On the number of intersections of self-repelling polymer chains

We give a detailed analysis of the intersection properties of polymers. Using the renormalization group we provide a full crossover function for the dependence of the number of intersections in a single polymer on chain length and excluded volume strength. We compare our results with Monte-Carlo data and with exact calculations for a random walk, finding good agreement in all respects. Restricting to the vicinity of the eight ternary fixed points we also calculate the number of intersections between two chains placed at a fixed distance, including the two halves of a block-copolymer. The analysis of these systems confirms the interpretation of the different contributions to the number of intersections in a single chain. Due to the highly nontrivial character of the correlations in a polymer chain the correction exponents in both cases however are different. None of the results can be extracted from any Flory-type estimate. Received: 1 April 1997 / Revised: 24 October 1997 / Accepted: 29 January 1998     

Statistics of the ensemble of primary structures for inhomogeneous polymer chains

We consider linear polymer chains of fixed lengthN made up randomly of two different kinds of monomers X and Y near the tricritical theta point. Two kinds of short-range repulsive volume-excluded interactions, X-X and Y-Y are considered. In addition, short-range attractive two-body X-Y interactions and threebody interactions are included in the theoretical treatment. We derive the distribution of the average size of the random coil over the ensemble of primary structures or X-Y sequences. The variance of thermal or configurational averages for the different sequences is calculated making use of field-theoretic methods.     

Adsorption of polymer chains at two impenetrable interfaces

We study the problem of adsorption of polymer chains in a system containing two impenetrable attracting interfaces within the mean-field approximation. We find the exact solution of this mean-field polymer adsorption problem that is controlled by a single scaling variable describing the coupling between the impenetrable interfaces due to the polymer chains. At the saturation point we obtain the total number of adsorbed monomers, the total energy of the system and the force acting between impenetrable interfaces that is turned to be strictly attractive and monotonously decaying towards zero for increasing distance between the interfaces.     

On Edwards' model for polymer chains

We obtain an existence and uniqueness theorem for the self-consistent potential in Edwards' model for polymer chains, and confirm the asymptotic analysis proposed by him on the basis of WKB arguments.     

Surface structures of equilibrium restricted curvature model on two fractal substrates

With the aim to probe the effects of the microscopic details of fractal substrates on the scaling of discrete growth models, the surface structures of the equilibrium restricted curvature(ERC) model on Sierpinski arrowhead and crab substrates are analyzed by means of Monte Carlo simulations. These two fractal substrates have the same fractal dimension df, but possess different dynamic exponents of random walk zrw. The results show that the surface structure of the ERC model on fractal substrates are related to not only the fractal dimension df, but also to the microscopic structures of the substrates expressed by the dynamic exponent of random walk zrw. The ERC model growing on the two substrates follows the well-known Family–Vicsek scaling law and satisfies the scaling relations 2α + df≈ z ≈ 2zrw. In addition, the values of the scaling exponents are in good agreement with the analytical prediction of the fractional Mullins–Herring equation.     

Critical exponents for numbers of differently anchored polymer chains on fractal lattices with adsorbing boundaries

We study the problem of polymer adsorption 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. Members of the SG family are enumerated by an integerb (2b), and it is assumed that one side of each SG fractal is an impenetrable adsorbing boundary. We calculate the critical exponents ₁, ₁₁, and _s , which, within the self-avoiding walk model (SAW) of the polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends anchored to the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method for 2b8 and the Monte Carlo renormalization group (MCRG) method for a sequence of fractals with 2b80, we obtain specific values for these exponents. The obtained results show that all three critical exponents ₁, ₁₁, and _s , in both the bulk phase and crossover region are monotonically increasing functions withb. We discuss their mutual relations, their relations with other critical exponents pertinent to SAWs on the SG fractals, and their possible asymptotic behavior in the limitb, when the fractal dimension of the SG fractals approaches the Euclidean value 2.     

半刚性高分子链螺旋结构诱导纳米棒的有序结构

在自然界中, 螺旋结构广泛存在. 在熵的驱动下, 高分子链能在某些特殊情形下形成螺旋结构. 采用分子动力学方法研究了高分子链诱导纳米棒的自组装行为, 发现纳米棒/高分子链体系的构象与纳米棒的数量、高分子链的刚性等密切相关. 当纳米棒与高分子链之间存在适度吸附能时, 纳米棒能够形成三种完全不同的构象, 特别是在半刚性高分子链诱导下纳米棒能够形成线型排列. 研究结果对新型材料制备具有一定指导意义.     

On Edwards' model for long polymer chains

An existence theorem is proved for a probability measure on continuous paths in space, proposed by Edwards as a stochastic model for the geometric properties of long polymer chains.     

Random walk statistics on fractal structures

We consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds: range, renewal theory, mean first passage time, etc. Asymptotic behaviors are shown to be controlled by the fractal (¯d) and spectral (¯d) dimensionalities of the considered structure. A simple decimation procedure giving the value of (¯d) is outlined and illustrated in the case of the Sierpinski gaskets. Recent results for the trapping problem, the self-avoiding walk, and the true-self-avoiding walk are briefly reviewed. New numerical results for diffusion on percolation clusters are also presented.     

