Assembly of diblock copolymer functionalized spherical nanoparticles as a function ofcopolymer compositionCarla E. Estridge and Arthi Jayaraman Citation: The Journal of Chemical Physics 140, 144905 (2014); doi: 10.1063/1.4870592 View online: http://dx.doi.org/10.1063/1.4870592 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ordered structures of diblock nanorods induced by diblock copolymers J. Chem. Phys. 139, 104901 (2013); 10.1063/1.4819775 Preparation, properties and self-assembly behavior of PTFE based core-shell nanospheres AIP Conf. Proc. 1459, 61 (2012); 10.1063/1.4738398 Dissipative particle dynamics simulations of polymer-protected nanoparticle self-assembly J. Chem. Phys. 135, 184903 (2011); 10.1063/1.3653379 Coarse-grained molecular dynamics simulation on the placement of nanoparticles within symmetric diblockcopolymers under shear flow J. Chem. Phys. 128, 164909 (2008); 10.1063/1.2911690 A theoretical study for nanoparticle partitioning in the lamellae of diblock copolymers J. Chem. Phys. 128, 074901 (2008); 10.1063/1.2827470

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THE JOURNAL OF CHEMICAL PHYSICS 140, 144905 (2014)

Assembly of diblock copolymer functionalized spherical nanoparticlesas a function of copolymer composition

Carla E. Estridge1 and Arthi Jayaraman2,a)

1Department of Chemistry and Biochemistry, University of Colorado, 215 UCB, Boulder,Colorado 80309, USA2Department of Chemical and Biological Engineering, University of Colorado, 596 UCB Boulder,Colorado 80309, USA

(Received 9 January 2014; accepted 25 March 2014; published online 14 April 2014)

In this work, we use coarse-grained molecular dynamics simulations to study spherical nanoparti-cles functionalized with AB diblock copolymer chains at low grafting density, to obtain a designlibrary linking copolymer composition, monomer-monomer interaction strengths, graft lengths, par-ticle sizes, and monomer solvent-philicity to the two stages of nanoparticle assembly: the initial for-mation of patches within the copolymer-grafted particles from attractive monomers aggregating, andthen the (equilibrium) assembled cluster formation. With regards to patch formation, as the fractionof the solvent-phobic (A or B) block decreases along the graft and the interaction strength decreases,the propensity to form patches on the particles decreases. As the fraction of the solvent-phobic blockincreases, the time for patch formation decreases, in particular if the inner A block is solvent-phobic.As the ratio of graft length to particle size increases the propensity to form fewer patches increasesdue to inter-graft monomer aggregation. For all compositions, the assembled clusters formed in B-selective solvents (solvent-phobic A block) have a higher fraction of particles at smaller inter-particledistances than in A-selective solvents (solvent-phobic B block). In an A-selective solvent at low in-teraction strengths, as the graft length to particle diameter ratio is increased, the tendency to formisotropic clusters at equilibrium is increased, and intermediate anisotropic chain-like assembly isobserved. When these anisotropic intermediate states are observed, they accelerate the formationof equilibrium isotropic clusters. With increasing strength of interaction between solvent-phobic Bmonomers, the intermediate states disappear from the assembly pathway. At low and intermediateinteraction strengths, as fraction of A block increases, the clusters’ outer shell of solvent-philic Amonomers which surrounds the patch of solvent-phobic B monomers becomes dense, hindering ad-dition of more particles to the cluster leading to smaller overall clusters. In a B-selective solvent,predominantly anisotropic clusters form, and show an increase in shape anisotropy with increasingmonomer interaction strength. In both cases of monomer solvent-philicity, with an increase in thegraft length to particle diameter ratio we see a decrease in anisotropic cluster formation. And, inboth cases of monomer solvent philicity, with increasing monomer interaction strength the averagecluster size and tendency to form anisotropic clusters is increased. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4870592]

I. INTRODUCTION

Assembly of nanoparticles is extremely important in thedesign of novel materials with targeted properties in a vari-ety of applications, such as microelectronics, photovoltaics,metamaterials, and medicine. Surface functionalization ofnanoparticles with ligands including macromolecules suchas polymers,1–23 proteins (see, for example, Refs. 24–26),DNA,27–36 and small molecules37–39 has been demonstratedto be an excellent route to control and tune the assemblyof nanoparticles. The ligands that are grafted/attached to theparticle surface mediate the interactions of the nanoparti-cles with both the solvent and other grafted nanoparticlesin the system. Past theoretical and experimental work onhomopolymer-grafted nanoparticles have found the chemistry

a)Author to whom correspondence should be addressed. Electronic mail:arthi.jayaraman@colorado.edu

of the grafted polymers, solvent and nanoparticles, the poly-mer grafting density, molecular weight and polydispersity ofthe homopolymer, and the particle size play a critical rolein the assembly of the nanoparticles.8, 40–45 By introducingchemical heterogeneity into the grafted polymers, either byusing chemically distinct homopolymers or copolymers, it ispossible to precisely tune strength and directionality into theeffective interactions between the polymer grafted particlesin the medium.19, 46–48, 59 In the case of chemically distinct ho-mopolymer grafts, simulations have shown that the introduc-tion of anisotropy to grafted spherical nanoparticle systemsby the use of two chemically distinct homopolymer graftsleads to assembled structures that take on seven distinct or-dered phases.49 In experiments, it has been shown that goldnanoparticles with a mixture of hydrophobic and hydrophilichomopolymer grafts will migrate to the interfaces in a mi-crophase separated diblock copolymer matrix to decrease theinterfacial tension between the copolymer domains, and to

0021-9606/2014/140(14)/144905/13/$30.00 © 2014 AIP Publishing LLC140, 144905-1

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144905-2 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

allow homopolymer grafts to take on conformations wherethey can interact with the domain for which they have favor-able enthalpic interactions.4, 6, 15, 50–55

In the case of copolymer grafts, using self-consistent fieldtheory (SCFT) Vorselaars et al.56 have studied a single spher-ical nanoparticle functionalized with a dense layer of diblockcopolymer grafts in order to understand the domains formedwithin the grafted layer on the particle due to aggregation ofthe two monomers types within the grafts. Using Monte Carlo(MC) simulations, Seifpour and Jayaraman48 have studied theeffect of monomer sequence and chemistry on the copolymergraft conformations on spherical nanoparticles at low and in-termediate grafting densities. They have found that monomersequence, nanoparticle size, and grafting density dictate graftconformations, inter-chain and intra-chain monomer aggrega-tion, and the grafted particles’ radius of gyration varies non-monotonically with increasing graft blockiness. Zhu et al.57

