Distribution of end-to-end distance of a grafted compact polymer chain confined in two parallels

The end-to-end distance distribution of a grafted compact polymer chain confined between two parallels is studied by using the PERM (pruned-enriched-Rosenbluth method) algorithm. Because of the difference in different directions of confined polymer chains, in this paper we just consider the distribution function P (x) in the x-axis direction, and it can be expressed as In[p(x)/P-m (x)]/ND-2/3 = a(0) + a(1) u + a(2) u(2) + a(3) u(3), here u = x/ND-2/3, N is the chain length, P-m(x) is the maximum value of P(x), D + 1 is the distance between the two parallels. On the other hand, we calculate the Shannon entropy according to P (x), which is very sensitive to the variation of the distance of two parallels, and it can be used to describe the restriction of the polymer chain. With a given chain length N, the Shannon entropy of the distribution P (x) decreases rapidly first and then tends to a fixed value with increasing D, by which we can find a transition point D-c When D > D-c the Shannon entropy tends to a constant, and at the same time the restriction of polymer chain performed by the two parallels can be ignored. We also find that the relationship between D-c and N as D-c-N-lambda, here lambda = 0.543.