Fluctuation Effects on the Brush Structure of Mixed Brush

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Fluctuation Effects on the Brush Structure of Mixed Brush Nanoparticles in Solution Jason P. Koski, and Amalie L. Frischknecht ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.7b08271 • Publication Date (Web): 18 Jan 2018 Downloaded from http://pubs.acs.org on January 20, 2018

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Fluctuation Effects on the Brush Structure of Mixed Brush Nanoparticles in Solution Jason P. Koski∗ and Amalie L. Frischknecht Sandia National Laboratories, Albuquerque, New Mexico 87185, USA E-mail: [email protected]

Abstract A potentially attractive way to control nanoparticle assembly is to graft one or more polymers on the nanoparticle, to control the nanoparticle-nanoparticle interactions. When two immiscible polymers are grafted on the nanoparticle, they can microphase separate to form domains at the nanoparticle surface. Here we computationally investigate the phase behavior of such binary mixed brush nanoparticles in solution, across a large and experimentally relevant parameter space. Specifically, we calculate the mean-field phase diagram, assuming uniform grafting of the two polymers, as a function of the nanoparticle size relative to the length of the grafted chains, the grafting density, the enthalpic repulsion between the grafted chains, and the solvent quality. We find a variety of phases including a Janus phase and phases with varying numbers of striped domains. Using a non-uniform, random distribution of grafting sites on the nanoparticle instead of the uniform distribution leads to the development of defects in the mixed brush structures. Introducing fluctuations as well leads to increasingly defective structures for the striped phases. However, we find that the simple Janus phase is preserved in all calculations, even with the introduction of non-uniform grafting and fluctuations. We conclude that the formation of the Janus phase is more realistic experimentally than is the formation of defect-free multivalent mixed brush nanoparticles.

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Keywords: polymers, polymer nanocomposites, mixed brush, brush structure, field theory, thermal fluctuations, non-uniform grafting

Grafting multiple chemically distinct polymers to a surface to create a mixed brush provides a practical route to construct materials with tunable surface functionality that hold promise in applications such as colloid stability, wettability, and adhesion. 1–4 While the behavior of planar mixed brushes has been studied extensively, mixed brushes on surfaces with high curvature, such as nanoparticles, have only recently started to receive attention. While the most comprehensively studied polymer nanocomposite (PNC) system is homopolymer-grafted nanoparticles in a chemically identical polymer matrix, 5 using a mixed brush on the particles provides an extra degree of control in tuning the distribution of nanoparticles to optimize the target property or application. Ferrier et al. 6 showed that mixed brushes can considerably improve the dispersion of gold nanorods in a high molecular weight matrix polymer over the analogous athermal homopolymer brush case, thus improving the degree of control of the optical properties in these systems. Mixed brush nanoparticles can also exhibit interesting self-assembly behavior. The assembly of nanoparticles functionalized with multiple ligands or surfactants has been extensively studied experimentally and through simulations. 7–9 The case of long polymers grafted onto nanoparticles, where the radius of gyration Rg of the polymer is of similar magnitude as the particle radius R p , has been less explored. Previous experimental work in this regime has shown that the grafted nanoparticles can form complex structures that include nano rod-like or “worm-like” assemblies 10 and nanovesicles 11 using 2 nm and 14 nm diameter particles, respectively. These nanoparticle assemblies could be used as ion or electron channels in solution or as drug delivery vesicles. Understanding this assembly behavior theoretically can help refine the development of previously observed structures and design desired nanoparticle assemblies. Towards this goal, in this paper we calculate the structure of a binary mixed polymer brush on

