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THE JOURNAL OF 131, 114907 ͑2009͒

Phase transitions of a single chain: A Wang–Landau simulation study ͒ Mark P. Taylor,1,a Wolfgang Paul,2 and Kurt Binder2 1Department of Physics, Hiram College, Hiram, Ohio 44234, USA 2Institut für Physik, Johannes-Gutenberg-Universität, Staudinger Weg 7, D-55099 Mainz, Germany ͑Received 11 June 2009; accepted 21 August 2009; published online 21 September 2009͒

A single flexible homopolymer chain can assume a variety of conformations which can be broadly classified as expanded coil, collapsed globule, and compact . Here we study transitions between these conformational states for an interaction-site polymer chain comprised of N=128 square-well-sphere with hard-sphere diameter ␴ and square-well diameter ␭␴. Wang– Landau sampling with bond-rebridging Monte Carlo moves is used to compute the for this chain and both canonical and microcanonical analyses are used to identify and characterize transitions in this finite size system. The -interaction range ͑i.e., T-␭͒ is constructed for ␭Յ1.30. Chains assume an expanded coil conformation at high and a crystallite structure at low temperatures. For ␭Ͼ1.06 these two states are separated by an intervening collapsed globule phase and thus, with decreasing temperature a chain undergoes a continuous coil-globule ͑collapse͒ transition followed by a discontinuous globule- ͑freezing͒ transition. For well diameters ␭Ͻ1.06 the collapse transition is pre-empted by the transition and thus there is a direct first-order coil-crystal . These results confirm the recent prediction, based on a lattice polymer model, that a collapsed globule state is unstable with respect to a phase for flexible with sufficiently short-range -monomer interactions. © 2009 American Institute of Physics. ͓doi:10.1063/1.3227751͔

I. INTRODUCTION theory,” and thus the polymer collapse transition is thought to be a true second-order phase transition in this limit.2 How- Polymer chain can undergo significant, and in ever, for biological such as which some cases cooperative, conformational changes in response to changes in local environment.1,2 Perhaps the best known have well defined sequences and, compared to synthetic example of a large-scale conformational transition exhibited polymers, typically small sizes, no thermodynamic limit ex- by a single polymer chain is the polymer collapse or coil- ists. For such inherently finite size systems it is still possible globule transition which occurs in dilute polymer-solvent to define and rigorously study phase transformations. In such systems when solvent conditions are changed from “good” to cases it can be advantageous to use the microcanonical en- “poor.”3,4 This conformational transition occurs in a continu- semble where phase changes can be identified from the cur- 8 ous fashion, displaying no coexistence between distinct vature properties of the microcanonical function. “coil” and “globule” states, and thus is analogous to a While the specific details of conformational transitions in second-order phase transition. The helix-coil transition found are strongly dependent on the heteropolymer in some synthetic polymers as well as proteins and nucleic nature of these macromolecules, the more universal aspects acids is another example of a continuous single-chain con- of such transitions are perhaps best explored using the pro- formational transition. In contrast, reversible folding totypical model of a simple flexible homopolymer chain. provides an example of a discontinuous single-chain confor- Previous on phase transitions of single flexible mational transition, analogous to a first-order phase transi- square-well ͑SW͒ and Lennard-Jones ͑LJ͒ homopolymer tion, in which there is coexistence between a folded native chains has identified both a continuous collapse and a dis- state and an ensemble of unfolded denatured states.5 continuous freezing transition.9–13 Thus, a finite length chain In the usual rigorous theory of phase transitions, nonana- displays phase behavior analogous to a bulk simple , lyticities in the thermodynamic functions appear only in the where chain collapse and freezing are akin to the -liquid thermodynamic ͑i.e., infinite size͒ limit6 and thus, when and liquid-solid transitions, respectively. In simple liquid studying such transitions using computer simulations, one systems, phase behavior is known to be sensitive to the range 7 must make use of finite size scaling techniques. In the case of intermolecular interaction. In particular, systems with very of single chain molecules, even for a very high molecular short-range attractive interactions exhibit no stable liquid weight polymer, the system is always far from the thermo- phase, but rather undergo a direct freezing transition from a dynamic limit. Of course one can invoke such a limit “in noncondensed gas/fluid phase.14–16 Thus one might suspect that for a sufficiently short-range interaction, a flexible chain ͒ a Electronic mail: [email protected]. would also undergo a direct transition from the

0021-9606/2009/131͑11͒/114907/9/$25.00131, 114907-1 © 2009 American Institute of Physics

Downloaded 21 Sep 2009 to 143.206.80.1. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 114907-2 Taylor, Paul, and J. Chem. Phys. 131, 114907 ͑2009͒ expanded coil to the crystal state. However, it is not imme- ␧ N−1 diately obvious that this is true since chain connectivity se- g͑E ͒ = ͵ ͵ ͟ s͑r ͒ n V ¯ k,k+1 verely limits the maximum entropy of the expanded phase k=1 ͑compared to the phase of a simple fluid͒ and thus, N ϫ␦ͩ ͑ ͒ͪ ជ ជ ͑ ͒ there may be insufficient driving force to induce a direct En − ͚ u rij dr1, ...,drN, 2 expanded-to-crystal transition.5 iϽj+1 We have recently resolved this latter question, showing where the distribution function s͑r͒=␦͑r−L͒/4␲L2 imposes that a finite length chain with sufficiently short-range inter- the fixed bond length constraint and V is the total volume actions does undergo a direct transition from an expanded 17 accessible to the chain. Although this density of states func- coil to a compact crystallite. This direct freezing transition tion, and a related microcanonical partition function, can be was found for SW chains with lengths N=128 and 256 and computed exactly for short ͑NՅ6͒ chains,21–23 for longer SW interaction range of ␭=1.05. In the present work we chains the required high-dimensional integrals are impracti- focus on the case of N=128 and construct a phase diagram cal to compute numerically and one must resort to simulation for chains with ␭Յ1.30. This phase diagram exhibits a tri- methods. critical point near ␭=1.06, where the continuous coil-globule Here we construct the single chain density of states func- ͑collapse͒ transition and the discontinuous globule-crystallite tion for a flexible SW chain using the Wang–Landau ͑WL͒ ͑freezing͒ transition merge into a single discontinuous coil- algorithm.24,25 In this approach one generates a sequence of crystallite transition. The disappearance of the globule phase chain conformations using a set of Monte Carlo ͑MC͒ from the phase diagram of a single flexible polymer chain 18 moves; however, rather than accepting new conformations was first observed by Rampf et al. for the lattice bond- with the usual temperature dependent Metropolis criterion, fluctuation polymer model in the long chain limit. These au- one uses the following temperature independent multicanoni- thors predicted a similar behavior for finite length chains cal acceptance probability: with interactions of a shorter range than was possible in the 19,20 bond-fluctuation model and our recent results have con- w g͑E ͒ P ͑a → b͒ = minͩ1, b a ͪ, ͑3͒ firmed this prediction. acc ͑ ͒ wag Eb

