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Theoretical and experimental work in polymer nanocomposites has established that adding nanoscale additives, such as silica particles, metal particles, carbon nanotubes, or layered silicates to a polymer matrix enhances the inherent properties of the polymer due to synergistic interactions between the polymer matrix and the nanoparticles. Conventionally, it is accepted that isotropic dispersion of the nanoparticles is essential for improved properties such as reduced permeability, increased mechanical strength, improved thermal resistance, and so forth. Conversely, in the case of polymer nanocomposites that can be used for photonics, solar or photovoltaic (alternative energy), and electronics applications, precise assembly and ordering of nanoparticles mediated by a polymer matrix is needed. Irrespective of the application, controlling the morphology of the nanoparticles within the polymer matrix is highly desirable to design and engineer materials with optimal target macroscopic properties. One way to control morphology within a polymer nanocomposite is by functionalizing or grafting the nanoparticle surface with polymers1–41 that can then manipulate the interfacial interactions between the nanoparticles and the medium (a small molecule solvent or a polymer matrix of same or different chemistry as the graft polymer) that the particles are placed in, and thus control their spatial arrangement. A recent comprehensive review by Green covers major theoretical and experimental advances specifically in the area of polymer grafted nanoparticles in a homopolymer matrix.13 Advances in synthetic ability to create designer polymer functionalized particles of desired polymer and particle chemistry, polymer molecular weight, and grafting density have motivated many theoretical and computational studies to explore the effects of this vast set of molecular-level parameters on assembly/dispersion of the polymer grafted nanoparticles in a medium. The fundamental insight provided by theory and simulation studies on the effects of polymer and particle chemistry, graft and matrix molecular weight, and grafting density on the composite morphology and phase behavior provide valuable guidance to ongoing synthetic efforts in experimental groups. The focus of this feature article is to present such theoretical and computational contributions from our group, and to briefly review other recent advances, theoretical and experimental, in this field of polymer functionalized nanoparticles based nanocomposites.
Computational6, 7, 9, 12, 16, 18–21, 24–27, 33 and experimental3, 4, 10, 13, 14, 22, 28–31, 35, 40 studies on nanocomposites, consisting of homopolymer-functionalized nanoparticles in a polymer matrix, have demonstrated that the chemistry and relative molecular weights of the graft and matrix polymers, grafting density, and nanoparticle size are parameters that play a critical role in dictating the spatial organization of the nanoparticles. For example, experimental studies30, 42, 43 have achieved the migration of the polymer-grafted nanoparticles from one domain to another domain in the matrix by thermally changing the chemistry of the grafted homopolymers on the nanoparticle, and thus the compatibility of the grafted polymer and matrix. Another important parameter that dictates the effective interparticle interaction, and therefore the particle assembly, is the polymer grafting density, defined as the number of grafted polymers per unit surface area of the nanoparticle. At high grafting density, the grafted polymers extend due to crowding and form a brush-like conformation on the particle surface. Particles carrying a densely grafted homopolymer brush placed in a homopolymer matrix whose chemistry is identical to the grafted polymer disperse (aggregate) if the molecular weight of the matrix homopolymer is lower (higher) than that of the grafted homopolymer.22, 31, 36, 37 At low grafting density20, 33, 44, 45 the grafted polymers do not face any crowding from monomers of adjacent chains and as a result do not stretch into brush-like conformations. The surface of the nanoparticle that is exposed versus that covered by the grafted monomers dictates the effective interparticle interactions.20, 27 Such homopolymer-grafted nanoparticles at low grafting densities have been shown to assemble into a variety of nanostructures in solvent34, 38, 46–51 and in matrix.22
Although all of the above studies have established the effect of various molecular parameters on the behavior of relatively monodisperse homopolymer-grafted nanoparticles either in solvent or in a polymer matrix, the effect of polymer functionalization with chemical heterogeneity (e.g., copolymers)17, 52–61 or physical heterogeneity (e.g., distribution of molecular weights)1, 2, 5, 9 on spatial organization of nanoparticles in a solvent or polymer matrix has been studied to a much smaller extent. Our focus in the past few years has been on understanding how heterogeneous polymer ligands, such as copolymers and polydisperse homopolymers, bring new phase behavior in polymer nanocomposites, as compared to monodisperse homopolymer ligands. This feature article presents a short review of recent work in the area of copolymer functionalized nanoparticles followed by a review of polydisperse homopolymer functionalized nanoparticles. The article concludes with a section on future directions/trends for both theory and experiments in this field of polymer functionalized particles.
