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

Polymer Grafted Nanoparticles: Effect of Chemical and Physical

Heterogeneity in Polymer Grafts on Particle Assembly and Dispersion

Arthi Jayaraman

Department of Chemical and Biological Engineering, University of Colorado-Boulder, UCB 596, Boulder, Colorado 80309

Correspondence to: A. Jayaraman (E-mail: [email protected])

Received 17 December 2012; accepted 15 January 2013; published online 7 February 2013

DOI: 10.1002/polb.23260

ABSTRACT: Macroscopic properties of polymer nanocomposites

depend on the microscopic composite morphology of the con-

stituent nanoparticles and polymer matrix. One way to control

the spatial arrangement of the nanoparticles in the polymer ma-

trix 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 interac-

tions 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 fea-

ture 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. VC 2013 Wiley Periodicals, Inc. J.

Polym. Sci., Part B: Polym. Phys. 2013, 51, 524–534

KEYWORDS: copolymer; molecular modeling; polydispersity;

polymer nanocomposites

INTRODUCTION Theoretical and experimental work in poly-mer nanocomposites has established that adding nanoscaleadditives, such as silica particles, metal particles, carbonnanotubes, or layered silicates to a polymer matrix enhancesthe inherent properties of the polymer due to synergisticinteractions between the polymer matrix and the nanopar-ticles. Conventionally, it is accepted that isotropic dispersionof the nanoparticles is essential for improved propertiessuch as reduced permeability, increased mechanical strength,improved thermal resistance, and so forth. Conversely, in thecase of polymer nanocomposites that can be used for pho-tonics, solar or photovoltaic (alternative energy), and elec-tronics applications, precise assembly and ordering of nano-particles mediated by a polymer matrix is needed.Irrespective of the application, controlling the morphology ofthe nanoparticles within the polymer matrix is highly desira-ble to design and engineer materials with optimal targetmacroscopic properties. One way to control morphologywithin a polymer nanocomposite is by functionalizing orgrafting the nanoparticle surface with polymers1–41 that canthen manipulate the interfacial interactions between thenanoparticles and the medium (a small molecule solvent ora polymer matrix of same or different chemistry as the graftpolymer) that the particles are placed in, and thus controltheir spatial arrangement. A recent comprehensive review byGreen covers major theoretical and experimental advancesspecifically in the area of polymer grafted nanoparticles in a

homopolymer matrix.13 Advances in synthetic ability to cre-ate designer polymer functionalized particles of desired poly-mer and particle chemistry, polymer molecular weight, andgrafting density have motivated many theoretical and com-putational studies to explore the effects of this vast set ofmolecular-level parameters on assembly/dispersion of thepolymer grafted nanoparticles in a medium. The fundamentalinsight provided by theory and simulation studies on theeffects of polymer and particle chemistry, graft and matrixmolecular weight, and grafting density on the compositemorphology and phase behavior provide valuable guidanceto ongoing synthetic efforts in experimental groups. Thefocus of this feature article is to present such theoretical andcomputational contributions from our group, and to brieflyreview other recent advances, theoretical and experimental,in this field of polymer functionalized nanoparticles basednanocomposites.

Computational6,7,9,12,16,18–21,24–27,33 and experimen-tal3,4,10,13,14,22,28–31,35,40 studies on nanocomposites, consist-ing of homopolymer-functionalized nanoparticles in apolymer matrix, have demonstrated that the chemistry andrelative molecular weights of the graft and matrix polymers,grafting density, and nanoparticle size are parameters thatplay a critical role in dictating the spatial organization of thenanoparticles. For example, experimental studies30,42,43 haveachieved the migration of the polymer-grafted nanoparticlesfrom one domain to another domain in the matrix by

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thermally changing the chemistry of the grafted homopoly-mers on the nanoparticle, and thus the compatibility of thegrafted polymer and matrix. Another important parameterthat dictates the effective interparticle interaction, and there-fore the particle assembly, is the polymer grafting density,defined as the number of grafted polymers per unit surfacearea of the nanoparticle. At high grafting density, the graftedpolymers extend due to crowding and form a brush-like con-formation on the particle surface. Particles carrying adensely grafted homopolymer brush placed in a homopoly-mer matrix whose chemistry is identical to the grafted poly-mer disperse (aggregate) if the molecular weight of thematrix homopolymer is lower (higher) than that of the graftedhomopolymer.22,31,36,37 At low grafting density20,33,44,45 thegrafted polymers do not face any crowding from monomersof adjacent chains and as a result do not stretch into brush-like conformations. The surface of the nanoparticle that isexposed versus that covered by the grafted monomersdictates the effective interparticle interactions.20,27 Suchhomopolymer-grafted nanoparticles at low grafting densitieshave been shown to assemble into a variety of nanostruc-tures in solvent34,38,46–51 and in matrix.22

Although all of the above studies have established the effectof various molecular parameters on the behavior of relativelymonodisperse homopolymer-grafted nanoparticles either insolvent or in a polymer matrix, the effect of polymer function-alization with chemical heterogeneity (e.g., copolymers)17,52–61

or physical heterogeneity (e.g., distribution of molecularweights)1,2,5,9 on spatial organization of nanoparticles in a sol-vent or polymer matrix has been studied to a much smallerextent. Our focus in the past few years has been on under-standing how heterogeneous polymer ligands, such as copoly-mers and polydisperse homopolymers, bring new phase behav-ior in polymer nanocomposites, as compared to monodispersehomopolymer ligands. This feature article presents a shortreview of recent work in the area of copolymer functionalizednanoparticles followed by a review of polydisperse homopoly-mer functionalized nanoparticles. The article concludes with asection on future directions/trends for both theory and experi-ments in this field of polymer functionalized particles.