have used both SCFT and density functional theory (DFT)to study assembly in systems of spherical nanoparticles eachfunctionalized with a single diblock copolymer graft. Theyvaried the chemical interaction of the particle surface withthe graft and the composition of the diblock copolymer graftand observed that when the particle has a chemically neutralinteraction with the graft the grafted particles took on typi-cal diblock copolymer morphologies, and when the particlehad a repulsive interaction with the graft they observed struc-tures that are not typical of diblock copolymer melts. Nei andco-workers58 have used both experimental methods and dissi-pative particle dynamic (DPD) simulations to study the as-sembly of block copolymer functionalized nanoparticles inselective solvents. They observed that with increasing ratioof particle size to molecular weight of the solvent-phobic in-ner block, the assemblies increased in size from single par-ticle micelles, to small clusters, and finally to large vesi-cles. Martin, Seifpour, and Jayaraman18, 47 have used MonteCarlo (MC) simulations to study the effect of AB copoly-mer graft sequence on assembly of nanoparticles with mul-tiple copolymer-grafted chains in the presence of varying likemonomer (A-A, or B-B) or unlike monomer (A-B) interac-tions. The introduction of A-B repulsion showed a strongereffect on the assemblies of nanoparticles with grafts of alter-nating sequences in comparison to nanoparticles with diblockcopolymer grafts. Additionally, diblock copolymer graftednanoparticles tended to assemble into anisotropic assembliesdespite isotropic grafting of the copolymers in the presence ofA-B repulsions. Following this work, Martin, Mckinney, andJayaraman18 used MC simulations to study the effect of graftblockiness on copolymer grafted nanoparticle assembly. Theyfound that with increasing blockiness of the grafted polymerchain, in the presence of negligible A-B repulsion, the averagecoordination number increased for A-A or B-B attractions.However, the coordination number is constant for A-A andB-B attractions. These studies have established that copoly-mer graft sequence and the interaction of chemically distinctportions of the graft with one another and the nanoparticlesize (or curvature) affect the overall assembled structures ofthe copolymer-grafted nanoparticles.

In this paper, we use coarse-grained molecular dynamics(MD) simulations to study diblock copolymer grafted spheri-

cal nanoparticles in implicit solvent in order to determine howdiblock copolymer graft composition (the relative sizes of thetwo blocks that compose the polymer) affects graft conforma-tions and patch formation at the beginning of the assemblyprocess, intermediate metastable assembled states, and equi-librium assembled states, while also presenting correspond-ing relative timescales of the various stages in the assem-bly process. We also vary the solvent selectivity, the strengthof monomer-monomer attractions, and graft length to par-ticle size ratio to quantify their impact along with copoly-mer composition on the monomer concentration profiles andpatch formation at early stages of the simulation, the shapeand size of the assembled cluster, the inter-particle distanceswithin the assembled cluster and timescales of assembly. Thecomputational results presented in this paper serve to provideguidelines to synthetic chemists on what functionalization tohave on the nanoparticles to produce targeted nanoparticleassemblies.

This paper is organized as follows. In Sec. II, we providedetails of our model, simulation techniques, and our analy-sis methods. Section III provides a detailed discussion of ourresults, presenting the effects of graft composition, solvent-philicity, monomer-monomer interaction strength, and graftlength on the various stages of assembly and structures ofthe assembled nanoparticles. In our final section, we concludewith a discussion of trends observed in this study and their ex-perimental relevance.

II. APPROACH

A. Model

We model each diblock copolymer grafted nanoparti-cle as a hard spherical particle of diameter D (in units ofmonomer diameter σ∼1 nm) with six grafted AB diblockcopolymers of length NA+B. We choose to use six grafts onparticles of sizes ranging from 4 to 12 σ to model an experi-mentally accessible low grafting density regime. Each copoly-mer is modeled as a freely jointed chain with monomers ofdiameter σ . The first monomer of each copolymer graft isgrafted 0.5 σ from the surface at six equally spaced pointson the nanoparticle. The nanoparticle and the first monomerof each graft are treated as a rigid body to ensure grafts do notmigrate on the surface of the particle. In all cases the blockgrafted directly to the surface of the particle, the inner blockof the diblock copolymer, is made up of A monomers and thenumber of monomers in the inner block is given by fA∗NA+B,where fA is the fraction of the grafted polymer made up of Amonomers.

Particle-particle, particle-monomer, and chemically dis-tinct monomer-monomer (A-B) interactions are modeled us-ing the Weeks-Chandler-Andersen59 (WCA) potential. Themonomer-monomer interactions between non-bonded, chem-ically identical monomers (A-A, B-B) are modeled usinga standard 6-12 Lennard-Jones (LJ) potential.60 Bondedmonomers along the copolymer graft are modeled using a har-monic potential,

Ubond (r) = k(r − Ro)2, (1)

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144905-3 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

where parameters k = 30ε/σ 2 and R0 = 1.0 σ are chosen sothat unphysical bond overlaps or crossings are eliminated.

B. Simulation protocol

We conduct Langevin dynamics simulation61 of a systemof diblock copolymer grafted particles. The trajectory of thesystem is computed using the Langevin equation

mi

d2r

dt2= −∇jUi − mi

γ

dri

dt+ fij , (2)

where t is the time, mi is the mass of ith monomer, γ is thefriction coefficient chosen in order to simulate diffusion ofnanoparticles in a small molecule solvent such as water, Ui isthe net potential acting on the ith monomer, fij is the j com-ponent of the random force acting on the ith monomer obey-ing fluctuation-dissipation theorem with a magnitude given by√

kBTmi/γ dt (kBT is the Boltzmann constant times the tem-

perature). We use the LAMMPS61 package in a NVT ensem-ble to solve the above equations of motion for our system witha time step �t = 0.001τ , where τ = σ (m/ε)1/2 is the standardLJ unit of time with ε in units of kBT. Ten separate simula-tion trials of each system are carried out to ensure good sam-pling of the assembled states. Each simulation is conductedwith periodic boundary conditions applied in all directionsand run in two stages: initialization and equilibration. Duringthe initialization, nanoparticles are placed on a grid withinour simulation box. Grafts are grown radially outward fromthe poles of each particle with a spacing of 1 σ with the initialmonomer being 0.5 σ from the surface particle. In this initial-ization stage the system is subjected to NVT dynamics at areduced temperature T∗ = 1.0 with a purely WCA potentialbetween all monomers in the system for 1 × 105 time stepswith initial velocities assigned from a Boltzmann distribution.This initialization allows us to avoid any bias that may comefrom the initial configuration of the system.

After the random initial configurations are generated, inthe equilibration stage we turn on attractive interactions be-tween like graft monomers by changing the appropriate po-tentials from WCA potential to an attractive LJ potential asstated in the model section (see Sec. II A). The system is equi-librated for 4 × 107 time steps, which we find to be sufficienttime for these systems to sample their equilibrium configura-tions. In Sec. I of the supplementary material,63 we discusshow we ensure that the cluster assembly data we present areindeed equilibrium assembly. We observe that the temperatureof the system is held within 5% of our desired dimensionlesstemperature, T∗ = 1.0. For each of the ten simulation trials,at early times in the simulations (< ∼10 000 time steps) dataare collected every 1 000 time steps allowing for analysis ofearly time monomer configurations. During the remainder ofthe simulation data is collected every 100 000 time steps.

