Effect of matrix bidispersity on the morphology of polymer-grafted nanoparticle-filled polymer nanocomposites

Effect of Matrix Bidispersity on the Morphology of Polymer-Grafted

Nanoparticle-Filled Polymer Nanocomposites

Tyler B. Martin, Arthi Jayaraman

Department of Chemical and Biolgical Engineering, UCB 596 University of Colorado at Boulder, Boulder, Colorado 80309

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

Received 29 April 2014; revised 15 May 2014; accepted 16 May 2014; published online 00 Month 2014

DOI: 10.1002/polb.23517

ABSTRACT: We present a simulation study showing the effect

of bidispersity in matrix homopolymer length on the wetting/

dewetting of homopolymer-grafted nanoparticles and the mor-

phology of polymer nanocomposites where the graft and

matrix polymer chemistries are identical. In a bidisperse matrix

with equal number of short and long chains and average

matrix length greater than the monodisperse graft length, the

densely grafted polymer layer is preferentially wet by the short

chains and relatively dewet by the long chains. This is driven

by a larger gain in entropy of mixing between graft and matrix

for short matrix chains than long matrix chains. Despite the

preferential wetting of the short and dewetting of long chains,

matrix length bidispersity does not significantly change the

overall wetting of the grafted layer. Unlike graft length bidis-

persity that significantly improves particle dispersion, matrix

length bidispersity slightly increases particle aggregation in the

polymer matrix. VC 2014 Wiley Periodicals, Inc. J. Polym. Sci.,

Part B: Polym. Phys. 2014, 00, 000–000

KEYWORDS: dispersion; molecular simulation; nanocomposites;

polydispersity; polymer-grafted nanoparticle; polymer nano-

composite; simulation

INTRODUCTION In order to engineer polymer nanocomposites(PNCs) with desirable macroscopic properties, the ability tocontrol the microscopic arrangement, or morphology, of thematrix and fillers (additives) is important.1,2 PNC morphologiesfall into two broad classes: aggregated morphologies wherethe fillers assemble into a variety of isotropic and anisotropicstructures,3,4 and dispersed morphologies where the averagefiller–filler distance is maximized.5–9 By grafting ligands to thesurface of the fillers (e.g., nanoparticles), the filler–filler and fil-ler–matrix interactions, and thereby the morphology of thePNCs can be tuned. For a homopolymer matrix filled withhomopolymer-grafted nanoparticles, where the matrix andgraft chemistry are identical and the grafting density is high, ithas been shown that the ratio of the matrix to graft molecularweight (Nmatrix/Ngraft) dictates whether grafted particles staydispersed or aggregated.7 Specifically, for spherical nanopar-ticles where the radius of the particle is similar to the radiusof gyration of the grafted chains, one observes particle aggre-gation when Nmatrix/Ngraft is greater than �4–5 and dispersionfor smaller Nmatrix/Ngraft. The underlying molecular mechanismfor particle dispersion and aggregation has been suggested tobe a transition from a wet brush to dry brush.7

Exploiting this idea of wetting/dewetting driven dispersion/aggregation, we recently investigated how polydispersity in

grafted chain lengths affects the wetting of grafted nanopar-ticles by a monodisperse polymer matrix. We predicted, usingtheory that in the regimes where particle aggregation isexpected, graft length polydispersity stabilized the dispersedphase of the grafted particles by promoting wetting of thedensely grafted polymer layer by the matrix. Due to increasedwetting of the grafted layer, the mid-range attraction in theparticle–particle potential of mean force is eliminated, therebyremoving the tendency for aggregation of the particles.10–12

We also found that the ideal graft length distribution for pro-moting particle dispersion is one that provides (a) maximummonomer crowding near the surface of the nanoparticle, inorder to shield short-range particle–particle attraction andsterically repel other approaching grafted nanoparticles, and(b) minimal crowding near the edge of the grafted layer inorder to maximize wetting of the grafted layer by the matrixchains.13 In agreement with our theoretical predictions ofgraft polydispersity-stabilized particle dispersion, experimen-tal work by Schadler et al. shows that a “bimodal”-graftedchain length distribution greatly improves the dispersion ofgrafted nanoparticles in a polymer matrix over that observedwith a monodisperse graft length distribution.14,15

While the above theoretical and experimental studies havefocused on the effect of “graft” length polydispersity on the

Additional Supporting Information may be found in the online version of this article.

