Dispersion of Polymer-Grafted Nanorods in Homopolymer Films

Nadia M. KrookJamie FordManuel MaréchalPatrice RannouJeffrey S. MethChristopher B. MurrayRussell J. Composto. ACS Macro Letters 2018 7 (12), 1400- ...
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Dispersion of Polymer-Grafted Nanorods in Homopolymer Films: Theory and Experiment Amalie L. Frischknecht,*,† Michael J. A. Hore,‡,∥ Jamie Ford,§ and Russell J. Composto‡ †

Center for Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States Department of Materials Science and Engineering and the Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States, and § Penn Regional Nanotechnology Facility, University of Pennsylvania, Philadelphia, Pennsylvania, 19104, United States ‡

S Supporting Information *

ABSTRACT: An understanding of the dispersion of nanoparticles into polymer melts is needed in order to control material properties of polymer nanocomposites. Here we study the dispersion of polymer-grafted nanorods in homopolymer melts of the same chemistry, using both experiment and theory. The theoretical calculations are performed over the range of experimental system parameters. Polymer-grafted gold nanorods (Au NRs) were found to be dispersed when the matrix chain lengths were small relative to the brush chain lengths, and aggregated at higher matrix chain lengths. Both classical density functional theory (DFT) and self-consistent field theory (SCFT) are used to calculate the structure of a polymer brush around an isolated NR in a polymer melt. Both theories predict a gradual transition from a “wet” to a “dry” brush as the grafting density, the NR radius, and/or the ratio of matrix to brush chain lengths is increased. DFT calculations of the interaction free energy between two NRs find an attractive well at intermediate NR separations, with a repulsive barrier at closer NR separations. The strength of the attraction increases as the brushes become more dry. Including the van der Waals attractions between the NRs gives an estimate of their total interaction free energy, which can be used to predict at which values of the system parameters the NRs are dispersed or aggregated. A dispersion map shows good agreement between DFT calculations and experimental observations of dispersed and aggregated nanorods.

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INTRODUCTION In order to exploit the unique properties of nanoparticles in devices, they must be incorporated into a host material in a controllable manner. Polymers are an attractive choice for the matrix material due to their processability. The controlled dispersion of nanoparticles into polymers has been extensively investigated in recent years.1,2 Variables that affect dispersion include nanoparticle size, shape, interactions with the polymer and with other particles, and surface functionalization. One strategy to both prevent the nanoparticles from aggregating together (i.e., disperse) and to control the spacing between neighboring nanoparticles is to coat them with a polymer brush. In principle, the polymer brush prevents the particles from approaching too closely and hence can overcome the attractive van der Waals and depletion interactions between particles. The dispersion or aggregation of polymer brush-coated nanoparticles in a homopolymer matrix depends strongly on the interactions between the brush and the matrix. When the matrix chains wet the brush, there is a repulsive force between the two brushes which promotes dispersion of the particles. This can be achieved by employing polymers with strong attractive enthalpic interactions.3 When the brush and matrix chains are chemically identical, the interactions are dominated by entropic effects. Both the matrix and brush chains lose configurational entropy when the matrix chains penetrate the © 2013 American Chemical Society

brush. This loss can overcome the mixing entropy and lead to the exclusion of the matrix chains from the brush, which becomes “dry.” This phemonenon is known as autophobic dewetting.4 In the dry brush case, there is a positive interfacial tension between the brush and the matrix polymers, which leads to an attraction between polymer brush-coated particles in a chemically identical polymer melt. It is attractive to characterize the wet-to-dry transition, and hence the dispersed-to-aggregated state of brush-coated particles, by a simple relationship between the degrees of polymerization N of the brush and P of the matrix chains. Early experimental work on autophobic dewetting of brushes attached to planar surfaces found an evolution from a wet to a dry brush as P increased above ≈5N.5 Initial self-consistent field theory (SCFT) calculations on planar (flat) brushes predicted the wetting/dewetting boundary to be at P ≈ N.6 However, the grafing density σ also affects the location of the boundary, and additional SCFT calculations suggested the transition to a dry brush occurs for σ > α−2, where α = P/N.7 Furthermore, this picture is overly simplified; the transition between wet and dry brushes is a gradual one and in fact cannot Received: November 29, 2012 Revised: February 8, 2013 Published: March 26, 2013 2856

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dewetting transition is indeed shifted to larger values of α for spherical nanoparticles as compared to flat surfaces. The case of cylindrical curvature has been less studied. Gold nanorods (NRs) are of interest due to their optical properties, since controlled dispersion and spacing of Au nanorods would allow tuning of their surface plasmon resonances. Very recently, we reported that PS functionalized gold NRs in PS aggregate for P > 2N.27 Preliminary results showed the same behavior for PEO-functionalized Au NRs in PEO homopolymer. The only previous theoretical work on brush-coated NRs in a homopolymer melt were classical density functional theory (DFT) calculations of the interaction free energy between two parallel NRs as a function of the distance between them.28 As in the case of flat brushes, an attractive well was found for all cases but was very small for low values of σ and α. In this paper, we revisit these previous DFT calculations, focusing on values of the grafting density σ*, the nanorod radius Rrod, and the ratio of brush to matrix molecular weights P/N = α relevant to our recent experiments on PS-Au NRs in PS.27 We also present new data for the dispersion of Au NRs functionalized with poly(ethylene oxide) (PEO) in PEO. In particular, we are interested in defining the relevant variables that are most important for determining dispersion or aggregation of the NRs. We first examine the nature of the wet-to-dry transition for polymer brushes on an isolated NR in the homopolymer matrix, as a function of σ*, Rrod, and α. As a check on the robustness of the results, the brush profiles from DFT are compared with those calculated from SCFT for the same systems. We find excellent qualitative agreement between the two theories for the brush properties. Both theories predict a gradual wet to dry brush transition, which depends on all three parameters (σ*, Rrod, and α). We then use the DFT to calculate the interaction energies between two NRs as a function of the distance between them. In most cases the polymer-mediated free energy between the NRs has an attractive well, which deepens with increasing σ*, Rrod, and α. The attractions can be sufficiently strong to cause NR aggregation, particularly when the van der Waals attraction between the NRs is included. A dispersion map shows that our experimental data and DFT results are in good agreement. These results provide experimentalists with guidelines for selecting materials parameters, σ*, Rrod, and α, that produce dispersed or aggregated NRs in a polymer matrix.

