Density functional theory for a primitive model of nanoparticle-block copolymer mixtures

Amphiphilic block copolymers provide useful templates for fabrication of nanostructured materials that are appealing for a wide variety of applications. The preparation of polymer-particle hybrid materials requires a good understanding of the chemical nature and topology of the amphiphilic molecules as well as their interactions with the embedded nanoparticles. This article reports a density functional theory (DFT) for a coarse-grained model of block copolymer-nanoparticle mixtures that is able to account for the properties of particles and copolymers within a self-consistent framework. It predicts various well-organized structures that can be effectively controlled by adjusting the polymer chain length and polymer-particle interactions. Illustrative examples based on relatively short chains suggest that, in qualitative agreement with experiments, large particles tend to be excluded from a polymer brush near a solid substrate, whereas smaller particles may be dissolved. The DFT is able to capture the dispersion of large particles in the microdomain of block copolymer that is energetically favorable, but localization of smaller particles at the microdomain interfaces.     

Adsorption of copolymers in a selective nanoslit: a hybrid density functional theory

A hybrid density functional theory (DFT) is developed for adsorption of copolymers in a selective nanoslit. The DFT incorporates a single-chain simulation for the ideal-gas free energy functional with two weighted density approximations for the residual free energy functional. The theory is found to be insensitive to the width parameter used in the weighted density. Theoretical predictions are in excellent agreement with simulation results in the segment density profiles and the adsorption configurations including tail, loop, and train for copolymers with various sequences over a wide range of surface affinity. The bridge conformation is also observed in multiblock copolymers. Ordered assembly is facilitated in copolymers with longer chain/block and at stronger attraction between segment B and the slit wall. While diblock copolymer shows the longest tail, alternating copolymer has the shortest. As the attraction between segment B and the slit wall increases, the average size and fraction decrease for tail, but increase for loop and train.     

Density functional theory of homopolymer mixtures confined in a slit

A density functional theory (DFT) is developed for polymer mixtures with shorted-ranged attractive interparticle interactions confined in a slit. Different weighting functions are used separately for the repulsive part and the attractive part of the excess free energy functional by applying the weighted density approximation. The predicted results by DFT are in good agreement with the corresponding simulation data indicating the reliability of the theory. Furthermore, the center-of-mass profiles and the end-to-end distance distributions are obtained by the single chain simulation; the predictions also agree well with simulation data. The results reveal that both the attraction of the slit wall and the temperature has stronger effect on longer chains than on shorter ones because the intrasegment correlation of chains increases with increasing chain length.     

Phase behavior of a blend of polymer-tethered nanoparticles with diblock copolymers

Using the self-consistent field theory (SCFT), we investigate the phase behavior of a mixture of diblock copolymers and nanoparticles with monodisperse polymer chains tethered to their surfaces. We assume the size of the nanoparticles to be much smaller than that of the attached polymer chains and therefore model the particles with their grafted polymer "shell" as star polymers. The polymer chains attached to the particles are of the same species as one of the blocks of the symmetric diblock copolymer. Of primary interest is how to tune the shell of the particle by changing both the length and number of tethered polymers in order to achieve higher loading of nanoparticles within an ordered structure without macrophase separation occurring. We find that the phase behavior of the system is very sensitive to the size of the particle including its tethered shell. The region of microphase separation is increased upon decreasing the star polymer size, which may be achieved by shortening and/or removing tethered polymer chains. To explore the possible structures in these systems we employ SCFT simulations that provide insight into the arrangement of the different species in these complex composites.     

Hybrid density functional theory for homopolymer mixtures confined in a selective nanoslit

By integrating polymer density function theory (DFT) and single-chain molecular simulation, a hybrid DFT is developed for homopolymer mixtures confined in a selective nanoslit. Two weighting functions are adopted separately in the polymer DFT for repulsive and attractive contributions to the excess free energy functional. The theoretical results agree well with simulation data for the density profiles, configurations (tail, loop and train), adsorption amounts, layer thicknesses, and partition coefficients. The polymer-slit interaction is found to have a large effect on the density profiles and partition coefficients but is found to have a small effect on the average sizes and percentages of the configurations. Nearly half of the polymer segments form tails, and the other half form trains. In addition, bridges are observed to form for sufficiently long polymer chains. As the length difference between two polymers increases, the effect of chain connectivity becomes increasingly important.     