where wa and wb are conformation dependent weight factors II. CHAIN MODEL AND SIMULATION METHODS which ensure microscopic reversibility for the given MC move.26,27 If g͑E ͒ were known, this approach would yield a In this work we study a flexible interaction-site ho- n uniform exploration of all accessible energy states of the mopolymer chain comprised of N spherically symmetric system ͑and thus, presumably of all configurational phase monomers connected by “universal joints” of fixed bond space͒. In the WL approach g͑E ͒ is constructed in a dy- length L. The monomers are sequentially labeled 1 through N n namic and iterative fashion, where smaller scale refinements and monomer i is located by the vector rជ . Nonbonded mono- i are made at each level of the iteration. Thus, at the mth level mers i and j ͉͑i− j͉Ͼ1͒, separated by the distance r =͉rជ ij i of iteration, after every attempted MC move the current ជ ͉ −rj , interact via the SW potential ͑ ͒ g En value for the resulting energy state is updated with a Ͻ Ͻ ␴ Ͼ ͑ ͒→ ͑ ͒ ϱ 0 rij modification factor fm 1 via g En fm g En and a state ͑ ͒ u͑r ͒ = − ␧ ␴ Ͻ r Ͻ␭␴ ͑1͒ visitation histogram H En is simultaneously updated via ij Ά ij · ͑ ͒→ ͑ ͒ ͑ ͒ Ͼ␭␴ H En H En +1. This H En histogram is periodically 0 rij , ͓ ͑ ͒ checked for “flatness” i.e., each entry in H En is within p ͑ ͔͒ where ␴ is the hard-sphere diameter and ␭␴ is the SW diam- percent of the overall average value of H En indicating an eter. The well depth ␧ sets the energy scale and can be used approximate equal visitation of all states. When flatness is ␧ ͱ/ ء to define a dimensionless temperature T =kBT , where kB is achieved the modification factor is reduced via fm+1= fm, the Boltzmann constant. Here we will consider the phase the visitation histogram is reset to zero for all states, and the behavior of a SW chain of length N=128 and bond length ͑m+1͒st level of iteration is begun. One difficulty with the L=␴ for a range of ␭ values. Since we deal with an isolated application of the above algorithm to chain molecules is the chain the above potential is to be treated as an effective possible existence of “bottlenecks” in configuration space potential which implicitly includes the effect of solvent, al- which can lead to the generation of highly asymmetric his- though here we take this potential to be temperature indepen- tograms which would take a prohibitively large amount of dent. This chain model has a discrete potential energy spec- time to relax. To overcome this problem we also monitor the ␧ ͑ ͒ ͑ trum given by En =−n , where n is the number of SW H En histogram for uniform growth i.e., each entry has overlaps in the chain configuration, and the ground state en- increased by an amount within p percent of the overall aver- ergy of the chain will depend on the interaction range ␭. age increase since the last check͒ and use this as an alternate The thermodynamic properties of the above chain model flatness criterion. ͑ ͒ can be completely determined from the single chain density In this work we start with an initial guess of g En =1 for ͑ ͒ 1 of states function g En . This function gives the volume of all states, take f0 =e , p=20, and check for flatness and uni- configurational phase space associated with each energy state form growth every 104 and 5ϫ107 MC cycles, respectively. ͑ En and can be written relative to the value for an ideal freely Unlike most previous applications of the WL algorithm ͒ 21 ͑ Ϸ −6͒ jointed chain as which find that m=20 levels of iteration f20−1 10 are

Downloaded 21 Sep 2009 to 143.206.80.1. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 114907-3 Phase transitions of a single polymer chain J. Chem. Phys. 131, 114907 ͑2009͒

