On the number of contacts of a floating polymer chain cross-linked with a surface adsorbed chain on fractal structures

We study the interaction problem of a linear polymer chain, floating in fractal containers that belong to the three-dimensional Sierpinski gasket (3D SG) family of fractals, with a surface-adsorbed linear polymer chain. Each member of the 3D SG fractal family has a fractal impenetrable 2D adsorbing surface, which appears to be 2D SG fractal. The two-polymer system is modelled by two mutually crossing self-avoiding walks. By applying the Monte Carlo Renormalization Group (MCRG) method, we calculate the critical exponents $\phi$, associated with the number of contacts of the 3D SG floating polymer chain, and the 2D SG adsorbed polymer chain, for a sequence of SG fractals with $2\le b\le 40$. Besides, we propose the codimension additivity (CA) argument formula for $\phi$, and compare its predictions with our reliable set of the MCRG data. We find that $\phi$ monotonically decreases with increasing $b$, that is, with increase of the container fractal dimension. Finally, we discuss the relations between different contact exponents, and analyze their possible behaviour in the fractal-to-Euclidean crossover region $b\to\infty$.Comment: 15 pages, 3 figure

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 $\omega$ 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 hold for fractal lattices too.Comment: 22 pages, 6 figures; final versio

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>lt), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the surface critical exponents ;11;1, and ;be 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. © 2014 Elsevier B.V. All rights reserved

Statistics of close-packed dimers on fractal lattices

© 2020 Elsevier B.V. We study the model of close-packed dimers on planar lattices belonging to the family of modified rectangular (MR) fractals, whose members are enumerated by an integer p≥2, as well as on the non-planar 4-simplex fractal lattice. By applying an exact recurrence enumeration method, we determine the asymptotic forms for numbers of dimer coverings, and numerically calculate entropies per dimer in the thermodynamic limit, for a sequence of MR lattices with 2≤p≤8 and for 4-simplex fractal. We find that the entropy per dimer on MR fractals is increasing function of the scaling parameter p, and for every considered p it is smaller than the entropy per dimer of the same model on 4-simplex lattice. Obtained results are discussed and compared with the results obtained previously on some translationally invariant and fractal lattices

Semi-flexible compact polymers in two dimensional nonhomogeneous confinement

© 2019 IOP Publishing Ltd. We have studied the compact phase conformations of semi-flexible polymer chains confined in two dimensional nonhomogeneous media, modelled by fractals that belong to the family of modified rectangular (MR) lattices. Members of the MR family are enumerated by an integer p  and fractal dimension of each member of the family is equal to 2. The polymer flexibility is described by the stiffness parameter s, while the polymer conformations are modelled by weighted Hamiltonian walks (HWs). Applying an exact recurrence equations method, we have found that partition function Z N for closed HWs consisting of N steps scales as , where constants and depend on both p  and s. We have calculated numerically the stiffness dependence of the polymer persistence length, as well as various thermodynamic quantities (such as free and internal energy, specific heat and entropy) for a large set of members of the MR family. Analysis of these quantities has shown that semi-flexible compact polymers on MR lattices can exist only in the liquid-like (disordered) phase, whereas the crystal (ordered) phase has not appeared. Finally, behavior of the examined system at zero temperature has been discussed

Force-induced desorption of self-avoiding walks on Sierpinski gasket fractals

In this work we investigate force-induced desorption of linear polymers in good solvents in non-homogeneous environment, by applying the model of self-avoiding walk on two- and three-dimensional fractal lattices, obtained as generalization of the Sierpinski gasket fractal. For each of these lattices one of its boundaries represents an adsorbing wall, whereas along one of the fractal edges, not lying in the adsorbing wall, an external force acts on the self-avoiding walk. The hierarchical nature of the lattices under study enables an exact real-space renormalization group treatment, which yields the phase diagram of polymer critical behavior. We show that for this model there is no low-temperature reentrance in the cases of two-dimensional lattices, whereas in all studied three-dimensional cases the force-temperature dependance is reentrant. We also find that in all cases the force-induced desorption transition is of first order

On the number of contacts of two polymer chains situated on fractal structures

We study the critical behavior of the number of monomer-monomer contacts for two polymers in a good solvent. Polymers are modeled by two self-avoiding walks situated on fractals that belong to the checkerboard (CB) and X family. Each member of a family is labeled by an odd integer b, $3\le b\le\infty$ . By applying the exact Renormalization Group (RG) method, we establish the relevant phase diagrams whereby we calculate the contact critical exponents $\varphi$ (for the CB and X fractals with b=5 and b=7). The critical exponent $\varphi$ is associated with power law of the number of sites at which the two polymers are touching each other. Copyright Springer-Verlag Berlin/Heidelberg 2004