Impact of Core Size Distributions on Polymer-Grafted Nanoparticles - PDF

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This is an open access article published under an ACS AuthorChoice License, which permits copying and redistribution of the article or any adaptations for non-commercial purposes. Article pubs.acs.org/Macromolecules Impact of the Distributions of Core Size and Grafting Density on the Self-Assembly of Polymer Grafted Nanoparticles Nirmalya Bachhar,† Yang Jiao,‡ Makoto Asai,† Pinar Akcora,*,‡ Rajdip Bandyopadhyaya,§ and Sanat K. Kumar*,† † Department of Chemical Engineering, Columbia University, New York, New York 10027, United States ‡ Department of Chemical Engineering & Materials Science, Stevens Institute of Technology, Hoboken, New Jersey 07030, United States § Department of Chemical Engineering, Indian institute of Technology Bombay, Powai, Mumbai 400076, India See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. * S Supporting Information ABSTRACT: It is now well-accepted that hydrophilic nanoparticles (NPs) lightly Downloaded via BROCK UNIV on October 18, 2024 at 18:30:45 (UTC). grafted with polymer chains self-assemble into a variety of superstructures when placed in a hydrophobic homopolymer matrix or in a small molecule solvent. Currently, it is thought that a given NP sample should only assemble into one kind of superstructure depending on the relative balance between favorable NP core−core attractions and steric repulsion between grafted polymer chains. Surprisingly, we find that each sample shows the simultaneous formation of a variety of NP-assemblies, e.g., well-dispersed particles, strings, and aggregates. We show through the generalization of a simple geometric model that accounting for the distributions of the NP core size and the number of grafted chains on each NP (which is especially important at low coverages) allows us to quantitatively model the aggregate shape distribution. We conclude that, in contrast to molecular surfactants with well-defined chemistries, the self-assembly of these NP analogues is dominated by such fluctuation effects. INTRODUCTION The self-assembly of molecular surfactants is uniquely discussed in the Experimental Section). Asai et al.2 have further shown that a simple geometrical model, where a PGNP7 is determined by their chemical structure. Israelachvili,1 for mapped into a Janus particle,10 with one lyophilic and one example, popularized this concept and emphasized the lyophobic patch, can be used to explain the molecular origins of importance of a geometrical quantity, the packing parameter, each of these experimentally observed morphologies. in determining the specific self-assembled structure that forms at Thus, it is now believed that a given PGNP sample should a given state point. These ideas assume that the underlying shape form a single type of self-assembled structure. We have realized and size of all the surfactants (amphiphiles) in a given sample are by a careful reanalysis that each sample instead shows the identical, a fact that is guaranteed for the case of molecular formation of a variety of structures (strings, clumps, aggregates, surfactants with defined chemistries. well-dispersed NPs) even though only one typically is dominant. By analogy, it is now thought that the aggregation behavior of While it is possible that these are kinetically trapped structures, polymer-grafted nanoparticles (PGNPs) is controlled by their such distributions do not disappear with annealing time. In amphiphilicity,3,4 i.e., by the balance between the effective NP addition, we find that these assemblies form in both polymeric core−core attractions and steric repulsion afforded by the matrices and when the NPs are dispersed in solvent and then polymeric tethers.5,6 Thus, it is expected that these NP-based drop cast on TEM grids (followed by solvent evaporation). Such building blocks should also form uniquely shaped self-assembled aggregates are also found in aqueous dispersions of dextran- structures in a given medium.7−9 In particular, Akcora et al.7 have coated iron oxide NPs.11 We propose that the different assembly shown that the different morphological structures that result shapes occur naturally because the NP cores have a size when the PGNPs are placed in a matrix polymer, with