Flory theory of randomly branched polymers

Water Sorption and Structural Properties of Human Airway Mucus in Health and Muco-Obstructive Diseases

Muco-obstructive diseases change airway mucus properties, impairing mucociliary transport and increasing the likelihood of infections. To investigate the sorption properties and nanostructures of mucus in health and disease, we investigated mucus samples from patients and cell cultures (cc) from healthy, chronic obstructive pulmonary disease (COPD), and cystic fibrosis (CF) airways. Atomic force microscopy (AFM) revealed mucin monomers with typical barbell structures, where the globule to spacer volume ratio was the highest for CF mucin. Accordingly, synchrotron small-angle X-ray scattering (SAXS) revealed more pronounced scattering from CF mucin globules and suggested shorter carbohydrate side chains in CF mucin and longer side chains in COPD mucin. Quartz crystal microbalance with dissipation (QCM-D) analysis presented water sorption isotherms of the three types of human airway mucus, where, at high relative humidity, COPD mucus had the highest water content compared to cc-CF and healthy airway mucus (HAM). The higher hydration of the COPD mucus is consistent with the observation of longer side chains of the COPD mucins. At low humidity, no dehydration-induced glass transition was observed in healthy and diseased mucus, suggesting mucus remained in a rubbery state. However, in dialyzed cc-HAM, a sorption-desorption hysteresis (typically observed in the glassy state) appeared, suggesting that small molecules present in mucus suppress the glass transition.

Topology-Based Detection and Tracking of Deadlocks Reveal Aging of Active Ring Melts

Connecting the viscoelastic behavior of stressed ring melts to the various forms of entanglement that can emerge in such systems is still an open challenge. Here, we consider active ring melts, where stress is generated internally, and introduce a topology-based method to detect and track consequential forms of ring entanglements, namely, deadlocks. We demonstrate that, as stress accumulates, more and more rings are co-opted in a growing web of deadlocks that entrap many other rings by threading, bringing the system to a standstill. The method ought to help the study of topological aging in more general polymer contexts.

DNA supercoiling in bacteria: state of play and challenges from a viewpoint of physics based modeling

DNA supercoiling is central to many fundamental processes of living organisms. Its average level along the chromosome and over time reflects the dynamic equilibrium of opposite activities of topoisomerases, which are required to relax mechanical stresses that are inevitably produced during DNA replication and gene transcription. Supercoiling affects all scales of the spatio-temporal organization of bacterial DNA, from the base pair to the large scale chromosome conformation. Highlighted in vitro and in vivo in the 1960s and 1970s, respectively, the first physical models were proposed concomitantly in order to predict the deformation properties of the double helix. About fifteen years later, polymer physics models demonstrated on larger scales the plectonemic nature and the tree-like organization of supercoiled DNA. Since then, many works have tried to establish a better understanding of the multiple structuring and physiological properties of bacterial DNA in thermodynamic equilibrium and far from equilibrium. The purpose of this essay is to address upcoming challenges by thoroughly exploring the relevance, predictive capacity, and limitations of current physical models, with a specific focus on structural properties beyond the scale of the double helix. We discuss more particularly the problem of DNA conformations, the interplay between DNA supercoiling with gene transcription and DNA replication, its role on nucleoid formation and, finally, the problem of scaling up models. Our primary objective is to foster increased collaboration between physicists and biologists. To achieve this, we have reduced the respective jargon to a minimum and we provide some explanatory background material for the two communities.

Entanglement Length Scale Separates Threading from Branching of Unknotted and Non-concatenated Ring Polymers in Melts