be characterized by a simple criterion. This was shown for the case of flat brushes by Matsen and Gardiner,8 who performed SCFT calculations of the interaction energy between two flat brushes in a homopolymer melt. They demonstrated that there is always a positive interfacial tension γ between the brush and the melt. However, γ and hence the attraction between the surfaces is very weak for low matrix molecular weights and/or low grafting densities. The boundary between wet and dry brushes, as determined by the interpenetration of the matrix chains into the brush, scales approximately as σ ∼ α−0.5 for α < 1, and slowly saturates to become independent of α at larger values of α. Alternatively, the transition between wetting and dewetting based on the surface tension scales roughly as σ ∼ α−0.7 for α < 3, and then again saturates for larger α. Thus, for reduced grafting densities on the order of σ ∼ 1, these SCFT results predict dewetting of the matrix chains from brushes on a flat surface for roughly P > N1.4 at moderate values of α. The gradual nature of the brush dewetting was also found by Maas et al.9 and confirmed experimentally.10 For brush-coated nanoparticles, the size of the particle is typically on the same order as the size of the polymers and so curvature is important. The brush chains have more volume to explore around a curved particle, which leads to less extended brushes compared to a flat brush at the same grafting density. One might expect then that dewetting of the brush and hence particle aggregation should occur for larger values of σ and α on curved surfaces as compared to flat ones. This transition has yet to be systematically explored as a function of all the relevant variables. Most experimental work to date has been on spherical silica nanoparticles grafted with polystyrene (PS) in PS. For this system, Bansal et al.11 found that particles of radius R/Rg = 1.6 (where Rg is the radius of gyration of the grafted chains) and grafting density 0.27 chains/nm2 form good dispersions for chain length ratios up to α = 2.3. Chevigny et al.12 found that particles with R/Rg ∼ 2.2−3.6 and grafting densities of 0.15− 0.19 chains/nm2 aggregate for α > 4. Very recently, Sunday et al.13 performed an extensive study of PS-grafted nanoparticles in PS at fixed grafting chain length and nanoparticle size of R/ Rg = 1.3, for varying grafting densities and α. They found aggregates for σ > α−0.7, with aggregation occurring for α > 4.3 at σ = 0.27 chains/nm2 and falling to α > 1.3 at higher grafting densities of σ = 0.70 chains/nm2. Kim and Green14 found evidence that PS functionalized spherical gold nanoparticles in PS aggregate for α > 3, although that study was in a thin film rather than bulk. There have been many previous simulation and theoretical studies of polymer-grafted spherical nanoparticles in polymer matrices (see, e.g., refs 15−26.). A few have focused on the regime of interest here, namely nanoparticles grafted with homopolymers at high enough density to be in the polymer brush regime, immersed in a chemically identical homopolymer melt.22−26 In particular, Jayaraman and Schweizer24 used PRISM theory to calculate the potential of mean force (PMF) between two brush-grafted particles and found that for small particles at moderate grafting density, the PMF is repulsive for α = 1 and has only a weak attraction for α = 6.25. Using SCFT, Xu et al.22 calculated the interactions between nanoparticles for α = 1, and found an attraction for sufficiently large particle radius and grafting density. Trombly and Ganesan25 performed more extensive SCFT calculations for spherical nanoparticles as a function of grafting density, nanoparticle radius, and α. They found that the wetting-

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METHODS

Classical Density Functional Theory. The DFT calculations are performed on essentially the same model system as previously described.28 Both the brush and matrix polymers are modeled as freely jointed chains of spherical interaction sites of diameter a. The NRs have radius Rrod and are immersed in a homopolymer melt with chains of length P. The brush chains of length N have a sticky site at one end of the chain which is attracted to the surface of the NRs. This causes the brush chains to adsorb onto the rods and form a polymer brush. The NRs are treated as external surfaces, which serve to exclude the polymers. There is no direct interaction between the NRs. The interaction between the NRs and the polymers is through a LennardJones interaction of the form

Vα(r ) =

12 ⎡ 12 ⎛ a ⎞6 ⎤ ⎛ a ⎞6 ⎤ 4ε ⎡⎛⎜ a ⎞⎟ 4ε ⎢⎛ a ⎞ ⎢ −⎜ ⎟⎥− ⎜ ⎟ − ⎜ ⎟ ⎥, ⎝ r ⎠ ⎦ kBT ⎢⎝ rcα ⎠ kBT ⎣⎝ r ⎠ ⎝ rcα ⎠ ⎥⎦ ⎣

r < rcα Vα(r ) = 0, r > rcα (1) 2857

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Here r is the radial distance between a polymer segment α and the surface of the NR, kB is Boltzmann’s constant and T is temperature. The cutoff distance is given by rcα = 21/6a for all monomers except the sticky end, which has a cutoff of rce = 2a. This results in all the monomers having a purely repulsive interaction with the NRs except for the sticky end, which has an attractive well of depth ε/kBT in its interaction with the NRs. The interaction between monomers is also given by a purely repulsive Lennard-Jones interaction of the same form, with a cutoff of 21/6a. The DFT is formulated in an open ensemble, so that in the bulk regions far from the NRs, the system consists of a melt blend of brush and matrix polymers at a constant bulk density ρα. These bulk densities set the values of the chemical potentials. In the calculations described here, the brush chain monomers had a bulk density of ρba3 = 0.01 and the matrix chain monomers a bulk density of ρma3 = 0.85, for a total bulk monomer density of ρ0a3 = 0.86. This is higher than the density of ρ0a3 = 0.8 used in our previous calculations28 and is more likely to be representative of a polymer melt. We note that the DFT employed here, unlike typical SCFTs, is a compressible theory. Previous DFT calculations on flat brushes showed that as the matrix polymer density is increased, the brush becomes more compressed, and the attractive minimum occurs at smaller surface separations.29 For the NRs, the attractive well depth increases with increasing ρ0. Calculations are performed in a two-dimensional box on a Cartesian grid with a mesh size of 0.2a. A larger mesh size was used here than in our previous calculations28 due to the larger system sizes considered. One half of a NR is placed at the center of the x-axis, at x = xc and y = 0. Reflective boundary conditions are applied in all directions, so that it is only necessary to calculate one-quarter of the system. This also results in infinitely long NRs which are parallel to each other. Initial calculations are performed in a sufficiently large box that the two NRs (the one placed at xc and its reflected image) do not influence each other. We use the Chandler−McCoy−Singer (CMS) version of DFT.30−32 Details of the theory, along with the numerical methods used to solve the theory, have been enumerated elsewhere.33−35 Briefly, the basic quantities in the theory are the inhomogeneous site density profiles ρα(r), where ρα(r) is the density of site type α at r. The basic idea in CMS-DFT is to replace the interacting system of interest with a reference system of ideal, noninteracting chains, in a medium induced potential Uα(r) which captures the effects of the site interactions. The grand potential free energy Ω of the inhomogeneous system of interest is measured relative to the free energy Ωb of the bulk, homogeneous polymer blend which serves as the reservoir for the inhomogeneous system. The grand potential free energy difference ΔΩ = Ω − Ωb is expanded in a Taylor series about the noninteracting reference system, truncated at second order. A functional minimization of ΔΩ with respect to the ρα(r) leads to the DFT equations to be solved.28 These equations are implemented in the fluids−DFT code Tramonto36 and are solved using a Newton’s method as described in detail elsewhere.33−35 Calculations for this work were performed on the Sandia redsky cluster, utilizing between four and 10 8-processor nodes. An absolute convergence tolerance of 10−6 was used for the solution of the DFT equations. Once the DFT equations have been solved, the equilibrium grand potential surface free energy Ωs = ΔΩ can be calculated from the converged density profiles. All lengths are measured in units of the radius of gyration Rg = a(N/6)1/2 of the brush chains, and energies are measured in units of kBT. As discussed in the Introduction, there are three critical system parameters: the grafting density σ, the NR radius Rrod/Rg, and the ratio of brush to matrix chain lengths α = P/N. The grafting density is not an input to the DFT but is instead calculated from the converged density profiles. The number of brush chains adsorbed to one NR can be found from the excess adsorption Γe of the sticky end site: Γe =

∫ dr (ρe (r) − ρbe )

per chain, this quantity measures the number of adsorbed chains per unit length of the NR. The number of adsorbed chains per area is then approximately σ = Γe/πD, where here we assume that the NRs are smooth, while in our numerical calculations they are cylindrical only to within the limits of the mesh size. In the brush regime, the number of adsorbed chains σ scales approximately linearly with the sticky end energy ε.28 In this paper, we will refer to σ as the grafting density, although in the DFT the chains are end-adsorbed rather than truly grafted to the NRs. For the properties of the brush around a single, isolated NR, this distinction is not important. As two NRs are brought together in close proximity, a small fraction of the adsorbed chains leave the NR surface so the effective grafting density decreases slightly, as detailed previously.28 Additionally, because they are physisorbed, the brush chains are mobile on the NR surface, which leads to a somewhat asymmetric distribution of the sticky groups on the NR surface as the two NRs are brought close together.28 Because some of the brush chains can escape the region between the two approaching NRs to lower their free energy, either by desorption or by moving to the other side of the nanorod, the interaction free energies predicted by the DFT should be somewhat smaller in magnitude than if the chains were covalently bonded to specific surface sites on the NR. These effects are not large, as there is always a significant brush chain density between the two NRs at all NR separations. We characterize the grafting density by a dimensionless, reduced grafting density σ* ≡

