Distributions of polymers in disordered structures

We study the behavior of linear polymers (modeled by self-avoiding random walks of N steps) on random fractal structures, both in Euclidean metric (r space) and ''chemical'' metric (l space). The chemical distance l between two sites on the structure at distance r from each other is the length of the shortest path connecting them, l∼rdmin, where dmin is the fractal dimension of the shortest path. We consider the average probabilities P(l,N) and P(r,N) to find a polymer of N monomers having a chemical end-to-end distance l or an Euclidean end-to-end distance r, respectively. We also study the first moments l(N) and r(N) of these probabilities. Our numerical results, obtained for two-dimensional percolation clusters at criticality, show that the fluctuations in l space are considerably smaller than in r space, suggesting that the l metric is more appropriate for studying structural properties of polymers in disordered media. We find l(N)∼Nlν, with νl=0.87±0.02, and P(l,N)∼(1/l)ylg when y<0.35, and P(l,N)∼(1/l)ylg′ exp(-bylδ) when y>0.35, where yl/Nlν, gl=2.5±0.2, gl′=3.0±0.2, and δl=1/(1-νl). From this form, we show analytically that P(r,N)∼(1/r)xrg exp(-cxrδ), where xr/Nrν, νr=νl/dmin, gr=gldmin, and δr=1/(1-νr). Here dmin1.13. These predictions are supported by the numerical simulations in r space, which yield νr=0.76±0.02 and gr=2.9±0.2. We also derive analytically the way the number of configurations taken in the averages affects the exponent δr. (c) 1995 The American Physical Society