Model Of Branched Polymers Research Articles

Improving convergence of generalised Rosenbluth sampling for branched polymer models by uniform sampling

Sampling with the generalised atmospheric Rosenbluth method (GARM) is a technique for estimating the distributions of lattice polymer models that has had some success in the study of linear polymers and lattice polygons. In this paper we will explain how and why such sampling appears not to be effective for many models of branched polymers. Analysing the algorithm on a simple binary tree, we argue that the fundamental issue is an inherent bias towards extreme configurations that is costly to correct with reweighting techniques. We provide a solution to this by applying uniform sampling methods to the atmospheres that are central to GARM. We caution that the ensuing computational complexity often outweighs the improvements gained.

Micellar structure of hydrophobically modified polysaccharides in aqueous solution

This article reviews previous studies on micellar structures formed by hydrophobically modified polysaccharides in aqueous solutions by static and dynamic light scattering, small angle X-ray and neutron scattering, and fluorescence from pyrene solubilized in polymer solution. The experimental results are consistently explained by the full or loose flower necklace model for pullulan bearing octenyl groups and amylose bearing dodecyl groups and by the randomly branched polymer model, which is often called a “nanogel”, for pullulan bearing cholesteryl groups. We discuss the micellar structures of hydrophobically modified polysaccharides as well as of an amphiphilic alternating vinyl polymer bearing dodecyl groups in regard to the degree of substitution as well as the chemical structures of the hydrophobic moiety and backbone chain.

Dimensional reduction by conformal bootstrap

Abstract Dimensional reductions in the branched polymer model and the random field Ising model (RFIM) are discussed by a conformal bootstrap method. Small minors are applied for the evaluations of the scale dimensions of these two models and the results are compared to the $D'=D-2$D Yang–Lee edge singularity and to the pure $D'=D-2$D Ising model, respectively. For the former case, the dimensional reduction is shown to be valid for $3 \le D \le 8$ and, for the latter case, the deviation from the dimensional reduction can be seen below five dimensions.

Sizes of Long RNA Molecules Are Determined by the Branching Patterns of Their Secondary Structures

Long RNA molecules are at the core of gene regulation across all kingdoms of life, while also serving as genomes in RNA viruses. Few studies have addressed the basic physical properties of long single-stranded RNAs. Long RNAs with nonrepeating sequences usually adopt highly ramified secondary structures and are better described as branched polymers. To test whether a branched polymer model can estimate the overall sizes of large RNAs, we employed fluorescence correlation spectroscopy to examine the hydrodynamic radii of a broad spectrum of biologically important RNAs, ranging from viral genomes to long noncoding regulatory RNAs. The relative sizes of long RNAs measured at low ionic strength correspond well to those predicted by two theoretical approaches that treat the effective branching associated with secondary structure formation—one employing the Kramers theorem for calculating radii of gyration, and the other featuring the metric of maximum ladder distance. Upon addition of multivalent cations, most RNAs are found to be compacted as compared with their original, low ionic-strength sizes. These results suggest that sizes of long RNA molecules are determined by the branching pattern of their secondary structures. We also experimentally validate the proposed computational approaches for estimating hydrodynamic radii of single-stranded RNAs, which use generic RNA structure prediction tools and thus can be universally applied to a wide range of long RNAs.

Branched polymer models and the mechanism of multilayer film buildup

The “in and out diffusion” hypothesis does not provide a conclusive explanation of the buildup displayed by some polyelectrolyte multilayer film systems. Here, we report initial tests of an alternative hypothesis, on which the completion of each adsorption cycle results in an increase in the number of polymer binding sites on the film surface. Polycationic dendrimeric peptides, which can potentially bind several oppositely-charged peptides each, have been designed, synthesized and utilized in comparative film buildup experiments. Material deposited, internal film structure and film surface morphology have been studied by ultraviolet spectroscopy (UVS), circular dichroism spectroscopy (CD), quartz crystal microbalance (QCM) and atomic force microscopy (AFM). Polycations tended to contribute more to film buildup than did polyanions on quartz but not on gold. Increasing the number of branches in the dendrimeric peptides from 4 to 8 reproducibly resulted in an increase in the film growth rate on quartz but not on gold. Peptide backbones tended to adopt a β-strand conformation on incorporation into a film. Thicker films had a greater surface roughness than thin films. The data are consistent with film buildup models in which the average number of polymer binding sites will increase with each successive adsorption cycle in the range where exponential growth is displayed.

