Role of Grafting Mechanism on the Polymer Coverage and Self

Jun 15, 2017 - Thus, for the small mean number of grafts that can be typically achieved by the “grafting to” process, the distribution of the numb...
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Role of Grafting Mechanism on the Polymer Coverage and Self-Assembly of Hairy Nanoparticles Makoto Asai, Dan Zhao, and Sanat K. Kumar* Department of Chemical Engineering, Columbia University, New York, New York 10027, United States ABSTRACT: It is now well-accepted that controlling the spatial dispersion of nanoparticles (NPs), which can be achieved by grafting them with polymers of different chain lengths and grafting densities, is central to optimizing the thermomechanical properties of the resulting polymer nanocomposites. In general, there are two methods for creating such polymer-grafted NPs: “grafting to” and “grafting from”. The conventional wisdom is that the “grafting from” mechanism, where monomer-sized initiator/functional groups are attached to the surface followed by growing the chains, allows for higher polymer grafting densities and hence a more uniform polymer coverage of the NP surface. Here, we perform calculations and instead show that the “grafting to” strategy surprisingly leads to a more uniform polymer coverage of the NP surface at a given grafting density since the brush is formed while respecting the excluded volume constraints of the previously grafted chains. This conclusion is especially clear in the limit of low-tomoderate grafting density. Thus, at a given grafting density, the “grafting to” mechanism leads to an enhanced miscibility of the NPs in the matrix (which has the same chemistry as the grafts) and lower propensity to create self-assembled structures. Another important factor is that the dispersity in the number of grafted chains on the NPs is also smaller in the case of “grafting to” systems, thus leading to better defined materials. These two conclusions imply that the “grafting to” mechanism may provide better control over the NP dispersion state and hence the thermomechanical properties of polymer nanocomposites. KEYWORDS: polymer grafted nanoparticle, grafting process, self-assembly, surface coverage, excluded volume effect, anisotropy, distribution of grafting density immobilized.12 It has been generally believed that the “grafting from” method is favorable in the context of sterically stabilizing NPs in a polymer matrix since it allows for higher grafting densities and hence larger polymer coverages of the NP surface. In contrast, recent experimental work by Zhao et al.13 surprisingly showed that the physical adsorption strategy (or essentially the “grafting to” process) leads to improved miscibility, relative to “grafting from” NPs at comparable grafting densities, when both types of PGNPs are placed in a matrix with the same chemistry as the grafts. To rationalize these results, it was proposed that, while (monomer sized) grafting agents can be randomly located on the NP surface in the “grafting from” method, the “grafting to” protocol must respect the excluded volume effect of the already adsorbed polymer chains. Thus, at a given grafting density, the “grafting to” method provides a larger and more uniform polymeric

D

eveloping techniques for spontaneously assembling nanoparticles (NPs) into a variety of superstructures is a popular topic in the field of nanotechnology.1−5 Here, we focus on polymer-grafted NPs (PGNPs) at low-tomoderate grafting densities to achieve this goal.6,7 It is believed that the grafted polymer chains convert the NPs into surfactants with a hydrophilic patch (the “uncovered” NP surface) and a hydrophobic patch (where the NP surface is covered by the polymer chains).8 There are two primary methods used for synthesizing these PGNPs. In the “grafting from” scheme, monomer-sized grafting agents are attached onto the NP surface. The polymer chains are then grown from the NP surface by typical polymerization methods such as reversible addition−fragmentation chain-transfer polymerization (RAFT), atom-transfer radical polymerization (ATRP), or nitroxide-mediated polymerization.9−11 In contrast, in the “grafting to” protocol, preformed polymers are directly endgrafted onto NP surfaces. In a related method, workers have adsorbed one block of a diblock copolymer on to the NP surface, whereby the polymer chain becomes effectively © 2017 American Chemical Society

