Dense Polymer Solutions Research Articles

Density Functional Theory for Polymer Phase Separations Induced by Coupling of Chemical Reaction and Elastic Stress

AbstractA density functional theory is applied to phase separation dynamics influenced by crosslinking reactions in dense polymer solutions. The crosslinking reaction is modeled by a change from non‐crosslinked polymers comprising transient network (TN) to crosslinked polymers participating in the percolated permanent network (PN). Deformed TN polymers are considered to relax to the isotropic equilibrium state according to the Maxwellian linear viscoelastic constitutive equation, which is used in the modeling of viscoelastic phase separations. The PN is modeled by a linear elastic constitutive model. When TN polymers are taken into the PN by crosslinking reaction, the instantaneous deformation of the TN polymers are frozen, and such frozen deformations are accumulated as time goes on. A series of simulations is performed using this model, so that two specific features of the viscoelastic and reactive phase separation are obtained, i.e., 1) two‐stage phase separation process that leads to a domain structure with two different characteristic length scales, and 2) fixing the phase‐separated structure before reaching the macrophase separation.

Dynamical Entanglement and Cooperative Dynamics in Entangled Solutions of Ring and Linear Polymers.

Understanding how entanglements affect the behavior of polymeric complex fluids is an open challenge in many fields. To elucidate the nature and consequence of entanglements in dense polymer solutions, we propose a novel method: a "dynamical entanglement analysis" (DEA) to extract spatiotemporal entanglement structures from the pairwise displacement correlation of entangled chains. By applying this method to large-scale molecular dynamics simulations of linear and unknotted, nonconcatenated ring polymers, we find a strong and unexpected cooperative dynamics: the footprint of mutual entrainment between entangled chains. We show that DEA is a powerful and sensitive probe that reveals previously unnoticed and architecture-dependent spatiotemporal structures of dynamical entanglement in polymeric solutions. We also propose a mean-field approximation of our analysis that provides previously under-appreciated physical insights into the dynamics of generic entangled polymers. We envisage DEA will be useful to analyze the dynamical evolution of entanglements in generic polymeric systems such as blends and composites.

Grand-canonical polymers under confinement: Dense solutions.

We theoretically study dense polymer solutions under open (capillary and slit) and closed (box) confinement. The theory is formulated for grand-canonical polymers and corrections to the self-consistent mean-field results are discussed. In contrast to the mean-field prediction, we found that the partition function of a labeled chain is affected by confinement even under neutral von Neumann boundary conditions and the chain length distribution is biased to short chains. As the container size increases, the contribution of the transverse excited states to the free energy of a labeled chain is found to approach its bulk value nonmonotonically (through an extremum) for the box and the capillary confinement but not for the slit. So does the confinement free energy of a labeled chain. The confinement energy of the solution is well behaved for open confinement but formally diverges for a closed box in the limit that the average chain length goes to infinity. Counted per chain, the confinement energy of the dense solution is qualitatively weaker than for a single ideal chain under similarly strong confinement (by one power in transverse container size). The container boundary contributes a surface tension to the free energy, which makes the effective monomer-wall affinity more repulsive. This correction increases with the average chain length. If present, edge or vertex singularities also contribute to the grand potential of the solution.

Osmotic pressure of confined square lattice self-avoiding walks

Flory–Huggins theory (Flory 1942 J. Chem. Phys. 10 51–61; Huggins 1942 J. Am. Chem. Soc. 64 2716–8) is a mean field theory for modelling the free energy of dense polymer solutions and polymer melts. In this paper we use Flory–Huggins theory as a model of a dense two-dimensional self-avoiding walk compressed in a square in the square lattice. The theory describes the free energy of the walk well, and we estimate the Flory interaction parameter of the walk to be .

New regime in polyelectrolyte solutions

Usually polyelectrolyte solutions present a peak in the polyion/polyion structure factor for some value of the momentum transfer (scattering vector). We describe a new regime in dense polymer solutions, typically above , where is proportional to the concentration c of the solution. It is however not observed for all polyelectrolytes but only for those of intermediate stiffness, usually together with an ultimate regime where is proportional to and local orientational order prevails. This regime is robust against sample preparation and stable over time. We can understand it as jammed with randomly oriented persistent segments. This may explain why fully developed nematic order is so difficult to observe in synthetic polyelectrolytes.

