Modeling the Morphology and Phase Behavior of One-Component Polymer-Grafted Nanoparticle Systems

Nanocrystal Assemblies: Current Advances and Open Problems

We explore the potential of nanocrystals (a term used equivalently to nanoparticles) as building blocks for nanomaterials, and the current advances and open challenges for fundamental science developments and applications. Nanocrystal assemblies are inherently multiscale, and the generation of revolutionary material properties requires a precise understanding of the relationship between structure and function, the former being determined by classical effects and the latter often by quantum effects. With an emphasis on theory and computation, we discuss challenges that hamper current assembly strategies and to what extent nanocrystal assemblies represent thermodynamic equilibrium or kinetically trapped metastable states. We also examine dynamic effects and optimization of assembly protocols. Finally, we discuss promising material functions and examples of their realization with nanocrystal assemblies.

Characterizing the shear response of polymer-grafted nanoparticles

Grafting polymer chains to the surface of nanoparticles overcomes the challenge of nanoparticle dispersion within nanocomposites and establishes high-volume fractions that are found to enable enhanced material mechanical properties. This study utilizes coarse-grained molecular dynamics simulations to quantify how the shear modulus of polymer-grafted nanoparticle (PGN) systems in their glassy state depends on parameters such as strain rate, nanoparticle size, grafting density, and chain length. The results are interpreted through further analysis of the dynamics of chain conformations and volume fraction arguments. The volume fraction of nanoparticles is found to be the most influential variable in deciding the shear modulus of PGN systems. A simple rule of mixture is utilized to express the monotonic dependence of shear modulus on the volume fraction of nanoparticles. Due to the reinforcing effect of nanoparticles, shortening the grafted chains results in a higher shear modulus in PGNs, which is not seen in linear systems. These results offer timely insight into calibrating molecular design parameters for achieving the desired mechanical properties in PGNs.

Solution Self-Assembly of Amphiphilic Tadpole-like Giant Molecules Constructed by Monotethering Diblock Copolymer Chain onto a Nanoparticle

The self-assembly behavior of a tadpole-like giant molecule (TGM) constructed from a hydrophobic nanoparticle (NP) monotethered by a single amphiphilic AB diblock copolymer chain was investigated by combining self-consistent field theory and density functional theory in solution. The effects of the hydrophobicities of the B blocks and NPs (i.e., solvent properties) on the self-assembly behavior of the TGMs were investigated in the cases of weak and strong intramolecular interactions (i.e., incompatibilities) between the components of giant molecules, respectively. Besides conventional ordered aggregates (such as spheres, rings, and vesicles) with hydrophobic B-cores covered by NP shells, several aggregates with novel hierarchical structures, including vesicles with NP-inserted hydrophobic walls, bead-string-like micelles, and long cylindrical micelles with NP bumps, were obtained by tuning the solvent properties under different intramolecular interactions. Noteworthy that the simulation results show that the arrangement of the NP bumps on the long cylindrical micelles may have a certain degree of helicity, which means that these micelles may have some unique electromagnetic features such as circular dichroism. Phase diagrams as a function of the hydrophobicities of the B blocks and NPs were constructed to show the formation conditions of these novel structures. These findings can not only offer new insights into understanding of the self-assembly behavior of the TGM in solution but also provide useful guidance for simple and efficient regulation of the morphology, as well as the NP distribution and arrangement of the ordered aggregates in experiments.

Hybrid Nanoparticles at Fluid-Fluid Interfaces: Insight from Theory and Simulation

Hybrid nanoparticles that combine special properties of their different parts have numerous applications in electronics, optics, catalysis, medicine, and many others. Of the currently produced particles, Janus particles and ligand-tethered (hairy) particles are of particular interest both from a practical and purely cognitive point of view. Understanding their behavior at fluid interfaces is important to many fields because particle-laden interfaces are ubiquitous in nature and industry. We provide a review of the literature, focusing on theoretical studies of hybrid particles at fluid-fluid interfaces. Our goal is to give a link between simple phenomenological models and advanced molecular simulations. We analyze the adsorption of individual Janus particles and hairy particles at the interfaces. Then, their interfacial assembly is also discussed. The simple equations for the attachment energy of various Janus particles are presented. We discuss how such parameters as the particle size, the particle shape, the relative sizes of different patches, and the amphiphilicity affect particle adsorption. This is essential for taking advantage of the particle capacity to stabilize interfaces. Representative examples of molecular simulations were presented. We show that the simple models surprisingly well reproduce experimental and simulation data. In the case of hairy particles, we concentrate on the effects of reconfiguration of the polymer brushes at the interface. This review is expected to provide a general perspective on the subject and may be helpful to many researchers and technologists working with particle-laden layers.

