Polymer Reference Interaction Site Model Theory Research Articles

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.

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.

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.

Integral equation prediction of structure of nanocomposites with polymer-grafted nanoparticles near solid surface

In this work, the inhomogeneous polymer reference interaction site model (PRISM) integral equation theory is extended to investigate the microscopic structure of a new system of nanocomposites with polymer-grafted nanoparticles near solid surface. The chemical and morphologic details of polymers are described by a semiflexible chain model, which can be utilized to capture the intramolecular interactions between different segments. Based on the novel bridge functionals of different segments constructed from the corresponding chemical potentials, the modified hypernetted chain approximation is combined with the inhomogeneous PRISM equation to obtain the detailed density distributions of nanoparticle-polymer blends near substrate. The effects of particle volume fraction, nanoparticle size, attractive interaction strength between different segments and chain stiffness on the density profiles of single tethered polymer nanocomposites (PNCs) near surface are systematically investigated to capture the contribution of the packing and configurational entropies. The end effects of polymer chains and effects of solid wall confinement are also considered. It is found that the particle size and particle volume fraction play the most significant roles in the density profiles of the single tethered PNCs among the present investigated model parameters. The increase of the particle volume fraction leads to a monotonic increasing variation behavior of the density profiles of single tethered nanoparticles as the particle diameter is relatively small. With increasing the nanoparticle size, the intensity of the density profiles of nanoparticles decreases prominently. The detailed packing structure of polymer-nanoparticle blends is also influenced by the polymer chain stiffness and particle-monomer interactions, where these parameters of structure and chemistry can enable the density profiles of the single tethered PNCs to show similar variation trends. The present inhomogeneous PRISM theory can provide a detailed description of packing and conformational entropies on the density profiles of different segments in nanocomposites with polymer-grafted nanoparticles. It could also be easily extended to some real systems with more complicated architecture of polymer-grafted nanoparticles under confinement and anticipated to assist in developing some predictive approaches for design control of PNCs near substrate.

Phase separation of comb polymer nanocomposite melts.

In this work, the spinodal phase demixing of branched comb polymer nanocomposite (PNC) melts is systematically investigated using the polymer reference interaction site model (PRISM) theory. To verify the reliability of the present method in characterizing the phase behavior of comb PNCs, the intermolecular correlation functions of the system for nonzero particle volume fractions are compared with our molecular dynamics simulation data. After verifying the model and discussing the structure of the comb PNCs in the dilute nanoparticle limit, the interference among the side chain number, side chain length, nanoparticle-monomer size ratio and attractive interactions between the comb polymer and nanoparticles in spinodal demixing curves is analyzed and discussed in detail. The results predict two kinds of distinct phase separation behaviors. One is called classic fluid phase boundary, which is mediated by the entropic depletion attraction and contact aggregation of nanoparticles at relatively low nanoparticle-monomer attraction strength. The second demixing transition occurs at relatively high attraction strength and involves the formation of an equilibrium physical network phase with local bridging of nanoparticles. The phase boundaries are found to be sensitive to the side chain number, side chain length, nanoparticle-monomer size ratio and attractive interactions. As the side chain length is fixed, the side chain number has a large effect on the phase behavior of comb PNCs; with increasing side chain number, the miscibility window first widens and then shrinks. When the side chain number is lower than a threshold value, the phase boundaries undergo a process from enlarging the miscibility window to narrowing as side chain length increases. Once the side chain number overtakes this threshold value, the phase boundary shifts towards less miscibility. With increasing nanoparticle-monomer size ratio, a crossover of particle size occurs, above which the phase separation is consistent with that of chain PNCs. The miscibility window for this condition gradually narrows while the other parameters of the PNCs system are held constant. These results indicate that the present PRISM theory can give molecular-level details of the underlying mechanisms of the comb PNCs. It is hoped that the results can be used to provide useful guidance for the future design control of novel, thermodynamically stable comb PNCs.

Structure and effective interactions of comb polymer nanocomposite melts.

