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Modeling the Morphology and Phase Behavior of One-Component Polymer-Grafted Nanoparticle Systems Valeriy V. Ginzburg* Materials Science and Engineering, The Dow Chemical Company, Building 1702, Midland, Michigan 48674, United States S Supporting Information *

ABSTRACT: Polymer-grafted or “hairy” nanoparticles (HNP) represent an important and relatively new class of materials. Traditionally, polymers or oligomers were grafted onto the particle (silica, metal, or semiconductor) surface to improve the dispersion of particles in a polymer matrix. Recently, the scope of research has broadened substantially, as it was shown that such nanoparticles can form anisotropic structures ranging from self-assembled wires to sheets to networks. Furthermore, it has been demonstrated that one could make hybrid polymer−inorganic materials with HNPs alone, without using a separate matrix polymer. Such one-component hybrid materials are not prone to macroscopic phase separation and can, in principle, have a variety of interesting microphase-separated, anisotropic morphologies, similar to surfactants or block copolymers. Here, we develop a new selfconsistent field theory describing the behavior of such one-component HNP systems and apply it to predict morphology as a function of the ligand molecular weight and grafting density. As in the case of block copolymers, we observe lamellar, cylindrical, and spherical morphologies and elucidate phase boundaries as a function of the core (nanoparticle) volume fraction and the ratio of the particle radius to the ligand radius of gyration. We also observe the formation of a novel phase, labeled as “sheets”, where the lamellar-like ordering of particle-rich and ligand-rich layers is additionally characterized by the hexagonal ordering of the particles within the particle-rich layer. Our theoretical approach can be easily extended to other HNPs, including those with mixed ligands and block copolymer ligands.

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INTRODUCTION In recent years, polymer-grafted nanoparticles (PGNP), alternatively described as polymer-brushed nanoparticles (PBNP) or hairy nanoparticles (HNP), have been the subject of increased interest in the polymer science field.1−5 Initially, grafted ligands were primarily used as tools to improve nanoparticle dispersion in polymer matrices.6−11 Thus, grafting organic ligands on the particles became necessary to form functional nanocomposites in various applications, from quantum dot (QD) containing resins to polymer−clay thermoplastic composites. However, several years ago, it was shown that the behavior of HNPs in polymers was much richer than originally thought, with anisotropic self-assembly caused by a complicated interplay between the enthalpy of particle− polymer repulsion and the entropy of ligand and matrix chain conformational arrangements.4,12−16 The ligand “cloud” around the nanoparticleeven if originally isotropiccan “polarize” in the presence of other HNPs, forming structures such as “wires” and “sheets”, in addition to more familiar morphologies such as uniformly dispersed particles or, at the other extreme, large aggregates completely phase-separated from the matrix. In principle, it is not even necessary to have a separate polymer matrix. With the versatility of chemical “grafting to” and “grafting from” methods, one can create HNPs with any polymer-to-filler volume ratio, from very high (with higher© XXXX American Chemical Society

molecular-weight ligands) to very low (with low-molecularweight ligand and/or very low grafting density). By making HNPs in solution and then casting films out of those solutions, one can create “one-component nanocomposites”, also referred to as “nanoparticle organic hybrid materials” (NOHM).5,17−23 Since these materials are one-component, theylike block copolymersshould generally exhibit microphase separation, as opposed to macrophase separation. Changing the ligand characteristics (e.g., using block copolymer ligands or mixture of ligands of various molecular weights and chemical compositions) could expand the complexity of possible morphologies, while changing the particle nature (metal, semiconductor, or polymeric) could determine which application the composite would be best suited for. Thus, HNPs could become ideal “building blocks” for versatile nanostructure engineering, as envisioned, e.g., by Cheng et al.24−27 and Glotzer et al.28−33 Experimentally, researchers have observed three-dimensional crystals such as face-centered cubic (FCC)19 as well as local anisotropic ordering5 and disordered liquid.23 Other morphologies likely can be found as well; to understand them, we need the right theoretical description. Received: September 5, 2017 Revised: October 31, 2017