COPOLYMER FUNCTIONALIZED NANOPARTICLES

Copolymer functionalization, as opposed to homopolymerfunctionalization, creates additional tuning parameters ofgraft sequence and monomer chemistry (or interactions)

which provide further control over the morphology of poly-mer grafted nanoparticles in a given medium. Copolymer-grafted nanoparticles have been synthesized successfully byvarious experimental groups, either using atom transfer radi-cal polymerization to grow copolymers from surface of silicananoparticles62 or using Z-supported reversible addition-fragmentation chain transfer (RAFT) polymerization tosynthesize diblock copolymer-grafted particles.63 These advan-ces in synthesis of polymer functionalized nanoparticles havemotivated recent theory17,54,64 and simulations55,57,58,65 tostudy how copolymer functionalization on nanoparticlesaffects assembly/dispersion in solvents or polymer matrix.

Vorselaars et al.54 have used (self-consistent field theory) SCFTto study dense layers of diblock copolymers grafted onto a sin-gle spherical nanoparticle at high grafting density to under-stand the domains formed by the two monomers within thegrafted layer. They found various domain shapes on the parti-cle surface depending on the composition of the copolymer,and discussed the stability of the various morphologies onthese highly curved nanoparticle surfaces, in contrast to flatsurfaces (zero curvature). Zhu et al.64 have employed bothSCFT and DFT (density functional theory) to study a dense sys-tem of nanoparticles each with a single diblock copolymergraft. When the particle surface is chemically neutral to thegrafted chain, they observed typical block copolymer morphol-ogies (i.e., cylinders and lamellae) determined by both thecomposition of the copolymer and the particle size. When theparticle surface is repulsive to both blocks of the copolymer,they observed hierarchical morphologies, such as ‘‘lamellaewith cylinders at interfaces’’ not typically observed with blockcopolymer melts. Chan et al.57 have used molecular simula-tions to compare the assembled phases observed in a melt ofcubic nanoparticles each grafted with a single diblock copoly-mer to that seen in melts of diblock and triblock copolymers(no particles). They contrast the effect the bulky and rigidcubic nanoparticle has on the curvatures within the assembledphases to the effect the middle linear block in a triblock copol-ymer system has on its assembled phases. Although the abovestudies focused on either a single copolymer grafted nanopar-ticle at high grafting density54 or grafted nanoparticles with asingle grafted chain57,64, our recent work have focused on con-ducting systematic studies of single to many copolymer graftedspherical nanoparticles with varying copolymer grafting den-sities (low to intermediate), copolymer sequence and chemis-try (or interactions), and particle sizes using a combination oftheory17 and simulation techniques.55,58,65

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.

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Our first goal was to quantify the effect of the monomersequence and chemistry on the chain conformations of thecopolymers grafted on spherical nanoparticles (in the 2–10nm range) at low to intermediate grafting density. We con-ducted Monte Carlo (MC) simulations of a single sphericalnanoparticle grafted with AB copolymers with either alter-nating, multiblock or diblock sequences in an implicit solventwith monomer–monomer attractive interactions between A–A or B–B monomer pairs.65 These interactions were chosento implicitly model the effect of an A- or B-selective solvent.For example, in an A-selective solvent, one would expect theB monomers to have an effective attraction caused by thesolvent molecules preferentially locating themselves aroundA-monomers. Thus, a system with B–B attractions wouldmimic the implicit effect of an A-selective solvent. Similarly,a system with A–A attractions would mimic the impliciteffect of an B-selective solvent. Through visual analysis andcalculation of monomer contacts, monomer concentrationprofiles from the particle surface, and grafted chains’ radii ofgyration we concluded that on a copolymer grafted nanopar-ticle the monomer sequence, particle diameter, and graftingdensity dictate whether (a) the grafted copolymer chains ag-gregate to bring attractive monomers from multiple graftedchains together (interchain and intrachain monomer aggrega-tion) if the enthalpy gained by doing so offsets the entropicloss caused by stretching of chains, or (b) each grafted chainfolds onto itself to bring its attractive monomers together(only intrachain monomer aggregation) if the entropic lossfrom interchain aggregation cannot be overcome by theenthalpic gain. For six AB copolymer chains, each containing24 Kuhn segments or ‘‘monomers,’’ grafted on a sphericalparticle of diameter D ¼ 4d, where d is the size of a Kuhnsegment (�1 nm), interchain and intrachain monomer aggre-gation occurred, and the radius of gyration varied non-monotonically with increasing blockiness of the monomersequence. This is because as the blockiness of the monomersequence increases the copolymer chains have larger blocksof like-monomers that can easily bring like-attractive mono-mers from other grafted chains or within themselves to-gether, with the exception of the alternating copolymerwhich has the most frustrated AB sequence. At larger parti-cle diameters with six chains of chain length 24 monomers,the grafted chains transition to purely intrachain monomeraggregation because it is entropically unfavorable for the co-polymer chains to stretch to make interchain monomer con-tacts with a neighboring grafted chain. At these higher parti-cle diameters the radius of gyration of the graftedcopolymers varies monotonically with monomer sequencedue to intrachain monomer aggregation, because as thesequence becomes blockier (like monomers are grouped to-gether), the copolymer chain has to fold less compactly tomaximize the enthalpically favorable contacts while main-taining high conformational entropy. For AnBn diblock copol-ymer grafts and constant particle size, the solvent quality ormonomer–monomer attractions—A–A or B–B—affected thechain conformations significantly. Since the copolymer chainswere grafted through the A-block, the A monomers areplaced closer to the particle surface. Thus,