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a single nanoparticle in solution, for RP /Rg ≈ 1. When two immiscible polymers are covalently grafted to a surface, they microphase separate from each other and form domains on the nanoscale. It is informative to review the behavior of planar mixed brushes, which represent the limit of RP /Rg >> 1. Self-consistent field theory (SCFT) and simulation studies 12–17 of planar, binary mixed brushes have shown that the two polymers can phase separate perpendicular (vertically) or parallel (laterally) to the surface, depending on the system parameters. Experimental work has verified both kinds of phase separation in mixed brushes. 13,14,18 For mixed brushes that laterally phase separate, the nanoscale patterns that form typically have poor long-range order, unlike the perfect long range order originally predicted by SCFT. The origin of this lack of long range order has been shown to be randomness in the spatial locations of the grafting sites for the two polymers on the surface. 19 In particular, using a random, non-uniform grafting site distribution in SCFT calculations leads to less long range order in the system 20,21 . An additional aspect to consider in the SCFT studies is the suppression of fluctuations via the mean-field approximation. Monte Carlo single-chain-in-mean-field (MC-SCMF) studies demonstrate that the introduction of thermal fluctuations exacerbates the effect of the non-uniform grafting distribution. 16,19 Lateral phase separation is of interest for forming nanoparticles with different polymer domains arranged spatially around the surface of the nanoparticle. In principle, well-ordered polymer domains could lead to specific nanoparticle assemblies. Previous theoretical work on mixed brush nanoparticles 22–27 has shown the emergence of interesting phases that range from a simple Janus phase (where each polymer forms a domain on half the particle), to structures with multiple striped or spotted domains, referred to as multivalent particles. However, these studies either focused on a limited parameter space or used overly simplified approximations. Kim et al. 22 used Monte Carlo simulations to model a specific mixed brush nanoparticle in solution to complement experiments. Koski et al. 23 used a combination of SCFT and fluctuating field theory simulations 28 to study the structure and interfacial activity of complex grafted nanoparticles, including mixed brush particles, at the interface of a polymer blend. More extensive parameter space studies analyzed the mixed brush structure on both spheres 24,25 and nanorods 26 in solution as well as the assembly of neat

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mixed brush nanoparticles (no solvent) into lattice structures. 27 Though Roan 24 performed a few calculations with non-uniform grafting, the remaining studies used strictly uniform grafting distributions of the polymers. Experimentally, mixed brush particles will have non-uniform, random grafting sites. This non-uniformity in the grafting sites is likely to introduce defects in the brush structure, in analogy to the behavior of planar mixed brushes. These defects, in turn, are likely to lead to less long-range order in mixed brush nanoparticle assemblies, such as the lattice structures that Chen et al. 27 predict. Furthermore, the previous studies used a mean-field approximation, even for the systems in solution, where it is understood that fluctuations are prominent and that the mean-field approximation is inaccurate. 28 Here we investigate the effects of fluctuations and of non-uniform grafting site distributions on the brush structure of a single particle using a combination of SCFT and a fluctuating dynamic mean-field theory 29–31 (DMFT). The DMFT method used in this study is similar to the MC-SCMF model used to study planar mixed brushes with the small caveat that we use Langevin dynamics rather than Monte Carlo moves to update the particle positions; this is largely irrelevant in the analysis of equilibrium phenomena. We first calculate the phase behavior of the mixed brush particles using the uniform grafting and the mean-field approximations. To test the impact of these approximations and to more accurately model experiments, we then systematically relax these approximations. Specifically, we first conduct SCFT calculations with a non-uniform grafting distribution. We then introduce fluctuations via the DMFT method to relax the mean-field approximation. A visualization of uniform versus non-uniform grafting is shown in Fig. 1b and each of these implementations are discussed further in the Results section. The uniform grafting, SCFT calculations predict that the mixed brush nanoparticles form a variety of well-ordered phases, depending on the parameters. Introducing non-uniform grafting site distributions leads to defects in the mixed brush structures. These defects are more prominent in the multivalent phases formed with relatively larger particles and larger grafting densities. Fluctuations lead to decorrelations in the brush structure, even at low temperature conditions, and exacerbate the effects of the non-uniform grafting leading to more defective and less regular phases. Significantly,

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the formation of the Janus phase is not affected by either non-uniform grafting or fluctuations.

