Polymer-Grafted Nanoparticles in Polymer Melts: Modeling Using the

Dec 12, 2013 - In Encyclopedia of Polymer Blends; Isayev , A. I., Ed.; Wiley-VCH: Weinheim, ...... Fleer , G. J. Polymers at Interfaces, 1st ed.; Chap...
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Polymer-Grafted Nanoparticles in Polymer Melts: Modeling Using the Combined SCFT−DFT Approach Valeriy V. Ginzburg* Research and Development, The Dow Chemical Company, Building 1702, Midland, Michigan 48674, United States ABSTRACT: Nanoparticles (silica, carbon black, etc.) are often used as fillers to improve physical (thermal conducticity, coefficient of thermal expansion) and mechanical (modulus, strength) properties of polymer materials. In many cases, however, lack of nanoparticle dispersion in the polymer limits the utility of a resulting nanocomposite material. To improve dispersion, one often grafts organic chains (“ligands”) onto the surface of the particles; if the ligands are chemically miscible with the matrix polymer, it helps the particles to disperse more uniformly. The relationship between ligand length and grafting density, on the one hand, and the nanocomposite morphology, on the other, has, in recent years, been investigated in several experimental, theoretical, and computational studies. Those studies, however, primarily considered the limit of very small particle loadings (usually less than 3−5%). In this paper, we adapt our earlier formalism combining self-consistent field theory (SCFT) for polymers with density functional theory (DFT) for the particles; the modified formalism explicitly incorporates the grafted chains into SCFT. We then perform several simulations to study the dependence of morphology on the length and density of grafted chains as well as the nanoparticle loading. The results are in qualitative agreement with predictions of earlier theories in the limit of lower particle loadings and predict new morphologies (“bundles of wires”) for the case of larger particle loadings. The method can be easily extended to more complex cases (for example, where the matrix and/or ligand itself is a blend or block copolymer).

1. INTRODUCTION In recent years, the use of nanoparticle additives to polymers has become widespread; it has been shown that the addition of nanoparticles can substantially change mechanical (modulus, strength, toughness) and physical (thermal conductivity, coefficient of linear thermal expansion, etc.) properties of the polymer matrix.1−6 One of the most challenging issues, both from the practical standpoint and from theoretical point of view, is that of nanoparticle dispersion in the polymer matrix. Our ability to optimize and tailor specific properties is often dictated by the morphology of the particles and the way they distribute themselves across the composite. If the particles agglomerate into large aggregates, the resulting material often ends up having inferior properties; if, on the other hand, the particles are distributed uniformly, overall properties could dramatically improve. Additionally, particles could organize themselves into various anisotropic aggregates, imparting new physical and mechanical properties to the resulting composite. What, then, are the factors determining nanoparticle dispersion and morphology? It has been known for long time that when large colloidal particles are introduced into polymer melt, they experience both entropic and enthalpic-driven attraction.7−10 The enthalpic term comes from the chemical dissimilarity between the matrix polymer and the particles, while the entropic term comes from the fact that matrix polymer chains lose entropy when confined between the surfaces of two adjacent particles. To counter this attraction, one could use oligomeric chains grafted onto nanoparticle surfacesthe addition of such chains reduces effective particle/polymer © 2013 American Chemical Society

surface tension and promotes nanoparticle dispersion. Using self-consistent field theory (SCFT), integral equation approaches, or scaling models, various authors determined the effective particle−particle interaction potential as a function of particle radius, matrix polymer chain length, grafted oligomer density, and chain length and established conditions corresponding to aggregating particles and dispersed particles.11−19 Usually, in the absence of grafted brushes, particles are supposed to aggregateeven if there is no enthalpic-driven attraction, entropic forces should still be sufficient to cause aggregation. In recent years, however, more and more data came to light showing that for nanoparticles this picture is not complete. On the theoretical side, Khounlavong, Ganesan, and Pryamitsyn demonstrated the critical role of multibody interactions even at low particle loadings, thus showing the insufficiency of a simple potential of the mean-force approach.20 On the experimental side, the work of Mackay et al.21 demonstrated that when enthalpic interaction between matrix and particles is neutralized (particles were made out of the same polymer as the matrix), the particles tend to aggregate when the particle radius (Rp) is greater than the matrix polymer radius of gyration (Rg); however, the particles tend to disperse uniformly if Rp < Rg. It was also shown that nanocomposites with small particles have other interesting properties; for example, under some Received: October 25, 2013 Revised: November 27, 2013 Published: December 12, 2013 9798

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attached to it. One would imagine, then, that combining either SCFT-DFT or SCFT-BD with a proper treatment of grafted chains can provide new insights into the assembly of polymergrafted nanoparticles in polymer matrices (homopolymer, block copolymer, or blend). Here, we propose a simple extension of the SCFT-DFT approach in which the grafted oligomers (“ligands”) are treated explicitly in a mean-field fashion. The model is applied to the case where spherical nanoparticles having radius of 5.5 nm (those could correspond to silica, gold, silver, or quantum dots, depending on the application) are dispersed in a homopolymer matrix (polymer molecular weight on the order of 50 kg/mol). By varying the grafting density and the particle loading, we can guide the assembly of nanoparticles into various structures and thus impact the overall properties of the composite. The method could be extended to other, more complex, mixtures as well.