Field theoretic simulations of polymer nanocomposites

Polymer field theory has emerged as a powerful tool for describing the equilibrium phase behavior of complex polymer formulations, particularly when one is interested in the thermodynamics of dense polymer melts and solutions where the polymer chains can be accurately described using Gaussian models. However, there are many systems of interest where polymer field theory cannot be applied in such a straightforward manner, such as polymer nanocomposites. Current approaches for incorporating nanoparticles have been restricted to the mean-field level and often require approximations where it is unclear how to improve their accuracy. In this paper, we present a unified framework that enables the description of polymer nanocomposites using a field theoretic approach. This method enables straightforward simulations of the fully fluctuating field theory for polymer formulations containing spherical or anisotropic nanoparticles. We demonstrate our approach captures the correlations between particle positions, present results for spherical and cylindrical nanoparticles, and we explore the effect of the numerical parameters on the performance of our approach.

Polydispersity for Tuning the Potential of Mean Force between Polymer Grafted Nanoparticles in a Polymer Matrix

We present an integrated theory and simulation study of polydisperse polymer grafted nanoparticles in a polymer matrix to demonstrate the effect of polydispersity in graft length on the potential of mean force between the grafted nanoparticles. In dense polymer solutions, increasing polydispersity in graft length reduces the strength of repulsion at contact and weakens the attractive well at intermediate interparticle distances, completely eliminating the latter at high polydispersity index. The reduction in contact repulsion is attributable to polydispersity relieving monomer crowding near the particle surface, especially at high grafting densities. The elimination of the midrange attractive well is attributable to the longer grafts in the polydisperse graft length distribution that introduce longer range steric repulsion and alter the wetting of the grafted layer by matrix chains. Dispersion of the grafted particles is stabilized by increased penetration or wetting of the polydisperse grafted layer by the matrix chains. This work demonstrates that at high grafting densities, polydispersity in graft length can be used to stabilize dispersions of grafted nanoparticles in a polymer matrix at conditions where monodisperse grafts would cause aggregation.

Flashing subdiffusive ratchets in viscoelastic media

We study subdiffusive ratchet transport in periodically and randomly flashing potentials. A central Brownian particle is elastically coupled to the surrounding auxiliary Brownian quasi-particles, which account for the influence of the viscoelastic environment. Similar to standard dynamical modeling of Brownian motion, the external force influences only the motion of the central particle, not affecting directly the environmental degrees of freedom. Just a handful of auxiliary Brownian particles suffices to model subdiffusion over many temporal decades. Time modulation of the potential violates the symmetry of thermal detailed balance and induces an anomalous subdiffusive current which exhibits a remarkably small dispersion at low temperatures, as well as a number of other surprising features such as saturation at low temperatures, and multiple inversions of the transport direction upon a change of the driving frequency in the non-adiabatic regime. It is shown that the subdiffusive current is finite at zero temperature for random flashing and can be finite for periodic flashing for a certain frequency window. Our study generalizes classical Brownian motors towards operating in sticky viscoelastic environments such as the cytosol of biological cells or dense polymer solutions.

Polymerlike statistical characterization of two-dimensional granular chains

Statistical behaviors of packing collections of granular chains in a two-dimensional container have been investigated experimentally. On compaction from their own gravity, the longer chains pack into a structure with lower packing density due to the prevalence of backbone loops. The packing of chains can be considered as the jamming of the granular system. The structure factor of packing chains shows scaling behavior g(q)∼q(-2) in good agreement with dense polymer solutions. In addition, we compute various probability distributions of distances and estimate three crucial contact exponents, finding that the scaling behavior from granular chains is in accord with the theoretical expectation of polymers. Finally, an orientational anticorrelation of granular chains is observed by bond-bond correlation function, which agrees with the results in the two-dimensional model of compact polymers.

The structure factor of dense two-dimensional polymer solutions

According to the generalized Porod law the intramolecular structure factor F ( q ) of compact objects with surface dimension d s scales as F ( q ) / N ≈ 1 / ( R ( N ) q ) 2 d − d s in the intermediate range of the wave vector q with d being the dimension of the embedding space, N the mass of the objects and R ( N ) ∼ N 1 / d their typical size. By means of molecular-dynamics simulations of a bead-spring model with chain lengths up to N = 2048 it is shown that dense self-avoiding polymers in strictly two dimensions ( d = 2 ) adopt compact configurations of surface dimension d s = 5 / 4 . In agreement with the generalized Porod law the Kratky representation of F ( q ) thus reveals a nonmonotonous behavior with q 2 F ( q ) ∼ 1 / ( N 1 / 2 q ) 3 / 4 . Using a similar data analysis we argue briefly that melts of non-concatenated rings in three dimensions become marginally compact with d s = d = 3 , i.e. q 2 F ( q ) ∼ N 0 / q , for asymptotically long chains.