B-selective implicit solvent (or A–A attractions) led to Amonomers aggregating close to the particle surface andshielding the particle surface. In contrast, A-selective implicitsolvent (or B–B attractions) led to the B monomers, whichare topologically placed farther from the surface in thegrafted chain, to aggregate farther from the particle surfaceexposing the nanoparticle surface. This complex interplay ofmonomer sequence, monomer attractions, ratio of graftedchain length to particle diameter and their non-trivial effectson the grafted chain conformations on the nanoparticle sur-face motivated us to study how these parameters affect thepotential of mean force (PMF) or effective interactionsbetween two copolymer-grafted nanoparticles in a medium.

As our system of interest was polymer nanocomposites withpolymer grafted nanoparticles, we set out to calculate thePMF between copolymer-grafted nanoparticles in a homopol-ymer matrix, at low grafting density, as a function of mono-mer sequence (alternating versus diblock) in the graftedchain, molecular weight of the grafted and matrix chains,particle size, matrix packing fraction (dense polymer solu-tions to melts) and monomer attractions. Using a self-consistent Polymer Reference Interaction Site Model(PRISM)-MC17 approach, we studied AB copolymer-graftedspherical nanoparticles at low grafting density (with the co-polymer chains grafted on particle through A monomer)placed in A or B homopolymer matrix. We use a self-consist-ent approach where the grafted chain conformations that areinputs to PRISM theory, originally developed by Schweizerand Curro,66 are provided by MC simulations of a single co-polymer-grafted nanoparticle in an external medium-inducedsolvation potential obtained from PRISM theory. We do notpresent the details of this self-consistent PRISM-MCapproach here, and instead direct the reader to a recentreview of this approach and description of the method as itis applied to study polymer grafted nanoparticles.7 We foundthat the PMF between two alternating AB copolymer graftedparticles in a homopolymer matrix is insensitive to thechemistry of the homopolymer matrix (A- or B-homopoly-mer) in the case of a weak A–B v parameter [Fig. 1(a)]. TheA–B v parameter is a collective measure of relative A-A, B-Band A-B attractions defined as v � eAB � 1/2(eAA þ eBB)where eij is the strength of attraction between i and j mono-mers. The Inset of Figure 1(a) shows a schematic of thearrangement of A and B monomers on the particle surfaceas seen in either A- or B-polymer matrix. Additionally, in adense solutions of A or B homopolymer matrix in the case ofa weak A–B v parameter, the behavior of the alternatinggrafted particles is similar to the behavior of particles withathermal homopolymer grafts [solid black line in Fig. 1(a)].The weak A–B v coupled with the frustrated—ABAB—mono-mer arrangement do not allow the alternating grafted chainto assume a compact structure because the loss in conforma-tional entropy upon doing so is not overcome by theenthalpic gain, and as a result the grafts assume a configura-tion that is similar to the homopolymer case.

In contrast, the AB diblock grafted particles exhibit behaviorthat is strongly dependent on the matrix chemistry even at


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weak A–B v parameter [Fig. 1(b)]. The formation of aggre-gates of blocks of A or B blocks [as shown in Fig. 1(b)insets] and how they aid or hinder the matrix-inducedattractive interactions between the particles dictate the mag-nitude, nature and location of attraction/repulsion in thePMF between two grafted particles. For example, whenplaced in homopolymer B matrix in the presence of weakA–B v parameter (eAA ¼ 0.2kT) the attractive A monomers inthe block closer to the particle surface aggregate and form aprotective shell on the particle surface, away from B matrixchains. This shell of A monomers on the particle surfacesterically hinders matrix-induced direct contact with anothergrafted particle. When placed in a homopolymer A matrixwith eBB ¼ 0.2kT, the block of attractive B monomers formsaggregates away from the nanoparticle surface and the athe-rmal A monomers do not form a shell on the surface, thusleading to matrix-induced depletion-like attraction at contactand B-aggregate-induced steric repulsion at larger interpar-ticle distances [Fig. 1(b)].

We also observed that increasing matrix packing fractionsreduced the dependence of the PMF on the monomersequence and intermonomer attractions because at high ma-trix packing fractions, the matrix-induced depletion-likeforces exerted on the grafted particles dominate and over-come the steric hindrance caused by the conformations ofthe grafts. This leads to an attractive PMF between the co-polymer grafted particles in a dense polymer matrix, forboth monomer sequences. At constant graft length (Ng),when the matrix chain length (Nm) was varied (Nm < Ng, Nm

¼ Ng, Nm > Ng), the PMF at contact changed only by 0.1–0.2kT for both alternating and diblock grafts, while maintain-ing the same quality (i.e., repulsion or attraction) at constant

packing fraction. This minimal matrix length effects for theselightly grafted copolymer grafted particles is in agreementwith what had been seen before for homopolymer graftedparticles at low grafting density in a chemically identicalhomopolymer matrix.20,27 At constant graft length and num-ber of grafts, as particle size was increased we observed: (a)a greater portion of the particle surface was exposed to thesurroundings unobstructed by the monomers of the graftedchains; (b) the role of monomer sequence gradually vanishedas evidenced by the similar PMF plots for the alternatingand diblock grafted particles at larger particle sizes due tothe interactions being dominated by core–core contactsrather than grafted copolymer shielded interactions; (c) ma-trix chemistry affected the PMF to a smaller extent asobserved by the reduced difference between the respectivePMF values at contact for the diblock grafts in A and B ma-trix. Finally, an increase in the A–B v parameter has a minoreffect on the PMF for large particles and short graft chainlengths because of the small number of attractive unitswithin the grafted chain; furthermore, the matrix chemistryhas little effect on the PMF at short graft lengths. As thegraft length increases an intermediate A–B v parameter ispotent enough to appreciably alter the behavior of thediblock grafted particles depending on which block isattractive.