the same distribution. We convolve this effect with the well-established chemistry as the brush chains, can be broadly classified as notion that the number of grafted chains on each NP is not disperse (isolated NPs), string-shaped (1D), sheet-like (2D), constant, but rather has a well-defined distribution. When we and spherical aggregates (3D), with different structures being incorporate both of these effects into the geometrical model of formed by varying the grafting density and their chain length Asai et al.,2 we can nearly quantitatively describe the shape (molecular weight) and the embedding medium’s molecular weight. Our current understanding of the different self- Received: May 25, 2017 assembled structures formed by these systems is schematically Revised: July 12, 2017 shown in Figure 1 (detailed classifications of each structure are Published: September 22, 2017 © 2017 American Chemical Society 7730 DOI: 10.1021/acs.macromol.7b01093 Macromolecules 2017, 50, 7730−7738 Macromolecules Article Figure 1. Different aggregate morphologies with decreasing polymer surface coverage, following Asai et al.:2 (a) isolated particle with full surface coverage (coordination number 0); (b) small aggregates including dimers, trimers, and tetramers (coordination number 1−3); (c) one-dimensional linear aggregate (coordination number 2); and (d) two-dimensional and higher order aggregates (coordination number ≥4). Figure 2. Representative TEM images showing different morphologies for different grafting densities in a 124 kDa polystyrene matrix. The top panels (a−c) are for polystyrene grafted (Mw = 43 kDa) iron oxide nanoparticles having grafting density (ρg): (a) 0.017, (b) 0.044, and (c) 0.066 chains/nm2. The bottom panels (d, e) show polystyrene grafted (Mw = 124 kDa) iron oxide nanoparticles having ρg: (d) 0.013 and (e) 0.052 chains/nm2; in the same 124 kDa polystyrene matrix. distributions seen in the experiments. We therefore propose that the self-assembly of these PGNPs corresponds to a linear EXPERIMENTAL SECTION We examine the self-assembly and resulting morphologies of polystyrene-grafted iron oxide (Fe3O4) nanoparticles (NPs), both in a superposition of the structures formed by each member of this polystyrene homopolymer matrix and in a toluene dispersion. We also NP population (that is characterized by the core size and the studied an aqueous dispersion of dextran-coated (physically adsorbed) iron oxide NPs (reported in the Supporting Information). number of grafted chains). We emphasize that these effects are Synthesis of Polystyrene Tethered Iron Oxide (Fe3O4) inherent to this class of NPs and that they are very different in Nanoparticles and Preparation of Pure Grafted Particles and Their Composites for Imaging. Oleic acid and oleylamine-stabilized origin from the case where the grafts have a distribution of chain Fe3O4 nanoparticles were synthesized by the thermal decomposition lengths.12 method, following the protocol reported by Sun et al.13 Polystyrene 7731 DOI: 10.1021/acs.macromol.7b01093 Macromolecules 2017, 50, 7730−7738 Macromolecules Article Figure 3. Representative TEM images showing different morphologies in toluene dispersion. The top panels (a−c) show TEM images of polystyrene grafted (Mw = 43 kDa) iron oxide nanoparticles having grafting density (ρg): (a) 0.017, (b) 0.044, and (c) 0.066 chains/nm2. The bottom panels (d, e) show TEM images of polystyrene grafted (Mw = 124 kDa) iron oxide nanoparticles having ρg: (d) 0.013 and (e) 0.052 chains/nm2. grafted Fe3O4 nanoparticles were prepared by grafting-to and grafting- solution (where such dynamic effects should be unimportant) give us from methods as reported in our previous work.5 some assurance. So, in summary, there are potential kinetic effects in Grafted nanoparticles in toluene were slowly added to the both cases, but since all three systems yield the same findings, we believe polystyrene matrix dissolved in toluene (5 wt % of PGNPs with respect that our findings are a robust facet of their behavior. to matrix polymer or toluene dispersion), sonicated, and cast to form To lend credence to our argument against kinetic effects, we estimate films. Bulk films were annealed at 150 °C for 7 days in a vacuum oven the displacement of particles following the Stokes−Einstein equation, and