COPOLYMER FUNCTIONALIZED NANOPARTICLES
Copolymer functionalization, as opposed to homopolymer functionalization, creates additional tuning parameters of graft sequence and monomer chemistry (or interactions) which provide further control over the morphology of polymer grafted nanoparticles in a given medium. Copolymer-grafted nanoparticles have been synthesized successfully by various experimental groups, either using atom transfer radical polymerization to grow copolymers from surface of silica nanoparticles62 or using Z-supported reversible addition-fragmentation chain transfer (RAFT) polymerization to synthesize diblock copolymer-grafted particles.63 These advances in synthesis of polymer functionalized nanoparticles have motivated recent theory17, 54, 64 and simulations55, 57, 58, 65 to study how copolymer functionalization on nanoparticles affects assembly/dispersion in solvents or polymer matrix.
Vorselaars et al.54 have used (self-consistent field theory) SCFT to study dense layers of diblock copolymers grafted onto a single spherical nanoparticle at high grafting density to understand the domains formed by the two monomers within the grafted layer. They found various domain shapes on the particle surface depending on the composition of the copolymer, and discussed the stability of the various morphologies on these highly curved nanoparticle surfaces, in contrast to flat surfaces (zero curvature). Zhu et al.64 have employed both SCFT and DFT (density functional theory) to study a dense system of nanoparticles each with a single diblock copolymer graft. When the particle surface is chemically neutral to the grafted chain, they observed typical block copolymer morphologies (i.e., cylinders and lamellae) determined by both the composition of the copolymer and the particle size. When the particle surface is repulsive to both blocks of the copolymer, they observed hierarchical morphologies, such as “lamellae with cylinders at interfaces” not typically observed with block copolymer melts. Chan et al.57 have used molecular simulations to compare the assembled phases observed in a melt of cubic nanoparticles each grafted with a single diblock copolymer to that seen in melts of diblock and triblock copolymers (no particles). They contrast the effect the bulky and rigid cubic nanoparticle has on the curvatures within the assembled phases to the effect the middle linear block in a triblock copolymer system has on its assembled phases. Although the above studies focused on either a single copolymer grafted nanoparticle at high grafting density54 or grafted nanoparticles with a single grafted chain57, 64, our recent work have focused on conducting systematic studies of single to many copolymer grafted spherical nanoparticles with varying copolymer grafting densities (low to intermediate), copolymer sequence and chemistry (or interactions), and particle sizes using a combination of theory17 and simulation techniques.55, 58, 65
Our first goal was to quantify the effect of the monomer sequence and chemistry on the chain conformations of the copolymers grafted on spherical nanoparticles (in the 2–10 nm range) at low to intermediate grafting density. We conducted Monte Carlo (MC) simulations of a single spherical nanoparticle grafted with AB copolymers with either alternating, multiblock or diblock sequences in an implicit solvent with monomer–monomer attractive interactions between A–A or B–B monomer pairs.65 These interactions were chosen to implicitly model the effect of an A- or B-selective solvent. For example, in an A-selective solvent, one would expect the B monomers to have an effective attraction caused by the solvent molecules preferentially locating themselves around A-monomers. Thus, a system with B–B attractions would mimic the implicit effect of an A-selective solvent. Similarly, a system with A–A attractions would mimic the implicit effect of an B-selective solvent. Through visual analysis and calculation of monomer contacts, monomer concentration profiles from the particle surface, and grafted chains' radii of gyration we concluded that on a copolymer grafted nanoparticle the monomer sequence, particle diameter, and grafting density dictate whether (a) the grafted copolymer chains aggregate to bring attractive monomers from multiple grafted chains together (interchain and intrachain monomer aggregation) if the enthalpy gained by doing so offsets the entropic loss caused by stretching of chains, or (b) each grafted chain folds onto itself to bring its attractive monomers together (only intrachain monomer aggregation) if the entropic loss from interchain aggregation cannot be overcome by the enthalpic gain. For six AB copolymer chains, each containing 24 Kuhn segments or “monomers,” grafted on a spherical particle of diameter D = 4d, where d is the size of a Kuhn segment (∼1 nm), interchain and intrachain monomer aggregation occurred, and the radius of gyration varied non-monotonically with increasing blockiness of the monomer sequence. This is because as the blockiness of the monomer sequence increases the copolymer chains have larger blocks of like-monomers that can easily bring like-attractive monomers from other grafted chains or within themselves together, with the exception of the alternating copolymer which has the most frustrated AB sequence. At larger particle diameters with six chains of chain length 24 monomers, the grafted chains transition to purely intrachain monomer aggregation because it is entropically unfavorable for the copolymer chains to stretch to make interchain monomer contacts with a neighboring grafted chain. At these higher particle diameters the radius of gyration of the grafted copolymers varies monotonically with monomer sequence due to intrachain monomer aggregation, because as the sequence becomes blockier (like monomers are grouped together), the copolymer chain has to fold less compactly to maximize the enthalpically favorable contacts while maintaining high conformational entropy. For AnBn diblock copolymer grafts and constant particle size, the solvent quality or monomer–monomer attractions—A–A or B–B—affected the chain conformations significantly. Since the copolymer chains were grafted through the A-block, the A monomers are placed closer to the particle surface. Thus, B-selective implicit solvent (or A–A attractions) led to A monomers aggregating close to the particle surface and shielding the particle surface. In contrast, A-selective implicit solvent (or B–B attractions) led to the B monomers, which are topologically placed farther from the surface in the grafted chain, to aggregate farther from the particle surface exposing the nanoparticle surface. This complex interplay of monomer sequence, monomer attractions, ratio of grafted chain length to particle diameter and their non-trivial effects on the grafted chain conformations on the nanoparticle surface motivated us to study how these parameters affect the potential of mean force (PMF) or effective interactions between two copolymer-grafted nanoparticles in a medium.