C. Parameters

We simulate diblock copolymer grafted particles, witheach particle having six grafts, in an implicit solvent. We vary:composition fA, the fraction of the AB copolymer graft com-

fA

Solvent Selectivity

A B

NA+B D

Interaction Strength (KT)

0.75

0.50

0.25

1 5 10 2 3

6

InSt

10

D

N A+B

(a)

(b)

Initial Configuration

Patch Formation

Aggregation and Rearrangement

Final Cluster

FIG. 1. (a) Schematic representation of the design parameters in this paper:fA, the fraction of a graft made up of A monomers, solvent selectivity or themonomer type that is solvent-philic, interaction strength (in kBT), the well-depth in the potential between two non-bonded solvent-phobic monomers (1kBT is referred to as low, 5 kBT as intermediate and 10 kBT as high), andNA+B/D, the ratio of grafted chain length to the diameter of the particle. (b)Snapshots from simulation of cluster assembly to show the focus of this paperon characterization of assembly stages in dashed boxes, patch formation, andthe final cluster.

posed of A monomers, NA+B/D, the ratio of the graft lengthto particle diameter, interaction strength, the well depth ofthe attractive LJ interactions between like monomers, andsolvent selectivity, the monomer type that is solvent-philic.When an A monomer is solvent-philic the attractive LJ in-teractions between B monomers is ten times stronger thanthe interaction between A monomers. These parameters areshown in the parameter space in Figure 1(a). The fractionof the graft made up of A monomers is varied, 0.25, 0.50,or 0.75, with the A block always being grafted to the parti-cle surface and referred to as the inner block. The ratio ofNA+B/D is varied between 2.0, 3.0, and 6.0 monomers/σ ,where 2.0 corresponds to a graft length of 24 monomersand particle diameter of 12 σ , 3.0 to a graft length of 12monomers and particle diameter of 4 σ , and 6.0 a graft lengthof 24 monomers and particle diameter of 4 σ . The interactionstrength between like monomers is varied between low inter-action strength corresponding to εii = 1.0 for solvent-phobicand 0.1 for solvent-philic monomers, intermediate interactionstrength corresponding to εii = 5.0 for solvent-phobic and0.5 for solvent-philic monomers, and high interaction strengthcorresponding to εii = 10.0 for solvent-phobic monomersand 1.0 for solvent-philic monomers. Solvent-philicity is des-ignated as either A or B and, as stated above, is modeledthrough the magnitudes of the monomer-monomer attractionstrengths. We use the term “A selective solvent” to refer toa solvent in which A monomers are solvent-philic, and “Bselective solvent” for a solvent in which B monomers aresolvent-philic. We simulate a fixed concentration of diblockcopolymer grafted nanoparticles, 10 particles in a 100 × 100× 100 σ 3 simulation box. In specific cases, discussed inSec. II of the supplementary material,63 we test for sys-tem size effects (e.g., choice of 10 particles) on the as-sembled structures observed by comparing the results fromthese 10 particle systems to 100 particle systems, at sameconcentration.

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144905-4 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

D. Analysis

Our analyses focus on characterizing the chain conforma-tions and particle arrangements in two distinct stages of thenanoparticle assembly (see Figure 1(b)): (a) patch formationof solvent-phobic monomers and (b) the final cluster assem-blies. Patches are defined as a collection of solvent-phobicmonomers located within a cutoff distance of one another,corresponding to the first minimum in the monomer-monomerradial distribution. Clusters are defined as a collection of par-ticles located within the cutoff distance of one another, cor-responding to the first minimum of the particle-particle radialdistribution. For each stage we choose a variety of methods toquantify the structures observed. For initial patch formationwe calculate the concentration profile for A or B monomersas a function of the radial distance, r, from the particle surfaceto which they are grafted,

C(r) = lim�r→0

p(r)

4πr2�r, (3)

where p(r) is the average number of monomers found at aradial distance between r and r+�r. We also measure the en-semble average radius of gyration, 〈R2

g〉1/2defined as the av-

erage distance of the monomers from the center of mass ofthe grafted particle:

〈R2g〉1/2 =

⟨1

N

∑N

i=1(Ri − Rcom)2

⟩1/2

, (4)

where N is the number of monomers (6 grafts ∗NA+B

monomers/graft) and Ri is the position of the ith monomer.Both C(r) and 〈R2

g〉1/2vary with the composition of grafts,

the interaction strength, NA+B/D and the choice of monomersolvent-philicity, and give us a description of the monomerarrangement around the particle surface and the size of thecopolymer grafted particle as a function of these parameters.

We characterize the particle assemblies by measuring theinter-particle distances of particles found within the samecluster, and the cluster sizes. If particles are found withinthe set cutoff determined by the particle center-center radialdistribution function we check for attractive monomer con-tacts between grafts on those (distinct) particles. If we ob-serve particle assembly, we measure the number of particlesdetermined to be in a cluster, and the center-to-center dis-tance of particles within the cluster. For these analyses weplot histograms of all center-to-center distances and clustersizes measured in all 10 trials to show the range of distancesand cluster sizes observed for a particular set of parameters.

Throughout the simulations, we also monitor the amountof time it takes to form patchy particles and final clusterassemblies as a function of the parameters varied. In someparameter sets intermediate structures are observed. Theseintermediate structures are distinctly different in spatial ar-rangement from the final cluster assemblies (see Figure 1(b)).Intermediate structures vary depending on the simulation pa-rameters and are identified by monitoring renderings of par-ticle trajectories throughout the course of the simulation. Todetermine the assembly time, we measure the number of timesteps it takes for particles to go from their initial conforma-tion to patchy particles, from patchy particles to intermedi-

ate (metastable) assembled states (in cases where these inter-mediates exist) and the intermediate (metastable) assembledstates to the final equilibrium assembled structures. For ex-ample, if we are interested in measuring the amount of timeneeded to form patches of attractive monomers we monitorthe number of time steps necessary for one or two distinctpatches (depending on the system parameters) of attractivemonomers to form using our clustering algorithm with a cut-off chosen to correspond to monomer clustering. Patch for-mation times are averaged over all particles in the system forten separate simulation trials. Times for intermediate structureformation and final cluster assemblies are also averaged overten simulation trials of the grafted particles.

III. RESULTS

A. Effect of graft composition on patch formation

First, we consider the effect of graft compositionon patch formation within the copolymers of the graftednanoparticles at the early stages of the simulation beforeparticles assemble into clusters.

Figure 2(a) shows snapshots of individual particle con-formations as a function of solvent selectivity and interac-tion strength at all graft compositions for particles of size D= 4 σ and six grafts of length of NA+B = 12. Low inter-action strength: In a B selective solvent at low interactionstrength patch formation is absent for fA = 0.25 and 0.50 be-cause the solvent-phobic inner A block of the graft has fewA monomers (3 and 6, respectively) and is grafted to the par-ticle, hindering formation of inter-graft monomer aggregatesor patches. When fA is increased to 0.75, the driving force foraggregation is increased as the inner attractive block is larger,and the grafts form two distinct patches near the surface ofthe particle by bringing solvent-phobic A monomers on mul-tiple grafts together into the two patches. In an A selectivesolvent at low interaction strength increasing the fA fractiondecreases patch formation of solvent-phobic B monomers.For fA = 0.25 and 0.5, the outer B block of the copolymerhas sufficient B monomers allowing the enthalpic gain frombringing attractive B monomers together into patches to be-come large enough to overcome the chain conformational en-tropy loss upon patch formation. When fA = 0.75, the num-ber of B monomers in each graft is reduced to 3, and theenthalpic gain upon patch formation is likely smaller thanthe entropic loss, and patches of B monomers do not form.If we compare A and B selective solvents, when fA = 0.50,there is an asymmetric effect on patch formation based onthe location of the solvent-phobic block (inner or outer). AtfA = 0.50 in an A selective solvent (when outer B blockis solvent-phobic) patches form, while in a B selective sol-vent (when inner A block is solvent-phobic) patches do notform. Intermediate interaction strength: All combinations offA and solvent selectivity exhibit formation of (mostly two)patches of attractive monomers. By increasing the interac-tion strength we have increased the enthalpic gain of patchformation and in doing so increased the propensity for patchformation. The two cases at low interaction strength that didnot show patch formation, fA = 0.25 B selective solvent andfA = 0.75 A selective solvent, now show patch formation.

