Collapse transitions in a flexible homopolymer chain: Application of the Wang-Landau algorithm

The thermodynamic behavior of a continuous homopolymer is described using the Wang-Landau algorithm for chain lengths up to $N=561$. The coil-globule and liquid-solid transitions are analyzed in detail with traditional thermodynamic and structural quantities. The behavior of the coil-globule transition is well within theoretical and computational predictions for all chain lengths, while the behavior of the liquid-solid transition is much more susceptible to finite-size effects. Certain “magic number” lengths $\left(N=13,55,147,309,561\right)$, whose minimal energy states offer unique icosahedral geometries, are discussed along with chains residing between these special cases. The low temperature behavior near the liquid-solid transition is rich in structural transformations for certain chain lengths, showing many similarities to the behavior of classical clusters with similar interaction potentials. General comments are made on this size dependent behavior and how it affects transition behavior in this model.

Figure 1

(Color online) Bonded and nonbonded potentials versus the distance between two monomers. A harmonic potential is plotted as a reference for the bonded potential (FENE-Lennard-Jones combination). The nonbonded potential is plotted to a distance of only 1.5 for clarity (the actual cutoff occurs at 3).

Figure 2

(Color online) The relative density of states versus energy per monomer for a number of chain lengths. The maximum of each curve is set to zero for plotting purposes. Values for the highest and lowest energy states for the longest chain $\left(N=500\right)$ differ by $\approx 1000$ orders of magnitude. (Each curve represents a single run and the majority of symbols were omitted for clarity.)

Figure 3

(Color online) The radius of gyration versus energy per monomer for chain lengths 100, 200, 300, 400, and 500. All curves are monotonic at both low and high $E/N$. (The majority of symbols were omitted for clarity and error bars are smaller than the symbol size.)

Figure 4

(Color online) The heat capacity for chain lengths 100, 200, 300, 400, and 500. Two noticeable features appear: large peaks at low $T$ representing the liquid-solid transition, and shoulders at high $T$ representing the coil-globule transition. Each curve is the average of at least 20 independent simulations, where the error bars are calculated using a Jackknife analysis $\left(dT=0.001\right)$.

Figure 5

(Color online) The radius of gyration (top) and its derivative (bottom) versus temperature for chain lengths 100, 200, 300, 400, and 500. Features that were seen in the heat capacity are also present in this figure. The derivative indicates the coil-globule transition by large, broad peaks at high $T$. The liquid-solid transition is also represented at low $T$, but the peaks are much smaller in height and less distinguishable. (The majority of symbols were omitted for clarity and error bars are smaller than the symbol size, $dT=0.001$.)

Figure 6

(Color online) The core density (top) and its derivative (bottom) versus temperature for chain lengths 100, 200, 300, 400, and 500. Features that were present in previous figures are also found here, namely, peaks at low $T$ indicative of the liquid-solid transition and broad peaks (similar to the shoulder seen in the specific heat) at high $T$ indicative of the coil-globule transition. (The majority of symbols were omitted for clarity and error bars are smaller than the symbol size, $dT=0.001$.)

Figure 7

(Color online) Transition temperatures $\left(T_{\text{tr}}\right)$ from the specific heat (top), along with the corresponding specific heat $\left(C_V/N\right)$ values (bottom), plotted versus chain length $N$. The coil-globule (circles), the liquid-solid (squares), and solid-solid (triangles) transitions are all represented. The coil-globule and liquid-solid transitions are present in nearly all chain lengths, while the solid-solid transition appears between $30\le N\le 54$. The small gap between $42\le N\le 54$ is a result of the difficulty in distinguishing between the liquid-solid and solid-solid transition. This plot has striking similarities compared to recent studies of classical Lennard-Jones clusters (Ref. [28, 29, 32]). (Error bars are on the order of or smaller than the symbols.)

Figure 8

(Color online) The specific heat versus temperature for the magic set of chain lengths, where minimum energy configurations for each chain correspond to an icosahedral geometry. These chain lengths offer two generic features: a shoulder in $C_V/N$ indicative of the coil-globule transition (high temperatures) and a first-order like peak indicative of a liquid-solid transition (low temperatures). They also offer a basis for understanding the behavior of chain lengths between two magic lengths.

Figure 9

(Color online) Illustration of Mackay and Anti-Mackay packing, along with minimum energy states for $N=13$, $N=40$, and $N=55$. $N=13$ is the first icosahedral geometry in the magic set, and $N=55$ is the next. The $N=40$ state represents a Mackay ground state, where a $N=13$ icosahedral core can be found at its center. This illustration represents a class of minimum energy states where the outer layer surrounding the icosahedral core begins forming the overlayer for the next magic number chain length. (The central monomer is red, the first monomer layer is yellow, and the outer monomer layer is blue.)

Figure 10

(Color online) The heat capacity (top) and the derivative of the radius of gyration (bottom) versus temperature for chain lengths between $13\le N\le 55$. These chain lengths offer unique behavior in the liquid-solid regime, where for some chains two peaks can be seen in the heat capacity and a peak and shoulder in the radius of gyration. Each curve is shifted by an arbitrary amount for visual clarity. $\left(dT=0.001\right)$

Figure 11

(Color online) Analysis of the liquid-solid and coil-globule transitions using transition temperatures provided by features in the specific heat. The liquid-solid transition data (squares) are plotted with the lower-horizontal axis, and the coil-globule transition data (circles) are plotted with the upper-horizontal axis. Dashed lines represent the fits from Eqs. (8, 9).