ineach path a given hexagon appears at most once!, and joinpoints 1 and 2. Therefore the backbone does not include
FIG. 13. Path crossing configurations for~a! a dashed line and asolid line and~b! two dashed lines and two solid lines. The proabilities of the configurations yield the fractal dimensions of texternal perimeter of the hull~a! and of the cutting hexagons of thbackbone~b!. The dashed region indicates the percolating clusteoccupied hexagons, while the double dashed region of~b! shows adangling end, a part of the cluster which does not belong tobackbone.
FIG. 14. ~a! Schematic representation of a percolation cluswith dangling ends~dashed areas!. When these are eliminated onremains with the cluster backbone~b!. The dashed segments cut thcluster in corresponding to the so-called ‘‘red’’ hexagons.
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so called ‘‘dangling ends,’’ i.e., those branches of the clusconnected to the main body by narrow bridges~i.e., by re-gions in which only one occupied hexagon is available, ming it impossible for a self-avoiding path of hexagonspenetrate and exit at the same time!. An example of danglingend is also schematically shown by the double dashed areFig. 13~b!.
The two points we fix on the contour clearly divide intwo sides the perimeter of the backbone. Even if in this cit is not possible to give a simple expression for the effectinteractions determining the shape of the two backbosides, we expect them to be essentially local, as in the casthe hull, and to act differently according to whether thinvolve close encounters of the same side, or between difent sides. This is consistent with the idea that the two siof the backbone perimeter could represent the statisticsring version of the diblock copolymer at the transition, ttwo parts corresponding, respectively, to blocksA andB.
To calculate the fractal dimension of the external perieter of the backbone one can use Eqs.~14! and ~15!, takingtwo continuous paths and a dashed path. This configuraclearly identifies the perimeter of the backbone. Indeed,two continuous paths guarantee that occupied hexagonside the interior circle belong to a whole path connecting tinfinitely distant points. At the same time, a dashed pimplies that the vacant hexagons facing the occupied obelong to the exterior of the cluster, and thus, are also paits backbone.
Therefore, we now takel 53 for the exponents defined iEq. ~15!. In this case we find that the external perimeterthe backbone scales asLbb;R22x35R4/3, which implies afractal dimensionD354/3 @30#. This dimension is consistenwith that found for the diblock copolymer at the zippintransition. Furthermore, it is natural to associate the switing on of effective attractive interactions between the tbackbone sides to the existence of narrow bottlenecks inbackbone itself~corresponding to only one hexagon!. Theseare the so-called cutting or ‘‘red’’ hexagons of the backbo@17#, which are visited by the two blocks simultaneously.order to determine their fractal dimension one has to csider a percolative configuration with two continuous atwo dashed paths joining the circles, as sketched in F13~b!. These paths identify a dimensionD4522x453/4.Thus, for a backbone with external perimeter equal toN andan average ofNAB contacts we findN;RD3 andNAB;RD4.Consequently, the average number of contacts betweenbackbone sides grows likeNAB;ND4 /D35N9/16. By identi-fying the external perimeter of the backbone with the ridiblock copolymer at its transition, one eventually findsnc53/4 andf59/16. The numerical determinations ofnc andf at the zipping transition are remarkably consistent wthese values, making the conjecture extremely plausible@31#.
VI. CONCLUSIONS
In this paper we studied the phase diagram for the clapse transition of a diblock copolymer with attractive inteactions between monomers of different species and a trcontact interactiond, which, according to its sign, may eithe
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BAIESI, CARLON, ORLANDINI, AND STELLA PHYSICAL REVIEW E 63 041801
favor, or unfavor, compactification. In the region of negatid we find a first order transition from a swollen to a compaspiral phase, while in the positived region there is a sequence of a continuous zipping transition and a collapsefirst order type to compact conformations at a lower tempeture.
Our exact enumerations and Monte Carlo simulatioyield numerical estimates of the critical exponentsnc andfof the zipping transition, which are consistent with thosecould conjecture using recent results for the fractal dimsions of the percolation cluster backbone, from whichexpectnc53/4 andf59/16. The numerically determined exponents, therefore, support the hypothesis that the transadmits a description in terms of percolative stochastic geetry: the two blocks of the copolymer have the same frageometry as the two sides of a cluster backbone, and tcontacts correspond to the cutting hexagons, or links ofsame backbone. This is, to our knowledge, the secondample of a percolative representation for a polymer conmational transition in two dimensions, besides that of theQpoint. The common percolative roots of these transitions sgests the possibility of a deep link between them, whought to be elucidated by further studies.
The results obtained for the various transitions appeain the phase diagram help in clarifying the nature of tadsorptionlike collapse occurring atd50 and first detected inRef. @15#. In spite of the fact that most tests are not ableput into clear evidence the existence of two successive tsitions, the only multicritical behavior which can be charaterized as coming from the high temperature region sedefinitely to belong to the universality class of the continous zipping transition identified for positive values ofd. Be-sides the compatibility of the exponent estimates, a vstrong support for such conclusion comes from our deternation of the universal amplitude ratio between the squaend-to-end distance and the radius of gyration of the pomer.
Other interesting aspects of the phase diagram calculin this paper are the first order swollen-collapsed and zippcollapsed transitions found, respectively, for negative apositive values ofd. In particular the latter resembles thtransition from a swollen to a spiral state found inorientedpolymers@16#, i.e., chains to which an overall orientationassigned, and where different energies are associated
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contacts between parallel or antiparallel segments ofchain. In fact the analogy between the diblock copolymera zipped state and an oriented polymer is very appropriatethe zipped diblock parallel contacts are ofAB type, whileantiparallel ones are contacts between monomers of etype. Different energies are clearly associated with the ttypes of contacts.
It is worthwhile to recall that simple polymer models witsome sort of zipping transition already have attracted soattention in recent literature@32,33#, mainly because of therelevance that such transition can have for biopolymers.bertet al. @32# considered a diblock formed by two strandsoppositely charged monomers interacting with each otthrough long range Coulomb forces and found evidencethe existence of a zipping transition followed by a collapselower temperatures. Causoet al. @33# considered a simplemodel for the DNA denaturation transition, in which only thmonomers which are at equal distances along the sequfrom the center of the chain interact. They found evidencea first order transition, from a swollen phase to a zippphase. By its construction their model has no other trantions to a compact state. In their case the first order zippseems to be due to the selective interactions of monomalong the chain. Also, in our model, if we turn on interations only betweenAB monomers at equal distances frothe center, we find evidence of a first order zipping trantion.
Finally, we point out that there are several possible extsions of this work. First of all, it would be interesting tgeneralize the model to three dimensions and to investigthe properties of the zipping transition in that case@15#. An-other open issue is the effect of disorder on the interacbetween monomers for the zipping transition, which wouallow one to understand the behavior of models of polymmore relevant for applications to chemistry or biology thansimple diblock.
ACKNOWLEDGMENTS
We thank F. Seno for discussions and collaboration inearly stages of this work. Financial support by MURSthrough COFIN 1999 and INFM through PAIS 1999gratefully acknowledged. A.L.S. acknowledges partial suport from European Network Contract No. ERBFMRXCT980183.
Btt.
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ZIPPING AND COLLAPSE OF DIBLOCK COPOLYMERS PHYSICAL REVIEW E63 041801
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N→`FN(b,d) rigorously is an open problem, sinc
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Q/rgQ'5.6.
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