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0.25 0.5 0.752

3

4

5

6

7AA = 1.0

BB = 0.1

AA = 0.1

BB = 1.0

AA = 5.0

BB = 0.5

AA = 0.5

BB = 5.0

AA = 10

BB = 1.0

AA = 1.0

BB = 10

fA = 0.25 fA = 0.50 fA = 0.75

1 2 3 4 50

1

2

3

4

5

6

1 2 3 4 5 1 2 3 4 5

1 2 3 4 50

1

2

3

4

5

6

1 2 3 4 5 1 2 3 4 5

1 2 3 4 50

1

2

3

4

5

6

1 2 3 4 5 1 2 3 4 5

0.25 0.5 0.752

3

4

5

6

7

0.25 0.5 0.752

3

4

5

6

7

<C(r

)>

n + Rm) n + Rm) n + Rm)

fA

fA

n + Rm) n + Rm) n + Rm)

fA

<C(r

)>

n + Rm) n + Rm) n + Rm)

<C(r

)>

<Rg

2 >1/

2<R

g2 >

1/2

<Rg

2 >1/

2

fA = 0.25 fA = 0.50 fA = 0.75(a)

(b) (c) (d)

(e) (f) (g)

(h) (i) (j)

(k)

(l)

(m)

FIG. 2. Characterization of patchy particles with NA+B/D = 3.0. (a) Simulation snapshots from a single trial showing representative patch formation of a singlediblock copolymer grafted particle in a (100 σ )3 box containing 10 particles. (b)–(j) Monomer concentration profiles for particles of low (b)–(d), intermediate(e)–(g), and high (h)–(j) interaction strength are shown. Concentration profiles are shown as average grafted monomer concentrations 〈C(r)〉 as a function ofdistance from the central nanoparticle where Rn is the radius of the nanoparticle and Rm is the radius of a single monomer. Monomer A concentrations arerepresented using solid lines with open symbols and monomer B concentrations are represented using dashed lines with filled symbols. Line color indicates

solvent selectivity, where A selective is blue and B selective is red. (k)–(m) Ensemble average of the radius of gyration, 〈R2g〉1/2

of individual grafted particlesat low (k), intermediate (l), and high (m) interaction strengths. The line color indicates the solvent selectivity, an A selective solvent is blue (circular markers)and a B selective solvent is red (triangular markers).

Additionally, at fA = 0.75 and intermediate interactionstrengths the grafts form a single patch of attractive solvent-phobic B monomers due to the large inner A block and suf-ficient attraction between B monomers, that facilitates inter-graft B monomer aggregation. High interaction strength: Inthis case, both A and B monomers have attractions on the or-der of or larger than thermal fluctuations, therefore both theA and B monomers can form patches. In both A and B se-lective solvents for fA = 0.25 and 0.5, we observe the forma-tion of both A and B patches. For fA = 0.75 in an A selectivesolvent, in contrast to the corresponding intermediate inter-action strength where we observed attractive monomers in asingle patch, due to the increase in interaction strength of theinner A block as well as the outer B block, it is more en-thalpically favorable to form two patches as that increases thetotal number of A-A and B-B monomer interactions. In thiscase, the two patch configuration is more enthalpically favor-able than a single patch because a single patch restricts theinner A block from forming favorable A-A contacts as the Ablock in the graft must circumnavigate the particle in order to

bring the B monomers together. Interestingly, for fA = 0.75in B-selective solvent, we see only patches of A monomersand do not see patches of B monomers; this is different fromthe corresponding system in A selective solvent where we seeboth A and B monomers patches. The strong attraction be-tween the A monomers in the inner block is more significantthan the B monomers, and as a result the (spatially restricted)A monomer aggregation takes precedence over B monomeraggregation.

In Figures 2(b)–2(j), we quantify the effects shown vi-sually in the simulation snapshots by presenting the aver-age monomer concentration profile from 10 particles in 10different trials. Since the A monomers form the inner blockand the B monomers form the outer block, the peak in themonomer concentration profiles of B monomers is always far-ther from the particle surface than the peak in the concentra-tion profiles of the A monomers. As fA increases, the locationof the peak in the B monomer concentration profiles shiftsto larger distances from the particle surface and its magni-tude decreases, and the maximum value of the A monomer

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144905-6 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

profile increases. Low interaction strength: In Figures 2(b)–2(d), when we compare A monomers in an A and B selectivesolvent, for fA = 0.25 and fA = 0.50 there is not a differencein the location or magnitude of the peak in the concentrationprofile, but for fA = 0.75 the concentration of A monomers hasa larger maximum close to the particle surface in a B selectivesolvent due to the aggregation of A monomers into patches.When we compare B monomers in an A selective solvent tothose in a B selective solvent, at fA = 0.25 and 0.50, we ob-serve a larger concentration of B monomers close to the parti-cle surface in the A selective solvent than B selective solventbecause they form B patches in an A selective solvent but notin the B selective solvent, and these B patches bring a largeconcentration of the B monomers closer to the surface. Whenthe grafts are composed of mostly A monomers, fA = 0.75,there is no difference between B monomer concentrations inA and B selective solvents as they do not form patches. Inter-mediate and high interaction strengths: When the interactionstrength is increased to the intermediate and high interactionstrength, we see the same general trends in the concentrationprofiles for both A and B monomers, as seen in low interac-tion strengths (Figures 2(e), 2(f), and 2(j)).

In Figure 2(k), the radius of gyration at low interac-tion strengths shows quantitatively the patch formation weobserve through simulation snapshots. Grafted particles thatform patches have a smaller radius of gyration than those thatdo not. At intermediate interaction strengths, in a B selectivesolvent (Figure 2(l)) we observe a slight decrease in Rg withincreasing fA due to the increasing number of A monomersinteracting in the patches formed, which confines monomersfound in the patch close to the particle surface. In an A selec-tive solvent, when fA is increased from 0.50 to 0.75 we see adecrease in Rg due to the formation of a single B monomerpatch in the case of fA = 0.75. At high interaction strength(Figure 2(m)), the average radii of gyration is smaller thanthe corresponding systems at intermediate and low interac-tion strength, because both A and B monomers form patchesin both A and B selective solvents leading to overall smallerpolymer grafted particles.

