Scaling of Polymers in Aligned Rods

We study the behavior of self-avoiding polymers in a background of infinitely long vertically aligned rods that are either frozen in random positions or free to move horizontally. We find that in both cases the polymer chains are highly elongated, with vertical and horizontal size exponents that differ by a factor of 3. Though these results are different than previous predictions, our results are confirmed by detailed computer simulations.

Figure 1

Mean monomer separation squared as a function of arclength defined in Eq. (4), both the total (upper curve) and the $x\mathrm{\text{−}}y$ projection (lower curve) for an SAW of 4096 steps and vertical rods with $\rho =0.32$ with mobile rods. The slope of the best fit line for the upper graph is $2\times 0.974$.

Figure 2

Radius of gyration squared versus chain length on a log-log plot for an SAW in annealed vertical rods. The best fit line is $2\times 0.82$ for the points at the highest rod filling fractions (0.32). The $+$’s show the radius of gyration in the $x\mathrm{\text{−}}y$ plane for the same conditions. The *’s and □’s are the same data for filling fractions of 0.16 and 0.08, respectively.

Figure 3

Radius of gyration squared versus chain length on a log-log plot for an SAW in quenched vertical rods. The $\times$ and square symbols denote a rod filling fraction of 0.16 and 0.08, respectively. The best fit line is $2\times 1.01$ for the points at filling fractions of 0.16. The $+$’s and *’s show the radius of gyration in the $x\mathrm{\text{−}}y$ plane for the same conditions, filling fraction of 0.16 and 0.08, respectively.

Figure 4

Horizontal radius of gyration squared versus horizontal system size. The horizontal axis is 100 times the inverse width.