Results and Discussion In all calculations, we place an explicit particle grafted with two types of polymer, modeled as discrete Gaussian chains, in the center of a 3D box with periodic boundary conditions, filled with an explicit solvent (see Fig. 1a). The nanoparticle is immobilized and has enthalpically neutral interactions with the other components. We assume a symmetric brush composition in which the polymers have the same length and there are an equal number nA of A and nB of B grafted polymers (nA = nB ). The A and B monomers interact with a Flory interaction of strength χAB , while the solvent is nonselective with χAS = χBS = χS . As mentioned previously, Fig. 1b shows a visualization of uniform vs. non-uniform grafting. In the pure field-based SCFT, the grafting distribution is a probability distribution for the lateral position of the first segment of the polymer on the surface of the nanoparticle. This easily implemented as a uniform distribution at the surface. 12 In the DMFT method however, we have access to the explicit monomer coordinates and it is trivial to place the first segment of each grafted chain in a random (non-uniform) manner on the surface of the particle. In each type of grafting, the grafting distribution is defined so that the grafted segments are not free to move laterally on the surface of the nanoparticle. We investigate the mixed brush phase behavior as a function of the radius RP of the nanoparticle relative to the radius of gyration Rg of the grafted chains, the grafting density (σ ), the polymer incompatibility (χAB ), and the solvent quality (χS ). To most efficiently sample the parameter space and test the impact of varying each parameter, we first conduct calculations using a uniform grafting site distribution and a mean-field approximation (SCFT). These calculations also set a baseline to compare the effect of the uniform grafting and mean-field approximations on the mixed brush structure as we then systematically relax each approximation. In calculations with non-uniform grafting, a different grafting site distribution is used in each SCFT calculation or DMFT trajectory. For simulations that include fluctuations using the DMFT model, the noise used in the Langevin

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dynamics is randomized in each trajectory.

Figure 1: (a) Schematic of mixed brush nanoparticle in explicit solvent. The red circles represent the A grafted polymers, the blue circles are the B grafted polymers, and the yellow circles denote the explicit solvent S. The larger gray circle is the nanoparticle. (b) 2D projections of a uniform and non-uniform grafting distribution for the A grafted polymers. The grafting distributions are normalized so that the integral over all space is equal to 1. The color bar denotes the probability of the first segment of a grafted chain being placed at that location. Figure 2 shows the phase diagram of the structures with the lowest free energy in the parameter regime tested using SCFT with the uniform grafting approximation. Other SCFT studies 23–27 have observed the structures shown in Fig. 2, using the same approximations. At each state point, the mixed brush structure is initialized to form structures described by the spherical harmonics basis set, with the additional inclusion of the AB† and ABA† phases (see Methods). We then run a SCFT calculation for each initial condition and determine which structure has the lowest free energy. The isosurfaces shown in Fig. 2 and throughout the paper represent the spatial arrangement of the polymer brush where the red isosurface denotes the density of A chains and the blue isosurface denotes the density of B chains. For the disordered phase, a purple isosurface is used as there is an equal density of A and B polymer at every point in the brush layer. The nomenclature of each phase is given in Fig. 2. The † phases represent stable phases that appear between the phases that are rotationally symmetric about the azimuthal angle. With the uniform grafting approximation, each SCFT calculation is deterministic and leads to an efficient description of the phase behavior with direct access to the free energy, F. At each state point, the Boltzmann weight Pi of each phase i was determined relative to all of the calculated 6


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phases, where Pi = exp(−β Fi )/ ∑ j exp(−β Fj ). Here, β is the inverse of the thermal energy, kB T . In the calculation of P, we note the ABA phase has the same free energy as the BAB phase and the ABA† phase has the same free energy as the BAB† phase. The BAB and BAB† phases are included in the calculation of P as they are distinct phases from the ABA and ABA† phases, respectively. It is important to note that these phases are distinct because they are equally likely to form experimentally. It is unclear how the presence of mixed brush particles with two equivalent phases (e.g. ABA and BAB) would affect nanoparticle assembly, such as the lattice structures predicted by Chen et al. 27 assuming only a single particle structure (e.g. the ABA phase). The blue region in Figure 2 represents both the ABA and BAB phases and the yellow region represents both the ABA† and BAB† phases. A cubic spline interpolation scheme was used to draw the phase regions and boundaries based on the value of Pi at each calculated state point. Figure 2 provides a wealth of information on the role of each parameter in the phase behavior of the mixed brush particles. In general, the trends in the phase behavior are a result of the grafted chains maximizing their chain entropy while minimizing unfavorable A − B contacts. As the number of A − B contacts in the brush increases, each chain sacrifices chain entropy to stretch and arrange next to a chain of its own type. One end of this spectrum is the disordered phase where the chain entropy and number of A − B contacts are at a maximum while at the other end of the spectrum is the AB or Janus phase where the chain entropy and number of A − B contacts are at a minimum. The visual representations of each phase shown at the top of Figure 2 are organized by increasing degree of order, by which we mean decreasing chain entropy and decreasing numbers of A − B interfaces on the nanoparticle. The competition between maximizing chain entropy and minimizing unfavorable A − B contacts is clearly illustrated in Fig. 3. Here we show that for decreasing χS , which is equivalent to increasing temperature, there is a transition from the AB phase to the ABA phase in Fig. 3a and a transition from the ABA to the ABAB phase in Fig. 3b, both of which are a result of increasing the chain entropy with increasing temperature. In analyzing the phase behavior of Fig. 2, we first focus on the trends as a function of RP /Rg . Two major trends occur as RP /Rg increases. First, the size of the disordered region shrinks because