Structure and transport properties of polymer grafted nanoparticles

We perform molecular dynamics simulations on a bead-spring model of pure polymer grafted nanoparticles (PGNs) and of a blend of PGNs with a polymer melt to investigate the correlation between PGN design parameters (such as particle core concentration, polymer grafting density, and polymer length) and properties, such as microstructure, particle mobility, and viscous response. Constant strain-rate simulations were carried out to calculate viscosities and a constant-stress ensemble was used to calculate yield stresses. The PGN systems are found to have less structural order, lower viscosity, and faster diffusivity with increasing length of the grafted chains for a given core concentration or grafting density. Decreasing grafting density causes depletion effects associated with the chains leading to close contacts between some particle cores. All systems were found to shear thin, with the pure PGN systems shear thinning more than the blend; also, the pure systems exhibited a clear yielding behavior that was absent in the blend. Regarding the mechanism of shear thinning at the high shear rates examined, it was found that the shear-induced decrease of Brownian stresses and increase in chain alignment, both correlate with the reduction of viscosity in the system with the latter being more dominant. A coupling between Brownian stresses and chain alignment was also observed wherein the non-equilibrium particle distribution itself promotes chain alignment in the direction of shear.     

Dissipative particle dynamics simulations of polymer-protected nanoparticle self-assembly

Dissipative particle dynamics simulations were used to study the effects of mixing time, solute solubility, solute and diblock copolymer concentrations, and copolymer block length on the rapid coprecipitation of polymer-protected nanoparticles. The simulations were aimed at modeling Flash NanoPrecipitation, a process in which hydrophobic solutes and amphiphilic block copolymers are dissolved in a water-miscible organic solvent and then rapidly mixed with water to produce composite nanoparticles. A previously developed model by Spaeth et al. [J. Chem. Phys. 134, 164902 (2011)] was used. The model was parameterized to reproduce equilibrium and transport properties of the solvent, hydrophobic solute, and diblock copolymer. Anti-solvent mixing was modeled using time-dependent solvent-solute and solvent-copolymer interactions. We find that particle size increases with mixing time, due to the difference in solute and polymer solubilities. Increasing the solubility of the solute leads to larger nanoparticles for unfavorable solute-polymer interactions and to smaller nanoparticles for favorable solute-polymer interactions. A decrease in overall solute and polymer concentration produces smaller nanoparticles, because the difference in the diffusion coefficients of a single polymer and of larger clusters becomes more important to their relative rates of collisions under more dilute conditions. An increase in the solute-polymer ratio produces larger nanoparticles, since a collection of large particles has less surface area than a collection of small particles with the same total volume. An increase in the hydrophilic block length of the polymer leads to smaller nanoparticles, due to an enhanced ability of each polymer to shield the nanoparticle core. For unfavorable solute-polymer interactions, the nanoparticle size increases with hydrophobic block length. However, for favorable solute-polymer interactions, nanoparticle size exhibits a local minimum with respect to the hydrophobic block length. Our results provide insights on ways in which experimentally controllable parameters of the Flash NanoPrecipitation process can be used to influence aggregate size and composition during self-assembly.     

Spherical polymer brushes under good solvent conditions: molecular dynamics results compared to density functional theory

A coarse grained model for flexible polymers end-grafted to repulsive spherical nanoparticles is studied for various chain lengths and grafting densities under good solvent conditions by molecular dynamics methods and density functional theory. With increasing chain length, the monomer density profile exhibits a crossover to the star polymer limit. The distribution of polymer ends and the linear dimensions of individual polymer chains are obtained, while the inhomogeneous stretching of the chains is characterized by the local persistence lengths. The results on the structure factor of both single chain and full spherical brush as well as the range of applicability of the different theoretical tools are presented. Finally, a brief discussion of the experiment is given.     