Controlling the anisotropic self-assembly of polybutadiene-grafted silica nanoparticles by tuning three-body interaction forces

Recently, the significant improvements in polymer composites properties have been mainly attributed to the ability of filler nanoparticles (NPs) to self-assemble into highly anisotropic self-assembled structures. In this work, we investigate the self-assembly of core-shell NPs composed of a silica core grafted with polybutadiene (PB) chains, generating the so-called "hairy" NPs (HNPs), immersed in tetrahydrofuran solvent. While uncoated silica beads aggregate forming uniform compact structures, the presence of a PB shell affects the silica NPs organization to the point that by increasing the polymer density at the corona, they tend to self-assemble into linear chain-like structures. To reproduce the experimental observations, we propose a theoretical model for the two-body that considers the van der Waals attractive energy together with the polymer-induced repulsive steric contribution and includes an additional three-body interaction term. This term arises due to the anisotropic distribution of PB, which increases their concentration near the NPs contact region. The resulting steric repulsion experienced by a third NP approaching the dimer prevents its binding close to the dimer bond and favors the growth of chain-like structures. We find good agreement between the simulated and experimental self-assembled superstructures, confirming that this three-body steric repulsion plays a key role in determining the cluster morphology of these core-shell NPs. The model also shows that further increasing the grafting density leads to low-density gel-like open structures.

Shape Transformations and Self-Assembly of Hairy Particles under Confinement

Molecular dynamics simulations are used to investigate the behavior of polymer-tethered nanoparticles between two inert or attractive walls. The confinement in pores creates new possibilities for controlling the shape transformation of individual hairy particles and their self-organization. We introduce a minimalistic model of the system; only chain-wall interactions are assumed to be attractive, while the others are softly repulsive. We show how the shape of isolated particles can be controlled by changing the wall separation and the strength of the interaction with the surfaces. For attractive walls, we found two types of structures, “bridges” and “mounds”. The first structures are similar to flanged spools in which the chains are connected with both walls and form bridges between them. We observed various bridges, symmetrical and asymmetrical spools, hourglasses, and pillars. The bridge-like structures can be “nano-oscillators” in which the cores jump from one wall to the other. We also study the self-assembly of a dense fluid of hairy particles in slit-like pores and analyze how the system morphology depends on interactions with the surfaces and the wall separation. The hairy particles form layers parallel to the walls. Different ordered structures, resembling two-dimensional crystalline lattices, are reported. We demonstrate that hairy particles are a versatile soft component forming a variety of structures in the slits.

Self-assembled morphologies of polyelectrolyte-grafted nanoparticles directed by oppositely charged polymer matrices

Self-assembled structure of polymer grafted nanoparticles is an interesting and growing subject in the field of hybrid electronics and high energy density materials. In light of this, the self-assembled morphologies of polyelectrolyte (PE) sparsely grafted nanoparticles tuned by oppositely charged matrix chains are studied using molecular dynamics simulations. Our focus is to elucidate the effect of matrix chain polymerization on modulating the stretching properties of tethered PE layers, on the self-assembled structuring of nanoparticles. Through varying the matrix chain length and stiffness as well as electrostatic interaction strength, rich phase behaviors of PE coated nanoparticles are predicted, including spherical micelle-like structures being preferred with short matrix chains and percolating network morphologies favored with long matrix chains, which is more pronounced with an enhanced matrix chain rigidness. To pinpoint the mechanisms of self-assembled structure formation, the thickness of grafted layers, the gyration radius of tethered chains, and pair correlation functions between nanoparticles are analyzed carefully. Additionally, electrostatic correlations, manifested as the bridging via matrix chains, are examined by identifying three states of matrix PE chains. Our simulation results may be useful for designing smart polymer nanocomposites based on PE coated nanoparticles.

Self-assembled structure of polymer grafted nanoparticles is an interesting and growing subject in the field of hybrid electronics and high energy density materials.

