Distribution function of the end-to-end distances of linear polymers with excluded volume effects

Abstract

The distribution function of t he absolute values of chain lengths of a polymer molec ule which displays t he excluded volume effect cannot assume a Ga ussia n form . This fact follows directly from t heoretical considerations based on the application of t he Central Limit Theorem t o the theory of Markov chains . In order t o determine the exact s hape of the polymer cha in-end distribution function we calculated its vario us momel:ts taken a bou t t he ori"in a nd t heir dependence on the number of polymer segmen ts, USll1g a Mo nte Carlo tech nique for gener ating poly m er chains on a lattice. The res ults obta ined from t he extrapolation of various combinations of t hese mome nts of the general form arc used to determine t he s hap e of t he poly mer distribu t ion fun ction. It is found that t he cha in-end distribution function can be approximated by t he following form :wit h t= 3.2 and a being a parameter, determina ble from the average m ean square cha in-end dis tances.

Structure factor of dilute ring polymers

“…5 we report the corresponding expressions for the monomer-monomer distribution function. We also verify that the phenomenological expression often used for the end-to-end distribution function for linear polymers [15][16][17][18] also provides a good approximation to the monomer-monomer distribution function considered here.…”

“…In three dimensions the end-to-end distribution function for linear polymers is well described by a very simple phenomenological expression [15][16][17][18][19], (5.13)…”

“…Indeed, we show that the structure factor is directly related to the energy-energy correlation function. Then, by extending the results available for the end-to-end distribution function [13][14][15][16][17][18][19][20], we compute its large-and small-momentum behavior. Moreover, by using an interpolation formula based on a dispersive approach [12,21], we obtain a general expression valid for all momenta, i.e.…”

Self‐Avoiding Walks on Lattices

“…Mazur" attempted to fit the three-dimensional endto-end distribution by an expression of the form ~n ( r > N C n r6 ~X P C-(r/Rn)'I (20) using the first few moments to determine E and 6. Although the best fit to the data corresponded to 6 = 3.2, E = 2, the moments were insensitive to 6 and E and an adequate fit could be obtained from 6 = E = 2.58.…”

“…Strong numerical evidence had been adduced by Domb, Gillis, and Wilmers" to indicate that the distributions Fisher established two rigorous bounds which suggest that the decay of P(0, r) with r is of the form exp [ -K ( e ) r ] . In order to obtain this form with a function of type (20) and ( R n 2 ) given by Eq. (17) it was necessary that the following consistency condition be satisfied : 012 + 116 = I…”