Self-avoiding walks on self-similar structures: finite versus infinite ramification

Ring polymers in crowded environment: conformational properties

We analyze the universal size characteristics of flexible ring polymers in solutions in presence of structural obstacles (impurities) in d dimensions. One encounters such situations when considering polymers in gels, colloidal solutions, intra- and extracellular environments. A special case of extended impurities correlated on large distances r according to a power law ~r(-a) is considered. Applying the direct polymer renormalization scheme, we evaluate the estimates for averaged gyration radius ⟨R(g ring)⟩ and spanning radius ⟨R(1/2 ring)⟩ of typical ring polymer conformation up to the first order of double ɛ = 4 - d, δ = 4 - a expansion. Our results quantitatively reveal an extent of the effective size and anisotropy of closed ring macromolecules in disordered environment. In particular, the size ratio of ring and open (linear) polymers of the same molecular weight grows when increasing the strength of disorder according to ⟨R(g ring)(2)⟩/⟨R(g chain)(2)⟩=½(1+(13/48)δ).

Equivalence of quenched and annealed averaging in models of disordered polymers

The equivalence of the influence of quenched and annealed disorder on the scaling properties of long flexible polymer chains is proved by analyzing the O(m)-symmetric field theory in the polymer (de Gennes) limit m → 0. Additional symmetry properties of the model in this limit are discussed.

Temperature-dependent structural behavior of self-avoiding walks on three-dimensional Sierpinski sponges

The statistics of self-avoiding random walks (SAWs), consisting of up to N=1280 steps, on deterministic fractal structures with infinite ramification, modeled by Sierpinski cubic lattices, in the presence of a finite temperature is investigated as a model for polymers absorbed on a disordered medium. Thereby, the three-dimensional Sierpinski sponge is defined by two types of sites with energy 0 and ϵ>0 , respectively, yielding a deterministic fractal energy landscape. The probability distribution function of the end-to-end distance of SAWs is obtained and its scaling behavior studied. In the limiting case of temperature T → ∞, the known behavior of SAWs on regular cubic lattices is recovered, while for T → 0 the resulting scaling exponents are confronted with previous calculations for much shorter linear chains based on the exact enumeration technique. For finite temperatures, the structural behavior of SAWs in three dimensions is compared to its two-dimensional counterpart and found to be intermediate between the two limiting cases (T → 0 and T → ∞, respectively), where the characteristic exponents, however, display a nontrivial dependence on temperature.

Multifractality of self-avoiding walks on percolation clusters

We consider self-avoiding walks on the backbone of percolation clusters in space dimensions d=2,3,4. Applying numerical simulations, we show that the whole multifractal spectrum of singularities emerges in exploring the peculiarities of the model. We obtain estimates for the set of critical exponents that govern scaling laws of higher moments of the distribution of percolation cluster sites visited by self-avoiding walks, in a good correspondence with an appropriately summed field-theoretical epsilon=6-d expansion [H.-K. Janssen and O. Stenull, Phys. Rev. E 75, 020801(R) (2007)10.1103/PhysRevE.75.020801].

Temperature-dependent structural behavior of self-avoiding walks on Sierpinski carpets

We study the temperature-dependent structural behavior of self-avoiding walks (SAWs) on two-dimensional Sierpinski carpets as a simple model of polymers adsorbed on a disordered surface. Thereby, the Sierpinski carpet defines two types of sites with energy 0 and >0 , respectively, yielding a deterministic fractal energy landscape. In the limiting cases of temperature T-->0 and T-->infinity , the known behaviors of SAWs on Sierpinski carpets and on regular square lattices, respectively, are recovered. For finite temperatures, the structural behavior is found to be intermediate between the two limiting cases; the characteristic exponents, however, display a nontrivial dependence on temperature.

Scaling of Hamiltonian walks on fractal lattices

We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e., self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on three-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG), and n -simplex fractal families. For GM, MSG and n -simplex lattices with odd values of n , the number of open HWs Z(N), for the lattice with N>>1 sites, varies as omega(N)}N(gamma). We explicitly calculate the exponent gamma for several members of GM and MSG families, as well as for n-simplices with n=3, 5, and 7. For n-simplex fractals with even n we find different scaling form: Z(N) approximately omega(N)mu(N1/d(f), where d(f) is the fractal dimension of the lattice, which also differs from the formula expected for homogeneous lattices. We discuss possible implications of our results on studies of real compact polymers.

Violation of the des Cloizeaux relation for self-avoiding walks on Sierpinski square lattices

The statistics of self-avoiding walks (SAWs) on deterministic fractal structures with infinite ramification, modeled by Sierpinski square lattices, is revisited in two and three dimensions using the reptation algorithm. The probability distribution function of the end-to-end distance of SAWs, consisting of up to 400 steps, is obtained and its scaling behavior at small distances is studied. The resulting scaling exponents are confronted with previous calculations for much shorter linear chains (20 to 30 steps) based on the exact enumeration (EE) technique. The present results coincide with the EE values in two dimensions, but differ slightly in three dimensions. A possible explanation for this discrepancy is discussed. Despite this, the violation of the so-called des Cloizeaux relation, a renormalization result that holds on regular lattices and on deterministic fractal structures with finite ramification, is confirmed numerically.