The collapse transition of a single polymer chain in two and three dimensions: A Monte Carlo study

Enhancing Polymer Blend Compatibility with Linear and Complex Star Copolymer Architectures: A Monte Carlo Simulation Study with the Bond Fluctuation Model

A Monte Carlo study of the compatibilization of A/B polymer blends has been performed using the bond fluctuation model. The considered compatibilizers are copolymer molecules composed of A and B blocks. Different types of copolymer structures have been included, namely, linear diblock and 4-block alternating copolymers, star block copolymers, miktoarm stars, and zipper stars. Zipper stars are composed of two arms of diblock copolymers arranged in alternate order (AB and BA) from the central unit, along with two homogeneous arms of A and B units. The compatibilization performance has been characterized by analyzing the equilibration of repulsion energy, the simulated scattering intensity obtained with opposite refractive indices for A and B, the profiles along a coordinate axis, the radial distribution functions, and the compatibilizer aggregation numbers. According to the results, linear alternate block copolymers, star block copolymers, and zipper stars exhibit significantly better compatibilization, with zipper stars showing slightly but consistently better performance.

A Tale of Two Chains: Geometries of a Chain Model and Protein Native State Structures

Linear chain molecules play a central role in polymer physics with innumerable industrial applications. They are also ubiquitous constituents of living cells. Here, we highlight the similarities and differences between two distinct ways of viewing a linear chain. We do this, on the one hand, through the lens of simulations for a standard polymer chain of tethered spheres at low and high temperatures and, on the other hand, through published experimental data on an important class of biopolymers, proteins. We present detailed analyses of their local and non-local structures as well as the maps of their closest contacts. We seek to reconcile the startlingly different behaviors of the two types of chains based on symmetry considerations.

Coil-globule transition in two-dimensional polymer chains in an explicit solvent

The structure of two-dimensional polymer chains in a solvent at different temperatures is still far from being fully understood. Computer simulations of high-density macromolecular systems require the use of appropriate algorithms, and therefore the simulations were carried out using the Cooperative Motion Algorithm. The polymer model studied was exactly two-dimensional, coarse-grained and based on a triangular lattice. The theta temperature and temperature of coil-to-globule transition, and critical exponents were determined. The differences between the structure of such a disk and that of a chain in a dense polymer liquid were shown.

Structure of Strongly Adsorbed Polymer Systems: A Computer Simulation Study

The structure of very thin polymer films formed by strongly adsorbed macromolecules was studied by computer simulation. A coarse-grained model of strictly two-dimensional polymer systems was built, and its properties determined by an efficient Monte Carlo simulation algorithm. Properties of the model system were determined by means of Monte Carlo simulations with a sampling algorithm that combines Verdier-Stockmayer, pivot and reputation moves. The effects of temperature, chain length and polymer concentration on the macromolecular structure were investigated. It was shown that at low temperatures, the chain size increases with the concentration, that is, inversely with high temperatures. This behavior should be explained by the influence of inter-chain interactions.

Simulation of Nondilute Dendrimer Systems with the Bond Fluctuation Model

Using the bond fluctuation model, we performed Monte Carlo simulations for solutions of generation 4 and 5 dendrimers with only an intermediate unit between the branching points at different concentrations, ranging from moderately dilute solutions to concentrated systems close to the melt behavior. This model may represent different real types of dendrimer families. We obtained the mean sizes, asphericities, displacement of units, scattering functions, radial distribution functions and structure factors. We compared the results obtained for the last two properties with much faster Monte Carlo simulations of point-like dendrimers using global potentials obtained through the study of binary interactions. The latter procedure provided good reproductions of these properties but failed in the reproduction of the scattering functions in the range of higher concentrations. In this range, the scattering function cannot be described as the product of the structure function and the form factor, because the intensity decreases when the density of the dendrimer units becomes more homogenous.