Current theories on the conformation and dynamics of unknotted and non-concatenated ring polymers in melt conditions describe each ring as a tree-like double-folded object. While evidence from simulations supports this picture on a single ring level, other works show pairs of rings also thread each other, a feature overlooked in the tree theories. Here we reconcile this dichotomy using Monte Carlo simulations of the ring melts with different bending rigidities. We find that rings are double-folded (more strongly for stiffer rings) on and above the entanglement length scale, while the threadings are localized on smaller scales. The different theories disagree on the details of the tree structure, i.e., the fractal dimension of the backbone of the tree. In the stiffer melts we find an indication of a self-avoiding scaling of the backbone, while more flexible chains do not exhibit such a regime. Moreover, the theories commonly neglect threadings and assign different importance to the impact of the progressive constraint release (tube dilation) on single ring relaxation due to the motion of other rings. Despite that each threading creates only a small opening in the double-folded structure, the threading loops can be numerous and their length can exceed substantially the entanglement scale. We link the threading constraints to the divergence of the relaxation time of a ring, if the tube dilation is hindered by pinning a fraction of other rings in space. Current theories do not predict such divergence and predict faster than measured diffusion of rings, pointing at the relevance of the threading constraints in unpinned systems as well. Revision of the theories with explicit threading constraints might elucidate the validity of the conjectured existence of topological glass.

Circular Polycatenanes: Supramolecular Structures with Topologically Tunable Properties

Polycatenanes, macrochains of topologically interlocked rings with unique physical properties have recently gained considerable interest in supramolecular chemistry, biology, and soft matter. Most of the work has been, so far, focused on linear chains and on their variety of conformational properties compared to standard polymers. Here we go beyond the linear case and show that, by circularizing such macrochains, one can exploit the topology of the local interlockings to store twist in the system, significantly altering its metric and local properties. Moreover, by properly defining the twist (Tw) and writhe (Wr) of these macrorings we show the validity of a relation equivalent to the Călugăreanu-White-Fuller theorem Tw+Wr=const, originally proved for ribbonlike structures such as double stranded DNA. Our results suggest that circular polycatenanes with storable and tunable twist can form a new category of highly designable multiscale structures with potential applications in supramolecular chemistry and material science.

Deconstruction and Reassembly of Renewable Polymers and Biocolloids into Next Generation Structured Materials

This review considers the most recent developments in supramolecular and supraparticle structures obtained from natural, renewable biopolymers as well as their disassembly and reassembly into engineered materials. We introduce the main interactions that control bottom-up synthesis and top-down design at different length scales, highlighting the promise of natural biopolymers and associated building blocks. The latter have become main actors in the recent surge of the scientific and patent literature related to the subject. Such developments make prominent use of multicomponent and hierarchical polymeric assemblies and structures that contain polysaccharides (cellulose, chitin, and others), polyphenols (lignins, tannins), and proteins (soy, whey, silk, and other proteins). We offer a comprehensive discussion about the interactions that exist in their native architectures (including multicomponent and composite forms), the chemical modification of polysaccharides and their deconstruction into high axial aspect nanofibers and nanorods. We reflect on the availability and suitability of the latter types of building blocks to enable superstructures and colloidal associations. As far as processing, we describe the most relevant transitions, from the solution to the gel state and the routes that can be used to arrive to consolidated materials with prescribed properties. We highlight the implementation of supramolecular and superstructures in different technological fields that exploit the synergies exhibited by renewable polymers and biocolloids integrated in structured materials.

Monte Carlo simulation of a lattice model for the dynamics of randomly branching double-folded ring polymers

Supercoiled DNA, crumpled interphase chromosomes, and topologically constrained ring polymers often adopt treelike, double-folded, randomly branching configurations. Here we study an elastic lattice model for tightly double-folded ring polymers, which allows for the spontaneous creation and deletion of side branches coupled to a diffusive mass transport, which is local both in space and on the connectivity graph of the tree. We use Monte Carlo simulations to study systems falling into three different universality classes: ideal double-folded rings without excluded volume interactions, self-avoiding double-folded rings, and double-folded rings in the melt state. The observed static properties are in good agreement with exact results, simulations, and predictions of Flory theory for randomly branching polymers. For example, in the melt state rings adopt compact configurations and exhibit territorial behavior. In particular, we show that the emergent dynamics is in excellent agreement with a recent scaling theory and illustrate the qualitative differences with the familiar reptation dynamics of linear chains.