Thermodynamic and structural properties of mixed colloids represented by a hard-core two-Yukawa mixture model fluid: Monte Carlo simulations and an analytical theory

The interaction between colloidal particles is well represented by a hard-core two-Yukawa potential. In order to assess the accuracy of theoretical predictions for the thermodynamic and structural properties of mixed colloids, standard Monte Carlo simulations are carried out for the hard-core two-Yukawa mixtures. In the simulations, one range parameter in the two-Yukawa potential is taken as 1.8 or 2.8647, and another is taken as 4, 8, or 13.5485. Both attractive and repulsive dominant cases of the potential outside the hard core are considered. The effects of temperature, density, composition, size and energy parameter ratios on internal energy, compressibility factor, and radial distribution function are investigated extensively. Theoretical calculations are performed in the framework of analytical solution for the Ornstein-Zernike equation with the first-order mean spherical approximation (FMSA). Our analysis shows that the FMSA is very accurate for the prediction of the compressibility factor of the hard-core two-Yukawa mixtures at all conditions studied. The FMSA generally predicts accurate internal energy, but overestimates the internal energy of the systems at lower temperatures. Furthermore, we found that a simplified exponential version of the FMSA predicts fairly good radial distribution function at contact for the mixed two-Yukawa fluids. The comparison of the theoretical compressibility factor with that from the Monte Carlo simulations suggests that the FMSA can be used to investigate the fluid-fluid equilibria of hard-core two-Yukawa mixtures.     

Tilt grain boundaries in a diblock copolymer ordered nanocomposite lamellar phase

A hybrid self-consistent field theory/density functional theory method is applied to predict tilt (kink) grain boundary structures between lamellar domains of a symmetric diblock copolymer with added spherical nanoparticles. Structures consistent with experimental observations are found and theoretical evidence is provided in support of a hypothesis regarding the positioning of nanoparticles. Some particle distributions are predicted for situations not yet examined by experiment.     

Size Effect on the Raman Spectra and Electronic Structure of the Glycine-alanine Oligopeptide Chains

A theoretical study on oligopeptide chains of glycine-alanine by density functional theory(DFT) is given in this paper. Raman spectra of the oligopeptide chains are examined. The geometric structures, frontier orbital, energy gap, atomic charge distribution, density of states and chemical activity of the side chain are studied at the B3LYP/6-31G(d) level. Results show that, with the number of residues increasing, vibrations of typical functional groups present Raman frequency shift, and the energy gap is gradually reduced. The HOMO and LUMO focus on the amino and carboxyl at the ends of oligopeptides. It is helpful for oligopeptides to self-assemble into chains. In addition, different residues(glycine or alanine) at the ends of chains result in the even-odd effect of orbital energy in the growth process. The size effects of physical and chemical properties only exist when the oligopeptides are shorter, and the phenomenon disappeared as the chain continues to grow.     

Mixing rules for binary Lennard–Jones chains: theory and Monte Carlo simulation

Theoretically-based van der Waals one-fluid (vdW1) mixing rules are derived for Lennard–Jones (LJ) chain mixtures. The rules provide equivalent one-fluid segment parameters for LJ size (σ) and energy () parameter as well as chain length (m) based on the parameters of the individual mixture components and the component mole fractions. The mixing rules are tested by performing Monte Carlo simulations of eight different binary mixtures and the equivalent vdW1 pure fluid, each at three densities. The simulations test the effects of changing LJ size parameter, LJ energy parameter and chain length individually and together. The effects of mole fraction and density are also examined. The mixing rules are tested for accuracy in predicting compressibility factors and radial distribution functions. It is found that the vdW1 rules provide excellent agreement when size and energy parameter are varied. Good agreement is found for mixtures with different chain lengths. The discrepancy is worst at very high densities when all component parameters are varied simultaneously.     

The effects of nanoparticles on the lamellar phase separation of diblock copolymers

The phase separation of diblock copolymers containing some energetically neutral/biased nanoparticles is studied by means of large-scale dissipative particle dynamics (DPD) simulations. The effects of the volume fraction of nanoparticles, the size of nanoparticles, and the interaction strength between nanoparticles and blocks on the lamellar phase separation of diblock copolymers are investigated. When these effects are up to a critical value, the diblock copolymer nanocomposites can form a new bicontinuous morphology, which is well consistent with the experimental results. It is also found that the degree of order of phase separation for a given system increases slightly and then decreases abruptly until the bicontinuous morphology is formed as the volume fraction of nanoparticles increases. Furthermore, we discuss the microphase transition through the position distributions of nanoparticles and present a phase diagram in terms of the nanoparticle volume fraction, size, and surface interaction strength.     