Influence of network defects on the conformational structure of nanogel particles: From "closed compact" to "open fractal" nanogel particles

We propose an approach to generate a wide range of randomly branched polymeric structures to gain general insights into how polymer topology encodes a configurational structure in solution. Nanogel particles can take forms ranging from relatively symmetric sponge-like compact structures to relatively anisotropic open fractal structures observed in some nanogel clusters and in some self-associating polymers in solutions, such as aggrecan solutions under physiologically relevant conditions. We hypothesize that this broad "spectrum" of branched polymer structures derives from the degree of regularity of bonding in the network defining these structures. Accordingly, we systematically introduce bonding defects in an initially perfect network having a lattice structure in three and two topological dimensions corresponding to "sponge" and "sheet" structures, respectively. The introduction of bonding defects causes these "closed" and relatively compact nanogel particles to transform near a well-defined bond percolation threshold into "open" fractal objects with the inherent anisotropy of randomly branched polymers. Moreover, with increasing network decimation, the network structure of these polymers acquires other configurational properties similar to those of randomly branched polymers. In particular, the mass scaling of the radius of gyration and its eigenvalues, as well as hydrodynamic radius, intrinsic viscosity, and form factor for scattering, all undergo abrupt changes that accompany these topological transitions. Our findings support the idea that randomly branched polymers can be considered to be equivalent to perforated sheets from a "universality class" standpoint. We utilize our model to gain insight into scattering measurements made on aggrecan solutions.

Hydrogen Bonding Sequence Directed Coil-Globule Transition in Water Soluble Thermoresponsive Polymers

The origin of the coil-globule transition for water-soluble thermoresponsive polymers frequently used in nanomaterials remains elusive. Using polypropylene oxide as an example we demonstrate by means of atomistic molecular dynamics simulations that temperature-induced increase in the sequence length of monomers that are not hydrogen bonded to water drives the coil-globule transition. Longer chains statistically exhibit longer sequences which serve as nucleation sites for hydrophobic cluster formation, facilitating chain collapse at lower temperature in agreement with experimental data.

Structure and conformational properties of ideal nanogel particles in athermal solutions

We investigate the conformational properties of "ideal" nanogel particles having a lattice network topology by molecular dynamics simulations to quantify the influence of polymer topology on the solution properties of this type of branched molecular architecture. In particular, we calculate the mass scaling of the radius of gyration (Rg), the hydrodynamic radius, as well as the intrinsic viscosity with the variation of the degree of branching, the length of the chains between the branched points, and the average mesh size within these nanogel particles under good solvent conditions. We find competing trends between the molecular characteristics, where an increase in mesh size or degree of branching results in the emergence of particle-like characteristics, while an increase in the chain length enhances linear polymer-like characteristics. This crossover between these limiting behaviors is also apparent in our calculation of the form factor, P(q), for these structures. Specifically, a primary scattering peak emerges, characterizing the overall nanogel particle size. Moreover, a distinct power-law regime emerges in P(q) at length scales larger than the chain size but smaller than Rg of the nanogel particle, and the Rg mass scaling exponent progressively approaches zero as the mesh size increases, the same scaling as for an infinite network of Gaussian chains. The "fuzzy sphere" model does not capture this feature, and we propose an extension to this popular model. These structural features become more pronounced for values of molecular parameters that enhance the localization of the branching segments within the nanogel particle.

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.

Flory theory of randomly branched polymers

Randomly branched polymer chains (or trees) are a classical subject of polymer physics with connections to the theory of magnetic systems, percolation and critical phenomena. More recently, the model has been reconsidered for RNA, supercoiled DNA and the crumpling of topologically-constrained polymers. While solvable in the ideal case, little is known exactly about randomly branched polymers with volume interactions. Flory theory provides a simple, unifying description for a wide range of branched systems, including isolated trees in good and θ-solvent, and tree melts. In particular, the approach provides a common framework for the description of randomly branched polymers with quenched connectivity and for randomly branching polymers with annealed connectivity. Here we review the Flory theory for interacting trees in the asymptotic limit of very large polymerization degree for good solvent, θ-solutions and melts, and report its predictions for annealed connectivity in θ-solvents. We compare the predictions of Flory theory for randomly branched polymers to a wide range of available analytical and numerical results and conclude that they are qualitatively excellent and quantitatively good in most cases.