Polymer Physics by Quantum Computing

Sampling equilibrium ensembles of dense polymer mixtures is a paradigmatically hard problem in computational physics, even in lattice-based models. Here, we develop a formalism based on interacting binary tensors that allows for tackling this problem using quantum annealing machines. Our approach is general in that properties such as self-avoidance, branching, and looping can all be specified in terms of quadratic interactions of the tensors. Microstates' realizations of different lattice polymer ensembles are then seamlessly generated by solving suitable discrete energy-minimization problems. This approach enables us to capitalize on the strengths of quantum annealing machines, as we demonstrate by sampling polymer mixtures from low to high densities, using the D-Wave quantum annealer. Our systematic approach offers a promising avenue to harness the rapid development of quantum machines for sampling discrete models of filamentous soft-matter systems.

Quantification of branching within high molecular weight polymers with polyester backbones formed by transfer-dominated branching radical telomerisation (TBRT)

New branched polymerisations offer previously inaccessible macromolecules and architectural understanding is important as it provides insight into the branching mechanism and enables the determination of structure-property relationships. Here we present a detailed inverse gated 13C NMR characterisation of materials derived from the very recently reported Transfer-dominated Branching Radical Telomerisation (TBRT) approach to quantify branching and provide an insight into cyclisation.

Topological tuning of DNA mobility in entangled solutions of supercoiled plasmids

Ring polymers in dense solutions are among the most intriguing problems in polymer physics. Because of its natural occurrence in circular form, DNA has been extensively used as a proxy to study the fundamental physics of ring polymers in different topological states. Yet, torsionally constrained-such as supercoiled-topologies have been largely neglected so far. The applicability of existing theoretical models to dense supercoiled DNA is thus unknown. Here, we address this gap by coupling large-scale molecular dynamics simulations with differential dynamic microscopy of entangled supercoiled DNA plasmids. We find that, unexpectedly, larger supercoiling increases the size of entangled plasmids and concomitantly induces an enhancement in DNA mobility. These findings are reconciled as due to supercoiling-driven asymmetric and double-folded plasmid conformations that reduce interplasmid entanglements and threadings. Our results suggest a way to topologically tune DNA mobility via supercoiling, thus enabling topological control over the (micro)rheology of DNA-based complex fluids.

Spectroscopic Investigation of Chlorin-Based Photosensitizers in Polymer Matrix

Chlorin e6 and its derivatives are the basis of a number of drugs used in medicine in the treatment of various diseases, including cancer, by photodynamic therapy. Nonpolar derivatives of Chlorin e6—dimethyl ether of Chlorin e6 (DME Ce6) and trimethyl ether of Chlorin e6 (TME Ce6)—are actively studied for application during photodynamic therapy. In this work, based on the electron optical absorption spectra, the interaction of photosensitizer molecules with branched star-like copolymer dextran-graft-polyacrylamide in anionic form was investigated and the possibility of using the latter as a carrier for drug delivery to tumor cells was suggested.

Effect of pH and Molecular Length on the Structure and Dynamics of Linear and Short-Chain Branched Poly(ethylene imine) in Dilute Solution: Scaling Laws from Detailed Molecular Dynamics Simulations

Atomistic molecular dynamics (MD) simulations are carried out to examine the effect of molecular weight M_w (= 0.6, 0.86, 1.12, and 2.15 kDa) and pH (or equivalently, degree of ionization, α⁺ = 0, 50, and 100%) on the structure, state of hydration, and dynamics of linear and branched poly(ethylene imine) (PEI) chains in infinitely dilute salt-free aqueous solutions. It is found that the degree of ionization is the key factor determining the type of molecular conformation adopted by PEI, regardless of molecular architecture and chain length, resulting in a stable trans conformation for fully ionized solutions and in a stable gauche⁺/gauche^- state for neutral or alternate ionized ones; in the latter case, a strong electrolyte behavior is verified for both linear and branched PEI. Linear PEI is observed to be significantly stiffer than branched PEI of the same molecular weight at 100% degree of ionization, but the effect subsides as the degree of ionization decreases. Also, linear PEI diffuses markedly slower than branched PEI of the same M_w. From the MD results, scaling exponents are deduced and reported for the conformation, solvent-accessible surface area, and dynamics of the two different PEI structures with M_w.

