Effect of homopolymer matrix on diblock copolymer grafted nanoparticle conformation and potential of mean force: A molecular simulation study

Effect of Homopolymer Matrix on Diblock Copolymer Grafted

Nanoparticle Conformation and Potential of Mean Force: A Molecular

Simulation Study

Carla E. Estridge,1,2 Arthi Jayaraman2,3

1Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 803092Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, Delaware 197163Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716

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

Received 23 September 2014; revised 6 November 2014; accepted 6 November 2014; published online 19 November 2014

DOI: 10.1002/polb.23637

ABSTRACT: We study the effect of homopolymer A or B matrix

on the conformations and effective interactions of AB diblock

copolymer grafted particles using coarse-grained molecular

dynamics simulations. In an A homopolymer matrix we

observe patchy conformations within the AB diblock copoly-

mer grafted layer, where the number of B patches is controlled

by the A-A attractive interaction strength. In a B homopolymer

matrix the grafted particle takes on a core-corona conforma-

tion, where the inner A block aggregates near the particle sur-

face and the outer B block forms a corona that interacts with

the B matrix. The potential of mean force (PMF) between two

particles in an A homopolymer matrix has a long-ranged

attractive well with a minima at intermediate distances corre-

sponding to the location of the outer B block patches. The PMF

between two particles in a B homopolymer matrix has an

attractive well at short interparticle distances corresponding to

the size of the inner A block. We isolate the contribution of the

homopolymer matrix on the PMF between the two diblock

copolymer grafted particles, by deducting the PMF in the

absence of a matrix, assuming the contributions of the grafted

particle and matrix to the PMF to be additive. VC 2014 Wiley

Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015, 53,

76–88

KEYWORDS: diblock copolymers; molecular dynamics; molecu-

lar modeling; nanocomposites

INTRODUCTION The modification of nanoscale filler materialsby functionalization of the filler (nanoparticle) surface withpolymers has emerged as one strategy for controlling inter-particle and particle-matrix interactions, and morphology inpolymer nanocomposites.1–4 Introduction of chemical hetero-geneity into the grafted layer or the polymer matrix providesadditional control over the composite morphology via tuna-bility provided by enthalpic interactions between the variouschemical species.5–15 For example, in the case of a nanopar-ticle grafted with a mixed layer of A and B homopolymers, atbrush-like grafting densities, the grafted A and B homopoly-mers adopt different conformations based on the chemistryof the medium the particles are placed in.8,16,17 As a result,the heterogeneous chemistry of the mixed homopolymergrafted layer has been shown to guide the assembly of nano-particles at interfaces in polymer blends, where the graftedlayer takes on surfactant-like conformations that act toreduce the interfacial tension within the polymerblend.6,18–22 Similarly, in the case of AB copolymer graftedparticles, the monomer sequence within an AB copolymer

graft impacts the assembly of nanoparticles in solution, withMonte Carlo (MC) simulations predicting that the diblockcopolymer grafted particles assemble into anisotropic clus-ters when A-B repulsions are significant, despite isotropicgrafting of the copolymers.14 Experiments have shown thatwith increasing ratio of particle size to graft copolymermolecular weight, particle assembly changes from single par-ticle micelles to small clusters to large vesicles.23 In previouswork, using molecular dynamics (MD) simulations we havestudied the effect of diblock copolymer composition on theassembly of spherical nanoparticles in solution.24 We havefound that at the early stages of assembly, the time necessaryfor patch formation decreases with an increase in the frac-tion of the graft composed of solvent phobic monomers, andfor all compositions, the final assembled clusters have ahigher probability of being anisotropic when the solvent isselective for the outer B block than those formed in a solventselective for the inner A block. Using self-consistent PRISMtheory-MC simulation, Nair and Jayaraman have shown thatwhen copolymer grafted particles are introduced to a

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

VC 2014 Wiley Periodicals, Inc.

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homopolymer matrix the potential of mean force (PMF)between alternating AB copolymer grafted particles is insen-sitive to matrix chemistry in the case of weak attractiveinteractions, while the shape of the PMF between diblockcopolymer grafted particles varies with matrix chemistry.10

While these past studies described above elucidate the effect ofmonomer interactions (or chemistry) on the conformations ofthe copolymer grafts,11,12,14,25 and its impact on the resultingassembly in implicit solvent or explicit small molecule sol-vent,9,13,24 most do not present behavior of copolymer graftedparticles in an explicit homopolymer matrix. The studies thatdiscuss copolymer grafted particles in homopolymer matri-ces10,26 do not isolate the role of matrix chemistry on the copol-ymer grafted particle shape, conformation, and effectiveinteractions. The focus of this article is to (a) understand homo-polymer matrix induced effects on diblock copolymer graftedparticle configuration (e.g. patchy versus core-corona shapes),and (b) isolate the contributions from the matrix on the PMFbetween copolymer grafted particles in a homopolymer matrix.The results presented in this work serve to guide the materialsscience community on understanding how to choose matrixchemistry and matrix packing fraction to achieve target spatialarrangements of copolymer grafted particles in the matrix.

This article is organized as follows. We first present thedetails of the coarse-grained model, our simulation method,and the analyses techniques. In the Results section we pres-ent the effect of the homopolymer matrix chemistry on thecopolymer grafted particle size and shape. We then presentthe effect of the homopolymer matrix on the potential ofmean force between copolymer-grafted particles at the dilutetwo-particle limit. We conclude by highlighting the keyresults and the implications of the results in various materi-als applications.

