ВЫСОКОМОЛЕКУЛЯРНЫЕ СОЕДИНЕНИЯ, Серия С, 2013, том 55, № 7, с. 829-844

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© 2013 t. Mark P. Taylor", Wolfgang Paul4, and Kurt Binderc

a Department of Physics, Hiram College, Hiram, OH 44234, USA b Institut für Physik, Martin-Luther-Universität, D-06099Halle (Saale), Germany c Institut für Physik, Johannes-Gutenberg-Universität, Staudinger Weg 7, D-55099 Mainz, Germany

e-mail: taylormp@hiram.edu

Abstract — The Wang-Landau (WL) algorithm is a Monte Carlo simulation technique providing a direct computation of the density of states of a many-body system. The temperature independent density of states function encodes all thermodynamic information about a system, and thus, its construction allows for a very efficient determination of phase behavior. Here we describe the application of the WL approach to continuum interaction-site polymer chains and compute single-chain phase diagrams for three specific models (flexible and semi-flexible homopolymer chains built from square-well sphere monomers and a flexible AB-het-eropolymer comprised of alternating square-well and hard-sphere monomers). We provide details on the implementation of the WL algorithm, the underlying Monte Carlo move set, and the subsequent thermodynamic and structural analysis required to characterize phase behavior.

DOI: 10.7868/S0507547513060172


Polymer chain conformation is of fundamental importance in many areas of polymer science. For example, in polymeric materials, bulk macroscopic properties (including mechanical, optical, rheological, and electrical) are often coupled to the average microscopic conformation of the constituent polymer molecules [1]. In biological systems, catalytic, regulatory, and signaling activites are often closely related to the specific shapes of individual macromolecules and confor-mational transitions of single macromolecules play a key role in the functioning of such systems [2, 3]. Similarly, the functionality of a number of recently developed environmentally responsive or "smart" materials is based on conformational changes of polymers tethered to surfaces or nanoparticles [4—6].

Conformational transitions of individual polymer chains can be viewed as small system analogs to phase transitions in bulk systems. For example, the polymer coil-globule transition, in which an expanded disordered coil collapses into a compact disordered globule, is similar to a gas-liquid condensation transition [7]. Similarly, the transition of a polymer chain from a disordered coil or globule conformation into an ordered three-dimensional structure, as occurs in protein folding, is analogous to a freezing transition [3]. These single chain transitions can be described using the language of conventional phase transitions. Thus the coil-globule transition is continuous in nature, making it analogous to a gas-liquid system precisely at the critical point. Likewise, the folding of most small proteins involves a free-energy barrier giving this transition a discontinuous or first-order character.

In this work we study such conformational phase transitions of single polymer chains. From a theoretical point of view, phase transitions are identified from singularities in thermodynamic functions [8]. Such singular behavior is only truly present in the thermodynamic (i.e., infinite size) limit and thus, in computer simulation studies, one often employs scaling techniques to extrapolate bulk system behavior from the analysis of finite-size systems [9]. For a single polymer chain, even for very large molecular weight, the system is always far from the thermodynamic limit. One can of course theoretically invoke an infinite chain length limit for a single polymer, in which case the polymer coil-globule transition is found to be a true second-order phase transition [7]. However, in the study of single polymer phase transitions this long chain limit is often not appropriate. For example, many biological mac-romolecules, such as proteins, have a fixed monomer sequence and size. In such inherently finite size systems there is no thermodynamic limit and "phase transitions" will necessarily be associated with finite, "rounded" peaks, rather than divergences, in thermo-dynamic functions.

One method for studying phase transitions in finite size systems is through microcanonical thermodynamics [10]. This approach is based on analysis of the microcanonical entropy function S(E). Inflection points and changes in curvature of this function provide signatures for continuous and discontinuous phase transitions, respectively [11]. Given S(E), or equivalently the microcanonical density of states g(E), one can also construct the canonical partition function and carry out a more conventional canonical

thermodynamic analysis. Using both thermodynamics approaches can be informative [12, 13]. For example, one can construct both temperature independent (mi-crocanonical) configurational probability landscapes and temperature dependent (canonical) free-energy landscapes to map out the details of phase transitions in configuration space [14].