Equilibrium size distribution and phase separation of multivalent, molecular assemblies in dilute solution

Multivalent molecules can bind a limited number of multiple neighbors via specific interactions. In this paper, we investigate theoretically the self-assembly and phase separation of such molecules in dilute solution. We show that the equilibrium size (n) distributions of linear or branched assemblies qualitatively differ; the former decays exponentially with the relative size n/N[combining macron] (N[combining macron] = n), while the latter decays as a power law, with an exponential cutoff only for n ⪆ N[combining macron]2 ≫ N[combining macron]. In some cases, finite, branched assemblies are unstable and show a sol-gel transition at a critical concentration. In dilute solutions, non-specific interactions result in phase separation, whose critical point is described by an effective Flory Huggins theory that is sensitive to the nature of these distributions.

Influence of crowding agents on the dynamics of a multidomain protein in its denatured state: a solvation approach

It is now well appreciated that the crowded intracellular environment significantly modulates an array of physiological processes including protein folding-unfolding, aggregation, and dynamics to name a few. In this work we have studied the dynamics of domain I of the protein human serum albumin (HSA) in its urea-induced denatured states, in the presence of a series of commonly used macromolecular crowding agents. HSA was labeled at Cys-34 (a free cysteine) in domain I with the fluorophore 6-bromoacetyl-2-dimethylaminonaphthalene (BADAN) to act as a solvation probe. In partially denatured states (2-6 M urea), lower crowder concentrations (~ < 125 g/L) induced faster dynamics, while the dynamics became slower beyond 150 g/L of crowders. We propose that this apparent switch in dynamics is an evidence of a crossover from soft (enthalpic) to hard-core (entropic) interactions between the protein and crowder molecules. That soft interactions are also important for the crowders used here was further confirmed by the appreciable shift in the wavelength of the emission maximum of BADAN, in particular for PEG8000 and Ficoll 70 at concentrations where the excluded volume effect is not dominant.

The Effect of Glycidyl Azide Polymer Grafted Tetrafunctional Isocyanate on Polytriazole Polyethylene Oxide-Tetrahydrofuran Elastomer and its Propellant Properties

A new energetic curing reagent, Glycidyl azide polymer grafted tetrafunctional isocyanate (N100-g-GAP) was synthesized and characterized by FT-IR and GPC approaches. Polytriazole polyethylene oxide-tetrahydrofuran (PTPET) elastomer was prepared by N100-g-GAP and alkynyl terminated polyethylene oxide-tetrahydrofuran (ATPET). The resulting PTPET elastomer was fully characterized by TGA, DMA, FTIR and mechanical test. The above analysis indicates that PTPET elastomers using N100-g-GAP as curing reagent have the potential for use in propellants. The overall formulation test of the composite propellants shows that this curing system can effectively enhance mechanical strength and bring a significant improvement in the interface interaction between the RDX & AP particles and binder matrix.

Influence of Branching on the Configurational and Dynamical Properties of Entangled Polymer Melts

We probe the influence of branching on the configurational, packing, and density correlation function properties of polymer melts of linear and star polymers, with emphasis on molecular masses larger than the entanglement molecular mass of linear chains. In particular, we calculate the conformational properties of these polymers, such as the hydrodynamic radius R h , packing length p, pair correlation function g ( r ) , and polymer center of mass self-diffusion coefficient, D, with the use of coarse-grained molecular dynamics simulations. Our simulation results reproduce the phenomenology of simulated linear and branched polymers, and we attempt to understand our observations based on a combination of hydrodynamic and thermodynamic modeling. We introduce a model of "entanglement" phenomenon in high molecular mass polymers that assumes polymers can viewed in a coarse-grained sense as "soft" particles and, correspondingly, we model the emergence of heterogeneous dynamics in polymeric glass-forming liquids to occur in a fashion similar to glass-forming liquids in which the molecules have soft repulsive interactions. Based on this novel perspective of polymer melt dynamics, we propose a functional form for D that can describe our simulation results for both star and linear polymers, covering both the unentangled to entangled polymer melt regimes.