conditions, polymer melt in the presence of small nanoparticles had lower viscosity than the same melt without the particles.22 Mackay et al. hypothesized that the particle−polymer phase separation can be hindered by the inability of spheres to occupy all the interstitial volume; as a result, if particles aggregate, some polymer chains need to fill the interstitial volume. Assuming the particle radius is close to Rg, those chains will be severely constrained and their entropy will be substantially lower than in the pure melt. One could conceivably hypothesize that those entropy losses could become larger than in the case of particles dispersed uniformly or near-uniformly. However, predicting such behavior theoretically required approaches that could correctly capture both polymer free energy and multiparticle effects. For the simple nanoparticle/homopolymer blends, one such approach was based on statistical mechanical density functional theory (DFT).23−25 Using DFT, Frischknecht and co-workers described, for example, polymer/nanoparticle segregation near a hard wall and predicted the particle crystallization in the near-wall region. However, DFT is somewhat difficult to expand to two- or three-dimensional problems if one needs to equilibrate the system on two separate length scales (grafted chains on the particle surface and the overall nanocomposite). For the case of polymer-grafted nanoparticles in polymer melts, additional complications arise from symmetry breaking. In a recent paper, Akcora et al.26 demonstrated that polymergrafted nanoparticles in polymer melts can form strongly anisotropic objects such as one-dimensional wires, twodimensional sheets, and continuous networks of wires as well as dispersed singlets, doubles, etc. Other researchers observed similar phenomena in various systems. 27−30 Based on experimental studies, approximate phase diagrams have been established. Kumar and co-workers investigated the factors leading to the formation of various anisotropic structures with the help of coarse-grained Monte Carlo simulations,26,31−33 while Ganesan et al. developed a simple analytical mean-field theory describing the transitions between various structures. In general, a good qualitative agreement between theory, simulation, and experiment is observed, providing researchers with design rules for making advanced nanocomposites. However, both theoretical models and simulations are generally working best in the limit of very dilute (low particle loading) mixtures. In addition, Monte Carlo simulations with multiple particles usually can be done only with implicit matrix polymers, and if the matrix is modeled explicitly, one could only simulate a single nanoparticle or a nanoparticle pair. With the theoretical model, again, the phase diagram is computed in the limit of very low nanoparticle loadings. The question remainshow can one model the mixtures with finite particle loading, e.g., in the 5−20 vol % range? One possible method of studying systems with finite particle loading could be found by using the techniques applied to diblock/nanoparticle mixtures. Those methods were based on the SCFT description of the polymers, combined with either Brownian dynamics (BD)34,35 or density functional theory (DFT)36−40 description of the particles. The two approaches are fairly similar in many respects, and both correctly capture salient features of copolymer/nanoparticle mixtures, such as segregation of nanoparticles toward favorable blocks and/or interfaces, depending on the particle size and surface chemistry. More recently, attempts were made to explicitly incorporate grafted chains into SCFT-DFT;40,41 in those studies, however, it was assumed that each particle has a single polymer chain

2. THE MODEL Our starting point is the Thompson−Ginzburg−Matsen− Balazs (TGMB)36,37 approach that combines self-consistent field theory (SCFT) to describe polymeric species and statistical-mechanical density functional theory (DFT) to describe the particles. The TGMB free energy, F, for a binary mixture of nanoparticles (particle species are labeled P) and homopolymer matrix (polymer species labeled A, degree of polymerization is denoted P) can be written as FP = f1 + f2 + f3 + f4 kBTρ0 V (1) f1 = − f2 =

⎛Q ⎞ ⎛Q ⎞ ln⎜ P ⎟ − (1 − ϕP) ln⎜ A ⎟ ⎝ V ⎠ α ⎝V ⎠

ϕP

1 V

(2a)

∫ dr[(χAP P)φA(r)φP(r) − ξ(r)(1 − φA(r)

− φP(r))]

(2b)

f3 =

1 V

∫ dr[−wA(r)φA(r) − wP(r)ρP (r)]

(2c)

f4 =

1 V

∫ dr[ρP (r)ΨCS(φP(r))]

(2d)

The relationship between local particle volume fraction, ϕP(r), particle center probability distribution, ρP(r), and “weighted” particle density, φP(r)), is given by φP(r) =

1 P

φP(r) =

1 23P

⎛

∫ dr′ ρP (r′)Θ⎜⎝1 − ⎛

|r − r′| ⎞ ⎟ RP ⎠

∫ dr′ ρP (r′)Θ⎜⎝1 −

|r − r′| ⎞ ⎟ 2RP ⎠

(3a)

(3b)

All lengths are scaled relative to the matrix radius of gyration, RgA = a(P/6)1/2, with a being the statistical segment length. The Heaviside function Θ(x) = 1 if x ≥ 0 and 0 otherwise. Here, φP is the particle volume fraction, P is the chain length of the matrix polymer, V is the volume of the system, ρ0 = 1/v0 is the inverse of the reference monomer volume, and α = vP/ (Pv0) is the particle-to-matrix polymer volume ratio. The Flory−Huggins parameter χ AP is the measure of the incompatibility between the particles and the matrix polymer 9799