SIMULATION

ModelWe model our AB diblock copolymer grafted spherical nanopar-ticles in a homopolymer matrix using a coarse-grained model(Fig. 1). The spherical nanoparticles of diameter D 5 4r arecomprised of a shell of smaller, noninteracting beads of diame-ter r (r � 1 nm) constrained as a rigid body,27 which preservesthe desired excluded volume of a solid nanoparticle whilereducing computational cost. On this nanoparticle of diameter4r we isotropically anchor 25 AB diblock-copolymer graftsresulting in a grafting density of 0.51 chains/r2. Each graft andmatrix polymer is modeled as a bead-spring chain28 composedof 24 beads of diameter 1r, with each bead in the chain repre-senting a group of monomers or Kuhn segment within the poly-mer. The bonded interactions within the polymer chains aremodeled using a harmonic spring represented by:

UbondðrÞ51

2kbondðr2r0Þ2 (1)

with bond length ro 5 1.4r and kbond 5 30 kBT/r2, and r

equal to the center-to-center distance between adjacent

beads in the chain. The grafted chains are symmetric diblockcopolymers composed of an “A” and a “B” block. In all casesthe A block is anchored directly to the nanoparticle surface.

FIGURE 1 (a) Pairs of monomers with athermal interactions and

pairs of unlike species (e.g. A-B) interact through a Weeks Chandler

Andersen (WCA) potential, while attractive A-A and B-B interactions

are modeled via Lennard Jones (LJ) potential with a well depth of

1.0 kT. (b) The nanoparticle is of diameter D 5 4r and each polymer

bead mimicking a Kuhn segment is of diameter D 5 1r, where r �1

nm. (c) Nanoparticles are grafted with 25 symmetric diblock copoly-

mer grafts of chain length 24. The inner block in the graft (the block

closer to the surface) is composed of 12 A beads, and the outer

block in the graft is composed of 12 B beads. (d) Three different

homopolymer matrix chemistries are investigated: Homopolymers

with athermal interactions (gray), and attractive A (blue), and B

(red) homopolymers. In all cases the matrix chain length is 24.

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

wileyonlinelibrary.com.]

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Particle-particle, particle-monomer, chemically distinctmonomer-monomer (A-B) interactions, and like monomerpairs that are athermal are modeled using the Weeks-Chandler-Andersen (WCA) potential.29

UWCAðrÞ54err

� �122

rr

� �6� �1e r < rWCA

UWCAðrÞ50 r > rWCA

8><>: (2)

where e 5 1.0 (always in units of kBT in this work) and rWCA 5

r*21/6. The attractive monomer-monomer interactions betweennonbonded, chemically identical monomers (A-A, B-B) aremodeled using a standard 6-12 shifted Lennard-Jones (LJ)potential.30

ULfðrÞ54e

�rr

�12

2

�rr

�6" #

r < rcut

ULfðrÞ50 r > rcut

8>><>>: (3)

where e 5 1.0 (in units of kBT) and rcut 5 2.5r.

Simulation MethodUsing the model described above, we conduct Brownian Dynam-ics (BD) simulations using the HOOMD-Blue software package31

on NVIDIA graphics processing units (GPUs). We first conductsimulations for systems where all interactions are attractiveunder an NPT ensemble with P* 5 0, until the system is equili-brated. We then use these NPT simulations to set our box size forsimulations in an NVT ensemble (see Supporting InformationTable S.1.). This allows us to maintain equal total volume frac-tions of 0.34 for all systems. The reported ensemble average datain this article is from these NVT simulations.

For single nanoparticle simulations, we create an initial con-figuration as follows: We create a polymer-grafted nanopar-ticle with the grafted chains grown radially outward fromeach grafting point. The simulation box is then filled with4500 matrix polymer chains of length 24, and the box iscompressed to the simulation box size chosen based on theinitial NPT simulations for all attractive systems, attaining atotal volume fraction in the simulation boxes equal to 0.34.Five independent initial configurations are generated foreach parameter set, and each simulation is run for 3 3 108

timesteps with system snapshots being output every 1 3

106 timesteps. For all analyses we use 100 snapshots col-lected from the equilibrated portion of the simulation.

To calculate the potential of mean force (PMF) between twografted nanoparticles, we use umbrella sampling32 alongwith the weighted histogram analysis method.33,34 Since ourgoal is to measure the PMF as a function of interparticle dis-tance, we define the distance between the two particle cen-ters as our reaction coordinate. We use a biasing potential inthe form of a harmonic bond between the two particles ateach value of the reaction coordinate to sample configura-tions at each interparticle distance. The interparticle distan-ces sampled range from 5r to 30r with a window spacing of1r. The force constant of the harmonic bond is 5 kT/r2, to

ensure significant overlap between adjacent windows. Dataare collected every 10,000 time steps within each window,which is longer than the correlation time of the interparticledistance time series and ensures statistical independence ofthe sampled points (Supporting Information Fig. S.1). Simula-tions are run for 1 3 108 time steps within each window,which is found to be sufficient time for the PMF to converge(Supporting Information Fig. S.2). The weighted histogramanalysis method33,34 is used to calculate the unbiased PMFbetween the two particles.

The two-particle umbrella sampling simulations are initial-ized in the same manner as the single particle simulationswith the addition of a second nanoparticle that is placed at adistance from the first nanoparticle equal to r0 of the har-monic biasing potential. The simulation box size is chosen sothat the two nanoparticles never interact with a periodicimage of themselves irrespective of the value of the reactioncoordinate.

AnalysesWe analyze the conformations of the grafted particle by cal-culating the average radius of gyration of the grafted parti-cle and of the grafted copolymers, and the radiallyemanating monomer concentration profile. The averageradius of gyration of the grafted chains (in units of r) isdefined as:

hR2gi

1=25

�1

nGC � NGraft

� �XnGC

i51

XNGraft

j51

ri;j2rCOM 2�1=2

(4)

where NGraft is the length of the grafted chains, nGC is thenumber of grafts per particle core, ri,j is the position of jthmonomer on the ith chain, and rcom is the center of mass ofthe ith chain. The average radius of gyration of the graftedparticle (in units of r) quantifies the average size of thegrafted particle, and is defined as:

hR2gi

1=2GP5

�1

nGC � NGraft

� �XnGCi51

ðRi2RcomÞ2�1=2

(5)

where nGC is the number of grafts per core, Ri is the positionof monomer i, and Rcom is the center of mass of the graftedparticle. We measure the monomer concentration profilesfrom the particle surface for the inner block (of the graft),outer block, and matrix chains as follows:

CiðrÞ5piðrÞ

4pr2Dr(6)

where pi(r) is the average number of i type monomersfound at a radial distance between r and r 1 Dr from theparticle surface, and i denotes inner block, outer block, ormatrix. Monomer concentration profiles and radii of gyra-tion vary with the interaction strength and matrix chemis-try, and provide a description of the monomerarrangement around the particle surface and the size ofthe copolymer grafted particle as a function of these vary-ing parameters.