B. Effect of graft length on patch formation

The number of patches formed in Figure 2 is clearly sen-sitive to the ratio of the length of the solvent-phobic portion ofthe grafts to the particle size. In this section, we present patchformation for particles of diameter D = 4σ and NA+B = 24,to understand the effect of increasing graft length to particlesize ratio on patch formation.

We compare smaller graft, NA+B = 12 (Figure 2), andlonger graft NA+B = 24 (Figure 3), with constant particle size(D = 4σ ) and find the following differences. For longer graftsat low interaction strengths there is only one combination offA and solvent selectivity that does not lead to patch forma-tion: fA = 0.25 in a B selective solvent. In that case, the in-ner A block is 6 monomers long when NA+B = 24, and likefA = 0.50 at NA+B = 12 (Figure 2), is not large enough tobring attractive A monomers together and overcome entropiclosses. And, for longer grafts at the intermediate interaction

strength A selective solvent a single patch of B monomers isformed on the particle at all values of fA because the graftedchains are long in comparison to the diameter of the particle,and as a result are likely to be able to take on conformationsthat allow inter-graft B monomer aggregation into one patchwithout the inner A block losing a significant amount of con-formational entropy. At higher interaction strengths, while weobserved two patches form for all fA and solvent selectivity forsmaller grafts, for longer grafts we see a single patch form inB selective solvent for all fA; this is due to large number ofstrongly attractive A and B monomers. And, in A selectivesolvent, we see several patches form at fA = 0.75, in contrastto two patches seen in the corresponding system at smallergraft length, due to the same reasons as outlined for two ver-sus one patch in Sec. III A.

The concentration profiles in Figures 3(b)–3(f) and3(j) follow the same trends we observed in the case ofshorter graft lengths. The few differences between the short(Figures 2(b)–2(f) and 2(j)) and long graft (Figures 3(b)–3(j))concentration profiles arise due to larger number of monomersin each block at the longer chain length (24 monomers) com-pared to the chain length plotted in the concentration profilesin Figure 2 (12 monomers). As a result, in Figure 3 the mag-nitude of the monomer concentration for each case, and thedistance from the surface of the particle at which monomerscan be found is larger than the corresponding profiles inFigure 2. The radii of gyration measured for each parameterset is shown in 3(k)–3(m). At this graft length of NA+B = 24,there is an overall increase in the radius of gyration as com-pared to corresponding systems with graft length NA+B = 12.This is attributed not only to increase in the overall number ofmonomers per grafted particle going from NA+B = 12 to 24,but also to different number of patches formed (Figure 3(a)).

While Figures 2 and 3 present the patch formation re-sults for NA+B/D = 3 and 6, Figure S.1 of the supplemen-tary material63 presents the corresponding results for NA+B/D= 2 (NA+B = 24, D = 12 σ ). In the case of NA+B/D = 2,the ability for inter-graft patch formation decreases in bothA and B selective solvents at all interaction strengths. Dueto the increase in particle diameter, grafts are not capable ofinter-graft contacts and the solvent-phobic block of each grafttends to collapse onto itself, forming enthalpically favorableintra-graft contacts. As a result, the number of patches formedin these systems is the same as the number of grafts attachedto the particle. The location of the A and/or B patches inthe graft on the particle, as quantified by the monomer con-centration profile, is more sensitive to the solvent selectivityand interaction strength at NA+B/D = 2, than at NA+B/D = 3(Figure 2) and 6 (Figure 3), because each copolymer graft col-lapses to varying extents independent of other grafts on theparticle.

C. Effect of graft composition on assembly at lowinteraction strength

In this section, we focus on how graft composition, fA,

affects the relative times of the various stages of assem-bly, and the (equilibrium) structures of the assembled grafted

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144905-7 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

AA = 1.0

BB = 0.1

AA = 0.1

BB = 1.0

AA = 5.0

BB = 0.5

AA = 0.5

BB = 5.0

AA = 10

BB = 1.0

AA = 1.0

BB = 10

fA = 0.25 fA = 0.50 fA = 0.75

1.0

0.1

0.1

.0

5

0

= 5.0

BB = 0.5

1.0

10

= 10

= 1.0

0.25 0.5 0.752

3

4

5

6

7

0.25 0.5 0.752

3

4

5

6

7

0.25 0.5 0.752

3

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1 2 3 4 50

1

2

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6

1 2 3 4 5 1 2 3 4 5

1 2 3 4 50

1

2

3

4

5

6

1 2 3 4 5 1 2 3 4 5

1 2 3 4 50

1

2

3

4

5

6

1 2 3 4 5 1 2 3 4 5

fA

fA

fA

<C(r

)>

<C(r

)>

<C(r

)>

<Rg

2 >1/

2<R

g2 >

1/2

<Rg

2 >1/

2

fA = 0.25 fA = 0.50 fA = 0.75(a)

(b) (c) (d)

(e) (f) (g)

(h) (i) (j)

(k)

(l)

(m)

n + Rm) n + Rm) n + Rm)

n + Rm) n + Rm) n + Rm)

n + Rm) n + Rm) n + Rm)

FIG. 3. Characterization of patchy particles with NA+B/D = 6.0. (a) Simulation snapshots from a single trial showing representative patch formation of asingle diblock copolymer grafted particle in a box containing 10 particles. (b)–(j) Monomer concentration profiles for particles of low (b)–(d), intermediate(e)–(g), and high (h)–(j) interaction strengths are shown. Concentration profiles are shown as average grafted monomer concentrations 〈C(r)〉 as a function ofdistance from the central nanoparticle, where Rn is the radius of the nanoparticle, and Rm is the radius of a single monomer. Monomer A concentrations arerepresented using solid lines with open symbols and monomer B concentrations are represented using dashed lines with filled symbols. Line color indicatessolvent selectivity, where A selective is blue and B selective is red. (k)–(m) Ensemble average of the radius of gyration, 〈R2

g〉0.5 of individual grafted particlesat low (k), intermediate (l), and high (m) interaction strengths. The line color indicates the solvent selectivity, an A selective solvent is blue (circular markers)and a B selective solvent is red (triangular markers).

particles for all values of fA and solvent selectivity for parti-cles of NA+B/D = 3.0 at low interaction strength.

Figure 4 depicts stages during particle assembly for theonly three cases that exhibit cluster formation at low inter-action strengths when D = 4 σ and NA+B = 12. We notethat all timescales throughout the paper are normalized bythe timescale for patch formation for D = 4 σ particles withgrafts of length NA+B = 12 and composition fA = 0.75 in aB selective solvent at low interaction strength. We chose thisspecific timescale to normalize the other time scales by be-cause this is one of the shortest time scales for patch forma-tion in the large range of system parameters.

