Polymer Reference Interaction Site Model Research Articles

Closure to the PRISM equation derived from nonlinear response theory.

Nonlinear response theory is employed to derive a closure to the polymer reference interaction site model equation. The closure applies to a liquid of neutral polymers at melt densities. It can be considered a molecular generalization of the mean spherical approximation (MSA) closure of Lebowitz and Percus to the atomic Ornstein-Zernike (OZ) equation and is similar in some aspects to the reference "molecular" MSA (R-MMSA) closure of Schweizer and Yethiraj to PRISM. For a model binary blend of freely-jointed chains, the new closure predicts an unmixing critical temperature, Tc, via the susceptibility route that scales linearly with molecular weight, N, in agreement with Flory theory. Predictions for Tc of the new closure differ greatest from those of the R-MMSA at intermediate N, the latter being about 40% higher than the former there, but at large N, both theories give about the same values. For an isotopic blend of polyethylene, the new and R-MMSA closures predict a Tc about 25% higher than the experimental value, which is only moderately less accurate than the prediction of atomic OZ-MSA theory for Tc of methane. In this way, the derivation and its consequences help to identify the ingredients in a theory needed to properly model the equilibrium properties of a polymeric liquid at both short and long lengthscales.

Coexistence of two coacervate phases of polyglycine in water suggested by polymer reference interaction site model theory.

Mixing Gibbs energy and phase equilibria of aqueous solutions of polyglycine were studied theoretically by means of polymer reference interaction site model integral equation theory combined with the Gibbs-Duhem method. In addition to the ordinary liquid-liquid phase separation between dilute and concentrated solutions, the theoretical calculation predicted the coexistence of two coacervate phases, namely, the lower- and higher-density coacervates. The relative thermodynamic stabilities of these two phases change with the polymerization degree of polyglycine. The higher-density coacervate phase was rapidly stabilized by increasing the polymer length, and the lower-density phase became metastable at large polymers. The hydrogen bonds between the peptide chains were strengthened, and water was thermodynamically destabilized in the higher-density coacervate. A possible relation with the formation of amyloid fibril within a liquid droplet is also discussed.

Liquid state theory study of the phase behavior and macromolecular scale structure of model biomolecular condensates.

Biomolecular condensates can form through the liquid-liquid phase separation (LLPS) of proteins and RNAs in cells. However, other states of organization, including mesostructured network microstructures and physical gels, have been observed, the physical mechanism of which are not well understood. We use the Polymer Reference Interaction Site Model liquid state integral equation theory to study the equilibrium behavior of (generally aperiodic in sequence) biomolecular condensates based on a minimal sticker-spacer associating polymer model. The role of polymer packing fraction, sequence, and the strength and range of intermolecular interactions on macromolecular scale spatial organization and phase behavior is studied for typical sticker-spacer sequences. In addition to the prediction of conventional LLPS, a sequence-dependent strongly fluctuating polymeric microemulsion homogeneous state is predicted at high enough concentrations beyond the so-called Lifshitz-like point, which we suggest can be relevant to the dense phase of microstructured biomolecular condensates. New connections between local clustering and the formation of mesoscopic microdomains, the influence of attraction range, compressibility, and the role of spatial correlations across scales, are established. Our results are also germane to understanding the polymer physics of dense solutions of nonperiodic and unique sequence synthetic copolymers and provide a foundation to create new theories for how polymer diffusion and viscosity are modified in globally isotropic and homogeneous dense polymeric microemulsions.

Effects of intramolecular chain conformation on the hydration and miscibility of polyethylene glycol in water studied by means of polymer reference interaction site model theory.

To examine the conventional idea that the gauche conformation of the OCCO dihedral angle promotes the dissolution of polyethylene glycol (PEG) in water through strong hydration, the thermodynamic properties of liquid mixtures of PEG and water were studied by means of polymer reference interaction site model (PRISM) theory. The intramolecular correlation functions required as input for PRISM theory were calculated by the generator matrix method, accompanied by changes in the distribution of dihedral angles. In the infinite dilution limit, the increased probability of gauche conformation of the OCCO dihedral angles stabilizes the hydration of PEG through enhanced hydrogen bonding between the ether oxygen of PEG and water. The mixing Gibbs energies of the liquid mixtures were also calculated in the whole concentration range based on the Gibbs-Duhem equation, as per our recent proposal. A liquid-liquid phase separation was observed when all the dihedral angles of PEG were in the trans conformation; for the liquid mixture to be miscible in the whole concentration range, the introduction of the OCCO gauche conformation was found to be indispensable. The above theoretical results support the conventional idea that the OCCO gauche conformation is important for the high miscibility of PEG and water.