Received: April 17, 2017 Accepted: June 15, 2017 Published: June 15, 2017 7028

DOI: 10.1021/acsnano.7b02657 ACS Nano 2017, 11, 7028−7035

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ACS Nano coverage of the NP surface than the “grafting from” method. This surprising conjecture remains to be verified. A further subtle point we propose here is that the maximum number of monomer-sized grafting agents that can be attached onto the NP surface, relevant to the “grafting from” strategy, should be much larger than that for long polymer chains (“grafting to”). Thus, for the small mean number of grafts that can be typically achieved by the “grafting to” process, the distribution of the number of chains in the “grafting from” case is much broader than the “grafting to” analog. This fact might also result in a larger fraction of NPs which have much smaller (larger) coverage in the “grafting from” protocol, thus yielding apparently less (more) miscible structures. Here, we focus on capturing both of these effects in a theoretical framework. We develop a model to account for the distributions of grafts on a NP. We also consider the grafting process using molecular dynamics (MD) simulations to understand the surface coverages for the “grafting from” and “grafting to” protocols at a given grafting density. In this manner, we are able to systematically understand the role of grafting protocols on the structure and hence the self-assembly of the PGNPs.

Table 1. Simulation Parameters and Results for the Chain Length of Grafts, N, the Number of NP, NNP, the Total Number of Grafts in the Simulation Cell, Np, and the Deduced Value of f max

dt

= k[{fmax − (f − 1)}Nf − (fmax − f )Nf + 1]

(1)

where Nf denotes the number of NPs that are grafted with f polymers, f max is the maximum number of polymers that can be attached to the particle surface, k is a kinetic constant, and t is the reaction time. As a result of successive repetition of this reaction the P(f) can be derived as P(f ) =

fmax ! f ! (fmax − f )!

N

NNP

Np/NNP

f max

7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0

1 10 20 30 50 80 100 130 150 200 300 500

125 125 125 125 125 125 125 125 125 125 125 125

700 80 80 50 50 20 20 20 20 20 20 20

680 90 55 51 40 30 24 20 19 15 11 8

⟨f⟩ stays fixed, f max is strongly dependent on N. In the N = 1 case, note also that the peak value of f is shifted to smaller values relative to the peak value of the “grafting to” chain distribution; this reflects the asymmetry of the P(f) in that it has a long tail in the high f end, while the lower end of the distribution is truncated at f = 0. Predicting f max from the Polymer Coverage of the NP Surface. Here we assume that (1) polymers are randomly grafted on the NP surface and that the graft points are fixed; and (2) the maximum coverage of the NP surface by polymers is equal to the random close packing fraction value (ηRCP = 0.862) for hard disks on a two-dimensional surface.15 To calculate the maximum number of chains that can be grafted to a NP, f max, we need to know how much of the polymer surface is “covered” by a grafted polymer. To this end, therefore, we need to properly calculate the overlap between two grafted chains on a NP. We use the equivalent sphere (ES) model8 which treats a polymer chain as a sphere. It should be noted that in the original ES model,8 which was designed to model the “grafting from” process, we assumed that two or more ESs are fully penetrable to each other. However, the ES experienced hardsphere-like interactions with the NPs. Thus, the original ES model gives f max = ∞. This is reasonable considering the fact that very high f max have been reported in the “grafting from” process. However, ESs should not be penetrable to each other, particularly when we consider the “grafting to” process.13 The γ*ES was previously defined as the coverage ratio afforded by a ES on the NP surface to another NP; here we explicitly use the fact that an ES is impenetrable to a NP. Similarly, to calculate f max, we need the coverage ratio γ*ES,chain afforded by a polymer on the NP surface in the presence of “another chain”, as shown in Figure 4a,b. This follows a simple extension of the geometric relation that was used to calculate γES * in the ES model:

RESULTS/DISCUSSION Distribution of the Number of Grafts per Particle. We model the grafting density distribution, P( f), following Hakem et al., who use the following sequential reaction:14 dNf + 1

