Assembly of Amphiphilic Block Copolymers and Nanoparticles in

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Assembly of Amphiphilic Block Copolymers and Nanoparticles in Solution: Coarse-Grained Molecular Simulation Study Daniel J. Beltran-Villegas† and Arthi Jayaraman*,†,‡ †

Colburn Laboratory, Department of Chemical and Biomolecular Engineering and ‡Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, United States S Supporting Information *

ABSTRACT: Controlled assembly of amphiphilic block copolymers (BCPs) and inorganic nanoparticles (NPs) into hybrid materials is desirable for a broad range of applications such as biological or nonbiological cargo delivery, imaging contrast agents, pollutant capture, chemical sensing, and separation/purification applications. There has been growing interest in changing solvent quality for BCPs by mixing solvents and utilizing the effective solvophobicity of the BCP block(s) to tailor the assembled structure, namely the size and shape, composition, and spatial arrangement of the components in the NP−BCP hybrid assemblies. In this work, we present a comprehensive coarsegrained molecular dynamics (MD) simulations study exploring the impact of varying solvophobicity on assembly of amphiphilic BCP and NP as a function of BCP composition and sequence and NP affinity to either or both block(s) of BCP. We quantify the solvophobicity marking the transition from disassembled solution to assembled state (e.g., micelles). We also quantify and visualize, as a function of varying solvophobicity, the shape and size of assembled structures with and without NPs, the amount of NP uptake, and the spatial arrangement of the NPs in the assembled NP−BCP structure.

1. INTRODUCTION Assemblies of amphiphilic block copolymers (BCPs) and nanoparticles (NPs) belong to a wider category of hybrid inorganic−organic materials that combine technologically desirable characteristics of NPs (e.g., quantum confinement,1 surface properties,2 dielectric contrast,3 and ferromagnetic behavior4,5) and block copolymers (e.g., biocompatibility,6 reversible self-assembly7). These hybrid NP−BCP assemblies also present structural features at a range of length scales4,7,8 and have found uses in biological or nonbiological cargo delivery,9 imaging contrast agents,4,10 pollutant capture,11 chemical sensing,12 and separation/purification13 applications. Hybrid NP−BCP materials are formed by directed assembly of NPs in the BCP matrix14,15 or concurrent self-assembly of both inorganic and organic components in solution.4,7 Because the focus of this paper is on concurrent self-assembly of NPs and amphiphilic BCPs as a function of BCP design (e.g., sequence and composition), solvophobicity, and NP affinity, we chose to highlight relevant past work in this area. We direct the readers to review articles that describe the experimental as well as theoretical efforts geared toward directed assembly of NPs in BCPs.16−24 Amphiphilic diblock copolymers self-assemble in solutions through aggregation of solvophobic blocks of the copolymer.8,25−31 They form a wide variety of assembled structures depending on the copolymer chain design, i.e., the solvophilic and solvophobic block composition and arrangement, solvent © XXXX American Chemical Society

quality, i.e., how unfavorable the solvent is to the solvophobic block, and concentration of BCPs in the solution.8,25−30,32,33 In the presence of NPs, depending on the relative affinities of NPs to the solvophilic or solvophobic blocks of BCP34 and their size,35,36 NPs locate themselves in the domains or interface of BCP assembled structures. NPs also alter the self assembly of BCPs at a fundamental level by swelling some blocks, thus changing the geometry of the chain and the resulting assembled NP−BCP structures.37 Extensive work has been done to establish how varying BCP-solvent pairs (see references in review articles8,25,27−30,32,38) as well as NP surface functionalization39−47 impact the morphology and eventual properties of these materials in specific applications. Solution processing techniques also tune assembly in NP−BCP systems.4,5,7,11 One example is assembly of NPs and BCPs in oil-in-water (spherical) nanoemulsions, where evaporation of a solvent4,48 causes the formation of mostly spherical, assembled NP−BCP structures. Another processing technique involves solvent mixing/annealing, where all components are initially dispersed in a good solvent and gradual introduction of a poor solvent for one or more components leads to exotic NP−BCP assembled structures like core−shell, spherical and worm-like micelles, and Special Issue: Emerging Investigators Received: October 25, 2017 Accepted: January 22, 2018

A

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lamellae.4,10,49 An even wider pool of structures is accessed by changing solvent quality to kinetically trap structures out of equilibrium conditions, resulting in exotic structures like helices50 or segmented micelles.51 With increasing complexity in the building blocks (e.g., ABC triblock copolymers) and processing conditions52 (e.g., sequential “switch” of affinity between blocks or shear flow53), the range of assembled structures, such as toroidal micelles,54 helical-cylinder micelles,55 stacked disk-like micelles,56 also becomes more sophisticated and difficult to predict a priori. Given the extensive design space comprised of BCP sequence and composition, solvent type(s), and mixing, NP size, shape, chemistry, and functionalization, molecular simulations become a useful tool for predicting assembly and guiding experiments under the unexplored conditions. So far, most of the theory/simulation and experimental studies of hybrid NP−BCP assemblies have primarily been for diblock copolymer sequences5,10,11,34,36,37,57−60 and triblock sequences.61 Furthermore, computational studies focus mainly on assembly at fixed solvophobicity conditions58−60 even though solvent mixing/annealing has a nonintuitive impact on the assembly of solutions of NPs and BCPs,5,11 as stated above. To better understand the molecular mechanisms underlying these varying solvophobicity effects on NP−BCP concurrent assembly as a function of BCP design and to provide extensive data required for machine learning purposes,62−65 there is a need for a comprehensive simulation study covering/spanning this large NP−BCP design space, which is addressed in this paper. In this work, we use coarse-grained molecular dynamics (MD) simulations to conduct a comprehensive study on the impact of varying solvophobicity on the assembly of amphiphilic BCPs (with varying BCP composition and sequence) and NPs (with varying size and affinity to either or both block(s) of the BCP). The range of BCP sequences we cover includes AB diblock and ABA triblock with symmetric and asymmetric compositions of solvophilic A and solvophobic B blocks. We first study the assembly of the BCP systems without NPs to validate our simulation techniques through direct comparison with known experimental results of analogous systems. We then study assembly in solutions of BCPs with NPs of varying sizes and degrees of affinity to either or both blocks at varying solvophobicity. We quantify the transition value of solvophobicity needed for micellization. We also quantify and visualize the (a) shape and size of assembled structures with and without NPs, (b) NP uptake in the assembled structure, and (c) spatial arrangement of the NPs in the assembled NP−BCP structure as a function of solvophobicity. The paper is organized as follows. In section 2, we present the details of the model and the simulation method as well as the analyses performed to evaluate NP−BCP assembly. In section 3, we present the results of the impact of amphiphilic BCP design (i.e., BCP composition and sequence) and NP design (i.e., affinity to either or both block(s) of BCP) on NP− BCP assembly with varying solvophobicity. In the last section, we present the main conclusions and implications of the results presented in this paper as well as our vision for future work.