The time needed for monomers patch formation is short-est for grafted particles when fA = 0.75 in a B selective sol-vent (Figure 4, bottom row). In this case, the A monomerslikely form patches more quickly because of their large num-ber (9 monomers) and close proximity to each other due totheir location as the inner block. In the case of fA = 0.25in an A selective solvent (Figure 4, top row), the number ofsolvent-phobic B monomers per graft is equal to the numberof solvent-phobic A monomers in the case of fA = 0.75 in a B

selective solvent (Figure 4, bottom row), however the solvent-phobic B monomers from distinct grafts are not placed asclosely together as A monomers. Thus, patch formation isslightly slower than fA = 0.75 in a B selective solvent be-cause each solvent-phobic B block has a larger volume to ex-plore and is not able to encounter a solvent-phobic B blockof monomers in another graft on the same particle as fastas in the case of solvent-phobic A monomer. Particles withfA = 0.50 in an A selective solvent (Figure 4, middle row)have fewer solvent-phobic monomers than in the other twocases shown in Figure 4, leading to smaller attractive forcesand slower patch formation. Based on these observation wecan conclude that decreasing the configuration space attrac-tive monomers may explore (e.g., if inner block is solvent-phobic) while increasing the number of attractive monomersper graft, leads to an overall decrease in patch formation time.

For these three parameter sets, the speed of patch for-mation does not predict the speed of cluster formation. Thesystem with the slowest patch formation (fA = 0.50 in anA selective solvent, in Figure 4, middle row), has the fastestoverall assembly time, partly due to the quick assembly into

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144905-8 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

t = 1.22 ± 0.09

t = 4.40 ± 0.11

t = 1.00 ± 0.08

t = 3840 ± 450

t = 280 ± 95.4

t = 942 ± 21.9

t = 1270 ± 80

t = 1980 ± 290

fA = 0.25 AA = 0.1 BB = 1.0

fA = 0.50 AA = 0.1 BB = 1.0

fA = 0.75 AA = 1.0 BB = 0.1

FIG. 4. Simulation snapshots from single representative-trials depicting stages of particle assembly leading to the final clusters. Timescales are an ensembleaverage of 10 trials and the standard error is given, and normalized as stated in the text. The system depicted here consists of 10 particles grafted with six ABcopolymer chains with NA+B/D = 3.0.

an intermediate metastable state that then rearranges to formthe final equilibrium cluster. In the middle row of Figure 4,patchy particles come together initially to form chain-shapedanisotropic clusters, which then rearrange to maximize thenumber of attractive contacts within the final cluster. This cantake place because the inner (solvent-philic) A block facil-itates the rearrangement of the attractive B patches. In thesame A selective solvent when the inner A block is shorter (fA= 0.25, Figure 4, top row), this chain formation and rear-rangement is not observed due to the conformational restric-tions the short inner A block imposes on the overall structure.When fA = 0.75 in a B selective solvent (Figure 4, bottomrow), we observe small (relatively) isotropic trimers formingat intermediate times, these trimers then stack to form the fi-nal anisotropic (equilibrium) structure. The process of trimerformation and stacking is observed in visual renderings of thesimulation trajectory. When particles are able to form distinct(isotropic or anisotropic) intermediate structures it speeds uptheir assembly times into equilibrium structures.

Figure 5 shows the distance between particles within thesame cluster, which is a relevant structural characterizationin nanoclusters used for optical applications.62 For fA = 0.25in an A selective solvent where long anisotropic clusters areformed (Figure 4, top row), multiple peaks in the inter-particledistance profile depict the broad range of inter-particle dis-tances in an anisotropic structure (Figure 5(a)). For fA = 0.50in an A selective solvent, where the rearrangement of interme-diates leads to an isotropic final cluster with a single patch ofB monomers in the center (Figure 4, middle row), the sin-gle peak in the inter-particle distance profile (Figure 5(b))

is indicative of isotropic clusters. For fA = 0.75 in a B se-lective solvent where trimer formation and stacking occurs(Figure 4, bottom row), the first peak corresponds to thedistances between particles interacting in a single layer oftrimers, and the second to the distance between layers oftrimers (Figure 5(c)). We also observe that particles thatform clusters in a B selective solvent, on average, have ashorter distance between particles because the solvent-phobicA monomers in the inner A block aggregate, bringing the par-ticles closer together so the A patches on these particles caninteract.

We also calculate the likelihood of forming a clusterof a specific size from overall cluster formation data. InFigure S.2a of the supplementary material,63 we show clustersize probabilities for grafted particles of fA = 0.25. We notethat while the data shown up to now is also observed in sys-tem of 100 particles at the same concentration as 10 particlesin a (100 σ )3 simulation box (Sec. II of the supplementarymaterial),63 the quantitative information in Figure S.2. of thesupplementary material is dependent on number of particlesin the simulation box. Nonetheless, one can glean from thisdata general qualitative trends that hold for systems at thisconcentration. In Figure S.2. of the supplementary material,63

we see that particles in B selective solvent (red bar) do not ag-gregate (shown as 100% of “clusters” of size 1 particle in thecluster size probability histogram). When these particles areintroduced to an A selective solvent (blue bar in Figure S.2aof the supplementary material63) they have a high probabilityof forming a large cluster of all the particles in the system, anda non-zero probability of forming smaller clusters of 4 and 6

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144905-9 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

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5 10 15 5 10 15

Intermediate Interaction Strength

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(g)

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ctio

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arti

cles

5 10 15

Low Interaction Strength

(a)

(b)

(c)

(d)

(e)

(f)

FIG. 5. The distribution of particle center-to-center distances are shown for(NA+B = 12 D = 4 σ ) grafted particles within the same cluster for fA = 0.25((a), (d), (g)), fA = 0.50 ((b), (e), (h)), and fA = 0.75 ((c), (f), (i)) at low,intermediate, and high interaction strengths for NA+B/D = 3.0. Blue lineswith triangular markers represent clusters formed in an A selective solventand red lines with circular markers represent clusters formed in a B selectivesolvent.

particles. When fA is increased to 0.50 (Figure S.2b of the sup-plementary material)63 in an A selective solvent, the clustersize distribution shifts to smaller sizes than that at fA = 0.25,while in a B selective solvent the particles remain dispersed asseen for fA = 0.25. In an A selective solvent at fA = 0.50, theseparticles form isotropic clusters with a single B patch on theinterior and an exterior shell of A monomers. This solvent-philic A exterior shields the solvent-phobic B center making

it difficult for clusters to grow once the chain rearranges. Inthe one case where a cluster of ten particles formed (shown inFigure 4, row 2), we saw two smaller clusters of four and sixnanoparticles form linked by one single graft. Finally, we ob-serve that grafted particles with fA = 0.75 (Figure S.2c of thesupplementary material)63 do not form clusters in A selectivesolvent but have a broad cluster size distribution ranging to allten particles in an A selective solvent.

D. Effect of increasing interaction strengths onassembly

Table I presents the relative timescales for patch andcluster formation at both intermediate and high interactionstrengths for graft length, NA+B = 12 D = 4σ . For almostall fA in both A and B selective solvents as the interactionstrength increases the patch formation time decreases; one ex-ception to this is fA = 0.75 and interaction set εAA = 1.0, εBB

= 0.1 where there is an insignificant change in patch forma-tion times with interaction strength.