Rn(σ)

e−fmax v(e v − 1) f (2)

where ν  − ln(1 − ε), ε  ⟨f⟩/f max, and ⟨f⟩ is the average number of grafted polymers per NP. We used MD simulations to mimic the “grafting to” protocol to directly measure P(f) (See Methods/Experimental Section). Polymer chains are represented using the Kremer−Grest model. A new bond is formed between a chain and the NP when only one of the terminal polymer beads is close enough to the NP surface; this bonding site on the NP surface is then immobile. (No loops are allowed to form.) We vary the chain length of the graft, N, from 1 to 500 in a series of simulations, Table 1. The N = 1 case is representative of the “grafting from” scenario. Figure 1 shows examples of the time evolution of P(f), which is well fit by eq 2 except at the earliest times, and yields both f and f max (the large fluctuations in f max at early times are simply a consequence of the inaccuracy of eq 2 under these conditions). In Figure 1a−c, the peak position in P( f) shifts to larger f with increasing simulation time, t. Simultaneously, ⟨f⟩ increases and f max converges to its asymptotic value, Figure 1d. Figure 2 illustrates the Ndependence of f max. It is apparent, in agreement with intuition, that the f max decreases monotonically with increasing N. Perhaps more interesting are our results in Figure 3a,b which illustrate the fact that, at a given ⟨f⟩, the distribution of attached monomers is much broader than the distributions of grafted long chains. This result simply reflects the fact that, although

* γES,chain =

⎧ 1⎪ ⎨1 − 2⎪ ⎩

⎛ RES ⎞2 ⎫ ⎪ 1−⎜ ⎟ ⎬ ⎝ RES + R n ⎠ ⎪ ⎭

(3)

We postulate that the radius of the ES RES follows RES = βRg, where Rg is the chain radius of gyration, and we use β = 1 in the absence of any more information. We then get the maximum number of grafted chains following f max = ηRCP/γ*ES,chain. To compare these predictions to simulations, we use Rg = 0.33N0.70 (N ≤ 50) and Rg = 0.49N0.60 (N > 50) for the Kremer−Grest 7029

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Figure 1. Examples of the time evolution of the distribution of the number of grafted chains per NP in the “grafting to” scenario. (a−c) P( f) at t = 2250, 6000, and 9000, respectively, for chains of length N = 200. (d) Time dependence of f max and ⟨f⟩ for N = 200.

represent the curved brush.16,17 We calculate the overlapping volume between two grafted polymers on a single NP that are separated by distance L (see Figures 4c and 5). Then, the number of overlapping monomers Ψ can be derived as follows: ⎧ ⎛ L − R g ⎞ D⎛ DR g + L ⎞ N⎪ L2D2 ⎟⎟ ⎜ Ψ = ⎨1 − + ⎜⎜ ⎟ 2⎪ 2R gL(D2 − 1) ⎝ R g ⎠ ⎝ D2 − 1 ⎠ ⎩ ⎫ (L2 − 2R g 2)D ⎪ ⎬ − 2R gL(D2 − 1) ⎪ ⎭ R g < L ≤ 2R g (4-1) Figure 2. N-dependence of f max (or the corresponding maximum grafting density σmax) in the “grafting to” scenario. The solid, broken, and dashed lines represent theoretical curves predicted by the ES, the mES model without thermal fluctuation, and the mES model with thermal fluctuation, respectively. The dashed arrow indicates the crossover length, Nc, delineating regimes where the ES and mES models are applicable. The inset shows the relation between the MSE and ΔF.