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Note that the ligand volume fraction, φL, can be related to the grafting density, σ, in the following fashion, φL = φP(3/ RP)σNv0. We now describe the ligands using approach similar to the one first proposed by Fleer and co-workers.48−50 Specifically, we note that one end of each ligand must be close to the particle surface, while the other end could be located everywhere. Therefore, we suggest that the propagators for the ligand, qL and q†Lobey the following equations:

and can, in principle, be related to the polymer/particle surface tension. The free energy is a functional of density fields ϕA(r) and ρP(r), conjugate chemical potential fields wA(r) and wP(r), and the Lagrange multiplier (“pressure”) ξ(r). The first term, f1, describes the “ideal mean-field” free energies of each component given external chemical potential fields w. The partition functions of individual components, QP and QA, are defined below. The second term, f 2, contains the (local) interactions among various species, described in a traditional Flory−Huggins42,43 fashion, as well as the incompressibility constraint. The third term, f 3, contains the terms with chemical potential fields w. Finally, the fourth term, f4, contains the nonideal hard-sphere interactions, summed up via the “smoothed density approximation” of the hard-sphere density functional theory (DFT) due to Tarazona.44−46 In the Tarazona DFT, the nonideal free energy of the hard-sphere fluid is given by the Carnahan−Starling47 equation of state:

ΨCS(x) =

4x − 3x 2 (1 − x)2

∂qL(r, s) ∂s ∂qL†(r, s) ∂s

Partition functions of individual components are given by

∫ q(r, 1) dr

(5a)

QP =

∫ exp{−wP(r)} dr

(5b)

Here, the propagator q(r,s) (where 0 < s < 1 is the index denoting the position along the matrix chain) is given by the modified diffusion equation ∂q(r, s) = [∇2 − wA ]q(r, s) ∂s

= −[∇2 − wL]qL†(r, s)

where 0 ≤ s ≤ κL and q†L(r,κL) = 1; qL(r,0) = ϕP(r)[1 − ϕP(r)]. It is important to note here that the above assumption implies that the grafted chains are “weakly bound” to the particlesthey are not anchored to specific location on the particle surface but could rearrange themselves to reduce the overall free energy of the system. The validity of this approximation depends on the nature of the chemical bond between particle and grafted oligomer. To solve for all fields and densities, we use the standard SCFT approach and obtain the following set of self-consistency equations:51−53,36

(4)

QA =

= [∇2 − wL]qL(r, s)

wA(r) = χAP PφP(r) + χAL PφL(r) + ξ(r) = = μA (r)

(9a)

wL(r) = χAL PφA (r) + χLP PφP(r) + ξ(r) = = μL (r)

(9b)

|r − r′| ⎞ ⎟ RP ⎠ 1 [χAP PφA (r′) + χLP PφL(r′) + ξ(r′)] + 3 dr′ 2P ⎛ |r − r′| ⎞ Θ⎜1 − ⎟[ρ (r′)Ψ′CS( φP(r′))] = = μP (r) 2RP ⎠ P ⎝

(9c)

wP(r) = ΨCS( φP(r)) +

(6)

with the boundary condition q(r,0) = 1. Next, let us consider a ternary system, where in addition to the matrix homopolymer, A, and particles, P, there are also ligands, L. Each ligand has chain length N. The ligand volume fraction is denoted φL. We can easily modify the free energy contributions (eqs 2a−2d) as follows:

1 P

⎛

∫ dr′ Θ⎜⎝1 −

∫

⎛Q ⎞ ⎛Q ⎞ ϕ f1 = − ln⎜ P ⎟ − L ln⎜ L ⎟ − (1 − ϕP − ϕL) α ⎝V ⎠ κL ⎝ V ⎠ ⎛Q ⎞ ln⎜ A ⎟ ⎝ V ⎠ (7a) ϕP

ρP (r) =

ϕP V exp[−wP(r)] α QP

φA (r) = [1 − ϕP − ϕL]

V QA

∫0

(9d) 1

q(r, s)q(r, 1 − s) ds (9e)

1 f2 = V

∫ dr[(χAP P)φA(r)φP(r) + (χAL P)φA(r)φL(r)

φL(r) = ϕL

+ (χLP P)φL(r)φP(r) − ξ(r)(1 − φA (r) − φP(r))] (7b)

f3 =

1 V

1 V

(7d)

Here, κL = N/P, and the partition function QL is given by QL =

∫ qL(r, κL) dr

qL(r, s)qL†(r, s) ds

(9f) (9g)

Equations 9a−9g, together with definitions of functions ϕP(r) and φP(r) (eqs 3a and 3b), constitute full set of self-consistency equations that need to be solved iteratively. The solution algorithmadapted from the approach of Drolet and Fredrickson52 and the original TGMB paper36is as follows. First, the chemical potential and pressure fields are initialized. Next, using eqs 9d−9f, together with eqs 3a and 3b, density fields for all species are calculated. Then, new chemical potential fields are generated using the Drolet−Fredrickson prescription

