Non-extensivity of the chemical potential of polymer melts

Abstract

Following Flory's ideality hypothesis, the chemical potential of a test chain of length n immersed into a dense solution of chemically identical polymers of length distribution P(N) is extensive in n . We argue that an additional contribution deltamu(c)(n) approximately +1/rho (sqrt[n]) arises (rho being the monomer density) for all P(N) if n " <N> which can be traced back to the overall incompressibility of the solution leading to a long-range repulsion between monomers. Focusing on Flory-distributed melts, we obtain deltamu(c)(n) approximately equal to (1 - 2n/ <N> ) / rho (sqrt[n]) for n " <N>(2) , hence, deltarho(c)(n) approximately equal to -1/rho (sqrt[n]) if n is similar to the typical length of the bath <N> . Similar results are obtained for monodisperse solutions. Our perturbation calculations are checked numerically by analyzing the annealed length distribution P(N) of linear equilibrium polymers generated by Monte Carlo simulation of the bond fluctuation model. As predicted we find, e.g., the non-exponentiality parameter K (p) = 1 - <N(P)> (p)! <N>(p) to decay as K (p) approximately equal to 1/ (sqrt[<N>]) for all moments p of the distribution.

Scale-Free Static and Dynamical Correlations in Melts of Monodisperse and Flory-Distributed Homopolymers

“…Summarizing recent theoretical and computational results we have argued in the present contribution that due to the incompressibility constraint both static [80,81,82,83,84,85,86,87,88,89] and dynamical [90,91] correlations of the composition fluctuations, i.e. of the density and displacement fields of marked chains and subchains, must arise for distances r ≫ ξ and for corresponding arc-lengths s ≫ g.…”

“…of dense BFM solutions such as the chemical potential µ N in EP systems in Sec. 4.6 [88]. We verify in Sec.…”

“…4.5 that the chemical potential µ N is essentially extensive with respect to the chain length, as expected from Flory's ideality hypothesis. In order to demonstrate that the small deviations predicted by the perturbation theory [88] exist, we have investigated in Sec. 4.6 essentially Flory-distributed ( Fig.…”

“…We have seen in Sec. 4.5 that the chemical potential µ N of a test chain of length N plugged into a melt of chemically identical monodisperse polymers of the same length increases essentially linearly with N. Focusing now on classical BFM systems where ε = ∞ and setting temperature to unity (T = 1), we show how small non-linear deviations may be captured numerically [88]. More generally, the challenge is to characterize the chemical potential µ n of a test chain of length n immersed into a bath of N-chains of an arbitrary (normalized) length distribution p N .…”

“…A more precise calculation for monodisperse chains using the full Debye function is given in Ref. [88].…”