A

DOI: 10.1021/acs.macromol.7b01922 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Theory and modeling of polymer/nanoparticle composites has been a very active field for several decades.4,7,34−40 Theoretical and computational methods used include strong segregation theory,16 Monte Carlo simulations,16 molecular dynamics, 15 polymer reference interaction site model (PRISM),35,41−45 and self-consistent field theory (SCFT) based approaches.46−49 For the pure HNP systems and their solutions (“giant surfactants”), particle-based simulations of Glotzer and co-workers offered many interesting insights into their morphologies and phase behavior.11,31,32,50 The transition between FCC crystal and disordered liquid in one-component HNP systems was investigated by Choi et al.23 using scaling theory and by Koch and co-workers51−55 by means of density functional theory (DFT). Chremos and co-workers modeled the HNP behavior using coarse-grained molecular dynamics (CG-MD).56−59 They also compared CG-MD with DFT54 and found that in DFT (and similar field-based theories) the grafted brush stretching is somewhat underestimated. Jayaraman and Schweizer used PRISM approach to compute pair correlation functions and determine microphase separation boundaries for one- and multitethered nanoparticles.43−45 Lee et al.60 utilized the self-consistent-field/density functional theory (SCF-DFT) approach61,62 to predict microphase-separated lamellar and hexagonal morphologies of “tadpole” (coil−sphere) diblock copolymers (one-tethered nanoparticles). A slightly different version of the SCF-DFT approach was developed by Hur et al.63 for the case of multiple ligands. Here, we adapt the SCFDFT approach of refs 60−62 to model the phase behavior of one-component HNP assemblies where the nanoparticles can have arbitrary number of grafted ligands. We use this model to investigate how the HNP morphologies depend on the ligand grafting density and chain length and sketch a simple HNP phase map.

Q HNP =

1 V

(2a)

∫ dr′ ρP (r′)Θ(2RP − |r − r′|)

(2b)

∫ dr ρP (r)

(2c)

4x − 3x 2 (1 − x)2

(2d)

∫ dr (ZL(r))n exp[−wP(r)]

(3)

where ZL(r) =

∫ dr′ q(r′, 1)δ(|r − r′| − RP) (4)

A

Here A is the surface area of the particle, and δ(x) is the standard (one-dimensional) Dirac delta function. The propagator q(r,s) is computed by solving the so-called modified diffusion equation (MDE) ∂q(r, s) = (R g 2∇2 − wL(r))q(r, s) ∂s

(5) Na 2

with the “initial condition” q(r,0) = 1, and R g 2 = 6 , where a is the Kuhn length. The index s runs from 0 to 1, with 0 corresponding to the free end and 1 corresponding to the bound end; thus, q(r,1) is related to the probability of finding the bound end of the ligand at point r. The integral in eq 4 is the sum over all configurations of a ligand grafted on a surface of a particle whose center is located at point r. Equations 15 describe the HNP system. We note here that it is straightforward to extend this formalism to other HNPbased systems (e.g., mixtures of HNPs with homopolymers, mixtures of HNPs with block copolymers, HNPs with block copolymer ligands, HNPs with mixed homopolymer ligands resulting in patchy particles, etc.) For the time being, we concentrate on the HNPs with homopolymer ligands. To calculate the morphologies of these systems, we need to find the saddle point of the free energy functional with respect to the densities φL(r) and ρP(r), as well as the conjugate fields wL(r) and wP(r), and the Lagrange multiplier (“pressure”) field ξ(r). The self-consistency equations, again, are very similar to those obtained by Thompson et al.61,62

⎛Q ⎞ FNv = −nHNP ln⎜ HNP ⎟ ⎝ V ⎠ kBTV 1 + dr[(χN ){φL(r) − ϕL}{φP (r) − ϕP } V

∫

∫

∫

1 2 Nv 3

∫ dr′ ρP (r′)Θ(RP − |r − r′|)

Here Θ(x) is the Heaviside step function: Θ(x) = 0, if x < 0, Θ(0) = 0.5, and Θ(x) = 1, if x > 0. The particle center probability, ρP(r), and the overall number of particles per unit volume, nHNP, are multiplied by the ligand volume, Nv, following the notation of the earlier papers. The Flory− Huggins parameter χ describes the local interaction between the ligand and the particle species. Finally, we need to express the partition function of the hairy nanoparticles, QHNP, as functional of the fields wL and wP. For that we assume that the n ligands grafted on the particle surface are mobile in a sense that their first segments are able to travel freely on the nanoparticle surface but unable to leave it. In that case, the partition function can be written as