spring66,67 chains with solvophilic (A) and solvophobic (B) beads of diameter σ, where each bead represents a Kuhn segment for the solvophilic and solvophobic polymer chemistries. The harmonic spring potential in the BCPs has an equilibrium bond length of σ and force constant of 50ε/σ2, where ε is the energy unit in the simulation, which can be related to real units by mapping to the generic model to a specific polymer chemistry. We model each NP with a single particle (C) bead with bead diameters dC = 1.0σ and dC = 2.15σ. We model the solvent(s) implicitly with the solvophobicity captured through the effective attractions between B beads of the BCP. This effective solvent-induced attraction between pairs of nonbonded solvophobic B beads is modeled using Lennard-Jones interaction potential with a cutoff distance of 2.5σ and well-depth εBB as a measure of solvophobicity. We note that, although explicit solvent representation is more realistic, it decreases the computational efficiency and impedes studies of large experimentally relevant length and time scales. Implicit solvent representation facilitates faster study of assembly of BCPs and NPs over a large design space. Implicit solvent representation captured through effective B−B attractive interactions can be connected to specific chemistries of polymers and solvents by calculating the potential of mean force (PMF) between two specific polymer Kuhn segments in explicit solvent or solvent mixture using atomistic simulations with enhanced sampling techniques (see, for example, ref 68). The features of the PMF (e.g., close-range repulsion, midrange attraction) can then be fit to LennardJones B−B interaction potentials that we use in our generic coarse-grained model. The implicit solvent coarse-grained model can also be mapped to an equivalent explicit solvent coarse-grained model; in the latter case, the solvent quality is captured by the interaction between solvent beads and polymer beads. The procedure to map an implicit to an explicit solvent model has been pursued in the literature usually by matching pair correlation functions between different segments in the implicit vs explicit solvent system.69,70 In the Supporting Information (SI) section S1, we present a simpler mapping between implicit and explicit solvent models based on equating the Flory−Huggins interaction parameter between the polymer and the solvent. We show that, at least for one sequence and the system size we tested, the implicit and explicit model results show agreement in the micellization behavior proving the equivalence of our implicit solvent system to an explicit solvent description (Figure S1). To model the affinity of the NP to each or both chemistries in the BCP (i.e., A−C and B−C pair interactions), we use Lennard-Jones interaction potentials with a cutoff distance of 1.5σ + dC and vary the well-depth of these potentials, εAC and εBC. Past computational71,72 and experimental73,74 work on functionalized NPs has shown that the functionalization on the NP can be designed to tailor the effective interactions between the functionalized NP and the medium in which the particles are placed. Additionally, one could calculate the effective interactions between the particle and medium for specific polymer and NP chemistries to our model interaction potentials. To capture the realistic nature of interactions in amphiphilic BCPs and NPs in solution, where the dominant interactions driving assembly and NP arrangement are B−B, A−C, and B− C attractive interactions, we model other nonbonded pairwise interactions, A−A, A−B, and C−C, using purely repulsive Weeks Chandler Andersen75 (WCA) potentials with a cutoff

2. METHODS 2.1. Model. We use a generic coarse-grained model to represent amphiphilic AB block copolymers (BCPs) and NPs in solution. The BCPs are modeled as linear flexible bead− B

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Figure 1. Design parameters explored in this study. (a) Schematics of different block copolymer compositions and sequences explored in this work. Red beads represent solvophilic (A) beads; green beads represent solvophobic (B) beads. Shorthand notation for each case is shown alongside schematics. (b) Schematics of nanoparticle (yellow C beads) sizes used. (c) Schematic representation of interaction parameters between block copolymer beads and nanoparticles (best viewed in color).

Table 1. Total Number of Beads for Simulations Run for Different NP Sizes, NP Loading, and Simulation Box Size dC = 1.0σ

dC = 2.15σ

simulation box size

ϕC

Nchains

NBCP

NC

Ntot

Nchains

NBCP

NC

Ntot

regular, Lx = Ly = Lz = 67σ

0 0.1 0.25 0 0.1 0.25

600 540 450 2,400 2,160 1,800

14,400 12,960 10,800 57,600 51,840 43,200

0 1,440 3,600 0 5,760 14,400

14,400 14,400 14,400 57,600 57,600 57,600

600 540 450 2,400 2,160 1,800

14,400 12,960 10,800 57,600 51,840 43,200

0 144 360 0 576 1,440

14,400 13,104 11,160 57,600 52,416 44,640

large, Lx = Ly = Lz = 106.4σ

All εij values in this work are presented in reduced units of energy and normalized by the same ε to facilitate comparison; as stated above, if one wishes to link a well-depth εBB of 0.4, say, to a real energy value, they can map real polymer chemistries to our generic coarse-grained model. The NP loading in the BCP solution is denoted by ϕC, which is defined as

distance of 21/6σ for A−A and A−B interactions and 21/6dC for C−C interactions. 2.2. Design Space with Varied Parameters. We maintain BCP chain length to be 24 Kuhn segments and vary BCP composition and sequence. The two BCP compositions considered are symmetric, i.e., 12A and 12B beads, solvophilic heavy (A-heavy) asymmetric, i.e., 18A and 6B beads, or solvophobic heavy (B-heavy) asymmetric, i.e., 6A and 18B beads. The BCP sequence denotes the relative arrangements of A and B beads. Figure 1a shows the different compositions and sequences explored in this work. These compositions and sequences will be referred to in the following sections with the shorthand shown in Figure 1, i.e., AX-b-BY for diblock and AX-bBY-b-AX for ABA triblock sequences, where the subscripts X and Y refer to the number of A or B Kuhn segments or beads in each block. Figure 1b shows the different NP sizes used, i.e. dC = 1.0σ and dC = 2.15σ. The small, dC = 1.0σ, NPs are equal to the Kuhn segment size of the polymers. The larger NPs of size dC = 2.15σ have a volume that is ten times the volume of dC = 1.0σ particles and is 0.78-times the radius of gyration of the polymer at low εBB. We increase the B block solvophobicity by increasing values of εBB from 0.1 to values above 1.0 through a gradual stepwise procedure described in the next section. The affinity of the NP to one or both blocks of BCP is varied as εAC = 0.25, 0.50, and 1.00 and εBC = 0.25, 0.50, and 1.00. Figure 1c shows εBB, εAC, and εBC interactions in the schematic.

ϕC =

VCNC VBCPNBCP + VCNC

(1)

where NC is the number of NP beads, VC is the volume of an NP bead, NBCP is the number of copolymer beads, and VBCP is the volume of a polymer bead. We explore three ϕC = 0.0 (no NP), 0.1, and 0.25. For all cases, we maintain the simulation box total packing fraction, η = 0.025, where η is defined as η=

VBCPNBCP + VCNC Vtot

(2)

where Vtot is the volume of the simulation box. We choose a simulation box size of 67σ to ensure no selfinteractions of BCP chains across the box size through periodic images. We ensure there are no system size effects by also conducting (for a few select systems) simulations with a larger box size of 106.4σ. The total number of beads for each simulation box is summarized in Table 1. 2.3. Simulation Details. We use the LAMMPS76 package to simulate the assembly of amphiphilic block copolymers and C