6 σN1/2 aρ0

(3)

where ρ0 is the bulk monomer density. We note that obtaining a specific value of σ* (which depends on ε) requires fitting a curve to σ*(ε) for a variety of ε values, and then targetting that value of σ* in each DFT calculation. Finally, all calculations presented here are for systems with high enough grafting density to be in the polymer brush regime; the average distance between adsorbed (grafted) polymers is smaller than the end-to-end distance of the brush chains. Self-Consistent Field Theory. Self-consistent field theory (SCFT) calculations of the polymer brush and matrix density profiles as a function of σ*, Rrod, and α were performed to compare with those obtained from DFT. With SCFT, polymers are modeled as Gaussian chains described by continuous space curves R(s), where s denotes the position along the contour. Here, the density profiles for a chemically similar polymer brush and matrix are determined for a single NR with radius Rrod in a system with volume V = 40 × 40Rg2, where Rg = a(N/ 6)1/2. The polymer brush has a reduced grafting density σ* given by eq 3. Unlike with DFT, the polymer brush chains in the SCFT calculations are grafted to the NR surface and cannot desorb, although the chains are mobile on the NR surface as in the DFT. For the single rod calculations performed here with SCFT, this mobility is irrelevant as the brush profile is radially symmetric. Our numerical approach is similar to that used in our recent publication37 except that because the brush and matrix are chemically similar, the partition function is given as

A=

nb

nm

i=0

j=0

∫ ∏ +R i ,b(s) ∫ ∏ +R j ,m(s) exp(− βU0[R i , b(s)] − βU0[R j , m(s)])δ[1 − ϕb̂ − ϕm̂ ]

(4)

where ϕ̂ b and ϕ̂ m are the local microscopic densities of the brush and matrix, respectively. U0 is an entropic harmonic potential between adjacent monomers and is defined as

U0[R i , k(s)] = (2)

Here ρbe is the bulk density of sticky sites and the integral is over the region adjacent to one of the NRs. Since there is only one sticky site

3 2a 2

∫0

Nk

ds

∂R i , k(s) ∂s

2

(5)

where Nk is the length of a chain of species k (i.e., brush or matrix) and a is the statistical step length. 2858

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The partition function in eq 4 is transformed using a delta functional transformation into one described by a single auxiliary field ω+, A=

∫ +ω+ exp(−/[ω+(r)])

The implications for dispersion or aggregation of the NRs are discussed and compared to our experimental observations. Experiment. Polystyrene Nanocomposites. Recently27 the dispersion of PS(N)-Au NRs in PS(P) films was investigated as a function of the ratio of matrix chain length P to brush chain length N, α = P/N, using a combination of TEM, UV−visible spectroscopy, and Monte Carlo simulations. The dimensionless parameters characterizing these systems are shown in Table 1.

(6)

where the effective Hamiltonian is

/ 1 =− kBTC V

∫ dr {iω+(r)} − ϕb ln Q b[iω+] − (1 − ϕb)α−1

ln Q m[iω+]

Table 1. Parameters for PS-Au(N):PS(P) Films Studied in Reference 17a

(7)

The constant C = ρ0V/N. Qb and Qm are the single chain partition functions of the brush and matrix, respectively. The details of the numerical method employed to calculate the polymer brush density profiles for a polymer-grafted NR can be found in a recent publication by Hore and Composto.37 The SCFT calculations were performed on a six core Linux workstation. A typical run required approximately 3 h to complete. Experiment. Polymer thin films consisting of either poly(ethylene oxide)-functionalized Au nanorods (PEO-Au NRs) in a PEO matrix or polystyrene-functionalized Au nanorods (PS-Au NRs) in a PS matrix were prepared on silicon substrates and imaged on an FEI Quanta ESEM at an operating voltage of 10 kV and a JEOL JEM 2010 TEM at 200 kV. The Au NRs were synthesized using a seed-mediated growth method in the presence of cetyltrimethylammonium bromide (CTAB).38 A more detailed discussion of our standard synthesis procedure can be found in recent publications.3,27 Chemicals were obtained from Sigma-Aldrich and used as received. During a typical synthesis process, a seed solution was prepared with 7.5 mL of a 0.1 M CTAB solution, 250 μL of a 0.01 M HAuCl4 solution, and 600 μL of an ice cold 0.01 M NaBH4 solution. The solutions were combined and the resulting seed solution color changed from a deep yellow to light brown, signifying the formation of 2 to 3 nm Au seed crystals. The seed solution was incubated in a water bath at 30 °C for at least 2 h prior to use to ensure that no excess reducing agent was present. Au NRs were synthesized by combining 0.420 mL of seed solution with a growth solution containing 40 mL of 0.1 M CTAB, 1.7 mL of 0.01 M HAuCl4, 0.250 mL of a 0.01 M AgNO3, and 0.270 mL of 0.1 M Lascorbic acid. The clear growth solutions turned purple-red in color after 5−10 min, signifying the growth of NRs. The resulting nanorods were approximately 12 × 42 nm (diameter × length) in size. Functionalization of the NRs with thiol-terminated PEO and PS is detailed in our recent publications.3,27 Thiol-terminated PEO and PS were obtained from Polymer Source, Inc. For these studies, thiolterminated PEO with molecular weights of Mn = 5000 and 10 000 g/ mol and polydispersities Mw/Mn = 1.08 were used. Thiol-terminated PS with Mn = 5000, 11 500, and 20 000 g/mol and Mw/Mn = 1.10, 1.08, and 1.08, respectively, were used. Briefly, functionalization was performed by first centrifuging and concentrating 120 mL of the synthesized nanorods in 500 mL of ultrapure water. The concentrated nanorod solution was then added to 8 mL of a 2 mM solution of either thiol-PEG in water or thiol-PS in tetrahydrofuran (THF), and stirred for 24 h. PEO and PS matrix polymers were obtained from Polymer Source, Inc. Thin nano composite films were spin coated onto clean silicon from a 1 wt % polymer (PEO or PS)/toluene solution. The resulting films had thicknesses of d ≈ 30 nm (PEO) and d ≈ 20 nm (PS), as determined by ellipsometry.

N

Rg (nm)

Rrod/ Rg

L/Rg

σ (chains/ nm2)

σ*

α = P/N

48 110

1.9 2.9

3.16 2.08

22.11 14.58

0.53 0.28

2.38 1.91

192

3.8

1.58

11.05

0.15

1.35

0.54, 3.6, 30.4 0.23, 1.6, 3.5, 5.3, 13.3 0.14, 0.9, 2.0, 3.1, 7.6

Values of α in bold denote fully dispersed systems (>90% individual NRs). a

Here N is the number of monomers in each brush chain, Rg is the unperturbed (bulk) radius of gyration of the brush chains, Rrod is the radius of the NRs, L is their length, and σ is the grafting density. To compare with the theory results, the dimensionless grafting density σ* was calculated from eq 3, using a bulk monomer density for PS of ρ0 = 0.969 g/cm3 and a segment size of a = 0.67 nm.39,40 For α < 2, we observed the PS-Au NRs to be uniformly dispersed throughout the flim with more than 90% of the NRs found isolated from one another. On the other hand, as α increases above 2, the NRs were found both as isolated NRs and in aggregates consisting of side-byside aligned NRs with a very regular NR separation. For the case of α > 2, approximately 60 to 80% of the NRs were individual, isolated NRs whereas the remainder were found within the aggregates. Finally, for α > 7, less than 50% of the NRs were dispersed and the majority of the NRs was found in aggregates. In addition, as N increased, the NR separation within the aggregates increased from approximately Rrod (6 nm, N = 48) to about 3Rrod (18 nm, N = 191). The preferential sideby-side alignment and regular NR separations within the aggregates are of potential importance for sensitive single molecular assays based on surface-enhanced Raman spectroscopy (SERS), for example. On the basis of Monte Carlo simulations using input parameters determined by our preliminary classical DFT calculations, the aggregation of NRs for α > 2 was attributed to depletion-attraction forces from autophobic dewetting of the brush and matrix. Poly(ethylene oxide) Nanocomposites. To show the generality of nanorod dispersion behavior, we investigated a second nanocomposite film, namely PEO(N)-Au NRs in PEO(P). The dimensionless parameters of the PEO composites are summarized in Table 2. Here σ* was calculating using ρ0 =

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Table 2. Parameters for PEO-Au(N):PEO(P) Filmsa

RESULTS In this section, we first summarize our previous experimental findings for PS(N)-Au NRs in PS(P) films, and then present new results for PEO(N)-Au NRs in PEO(P) films. Next we describe calculations of the brush profiles around single, isolated NRs from both DFT and SCFT. The interaction free energies between two parallel NRs are calculated using DFT.