When comparing the cluster formation, first, we do notobserve intermediate assembly states at intermediate and highinteraction strengths, as observed at low interaction strengths(Figure 4) where the intermediate states facilitate and accel-erate cluster formation. For the cases at intermediate and highinteraction strengths where intermediate states were removedfrom the assembly pathway due to the increase in interactionstrength, we see an overall increase in final cluster formationtime. For fA = 0.25 in an A selective solvent, there are nointermediate states, and individual particles aggregate with-out significant rearrangement, so by increasing the interactionstrength we have not significantly changed the pathway or thetimescales to cluster formation. In general, in B selective sol-vents at intermediate and high interaction strengths the timefor cluster formation is highest when fA = 0.25 because of thefew solvent-phobic A monomers being constrained close tothe particle surface and decreasing the effective reach of theattractive, solvent-phobic A patch on one particle to aggregatewith A patches on other particles.

In Figures 5(d)–5(i), we plot inter-particle distances forclusters formed at intermediate and high interaction strengths.At intermediate attraction strengths (Figures 5(d)–5(f)) for allfA, the clusters formed in B selective solvents (red circle) havea higher fraction of particles at smaller inter-particle distances

TABLE I. Normalized patch formation and final cluster assembly time-steps for all monomer interaction strengths, solvent selectivities, and copolymercompositions for NA+B/D = 3.0.a

Patch formationlow interaction

strength

Cluster formationlow interaction

strength

Patch formationintermediate

interaction strength

Cluster formationintermediate

interaction strength

Patch formationhigh interaction

strength

Cluster formationhigh interaction

strength

fA = 0.25 A selective solvent 1.22 ± 0.09 3481 ± 450 0.82 ± 0.06 3380 ± 220 0.69 ± 0.05 5334 ± 320fA = 0.50 A selective solvent 4.40 ± 0.11 2054 ± 170 1.07 ± 0.07 4069 ± 290 0.55 ± 0.15 4320 ± 394fA = 0.75 A selective solvent . . . . . . 1.15 ± 0.07 4153 ± 340 0.78 ± 0.07 4300 ± 390fA = 0.25 B selective solvent . . . . . . 1.19 ± 0.11 3380 ± 310 0.645 ± 0.08 4600 ± 590fA = 0.50 B selective solvent . . . . . . 1.12 ± 0.10 2700 ± 308 0.645 ± 0.09 3608 ± 330fA = 0.75 B selective solvent 1.00 ± 0.08

∗2923 ± 310 1.14 ± 0.08 2990 ± 310 0.943 ± 0.29 3180 ± 215

aThe time steps shown are ensemble averages with standard error given of 10 individual trials of 10 particles in a 100 × 100 × 100 σ 3 box. Times are normalized by the shortesttimescale denoted by ∗, which is the time for patch formation in NA+B/D = 3.0, fA = 0.75, εAA = 1.0, εBB = 0.1.

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144905-10 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

fA = 0.25 AA = 0.1 BB = 1.0

fA = 0.50 AA = 0.1 BB = 1.0

fA = 0.75 AA = 0.1 BB = 1.0

t = 2260 ±176

t = 1867 ± 178

t = 1388 ± 272

t = 338 ± 90

t = 320 ± 70

t = 253 ± 90

t = 2.22 ± 0.7

t = 1.43 ± 0.6

t = 1.23 ± 0.7

FIG. 6. Simulation snapshots from single representative-trials depicting stages of particle assembly leading to the final clusters. Timescales are an ensembleaverage of 10 trials and the standard error is given. The system depicted here consists of 10 particles grafted with six AB copolymer chains with NA+B/D = 6.0.

than in A selective solvents (blue triangle). This is because ina B selective solvent the inner A block of the graft is solvent-phobic, thus particles must be closer to one another in orderto make enthalpically favorable contacts. We note that in aB selective solvent clusters take on long wire or chain-likestructures (see Figure S.3 of the supplementary material),63

which are of interest in optical applications.62

E. Effect of graft length on assembly

In Figure 6, we present snapshots and normalized as-sembly times for grafted particles with grafts of length NA+B

= 24 for each fA value in an A selective solvent at a low in-teraction strength. Unlike particles with grafts of length NA+B

= 12, the chain-like intermediate states are present in all casesfor grafts of length NA+B = 24, when particles are in an A se-lective solvent. For all values of fA these intermediates formin approximately the same amount of time, but the amountof time it takes for chain rearrangement into the final clus-ter decreases with increasing fA. We attribute this decrease inrearrangement time to the larger size of the inner A block.As the length of the solvent philic inner A block increasesthe chains of grafted particles more easily find configura-tions where solvent-phobic patches of B monomers can mergetogether.

Additionally, for all three cases shown, fA = 0.25, 0.50,and 0.75 in an A selective solvent, chain rearrangement leadsto equilibrium isotropic clusters where all B (solvent-phobic)monomers interact in one large aggregate/patch located at thecenter and A monomers shell that acts as a shield, so the addi-

tion of more particles is not possible once rearrangement takesplace. We also see cluster formation time decreases with theincrease in graft length for particles in both A and B selectivesolvents (comparing corresponding cases in Figures 4 and 6).In the case of longer graft length, the particles have a largerreach within the simulation box as can be seen by the largerradius of gyration observed in Sec. III B. The larger overallsize of each grafted particle leads to an increased probabil-ity of two particles interacting within the simulation box ina time step, which decreases the amount of time needed forfinal cluster formation.

While Figure 6 shows the final cluster assemblies in Aselective solvent the snapshots of final cluster assemblies in Bselective solvent are shown in Figure S.4 of the supplemen-tary material.63 When clusters form in a B selective solvent atNA+B = 24 we do not observe intermediates and the amountof time necessary to form the final clusters is slightly longerthan cluster formation times in an A selective solvent.

We plot inter-particle distances profiles for grafted par-ticles with grafts of length NA+B = 24 for each fA valuein an A and B selective solvent in Figure 7. With increas-ing fA at low interaction strength (Figures 7(a)–7(c)) clustersthat form in an A selective solvent have similar distance pro-files, and the same characteristic shape of isotropic clustersthat we discussed earlier. In the B selective solvent, while noclusters form at fA = 0.25, as fA increases the inter-particledistances shift to higher values because with an increase infA we also observe a slight increase in overall cluster size(Figures S.5b and S.5c of the supplementary material) anddue to the anisotropic shape of the clusters, this increasein size leads to an increase in inter-particle distances. In

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144905-11 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

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(g)

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FIG. 7. The distribution of particle center-to-center distances are shown for(NA+B = 24 and D = 4 σ ) particles within the same cluster for fA = 0.25((a), (d), (g)), fA = 0.50 ((b), (e), (h)), and fA = 0.75 ((c), (f), (i)) at low, in-termediate, and high interaction strengths. Blue lines with triangular markersrepresent clusters formed in an A selective solvent and red lines with circularmarkers represent clusters formed in a B selective solvent.

Fig. S.2 of the supplementary material,63 we discuss our ob-servations of cluster size formation when we conduct simula-tions of 100 at the same concentration as the 10 particle sim-ulations we have discussed so far. In both cases, 10 and 100particles, isotropic clusters are predominantly composed of5–7 nanoparticles and systems that formed isotropic clustersin the 10 nanoparticle simulations formed isotropic clusterswhen 100 nanoparticles are present.