Adsorption of Polymer-Tethered Particles on Solid Surfaces

We explore the behavior of polymer-tethered particles on solid surfaces using coarse-grained molecular dynamics simulations. Segment-segment, segment-core, and core-core interactions are assumed to be purely repulsive, while the segment-substrate interactions are attractive. We analyze changes in the internal structure of single hairy particles on the surfaces with the increasing strength of the segment-substrate interactions. For this purpose, we calculate the density profiles along the x, y, z axes and the mass dipole moments. The adsorbed hairy particles are found to be symmetrical in a plane parallel to the substrate but strongly asymmetric in the vertical direction. On stronger adsorbents, the particle canopies become flattened and the cores lie closer to the wall. We consider the adsorption of hairy nanoparticles dispersed in systems of different initial particle densities. We show how the strength of segment-substrate interactions affects the structure of the adsorbed phase, the particle-wall potential of the average force, the excess adsorption isotherms, and the real adsorption isotherms.

Building blocks for autonomous computing materials: Dimers, trimers, and tetramers

Autonomous computing materials for data storage and computing offer an opportunity for next generation of computing devices. Patchy nanoparticle networks, for example, have been suggested as potential candidates for emulating neuronal networks and performing brain-like computing. Here, we use molecular dynamics (MD) simulations to show that stable dimers, trimers, and tetramers can be built from citrate capped gold nanoparticles (cit-AuNPs) linked by poly(allylamine hydrochloride) (PAH) chains. We use different lengths of PAHs to build polymer-networked nanoparticle assemblies that can emulate a complex neuronal network linked by axons of varying lengths. We find that the tetramer structure can accommodate up to 11 different states when the AuNP pairs are connected by either of two polymer linkers, PAH₂₀₀ and PAH₃₀₀. We find that the heavy AuNPs contribute to the assembly's structure stability. To further illustrate the stability, the AuNP-AuNP distances in dimer, trimer, and tetramer structures are reduced by steering the cit-AuNPs closer to each other. At different distances, these steered structures are all locally stable in a 10 ns MD simulation time scale because of their connection to the AuNPs. We also find that the global potential energy minimum is at short AuNP-AuNP distances where AuNPs collapse because the -NH₃ ⁺ and -COO^- attraction reduces the potential energy. The stability and application of these fundamental structures remain to be further improved through the use of alternative polymer linkers and nanoparticles.

Body centered tetragonal nanoparticle superlattices: why and when they form?

Body centered tetragonal (BCT) phases are structural intermediates between body centered cubic (BCC) and face centered cubic (FCC) structures. However, BCC ↔ FCC transitions may or may not involve a stable BCT intermediate. Interestingly, nanoparticle superlattices usually crystallize in BCT structures, but this phase is much less frequent for colloidal crystals of micrometer-sized particles. Two origins have been proposed for the formation of BCT NPSLs: (i) the influence of the substrate on which the nanoparticle superlattice is deposited, and (ii) non-spherical nanoparticle shapes, combined with the fact that different crystal facets have different ligand organizations. Notably, none of these two mechanisms alone is able to explain the set of available experimental observations. In this work, these two hypotheses were independently tested using a recently developed molecular theory for nanoparticle superlattices that explicitly captures the degrees of freedom associated with the ligands on the nanoparticle surface and the crystallization solvent. We show that the presence of a substrate can stabilize the BCT structure for spherical nanoparticles, but only for very specific combinations of parameters. On the other hand, a truncated-octahedron nanoparticle shape strongly stabilizes BCT structures in a wide region of the phase diagram. In the latter case, we show that the stabilization of BCT results from the geometry of the system and it does not require different crystal facets to have different ligand properties, as previously proposed. These results shed light on the mechanisms of BCT stabilization in nanoparticle superlattices and provide guidelines to control its formation.

Viscosity of polyelectrolyte-grafted nanoparticle solutions

The effect of charges and hydrogen bonding on viscosity in solutions containing polyelectrolyte-grafted nanoparticles (PENP) has been investigated using molecular dynamics (MD) simulations. The electrostatic interaction between the charged monomers on the grafted chains, which increases with the degree of ionization, causes the grafted polymers to stretch and increases the hydrodynamic size of the nanoparticles. The viscosity of the solution is partially governed by the balance between the entanglement of grafted chains and the electrostatic repulsion. Moreover, the charge-assisted hydrogen bonds between the monomers of different particles further enhance the viscosity of the solution. For shorter grafted chains, a majority of hydrogen bonds are formed within the same particle and thus show no significant enhancement in viscosity. The addition of polymer chains with hydrogen bonding sites has been shown to bridge multiple nanoparticles, creating a network structure, that increases viscosity. The chain stiffness has been shown to have a direct correlation with bridging and thus the viscosity of the solution.

