Effect of solvent quality on the dispersibility of polymer-grafted spherical nanoparticles in polymer solutions

In this work, lattice-based self consistent field theory is used to study the structural properties of individual polymer-grafted spherical nanopartices and particle-particle interactions in polymer melts and solutions under variable solvent conditions. Our study has focused on the depth of the minimum in the potential of mean force between the two brush-coated nanoparticles, if such a minimum occurs, and we have also addressed the corresponding radial density profiles of free and grafted chains around a single nanoparticle, in an attempt to clarify the extent of correlation between the depth of the minimum, W(min), and the parameter δ characterizing the interpenetration between the profiles of free and grafted chains. Although one cannot establish a simple one-to-one correspondence between W(min) and δ, we do find common trends, in particular, if the solvent conditions for free and grafted chains differ: varying the volume fraction of the free chains, δ typically exhibits a broad minimum, corresponding to a region where the magnitude of W(min) exceeds thermal energy k(B)T, leading to particle aggregation.     

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.     

Steric and bridging interactions between two plates induced by grafted polyelectrolytes

If colloidal particles are grafted with a polymer, then the grafted chains can provide steric repulsion between them. If some of the grafted polymer chains are also adsorbed to a second particle, then a bridging force is generated as well. For uncharged plates and polymer, the following contributions to the free energy of the system have been taken into account in the calculation of the interaction force: (i) the Flory-Huggins expression for the mixing free energy of the grafted chains with the liquid; (ii) the entropy loss due to the connectivity of the polymeric segments; (iii) the van der Waals interactions between the segments and the plates; and (iv) the free energy of adsorption of the polymer segments of the grafted chains on the other plate. For charged plates, the electrostatic free energy as well as the free energy of the electrolyte are included in the total free energy of the system. By minimizing the free energy with respect to the segment concentration and, when it is the case, with respect to the electrical potential, equations for the segment number density distribution and for the electrical potential are obtained, on the basis of which the interactions between two plates grafted with polymer chains that can be also adsorbed on the other plate were calculated. The interaction thus obtained includes steric and bridging forces.     

Control of carbon nanotube-surface interactions: the role of grafted polymers

We have investigated the interactions between single-walled carbon nanotubes, coated with polymer chains end-grafted to the tubes, and planar surfaces. By proper functionalization of the grafted polymers' free ends, we show how to obtain an attractive interaction that can be used to immobilize the tube at a desired distance from the surface. We demonstrate how the strength and distance of the minimal interaction can be controlled by the proper choice of polymer chain length, surface coverage, and type of functional end-group.     

Molecular dynamics simulations of grafted polyelectrolytes on two apposing walls

Molecular dynamics simulations of polyelectrolytes grafted to two apposing surfaces were performed. Bead-spring polymer models are used to treat flexible chains [e.g., sodium poly(styrene sulfonate)] and stiff chains (double-stranded DNA). The counterions are explicitly treated. The effect of the surface density of the grafted polymer, the chain length, and the gap width on the structure and the pressure were studied. Results are compared to experimental measurements and to simulations of polyelectrolyte brushes on a single surface. The density profiles exhibit a maximum not found in single surface data. The maximum is due to the brushes shrinking to avoid interpenetration.     

Mixtures of liquid-crystalline polymers

A density-functional expansion method is used to derive the free energy of a polymer mixture. The expression obtained includes the entropy of mixing, the entropy of configuration of the chains and the interactions (both isotropic and anisotropic ones). The chains are modelled as interacting elastic lines (bend curvature). The method is very general, and we only focus our attention on binary mixtures. The phase diagram and the order parameters are calculated. We show some results for two types of mixtures: a nematic polymer in a non-mesomorphic particle (polymer or solvent) and in another nematic liquid crystal (small-molecule or polymer). We discuss the influence of the molecular weights, the persistence length and the interactions on the phase separation.     