ultramicrotomed into 50−80 nm slices with a diamond knife at x = 6Dpt , where t and Dp are time and diffusivity of the particle in the room temperature and examined by a transmission electron microscope liquid; we estimate Dp = kBT/6πηr, where kB, T, r, and η are the (FEI CM20 FE S/TEM) operated at 200 keV. Many TEM images were Boltzmann constant, temperature, dynamic viscosity, and the NP radius. captured over different areas and representative images presented in It is clear from the Table 1 that the NPs should move many times their Figure 2. The dispersion of grafted NPs in toluene was analyzed by casting a drop of solution on a Formvar-coated grid. Toluene evaporated in several minutes on the grid. These samples are mentioned as Table 1. List of Kinetic Effect Parameters and Results “nanoparticles in toluene dispersion” in the paper (Figure 3). parameters values reference/remarks ImageJ software was used for the manual analysis of the TEM T toluene 298 K our experiment micrographs. The micrographs were divided into small areas for counting. The image contrast was increased in some cases for better polystyrene 423 K melt visualization. Any overlap of two or more nanoparticles (below or above η toluene 0.56 × 10−3 Pa·s Krall et al.2 the focal plane) increases the contrast; in such cases the particles are polystyrene 0.251 × 105 Pa·s Rudin and Chee3 assumed to be touching, and the coordination number is calculated melt accordingly. If the particles having a significant gap (>0.5 nm) between ⟨r⟩ 3.5 nm our experiment them, which can be characterized by a strong contrast difference Dp toluene 1.5 × 10−10 m2 s−1 following Stokes− between two particles, the particles are assumed to be not touching. Einstein equation Converting the micrographs into binary images was not helpful as the polystyrene 3.5 × 10−18 m2 s−1 resulting images smudge aggregates into one large particle. A better melt technique is yet to be developed for such systems. For the statistical t toluene 5 min our experiment analysis, in each system, the TEM micrographs were divided into 3−4 polystyrene 7 days groups. The mean of the groups was reported, and the standard melt deviations were plotted as the error bar on the experimental data. For the x toluene 0.53 mm x/⟨r⟩ = 1.5 × 105 nanocomposite system and toluene dispersion ∼2000 and ∼1500 polystyrene 3.56 μm x/⟨r⟩ = 1.0 × 103 particles, respectively, were analyzed. melt Kinetic Effects. There are potential kinetic effects in both experimental situations considered here (and in the dextran-coated NPs discussed in the Supporting Information). In the solution case, we diameter (and also many times the inter-NP distance) during our take a PGNP dispersion in toluene and drop cast it on a TEM grid, annealing protocols hence again verifying that kinetic effects probably where the solvent evaporates. There is a possibility that the NP are not critical here. structures in this case form due to solvent evaporation. This concern is Structural Classification. We have analyzed the coordination alleviated by the fact that the NPs form a similar distribution of number of each NP in a sample (present either as an isolated particle or aggregates in a polymer melt host. While the slow NP dynamics could as a part of any aggregate) and categorized it into one of four different play a role in the melt case, the fact that we see the same structures in classes. 7732 DOI: 10.1021/acs.macromol.7b01093 Macromolecules 2017, 50, 7730−7738 Macromolecules Article Table 2. List of Experimental Parameters grafting density, ⟨ρg⟩ Radius, μp std. deviation, σp Figure Figure matrix/dispersant grafted polymer (chains/nm2) (nm) (nm) (TEM) (results) polystyrene matrix (Mw ∼ 124 polystyrene (Mw ∼ 0.017 7 0.7 2a 4a kDa) 43 kDa) 0.044 7 0.7 2b 4b 0.066 7 0.7 2c 4c polystyrene (Mw ∼ 0.017 7 0.7 2d 4d 124 kDa) 0.052 10 1.37 2e 4e toluene dispersion polystyrene (Mw ∼ 0.017 6.8 1.9 3a 4a 43 kDa) 0.044 6.8 1.9 3b 4b 0.066 6.8 1.9 3c 4c polystyrene (Mw ∼ 0.013 7 0.7 3a 5a 124 kDa) 0.052 10 1.37 3b 5b aqueous dispersion dextran (Mw ∼ 60 kDa) 0.27 4.5 0.55 S1a S2a 0.35 3.0 0.3 S1b S2b dextran (Mw ∼ 100 kDa) 0.26 4.7 0.6 S1c S2c 0.35 3 0.3 S1d S2d Figure 4. Morphological distribution of aggregates in polystyrene matrix (Mw = 124 kDa). The top panels (a−c) show the distribution of