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Figure 2: Phase diagram of mixed brush nanoparticles as a function of RP /Rg , σ , χAB N, and χS . Each square symbol in the graphs represent a calculated state point. Visual representations of the lowest free energy phases are shown at the top where the red isosurface denotes the A chains, the blue isosurface denotes the B chains, and the background color corresponds to the colored regions in the phase diagram. The purple isosurface in the leftmost figure denotes a disordered phase where the A and B densities are the same at each point in the grafted brush. From left to right, the nomenclature of each phase is as follows: Disordered, ABAB, ABA† , ABA, AB† , AB. The AB phase is also referred to as the Janus phase.

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Figure 3: Lowest free energy structures under the mean-field and uniform grafting approximations for varying values of χS with (a) RP /Rg = 2/3 and (b) RP /Rg = 1. σ = 3.22 chains/b2 and χAB N = 28.3. there are more grafted chains on larger particles (with the same grafting density), which leads to more unfavorable A − B contacts, driving the grafted chains to order. Second, as RP /Rg is increased, the Janus region disappears entirely while the more striped ABA, ABA† , and ABAB phases become more prominent. This is because the chains cannot as easily wrap around the surface of the particle to reduce the unfavorable A−B contacts when the particle size grows relative to the length of the grafted chains. The trends in grafting density are analogous to the trends in RP /Rg . For low grafting densities, there is not enough polymer on the surface of the particle for the chains to need to rearrange and form an ordered structure. As the grafting density is increased, the number of A − B contacts increases, which drives the ordering of the mixed brush and reduces the size of the disordered region. In regions of the phase diagram with order-to-order transitions, at lower grafting densities the brush is not crowded and the polymers are able to easily arrange to reduce the A − B contacts. At higher grafting densities, the brush becomes more crowded and each chain is relatively more constricted to its grafting location. As a result, increasing the grafting density in an order-to-order

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transition region leads to structures with less order (more domains). Increasing either χAB N or χS (decreasing temperature) leads to increasingly ordered structures, though the mechanisms are different. At a higher χAB N value, there is an increased repulsion between polymer A and polymer B so that there is an increased drive for the polymers to sacrifice chain entropy to reduce the number of less favorable A − B contacts. At higher χS values, the solvent is progressively expelled from the brush (see Fig. 5 below) so that the brush layer contracts and results in more A − B contacts. This in turn leads to the mixed brush forming more ordered phases with fewer domains. Next, we relax both the uniform grafting and mean-field approximations. Figure 4 shows uniform grafting SCFT, non-uniform grafting SCFT, and non-uniform grafting DMFT calculations, for varying RP /Rg and σ values. In all cases, we use the highest χAB N and χS values included in Fig. 2, where fluctuation effects should be less than at lower χAB N and χS values. In the right-hand column of Fig. 4 we include nanoparticles with a larger radius than shown in the phase diagram in Fig. 2. The top rows of Figs. 4a and 4b show the lowest free energy phases using the uniform grafting and mean-field approximations. For all other calculations shown in Fig. 4 we ran 10 trajectories at each condition and show three representative images of the mixed brush structures. For small nanoparticles with RP /Rg = 1/3 which form an AB Janus phase, we find that introducing non-uniform grafting and fluctuations has only minimal effects on the brush structure, at both grafting densities. When the grafting density is small, fluctuations lead to some decorrelation in the structure, resulting in the nanoparticle surface being exposed at the A − B interface of the Janus phase. When the grafting density is large, the non-uniform grafting distribution and fluctuations lead to slight imperfections relative to the uniform grafting, mean-field result. Overall, the Janus phase is consistently observed when RP /Rg is small even with a non-uniform graft site distribution and fluctuation effects. The effects of non-uniform grafting and fluctuations are greater when RP /Rg is large, and these effects are further exacerbated when both RP /Rg and σ are large. The first important point is that non-uniform grafting leads to phases that are different than the lowest free energy structures pre-

