NONUNIVERSAL PROPERTIES OF SELF-INTERACTING POLYMER IN NON-HOMOGENEOUS ENVIRONMENT MODELED BY 3-SIMPLEX FRACTAL LATTICE

We have studied lattice self-avoiding polygons with attractive interaction between contacts which are nonconsecutively visited nearest neighboring sites. The lattice of choice is 3-simplex fractal lattice and the model represents a ring polymer in non-homogeneous solution whose quality depends on the interaction parameter. It has already been shown, by the renormalization group approach, that polymer on this lattice at any nonzero temperature can exist only in the extended phase. Universal critical exponents, which do not depend on the interaction strength, have also been determined. In this paper we are concerned with two nonuniversal quantities: the connectivity constant related with the free energy of the model and the mean number of contacts related with the internal energy. We have shown that the connectivity constant is an unboundedly increasing function of the interaction strength, while the mean number of contacts is an increasing function asymptotically approaching a limiting value equal to one half, which is the mean number of contacts in the case of Hamiltonian walks on the same lattice. This limiting value is expected, since in the limit of infinite interaction (or zero temperature) each self-avoiding walk on 3-simplex lattice becomes maximally compact and occupies all lattice points, i.e. becomes Hamiltonian walk.

Grid Based on the Sierpinski Fractal and an Assessment of the Prospects for its Application in Aircraft Parts

Modern geometric methods open up prospects for improving the shape and structure of parts. Such improvement can pursue the goals of increasing the strength with constant material consumption, or reducing the mass when it is not necessary to increase the strength. The meaning of geometric methods is to create a part shape the stresses arising in the part material under the action of applied loads are distributed most evenly. Such methods include the use of fractal geometry. This article presents the results of a study of a fractal lattice created on the basis of the Sierpinski triangle. Computer simulation in the SolidWorks, as well as strength studies of parts produced using additive technologies, allowed us to confirm a multiple increase in the strength of the fractal lattice with an increase in the number of fractal iterations. One of the most promising areas of application of fractal structures may be aviation technology. In this area, weight reduction is needful, and the complex shape of the parts is realized with the help of expensive production methods. For this reason, a number of experiments were conducted within the framework of the study, the purpose of which was to test the feasibility of using fractal gratings to reduce the weight of aircraft parts, using the example of the fork of the front landing gear of the combat training aircraft Yak-130.

ENUMERATION OF DIMER CONFIGURATIONS ON A FRACTAL LATTICE

In this paper, we present a solution to the close-packed dimer problem on a fractal lattice. The dimer model is canonical model in statistical physics related with many physical phenomena. Originally, it was introduced as a model for adsorption of diatomic molecules on surfaces. Here we assume that the two dimensional substrate on which the adsorption occurs is nonhomogeneous and we represent it by the modified rectangular (MR) fractal lattice. Self-similarity of the fractal lattice enables exact recursive enumeration of all close-packed dimer configurations at every stage of fractal construction. Asymptotic form for the overall number of dimer coverings is determined and entropy per dimer in the thermodynamic limit is obtained.

SEMI-FLEXIBLE COMPACT POLYMERS IN A DISORDERED ENVIRONMENT

Hamiltonian cycles with bending rigidity are studied on the first three members of the fractal family obtained by generalization of the modified rectangular (MR) fractal lattice. This model is proposed to describe conformational and thermodynamic properties of a single semi-flexible ring polymer confined in a poor and disordered (e.g. crowded) solvent. Due to the competition between temperature and polymer stiffness, there is a possibility for the phase transition between molten globule and crystal phase of a polymer to occur. The partition function of the model in the thermodynamic limit is obtained and analyzed as a function of polymer stiffness parameter s (Boltzmann weight), which for semi-flexible polymers can take on values over the interval (0,1). Other quantities, such as persistence length, specific heat and entropy, are obtained numerically and presented graphically as functions of stiffness parameter s.