Polymer Collapse Research Articles

Universality for interacting oriented self-avoiding walk: A transfer matrix calculation

In this paper we perform transfer matrix calculations to study the two-dimensional problem of oriented self-avoiding walks on a square lattice where nearest neighbor interactions depend on the relative orientation (parallel or antiparallel) between different parts of the path. Our main purpose is to verify a conformal field theory conjecture that the entropic exponent $\ensuremath{\gamma}$, which is related to the number of such walks, varies continuously with the energy of the parallel interaction. We find no evidence of this behavior, but see many unexpected features for the model, such as the existence of a line of $\ensuremath{\theta}$ points belonging to the universality class of polymer collapse on a Manhattan lattice. Finally we study the phase diagram in the parallel and antiparallel interaction planes and we conjecture a crucial role for the standard $\ensuremath{\theta}$ point.

Gaussian self-consistent approach to multichain polymer systems

Here we extend the Gaussian self-consistent method that has been successfully applied to the single polymer collapse problem to a polymer solution of many identical chains earlier. This method permits a complete study of the equilibrium thermodynamic properties as well as of the kinetic transformations. We discuss various aspects of the aggregation and collapse phenomena paying particular attention to the interplay between the intra- and inter-molecular degrees of freedom. We show that, at equilibrium, in part of the parameter space our equations may be reduced to those of the Flory-Huggins mean field theory. In kinetics a new phenomenon, notably the existence of a long-lived metastable state of “mesoglobules”, is found.

Collapse of a ring polymer: Comparison of Monte Carlo and Born–Green–Yvon integral equation results

The equilibrium properties of an isolated ring polymer are studied using a Born–Green–Yvon (BGY) integral equation and Monte Carlo simulation. The model polymer is composed of n identical spherical interaction sites connected by universal joints of bond length σ. In particular, we study rings composed of up to n=400 square-well spheres with hard-core diameter σ and well diameter λσ (1⩽λ⩽2). Intramolecular site–site distribution functions and the resulting configurational and energetic properties are computed over a wide range of temperatures for the case of λ=1.5. In the high temperature (good solvent) limit this model is identical to a tangent-hard-sphere ring. With decreasing temperature (worsening solvent) both the radius of gyration and the internal energy of the ring polymer decrease, and a collapse transition is signaled by a peak in the single ring specific heat. In comparison with the Monte Carlo calculations, the BGY theory yields quantitative to semiquantitative results for T≳Tθ and is qualitatively accurate for T≲Tθ, where Tθ is the theta temperature. The thermal behavior of an isolated square-well ring is found to be quite similar to the behavior of an isolated square-well chain. The BGY theory indicates that rings and chains have comparable theta and collapse transition temperatures. In the low temperature limit (collapsed state) the microscopic structure of rings and chains becomes nearly identical.

DNA condensation by multivalent cations.

In the presence of multivalent cations, high molecular weight DNA undergoes a dramatic condensation to a compact, usually highly ordered toroidal structure. This review begins with an overview of DNA condensation: condensing agents, morphology, kinetics, and reversibility, and the minimum size required to form orderly condensates. It then summarizes the statistical mechanics of the collapse of stiff polymers, which shows why DNA condensation is abrupt and why toroids are favored structures. Various ways to estimate or measure intermolecular forces in DNA condensation are discussed, all of them agreeing that the free energy change per base pair is very small, on the order of 1% of thermal energy. Experimental evidence is surveyed showing that DNA condensation occurs when about 90% of its charge is neutralized by counterions. The various intermolecular forces whose interplay gives rise to DNA condensation are then reviewed. The entropy loss upon collapse of the expanded wormlike coil costs free energy, and stiffness sets limits on tight curvature. However, the dominant contributions seem to come from ions and water. Electrostatic repulsions must be overcome by high salt concentrations or by the correlated fluctuations of territorially bound multivalent cations. Hydration must be adjusted to allow a cooperative accommodation of the water structure surrounding surface groups on the DNA helices as they approach. Undulations of the DNA in its confined surroundings extend the range of the electrostatic forces. The condensing ions may also subtly modify the local structure of the double helix.