The Rabl configuration limits topological entanglement of chromosomes in budding yeast

The three dimensional organization of genomes remains mostly unknown due to their high degree of condensation. Biophysical studies predict that condensation promotes the topological entanglement of chromatin fibers and the inhibition of function. How organisms balance between functionally active genomes and a high degree of condensation remains to be determined. Here we hypothesize that the Rabl configuration, characterized by the attachment of centromeres and telomeres to the nuclear envelope, helps to reduce the topological entanglement of chromosomes. To test this hypothesis we developed a novel method to quantify chromosome entanglement complexity in 3D reconstructions obtained from Chromosome Conformation Capture (CCC) data. Applying this method to published data of the yeast genome, we show that computational models implementing the attachment of telomeres or centromeres alone are not sufficient to obtain the reduced entanglement complexity observed in 3D reconstructions. It is only when the centromeres and telomeres are attached to the nuclear envelope (i.e. the Rabl configuration) that the complexity of entanglement of the genome is comparable to that of the 3D reconstructions. We therefore suggest that the Rabl configuration is an essential player in the simplification of the entanglement of chromatin fibers.

Many-body contacts in fractal polymer chains and fractional Brownian trajectories

We calculate the probabilities that a trajectory of a fractional Brownian motion with arbitrary fractal dimension d_{f} visits the same spot n≥3 times, at given moments t_{1},...,t_{n}, and obtain a determinant expression for these probabilities in terms of a displacement-displacement covariance matrix. Except for the standard Brownian trajectories with d_{f}=2, the resulting many-body contact probabilities cannot be factorized into a product of single-loop contributions. Within a Gaussian network model of a self-interacting polymer chain, which we suggested recently [K. Polovnikov et al., Soft Matter 14, 6561 (2018)1744-683X10.1039/C8SM00785C], the probabilities we calculate here can be interpreted as probabilities of multibody contacts in a fractal polymer conformation with the same fractal dimension d_{f}. This Gaussian approach, which implies a mapping from fractional Brownian motion trajectories to polymer conformations, can be used as a semiquantitative model of polymer chains in topologically stabilized conformations, e.g., in melts of unconcatenated rings or in the chromatin fiber, which is the material medium containing genetic information. The model presented here can be used, therefore, as a benchmark for interpretation of the data of many-body contacts in genomes, which we expect to be available soon in, e.g., Hi-C experiments.

Generalized Flory Theory for Rotational Symmetry Breaking of Complex Macromolecules

We report on spontaneous rotational symmetry breaking in a minimal model of complex macromolecules with branches and cycles. The transition takes place as the strength of the self-repulsion is increased. At the transition point, the density distribution transforms from isotropic to anisotropic. We analyze this transition using a variational mean-field theory that combines the Gibbs-Bogolyubov-Feynman inequality with the concept of the Laplacian matrix. The density distribution of the broken symmetry state is shown to be determined by the eigenvalues and eigenvectors of this Laplacian matrix. Physically, this reflects the increasing role of the underlying topological structure in determining the density of the macromolecule when repulsive interactions generate internal tension. Eventually, the variational free energy landscape develops a complex structure with multiple competing minima.

Influence of knot complexity on glass-formation in low molecular mass ring polymer melts

We perform molecular dynamics simulations on a coarse-grained polymer melt to study the dynamics of glass-formation in ring polymer melts of variable knot complexity. After generating melts of non-concatenated polymeric rings having a range of minimum crossing number values, mc, we compute the coherent intermediate scattering function, the segmental α-relaxation time, fragility, and the glass transition temperature as a function of mc. Variation of knot complexity is found to have a pronounced effect on the dynamics of polymer melts since both molecular rigidity and packing are altered, primary physical factors governing glass-formation in polymeric materials.