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morphology of PNCs, to the best of our knowledge, the effectof “matrix” length polydispersity on the morphology ofhomopolymer-grafted particle-filled PNCs has not been inves-tigated. One past study worth noting is that by Broseta et al.who studied the effect of polydispersity on morphology ofstrongly segregating homopolymer blends. They found thatthe polydispersity reduced the interfacial tension of the homo-polymers, with the shorter polymer chains segregating to theinterface.16 Despite the significance of their results, since theirstudy focused on entropically driven segregation of shortchains to flat interfaces formed by two strongly (enthalpically)segregating polymers, one cannot extend their prediction to ablend of polymer-grafted particles and polymer matrix wherethe graft–matrix polymer chemistries are the same (noenthalpically driven segregation), and the graft–matrix interfa-ces are curved. In this article, we present a computationalstudy of the effect of matrix length polydispersity on the wet-ting/dewetting phenomena and the morphology ofhomopolymer-grafted nanoparticle-filled PNCs, which, to thebest of our knowledge, has not been studied before.

APPROACH

We simulate the PNCs in this study using Brownian dynam-ics (BD)17 with coarse grained models of grafted and matrixchains. The details of the computational approach—model,simulation, and parameters varied—are as follows.

ModelWe model the grafted and matrix chains as freely jointed chainsof beads of diameter 1d, where d is the size of a Kuhn segmentin the polymers (graft or matrix). The polymer beads that rep-resent Kuhn segments are connected by harmonic bonds:

Ubond rð Þ5k r2r0ð Þ2 (1)

where k is the bond strength, and r0 is the resting distance ofthe bond. We model the nanoparticles as rigid bodies11 of over-lapping spheres of diameter 1d. These spheres of the nanopar-ticle serve as both the surface of the particle and as graftingsites. We ensure that both the surface beads and the graftingsites are evenly distributed across the particle for desired arbi-trary diameters (D) and grafting densities (chains/d2).

We maintain athermal interactions between all entities tomimic PNCs with identical graft and matrix chemistry, andwith effective shielding of particle–particle attraction at highgrafting density. Our past work has shown that the stericrepulsion due to the polymers at high grafting density effec-tively shields the particle–particle attraction13. To modelthese athermal interactions, all pairs of nonbonded beads inthe system interact via purely repulsive Week-Chandler-Andersen potentials18:

UWCA rð Þ54e

rr

� �122

rr

� �6� �1e; r � 2

16r

0:0; r > 216r

8><>: (2)

where r is the bead–bead distance, e is the steepness of therepulsive potential, and r is the contact distance for the twobeads.

Simulation and TheoryWe conduct BD simulations using the HOOMD-blue plat-form17. We initialize our systems as follows: we first build agrafted nanoparticle, with grafted chains extending radiallyfrom the particle surface, in the absence of any matrixchains. A short simulation with strong Lennard–Jones mono-mer–monomer and monomer–particle attraction is then runto make the grafted chains take up a compact conformation.Note that this is the only time we use attractive nonbondedinteractions in our simulation. Copies of this compact graftednanoparticle, along with the desired number of short andlong matrix chains, are then randomly placed in a large cubicbox. The randomly placed molecules are integrated using aBrownian dynamics integrator for 0.5e6 time steps to bothmix and relax the grafted and matrix chains. The box is thencompressed to the desired volume fraction over 0.5e6 steps,and then mixed again for 0.5e6 steps at the compressedstate. Finally, the production simulation runs for 100–200million time steps where snapshots of the system are savedevery 0.5e6 time steps. The analysis is conducted using thesesnapshots and ensemble average results obtained from fiveindependent simulations are reported.

One of the simulation challenges in such densely packed sys-tems is ensuring that the ensemble average results areindeed representing equilibrium states. In the SupportingInformation, we show the evolution of the average meansquare internal distances of our matrix chains to demon-strate that the polymer chains relax on a timescale that isfaster than our sampling period (Supporting Information Fig.S1).19 Additionally, in Supporting Information Figure S2, weshow pair correlation functions for five independent simula-tion trials with randomly initialized structures of the samecomposite, and demonstrate that we are are reaching statisti-cally similar equilibrium results. We also use liquid statetheory based calculations8,20,21 (described next), which aredevoid of these equilibration issues, and find qualitativeagreement between the theoretical and simulation results,further confirming that we are indeed reporting equilibriumbehavior in the simulations.

