Crystallization in melts of short, semiflexible hard polymer chains: An interplay of entropies and dimensions

What is the thermodynamic driving force for the crystallization of melts of semiflexible polymers? We try to answer this question by employing stochastic approximation Monte Carlo simulations to obtain the complete thermodynamic equilibrium information for a melt of short, semiflexible polymer chains with purely repulsive nonbonded interactions. The thermodynamics is obtained based on the density of states of our coarse-grained model, which varies by up to 5600 orders of magnitude. We show that our polymer melt undergoes a first-order crystallization transition upon increasing the chain stiffness at fixed density. This crystallization can be understood by the interplay of the maximization of different entropy contributions in different spatial dimensions. At sufficient stiffness and density, the three-dimensional orientational interactions drive the orientational ordering transition, which is accompanied by a two-dimensional translational ordering transition in the plane perpendicular to the chains resulting in a hexagonal crystal structure. While the three-dimensional ordering can be understood in terms of Onsager theory, the two-dimensional transition can be understood in terms of the liquid-hexatic transition of hard disks. Due to the domination of lateral two-dimensional translational entropy over the one-dimensional translational entropy connected with columnar displacements, the chains form a lamellar phase. Based on this physical understanding, orientational ordering and translational ordering should be separable for polymer melts. A phenomenological theory based on this understanding predicts a qualitative phase diagram as a function of volume fraction and stiffness in good agreement with results from the literature.

Figure 1

The inset shows the microcanonical entropy per chain as a function of energy per chain for chains of length $N=10$ in systems of linear dimension $L=10$ (black line) and $L=20$ (red line). The main figure shows the specific heat as a function of temperature derived from the entropy results.

Figure 2

Left: snapshot of the disordered state at $E=-426$ (only 5% of the chains are shown). Right: snapshot of the ordered state at $E=-5747=0.998E_{\min }$, where only chains aligned within a slice that is one chain thick are displayed. Different colors are used to distinguish different chains.

Figure 3

Orientational order parameters for the complete box, $m_b$, and individual chains $m_c$ (see text), as a function of temperature (left ordinate). Chain extension $R=\sqrt{\langle R_E^2\rangle }$ and its approximation by the freely rotating chain model (frc, right ordinate).

Figure 4

Inset: projections of the center of mass of all chains in the ordered phase onto a plane perpendicular to the director for the system with $L=20$. Main panel: Temperature dependence of the hexagonal order parameter in the plane perpendicular to the director. The arrow indicates the temperature of the orientational ordering transition in Fig. 3.

Figure 5

Top: Two times the probability to find a chain end as a function of position along the director (red dashed line, factor 2 to facilitate comparison with the other data set) and monomer density (black full line). The location of the lamellar boundaries is clearly visible. Bottom: Snapshot of a low energy configuration for the $10\times 10\times 80$ system for which the data on the top were obtained.

Figure 6

Suggested phase diagram in the plane of volume fraction and Kuhn length. The dividing line between isotropic and orientationally ordered states is given by $\varphi =1.4/({\mathrm{l}}_K/d)$. At a given chain length only a range of Kuhn lengths is accessible. The dotted green lines are obtained from the coexisting area fractions of 2D liquid and hexatic states (see text). The diamond is the isotropic-nematic transition density found from simulations of a closely related model [38].

Figure 7

Height map of the logarithm of the two-dimensional density of states $lng(n_a,n_c)$ in the $(n_a,n_c)$ plane. Note that the density of states now even varies over about 6000 orders of magnitude.

Figure 8

Height map of the canonical specific heat in the $(T/\varepsilon ,{\varepsilon }_c/\varepsilon )$ plane (red indicates the largest values, dark blue the smallest). The position of the narrow singular peak in the specific heat indicates the crystallization temperature as in Fig. 1.