N

Rg (nm)

Rrod/ Rg

L/Rg

σ (chains/ nm2)

σ*

α = P/N

80 161

2.92 4.14

2.05 1.45

14.38 10.14

0.5 0.5

0.94 1.33

0.43, 1.58, 19.2 0.21, 0.78, 9.5

Values of α in bold denote fully dispersed systems (> 90% individual NRs). a

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Figure 1. PEO(80)-Au nanorod morphology in PEO thin films. SEM images for α = P/N = (a) 0.43 and (b) 1.58. (c) TEM image for α = 19.2. The scale bars are 500 nm. Insets show a magnified region of the images.

the range of σ* = 1.0−3.0, Rrod/Rg = 1, 2, and 3.16, and α = 1, 2, 3, and 4. Larger values of α become difficult since the computational cost of the DFT increases with increasing polymer chain lengths. In the DFT the brush chain length was kept fixed at N = 40, in order to maintain a tractable system size. We note that our previous DFT calculations28 covered a similar range of grafting densities but employed N = 20 and a smaller range of the other two parameters, namely radii of Rrod/ Rg = 0.82, 1.1, and 1.37, and α = 1, 1.5, and 2. We also performed selected SCFT calculations of an individual brush-grafted NR in a homopolymer matrix at the same values of σ*, Rrod/Rg, and α, to compare with the DFT results for the brush profiles. SCFT is a complementary theory to DFT, and as applied here involves somewhat less molecular detail since the chains are modeled as Gaussian curves rather than as freely jointed chains with an explicit monomer length scale as in the DFT. On the length scale of Rg, we anticipate that SCFT and DFT will yield similar results. We first examine the density profiles for the brush and matrix chains around a single, isolated NR in the homopolymer matrix. The density profiles ρb(r) of the brush chains calculated by the DFT show oscillatory behavior near the NR due to the finite bead size (a), while the profile becomes smooth at larger distances from the NR. The profiles obtained from SCFT do not contain this oscillatory behavior because the polymer chains are Gaussian curves, with no monomer length scale. Typical DFT density profiles for the brushes are shown in Figure 2 of ref 18. Here we instead calculate the volume fractions ϕb and ϕm of the brush and matrix chains, respectively, as ϕb(r) = ρb(r)/(ρb(r) + ρm(r)) and ϕm(r) = 1 − ϕb(r). Although the oscillatory nature of the DFT density profiles are obscured, the volume fraction profiles are a natural output of the SCFT and allow for better comparison between theories. For both DFT and SCFT, Figures 3−5 show representative volume fraction profiles as a function of radial distance from the NR surface, upon varying each parameter while keeping the other two fixed. Figure 3 shows the ϕb and ϕm as σ* varies from 0.6 to 2.48, for fixed Rrod/Rg = 3.16 and α = 3. At low grafting densities, the brush volume fraction decreases strongly as a function of distance from the surface of the NR, and the matrix chains wet the brush substantially. As the grafting density increases, the brush profile transitions to a more parabolic shape, with a relatively more flat region near the NR. The brush thickness increases, and the matrix chains become excluded from the interior of the brush near the NR. Correspondingly, the volume fraction of the brush at the NR surface increases, from ∼0.6 to ∼1, for both the DFT and SCFT profiles. The brush thus gradually transitions from being “wet” to “dry”. The

1.064 g/cm3 and a = 0.72 nm for PEO.39,40 Shown in Figure 1 are electron microscopy images for PEO(80)-Au NRs in PEO films at three values of α. For α = 0.43 (Figure 1a), the NRs are dispersed and isolated from one another. For α = 1.58, a fraction of the NRs is found in aggregates containing approximately 3 to 4 NRs, while a large fraction of isolated NRs is also observed. Similar morphologies are observed upon increasing the length of the PEO brush to N = 161. As with PS−PS composites, the NRs align side-by-side in the aggregates with a regular NR spacing. Finally, for α = 19.2, nearly all the NRs are found within aggregates. The dispersion of Au NRs within both the PS-Au:PS (squares) and PEO-Au:PEO (triangles) systems is summarized in the dispersion map in Figure 2. Independent of the chemical

Figure 2. Dispersion map for PS-Au:PS (squares) and PEO-Au:PEO (triangles) nanocomposites. Solid symbols represent aggregated systems while open symbols represent dispersed systems. The dashed line is α = P/N = 2. For α > 2, nanorods are aggregated within the polymer matrix, whereas they mainly disperse for α < 2. The striped point at N = 48, P = 26 represents a partially aggregated morphology.

composition of the brush and matrix, NRs are found to aggregate for α > 2 and generally disperse for α < 2. As will be demonstrated in the following sections, because the transition from a wet to dry brush is gradual and the magnitude of the depletion-attractions between NRs depends on σ*, Rrod/Rg, and α, the boundary between dispersion and aggregation in the dispersion map is diffuse. DFT and SCFT Results. Volume Fraction Profiles: Effect of σ*, Rrod/Rg, and α. Parameters for the DFT calculations were chosen to cover the range of experimental values in Table 1 and Table 2. Thus, we consider nanorods with grafting densities in 2860

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Figure 3. Volume fraction profiles from DFT and SCFT for the brush ϕb (solid curves) and matrix ϕm (dashed curves) chains, for α = 3, Rrod/Rg = 3.16, and grafting densities of σ* = 0.6 (red), 1 (blue), 1.66 (black), 2.08 (green), and 2.48 (tan). As σ* increases, the brush thickness strongly increases.

Figure 4. Volume fraction profiles from DFT and SCFT for the brush (solid curves) and matrix (dashed curves) chains, for α = 2, σ* = 2.35, and Rrod/Rg = 1 (red), 2 (blue), and 3.16 (black). As Rrod/Rg increases, the brush becomes less wet.