Interfaces in polymer blends

We investigate the structure and thermodynamics of interfaces in dense polymer blends using Monte Carlo (MC) simulations and self‐consistent field (SCF) calculations. For structurally symmetric blends we find quantitative agreement between the MC simulations and the SCF calculations for excess quantities of the interface (e.g., interfacial tension or enrichment of copolymers at the interface). However, a quantitative comparison between profiles across the interface in the MC simulations and the SCF calculations has to take due account of capillary waves. While the profiles in the SCF calculations correspond to intrinsic profiles of a perfectly flat interface the local interfacial position fluctuates in the MC simulations. We test this concept by extensive Monte Carlo simulations and study the cross‐over between “intrinsic” fluctuations which build up the local profile and capillary waves on long (lateral) length scales. Properties of structurally asymmetric blends are exemplified by investigating polymers of different stiffness. At high incompatibilities the interfacial width is not much larger than the persistence length of the stiffer component. In this limit we find deviations from the predictions of the Gaussian chain model: while the Gaussian chain model yields an increase of the interfacial width upon increasing the persistence length, no such increase is found in the MC simulations. Using a partial enumeration technique, however, we can account for the details of the chain architecture on all length scales in the SCF calculations and achieve good agreement with the MC simulations. In blends containing diblock copolymers we investigate the enrichment of copolymers at the interface and the concomitant reduction of the interfacial tension. At weak segregation the addition of copolymers leads to compatibilization. At high incompatibilities, the homopolymer‐rich phase can accommodate only a small fraction of copolymer before the copolymer forms a lamellar phase. The analysis of interfacial fluctuations yields an estimate for the bending rigidity of the interface. The latter quantity is important for the formation of a polymeric microemulsion at intermediate segregation and the consequences for the phase diagram are discussed.     

Simulation of interaction forces between nanoparticles in the presence of Lennard-Jones polymers: freely adsorbing homopolymer modifiers

The force between two nanoscale colloidal particles dispersed in a solution of freely adsorbing Lennard-Jones homopolymer modifiers is calculated using the expanded grand canonical Monte Carlo simulation method. We investigate the effect of polymer chain length (N), nanoparticle diameter (sigma(c)), and colloid-polymer interaction energy (epsilon(cp)) on polymer adsorption (Gamma) and polymer-induced forces (F(P)(r)) between nanoparticles in the full thermodynamic equilibrium condition. There is a strong correlation between polymer adsorption and the polymer-mediated nanoparticle forces. When the polymer adsorption is weak, as in the case of smaller diameters and short polymer chain lengths (sigma(c) = 5, N = 10), the polymers do not have any significant effect on the bare nanoparticle interactions. The adsorbed amount increases with increasing particle diameter, polymer chain length, and colloid-polymer interaction energy. In general, for strong polymer-particle adsorption the polymer-governed force profiles between nanoparticles show short-range repulsion and long-ranged attraction, suggesting that homopolymers would not be ideal for achieving stabilization in nanoparticle dispersions. The attraction is likely due to bridging, as well as polymer segment-segment interactions. The location and magnitude of attractive minimum in the force profile can be controlled by varying N and epsilon(cp). The results show partial agreement and some marked differences with previous theoretical and experimental studies of forces in the limit of flat walls in an adsorbing polymer solution. The difference could be attributed to incorporation of long-ranged colloid-polymer potential in our simulations and the influence of the curvature of the nanoparticles.     

A comparison of implicit- and explicit-solvent simulations of self-assembly in block copolymer and solute systems