Local loop opening in untangled ring polymer melts: a detailed "Feynman test" of models for the large scale structure

The conformational statistics of ring polymers in melts or dense solutions is strongly affected by their quenched microscopic topological state. The effect is particularly strong for untangled (i.e. non-concatenated and unknotted) rings, which are known to crumple and segregate. Here we study these systems using a computationally efficient multi-scale approach, where we combine massive simulations on the fiber level with the explicit construction of untangled ring melt configurations based on theoretical ideas for their large scale structure. We find (i) that topological constraints may be neglected on scales below the standard entanglement length, Le, (ii) that rings with a size 1 ≤ Lr/Le ≤ 30 exhibit nearly ideal lattice tree behavior characterized by primitive paths which are randomly branched on the entanglement scale, and (iii) that larger rings are compact with gyration radii Rg2(Lr) ∝ Lr2/3. The detailed comparison between equilibrated and constructed ensembles allows us to perform a "Feynman test" of our understanding of untangled rings: can we convert ideas for the large scale ring structure into algorithms for constructing (nearly) equilibrated ring melt samples? We show that most structural observables are quantitatively reproduced by two different construction schemes: hierarchical crumpling and ring melts derived from the analogy to interacting branched polymers. However, the latter fail the "Feynman test" with respect to the magnetic radius, Rm, which we have defined based on an analogy to magnetostatics. While Rm is expected to vanish for double-folded structures, the observed values of Rm2(Lr) ∝ Rg2(Lr) provide a simple and computationally convenient measure of the presence of a non-negligible amount of local loop opening in crumpled rings.

Conformational statistics of randomly branching double-folded ring polymers

The conformations of topologically constrained double-folded ring polymers can be described as wrappings of randomly branched primitive trees. We extend previous work on the tree statistics under different (solvent) conditions to explore the conformational statistics of double-folded rings in the limit of tight wrapping. In particular, we relate the exponents characterizing the ring statistics to those describing the primitive trees and discuss the distribution functions [Formula: see text] and [Formula: see text] for the spatial distance, [Formula: see text], and tree contour distance, L, between monomers as a function of their ring contour distance, [Formula: see text].

The Ising model in swollen vs. compact polymers: Mean-field approach and computer simulations

We study the properties of the classical Ising model with nearest-neighbor interaction for spins located at the monomers of long polymer chains in 2 and 3 dimensions. We compare results for two ensembles of polymers with very different single chain properties: 1) swollen, self-avoiding linear polymer chains in good solvent conditions and 2) compact, space-filling randomly branching polymers in melt. By employing a mean-field approach and Monte Carlo computer simulations, we show that swollen polymers cannot sustain an ordered phase. On the contrary, compact polymers may indeed produce an observable phase transition. Finally, we briefly consider the statistical properties of the ordered phase by comparing polymer chains within the same universality class but characterized by very different shapes.

Effective Hamiltonian of topologically stabilized polymer states

Topologically stabilized polymer conformations in melts of nonconcatenated polymer rings and crumpled globules are considered to be a good candidate for the description of the spatial structure of mitotic chromosomes. Despite significant efforts, the microscopic Hamiltonian capable of describing such systems still remains unknown. We describe a polymer conformation by a Gaussian network - a system with a Hamiltonian quadratic in all coordinates - and show that by tuning interaction constants, one can obtain equilibrium conformations with any fractal dimension between 2 (an ideal polymer chain) and 3 (a crumpled globule). Monomer-to-monomer distances in topologically stabilized states, according to available numerical data, fit very well the Gaussian distribution, giving an additional argument in support of the quadratic Hamiltonian model. Mathematically, the polymer conformations are mapped onto the trajectories of a subdiffusive fractal Brownian particle. Moreover, we explicitly show that the quadratic Hamiltonian with a hierarchical set of coupling constants provides the microscopic background for the description of the path integral of the fractional Brownian motion with an algebraically decaying kernel.

A comparative study of thermodynamic, conformational, and structural properties of bottlebrush with star and ring polymer melts

