Control of Aggregation of Nanoparticles by Double-Hydrophilic Block

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J. Phys. Chem. B 2007, 111, 7735-7741

7735

Control of Aggregation of Nanoparticles by Double-Hydrophilic Block Copolymers: A Dissipative Particle Dynamics Study Jianhua Huang*,† and Yongmei Wang‡ Department of Chemistry, Zhejiang Sci-Tech UniVersity, Hangzhou 310018, China, and Department of Chemistry, The UniVersity of Memphis, Memphis, Tennessee 38152 ReceiVed: January 9, 2007; In Final Form: May 10, 2007

Double-hydrophilic block copolymer (DHBC)-directed mineralization is investigated by dissipative particle dynamics (DPD) simulation. By mineralization, we refer to the formation of inorganic crystals from the solution. In the current study, the DHBCs are modeled as chains of A and B blocks with repulsion between unlike blocks, while the mineralization is approximated by aggregation of hydrophobic nanoparticles from the solution. Depending on the relative concentrations of nanoparticles and DHBC, dispersed spherical aggregates, hexagonally packed cylinders, and ordered lamellae structures are obtained. The structures formed are seen to be controlled by competing forces between aggregation of nanoparticles, the interaction of DHBC with nanoparticles, and the self-assembly of DHBC in the solution. The time evolutions of hexagonally packed cylinders and ordered lamellae are studied. For the development of cylinders, nanoparticles first aggregate into orientationally disordered small cylinders, then these cylinders slowly grow into hexagonally packed long cylinders. For the development of ordered lamellae, nanoparticles first form a disordered structure, then grow into disordered lamellae, and at last evolve into ordered lamellae. The simulation demonstrates that addition of DHBC can effectively control the aggregation of inorganic particles and lead to formation of a variety of nanostructures.

1. Introduction Minerals refer to naturally occurring inorganic crystals formed either through geological processes or through biological processes. Biominerals found in nature, such as seashell, pearl, bone, etc., exhibit remarkably intricate structures with patterning organized from the nanoscale to the macroscale. These hierarchical structures are further sculpted into complex formss spheroids, spirals, and skeletonssthat lead to unique mechanical and optical properties. These natural systems provide inspirations to synthetic chemists trying to produce biomimetic systems with nanostructured inorganic materials.1 Biomimetic mineralization is an emerging field where synthetic chemists seek to copy Nature’s process to produce advanced materials.2 Biominerals are usually produced in the presence of organic scaffold and soluble molecules that direct and control the crystallization growth. These processes can be mimicked successfully by making use of appropriate organic agents, typically surfactants or polymers, to orchestrate the positioning, interconnection, and stabilization of inorganic building blocks at the nano-, meso-, micro-, and macroscopic levels. Inorganic materials with complex forms have been chemically synthesized by replication of self-organized organic assemblies such as micelles, vesicles, and foams. Hollow silica tubes,3 cylinders coated with iron oxide,4 and helical strings of gold5 crystal have been prepared. The key factors in achieving hierarchical structures with complex forms are the controlled interactions between the inorganic building blocks and organic scaffolds.1 Block copolymers, in particular, double-hydrophilic block copolymers (DHBC), * To whom correspondence should be addressed. E-mail: jhhuang@ zstu.edu.cn. Phone: +86+571-86843228. † Zhejiang Sci-Tech University. ‡ The University of Memphis.