∫ dr[−wA(r)φA(r) − wL(r)φL(r) − wP(r)ρP (r)]

∫ dr[ρP (r)ΨCS(φP(r))]

∫0

φA (r) + φL(r) + φP(r) = 1

(7c)

f4 =

V QL

κL

(8) 9800

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Article

(10)

where t is the iteration number (“effective time”), i = A, L, or P, and μi are the “new chemical potentials” calculated using eqs 9a−9c. Finally, we update the pressure field, ξ, using the following formula: ξ t + 1(r) = ξ t (r) + ε[φA (r) + φL(r) + φP(r) − 1]

(11)

The equations are solved on a rectangular grid, and periodic boundary conditions are applied on all three directions. To solve the modified diffusion equations for all the propagators, we utilize the pseudospectral approach of Rasmussen and Kalosakas.54 The simulations are carried out on a 64 × 64 × 64 grid, with mesh size dx = 0.25RgA. We use a coarse-graining in which one repeat unit corresponds to v0 = 1000 g/mol, and the corresponding statistical segment length a = 2.1 nm. The matrix polymer is assumed to have P = 50, corresponding to Mn = 50 kg/mol. The grafted chains have N = 5, 15, and 50 (5, 15, and 50 kg/mol, respectively). The matrix radius of gyration is estimated to be 6.06 nm, and the mesh size dx = 1.515 nm. Finally, we assume that ligand and matrix polymers are chemically identical, so χAL = 0.0, and set χAP = χLP = 0.308. Each simulation was run for 10 000 iterations. The convergence was monitored based on the incompressibility criterion, ⟨|ϕA(r) + ϕL(r) + ϕP(r) − 1|⟩ < 2 × 10−3, and the requirement that the free energy change per iteration was sufficiently small, |δF| < 10−5. The iteration parameters were as follows: λA = λL = 0.03, λP = 0.025, ε = 0.90. The results are described in the next section.

Figure 1. Unmodified nanoparticles in the homopolymer matrix. The red coloring corresponds to points where ϕp is close to 1, white corresponds to ϕp > 0.3, and the silver surfaces separate regions with ϕp > 0.05 (inside) from the particle-free zones.

Table 1. Compositions of Composite Mixturesa

3. RESULTS AND DISCUSSION To validate our method, we first considered the dispersion of unmodified nanoparticles in the homopolymer matrix. The nanoparticle radius was set at Rp = 5.5 nm. The matrix polymer is assumed to have the radius of gyration Rg = 6.06 nm, so the crucial Rp/Rg ratio equals 0.91. We performed simulations for the case of 50, 100, and 150 particles (volume fractions of 5.65%, 11.3%, and 16.95%, respectively) dispersed in the simulation box with dimensions 97 nm × 97 nm × 97 nm (64 lattice points in each direction, with mesh size dx = 0.25Rg = 1.51 nm). In Figure 1, the particle density map for the 5.65% nanocomposite is shown. One can see clear evidence of phase separation between particle-rich and particle-free regions. The particle-rich region is a spherical “droplet” where particles are packed together, with some matrix polymer filling the interstitial areas. This picture is similar to the model proposed heuristically by Mackay and co-workers.21 The inability of spherical particles to pack efficiently (beyond the face-centered cubical lattice packing fraction of 0.74) requires that there is substantial penetration of the matrix polymer in the particlerich region. On the other hand, the polymer-rich phase certainly does not require any nanoparticle presence, and thus ϕp is very close to zero outside the particle-rich cluster. Generally, one would expect that the particles arrange themselves into a body-centered or face-centered cubic lattice; however, achieving this would require a more accurate DFT than the one used in the current approach. Now, let us consider nanoparticles with grafted ligands. We vary the ligand length, total volume fraction of the ligands, and the particle volume fraction, as described in Table 1. As shown in several earlier publications,26,31,32 both experimental and computational, such variation of the grafted chain length and

run

N(ligand)

no. of particles

graft density (ligands/nm2)

VF(particles)

VF(ligand)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

5 5 5 5 5 5 5 5 5 15 15 15 15 15 15 15 15 15 50 50 50 50 50 50 50 50 50

50 100 150 50 100 150 50 100 150 50 100 150 50 100 150 50 100 150 50 100 150 50 100 150 50 100 150

0.1837 0.0863 0.0539 0.3674 0.1727 0.1078 0.5511 0.2590 0.1617 0.0612 0.0288 0.0180 0.1225 0.0576 0.0359 0.1837 0.0863 0.0539 0.0184 0.0086 0.0054 0.0367 0.0173 0.0108 0.0551 0.0259 0.0162

0.0565 0.1130 0.1695 0.0565 0.1130 0.1695 0.0565 0.1130 0.1695 0.0565 0.1130 0.1695 0.0565 0.1130 0.1695 0.0565 0.1130 0.1695 0.0565 0.1130 0.1695 0.0565 0.1130 0.1695 0.0565 0.1130 0.1695

0.0472 0.0444 0.0415 0.0944 0.0887 0.0831 0.1415 0.1331 0.1246 0.0472 0.0444 0.0415 0.0944 0.0887 0.0831 0.1415 0.1331 0.1246 0.0472 0.0444 0.0415 0.0944 0.0887 0.0831 0.1415 0.1331 0.1246

a Here, N(ligand) is the length of grafted chains, no. of particles is the number of spherical nanoparticles (Rp = 5.5 nm) per simulation box; graft density is grafting density (ligands per 1 nm2 of particle surface); VF(particles) is the nanoparticle volume fraction; VF(ligand) is the volume fraction of grafted chains.

