Multifractality of Brownian motion near absorbing polymers

Shape analysis of random polymer networks

We propose the model of a random polymer network, formed on the base on Erdös-Rényi random graph. In the language of mathematical graphs, the chemical bonds between monomers can be treated as vertices, and their chemical functionalities as degrees of these vertices. We consider graphs with fixed number of verticesN= 5 and variable parameterc(connectedness), defining the total number of linksL=cN(N- 1)/2 between vertices. Each link in such graphs is treated as a Gaussian polymer chain. The universal rotationally invariant size and shape characteristics, such as averaged asphericity and size ratio of such structures are obtained both numerically by application of Wei's method and analytically within the continuous chain model. In particular, our results quantitatively indicate an increase of asymmetry of polymer network structure when its connectednesscdecreases.

Universal shape characteristics for the mesoscopic star-shaped polymer via dissipative particle dynamics simulations

In this paper we study the shape characteristics of star-like polymers in various solvent quality using a mesoscopic level of modeling. The dissipative particle dynamics simulations are performed for the homogeneous and four different heterogeneous star polymers with the same molecular weight. We analyse the gyration radius and asphericity at the poor, good and θ-solvent regimes. Detailed explanation based on interplay between enthalpic and entropic contributions to the free energy and analyses on of the asphericity of individual branches are provided to explain the increase of the apsphericity in θ-solvent regime.

Anisotropy of a cubic ferromagnet at criticality

Critical fluctuations change the effective anisotropy of cubic ferromagnet near the Curie point. If the crystal undergoes phase transition into orthorhombic phase and the initial anisotropy is not too strong, reduced anisotropy of nonlinear susceptibility acquires at T_{c} the universal value δ_{4}^{*}=2v^{*}/3(u^{*}+v^{*}) where u^{*} and v^{*} are coordinates of the cubic fixed point on the flow diagram of renormalization group equations. In the paper, the critical value of the reduced anisotropy is estimated within the pseudo-ε expansion approach. The six-loop pseudo-ε expansions for u^{*}, v^{*}, and δ_{4}^{*} are derived for the arbitrary spin dimensionality n. For cubic crystals (n=3) higher-order coefficients of the pseudo-ε expansions obtained turn out to be so small that use of simple Padé approximants yields reliable numerical results. Padé resummation of the pseudo-ε series for u^{*}, v^{*}, and δ_{4}^{*} leads to the estimate δ_{4}^{*}=0.079±0.006, indicating that detection of the anisotropic critical behavior of cubic ferromagnets in physical and computer experiments is certainly possible.

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.

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.

Unbinding of mutually avoiding random walks and two-dimensional quantum gravity

We analyze the unbinding transition for a two-dimensional lattice polymer in which the constituent strands are mutually avoiding random walks. At low temperatures the strands are bound and form a single self-avoiding walk. We show that unbinding in this model is a strong first order transition. The entropic exponents associated with denaturated loops and end-segment distributions show sharp differences at the transition point and in the high temperature phase. Their values can be deduced from some exact arguments relying on a conformal mapping of copolymer networks into a fluctuating geometry, i.e., in the presence of quantum gravity. An excellent agreement between analytical and numerical estimates is observed for all cases analyzed.

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 .

Two-dimensional copolymers and multifractality: comparing perturbative expansions, Monte Carlo simulations, and exact results

We analyze the scaling laws for a set of two different species of long flexible polymer chains joined together at one of their extremities (copolymer stars) in space dimension D=2. We use a formerly constructed field-theoretic description and compare our perturbative results for the scaling exponents with recent conjectures for exact conformal scaling dimensions derived by a conformal invariance technique in the context of D=2 quantum gravity. A simple Monte Carlo simulation brings about reasonable agreement with both approaches. We analyze the remarkable multifractal properties of the spectrum of scaling exponents.