Static Properties of Polymer Melts in Two Dimensions

TL;DR

This study uses molecular dynamics simulations to analyze the static properties of two-dimensional polymer melts, confirming theoretical predictions about chain configurations, contact exponents, and fractal contours.

Contribution

It provides the first detailed numerical verification of Duplantier's predictions for contact exponents and fractal contours in 2D polymer melts.

Findings

Chains adopt compact, spherical configurations with size scaling as N^{1/2}.

Contact exponents match Duplantier's theoretical predictions.

The perimeter of chain contours exhibits fractal behavior with dimension 5/4.

Abstract

Self-avoiding polymers in strictly two-dimensional ($d=2$) melts are investigated by means of molecular dynamics simulation of a standard bead-spring model with chain lengths ranging up to N=2048. % The chains adopt compact configurations of typical size $R(N) \sim N^{\nu}$ with $\nu=1/d$. % The precise measurement of various distributions of internal chain distances allows a direct test of the contact exponents $\Theta_0=3/8$, $\Theta_1=1/2$ and $\Theta_2=3/4$ predicted by Duplantier. % Due to the segregation of the chains the ratio of end-to-end distance $\Rend(N)$ and gyration radius $\Rgyr(N)$ becomes $\Rend^2(N)/\Rgyr^2(N) \approx 5.3 < 6$ for $N \gg 100$ and the chains are more spherical than Gaussian phantom chains. % The second Legendre polynomial $P_2(s)$ of the bond vectors decays as $P_2(s) \sim 1/s^{1+\nu\Theta_2}$ measuring thus the return probability of the chain after $s$ steps. % The irregular chain contours are shown to be characterized by a perimeter length $L(N) \sim R(N)^{\dc}$ of fractal line dimension $\dc = d-\Theta_2 =5/4$. % % In agreement with the generalized Porod scattering of compact objects with fractal contour the Kratky representation of the intramolecular structure factor $F(q)$ reveals a strong non-monotonous behavior with $q^dF(q) \sim 1/(q R(N))^{\Theta_2}$ in the intermediate regime of the wave vector $q$. This may allow to confirm the predicted contour fractality in a real experiment.