Correlations in Hard- and Soft-Core Generic Polymer Models

Generic polymer models capturing the chain connectivity and the non-bonded excluded-volume interactions between polymer segments can be classified into hard- and soft-core models depending on their non-bonded pair potential. Here we compared the correlation effects on the structural and thermodynamic properties of the hard- and soft-core models given by the polymer reference interaction site model (PRISM) theory, and found different behaviors of the soft-core models at large invariant degree of polymerization (IDP) depending on how IDP is varied. We also proposed an efficient numerical approach, which enables us to accurately solve the PRISM theory for chain lengths as large as 106.

Semiflexible polymer nanocomposites: Role of stiffness on structure and macrophase separation

Polymer nanocomposites are lighter in weight as compared to the traditionally bigger size filler-loaded composites. Moreover, polymer nanocomposites show enhancement in properties with small fractions of nanoparticles. In certain cases, a high fraction of nanoparticles is needed to incorporate the required properties significantly. In most of the theoretical and simulation studies of polymeric systems, the polymer chain is modeled as ideal freely-jointed chains, thereby neglecting the effects of chain stiffness. We study a system of semiflexible polymers containing 50% volume fraction of nanoparticles using Polymer Reference Interaction Site Model (PRISM) theory. In the absence of chemical anisotropy, these systems undergo a disordered fluid phase to a macrophase-separated state. The phase transition of these systems is favored by an increase in the persistence length of the polymer chain. At lower values of persistence lengths, a slight increase in the persistence length results in a sharp decrease in the critical packing fraction. However, for longer persistence lengths, an increase in the persistence length barely affects the critical packing fraction. Studies of the local structure of these systems show that an increase in persistence length increases the polymer–polymer contacts and polymer-nanoparticle contacts. On the contrary, nanoparticle-nanoparticle contact initially increases suddenly, then decreases slightly at longer persistence lengths.

Electrostatic Repulsion Slows Relaxations of Polyelectrolytes in Semidilute Solutions

We investigate the structure and dynamics of unentangled semidilute solutions of sodium polystyrenesulfonate (NaPSS) using small-angle neutron scattering (SANS) and neutron spin-echo (NSE) spectroscopy. The effects of electrostatic interactions and chain structure are examined as a function of ionic strength and polymer concentration, respectively. The SANS profiles exhibit a characteristic structural peak, signature of polyelectrolyte solutions, that can be fit with a combination of a semiflexible chain with excluded volume interactions form factor and a polymer reference interaction site model (PRISM) structure factor. We confirm that electrostatic interactions vary with ionic strength across solutions with similar geometries. The segmental relaxations from NSE deviate from theoretical predictions from Zimm and exhibit two scaling behaviors, with the crossover between the two regimes taking place around the characteristic structural peak. The chain dynamics are suppressed across the length scale of the correlation blob, and inversely related to the structure factor. These observations suggest that the highly correlated nature of polyelectrolytes presents an additional energy barrier that leads to de Gennes narrowing behavior.

Shear-induced nanostructural changes in micelles formed by sugar-based surfactants with varied anomeric configuration