We use a variant of the self-consistent Polymer ReferenceInteraction Site Model theory and Monte Carlo simulation(PRISM-MC),21,22 in which we use BD rather than Monte Carloto calculate the intramolecular structure factor. PRISM theoryconsists of a matrix of Ornstein-Zernike-like integral equationsthat relate the total site–site intermolecular pair correlationfunction, hij(r), to the intermolecular direct correlation func-tion, cij(r), and intramolecular pair correlation function, xij(r).The PRISM equations in Fourier space are as follows:

H qð Þ5X qð Þ C qð Þ X qð Þ1H qð Þ½ � (3)

Hij qð Þ5qiqj hij qð Þ (4)

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Xij qð Þ5qXNi

a51

XNj

b51

Xaibj(5)

where H(q), C(q), and X(q) in this study are matrices of size 43 4 for the following four types of sites: graft monomers (A),particle (B), short matrix monomers (C), and long matrixmonomers (D), with the matrix elements defined in eqs 4 and5. We note that despite the chemistry of the graft and matrixbeing the same, these sites are physically identified separatelyas graft or matrix. In the above equations, Ni and qi are,respectively, the number and number density of site i, q is themolecular number density, and Xij(q) the intramolecular paircorrelation function between sites i and j within a certain mol-ecule in Fourier space. To solve eq 3, we use closure relationsconnecting the real space cij(r), hij(r) (5 gij(r)2 1) and inter-action potentials Uij(r). Previous work on a mixture of nano-particles and polymers23–32 shows that the Percus–Yevick (PY)closure for polymer–polymer and polymer–particle, and thehypernetted chain (HNC) closure for particle–particle providesgood agreement between theory and simulations. Thus, wehave used the same combination of atomic closures, since thiswork also consists of polymers and nanoparticles. Given thatrij is the distance of closest approach between sites i and j,that is, rij5 d for monomer–monomer pairs and rij5 (D1 d)/2 (as stated earlier, d and D are the monomer and particlediameters, respectively) and particle–monomer pairs, theimpenetrability condition applies inside the hard core:

gij rð Þ50; r < rij (6)

Outside the hard core, the PY approximation describes thedirect correlation function between all pairs of sites (exceptparticle–particle):

cij rð Þ5 12ebUij rð Þ� �

gij rð Þ; r > rij (7)

and the HNC closure handles the particle–particle direct cor-relation function:

cBB5hBB rð Þ2ln gBB rð Þð Þ2bUBB rð Þ; r > D (8)

To efficiently solve this system of coupled nonlinear integralequations, we employ the KINSOL package33 using a General-ized Minimal Residual Newton-Krylov method along with theline search optimization strategy. Attaining convergence forcomplex nonlinear integral equations is much easier withKINSOL as compared to the Picard technique, the methodused in prior PRISM theory work. The solution of the PRISMequations yields pair correlation functions, gij(r), and partialcollective structure factors, Sij(q). We note that some sets ofparameters, especially those involving larger particle sizes orlonger polymer chains (graft or matrix) do not yield any sol-utions due to numerical issues.

We use a self-consistent approach linking PRISM theory andBD simulations, where the chain conformations (via theintramolecular pair correlation function Xij) input to PRISMare provided by BD simulations of a single polymer-grafted

nanoparticle or a single matrix chain in an external medium-induced potential obtained from PRISM theory. The interde-pendence of the chain conformations and the medium-induced potential gives rise to the self-consistency of thePRISM-BD method. The self-consistent loop starts with threedifferent BD simulations of (i) a single grafted particle (inthe absence of any matrix chains), (ii) a single long matrixchain (in the absence of grafted particles), and (iii) a singleshort matrix chain (in the absence of grafted particles). Themodel and interactions used in these BD simulations is iden-tical to the one described before. The grafted and matrixchains are integrated for 1e6 time steps with snapshotsbeing collected every 0.25e5 time steps. Using these snap-shots, the intramolecular structure factors: xAA (graft–graft),xAB (graft–particle), xCC (short matrix–short matrix), xDD

(long matrix–long matrix) are calculated. Next, the PRISMequations are solved with the xij and intermolecular poten-tials Uij as input. Using the results from the PRISM calcula-tion, the pairwise-decomposed medium-induced solvationpotential, Dwij(r) is calculated; this describes the interactionbetween any two sites i and j as mediated by all the remain-ing sites in the system, that is, including the matrix, graftsand particles themselves. The form of the solvation potentialdepends on the approximation used in its derivation34–39andwe use the PY form:

DwPYij rð Þ5 2kBTln 11cik rð Þ � skk0 rð Þ � ck0 j rð Þ

� �(9)

where “*” in eq 9 denotes a convolution integral in spatialcoordinates, kB is the Boltzmann constant, and T is the tem-perature. The solvation potential Dwij(r) is then used in allfollowing BD simulations of a single polymer-grafted particle,a single short matrix chain, or a single long matrix chain,completing the self-consistent loop. In all BD simulationsafter the first self-consistent loop, the nonbonded interac-tions between sites are a sum of the WCA potential and thesolvation potential from the previous PRISM calculation:

Utotij rð Þ5UWCA;ij rð Þ1DwijðrÞ (10)

Note that, while the WCA forces are calculated internally inHOOMD-blue, a table of forces must be created for the solva-tion potential force using finite-difference approximations.The self-consistent PRISM-BD iterations are continued untilconvergence of Dwij(r) is achieved. To define the convergencecriteria, we first specify “error” of loop n as:

En5Xi;j

Xr

Dwn11ij rð Þ2Dwn

ij rð Þh i2

(11)

where the summation of i and j is over the site-pairs thatare relevant to the BD simulations: AA AB, CC, DD. The self-consistent loop converges on when En/E0< 0.01 for threeconsecutive loops.

ParametersThe system presented in this article consists ofhomopolymer-grafted spherical nanoparticles in an explicit



homopolymer matrix where the grafted and matrix polymersare either monodisperse or bidisperse. The bidispersity ratiobX5NX,long/NX,short (where X5 graft or matrix) and averagematrix (Nmatrix,avg) and graft (Ngraft,avg) lengths are varied totest the effect of varying extents of bidispersity. The sphericalnanoparticle has diameter D5 5d (where d is the diameter ofa Kuhn segment or “monomer”) and the polymer graftingdensity is kept in the brush-like regime at 0.65 chains/d2. Thetotal system packing fraction g is 0.1 to mimic a dense solu-tion of matrix polymers. The filler fraction of the grafted par-ticles, /, defined as the fraction of the total occupied volumethat is occupied by particles or grafted monomers, is variedfrom low / (/ 5 0.02) to high / (/ 5 0.3) during the study.In the Results section, we only present results at filler fractionof 0.3 as the qualitative trends described are seen both atdilute filler fraction and high filler fraction. In Figure 1, weshow simulation snapshots of the system at / 5 0.3. Simula-tion snapshots for / 5 0.1 are provided in Supporting Infor-mation Figure S3.

RESULTS

In Figure 2(a), we show the pair correlation functionsbetween the particle centers and monodisperse matrix mono-mers (gPM, solid black rightward triangles, Nmatrix5 30), shortmatrix chain monomers (gPS, red upward triangles,Nmatrix,short5 30), and long matrix chain monomers (gPL, greendownward triangles, Nmatrix,short5 60) of a bidisperse matrix.

In the bidisperse matrix, the short chains show a higher cor-relation with the particle centers at short distances than thelong chains in the same system. This means that the shortmatrix chains “preferentially wet” the grafted layer over thelong chains. In the monodisperse matrix, it is well under-stood that, at constant Ngraft, wetting of the grafted layershould decrease with increasing matrix chain length.7,10,40

Since both Nmatrix,short and Nmatrix,long are greater than Ngraft,the matrix chains should dewet the grafted layer. Further-more, since Nmatrix,long>Nmatrix,short it might seem unsurpris-ing that the short chains preferentially wet the grafted layermore than the long chains. However, in Figure 2(a), if wenow compare the short chains of the bidisperse pair (openred upward triangles) to a monodisperse matrix of the samelength as the short chains (solid rightward triangles), theshort chains in the bidisperse environment show a highercorrelation with the particles at short distances than in amonodisperse environment. This demonstrates that it is thematrix length bidispersity (simultaneous presence of shortand long chains) that is driving the preferential wetting ofthe grafted layer by the short chains. In Figure 2(b), we com-pare the same bidisperse system to a monodisperse matrix(solid black leftward triangles, Nmatrix 5 60) of the samelength as the long bidisperse chains. The long chains in thebidisperse environment dewet the grafted layers more thanthe monodisperse matrix of the same length as the longchains, showing that matrix bidispersity drives bothincreased wetting of the grafted layer by short matrix chains,and decreased wetting by long matrix chains. See Supporting

Information Figure S4 for additional simulation snapshotsthat (to some extent) visually depict the extent of wetting ofthe grafted layer by the matrix chains.