)

− ξ(r)(1 − φL(r) − φP (r))] 1 − dr[wP(r)ρP (r) + wL(r)φL(r)] V 1 + dr[ρP (r)ΨHS(φP̿ (r))] V

φP̿ (r) =

ΨHS(x) =

METHOD AND IMPLEMENTATION We start from a generic version of the SCF-DFT mean-field free energy similar to that of Thompson et al.60−62 The system we consider is a one-component “melt” of polymer-grafted or “hairy” nanoparticles (HNP), occupying total volume V. Each HNP consists of the “core” (spherical nanoparticle with radius 4 RP and volume VP = 3 πRP 3) and the “shell” (n homopolymer ligands with chain length N). The particle species is denoted as P and ligand as L. The overall volume of each HNP is given by VHNP = VP + nNv, where v is the monomer (repeat unit) volume. The overall particle volume fraction, 4 4 ϕP = 3 πRP 3 / 3 πRP 3 + nNv , and the overall ligand volume fraction ϕL = 1 − ϕP. The free energy is then written as46,61,62

)(

1 Nv

nHNP =

■

(

φP (r) =

(1)

wL(r) = (χN ){φP (r) − ϕP } + ξ(r)

Here, the following definitions are used: B

(6a)

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Macromolecules = (r)) + wP(r) = ΨCS(φ P

1 N

⎧ V [φL(r)]new = [φL(r)]old + λ⎨ϕL ⎩ QL

∫ dr′[(χN ){φL(r′) − ϕL}

⎪

⎪

1 dr′ + ξ(r′)]Θ(RP − |r − r′|)+ 3 2 Nv = (r′))]Θ(2R − |r − r′|) × [ρP (r′)Ψ′CS(φ P P

∫

ρP (r) = nHNP

φL(r) = ϕL

V Q HNP

V QL

∫0

1

× (6b)

(6d) (6e)

The local particle density φP(r) is computed by integrating particle center probability ρP(r) according to eq 2a. To complete the set of equations, we need to determine the ligand partition function QL and the conjugate propagator q†L(r,s). The single-ligand partition function is given by

∫ qL†(r,0) dr

The propagator equation ∂qL†(r,s) ∂s

(7)

is described by the modified diffusion

= −(R g 2∇2 − wL(r))qL†(r,s)

(8a)

with the “initial condition” reflecting the fact that the “bound” end of each ligand must be on the particle surface which it shares with (n − 1) other ligands

∫ dr′(ZL(r′))n−1 exp[−wP(r′)]δ(|r − r′| − RP)

qL†(r,1) = A−1

(8b)

In the limit of an infinitely small particle (RP → 0), eq 8b would reduce to the familiar expression for the propagator of an n-arm star.64 To find numerical solutions of the self-consistency equations, we discretize them on a cubic grid, with mesh size dx. The Θ and δ functions are approximated as follows: Θ(RP − |r|) ≈

δ(|r| − RP) ≈

⎛ |r| − RP ⎞⎤ 1⎡ ⎟⎥ ⎢1 − tanh⎜K ⎝ 2⎣ d x ⎠⎦

(9a)

⎧ ⎛ |r| − R ⎞2 ⎫ K P ⎟⎬ exp⎨−K ⎜ π dx ⎩ ⎝ dx ⎠ ⎭ ⎪

⎪

⎪

⎪

χ=v

(9b)

⎧ V [ρP (r)]new = [ρP (r)]old + λP ⎨nHNP Q ⎩ HNP ⎪

⎪

⎪

⎪

⎪

⎪

(10b) (10c)

( eP − eL )2 (δP − δL)2 =v RT RT

(11)

where eP and eL denote cohesive energy densities (CED, usually expressed in units of J/cm3) of particle and ligand, respectively, and δL , P = eL , P are the respective solubility parameters. Cohesive energy densities of inorganic particles are much greater than those for typical polymers, and thus the polymer− particle interaction parameters are expected to be large (see Supporting Information). As χ is increased, the segregation between each particle and its surroundings becomes stronger, even though at high particle volume fractions, particles can still “fuse” into wires or sheets.46 Here, we used χ chosen in a way that for all compositions χN is much greater than the corresponding χNODT for flexible block copolymers (see

where K is sufficiently large to ensure that the transition from the “inside” to the “outside” occurs over a single layer of grid points (in our calculations, we used K = 1.5). We solve eqs 6a−6e in an iterative manner, using Picard algorithm to update the densities, as follows:

⎫ × exp[−wP(r)](ZL(r)n ) − [ρP (r)]old ⎬ ⎭

⎫ qL(r, s)qL†(r,s) ds − [φL(r)]old ⎬ ⎭

After updating the densities and pressures according to eqs 10a−10c, we then compute the new fields wP and wL according to eqs 6a and 6b and repeat the process until convergence. The Rasmussen−Kalosakas65 pseudospectral approach is employed to speed up the calculations. The use of pseudospectral method and utilizing Fourier transforms for the computation of convolution integrals makes it possible to solve the ligand propagator MDE (eqs 5, 8a, and 8b) relatively fast, so that all computations can be performed within hours (exact time depends on the specific hardware, size of the simulation box, and the number of discretization points used to describe the ligand chains). We use convergence criterion ⟨|φL(r) + φP(r) − 1|⟩ < 10−3. Other simulation parameters are as follows: λ = 0.025, λP = 0.007, and ε = 30.0. Although the approach to finding the locally equilibrium morphology in SCF-DFT is similar to that in a pure SCFT, there are some notable challenges specific to hybrid nanoparticle−polymer systems. In SCF-DFT, particles are not introduced as solid objects whose interior is impenetrable to the polymers; rather, regions with high φP (“particles”) and low φP (“polymers”) emerge from the calculations. The resulting morphology is usually more or less smeared, representing ensemble-averaged particle density profiles, rather than instantaneous configurations. While so-called “hybrid” models66−70 preserve the particle shape and disallow particle− particle or particle−polymer overlap at any given moment, they often suffer from significant perturbations in the polymer field when particles are moved and require lengthy equilibration process. Here, we attempt to utilize SCF-DFT while minimizing the particle deformation and overlap. We follow the ideas of Chen et al.,71 who proposed to add a “particle cohesive energy” term in the free energy. The addition of this term results in sharpening the peaks of ρP(r) and preserving the spherical shapes of individual particles. For the case of a twospecies system, increasing the cohesive energy density of the particle relative to the polymer is equivalent to increasing the Flory−Huggins parameter. Indeed, the Flory−Huggins parameter χ is related to the cohesive energy densities of the polymer and the particle via72

(6c)

qL(r, s)qL†(r,s) ds

q†L(r,s)

1

[ξ(r)]new = [ξ(r)]old + λε{φL(r) + φP (r) − 1}

exp[−wP(r)](ZL(r))n

φL(r) + φP (r) = 1

QL =

∫0

(10a) C

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Figure 1. Various HNPs having the same core volume fraction (φP) of 0.5. Left to right: one grafted ligand with N = 120; two grafted ligands with N = 60; four grafted ligands with N = 30; and eight grafted ligands with N = 15.

Figure 2. (a) Three-dimensional view of lamellar morphology, with red representing particle-rich layers. The surfaces correspond to ϕ = 0.5. The blue arrow goes from the center of the particle-rich layer to the center of the ligand layer. (b) Density profiles of particle (red) and ligand (blue) species along the direction of the blue arrow. Here, n = 8 and N = 15.

further analysis, we also define three important parameters: the n grafting density σ = 4πR 2 , the ratio of the particle radius to the

Supporting Information). Thus, we expect that all calculations below correspond to strong segregation behavior and the morphology is then χN-independent. The calculation of the phase diagram is done using a realspace approach (see, e.g., Drolet and Fredrickson73), with the Rasmussen−Kalosakas65 pseudospectral approach to solving the modified diffusion equation (MDE). In this initial study, we considered three classical morphologies ubiquitous in block copolymers, i.e., spherical (cubic), cylindrical (hexagonal), and lamellar. For each composition, we computed the free energy of the face-centered cubic (FCC) structure (if stable) and also found the free energy minima of the cylindrical and lamellar structures with respect to the unit cell dimensions (for more details, see the Supporting Information). Interestingly, we found that in addition to the classical phases there is also a “layered sheet” phase, similar to lamellar but with 2D-hexagonal ordering of the particles within the particle-rich layer. The detailed description of those morphologies is given below.