As our system of interest was polymer nanocomposites with polymer grafted nanoparticles, we set out to calculate the PMF between copolymer-grafted nanoparticles in a homopolymer matrix, at low grafting density, as a function of monomer sequence (alternating versus diblock) in the grafted chain, molecular weight of the grafted and matrix chains, particle size, matrix packing fraction (dense polymer solutions to melts) and monomer attractions. Using a self-consistent Polymer Reference Interaction Site Model (PRISM)-MC17 approach, we studied AB copolymer-grafted spherical nanoparticles at low grafting density (with the copolymer chains grafted on particle through A monomer) placed in A or B homopolymer matrix. We use a self-consistent approach where the grafted chain conformations that are inputs to PRISM theory, originally developed by Schweizer and Curro,66 are provided by MC simulations of a single copolymer-grafted nanoparticle in an external medium-induced solvation potential obtained from PRISM theory. We do not present the details of this self-consistent PRISM-MC approach here, and instead direct the reader to a recent review of this approach and description of the method as it is applied to study polymer grafted nanoparticles.7 We found that the PMF between two alternating AB copolymer grafted particles in a homopolymer matrix is insensitive to the chemistry of the homopolymer matrix (A- or B-homopolymer) in the case of a weak A–B χ parameter [Fig. 1(a)]. The A–B χ parameter is a collective measure of relative A-A, B-B and A-B attractions defined as χ ∼ εAB − 1/2(εAA + εBB) where εij is the strength of attraction between i and j monomers. The Inset of Figure 1(a) shows a schematic of the arrangement of A and B monomers on the particle surface as seen in either A- or B-polymer matrix. Additionally, in a dense solutions of A or B homopolymer matrix in the case of a weak A–B χ parameter, the behavior of the alternating grafted particles is similar to the behavior of particles with athermal homopolymer grafts [solid black line in Fig. 1(a)]. The weak A–B χ coupled with the frustrated—ABAB—monomer arrangement do not allow the alternating grafted chain to assume a compact structure because the loss in conformational entropy upon doing so is not overcome by the enthalpic gain, and as a result the grafts assume a configuration that is similar to the homopolymer case.
In contrast, the AB diblock grafted particles exhibit behavior that is strongly dependent on the matrix chemistry even at weak A–B χ parameter [Fig. 1(b)]. The formation of aggregates of blocks of A or B blocks [as shown in Fig. 1(b) insets] and how they aid or hinder the matrix-induced attractive interactions between the particles dictate the magnitude, nature and location of attraction/repulsion in the PMF between two grafted particles. For example, when placed in homopolymer B matrix in the presence of weak A–B χ parameter (εAA = 0.2kT) the attractive A monomers in the block closer to the particle surface aggregate and form a protective shell on the particle surface, away from B matrix chains. This shell of A monomers on the particle surface sterically hinders matrix-induced direct contact with another grafted particle. When placed in a homopolymer A matrix with εBB = 0.2kT, the block of attractive B monomers forms aggregates away from the nanoparticle surface and the athermal A monomers do not form a shell on the surface, thus leading to matrix-induced depletion-like attraction at contact and B-aggregate-induced steric repulsion at larger interparticle distances [Fig. 1(b)].