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RESULTS

Effect of Matrix Chemistry on Grafted Particle Size andShapeHomopolymer A MatrixFigure 2 shows the concentration profiles of inner (A) andouter (B) block monomers as well as A matrix monomersfrom the particle surface. When B-B interactions in the graftouter B block are attractive, while all other pairwise interac-tions are athermal, there is no significant effect of B-B attrac-tions on the athermal inner A block monomer concentrationor the A matrix monomer concentration within the graftedlayer of the particle [Fig. 2(a)]. In contrast, the maximumvalue in the outer B block monomer concentration shiftscloser to the particle surface [red right triangles in Fig. 2(b)]than the case with all athermal interactions [black circles inFig. 2(b)]. The average radius of gyration of the particle isreduced (Table 1) from 6.91 6 0.04r to 5.76 6 0.03r. This

is due to aggregation of B monomers in the grafted layer. Inall simulation trials of this case, we see that the particlestake on patchy conformations where the outer B blockaggregates into two distinct patches. This two patch confor-mation [eBB 5 1.0 in Fig. 2(c) right], as opposed to single ormany patch conformations, likely occurs due to the competi-tion between maximizing the enthalpic gain from B-B con-tacts, while minimizing the configurational entropy loss ofthe inner A block due to patch formation and maximizingthe configurational entropy of the matrix chains by gainingfree volume.

When A-A interactions, both in the matrix and the graftinner A block, are attractive, while outer B block interactionsare athermal, [blue upward triangles in Fig. 2(d)] there is adecrease in graft A monomer concentration close to the par-ticle surface compared with the all athermal case [circles inFig. 2(a), and for convenience, see the reorganized plots in

FIGURE 2 Single diblock-copolymer grafted nanoparticle in A homopolymer matrix. Matrix chemistry matches graft inner block

chemistry. Particle of diameter D 5 2Rp 5 4r, monomers of diameter d 5 2Rm 5 1r, grafting density 5 0.51 chains/r2, volume frac-

tion u 5 0.34, Ngraft 5 24, Nmatrix 5 24. (a) Average concentration profiles of graft inner block and matrix versus bead distance (in

units of r) from nanoparticle surface and (b) average concentration profiles of graft outer block versus bead distance (in units of r)

from nanoparticle surface for single particles in an athermal homopolymer matrix. (c) Representative simulation snapshots depict-

ing grafted particle conformations for diblock copolymer grafted particles in an athermal matrix. Symbol shape corresponds to

symbols used in concentration profile above. (d) Average concentration profiles of graft inner block and matrix versus bead dis-

tance (in units of r) from nanoparticle surface, and (e) average concentration profiles of graft outer block versus bead distance (in

units of r) from nanoparticle surface for single particles in an attractive A homopolymer matrix (f) Representative simulation snap-

shots depicting grafted particle conformations for diblock copolymer grafted particles in an attractive A homopolymer matrix.

Symbol shape corresponds to symbols used in concentration profile above. Line color denotes bead interactions, where black is

athermal, red is attractive B monomers, and blue is attractive A monomers (best viewed in color). [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com.]




Supporting Information Fig. S.3]. This is because the attrac-tive interactions between the inner A block monomers andthe A matrix cause the inner A block to swell, thus decreas-ing the inner A block concentration near the particle; this isalso seen in the surface area of the inner A block in contactwith the A matrix increasing from 910r2 for athermal Amatrix to 1025r2 for attractive A matrix (Fig. 3). The Amatrix concentration profile shows that there is a slightlylower concentration of the A matrix monomers within thegrafted layer than the all athermal case (Supporting Informa-tion Fig. S.3), as the A matrix increasing monomers depletethe grafted layer to form attractive A-A contacts outside thegrafted layer within the A matrix. If the B-B interactions arealso attractive, the outer B block concentration [red crossesin Fig. 2(e)] increases near the particle surface due to Bmonomer aggregation, and the A matrix concentration [bluecrosses in Fig. 2(d)] in the grafted layer decreases likely dueto the aggregated graft A and B monomers reducing accessto the particle surface. Correspondingly, the average radius

of gyration is the smaller (5.56 6 0.04r) in this case com-pared with the case where A monomers in the grafted layerand matrix are attractive while B monomers are athermal(5.89 6 0.01r) (Table 1). However, the case where all A andB monomers in the grafted layer are attractive has a largerradius of gyration when the particle is in an attractive Amatrix (5.89 6 0.01r) than in an athermal A matrix (5.09 6

0.01r). This is due to the enthalpically favorable interactionsbetween the inner A block monomers and the attractive Amatrix causing the copolymer grafted particle to swell.

As the interactions involving A matrix monomers changesfrom being athermal to attractive, the number of B monomerpatches decreases [as seen in simulation snapshots in Fig.2(f)]. In an athermal A matrix, in all trials, two patches of Bmonomers form in the grafted layer. When the A matrix hasattractive A-A interactions, the outer B block is driven toform a single patch in all trials. This could be because the Amatrix drives B monomers that are chemically different fromthe matrix to aggregate to maximize the interactionsbetween the inner A block and the A matrix. By taking on asingle patch conformation the grafted layer increases thesurface area of inner A block monomers accessible to the Amatrix, thus increasing enthalpically favorable contactsbetween the two (Fig. 3).