Nanopore gates via reversible crosslinking of polymer brushes: a theoretical study

Polymer-brush-modified nanopores are synthetic structures inspired by the gated transport exhibited by their biological counterparts. This work theoretically analyzes how the reversible crosslinking of a polymer network by soluble species can be used to control transport through nanochannels and pores. The study was performed with a molecular theory that allows inhomogeneities in the three spatial dimensions and explicitly takes into account the size, shape and conformations of all molecular species, considers the intermolecular interactions between the polymers and the soluble crosslinkers and includes the presence of a translocating particle inside the pore. It is shown than increasing the concentration of the soluble crosslinkers in bulk solution leads to a gradual increase of its number within the pore until a critical bulk concentration is reached. At the critical concentration, the number of crosslinkers inside the pore increases abruptly. For long chains, this sudden transition triggers the collapse of the polymer brush to the center of the nanopore. The resulting structure increases the free-energy barrier that a translocating particle has to surmount to go across the pore and modifies the route of translocation from the axis of the pore to its walls. On the other hand, for short polymer chains the crosslinkers trigger the collapse of the brush to the pore walls, which reduces the translocation barrier.

Synthesis of Well-Defined Polyolefin Grafted SiO2 Nanoparticles with Molecular Weight and Graft Density Control

Recent advances in surface-initiated polymerization have given rise to a range of brush nanocomposites and hybrid functional materials. However, the synthesis of pure polyolefin-grafted nanocomposites by surface-initiated ring-opening metathesis polymerization (SI-ROMP) is a significant challenge due to the particle aggregation and irreversible particle coupling. This study presents a synthetic approach toward well-defined poly(cyclooctene)- and polyethylene-grafted nanoparticles by tethering Grubbs third generation catalyst on the particle surface and initiating the polymerization in a rapid manner. This work also serves as a template to prepare other hairy nanoparticles and functions as a basis toward understanding their thermomechanical behaviors.

Assembly of nanocrystal clusters by solvent evaporation: icosahedral order and the breakdown of the Maxwell regime

We carry out molecular dynamics simulations of N gold alkylthiolated nanocrystals (0 ≤ N ≤ 29) contained in liquid droplets of octane, nonane and decane coexisting with its vapor. The equilibrium structures that result when all the solvent dries up consist of highly symmetric nanocrystal clusters with different degrees of icosahedral order that are thoroughly characterized. We show that the relaxation times follow two regimes, a first for small nanocrystal packing fraction, dominated by the diffusion of vapor molecules (Maxwell regime, relaxation times independent of N) and another, for larger packing fractions, where the solvent diffuses through the cluster (with relaxation times growing like N2/3). We discuss the connection to the assembly of superlattices, prediction of lattice constants and evaporation models.

The Phase Behavior of Nanoparticle Superlattices in the Presence of a Solvent

Superlattices of nanoparticles coated by alkyl-chain ligands are usually prepared from a stable solution by evaporation, therefore the pathway of superlattice self-assembly critically depends on the amount of solvent present within it. This work addresses the role of the solvent on the structure and the relative stability of the different supercrystalline phases of single-component superlattices (simple cubic, body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed). The study is performed with a molecular theory for nanoparticle superlattices introduced in this work, which predicts the structure and thermodynamics of the supercrystals explicitly treating the presence and molecular details of the solvent and the ligands. The theory predicts a FCC-BCC transition with decreasing solvent content due to the competition between the translational entropy of the solvent and the entropy and internal energy of the ligands. This result provides an explanation for recent experimental observations by in situ X-ray scattering, which reported a FCC-BCC transition during solvent evaporation. The theory also predicts the effects of the length and surface coverage of the ligands and the radius of the core on the phase behavior in agreement with experimental evidence and previous molecular dynamics simulations. These results validate the use of the dimensionless softness parameter λ (ratio of ligand length to core radius) to predict the phase behavior of wet superlattices. Our results stress the importance of explicitly considering the presence of the solvent in order to reach a complete picture of the mechanisms that mediate the self-assembly of nanoparticle superlattices.