In this work, the structure and effective interactions of branched comb polymer nanocomposite (PNC) melts are investigated by using the polymer reference interaction site model (PRISM) integral equation theory. It is observed that the nanoparticle contact (bridging) aggregation is formed when the nanoparticle-monomer attraction strength is relatively weak (large) in comb PNCs. The organization states of aggregation for the moderate nanoparticle-monomer attraction strength can be well suppressed by the comb polymer architecture, while the bridging structure for relatively large attraction is obviously promoted. With the increase of the particle volume fraction, the organization states of bridging-type structure become stronger and tighter; however, this effect is weaker than that of the nanoparticle-monomer attraction strength. When the particle volume fraction and moderate nanoparticle-monomer attraction strength are fixed, the effects of degree of polymerization, side chain number, side chain length, and nanoparticle-monomer size ratio on the organization states of PNC melts are not prominent and the nanoparticles can well disperse in comb polymer. All the observations indicate that the present PRISM theory can give a detailed description of the comb PNC melts and assist in future design control of new nanomaterials.

Modified PRISM theory for confined polymers

We propose a modified polymer reference interaction site model (PRISM) to describe the interfacial density profiles of polymers in contact with planar and curved solid surfaces. In the theoretical approach, a bridge function derived from density functional method is included. In description of hard-sphere polymer at planar and curved surfaces with an arbitrary external field, the effect of modification has been validated by the available simulation data, except for low density system. When extended to confined real systems, the modified theoretical model also shows an encouraging prospect in description of the interfacial structure and properties.

Integrating PRISM theory and Monte Carlo simulation to study polymer-functionalised particles and polymer nanocomposites

Polymer nanocomposites consist of nanoscale additives embedded in a polymer matrix and are widely used in the automobiles, optics and microelectronics industries. Since the composition and the morphology of the polymer nanocomposite impact its macroscopic properties, significant efforts have been made to understand how parameters, such as polymer and nanoparticle chemistries, molecular weight of the matrix polymers and nanoparticle size, help tune the morphology. Theory and simulations have proven to be useful tools in this field due to their ability to link molecular level interactions, the morphology and the macroscopic properties. Due to the computational intensity of molecular simulations of a dense polymer matrix, there has been a strong effort on the theoretical front to develop methodologies that map out equilibrium structure and phase behaviour of polymer nanocomposites over a large parameter space. In this paper, we review the details of the self-consistent polymer reference interaction site model (PRISM)–Monte Carlo (MC) simulation method which integrates theory and simulation to study phase behaviour in polymer nanocomposites. We discuss two specific cases of polymer nanocomposites containing polymer-grafted nanoparticles with chemical and physical heterogeneity in grafts in which this self-consistent PRISM–MC approach has been used to study effective inter-filler interactions and phase behaviour.

X-ray Scattering of Vinyl Polyolefin Liquids and Random Copolymers: Theory and Experiment

Polymer reference interaction site model (PRISM) calculations and wide-angle X-ray experiments were performed on vinyl polyolefin melts of polyethylene (PE), isotactic polypropylene (iPP), and poly(1-butene) (P1B). PRISM calculations were also carried out on poly(1-hexene) (P1H). For PE and iPP melts a single main peak in the scattering pattern was observed near wave vectors k ∼ 1 A−1; for P1B and P1H we observed this main peak to split with the development of a prepeak that moves to lower wave vectors as the side chain size increased. By calculation of the various partial structure factors, we found that the prepeak formation is due to a combination of two effects: (1) an increase in the average distance between the chain backbones due to screening of the side chains and (2) increased intermolecular ordering of the pendant methyl groups as the side chains become longer. The scattering of random copolymers (PEP, PEB, PEH) of PE with iPP, P1B, and P1H, respectively, was also studied with PRISM theory as a ...

Description of the Structure and Properties of Atactic Polystyrene Melt Using Integral Equation Theory

AbstractThe polymer reference interaction site model (PRISM) integral equation theory was used to describe the structure and thermodynamic properties of atactic polystyrene (aPS) melt, in which the monomer of aPS is represented with an eight‐site model to characterize its microstructure. The intramolecular structure factors needed in the PRISM calculations were obtained from single chain MD simulations. The calculated results indicate that the results by the integral equation method agrees well with experiments, and can reflect the fine microscopic structure of real aPS melt. This work shows that the PRISM theory is a powerful tool for investigating the structure and properties of complex polymers.