A constitutive model for nonlinear viscoelastic behavior of polymers

AbstractA new constitutive model for the nonlinear behavior of noncrosslinked polymers with infinitesimal strains is derived. It generalizes a model of adaptive links for the viscoelastic behavior of crosslinked polymers. According to the model, a viscoelastic medium is treated as a set of elastic springs that replace each other. The springs model chemical links between polymer molecules, which arise and collapse because of the micro‐Brownian motion, a law for replacing the springs determines the stress relaxation in a viscoelastic medium. Unlike crosslinked polymers, noncrosslinked polymers demonstrate steady‐state creep flow after some transition period. To describe both the transition process and the steady creep, the model distinguishes two different types of adaptive links. Links of the first type are ruptured under loading, whereas links of the second type replace one another. Both these processes (destruction and replacement) are treated as kinetic, and equations of chemical kinetics are introduced for their description. The nonlinearity of the model arises because of a dependence of rates of kinetic processes on the stress intensity. The constitutive equations derived are testified by comparing theoretical results with data for polypropylene fibers. For the verification we use data presented by three independent sources. The results demonstrate good agreement between experimental observations and their theoretical predictions. Finally, the effect of molecular weight of polymers on parameters of the model is studied.

The kinetics laws for polymer collapse and biopolymer folding

AbstractWe discuss some recent ideas relating to the folding and other conformational transitions of polymers. In particular we emphasize that it is possible to conceive the accompanying kinetic processes as a problem in non‐equilibrium statistical mechanics. In this sense it should be possible to develop such methods to deduce the kinetic laws in the vicinity of various conformational transitions. To establish the ideas we first study the well known problem of the collapse transition of a homopolymer. We then turn to conformational transitions in periodic and random copolymers. We offer a brief survey of the status of these ideas in applications to biopolymeric conformational kinetics.

Phase diagram of branched polymer collapse.

The phase diagram of the collapse of a two-dimensional infinite branched polymer interacting with the solvent and with itself through contact interactions is studied from the $q\to 1$ limit of an extension of the $q-$ states Potts model. Exact solution on the Bethe lattice and Migdal-Kadanoff renormalization group calculations show that there is a line of $\theta$ transitions from the extended to a single compact phase. The $\theta$ line, governed by three different fixed points, consists of two lines of extended--compact transitions which are in different universality classes and meet in a multicritical point. On the other hand, directed branched polymers are shown to be completely determined by the strongly embedded case and there is a single $\theta$ transition which is in the directed percolation universality class.

`Smart' self-avoiding trails and the collapse of chain polymers in three dimensions

Recently, it was shown that certain `smart' self-avoiding trails have end-to-end distances in three dimensions. Also, the corrections to scaling of and the specific heat are qualitatively very similar to those for polymers exactly at the point. The question was thus posed whether they are in the same universality class as linear polymers at the temperature. We show that this is not the case since these trails show a first-order transition, instead of the second-order transition at the usual point. We argue that this is due to the fact that the `smartness' of these trails implies that the renormalized n-body interactions vanish identically for any finite . We conjecture that the qualitative similarity with recent simulations of polymers indicates that for n-body interactions the renormalized three-body interaction is small in real polymers.

Control of protein–ligand recognition using a stimuli-responsive polymer

Stimuli-responsive polymers exhibit reversible phase changes in response to changes in environmental factors such as pH or temperature. Conjugating such polymers to antibodies and proteins provides molecular systems for applications such as affinity separations, immunoassays and enzyme recovery and recycling. Here we show that conjugating a temperature-sensitive polymer to a genetically engineered site on a protein allows the protein's ligand binding affinity to be controlled. We synthesized a mutant of the protein streptavidin to enable site-specific conjugation of the responsive polymer near the protein's binding site. Normal binding of biotin to the modified protein occurs below 32 degrees C, whereas above this temperature the polymer collapses and blocks binding. The collapse of the polymer and thus the enabling and disabling of binding, is reversible. Such environmentally triggered control of binding may find many applications in biotechnology and biomedicine, such as the control of enzyme reaction rates and of biosensor activity, and the controlled release of drugs.