The density of states functions for interacting many-body systems can be directly computed via modern computer simulation techniques such as the Berg and Neuhaus multicanonical method [15] or the iterative Wang-Landau (WL) algorithm [16, 17]. We have recently used the WL algorithm to compute g(E) for a flexible homopolymer chain built from tangent square-well spheres [12, 13]. This is a coarse-grained polymer model in which the square-well potential is to be considered an effective potential that averages over chemical details and accounts for local solvent effects. Such simple models capture many features of polymeric systems without incurring the extreme computational costs of atomistic simulations [18]. One can always systematically increase the complexity of simple interaction-site models to bring them closer to full atomistic detail [19], but we believe there is still much basic polymer physics that can be learned from simple models. The square-well-sphere chain is a continuum model, however, it has a discrete set of energy states (as in a lattice model) which facilitates the calculation of g(E). We note that the WL method has also been applied to continuum homopolymer models with continuous energy spectra [20, 21].

Using WL density of states results we have previously mapped out the phase behavior of the N = 128 tangent SW-homopolymer chain [13]. For a sufficiently large square-well diameter or interaction range this chain undergoes, with decreasing temperature, a continuous coil-globule transition followed by a discontinuous freezing transition. With decreasing square-well diameter the temperature range for a stable globule phase decreases and, for a critical interaction range, the globule phase disappears. For shorter range interactions the chain undergoes a first-orderlike transition from an expanded coil state to a frozen crystallite. The thermodynamics of this later direct freezing transition are identical to the thermodynamics characterizing the all-or-none folding of small proteins [3].

In this work we provide details on the implementation of the WL approach to continuum polymer chains and present results for three different polymer models. The underlying Monte Carlo move set is critical to the success of the WL approach since the simulation must access all regions of configuration space including highly compact and crystal states and rare transition state structures. We have found that bridging (or cut-and-paste) moves, especially those that redefine an end-site of the chain, are important for realizing such chain configurations. Unfortunately, these types of

moves are not generally applicable to heteropolymers, however, as we show here, they can be adapted to the study of regular heteropolymers. To analyze phase behavior one also typically requires some structural analysis in addition to the thermodynamics directly given by g(E). This is often best done in subsequent "production" simulations that explore configurational phase space using the density of states constructed with the WL algorithm [22]. Such production simulations can also be used to construct multidimensional configurational probability landscapes [14].


Interaction-Site Chain Model

In this work we study a single flexible interaction-site polymer chain comprised of N spherical sites or monomers connected by "universal joints" of fixed bond length L. Monomers are sequentially numbered 1 through N and monomer i is located at position vector r. All pairs of non-bonded sites i and j (| i - j| > 1) interact via a spherically symmetric site-site potential ulj(r) where r = ry = |r - j. For a chain in contact with a thermal reservoir at fixed temperature T, the single-chain canonical partition function can be written as [23-25]

Zn (T) = J... j" S-un /kBTdfi2...dfN-1,n (1)

where SN = HNi1 +1 - L)/4tcL2 is the product of intramolecular distribution functions imposing the


■ i Uy is

the total potential energy of a given chain configuration, kB is the Boltzmann constant, and we take site 1 as the coordinate origin for the integration. The above configurational partition function is simply a Boltz-mann weighted average over all possible chain conformations and is defined such that ZN (T) = 1 in the limit of a freely-jointed or non-interacting chain.

The single chain partition function ZN(T) contains all thermodynamic information about the chain molecule. Thus, for example, one might compute ZN (T) for a range of temperatures and then search for maxima in the single chain specific heat Cv =

= kBp2d2 ln ZN(T)/dp2, where p = 1/kBT, to locate phase transitions. An alternate approach to constructing the single-chain partition function ZN(T) is provided by formally rewriting the 3(N - 1)-dimensional integral over cha

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