Self-avoiding walks on diluted networks

Self-avoiding walk on a square lattice with correlated vacancies

The self-avoiding walk on the square site-diluted correlated percolation lattice is considered. The Ising model is employed to realize the spatial correlations of the metric space. As a well-accepted result, the (generalized) Flory's mean-field relation is tested to measure the effect of correlation. After exploring a perturbative Fokker-Planck-like equation, we apply an enriched Rosenbluth Monte Carlo method to study the problem. To be more precise, the winding angle analysis is also performed from which the diffusivity parameter of Schramm-Loewner evolution theory (κ) is extracted. We find that at the critical Ising (host) system, the exponents are in agreement with Flory's approximation. For the off-critical Ising system, we find also a behavior for the fractal dimension of the walker trace in terms of the correlation length of the Ising system ξ(T), i.e., D_{F}^{SAW}(T)-D_{F}^{SAW}(T_{c})∼1/sqrt[ξ(T)].

Surface growth on percolation networks by a conserved-noise restricted solid-on-solid growth model

Surface growth by the conserved-noise restricted solid-on-solid model is investigated on diluted lattices, i.e., on percolation networks that are embedded in two spatial dimensions. The growth exponent β and the roughness exponent α are defined, respectively, by the mean-square surface width via W(2)(t)∼t(2β) and the mean-square saturated width via W(sat)(2)(L)∼L(2α), where L is the system size. These are measured on both an infinite network and a backbone network and the results are compared with power-counting predictions obtained using the fractional Langevin equation. While the Monte Carlo results on deterministic fractal substrates show excellent agreement with the predictions [D. H. Kim and J. M. Kim, Phys. Rev. E 84, 011105 (2011)], the results on critical percolation networks deviate by 8%-12% from these predictions.

Asymptotic scaling behavior of self-avoiding walks on critical percolation clusters

We study self-avoiding walks on three-dimensional critical percolation clusters using a new exact enumeration method. It overcomes the exponential increase in computation time by exploiting the clusters' fractal nature. We enumerate walks of over 10^{4} steps, far more than has ever been possible. The scaling exponent ν for the end-to-end distance turns out to be smaller than previously thought and appears to be the same on the backbones as on full clusters. We find strong evidence against the widely assumed scaling law for the number of conformations and propose an alternative, which perfectly fits our data.

Linear polymers in disordered media: the shortest, the longest, and the mean self-avoiding walk on percolation clusters

Long linear polymers in strongly disordered media are well described by self-avoiding walks (SAWs) on percolation clusters and a lot can be learned about the statistics of these polymers by studying the length distribution of SAWs on percolation clusters. This distribution encompasses 2 distinct averages, viz., the average over the conformations of the underlying cluster and the SAW conformations. For the latter average, there are two basic options, one being static and one being kinetic. It is well known for static averaging that if the disorder of the underlying medium is weak, this disorder is redundant in the sense the renormalization group; i.e., differences to the ordered case appear merely in nonuniversal quantities. Using dynamical field theory, we show that the same holds true for kinetic averaging. Our main focus, however, lies on strong disorder, i.e., the medium being close to the percolation point, where disorder is relevant. Employing a field theory for the nonlinear random resistor network in conjunction with a real-world interpretation of the corresponding Feynman diagrams, we calculate the scaling exponents for the shortest, the longest, and the mean or average SAW to 2-loop order. In addition, we calculate to 2-loop order the entire family of multifractal exponents that governs the moments of the the statistical weights of the elementary constituents (bonds or sites of the underlying fractal cluster) contributing to the SAWs. Our RG analysis reveals that kinetic averaging leads to renormalizability whereas static averaging does not, and hence, we argue that the latter does not lead to a well-defined scaling limit. We discuss the possible implications of this finding for experiments and numerical simulations which have produced widespread results for the exponent of the average SAW. To corroborate our results, we also study the well-known Meir-Harris model for SAWs on percolation clusters. We demonstrate that the Meir-Harris model leads back up to 2-loop order to the renormalizable real-world formulation with kinetic averaging if the replica limit is consistently performed at the first possible instant in the course of the calculation.

Star copolymers in porous environments: scaling and its manifestations

We consider star polymers, consisting of two different polymer species, in a solvent subject to quenched correlated structural obstacles. We assume that the disorder is correlated with a power-law decay of the pair-correlation function g(x)~x(-a). Applying the field-theoretical renormalization group approach in d dimensions, we analyze different scenarios of scaling behavior working to first order of a double ɛ=4-d, δ=4-a expansion. We discuss the influence of the correlated disorder on the resulting scaling laws and possible manifestations such as diffusion-controlled reactions in the vicinity of absorbing traps placed on polymers as well as the effective short-distance interaction between star copolymers.

Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold

Continuing the program begun by the authors in a previous paper, we develop an exact low-density expansion for the random minimum spanning tree (MST) on a finite graph and use it to develop a continuum perturbation expansion for the MST on critical percolation clusters in space dimension d . The perturbation expansion is proved to be renormalizable in d=6 dimensions. We consider the fractal dimension D(p) of paths on the latter MST; our previous results lead us to predict that D(p)=2 for d>d(c)=6 . Using a renormalization-group approach, we confirm the result for d>6 and calculate D(p) to first order in epsilon=6-d for d<6 using the connection with critical percolation, with the result D(p)=2-epsilon/7+O(epsilon(2)) .

Distribution functions in percolation problems

Percolation clusters are random fractals whose geometrical and transport properties can be characterized with the help of probability distribution functions. Using renormalized field theory, we determine the asymptotic form of various such distribution functions in the limits where certain scaling variables become small or large. Our study includes the pair-connection probability, the distributions of the fractal masses of the backbone, the red bonds, and the shortest, the longest, and the average self-avoiding walk between any two points on a cluster, as well as the distribution of the total resistance in the random resistor network. Our analysis draws solely on general, structural features of the underlying diagrammatic perturbation theory, and hence our main results are valid to arbitrary loop order.

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].

Scaling behavior of linear polymers in disordered media

It has long been known that the universal scaling properties of linear polymers in disordered media are well described by the statistics of self-avoiding walks (SAWs) on percolation clusters and their critical exponent nu(SAW), with the SAW implicitly referring to the average SAW. Hitherto, static averaging has been commonly used, e.g., in numerical simulations, to determine what the average SAW is. We assert that only kinetic, rather than static, averaging can lead to asymptotic scaling behavior and corroborate our assertion by heuristic arguments and a renormalizable field theory. Moreover, we calculate to two-loop order nu(SAW), the exponent nu(max) for the longest SAW, and a family of multifractal exponents nu(alpha).

Entropy-induced separation of star polymers in porous media

We present a quantitative picture of the separation of star polymers in a solution where part of the volume is influenced by a porous medium. To this end, we study the impact of long-range-correlated quenched disorder on the entropy and scaling properties of f-arm star polymers in a good solvent. We assume that the disorder is correlated on the polymer length scale with a power-law decay of the pair correlation function g(r) approximately r-a. Applying the field-theoretical renormalization group approach we show in a double expansion in epsilon=4-d and delta=4-a that there is a range of correlation strengths delta for which the disorder changes the scaling behavior of star polymers. In a second approach we calculate for fixed space dimension d=3 and different values of the correlation parameter a the corresponding scaling exponents gammaf that govern entropic effects. We find that gammaf-1, the deviation of gammaf from its mean field value is amplified by the disorder once we increase delta beyond a threshold. The consequences for a solution of diluted chain and star polymers of equal molecular weight inside a porous medium are that star polymers exert a higher osmotic pressure than chain polymers and in general higher branched star polymers are expelled more strongly from the correlated porous medium. Surprisingly, polymer chains will prefer a stronger correlated medium to a less or uncorrelated medium of the same density while the opposite is the case for star polymers.

Where two fractals meet: the scaling of a self-avoiding walk on a percolation cluster

The scaling properties of self-avoiding walks on a d -dimensional diluted lattice at the percolation threshold are analyzed by a field-theoretical renormalization group approach. To this end we reconsider the model of Phys. Rev. Lett. 63, 2819 (1989)] and argue that via renormalization its multifractal properties are directly accessible. While the former first order perturbation did not agree with the results of other methods our analytic result gives an accurate description of the available MC and exact enumeration data in a wide range of dimensions 2</=d</=6 .

Polymers in long-range-correlated disorder

We study the scaling properties of polymers in a d-dimensional medium with quenched defects that have power law correlations approximately r(-a) for large separations r. This type of disorder is known to be relevant for magnetic phase transitions. We find strong evidence that this is true also for the polymer case. Applying the field-theoretical renormalization group approach we perform calculations both in a double expansion in epsilon=4-d and delta=4-a up to the one-loop order and second in a fixed dimension (d=3) approach up to the two-loop approximation for different fixed values of the correlation parameter, 2<or=a<or=3. In the latter case the numerical results need appropriate resummation. We find that the asymptotic behavior of self-avoiding walks in three dimensions and long-range-correlated disorder is governed by a set of separate exponents. In particular, we give estimates for the nu and gamma exponents as well as for the correction-to-scaling exponent omega. The latter exponent is also calculated for the general m-vector model with m=1,2,3.

The Collapse of Free Polymer Chains in a Network

The conformation of linear polymer chains trapped in a matrix of cross-linked polymer has been measured by neutron scattering. Three regimes were found depending on the length of the linear chain, Nl, with respect to the mesh size of the network, N(c). When N(c) > Nl, the radius of gyration, R(g), of the linear chain is the same as that observed in the uncrosslinked melt. When N(c) < Nl, R(g) shrinks according to the scaling relation R(g)(-1) approximately N(c)(-1) that has been predicted for isolated polymer chains trapped in a field of random obstacles. When N(c) << Nl the linear chains are observed to segregate.