At low interaction strength (Figures S.5a–S.5c of the sup-plementary material)63 as fA increases, the size of clustersobserved in an A selective solvent decreases and the size ofclusters observed in a B selective solvent increases. Clustersformed in an A selective solvent are isotropic with one largeaggregate of solvent-phobic B monomers at the center andan outer shell of A monomers as seen in Figure 6. As fAincreases, in A selective solvent, the clusters’ outer shell ofA monomers becomes more dense blocking the addition ofmore particles to the cluster leading to smaller overall clus-ters. The opposite trend is true in a B selective solvent. AsfA increases in a B selective solvent the number of solvent-phobic A monomers increases, and as a result the propensityto form large clusters increases.

Table II summarizes the timescales for patch forma-tion and cluster formation for particles with NA+B/D = 6.0.At graft length 24, there is no significant change in patchformation time as compared to corresponding cases withgraft length 12 (Table I). There is an overall decrease intimescales for final cluster formation when comparing par-ticles of NA+B/D = 6.0 to NA+B/D = 3.0, except for thecase of particles at an intermediate interaction strength in aB selective solvent where the average cluster assembly timesincreased for particles of NA+B/D = 6.0 in comparison toNA+B/D = 3.0. The assembly time for NA+B/D = 6.0 at in-termediate interactions strengths is also longer when com-pared to systems with low and high interaction strengths withthe same graft length. In this unique case of the interme-diate interaction strength in B-selective solvent, the solventA monomers aggregate with attractive interactions strongerthan thermal fluctuations and once assembled reconfigura-tions are hindered, making assembly timescales longer whencompared to the low interaction strength. The timescale ofassembly is longer than systems at high interaction strengthsbecause at intermediate interaction strengths B-selective sol-vent the B monomers interact with attractions less than theenergy from thermal fluctuations, but at higher interactionstrengths the outer B block has attractions on the order of ther-mal fluctuations so the attractions of both the inner A blockand the outer B block drive aggregation making the timescalefaster.

Finally, the effect of particle diameter on clusterassembly can be found in S.IV of the supplementarymaterial.63

TABLE II. Normalized patch formation and final cluster assembly time-steps for all monomer interaction strengths, solvent selectivities, and copolymercompositions for NA+B/D = 6.0.a

Patch formationlow interaction

strengthCluster formation low

interaction strength

Patch formationintermediate

interaction strength

Cluster formationintermediate

interaction strengthPatch formation highinteraction strength

Cluster formationhigh interaction

strength

fA = 0.25 A selective solvent 1.23 ± 0.07 2599 ± 266 1.02 ± 0.08 3001 ± 312 0.60 ± 0.05 3990 ± 340fA = 0.50 A selective solvent 2.22 ± 0.7 2089 ± 248 1.01 ± 0.05 3217 ± 413 0.57 ± 0.05 3450 ± 490fA = 0.75 A selective solvent 1.43 ± 0.6 1642 ± 362 1.11 ± 0.07 4022 ± 317 0.62 ± 0.07 2868± 440fA = 0.25 B selective solvent . . . . . . 1.03 ± 0.06 4433 ± 690 0.53 ± 0.07 4060 ± 290fA = 0.50 B selective solvent 1.37 ± 0.7 2742 ± 460 0.94 ± 0.06 5360 ± 395 0.54 ± 0.07 2340 ± 370fA = 0.75 B selective solvent 1.22 ± 0.08 2300 ± 210 1.07 ± 0.07 5730 ± 380 0.51 ± 0.03 1771 ± 180

aMonomer interaction strengths and graft A content are as labeled. The time steps shown are ensemble averages with standard error given of 10 individual trials of 10 particles in a100 × 100 × 100 σ 3 box. Times are normalized by the timescale for patch formation in NA+B/D = 3.0, fA = 0.75, εAA = 1.0, εBB = 0.1).

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144905-12 C. E. Estridge and A. Jayaraman J. Chem. Phys. 140, 144905 (2014)

IV. CONCLUSION

Using molecular dynamics simulations we have studiedAB diblock copolymer-grafted particles in an implicit sol-vent to elucidate the effect of varying graft composition, graftlength, and particle size in different solvent selectivities andmonomer interaction strengths on formation of patchy parti-cles and their assembly. This study has lead to the followingdesign guidelines: (i) Increasing the size of the solvent-phobicblock of the diblock copolymer facilitates patch formation onthe particles, and decreases the time needed to form patches.Increasing NA+B/D facilitates inter-graft patch formation, anddecreases the number of patches formed on the particle forall copolymer compositions, solvent selectivity, and interac-tion strength. (ii) At low interaction strengths, the patchyparticles assemble into isotropic clusters in an A selectivesolvent with the cluster formation times decreasing with in-creasing fraction of the inner solvent-philic block and a largeenough outer block. In a B selective solvent at lower values ofNA+B/D, there is a high probability of anisotropic cluster for-mation (e.g., nano-wire type structures) for all graft composi-tions at intermediate and high interaction strengths. At higherNA+B/D, these nanowire structures no longer exist at low in-teraction strengths, and relatively isotropic clusters are morelikely. (iii) In A selective solvent, clusters form with a singleaggregate of attractive B monomers at the center of the clusterwith solvent-philic A monomers acting to shield B monomers.This type of assembly is not possible in a B selective sol-vent. Additionally, equilibrium structures formed in B selec-tive solvents are found to have smaller inter-particle distanceson average due to the location of solvent-phobic A monomersas the inner block. (iv) At large values of NA+B/D there isa higher likelihood of metastable intermediates (e.g., NA+B

= 6.0 shows chain conformations for all values of fA inan A selective solvent) and these intermediates decreasethe time for nanoparticles to assemble into their final equi-librium structures. (v) The monomer-monomer interactionstrength, which one can choose by choosing the two monomerchemistries and/or solvent chemistry, affects short timemonomer arrangements as well as equilibrium cluster assem-blies. The equilibrium structures show an increase in the av-erage cluster size and a larger distribution of inter-particledistances with increasing interaction strength in all cases ofcopolymer graft composition and solvent selectivity. Thistrend is more pronounced with an increase in graft length.

These results present guidelines linking the molecularfeatures of the copolymer grafts, that can be tuned duringsynthesis, to the resulting copolymer grafted particle confor-mation and the resulting particle assembly. Specifically, ourresults build a knowledge base of assembly pathways, trendsin relative time scales of patch formation, intermediate states,and final equilibrium cluster as a function of design parame-ters –copolymer composition, graft length to particle size ra-tio, solvent selectivity and strength. These trends guide ma-terials scientists on what design parameters to choose to beable to trap/freeze the particles in desired assembled states.Since much of the past work has been focused on homopoly-mer grafted particles, the results from this work shoulddrive the materials community to create designer copoly-

mer grafted particles for target assemblies and novel materialproperties.

ACKNOWLEDGMENTS

This work acknowledges financial support by the Depart-ment of Energy under Grant No. DE-SC0003912. This workalso acknowledges the use of the Janus supercomputer, whichis supported by the National Science Foundation (Award No.CNS-0821794) and the University of Colorado Boulder. TheJanus supercomputer is a joint effort of the University of Col-orado Boulder, the University of Colorado Denver, and theNational Center for Atmospheric Research.

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