polystyrene- grafted (Mw = 43 kDa) iron oxide nanoparticles having grafting density (ρg): (a) 0.017, (b) 0.044, and (c) 0.066 chains/nm2. The bottom panels (d, e) show the distribution of polystyrene-grafted (Mw = 124 kDa) iron oxide nanoparticles having ρg: (d) 0.017 and (e) 0.052 chains/nm2. A total of ∼2000 nanoparticles were analyzed for each case (a)−(e) to ensure statistical significance of mean and standard deviation values. (a) Dispersed NP: Isolated particles with coordination number 0 have higher surface coverage in order to arrest further aggregation, (Figure 1a). which justifies this classification. (b) Clumps: A NP is assigned to the “clump” population in three (c) Strings: A particle belongs to the “string” population (Figure 1c) if different situations: (i) if it is a part of an aggregate of 2 or more NPs, it is part of a linear or branched chain (coordination number 2). where the particle coordination number is in the range 1−3 (Figure 1b); Therefore, the end particles of a string and the particle at the branching (ii) if it is the end particle of a linear string, i.e., coordination number 1; junction of a branched string belong to the clump population. or (iii) if it is the peripheral particle of a large aggregate with (d) Aggregates: A NP is part of the “aggregate” population if it belongs coordination numbers 1−3. The NP terminating any structure has to to a large aggregate containing 4 or more particles having coordination 7733 DOI: 10.1021/acs.macromol.7b01093 Macromolecules 2017, 50, 7730−7738 Macromolecules Article Figure 5. Morphological distribution of aggregates in toluene dispersion. The top panels (a−c) show the distribution of polystyrene-grafted (Mw = 43 kDa) iron oxide nanoparticles having grafting density (ρg): (a) 0.017, (b) 0.044, and (c) 0.066 chains/nm2. The bottom panels (d, e) show the distribution of polystyrene-grafted (Mw = 124 kDa) iron oxide nanoparticles having ρg: (d) 0.017 and (e) 0.052 chains/nm2. A total of ∼1500 nanoparticles were analyzed for each case (a)−(e). number 4 or more. This assignment combines two-dimensional sheet- generated a morphology diagram where different structures are like structures (Figure 1d) and three-dimensional disordered formed in distinct regions of any f vs α plot. This theory,2 which aggregates2 into one classification. assumes that all NPs in a sample are identical, predicts the These definitions are constructed following the approach of Asai et formation of a single aggregate shape at each state point, in al.,2 which utilizes the available surface area for NP−NP aggregation disagreement with current experimental results in Figures 4, 5, driven by core−core attractions. and S2. We postulate that different structures occur in a single system RESULTS AND DISCUSSION since the NPs possess a size distribution coupled to a distribution Jiao and Akcora5 showed a transition from larger to smaller of the number of grafted polymer chains on the NP surface. The strings/aggregates with increases in average grafting density, core iron oxide NP size histogram (Figure 6a) corresponds to the ⟨ρg⟩, of polystyrene-grafted iron oxide NPs, in both a polystyrene TEM images shown in Figures 2a−c. Similar histograms were matrix (Figure 2) and toluene dispersion (Figure 3). We have calculated for Figures 3 and S1 (not shown). The histogram was also synthesized aqueous dispersions of in situ dextran adsorbed converted to a probability density function and fit to a normal iron oxide NPs.16,17 The list of all experimental systems studied is distribution; the mean radius of the core particle (μp) and given in Table 2. standard deviation (σp) being the two parameters (the fit values The most striking feature of each TEM derived plot is the are in Table 2). existence of a spectrum of assemblies going from well-dispersed Since the NP radius (r) has a distribution (Figure 6a), α NPs to random, large NP aggregates in each sample (the (which is a function of r) should also have a distribution. histograms of morphologies are shown in Figures 4, 5, and S2). ⎡ re 2 ⎤ This result is in sharp contrast to expectations from molecular re ⎢ exp⎢ ( α − μp )⎥ surfactants where only a single type of aggregate (e.g., spherical pα (α) = ⎥ micelles, worm-like cylinders, or lamellar structures) is expected α 2σp 2π ⎢ 2σp2 ⎥ ⎣ ⎦ (1) at a state point. Asai et al.’s2 theory shows that the formation of a given self- which is also a restatement of the NP size distribution in terms of assembled structure is controlled by two parameters: (i) a α. One such representative plot of α generated following this geometrical parameter α (=re/r), which is the ratio of the procedure is shown in Figure 6b. equivalent polymer radius (re, which is different from the radius Hakem et al.18 showed that there exists a distribution of the of gyration of the free polymer) to the NP radius, and (ii) the number of ligands on the NP surface, pf(f |f max). f is the number of number of polymers attached on the NP surface ( f). They grafted chains per NP, and f max is the maximum number of chains 7734 DOI: 10.1021/acs.macromol.7b01093 Macromolecules 2017, 50, 7730−7738 Macromolecules Article Figure 6. Relevant sample distributions and morphology diagram of different aggregation morphologies of polystyrene (Mw = 43 kDa, ⟨ρg⟩ = 0.017 chains/nm2) grafted Fe3O4 nanoparticles. (a) Experimental histogram data of NP diameter from ref 5, fitted to a normal probability density function. (b) Probability distribution of α. (c) Probability distribution of f max and (d) probability distribution of f. (e) Phase diagram of different morphologies are shown for samples in Figure 2e. Different morphological phase boundaries are marked by black, red, and green curves. The contour map is a representation of the bivariate probability distribution for all combinations of α and f. that can be grafted on a single NP. Since pf( f|f max) is hard to contour map, for one single specific case of polystyrene (43 kDa, characterize experimentally, we follow the ideas of Hakem et ⟨ρg⟩ = 0.017 chains/nm2) grafted iron oxide nanoparticles in a al.,18 who have validated the following conditional probability polystyrene matrix (124 kDa). The contour maps of the bivariate density form of pf( f |f max) against experiment: probability distribution for each NP size class and each number of grafting chain-class are integrated to yield fractions of each fmax ! pf (f |fmax ) = e−fmax v(e v − 1) f morphology type, shown as relative percentages in Figures 4, 5, f ! (fmax − f )! (2) and S2. The results clearly show that in a polymer matrix, where the aggregation is thermodynamically stable (annealed for 7 where v = −ln(1 − ε) and ε = ⟨f⟩/f max. ⟨f⟩ is the mean value of f, days), we have a distribution of different self-assembled which is proportional to the surface area of a particular NP of structures. These facts are captured by our model. Even in radius r. Thus, for a particular NP of radius, r, f max = (1/ solvent dispersion (organic or aqueous) the experimental results ε)4πr2⟨ρg⟩. By knowing the size distribution of the NPs, we can match closely with predictions, verifying the robustness of the thus write the probability density function of f max model. ⎛ To elaborate the link between the two individual parameters 21⎞ p f (fmax ) = p f ⎜4π ⟨ρ ⟩r ⎟ (namely r and f) and the resultant aggregate structure further, we max max ⎝ g ε⎠ have carried out a sensitivity analysis involving the effect of each ⎡ ⎛ ⎞2 ⎤ of these two parameters individually. Five of the representative ⎢ ⎜ εfmax − μp ⎟ ⎥ results are given in Figure 7. Sensitivity analyses for all systems 1 ε ⎢ ⎝ 4π ⟨ρg ⟩ ⎠ ⎥ are shown in Figures S3−S5 of the Supporting Information. = exp⎢ − 2 ⎥ πσ 32fmax ⟨ρg ⟩ ⎢ 2σp ⎥ Examining all the subplots in Figure 7, one can infer that the ⎢ ⎥ distribution in r, the core radius, greatly influences the shape ⎣ ⎦ (3) distribution of aggregates (red and blue bars in Figures 7a−e), whereas the distribution in f, the surface coverage, plays a which is nothing but a restatement of the NP size distribution in secondary role (green bars in Figures 7a−e). Both α and f are terms of f max. A representative plot of the probability of f max is functions of r whereas only f is also a function of ρg. Thus, this shown in Figure 6c. Thus, using eqs 2 and 3, pf,f max(f) = pf(f |f max) sensitivity analysis nicely delineates the role of each distribution pf max(f max). Figure 6d shows that this yields a distribution of the and identifies the more critical parameter (among the two) in this number of attachment sites on the NP. case. For fixed values of f and α (without