sufficient for convergence ͑with Ref. 27 being an exception͒, TABLE I. Estimated ground state energies Egs and actual minimum energies ␭␴ for the present study of a continuum chain molecule we find Emin used in WL sampling for N=128 SW chains with well diameter . that mՆ26 ͑f −1Ϸ10−8͒ is required to achieve conver- 26 ␭ −E /␧ ͑estimated͒ −E /␧ ͑WL sampling͒ gence and in most cases we continue the simulations to m gs min ͑ Ϸ −9͒ ͑ ͒ =30 f30−1 10 . Note that in the above algorithm g En 1.02 447 439 grows continuously throughout the calculation and that we 1.05 451 440 ͑ ͒ 1.10 451 447 only obtain g En up to an arbitrary multiplicative constant ͓which drops out in the Eq. ͑3͒ acceptance criterion͔.To 1.15 464 450 1.20 495 475 avoid numerical difficulties we actually compute the loga- ͑ ͒ 1.25 517 500 rithm of g En and at every flatness check we uniformly re- 1.30 538 524 ͓ ͑ ͔͒ duce all entries in the ln g En estimate by the current maxi- mum entry. In our WL simulations a single MC cycle consists of, on low energy cutoffs in g͑E͒ as the resulting low temperature average, the following set of attempted moves ͑with moves can be strongly affected.18,29 We discuss 28 being chosen at random͒: N−2 single-bead crankshaft this issue in regard to our own results later in the . moves in which a randomly chosen interior site i ͑1Ͻi Ͻ ͒ ͑ ͒ ͑ N is rotated through a small angle about the i−1 - i III. RESULTS +1͒ axis; two random end bead rotations; two moves in which one of the chain ends is removed and ran- A. Density of states domly reattached at the other end of the chain; one pivot In Table I we give our ground state energy estimates for move in which a portion of the chain ͑i.e., sites ͓1,i−1͔ or N=128 SW chains for several well diameters ␭ in the range ͓i+1,N͔͒ undergoes a three-dimensional Euler rotation about 1.02Յ␭Յ1.30. As expected, this ground state energy de- a randomly chosen pivot site i ͑1ϽiϽN͒; and N “end- creases with increasing interaction range ␭.21 We also pro- bridging” moves where we select at random end site 1 ͑or vide in Table I the lowest energy value used in our full WL N͒, identify all interior sites iϾ3 ͑or iϽN−2͒ within dis- simulations for this ␭. As noted above, these low energy tance 2L of this end site, and selecting one of these at ran- cutoffs are determined from the H͑E͒ visitation histogram dom, join sites i and 1 ͑or N͒ via removal and reinsertion ͑at generated in our preliminary WL run. These histograms typi- a randomly chosen azimuthal angle͒ of site i−1 ͑or i+1͒. cally show a sharp dropoff beyond some low energy and we The weight factor for this later bond-bridging move is wa set our energy cutoff just above this dropoff point. =baJa, where ba is the number of possible bridging sites i In Fig. 1 we show our density of states results, presented / ͑ / ͒ present in state a and Ja =1 r1i or 1 riN is a Jacobian factor in logarithmic form relative to the density of states for en- arising from the fixed bond length restriction.26 Weight fac- ergy E=0,asln͓g͑E͒/g͑0͔͒ for the ␭ values given in Table I. tors for the other MC moves are all unity. We have verified These results have been obtained using two energy windows ෈͓ ␧͔ ͓ ␧ ͔ the WL algorithm with this MC move set for the SW chain with the ranges E 0,Ejoin−50 and Ejoin+50 ,Emin , ͑ ͒ model by direct comparison with exact results for g En for where Ejoin is the energy at which the results of the two short SW chains ͑NՅ6͒ for a range of ␭ ͑1.01Յ␭Յ1.9͒ windows are joined which, in most cases, is set to either ͑Ref. 21͒ and by comparison with the Metropolis MC results −250␧ or −350␧. In all cases, comparison of the derivative of Zhou et al.10 for SW chains with NՅ64 and ␭=1.5. For d ln g͑E͒/dE from the two windows shows a very good longer chains we find the bond-bridging move to be essential agreement in the neighborhood of the join point.30 The re- in order to sample compact low energy states. Also for sults shown in Fig. 1 are the average from two to four inde- longer chains we find it most efficient to carry out a set of pendent simulations for each energy window. Note that in the WL simulations for a set of overlapping energy windows.18 MC moves taking the system out of the energy window are rejected with no updating of g͑E͒ or H͑E͒͑note, however, 0 that moves resulting in a hardcore overlap are always re-  = 1.30 jected with updating͒. Results from these energy windows -300

are joined by matching g͑E͒ values at the center of the over- 0)] 1.25 ( lap region ͑which typically has a width of 100␧͒. The quality g )/

E -600

of such joins is assessed by the smoothness of the derivative ( of g͑E͒. Before we begin a full WL run we carry out a g 1.20 n[ l preliminary run at low energies, without a low energy cutoff, 1.15 1.10 -900 to estimate the ground state energy of the chain. The H͑E͒ 1.05 ͑ SW Chain generated in this initial WL sampling which is restricted to 1.02 N = 128 level m=0 and is terminated by hand͒ also allows us to es- -1200 timate the lowest energy that can be included in our full WL -500 -400 -300 -200 -100 0 run and still allows the WL method to converge in a “rea-  /  sonable” amount of time. For longer chains our lowest en- FIG. 1. of the density of states ln͓g͑E͔͒ ͓i.e., the microcanonical ergy window typically extends to within a few percent of the ͑ ͒/ ͔ entropy S E kB , relative to the value at E=0, vs energy E for a SW chain ground state energy. Care must be taken in imposing such of length N=128 and SW diameter ␭␴ as indicated.

Downloaded 21 Sep 2009 to 143.206.80.1. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 114907-4 Taylor, Paul, and Binder J. Chem. Phys. 131, 114907 ͑2009͒ case of ␭=1.02 the density of states function covers approxi- 0.02 1.04 1.05 )  = 1.05

(a) T mately 500 orders of magnitude. Although the results shown , E

1000 ( in Fig. 1 appear to be relatively featureless, these density of 1.06 P T* = 0.446 1.02 0.01 states functions in fact encode the complete thermodynamics 1.08 8 of the chain molecules. In Secs. III B and III C we employ a B k 100

͑ ͒ N canonical and a microcanonical analysis of our g En results 0.00 / -400 -200 0 E/ to determine the phase behavior of the N=128 SW chain as a ) T  = 1.10 ) ء ␭ function of interaction range and temperature T . Our den- C SW Chain 10 sity of states results for these N=128 chains are highly re- N = 128 producible although, as in Ref. 17, for smaller ␭ we do find some hysteresislike effects in a small energy range where 1 crystal occurs. In all cases presented here, transi- 0.3 0.4 0.5 0.6 0.7 0.8 tion temperatures obtained from independent runs agree to 100 ء ␦ Ϯ  = 1.15 within T = 0.006. (b) 0.02 T* = 0.714 ) T ,

 E 0.526

= 1.15 ( P B. Canonical analysis 1.20 0.01

B 10 From the single chain density of states g͑E ͒ one can 1.25 n 0.00

/Nk -400 -200 0 construct the partition function E/ )

T SW Chain / ( ͑ ͒ ͑ ͒ −En kBT ͑ ͒ 1.30 Z T = ͚ g En e 4 C N = 128 n 1 and the canonical probability function