Figure 5. Volume fraction profiles from DFT and SCFT for the brush (solid curves) and matrix (dashed curves) chains, for Rrod/Rg = 3.16, σ* = 2.38, and α = 1 (red), 2 (blue), 3 (black), and 4 (green). As α increases from 1 to 4 (wet to dry), the brush profiles show very little change.

trends for other values of Rrod and α as a function of σ* are qualitatively the same. Figure 4 shows that the NR radius also has a strong influence on the brush profiles. For σ* = 2.35 and α = 2, the brush becomes less extended and the amount of matrix chain penetration increases as Rrod/Rg decreases from 3.16 to 1. This increase in wetting results from an increase in the volume

around a smaller radius nanorod, so that the brush chains are less crowded and less extended at the same areal grafting density compared to a larger radius NR or flat surface. The brush is therefore more wet as the radius becomes smaller. In Figure 5 we show the volume fraction profiles at fixed σ* = 2.38 and Rrod/Rg = 3.16, for α values from 1 to 4. Because the experimental aggregation behavior of nanoparticles is strongly 2861

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affected by α (Figure 2), a strong effect of α on the brush profiles might be expected. Surprisingly, Figure 5 shows that varying the ratio of chain lengths from α = 1 to α = 4 leads to only small changes in the brush profiles. We will show below that this variation in α ranges from below to above the transition from dispersion to aggregation of the NRs. Thus, comparing changes in volume fraction profiles does not provide much insight into the wet to dry brush “transition” for NRs. As noted later, the rod−rod interaction energy is a better predictor of NR dispersion (Figure 11). Comparison of DFT and SCFT. Although the DFT and SCFT profiles clearly show the same trends as a function of σ*, Rrod/Rg, and α (Figures 3−5), the two theories exhibit a few systematic, quantitative differences. In particular, the DFT predicts more extended brushes and less matrix penetration of the brush at a given value of σ* compared to the SCFT. There are several possible reasons for this difference. In the DFT the polymers are modeled as finite length chains of discrete spherical interaction sites, where here we have chosen to use N = 40 sites for the brush chains. Using the same set of reduced parameters (σ*, Rrod/Rg, α) but a longer brush chain length of N = 60 leads to a somewhat more compressed brush, as shown in Figure S1 in the Supporting Information. The chain lengths are not independent variables in the SCFT, but the SCFT is most valid for long chains and assumes infinite extensibility. The agreement between the two theories should therefore increase as N increases in the DFT, and the more compressed DFT brush profile for N = 60 in Figure S1 supports this hypothesis. Experimentally, the shortest brush chain lengths are relatively short, on the order of 48 monomers in the PS-Au(N) NR case, so the DFT may be more accurate for these shorter chains while the SCFT may be more accurate for some of the other systems with longer grafted chains. A probably more significant difference between the two theories is that the DFT is a compressible theory, while the SCFT assumes that the system is incompressible. Because of the finite compressibility in the DFT, the total density of the brush and matrix chains around the NR in the DFT is not constant, and in fact the monomer density is somewhat higher in the brush than it is in the bulk matrix polymer away from the NR. This extra density could lead to the brush being more extended in the DFT than in the SCFT. Additionally, the DFT includes an extra variable, the bulk monomer density ρ0. It is not a priori clear what value this parameter should have in order to compare the DFT results to SCFT; typical “melt” densities range from ρ0a3 = 0.8−0.85 but could perhaps be even higher. As noted in the Methods, increasing the bulk density ρ0 in the DFT calculations also results in more compressed (less extended) brushes. We have compared DFT and SCFT results at the same values of σ*, but σ* includes the bulk density ρ0 and so it is not completely clear that the same σ* corresponds to exactly the same “effective” grafting density in both theories. Indeed, the shapes of the volume fraction profiles from both theories, along with their change with radius, agree quantitatively if σ* is shifted by a constant, as shown in Figure S2 of the Supporting Information. Thus, the major features of the brush profiles, as well as the effects of the main parameters on the changes in the profiles, are the same in both theories, while compressibility and finite chain length effects lead to some quantitative differences. The trends found here appear relatively robust, given that two complementary theories give such similar results. We focus on

DFT results for the remainder of the paper, but would expect to obtain similar results from further SCFT calculations. Brush and Matrix Characteristics from DFT. The brush profiles can be characterized more quantitatively by the brush height (thickness), the width of the interface between the brush and matrix chains, and the degree of penetration of the matrix chains into the brush. Following Matsen and Gardiner,8 the height hb of the cylindrical brush is defined as the radial distance from the NR surface where the brush volume fraction has fallen to 1/2 its maximum value:

ϕb(hb) =

1 ϕ (0) 2 b

(8)

Although alternative measures of the brush height can be defined, such as moments over the brush profile, hb as defined here gives the most consistent results for the parameter ranges studied. The interfacial brush width wb can be defined as wb =

ϕb(0) |ϕb′(hb)|

(9)

where |ϕb′(hb)| is the magnitude of the derivative of the brush profile at the height hb. Finally, the penetration of the matrix chains into the brush is quantified by the matrix volume fraction at the NR surface, ϕm(0). The height, width, and surface matrix volume fraction are shown as a function of grafting density σ* for fixed α = 3 in Figure 6, and as a function of the chain length ratio α for fixed σ* = 2.38 in Figure 7. Results for a brush on a flat surface (black triangles) are included for comparison. Similar behavior is seen for other values of α (Figure 6) and σ* (Figure 7). As expected, the brush height hb increases monotonically with increasing grafting density, and also with increasing NR radius (decreasing curvature). More quantitatively, the brush height scales with the grafting density as hb/Rg ∼ (σ*)β, with exponents of β = 0.72, 0.77, and 0.82 for Rrod/Rg = 1, 2, and 3.16, respectively. As expected, for the flat brush the height is linearly proportional to the grafting density, hb/Rg ∼ σ*. In Figure 6a, the height of the planar brush increases by nearly 3× over the range of σ*, whereas for Rrod/Rg = 1 (smallest NR), this increase is only 2×. These results show that brush height can be increased by either using large NRs or high grafting densities. The width of the brush is approximately constant for smaller values of σ* and then decreases at larger σ* (Figure 7b). Variations in wb at the lower values of σ* are smaller than the mesh size in the DFT calculations and are not likely to be significant. The value of σ* at which the width starts decreasing monotonically depends on NR radius, decreasing with increasing Rrod. A visual inspection shows this change in slope in wb occurs roughly when the brush profile transitions to a more parabolic shape, with a higher brush fraction at the NR surface (cf., Figure 3). If the brush width is normalized by the height, then wb/hb decreases relatively smoothly with increasing σ* (see Figure S3 in the Supporting Information). The fraction of matrix chains at the NR surface decreases substantially with increasing σ* (note the semilog scale in Figure 6c). These changes can be seen in the relatively strong dependence of the brush profiles on σ* in Figure 3. As seen in Figure 7, the brush height is nearly constant at fixed σ* as a function of α, consistent with the nearly overlapping brush profiles in Figure 5. What is difficult to see in Figure 5 but is revealed here is that the brush width wb does decrease with increasing α. The penetration of the matrix 2862

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Figure 7. (a) Brush height hb, (b) brush width wb, and (c) matrix penetration ϕm(0), as a function of α for σ* = 2.38. Symbols as in Figure 6.

Figure 6. (a) Brush height hb, (b) brush width wb, and (c) matrix penetration ϕm(0), as a function of the grafting density for α = 3. NR radii are Rrod/Rg = 1 (red circles), 2 (blue diamonds), and 3.16 (green squares). The values for a flat brush are shown in black triangles.

interacting NRs, minus the free energy for two isolated NRs in the matrix:

chains into the brush decreases significantly with increasing α, particularly for lower values of α. The small decrease in width as a function of α is similar to that calculated for spherical nanoparticles using SCFT (compare Figure 2 of ref 16 with Figure S4 in the Supporting Information, which shows wb/hb as a function of α). The trends in the brush profiles as a function of α are also similar to those calculated previously for flat brushes by SCFT.8 The largest decrease in both wb and ϕm(0) occur between α = 1 and α = 2. For larger α, the brush width and penetration of the matrix chains into the brush appear to begin to saturate; based on the SCFT calculations for flat brushes up to α = 10, we would expect these parameters to become independent of α for sufficiently large α.8 Rod−rod interactions. Experimentally NRs aligned sideby-side are the dominant feature within aggregates (i.e., for α > 2). We restrict the DFT calculations to this geometry and calculate the free energy between two parallel NRs as a function of the distance r between their centers. The relevant interaction free energy Ω is the total free energy, per unit length, of the two

Ω = Ω tot − 2Ωs

(10)

In this section, Ω is plotted as a function of the distance H = r − 2Rrod between the NR surfaces. At large NR separations the free energy is zero because the rods are too far apart to influence each other. At intermediate distances there is often an attractive well, and at shorter distances the energy is repulsive as the NRs approach more closely and the brushes become compressed. In our previous work, we found a small attraction between the rods for all parameters investigated.28 Here, in some cases for α = 1 we do not find an attraction, but this is likely due to numerical errors associated with the large mesh size (0.2 a). As shown below, an attraction is always observed for α > 1. Also, recall that the brush chains in the DFT are physisorbed on the NRs (see Methods) and are therefore mobile on the NR surface. The experimental systems are not thermally annealed so the thiols are likely not mobile on the Au NRs in these particular experiments. The DFT may therefore 2863

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Figure 8. Free energy per unit length between nanorods for α = 3, for (a) σ* = 1.91 and (b) σ* = 2.38. For comparison, the free energy for σ* = 1.91 and Rrod/Rg = 2 from part a is reproduced as the dashed blue curve in part b.