Homopolymer B MatrixWhen the A-A interactions in the inner A block are attractivewhile all other interactions are athermal, the inner A blockconcentration profile [blue triangle in Fig. 4(a)] increasesslightly at distances close to the particle surface comparedwith all athermal interactions case [black circles in Fig. 4(a)].This is due to the enthalpically driven A monomer aggrega-tion in the grafted layer, which also shifts the maxima in theouter B block concentration [black triangle of outer block inFig. 4(b)] to distances farther from the particle surface.There is no significant difference between the B matrix con-centration profiles [Fig. 4(a)] as the matrix is chemically sim-ilar to the outer B block, and does not have any energeticreasons to interact with the aggregated inner A block.

When the B-B interactions in the matrix and outer B blockare attractive, while all other interactions are athermal, theconcentration profiles of the inner A and outer B blockmonomers shift to distances farther from the particle surface[upward triangles in Fig. 4(d,e)] and the particle radius ofgyration increases significantly from 6.916 0.04r for the all

TABLE 1 Average Radii of Gyration (in Units of r) for Single Grafted Particles in a Homopolymer Matrix

Athermal Matrix A Matrix B Matrix

Athermal eAA5 10 eBB51.0

eAA5 10

eBB5 1.0 eAA5 10

eAA510

eBB5 1.0 eBB51.0

eAA5 10

eBB5 1.0

<R2g >GP

0.5 6.9160.04 5.576 0.02 5.766 0.03 5.096 0.01 5.896 0.01 5.566 0.04 7.116 0.01 6.346 0.02

<R2gA >

0.5 2.1960.02 1.96 6 0.02 2.1310.02 2.056 0.02 2.27 6 0.03 2.336 0.03 2.356 0.03 2.096 0.02

<R2gB >

0.5 2.146 0.02 2.09 6 0.03 1.89 6 0.02 2.036 0.03 2.18 60.03 2.046 0.03 2.326 0.03 2.286 0.02

FIGURE 3 Matrix accessible surface area for each block of the

grafted layer within a homopolymer A matrix. The first column

(eBB 5 1.0) corresponds to the case where the inner A block

and A matrix have athermal interactions. The second column

(eAA 5 1.0) corresponds to the case where the outer B block

has athermal interactions, and inner A block and A matrix is

attractive. Lastly, the third column corresponds to the case

where inner A block, outer B block and A matrix are all attrac-

tive. For all cases, the particle is of diameter D 5 4r, grafting

density 5 0.51 chains/r2, and Ngraft 5 24, Nmatrix 5 24, and total

volume fraction u 5 0.34 (best viewed in color). [Color figure

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



POLYMER SCIENCE



athermal case to 7.11 6 0.01r (Table 1). This is due toattractive B-B interactions between the outer B block and Bmatrix. The outer B block extends to distances farther fromthe particle surface, which facilitates attractive interactionsbetween B block and the B matrix (see individual block radiiof gyration data in Table 1). We also observe a decrease inthe B matrix concentration near the particle surface com-pared with the all athermal case (Supporting InformationFig. S.3.). This is because, unlike the all athermal case, in thiscase there is an enthalpic driving force for the B matrix tostay in the outer B block region of the grafted layer ratherthan penetrate to the inner A block region, because of the B-B attractions between outer B block and B matrix. Now, ifthe A-A interactions are also attractive in addition to the B-Battractive interactions, the radius of gyration decreases to6.34 6 0.02r (Table 1) and A monomer concentration [bluecrosses in Fig. 4(d)] near the particle surface increases, dueto aggregation of A monomers. The attractive B matrix [red

crosses in Fig. 4(d)] increases in concentration at short dis-tances from the particle surface as compared with the casewhere the inner A block is athermal. This is because theinner A block aggregates close to the particle surface pullingthe attractive outer B block to be closer to the particle sur-face, which in turn creates an enthalpic drive for the Bmatrix to follow the grafted A block and be closer to the par-ticle surface.

Key General Trends in Homopolymer A Matrix VersusHomopolymer B MatrixIn the case of a homopolymer matrix that matches the innerblock chemistry (A), the diblock copolymer grafted particlestake on patchy conformations with the outer B block mono-mers aggregating into patches (Fig. 2 snapshots). In contrast,in the case of a homopolymer matrix that matches the outerblock chemistry (B), the diblock copolymer grafted particlestake on core-corona conformations, where the inner A block

FIGURE 4 Single diblock-copolymer grafted nanoparticles in B homopolymer matrix. Matrix chemistry matches graft outer block

chemistry. Particle of diameter D 5 2Rp 5 4r, monomers of diameter d 5 2Rm 5 1r, grafting density 5 0.51 chains/r2, volume frac-

tion u 5 0.34, Ngraft 5 24, Nmatrix 5 24. (a) Average concentration profiles of graft inner block and matrix versus bead distance (in

units of r) from nanoparticle surface, and (b) average concentration profiles of graft outer block versus bead distance (in units of r)

from nanoparticle surface for single particles in an athermal homopolymer matrix. (c) Representative simulation snapshots depicting

grafted particle conformations for diblock copolymer grafted particles in an athermal matrix. Symbol shape corresponds to symbols

used in concentration profile above. (d) Average concentration profiles of graft inner block and matrix versus bead distance (in units

of r) from nanoparticle surface, and (e) average concentration profiles of graft outer block versus bead distance (in units of r) from

nanoparticle surface for single particles in an attractive B homopolymer matrix. (f) Representative simulation snapshots depicting

grafted particle conformations for diblock copolymer grafted particles. Symbol shape corresponds to symbols used in concentration

profile above. Line color denotes bead interactions, where black is athermal, red is attractive B monomers, and blue is attractive A

monomers. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]




aggregates near the particle surface to form the core, andthe outer B block forms a corona which interacts with thematrix (Fig. 4 snapshots). Therefore, the matrix chemistrytunes the grafted chain conformations, and thus the shape ofthe copolymer grafted particle.