The above studies of the grafted chain conformations andeffective interactions between copolymer grafted particlesshow that monomer sequence and chemistry in the graftedcopolymers strongly influence monomer arrangement on thenanoparticles, imparting a unique sequence-dependent‘‘patchiness’’ onto the nanoparticles. This unique ‘‘patchiness’’that arises from sequence-dependent folding of the grafted

FIGURE 1 PMF (in kT) versus interparticle distances (in units of d, where d is the size of a Kuhn segment) for a system of infinitely

dilute volume fraction of a) alternating and b) diblock AB copolymer grafted nanoparticles in a dense solution of A- or B-homopol-

ymer matrix. The copolymer grafted nanoparticles have six grafts each of length 24 Kuhn segments and the particle diameter is

equal to 4d. The red circles correspond to system where eAA ¼ 0.2kT, blue crosses correspond to system where eBB ¼ 0.2kT and

solid black line correspond to a system with purely athermal interactions. For more details the reader is directed to Ref. 17.



copolymers motivated our next set of studies focused onunderstanding how the ‘‘patchiness’’ affected assembly ofmultiple copolymer grafted nanoparticles, as a function ofthe monomer sequence and monomer chemistry. We usedMC simulations to demonstrate how the monomer sequencein the grafted copolymers can be a tuning parameter todirect assembly of nanoparticles, and the shapes, sizes andstructures of the assembled nanoclusters.55,58 We studiedspherical nanoparticles grafted with AB copolymers witheither alternating or diblock sequence55 and a range of like-monomer (A–A and/or B–B) attractive interactions in thepresence of either relatively strong or negligible unlike-monomer (A–B) repulsive interaction. In the presence ofnegligible A–B repulsions the alternating sequence producesnanoclusters that are relatively isotropic regardless ofwhether A–A or B–B monomers are attractive, while thediblock sequence produces nanoclusters that are smaller andmore compact when the block closer to the surface (A–A) isattractive and larger loosely held together clusters when theouter block (B–B) is attractive. Particle size and graft lengthbalance enthalpic gain and entropic losses coming frominterparticle interchain contacts and/or intergrafted andintragrafted chain contacts within the same grafted particle,and in turn dictate the shape and size of the cluster. In thepresence of strong A–B repulsions, the alternating sequenceleads to either particle dispersion or smaller clusters thanthose at negligible A–B repulsions because the alternatingsequence causes A-B repulsive contacts when trying to makeA-A and B-B attractive contacts. In contrast, for the diblocksequence, the presence or absence of A–B repulsions did notalter the cluster characteristics because of topologicallyseparated A and B blocks, and in turn fewer A-B repulsivecontacts than alternating sequence. Additionally, diblock co-polymer grafted particles tend to assemble into anisotropicshapes despite the isotropic grafting of the copolymer chainson the particle surface because of fewer patches formedfrom 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 arestronger than B–B attractions, diblock copolymer graftedparticles form long ‘‘caterpillar-like’’ structures with largeparticle diameters, and short nanowires with small particlediameters. In the dilute concentration regime, a smallincrease in the particle concentration does not change thecluster characteristics arising from each copolymer sequence,confirming that at a constant grafting density, particle size,and graft length the structure within a cluster is primarilygoverned by the copolymer sequence induced patchinessimparting a ‘‘valency’’ to the nanoparticle ‘‘atom.’’

To go beyond the alternating sequence and diblock sequencewe conducted a following study58 to investigate the effect ofblockiness [Fig. 2(a)], defined as the length of contiguousblocks of like-monomers, ranging from the alternatingsequence to diblock sequence. We used MC simulations tostudy the assembly of copolymer grafted nanoparticles withincreasing blockiness in the grafted monomer sequence, in animplicit solvent. When A–B repulsion is negligible, with

increasing blockiness at constant graft length, the cluster sizeand average coordination number decrease in the presence ofA–A or B–B attractions [Fig. 2(b)], and are approximately con-stant in the presence of A–A and B–B attractions (not shownhere, but presented in ref. 58). When A–B repulsion is strong,the cluster size and average coordination number increasewith increasing blockiness for small and large particles [Fig.2(c)]. This is explained by the higher number of possible re-pulsive A–B contacts within the assembly of particles directedby grafted copolymers of low blockiness (e.g., A2B2) therebyreducing the tendency of those low blocky copolymer graftedparticles to assemble [<Z> � 0 and <N> � 1 in Fig. 2(c)].

