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 $R(N)\sim N^{\nu }$ with $\nu =1\mathrm{/}d$. % The precise measurement of various distributions of internal chain distances allows a direct test of the contact exponents ${\mathrm{\Theta }}_0=3\mathrm{/}8$, ${\mathrm{\Theta }}_1=1\mathrm{/}2$ and ${\mathrm{\Theta }}_2=3\mathrm{/}4$ predicted by Duplantier. % Due to the segregation of the chains the ratio of end-to-end distance $\text{Rend}(N)$ and gyration radius $\text{Rgyr}(N)$ becomes ${\text{Rend}}^2(N)\mathrm{/}{\text{Rgyr}}^2(N)\approx 5.3<6$ 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 $P_2(s) \sim 1/s^{1+\nu\Theta_2}$measuringthusthereturnprobabilityofthechainafter$s$ steps. % The irregular chain contours are shown to be characterized by a perimeter length $L(N)\sim R(N{)}^{\text{dc}}$ 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 $q^{d}F(q)\sim 1\mathrm{/}(qR(N){)}^{{\mathrm{\Theta }}_2}$ in the intermediate regime of the wave vector $q$. This may allow to confirm the predicted contour fractality in a real experiment.