Free energy barrier for compact chains escaping from a small sphere

In this paper, the process of compact polymer chains escaping from a small sphere to a large one in the view of thermodynamics is investigated in detail based on the pruned-enriched-Rosenbluth method (PERM), which is quite efficient for the three-dimensional polymers on the simple-cubic lattice. In our simulation, three representative states of a polymer chain during the escaping process are studied, and some statistical properties of the chain size and the chain shape, such as mean-square radius of gyration per bond 〈S2〉/N and the shape factor 〈δ∗〉 are investigated. Our aim is to illuminate how the size and shape of the compact chains change during the escaping process. The changes of 〈S2〉/N and 〈δ∗〉 are not monotone and it is due to the fact that the chain should stretch itself in the escaping process. In the meantime, some thermodynamic properties are also calculated here. The hole is designed to be small enough to allow only one monomer at a time and it thus reduces the number of allowed chain conformations and breaks contacts between monomers at the beginning of the process. Additionally, we discuss the free energy barrier per bond H2 − H1 = ΔH of a compact chain, and here H2 is the maximum free energy per bond during the process and H1 is the minimum one when the compact chain is within the small sphere. Averaging free energy barrier over chain length N is convenient for the comparison with different chain lengths. ΔH as a function of chain length N and radius r1 of the small sphere is also studied and our result shows that ΔH for longer chains is lower means that it is relatively easier for each bond in longer chains to surmount the free energy barrier to escape. Some discussions about the self-avoiding walk (SAW) and swollen chains are also made for the comparison, and our results also show that the restriction of the small sphere on the SAW and the swollen chains is more effective because of their relatively looser intrinsic structure.