We have developed explicit- and implicit-solvent models for the flash nanoprecipitation process, which involves rapid coprecipitation of block copolymers and solutes by changing solvent quality. The explicit-solvent model uses the dissipative particle dynamics (DPD) method and the implicit-solvent model uses the Brownian dynamics (BD) method. Each of the two models was parameterized to match key properties of the diblock copolymer (specifically, critical micelle concentration, diffusion coefficient, polystyrene melt density, and polyethylene glycol radius of gyration) and the hydrophobic solute (aqueous solubility, diffusion coefficient, and solid density). The models were simulated in the limit of instantaneous mixing of solvent with antisolvent. Despite the significant differences in the potentials employed in the implicit- and explicit-solvent models, the polymer-stabilized nanoparticles formed in both sets of simulations are similar in size and structure; however, the dynamic evolution of the two simulations is quite different. Nanoparticles in the BD simulations have diffusion coefficients that follow Rouse behavior (D ∝ M(-1)), whereas those in the DPD simulations have diffusion coefficients that are close to the values predicted by the Stokes-Einstein relation (D ∝ R(-1)). As the nanoparticles become larger, the discrepancy between diffusion coefficients grows. As a consequence, BD simulations produce increasingly slower aggregation dynamics with respect to real time and result in an unphysical evolution of the nanoparticle size distribution. Surface area per polymer of the stable explicit-solvent nanoparticles agrees well with experimental values, whereas the implicit-solvent nanoparticles are stable when the surface area per particle is roughly two to four times larger. We conclude that implicit-solvent models may produce questionable results when simulating nonequilibrium processes in which hydrodynamics play a critical role.     

Nanoparticle Dispersion and Glass Transition Behavior of Polyimide-grafted Silica Nanocomposites

How to control the spatial distribution of nanoparticles to meet different performance requirements is a constant challenge in the field of polymer nanocomposites.Current studies have been focused on the flexible polymer chain systems.In this study,the rigid polyimide(PI) chain grafted silica particles with different grafting chain lengths and grafting densities were prepared by "grafting to" method,and the influence of polymerization degree of grafted chains(N),matrix chains(P),and grafting density(a) on the spatial distribution of nanoparticles in the PI matrix was explored.The glass transition temperature(Tg) of PI composites was systematically investigated as well.The results show that silica particles are well dispersed in polyamic acid composite systems,while aggregation and small clusters appear in PI nanocomposites after thermal imidization.Besides,the particle size has no impact on the spatial distribution of nanoparticles.When σ·N0.5<<(N/P)2,the grafted and matrix chains interpenetrate,and the frictional resistance of the segment increases,resulting in restricted relaxation kinetics and Tg increase of the PI composite system.In addition,smaller particle size and longer grafted chains are beneficial to improving Tg of composites These results are all propitious to complete the microstructure control theory of nanocomposites and make a theoretical foundation for the high performance and multi-function of PI nanocomposites.     

Completely dispersible PEGylated gold nanoparticles under physiological conditions: modification of gold nanoparticles with precisely controlled PEG-b-polyamine

A novel water-soluble, biocompatible polymer, poly(ethylene glycol)-block-poly((2-N,N-dimethylamino)ethyl methacrylate) (PEG-b-PAMA), possessing controlled molecular weight with a narrow molecular weight distribution, was synthesized by the atom-transfer radical polymerization (ATRP) method. PEG-b-PAMA having a short PAMA chain length was successfully synthesized under suitable polymerization conditions. Gold nanoparticles (GNPs) were modified using PEG-b-PAMA prepared under a variety of PEGylation conditions. Under alkaline conditions (pH >10) and an [N]/[GNP] ratio of more than 3300, the PEGylated GNPs (PEG-GNPs) showed complete dispersion stability, avoiding coagulation. The amino groups of the PAMA segment of the block copolymers were completely deprotonated above pH 10. This means that PEG-b-PAMA interacted with the GNP surface via multipoint coordination of the tertiary amino groups of PAMA, not electrostatically. The effect of the number of amino groups in the PAMA segment on GNP surface modifications was investigated by zeta potential and dynamic light scattering (DLS) measurements. When the PEG-GNPs were prepared in excess polymer solution, almost the same diameter was observed regardless of the PAMA chain length. After the PEG-GNPs were purified by centrifugation, the zeta potentials of all PEG-GNPs were shielded to almost 0 mV, indicating the effective modifications of the GNP surface by PEG-b-PAMA regardless of the chain length. However, the particle size and particle size distribution of the purified PEG-GNPs were strongly affected by the PAMA chain length. PEG-GNPs with longer PAMA segments underwent coagulation after purification, whereas PEG-GNPs with shorter PAMA segments increased their dispersion stability. The experimental results of the thermal gravimetric analysis confirmed that the PEG density on the GNP surface increased as the AMA units decreased to 3. Thus, the dispersion stability depended significantly on the PEG density on the GNP surface. GNPs modified with PEG-b-PAMA having short AMA units showed excellent dispersion stability under a variety of pH conditions. The excellent dispersion stability of the obtained PEG-GNP was also confirmed both in bovine serum albumin (BSA) solution and 95% human serum.     