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quasi-two-dimensional structures (vesicle-like structure for low particle loading; filled cylinder for intermediate particle loading; bundle of wires for higher particle loading). For the intermediate ligand length (N = 15, N/P = 0.3, Figure 2b), each particle has, on average, close to two nearest neighbors, so that the particles now form “strings”. Again, for low particle concentration, the strings “roll on themselves” to form vesiclelike structure, while for larger particle concentrations, they form bundles. Finally, at larger ligand length (N = 50, N/P = 1, Figure 2c), the particles are well-separated by their shells; at low particle loading, the particle/ligand mixed phase forms compact, slightly anisotropic droplets, while at higher particle loadings, weakly bundled wires are again observed. In general, the arrangement of particles into such anisotropic structures is in a good qualitative agreement with the simulations of Moll et al.55 and the scaling theory of Pryamitsyn et al.31 Next, consider the dependence of morphology on the thickness of grafted shell and the grafted chain length. As expected, adding ligands causes the particles to become more separated, though the structure of their arrangements often remains intact. In the case of short ligands and low particle loadings (Figure 2a), as the ligand volume fraction increases, the morphology changes from “raspberry-like” vesicle to “smooth” vesicle to homogeneous ligand-particle phase separated from the matrix phase. In the case of intermediate ligands (Figures 2b), there is a similar change from vesicle to a closed droplet. For the large ligands (Figures 2c), it is more difficult to observe morphological transitions, as the particles seem to be relatively well-separated even at lower ligand loading. One can analyze the morphology in more detail by looking at cross sections of the density maps. Thus, in Figure 3, we plot

the grafting density could cause the formation of various structures, both isotropic and anisotropic. Some of these structures are shown in Figures 2a−c, depicting color maps of

Figure 3. Ligand density mapscross sections perpendicular to the direction of the “wires”. For all three cases, particle volume fraction is 5.65%, and ligand volume fraction is 4.72%. The color scale ranges from 0 (dark blue) to 0.35 (dark red).

the density of the ligands in the XZ-plane. In all cases (short, intermediate, and long ligands), the particle-ligand phase forms compact aggregates; however, the ligand density maps are very different. In the case of short ligands, one could see some evidence of short-range crystalline order, where the particles and their shells are fairly tightly packed. In the case of intermediate ligands, there is evidence of anisotropythe particles form wires going in the Y-direction (normal to the plane), and the ligands are pushed into regions between the adjacent wires. Finally, in the case of long ligands, the shell is so much swollen by the matrix polymers that the interaction between the “wires” substantially weakens, and they are substantially further apart. Similar trends can be deduced by looking at the cross sections of the matrix density maps (Figure 4). In the case of the short ligands, the “wires” in a bundle are arranged in a well-defined hexagonal array; in the case of intermediate ligands, the “wires” still show partial “crystallinity”,

Figure 2. Particle density map diagrams: (a) ligand length N = 5; (b) N = 15; (c) N = 50. The colors are similar to Figure 1: red coloring corresponds to points where ϕp is close to 1, white corresponds to 0.3 < ϕp < 0.7, and blue corresponds to ϕp close to 0.

particle densities (red: particle-rich; blue: no particles; white: surfaces). The investigation of these phase diagrams allows one to discern several trends. First, consider the dependence of the morphology on the particle volume fraction at a given ligand volume fraction (e.g., 10 vol % ligand relative to overall polymer content). In general, we can see that the local arrangement of particles is not changing very much as more particles are added. For the case of short ligand (N = 5, N/P = 0.1, Figure 2a), each particle has, on average, three or more nearest neighbors; as a result, we see 9802

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(Figure 5c), the particles are well-separated, and the correlation between them is substantially weakened. Let us now consider the overall composite morphologies. To better visualize them, we replicate our simulation box in all three dimensions and plot the resulting 128 × 128 × 128 boxes (this is permitted because of periodic boundary conditions). The results are shown in Figure 6. For the case of short ligands Figure 4. Matrix polymer density mapscross sections perpendicular to the direction of the “wires”. For all three cases, particle volume fraction is 11.3%, and ligand volume fraction is 9.43%. The color scale ranges from 0 (dark blue) to 1 (dark red).

and in the case of long ligands, the “wires” are not ordered at all. We can schematically depict the role of grafted chains in determining nanocomposite morphology as shown in Figure 5.

Figure 5. Schematic sketch describing how the addition of grafted chains modifies the morphology of a nanocomposite. The figures on the left correspond to particles without ligand at the same particle volume fraction. (a) Short ligands. (b) Intermediate ligands. (c) Long ligands. Here, red circles represent the particles, orange, yellow, and white regions represent the mixed shells (ligand and matrix polymers combined together), and blue represents the matrix.