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NPs in the canonical (NVT) ensemble. We randomly place BCP chains initially in rodlike extended state and NPs (when added) on a cubic lattice in a large simulation box (i.e., at low total packing fraction η). First, we let the system relax from this initial configuration and then compress the system to a simulation box size to achieve a volume fraction of η = 0.025 while maintaining temperature T* = 4 and εBB = 0.1 for a total of 600,000 time steps, where a time step corresponds to Δt = 0.005. Then, over an additional 600,000 time steps, we lower the temperature from T* = 4 to T* = 1. After another 600,000 time steps at T* = 1, we start a stage-wise gradual increase in εBB using the expression εBB,i+1= εBB,i + ΔεBB, where εBB,i is the εBB in the ith stage and ΔεBB = 0.009. In each stage, the value of εBB is kept constant, and the simulation is run for 106 time steps for all cases explored, which we ensure is enough to equilibrate the system. In each stage, the simulation box configurations are saved every 100,000 time steps, ensuring the sampled configurations are uncorrelated. This particular choice of annealing schedule quantified by stage length, i.e., the number of time steps over which εBB is kept constant and the height, i.e., ΔεBB, is varied to ensure that the equilibrium structure in each stage is reproduced irrespective of ΔεBB or stage length (in time steps). Lack of reproducibility would suggest that the system is likely kinetically trapped, and a slower annealing (i.e., smaller ΔεBB) and/or longer stages at each εBB are needed to ensure escape from kinetically trapped/metastable states. Computationally, a gradual increase in εBB mimics experiments where a second solvent that is poorer in quality to the solvophobic block than the initial good solvent is gradually added to increase the solvophobicity. As the mixture of solvents becomes more unfavorable to the solvophobic B block, the BCPs undergo micellization through aggregation of the B block. 2.4. Analyses. MD simulations are analyzed to identify micelles/clusters, quantify the structure of the BCP-NP clusters, and determine the NP uptake and spatial arrangement within the clusters as the fraction of all NPs located in the core, corona, and interfacial regions of the clusters. 2.4.1. Cluster Identification and Cluster Analysis. To identify BCP clusters at high εBB, we first find groups of BCP chains. To find groups of BCP chains, we initially calculate the radial distribution function between centers of mass of solvophobic B block (i.e., gBB,COM (r)) at high εBB, where we see clusters (visually). Then, we define any two BCP chains to be part of the same cluster if they are neighbors, defined as chains whose respective centers of mass of the solvophobic B block are less than a coordination threshold 5σ, the first valley of the gBB,COM (r) at high εBB. We then define a cluster when each chain in that cluster has at least five other neighboring chains. The details of how we select a critical number of five neighbors is in Supporting Information (SI) section S2 and Figure S2. After pairs of chains are identified as part of the same cluster, we use a “friends-of-friends” algorithm77 to identify all clusters. Once all the clusters are identified, the number of clusters (different from unimers) denoted as n and the size of each cluster (i.e., number of chains in the cluster), denoted as Nclus,i for the ith cluster, are recorded. Using the n number of clusters and simulation box volume Vtot, we calculate the cluster density/concentration as ρclus =

n Vtot

Using the individual cluster sizes (i.e., number of chains in each cluster) and number of clusters, we also calculate the average cluster size as n

∑i = 1 Nclus, i

⟨Nclus⟩ =

(4)

n

We also calculate cluster size distributions, P(Nclus), by constructing histograms of Nclus,i. 2.4.2. Relative Shape Anisotropy, κ2. Relative shape anisotropy quantifies the shape of the cluster and is estimated from the gyration tensor of a cluster as follows. The radius of gyration tensor is defined as N

[R g]ilm =

24

cm ∑ j =clus,1 i ∑k = 1 (rijkl − rilcm)(rijkm − rim )

Nclus, i

(5)

where rijkl is the lth component (e.g., “x”, “y”, or “z”) of the position vector of bead k belonging to copolymer chain j on cluster i and rilcm is the lth component of the center of mass position vector for cluster i. The eigenvalues of the gyration tensor, λ1i, λ2i, and λ3i, correspond to mass distribution moments along each of the principal directions, i.e., eigenvectors v1i, v2i, and v3i. The cluster radius of gyration, Rgi, is given by R gi =

λ1i + λ 2i + λ3i

The relative shape anisotropy, κ i2 = 1 − 3

(6)

κ2i,

is given by

λ1iλ 2i + λ1iλ3i + λ 2iλ3i (λ1i + λ 2i + λ3i)2

(7)

and the average cluster relative shape anisotropy is given by n

⟨κ 2⟩ =

∑i = 1 κi2 n

(8)

The value of κ2 gives an indication to the shape of clusters with ⟨κ2⟩ ∼ 0 for spherical clusters, ⟨κ2⟩ ∼ 1 for linear or 1dimensional clusters, and ⟨κ2⟩ ∼ 0.25 for planar or 2dimensional clusters. 2.4.3. NP Uptake, f tot, and NP Location within the Clusters. Average NP uptake, f tot, is defined as

ftot =

NC,clus NC

(9)

where NC is the total number of NPs in the simulation box and NC,clus is the total number of NPs that are in clusters. An NP is said to be in a cluster if it is within 0.5σ + dC distance from any copolymer bead in that cluster. Additionally, we assign NPs to corona, core, or interface depending on the types of BCP beads within the 0.5σ + dC shell around the NP. If more than 75% of copolymer beads in the 0.5σ + dC shell around the NP are A (solvophilic) beads, then the NP is said to be located in the corona; if more than 75% of those beads are B (solvophobic) beads, then the NP is said to be located in the core. If a mixture of A and B beads are found within the shell, i.e., 25−75% A and rest B, then the NP is located at the interface. We note that this choice of 25%/75% threshold is arbitrary. We test 20%/80% and 30%/70% thresholds for NP location and find only small quantitative differences and no qualitative differences in our results. The average number of NPs in the corona per micelle, ⟨NC,corona⟩, is defined as

(3) D

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Figure 2. Cluster formation for A12-b-B12 chains as a function of solvophobicity parameter εBB. (a) Cluster density, (b) average cluster size, and (c) average relative shape anisotropy as a function of εBB. (d−h) Cluster size distributions and as well as representative snapshots of clusters of average size at εBB values marked with vertical lines in (a−c).

⟨NC,corona⟩ =

NC,corona n

p=

(10)

where NC,corona refers to the total NPs in the corona as specified above. In a similar manner, we define the average number of NPs in the core and core−corona interface per micelle, ⟨NC,core⟩ and ⟨NC,int⟩, respectively, as ⟨NC,core⟩ =

(11)

and

⟨NC,int⟩ =

NC,int n

(13)

where VB corresponds to the volume of the solvophobic block per chain, lB is the length of the solvophobic block, and aAB is the core−corona interface area per chain with all quantities corresponding to BCPs assembled in clusters. The value of p is directly correlated with the shape of the micelle formed. Spheres have p = 1/3; cylinders have 1/3 < p < 1/2, and vesicles have p > 1/2. As p increases, the curvature of core− corona interface in the micelles decreases. In the present work, we use the packing factor as a qualitative argument that explains micelle structure changes we observe.