To see how the behavior in Figure 2(a,b) changes withincreasing matrix bidispersity, in Figure 2(c,d), we plotthe particle–short matrix pair correlation functions [gPS in

FIGURE 1 A representative simulation snapshot of a PNC with

polymer grafted particles of diameter D 5 5d grafted with

homopolymers (blue chains) with a monodisperse length dis-

tribution (Ngraft 5 20) at grafting density 0.65 chains/d2 and filler

fraction / 5 0.3 in a bidisperse homopolymer matrix of lengths

20 (red chains) and 60 (green chains). The matrix chains are

removed in the bottom image to show the grafted particles

only. While the red, blue, and green colors are used to distin-

guish the grafts, the short and long matrix chains in the snap-

shots, the grafts and all matrix polymers are of the same

chemistry and modeled using athermal interactions. [Color fig-

ure can be viewed in the online issue, which is available at

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Fig. 2(c)] and the particle–long matrix correlation functions[gPL in Fig. 2(d)] for three bidispersity ratios bmatrix5

Nmatrix,long/Nmatrix,short5 1.5, 3.0, 7.0 at constant Nmatrix,avg5

40. With increasing bmatrix, the preferential wetting of thegrafted layer by the short matrix chains becomes more pro-nounced, while the dewetting of the grafted layer by the longchains is relatively less affected. To address any questions thatmay arise regarding efficient sampling and equilibration, inSupporting Information Figure S5, we present a comparison ofthe graft–short matrix pair correlation functions (gGS) and thegraft–long matrix correlation functions (gGL) from both simula-tions and liquid-state theory. From both methods, we observethe grafted layer being preferentially wet by the short chains.As liquid-state theory is an equilibrium theory, these resultsconfirm that our simulation results are indicative of the equi-librium behavior of the system. Also to confirm that our

results are not strongly dependent on chosen filler fraction(defined as the fraction of the total occupied volume that isoccupied by particles or grafted monomers), we also studied /ranging from 0.02 to 0.3 and found the preferential wetting ofthe grafted layer by the short-matrix is present at all / withinthis range (Supporting Information Fig. S6.)

Since we maintain athermal interactions using Weeks-Chandler-Andersen (WCA) pair-wise potentials between allmonomers, the driving forces that cause the short-matrixchains to preferentially wet the grafted layer are purelyentropic in nature. We show the average entropy change ofwetting (DSwetting) for a monomer belonging to either a shortor long chain in a bidisperse matrix or any chain in a mono-disperse matrix. We estimate these values via Flory–Hugginsmixing theory41:

FIGURE 2 Particle–monomer pair correlation functions gij(r) versus particle–monomer distance, r 2 (D 1 d)/2, in units of d, for par-

ticles of diameter D 5 5d grafted with homopolymers with a monodisperse length distribution (Ngraft 5 20) at 0.65 chains/d2 and fil-

ler fraction / 5 0.3 in a bidisperse or monodisperse homopolymer matrix. In subplots a and b, the symbols indicate the pair

correlation between the particles and the monomers of a monodisperse matrix of Nmatrix 5 30, bmatrix 5 1.0 (solid black rightward

triangles) or Nmatrix 5 60, bmatrix 5 1.0 (solid black leftward triangles), or the monomers belonging to the short (open red upward tri-

angles, Nshort 5 30) or long (open green downward triangles, Nlong 5 60) chains of a bidisperse matrix with bmatrix 5 3.0 and

Nmatrix,avg 5 45. In subplots c and d, the symbols indicate the pair correlation between the particles and the monomers belonging

to short (subplot c) or long (subplot d) chains of a bidisperse matrix with Nmatrix,avg 5 40 and bmatrix 5 1.5 (black squares), 3.0 (blue

leftward triangles), or 7.0 (pink rightward triangles). [Color figure can be viewed in the online issue, which is available at wileyonli-

nelibrary.com.]





1

kB

� �DSwetting5 2

1

NXln

qX;wet

qX ;unwet(12)

where X is either short, long, or monodisperse; NX is thelength of the X matrix component, qX,wet is the ratio ofthe volume of X monomers that have wet the grafted layerto the volume of all (graft, short and long matrix) monomerswithin the grafted layer, qX,unwet is the ratio of the volume ofX monomers that are outside the grafted layer to the volumeof all monomers outside the grafted layer, and kB is theBoltzmann constant. All the volume fractions used in eq 12are calculated from our BD simulations. We define a mono-mer as having “wet” the grafted layer when it is within thetrial-averaged root-mean-square brush height (SupportingInformation Fig. S7) for the system. DSwetting represents thegain in mixing entropy upon a matrix monomer leaving thebulk matrix and entering (wetting) the grafted layer.