Nature of the collapse transition in interacting self-avoiding trails

We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice of general coordination q and on a Husimi lattice built with squares and coordination q=4. The exact grand-canonical solutions of the model are obtained, considering that up to K monomers can be placed on a site and associating a weight ω_{i} with an i-fold visited site. Very rich phase diagrams are found with nonpolymerized, regular polymerized, and dense polymerized phases separated by lines (or surfaces) of continuous and discontinuous transitions. For a Bethe lattice with q=4 and K=2, the collapse transition is identified with a bicritical point and the collapsed phase is associated with the dense polymerized (solidlike) phase instead of the regular polymerized (liquidlike) phase. A similar result is found for the Husimi lattice, which may explain the difference between the collapse transition for ISATs and for interacting self-avoiding walks on the square lattice. For q=6 and K=3 (studied on the Bethe lattice only), a more complex phase diagram is found, with two critical planes and two coexistence surfaces, separated by two tricritical and two critical end-point lines meeting at a multicritical point. The mapping of the phase diagrams in the canonical ensemble is discussed and compared with simulational results for regular lattices.

A Replica Exchange Molecular Dynamics Simulation of a Single Polyethylene Chain: Temperature Dependence of Structural Properties and Chain Conformational Study at the Equilibrium Melting Temperature

The conformational properties of a finite length polyethylene chain were explored over a wide range of temperatures using a replica exchange molecular dynamics simulation providing high quality simulation data representative for the equilibrium behavior of the chain molecule. The radial distribution function (RDF) and the structure factor S(q) of the chain as a function of temperature are analyzed in detail. The different characteristic peaks in the RDF and S(q) were assigned to specific distances in the chain and structural changes occurring with the temperature. In S(q), a peak characteristic for the order in the solid state was found and used to determine the equilibrium melting temperature. A detailed scaling analysis of the structure factor covering the full q range was performed according to the work of Hammouda. In the Θ region, a quantitative analysis of the full structure factor was done using the equivalent Kuhn chain, which enabled us to assign the Θ region of our chain and to demonstrate, in our particular case, the failure of the Gaussian chain approach. The chain conformational properties at the equilibrium melting temperature are discussed using conformational distribution functions, using the largest principal component of the radius of gyration and shape parameters as order parameters. We demonstrate that for the system studied here, the Landau free energy expression based on this conformational distribution information leads to erroneous conclusions concerning the thermodynamic transition behavior. Finally, we focus on the instantaneous conformational properties at the equilibrium melting temperature and give a detailed analysis of the conformational shapes using different shape parameters and a simulation snapshot. We show that the chain does not only take the lamellar rod-like and globular conformational shapes, typical of the solid and liquid states, but can also explore many other conformational states, including the toroidal conformational state. It is the first demonstration that a flexible molecule like PE can also take a toroidal conformational state, which is normally linked to stiffer chains.

Macromolecular knot in good and poor solvents: a Monte Carlo simulation

The probability and dimension of the simple macromolecular knots over a wide range of temperatures corresponding from good to poor solvents are investigated by Monte Carlo simulation. Macromolecular knots are modeled as rings of self-avoiding walks on a simple cubic lattice with the nearest neighbor attractions. We found that there is a minimum probability for the unknotted ring at a certain temperature. The size dependence of trivial, trefoil, and figure-eight knots on chain lengths and temperatures is presented. The simulation results for the size dependence on the knot's complication in different solvents are in good qualitative agreement with prediction of the scaling model proposed by Grosberg et al. The critical exponent for long chain is independent of the knot types based on the simulation results, although the mean square radius of gyration is influenced significantly by the knot types for a shorter length macromolecular ring. We calculated the ratio of the topological invariant p of trefoil knot and figure-eight knot and found that the ratio is approaching to 1.3 with the increasing of the chain length.

Solvation potentials for flexible chain molecules in solution: on the validity of a pairwise decomposition

The effects of a solvent on the conformation of a flexible n-site solute molecule can be described formally in terms of an n-body solvation potential. Given the practical difficulty in computing such multibody potentials, it is common to carry out a pairwise decomposition in which the n-body potential is approximated by a sum of two-body potentials. Here we investigate the validity of this two-site approximation for short interaction-site chain-in-solvent systems. Using exact expressions for the conformation of an isolated chain, we construct a mapping between the full chain-in-solvent system and its solvation potential representation. We present results for both hard-sphere and square-well systems with n=5 that show that the two-site approximation is sufficient to completely capture the effects of an explicit solvent on chain conformation for a wide range of conditions (which include varying the solvent diameter in the hard-sphere system and varying the chain-solvent coupling in the square-well system). In all cases, a set of two-site potentials (one for each distinct site-site pair) is required. We also show that these two-site solvation potentials can be used to accurately compute a multisite intramolecular correlation function.