Chain dimension of polystyrene-like hyperbranched polymers with various chain end groups in THF

A hyperbranched polymer (HPS-DC) synthesized using N,N-diethylamino -dithiocarbamoylstyrene as an inimer and a hyperbranched poly(4-methylstyrene) (HPS-H) prepared by hydrogenation of the HPS-DC are characterized in tetrahydrofuran (THF) solutions by static and dynamic light scattering measurements. The experimental values of z-averaged mean-square radius of gyration (<S2>z1/2) and hydrodynamic radius (RH) of the HPS-H have smaller dimensions than those of the linear polystyrene (PS) at the same degree of polymerization (Nw), whereas the RH of the HPS-DC in the region of Nw < 700 is larger than that of the linear PS at the same Nw. The Nw-dependence of <S2>z1/2 and RH for the HPS-H are described by a randomly trifunctional branched polymer model with the molecular weight per a branch point (4.55 × 103 g/mol) and the chain dimensions of the linear PS in THF at 25 °C.

Structural analysis of aggregates formed by a thermoresponsive homopolymer in dilute aqueous solutions

AbstractThe aqueous solution of a thermoresponsive polymer, poly[2‐(2‐ethoxy) ethoxyethyl vinyl ether] poly(EOEOVE), contains a tiny amount of large polymer aggregates at low polymer concentrations far below the lower critical solution temperature (∼40 °C). The molar mass Mw,slow, radius of gyration 〈S2〉, and hydrodynamic radius RH,slow of the aggregating component of poly(EOEOVE) were obtained by simultaneous static and dynamic light scattering as functions of the polymer concentration and temperature, while the weight fraction wslow of the component was estimated by size‐exclusion chromatography. The Mw,slow dependencies of 〈S2〉 and RH,slow, as well as the ratio 〈S2〉/RH,slow, indicated that the poly(EOEOVE) aggregate takes a sparsely branched polymer‐like conformation. We have analyzed the structure of the aggregate, using the branched polymer model of random type. The Mw,slow dependence of 〈S2〉 obtained was favorably compared with this model with reasonable structural parameters. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1179–1187, 2006

Dimensional Reduction and Crossover to Mean-Field Behavior for Branched Polymers

This article will review recent results on dimensional reduction for branched polymers, and discuss implications for critical phenomena. Parisi and Sourlas argued in 1981 that branched polymers fall into the universality class of the Yang-Lee edge in two fewer dimensions. Brydges and I have proven in [math-ph/0107005] that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions (which is in the Yang-Lee or $i \phi^3$ class). I will describe how this equivalence arises from an underlying supersymmetry of the branched polymer model. - I will also use dimensional reduction to analyze the crossover of two-dimensional branched polymers to their mean-field limit, and to show that the scaling is given by an Airy function (the same as in [cond-mat/0107223]).

The Hausdorff dimension in polymerized quantum gravity

We calculate the Hausdorff dimension, d H , and the correlation function exponent, η, for polymerized two dimensional quantum gravity models. If the non-polymerized model has correlation function exponent η 0>3 then d H = γ −1 where γ is the susceptibility exponent. This suggests that these models may be in the same universality class as certain non-generic branched polymer models.

The spectral dimension of non-generic branched polymer ensembles

We show that the spectral dimension on non-generic branched polymer models with susceptibility exponent γ>0 is given by d s =2/(1+ γ). For those models with γ<0 we find that d s =2.

Correlations in fluctuating geometries

We compare two definitions of connected correlation functions in fluctuating geometries. We show results of the MC simulations for 4D dynamical triangulation in the elongated phase and compare them with the exact calculations of correlation functions in the branched polymer model.

Long range correlations in branched polymers

We study the correlation functions in the branched polymer model. Although there are no correlations in the grand canonical ensemble, when looking at the canonical ensemble we find negative long range power like correlations. We propose that a similar mechanism explains the shape of recently measured correlation functions in the elongated phase of 4d simplicial gravity [B. de Bakker and J. Smit, Nucl. Phys. B 454 (1995) 343].

Branched polymers on Bethe and Husimi lattices.

We study a branched polymer model defined on a lattice with coordination number q. We introduce q different site fugacities, corresponding to all possible incoming polymer bonds on a site. The model is solved exactly on the treelike Bethe and Husimi lattices. In particular, the interesting case of two nonzero fugacities is treated. The model presents qualitatively different phase diagrams when the coordination number of the lattice is changed. The system has in some cases one polymerized phase, but in other cases a second polymerized phase appears. The conditions for the existence of this second poly- merized phase are established.

Matter fields with c > 1 coupled to 2d gravity

We solve a class of branched polymer models coupled to spin systems and show that they have no phase transition and are either always magnetized or never magnetized depending on the branching weights. By comparing these results with numerical simulations of two-dimensional quantum gravity coupled to matter fields with central charge c we provide evidence that for c sufficiently large ( c ⩾ 12) these models are effectively described by branched polymers. Moreover, the numerical results indicate a remarkable universality in the influence on the geometry of surfaces due to the interaction with matter. For spin systems this influence only depends on the total central charge.

Competition between self-attraction and adsorption in branched polymers situated on a fractal lattice

The authors have studied the branched polymer model, situated on the Sierpinski gasket lattice, with competing monomer-monomer and monomer-surface interactions. They have determined critical properties of this model within an exact renormalization group approach. Their results reveal that the corresponding model system possesses a rich phase diagram that is typical for polymer surface physics.