turn out to be an attractive patterning agent.6 DHBCs consist of two hydrophilic blocks, for example, poly(ethylene oxide)block-(polystyrene sulfonic acid), and both blocks are soluble in water but have different interaction strength with inorganic minerals or surfaces. DHBCs themselves do not form micelles, but are very effective in controlling the crystallization of inorganic particles.7 DHBCs that controlled or directed formation of CaCO3 pancakes,8 BaCrO4 fiber bundles,9 peanut-like calcite particles,1 and unconventional crystal superstructures of BaSO410 have been reported. The success of the approach relies on synergistic co-assembly of DHBCs with inorganic crystals, as opposed to replication of stable, pre-organized, self-assembled organic templates. Experiments showed that not only the shape of primary nanocrystals, but also the complicated superstructure, can be effectively controlled by DHBC.1,6,8 However, despite a large amount of data that show altered morphologies of crystals in the presence of DHBCs, the mechanistic understanding of the DHBC controlled or directed crystal growth is still lacking.6 Computer simulations can provide fundamental insight into the polymer controlled/directed mineralization process and knowledge on important parameters governing the fabrication of inorganic particles with complex structures. So far, however, no such computer simulation efforts have been reported. The current report is our first attempt to model the polymer-directed mineralization process. Since mineralization is the formation of inorganic crystals from solution, an appropriate simulation model for polymer-directed mineralization should model the crystal formation of the solute in the presence of DHBC. The crystallization process contains two major steps, nucleation and subsequent growth. Nucleation is the step where the solute molecules dispersed in solvent form aggregates (i.e., nuclei). These nuclei are unstable because of their high surface energies and grow subsequently after their formation. The growth of

10.1021/jp070160y CCC: $37.00 © 2007 American Chemical Society Published on Web 06/19/2007

7736 J. Phys. Chem. B, Vol. 111, No. 27, 2007 crystal is influenced by many factors, among these the surface energy is an important factor. Not surprisingly then, surfactants or surface active polymers can strongly influence the crystallization process.2 Simple Lennard-Jones fluids, when cooled, were shown to form crystals.11 However, modeling the crystallization process from solution at the molecular level is very challenging. At present, brute force molecular dynamics simulations are unable to simulate crystallization from solution.12 While we are motivated to simulate polymer-directed mineralization, in the present study, however, we approximate the mineralization process by simple aggregation of insoluble particles. We have used dissipative particle dynamics (DPD) simulations since DPD is a versatile method capable of simulating a variety of systems, e.g., polymer blends,13 block copolymers,14 colloidal solutions,15 and polymer-nanoparticle composites.16 In our simulations, we have used two types of DPD particles, representing a solvent component and a solute component. The solvent and solute DPD particles are distinguished from each other by their different mutual interactions. Basically, the solute repels the solvent and forms the aggregate. We name our solutes as nanoparticles since DPD is a coarse-grained method and one DPD particle typically represents a cluster of solutes with a size in the nanoscale range. Others have modeled nanoparticles by fusing several DPD particles together as one rigid body.16 However, a single DPD particle has also been used to represent nanoparticles.17 Additionally, in our simulations DHBCs are modeled as chains composed of A and B blocks. In the absence of DHBC, the solute particles form one large aggregate. The presence of DHBC, however, alters the aggregation process and leads to the formation of various dispersed nanostructures. Although the aggregation process may not represent the crystallization process, the knowledge learned on how DHBC alters the aggregation process is relevant to the understanding of polymerdirected mineralization since both processes involve the alteration of surface free energies of clusters. Moreover, DPD simulation has been recently shown to be able to simulate the crystallization process when the conservative force used is changed from soft repulsive to soft repulsive plus an attractive tale.18 In the future, we plan to employ the latter approach to better mimic the mineralization process. One should note that our study differs from earlier studies which examined the dispersion of nanoparticles in the neat block copolymer.17,19-21 The system examined in earlier studies is a two-component system, i.e., copolymers plus nanoparticles, while our system is a ternary system: solvent, copolymers, plus nanoparticles. Earlier studies focused on the phase behavior of block copolymer/nanoparticle composites and the distribution of nanoparticles in the microdomain of segregated block copolymers.17,19-21 We are interested in how the addition of DHBC affects the aggregation of nanoparticles, motivated by experimental works in biomimetic mineralization.2 In the present work, we adopted dissipative particle dynamics (DPD) simulation to study the aggregation of nanoparticles in the presence of DHBC. As the volume fraction of DHBC increases, the aggregation process changes from the formation of one large aggregate to the formation of various ordered nanostructures: dispersed aggregates, hexagonally packed cylinders, and lamellae. The current study captures the basic feature of polymerdirected mineralization and provides a first glimpse to the polymer-controlled/directed mineralization at the molecular level. 2. Simulation Details 2.1. The DPD Algorithm. The DPD method was developed by Hoogerbrugge and Koelman22 and cast in the present form