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Figure 4: Visualizations of mixed brush structures using uniform grafting and a mean-field approximation (top rows), non-uniform grafting and a mean-field approximation (middle rows), and non-uniform grafting and fluctuations (bottom rows) for RP /Rg = 1/3 (left column), RP /Rg = 1 (middle column), RP /Rg = 5/3 (right column), σ = 1.07 chains/b2 (a), and σ = 3.22 chains/b2 (b). In all calculations, χAB N = 28.3 and χS = 1. The grafted A and B chain isosurfaces for RP /Rg = 1/3 and σ = 1.07 chains/b2 are generated with a volume fraction threshold of 0.25 while all other isosurfaces are generated with a volume fraction threshold of 0.5. For the uniform grafting, mean-field calculations, the lowest free energy structure is shown while in the other calculations, 3 representative images are shown from 10 trajectories.

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dicted from the uniform grafting SCFT calculations. As RP /Rg and σ are increased, introducing non-uniform grafting and fluctuations lead to increasing numbers of defects as compared with the uniform grafting SCFT phases. Thus, Fig. 4 demonstrates that when RP /Rg and σ are relatively large, the brush structure is more dependent on the details of the grafting site distribution. As RP /Rg and σ are increased, the grafted chains are more constricted to their grafting region because they are less able to wrap around the surface of the particle to reduce unfavorable A − B contacts. This effect is less for relatively small RP /Rg or σ values, which is why the mixed brush is consistently able to form the lowest free energy structure, the Janus phase, even with non-uniform grafting. Furthermore, when RP /Rg is large but σ is small, we observe that the structures form more minor defects relative to brushes with both large RP /Rg and large σ . To further explore the effects of fluctuations, we consider the effects of varying χAB N and χS . Figure 5 shows the volume fractions of the polymer brush (both A and B chains) and the solvent as a function of the radial distance from the surface of the nanoparticle, for varying values of χAB N and χS both for the uniform grafting SCFT calculations and the DMFT calculations. In each calculation, RP /Rg = 1/3 and σ = 2.15 chains/b2 . The DMFT profiles are averaged over three independent trajectories. First we note that the volume fractions change minimally as a function of χAB N even when the lowest free energy structures are different. For example, the radial profiles in Fig. 5b and Fig. 5h are minimally different even though the brush in Fig. 5b is disordered while the brush in Fig. 5h is in the Janus AB phase. This makes sense since we are plotting the total volume fraction of the brush polymers, which should only be strongly affected by the solvent quality. Indeed, as χS is increased the solvent is progressively expelled from the brush region and the brush becomes more compact. Comparing the profiles with and without fluctuations, we see that in all cases when fluctuations are included, the solvent penetrates the brush further, resulting in the polymer brush extending out further from the surface. To quantify the effects of the fluctuations further, we calculate the autocorrelation of the A monomer density, hρA (r0 )ρA (r00 )i, which simply calculates the correlation between the density of A monomers at point r0 in the brush with the density at all other points r00 ; the result is a function

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Figure 5: Volume fraction of the total polymer brush (black) and solvent (red) as a function of the radial distance from the nanoparticle surface for (a)-(c) χAB N = 9.5, (d)-(f) χAB N = 18.9, and (g)-(i) χAB N = 28.3 with (a), (d), (g) χS = 0.5, (b), (e), (h) χS = 0.75, and (c), (f), (i) χS = 1.0. The solid lines denote SCFT calculations while the dashed lines are the average of 3 independent DMFT simulations. RP /Rg = 1/3 and σ = 2.15 chains/b2 in all cases.