Effective interaction parameter of linear/star polymer blends and comparison with that of linear/linear and star/star blends

The authors present a detailed study of the microscopic parameters, which control the miscibility in binary linear/star polymer blends. The effective interactions of linear/star polymer blends are studied by means of Monte Carlo simulations and comparison is made with linear/linear and star/star blends, which they also determined. Using the bond fluctuation model on a simple cubic lattice, the authors are able to simulate symmetric linear/linear, star/star, and, for the first time, linear/star blends with a moderate number of arms. The simulations were performed at a volume fraction of occupied lattice sites phi=0.5, which corresponds to dense polymer mixtures for this algorithm. In particular, we study star/star blends with 4, 8, and 12 arms and the respective linear/linear blends as well as linear/star blends, all having the same total number of units equal to 73 and 121. The authors find that linear/star blends are more miscible than the corresponding linear/linear blends, which is in agreement with recent experimental and theoretical results. They find that linear/star mixtures are less miscible than star/star blends, a result which is also verified by theoretical findings.

Simulation study of the coil-globule transition of a polymer in solvent

Molecular dynamics simulations are used to study the coil-globule transition for a system composed of a bead-spring polymer immersed in an explicitly modeled solvent. Two different versions of the model are used, which are differentiated by the nature of monomer-solvent, solvent-solvent, and nonbonded monomer-monomer interactions. For each case, a model parameter lambda determines the degree of hydrophobicity of the monomers by controlling the degree of energy mismatch between the monomers and solvent particles. We consider a lambda-driven coil-globule transition at constant temperature. The simulations are used to calculate average static structure factors, which are then used to determine the scaling exponents of the system in order to determine the theta-point values lambdatheta separating the coil from the globule states. For each model we construct coil-globule phase diagrams in terms of lambda and the particle density rho. The results are analyzed in terms of a simple Flory-type theory of the collapse transition. The ratio of lambdatheta for the two models converges in the high density limit exactly to the value predicted by the theory in the random mixing approximation. Generally, the predicted values of lambdatheta are in reasonable agreement with the measured values at high rho, though the accuracy improves if the average chain size is calculated using the full probability distribution associated with the polymer-solvent free energy, rather than merely using the value obtained from the minimum of the free energy.

Design of random copolymers with statistically controlled monomer sequence distributions via Monte Carlo simulations

We use Monte Carlo simulations to model the formation of random copolymers with tunable monomer sequence distributions. Our scheme is based on the original idea proposed a few years ago by Khokhlov and Khalatur [Physica A 249, 253 (1998); Phys. Rev. Lett. 82, 3456 (1999)], who showed that the distribution of species B in A-B random copolymers can be regulated by (a) adjusting the coil size of a homopolymer A and (b) chemically modifying ("coloring") monomers that reside at (or close to) the periphery of the coil with species B. In contrast to Khokhlov and Khalatur's work, who modeled the polymer modification by performing the coloring instantaneously, we let the chemical coloring reaction progress over time using computer simulations. We show that similar to Khokhlov and Khalatur's work, the blockiness (i.e., number of consecutive monomers) of the B species along the A-B copolymer increases with increasing degree of collapse of the parent homopolymer A. A simple analysis of the A-B monomer sequences in the copolymers reveals that monomer sequence distributions in homopolymers "colored" under collapsed conformations possess certain degrees of self-similarity, while there is no correlation found among the monomer sequence distributions formed by coloring homopolymers with expanded conformations.

Slow relaxation and solvent effects in the collapse of a polymer

Using molecular dynamic simulations we study the quench of a homopolymer chain into a poor solvent at finite temperature. We show that, depending on the quench depth, there are different relaxation pathways to the collapsed state. Solvent effects are introduced through an effective Lennard-Jones potential depending on the local monomer density. The various relaxation regimes are characterized by the contact correlation function. As the quench depth increases, the system evolves towards a glassy state, and the relaxation dynamics continuously changes from an exponential to a stretched exponential law. The characteristic relaxation time diverges at low temperature following an Arrhenius law, like in the case of strong glasses. We found that the stretching exponent depends on aging in a nonuniversal way. The solvent modifies the globular state by diminishing the effects of frustration and glassy behavior.