Polymer Chain Conformation on Deuterated Polystyrene Nanoparticles Investigated by SANS

We previously reported that grafted polystyrene (PS) chains on silica nanoparticles at a low grafting density show similar conformations to free PS chains in the same solvent, THF (diameter ?50 nm, Colloid.poly.Sci. (2013), 291, 9, 2087–2099). As an extension of our previous study we choose an organic nanoparticle (deuterated polystyrene, dPS) instead of inorganic nanoparticle to see the impact of the substrate material on chain conformation. Additionally, a wider range of molecular weights were prepared to investigate the conformation feature of grafted PS chains more in detail. Small angle neutron scattering (SANS) experiments were performed to characterize PS grafted dPS particles in good solvent condition, with deuterated toluene and deuterated THF as solvent. To get insight into the conformation of the grafted PS layer we apply a scaling law describing the dimension of free PS polymer in good solvent condition to the obtained thickness of the grafted PS layer. We find an overall agreement with the scaling law where the thickness of the grafted PS layer is slightly larger than 2R_g of the free polymer chains in the respective solvent giving hint for semi dilute polymer brush (SDPB) situation.     

Adsorption of enantiomeric poly(lactide)s on surface-grafted poly(L-lactide)

We show that resistance of densely grafted polymer layers to adsorption of chemically identical free chains, which is known to be caused by entropic expulsion of free chains from the grafted layer, can be suppressed using the grafted and free chains of opposite stereoconfiguration. Specifically, we study adsorption of poly(L-lactide) (PLLA) and its enantiomer poly(D-lactide) (PDLA) onto layers of surface-grafted PLLA in acetonitrile and chloroform by infrared spectroscopy (IR). The grafted layers with thicknesses ranging from 7 to 35 nm are produced by ring-opening polymerization of L-lactide from hydroxyl end-groups of a self-assembled monolayer on gold. The IR data indicate that adsorption on the bare gold surface is the same for the L- and D-form of the polymer. However, covering the gold with the surface-grafted PLLA produces a significant decline in the adsorption of free PLLA and, by contrast, a strong enhancement in the adsorption of free PDLA. In addition, the IR data indicate that the adsorbed PDLA chains are stereocomplexed with the grafted PLLA chains. Thus, entropic expulsion of free chains from the grafted layer, which is responsible for the resistance of surface-grafted PLLA to adsorption of free PLLA, is suppressed in the case of free PDLA by stereocomplexation between the grafted and free chains.     

Brownian dynamics simulation of grafted polymer brushes

We present results of computer simulations by the method of Brownian dynamics of polymeric brushes attached to impenetrable planes. For testing both model and method we have used one polymer brush attached to a repulsive plane and compare some results with Monte Carlo results of Lai and Binder on the bond fluctuation model. We have also studied two polymeric brushes attached to two parallel planes at different distances between planes, and investigate the interplay between the interpenetration of the brushes and the configurational properties of the grafted chains.     

Effect of grafting on nanoparticle segregation in polymer/nanoparticle blends near a substrate

Nanoparticles in polymer films have shown the tendency to migrate to the substrate due to an entropic-based attractive depletion interaction between the particles and the substrate. It is also known that polymer-grafted nanoparticles show better dispersion in a polymer matrix. Here, molecular dynamics simulations are employed to study the effect of grafting on the nanoparticle segregation to the substrate. The nanoparticles were modeled as spheres and the polymers as bead-spring chains. The polymers of the grafts and the matrix are identical in nature. For a purely repulsive system, the nanoparticle density near the surface was found to decrease as the length of grafted chains and the number of grafts increased and in the bulk, the nanoparticles are well-dispersed. Whereas, in case of attractive systems with interparticle interactions on the order of thermal energy, the nanoparticles segregated to the substrate even more strongly, essentially forming clusters on the wall and in the bulk. However, due to the presence of grafted chains on the nanoparticles, the clusters formed in the bulk are structurally anisotropic. The effect of grafts on nanoparticle segregation to the surface was found to be qualitatively similar to the purely repulsive case.     

Static and dynamic properties of the interface between a polymer brush and a melt of identical chains

Molecular-dynamics simulations of a short-chain polymer melt between two brush-covered surfaces under shear have been performed. The end-grafted polymers which constitute the brush have the same chemical properties as the free chains in the melt and provide a soft deformable substrate. Polymer chains are described by a coarse-grained bead-spring model, which includes excluded volume and backbone connectivity of the chains. The grafting density of the brush layer offers a way of controlling the behavior of the surface without altering the molecular interactions. We perform equilibrium and nonequilibrium molecular-dynamics simulations at constant temperature and volume using the dissipative particle dynamics thermostat. The equilibrium density profiles and the behavior under shear are studied as well as the interdigitation of the melt into the brush, the orientation on different length scales (bond vectors, radius of gyration, and end-to-end vector) of free and grafted chains, and velocity profiles. The obtained boundary conditions and slip length show a rich behavior as a function of grafting density and shear velocity.     