Ψ=

N ⎛⎜ L ⎞⎟ N 1 − + ⎜ ⎟ D 2⎝ 2R g ⎠ 4R g L(D2 − 1) [R gD − 1{DR g 2 − (D + 1)L2} − 2(R g − L)D (DR g + L)] (0 < L ≤ R g)

(4-2)

Ψ(L) = N

chains. Figure 2 shows that this model works well to predict f max in the small N limit, but that it fails for larger N. We conjecture that this is because the chain-level ES description is too crude to capture overlap at the level of two monomers; this failure motivates us to develop the more self-consistent modified equivalent sphere (mES) theory, described below. To improve the ES model, we developed the mES model, which properly captures monomer−monomer overlap across two grafted chains by using the Daoud−Cotton model to

(L = 0)

(4-3)

where D is the fractal dimension of the polymer. The detailed derivation of eqs 4-1, 4-2, and 4-3 is provided in the Methods/ Experimental Section section. From the definition of γmES,chain * : * γmES,chain = 7030

1 8πR g 2

∫0

2π

dϕ

∫0

2R g

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Figure 3. P(f) results for ⟨f⟩ = 6. (a) Examples of P( f) for N = 1 and 200. (b) N-dependence of the normalized variance DN.

Figure 4. Schematic representation of how a grafted chain is considered. (a) Real chain model. (b) ES model (the grafted chain is treated as a hard sphere). (c) mES model (the grafted chain is described in the context of Daoud−Cotton model). Color (red) indicates the distribution of Ψ. Color (blue) shows polymer density.

Figure 5. Schematic illustration of two grafted polymers in overlap. Overlap calculation for two polymeric grafts as a function of their separation distance, L. (a) Rg < L ≤ 2Rg. (b) 0 < L ≤ Rg. (c) L = 0.

⎧1, Ψ ≥ 1 Ψ̅ (L) ≡ ⎨ ⎩ 0, Ψ < 1

to set a cutoff free-energy difference that is allowable due to such thermal fluctuation effects, we select |ΔF| ≈ 6kBT, see min below, to find the chain size Rmin g . Using Rg , the radius of gyration that includes fluctuation effects, instead of Rg in eqs 41, 4-2, 4-3, and 5, gives a more accurate means of calculating f max. The results of these calculations, using the mES model (without/with thermal fluctuation) are shown in Figure 2. It is clear that, as expected, the mES model works better when we consider longer chains. Also, the inset of Figure 2 shows the relation between the mean squared error (MSE) between the theoretical predictions and simulation results in the range N ≥ 80. As we noted, the MSE reaches a minimum around ΔF ≈ 6 kBT. It is important to note that the grafted points are assumed to be immobile in this calculation, and thus we used the random-close-packing fraction on a curved surface as the maximum packing fraction. There are alternative cases where the grafting points are mobile on the NP surface. In such

(6)

We then calculate f max using the relationship f max = ηRCP/ γ*mES,chain. Note here that we use a strict hard-sphere definition of overlap between two monomers, but in reality the monomers have some softness, e.g., the repulsive part of the LJ potential. To this end, we define the chain free energy in the Flory approximation: F (R ) 3R2 υN 2 = + kBT 2Nb2 2R3

(7)

where R is the radius of graft with fluctuation effects, while Rg = bNυ (υ = 3/5 is the Flory exponent in good solvent, and b is Kuhn length) is the unperturbed chain size. We calculate the free-energy difference, ΔF, when the grafts are deformed from Rg (equilibrium state) to R (perturbed state). Since it is natural 7031

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Figure 6. The coverage ratio by polymers on the NP surface. (a) The fraction S* of the NP surface that is covered by spheres to a second approaching NP. We numerically generate f spheres with radius R on the NP surface and directly measure S*. The basic calculation method and geometric concept are described in ref 8. Here, R can be regarded as the effective radius of a polymer where a second approaching NP cannot overlap. Two calculations are made, one with and the other without excluded volume between the spheres at two different values of α = R/Rn and different f values. (b) Data from the “grafting to” scenario, which are modeled by accounting for excluded volume between the *1.17)}1.63, while the data from “grafting from” scenario (not shown) ignores excluded volume between the spheres, S* = {1 − exp(−2.57 fγES * ) . Here, γES * indicates the coverage ratio by a sphere to a second approaching NP. (c and d) Data when spheres, following S* = 1 − exp(−fγES Kremer-Grest chains of length 100 and 200 are grafted to the NPs. The predictions have the same form as that used to model the grafted spheres with excluded volume. We calculated γ* at given N and f using the mES model [unpublished].