Effect of bidispersity in grafted chain length on grafted chain conformations and potential of mean force between polymer grafted nanoparticles in a homopolymer matrix

In efforts to produce polymeric materials with tailored physical properties, significant interest has grown around the ability to control the spatial organization of nanoparticles in polymer nanocomposites. One way to achieve controlled particle arrangement is by grafting the nanoparticle surface with polymers that are compatible with the matrix, thus manipulating the interfacial interactions between the nanoparticles and the polymer matrix. Previous work has shown that the molecular weight of the grafted polymer, both at high grafting density and low grafting density, plays a key role in dictating the effective inter-particle interactions in a polymer matrix. At high grafting density nanoparticles disperse (aggregate) if the graft molecular weight is higher (lower) than the matrix molecular weight. At low grafting density the longer grafts can better shield the nanoparticle surface from direct particle-particle contacts than the shorter grafts and lead to the dispersion of the grafted particles in the matrix. Despite the importance of graft molecular weight, and evidence of non-trivial effects of polydispersity of chains grafted on flat surfaces, most theoretical work on polymer grafted nanoparticles has only focused on monodisperse grafted chains. In this paper, we focus on how bidispersity in grafted chain lengths affects the grafted chain conformations and inter-particle interactions in an implicit solvent and in a dense homopolymer polymer matrix. We first present the effects of bidispersity on grafted chain conformations in a single polymer grafted particle using purely Monte Carlo (MC) simulations. This is followed by calculations of the potential of mean force (PMF) between two grafted particles in a polymer matrix using a self-consistent Polymer Reference Interaction Site Model theory-Monte Carlo simulation approach. Monte Carlo simulations of a single polymer grafted particle in an implicit solvent show that in the bidisperse polymer grafted particles with an equal number of short and long grafts at low to medium grafting density, the short grafts are in a more coiled up conformation (lower radius of gyration) than their monodisperse counterparts to provide a larger free volume to the longer grafts so they can gain conformational entropy. The longer grafts do not show much difference in conformation from their monodisperse counterparts at low grafting density, but at medium grafting density the longer grafts exhibit less stretched conformations (lower radius of gyration) as compared to their monodisperse counterparts. In the presence of an explicit homopolymer matrix, the longer grafts are more compressed by the matrix homopolymer chains than the short grafts. We observe that the potential of mean force between bidisperse grafted particles has features of the PMF of monodisperse grafted particles with short grafts and monodisperse grafted particles with long grafts. The value of the PMF at contact is governed by the short grafts and values at large inter-particle distances are governed by the longer grafts. Further comparison of the PMF for bidisperse and monodisperse polymer grafted particles in a homopolymer matrix at varying parameters shows that the effects of matrix chain length, matrix packing fraction, grafting density, and particle curvature on the PMF between bidisperse polymer grafted particles are similar to those seen between monodisperse polymer grafted particles.     

Electrohydrodynamics of spherical polyampholyte‐grafted nanoparticles: Multiscale simulations by coupling of molecular dynamics and lattice‐boltzmann method

We perform multiscale simulations based on the coupling of molecular dynamics and lattice‐Boltzmann (LB) method to study the electrohydrodynamics of a polyampholyte‐grafted spherical nanoparticle. The long‐range hydrodynamic interactions are modeled by coupling the movement of particles to a LB fluid. Our results indicate that the net‐neutral soft particle moves with a nonzero mobility under applied electric fields. We systematically explore the effects of different parameters, including the chain length, grafting density, electric field, and charge sequence, on the structures of the polymer layer and the electrophoretic mobility of the soft particle. It shows that the mobility of nanoparticles has remarkable dependence on these parameters. We find that the deformation of the polyampholyte chains and the ion distribution play dominant roles in modulating the electrokinetic behavior of the polyampholyte‐grafted particle. The enhancement or reduction in the accumulation of counterions around monomers can be attributed to the polymer layer structure and the conformational transition of the chains in the electric field. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 1435–1447