Mesoscale simulations of the behavior of charged polymer brushes under normal compression and lateral shear forces

Dissipative particle dynamics (DPD) was used to investigate the behavior of two opposing end-grafted charged polymer brushes in aqueous media under normal compression and lateral shear. The effect of polymer molecular weight, degree of ionization, grafting density, ionic strength, and compression on the polymer conformation and the resulting shear force between the opposing polymer layers were investigated. The simulations were carried out for the poly(tert-butyl methacrylate)-block-poly(sodium sulfonate glycidyl methacrylate) copolymer, referred as PtBMA-b-PGMAS, end-attached to a hydrophobic surface for comparison with previous experimental data. Mutual interpenetration of the opposing end-grafted chains upon compression is negligible for highly charged polymer brushes for compression ratios ranging from 2.5 to 0.25. Under electrostatic screening effects or for weakly charged polymer brushes, a significant mutual interpenetration was measured. The variation of interpenetration thickness with separation distance, grafting density, and polymer size follows the same scaling law as the one observed for two opposing grafted neutral brushes in good solvent. However, compression between two opposing charged brushes results in less interpenetration relative to neutral brushes when considering equivalent grafting density and molecular weight. The friction coefficient between two opposing polymer-coated surfaces sliding past each other is shown to be directly correlated with the interpenetration thickness and more specifically to the number of polymer segments within the interpenetration layer.     

A Langevin dynamics simulation study of the tribology of polymer loop brushes

The tribology of surfaces modified with doubly bound polymer chains (loops) has been investigated in good solvent conditions using Langevin dynamics simulations. The density profiles, brush interpenetration, chain inclination, normal forces, and shear forces for two flat substrates modified by doubly bound bead-necklace polymers and equivalent singly bound polymers (twice as many polymer chains of 12 the molecular weight of the loop chains) were determined and compared as a function of surface separation, grafting density, and shear velocity. The doubly bound polymer layers showed less interpenetration with decreasing separation than the equivalent singly bound layers. Surprisingly, this difference in interpenetration between doubly bound polymer and singly bound polymer did not result in decreased friction at high shear velocity possibly due to the decreased ability of the doubly bound chains to deform in response to the applied shear. However, at lower shear velocity, where deformation of the chains in the flow direction is less pronounced and the difference in interpenetration is greater between the doubly bound and singly bound chains, some reduction in friction was observed.     

Polymer grafted nanoparticles: Effect of chemical and physical heterogeneity in polymer grafts on particle assembly and dispersion

Macroscopic properties of polymer nanocomposites depend on the microscopic composite morphology of the constituent nanoparticles and polymer matrix. One way to control the spatial arrangement of the nanoparticles in the polymer matrix is by grafting the nanoparticle surfaces with polymers that can tune the effective interparticle interactions in the polymer matrix. A fundamental understanding of how graft and matrix polymer chemistries and molecular weight, grafting density, and nanoparticle size, and chemistry affect interparticle interactions is needed to design the appropriate polymer ligands to achieve the target morphology. Theory and simulations have proven to be useful tools in this regard due to their ability to link molecular level interactions to the morphology. In this feature article, we present our recent theory and simulation studies of polymer grafted nanoparticles with chemical and physical heterogeneity in grafts to calculate the effective interactions and morphology as a function of chemistry, molecular weights, grafting densities, and so forth. © 2013 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2013     

Interactions between polymer brushes in solvents of variable quality: a density functional theory study

We present a density functional theory study of interactions between sterically stabilized colloidal particles in solvents of variable quality. Both flat and spherical polymer brushes are considered, as well as both monatomic and polymeric solvents. It is shown that the interaction between sterically stabilized particles can be tuned from repulsive to attractive by varying the solvent quality, the relative length of free and grafted chains, and by employing a mixed brush consisting of both well and poorly solvated chains.     