any distributions), Finally, we note that there are two fit parameters re and ε that following Asai et al.,2 one expects a single particle aggregate are used to achieve agreement with the experimental data. From a shape. However, since we have distributions in both f and α, we geometric viewpoint the effective radius (re) of an individual will have an entire spectrum of structures. Figure 6e shows this polymer on the nanoparticle decreases as the number of grafting through a representative probability distribution, plotted as a chain per particle ( f) increases; this corresponds to an increased 7735 DOI: 10.1021/acs.macromol.7b01093 Macromolecules 2017, 50, 7730−7738 Macromolecules Article Figure 7. Distributions showing relative frequencies of different morphologies of aggregates to assess the role of distributions in r and f: (a) graft polystyrene (Mw ∼ 43 kDa, ρg = 0.017 chains/nm2); (b) graft polystyrene (Mw ∼ 124 kDa, ρg = 0.013 chains/nm2); (c) graft polystyrene (Mw ∼ 43 kDa, ρg = 0.017 chains/nm2); and (d) graft polystyrene (Mw ∼ 124 kDa, ρg = 0.013 chains/nm2). (a) and (b) are in polystyrene matrix, and (c) and (d) are in toluene dispersion. The different bar charts represent models with different choice of parameters as discussed in the text. The black bar represents the experimental distribution of particle aggregate shapes. The red bar represents the model accounting for distribution of both r and f. The blue bar represents the model accounting for distribution in r but constant mean f. The green bar represents the model accounting for distribution in f but constant average r, and finally the cyan bar represents the model accounting for both average r and f following Asai et al.2 Figure 8. re and ε obtained from model fits to experiment plotted as a function of ρg for brushes of Mw (a) ∼43 kDa and (b) ∼124 kDa in a polystyrene matrix and in toluene dispersion. The lines are drawn as a guide to the trends. overlap between adjacent grafted chains. Similarly, when ρg → (Figure 8a). (We remind the reader that we can vary the solvent ρg,max, ε → 1. Our fitted parameter values show these expected quality of the matrix by varying its molecular weight relative to trends (Figure 8). The data show that when the tethered the brush; when the brush becomes significantly shorter than the polymers are in a good solvent, such as a 124 kDa brush in a 124 matrix, then the solvent condition is poor, while good solvent kDa polystyrene matrix or in toluene (Figure 8b), very good conditions are applicable otherwise.) This shows that although agreement is obtained for effective polymer radius (re) in the two the model can predict the morphological distribution even in the cases. However, in the case of poor solvent the match is not close case of poor solvent conditions, the parameter values required to 7736 DOI: 10.1021/acs.macromol.7b01093 Macromolecules 2017, 50, 7730−7738 Macromolecules Article completely capture the phenomenon no longer are physically et al.25 have experimentally shown that the toughness of a meaningful. nanocomposite can be increased by increasing the nanoparticle DISCUSSION While our current work emphasizes the importance of NP size loading. However, their study incorporates particles which are much larger than the range of effective improvement of toughness as theoretically shown by Zappalorto et el.24 These distributions (and the distribution of the number of grafts per studies suggest that there is an incentive to model the NP) on the self-assembled structures that form, it is important to nanoparticle aggregation in nanocomposites to better engineer note that the impact of particle size on complex morphology sample toughness. Our groups are currently working on this development is known in other fields. For example, size topic. segregation is common in the preparation of TEM grids of the product of NP synthesis. Size separation and size-dependent packing are known in granular materials.19−22 The physics of CONCLUSIONS While the phenomenon of NP self-assembly has been studied for these processes are extensively discussed. Thus, our work follows a long time, the effect of a grafted/adsorbed polymer on the in a long lineage that emphasizes that sample imperfections can shape and structures of the resulting aggregates has not been sometimes