1 / P͑E ,T͒ = g͑E ͒e−En kBT. ͑5͒ 0.0 0.4 0.8 1.2 1.6 2.0 n Z͑T͒ n  ء /͒ ͑ From these we obtain the FIG. 2. Specific per monomer C T NkB vs temperature T foraSW chain of length N=128 and SW diameter ␭␴ as indicated for ͑a͒ ␭Յ1.10 ͑ ͒ ͑ ͒͑͒ ͑ ͒ ␭Ն ͑ ͒ F T =−kBT ln Z T 6 and b 1.15. Insets: canonical probability function P E,T vs energy E which ,ءfor ͑a͒ ␭=1.05 and ͑b͒ ␭=1.15 at the indicated temperatures T and the internal energy correspond to the peaks in the associated C͑T͒ curve. Solid curves show the equally weighted bimodal distributions characterizing a first-order transition ͑ ͒ ͗ ͘ ͑ ͒ ͑ ͒ ␭ U T = En = ͚ EnP En,T . 7 and the dashed curve for =1.15 shows the distribution at the continuous n coil-globule transition. ͑ ͒ Since the WL simulation method only gives g En up to an arbitrary multiplicative constant, we only know F͑T͒ up to transition and the low temperature peak with the chain freez- an arbitrary additive linear function of T. However, the in- ing transition.9–12 ͑The absence of a freezing peak in the case ternal energy and other averages over functions of En are of ␭=1.30 will be further discussed below͒. The inset in Fig. determined absolutely. To locate phase transitions within the 2͑b͒ shows that for ␭=1.15 the collapse and freezing transi- canonical ensemble one typically examines the specific heat tions are characterized, respectively, by unimodal and bimo- function dal P͑E,T͒ probability distributions. Thus we have a con- dU 1 tinuous chain collapse transition followed by a discontinuous C͑T͒ = = ͑͗E2͘ − ͗E ͘2͒. ͑8͒ chain freezing transition. With increasing ␭ the freezing tran- dT k T2 n n B sition weakens, with the two peaks of the bimodal P͑E,T͒ For a finite size system, maxima in C͑T͒ can be associated distribution merging, leading to a unimodal probability dis- with phase transitions or other structural tribution at “freezing” for ␭=1.25 and no clear freezing peak rearrangements.9,10,27 A first-order transition can be identified in C͑T͒ for ␭=1.30. In the following sections we will make ͑ ͒ by a bimodal P En ,T probability distribution with equal use of a microcanonical analysis to further characterize the weight in the peaks which correspond to the two coexisting low temperature behavior of these chains with larger ␭. states. For SW diameters less than ␭=1.10, shown in Fig. 2͑a͒, In Fig. 2 we show the canonical specific heat C͑T͒ ob- the C͑T͒ curves undergo an interesting change in structure. tained from our density of states results for a range of well The collapse-peak plateau becomes a shoulder for 1.06Յ␭ diameters ␭. For larger values of ␭ ͓␭Ն1.15, shown in Fig. Յ1.08 and then disappears for ␭Յ1.05 leaving a very nar- 2͑b͔͒ these curves all display a high temperature peak and row spike at the freezing transition. The P͑E,T͒ probability broad plateau followed ͑with the exception of ␭=1.30͒ by a distribution for ␭=1.05 at the freezing temperature, shown in stronger low temperature peak. As ␭ is decreased, these two the inset of Fig. 2͑a͒, is strongly bimodal with two very well peaks increase in height and the plateau region narrows, with separated peaks. Thus for a sufficiently short-range the high temperature peak moving to lower T and the low monomer-monomer interaction the N=128 SW chain under- temperature peak moving to higher T. We can identify the goes a very strong first-order freezing transition directly from high temperature peak with the chain collapse or coil-globule a high-energy ͑and, as shown below, expanded͒ state.17

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3.0 These results suggest the disappearance of a distinct chain 20 collapse transition for chains with short-range interactions as

19,20 B predicted by Paul and co-workers and discussed in Sec. I. dE 2.5 10 Nk )/ )/ E E (

In the next section we focus on this issue using an alternate, ( c 0 microcanonical thermodynamic analysis. dS 2.0 ) B

k -10

/ -400 -200 0   / 

C. Microcanonical analysis =( 1.5 1 –

)] SW Chain A weakness of the above canonical analysis is the inabil- E ( 1.0 N = 128

ity to definitely locate the disappearance of the collapse tran- T* [  = 1.15 sition since the specific heat collapse peak smoothly merges into the freezing peak. However, as we have recently 0.5 -500 -400 -300 -200 -100 0 shown,17 it is possible to clearly distinguish the collapse and  /  freezing transitions via analysis within the . Such an analysis is based on the curvature of the FIG. 3. Derivative of the microcanonical entropy for N=128, ␭=1.15. The microcanonical entropy function dashed tie line, obtained by an equal area construction, locates the transition temperature and coexisting states for the first-order freezing transition. The ͑ ͒ ͑ ͒͑͒ filled circle at E=−158␧ locates the continuous coil-globule transition. Inset: S E = kB ln g E 9 microcanonical specific heat c͑E͒ vs energy E. The isolated peak, marked by ͑shown in Fig. 1͒ and has proven very useful in studying the open circle, locates the coil-globule transition energy. phase transitions in finite size systems.8,31–33 Discontinuous or first-order phase transitions are particu- lapse peaks in the canonical specific heat function C͑T͒, 0.526͑1͒= ءlarly distinct in the microcanonical analysis, being signaled shown in Fig. 2͑b͒ for ␭=1.15, are located at T ͑ ͒ by a so-called “convex intruder” in S E . This feature results and 0.714͑2͒, respectively. in a characteristic Maxwell-type loop in the microcanonical The usefulness of the microcanonical analysis is made temperature given by evident in Fig. 4 where we show the inverse caloric curves T͑E͒ = ͓dS/dE͔−1. ͑10͒ for three cases in which the canonical analysis is unable to definitively locate a coil-globule transition due to the merg- The coexisting states and transition temperature for this dis- ing of the collapse and freezing specific heat peaks ͓see Fig. continuous transition can be obtained from either a common 2͑a͔͒. Each curve in Fig. 4 shows a very pronounced ͑ ͒ tangent construction on the S E curve or via an equal area Maxwell-type loop indicating a strong first-order transition. −1͑ ͒ ͑ construction on the T E curve also known as the inverse Equal area constructions on these loops, marked by the ͒ caloric curve . A continuous phase transition is signaled by dashed tie lines, give the temperatures and coexisting energy ͑ the existence of an isolated inflection point which becomes states at the transition. In addition to a first-order freezing ͒ ͑ ͒ a saddle point in the thermodynamic limit in T E . In a finite transition, for each case shown in Fig. 4 there is also a con- system such a point will produce a finite peak in the micro- tinuous transition signaled by an inflection point in T͑E͓͒or canonical specific heat given by equivalently by an isolated peak in c͑E͔͒ and indicated by −1 d2S −1 the filled circles. In the case of ␭=1.05 this continuous tran- c͑E͒ = ͓dT/dE͔−1 = ͩ ͪ . ͑11͒ T2͑E͒ dE2 3.0 For a first-order phase transition, the loop in the T͑E͒ caloric  = 1.05 curve produces a very distinct signature in this specific heat dE shifted up function, consisting of a negative region bounded by two )/ by 0.25 E poles ͑where dT͑E͒/dE=0͒, demarking an energy region ( 2.5

dS 1.06 which is thermally unstable. ) B k / In Fig. 3 we show an example of this microcanonical 

analysis for the case of ␭=1.15. The inverse caloric curve =( 2.0

1 1.08 −1͑ ͒ –

T E , shown in the main figure, is used to locate discon- )] E tinuous transitions and the microcanonical specific heat c͑E͒, ( shifted down T*