Finally, the extent to which monomer–particle attractiveinteractions change the above trends was found to be highlydependent on the relative strength of monomer–particle tomonomer–monomer interactions, in addition to the ratio ofparticle size to graft length, and the grafting density. At lowgrafting density, the effect of monomer–particle interaction isexpected to be higher than intermediate-high grafting den-sity as there is a larger portion of the particle surface that isexposed compared to intermediate and high grafting density.However, even at the low grafting density, if the monomer–monomer attraction strength is comparable to particle–

FIGURE 2 (a) Schematic showing the sequences with decreas-

ing blockiness from top to bottom present in the copolymer

grafts. Ensemble average coordination number <Z> defined as

number of neighbors each particle has in an assembled cluster,

and ensemble average number of particles in an assembled

cluster <N> as a function of sequence blockiness in a system

of dilute concentration of AB copolymer grafted nanoparticles

with six grafts of length 24 Kuhn segments on spherical par-

ticles of diameters �4 nm, in the presence of (b) only B––B

attractions while other pairs of interactions are athermal, and

(c) B–B attractions and A–B repulsions while other pairs of

interactions are athermal. For other characterizations and pa-

rameters the reader is directed to Refs. 55 and 58


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monomer attraction, and if the number of monomers (orlength of the graft polymers) around the nanoparticle islarge enough, the number of pair-wise monomer–monomerinteractions will be significant and compete with monomer–particle interactions. For graft lengths on the order of 25Kuhn segments and particle sizes of 4–12 nm, at low graft-ing density (and cubic lattice) we found the monomer–parti-cle interactions to significantly affect the assembly onlywhen the monomer–particle interactions are �16 times thatof the monomer–monomer attraction strength (see Support-ing Information in ref. 58). We note that we have not investi-gated the effect of direct particle–particle interactions on theabove trend; we expect the effect to be similar to that ofmonomer–particle interactions discussed in ref. 58, in that atlow grafting density, with chains having �25 Kuhn segmentsand particle size 4–12 nm, particle–particle interactions willaffect assembly only when the particle–particle interactionsare much stronger than the monomer–monomer interactions.In this regard, we direct readers to recent studies onpolymer grafted magnetic nanoparticles where the dipolarnature of particle–particles interactions can compete withthe homopolymer–homopolymer interactions, as shown byJiao and Akcora.4

In summary, chemical heterogeneity in polymer grafts canbe a valuable design parameter to tune the effective inter-particle interactions and in turn assembly in a small mole-cule solvent and polymer matrix. The complex interplay ofthe copolymer sequence and chemistry, particle size relativeto the grafted chain molecular weight, and grafting densityalong with the medium chemistry dictate the monomerarrangement around the grafted nanoparticle surface andas a result the effective interparticle interactions andassembly.

POLYDISPERSE HOMOPOLYMER FUNCTIONALIZED

NANOPARTICLES

A recent comprehensive review13 on homopolymer graftednanoparticle-based polymer nanocomposites presents exten-sive theoretical and experimental work (see references citedin ref. 13) that have shown the critical role that the molecu-lar weights of the grafted and matrix polymer play in dictat-ing interparticle interactions, both at high and low graftingdensity. As stated in the introduction, at high graftingdensity, where the grafted chains are in the ‘‘strong brush’’regime, nanoparticles disperse (aggregate) if the graft molec-ular weight is higher (lower) than matrix molecular weightwith dispersion and aggregation being driven by wetting anddewetting of the grafted layer by matrix chains, respectively.At low grafting density, larger graft molecular weight chainscan better shield nanoparticles from direct particle–particlecontacts and lead to dispersion of grafted particles in thepolymer matrix. Despite the importance of graft molecularweight for controlling the morphology, experimental andtheoretical studies on polymer grafted nanoparticles havenot investigated how polydispersity in the chains grafted onnanoparticles affects the morphology of the particles in apolymer matrix.

Polydispersity in polymer chains is measured in the form ofa polydispersity index (PDI) which is the ratio of weightaveraged molecular weight and number average molecularweight. Studies on polydisperse chains (PDI greater than 1)grafted on flat surfaces have shown that polydispersityaffects the grafted layer thickness, free end monomer distri-bution, grafted chains’ conformation, and in turn their wett-ability by matrix chains.67–70 For densely grafted flatbrushes, as grafted chain polydispersity increases the graftedmonomer density profile from the surface changes from hav-ing a parabolic shape (in monodisperse systems) to a con-cave shape, and the average stretching of the chains withinthe grafted brush decreases. This is because as polydisper-sity increases, the grafted chains have the ability to redistrib-ute their monomers into less stretched conformations thatare entropically favorable. Despite reduced chain stretchingdue to presence of long and short chains in a truly polydis-perse distribution, there is an overall increase in the graftedlayer thickness. At constant number average molecularweight, the difference between the polydisperse grafted layerthickness and monodisperse grafted layer thickness has beenshown to scale with extent of polydispersity as q1/2, where q¼ PDI � 1. Increasing polydispersity has also been shown toreduce the width of the free end monomer distributionwithin the grafted layer.71 A special case of polydisperse sys-tems is the one with a bidisperse or bimodal chain lengthdistribution. Bidisperse flat brushes (made of long and shortchains) has also received some attention in theory,72–74