Stochastic entangled chain dynamics of dense polymer solutions

We propose an adjustable-parameter-free, entangled chain dynamics model of dense polymer solutions. The model includes the self-consistent dynamics of molecular chains and solvent by describing the former via coarse-grained polymer dynamics that incorporate hydrodynamic interaction effects, and the latter via the forced Stokes equation. Real chain elasticity is modeled via the inclusion of a Pincus regime in the polymer's force-extension curve. Excluded volume effects are taken into account via the combined action of coarse-grained intermolecular potentials and explicit geometric tracking of chain entanglements. We demonstrate that entanglements are responsible for a new (compared to phantom chain dynamics), slow relaxation mode whose characteristic time scale agrees very well with experiment. Similarly good agreement between theory and experiment is also obtained for the equilibrium chain size. We develop methods for the solution of the model in periodic flow domains and apply them to the computation of entangled polymer solutions in equilibrium. We show that the number of entanglements Π agrees well with the number of entanglements expected on the basis of tube theory, satisfactorily reproducing the latter's scaling of Π with the polymer volume fraction φ. Our model predicts diminishing chain size with concentration, thus vindicating Flory's suggestion of excluded volume effects screening in dense solutions. The predicted scaling of chain size with φ is consistent with the heuristic, Flory theory based value.

Distance Dependence of Angular Correlations in Dense Polymer Solutions

Angular correlations in dense solutions and melts of flexible polymer chains are investigated with respect to the distance $r$ between the bonds by comparing quantitative predictions of perturbation calculations with numerical data obtained by Monte Carlo simulation of the bond-fluctuation model. We consider both monodisperse systems and grand-canonical (Flory-distributed) equilibrium polymers. Density effects are discussed as well as finite chain length corrections. The intrachain bond-bond correlation function $P(r)$ is shown to decay as $P(r) \sim 1/r^3$ for $\xi \ll r \ll \r^*$ with $\xi$ being the screening length of the density fluctuations and $r^* \sim N^{1/3}$ a novel length scale increasing slowly with (mean) chain length $N$.

Intramolecular Form Factor in Dense Polymer Systems: Systematic Deviations from the Debye Formula

We discuss theoretically and numerically the intramolecular form factor $F(q)$ in dense polymer systems. Following Flory's ideality hypothesis, chains in the melt adopt Gaussian configurations and their form factor is supposed to be given by Debye's formula. At striking variance to this, we obtain noticeable (up to 20%) non-monotonic deviations which can be traced back to the incompressibility of dense polymer solutions beyond a local scale. The Kratky plot ($q^2F(q)$ {\it vs.} wavevector $q$) does not exhibit the plateau expected for Gaussian chains in the intermediate $q$-range. One rather finds a significant decrease according to the correction $\delta(F^{-1}(q)) = q^3/32\rho$ that only depends on the concentration $\rho$ of the solution, but neither on the persistence length or the interaction strength.

Polydisperse star polymer solutions

We analyze the effect of polydispersity in the arm number on the effective interactions, structural correlations, and phase behavior of star polymers in a good solvent. The effective interaction potential between two star polymers with different arm numbers is derived using scaling theory. The resulting expression is tested against monomer-resolved molecular dynamics simulations. We find that the theoretical pair potential is in agreement with the simulation data in a much wider polydispersity range than other proposed potentials. We then use this pair potential as an input in a many-body theory to investigate polydispersity effects on the structural correlations and the phase diagram of dense star polymer solutions. In particular, we find that a polydispersity of 10%, which is typical in experimental samples, does not significantly alter previous findings for the phase diagram of monodisperse solutions.