Figure 6. Overall morphology phase diagrams. (a) Short ligand (N = 5). (b) Intermediate ligands (N = 15). (c) Long ligands (N = 50). See text for more details.

(Figure 6a), particle aggregate into spherical domains at low loadings and then form sheets as the particle loading increases. At intermediate particle loading and high ligand volume fraction, the sheets roll into cylinders. For the case of intermediate ligands (Figure 6b), spherical domains are observed at low particle loading, and sheets and cylinders at intermediate particle loading; at high particle loading, the sheets become perforated and connected into a continuous network. Finally, for the case of large ligands (Figure 6c), the separate sheets and cylinders disappear, and we observe a direct

In the absence of grafted chains, particles arrange into strings, and a string then collapses onto itself, forming a “globule”. Because of the incompressibility, there has to be a substantial amount of matrix polymer around the particles. The grafting of short-chain ligands onto the particles actually helps expel the matrix polymer from the particle-rich area, potentially improving the local ordering (Figure 5a). As the ligand chain length is increased, the grafted chains become more flexible and can shift around, promoting and stabilizing anisotropic arrangements (Figure 5b). Finally, for long grafted chains 9803

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(4) Ginzburg, V., Nanoparticle/polymer blends: Theory and modeling. In Encyclopedia of Polymer Blends; Isayev, A. I., Ed.; WileyVCH: Weinheim, 2011; Vol. 1, pp 233−267. (5) Winey, K. I.; Vaia, R. A. Polymer nanocomposites. MRS Bull. 2007, 32 (4), 314−319. (6) Rozenberg, B. A.; Tenne, R. Polymer-assisted fabrication of nanoparticles and nanocomposites. Prog. Polym. Sci. 2008, 33 (1), 40− 112. (7) Vaia, R. A.; Giannelis, E. P. Lattice model of polymer melt intercalation in organically-modified layered silicates. Macromolecules 1997, 30 (25), 7990−7999. (8) Balazs, A. C.; Singh, C.; Zhulina, E.; Lyatskaya, Y. Modeling the phase behavior of polymer/clay nanocomposites. Acc. Chem. Res. 1999, 32 (8), 651−657. (9) Balazs, A. C.; Bicerano, J.; Ginzburg, V. V., Polyolefin/clay nanocomposites: Theory and simulation. In Polyolefin Composites; John Wiley & Sons, Inc.: New York, 2007; pp 415−448. (10) Jog, P.; Ginzburg, V.; Srivastava, R.; Weinhold, J.; Jain, S.; Chapman, W. G. Application of mesoscale field-based models to predict stability of particle dispersions in polymer melts. In Advances in Chemical Engineering; West, D. H., Ed.; Academic Press/Elsevier: Amsterdam, 2010; Vol. 39, pp 131−164. (11) Gast, A. P.; Leibler, L. Interactions of sterically stabilized particles suspended in a polymer solution. Macromolecules 1986, 19 (3), 686−691. (12) Ferreira, P. G.; Ajdari, A.; Leibler, L. Scaling law for entropic effects at interfaces between grafted layers and polymer melts. Macromolecules 1998, 31 (12), 3994−4003. (13) Borukhov, I.; Leibler, L. Stabilizing grafted colloids in a polymer melt: Favorable enthalpic interactions. Phys. Rev. E 2000, 62 (1), R41− R44. (14) Borukhov, I.; Leibler, L. Enthalpic stabilization of brush-coated particles in a polymer melt. Macromolecules 2002, 35 (13), 5171− 5182. (15) Hall, L. M.; Jayaraman, A.; Schweizer, K. S. Molecular theories of polymer nanocomposites. Curr. Opin. Solid State Mater. Sci. 2010, 14 (2), 38−48. (16) Jayaraman, A.; Schweizer, K. S. Effective interactions and selfassembly of hybrid polymer grafted nanoparticles in a homopolymer matrix. Macromolecules 2009, 42 (21), 8423−8434. (17) Jayaraman, A.; Schweizer, K. S. Liquid state theory of the structure and phase behaviour of polymer-tethered nanoparticles in dense suspensions, melts and nanocomposites. Mol. Simul. 2009, 35 (10−11), 835−848. (18) Jayaraman, A.; Schweizer, K. S. Effective interactions, structure, and phase behavior of lightly tethered nanoparticles in polymer melts. Macromolecules 2008, 41 (23), 9430−9438. (19) Jayaraman, A.; Nair, N. Integrating PRISM theory and Monte Carlo simulation to study polymer-functionalised particles and polymer nanocomposites. Mol. Simul. 2012, 38 (8−9), 751−761. (20) Khounlavong, L.; Pryamitsyn, V.; Ganesan, V., Many-body interactions and coarse-grained simulations of structure of nanoparticle-polymer melt mixtures. J. Chem. Phys. 2010, 133, (14). (21) Mackay, M. E.; Tuteja, A.; Duxbury, P. M.; Hawker, C. J.; Van Horn, B.; Guan, Z. B.; Chen, G. H.; Krishnan, R. S. General strategies for nanoparticle dispersion. Science 2006, 311 (5768), 1740−1743. (22) Tuteja, A.; Duxbury, P. M.; Mackay, M. E. Multifunctional nanocomposites with reduced viscosity. Macromolecules 2007, 40 (26), 9427−9434. (23) McGarrity, E. S.; Frischknecht, A. L.; Mackay, M. E., Phase behavior of polymer/nanoparticle blends near a substrate. J. Chem. Phys. 2008, 128, (15). (24) McGarrity, E. S.; Frischknecht, A. L.; Frink, L. J. D.; Mackay, M. E. Surface-induced first-order transition in athermal polymer-nanoparticle blends. Phys. Rev. Lett. 2007, 99 (23), 238302. (25) McGarrity, E. S.; Duxbury, P. M.; Mackay, M. E.; Frischknecht, A. L. Calculation of entropic terms governing nanoparticle selfassembly in polymer films. Macromolecules 2008, 41 (15), 5952−5954.