The matrix chemistry tunes the overall grafted particle sizeas well. In the case of homopolymer A matrix, the graftedparticles take on a smaller size in the presence of A-A matrixattractions as compared with all athermal interactions. In thecase of homopolymer B matrix, the grafted particles takeon larger size in the presence of B-B matrix attractions ascompared with all athermal interactions.

The matrix chemistry also dictates the extent of matrixmonomer penetration into the grafted layer or “wetting” ofthe grafted layer that could impact the mechanical propertiesof such a composite. Irrespective of matrix chemistry (A or Bhomopolymer matrix), we observe relative depletion of thematrix from the grafted layer when the matrix monomersare attractive compared with when the matrix monomers areathermal. The depletion is stronger closer to the particle sur-face in the case of attractive homopolymer B matrix. Thedepletion is stronger at intermediate distances from particlesurface, where the patches of aggregated B monomers areformed, in the case of an attractive homopolymer A matrix.

Absence of Homopolymer MatrixTo isolate the role of matrix chains on the results presentedso far, we compare the average radius of gyration of thegrafted particles in an A or B homopolymer matrix (Table 1)to those in the absence of a matrix (Table 2).

In the case of athermal interactions, the particles have alarger radius of gyration in the absence of the matrix than inan athermal homopolymer matrix (9.91 6 0.03r vs. 6.91 6

0.04r). This is in agreement with past Monte Carlo simula-tion studies35 that show that in a purely athermal system,the matrix chains drive the grafted chains (with less confor-mational entropy than matrix chains) to take on more com-pact configurations to maximize the free volume of thematrix chains. In the case of attractive A-A interactions orattractive B-B interactions, the presence of the matrix (athe-rmal or attractive) leads to a more compact configuration(lower average radius of gyration) than in the absence of thematrix. When both copolymer grafted A-A and B-B interac-tions are attractive, the presence of an athermal matrixreduces the size of the particle more than in the absence ofa matrix, while the presence of an attractive (A or B) matrixincreases the size of the copolymer grafted particle over that

seen in the absence of the matrix. The increase in size inboth cases of an attractive matrix (A or B) is due to theattractive interaction of the matrix with one of the blocks inthe grafted layer. Furthermore, an attractive homopolymer Bmatrix increases the size more than an attractive homopoly-mer A matrix. This difference in size is explained by the con-formations the grafted particles take on to maximizeattractive interactions between the attractive grafted blockand chemically similar matrix. In an attractive A matrix, theparticles take on a patchy configuration, where both the Aand B block are conformationally hindered due to the aggre-gation of the outer B block. In an attractive B matrix, par-ticles take on a core-corona configuration, where the outer Bblock is conformationally unhindered leading to an overalllarger radius of gyration than a particle in an attractive Amatrix.

Effect of Matrix Chemistry on Potential of Mean ForceBetween Grafted ParticlesAthermal Homopolymer MatrixIn the purely athermal case, the potential of mean force(PMF) between two polymer grafted particles is repulsivefrom particle-particle contact to a distance equal to twicethe brush height (Fig. 5 circles), in agreement with previoustheoretical studies.36 This is due to steric repulsion arisingfrom the grafted monomers on the two grafted particles. Inthe absence of the matrix (Supporting Information Fig. S.4.)this steric repulsion is significantly stronger at contact, andlonger ranged compared with that in the presence of anathermal matrix. This is explained by depletion-attractioninduced by the athermal matrix, due to athermal matrixchains depleting the volume between the two particles atclose distances (Fig. 6). This depletion attraction37,38 is quan-tified in Supporting Information Figure S.4 as the differencebetween the PMF in the presence of the matrix and the PMFin the absence of the matrix.

Homopolymer A MatrixWhen B-B interactions are attractive and all other pair-wiseinteractions are athermal [black right triangles in Fig. 5(a)],the steric repulsion seen in the all athermal case [blackcircles in Fig. 5(a)] is replaced with an attractive well. Thedeepest point in the well occurs at intermediate interparticledistances, where the grafts’ outer B block on the two par-ticles are able to aggregate creating enthalpically favorable(attractive) contacts (quantified in Supporting InformationFig. S.5) that overcome the steric repulsion seen in the allathermal case. At shorter interparticle distances, the forcebetween the two particles is repulsive because the small dis-tance between the particles is conformationally restricting tothe inner A block, reducing the configurational entropy, eventhough the B monomers form energetically favorable con-tacts. As shown in the snapshots in Figure 5(b) an aggregateof attractive B monomers is able to form when the particlesare in contact and at intermediate distances, thus maintain-ing a negative PMF at all distances from contact to the dis-tance where the outer B blocks of the two particles are ableto merge.

TABLE 2 Average Radius of Gyration, in Units of r, for Single

Grafted Particles in the Absence of Matrix

Athermal eAA 5 1.0 eBB 5 1.0

eAA 5 1.0;

eBB5 1.0

<R2g >GP

1/2 9.91 6 0.03 7.53 6 0.02 8.38 6 0.03 5.31 6 0.02


POLYMER SCIENCE


When the A-A interactions in the matrix and the inner Ablock are attractive while all other interactions are athermal(blue upward triangle in Fig. 5), the shape of the PMF exhib-

its a stronger repulsive force at short distances comparedwith the all athermal case and the B-B attraction case, and adeeper attractive well at intermediate distances compared

FIGURE 5 (a) Potential of mean force, PMF, (in units of kT) as a function of nanoparticle surface to surface distance (r–D in units

of r) for two copolymer grafted particles in an A homopolymer matrix. Line color denotes matrix interactions of athermal (black)

or attractive (blue). Marker shape denotes interaction set: all athermal interactions (circle), athermal inner A block and attractive

outer B block in an athermal A matrix (right triangle), attractive inner A block and athermal outer B block in an attractive A matrix

(upwards triangle), and an attractive inner A block and attractive B outer block in an attractive A matrix (crosses). Particle diameter

D 5 4r, grafting density 5 0.51 chains/r2, total volume fraction u 5 0.34, Ngraft5 24, Nmatrix5 24. (b) Simulation snapshots corre-

sponding to three regions in the two particle potential of mean force for each interaction set. The nanoparticle location is high-

lighted for clarity, and table background color denotes matrix chemistry. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

