Density fluctuations of polymers in disordered media

We study self-avoiding random walks in an environment where sites are excluded randomly, in two and three dimensions. For a single polymer chain, we study the statistics of the time averaged monomer density and show that these are well described by multifractal statistics. This is true even far from the percolation transition of the disordered medium. We investigate solutions of chains in a disordered environment and show that the statistics cease to be multifractal beyond the screening length of the solution.

Figure 1

Time averaged monomer density $\rho$ for a self-avoiding random walk in two dimensions: (a) $N=512$, $L=128$, and $n_o=0.1$; (b) $N=256$, $L=64$, and $n_o=0.3$. The horizontal axes are labeled $x$ and $y$.

Figure 2

Scaling of moments of the coarse grained density $T_q$ as defined in Eq. (3) as a function of the amount of coarse graining $m$ in two dimensions. The top line is data for $q=-1$, the middle, $q=2$, and the bottom is for $q=1$.

Figure 3

The results of simulations to determine the multifractal dimensions for different obstacle densities $n_0=0.1$ (triangles), $0.2$ (squares), and $0.3$ (diamonds) in two dimensions.

Figure 4

Scaling of moments of the coarse grained density $T_q$ as defined in Eq. (3) as a function of the amount of coarse graining $m$ in three dimensions for $n_0=0.3$. The top line is data for $q=-1$, the middle, $q=2$, and the bottom is for $q=1$.

Figure 5

The results of simulations to determine the multifractal dimensions for different obstacle densities $n_0=0.1$ (triangles), $0.2$ (squares), and $0.3$ (diamonds) in three dimensions.

Figure 6

Distribution of the TAMD natural logarithm of $\rho$ for $40$ chains, an obstacle density of $n_0=0.2$ in three dimensions with $N=128$, $L=32$. The vertical axis the natural logarithm of the probability distribution.