HypothesisThe self-assembly of long tail sugar-based surfactants into worm-like micelles has recently been demonstrated, and the rheological properties of such systems have been shown to be tuneable through subtle modifications of the molecular characteristics of the surfactant monomer. In particular, the anomeric configuration of the hexadecylmaltoside headgroup was shown to induce profound changes in the nanostructure and rheology of the system. The origin of such changes is hypothesised to arise from differences in the structure and relaxation of the micellar networks in the semi-dilute regime. ExperimentsHere we explore the molecular background to the flow properties of the two anomers of hexadecylmaltoside (α- and β-C16G2) by directly connecting their rheological behaviour to the micelle morphology. For this purpose, 1–3 plane rheo-small-angle neutron scattering measurements, using a Couette cell geometry, probed the structural changes in the micellar phase under shear. The effect of surfactant anomeric configuration, surfactant concentration, temperature and mixing ratio of the two anomers were investigated. The static micelle structure in the semi-dilute regime was determined using the polymer reference interaction site model. FindingsThe segmental alignment of the micellar phase was studied under several flow conditions, showing that the shear-thinning behaviour relates to the re-arrangement of β-C16G2 worm-like micelles, whilst shorter α-C16G2 micelles are considerably less affected by the flow. The results are rationalised in terms of micelle alignment and disruption of the entangled network, providing a detailed mechanism by which sugar-based surfactants control the rheology of the fluid. To further enable future studies, we provide the complete code for modelling micelle structure in the semi-dilute regime using the polymer reference interaction site model.

Phase behavior of polymer-nanorod composites: A comparative study using PRISM theory and molecular dynamics simulations.

Well-dispersed composites of polymer and nanorods have many emerging applications and, therefore, are an important area of research. Polymer reference interaction site model (PRISM) theory and molecular dynamics simulations have become powerful tools in the study of the structure and phase behavior of polymer nanocomposites. In this work, we employ both PRISM theory and molecular dynamics simulations to determine the structure and spinodal phase diagram of 1% volume fraction of nanorods in a polymer melt. We make quantitative comparisons between the phase diagrams, which are reported as a function of nanorod aspect ratio and polymer-nanorod interactions. We find that both PRISM theory and molecular dynamics simulations predict the formation of contact aggregates at low polymer-nanorod attraction strength (γ) and bridged aggregates at high polymer-nanorod attraction strength. They predict an entropic depletion-driven phase separation at low γ and a bridging-driven spinodal phase separation at high γ. The polymer and nanorods are found to form stable composites at intermediate values of the polymer-nanorod attraction strength. The fall of the bridging boundary and the gradual rise of the depletion boundary with the nanorod aspect ratio are predicted by both PRISM theory and molecular dynamics simulations. Hence, the miscible region narrows with increasing aspect ratio. The depletion boundaries predicted by theory and simulation are quite close. However, the respective bridging boundaries present a significant quantitative difference. Therefore, we find that theory and simulations qualitatively complement each other and display quantitative differences.

Interfacial Compatibilization in Ternary Polymer Nanocomposites: Comparing Theory and Experiments

In this work, we examine binary and ternary nanocomposites of poly(methyl methacrylate) grafted silica nanoparticles (PMMA-NP), in poly(styrene-ran-acrylonitrile) (SAN), and poly(methyl methacrylate) matrices as a platform to directly probe governing parameters guiding phase behavior and nanoparticle assembly in composite materials. Through the addition of PMMA matrix chains similar in molecular weight to the grafted PMMA chains and significantly smaller than the SAN matrix chains, we observe increased nanoparticle miscibility in off-critical compositions due to interfacial segregation of PMMA matrix chains. A simple interfacial model provides a general guideline for predicting the extent of compatibilization. Further insights on compatibilization behavior are provided by polymer particle pair correlation functions and structure factors obtained using polymer reference interaction site model theory calculations as well as polymer concentration profiles from molecular dynamics simulations. This study serves as a guideline to facilitate PNC processing and design of materials for a broad range of technological applications.

In Situ Nanostructural Analysis of Concentrated Wormlike Micellar Fluids Comprising Sodium Laureth Sulfate and Cocamidopropyl Betaine Using Small-Angle Neutron Scattering.