[ SW Chain shown in the inset, is used to locate continuous transitions. 1.5 by 0.25 The clear Maxwell-type loop in T−1͑E͓͒and the associated N = 128 poles in c͑E͔͒ indicates the discontinuous freezing transition -500 -400 -300 -200 -100 0 and an equal area construction across this loop finds the co-  /  /␧ ͑ ͒ ͑ ͒ existing energy states to be −E =299 1 and 379 1 at a ␭ ,FIG. 4. Derivative of the microcanonical entropy for N=128, =1.05, 1.06 ͒ ͑ ء transition temperature of T =0.526 1 . The weak isolated and 1.08, as indicated, vs energy E.AsinFig.3, the dashed tie lines locate peak in c͑E͒ indicates the continuous coil-globule transition, the transition temperatures and coexisting states for the first-order freezing which is located at energy −E/␧=158͑4͒, which corresponds transition and the filled circles locate the continuous coil-globule transition ͑ ͒ ␭ These microcanonical transition with uncertainty being indicated by the symbol sizes . For =1.05, the .0.718͑4͒= ءto temperature T latter transition is seen to fall within the freezing transition coexistence temperatures are in agreement with the results obtained from region. Note that the curves for ␭=1.05 and 1.08 have been shifted for the previous canonical analysis, where the freezing and col- clarity.

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100 1.2 100  = 1.05 SW Chain

2 N = 128 1.15 1.0  /

  ) = 1.25

2 1.25

T expanded (  2 g

/ 0.8 coil R 1.15  E

 10

2 T* collapsed g R 0.6 globule  0.0 0.5 1.0 1.5 2.0 1.05 10 T* 0.4 frozen SW Chain crystallite N = 128 0.2 -500 -400 -300 -200 -100 0 1.00 1.10 1.20 1.30 E /  Interaction Range  ͗R2͘ E FIG. 5. Microcanonical mean square radius of gyration g E vs energy ͑T ␭͒ N ␭ FIG. 6. Temperature-interaction range - phase diagram for the =128 for N=128 with =1.05, 1.15, and 1.25 as indicated. Inset: canonical mean SW chain. The filled circles locate the continuous coil-globule ͑collapse͒ ءR2͑T͒͘ T͗ square radius of gyration g vs reduced temperature . The filled transition and the filled squares locate the discontinuous globule-crystal or circles and open diamonds locate the collapse and freezing transitions, coil-crystal freezing transition. The connecting dashed and solid lines are respectively. simply interpolations between these points and uncertainties are smaller than the symbol size. The simulation snapshots show representative chain con- ␭ sition is seen to be located within the coexistence region of formations coexisting at the freezing transition for interaction ranges =1.05 ͑−E/␧=404 and 89͒ and ␭=1.15 ͑−E/␧=379 and 299͒, as indicated. the freezing transition. Thus, for this short-range interaction For short-range interactions ͑␭Յ1.05͒ the chain freezes directly from an the continuous collapse transition is pre-empted by the freez- expanded coil state while for longer range interactions ͑␭Ͼ1.06͒ the chain ing transition and there is a direct coil-crystallite transition. freezes from the collapsed globule state. For the case of ␭=1.06 the collapse transition essentially coincides with the onset of the coexistence region while for well diameters the microcanonical radius of gyration de- larger ␭ the collapse transition energy clearly moves outside ␭Ͼ creases monotonically with decreasing energy and there are the coexistence energy range. Thus for 1.06 there are no clear signals in chain size itself indicating a phase transi- distinct collapse and freezing transitions. tion. For the shorter range interaction ͑␭=1.05͒, the chain exhibits a surprising abrupt increase in average size as the /␧Ϸ ͗ 2͘ D. Chain dimensions and structure energy decreases near −E 200. This step in Rg E occurs within the two-phase region and coincides with the location In our above analysis of phase transitions of a single of the free energy ͓F͑E,T͒=−ln P͑E,T͒=E−TS͑E͔͒ barrier polymer chain we have assumed that the high temperature separating the coexisting coil and crystallite states ͑which ͑high energy͒ continuous transition corresponds to chain col- also approximately coincides with the upper energy bound of lapse and the low temperature ͑low energy͒ discontinuous 17 the thermally unstable region͒. An analysis of conforma- transition corresponds to chain freezing. To verify the nature tional snapshots at −E/␧Ϸ210 shows that this local maxi- of these transitions we have carried out additional structural mum in chain size is due to the formation of a transition state analysis, computing average chain dimensions and examin- structure consisting of a crystalline nucleus attached to one ing configurational snapshots. or more extended chain segments. To both compute chain dimensions and generate typical In the inset of Fig. 5 we show the canonical mean-square equilibrium configurations we carry out a multicanonical- radius of gyration as a function of temperature. These curves type simulation,34 using the acceptance probability given by all display the usual sigmoidal shape expected for the chain Eq. ͑3͒, with fixed g͑E͒ given by the final results of our WL 2 collapse transition and the filled circles indicate the collapse simulations. In the course of this simulation we determine transition temperatures for ␭=1.15 and 1.25. For ␭=1.15 the the microcanonical mean-square radius of gyration lower temperature freezing transition is evident as a small N ͗ 2͑ ͒͘ 1 step in Rg T at the freezing temperature. A much larger ͗ 2͘ ͗ 2 ͘ ͑ ͒ Rg E = 2 ͚ rij E, 12 step is seen for the case of ␭=1.05 where there is a direct N Ͻ i j freezing transition from the coil state ͑i.e., there is no sepa- where the angular brackets indicate an average over chain rate collapse transition͒. The open diamonds in the Fig. 5 conformations with energy E. The mean-square radius of gy- inset locate the freezing transition temperature, which is seen ͗ 2͑ ͒͘ ration at canonical temperature T is then simply given by to fall approximately in the middle of steps in Rg T . For ␭=1.25 the energy discontinuity at freezing is very small ͑or, ͗ 2͑ ͒͘ ͗ 2͘ ͑ ͒ ͑ ͒ Rg T = ͚ Rg EP E,T , 13 as discussed below, zero͒ and there is no apparent step in E ͗ 2͑ ͒͘ Rg T . where the canonical probability distribution P͑E,T͒ is given Representative chain conformations at the freezing tran- by Eq. ͑5͒. Both canonical and microcanonical radius of gy- sition are shown within Fig. 6 for the cases of ␭=1.05 and ration results are shown in Fig. 5 for N=128 SW chains with 1.15. These simulation snapshots are of conformations with SW diameters ␭=1.05, 1.15, and 1.25. For the two larger sizes approximately equal to the average chain sizes for the