Figure 7. Theoretical morphology diagrams as predicted for “grafting from” (dashed lines) and “grafting to” (solid lines) NPs. The experimental TEM micrographs are from ref 13.

f max ≈ 6.2(σmax ≈ 0.01 chains/nm2) when polystyrene-b-poly(2vinylpyridine) (PS-b-P2VP) block copolymer chains were adsorbed onto silica NP (Rn = 7 ± 2 nm) from methyl ethyl ketone (MEK) solution. The mES model without/with thermal fluctuation gives f max ≈ 8.6 and 10.0 (σmax ≈ 0.014 and 0.016 chains/nm2), respectively, where the PS size in MEK was

systems, since the packing efficiency strongly depends on the size ratio of mES sphere to the NP,18 the maximum packing density also depends on the size ratio. As a result, f max should be strongly influenced. We also compare the theoretical predictions for f max with the experimental results of Zhao et al.13 These workers reported 7032

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ACS Nano estimated using Rg = 0.017(N × 104.15)0.54.19 While this is a reasonable agreement, we note that the experimentally determined f max is smaller than the mES model predictions probably because the P2VP adsorbs at multiple points. Since the mES model assumes a single graft point, the excluded area of the different P2VP segments, not taken into account by mES model, might be a source of the disagreement. Another possibility is that the experimental results are still evolving; following Granick et al.,20,21 it is well-known that polymer adsorption becomes exponentially slowed down when one approaches full surface coverage and hence the approach to equilibrium becomes increasingly difficult. Calculation of NP Surface Covered due to Grafted Polymer on a Second Approaching NP. Finally, we enumerate the morphology diagrams for the “grafting to” and “grafting from” analogs using the previous work of Asai et al.8 An important component of making this prediction is to calculate the fraction of the NP surface that is covered by polymer in both cases with respect to the approach of a second NP. We make the following two assumptions: (i) If two ESs on the surface of the first NP are assumed to be fully penetrable to each other (but not to another approaching NP), then we believe that we are roughly equivalent to the “grafting from” scenario; and (ii) if we use the mESs model, where monomers are assumed to be impenetrable to each other as discussed above in the calculation of f max, then we propose that this corresponds to the “grafting to” situation. Figure 6 shows that, as expected, the “grafting from” scenario offers lower polymer coverage of the NP surface than the “grafting to” analog at fixed N and f. The “morphology diagram” (Figure 7) generated from this information8 shows that the “grafting to” scenario offers improved miscibility in all cases relative to the “grafting from” analog. Here we draw the phase boundaries for each morphology using geometric considerations based on the analogy with patchy particles suggested by Asai et al.8 The results in Figure 7 mirror what was experimentally found by Zhao et al.13 Clearly, this result follows from the fact that the “grafting to” scenario offers increased surface coverage of the NP than the “grafting from” scenario, implying that the grafting mechanism plays an important role in this context. In addition to improving the miscibility of the NPs, the “grafting from” scenario also yields a narrower distribution of P( f) and hence a narrower distribution of S* values. This result directly implies that the “grafting from” scenario yields a much broader distribution of self-assembled structures than the “grafting to” case (Figure 8).

Figure 8. Predicted distribution of various self-assembled structures from the “grafting from” and “grafting to” scenarios. The inset has the P( f) distribution that was used to make these predictions. Note that, as expected, the “grafting to” scenario offers a more monodisperse set of structures since the distribution of P(f) is narrower.

dNP−g/(Rn + σ/2), where dNP−g is the distance between the center of mass of the graft sites and the center of the NP, and σ is the size of the graft monomer. We also calculate an anisotropy ΔNP−P, which is defined as ΔNP−P  dNP−P/(Rn + Rg). Here, dNP−P is the distance between the center of mass of the grafted chains and the center of the NP, and dNP−P = 0 implies that the NP is covered isotropically by the polymer segments. Figure 9 shows that the ΔNP−g is unaffected by the

Figure 9. N-dependence of anisotropy Δ of PGNPs.