FIGURE 6 Quantifying the depletion of matrix monomers between the two grafted particles. (a) Schematic representation of the

cylindrical area used to measure the matrix density where L, the length of the cylinder, is equal to the particle surface to surface

distance of the two grafted particles, and H, the radius of the cylinder, is equal to the brush height plus the radius of the particle

calculated from single particle simulations. (b) Representative simulation snapshot of two grafted particles, grafts shown in gray,

and the matrix chains, shown in green, found in the cylindrical volume between the two particles. (c) Ratio of the matrix density

in the confined cylindrical region to the bulk matrix density as a function of particle surface-to-surface distance. Particle diameter

D 5 4r, grafting density 5 0.51 chains/r2, total volume fraction u 5 0.34, Ngraft 5 24, Nmatrix 5 24. [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com.]





with B-B attraction case. The attractive well is deeper due toa significant increase in the (magnitude of) average potentialenergy when the matrix and outer B block have attractiveinteractions (Supporting Information Fig. S.5). The attractivewell at intermediate distances occurs because the athermalB monomers are able to aggregate in the small regionbetween the two particles [snapshot in Fig. 6(b)], maximiz-ing the volume for the attractive A monomers in the graftinner block and A matrix. This well is shifted to larger dis-tances because the grafted particles take on larger conforma-tions in the attractive A matrix when the outer block isathermal 5.89 6 0.02r, than in the athermal matrix case5.57 6 0.01r (Table 1). As the particles get closer, the con-finement of the athermal B monomers between the particlesexerts steric repulsion at short distances.

Upon introducing B-B attraction, while maintaining an attrac-tive A-A interaction in the inner A block and A matrix (bluecrosses in Fig. 5), the attractive well depth shifts to slightlycloser interparticle distances, and the gradient of the PMF atshort distances is weaker. The steric repulsion is weaker atshort interparticle distances in this case compared with athe-rmal B-B interactions due to the favorable enthalpic drivecoming from attractive B-B contacts [red patch in bottomrow of Fig. 5(b)]. The shift in the attractive well to shorterinter particle distances is due to a reduction in grafted parti-cle size from 5.89 6 0.01r to 5.56 6 0.04r (Table 1), anddue to attractive A and B contacts in a small volume aroundthe grafted particles, that also increases the free volume forthe A matrix to make attractive A contacts.

Homopolymer B MatrixIn the case of attractive A-A interactions, while maintainingathermal pairwise interactions otherwise, the PMF betweenthe two copolymer grafted particles exhibits an attractionclose to the particle surface [black right triangles in Fig.7(a)] in contrast to the purely repulsive PMF for the com-pletely athermal case [black circles in Fig. 7(a)]. The favor-able enthalpic drive arising from attractive contacts betweenthe inner A blocks in both particles leads to this attractivewell [Supporting Information Fig. S.6(b)]. If B-B interactionsin the matrix and the outer B block are attractive, and A-Ainteractions are athermal, the attractive well becomes deepand long ranged [red upward triangles Fig. 7(a)]. This maybe explained by the differences in the extents to which thematrix depletes (dewets) the grafted layer (Supporting Infor-mation Fig. S.3). When the B matrix is attractive, it stronglydepletes the grafted layer of the particle as compared withan athermal B matrix. Furthermore, when the inner A blockis athermal, the depletion is the strongest [Supporting Infor-mation Fig. S.3(b) red triangles], as the attractive B matrixprefers to interact with the attractive B outer block, and inthis case we see the longest ranged attractive well in thePMF. Long ranged depletion of the matrix should inducelong-range matrix induced attraction between particles. Theslight difference in the well depth between the cases with A-A attraction [crosses in Fig. 7(a)] and athermal inner A block[red upwards triangles in Fig. 7(a)] at short particle surface-particle surface distances is caused by the particles beingable to make enthalpically favorable A contacts between theinner A block on the two particles as evidenced by the

FIGURE 7 (a) Potentials of mean force, PMF, (in units of kT) as a function of nanoparticle surface to surface distance (r–D in units

of r) for two copolymer grafted particles in a B homopolymer matrix. Line color denotes matrix interactions of athermal (black) or

attractive (red). Marker shape denotes interaction set: all athermal interactions (circle), athermal outer B block and attractive inner

block in an athermal matrix (right triangle), attractive inner block and athermal outer block in an attractive B matrix (upwards trian-

gle), and an attractive inner block and attractive outer block in an attractive matrix (crosses). Particle of diameter D 5 4r, grafting

density 5 0.51 chains/r2, total volume fraction u 5 0.34, Ngraft 5 24, Nmatrix 5 24. (b) Simulation snapshots corresponding to two

regions in the two particle potential of mean force for each interaction set. I depicts particle conformations when the PMF is at a

minimum, and II depicts particle conformations when particles do not interact. The nanoparticle location is highlighted for clarity,

and table background color denotes matrix chemistry. [Color figure can be viewed in the online issue, which is available at



POLYMER SCIENCE



increase in the (magnitude of) average potential energywhen attractive A block contacts can be made (SupportingInformation Fig. S.6).

Key General Trends in Homopolymer A Matrix VersusHomopolymer B MatrixWhen the matrix is chemically similar/same as the outerblock [Fig. 7(a)], the shape of the PMF is significantly differ-ent from the shapes of the PMF when the matrix is chemi-cally similar/same as the inner block [Fig. 5(a)]. As seen inthe snapshots in Figures 5(b) and 7(b), the difference inshape largely arises from the matrix induced patchy particleor core-corona configurations that the grafted particleadopts, as discussed in the previous section in detail.

The depth of the attractive well is much larger in an attrac-tive B matrix than in an attractive A matrix. This may be dueto the depletion of the matrix from the particle grafted layeras well as the location of the attractive block of the graftedlayer. There is a greater depletion of the matrix from thegrafted particle when the outer block and the matrix havethe same chemistry causing the matrix to induce a strongerattraction between the two particles.