We also observed that increasing matrix packing fractions reduced the dependence of the PMF on the monomer sequence and intermonomer attractions because at high matrix packing fractions, the matrix-induced depletion-like forces exerted on the grafted particles dominate and overcome the steric hindrance caused by the conformations of the grafts. This leads to an attractive PMF between the copolymer grafted particles in a dense polymer matrix, for both monomer sequences. At constant graft length (Ng), when the matrix chain length (Nm) was varied (Nm < Ng, Nm = Ng, Nm > Ng), the PMF at contact changed only by 0.1–0.2kT for both alternating and diblock grafts, while maintaining the same quality (i.e., repulsion or attraction) at constant packing fraction. This minimal matrix length effects for these lightly grafted copolymer grafted particles is in agreement with what had been seen before for homopolymer grafted particles at low grafting density in a chemically identical homopolymer matrix.20, 27 At constant graft length and number of grafts, as particle size was increased we observed: (a) a greater portion of the particle surface was exposed to the surroundings unobstructed by the monomers of the grafted chains; (b) the role of monomer sequence gradually vanished as evidenced by the similar PMF plots for the alternating and diblock grafted particles at larger particle sizes due to the interactions being dominated by core–core contacts rather than grafted copolymer shielded interactions; (c) matrix chemistry affected the PMF to a smaller extent as observed by the reduced difference between the respective PMF values at contact for the diblock grafts in A and B matrix. Finally, an increase in the A–B χ parameter has a minor effect on the PMF for large particles and short graft chain lengths because of the small number of attractive units within the grafted chain; furthermore, the matrix chemistry has little effect on the PMF at short graft lengths. As the graft length increases an intermediate A–B χ parameter is potent enough to appreciably alter the behavior of the diblock grafted particles depending on which block is attractive.
The above studies of the grafted chain conformations and effective interactions between copolymer grafted particles show that monomer sequence and chemistry in the grafted copolymers strongly influence monomer arrangement on the nanoparticles, imparting a unique sequence-dependent “patchiness” onto the nanoparticles. This unique “patchiness” that arises from sequence-dependent folding of the grafted copolymers motivated our next set of studies focused on understanding how the “patchiness” affected assembly of multiple copolymer grafted nanoparticles, as a function of the monomer sequence and monomer chemistry. We used MC simulations to demonstrate how the monomer sequence in the grafted copolymers can be a tuning parameter to direct assembly of nanoparticles, and the shapes, sizes and structures of the assembled nanoclusters.55, 58 We studied spherical nanoparticles grafted with AB copolymers with either alternating or diblock sequence55 and a range of like-monomer (A–A and/or B–B) attractive interactions in the presence of either relatively strong or negligible unlike-monomer (A–B) repulsive interaction. In the presence of negligible A–B repulsions the alternating sequence produces nanoclusters that are relatively isotropic regardless of whether A–A or B–B monomers are attractive, while the diblock sequence produces nanoclusters that are smaller and more compact when the block closer to the surface (A–A) is attractive and larger loosely held together clusters when the outer block (B–B) is attractive. Particle size and graft length balance enthalpic gain and entropic losses coming from interparticle interchain contacts and/or intergrafted and intragrafted chain contacts within the same grafted particle, and in turn dictate the shape and size of the cluster. In the presence of strong A–B repulsions, the alternating sequence leads to either particle dispersion or smaller clusters than those at negligible A–B repulsions because the alternating sequence causes A-B repulsive contacts when trying to make A-A and B-B attractive contacts. In contrast, for the diblock sequence, the presence or absence of A–B repulsions did not alter the cluster characteristics because of topologically separated A and B blocks, and in turn fewer A-B repulsive contacts than alternating sequence. Additionally, diblock copolymer grafted particles tend to assemble into anisotropic shapes despite the isotropic grafting of the copolymer chains on the particle surface because of fewer patches formed from A-monomer aggregates (in the case of A-A attractions) and B-monomer aggregates (in the case of B-B attractions). For constant graft length and when A–A attractions are stronger than B–B attractions, diblock copolymer grafted particles form long “caterpillar-like” structures with large particle diameters, and short nanowires with small particle diameters. In the dilute concentration regime, a small increase in the particle concentration does not change the cluster characteristics arising from each copolymer sequence, confirming that at a constant grafting density, particle size, and graft length the structure within a cluster is primarily governed by the copolymer sequence induced patchiness imparting a “valency” to the nanoparticle “atom.”
To go beyond the alternating sequence and diblock sequence we conducted a following study58 to investigate the effect of blockiness [Fig. 2(a)], defined as the length of contiguous blocks of like-monomers, ranging from the alternating sequence to diblock sequence. We used MC simulations to study the assembly of copolymer grafted nanoparticles with increasing blockiness in the grafted monomer sequence, in an implicit solvent. When A–B repulsion is negligible, with increasing blockiness at constant graft length, the cluster size and average coordination number decrease in the presence of A–A or B–B attractions [Fig. 2(b)], and are approximately constant in the presence of A–A and B–B attractions (not shown here, but presented in ref.58). When A–B repulsion is strong, the cluster size and average coordination number increase with increasing blockiness for small and large particles [Fig. 2(c)]. This is explained by the higher number of possible repulsive A–B contacts within the assembly of particles directed by grafted copolymers of low blockiness (e.g., A2B2) thereby reducing the tendency of those low blocky copolymer grafted particles to assemble [<Z> ∼ 0 and <N> ∼ 1 in Fig. 2(c)].