Huang and Wang by Espan˜ol.23 In DPD, fluid elements are coarse-grained into particles called DPD particles. These DPD particles interact with each other via pairwise forces that locally conserve momentum leading to a correct hydrodynamic description.24,25 The pairwise forces contain the conservative force F(C), random force F(R), and dissipative force F(D), therefore the net force on the ith particle, fi, is given by

fi )

(D) (R) (F(C) ∑ ij + Fij + Fij ) j*i

(1)

The three pairwise forces are given by:

ˆ ij F(C) ij ) aijw(rij)r

(2)

2 F(D) ˆ ij‚vij)rˆ ij ij ) -γw (rij)(r

(3)

and

F(R) ij )

σ w(rij)θijrˆ ij (∆t)1/2

(4)

where rij ) ri - rj, rˆ ij ) rij/rij, vij ) vi - vj, and ∆t is the iteration time step. θij is a symmetric random noise with zero mean and unit variance and is uncorrelated for different degrees of freedom and different times. σ is the amplitude of the thermal noise, and γ is a friction coefficient. The combined effect of the dissipative and random forces is that of a thermostat, leading to σ2 )2γkBT. The softness of the interaction is determined by the weight function, w(r), for which we adopt the commonly used choice, w(r) ) 1 - r/rc for r e rc, and w(r) ) 0 for r > rc, where rc is the cutoff radius. In eq 2, the interaction amplitude aij is the strength of the repulsive interaction between particles i and j and this can vary depending on particle types. In our simulations, we have used four types of DPD particles, representing solvent (s), nanoparticle (n), and A and B polymer segments. The relative interaction strength aij between different types of DPD particles is specified by eq 7 discussed in the next section. The particles move according to Hamilton’s equations,

∂ri ) vi, dt

∂vi mi ) fi dt

(5)

The positions and velocities of the particles are solved by using a modified velocity-Verlet algorithm proposed by Groot and Warren.13 2.2. Models for Copolymer and Nanoparticles. Our simulations are mainly performed in a cubic simulation box with size 20 × 20 × 20. The simulation box contains four types of DPD particles, representing solvent (s), nanoparticle (n), and A and B polymer segments that are connected to form the DHBC polymer chains. In our simulation, the DHBC chain studied is A3B3, which consists of three A segments and three B segments connected via a finitely extensible nonlinear elastic (FENE) potential:26 FENE Ui,i +1 ) kF ri,i+1 - req - (rmax - req)2 ln 1 2 rmax - req ∞

{

[ (

)] 2

for ri,i+1 < rmax for ri,i+1 g rmax (6)

where the equilibrium bond length req ) 0.7, the maximum bond length rmax ) 2.0, and the elastic coefficient kF ) 40.

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Figure 1. Equilibrium configurations of the DHBC/solvent two-component system at different concentrations of DHBC, Cp ) 0.5 (a), 0.8 (b), and 0.9 (c). Red and blue rods represent A and B blocks of copolymer, respectively. For clarity, solvent particles are not shown. Symbols are the same in the remaining figures.

The repulsive interaction parameter alk used for the four types of DPD particles is specified by

(

A alk ) B s n

A 25 30 20 25

B 30 25 25 15

s 20 25 25 50

n 25 15 50 25

)

(7)