DISCUSSION Our results above clearly illustrate that the “grafting to” scenario offers two advantages over the “grafting from” situation. First, it results in a narrower distribution in the number of grafted chains, P( f), presumably because of a decreased value of f max when chains are grafted to rather than grafted from. Second, it allows for an increased polymer coverage of the NP surface. These facts result in an improved miscibility of the NPs in a matrix (that is favorable to the chains) and also yield much more monodisperse structures. An aspect that we have ignored to this point are the results of Bozorgui et al.,22 who showed that another factor that governed the structures formed is the anisotropy in the polymer surface coverage of the NP. We define two types of anisotropies. The anisotropy of the graft sites, ΔNP−g, is defined as ΔNP−g 

polymer chain length N in the case of the “grafting from” scenario, while it decreases in the “grafting to” case. While ΔNP−P decreases with N in both the “grafting from” and the “grafting to” methods, the “grafting to” technique always results in more isotropically covered NP. We believe that this last piece of information must also be built into theories that model the effect of grafting mechanism on the self-assembled structures formed by polymer grafted NPs. We expect that the inclusion of these effects should increase the differences between the “grafting to” and “grafting from” scenarios, with the “grafting to” method yielding even more miscible NPs than that predicted in Figure 7. Including this aspect in a more complete theory remains an open challenge at this time and is the focus of ongoing research. 7033

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= 30ε/σ2. We use an expanded LJ potential for describing NP−NP and NP−polymer bead interactions:

Recently, Vaia and co-workers developed a theoretical model to describe the distribution of the number of grafted chains on a NP by using an analogy with micellization.23 While this approach is based on the “macroscopic” picture of the interaction between polymers and NP, our approach is based on a “microscopic” picture. Since their model considers the dense brush regime, f max is overestimated, and their predictions are different from those of our model. Interestingly, however, in the small f limit, both models are in good agreement in terms of the P(f) (not shown), because in this regime, the excluded volume effect plays a minimal role, and the ES model becomes applicable (see Figure 2). While we have pointed to the “grafting to” mechanism as a means of creating a more uniform polymer coverage of the NP surface, many factors, such as fabrication time (much longer for “grafting to”, e.g., refs 19 and 20), decreased design flexibility as N increases (i.e., lower maximum grafting density, Figure 2), and the dependence of the ratio of particle size to polymer size on assembly behavior, imply there is likely an optimal approach for different objectives. For example, the toughness of the nanoparticle−matrix interface, processability, and multifunctionality goals place different requirements on a minimum N and nanoparticle size. “Grafting to” of high N at high grafting densities may be impractical in many instances due to reaction time, throughput, reaction concentration, and maintaining colloidal stability. Additionally, synthetic chemists could develop approaches that lead to more uniform distribution of initiator density on the nanoparticle surface. Such factors and associated complexities need to be used to guide future investigations. Thus, the model presented here and the associated discussion highlight the importance of the uniformity of polymer graft distribution and point out the need for future concepts to increase grafting uniformity.

⎧ ⎡⎛ 12 ⎛ σ ⎞6 ⎤ σ ⎞⎟ ⎪ ⎟ ⎥, r ≤ r + Δ −⎜ ⎪ 4ε⎢⎜ c ⎝r − Δ⎠ ⎦ U (r ) = ⎨ ⎣⎝ r − Δ ⎠ ⎪ ⎪ 0, r > rc + Δ ⎩