Finally, the extent to which monomer–particle attractive interactions change the above trends was found to be highly dependent on the relative strength of monomer–particle to monomer–monomer interactions, in addition to the ratio of particle size to graft length, and the grafting density. At low grafting density, the effect of monomer–particle interaction is expected to be higher than intermediate-high grafting density as there is a larger portion of the particle surface that is exposed compared to intermediate and high grafting density. However, even at the low grafting density, if the monomer–monomer attraction strength is comparable to particle–monomer attraction, and if the number of monomers (or length of the graft polymers) around the nanoparticle is large enough, the number of pair-wise monomer–monomer interactions will be significant and compete with monomer–particle interactions. For graft lengths on the order of 25 Kuhn segments and particle sizes of 4–12 nm, at low grafting density (and cubic lattice) we found the monomer–particle interactions to significantly affect the assembly only when the monomer–particle interactions are ∼16 times that of the monomer–monomer attraction strength (see Supporting Information in ref.58). We note that we have not investigated the effect of direct particle–particle interactions on the above trend; we expect the effect to be similar to that of monomer–particle interactions discussed in ref.58, in that at low grafting density, with chains having ∼25 Kuhn segments and particle size 4–12 nm, particle–particle interactions will affect assembly only when the particle–particle interactions are much stronger than the monomer–monomer interactions. In this regard, we direct readers to recent studies on polymer grafted magnetic nanoparticles where the dipolar nature of particle–particles interactions can compete with the homopolymer–homopolymer interactions, as shown by Jiao and Akcora.4
In summary, chemical heterogeneity in polymer grafts can be a valuable design parameter to tune the effective interparticle interactions and in turn assembly in a small molecule solvent and polymer matrix. The complex interplay of the copolymer sequence and chemistry, particle size relative to the grafted chain molecular weight, and grafting density along with the medium chemistry dictate the monomer arrangement around the grafted nanoparticle surface and as a result the effective interparticle interactions and assembly.
A recent comprehensive review13 on homopolymer grafted nanoparticle-based polymer nanocomposites presents extensive theoretical and experimental work (see references cited in ref.13) that have shown the critical role that the molecular weights of the grafted and matrix polymer play in dictating interparticle interactions, both at high and low grafting density. As stated in the introduction, at high grafting density, where the grafted chains are in the “strong brush” regime, nanoparticles disperse (aggregate) if the graft molecular weight is higher (lower) than matrix molecular weight with dispersion and aggregation being driven by wetting and dewetting of the grafted layer by matrix chains, respectively. At low grafting density, larger graft molecular weight chains can better shield nanoparticles from direct particle–particle contacts and lead to dispersion of grafted particles in the polymer matrix. Despite the importance of graft molecular weight for controlling the morphology, experimental and theoretical studies on polymer grafted nanoparticles have not investigated how polydispersity in the chains grafted on nanoparticles affects the morphology of the particles in a polymer matrix.
Polydispersity in polymer chains is measured in the form of a polydispersity index (PDI) which is the ratio of weight averaged molecular weight and number average molecular weight. Studies on polydisperse chains (PDI greater than 1) grafted on flat surfaces have shown that polydispersity affects the grafted layer thickness, free end monomer distribution, grafted chains' conformation, and in turn their wettability by matrix chains.67–70 For densely grafted flat brushes, as grafted chain polydispersity increases the grafted monomer density profile from the surface changes from having a parabolic shape (in monodisperse systems) to a concave shape, and the average stretching of the chains within the grafted brush decreases. This is because as polydispersity increases, the grafted chains have the ability to redistribute their monomers into less stretched conformations that are entropically favorable. Despite reduced chain stretching due to presence of long and short chains in a truly polydisperse distribution, there is an overall increase in the grafted layer thickness. At constant number average molecular weight, the difference between the polydisperse grafted layer thickness and monodisperse grafted layer thickness has been shown to scale with extent of polydispersity as q1/2, where q = PDI − 1. Increasing polydispersity has also been shown to reduce the width of the free end monomer distribution within the grafted layer.71 A special case of polydisperse systems is the one with a bidisperse or bimodal chain length distribution. Bidisperse flat brushes (made of long and short chains) has also received some attention in theory,72–74 experiments,75–77 and simulation.78–80 These past studies on bidisperse brushes in good solvent suggest a two-layer structure within the grafted region—a lower