Static properties of polymer melts in two dimensions

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 $R(N)∼Nν$ with $ν=1/d$⁠. The precise measurement of various distributions of internal chain distances allows a direct test of the contact exponents $Θ0=3/8$⁠, $Θ1=1/2$⁠, and $Θ2=3/4$ predicted by Duplantier. Due to the segregation of the chains the ratio of end-to-end distance $Re(N)$ and gyration radius $Rg(N)$ becomes $Re2(N)/Rg2(N)≈5.3<6$ for $N⪢100$ and the chains are more spherical than Gaussian phantom chains. The second Legendre polynomial $P2(s)$ of the bond vectors decays as $P2(s)∼1/s1+νΘ2$⁠, thus measuring the return probability of the chain after $s$ steps. The irregular chain contours are shown to be characterized by a perimeter length $L(N)∼R(N)dp$ of fractal line dimension $dp=d−Θ2=5/4$⁠. 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 nonmonotonous behavior with $qdF(q)∼1/(qR(N))Θ2$ in the intermediate regime of the wave vector $q$⁠. This may allow to confirm the predicted contour fractality in a real experiment.