Figure 9. (a) Well-depth Emin and (b) its location Hmin, for α = 3 and for Rrod/Rg = 1 (red circles), 2 (blue diamonds), and 3.16 (green squares).

the location of the minimum Hmin. Both Emin and Hmin are shown as functions of σ* for α = 3 and all three NR radii in Figure 9. The values of Emin are relatively small compared to kT, but recall that these attractive energies are per unit length of the NRs. For the NRs of interest here (see values for L/Rg in Table 1 and Table 2), the total attractive interaction between two NRs will be Emin multiplied by their length, which will give factors of 10 or larger; thus the attractions are of order kT and larger. The values of σ* mostly cover the experimental values for both the PS and PEO systems. (For Rrod/Rg = 1 and low grafting densities, the NRs can approach closely enough that the minimum in their interaction curve occurs in the region of oscillatory behavior due to the finite size of the interaction sites in the polymer in the DFT. In this region it is not possible to determine Emin accurately.) Clearly, there is a larger change in both Emin and Hmin as Rrod/Rg is increased from 1 to 2, than there is as Rrod/Rg is increased further to 3.16. As σ* increases from 1.35 to 2.85, the attractive well gets deeper by nearly 50% for Rrod/Rg = 3.16. Correspondingly, the location of the minimum increases by a factor of 2. Such changes in separation between NRs would produce a significant weakening of the LSPR, which decays exponentially with distance. To compare the results for the NRs with the energy between two flat brushes, we plot the free energy per unit area in Figure 10, again for α = 3. Whereas all three NR energy curves are similar when normalized by unit area, the attractive energy between two planar brushes is much stronger and occurs at a larger separation between the brushes when compared at the

somewhat underestimate the strengths of both the repulsive and attractive parts of the interaction energies compared to experiment. Polymer-Mediated Interaction Energies between NRs. We first describe the interactions between NRs as a function of grafting density and radius, at fixed α = 3. This intermediate value of α is chosen because it provides a distinct attractive minimum in Ω(H). The free energy per unit length of the NRs at α = 3 is shown in Figure 8, for all three NR radii, and two different grafting densities. At σ* = 1.91 and Rrod/Rg = 1, NRs separated by more than about 4Rg do not strongly interact. This range increases to about 5Rg as Rrod/Rg increases to 3.16. At the higher grafting density of σ* = 2.38, the range of attractions is longer than at σ* = 1.91 when compared at the same NR radius Rrod. The depth of the attractive well grows deeper and moves to larger rod−rod separations for both larger NRs and higher grafting densities. This behavior is correlated with the changes in the brush structure; as shown in Figure 6, the brush width and the penetration of the matrix chains into the brush decrease with increasing Rrod or increasing σ*. Thus, as the brush becomes more “dry”, the depth of the attractive well increases. The minimum occurs at larger separations because the brushes also increase in height with increasing σ* and Rrod (Figure 6a), and hence the brushes on two approaching NRs begin to interact at larger NR separations H. The same trends were found previously for both NRs28 and spherical nanoparticles.25 A summary of the interaction curves for all the parameters studied is provided by the depth of the attractive well Emin and 2864

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the surface plasmon resonance under these conditions may be measurable. For brushes on flat surfaces, SCFT predicts that Hmin is a strongly decreasing function of α for smaller α, which then saturates at larger α.8 To summarize this section, DFT predicts that σ*, Rrod, and α all have significant effects on the strength of the polymermediated attraction between the NRs. Because of the variations in brush height, these parameters also affect the range of the attractive interactions and the location of the minimum in the attractive well. The changes in attractive interaction energy correlate with the changes in brush structure, and particularly with the amount of interpenetration of the brush by the matrix chains. These results are consistent with our previous DFT calculations performed for a different set of parameters.28 Previous SCFT calculations on polymer-grafted spherical nanoparticles in homopolymer melts found that the minima in the interaction free energies were correlated with the brush widths wb.25 The DFT for NRs does not predict such a correlation, but our calculations were also performed over a much smaller range of σ* and Rrod than the SCFT calculations, which may preclude finding such a correlation. Inclusion of van der Waals Attractions. To compare the DFT with our experimental results, a van der Waals interaction between parallel pairs of NRs should be included. As described previously,28 this interaction can be approximated by

Figure 10. Free energy per unit area ΩA between nanorods for α = 3 and σ* = 2.38, for radii Rrod/Rg = 1 (red), 2 (blue), and 3.16 (dashed green). The free energy between two flat brushes under the same conditions is shown in black.

same grafting density of σ* = 2.38. This observation again correlates with the brush profile, which is more extended and more dry on flat surfaces. We now turn to the interaction free energy as a function of chain length ratio α. Two representative cases are shown: interaction energies for Rrod/Rg = 2.0 and σ* = 1.91 in Figure 11a, and for Rrod/Rg = 3.16 and σ* = 2.38 in Figure 11b. In both cases the free energy is repulsive at all distances for α = 1, but has an attractive well for larger α. There is a large increase in the depth of the attractive well with increasing α, even though the brush profiles appear to be very similar for α from 1 to 4 (Figure 5). Also, the location of the minimum Hmin becomes smaller with increasing α. The same trends are observed for all the parameters studied, as summarized by the well depths Emin and locations of the minima Hmin shown in Figure 12 as a function of α. Values of α = 1 are not shown because an attractive well, if present, is not resolved. The increase in magnitude in Emin with α is larger for larger NR radii. For all radii, Hmin decreases with increasing α. At equilibrium, nanorods aggregate with a spacing equal to the location of the minimum in their interaction free energy; thus the DFT predicts that the NRs should aggregate together more closely with increasing α. Experimentally, the change in separation between α = 2 and 4 is difficult to measure directly by electron microscopy in a polymer matrix. However, shifts in

W (H ) = −

ALR rod1/2 24H3/2

(11)

where L is the length of the NR, H is the distance between NR surfaces, and A is the Hamaker constant. Typical values of A for polymers are in the range of 30−80 zJ (zepto Joules), while those for metals are in the range of 300−500 zJ.41,42 An estimate of the Hamaker constant for the NRs (medium 1) interacting across the polymer medium 2 can be obtained from41 A121 ≈ ( A11 −

A 22 )2

(12)

A typical value of A121 for gold NRs in a polymer matrix is 50 zJ ≈ 12 kBT at 298 K, which we use here. The total interaction free energy Ωtot between the NRs is then the sum of the polymer-mediated free energy and the van der Waals energy from eq 11, Ωtot(H) = Ω(H) + W(H). The effect of the van der Waals interactions on the rod−rod free energy is shown in Figure 13a for systems with varying σ*

Figure 11. Free energy per unit length of rod−rod interactions for (a) Rrod/Rg = 2 and σ* = 1.91 and (b) Rrod/Rg = 3.16 and σ* = 2.38. 2865

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Figure 12. (a) Well depth and (b) location of the minimum in the free energy, for Rrod/Rg = 1 (red circles), 2, (blue diamonds), and 3.16 (green squares). Solid curves and filled symbols are for σ* = 2.38, whereas the dashed curves and open symbols are for σ* = 1.91.