experiments,75–77 and simulation.78–80 These past studies onbidisperse brushes in good solvent suggest a two-layer struc-ture within the grafted region—a lower layer close to thegrafting surface containing both short and long chains, andan upper layer containing monomers from the long chains.Additionally, simulations have found that in the inner layerthe short chains are more compressed and long chains aremore stretched in the bidisperse brush as compared to theirconformations in the monodisperse brush,79 also confirmedby experiments.76,77 When short chains dominate the compo-sition of the bidisperse brush, the majority of the brushthickness is made of short chains and the long chains liewithin, and in a thin layer above the short chains ends.When long chains dominate the brush, short chains coil nearthe grafting surface. The relative thickness of the brush innerlayer was found to increase with grafting density when shortchains dominate the brush, but decrease with grafting den-sity when long chains dominate the brush, also in agreementwith experiments.77 Although the above studies show thatpolydispersity affects both the chain conformations and thebrush structure on flat surfaces, these effects have also beenshown to alter the interface between the grafted polymerand a matrix of free polymers.69 When graft–matrix interac-tion is repulsive, polydispersity does not affect the width ofthe interface between grafted brush and matrix.69 Whengraft–matrix interaction is attractive or athermal, there isincreased stretching of the brush chains into the matrix asthe polydispersity increases from 1.0 to 3.0, indicatingenhanced mixing between the matrix and highly polydispersegrafted chains.69



In terms of curvature, the case that is the extreme oppositeof a grafted flat brush (zero curvature) is a star polymerwith polymer arms grafted to an infinitely high curvaturecore. Theoretical calculations by Daoud and coworkers81 forthe effective force F between polydisperse star polymers(where the polymer arms are polydisperse) in a good solventshow a drastically different expression from that seen formonodisperse star polymers. For the monodisperse case, theeffective force between two star polymers is F(h)/kBT � f 3/2h�1,where f is the number of grafted chains and h is the height of thegrafted region, as calculated by Witten and Pincus.82 For thepolydisperse case with chain distribution defined as P(n) � n�a

for n < maximum graft length N and 1� a �2, P(n) ¼ 0 for n >N, Daoud et al.81 found F(h)/kBT � f 9/2(4�a) h�x where x ¼(13a � 7)/(8 � 2a).

Although the polydisperse flat brush and star polymerdiscussed so far are cases at the extremes of surface curva-tures, the system relevant to this article is polydispersechains grafted on nanoparticle surfaces with finite curvature[Fig. 3(a)] where the available volume per grafted chain on ananoparticle surface is higher than the flat surface but lowerthan that on an infinitely small core (star polymer).

Using MC simulations,5 we have studied a single sphericalnanoparticle grafted with polydisperse homopolymer chains,in an implicit solvent, at a purely athermal limit, for varyingpolydispersity indices (PDI ¼ 1 to 2.5) with same averagegraft length, varying particle diameter and grafting density.We showed that the conformations of the grafted chains in apolydisperse system deviates significantly from the monodis-perse counterpart, and approaches that of a single graftedchain 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 densityof 0.65 chains/d2; as PDI increases the scaling exponent of

grafted chains decreases from the value at monodisperse(PDI ¼ 1) state and approaches the value obtained for a sin-gle chain (dashed lines) grafted on the same particle surfacefor both D ¼ 5d and D ¼ 8d. The exponent in Figure 3(b)was calculated using a power law fit to chain length N andaverage squared radius of gyration <R2

g> data, when allgrafted chains were considered. To show how polydispersityaffects chains of the different lengths differently, we com-pared the radius of gyration of the short (less than averagegraft length) and long (greater than average graft length)chains to their monodisperse counterparts. The short chainsexhibit a more coiled-up chain conformation than theirmonodisperse counterparts to provide larger free volume tothe longer grafts so that the longer grafted chains can gainconformational entropy. The longer grafts do not show muchdifference in conformation from their monodisperse counter-parts at low grafting density, but at medium and high graft-ing density, they exhibit less stretched conformations thantheir monodisperse counterparts. In addition, at high graftingdensity in the polydisperse case, the longer grafts adopt astretched ‘‘stem’’ conformation near the surface and a relaxed‘‘crown’’ conformation farther from the particle surface.

These observations that the homopolymer chain conforma-tions on hard nanoparticles with finite curvature are signifi-cantly affected by polydispersity in the grafted chain lengthsleads to the question: Is the effect of polydispersity ongrafted chain conformations on nanoparticle surfaces largeenough to alter how matrix chains wet/dewet/deplete thegrafted layer? If yes, is this change in matrix wettability ofthe grafted layer predictable so that one could deliberatelyintroduce 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 distri-butions, placed in a homopolymer matrix.

FIGURE 3 (a) Simulation snapshot of a polydisperse homopolymer grafted spherical nanoparticle at a grafting density of 0.65 chains/

d2 with average graft length of 20 Kuhn segments (Kuhn segment diameter ¼ 1d or �1 nm) and PDI ¼ 2 on a particle of diameter D

¼ 5d. (b) Scaling exponent (m) versus PDI for polymers with number average chain length of 20 Kuhn segments grafted on particle

diameters D ¼ 5d and 8d at grafting density of 0.65 chains/d2. Dashed lines denote the scaling exponent obtained for a single chain

grafted on particles of diameters D ¼ 5d and 8d. This figure is taken from Ref. 5 published in J. Polym. Sci. Part B: Polym. Phys. 2012.