Coil-globule transition for regular, random, and specially designed copolymers: Monte Carlo simulation and self-consistent field theory

The coil-globule transition has been studied for A-B copolymer chains both by means of lattice Monte Carlo (MC) simulations using bond fluctuation algorithm and by a numerical self-consistent-field (SCF) method. Copolymer chains of fixed length with A and B monomeric units with regular, random, and specially designed (proteinlike) primary sequences have been investigated. The dependence of the transition temperature on the AB sequence has been analyzed. A proteinlike copolymer is more stable than a copolymer with statistically random sequence. The transition is more sharp for random copolymers. It is found that there exists a temperature below which the chain appears to be in the lowest energy state (ground state). Both for random and proteinlike sequences and for regular copolymers with a relatively long repeating block, a molten globule regime is found between the ground state temperature and the transition temperature. For regular block copolymers the transition temperature increases with block size. Qualitatively, the results from both methods are in agreement. Differences between the methods result from approximations in the SCF theory and equilibration problems in MC simulations. The two methods are thus complementary.

Self-generated disorder and structural glass formation in homopolymer globules

We have investigated the interrelation between spin glasses and structural glasses. Spin glasses in this case are p-spin interaction spin glasses (at p>2) or Potts glasses that contain quenched disorder, whereas the structural glasses are here exemplified by a homopolymeric globule, which can be viewed as a liquid of connected units on a nanoscale. It is argued that the homopolymeric globule problem can be mapped onto a disorder field theoretical model whose effective Hamiltonian resembles the corresponding one for the spin glass model. In this sense the disorder in the globule is self-generated (in contrast to spin glasses) and can be related to competing interactions (virial coefficients of different signs) and the chain connectivity. The work is aimed at giving a quantitative description of this analogy. We have investigated in the mean-field approximation the phase diagram of the homopolymeric globule where the transition line from the liquid to glassy globule is treated in terms of the replica symmetry breaking paradigm. The configurational entropy temperature dependence is also discussed.

Early stages of homopolymer collapse

Interest in the protein folding problem has motivated a wide range of theoretical and experimental studies of the kinetics of the collapse of flexible homopolymers. In this paper, a phenomenological model is proposed for the kinetics of the early stages of homopolymer collapse following a quench from temperatures above to below the straight theta temperature. In the first stage, nascent droplets of the dense phase are formed, with little effect on the configurations of the bridges that join them. The droplets then grow by accreting monomers from the bridges, thus causing the bridges to stretch. During these two stages, the overall dimensions of the chain decrease only weakly. Further growth of the droplets is accomplished by the shortening of the bridges, which causes the shrinking of the overall dimensions of the chain. The characteristic times of the three stages scale as N0, N(1/5), and N(6/5), respectively, where N is the degree of polymerization of the chain.

Theta-point behavior of diluted polymer solutions: can one observe the universal logarithmic corrections predicted by field theory?

In recent large-scale Monte Carlo simulations of various models of Theta-point polymers in three dimensions Grassberger and Hegger found logarithmic corrections to mean field theory with amplitudes much larger than the universal amplitudes of the leading logarithmic corrections calculated by Duplantier in the framework of tricritical O(n) field theory. To resolve this issue we calculate the universal subleading correction of field theory, which turns out to be of the same order of magnitude as the leading correction for all chain lengths available in present day simulations. Borel resummation of the renormalization group flow equations also shows the presence of such large corrections. This suggests that the published simulations did not reach the asymptotic regime. To further support this view, we present results of Monte Carlo simulations on a Domb-Joyce-like model of weakly interacting random walks. Again the results cannot be explained by keeping only the leading corrections, but are in fair accordance with our full theoretical result. The corrections found for the Domb-Joyce model are much smaller than those for other models, which clearly shows that the effective corrections of are not yet in the asymptotic regime. Altogether, our findings show that the existing simulations of Theta polymers are compatible with tricritical field theory, since the crossover to the asymptotic regime is very slow. Similar results were found earlier for self-avoiding walks at their upper critical dimension d(c)=4.