Downloaded 21 Sep 2009 to 143.206.80.1. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 114907-7 Phase transitions of a single polymer chain J. Chem. Phys. 131, 114907 ͑2009͒

TABLE II. Freezing and coil-globule transition temperatures for N=128 co-workers,19,20 the phase diagram for a single flexible poly- ␭␴ SW chains with well diameter . The numbers in parentheses are the mer chain contains a tricritical point where the continuous estimated uncertainty in the last digit shown. Freezing temperatures in square brackets correspond to cases where there is no clear Maxwell loop in coil-globule transition and the discontinuous globule-crystal T͑E͒ associated with the freezing peak in C͑T͒. transition merge into a single discontinuous coil-crystal tran- sition. For the N=128 SW chain this tricritical point is lo- ء ء ء ␭ Tf Tc−g cated at approximately ␭Ϸ1.06, T Ϸ0.465. ͑ ͒ In the Fig. 6 phase diagram we have terminated the 1.01 0.317 3 ¯ ͑ ͒ freezing line at ␭=1.22 because the exact nature and location 1.02 0.363 2 ¯ ͑ ͒ of the freezing transition for larger interaction range are not 1.03 0.397 1 ¯ ͑ ͒ clear. In fact, the Maxwell-type loop signaling the discon- 1.04 0.424 1 ¯ 1.05 0.446͑1͒ 0.426͑2͒ tinuous freezing transition is not present in our results for 1.06 0.465͑1͒ 0.463͑4͒ 1.22Յ␭Յ1.45 ͑although in all of these cases we have a well 1.07 0.481͑1͒ 0.495͑3͒ defined saddle point or nearly flat plateau in lieu of a loop͒. 1.08 0.495͑1͒ 0.516͑2͒ This difficulty in clearly identifying the freezing transition 1.10 0.515͑1͒ 0.576͑1͒ from our density of states results may be due, in part, to our ͑ ͒ ͑ ͒ 1.12 0.527 1 0.633 1 cutoff of the lowest energy states in our WL sampling. It is ͑ ͒ ͑ ͒ 1.14 0.530 1 0.687 1 known that low energy cutoffs can lead to incorrect low tem- 1.15 0.526͑1͒ 0.714͑2͒ 29 ͑ ͒ ͑ ͒ perature thermodynamic results and the WL results of Par- 1.16 0.521 1 0.743 2 35 1.18 0.493͑4͒ 0.800͑2͒ sons and Williams appear to miss the freezing transition of 1.20 0.447͑4͒ 0.861͑2͒ a LJ chain due to such an inappropriate energy cutoff. Alter- 1.22 ͓0.300͑4͔͒ 0.923͑2͒ natively, there may in fact be no true freezing ͑i.e., discon- 1.25 ͓0.342͑6͔͒ 1.010͑5͒ tinuous ordering͒ transition for N=128 SW chains with 1.30 ? 1.165͑5͒ 1.22Յ␭Յ1.45, although our C͑T͒ results do suggest some type of weak transition in the vicinity where we expect freez- ing. In the case of a simple monomeric SW liquid there is coexisting energy states at freezing ͑as indicated in Fig. 5͒. evidence that there is no freezing transition in the case of These energies are given by the tie-line analysis shown in ␭=1.40 and that the freezing temperature is strongly sup- Figs. 3 and 4 and also correspond to the most probable en- pressed for slightly larger well diameters.36 Thus, we may be ergies at freezing, given by the peaks in P͑E,T͒ shown in the seeing a reflection of this bulk fluid behavior in our finite Fig. 2 insets. For ␭=1.05 a well-ordered crystallite coexists length chains, but occurring for somewhat smaller well di- with a disordered expanded coil, while for ␭=1.15 a some- ameter. We do find a discontinuous freezing transition for what less ordered crystallite coexists with a disordered com- ␭=1.50 in agreement with the earlier findings of Zhou et pact globule. In general, a single snapshot is insufficient to al.10 and we note that these authors were similarly unable to summarize the large number of conformations sampled by a find evidence of a discontinuous freezing transition for their flexible chain at any energy. However, both visual examina- SW chain model with N=64 and ␭=1.30. tion and analysis of pair probability functions for a large number of independent chain conformations at each of the coexistence energies confirm that these snapshots are quite IV. DISCUSSION representative of the ensemble of structures at that energy. In this work we have examined conformational transi- We also note that with increasing well diameter the degree of tions of a single polymer chain and constructed the order in the coexisting crystallite state clearly decreases. temperature-interaction range phase diagram for a SW chain with length N=128. This phase diagram is comprised of re- E. Single chain phase diagram gions of expanded coil, collapsed globule, and compact crys- Combining our canonical and microcanonical calorimet- tallite structures. For interaction ranges ␭Ͼ1.06 there is a ric results, which precisely locate phase transitions, with our continuous coil-globule ͑collapse͒ transition at a temperature direct structural analysis of these phases, we are able to build somewhat lower than the SW monomer Boyle temperature a phase diagram for the single flexible polymer chain. In Fig. ͑defined below͒ and a discontinuous globule-crystallite 6 we show the temperature-interaction range ͑T-␭͒ phase ͑freezing͒ transition at lower temperature. For ␭Յ1.05 a mi- diagram for the N=128 SW chain for 1.01Յ␭Յ1.30 con- crocanonical analysis unambiguously shows that the collapse structed from the transition temperatures given in Table II. transition is pre-empted by the freezing transition and there For ␭Ͼ1.06 the high temperature expanded coil phase is is a direct transition from an expanded coil to a compact separated from the low temperature crystallite phase by an crystallite. Thus the SW chain phase diagram contains a tri- intervening collapsed globule phase. Thus, upon cooling, an critical point where the continuous collapse and discontinu- expanded chain undergoes a continuous coil-globule ͑col- ous freezing transitions merge into a single discontinuous lapse͒ transition followed by a discontinuous globule- coil-crystallite transition. These findings confirm the recent crystallite ͑freezing͒ transition. For ␭Ͻ1.06 the collapsed predictions of Paul et al.19 regarding the disappearance of the globule state is no longer thermodynamically stable and globule phase for flexible chain molecules with sufficiently there is a direct freezing transition from the expanded coil to short-range interactions and our Fig. 5 phase diagram has the the crystallite phase. Thus, as predicted by Paul and topology proposed by Binder and Paul.20 The phase behavior