Thermodynamic, conformational, and structural properties of bottlebrush polymer melts are investigated with molecular dynamics simulations and compared to linear, regular star, and unknotted ring polymer melts to gauge the influence of molecular topology on polymer melt properties. We focus on the variation of the backbone chain length, the grafting density along the backbone, and the length of the side chains at different temperatures above the melt glass transition temperature. Based on these comparisons, we find that the segmental density, isothermal compressibility, and isobaric thermal expansion of bottlebrush melts are quantitatively similar to unknotted ring polymer melts and star polymer melts having a moderate number ( f = 5 to 6) of arms. These similarities extend to the mass scaling of the chain radius of gyration. Our results together indicate that the configurational properties of bottlebrush polymers in their melt state are more similar to randomly branched polymers than linear polymer chains. We also find that the average shape of bottlebrush polymers having short backbone chains with respect to the side chain length is also rather similar to the unknotted ring and moderately branched star polymers in their melt state. As a general trend, the molecular shape of bottlebrush polymers becomes more spherically symmetric when the length of the side chains has a commensurate length as the backbone chain. Finally, we calculate the partial static structure factor of the backbone segments and we find the emergence of a peak at the length scales that characterizes the average distance between the backbone chains. This peak is absent when we calculate the full static structure factor. We characterize the scaling of this peak with parameters characterizing the bottlebrush molecular architecture to aid in the experimental characterization of these molecules by neutron scattering.

The Daoud and Cotton blob model and the interaction of star-shaped polymers

Since it was first proposed in 1982, the Daoud and Cotton (DC) model for star-shaped polymers was intensively used also for self-assembled copolymers and small colloids grafted with long polymers. We try to clarify the position of the DC model and focus on the star partition function which plays a central role in self-assembly and gives access to the star-star interaction. While the predicted star-star interaction agrees with scattering data by Likos et al. (Phys. Rev. Lett. 80, 4450 (1998)), an extensive simulation by Hsu et al. (Macromolecules, 37, 4658 (2004)) does not recover the prediction for the partition function. We try to reconcile this seemingly conflicting results. We discuss star-star interactions, star free energy in θ -solvents, mixing of A/B branches in copolymer stars, within or beyond the Daoud and Cotton blob model.

Confining annealed branched polymers inside spherical capsids

The Lifshitz equation for the confinement of a linear polymer in a spherical cavity of radius R has the form of the Schrödinger equation for a quantum particle trapped in a potential well with flat bottom and infinite walls at radius R. We show that the Lifshitz equation of a confined annealed branched polymer has the form of the Schrödinger equation for a quantum harmonic oscillator. The resulting confinement energy has a 1/R4 dependence on the confinement radius R, in contrast to the case of confined linear polymers, which have a 1/R2 dependence. We discuss the application of this result to the problem of the confinement of single-stranded RNA molecules inside spherical capsids.

Lattice animals in diffusion limited binary colloidal system

In a soft matter system, controlling the structure of the amorphous materials has been a key challenge. In this work, we have modeled irreversible diffusion limited cluster aggregation of binary colloids, which serves as a model for chemical gels. Irreversible aggregation of binary colloidal particles leads to the formation of a percolating cluster of one species or both species which are also called bigels. Before the formation of the percolating cluster, the system forms a self-similar structure defined by a fractal dimension. For a one component system when the volume fraction is very small, the clusters are far apart from each other and the system has a fractal dimension of 1.8. Contrary to this, we will show that for the binary system, we observe the presence of lattice animals which has a fractal dimension of 2 irrespective of the volume fraction. When the clusters start inter-penetrating, we observe a fractal dimension of 2.5, which is the same as in the case of the one component system. We were also able to predict the formation of bigels using a simple inequality relation. We have also shown that the growth of clusters follows the kinetic equations introduced by Smoluchowski for diffusion limited cluster aggregation. We will also show that the chemical distance of a cluster in the flocculation regime will follow the same scaling law as predicted for the lattice animals. Further, we will also show that irreversible binary aggregation comes under the universality class of the percolation theory.

Marginally compact hyperbranched polymer trees

Assuming Gaussian chain statistics along the chain contour, we generate by means of a proper fractal generator hyperbranched polymer trees which are marginally compact. Static and dynamical properties, such as the radial intrachain pair density distribution ρ_pair(r) or the shear-stress relaxation modulus G(t), are investigated theoretically and by means of computer simulations. We emphasize that albeit the self-contact density diverges logarithmically with the total mass N, this effect becomes rapidly irrelevant with increasing spacer length S. In addition to this it is seen that the standard Rouse analysis must necessarily become inappropriate for compact objects for which the relaxation time τ_p of mode p must scale as τ_p ∼ (N/p)^5/3 rather than the usual square power law for linear chains.