Absence of Homopolymer MatrixAssuming that the matrix induced potential is a perturbationto the potential arising from the direct grafted particle inter-actions, we calculate the matrix induced potential by sub-tracting from the PMF between grafted particles in thepresence of the matrix the PMF between the grafted par-ticles in the absence of the matrix.

In the athermal case, the presence of the matrix makes thePMF less repulsive due to the depletion attraction inducedby the matrix. The depletion potential calculated as the dif-ference between the PMF in the presence of matrix and thePMF in the absence of matrix as a function of interparticledistance is plotted in Supporting Information Figure S.4c.

When the inner A block interactions are attractive and allother interactions are athermal (Supporting Information Fig.S.7), the PMF between the grafted particles in the absence ofmatrix exhibits a repulsive PMF at large distances where thecoronas of the two particles begin to overlap, and the outerblock is conformationally restricted. At the point where theattractive inner A blocks begin to interact, the PMF becomesattractive; this point corresponds to the peak distance in thePMF in Supporting Information Figure S.7(b). The presenceof an athermal homopolymer matrix [Supporting InformationFig. S.7(a)] deepens the attraction in the PMF; this is due tothe matrix induced depletion attraction. If we compare thematrix-induced depletion potential for the all athermal case[solid line in Supporting Information Fig. S.7(c)] and graft A-A attraction [triangles in Supporting Information FigureS.7(c)] case in an athermal matrix, we see two differences:(i) at contact, the PMF of the all athermal case is purelyattractive and more negative than the graft A-A attractioncase, and (ii) at intermediate distances, where the outer(athermal) B block coronas of the two particles interact (r–D

5 7.2–15r), the PMF is steeper in the presence of graft A-Aattraction than athermal graft. These differences arise fromthe attractive inner A block monomers aggregating leadingto the core-corona conformation, which resembles a B homo-polymer grafted particle with graft length of 12 and lowergrafting density (effectively larger core than the particlediameter). Past studies have shown that a larger attractivecore, smaller graft length, and/or lower grafting densityincrease the matrix induced depletion that should drive par-ticle aggregation.36,39–44 This is in agreement with thesteeper PMF (larger attractive force) in the presence of graftA-A attraction compared with athermal A grafts.

When B-B interactions are attractive, and all other interactionsare athermal, in the absence of a matrix [Supporting Informa-tion Fig. S.8(b)], the PMF has an attractive well and repulsionat contact. The depth of the attractive well in the presence of ahomopolymer matrix [Supporting Information Fig. S.8(a)] islarger than in the absence of the matrix, due to the matrixinduced depletion attraction between the two particles [Sup-porting Information Fig. S.8(c)]. The repulsive peak at r–D 5

18r is likely because the enthalpic interaction between Bbeads is low compared with the configurational entropy lossassociated with patch formation, leading to a larger graftedparticles (Table 2), and as a result reduced volume (configura-tional entropy) for the matrix. The volume for the matrix isincreased when the outer B block patches from the two par-ticles begin to merge (r–D �15r); this merging also increasesthe number of enthalpically favorable B contacts betweenattractive B monomers, creating an increase in the (magnitudeof) average potential energy, and explains the presence of theattractive PMF well 7r < r–D <12r. The second repulsivepeak occurs at the distances where the outer B blocks merge,but the short interparticle distances force the grafts to take onconfigurations that constrain the inner A block and reduce thegraft configurational entropy. The attractive well found atshort inter particle distances may arise from the reduction intotal volume that the two particles occupy within the matrix,which maximizes the matrix configurational entropy.

When both A-A and B-B interactions are attractive, the PMF(Supporting Information Fig. S.9) shows signatures of thetwo previous cases: solely attractive A-A interactions, andsolely attractive B-B interactions. In Supporting InformationFigure S.9 the well at short interparticle distances arisesfrom the attractive interaction between the inner A blockmonomers, and the attractive well at intermediate distancescorresponds to the attractive interactions between the outerB block monomers. The repulsive peak between the twoattractive wells may arise from the configurations that thegrafted layer takes which reduces the number of enthalpi-cally favorable contacts within either the inner A blocks orouter B blocks on both particles.

Additional Variations: Interaction Strength and TotalVolume FractionSo far we have maintained the monomer attraction strengths inthe grafts and/or matrix at 1 kT and total volume fraction at 0.34



throughout the study. We now briefly describe how varying theinteraction strength (or alternatively the chemistry of the mono-mers) and volume fraction (or solution concentration) impact thetrends we have discussed so far. We do this for two cases—(i)attractive inner A block with athermal outer B block and athermalB matrix [Fig. 8(a)], and (ii) attractive outer B block with athermalinner A block and athermal A matrix [Fig. 8(b)].