Figure 13. Polymer-mediated free energy (dashed curves) and the sum of the polymer-mediated and van der Waals interactions (solid curves) as a function of H = r − 2Rrod. Curves are for (a) α = 3, Rrod/Rg = 3.16, and σ* = 1.35 (purple), 1.91 (red), and 2.38 (green) and (b) Rrod/Rg = 3.16, σ* = 2.38, and α = 1(blue), 2 (tan), 3 (green), and 4 (black). The free energy is per unit length of the NRs.

and in Figure 13b for systems with varying α. The van der Waals contribution W(H) to the attractive well only depends on Rrod (and not on σ* or α), and is shown as the thin black curve in each figure; W(H) diverges as H → 0. The polymermediated interaction energies Ω already discussed are shown as dashed curves in Figure 13, while the total free energies Ωtot are shown as the solid curves. First consider Figure 13a, which shows the interaction free energies for Rrod/Rg = 3.16 and α = 3 at three different grafting densities. As σ* increases, the location of the minimum in Ω(H) increases, as is shown in Figure 9b. Thus, the van der Waals energy makes a larger contribution to the depth of the attractive well in Ωtot for smaller σ* than for larger σ* (i.e., compare dashed and solid lines in Figure 13a); at larger σ*, the minimum Hmin in Ω occurs for larger H where the van der Waals contribution is smaller. This results in the total well depth being about the same for σ* = 1.91 and σ* = 2.38, though the polymer-mediated part Emin is slightly larger for σ* = 2.38 than for σ* = 1.91. These results show that inclusion of the van der Waals energy is significant; for example, the van der Waals term increases the attractive well depth by a factor of 1.5 for σ* = 1.91 in Figure 13a. Note that the polymer-mediated free energy Ω for σ* = 1.35 has a minimum at a small enough H that the van der Waals term overcomes the repulsive part of Ω(H), resulting in a large attraction in Ωtot at NR contact. Figure 13b shows the total interaction energies for Rrod/Rg = 3.16 and σ* = 2.38, for α from 1 to 4. In this case, Hmin for the

polymer-mediated interaction energy decreases with increasing α. The van der Waals contribution to the total well depth is therefore larger at larger α. In the case of α = 1, Hmin is sufficiently large that W(H = Hmin) is relatively small, so the van der Waals contribution leads to only a shallow attractive well at large H. NRs are strongly repulsive in this case. As α increases at fixed σ* and Rrod, the van der Waals contribution deepens the attractive well in Ωtot. Comparison of DFT and Experiment. Experimentally, the clearest signature of the NR interaction free energy is whether the NRs are dispersed or aggregated. One objective of the current study is to bridge experiment and theory. Namely, using DFT, can we extract a criterion for NR aggregation that can be compared with experimental results? Attractive interactions somewhat stronger than kBT are required for aggregation. Monte Carlo simulations of the NRs showed partially aggregated systems for a total interaction strength of 5 kBT and strongly aggregated systems for interactions of 10 kBT.27 One possible criterion for aggregation then is to assume that any system with a total attractive interaction >5kBT will be aggregated. At this point, the length of the NRs enters as an additional relevant variable, because the total interaction energy between the NRs will be the interaction free energy per unit length that we calculate from the DFT, multiplied by the experimental length of the rod. The experimental systems have a NR radius 2866

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Figure 14a shows a dispersion map as a function of Rrod/Rg and α, for grafting densities between σ* = 1.91 and σ* = 2.38. The experimental results are for the PS(48)-Au:PS and PS(110)-Au:PS systems in Table 1. Aggregated systems are shown in filled symbols and dispersed systems in open symbols. For the DFT results, systems with |Erod| < 5kBT are considered dispersed, and those with |Erod| > 5kBT are considered aggregated. For the plotted points, the DFT prediction of dispersion is the same for grafting densities between σ* = 1.91 and σ* = 2.38. The approximate location of the boundary, defined by Erod ≡ 5kBT, can be determined by interpolating the DFT total energies Etot as a function of Rrod/Rg at fixed α, or as a function of α at fixed Rrod/Rg. Performing this interpolation for σ* = 2.38 gives the crosses shown in Figure 14a. As can be seen from Table 3, the boundary for σ* = 1.91 would be slightly lower in α. The DFT agrees completely with the experiments for this range of variables. Over the experimental range of Rrod/ Rg, the NRs are aggregated for α > 2 and dispersed for α < 2, as found in Figure 2. For smaller NRs, the DFT predicts the transition to aggregated NRs occurs at higher α. The details of the transition do depend on the criterion for dispersion/ aggregation used in the DFT. If the boundary between dispersed and aggregated were instead 4 kBT, then for this range of σ* Figure 14a would stay the same except for two points: 1) the NRs with radius Rrod/Rg = 1 would be aggregated at α = 4, and 2) the system with Rrod/Rg = 3.16 and α = 2 would also be aggregated. A few additional DFT calculations were performed at a lower value of σ* = 1.35, corresponding to the PS(192)-Au:PS system and the PEO(161)-Au:PEO system (see Table 2). At these lower grafting densities, the polymer-mediated interaction energy curves are shifted to such small NR separation distances that the van der Waals attraction can overcome the repulsion, leading to attractions at all distances, as for the case of σ* = 1.35, Rrod/Rg = 3.16, and α = 3 shown in Figure 13a. The dispersion map for these lower σ* is shown in Figure 14b, using the same criterion of 5 kBT for the boundary in the DFT calculations. At the lower grafting density, both experiment and theory show a transition from dispersion to aggregation for 1 < α < 2 (or, for the PEO(80)-Au:PEO system at an even lower grafting density of σ* = 0.94, for 1 < α < 1.58), lower than at the higher grafting densities. This result indicates that the dispersion of NRs with a low grafting density is more difficult

of 6 nm and length of 42 nm. As N is changed in the experimental system, Rg changes and thus so do Rrod/Rg and L/ Rg as in Table 1 and Table 2. For the DFT results, we assume L/Rg = 22.1 for Rrod/Rg = 3.16, L/Rg = 14.58 for Rrod/Rg = 2.0, and L/Rg = 11 for Rrod/Rg = 1.0 (the values for the two PEO systems are similar to the second two PS systems). Small changes in the rod lengths do not affect our conclusions. To calculate the maximal interaction energy Erod, we multiply the minimum in Ωtot by the appropriate rod length. The resulting values of Erod are shown in Table 3, along with the minimum Emin in the polymer-mediated part of the free energy, and Hmin,tot, the location of the minimum in the total interaction free energy Ωtot. Table 3. DFT Interaction Energiesa

a

α

Rrod/Rg

σ*

Emin (kBT/Rg)

Erod (kBT)

Hmin,tot (Rg)

1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4

1 2 3.16 1 2 3.16 1 2 3.16 1 2 3.16 1 2 3.16 1 2 3.16 1 2 3.16 1 2 3.16

1.91 1.91 1.91 1.91 1.91 1.91 1.91 1.91 1.91 1.91 1.91 1.91 2.38 2.38 2.38 2.38 2.38 2.38 2.38 2.38 2.38 2.38 2.38 2.38

n/a n/a n/a −0.04 −0.08 −0.09 −0.12 −0.30 −0.35 −0.21 −0.40 −0.52 n/a n/a n/a −0.05 −0.11 −0.12 −0.15 −0.36 −0.41 −0.25 −0.47 −0.58

−0.6 −1.0 −1.5 −1.9 −3.2 −5.0 −3.4 −6.9 −11.7 −4.8 −8.6 −15.5 −0.5 −0.8 −1.4 −1.6 −3.1 −4.9 −3.0 −7.0 −11.8 −4.2 −8.7 −15.8

3.9 4.5 5.6 2.0 2.8 3.4 1.6 2.5 3.0 1.6 2.4 2.8 4.5 5.1 5.6 3.0 3.6 4.1 2.5 3.3 4.1 2.3 3.1 3.6

Values of Erod in bold indicate dispersed systems.