■

P

(unperturbed) radius of gyration of a single ligand chain, RP/Rg, and the volume fraction of the nanoparticle core, ϕP. The first two parameters are similar to those used in studies of HNPs in polymer matrices and solvents,15,16 while the third one is more akin to the composition parameter f in the theory of block copolymers.74 In choosing the parameter space for this investigation, we concentrated on the region of intermediate particle volume fractions, 0.2 < ϕP < 0.5. It is known that in colloid−polymer mixtures, at low ϕP, particles are dispersed uniformly, while at high ϕP, they form either gel or crystal, depending on the details of the polymer−particle interactions.75−78 In the case where there is a strong depletion attraction between the particles, there is also a broad two-phase coexistence region between particle-rich and particle-poor phases; however, such a macrophase separation is not possible for the single-component HNP system, as discussed above. Thus, we expect that the HNP material is in a homogeneous liquid state at low ϕP and in a gel or glassy state at high ϕP. At intermediate ϕP, formation of periodic soft-crystalline anisotropic structures is possible, and it is this region we would like to probe with SCF-DFT calculations. This expectation is reinforced by recent theoretical

RESULTS AND DISCUSSION

In this study, we fix the nanoparticle radius RP = 3.05 nm, so the nanoparticle volume VP = 120 nm3. The monomer volume is set to v = a3 = 1 nm3. The number of ligands per particle, n, and the ligand chain length, N, are varied as shown in Figure 1 and tabulated in Table S4 (Supporting Information). For D

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Figure 3. (a) Three-dimensional view of the “sheet” structure. The surfaces correspond to ϕ = 0.5. (b) Cross section through the center of the particle-rich domainthe color map representing the particle density in the XY-plane. Here, n = 10 and N = 30.

Figure 4. (a) Three-dimensional view of cylindrical morphology, with red representing particle-rich regions. The surfaces correspond to ϕ = 0.75. The blue arrow goes from the center of a cylinder to the point halfway between two adjacent cylinders. (b) Density profiles of particle (red) and ligand (blue) species along the direction of the blue arrow. Here, n = 11 and N = 15.

At lower grafting densities (higher particle volume fraction), nanoparticles attempt to reduce the overall interfacial energy of the system by organizing themselves into two-dimensional sheets or lamellae. The ligands are expelled from those monolayers and form separate particle-free domains, giving rise to lamellar morphology. For example, the system shown in Figure 2 has n = 8 and N = 15, corresponding to σ = 0.068 chains/nm2, RP/Rg = 1.93, and φP = 0.5. In rendering 3D images (Figure 2a and others below) the box size was doubled in each of the three directions. The lamellar period is approximately 2.4RP. The degree of interpenetration between the two species is substantially higher than it is for the case of flexible block copolymer with f = φP = 0.5 and similar degree of

studies showing formation of anisotropic structures in onecomponent hard core/soft shell sphere dispersions.79,80 The morphologies of the HNP melts obtained in the SCFDFT calculations are reminiscent of the theoretical and computational results of Pryamitsyn et al.16 (even though that work was devoted to solutions of HNPs, not melts, the main factors determining anisotropic assembly are very similar). The critical competition is between the interfacial energy of particle/polymer contacts (favoring the nanoparticle clustering and aggregation) and the conformational entropy of the ligands (favoring a more uniform dispersion). The result of this competition is different at different ligand lengths and grafting densities. We will now describe the various morphologies in more detail. E

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Figure 5. (a) Three-dimensional view of the FCC morphology, with red representing particle-rich regions. The surfaces correspond to ϕ = 0.5. The blue arrow goes from the center of a sphere to the point halfway between two spheres in the same layer. (b) Density profiles of particle (red) and ligand (blue) species along the direction of the blue arrow. Here, n = 24 and N = 15.