Network domain structure in phase-separating polymer solutions

Results from molecular dynamics simulations of phase separation in a quenched polymer solution are presented for both two- and three-dimensional systems. Simulations clearly demonstrate the existence of a networklike pattern formation at an early to intermediate time regime for sufficiently dense polymer solutions. This network structure, however, relaxes at late times as the minimization of the interface energy wins over the elastic stress of the network. A crossover of domain growth exponent from a smaller value at intermediate times to the Lifshitz–Slyozov value of 1/3 at late times is seen in three-dimensional (3D) simulations of network breakup. Our results for the late-time growth exponent and dynamical scaling of the structure factors strongly indicate that the late-time growth kinetics of quenched polymer solutions belong to the same universality class of small molecular mixtures.

The anomalous structure factor of dense star polymer solutions

The core-core structure factor of dense star polymer solutions in a good solvent is shown theoretically to exhibit an unusual behaviour above the overlap concentration. Unlike usual liquids, these solutions display a structure factor whose first peak decreases with increasing density while the second peak grows. The scenario repeats itself with the subsequent peaks as the density is further enhanced. For low enough arm numbers , various different considerations lead to the conclusion that the system remains fluid at all concentrations.

Networklike Pattern Formation in Phase Separating Polymer Solutions: A Molecular Dynamics Study

The dynamics of phase separation of a quenched polymer solution is studied using a stochastic molecular dynamics simulation. At early times, the elastic nature of the polymer chains generates a networklike domain morphology for sufficiently dense polymer solutions. However, at late times, this network structure breaks up into disconnected polymer-rich domains to minimize the interfacial energy. Our simulations explain why different experiments carried out to different time regimes seem to produce contradictory results, and strongly indicate that the true late-time growth kinetics of quenched polymer solutions belong to the same universality class of small molecular mixtures.

Tagged chain diffusion equation and effective friction tensor in dense polymer solutions

We start from the many-chain Smoluchowski equation that describes dynamics of dense polymer solutions and derive an effective diffusion equation for a tagged (probe) chain distribution function. In the tagged chain diffusion equation the effects of the inter-chain interactions are incorporated through an effective friction tensor. We propose a simple phenomenological formula for the friction tensor in which the friction on a given bead depends only on the positions of its two neighbors along the chain. This formula is used in conjunction with an exact lower bound for the center of mass self-diffusion coefficient. We show that the necessary condition for reproducing N−2 scaling of the self-diffusion coefficient is not weaker-than-linear dependence of the anisotropy on the chain length N. To check whether this also a sufficient condition, we perform a set of single chain Brownian dynamics simulations. We show that the linear chain length dependence of the anisotropy leads to Ds∼N−1.6 whereas the N2 and N3 scaling of the anisotropy result in the chain length dependence of D that is consistent with the observed N−2 behavior.

Dynamics of Selenium Chain Melts by MD Simulation

Abstract A molecular dynamics (MD) study of liquid selenium modelled by linear chains is presented. The simulated thermodynamic state corresponds to the experimental density of 3570 Kgm−3 at 873 K. We have studied melts with chains of different lengths, varying in the range of 10–80 atoms, just up to the limit where the melts relax in the temporary and spatial length window (101 ns and 28.6 A) of the numerical experiment. The flexibility of the chains at the thermodynamic state considered enabled us the observation of different spatio temporal dynamics regimes, which can be described by the Rouse model for dense polymer solutions.

Structure and dynamics of selenium chain melts: A molecular dynamics study

A molecular dynamics (MD) study of liquid selenium modeled by 16 linear chains of 40 monomers each is presented. The simulated thermodynamic state corresponds to the experimental density of 3570 Kg m−3 at 873 K. The structural and force constant data of the chains were obtained from previous studies of neutron diffraction experiments, lattice dynamics, and first principles calculations. The computed structural properties show a good agreement with available neutron scattering data. The flexibility of the chains and the high temperature thermodynamic state of the liquid enabled the observation of fast torsional motions and different spatiotemporal dynamic ranges, which can be described by the Rouse model for dense polymer solutions. We identify the crossover from an atomic to an intermediate or ‘‘universal’’ chain regime, and subsequently to global chain behaviors. The dynamics of the system is discussed in terms of time and space-dependent transport coefficients. The generated MD trajectory thus provides information on the single particle motions, the collective dynamics of one chain, and the dynamics of the global system. This separation is useful for understanding the low frequency collective motions which can be measured by inelastic neutron scattering. The spectra are interpreted in terms of existent dynamical models, which imply a degree of trapping of the atoms in some spatial regions of the liquid (‘‘chain cages’’) defined by atomic crosslinks, plus a slow diffusive process which modifies the shape of the cage according the renewal of the atomic crosslinks.