layer close to the grafting surface containing both short and long chains, and an upper layer containing monomers from the long chains. Additionally, simulations have found that in the inner layer the short chains are more compressed and long chains are more stretched in the bidisperse brush as compared to their conformations in the monodisperse brush,79 also confirmed by experiments.76, 77 When short chains dominate the composition of the bidisperse brush, the majority of the brush thickness is made of short chains and the long chains lie within, and in a thin layer above the short chains ends. When long chains dominate the brush, short chains coil near the grafting surface. The relative thickness of the brush inner layer was found to increase with grafting density when short chains dominate the brush, but decrease with grafting density when long chains dominate the brush, also in agreement with experiments.77 Although the above studies show that polydispersity affects both the chain conformations and the brush structure on flat surfaces, these effects have also been shown to alter the interface between the grafted polymer and a matrix of free polymers.69 When graft–matrix interaction is repulsive, polydispersity does not affect the width of the interface between grafted brush and matrix.69 When graft–matrix interaction is attractive or athermal, there is increased stretching of the brush chains into the matrix as the polydispersity increases from 1.0 to 3.0, indicating enhanced mixing between the matrix and highly polydisperse grafted chains.69
In terms of curvature, the case that is the extreme opposite of a grafted flat brush (zero curvature) is a star polymer with polymer arms grafted to an infinitely high curvature core. Theoretical calculations by Daoud and coworkers81 for the effective force F between polydisperse star polymers (where the polymer arms are polydisperse) in a good solvent show a drastically different expression from that seen for monodisperse star polymers. For the monodisperse case, the effective force between two star polymers is F(h)/kBT ∼ f3/2h−1, where f is the number of grafted chains and h is the height of the grafted region, as calculated by Witten and Pincus.82 For the polydisperse case with chain distribution defined as P(n) ∼ n−α for n < maximum graft length N and 1≤ α ≤2, P(n) = 0 for n > N, Daoud et al.81 found F(h)/kBT ∼ f9/2(4−α)h−ω where ω = (13α − 7)/(8 − 2α).
Although the polydisperse flat brush and star polymer discussed so far are cases at the extremes of surface curvatures, the system relevant to this article is polydisperse chains grafted on nanoparticle surfaces with finite curvature [Fig. 3(a)] where the available volume per grafted chain on a nanoparticle surface is higher than the flat surface but lower than that on an infinitely small core (star polymer).
Using MC simulations,5 we have studied a single spherical nanoparticle grafted with polydisperse homopolymer chains, in an implicit solvent, at a purely athermal limit, for varying polydispersity indices (PDI = 1 to 2.5) with same average graft length, varying particle diameter and grafting density. We showed that the conformations of the grafted chains in a polydisperse system deviates significantly from the monodisperse counterpart, and approaches that of a single grafted chain on the same particle size because of polydispersity-induced relief in monomer crowding in the grafted layer. Figure 3(b) shows the results at the highest grafting density of 0.65 chains/d2; as PDI increases the scaling exponent of grafted chains decreases from the value at monodisperse (PDI = 1) state and approaches the value obtained for a single chain (dashed lines) grafted on the same particle surface for both D = 5d and D = 8d. The exponent in Figure 3(b) was calculated using a power law fit to chain length N and average squared radius of gyration <R> data, when all grafted chains were considered. To show how polydispersity affects chains of the different lengths differently, we compared the radius of gyration of the short (less than average graft length) and long (greater than average graft length) chains to their monodisperse counterparts. The short chains exhibit a more coiled-up chain conformation than their monodisperse counterparts to provide larger free volume to the longer grafts so that the longer grafted chains 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 and high grafting density, they exhibit less stretched conformations than their monodisperse counterparts. In addition, at high grafting density in the polydisperse case, the longer grafts adopt a stretched “stem” conformation near the surface and a relaxed “crown” conformation farther from the particle surface.
These observations that the homopolymer chain conformations on hard nanoparticles with finite curvature are significantly affected by polydispersity in the grafted chain lengths leads to the question: Is the effect of polydispersity on grafted chain conformations on nanoparticle surfaces large enough to alter how matrix chains wet/dewet/deplete the grafted layer? If yes, is this change in matrix wettability of the grafted layer predictable so that one could deliberately introduce polydispersity to tailor inter–particle interactions? To answer these questions, we used a self-consistent PRISM-MC approach to study homopolymer-grafted nanoparticles, with bidisperse9 and polydisperse1 graft chain length distributions, placed in a homopolymer matrix.
