THE JOURNAL OF CHEMICAL PHYSICS 131, 114907 ͑2009͒ Phase transitions of a single polymer 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 crystallite. Here we study transitions between these conformational states for an interaction-site polymer chain comprised of N=128 square-well-sphere monomers with hard-sphere diameter ␴ and square-well diameter ␭␴. Wang– Landau sampling with bond-rebridging Monte Carlo moves is used to compute the density of states for this chain and both canonical and microcanonical analyses are used to identify and characterize phase transitions in this finite size system. The temperature-interaction range ͑i.e., T-␭͒ phase diagram is constructed for ␭Յ1.30. Chains assume an expanded coil conformation at high temperatures 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-crystal ͑freezing͒ transition. For well diameters ␭Ͻ1.06 the collapse transition is pre-empted by the freezing transition and thus there is a direct first-order coil-crystal phase transition. These results confirm the recent prediction, based on a lattice polymer model, that a collapsed globule state is unstable with respect to a solid phase for flexible polymers with sufficiently short-range monomer-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 molecules can undergo significant, and in ever, for biological macromolecules such as proteins 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 entropy 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 biopolymers 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 protein folding totypical model of a simple flexible homopolymer chain. provides an example of a discontinuous single-chain confor- Previous work 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 liquid, lyticities in the thermodynamic functions appear only in the where chain collapse and freezing are akin to the gas-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]. molecule 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 Binder 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 vapor 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.