dominate the behavior of the systems being well-studied or understood. There is no literature which investigated. addresses the question as to why a given polymer grafted NP It is important to note that our modeling approach separately sample forms a distribution of aggregate shapes in the same given delineates the structure formed by each member of the NP matrix/dispersion. In general, such distributions were neglected population; i.e., we independently characterize how each type of (or presented as a kinetic limitation), and only the mean shapes NP, characterized by a specified value of the core size and the were reported and understood.26−29 Our major result is to number of grafted chains, forms a superstructure. We then emphasize that, in contrast to conventional surfactants, the self- assume that the distribution of self-assembled structures seen in a assembly of polymer-grafted NPs is dominated by their core size sample corresponds to a linear superposition of such structures distributions coupled to the distribution of the number of grafted formed by each member of the population. The success of this chains on each NP. In particular, we show that distribution of approach effectively implies that there are no measurable self-assembled structures in a given specimen corresponds to a correlations between the self-assembly of particles of different linear superposition of the structures formed by each member of size or between particles with different numbers of grafted the NP population (that is uniquely characterized by its core size chains.a This simplification allows the physics of the system to be and the number of grafted chains). Since simple geometric described by existing treatments. If there was more complex models can be used to characterize these distributions, we believe cross-correlation, the predicted distributions would not match. that we have a unique means of understanding the self-assembly Our model, which emphasizes the importance of NP size in in this interesting class of materials and hence their properties. determining the type of self-assembled structure formed, would imply that the mean NP size in the different assemblies in a given ASSOCIATED CONTENT sample should be different. In our current experimental work, the particle size polydispersity is not high enough to identify each of * S Supporting Information the aggregate class size separately (Figure 9). However, our The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macro- mol.7b01093. Experimental method, TEM images, and morphological distribution of aggregates for dextran-coated iron oxide NPs in aqueous dispersion; sensitivity analysis for re and ε all systems (PDF) AUTHOR INFORMATION Corresponding Authors *(S.K.K.) E-mail: [email protected]. *(P.A.) E-mail: [email protected]. ORCID Nirmalya Bachhar: 0000-0001-9369-9655 Pinar Akcora: 0000-0001-7853-7201 Figure 9. Effect of size−polydispersity in the theoretical aggregation mean. Sanat K. Kumar: 0000-0002-6690-2221 Notes The authors declare no competing financial interest. model predicts that a similar mean NP size, but with higher polydispersity, should show such size segregation (Figure 9). Indeed, recent unpublished work from the Koberstein group ACKNOWLEDGMENTS supports this point. We leave this as an open question that should N.B. thanks Indo-US Science and Technology Forum (IUSSTF) be addressed by future experiments. and SERB for the fellowship grant (SERB Indo-US postdoctoral Adding nanoparticles to a polymer is known to improve the fellowship 2016/99). N.B. gratefully acknowledges the sophis- mechanical toughness of nanocomposites. Chen et al.23 have ticated analytical instrumental facility (SAIF) of the Indian shown that nanocomposite toughening may be strongly affected Institute of Technology Bombay for providing TEM facilities. by the size of the nanoparticles and by surface treatments.24 Brey P.A. acknowledges financial support from NSF-DMR CAREER 7737 DOI: 10.1021/acs.macromol.7b01093 Macromolecules 2017, 50, 7730−7738 Macromolecules Article grant (No. 1048865). S.K. acknowledges the National Science Particle and Particlelike Systems. J. Am. Chem. Soc. 2010, 132 (46), Foundation under Grant NSF-DMR 1709061. 16593−16598. (19) Wong, T. S.; Chen, T. 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