incompatibility χN, in excellent agreement with earlier study of Lee and co-workers.60 If one increases the grafting density of the ligands, the overall particle volume fraction is reduced, and one would expect a transition to a morphology where the particles are packed less tightly. While in flexible-chain block copolymers, such a morphology usually is a cylindrical one, here we also find a structure that can be labeled as “sheets”: the particle-rich and particle-free layer alternate similar to lamellar structure, but within the particle-rich layer, two-dimensional crystallization of spheres is taking place (Figure 3). The structure of this layer is similar to sheets observed in simulations of Akcora et al.15 (HNPs in polymer melts), Glotzer and co-workers28,50 (tethered nanospheres), and experiments of Jaeger and coworkers81 (nanoparticle membranes assembled at oil−water interfaces). Figure 3 shows the particle arrangement for the system with n = 10 and N = 30, corresponding to σ = 0.086 chains/nm2, RP/Rg = 1.36, and φP = 0.29. The arrangement of the particles within the layer is consistent with two-dimensional (monolayer) hexagonal crystal. The interparticle distance within the layer depends on the particle volume fraction and is significantly smaller than the distance between neighboring particles in adjacent layers. Most of the ligands are expelled from the particle-rich layers and form a particle-free layer between them, similar to the lamellar case. At lower particle volume fractions (higher grafting densities), a cylindrical morphology is observed. The particles now are arranged into strings, and the strings, in turn, are assembled into a 2D hexagonal crystalline structure. The system shown in Figure 4 has n = 11 and N = 15, corresponding to σ = 0.094 chains/nm2, RP/Rg = 1.93, and φP = 0.42. The intercylinder spacing is approximately 2.8RP. As in the lamellar case, there is substantial interpenetration between the particle core and the polymer matrix, once again in agreement with the results of Lee and co-workers.60 Further increase in the grafting density (decrease in the particle volume fraction) leads to the formation of threedimensional crystalline structures of spheres. Before discussing this morphology in more detail, we caution that the SCF-DFT free energy functional is not accurate enough to distinguish

between crystalline structures like FCC, BCC, and HCP. In this study, the FCC structure was used as a representative one. It is possible that the choice of FCC or BCC would give slightly different results, but the main featuretransition from cylinders to dispersed sphereswould remain. The topic of determining the exact crystalline morphologies of sphereforming block copolymers (including the so-called Frank− Kasper phases82) recently emerged as a new frontier in polymer science, and similar challenges would apply to HNP systems; it is, however, out of scope for this study. The morphology of an FCC crystal corresponding to the system with n = 24 and N = 15, corresponding to σ = 0.205 chains/nm2, RP/Rg = 1.93, and φP = 0.25, is shown in Figure 5. The distance between two neighboring spheres is approximately 3.0RP. The ligands are now distributed in a nearly isotropic fashion, so that each HNP can be modeled as a core− shell sphere, with hard core and soft shell. To our knowledge, three-dimensional assemblies of HNPs have not yet been probed with SCF-DFT but have been extensively studied using particle-based28,30−32,50 and integral equation-based41,42,45 approaches. To plot the phase map, we need to compare the free energies of all morphologies for each value of N and σ. The resulting equilibrium (lowest-free energy) morphologies are shown in Figure 6. From Figure 6, one can see that if the ligands are sufficiently long (large N), they cause segregation between particle-rich and particle-free regions. Then, at relatively low grafting density, the HNP system is arranged into lamellae; further increase in the grafting density leads to the formation of sheets with twodimensional crystallization within the particle-rich layers. As the grafting density is increased further, one can expect those layers to melt and change into isotropic, disordered liquid. In the case of shorter ligands, the sheet morphology is pre-empted by the cylindrical and FCC structures. We can gain further insights by replotting Figure 6 using (φP, RP/Rg) coordinates instead of (N, σ) coordinates (Figure 7). Such a representation would facilitate comparison between the behavior of tethered nanoparticles and conventional diblock copolymers and also allow comparison with experimental data. F