We first studied9 the bidisperse distribution to elucidate the effect of bidispersity on grafted chain conformations and the PMF between bidisperse polymer-grafted particles in a homopolymer matrix, and compared these trends to those seen for monodisperse polymer-grafted particles. Our model consisted of spherical nanoparticles grafted with equal number of short and long homopolymers chains placed in a matrix of homopolymer chains. To capture purely the entropic driving forces in these polydisperse systems, we maintained athermal interactions (i.e., hard-sphere potential) between all pairs of sites. The PMF between two grafted particles in a polymer matrix suggests two regimes: the shorter grafts dictate the value of the PMF at contact and at small interparticle distances, while longer grafts dictate the long-range characteristics of the PMF. This is expected, as past work on monodisperse grafted particles has shown that the length of the repulsive tail in the PMF is dependent on the graft length.83 However, most interestingly, at intermediate interparticle distances the presence of bidispersity in grafts, that is, short and long chains, eliminates the mild-attractive well seen for monodisperse short chains. This suggests that introduction of a few long chains among short chains can lead to dispersion of the polymer grafted particles that would have aggregated otherwise. We also saw that, upon increasing short graft length Ns, while keeping the long graft length constant, the differences between the bidisperse and monodisperse PMF profiles reduces. This is because as the short graft length increases, the graft conformations in the bidisperse case approach those of the monodisperse case. Finally, we found that grafting density and matrix packing fraction affect the PMF between bidisperse grafted particles in qualitatively the same manner as their monodisperse counterparts.
This begs the question: Is the improved stability of dispersion only due to a bidisperse polymer grafts or would this be seen with polydisperse (with a broad distribution of length) grafts as well? And, how does “polydisperse grafted polymer distribution” improve grafted particle dispersion in a matrix compared to a bidisperse grafted polymer length distribution? To address this, in a more recent study,1 we studied effects polydispersity in graft length on the PMF between the grafted nanoparticles, varying the PDI of the grafted chains from 1 to 2.5.
In dense polymer solutions, increasing polydispersity reduces the strength of repulsion at contact and weakens the attractive well at intermediate interparticle distances, completely eliminating the attractive well at intermediate interparticle distances at high PDI. Figure 4(a) shows the PMF only at intermediate interparticle distances as a function of small increments in PDI, demonstrating that the elimination of attractive well does not happen at small PDI, and that a critical PDI is needed to eliminate the attractive well completely. The elimination of the mid-range attractive well is due to the longer grafts in the polydisperse graft length distribution that introduce longer range steric repulsion, and the reduced crowding in the grafted layer that alters the wetting of the grafted layer by matrix chains. Calculation of the matrix-graft penetration depth showed an increased penetration or wetting of the polydisperse grafted layer by the matrix chains. Figure 4(b) shows most recent results84 directly comparing the PMF between grafted nanoparticles having polydisperse distribution to that with bidisperse distribution at PDI = 1.5, and monodisperse grafts with same average molecular weight. The results show that at a constant (and low) PDI = 1.5, the presence of a continuous distribution (polydisperse) of polymer graft lengths is able to eliminate the attractive well completely while a bidisperse distribution of polymer graft lengths shows a small attractive well, smaller in magnitude than monodisperse polymer grafts. We believe the reason for this trend is the presence of a few grafts in the polydisperse graft length distribution that are significantly longer than the long grafts in the bidisperse distribution. A more detailed comparison of polydisperse to bidisperse distribution is the focus of a current investigation in our group.
In summary, we predict that polydispersity in graft length can be used to stabilize dispersions of grafted nanoparticles in a polymer matrix at conditions where monodisperse grafts would cause aggregation. In agreement with these predictions, recent experiments by Rungta et al.,2 where silica particles with bimodal polystyrene grafts, synthesized using step-by-step RAFT polymerization, were found be better dispersed in a monodisperse polystyrene matrix as compared to silica particles with monodisperse polymer grafts of length comparable to the long polymers in the bidisperse distribution. These promising results from theory and experiments should direct more groups to treat polydispersity in homopolymer ligands as a design parameter for stabilizing dispersions of polymer grafted particle in a polymer matrix.
Most of this article has focused on how chemical and physical heterogeneity in polymer grafts affects the effective interactions and structure of polymer grafted nanoparticles in a homopolymer matrix. There is much interest in directing assembly of nanoparticles in a block copolymer matrix (see reviews in refs.85 and86). Although past studies have focused on morphology of homopolymer grafted particles in block copolymer domains, there have been significantly fewer investigations involving copolymer grafted particles in a block copolymer matrix. Recent theoretical studies by Ganesan and coworkers59, 60 on flat random copolymer brushes in contact with block copolymer matrix show that the interfacial interactions is modulated by the grafted random copolymers conformations which changes in response to the overlaying block copolymer film. Our work17 suggests that AB diblock copolymer grafted nanoparticles will exhibit different arrangements in the A and B domains of a block copolymer matrix due to their different effective interactions in A-matrix and B-matrix. Since homopolymer grafted particles have the ability to relieve the interfacial tension between A and B domains, which drives transitions in the block copolymer morphology87 and stabilizes bicontinuous network morphologies88, it would interesting to investigate if morphological transitions in block copolymer matrices can be tuned by tailoring the monomer sequences in the copolymer grafts.