NC,core n

VB lBaAB

(12)

3. RESULTS AND DISCUSSION 3.1. Assembled Structures of BCPs without NPs. In this section, we present results for BCP solutions without NPs to describe the key structural transitions as a function of εBB and BCP sequence along with validation for our simulation results through comparison to past experimental/computational studies. For brevity, we present only a few selected BCP sequences and compositions here. The results for the BCP sequences not presented in this section are described in the SI section S3 (Figures S3−S5).

where NC,core and NC,int are the total number of NPs in the core and at the interface, respectively. 2.4.4. Packing Factor, p, for Qualitative Assessment of Micelle Structures. The packing factor, p, relates the relative shape of an individual amphiphile that belongs to a micelle. It is a dimensionless number that takes into account both the size and shape of amphiphiles in micelle cores as well as the repulsion between the solvophilic “head” groups of said amphiphile.78 It is defined as E

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Figure 3. Cluster formation for A6-b-B18 chains as a function of solvophobicity parameter εBB. (a) Cluster density, (b) average cluster size, and (c) average relative shape anisotropy as a function of εBB. (d−h) Cluster size distributions and as well as representative snapshots of clusters of average size at εBB values marked with vertical lines in (a−c).

shape anisotropy (RSA) in Figure 2c, ⟨κ2⟩ ∼ 0.0 for A12-b-B12 confirms the spherical shape of the clusters. Panels d−h in Figure 2 show cluster size distributions, P(Nclus), for selected values of εBB to highlight the cluster formation at the different solvophobicities. Figure 2d shows results for εBB = 0.65, where ρclus is approximately halfway through the initial increase shown in Figure 2a. P(Nclus) shows mainly small (Nclus < 5) clusters. Panels e and f in Figure 2 show results at εBB = 0.70 and 0.725, respectively, where the ρclus reaches maximum value. P(Nclus) shows the small cluster peak at Nclus < 5 as well as development of midsize clusters (Nclus ∼ 25) and some large clusters (Nclus ∼ 35). Figure 2g shows results at εBB = 0.75, where ρclus decreases in Figure 2a and occurrences of Nclus ∼ 35 clusters increases. The Nclus ∼ 35 mode becomes the main peak in the P(Nclus) at high εBB as seen in Figure 2h. At higher εBB, P(Nclus) shows no further evolution; this corresponds to ρclus and Nclus reaching plateau values. In summary, A12-b-B12 BCPs chains assemble into a distribution of small and intermediate clusters at low εBB, which upon increasing εBB result in a narrower P(Nclus) of larger clusters. In other words, the unimers and small cluster population are being replaced by a single-mode distribution of clusters of ∼35 chains.

3.1.1. Symmetric AB Diblock BCP Chains, A12-b-B12. In the case of symmetric amphiphilic A12-b-B12 BCP solution, our simulations confirm the expected qualitative behavior that, as εBB increases, the solvophobic B blocks aggregate to form a core surrounded by a corona of solvophilic A blocks. Figure 2 quantifies this micellization/clustering behavior of A12-b-B12 chains as a function of εBB. In Figure 2, the cluster density, ρclus (Figure 2a), is used to characterize the value of εBB where the system transitions from free BCPs (or unimers) to assembled micellar structures. At low values of εBB, the BCPs are unassembled as there is not a sufficient thermodynamic driving force for the aggregation of the B blocks. As εBB increases, the solvophobic B blocks aggregate, and the average cluster size, ⟨Nclus⟩, becomes nonzero (Figure 2b). ρclus reaches a maximum value at εBB ∼ 0.725, and then with a small increase in εBB beyond 0.725, ρclus decreases at first and then reaches a plateau value. Concurrently, in Figure 2b, ⟨Nclus⟩ shows a monotonic increase with increasing εBB, reaching a plateau value of ⟨Nclus⟩ ∼ 30 chains per cluster at high values of εBB. At the high values of εBB, both ρclus and ⟨Nclus⟩ do not change and reach a plateau. The drop in ρclus in this range of εBB ∼ 0.72 and ∼0.76 suggests that either the smaller micelles fuse to form fewer larger micelles or larger micelles increase in aggregation number while taking in the chains present in the smaller micelles. The relative F

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Figure 4. Cluster formation results for A3-b-B18-b-A3 chains as a function of εBB. (a) Cluster density, (b) average cluster size, and (c) average relative shape anisotropy as a function of εBB. (d−h) Cluster size distributions and as well as representative snapshots of clusters of average size at εBB values marked with vertical lines in (a−c) with the x-axis in the (d−h) plots on a log scale.

The information on P(Nclus) change with increasing εBB also serves the purpose of calculating the transition solvophobicity εBBtr. Previous theoretical work on micellization defined the critical micelle conditions (concentration or εBB) as the point where P(Nclus) showed extrema, i.e., the conditions where a distinction between the unimer peak and fully formed micelle peak in the distribution is evident.79 In Figure 2d−f, we see mainly the unimer peak (unimers and intermediate-sized clusters), whereas in Figure 2g and h, a nonunimer peak is evident. We note that the transition between one- and two-peak P(Nclus) occurs roughly at the same εBB value where ρclus is maximum. We therefore use the εBB at the cusp of ρclus as εBBtr. Beyond the quantitative analysis described so far, in the inset images in Figures 2d−h, we present some visual characterization of the spherical clusters that are representative of the Nclus denoted above the image. The clusters have the typical core−corona micellar arrangement, where the corona effectively shields the core from interacting with other cores, being larger in size than the micelle core, commonly referred in the literature as a “hairy” spherical micelle.38 These images also confirm the RSA results of spherical micelles in Figure 2c. Although the micelle-level behavior described up to now is in qualitative agreement with experimental studies on symmetric diblock copolymer micelles, we also found quantitative

agreement in radial distribution functions and structure factors between our MD simulations and results from PRISM theory using the same model. These comparisons were described in a recent publication from our group.80 Briefly, we found good quantitative agreement in the radial distribution functions and structure factors at low εBB, where PRISM theory numerically converged to a solution. Furthermore, through extrapolation of the inverse of the structure factor peak, 1/SBB(k*), we found an expected transition εBB, which agreed with the εBBtr described here. Given this consistent qualitative agreement with past experimental38 and simulation81 observations at high εBB, and quantitative agreement with PRISM theory at low εBB, we are able to validate our chosen computational methodology. 3.1.2. Asymmetric Diblock BCP Chains, A6-b-B18. Figure 3 shows the clustering behavior of asymmetric diblock A6-b-B18 chains as a function of εBB. As in the symmetric diblock A12-bB12 case (Figure 2a), we observe an initial increase of ρclus with increasing εBB until a maximum is reached followed by a decrease until ρclus reaches a plateau (Figure 3a). As in the A12b-B12 case (Figure 2b), the ⟨Nclus⟩ increases monotonically with εBB (Figure 3b) and the clusters are spherical in nature (Figure 3c), but A6-b-B18 has a markedly larger value ⟨Nclus⟩ ∼75 compared to ⟨Nclus⟩ ∼ 35 in the A12-b-B12 case. Additionally, εBBtr for A6-b-B18 chains, marked by the cusp of ρclus, is lower G