Optimization of hydrophobic nanoparticles to better target lipid rafts with molecular dynamics simulations

Due to different interactions between lipids and proteins, a plasma membrane can segregate into different membrane domains. Among them, ordered functional membrane domains are defined as "lipid rafts", which play key roles in many biological processes (e.g., signal transduction, endocytosis, etc.) in the cell. Hence, it will be of much biological significance to monitor and even regulate the dynamics of lipid rafts. In this work, we designed a ligand-modified spherical nanoparticle with coarse-grained molecular dynamics simulations, which can be encapsulated into the hydrophobic region of the lipid membrane and specifically target either raft or non-raft membrane domains. The preferred localization of the nanoparticle can be tuned by adjusting ligand hydrophobicity, length and density. Generally, more hydrophobic nanoparticles tend to target the raft domain, while less hydrophobic nanoparticles prefer the non-raft domain. Besides, ligand length and density jointly determine the exposure of nanoparticle cores and thus affect the roles of ligands in nanoparticles' final localization. Our results may provide insights into the experimental design of functional nanoparticles, targeting the lipid raft and regulating its dynamics.

Entropic Effects in Solvent-Free Bidisperse Polymer Brushes Investigated Using Density Functional Theories

Solvent-free polymer-functionalized nanoparticles form a special type of colloid composed of inorganic cores self-suspended by their grafted coronas. In the absence of intervening solvent molecules, the fluidity of the system is provided by these tethered polymers as they fill the space. Here, we study the structure and interaction of neighboring polymer-grafted surfaces in the solvent-free condition using mean-field density functional theories. For opposing flat surfaces, the brush configuration and the associated energy landscape are semianalytically investigated given the incompressibility of the tethered entropic chains. The effect of brush polydispersity (including variations in both chain length and surface grafting density) is considered by two bidisperse models corresponding to different physical scenarios: one for opposing brushes uniformly mixed with two species at a fixed grafting density, and the other for opposing brushes with distinct chain lengths and grafting densities. The space-filling capabilities of the neighboring coronas differ not only by their ratio of radii of gyration for the composing polymers but also by their ratio of grafting densities. We show that the system energy depicts a steric repulsion as the brushes are compressed, which is typical for hairy particles in a solvent. However, as the interwall separation increases, the cooperative stretching of the chains leads to an entropic attraction between them, a unique characteristic of solventless systems. The corresponding brush profiles change from a bell-like shape to a more step-function-like feature as the interwall spacing increases significantly. The interwall separation associated with the overall free energy minimum therefore characterizes the favorable interparticle spacing for solvent-free polymer-functionalized particles. The limiting accessible parameter space of polymer sizes and grafting densities subjected to the space-filling constraint is comprehensively explored for representative interparticle spacing characterizing the compressed, relaxed, and stretched regimes for a given polymer species, respectively. Such information would be useful for guiding the design of experimental solvent-free polymer-functionalized nanoparticles.

Designing Superlattice Structure via Self-Assembly of One-Component Polymer-Grafted Nanoparticles

The control of the self-assembly of the nanocrystals into ordered structures has been extensively investigated, but fewer efforts have been devoted to studying one-component polymer-grafted nanoparticles (OPNPs). Herein, through coarse-grained molecular dynamics simulation, we design a novel nanoparticle (NP) grafted with polymer chains, focusing on its self-assembled structures. First, we examine the effects of length and density of grafted polymer chains by calculating the radial distribution function between NPs, as well as through direct visualization. We observe a monotonic change of the arranged morphology of grafted-NPs as a function of the density of grafted polymer chains, which indicates that the increase of the grafting density contributes to the order of the morphology. Meanwhile, we find that much longer grafted polymer chains worsen the regularity of the morphology. Then, we probe the influence of the stiffness of grafted polymer chains (denoted by K ranging from 0 to 500) on the order of grafted-NPs, finding that the order of the structure exhibits a nonmonotonic behavior as a function of K at moderate grafting density. For high grafting density, the order of the morphology is initially enhanced and becomes saturated as a function of K. For the effect of K on the stress-strain behavior, the system with the lowest order demonstrates the most remarkable reinforced mechanical behavior for both low and high grafting density. Last, we establish the phase diagram by varying the stiffness and density of the grafted polymer chains, which contains the amorphous, ordered, and superlattice structures, respectively. In general, our simulated results provide guidelines to tailor the self-assembly of the OPNPs by taking advantage of the length, density, and stiffness of grafted polymer chains.