Figure 14. Dispersion maps for experimental PS-Au:PS systems (squares), PEO-Au:PEO systems (triangles) and DFT results (circles), for (a) 1.91 ≤ σ* ≤ 2.38 and (b) 0.94 < σ* < 1.35. Aggregated systems are shown in solid symbols and dispersed systems in open symbols. Included in part a are points with Etot = 5kBT in the DFT (crosses). The dashed lines are guides to the eye. 2867

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Overall, the DFT predicts interaction energies between the NRs that are consistent with the aggregation/dispersion behavior observed experimentally. The calculated energies are approximate, especially given our simple estimation of the Hamaker constant and the van der Waals contribution to the total interaction free energy. Because of the approximations in the theory, more finely resolved predictions could be problematic. Nevertheless, the results show that the DFT calculations can be used as a guide to choosing parameters to obtain a desired aggregation state.

than at higher grafting density. This result might be somewhat counterintuitive, since the brush is more wet for lower grafting densities, leading in principle to better dispersion. However, for the variable ranges studied here, the brush height is less at the lower grafting densities, allowing the NRs to approach more closely, which leads to stronger van der Waals interactions. These stronger interactions then lead to aggregation at lower values of α. A more nuanced view of the aggregation behavior of the NRs can be obtained by dividing the systems into three categories rather than two. We previously characterized the PS-Au:PS systems as being dispersed if more than 90% of the NRs were isolated from each other, partially aggregated if between 50% and 90% of the NRs were isolated, and aggregated if less than 50% of the NRs were found isolated from each other.27 At equilibrium, one would expect that for sufficiently large NR interaction energies, all the NRs would be aggregated (i.e., phase separated from the polymer matrix), but this state may be kinetically difficult to achieve experimentally. The fraction of aggregated NRs probably indicates the relative strength of the NR interactions, with more highly aggregated systems representing stronger NR interaction energies. Previous Monte Carlo simulations of NRs found similar dispersion states as in the experiments, with fully dispersed, partially aggregated, and aggregated states for NR attractive interaction energies of 0, 5kBT, and 10kBT, respectively. From Table 3, we see that as σ*, Rrod/Rg, and α increase, the total NR interaction energy Erod gradually increases, with values ranging from nearly 0 to a maximum of nearly 16 kBT. We would thus expect that the systems near our dispersion/aggregation boundary, with Erod ≈ 5kBT, would be partially aggregated, and systems with energies nearer to 10 kBT and larger would be more fully aggregated. In the experiments, the highly aggregated systems all occurred for large α > 5, above the range of the DFT calculations. Additionally, the experiments of course include a finite volume fraction of NRs, whereas the DFT calculations are done for only two NRs. But the gradual nature of the transition from dispersed to aggregated can be seen both in the experiments and in the gradual increase in Erod in the DFT. A final point of comparison between the experiments and the DFT results is the average spacing between aggregated NRs. When aggregated side-by-side in equilibrium, the spacing between NRs should be approximately the spacing at the minimum in the total interaction free energy, which is Hmin,tot as tabulated in Table 3. From analysis of the electron microscopy images in ref 17, this spacing was estimated to be about Hmin,tot = Rrod = 3.2 Rg for PS(48)-Au:PS (Rrod/Rg = 3.16 and σ* = 2.38), and about Hmin,tot = 2Rrod = 4 Rg for PS(110)-Au:PS (Rrod/Rg = 2 and σ* = 1.91).27 For α = 4, the DFT predicts spacings of Hmin,tot = 3.6 Rg and 2.4 Rg, respectively. The smaller predicted spacing for PS(110)-Au:PS is because both the grafting density and the effective NR radius are smaller than for PS(48)-Au:PS, leading to a smaller predicted brush height and hence smaller NR spacing. The differences in estimates between theory and experiment could have many causes. The DFT is probably most accurate for systems with relatively short brush chains. The experimental spacings were measured for systems at high α (>4), and we know from Figure 12b that the spacing can decrease with increasing α; the calculated DFT spacing has not yet reached its asymptotic value at α = 4. Thus, we might expect the overestimation of Hmin,tot by DFT for the PS(48)-Au:PS system.

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CONCLUSIONS We investigated the dispersion behavior of polymer brushcoated nanorods in a homopolymer matrix using experiments, classical density functional theory, and self-consistent field theory calculations. The results show that all three major system variables, the reduced grafting density σ*, the NR radius Rrod/Rg, and the ratio of matrix to brush chain lengths α = P/N, are important for determining NR behavior. The NR length is also relevant but in the calculations here enters simply as a multiplier to the interaction energy between rods. Both DFT and SCFT calculations of the structure of the brush around a single isolated NR in the homopolymer matrix show that the transition from a “wet” to a “dry” brush is gradual as σ*, Rrod/ Rg, and α are increased. Changes in the brush profiles with α are surprisingly small, although the brush width and the amount of interpenetration of the matrix chains into the brush decrease with increasing α. The effects of σ*, Rrod/Rg, and α on the interaction free energy between two NRs, as calculated using DFT, are consistent with the changes in the brush profiles. In particular, the attractive well in the interaction free energy that occurs due to the autophobic dewetting of the brush by the matrix chains increases in strength as σ*, Rrod/Rg, and α increase and the brush becomes more dry. To compare with the experimentally observed aggregation behavior, we include an estimate of the van der Waals contribution to the interaction free energy. Using a dispersion map to summarize results, we find excellent agreement for whether the NRs are dispersed or aggregated between the DFT and the experiments, for the full parameter range covered by our study. To make this comparison, we assumed for the DFT results that aggregation occurs for NR interaction energies greater than 5 kBT, a somewhat arbitrary criterion that nonetheless captures a wide range of experimental results. The calculated boundary between dispersed and aggregated systems is not sharp, and depends on all the system parameters. For the experimental systems studied here, for NRs of radius 2−3 Rg and typical experimental grafting densities, we find that aggregation begins for α > 2 at the higher grafting densities and for 1 < α < 1.6 for somewhat lower grafting densities. Compared to similar results for flat planar brushes and spherical nanoparticles,8,25 it would appear that the transition to aggregation in NRs occurs for larger α than on flat surfaces and smaller α than for spherical nanoparticles, as would be expected based on the structure of the brushes on surfaces of different geometry. We have shown that the DFT calculations provide the design principles necessary for formulating NR assemblies which are dispersed or aggregated. Our results imply that to achieve good dispersion of NRs, it is best to have sufficiently small values of R/Rg and α to be in the dispersed part of the dispersion map. Grafting densities should be high enough to prevent the NRs from approaching too closely, but not so high as to promote 2868

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strong matrix chain dewetting from the brush. Alternatively, the parameters could be tuned to obtain controlled aggregation of the NRs at a desired spacing between the rods. This type of guidance could be used in the future to create practical devices which require a specific state of NR aggregation for their optical properties.

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ASSOCIATED CONTENT

S Supporting Information *

Plots of DFT volume fraction profiles and normalized brush widths. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Present Address

∥ (M.J.A.H) National Institute of Standards and Technology (NIST), Center for Neutron Research, 100 Bureau Drive, Gaithersburg, MD 20899.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. R.J.C. and M.J.A.H. acknowledge support from the National Science Foundation Polymer (DMR09-07493), CEMRI (DMR1120901), and IGERT (DGE02-21664) programs.

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