In Figure 3, we present the DSwetting for short-matrix (redupward triangles), long-matrix (green downward triangles),and monodisperse matrix (dashed lines), with the length of thematrix component shown next to the corresponding symbol/line. With increasing bmatrix, the DSwetting for the short matrixchains increases, while the DSwetting for the long matrix chainsdecreases. The DDSwetting between the short and long chains isthe effective driving force for preferential wetting of the graftedlayer by the short chains and this driving force increases withincreasing bmatrix. This is consistent with our observation that,with increasing bmatrix, there is increasing wetting of thegrafted layer by short matrix chains and (slightly) decreasingwetting by long matrix chains. Interestingly, the monodispersematrix with Nmatrix5 20 has a higher DSwetting than the bidis-perse short chains of the same length (Nmatrix,short5 20,bmatrix5 3.0). This is because the qunwet for the monodispersesystem is much greater than qunwet for the bidisperse system,due to the presence of long matrix chains in the bulk matrixfor the bidisperse case (Supporting Information Table S2). Thehigher qunwet means that the short chains in the monodisperseenvironment gain more entropy by wetting the grafted layerthan their bidisperse counterparts and, therefore, they have ahigher DSwetting. Note that it does not contradict our findings ofpreferential wetting of the grafted layer by short matrix chains,as it is the DDSwetting between the short and long chains thatdrives the preferential wetting and not the individual DSwettingof the short or the long matrix chains.

When compared to a monodisperse matrix of either short orlong chains, the bidisperse matrix has increased the wettingof the grafted layer by short matrix chains, but also slightlydecreased the wetting by the long matrix chains. Withincreased and decreased wetting of the grafted layer by dif-ferent components of the bidisperse matrix, how does thebidispersity affect the tendency for aggregation of the graftednanoparticles?

In Figure 4(a), we show the pair correlation functions betweenthe particles, gPP for monodisperse or bidisperse matrix sys-tems. The peak in gPP for bmatrix5 3.0, Nmatrix,avg5 40 (red

upward triangles) is slightly higher than the peak forbmatrix5 1.0 (monodisperse) Nmatrix5 40 (black circles) indi-cating that the bidisperse matrix increases the tendency foraggregation, when compared to a monodisperse system withthe same Nmatrix,avg. Furthermore, at constant Nmatrix,avg5 40 asbmatrix increases, we observe increasing tendency for aggrega-tion (Supporting Information Fig. S8), likely due to the lengthof the longer matrix chain increasing. In Figure 4(a), if wecompare the bmatrix5 3.0 with Nmatrix,long5 60 (red upward tri-angles), to a monodisperse matrix with Nmatrix5 60 (greendownward triangles), the peak in gPP is smaller for the bidis-perse matrix case, indicating that the bidisperse matrixincreases tendency for dispersion (slightly) more than a mono-disperse matrix with Nmatrix5Nmatrix,long. To understand theeffect of matrix bidispersity on particle aggregation/dispersion,we look at the total wetting of the grafted layer via the totalparticle–matrix pair correlation function [gPM, Fig. 4(b)], wherethe correlation of the short and long chains in the bidispersematrix are combined. The overall wetting of the grafted layershows only small variations between the monodisperse(circles, downward triangles) and bidisperse (upward trian-gles) matrix systems. While matrix bidispersity does increasethe wetting of the grafted layer by short matrix chains, it doesnot increase the overall wetting of the grafted layer.

In contrast to the bidisperse matrix, a system consisting ofbidisperse grafts with bgraft5 3.0, Ngraft,avg5 20 in a

FIGURE 3 Average per-monomer entropy of wetting versus

bmatrix for particles of diameter D 5 5d grafted with homopoly-

mers with monodisperse length distribution (Ngraft 5 20) at 0.65

chains/d2 and filler fraction /5 0.3 in a monodisperse or bidis-

perse matrix with Nmatrix,avg 5 40. The symbols correspond to

the entropy contribution from the short (red upward triangles)

or long (green downward triangles) matrix chains. The horizon-

tal dashed lines correspond to the entropy of wetting calcu-

lated for monodisperse matrices. The numbers next to the

symbols/lines indicate the matrix length for that datum. [Color

figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com.]