Concentrated wormlike micellar fluids form the basis for a vast array of formulated products, from liquid soaps and shampoos to drag reduction and drilling fluids. Typically, these systems are analyzed using bulk rheological measurements to determine their flow properties and cryo-microscopy to detect their nanostructure. Small-angle neutron scattering provides an opportunity to directly and nonperturbatively analyze nanostructure in situ but is complicated for concentrated systems by correlations from interparticle volume exclusion. Here, we use small-angle and ultra-small-angle neutron scattering to probe directly for the first time the nanostructure of concentrated wormlike micellar fluids composed of the widely used surfactant pair sodium laureth sulfate and cocamidopropyl betaine in aqueous electrolytes. Obtained data are analyzed using different approaches to determine scattering contributions from the wormlike particles themselves and interactions between them. It is found that approximating worms as locally rigid cylinders offers some insight into their aggregation dimensions at short length scales, and both volume exclusion and screened Coulombic interaction potentials describe interactions reasonably well. Using the semi-empirical polymer reference interaction site model (PRISM) gives excellent agreement with observed scattering, and physical insight obtained using this approach is discussed in detail. A drawback of this method is the significant complexity in coding the model in order to fit data, so to facilitate this for future researchers, we provide with this paper a fully operational, open-source code to utilize this model.

Theory of microstructure-dependent glassy shear elasticity and dynamic localization in melt polymer nanocomposites.

We present an integrated theoretical study of the structure, thermodynamic properties, dynamic localization, and glassy shear modulus of melt polymer nanocomposites (PNCs) that spans the three microstructural regimes of entropic depletion induced nanoparticle (NP) clustering, discrete adsorbed layer driven NP dispersion, and polymer-mediated bridging network. The evolution of equilibrium and dynamic properties with NP loading, total packing fraction, and strength of interfacial attraction is systematically studied based on a minimalist model. Structural predictions of polymer reference interaction site model integral equation theory are employed to establish the rich behavior of the interfacial cohesive force density, surface excess, and a measure of free volume as a function of PNC variables. The glassy dynamic shear modulus is predicted to be softened, reinforced, or hardly changed relative to the pure polymer melt depending on system parameters, as a result of the competing and qualitatively different influences of interfacial cohesion (physical bonding), free volume, and entropic depletion on dynamic localization and shear elasticity. The localization of polymer segments is the dominant factor in determining bulk PNC softening and reinforcement effects for moderate to strong interfacial attractions, respectively. While in the athermal entropy-dominated regime, the primary origin of mechanical reinforcement is the stress stored in the aggregated NP subsystem. The PNC shear modulus is often qualitatively correlated with the segment localization length but with notable exceptions. The present work provides the foundation for developing a theory of segmental relaxation, Tg changes, and collective NP dynamics in PNCs based on a self-consistent treatment of the cooperative activated motions of segments and NPs.

Dispersion and Aggregation of Polymer Grafted Particles in Polymer Nanocomposites Driven by the Hardness and Size of the Grafted Layer Tuned by Attractive Graft–Matrix Interactions

We use molecular dynamics (MD) simulations and Polymer Reference Interaction Site Model (PRISM) theory with a coarse-grained (CG) model to study polymer nanocomposites (PNCs) comprised of polymer grafted nanoparticles in a polymer matrix. Specifically, we describe the impact of increasing graft–matrix attraction on the PNC structure quantified in terms of the extent of interpenetration of matrix and graft chains (i.e., grafted layer wetting) and dispersion/aggregation of the grafted particles in the matrix via intermolecular pair correlation functions and structure factors. Past work on PNCs with attractive graft–matrix interactions had already established that grafted layer wetting–dewetting and dispersion-aggregation are two distinct phase transitions with the former being a continuous transition and the latter being a first-order transition with increasing graft–matrix attraction. In this paper, we go beyond that previous work and show that the dispersion and aggregation of polymer grafted particles in...

Analytic liquid state theory of the polymer-mediated depletion interaction between colloids beyond preaveraging approximation.

We develop a version of the polymer reference interaction site model of the polymer-mediated depletion interaction not relying on the conventional preaveraging of the polymer correlation functions (so-called "preaveraging of the polymer end effects"). The developed approach makes it possible to properly take into account the entropic interactions between the polymers and colloid surfaces, imposed by the polymer end effects. These entropic interactions are shown to give rise to an additional long-ranged contribution to the depletion potential that is comparable to its main osmotic part. The presence of the described entropic interactions essentially changes, in particular, the dependence of the second virial coefficient on the colloid radius. Based on the detailed comparison with the simulations in the work of Doxastakis et al. [J. Chem. Phys. 123, 034901 (2005)], we suggest that the described entropic effect gives an explanation for the fact that the range of the depletion potential obtained in these simulations is of the order of the polymer gyration radius even at relatively large polymer densities.