transition from spherical aggregates to a continuous network. This is consistent with the idea that the binding between the strings becomes less pronounced as the ligand chain length increases; thus, the strings can disperse more uniformly and associate into a percolated network instead of closed objects. We should note that the proposed model is mainly applicable to relatively high particle volume fractions where the effective particle−particle attraction is strong enough to cause formation of various aggregated shapes. At very low particle volume fractions and moderate-to-high density of long ligands, where particles are uniformly dispersed,26 the mean-field way of “grafting” the ligands to particles in the current model is no longer valid. However, under those conditions, consideration of pairwise interaction between adjacent particles should be sufficient to understand the transition from dispersed to agglomerated state.10,12,14,56 The above analysis was limited to a specific particle size and relatively narrow range of ligand lengths and concentrations. In the future, we expect to investigate the morphologies of other systems, with smaller or larger particles, as well as different ligand lengths and architectures. Additionally, we are planning to use the results of these simulations to investigate how the addition of nanoparticles influences physical and mechanical properties of nanocomposites by feeding the density profiles obtained here into a finite element (FE) type model.

4. CONCLUSIONS We investigated the morphology of polymer/nanoparticle composites where the nanoparticles had chemically grafted oligomer chains on their surfaces. By adapting the Thompson− Ginzburg−Matsen−Balazs (TGMB) mesoscale SCF−DFT model to explicitly account for grafted chain contributions, we generated several phase diagrams relating the morphology of the mixture to the nanoparticle volume fraction, ligand volume fraction, and ligand chain length. Simulations predicted the formation of various anisotropic and complex structures, from particle/ligand droplets and vesicles to cylinders and bundles of wires to sheets and continuous networks. The results obtained here are qualitatively similar to earlier particlebased simulation studies as well as experimental results. The new model is thus expected to become a useful tool to design new composites with specific targeted properties.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author thanks Drs. Jeffrey Weinhold and George Jacob (The Dow Chemical Company) for helpful discussions and comments.

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REFERENCES

(1) Kickelbick, G. Concepts for the incorporation of inorganic building blocks into organic polymers on a nanoscale. Prog. Polym. Sci. 2003, 28 (1), 83−114. (2) Paul, D. R.; Robeson, L. M. Polymer nanotechnology: Nanocomposites. Polymer 2008, 49 (15), 3187−3204. (3) Supova, M.; Martynkova, G. S.; Barabaszova, K. Effect of nanofillers dispersion in polymer matrices: A review. Sci. Adv. Mater. 2011, 3 (1), 1−25. 9804

dx.doi.org/10.1021/ma402210v | Macromolecules 2013, 46, 9798−9805

Macromolecules

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distribution, and adsorption isotherms. J. Phys. Chem. 1979, 83, 1619− 1635. (49) Scheutjens, J. M.; Fleer, G. J. Statistical theory of the adsorption of interacting chain molecules. 2. Train, loop, and tail size distribution. J. Phys. Chem. 1980, 84, 178−190. (50) Fleer, G. J. Polymers at Interfaces, 1st ed.; Chapman & Hall: London, 1993; p xviii, 502 pp. (51) Matsen, M. W.; Schick, M. Stable and unstable phases of a diblock copolymer melt. Phys. Rev. Lett. 1994, 72 (16), 2660−2663. (52) Drolet, F.; Fredrickson, G. H. Combinatorial screening of complex block copolymer assembly with self-consistent field theory. Phys. Rev. Lett. 1999, 83 (21), 4317−4320. (53) Fredrickson, G. H.; Ganesan, V.; Drolet, F. Field-theoretic computer simulation methods for polymers and complex fluids. Macromolecules 2002, 35 (1), 16−39. (54) Rasmussen, K. O.; Kalosakas, G. Improved numerical algorithm for exploring block copolymer mesophases. J. Polym. Sci., Part B: Polym. Phys. 2002, 40 (16), 1777−1783. (55) Moll, J. F.; Akcora, P.; Rungta, A.; Gong, S. S.; Colby, R. H.; Benicewicz, B. C.; Kumar, S. K. Mechanical reinforcement in polymer melts filled with polymer grafted nanoparticles. Macromolecules 2011, 44 (18), 7473−7477. (56) Xu, J. J.; Qiu, F.; Zhang, H. D.; Yang, Y. L. Morphology and interactions of polymer brush-coated spheres in a polymer matrix. J. Polym. Sci., Part B: Polym. Phys. 2006, 44 (19), 2811−2820.