Structural properties of crystallizable polymer melts: Intrachain and interchain correlation functions

We present results from constant pressure molecular-dynamics simulations for a bead-spring model of a crystallizable polymer melt. Our model has two main features, a chemically realistic intrachain rigidity and a purely repulsive interaction between nonbonded monomers. By means of intrachain and interchain structure factors we explore polymer conformation and melt structure above and below the temperature T{crys}{hom} of homogeneous crystallization. Here, we do not only determine average spatial correlations, but also site-specific correlations which depend on the position of the monomers along the polymer backbone. In the liquid phase above T{crys}{hom} we find that this site dependence can be well-accounted for by known theoretical approximations, the Koyama distribution for the intrachain structure and the polymer reference interaction site model (PRISM) for the interchain structure. This is no longer true in the semicrystalline phase. Below T{crys}{hom} short chains fully extend upon crystallization, whereas sufficiently long chains form chain-folded lamellae which coexist with amorphous regions. The structural features of these polymer crystals lead to violations of premises of the Koyama approximation or PRISM theory so that both theoretical approaches cannot be applied simultaneously. Furthermore, we find a violation of the Hansen-Verlet freezing criterion; our polymer melt crystallizes more easily than a simple liquid. This hints at the importance of the coupling between conformation (backbone rigidity) and density (packing constraints) for polymer crystallization.

Integral equation theory for atactic polystyrene melt with a coarse-grained model

In this work, an integral equation approach to investigate the atactic polystyrene (aPS) melt based on polymer reference interaction site model (PRISM) theory is proposed. The intramolecular structure factors, required as input to PRISM theory, are obtained from the semiflexible chain model. With a novel coarse-graining procedure and the explicit-atom molecular-dynamics (MD) simulations for aPS, the parameters needed for the coarse-grained model are obtained by using an automatic simplex optimization. These parameters can be used to describe the structure and thermodynamic properties of the complex aPS melt and good agreement is obtained between the theory and MD simulations. The proposed integral equation approach provides a basis for describing the structure and properties of PS nanocomposites where the application of molecular simulation is difficult.

Integral equation theory of random copolymer melts: Self-consistent treatment of intramolecular and intermolecular correlations

A self-consistent integral equation theory is presented for the conformational properties and spinodal lines of random copolymer melts. The theory combines field-theoretic methods with the polymer reference interaction site model (PRISM) theory. The many-chain problem is replaced by a single chain where the sites interact via a bare plus a self-consistently determined medium-induced potential, and the conformational properties are obtained using a variational method. The theoretical prediction for the spinodal line is qualitatively similar to that of non-self-consistent PRISM theory. The theory predicts macroscopic phase separation for all values of the monomer correlation strength, lambda. The inverse spinodal temperature is a nonmonotonic function of lambda with a maximum at lambda(max). For large values of lambda( approximately 1), the values of spinodal temperatures are almost identical to those of non-self-consistent PRISM theory. For low values of lambda, however, the theory predicts higher values for spinodal temperatures than non-self-consistent PRISM theory. The theory predicts significant changes in the mean-square end-to-end distance as the temperature is decreased.

Integral Equation Theory of Random Copolymer Melts

An integral equation theory is developed for the structure and thermodynamics of melts of random copolymers. The molecules are modeled as flexible chains with a random sequence of two types of sites, with square well potentials between sites. The polymer reference interaction site model (PRISM) integral equation theory is extended to random copolymers and used to calculate the static correlations and spinodal lines. For threadlike chains and with the incompressible random phase approximation (IRPA), the PRISM theory reduces to a replica field theory or Landau-type theory presented earlier by others. In this limit the theory predicts a macrophase separation for values of the monomer correlation strength, λ, greater than a critical value. For smaller values of λ a microphase separation is predicted with strong concentration fluctuations at finite wave vectors. With the hard-core interactions included, however, the predictions of the theory with different closure approximations are qualitatively different from each other. With atomic closure approximations the nature of the spinodal diagram is similar to the IRPA, although fluctuations are predicted to promote microphase separation. With molecular closure approximations the theory predicts that no microphase separation occurs in these systems and that there is a macrophase separation for all values of λ.

Structure of a freely jointed tangent hard-sphere chain in hard slit pores

A density functional theory (DFT), which is based both on the density functional expansion of the first-order direct correlational functional with respect to the bulk molecular density and on the contact-value theorem at the hard wall, has been developed for studying the structure of a freely jointed tangent hard-sphere chain in hard slit pores. Two different second-order direct correlation functions (DCFs) based on the Polymer Reference Interaction Site model (PRISM) coupled with the Percus–Yevick closure relation and the Wertheim's first-order perturbation theory modified by Yu and Wu (modified TPT1) were used as inputs. The calculated results show that (i) the PRISM theory is in excellent agreement with the computer simulations even for high densities, and that (ii) the calculated local density distribution strongly depends on the second-order DCFs as well as the DFT version employed.