Here the repulsion parameter between the same types of particles is set to 25, and the interactions between different types of particles are specified by the off-diagonal numbers. The mutual compatibility of the species depends on the value of alk between the lth and kth types. If alk >25, then the lth specie and the kth specie repel each other, and vice versa. Taking the solvent as water, then according to the interaction specified by eq 7, the solute nanoparticles are hydrophobic and they precipitate from the solvent in the absence of DHBC. Both A and B segments of the block copolymers are hydrophilic; however, A is more hydrophilic than B. Additionally, the B segments are more attracted to nanoparticles than the A segments. The A and B segments, however, repel each other. Previous simulations by Groot et al. have shown that the neat DHBC copolymers represented by A5B5 are microphase separated at T ) 1.14 In dilute solutions of DHBCs, however, these block copolymers do not form micelles. The exact phase diagram of the DHBC/ solvent system will be discussed shortly. We note that in the absence of solvent, our system returns to a two-component system that has been studied before. We intend to investigate general rules governing polymer-directed aggregation formation, thus at present we will not map these parameters to any particular system. The overall particle density F ) N/V is set as 6 throughout this study, here V is the box volume, N ) Ns + 6Np + Nn, Ns is the number of solvent particles, Np is the number of DHBC chains, and Nn is the number of nanoparticles. Concentrations of DHBC, Cp, and nanoparticles, Cn, are defined as Cp ) 6Np/N and Cn ) Nn/N, respectively. In the present work, we set the cutoff distance rc ) 1, energy scale kBT ) 1, and mass of DPD m ) 1. All results are in reduced DPD units, i.e., length in rc, energy in kBT, mass in m, and time τ ) mrc2/kBT. In the simulation, we use σ ) 3 and a time step ∆t ) 0.02 for integration. 3. Results and Discussion 3.1. Phase Diagram of the DHBC/Solvent Two-Component System. The DHBC/solvent two-component system forms a homogeneous solution for the concentration of DHBC, Cp up to 0.8. Figure 1 shows an example of structures where the concentration of DHBC Cp ) 0.5, 0.8, and 0.9. Microphase separation occurs when Cp exceeds 0.8 and a lamellar structure is observed for Cp ) 0.9.

Figure 2. Aggregates formed in pure nanoparticle/solvent systems at Cn ) 0.06 (a) and 0.1 (b). Gray beads represent nanoparticles. For clarity, solvent particles are not shown. The same symbols are used to represent nanoparticles in the following figures.

3.2. Aggregation of Nanoparticles in the Absence of DHBC. In the present work, nanoparticles are modeled as hydrophobic, so in the absence of DHBC, they aggregate into a big cluster to avoid solvent contact. Figure 2 presents the final structure of the cluster formed at the end of 5000 simulation times in a box of size 20 × 20 × 20 when the concentration of nanoparticles Cn is at 0.06 and 0.1, respectively. In the following, we monitor the growth of aggregate as a function of simulation time when the concentration of nanoparticles Cn is 0.06. Here two nanoparticles are said to form an aggregate whenever their distance is smaller than a certain value rmin, where rmin is set to be 1.0. Initially, nanoparticles form many small aggregates, as shown in Figure 3a, where the simulation time t ) 100. When the simulation time reaches 500, the number of aggregates decreases to 11 (see Figure 3b). The number of aggregates decreases continuously with t, as we observe in Figure 3c,d. At time t ) 2000, only two aggregates now exist (Figure 3e). These two aggregates exist for a long time, and eventually they form one large aggregate at time t ≈ 3700. From these snapshots, we speculate that the large aggregate is formed by fusion of small aggregates. The aggregation process can be followed by plotting the number of aggregates Nagg against the simulation time t, as shown in Figure 4. Figure 4 also gives the mass of the largest aggregate, Magg, as a function of time t. The number of nanoparticles in an aggregate is defined as the aggregate size. Figure 4 clearly shows that there is a rapid decrease in the number of aggregates and a simultaneous increase in aggregate size at initial times. Ultimately all nanoparticles form one big nearly spherical aggregate at simulation time t ≈ 4000. 3.3. Aggregation of Nanoparticles in the Presence of DHBC. To illustrate the influence of DHBC on the aggregation of nanoparticles, we have performed a series of simulations at a fixed Cn ) 0.06 with different DHBC concentrations. All simulations are performed for the same length, up to time t ) 5000. The final structures formed at the end of simulation are depicted in Figure 5. When a small portion of DHBC is added, for instance Cp ) 0.02, a big nearly spherical aggregate wrapped

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Huang and Wang

Figure 3. Aggregation kinetics of nanoparticles in pure nanoparticle/solvent system at time t ) 100 (a), 500 (b), 1000 (c), 1500 (d), 2000 (e) and 3700 (f). The concentration of the nanoparticle Cn ) 0.06.

