S. Elezović-Hadžić

Critical exponents of surface-interacting self-avoiding walks on a family of truncated n-simplex lattices

We study the critical behavior of surface-interacting self-avoiding random walks on a class of truncated simplex lattices, which can be labeled by an integer n ⩾ 3. Using the exact renormalization group method we have been able to obtain the exact values of various critical exponents for all values of n up to n = 6. We also derived simple formulas which describe the asymptotic behavior of these exponents in the limit of large n (n → ∞). In spite of the fact that the coordination number of the lattice tends to infinity in this limit, we found that most of the studied critical exponents approach certain finite values, which differ from corresponding values for simple random walks (without self-avoiding walk constraint).

Statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three-dimensional space

We study the adsorption problem of linear polymers, immersed in a good solvent, when the container of the polymer–solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals, embedded in the three-dimensional Euclidean space. Members of the SG family are enumerated by an integer b (2≤b<∞), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the surface critical exponents γ11,γ1, and γs which, within the self-avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with both, one, and no ends grafted to the adsorbing surface (adsorbing boundary), respectively. By applying the exact renormalization group method, for 2≤b≤4, we have obtained specific values for these exponents, for various types of polymer conformations. To extend the obtained sequences of exact values for surface critical exponents, we have applied the Monte Carlo renormalization group method for fractals with 2≤b≤40. The obtained results show that all studied exponents are monotonically increasing functions of the parameter b, for all possible polymer states. We discuss mutual relations between the studied critical exponents, and compare their values with those found for other types of lattices, in order to attain a unified picture of the attacked problem.

Pulling self-interacting linear polymers on a family of fractal lattices embedded in three-dimensional space

We have studied the problem of force pulling self-interacting linear polymers situated in fractal containers that belong to the Sierpinski gasket (SG) family of fractals embedded in three-dimensional (3D) space. Each member of this family is labeled with an integer b (2 ≤ b ≤ ∞). The polymer chain is modeled by a self-avoiding walk (SAW) with one end anchored to one of the four boundary walls of the lattice, while the other (floating in the bulk of the fractal) is the position at which the force is acting. By applying an exact renormalization group (RG) method we have established the phase diagrams, including the critical force–temperature dependence, for fractals with b = 2,3 and 4. Also, for the same fractals, in all polymer phases, we examined the generating function G1 for the numbers of all possible SAWs with one end anchored to the boundary wall. We found that besides the usual power-law singularity of G1, governed by the critical exponent γ1, whose specific values are worked out for all cases studied, in some regimes the function G1 displays an essential singularity in its behavior.

Statistics of semiflexible self-avoiding trails on a family of two-dimensional compact fractals

We have applied the exact and Monte Carlo renormalization group (MCRG) method to study the statistics of semiflexible self-avoiding trails (SATs) on the family of plane-filling (PF) fractals. Each fractal of the family is compact, that is, the fractal dimension df is equal to 2 for all members of the PF family, which are enumerated by an odd integer b, . Varying values of the stiffness parameter s of trails from 1 to 0 (so that when s decreases the trail stiffness increases) we calculate exactly (for 3 ≤ b ≤ 7) and through the MCRG approach (for b ≤ 201) the sets of the critical exponents ν (associated with the mean squared end-to-end distances of SATs) and γ (associated with the total number of different SATs). Our results show that critical exponents are stiffness dependent functions, so that ν(s) is a monotonically decreasing function of s, for each studied b, whereas γ(s) displays a non-monotonic behavior for some values of b. On the other hand, by fixing the stiffness parameter s, our results show clearly that for highly flexible trails (with s = 1 and 0.9) ν is a non-monotonic function of b, while for stiffer SATs (with s ≤ 0.7) ν monotonically decreases with b. We also show that γ(b) increases with increasing b, independently of s. Finally, we compare the obtained SAT data with those obtained for the semiflexible self-avoiding walk (SAW) model on the same fractal family, and for both models we discuss behavior of the studied exponents in the fractal-to-Euclidean crossover region .

A model of compact polymers on a family of three-dimensional fractal lattices

We study Hamiltonian walks (HWs) on the family of three-dimensional modified Sierpinski gasket fractals, as a model for compact polymers in nonhomogeneous media in three dimensions. Each member of this fractal family is labeled with an integer b ≥ 2. We apply an exact recursive method which allows for explicit enumeration of extremely long Hamiltonian walks of different types: closed and open, with end-points anywhere in the lattice, or with one or both ends fixed at the corner sites, as well as some Hamiltonian conformations consisting of two or three strands. Analyzing large sets of data obtained for b = 2, 3 and 4, we find that numbers ZN of Hamiltonian walks, on fractal lattice with N sites, for behave as ZN ~ ωNμNσ. The leading term ωN is characterized by the value of the connectivity constant ω > 1, which depends on b, but not on the type of HW. In contrast to that, the stretched exponential term μNσ depends on the type of HW through the constant μ 2. This differs from the formulae obtained recently for Hamiltonian walks on other fractal lattices, as well as from the formula expected for homogeneous lattices. We discuss the possible origins and implications of such a result.

Hamiltonian walks on Sierpinski and n-simplex fractals

We study Hamiltonian walks (HWs) on Sierpinski and n-simplex fractals. Via numerical analysis of exact recursion relations for the number of HWs we calculate the connectivity constant ω and find the asymptotic behaviour of the number of HWs. Depending on whether or not the polymer collapse transition is possible on a studied lattice, different scaling relations for the number of HWs are obtained. These relations are, in general, different from the well-known form characteristic of homogeneous lattices which has thus far been assumed to also hold for fractal lattices.

Critical behavior of interacting two-polymer system in a fractal solvent: an exact renormalization group approach

We study the polymer system consisting of two polymer chains situated in a fractal container that belongs to the three-dimensional Sierpinski gasket (3D SG) family of fractals. Each 3D SG fractal has four fractal impenetrable 2D surfaces, which are, in fact, 2D SG fractals. The two-polymer system is modeled by two interacting self-avoiding walks (SAW), one of them representing a 3D floating polymer, while the other corresponds to a chain confined to one of the four 2D SG boundaries (with no monomer in the bulk). We assume that the studied system is immersed in a poor solvent inducing the intra-chain interactions. For the inter-chain interactions we propose two models: in the first model (ASAW) the SAW chains are mutually avoiding, whereas in the second model (CSAW) chains can cross each other. By applying an exact renormalization group (RG) method, we establish the relevant phase diagrams for b = 2,3 and 4 members of the 3D SG fractal family for the model with avoiding SAWs, and for b = 2 and 3 fractals for the model with crossing SAWs. Also, at the appropriate transition fixed points we calculate the contact critical exponents, associated with the number of contacts between monomers of different chains. Throughout this paper we compare results obtained for the two models and discuss the impact of the topology of the underlying lattices on emerging phase diagrams.

Compact Polymers on Fractal Lattices

We study compact polymers, modelled by Hamiltonian walks (HWs), i.e. self‐avoiding walks that visit every site of the lattice, on various fractal lattices: Sierpinski gasket (SG), Given‐Mandelbrot family of fractals, modified SG fractals, and n‐simplex fractals. Self‐similarity of these lattices enables establishing exact recursion relations for the numbers of HWs conveniently divided into several classes. Via analytical and numerical analysis of these relations, we find the asymptotic behaviour of the number of HWs and calculate connectivity constants, as well as critical exponents corresponding to the overall number of open and closed HWs. The nonuniversality of the HW critical exponents, obtained for some homogeneous lattices is confirmed by our results, whereas the scaling relations for the number of HWs, obtained here, are in general different from the relations expected for homogeneous lattices.