Effect of Chain Architecture on Phase Behavior of Giant Surfactant Constructed from Nanoparticle Monotethered by Single Diblock Copolymer Chain

The phase behaviors of the giant surfactant constructed from a nanoparticle (NP) monotethered by a single AB diblock copolymer chain were investigated by combining self-consistent field theory and density functional theory. Three types of giant surfactants with different chain architectures were constructed via changing the location of NP on the diblock copolymer chain. The simulation results show that the introduction of the NP can induce phase separation of the originally disordered AB diblock copolymers, and phase diagrams as a function of the chain length ratio of A block and the attraction between A block and NP were constructed for the three giant surfactant systems. Via changing the location of NP from the end of B block to the AB-junction point and to the end of A block, the conformational entropies of the systems gradually decrease, leading to a significant difference in phase behaviors. When the NP is tethered to the end of B block, the giant surfactant system has the smallest phase-separation region in the phase diagram, and the resulting ordered structures have the smallest feature sizes. However, when the NP is tethered to the end of A block, the giant surfactant system has the largest phase-separation region, as well as the largest feature sizes of ordered structures. Moreover, the distributions of the NPs within microphase-separated domain can be well tailored by changing the chain length ratio of A block or the attraction between A block and NP in all of the three giant surfactant systems. These findings provide the guideline for the preparation of polymer-nanoparticle composites with controllable morphologies, desirable feature sizes, and precise NP distributions in experiments.

Reconfigurable Polymer Shells on Shape-Anisotropic Gold Nanoparticle Cores

Reconfigurable hybrid nanoparticles made by decorating flexible polymer shells on rigid inorganic nanoparticle cores can provide a unique means to build stimuli-responsive functional materials. The polymer shell reconfiguration has been expected to depend on the local core shape details, but limited systematic investigations have been undertaken. Here, two literature methods are adapted to coat either thiol-terminated polystyrene (PS) or polystyrene-poly(acrylic acid) (PS-b-PAA) shells onto a series of anisotropic gold nanoparticles of shapes not studied previously, including octahedron, concave cube, and bipyramid. These core shapes are complex, rendering shell contours with nanoscale details (e.g., local surface curvature, shell thickness) that are imaged and analyzed quantitatively using the authors' customized analysis codes. It is found that the hybrid nanoparticles based on the chosen core shapes, when coated with the above two polymer shells, exhibit distinct shell segregations upon a variation in solvent polarity or temperature. It is demonstrated for the PS-b-PAA-coated hybrid nanoparticles, the shell segregation is maintained even after a further decoration of the shell periphery with gold seeds; these seeds can potentially facilitate subsequent deposition of other nanostructures to enrich structural and functional diversity. These synthesis, imaging, and analysis methods for the hybrid nanoparticles of anisotropically shaped cores can potentially aid in their predictive design for materials reconfigurable from the bottom up.

Modeling individual and pairs of adsorbed polymer-grafted nanoparticles: structure and entanglements

We analyze the canopy structure and entanglement network of isolated polymer-grafted nanoparticles (PGNs) adsorbed on a surface. As expected, increasing the monomer-surface adsorption strength causes the polymer chains to spread out to increase contact with the surface, leading to a canopy shape that is in qualitative agreement with recent experimental results. We compare height profiles and other structural features of four PGN systems to show the separate and combined effects of increasing chain length and graft density. At moderate graft density and low surface attraction strength, nearby PGN canopies interpenetrate substantially and their combined shape is similar to that of a single PGN canopy. At high graft density or surface interaction, the interparticle spacing increases significantly. We use a geometrical primitive path analysis to calculate average entanglement properties including canopy-canopy entanglements between pairs of PGNs. The longer chain systems are well entangled at both graft densities considered, and we find that as the monomer-surface interaction strength is increased (and the interparticle distance increases), entanglements between the two PGNs are reduced. We find that the number of inter-PGN entanglements per chain is slightly larger at the lower graft density, likely because steric constraints at high graft density tend to reduce interparticle entanglements.