DOUBLE SCALING LIMIT IN O(N) VECTOR MODELS IN D DIMENSIONS

Using the standard 1/N expansion, we study O(N) vector models in D dimensions with an arbitrary potential. We limit ourselves to renormalizable theories. We show that there exists a value of the coupling constant corresponding to a critical point and that a double scaling limit can be performed as in D=0 and in the case of matrix models in D=0, 1. For D=1 the theory is renormalizable with an arbitrary potential and we find in general a hierarchy of critical theories labeled by an integer k. The universal partition function obtained in the double scaling limit is constructed. Finally, we show that the critical behaviour of those models is the same as a branched polymer model recently constructed by Ambjørn, Durhuus and Jónsson.

Exactly soluble addition and condensation models in coagulation kinetics

The kinetics of clustering through addition ( a k + a 1 → a k+1 ) and condensation ( a j + a k → a j+ k , j ≠ 1, k ≠ 1) for a model of cylindrically shaped monomeric units a 1 are studied, using Smoluchowski's coagulation equation, and analytic solutions for several limiting cases (flat disks and needles) with and without condensation reactions, were given. The condensation models of flat disks and needles include, respectively, the linear polymer model RA 2 and the branched polymer model A 2RB ∞ (with a gelation transition). If condensation reactions are inhibited, we obtain exactly soluble addition models with a monomer-cluster rate constant independent of or proportional to the cluster size. The monomers are (i) supplied in a given amount at the initial time; (ii) generated by a steady source; or (iii) supplied by an infinite reservoir that keeps the concentration of monomers, c 1( t), constant in time.

Branched polymer models

The importance of excluded volume in determinig the sizes of branched polymers is discussed in terms of several models. First, the branched chain analog of the rotational isomeric model is solved and gives improvement over earlier random flight models but, as was the case for linear chains, the asymptotic dependence of polymer size on monomer number (〈size〉≂ANν) remains the same, except that model differences are found in the preexponential factor A. Next, a hierarchy of subclasses of the full class of generally branched self-avoiding random walks is briefly described and compared to our recent Monte Carlo results on the full problem. Finally, simple heuristic arguments for excluded volume corrections are considered.

Rigorous results for branched polymer models with excluded volume

It is pointed out that on taking excluded volume into account for branched polymers there are modifications to previous work, such as that of Cayley and Polya on isomer enumeration. Indeed, it is argued that the classical Cayley–Polya count overestimates the corrected count by an arbitrarily large factor. Further, some limited rigorous results concerning the asymptotic behavior of isomer counts and of counts for polymer conformations are established.

Dilute solution properties of proteoglycan fractions from bovine nasal cartilage.

AbstractFresh proteogycans (adult bovine nasal cartilage) isolated from the densest portion of a dissociative density gradient had a weight‐average molecular weight of ca. 106 in 4M guanidine hydrochloride (GdnHCI) by light scattering. Fractions of such material obtained by elution with 4M GdnHCI from 2% agarose gel, both normal and cross‐linkd, has proteoglycan subunit molecular weights ranging from 0.8 to 2.6 × 106 and root‐mean‐square radii ranging from 35 to 52 nm in the same solvent. The protein molecular weight per proteoglycan subunit was about 1.2 × 105 and that of keratan sulfate about 1.8 × 105, both independent of total molecular weight. A random‐flight “graft copolymer” model having uniform side chains of chondroitin sulfate (40 disaccharides) and keratan sulfate (15 disaccharides) and a random‐coil polypeptide back bone was used to estimate the unperturbed radius, whihc was about 19 nm for a mol wt of 1.5 × 106. Experimental light‐scattering data for fractions were fitted very well by theoretical curves for the particale scattering factor for both linear and appropriate branched polymers. Examination of coil expansion on the basis of perturbation calculations for branched polymer models suggested that expansion did not account for the experimentally observed radii in terms of unperturbed radii calculated from the model. A possible explanation is that substantial local stiffening of the polypeptide chain due to substitution of side‐chain clusters increases the unperturbed radii. The intrinsic viscosity [η] is 4M GdnHCI ranged from 120 to 180 ml/g, and could be interpreted in terms of th eequivalent sphere model; the Flory number has approximately its normal value for flexible linear polymers. The treatment of the sedimentation coefficient by this is less successful, since the Man delkern‐Flory parameter β apparently increases with increasing molecular weight; average value are similar to those for flexible polymers, but the variation in β makes this method useful only for rough estimation of molecular weight of proteoglycans. Molecular weights of purified proteoglycans are the same in 0.2M NaCI as in 4M GdnHCI, while crude preparations gave higher molecular weights in 0.2M NaCI, probably because of association due to incomplete removel of “linking” proteins.