Here, we choose Δ = 4σ and Δ = 2σ for NP−NP and NP−polymer bead interactions, respectively. One end monomer of a grafted polymer can be coupled to the surface of the NP (grafting point) and fixed when the distance between the center of the bead and the center of the NP is less than Rn + σ, where Rn is the NP radius and σ is the monomer diameter on the chain. The f grafting points are randomly located on the surface in the “grafting from” scenario, while respecting the impenetrability of the monomers. NNP(= 53) NPs have different arrangements of grafting points. Molecular Dynamics Simulation. All simulations are carried out using the LAMMPS parallel MD package. NVT MD simulations are performed in an orthogonal cubic simulation box. Temperature (T) is set to 1.0ε/kB and is maintained by a Langevin thermostat with a damping constant Γ = 0.01σ−1(m/ε)−1/2, and kB is Boltzmann’s constant. The NPs’ positions are fixed, and only the dynamics of grafted polymers is calculated. The simulations are run for 106−107 time steps of length dt = 0.005 mσ 2/ε to equilibrate the system and then another 106−107 time steps for the grafting reaction. Derivation of Ψ. We will explain the derivation of Ψ. Here a polymer coil with the chain length N is represented by a sphere with radius of Rg, and it has fractal structure inside the sphere. The density of the polymer ρ(r) at distance r from the center of mass of the polymer can be described using the fractal dimension D of the chain as follows:25 ρ(r ) =

ND D − 3 r 4πR gD

(10) 0

0

0 2 r ρ(r)dr Rg

Here we used the mass balance condition: ∫ dϕ∫ sin θdθ∫

CONCLUSIONS In summary, we have shown that the “grafting to” mechanism covers more of the surface of a NP than the “grafting from” analog at the same grafting density. Similarly, the distribution of grafted chains is also narrower in the “grafting to” case. Finally, we show that the anisotropy of the NP surface coverage by the NP is also smaller in the “grafting to” situation. These three facts in conjunction imply that the “grafting to” method creates NPs that are more miscible with the matrix than the “grafting from” analog. Thus, we have shown that the grafting mechanism plays a singularly important role in determining the self-assembled structures that are formed by PGNPs. Hence we believe that the “grafting to” method is preferable in the context of many classes of polymer nanocomposites.

2π

π

= N. First, we consider the case where a polymer coil and another polymer coil are overlapped, but the mass center of each is located outside the other sphere as shown in Figure 5a. This condition corresponds to the following equation: Rg ≤ L ≤ 2Rg. In this case, Ψ can be described as

Ψ(L , R g) =

ND 4πR gD

∫0

2π

dϕ

∫0

θm

sin θdθ

∫r

Rg

r D − 1dr (11)

min

Here, the integration range is the overlapping range of two spheres, and the coordinates within the integration range are denoted by (r, θ, φ) (see Figure 4). Also, we define the maximum θ as θm. Below, the same definitions are made in all cases. Here, cos θm = L/2Rg, and

rmin = L cos θ − (L cosθ)2 − (L2 − R g 2) . As a result, we can derive eq 4-1. Next, let us consider the case where another polymer coil and the targeted polymer coil overlap with each other and the center of mass of each is located inside the other as shown in Figure 5b. This condition corresponds to the following equations: 0 < L < Rg. In this case, Ψ can be described as

METHODS/EXPERIMENTAL SECTION Simulation Model and Grafting Process. Grafted polymers are represented using the coarse-grained bead−spring model of Kremer and Grest.24 Each chain contains N beads of mass m = 1. All beads interact via the Lennard-Jones (LJ) potential: ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σ σ ⎪ ⎪ 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥, r ≤ rc ⎝r⎠ ⎦ Up(r ) = ⎨ ⎣⎝ r ⎠ ⎪ ⎪ r > rc 0, ⎩

(9)

Ψ(L , R n , R g) =

ND ⎧ ⎨ 4πR gD ⎩ +

(8)

∫0

∫0

2π

2π

dϕ +

∫0

dϕ

∫θ

θm

sin θdθ

π

m

sin θdθ

∫0

Rg

∫0

rmax

r D − 1dr

⎫ r D − 1dr ⎬ ⎭ (12)

where r is the distance between two beads, ε is the Lennard-Jones unit of energy, and σ is the bead diameter. We set rc = 21/6σ. Beads along the chain are connected by finitely extensible nonlinear elastic (FENE) springs UFENE(r) = −1/2klmax2 ln[1 − (r/lmax)2], with lmax = 1.5σ and k

Here, cos θm = L/2Rg and rmax = L cos θ − (L cosθ ) − (L − R g 2) . As a result, we derived eq 4-2. 2

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ACS Nano Finally, we consider the case where a polymer coil and the targeted polymer coil are overlapped perfectly as shown in Figure 5c. This condition corresponds to the following equations: L = 0. In this case, Ψ can be described as Ψ(L , R g) = N

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As a result, we derived eq 4-3.