Accelerated brush growth on nanoparticle surfaces by reversible addition‐fragmentation chain transfer polymerization

It is now well established that controlling the grafted chain lengths and densities on nanoparticle surfaces determines the effective interactions between particles, and their assembly. Here, we present unusual kinetic results for achieving grafted chain lengths longer than the free chains using reversible addition‐fragmentation chain transfer (RAFT) polymerization and discuss the limitations to obtaining polymer grafting density higher than ～0.06 chains/nm². We observe that surface initiated polymerization grows faster than the free chains in solution with high RAFT agent coverage (1.95 agents/nm²) on nanoparticles. The time‐dependence of graft density suggests that activation of the anchored chain transfer agent (CTA) is limited by the diffusion of macro‐radicals within growing grafts. Thus, radical transfer and exchange reactions become inefficient between grafts and free polymer, and convert the surface‐initiated RAFT mechanism to a free radical polymerization. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2014 , 52, 1700–1705     

Nanoparticle stability in semidilute and concentrated polymer solutions

The wetting of PDMS-grafted silica spheres (PDMS- g-silica) is connected to their depletion restabilization in semidilute and concentrated PDMS/cyohexane polymer solutions. Specifically, we found that a wetting diagram of chemically identical graft and free homopolymers predicts stability of hard, semisoft, and soft spheres as a function of the bulk free polymer volume fraction, graft density, and the graft and free polymer chain lengths. The transition between stable and aggregated regions is determined optically and with dynamic light scattering. The point of demarcation between the regions occurs when the graft and free polymer chains are equal in length. When graft chains are longer than free chains, the particles are stable; in contrast, the particles are unstable when the opposite is true. The regions of particle stability and instability are corroborated with theoretical self-consistent mean-field calculations, which not only show that the grafted brush is responsible for particle dispersion in the complete wetting region but also aggregation in the incomplete wetting region. Ultimately, our results indicate that depletion restabilization depends on the interfacial properties of the nanoparticles in semidilute and concentrated polymer solutions.     

Sterically stabilized lock and key colloids: a self-consistent field theory study

A self-consistent field theory study of lock and key type interactions between sterically stabilized colloids in polymer solution is performed. Both the key particle and the lock cavity are assumed to have cylindrical shape and their surfaces are uniformly grafted with polymer chains. The lock-key potential of mean force is computed for various model parameters, such as length of free and grafted chains, lock and key size matching, free chain volume fraction, grafting density, and various enthalpic interactions present in the system. The lock-key interaction is found to be highly tunable, which is important in the rapidly developing field of particle self-assembly.     

Modulating Orientational Order to Organize Polyhedral Nanoparticles into Plastic Crystals and Uniform Metacrystals

In nanoparticle self-assembly, the current lack of strategy to modulate orientational order creates challenges in isolating large-area plastic crystals. Here, we achieve two orientationally distinct supercrystals using one nanoparticle shape, including plastic crystals and uniform metacrystals. Our approach integrates multi-faceted Archimedean polyhedra with molecular-level surface polymeric interactions to tune nanoparticle orientational order during self-assembly. Experiments and simulations show that coiled surface polymer chains limit interparticle interactions, creating various geometrical configurations among Archimedean polyhedra to form plastic crystals. In contrast, brush-like polymer chains enable molecular interdigitation between neighboring particles, favoring consistent particle configurations and result in uniform metacrystals. Our strategy enhances supercrystal diversity for polyhedra comprising multiple nondegenerate facets.     

Structure of polymer-stabilized magnetic fluids: small-angle neutron scattering and mean-field lattice modeling

Small-angle neutron scattering and mean-field lattice modeling were used to characterize a class of water-based magnetic fluids tailored specifically to extract soluble organic compounds from water. The fluids consist of a suspension of approximately 7 nm magnetite (Fe3O4) nanoparticles coated with a bifunctional polymer layer comprised of an outer hydrophilic poly(ethylene oxide) (PEO) region for colloidal stability and an inner hydrophobic poly(propylene oxide) (PPO) region for solubilization of organic compounds. The inner region of the polymer shell is increasingly depleted of water as the fraction of PPO side chains increases. The incorporation of PPO side chains also leads to a small increase in interparticle attraction. The lattice model predicted a shell structure similar to that of a PEO-PPO-PEO triblock copolymer (Pluronic) micelle, with equivalent levels of hydration but with more PEO present in the PPO-rich regions, as the side chains grafted to the surface are less able to segregate than when in free micellar systems.