Theoretical work1, 9, 84 and recent experiments2 with polydisperse and bidisperse grafts on nanoparticles suggests that polydispersity in the grafted polymers can stabilize dispersions in a monodisperse polymer matrix better than the corresponding monodisperse polymer grafted particles. A natural extension of this work would be to study monodisperse/polydisperse polymer grafted particles placed in a polydisperse matrix. To conduct these studies in molecular simulations one would need a large simulation box to incorporate the broad distribution of matrix chains and finite number of polymer grafted particles, which would be computational intensive even with parallelized central processing unit (CPU)-based programs. In the past, to overcome these computational intensities, we adopted the self-consistent PRISM-MC approach.9, 17 In the case of a polydisperse matrix, it would be challenging to use the self-consistent PRISM-MC approach due to higher number of types of sites needed to represent each matrix chain length, and separate MC run for each matrix chain length, thus losing any computational advantage. With the recent progress in graphics processing units (GPUs), and simultaneous development of molecular simulation algorithms (e.g., HOOMD89 and LAMMPs90 packages) designed for or adapted to run optimally on GPUs, we now have the ability to simulate a finite volume fraction of polymer grafted nanoparticles in an explicit polydisperse polymer matrix (like a system shown in Fig. 5). What makes dynamic simulations of polymer systems perfect for implementation on GPUs is their computational complexity and parallelism. Although packages that were originally designed for parallel CPUs implementation, like LAMMPs, have now incorporated GPU compatibility, there are other packages, like HOOMD, that have explicitly been designed for GPU execution. With these developments, one is able to achieve large speedups compared to the same simulations on CPUs, making a variety of coarse-grained simulations in the field of polymer nanocomposites that would have been impractical before possible now. For example, in a recent study by Glotzer and coworkers,91 which aimed at studying the stability of the double gyroid phase in a melt of polymer tethered nanospheres with polydispersity in particle sizes, they found a 20–30 times speedup with GPUs over CPUs. They92 also comment that their system of polydisperse particles forced the speedup to be relatively lower than that possible in monodisperse polymer-nanoparticle systems, and with new advances in GPUs and CPUs one could achieve 80–120 times speedup with GPU based simulations over CPU based simulations. Our preliminary test for Brownian dynamics simulations (using HOOMD) in a box containing about 10 polymer grafted nanoparticles in an explicit polymer matrix, with particle sizes ∼5 nm and graft and matrix chain lengths of 10–30 Kuhn segments (∼28,000 coarse grained Kuhn segments) achieves computational speeds of 1400–1600 time steps per second on a single C2050 GPU processor! In addition to obtaining dynamic trajectories faster with GPU computing, data analysis of trajectories involving histograms also perform well on GPUs due to their intrinsic parallelism. The above developments in GPU-based computations should allow scientists to revisit simulations of polymer nanocomposites, especially those with polymer grafted nanoparticles in an explicit polymer matrix, that were too intensive with CPUs alone.
Finally, while this feature article focused solely on the structure and morphology with polymer grafted particle based nanocomposites, these morphological studies are all motivated by the common goal of controlling macroscopic properties of the polymer nanocomposite. It is therefore a natural next step for theoretical and experimental studies to elucidate how the chemical and physical heterogeneity in polymer grafts affects the processing and rheological behavior93 and resulting mechanical properties of these polymer nanocomposites.
The author acknowledges the partial financial support from Department of Energy under grant number DE-SC0003912 for the polydisperse grafted nanoparticle work, and the partial financial support from National Science Foundation under grant number CBET-0930940 for the copolymer grafted nanoparticle work. The author acknowledges C. Phillips, C. Iacovella and S. C. Glotzer for providing detailed information on the comparison of GPU and CPU computational speeds presented in the last section of this article. The author is grateful to T. B. Martin, E. Jankowski, A. Seifpour, N. Nair, and P. Dodd for scientific discussions and schematics in this article.
Arthi Jayaraman received her Bachelor's degree in Chemical Engineering from BITS, Pilani, India. She received her Ph.D. in Chemical Engineering from North Carolina State University. She conducted her postdoctoral research in Materials Science and Engineering at UIUC. In 2008, she joined the Department of Chemical and Biological Engineering at University of Colorado-Boulder, where she is currently Patten Assistant Professor. Her research interests lie in development and application of theory and simulation to study macromolecular materials.