Non-extensivity of the chemical potential of polymer melts

TL;DR

This paper demonstrates that the chemical potential of a test polymer chain in a dense polymer melt exhibits non-extensive behavior due to incompressibility effects, with analytical and numerical evidence supporting this long-range interaction.

Contribution

The study reveals a non-extensive correction to the chemical potential in polymer melts, challenging Flory's ideality hypothesis and providing analytical and simulation validation.

Findings

The correction to chemical potential scales as 1/√n, indicating non-extensivity.

Monte Carlo simulations confirm the decay of non-exponentiality parameter as 1/√<N>.

Analytical results apply to both polydisperse and monodisperse polymer solutions.

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 $\delta \mu_c(n) \sim +1/\rho\sqrt{n}$ arises ($\rho$ being the monomer density) for all $\P(N)$ if $n \ll <N>$ 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 $\delta \mu_c(n) \approx (1- 2 n/<N>) / \rho \sqrt{n}$ for $n \ll <N>^2$, hence, $\delta \mu_c(n) \approx - 1/\rho \sqrt{n}$ 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 $K_p \equiv 1 - <N^>/p!<N>^p$ to decay as $K_p \approx 1 / \sqrt{<N>}$ for all moments $p$ of the distribution.