On the theory of liquid-crystalline ordering of polymer chains with limited flexibility

We propose a universal approximate method which enables us to study the statistical mechanics of the liquid-crystalline ordering in the solutions of stiffchain macromolecules with limited flexibility of arbitrary nature at arbitrary solution concentrations and arbitrary temperatures (solvent quality). In the variables temperature and concentration we construct the full phase diagrams for the nematic transition for the cases of long rigid rods and semiflexible macromolecules with freely jointed, persistent, and rotational-isomeric flexibility mechanisms. The method proposed can be easily generalized for the study of thermotropic polymer liquid crystals, as well as more complex polymer chain models (e.g., copolymers containing stiff and flexible fragments in the chain).     

Radiation of some fractal structures

The fractal properties of a candle flame and fractal discharges are considered. The fractal concept is applied to ball lightning.     

Monte Carlo renormalization of hard sphere polymer chains in two to five dimensions

A renormalization group for polymer chains with hard-core interaction is considered, where a chain ofN ₀ links of lengthl ₀ and hard-core diameterh ₀ is mapped onto a chain ofN ₁=N ₀/s links of lengthl ₁ and hard-core diameterh ₁. The lengthl ₁ is defined in terms of suitable interior distances of the original chain, andh ₁ is found from the condition that the end-to-end distance is left invariant. This renormalization group procedure is carried through by various Monte-Carlo methods (simple sampling is found advantageous for short enough chains or high dimensionalities, while dynamic methods involving kinkjumps or reptation are used else). Particular attention is paid to investigate systematic errors of the method by checking the dependence of the results on bothN ₀ ands. It is found that for dimensionalitiesd=2, 3 only the nontrivial fixed-point is stable, where upon iteration the ratio _k =h _k /l _k tends to nonzero fixed-point value ^*, while ford=4,5 the method converges to the gaussian fixed point with ^*=0. Taking both statistical and systematic errors into account, we estimate the exponentv asv=0.74±0.01 (d=2) andv =0.59±0.01 (d=3). The results are consistent with the expected crossover exponents =1/2 (d=3) and =1 (d=2), respectively.     

Effects of interchain coupling on photoexcitation in two coupled polymer chains in the presence of an electric field

Within an extended Su-Schrieffer-Heeger model including interchain interactions and the extended Hubbard model (EHM), the dynamical relaxation of photoexcitations in the presence of an external electric field is investigated using a nonadiabatic evolution method. Under the action of the interchain interactions both intrachain excitons and interchain excitons are generated after photoexcitation in two coupled polymer chains. Our results show that the field required to dissociate the excitons depends sensitively on the strength of the interchain coupling. As the interchain coupling strength increases, the dissociation field decreases. By analyzing the relation between the yield of intrachain and interchain excitons and the interchain coupling strength, we explain the dependence between the dissociation field and the strength of the interchain interactions. The theoretical results are expected to provide useful predictions concerning which polymers with properly strong interchain interactions are likely to be most suitable for use in organic solar cells.     

On the critical properties of the Edwards and the self-avoiding walk model of polymer chains

We study the two- and three-dimensional, superrenormalizable Edwards model and the self-avoiding walk model of polymers. Using a Schwinger-Dyson equation and upper and lower bounds on correlations in terms of “skeleton diagrams” [6] we establish the existence of a non-trivial continuum limit in the two- and three-dimensional, superrenormalizable Edwards model. We also prove that perturbation theory is asymptotic for the continuum correlations of these models.A fairly detailed analysis of the approach to the critical point in the self-avoiding walk model is presented. In particular, we show that η<1. In dimension d?4, we discuss rigorous consequences of the conjecture that η is non-negative: among other implications, we derive that the continuum limit is trivial and that γ=1, in d?5 dimensions, and that corrections to mean-field scaling laws are at most logarithmic in four dimensions.     

On Edwards' model for polymer chains. III. Borel summability

The basic existence theory for Edwards' model of long polymer chains is completed.     

Negative Poisson coefficient of fractal structures

On the basis of a fractal model the macroscopic elastic properties of an inhomogeneous medium with random structure have been determined. It is shown that if the ratio of the bulk moduli of the phases K ₂/K ₁→0, then the percolation threshold p _c the Poisson coefficient is equal to 0.2. A study of the behavior of a two-phase medium with negative Poisson coefficient is carried out. Fiz. Tverd. Tela (St. Petersburg) 41, 2147–2153 (December 1999)     

Theta-point exponent for polymer chains on percolation fractals

We derive a new expression for the Flory exponent describing the average radius of gyration of polymer chains at the theta point. For this we make use of the appropriate distribution function for the radius of gyration. We start from Euclidean lattices and extend the results to percolation fractals, by taking into account the basic geometry and the topology of such structures. We show that such basic features have a very prominent effect on the Flory exponent of the chain polymer on fractals at the theta point.     

Interaction of electromagnetic waves with fractal structures

A review and analysis is presented of the state of theoretical and experimental investigations of the problem of the interaction of electromagnetic waves with different inhomogeneous media possessing fractal properties. Such problems are classified and the approaches to solving them are discussed, including those implemented by the present authors.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 76–87, October, 1993.     

Simulations of single polymer chains in the dense limit

We present simulation results for single α-thermal chain polymers in finite volumes. For this we use a recently proposed recursive implementation of the enrichment method. In 3 dimensions it allows the simulation of extremely long chains (up to N = 300000). It is much less efficient for d = 2, but we can also there extend considerably the previously accessible range of chain lengths and densities. We verify most tested scaling laws, and we point out similarities with complex optimization problems.