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monodisperse matrix (leftward triangles) exhibits a signifi-cantly reduced peak in gPP when compared to monodispersegrafted particles in monodisperse/bidisperse matrix. Thistrend is also consistent with what we observe from liquid-state theory calculations (Supporting Information Fig. S9)where the midrange attraction in the potential of mean forceis nearly eliminated for bidisperse grafts, but remains forbidisperse matrix. In Figure 4(b), we can see that the wet-ting of the grafted layer is increased for the system with

bidisperse grafts as compared to any of the other systemswith monodisperse grafts. Unlike bidispersity in the matrixchain lengths, bidispersity in the grafted chain lengths isable to increase the wetting of the grafted layer by thematrix chains, which in turn improves dispersion of the poly-mer grafted particles.

CONCLUSIONS

This article presents a coarse-grained computational study ofPNCs consisting of polymer grafted nanoparticles in a bidis-perse polymer matrix, with athermal polymer–polymer, poly-mer–particle, and particle–particle interactions. This studyfocuses on the effects of varying matrix bidispersity, averagematrix length, and filler volume fraction, while maintaininghigh grafting density and keeping the particle diameter andgraft length constant. Furthermore, in most cases, the matrixchain lengths are chosen to be larger than the graft chainlengths to explore the effect of matrix bidispersity on sys-tems where dewetting of the grafted layer by the matrixchains has been observed in previous studies with monodis-perse systems.

We find that the short-matrix chains show a higher correlationwith the particle than either the long matrix chains or mono-disperse matrix chains of the same length as the short chains.In other words the short matrix chains preferentially wet thegrafted layer on the particles in the presence of matrix bidis-persity. Due to athermal interactions, this preferential wettingof the grafted layer by the short chains is largely driven bythe disparity in mixing entropies between the grafted layerand the short or long matrix chains. Although the matrixbidispersity increases wetting of the grafted layer by shortmatrix chains, the overall wetting of the grafted layer islargely unaffected, as long chains dewet the grafted layermore than their monodisperse counterparts. As a result, atconstant Nmatrix,avg, increasing the matrix bidispersity slightlyincreases the tendency for aggregation of the grafted particles,in contrast to the bidispersity in the grafted layer which sta-bilizes dispersion of the grafted particles.

ACKNOWLEDGMENTS

The authors are grateful for supercomputing time on the Janussupercomputer, which was used for some of the work, and issupported by the National Science Foundation (award numberCNS-0821794) and the University of Colorado Boulder. TheJanus supercomputer is a joint effort of the University of Colo-rado Boulder, the University of Colorado Denver, and theNational Center for Atmospheric Research. They are also grate-ful for resources of the National Energy Research ScientificComputing Center, which was used for some of the work, and issupported by the Office of Science of the US Department ofEnergy under Contract DE-AC02–05CH11231. This workacknowledges financial support by Department of Energyunder Grant DE-SC0003912 (AJ) and National Science Founda-tion GRFAward Number DGE 1144083 (TBM).

FIGURE 4 Particle–particle (a) and particle–total matrix (b) pair

correlation functions versus particle–particle and particle–

monomer distance, in units of d for nanoparticles of diameter

D 5 5d grafted with homopolymers with a monodisperse or

bidisperse graft length distribution at filler fraction / 5 0.3 in a

bidisperse or monodisperse homopolymer matrix. The sym-

bols correspond to: bgraft 5 1.0, bmatrix 5 1.0, Ngraft 5 20,

Nmatrix 5 40 (black circles), bgraft 5 1.0, bmatrix 5 3.0, Ngraft 5 20,

Nmatrix,avg 5 40 (red upward triangles), bgraft 5 1.0, bmatrix 5 1.0,

Ngraft 5 20, Nmatrix 5 60 (green downward triangles), bgraft 5 3.0,

bmatrix 5 1.0, Ngraft,avg 5 20, Nmatrix 5 40 (blue leftward triangles).

[Color figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com.]




REFERENCES AND NOTES

1 V. Ganesan, A. Jayaraman, Soft Matter 2014, 10, 13–38.

2 S. K. Kumar, N. Jouault, B. Benicewicz, T. Neely, Macromole-

cules 2013, 46, 3199–3214.

3 P. Akcora, H. Liu, S. K. Kumar, J. Moll, Y. Li, B. C.

Benicewicz, L. S. Schadler, D. Acehan, A. Z. Panagiotopoulos,

V. Pryamitsyn, V. Ganesan, J. Ilavsky, P. Thiyagarajan, R. H.

Colby, J. F. Douglas, Nat. Mater. 2009, 8, 354–U121.

4 Y. Jiao, P. Akcora, Macromolecules 2012, 45, 3463–3470.

5 R. Krishnamoorti, MRS Bull. 2007, 32, 341–347.

6 C. Chevigny, F. Dalmas, E. Di Cola, D. Gigmes, D. Bertin, F.

Boue, J. Jestin, Macromolecules 2011, 44, 122–133.