(26) Akcora, P.; Liu, H.; Kumar, S. K.; Moll, J.; Li, Y.; Benicewicz, B. C.; Schadler, L. S.; Acehan, D.; Panagiotopoulos, A. Z.; Pryamitsyn, V.; Ganesan, V.; Ilavsky, J.; Thiyagarajan, P.; Colby, R. H.; Douglas, J. F. Anisotropic self-assembly of spherical polymer-grafted nanoparticles. Nat. Mater. 2009, 8 (4), 354−U121. (27) Lan, Q.; Francis, L. F.; Bates, F. S. Silica nanoparticle dispersions in homopolymer versus block copolymer. J. Polym. Sci., Part B: Polym. Phys. 2007, 45 (16), 2284−2299. (28) Wang, X.; Foltz, V. J.; Rackaitis, M.; Böhm, G. G. A. Dispersing hairy nanoparticles in polymer melts. Polymer 2008, 49 (26), 5683− 5691. (29) Chevigny, C.; Dalmas, F.; Di Cola, E.; Gigmes, D.; Bertin, D.; Boué, F. O.; Jestin, J. Polymer-grafted-nanoparticles nanocomposites: dispersion, grafted chain conformation, and rheological behavior. Macromolecules 2010, 44 (1), 122−133. (30) Sunday, D.; Ilavsky, J.; Green, D. L.; Phase, A. Diagram for polymer-grafted nanoparticles in homopolymer matrices. Macromolecules 2012, 45 (9), 4007−4011. (31) Pryamtisyn, V.; Ganesan, V.; Panagiotopoulos, A. Z.; Liu, H. J.; Kumar, S. K., Modeling the anisotropic self-assembly of spherical polymer-grafted nanoparticles. J. Chem. Phys. 2009, 131, (22). (32) Kalb, J.; Dukes, D.; Kumar, S. K.; Hoy, R. S.; Grest, G. S. End grafted polymer nanoparticles in a polymeric matrix: Effect of coverage and curvature. Soft Matter 2011, 7 (4), 1418−1425. (33) Meng, D.; Kumar, S. K.; Lane, J. M. D.; Grest, G. S. Effective interactions between grafted nanoparticles in a polymer matrix. Soft Matter 2012, 8 (18), 5002−5010. (34) Sides, S. W.; Kim, B. J.; Kramer, E. J.; Fredrickson, G. H. Hybrid particle-field simulations of polymer nanocomposites. Phys. Rev. Lett. 2006, 96 (25), 250601. (35) Raman, V.; Bose, A.; Olsen, B. D.; Hatton, T. A. Long-range ordering of symmetric block copolymer domains by chaining of superparamagnetic nanoparticles in external magnetic fields. Macromolecules 2012, 45, 9373−9382. (36) Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.; Balazs, A. C. Predicting the mesophases of copolymer-nanoparticle composites. Science 2001, 292 (5526), 2469−2472. (37) Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.; Balazs, A. C. Block copolymer-directed assembly of nanoparticles: Forming mesoscopically ordered hybrid materials. Macromolecules 2002, 35 (3), 1060−1071. (38) Lee, J. Y.; Shou, Z.; Balazs, A. C. Modeling the self-assembly of copolymer-nanoparticle mixtures confined between solid surfaces. Phys. Rev. Lett. 2003, 91 (13), 136103. (39) Lee, J. Y.; Shou, Z. Y.; Balazs, A. C. Predicting the morphologies of confined copolymer/nanoparticle mixtures. Macromolecules 2003, 36 (20), 7730−7739. (40) Lee, J. Y.; Balazs, A. C.; Thompson, R. B.; Hill, R. M. Selfassembly of amphiphilic nanoparticle-coil “tadpole” macromolecules. Macromolecules 2004, 37 (10), 3536−3539. (41) Zhu, X. M.; Wang, L. Q.; Lin, J. P.; Zhang, L. S. Ordered nanostructures self-assembled from block copolymer tethered nanoparticles. ACS Nano 2010, 4 (9), 4979−4988. (42) Flory, P. J. Thermodynamics of high polymer solutions. J. Chem. Phys. 1941, 9 (8), 660−660. (43) Huggins, M. L. Solutions of long chain compounds. J. Chem. Phys. 1941, 9 (5), 440−440. (44) Tarazona, P.; Evans, R. A simple density functional theory for inhomogeneous liquids - Wetting by gas at a solid liquid interface. Mol. Phys. 1984, 52 (4), 847−857. (45) Tarazona, P. Free-energy density functional for hard-spheres. Phys. Rev. A 1985, 31 (4), 2672−2679. (46) Tarazona, P. A density functional theory of melting. Mol. Phys. 1984, 52 (1), 81−96. (47) Carnahan, N. F.; Starling, K. E. Equation of state for nonattracting rigid spheres. J. Chem. Phys. 1969, 51 (2), 635−636. (48) Scheutjens, J. M.; Fleer, G. J. Statistical theory of the adsorption of interacting chain molecules. 1. Partition function, segment density 9805

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