Bond−Vector Correlation Functions in Dense Polymer Systems

Abstract

It is well-known that homopolymer chains do not exactly follow the Gaussian statistics even in the melt state. In particular, orientations of two bonds l 1 and l 2 of the same chain in a concentrated polymer system are always correlated even when they are separated by a long segment of s . 1 units: AEl 1 3 l 2 ae s = C(s) ∼ 1/s 3/2 . It is important to know how these orientational correlations are distributed in space, i.e. how AEl 1 3 l 2 ae depends on the distance r between the bonds. 1 An unexpected feature is revealed in the present paper: it is shown that the distancedependent bond-vector correlation function K(r) = AEl 1 3 l 2 ae r is extremely sensitive to the definition of r. Depending on the definition, K(r) for r . b (b is the monomer size) can be either significantly lower, or significantly higher than the position-averaged correlator C(s) with s ∼ r 2 /b 2 corresponding to a given distance r. We propose an "invariant" definition of the intrachain orientational correlation function and show that it is related to the formfactor of a single chain. A quantitative link between the orientational and positional correlations in polymer melts is thus discovered. We also have found a quantitative relationship between the intrachain and interchain correlation functions. It is shown that the inter-chain orientational correlation function (for bonds of different chains) is long-range and follows the 1/r 4 scaling law in the case of infinite chains.

Detailed analysis of Rouse mode and dynamic scattering function of highly entangled polymer melts in equilibrium

“…For small p/N (large N/p), one should expect that 4 sin 2 (pπ/(2N )) X 2 p reaches a plateau value b 2 . However, since at short length scale the local intramolecular correlations in the chains (the correlation hole effect) are important, a correction term [8,9,16] O((N/p) −1/2 ) is needed to be considered as follows,…”

Conformations of a Long Polymer in a Melt of Shorter Chains: Generalizations of the Flory Theorem

“…Recently, it was emphasized 12 , 13 that the intramolecular bond correlation function in dense melts decays as a power law in contrast to an exponential decay for chains without long-range correlations. This leads to a partial swelling of polymer chains even in monodisperse melts with the mean square radius of gyration of a chain with n monomers approximated 12 by with characteristic ratio The coefficient c = 0.656, root-mean-square bond length l = 2.636, and C ∞ = 1.52 were determined in ref ( 12 ) at simulation conditions identical to the present study.…”

The Effect of Bending Rigidity on Polymers

“…The traditional one is defined through the exponential decay of the orientational correlation function 〈cos θ( s )〉 (Equation ). Although for both random walk and self‐avoiding walk the orientational correlation function shows a power law decay behavior at a large length scale s > s *, this stays a good estimator considering that it can recover the stiffness parameter κ (Figure ) and that l p should not depend on the polymer length ( Figure blue solid line). Another way is to calculate l p from Equation or simply when L is large enough.…”

“…The continuous version of the Kratky‐Porod model is the worm‐like chain model, where the persistence length l p is defined through the exponential decay of the orientational correlation function: Here is the unit tangent vector to the chain at contour distance s , and is the position vector along the chain. Although chains in a dense melt or at the Θ‐point in solution behave like ideal chains without excluded volume effect, as the worm‐like chain does, recently it was shown that the orientational correlation function for chains in these conditions shows a power law decay s −3/2 instead of the above exponential decay for certain range of contour length 1 ≪ s ≪ N . For real chains Hsu et al .…”