POLYMER SCIENCE


We first studied9 the bidisperse distribution to elucidatethe effect of bidispersity on grafted chain conformationsand the PMF between bidisperse polymer-grafted particlesin a homopolymer matrix, and compared these trends tothose seen for monodisperse polymer-grafted particles. Ourmodel consisted of spherical nanoparticles grafted withequal number of short and long homopolymers chainsplaced in a matrix of homopolymer chains. To capturepurely the entropic driving forces in these polydisperse sys-tems, we maintained athermal interactions (i.e., hard-spherepotential) between all pairs of sites. The PMF between twografted particles in a polymer matrix suggests two regimes:the shorter grafts dictate the value of the PMF at contactand at small interparticle distances, while longer grafts dic-tate the long-range characteristics of the PMF. This isexpected, as past work on monodisperse grafted particleshas shown that the length of the repulsive tail in the PMFis dependent on the graft length.83 However, most interest-ingly, at intermediate interparticle distances the presence ofbidispersity in grafts, that is, short and long chains, elimi-nates the mild-attractive well seen for monodisperse shortchains. This suggests that introduction of a few long chainsamong short chains can lead to dispersion of the polymergrafted particles that would have aggregated otherwise. Wealso saw that, upon increasing short graft length Ns, whilekeeping the long graft length constant, the differencesbetween the bidisperse and monodisperse PMF profilesreduces. This is because as the short graft length increases,the graft conformations in the bidisperse case approachthose of the monodisperse case. Finally, we found thatgrafting density and matrix packing fraction affect the PMFbetween bidisperse grafted particles in qualitatively thesame manner as their monodisperse counterparts.

This begs the question: Is the improved stability of disper-sion only due to a bidisperse polymer grafts or would thisbe seen with polydisperse (with a broad distribution oflength) grafts as well? And, how does ‘‘polydisperse graftedpolymer distribution’’ improve grafted particle dispersion ina matrix compared to a bidisperse grafted polymer lengthdistribution? To address this, in a more recent study,1 westudied effects polydispersity in graft length on the PMFbetween the grafted nanoparticles, varying the PDI of thegrafted chains from 1 to 2.5.

In dense polymer solutions, increasing polydispersityreduces the strength of repulsion at contact and weakensthe attractive well at intermediate interparticle distances,completely eliminating the attractive well at intermediateinterparticle distances at high PDI. Figure 4(a) shows thePMF only at intermediate interparticle distances as a func-tion of small increments in PDI, demonstrating that the elim-ination of attractive well does not happen at small PDI, andthat a critical PDI is needed to eliminate the attractive wellcompletely. The elimination of the mid-range attractive wellis due to the longer grafts in the polydisperse graft lengthdistribution that introduce longer range steric repulsion, andthe reduced crowding in the grafted layer that alters thewetting of the grafted layer by matrix chains. Calculation ofthe matrix-graft penetration depth showed an increased pen-etration or wetting of the polydisperse grafted layer by thematrix chains. Figure 4(b) shows most recent results84

directly comparing the PMF between grafted nanoparticleshaving polydisperse distribution to that with bidisperse dis-tribution at PDI ¼ 1.5, and monodisperse grafts with sameaverage molecular weight. The results show that at a con-stant (and low) PDI ¼ 1.5, the presence of a continuous dis-tribution (polydisperse) of polymer graft lengths is able to

FIGURE 4 (a) Mid-range attractive well in the potentials of mean force, PMF (in units of kT) versus interparticle distance, r–D (in

units of d, where d is the size of a Kuhn segment), between nanoparticles of size D ¼ 5d, grafted with polydisperse chains at a

grafting density of r ¼ 0.65 chains/d2 with PDI ¼ 1.00 (circles), PDI ¼ 1.05 (upward facing triangles), PDI ¼ 1.10 (squares), PDI ¼1.15 (rightward facing triangles), PDI ¼ 1.20 (diamonds), and PDI ¼ 1.40 (downward facing triangles) with Ng,avg ¼ 20 in a dense

solution (g ¼ 0.1) of monodisperse homopolymer matrix chains with Nmatrix ¼ 40. (b) Potentials of mean force, PMF (in units of

kT) versus interparticle distance, r–D (in units of d), between nanoparticles of size D ¼ 5d, grafted with monodisperse (no symbol),

polydisperse at PDI ¼ 1.5 (filled symbols), and bidisperse at PDI ¼ 1.5 (open symbols) grafted chains at a grafting density of r ¼0.65 chains/d2 with Ng,avg ¼ 20 in a dense solution (g ¼ 0.1) of monodisperse homopolymer matrix chains with Nmatrix ¼ 40.



eliminate the attractive well completely while a bidispersedistribution of polymer graft lengths shows a small attractivewell, smaller in magnitude than monodisperse polymergrafts. We believe the reason for this trend is the presenceof a few grafts in the polydisperse graft length distributionthat are significantly longer than the long grafts in the bidis-perse distribution. A more detailed comparison of polydis-perse to bidisperse distribution is the focus of a currentinvestigation in our group.

In summary, we predict that polydispersity in graft lengthcan be used to stabilize dispersions of grafted nanoparticlesin a polymer matrix at conditions where monodisperse graftswould cause aggregation. In agreement with these predic-tions, recent experiments by Rungta et al.,2 where silica par-ticles with bimodal polystyrene grafts, synthesized usingstep-by-step RAFT polymerization, were found be better dis-persed in a monodisperse polystyrene matrix as comparedto silica particles with monodisperse polymer grafts oflength comparable to the long polymers in the bidispersedistribution. These promising results from theory and experi-ments should direct more groups to treat polydispersity inhomopolymer ligands as a design parameter for stabilizingdispersions of polymer grafted particle in a polymer matrix.