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of the distance r = |r0 − r00 | (the Fourier transform of the autocorrelation function is proportional to the structure factor). This density autocorrelation function provides a metric to directly compare the mean-field and fluctuating models. In these simulations, we average over three trajectories from both the non-uniform grafting SCFT and the non-uniform grafting DMFT calculations. We focus on RP /Rg = 1/3 and σ = 3.22 chains/b2 where we will confidently form the Janus phase in the ordered regime regardless of how high χAB N or χS are. As a result, we can most easily decouple the effect of fluctuations rather than comparing the A density autocorrelation between different brush structures. 0.06

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Figure 6: Autocorrelation of the A density as a function of distance for χAB N = 18.9 (a), χAB N = 28.3 (b), and χAB N = 56.3 (c) with χS = 2.0 (black), χS = 0.75 (red), and χS = 0.25 (blue). The solid lines denote an average of 3 non-uniform grafted mean-field calculations while the dashed line represents an average of 3 non-uniform grafted fluctuating simulations. RP /Rg = 1/3 and σ = 3.22 chains/b2 in all cases.

Figure 6 shows the density autocorrelation as a function of χAB N and χS . When χS is very low, we see the density autocorrelation of the mean-field and fluctuating models is the same for all χAB N values. This is because the brush is in the disordered phase so there is little correlation in the brush structure, and the differences with and without fluctuations are minimal. These calculations validate the equivalency of both methods. When the mixed brush structure is in the ordered regime, fluctuations lead to less correlation in the A monomer density. This is especially prevalent at lower χAB N and χS values. However, even at the highest combination of χAB N and χS , fluctuations still play a role in decorrelating the structure. This is expected as thermal fluctuations will be apparent until the limits of χAB N = ∞ and χS = ∞ are reached, which is equivalent to zero temperature. These results indicate that fluctuations are important at experimentally relevant parameters. 14


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We have shown that non-uniform grafting and fluctuations lead to defects in the mixed brush for phases that occur at larger RP /Rg and σ values at χAB N = 28.3 and χS = 1. This effect is going to be more pronounced for lower χAB N and χS values where defects will be more abundant in the striped and patchy phases. Non-uniform grafting and fluctuations destabilize the formation of these complex phases of mixed brush nanoparticles because there are many more locally stable structures. Our results suggest that creating well-ordered, multivalent mixed brush nanoparticles is unrealistic experimentally. Though higher χAB N and χS values decrease the effects of fluctuations, the non-uniform grafting SCFT results from Fig. 4 suggest that the inevitable variations in grafting distribution still limit the formation of well-ordered, monodisperse mixed brush phases. Furthermore, higher χAB N and χS will increase the energy barrier between locally stable phases and the globally stable phase such that the mixed brush structures will become even more sensitive to the graft site distribution. We further note that in regimes where striped or patchy phases are found, in an experiment the non-uniform grafting and fluctuations would mean that each nanoparticle would likely have a different distribution of defects, which in turn would complicate the nanoparticle self assembly. However, we predict that the Janus phase is robust to the effects of non-uniform grafting density and fluctuations, and should therefore form experimentally at the appropriate values of R p /Rg and σ , for sufficiently high χAB N and χs to form an ordered phase. A subtle point worth mentioning is that in certain conditions (e.g. very large χAB N and R p /Rg ), the Gaussian bonding potential between statistical segments becomes inaccurate as the polymer chains will stretch near or beyond their contour length. In these conditions, the formation of the Janus phase would be limited based on the polymer’s ability to physically wrap around the nanoparticle. Nevertheless, we believe the data presented in Figures 2-6 is well below this extreme as χAB N < 60 and R p ∼ Rg . To our knowledge, the only clear experimental result in a similar parameter regime as our calculations is by Kim et al., 22 who studied Au nanoparticles grafted with deuterated polystyrene and poly(methyl methacrylate) in toluene, a neutrally good solvent for both polymers. They use small-angle neutron scattering to show that the mixed brush nanoparticles form a disordered or partially ordered state

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at lower brush molecular weight, and a Janus phase as the molecular weight is increased to the regime where R p ∼ Rg . The experiments have additional complicating factors, namely that the interactions between the nanoparticle and polymers are attractive while the experiments are carried out at lower temperatures than our calculations; Kim et al. argue that the nanoparticle attraction causes the polymer brush to form a dense wetting layer, which raises the effective χ to allow phase separation. Nevertheless these experiments demonstrate the formation of a Janus phase on a mixed brush nanoparticle in a parameter range similar to our predictions.