Flocculation and Coagulation of Ca- and Mg-Saturated Montmorillonite in the Presence of a Neutral Polysaccharide

AbstractThe objective of this study was to observe flocculation of montmorillonite in the presence of a glucose polymer (dextran) and to observe the effect of saturating cation and coagulant addition on the flocculation process. Flocculation of montmorillonite was dependent on polymer molecular weight, polymer/clay ratio (w/w), nature of exchangeable cation, and ionic strength of the suspension to which the polymer was added. The T500 dextran (molecular weight = 5 × 105) caused enhanced sedimentation of Ca-montmorillonite suspension at a polymer/clay ratio of ≤0.01. Increasing the polymer concentration above this level stabilized the suspension such that sedimentation was less than or equal to that of the control. The T2000 dextran (molecular weight = 2 × 106) caused a similar increase in the sedimentation of Ca-montmorillonite at polymer/clay ratios of <0.1. The ability of the T2000 polymer to cause flocculation at greater polymer/clay ratios as compared to the T500 polymer was attributed to the lower osmotic pressure between clay particles for equal concentrations of the two polymers. Flocculation of Ca-montmorillonite by dextran was enhanced when the clay had initially been coagulated by the addition of salt. Reduction of the diffuse double layer upon addition of salt permitted the polymer to extend beyond the electrostatic barrier of the clays. Dextran was not able to flocculate Mg-montmorillonite suspensions with or without the presence of coagulant. The displacement of water molecules at the clay surface rather than within the hydration shell of the more highly polarizing Mg cations by polymer segments resulted in a greater polymer collapse on the clay surface leaving fewer and shorter polymer loops and tails available for contacting adjacent clay particles.

On the analyticity properties of scaling functions in models of polymer collapse

We consider the mathematical properties of the generating and partition functions in the two-variable scaling region about the tricritical point in some models of polymer collapse. We concentrate on models that have a similar behaviour to that of interacting partially-directed self-avoiding walks (IPDSAW) in two dimensions. However, we do not restrict the discussion to that model. After describing the properties for a general class of models, and stating exactly what we mean by scaling, we prove the following theorem: If the generating function of finite-size partition functions has a tricritical cross-over scaling form around the theta -point, and the associated tricritical scaling function, g, has a finite radius of convergence, then the partition function has a finite-size scaling form and importantly the finite-size scaling function, f, is an entire function. In the IPDSAW case we have an explicit representation of the finite-size scaling function. We point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta -point models. We discuss the result in relation to the Edwards model of polymer collapse for which it has recently been argued that the finite-size scaling functions are not entire.

Random matrix solution of a polymer collapse model

A polymer folding model on the square lattice is constructed with attractive contact interactions of strength 1/c^2, 0<c<1. The corresponding model on a dynamical random lattice, with freely fluctuating co-ordination number at each vertex, is formulated as a random two-matrix model and an expression for the partition function of a length-L chain is derived. Numerical estimates and analytical evaluation for L \to \infty shows a third-order collapse transition at c=\sqrt{2}-1. Geometrical critical exponents are computed in each phase and interpreted. The Knizhnik-Polyakov-Zamolodchikov 2D quantum gravity scaling relations are used to predict the corresponding behaviour on the regular lattice, which lies in a different universality class from the percolation Theta-point of Duplantier and Saleur.

Applications of parallel computing to biological problems.

Parallel computers should provide the greatest processing power and memory for scientific simulations in the coming decades. This review discusses general strategies and specific algorithms for the use of various parallel architectures in simulations of biological and artificial polymers. General strategies include space partitioning (domain decomposition cell methods) and distributed independent simulations. Specific algorithms include cellular automata for efficient abstract polymer simulation. One algorithm, the two-space algorithm, is particularly efficient both for parallel and serial computation. Three applications, 2D melts, gel electrophoresis, and polymer collapse, are described. Simulations of high-density melts in 2D show that contrary to expectations, polymers do not completely segregate at the highest densities; instead, polymer interpenetration is significant. Preliminary simulations of gel electrophoresis show its behavior in the diffusive regimen and demonstrate the use of Cellular Automaton Machines (CAMs). Polymer collapse is studied in the regime of large departures from good solvent conditions. In this regime, kinetics plays a significant role. Collapse is dominated (nucleated) by migration of the chain ends.

Motion of polymer ends in homopolymer and heteropolymer collapse

To investigate the polymer coil-to-globule transition we performed simulations for the kinetics of homopolymer and heteropolymer collapse. Our stimulations made use of abstract models of long flexible polymers to obtain extensive statistical sampling. For a variety of these models, the simulations suggest that collapse of long polymers is dominated by diffusion of the polymer ends, which accrete monomers and small aggregates. The growth of the end aggregate was found to be nearly linear in time for homopolymers and largely unaffected by variations in microstructure. In contrast, for heteropolymers the presence of non-aggregating (hydrophilic) monomers dramatically slows and alters the growth of the end mass. In models simulated, the end mass grows roughly as the cube root of time, but still dominates aggregation along the contour. In a model where only pairwise bonding is allowed, the collapse is uniform since more flexible end motion does not result in continued end accretion. The possible significance of our results for biopolymer kinetics is discussed.