Downloaded 21 Sep 2009 to 143.206.80.1. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 114907-8 Taylor, Paul, and Binder J. Chem. Phys. 131, 114907 ͑2009͒ of the finite SW chain is thus similar to that of a monomeric ordered and the other disordered, but both at the same en- SW fluid for which the gas-liquid transition is ergy. In order to further elucidate this configurational barrier pre-empted by a gas-solid freezing transition when ␭Յ1.25 we plan to carry out additional multicanonical simulations resulting in the loss of a stable liquid phase.16 This absence using our WL g͑E͒ functions to construct 2D free energy of a stable liquid phase is a general feature of simple landscapes that depend on both the energy E and either av- ͑or colloids͒ with very short-range interactions.14,15 ͗ 2͘ erage chain size Rg E or a local order parameter Q. We may The results presented here have been restricted to a poly- also attempt to carry out 2D WL simulations to construct mer chain of length N=128. Although such modest length joint density of states functions38,39 such as g͑E,␭͒, chains are relevant to the study of small proteins, for many ͑ ͗ 2͘ ͒ ͑ ͒ g E, Rg E ,org E,Q . Adding a dimension to the WL sam- other applications it will be of interest to extend our results pling is known to allow for easier movement through con- to longer chains. For example, as has been pointed out pre- figuration space, providing alternate pathways around viously, the phase behavior of a single long polymer chain 19,20 barriers. can be mapped onto that of a polymer and thus Although the full nature of globule-crystallite freezing our single chain results may provide insight into the behavior transition requires further characterization, the direct coil- of a many chain system. Thus, in addition to our rather ex- crystallite freezing transition of the SW chain found for ␭ tensive set of simulations for SW chains with N=128, we Յ1.05 is clearly discontinuous and provides an interesting have also carried out a smaller set of simulations for SW example of phase coexistence in a finite size system. The chains of lengths N=32, 64, and 256. These data have al- transition is characterized not only by coexistence between lowed us to make a preliminary finite size scaling analysis of the expanded coil and compact crystallite structures, but at the SW chain phase behavior. In general, it is observed that the transition state itself there is an apparent coexistence with increasing length the SW chain collapse and freezing transitions systematically move to higher temperatures, simi- within the single chain between a crystal nucleus domain and lar to the scaling behavior previously reported for the lattice one or more extended disordered chain sections. bond-fluctuation chain model.18,19 In the N→ϱ long chain The of the frozen SW chain has not yet been completely determined, although we have established ءlimit the SW chain tricritical point moves to ␭Ϸ1.15, T ͑␭Յ ͒ Ϸ0.85 and ͑for all ␭Ͼ1.15͒ the coil-globule transition tem- that for the short-range interactions 1.05 the chains ͑ ͒ perature moves very close to the SW monomer Boyle tem- freeze into a crystallite with defects and grain boundaries -with hexagonal-close-packed ͑hcp͒ structure and this struc ء perature TB, given by ture persists ͑becoming more uniform͒ down to the ground ͒ ͑ ␭3͒/ ͑ ء / 1 TB =−ln 1−1 . 14 state energy. In contrast, in the case of a bulk SW monomer system with short-range interactions, the ground state is A complete scaling analysis of the flexible SW chain will be thought to have a face-centered-cubic ͑fcc͒ structure36 and the subject of a future publication. for ␭Յ1.05 the system undergoes a solid-solid transition at One interesting observation made in the present study is 40 that the progress of the WL simulation itself can reveal in- high . For somewhat longer range interactions the formation about transition pathways in the system. In par- solid-state phase behavior of the bulk SW system is even more complex, being ␭ dependent and including numerous ticular, in all cases in which we find a discontinuous freezing 36,41,42 transition we have observed a configurational barrier within solid-solid transitions. While one cannot make direct the WL simulation. That is, as the simulation proceeds there comparison between the phase behavior of the bulk SW ͑ ͒ is a thorough exploration of the set of energy states above monomer system and an isolated i.e., in vacuum finite and below a narrow band of “barrier” states, but crossing of length SW polymer chain, one might expect that the phase this barrier is a rare event. This type of dynamics is also behavior of a long single chain will be similar to that of the observed in our multicanonical simulations, which are run bulk system at low pressures. The solid-state phase behavior using our converged g͑E͒ functions, and has been previously of the bulk SW monomer system has been particularly well 37 ␭ ͑ ͒ discussed by Neuhaus and Hager in the context of conden- studied for the two well diameters =1.50 Ref. 41 and 42 ␭ sation and freezing in the two-dimensional ͑2D͒ . 1.43. For the =1.50 SW system at low pressures the ini- As noted above, the observed configurational barrier appears tial freezing transition is into a hcp structure and upon fur- to be associated with a nucleation event, characteristic of a ther cooling the system undergoes a hcp-fcc solid-solid tran- first-order phase transition, leading to the formation of a par- sition. For the ␭=1.43 SW system at low the initial tially crystalline transition state structure. Interestingly, this freezing transition is into a body-centered-cubic ͑bcc͒ phase type of configurational barrier is also observed in those cases followed by a lower temperature bcc-hcp solid-solid phase where we expect, but do not find a discontinuous freezing transition. These results suggest that a solid-solid transition transition ͓i.e., no Maxwell loop in T͑E͔͒. Such behavior is may also occur in the SW chain. There is evidence for this in found for 1.25Յ␭Յ1.45 where we see a very pronounced the case of ␭=1.20 where the presence of a second low tem- inflection point in T͑E͒, indicating a continuous transition, in perature peak in C͑T͓͒Fig. 2͑b͔͒ and the pronounced low the energy range where we would have expected a Maxwell energy kink in S͑E͓͒Fig. 1͔ may indeed indicate such a loop associated with a discontinuous globule-crystallite transition. We find a similar second low temperature peak in freezing transition. The existence of a configurational barrier C͑T͒ for ␭=1.18 but not for ␭=1.22. Alternatively, these in these cases suggests that we may actually have phase co- additional low temperature peaks in C͑T͒ may be due to existence between two distinct populations of structures, one simple excitations from the ground state structure rather than