Conclusion We investigated the phase behavior of binary mixed brush nanoparticles in solution as a function of nanoparticle size relative to the grafted chains’ radius of gyration (RP /Rg ), grafting density (σ ), the enthalpic repulsion between the polymer chains (χAB ), and the solvent quality (χS ), for nanoparticles grafted with equal numbers of equal length A and B chains. We calculated a phase diagram using self-consistent field theory and assuming that the chains are grafted uniformly on the nanoparticle surface. The mixed brush forms increasingly multivalent phases with increasing RP /Rg and σ and with decreasing χAB and χS , due to a balance between maximizing chain entropy and minimizing unfavorable A − B contacts. However, using a non-uniform grafting distribution, even within the mean-field SCFT, shows that the mixed brush structures develop defects and do not form perfect striped phases. Calculations performed using DMFT that also include fluctuations show that defects are increased. Experimentally, both random, non-uniform grafting and fluctuations are expected to be important, especially for nanoparticles in solution. We show that the simple Janus phase is preserved in all calculations, even with the introduction of non-uniform grafting and fluctuations. We conclude that the formation of the Janus phase is more realistic experimentally than is the formation of defect-free multivalent mixed brush nanoparticles.

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Methods Model The model and method derivations follow closely those of our previous works for both the SCFT 23,32,33 and DMFT 30,31,33 methods. We place an explicit particle grafted with A and B type discrete Gaussian polymer chains in an explicit solvent, S, as shown in Figure 1a. All calculations are in 3D with periodic boundary conditions. The nanoparticle is immobilized in the center of the box and has enthalpically neutral interactions with the other components. The nanoparticle density is given by |r − rC | − RP ρ0 erfc . ρˆ P (r) = 2 ξ

(1)

Here, rC is the center of the box, RP is the radius of the particle, and ξ is the length scale on which the particle density goes from ρ0 inside the particle core to 0 outside the core. Grafted chains are attached to the nanoparticle surface with either a uniform distribution given by " 2 # 1 3(|r − rC | − RP − ξ ) Γσ K,uni (r) = exp − , σ0 2b2

(2)

or a random distribution given by 1 Γσ K,rand (r) = σ0

" # 3(|r − [rC + (RP + ξ ) · ui ]|) 2 , ∑ exp − 2 2b i nK

(3)

R

where σ0 is defined as dr Γσ (r) = 1, n is the number of polymer chains, K denotes polymer type A or B, and ui is a randomly determined orientation vector of magnitude 1. The polymer chains are connected via a Gaussian bonding potential nK NK −1

βU0 =

∑∑ i

j

17

3|ri, j − ri, j+1 |2 , 2b2


(4)

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where b is the statistical segment length of the polymer. Here, we assume that bA = bB = b. A Helfand compressibility potential is used to enforce an energetic penalty for deviations away from the average system density, ρ0 , given by

βU1 =

κ 2ρ0

Z

dr [ρˆ + (r) − ρ0 ]2 ,

(5)

where ρˆ + = ρˆ A + ρˆ B + ρˆ S + ρˆ P is the spatially varying total microscopic density, ρˆ I is the microscopic density of component I, and κ controls the strength of the density fluctuations. Finally, a repulsive Flory-like potential is used to include enthalpic repulsions between chemically dissimilar components

βU2

χAB = ρ0

Z

χS dr ρˆ A (r)ρˆ B (r) + ρ0

Z

dr ρˆ T (r)ρˆ S (r),

(6)

where ρˆ T = ρˆ A + ρˆ B . χAB and χS are the Flory parameters governing the strength of the interaction between the two polymers and between the solvent/total-polymer, respectively.

Self-Consistent Field Theory Hubbard-Stratonovich transformations are then used to decouple the intermolecular interactions and transform the relevant degrees of freedom from particle coordinates to chemical potential fields 34,35 . The result of the particle-to-field transformation is the field-theoretic partition function, Z

Z

= z1

nP

dr

Z

Dw+

Z

(±) DwAB

Z

(±)

DwT S e−H [{w}] ,

(7)

where z1 contains the numerical prefactors such as the thermal de Broglie wavelengths and the normalization constants from the Gaussian functional integrals used to de-couple the particle interactions, H [{w}] is the effective Hamiltonian of the system, and {w} is the set of chemical

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(−)

(+)

(+)

(−)

potential fields for our system, w+ , wAB , wAB , wT S , wT S . The effective Hamiltonian is given by ρ0 H [r , {w}] = 2κ nP