Structure-based coarse graining of diblock copolymer melts using PRISM theory

We applied our recently proposed systematic and simulation-free strategy for the structure-based coarse graining of multicomponent polymeric systems (Wang, Q., Polymer2017,117, 315) to diblock copolymer melts, where we use the well-developed polymer reference interaction site model (PRISM) theory, instead of the commonly used many-chain molecular simulations, to obtain the structural and thermodynamic properties of both the original and coarse-grained (CG) systems, and to quantitatively examine how the effective non-bonded pair potentials between CG segments and the thermodynamic properties of CG systems vary with the coarse-graining level. We proved that our strategy does not change the spinodal curve, regardless of the original model system, closures, and coarse-graining levels for the two blocks. Using a simple original model system for symmetric diblock copolymer melts, we coarse-grained each block as N/2 segments and examined CG systems with N ranging from 2 to 100. We found that coarse graining (i.e., decreasing N) increases both the peak value and peak location of the partial structure factor characterizing the composition fluctuations in the CG system. Contrary to the common practice in the literature, CG potentials obtained from short-chain systems cannot be directly used for long-chain systems; this is in fact the transferability problem in coarse graining (Louis, A. A., J. Phys.: Condens. Matter2002,14, 9187). We also found that the structure-based coarse graining cannot give thermodynamic consistency at any coarse-graining level, consistent with our previous work on homopolymer melts (Yang, D. L.; Wang, Q., J. Chem. Phys.2015,142, 054905; Wang, Q.; Yang, D. L., Polymer2017,111, 103) and binary polymer blends (Wang, Q., Polymer2017,117, 315).

Molecular Dynamics Simulation and PRISM Theory Study of Assembly in Solutions of Amphiphilic Bottlebrush Block Copolymers

We use a coarse-grained model with both Polymer Reference Interaction Site Model (PRISM) theory and molecular dynamics (MD) simulations to study self-assembly of amphiphilic bottlebrush block copolymers (BCPs) in solution as a function of increasing solvophobicity of the solvophobic blocks. First, we evaluate the ability of PRISM theory to describe and predict structure (e.g., intermolecular pair correlation functions and structure factors) and thermodynamics (e.g., disorder to order/assembled state transition solvophobicity and critical micelle concentration) for solutions of bottlebrush BCPs for varying BCP sequence, composition and solution concentration. Direct comparison of intermolecular pair correlation functions and structure factors from PRISM theory with that from MD simulations shows excellent qualitative, with some quantitative, agreement. Additionally, PRISM theory results at low solvophobicities present signatures of the structures observed in MD simulations at higher solvophobicities. Compa...

Integral Equation Prediction of the Structure of Alternating Copolymer Nanocomposites near a Substrate.

The packing structure and phase behavior of polymer-nanoparticle mixtures under confinement play an important role in developing strategies for rational design of nanomaterials. However, understanding the microscopic dispersion and aggregation mechanism of polymer nanocomposites is a great challenge through experimental techniques. In this work, the microscopic structure of alternating copolymer nanocomposites (ACNs) near a substrate is investigated systematically through extension of the inhomogeneous polymer reference interaction site model (PRISM) theory. In order to characterize the flexibility and internal chain stiffness of copolymers, a semiflexible chain model is introduced to describe the intramolecular correlations between different monomers. Based on the bridge functionals derived from the fluids density functional theory, the modified hypernetted chain closure is integrated with the PRISM equation to form a full theoretical framework to capture the density distributions of ACNs. The influence of the particle volume fraction, nanoparticle diameter, and adsorption strengths between different interaction sites on the packing structure of ACNs under confinement is analyzed and discussed in detail. With the increase of the particle volume fraction, the size asymmetry between nanoparticles and copolymer monomers can greatly influence the density profiles of ACNs near a substrate. Increasing the nanoparticle diameter, the density distribution of nanoparticles experiences a process from absorbing onto the solid surface to segregating from the wall to larger distances. With increasing the adsorption strength between copolymers and nanoparticles, the density distribution of nanoparticles decreases, which is similar to the case of nanoparticles containing attractive interactions. All these characteristics of ACNs show that the current inhomogeneous PRISM theory can give a detailed description of the packing behavior of different segments. Predictive approaches could be desired and developed for design control of alternating copolymer nanocomposites under confinement.