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system prefers to segregate the particle-rich and particle-free regions as either lamellar or sheet morphology. The transition from sheet to lamellar shifts to higher φP as RP/Rg is increased. At higher RP/Rg, the one-dimensional segregation between particle-rich and particle-free regions becomes less favorable, and the sheet morphology is pre-empted by morphologies with higher average interfacial curvature, hexagonal cylindrical, or FCC. It is important to caution that the current phase map is based on the specific choice of the particle size and the specific sphere free energy functional; thus, one can expect that the phase boundaries could be refined by future studies. We can also compare our phase map to some recent experimental studies (see Table 1 for detailed description of the nanocomposite particles). Koerner et al.5 observed the formation of anisotropic structures for their HNP system with the “intermediate” grafting density (φP = 0.18, RP/Rg = 0.82). According to SCF-DFT predictions, this HNP system is indeed expected to exhibit anisotropic ordering (“sheet” morphology to be more precise), as marked by the gray star labeled “1”. Goel et al.19 prepared a number of HNP systems forming a weak FCC crystalall those systems would be located in the upper left corner of our phase map (large RP/Rg and/or small φP), as schematically depicted by the blue star labeled “2” and the two arrows pointing leftward and upward. As was mentioned above, several earlier theoretical models analyzed in detail the transition from crystal (FCC) to isotropic liquid.23,51,53 Within those theories, it was predicted that the FCC crystal would “melt” upon increase in the degree of polymerization N and grafting density σ. As the “reduced” grafting density, σ* = σNa2, is increased, the polymer layer undergoes transitions from “mushroom” to “semidilute polymer brush” (SDPB) to “concentrated polymer brush” (CPB). Upon further increase in σ*, the polymer shell becomes layered, with the CPB layer surrounding the core and the SDPB layer facing the neighboring particles. As proposed e.g., by Choi et al.,23 the melting of the FCC crystals takes place where the volume fraction of SDPB becomes greater than the “free volume” of the FCC lattice (∼0.26). However, exact calculation of the CPB and SDPB volumes is challenging even in the limit of isolated HNP in a good solvent or polymer melt. Previous theoretical models were based on the scaling theory of Daoud and Cotton,83 originally developed for solutions of star polymers, and appropriately adapted to the HNP case.84 Other authors recently demonstrated that the use of selfconsistent field theory (SCFT) following the approach of Wijmans and Zhulina85 can result in a reasonably good agreement with experiment.86,87 Our prediction for the FCC melting (dashed thick blue line in Figure 7) can be ⎛ R ⎞* approximated by the following expression, ⎜ RP ⎟ ≈ 3.0ϕP1/2 , ⎝ g⎠

Figure 6. Phase map for the HNP melt. Here, N is ligand length and σ is grafting density. Symbols represent simulation results, with equilibrium (minimum free energy) morphologies shown as follows: yellow = lamellar, gray = sheets, orange = cylindrical, and blue = FCC spherical. Dashed lines are boundaries of the region studied in these simulations, while solid lines are approximate locations of transitions between various phases.

Figure 7. Phase map of the HNP one-component systems expressed in terms of φP and RP/Rg. Symbols are labeled as in Figure 6. Solid lines are guide to the eye depicting approximate phase boundaries. Stars with numbers represent the phase map locations of HNP systems from refs 5 and 19 (see text for more details). The dashed blue line is the predicted FCC melting transition.

Within the SCF-DFT theory in the strong segregation regime, HNP systems having the same scaled parameters (φP, RP/Rg) are expected to have the same morphologies; thus, we can compare results for systems with different core radii and superimpose them on the same phase map. Examination of the phase map of Figure 7 shows several interesting features. For small RP/Rg (longer ligands), the

or, alternatively, N * =

2 3

2

( Ra ) ϕP−1 P

for the transition chain

length assuming a given core volume fraction and nanoparticle radius. This scaling is qualitatively similar to the analysis of

Table 1. Experimental HNP Systems Referenced in This Study HNP system, as labeled in the original paper

core material

core radius (nm)

polymer shell

polymer MW (kg/mol)

grafting density (chains/nm2)