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than that of A12-b-B12. These trends are expected as the A6-b-B18 copolymer is richer in the solvophobic B block than A12-b-B12 and thus needs lower solvophobicity to assemble. There are additional differences between A12-b-B12 and A6-bB18 sequences of P(Nclus). At low εBB, P(Nclus) shows primarily small spherical clusters in both cases (Figures 2f and 3f). An increase in εBB close to the cusp in ρclus (Figure 3e and f) leads to clusters of Nclus ∼ 40 forming. As the εBB increases further, P(Nclus) shifts to Nclus ∼ 50 (Figure 3h) and eventually to a broad distribution centered around Nclus ∼ 80. The shift to increasingly larger Nclus and diminishing number of smaller clusters with increasing εBB is consistent with the idea of fusion of smaller clusters to form larger ones or growth of larger clusters at the expense of smaller ones. In contrast, in A12-b-B12 chain clusters, P(Nclus) shown in Figure 2f−h is centered on a fixed value of Nclus ∼ 35 consistent with lack of cluster growth with increasing εBB. Connecting our observations to past studies, this shift in micelle size distribution toward larger micelles with increasing εBB has been reported in past Monte Carlo simulation studies of model solvophobic block-rich diblock copolymers,82,83 where smaller micelles are expected when solvent selectivity is low and micelles grow as solvent selectivity increases.8 Past experimental work on polystyrene(PS)-b-poly(acrylic acid)(PAA) in aqueous solvent also shows an increase in ⟨Nclus⟩ with decreasing length of the PAA solvophilic block as well as with an increase in the length of the PS solvophobic block, consistent with our findings.84,85 The increase in ⟨Nclus⟩ compared to A12-b-B12 is expected for two reasons: (1) a shorter A block for a fixed B block length means a smaller core−corona interface area per chain leading to more BCPs packed in a spherical micelle, and (2) a longer B block with a fixed A block length means an increase in the volume per chain in a core with the core− corona interface area per chain changing little leading to more BCPs packed in a spherical micelle. By increasing the B block and reducing the A block, both effects lead to the observed result of higher ⟨Nclus⟩.86 Our results in Figure 3 show that the εBBtr for the A6-b-B18 copolymer decreases with respect to A12-bB12 BCPs. This decrease in εBBtr is consistent with previous Monte Carlo simulation work that showed BCP chains of equal length but with a larger portion of the solvophobic component micellized at lower values of solvent selectivity.82,83 3.1.3. Asymmetric ABA Triblock BCP Chains, A3-b-B18-b-A3. Going beyond the diblock copolymers, in Figure 4 we show the clustering behavior of A3-b-B18-b-A3 triblock copolymer chains as a function of εBB. We compare results for the A3-b-B18-b-A3 solution to that from A6-b-B18 solution as both BCPs have the same composition (25% A and 75% B) and vary only in sequence. Comparing the clustering behavior shown in Figure 4a and b with that in Figure 3a and b, we see that the increase, decrease, and plateauing in ρclus (Figure 4a) for the triblock occurs at larger values and over a wider range of εBB than for the diblock. ⟨Nclus⟩ reaches a higher value (Figure 4b) of ⟨Nclus⟩ ∼ 90 and at larger values of εBB than the diblock sequence (⟨Nclus⟩ ∼ 75 in Figure 3b). Lastly, the value of RSA for the clusters of A3-b-B18-b-A3 (Figure 4c) suggest nonspherical shapes unlike the spherical shapes of the diblock sequence. To show that there is precedence of triblock BCPs needing higher εBBtr than their diblock analogue, we cite past experimental studies87 of hydrophilic poly(N,N-dimethylacrylamide) (DMA) and poly(N-isopropylacrylamide) (PNIPAM) in water. These studies have found that triblock copolymers with the same composition as diblocks require a higher

micellization temperature for otherwise equal conditions. Because PNIPAM has a lower critical solution temperature (LCST) phase behavior, we can equate higher temperatures for PNIPAM in water to conditions that lead to higher solvophobicity; thus, these experimental results are in qualitative agreement with our diblock vs triblock trends. We also note that this trend of triblocks needing higher εBBtr than their diblock counterpart is also seen for 50% A and 50% B composition; in Figure S3 we show results for A6-b-B12-b-A6 chains that exhibit a higher εBBtr than that of their diblock counterpart (A12-b-B12). Like the diblock systems, εBBtr, defined as the first occurrence of extrema in P(Nclus), occurs at the cusp of ρclus vs εBB for this BCP sequence as well. The increase in ⟨Nclus⟩ going from diblock to triblock is caused by the change from spherical micelles in A6-b-B18 systems to cylindrical micelles in A3-b-B18-b-A3. The P(Nclus) values for A3-b-B18-b-A3 chains (Figure 4d−h) and A6-b-B18 (Figure 3d−h) have quantitative differences; we note the log scale of the x-axis in Figure 4d−h compared to the linear scale in Figure 3d−h. In Figure 4, the population of small clusters seen at low εBB give way to a well-defined P(Nclus) average around Nclus ∼ 35 at moderate εBB (∼0.80). The size distribution continues changing to include larger clusters at the highest εBB (1.20). The renderings presented alongside P(Nclus) in Figure 4 show that clusters tend to be mainly spherical up to the εBB producing an ⟨Nclus⟩ ∼ 35 mode but are cylindrical at larger εBB. The above transition from spherical to cylindrical micelles at larger εBB has been shown experimentally88,89 upon changing the solvent quality, which alters the surface tension between the solvophobic block and the solvent. An increase in surface tension tends to flatten the micelle surface, causing spherical micelles to initially grow in cluster size (⟨Nclus⟩ and micelle diameter) and subsequently transition to cylindrical (and eventually lamellar) morphologies. In our comparison between A12-b-B12 BCP micelles and A6-b-B18 BCP micelles, we mention that the longer B block and shorter A block lengths are responsible for the larger ⟨Nclus⟩ in the A6-b-B18 system. We can apply a similar argument to the morphology observed in A3-bB18-b-A3 BCPs. Because triblock copolymers (Am-b-Bn-b-Am) tend to behave similar to diblock copolymers of half the solvophobic block length (Am-b-B0.5n),30,90 the A3-b-B18-b-A3 chains are effectively two A3-b-B9 BCPs linked together. A3-b-B9 BCPs, compared with A6-b-B18, have a shorter A block, which reduces the core−corona interface area per chain, increasing the packing factor and leading to flatter interfaces, and a shorter B block, which reduces the solvophobic block volume per chain, reducing the packing factor forming more curved interfaces. The net effect we observe is a flatter interface, meaning the flattening due to the shorter A block is more significant than the curving due to the shorter B block in this particular case. In an MD simulation study of model amphiphiles of variable solvophilic and solvophobic blocks, a tendency to flatten interfaces was observed with decreasing overall chain length with fixed composition,91 consistent with our results. 3.2. Assembly of BCPs and NPs as a Function of NP− BCP Affinity. In this section, we present the concurrent selfassembly of BCPs with NPs as a function of εBB and NP affinity. Considering the large parameter space (e.g., BCP sequence and composition, NP affinity to A, εAC, and B, εBC, blocks, NP loading, ϕC, and εBB), we choose to highlight here only the most unique NP−BCP assembly results. The complete set of H

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Figure 5. Results for A6-b-B18 chains and nanoparticles of size dC = 1.0σ as a function of εBB for a nanoparticle loading of ϕC = 0.10. (a, f, and k) Average cluster size. (b, g, and l) Fraction of all nanoparticles in the system that belong to micelles. (c−e, h−j, and m−o) Average number per micelle of nanoparticles located in the corona (c, h, and m), core (d, i, and n), and core−corona interface (e, j, and o).