PyPRISM: A Computational Tool for Liquid-State Theory Calculations of Macromolecular Materials

The Polymer Reference Interaction Site Model (PRISM) theory describes the equilibrium spatial-correlations of liquid-like polymer systems including melts, blends, solutions, block copolymers, ionomers, liquid crystalline polymers, and nanocomposites. Using PRISM theory, one can calculate thermodynamic (second virial coefficients, Flory-Huggins χ interaction parameters, potentials of mean force) and structural (pair correlation functions, structure factors) data for these macromolecular materials. Here, we present a Python-based, open-source framework, pyPRISM, for conducting PRISM theory calculations. This framework aims to simplify PRISM-based studies by providing a user-friendly scripting interface for setting up and numerically solving the PRISM equations. pyPRISM also provides data structures, functions, and classes that streamline PRISM calculations, allowing pyPRISM to be extended for use in other tasks, such as the coarse-graining of atomistic simulation force-fields or the modeling of experimental scattering data. The goal of this framework is to reduce the barrier to correctly and appropriately using PRISM theory and to provide a platform for rapid calculations of the structure and thermodynamics of polymeric fluids and nanocomposites.

PRISM Theory Study of Amphiphilic Block Copolymer Solutions with Varying Copolymer Sequence and Composition

We present a comparison of Polymer Reference Interaction Site Model (PRISM) theory and molecular dynamics (MD) simulations for studying amphiphilic block copolymers in solution. We use a generic coarse-grained model to represent amphiphilic A–B block copolymers in implicit solvent with the solvophobicity of the B segments captured using effective B–B pairwise attraction modeled using the Lennard-Jones potential. We study the assembly of the amphiphilic A–B block copolymer as a function of solvophobicity for varying copolymer sequences (diblock and triblock) and composition (solvophobic-rich or solvophilic-rich). The PRISM theory equation along with the atomic Percus–Yevick closure is solved to obtain the intermolecular pair correlations in real space, g(r), and structure factors in Fourier space, S(k), for block copolymer solutions at increasing values of solvophobicity. The real-space intermolecular pair correlation functions and structure factors from PRISM theory and from MD simulations are compared di...

Effect of Polymer Architecture on the Structure and Interactions of Polymer Grafted Particles: Theory and Simulations

We use Langevin dynamics simulations and Polymer Reference Interaction Site Model (PRISM) theory to study polymer grafted nanoparticles specifically to explain the impact of comb polymer architecture on the grafted layer structure and effective interparticle interactions in solvent and in matrix polymer. First, we use simulations to study a single particle grafted with comb polymers with varying comb polymer design (i.e., spacing and length of side chains along the comb polymer backbone), grafting density (i.e., polymer chains/particle surface area), and particle curvature in implicit solvent at the athermal limit. We find that increasing side chain length or decreasing side chain spacing along the comb polymer effectively swells and extends the polymer backbone due to the increasing side chain monomer crowding. For particles at finite curvature with increasing side chain monomer crowding, the monomer concentration profile of the comb polymer backbone at short distances from the surface resembles the concentration profile of a semiflexible linear polymer and at farther distances resembles that of flexible linear polymers grafted to a flat surface. As the particle curvature decreases to zero (i.e., flat surface), increasing side chain crowding has a simpler effect of expanding the grafted layer without changing the overall shape of the concentration profile. To understand how architecture affects the interactions of the comb polymer grafted particles, we use PRISM theory to calculate the potential of mean force (PMF) between comb polymer grafted particles in implicit solvent, explicit solvent, and explicit matrix of athermal linear polymers. On the basis of the PMFs calculated for a wide range of design parameters (grafting density, comb polymer design), we find that, compared to linear polymers, the comb polymers exhibit stronger effective attraction in the PMF between the grafted particles in both small molecule solvent and polymer matrix due to the increased crowding in the grafted layer from the comb polymer side chains. Interestingly, the PMF between the grafted particles in a small molecule solvent is more sensitive to the comb polymer design (i.e., side chain length and spacing) than the PMF between the grafted particles in a polymer matrix.