estimated φP

estimated RP/Rg

morphology

ref

PBA-4K PBA-25K SiPS0p05

SiO2 SiO2 SiO2

8.25 8.25 8.0

PBA PBA PS

3.9 24.7 120

0.8 0.8 0.05

0.33 0.08 0.18

4.65 1.85 0.82

FCC FCC aniso

19 19 5

G

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Macromolecules Choi et al.,23 although detailed comparison is beyond the scope of this paper. One crucial question is whether the predicted anisotropic microphase-separated phases could be observed experimentally. Unlike in the case of block copolymers, microphase-separated ordered phases like hexagonal cylinders or lamellae have yet to be seen, even though anisotropic aggregates with short-range order are now known. It is possible that the HNP assemblies are not able to fully equilibrate and the morphologies observed in experiments are metastable or kinetically trapped. Alternatively, it is possible that the inability of the HNPs to organize into long-range ordered anisotropic structures is due to polydispersity (with regard to both particle size and ligand length). A recent study by Bachhar et al.88 shows that in polydisperse HNP systems (at least in solution or in a polymer matrix) there is a tendency to fractionate, rather than organize based on the “average” particle size and ligand length. On the basis of the modeling, one can hypothesize that “single-component nanocomposites” consisting of nanoparticles with one or more grafted polymer ligands offer wide variety of complex and potentially unexplored structures. In the simplest case where the ligands are pure monodisperse oligomers or homopolymers (and, in principle, we can graft linear block copolymers, star-polymers, blends of ligands, etc.), the morphology map is somewhat similar to that of diblock copolymers. Furthermore, the characteristic dimensions of those structures should be smaller than those of conventional block copolymers. For example, a lamellar period of a 104 kg/ mol PS-b-PMMA is approximately 48 nm,89 and even for higher-χ polymers such as PS-b-PDMS, structures with characteristic dimensions below 10 nm are difficult to prepare. For the HNP materials, characteristic dimensions are about 2− 3 times the particle radius, so sub-10 nm lamellar domains could be obtained in a fairly straightforward way. Furthermore, the two domain types (particle-rich and particle-free) will have extremely different properties (mechanical, thermal, physical, electrical, and optical). And finally, one could potentially find ways of making different types of porous materials by removing either the organic portion (perhaps by high-temperature treatment that should also ensure that particles have sufficiently sintered) or the particles themselves. The SCF-DFT approach, while fast and simple, has some limitations. As already mentioned, the free energy functional is based on the ensemble-averaged description of the particle density, rather than instantaneous particle positions. As a result, the detailed investigation of particle arrangements, e.g., within the monolayers, is not straightforward. Moreover, within this approach, one is likely limited to exploration of “intermediate” (φP ∼ 0.2−0.5) core volume fractions. At small φP, the particles become more delocalized, and density fluctuations (not captured in SCF-DFT) would become the dominant contribution to thermodynamics. At large φP, while the overall approach could still work, one would need a more elaborate density functional and substantially higher spatial resolution. Likewise, the model is also somewhat limited in terms of RP/Rg range. For very large or very small RP/Rg, the need to increase the spatial resolution so as to accurately describe either polymers or the particle would make computations expensive and impractical. Even so, the SCF-DFT approach can provide initial screening, to be followed by more extensive particlebased simulations. Furthermore, a full exploration of a block copolymer phase diagram using SCFT (or, in this case, SCFDFT) requires extensive optimization of the unit cells for

various candidate morphologies, followed by very accurate free energy calculation and selection of the lowest-free energy morphology for each point in the phase space.90,91 In particular, it is likely that gyroid, double-diamond, or other bicontinuous structures could be found at least for some parts of the phase map. Such a program is beyond the scope of the current paper and will be explored in the future.

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CONCLUSIONS We developed a new modeling framework aimed at predicting the morphology of polymer-grafted or “hairy” nanoparticles. The model is based on the SCF-DFT approach originally developed by Thompson et al.61 and expanded to account explicitly for the ligand configurations. The simulations demonstrated substantial similarity between the HNP onecomponent systems and the block copolymer melts, with spherical, cylindrical, and lamellar morphologies present in both material types. The simulation results suggest that the arrangement of nanoparticles in single-component HNP nanocomposites can be isotropic or nearly isotropic (FCC), quasi-one-dimensional (cylindrical), or quasi-two-dimensional (sheets/lamellar). The anisotropic morphologies are formed as the ligands rearrange along the surfaces of the particles, thus allowing more intimate particle−particle contacts. Note that we assumed a weak bonding between the ligand and the particle surface; it is possible that the phase diagram would change if the ligands are “locked” in their positions (although one would assume that the importance of the ligand binding strength should diminish as the ligand chain length is increased). The method proposed here can be extended to a number of other HNP systems (for example, considering the cases of block copolymer ligands or mixed homopolymer ligands). This will be the subject of future studies.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b01922. Selection of the Flory−Huggins parameters, morphology optimization procedure (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Valeriy V. Ginzburg: 0000-0002-2775-5492 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by The Dow Chemical Company. The author thanks Prof. Arthi Jayaraman (University of Delaware) and Drs. Jeffrey Weinhold, YuanQiao Rao, and Peter Trefonas (Dow) for stimulating discussions. I am also thankful to anonymous reviewers for many helpful suggestions.

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