have attractive “anchor points” that reduce the corona size, decreasing the effective core−corona interface area per chain (aAB). The simultaneous increase in VB and decrease in aAB causes the interface to flatten but apparently not large enough to cause micelle shape change with only spherical micelles observed for the εBB we explore in Figure 5. Figure 5 (second row) shows the fraction of NPs taken into the micelles, f tot. We see that f tot follows a nonmonotonic trend with increasing εBB with the f tot maximum occurring close to the εBBtr. We attribute the increase in f tot at low εBB followed by a decrease in f tot at high εBB to the competition between forming NP−B and B−B bead contacts. As εBB becomes larger than εBC, the NP−B contacts are replaced by B−B contacts. We also see f tot increase with increasing εAC and εBC because the NPs are more attracted to A and B blocks in the BCPs, and NP−A and NP−B are able to replace some of the B−B contacts. For better insight on why increasing εBC has a larger effect on the f tot than that of increasing εAC, we calculate the quantity of NPs in core, core−corona interface, and corona (Figure 5c−e, h−j, and m−o). The number of NPs in the core tends to be greater than that of corona and core−corona interface as the

results for the whole design space is shown in the SI section S4 (Figures S6−S21). 3.2.1. Nanoparticle Size dC = 1σ. Figure 5 shows our results for concurrent assembly of dC = 1.0σ NPs and A6-b-B18 chains as a function of εBB and different degrees of NP affinity to either BCP block, i.e., εAC and εBC, at NP loading ϕC = 0.10. We identify εBBtr as the cusp in ρclus vs εBB plot, which coincides with the εBB marking the first occurrence of extrema in P(Nclus)79 as shown in section 3.1. We see essentially the same εBBtr at ∼0.65 regardless of NP affinity. The εBBtr values for all cases are provided in the SI section S5 (Figures S22−S24). The value of εBBtr in the no NP case is ∼0.625, suggesting that the presence of NPs with the affinities in the range of 0.25−1.0 and dC = 1.0σ do not significantly change the thermodynamic driving force for aggregation. Inclusion of NPs increases ⟨Nclus⟩ especially at high εBB; with increasing A−C and B−C attraction, we see ⟨Nclus⟩ > 70. As εBC increases, NPs are attracted to the B blocks (i.e., micelle core) and effectively increase the chain length of solvophobic blocks, leading to a larger core volume per chain (VB). As εAC increases, NPs are attracted to the A block (i.e., micelle corona). The otherwise purely repulsive (or well-solvated) A blocks now I

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Figure 6. Results for A3-b-B18-b-A3 chains and NPs of size dC = 1.0σ as a function of εBB for an NP loading of ϕC = 0.10. (a, f, and k) Average cluster size. (b, g, and l) Fraction of all nanoparticles in the system that belong to micelles. (c−e, h−j, and m−o) Average number per micelle of nanoparticles located in the corona (c, h, and m), core (d, i, and n), and core−corona interface (e, j, and o).

follows similar trends as A6-b-B18 chains, namely an increase in ⟨Nclus⟩ with increasing εAC and εBC. The main difference we observe between Figures 5 and 6 is the size of clusters, ⟨Nclus⟩, where A6-b-B18 NP−BCPs have a maximum of ⟨Nclus⟩ ∼ 90 (see Figure 5a, f, and k) and A3-b-B18-b-A3 NP−BCPs form larger micelles (⟨Nclus⟩ ranges from ∼90 to ∼250 as seen in Figure 6a, f, and k). These larger micelles can accommodate more NPs per micelle. The cluster formation of A3-b-B18-b-A3 and A6-b-B18 BCPs in the absence of NPs showed A6-b-B18 forming smaller micelles with A6-b-B18 having ⟨Nclus⟩ ∼ 75 (Figure 3b) and A3-b-B18-b-A3 having ⟨Nclus⟩ ∼ 90 (Figure 4b). In the presence of NPs, the ⟨Nclus⟩ ∼ 250 cluster structure is a single large cylinder formed instead of a distribution of spherical and cylindrical micelles seen for the no-NP case in section 3.1.3. The shift from multiple short cylinders to a single long cylinder can be attributed to an increase in the packing factor p with NP addition. f tot and NP location for A3-b-B18-bA3 chains show the same trends as for A6-b-B18 chains with changing εAC, εBC, and εBB. The εBBtr ∼ 0.7 does not change significantly with NP addition from the no-NP case or with εAC and εBC, suggesting NPs do not influence micellization

denser cores provide more points of energetically favorable contacts to NPs than those of the corona or core−corona interface. Thus, the effect of increasing εBC on f tot is larger than that of increasing εAC. We note additional key trends: (1) A higher NP affinity to a given BCP block results in a higher number of NPs located in the domain made of that block, which is expected, as noted by Luo and co-workers.34 (2) As affinity to both blocks increases, more number of NPs go to the interface. This is because an overall increase in NP content in the core and/or corona leads to an increase in the number of NPs at the outer edge of the core or the inner edge of the corona. (3) At high εBB, where ⟨Nclus⟩ reaches a plateau in Figure 5a, f, and k, the number of NPs in the core decreases while the number in the corona and core−corona interface remain constant with increasing εBB. This is because, at such high εBB, the B block beads prefer to make more B−B contacts than NP−B contacts in the core. To the best of our knowledge, trends 2 and 3 have not been reported in the past in either simulation/theory or experimental work of NP−BCPs. Figure 6 shows our results for A3-b-B18-b-A3 chains and NPs analogous to results for A6-b-B18 in Figure 5 for the same size and composition of NPs. ⟨Nclus⟩ for A3-b-B18-b-A3 chains J

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Figure 7. Micelle structure change with NP addition for small, dC = 1.0σ, NPs for select block copolymers ((a−c) A6-b-B18 BCP, (d−f) A3-b-B18-b-A3 BCP) and conditions ((a, d) no NP systems, (b, e) ϕC = 0.10, (c, f) ϕC = 0.25).

alongside images of BCPs with NPs at ϕC = 0.1 (Figure 7b) and ϕC = 0.25 (Figure 7c) for high NP affinity to BCPs (εAC = εBC = 1.0). The micelles observed without NPs and at ϕC = 0.1 NPs are spherical with a higher ⟨Nclus⟩ in the latter case. At ϕC = 0.25, instead of spherical micelles we see a solution of cylindrical micelles in agreement with the packing factor increase argument. For A3-b-B18-b-A3 chains without NPs at high εBB, we see cylindrical and spherical micelles (Figure 7d). At ϕC = 0.1, the cylindrical micelles have a larger aspect ratio than those of the no NP case. At ϕC = 0.25, we see a branched cylindrical structure. Both observed structural transitions follow previously published trends37,60 for amphiphilic BCPs and NPs as well as our expectation from the packing factor increase argument. Although lamellae are not observed in the parameter range studied here for NPs of size dC = 1.0σ, we presume the structure will be seen for higher ϕC and/or higher NP affinity. Interestingly, with the inclusion of dC = 1.0σ NPs, despite changes in micelle structure, the conformation of individual polymer chains inside micelles remains unaffected. In Figures S25 and S26, we show entire chain and individual (A or B) block radii of gyration, Rg, and relative shape anisotropy, κ2, for polymers belonging to micelles. We do not see any appreciable change in either quantity with any amount of ϕC, εAC, and εBC regardless of the micelle size or structure. Micelle core swelling due to inclusion of NPs does not apparently include a change in chain conformation for the range of conditions studied here. 3.2.2. Nanoparticle Size dC = 2.15σ. Panels a−e in Figure 8 compare results for NP size dC = 2.15σ and dC = 1σ at NP affinity εAC = 0.50 and εBC = 1.00. We choose to show these results here to highlight sample trends observed for concurrent NP−BCP assembly with the relatively larger NPs. The first difference between smaller and larger NPs is the εBBtr, shown in Figure 8a, where micelles form at lower εBB for dC = 2.15σ. The trend of lower εBBtr for larger NPs can also be seen in Figure