When the interaction strength is reduced from 1.0 kT to 0.2kT there is little difference in the shape and depth of thePMF, irrespective of the homopolymer matrix chemistry. Thisshows that the strength of the enthalpic interactions is notthe dominant driving force in the system as compared withwhether the matrix is chemically identical to the inner orouter block of the copolymer graft. In contrast, when thetotal volume fraction is reduced from 0.34 to 0.1 there is asignificant change in the PMF. Going from 0.34 to 0.1 theattractive well in the PMF is either (a) completely removedfor the case where the inner A block is attractive and theouter B block and B matrix are athermal [Fig. 8(b) squares]or (b) the depth is significantly reduced when the outer Bblock is attractive and the inner A block and A matrix areathermal [Fig. 8(a) squares]. This is in agreement with poly-mer matrix induced depletion attraction weakening as thematrix volume fraction decreases.45 This result shows thatthe matrix-induced depletion potential between the twografted particles, driven by the need for larger free volume(and thus configurational entropy), more significantly affectsthe PMF between two copolymer grafted particles in athe-rmal matrix conditions. Most interestingly, the extent towhich the volume fraction affects the PMF is larger whenthe inner A block is attractive than when the outer B blockis attractive. When the inner A block is attractive the well isremoved from the PMF entirely upon reduction of the vol-ume fraction, while a shallow well is still present in the caseof an attractive outer B block. This may be explained by thesignificant differences in the grafted layer conformationsbetween the two cases, core-corona versus patchy. With anattractive inner block and an outer block that matches thematrix chemistry, the particle takes on a core-corona confor-mation. When particles with core-corona conformationsapproach one another the outer B block loses configurationalentropy due to crowding at distances where the two graftedlayers begin to interact. At low total volume fractions thisconfigurational entropy loss may not be overcome by thematrix induced depletion attraction between the two par-ticles leading to the removal of the attractive well from thePMF entirely. Unlike the core-corona conformation, whenparticles take on patchy conformations in an A matrix, theentire graft loses significant configurational entropy to createpatches of outer B block monomers. Because the graftedlayer has already given up this configurational entropy theentropic penalty due to bringing the particles together maybe smaller for the patchy conformations than that for thecore-corona conformations. With a smaller entropic penaltythe matrix induced depletion attraction may be strongenough to create a small attractive well between theparticles.

CONCLUSIONS

Using molecular dynamics simulations of diblock copolymergrafted nanoparticles in a homopolymer matrix we study the

FIGURE 8 Effects of monomer attraction strength and total volume

fraction on potential of mean force between two diblock copolymer

grafted particles in an A/B homopolymer matrix. Particles of diame-

ter D 5 4r, grafting density 5 0.51 chains/r2, Ngraft 5 24, Nmatrix 5

24. The interaction strength, eii, for attractive beads is varied from

0.2 kT to 1.0 kT and the total volume fraction, /, is varied from 0.10

to 0.34, as labeled. (a) Potential of mean force between two diblock-

copolymer grafted particles when the A matrix and inner A block

are athermal and the outer B block has attractive interactions. Inset

axis labels are the same as main figure axis. (b) Potential of mean

force between two diblock-copolymer grafted particles when the B

matrix and outer B block are athermal and the inner A block has

attractive interactions. [Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com.]


POLYMER SCIENCE



effect of matrix chemistry on the conformations of singlegrafted nanoparticles and elucidate the matrix-inducedeffects on the PMF between two particles.

We find that the matrix chemistry tunes the size and shapeof single grafted particles. When the matrix chemistrymatches the inner block chemistry of the grafted layer, thegrafted particles take on patchy configurations with theouter block aggregating to form patches. When the matrixchemistry matches the outer block chemistry, the graftedparticles take on core-corona conformations with the innerblock aggregating near the particle surface forming the coreand the outer block extending into the matrix creating thecorona. The size of the grafted particle is controlled throughthe strength of interactions between the matrix and itschemically identical block within the grafted layer. When allgrafted monomer interactions are athermal the grafted par-ticles take on smaller conformations when in an athermalmatrix than in the absence of a matrix. However, whenattractions between the grafted layer and the matrix arepresent, the particles swell to increase the number ofenthalpically favorable contacts and therefore take on largerconformations when in a matrix than in the absence of amatrix. This swelling is due to enthalpic interactionsbetween the grafted layer and the matrix and is greaterwhen the grafts outer block and matrix chemistry are thesame than when the grafts inner block and matrix chemistryare the same.

We calculate the PMF between two copolymer-grafted par-ticles in a homopolymer matrix in order to understand therole of the matrix induced effective interactions between theparticles. The location of the attractive well between twoparticles is tuned with matrix chemistry. When particles arein a matrix chemically identical to the inner block of thegrafted layer, the attractive well is found at farther distancesthan when the particles are in a matrix chemically identicalto the outer block. Irrespective of homopolymer A or Bmatrix, when matrix monomer attractions are present thedepth of the PMF between the two particles increases due toa corresponding increase in the dewetting of matrix from thegrafted layer. This depletion is much stronger when thematrix is the same chemistry as the outer block.

We also find that the presence of an attractive well in thetwo-particle PMF is strongly dependent on the total volumefraction. With a decrease in the total volume fraction from0.34 to 0.10, the attractive well is (a) completely removedfrom the PMF when the particles are in an athermal matrixthat matches the outer block chemistry and the inner blockis attractive, and (b) significantly reduced in depth when theparticles are in an athermal matrix that matches the innerblock chemistry and the outer block is attractive. The differ-ences in the extent to which the total volume fraction affectsthe PMF may be caused by the differences in particle confor-mations within an A or B matrix. In contrast to packing frac-tion, the magnitude of the attraction strength betweengrafted monomers does not strongly affect the depth, range,or location of the attractive well in either an A or B matrix.

The PMFs between two diblock copolymer grafted particles ina dilute particle limit in polymer matrices help us understandthe role that the matrix plays in the effective interactionsbetween particles. Although it is possible that many bodyinteractions may become important at high concentrations inthis regime, we are able to elucidate qualitative trends onhow matrix volume fraction, chemistry, and enthalpic interac-tions tune two body interactions between particles.

This work can be used to guide material engineers in control-ling the size, shape, and interactions of diblock copolymergrafted particles in polymer matrices. By changing the chemistryof the matrix we have elucidated how particle shape, size, andeffective interparticle attraction strength are tuned. The strengthand location of attraction and repulsion in the potential ofmean forces can be valuable in choosing monomer chemistriesto tailor the assembly of the grafted particles in a matrix.

ACKNOWLEDGMENTS

This work was supported by Department of Energy underGrant DE-SC0003912. The majority of this research used GPUresources of the National Energy Research Scientific Comput-ing Center, which is supported by the Office of Science of theU.S. Department of Energy under Contract DE-AC02-05CH11231. We thank E. Jankowski for helpful discussionsregarding set up of HOOMD package for the single particlesimulations (Introduction section) in early stages of this work.

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Documents

Effect of homopolymer matrix on diblock copolymer grafted nanoparticle conformation and potential of mean force: A molecular simulation study