FUTURE DIRECTIONS

Most of this article has focused on how chemical and physi-cal heterogeneity in polymer grafts affects the effective inter-actions and structure of polymer grafted nanoparticles in ahomopolymer matrix. There is much interest in directing as-sembly of nanoparticles in a block copolymer matrix (seereviews in refs. 85 and 86). Although past studies havefocused on morphology of homopolymer grafted particles inblock copolymer domains, there have been significantlyfewer investigations involving copolymer grafted particles ina block copolymer matrix. Recent theoretical studies byGanesan and coworkers59,60 on flat random copolymerbrushes in contact with block copolymer matrix show thatthe interfacial interactions is modulated by the grafted ran-dom copolymers conformations which changes in responseto the overlaying block copolymer film. Our work17 suggeststhat AB diblock copolymer grafted nanoparticles will exhibit

different arrangements in the A and B domains of a block co-polymer matrix due to their different effective interactions inA-matrix and B-matrix. Since homopolymer grafted particleshave the ability to relieve the interfacial tension between Aand B domains, which drives transitions in the block copoly-mer morphology87 and stabilizes bicontinuous network mor-phologies88, it would interesting to investigate if morphologi-cal transitions in block copolymer matrices can be tuned bytailoring the monomer sequences in the copolymer grafts.

Theoretical work1,9,84 and recent experiments2 with polydis-perse and bidisperse grafts on nanoparticles suggests that pol-ydispersity in the grafted polymers can stabilize dispersions ina monodisperse polymer matrix better than the correspondingmonodisperse polymer grafted particles. A natural extension ofthis work would be to study monodisperse/polydisperse poly-mer grafted particles placed in a polydisperse matrix. To con-duct these studies in molecular simulations one would need alarge simulation box to incorporate the broad distribution ofmatrix chains and finite number of polymer grafted particles,which would be computational intensive even with parallelizedcentral processing unit (CPU)-based programs. In the past, toovercome these computational intensities, we adopted the self-consistent PRISM-MC approach.9,17 In the case of a polydis-perse matrix, it would be challenging to use the self-consistentPRISM-MC approach due to higher number of types of sitesneeded to represent each matrix chain length, and separate MCrun for each matrix chain length, thus losing any computationaladvantage. With the recent progress in graphics processingunits (GPUs), and simultaneous development of molecular sim-ulation algorithms (e.g., HOOMD89 and LAMMPs90 packages)designed for or adapted to run optimally on GPUs, we nowhave the ability to simulate a finite volume fraction of polymergrafted nanoparticles in an explicit polydisperse polymermatrix (like a system shown in Fig. 5). What makes dynamicsimulations of polymer systems perfect for implementation onGPUs is their computational complexity and parallelism.Although packages that were originally designed for parallelCPUs implementation, like LAMMPs, have now incorporatedGPU compatibility, there are other packages, like HOOMD, thathave explicitly been designed for GPU execution. With thesedevelopments, one is able to achieve large speedups com-pared to the same simulations on CPUs, making a variety of

FIGURE 5 Schematic of a polymer nanocomposite with a finite volume fraction of polymer grafted nanoparticles in an explicit

polymer matrix.


POLYMER SCIENCE


coarse-grained simulations in the field of polymer nanocom-posites that would have been impractical before possiblenow. For example, in a recent study by Glotzer and co-workers,91 which aimed at studying the stability of the dou-ble gyroid phase in a melt of polymer tethered nanosphereswith polydispersity in particle sizes, they found a 20–30times speedup with GPUs over CPUs. They92 also commentthat their system of polydisperse particles forced thespeedup to be relatively lower than that possible in mono-disperse polymer-nanoparticle systems, and with new advan-ces in GPUs and CPUs one could achieve 80–120 timesspeedup with GPU based simulations over CPU based simu-lations. Our preliminary test for Brownian dynamics simula-tions (using HOOMD) in a box containing about 10 polymergrafted nanoparticles in an explicit polymer matrix, withparticle sizes �5 nm and graft and matrix chain lengths of10–30 Kuhn segments (�28,000 coarse grained Kuhn seg-ments) achieves computational speeds of 1400–1600 timesteps per second on a single C2050 GPU processor! In addi-tion to obtaining dynamic trajectories faster with GPU com-puting, data analysis of trajectories involving histograms alsoperform well on GPUs due to their intrinsic parallelism. Theabove developments in GPU-based computations shouldallow scientists to revisit simulations of polymer nanocom-posites, especially those with polymer grafted nanoparticlesin an explicit polymer matrix, that were too intensive withCPUs alone.

Finally, while this feature article focused solely on the struc-ture and morphology with polymer grafted particle basednanocomposites, these morphological studies are all moti-vated by the common goal of controlling macroscopic prop-erties of the polymer nanocomposite. It is therefore a naturalnext step for theoretical and experimental studies to eluci-date how the chemical and physical heterogeneity in poly-mer grafts affects the processing and rheological behavior93

and resulting mechanical properties of these polymernanocomposites.

ACKNOWLEDGMENTS

The author acknowledges the partial financial support fromDepartment of Energy under grant number DE-SC0003912for the polydisperse grafted nanoparticle work, and thepartial financial support from National Science Foundationunder grant number CBET-0930940 for the copolymergrafted nanoparticle work. The author acknowledges C.Phillips, C. Iacovella and S. C. Glotzer for providing detailedinformation on the comparison of GPU and CPU computa-tional speeds presented in the last section of this article.The author is grateful to T. B. Martin, E. Jankowski, A. Seif-pour, N. Nair, and P. Dodd for scientific discussions andschematics in this article.

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Documents

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