thermodynamics significantly for A3-b-B18-b-A3 BCPs for dC = 1.0σ, ϕC = 0.10, and NP affinities in the range of 0.25−1.00. So far, we have discussed results for NP volume fraction of ϕC = 0.10. Results at higher volume fraction, ϕC = 0.25, specifically the f tot and average number of NPs in core, corona, and core−corona interface for A6-b-B18 and A3-b-B18-b-A3 chains, are described in SI section S4. We see the same qualitative behavior of f tot and NP location with increasing εAC, εBC, and εBB as seen for ϕC = 0.10 with a noticeable increase in the value of f tot for ϕC = 0.25 as compared to ϕC = 0.10 for both sequences. The quantitative increase in f tot with increasing ϕC is a consequence of the equilibrium between free and associated NPs (i.e., NPs within the micelles), where a higher overall concentration of NPs leads to an increase in NPs in micelles. For other BCP sequences and compositions, in the range of conditions studied in the present work, we see similar trends with addition of NPs of size dC = 1.0σ at ϕC = 0.10 and ϕC = 0.25, as seen for both A6-b-B18 and A3-b-B18-b-A3 BCP chains in terms of f tot and NP location in micelles. These results are summarized in the figures in section S4 in the SI. εBBtr does not change significantly for the BCP sequences for NP of size dC = 1.0σ at higher ϕC, as shown in section S5 of the SI. In Figure 7, representative micelle structures for A6-b-B18 and A3-b-B18-b-A3 chains with increasing ϕC at high εBB are presented. In the preceding paragraphs, we have made the case for NPs changing the way BCPs assemble by changing the packing factor, p, by swelling the solvophobic block as NPs are attracted to the B block and reducing the core−corona interface area as NPs are attracted to the A block. The expectation is that core−corona interfaces should become progressively flatter as more NPs are added, following the sequence of sphere to cylinder to branched cylinder to vesicle sometimes called the “canonical” structure set.88 For A6-b-B18 chains, we show the high εBB (1.3) structure seen without NPs (Figure 7a) K

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Figure 8. Results for A6-b-B18 chains and NPs of size dC = 2.15σ as a function of εBB for an NP loading of ϕC = 0.10, εAC = 0.5, and εBC = 1.0. (a) Average cluster size. (b) Fraction of all NPs in the system that belong to micelles. (c−e) Average number per micelle of NPs located in the corona, core, and core−corona interface, respectively. For (a−e), data in pink triangles correspond to dC = 2.15σ and data in blue diamonds corresponds to the same simulation conditions for dC = 1.00σ presented for comparison. (f−i) For εBB = 0.7, 0.94, 1.03, and 1.10, respectively, density profiles as a function of radial distance from the center of spherical micelles for solvophilic A blocks (red lines), solvophobic B blocks (green lines), and NPs (C, yellow lines).

S24. In Figure S24, we also see that εBBtr decreases for increasing εBC for the range of BCP sequences studied here. The noticeable decrease in εBBtr with increasing εBC is attributed to larger NPs providing increased energetic driving force to favor micellization at lower εBB. Not surprisingly, increasing εAC does not appear to change εBBtr because the main driving force for BCP aggregation is the B block being in poor solvent conditions. The larger effect seen in εBBtr for larger NPs than the smaller NPs is attributed to the overall larger f tot, as discussed next, as well as the fact that each of the larger NPs can interact with more BCP beads than smaller NPs can. Another difference in micellization between smaller and larger NPs is f tot as a function of εBB, shown in Figure 8b, where f tot after micelles are formed is constant and close to unity; in other words, all NPs in the simulation box take part in the micelles, whereas for dC = 1.0σ, f tot decreases after micelles are formed. There are also differences in the location of NPs; whereas the number of larger NPs in the core decreases like we see in smaller NPs, the larger NPs do not get expelled from the micelles as the smaller NPs do, but rather increasingly

accumulate at the core−corona interface, as shown in Figure 8e. On the one hand, the larger NPs are capable of making more contacts with BCP beads than that the smaller NPs, thus, having a higher enthalpic gain to their free energy by remaining in micelles. On the other hand, after being expelled from the micelle core, larger NPs still have a significant enthalpic gain being at the core−corona interface where a larger number of contacts with both solvophobic B and solvophilic A beads is possible. Zhang and co-workers60 have noted this trend of larger NPs locating at core−corona interfaces more readily than smaller NPs in a self-consistent field theory/density functional theory study of NP−BCP. A visual aid to the location of NPs as a function of increasing εBB is shown in Figures 8f−i, where we show the concentration of different species (solvophilic A, solvophobic B, or NPs C) as a function of radial distance from the center of a micelle. The concentration of A and B beads mark the extent of the corona and core of the micelle, respectively, and the concentration of C beads marks the location of NPs within the micelle. There is a slight decrease in core extent, i.e., the position where the L

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Figure 9. Micelle structure change with NP addition for large, dC = 2.15σ, NPs for select block copolymers and conditions.

density of B beads disappears, as we go from the lowest εBB shown, i.e., εBB = 0.7 in Figure 8f, to higher εBB, shown in Figure 8g−i. The corona, on the other hand, shows little change as εBB increases. At εBB = 0.7, close to the maximum NP content in the core, we find that NPs are uniformly distributed along the micelle core. With increasing εBB, NPs are gradually being concentrated at the core−corona interface region and excluded from the center of the micelle core. NP content in the corona for dC = 2.15σ, shown in Figure 8c and shown schematically in Figure 8f−i, does not change significantly with increasing εBB. A similar trend is observed for dC = 1.0σ. We attribute the lack of change in NP content in the corona for increasing εBB to the fact that NPs interact with solvophilic A beads only through their affinity, εAC, which does not compete with contacts due to εBB in our simulations, and the corona itself is not changing shape. We expect to see the formation of larger micelles, and possibly micelle shape changes, following the canonical structure set of spheres to cylinders to branched cylinders to vesicles upon addition of larger NPs based on the packing factor increase with NP addition argument presented in the previous sections. We confirm this expectation in the renderings of BCP and NP of size dC = 2.15σ in Figure 9, where we show the key structural changes observed in the

parameter space explored. The simple symmetric diblock BCP A12-b-B12, which only show spherical structures in the absence of NPs (Figure 9a) and for all NP affinities and ϕC for dC = 1.0σ (not shown), exhibits growth in size for ϕC = 0.1 at εAC = εBC = 1.0 for dC = 2.15σ (Figure 9b) while maintaining a spherical shape. At ϕC = 0.25, we see the formation of fused spherical micelles at εAC = εBC = 1.0 (Figure 9c). For A6-b-B18 chains, we expect to see the formation of larger spherical and cylindrical micelles upon addition of larger NPs based on the packing factor increase argument; we confirm this expectation in Figure 9e and f for ϕC = 0.1 and ϕC = 0.25, respectively. The cylindrical micelles formed for dC = 1.0σ, shown in Figure 7c, are shorter than the cylinder formed for dC = 2.15σ, shown in Figure 9f. Finally, for A3-b-B18-b-A3 chains, we compare the short cylinders observed in the absence of NPs in Figure 9g with structures formed at ϕC = 0.1 and 0.25 in Figure 9h and i, respectively. We see the formation of large cylindrical micelles, in Figure 9h, and the formation of a vesicle shown in Figure 9i. Structural changes with larger NPs are more prominent than those with smaller NPs shown in Figure 7. We attribute the more drastic shape changes to a higher number of NPs in micelles (higher f tot) seen in Figure 8b (and in additional results for multiple BCP sequences and large NPs in section S4 of the SI) that leads to both higher swelling of the M