Z

2

dr w+ (r) − i

Z

dr w+ (r) (ρ0 − ρˆ P (r))

Z

Z

ρ0 ρ0 (+) (−) dr wAB (r)2 + dr wAB (r)2 χAB χAB Z Z ρ0 ρ0 (+) (−) 2 + dr wT S (r) + dr wT S (r)2 χT S χT S

+

−nA −nB

Z

dr Γσ A (r) ln qA [NG , r; µA ] Z

dr Γσ B (r) ln qB [NG , r; µB ] − nS ln QS [µS ],

(8)

where qA (NG , r), qB (NG , r), and QS are the partition functions of a single A chain, B chain, and solvent molecule, respectively. qA (NG , r), qB (NG , r), and QS can be determined from natural extensions of the detailed expressions in equations (S.9)–(S.11) as well as (S.26) and (S.27) in Koski et al. 23 In particular, note that the boundary condition on the grafted chain propagators that enforces the fixed location of the grafted segments follows equation (S.27) in Ref. 23. This is similar to the implementation of a fixed grafting location in Refs. 12 and 15. In calculating the SCFT phase diagram, we initialize the fields with a real-spaced spherical harmonic basis set in order to form specific phases. The spherical harmonics are defined by 36

Ylm (θ , φ ) =

   N m Pm (cos(θ )) cos(mφ )

m≥0

  N m Pm (cos(θ )) sin(|m|φ ), l l

m = 0, < θk,s (tn )θk0 ,s0 (tn0 ) > =

2δt δ 0δ 0δ 0. Dβ n,n k,k s,s

(18) (19)

Numerical Parameters and Theory-Experiment Mapping In all calculations, lengths are in units of the statistical segment length of the polymer, b. We choose a reference volume of v0 = 0.2057b3 and N = 60 discrete Gaussian beads in each polymer chain. We use a cubic box of 45b to a side with 81 discretization points in each respective direction. In all calculations κ = 5 and ξ = a = 0.5b. In the DMFT simulations, δt = 0.02, D = 1 for the polymers and solvent, and D = 0 for the nanoparticle to immobilize it in the center of the box. To emphasize we are in an experimentally relevant regime, our model parameters can be mapped to a polystyrene-poly(methyl methacrylate) mixed brush grafted nanoparticle. Setting v0 = 0.2057b3 corresponds to 1b = 2.073nm, such that N = 60 kg/mol, 2.2 nm ≤ RP ≤ 6.6 nm, 0.25 chains/nm2 ≤ σ ≤ 0.75 chains/nm2 , and χAB N = 18.9 equates to a temperature of 100o C. This example mapping illustrates the experimentally relevant regime of the parameters tested, but the model is general, with the theory-experimental mapping depending on the A − B polymer system of interest. χS = 0.5 represents a theta-solvent while χS > 0.5 is a poor solvent. The important quantity in mapping the χS parameter to experiments is the excluded volume parameter u0 where

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u0 = v0 (1 − 2χS ).

Acknowledgements This work was performed 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 multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. We would also like to thank Dale Huber, Rob Riggleman, and Ben Lindsay for helpful discussions.

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For Table of Contents Use Only Fluctuation Effects on the Brush Structure of Mixed Brush Nanoparticles in Solution In this study, we computationally investigate the phase behavior of binary mixed brush nanoparticles in solution, across a large and experimentally relevant parameter space. Specifically, we calculate the mean-field phase diagram, assuming uniform grafting of the two polymers, as a function of the nanoparticle size relative to the length of the grafted chains, the grafting density, the enthalpic repulsion between the grafted chains, and the solvent quality. We find a variety of phases including a Janus phase and phases with varying numbers of striped domains. Using a non-uniform, random distribution of grafting sites on the nanoparticle instead of the uniform distribution leads to the development of defects in the mixed brush structures. Introducing fluctuations as well leads to increasingly defective structures for the striped phases. However, we find that the simple Janus phase is preserved in all calculations, even with the introduction of non-uniform grafting and fluctuations. We conclude that the formation of the Janus phase is more realistic experimentally than is the formation of defect-free multivalent mixed brush nanoparticles. Jason P. Koski, Amalie L. Frischknecht

Table of Contents Graphic

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Fluctuation Effects on the Brush Structure of Mixed Brush

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