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Sanat K. Kumar: 0000-0002-6690-2221 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS The authors thank the National Science Foundation for financial support of this work. S.K.K. acknowledges the National Science Foundation through grant DMR-1709061. The authors thank Dr. Richard Vaia for helpful discussions. REFERENCES (1) Belkin, M.; Snezhko, A.; Aranson, I.; Kwok, W.-K. Driven Magnetic Particles on a Fluid Surface: Pattern Assisted Surface Flows. Phys. Rev. Lett. 2007, 99, 158301. (2) Tang, Z.; Zhang, Z.; Wang, Y.; Glotzer, S. C.; Kotov, N. A. SelfAssembly of CdTe Nanocrystals into Free-Floating Sheets. Science 2006, 314, 274−278. (3) Van Workum, K.; Douglas, J. F. Symmetry, Equivalence, and Molecular Self-Assembly. Phys. Rev. E 2006, 73, 031502. (4) Sacanna, S.; Pine, D. J. Shape-Anisotropic Colloids: Building Blocks for Complex Assemblies. Curr. Opin. Colloid Interface Sci. 2011, 16, 96−105. (5) Green, D. L.; Mewis, J. Connecting the Wetting and Rheological Behaviors of Poly(dimethylsiloxane)-Grafted Silica Spheres in Poly (dimethylsiloxane) Melts. Langmuir 2006, 22, 9546−9553. (6) Krishnamoorti, R.; Vaia, R. A. Polymer Nanocomposites. J. Polym. Sci., Part B: Polym. Phys. 2007, 45, 3252−3256. (7) 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, 354−U121. (8) Asai, M.; Cacciuto, A.; Kumar, S. K. Quantitative Analogy between Polymer-Grafted Nanoparticles and Patchy Particles. Soft Matter 2015, 11, 793−797. (9) Li, C. Z.; Benicewicz, B. C. Synthesis of Well-Defined Polymer Brushes Grafted onto Silica Nanoparticles via Surface Reversible Addition-Fragmentation Chain Transfer Polymerization. Macromolecules 2005, 38, 5929−5936. (10) von Werne, T.; Patten, T. E. Atom Transfer Radical Polymerization from Nanoparticles: A Tool for the Preparation of Well-Defined Hybrid Nanostructures and for Understanding the Chemistry of Controlled/“Living” Radical Polymerizations from Surfaces. J. Am. Chem. Soc. 2001, 123, 7497−7505. (11) Bartholome, C.; Beyou, E.; Bourgeat-Lami, E.; Chaumont, P.; Zydowicz, N. Nitroxide-Mediated Polymerizations from Silica Nanoparticle Surfaces: “Graft From” Polymerization of Styrene using a Triethoxysilyl-Terminated Alkoxyamine Initiator. Macromolecules 2003, 36, 7946−7952. (12) Jouault, N.; Lee, D.; Zhao, D.; Kumar, S. K. Block-CopolymerMediated Nanoparticle Dispersion and Assembly in Polymer Nanocomposites. Adv. Mater. 2014, 26, 4031−4036. (13) Zhao, D.; Di Nicola, M.; Khani, M. M.; Jestin, J.; Benicewicz, B. C.; Kumar, S. K. Role of Block Copolymer Adsorption versus Bimodal 7035

DOI: 10.1021/acsnano.7b02657 ACS Nano 2017, 11, 7028−7035