7 P. F. Green, Soft Matter 2011, 7, 7914–7926.

8 L. M. Hall, A. Jayaraman, K. S. Schweizer, Curr. Opin. Solid

State Mater. Sci. 2010, 14, 38–48.

9 R. Hasegawa, Y. Aoki, M. Doi, Macromolecules 1996, 29,

6656–6662.

10 T. B. Martin, P. M. Dodd, A. Jayaraman, Phys. Rev. Lett.

2013, 110, 018301.

11 T. B. Martin, A. Jayaraman, Soft Matter 2013, 9, 6876–6889.

12 N. Nair, N. Wentzel, A. Jayaraman, J. Chem. Phys. 2011,

134, 194906.

13 T. B. Martin, A. Jayaraman, Macromolecules 2013, 46, 9144–

9150.

14 Y. Li, T. M. Krentz, L. Wang, B. C. Benicewicz, L. S.

Schadler, ACS Appl. Mater. Interfaces 2014.

15 P. Tao, Y. Li, R. W. Siegel, L. S. Schadler, J. Mater. Chem. C

2013, 1, 86–94.

16 D. Broseta, G. H. Fredrickson, E. Helfand, L. Leibler, Macro-

molecules 1990, 23, 132–139.

17 J. A. Anderson, C. D. Lorenz, A. Travesset, J. Comput. Phys.

2008, 227, 5342–5359.

18 J. D. Weeks, D. Chandler, H. C. Andersen, J. Chem. Phys.

1971, 54, 5237-1.

19 R. Auhl, R. Everaers, G. S. Grest, K. Kremer, S. J. Plimpton,

J. Chem. Phys. 2003, 119, 12718–12728.

20 K. S. Schweizer, J. G. Curro, Phys. Rev. Lett. 1987, 58, 246–249.

21 A. Jayaraman, N. Nair, Mol. Simul. 2012, 38, 751–761.

22 N. Nair, A. Jayaraman, Macromolecules 2010, 43, 8251–

8263.

23 J. B. Hooper, K. S. Schweizer, Macromolecules 2005, 38,

8858–8869.

24 J. B. Hooper, K. S. Schweizer, Macromolecules 2006, 39,

5133–5142.

25 J. B. Hooper, K. S. Schweizer, T. G. Desai, R. Koshy, P.

Keblinski, J. Chem. Phys. 2004, 121, 6986–6997.

26 L. M. Hall, B. J. Anderson, C. F. Zukoski, K. S. Shweizer,

Macromolecules 2009, 42, 8435–8442.

27 L. M. Hall, K. S. Schweizer, J. Chem. Phys. 2008, 128,

234901.

28 L. M. Hall, K. S. Schweizer, Soft Matter 2010, 6, 1015–1025.

29 A. Jayaraman, K. S. Schweizer, J. Chem. Phys. 2008, 128,

164904.

30 A. Jayaraman, K. S. Schweizer, Langmuir 2008, 24, 11119–

11130.

31 A. Jayaraman, K. S. Schweizer, Macromolecules 2008, 41,

9430–9438.

32 A. Jayaraman, K. S. Schweizer, Macromolecules 2009, 42 pp

8423–8434.

33 A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R.

Serban, D. E. Shumaker, C. S. Woodward, ACM Trans. Math.

Softw. 2005, 31, 363–396.

34 K. S. Schweizer, J. G. Curro, Macromolecules 1988, 21,

3070–3081.

35 J. Melenkevitz, K. S. Schweizer, J. G. Curro, Macromole-

cules 1993, 26, 6190–6196.

36 C. J. Grayce, K. S. Schweizer, J. Chem. Phys. 1994, 100,

6846–6856.

37 C. J. Grayce, A. Yethiraj, K. S. Schweizer, J. Chem. Phys.

1994, 100, 6857–6872.

38 P. G. Khalatur, A. R. Khokhlov, Mol. Phys. 1998, 93, 555–572.

39 K. S. Schweizer, J. G. Curro, Adv. Polym. Sci. 1994, 116,

319–377.

40 D. M. Trombly, V. Ganesan, J. Chem. Phys. 2010, 133,

154904.

41 P. J. Flory, J. Chem. Phys. 1949, 17, 303–310.


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Effect of matrix bidispersity on the morphology of polymer-grafted nanoparticle-filled polymer nanocomposites