Models of polymer collapse in three dimensions: Evidence from kinetic growth simulations.

We present simulational evidence that kinetically grown tricolor walks and kinetically grown trails on selected lattices in three dimensions are equivalent to interacting walks and trails at their respective collapse temperatures, all of which model polymers in dilute solution at the \ensuremath{\theta} point. We discuss the relation of these models to the canonical model of a single self-attracting polymer in dilute solution: the self-avoiding walk with nearest-neighbor interactions. The main results concern the divergence of the specific heat and these differ from the predictions of the (three-parameter) Edwards model. The behavior of the kinetic trail simulations also raises doubts about the current field theoretic description of collapse in interacting trails.

Phase separation in binary hard-core mixtures

We report the observation of a purely entropic demixing transition in a three-dimensional binary hard-core mixture by computer simulations. This transition is observed in a lattice model of a binary hard-core mixture of parallel cubes provided that the size asymmetry of the large and small particles is sufficiently large (≥3, in the present case). In addition, we have performed simulations of a single athermal polymer in a hard-core solvent. As we increase the chemical potential of the solvent, we observe a purely entropy-driven collapse of the polymer: the scaling of the radius of gyration Rg of the polymer with the number of segments N changes from that of a polymer in a good solvent to that of a collapsed polymer. Both for the study of the hard-core demixing and of the polymer collapse, it was essential to use novel collective Monte Carlo moves to speed up equilibration. We show that in the limit σ1/σ2→0, the pair distribution function for an off-lattice binary hard-core mixture of parallel cubes with side lengths σ1 and σ2 diverges at contact for the large particles. For the lattice system, we calculated the pair distribution functions g(r) up to the fourth virial coefficient. The difference in g(r) at contact for a binary system and a pure system at the same packing fraction gives a rough criterion, whether the mixture phase separates.

Integral equation theory of polymers: Translational invariance approximation and properties of an isolated linear polymer in solution

In this paper, we continue investigations on the solution methods for the generalized Percus–Yevick equations for the pair correlation functions of polymers, which were formulated in the previous papers of this series [J. Chem. Phys. 99, 4084, 4103 (1993)]. Previously, they were reduced to recursive integral equations and solved numerically. In this paper, a translational invariance approximation is used to reduce the number of integral equations to solve. In this approximation, only N integral equations out of N2 integral equations are required for a polymer consisting of N beads (monomers). The behavior of an isolated polymer is studied with three different potential models, a soft sphere, a hard sphere, and a Lennard-Jones potential. The main motivation for considering these three potential models is in testing the idea of universality commonly believed to hold for some properties of polymers. We find that the universality holds for the power law exponent for the expansion factor of polymers at high temperatures. The end-to-end distance distribution functions, intermediate distribution functions, chemical potentials, the density distributions, and various expansion factors of the polymer chain are computed from the solutions of the integral equations in the case of coiled, ideal, and collapsed states of the polymer. The expansion factors in the collapsed regime are found to obey power laws with respect to the length of the polymer and [B(T)−B(θ̄)], where B(T) is the second virial coefficient and θ̄ is a modified θ temperature. The values of these exponents approach those from the known theories of polymer collapse as the chain length becomes long and the ratio of bond length to bead radius becomes large.

Interacting partially directed walks: A model of polymer collapse

A survey of the results that have been obtained to date on the partially directed walk model of the polymer collapse transition is presented.

Evidence for entropy-driven demixing in hard-core fluids.

We report the first observation, by computer simulation, of a purely entropic demixing transition in a three-dimensional binary hard-core mixture. This transition is observed in a mixture of large and small cubes. We also find evidence for demixing in other hard-core fluids and, in the case of an athermal polymer solution, we observe a purely entropy-driven polymer collapse. For the study of both the hard-core demixing and polymer collapse, it was essential to use novel collective Monte Carlo moves.

Monte Carlo simulation of polymers at interfaces

Polymers at interfaces pose challenging problems to statistical physics because their configurations often differ greatly from the bulk. Computer simulation of coarse-grained models then gives valuable insight and allows stringent tests of various theoretical predictions. Three examples are briefly treated: chain configurations of B-chains in the surface-enriched B-rich layer of an (AB) binary polymer mixture; “frustrated” lamellar ordering in ultra-thin block-copolymer films; and the collapse of polymer brushes in bad solvents.