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followed in certain cases by a change in micelle structure, consistent with the flattening of the core/corona interface with NP addition leading to a “canonical” progression of micelle shapes, i.e., from spheres to cylinders to lamellae, consistent with a qualitative argument of increasing the packing factor due to NP swelling micelle cores and reducing core−corona interface per chain by making coronas more compact. Although previous experimental and simulation work has reported similar structural transitions (namely sphere to vesicle37,60), no single study was attempted to summarize or generalize those findings upon changing both BCP sequence and composition, as well as NP size, loading, and affinity to BCP blocks. Finally, NP addition and increasing NP affinity to polymers resulted in an effective increase in solvophobicity of BCPs, noticeable in NP− BCP with large NPs; this leads to a decrease in transition solvophobicity, providing another design consideration for tuning solvent conditions for achieving NP−BCP assemblies. In summary, this paper presents a materials library that links molecular design of BCPs and NPs to assembled NP−BCP micelle structure, specific location, and quantity of NPs in the micelles and the effective solvophobicity that causes assembly in solution. Such a study serves as an example of the power of molecular simulations within the materials genome initiative, and how it serves to accelerate materials design for a wide range of applications. For instance, in the field of imaging contrast agents,4 it is desirable that micelles remain spherical to allow the flow of hybrid assemblies in the bloodstream. Knowledge of the limits of spherical micelle stability with inclusion of NPs with a given affinity to BCP block can prevent the undesirable formation of cylinders or lamellae. Control of the location of NPs can be also harnessed for practical applications. In pollutant capture,11 for example, NP−BCP hybrid assemblies are first formed, and subsequently either core or corona are cross-linked to keep micelle shape fixed; in either option, control of the location of NPs ensures that the cross-linking takes care of keeping NPs as part of the assembly. Although many past experimental studies have shown successful applications of NP−BCP assemblies,4,11 for specific chemistries, this more general study serves as a platform onto which many different chemistries can be mapped, guiding other computational and experimental studies in the future.

core and a decrease of the core−corona interface area per chain, consistent with our packing factor increase with NP addition argument. It is worth noting that individual BCP chain conformations remain unaffected by the presence and affinity of these larger NPs despite changes in micelle structure. The radii of gyration and relative shape anisotropies of chains and individual solvophilic or solvophobic blocks belonging to micelles do not change, as shown in Figures S25 and S26 as exemplary cases for A6-b-B18 chains and dC = 1.0σ NPs, but trends hold for other BCP sequences and NP size of dC = 2.15σ.

4. CONCLUSIONS In this paper, we have presented a coarse-grained molecular dynamics simulation study of concurrent self-assembly of amphiphilic block copolymers (BCPs) and nanoparticles (NPs) in solution as a function of increasing solvophobicity over a large design space of BCP sequence and composition, NP size, NP concentration, and NP affinity to the solvophilic and solvophobic blocks in the BCP. Our primary focus has been to elucidate the effect of adding NPs on the structure of micelles formed, NP uptake and location within the micelles, and, if any, changes to self-assembly thermodynamics (e.g., transition solvophobicity for assembly). We have considered two BCP sequences, diblock and triblock, with BCP composition varying between symmetric and asymmetric (solvophobic-rich and solvophilic-rich). The NP design space was comprised of sizes the same as the Kuhn segment of the polymer as well as sizes with 10 times the volume of the Kuhn segment. Additionally, we have varied the NP affinity to either/both blocks to be weak or commensurate or much stronger than the BCP transition solvophobicity. First, in the absence of NPs, we quantified the structure (cluster density, number of BCPs in micelles, shape of micelles) and thermodynamics (transition solvophobicity for assembly). These results were presented to provide both the validation for the methodology through comparison to past studies as well as to establish the baseline behavior in the absence of NPs. With increasing NP affinity to polymers and NP concentration, we have, not surprisingly, observed an increase in NP uptake for all BCP sequences and compositions considered in this work. Additionally, an increase in NP affinity to either solvophilic or solvophobic blocks leads to an increase in the number of NPs being part of the micelle corona or core, respectively. However, a less obvious observation was the decrease in NP content in micelle cores with increasing solvophobicity for all NP affinities. The fate of NPs vacated from micelle cores changes depending on the NP size. In systems with NPs of size comparable with the polymer Kuhn segment length, the NPs vacated from micelle cores are expelled from micelles altogether, keeping the number of NPs at the corona or the core−corona interface constant. In contrast, in systems with larger NPs (with volume 10-times the volume of the Kuhn segment), the NPs vacated from micelle cores build up at the core−corona interface, keeping the number of NPs at the corona and the total number of NPs in the micelle constant. Although some of these trends have been reported in both experimental35 and theoretical92 works, the effect of increasing solvophobicity with competing NP affinity to BCPs, to the best of our knowledge, has not been studied for such a large design space of BCPs and NPs. Our results also showed an increase in cluster size (the number of BCP chains in a micelle) with increasing NP uptake for all systems considered. The increase in cluster size is

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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.jced.7b00925. Equivalence between implicit and explicit solvent models, details on cluster determination, additional results for BCPs with and without NPs, transition solvophobicities for different NP−BCP systems, BCP chain conformation within micelles, summary of observed NP−BCP structures, equivalence of the generic model used in this work with a specific chemistry as an example, and videos for A12-b-B12, A6-b-B18, and A3-b-B18-b-A3 BCPs without NPs (V1, V2, V3, respectively) and with dC = 2.15σ at ϕC = 0.25 NPs at εAC = εBC = 1.0 (V4, V5, V6, for A 12 -b-B12 , A 6-b-B 18 , and A 3-b-B 18 -b-A 3 BCPs, respectively) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. N

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Arthi Jayaraman: 0000-0002-5295-4581 Funding

We acknowledge financial support from the National Science Foundation under grant number NSF DMREF-1629156. This work also used the Extreme Science and Engineering Discovery Environment (XSEDE) Stampede cluster at the University of Texas through allocation MCB100140, which is supported by National Science Foundation grant number ACI-1548562. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

We thank our collaborators K. Wooley, D. Pochan, W. Johnson and their research groups for valuable feedback on this work. This research was supported in part through the use of Information Technologies (IT) resources at the University of Delaware, specifically the high-performance computing resources of the Farber supercomputing cluster.

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