Downloaded 21 Sep 2009 to 143.206.80.1. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 114907-9 Phase transitions of a single polymer chain J. Chem. Phys. 131, 114907 ͑2009͒ global structural transformations.27 A more detailed analysis Liu, S. Garde, and S. Kumar, ibid. 123, 174505 ͑2005͒. 17 ͑ ͒ of the crystal structure and solid-state phase behavior of the M. P. Taylor, W. Paul, and K. Binder, Phys. Rev. E 79, 050801 R ͑2009͒. SW chains are currently being pursued. 18 F. Rampf, W. Paul, and K. Binder, Europhys. Lett. 70,628͑2005͒;F. Rampf, K. Binder, and W. Paul, J. Polym. Sci., Part B: Polym. Phys. 44, ACKNOWLEDGMENTS 2542 ͑2006͒. 19 W. Paul, T. Strauch, F. Rampf, and K. Binder, Phys. Rev. E 75, We thank David Landau, Jutta Luettmer-Strathmann, and 060801͑R͒͑2007͒. Peter Virnau for helpful discussions and Brad Goodner for 20 K. Binder and W. Paul, Macromolecules 41, 4537 ͑2008͒. 21 M. P. Taylor, J. Chem. Phys. 118, 883 ͑2003͒. making available the computing resources of the Hiram 22 ͑ ͒ ͑ J. E. Magee, L. Lue, and R. A. Curtis, Phys. Rev. E 78, 031803 2008 . College Center for Deciphering ’s Languages funded by 23 M. P. Taylor, K. Isik, and J. Luettmer-Strathmann, Phys. Rev. E 78, HHMI Undergraduate Science Grant No. 52005125͒. Finan- 051805 ͑2008͒. cial support through the Deutsche Forschungsgemeinschaft 24 F. Wang and D. P. Landau, Phys. Rev. Lett. 86, 2050 ͑2001͒; Phys. Rev. ͑ ͒ E 64, 056101 ͑2001͒. Grant No. SFB 625/A3 and the National Science Founda- 25 ͑ ͒ ͑ ͒ D. P. Landau, S.-H. Tsai, and M. Exler, Am. J. Phys. 72,1294 2004 . tion Grant No. DMR-0804370 and a sabbatical leave from 26 F. A. Escobedo and J. J. de Pablo, J. Chem. Phys. 102, 2636 ͑1995͒. Hiram College are also gratefully acknowledged. 27 T. Wüst and D. P. Landau, Phys. Rev. Lett. 102, 178101 ͑2009͒. 28 D. P. Landau and K. Binder, Monte Carlo Simulations in Statistical Phys- 1 ͑ ͒ P. J. Flory, Principles of Polymer ͑Cornell University Press, ics Cambridge University Press, Cambridge, 2000 . 29 Ithaca, 1953͒. D. T. Seaton, S. J. Mitchell, and D. P. Landau, Braz. J. Phys. 38,48 2 A. Yu. Grosberg and A. R. Khokhlov, of Macromol- ͑2008͒. 30 ecules ͑AIP, New York, 1994͒. More generally we find a very good agreement in d ln g͑E͒/dE across the 3 W. H. Stockmayer, Macromol. Chem. Phys. 35,54͑1960͒. entire overlap region with the exception of the extreme upper edge, where 4 I. Nishio, S.-T. Sun, G. Swislow, and T. Tanaka, Nature ͑London͒ 281, this derivative in the low energy window is clearly distorted. 31 208 ͑1979͒; G. Swislow, S.-T. Sun, I. Nishio, and T. Tanaka, Phys. Rev. C. Junghans, M. Bachmann, and W. Janke, Phys. Rev. Lett. 97, 218103 Lett. 44,796͑1980͒. ͑2006͒. 32 5 A. V. Finkelstein and O. B. Ptitsyn, Protein Physics ͑Academic, New W. Paul, F. Rampf, T. Strauch, and K. Binder, Comput. Phys. Commun. York, 2002͒. 179,17͑2008͒. 33 6 R. K. Pathria, ͑Pergamon, New York, 1985͒. J. Hernandez-Rojas and J. M. Gomez Llorente, Phys. Rev. Lett. 100, 7 K. Binder and D. W. Heermann, Monte Carlo Simulations in Statistical 258104 ͑2008͒. 34 Physics ͑Springer, New York, 1997͒. A. Mitsutake, Y. Sugita, and Y. Okamoto, Biopolymers 60,96͑2001͒. 35 8 D. H. E. Gross, Microcanonical Thermodynamics ͑World Scientific, Sin- D. F. Parsons and D. R. M. Williams, J. Chem. Phys. 124, 221103 gapore, 2001͒. ͑2006͒; Phys. Rev. E 74, 041804 ͑2006͒. 9 Y. Zhou, C. K. Hall, and M. Karplus, Phys. Rev. Lett. 77,2822͑1996͒. 36 E. Velasco, P. Tarazona, M. Reinaldo-Falagan, and E. Chacon, J. Chem. 10 Y. Zhou, M. Karplus, J. M. Wichert, and C. K. Hall, J. Chem. Phys. 107, Phys. 117, 10777 ͑2002͒. 10691 ͑1997͒. 37 T. Neuhaus and J. S. Hager, J. Stat. Phys. 113,47͑2003͒. 11 H. Liang and H. Chen, J. Chem. Phys. 113, 4469 ͑2000͒. 38 C. Zhou, T. C. Schulthess, S. Torbrügge, and D. P. Landau, Phys. Rev. 12 M. P. Taylor, J. Chem. Phys. 114,6472͑2001͒. Lett. 96, 120201 ͑2006͒. 13 F. Calvo, J. P. K. Doye, and D. J. Wales, J. Chem. Phys. 116, 2642 39 J. Luettmer-Strathmann, F. Rampf, W. Paul, and K. Binder, J. Chem. ͑2002͒. Phys. 128, 064903 ͑2008͒. 14 A. P. Gast, C. K. Hall, and W. B. Russel, J. Interface Sci. 96, 251 40 P. Bolhuis, M. Hagen, and D. Frenkel, Phys. Rev. E 50, 4880 ͑1994͒. ͑1983͒. 41 D. A. Young and B. J. Alder, J. Chem. Phys. 73, 2430 ͑1980͒. 15 M. H. J. Hagen and D. Frenkel, J. Chem. Phys. 101, 4093 ͑1994͒. 42 J. Serrano-Illan, G. Navascues, and E. Velasco, Phys. Rev. E 73, 011110 16 D. L. Pagan and J. D. Gunton, J. Chem. Phys. 122, 184515 ͑2005͒;H. ͑2006͒.

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