Figure 4. Change of the number of nanoparticle aggregates Nagg and the mass of the largest aggregate Magg with simulation time t in the absence of DHBC. The concentration of nanoparticles Cn ) 0.06.

by DHBC is formed (see Figure 5a). Here, A and B blocks of DHBC are both hydrophilic but the B block is more attracted to the nanoparticles than the A block. Hence the B block is seen to be adsorbed on the surface of the aggregate while the A block extends to the solution. The adsorption of DHBC on the surface of the aggregate lowers the interfacial energy of the aggregate since DHBCs are hydrophilic. With an increase in the concentration of DHBC, the final structures formed now consist of several smaller aggregates, as depicted in Figure 5bd. These small aggregates are randomly dispersed in the solution. The size of the aggregates decreases with the increase of Cp. Formation of small aggregates provides more surface area on which DHBC chains can be adsorbed. At even high concentrations of DHBC, nanoparticles no longer form aggregate (defined by their distance being less than rc), rather these nanoparticles assemble into ordered structures. At Cp ) 0.5, the cylindrical structure is obtained as shown in Figure 5f, where all cylinders are roughly parallel to each other. Recall that in the absence of nanoparticles, the pure DHBC solution is homogeneous at Cp ) 0.5 (see Figure 1a). The ordered structure formed at Cp ) 0.5 and Cn ) 0.06 is a result of synergistic co-assembly of DHBC and the nanoparticles. With a further increase of Cp to 0.9, a lamellae structure emerges, as shown in Figure 5g. This lamellae structure can be identified with the lamellae structure formed in the pure DHBC solution at Cp ) 0.9 (see Figure 1c). Apparently in Figure 5g the structure formed is now mainly controlled by the self-assembly of the DHBC, not so much

dependent on the co-assembly of DHBC and the nanoparticles. Therefore, we see that the final structures formed in our systems are governed by different factors depending on the conditions used. At low Cp, the final structure is primarily controlled by the aggregation of the nanoparticles. At an intermediate value of Cp, co-assembly of DHBC and the nanoparticles determines the final structure. At even higher Cp values, the self-assembly of DHBC determines the final structure. Therefore, DHBCdirected aggregation of nanoparticles exhibits a rich variety of structures that can be controlled by varying the concentration of DHBC. A similar feature is observed by Glotzer’s group,27,28 who studied the ordering and packing of tethered nanoparticles of different shapes. In their studies, the obtained morphologies in certain cases can be predicted based on the concepts of block copolymer microphase separation and/or liquid-crystal phase ordering, whereas in other cases, the unique packing constraints introduced by nanoparticle geometry and by nanoparticle-tether topology lead to structures far richer than those found in conventional block copolymer, surfactant, and liquid crystal systems.27,28 In short, we observe three typical structures: nearly spherical aggregate when Cp < 0.5, hexagonally packed cylinders when Cp is about 0.5, and ordered lamellae when Cp reaches 0.9. We observe a disordered structure in the region of Cp from about 0.5 to 0.9. One typical snapshot of the disordered structure is given in Figure 6. To differentiate the cylindrical structure from the lamellae structure, we calculate a two-dimensional (2D) structure factor for nanoparticles, which is defined as29,30

S(k B) )

Nn

1 Nn

2

exp(ik B‚b r j)|2 ∑ j)1

|

(8)

where b rj is the position vector of the ith nanoparticle, Nn is the total number of nanoparticles used in calculation, and the wave vector B k is two-dimensional (2D). Here, the structure factor is normalized at k ) 0, S(0) ) 1. Similar 2D structure factors for only the A blocks or the B blocks can be calculated with eq 3, except b rj now refers to the beads on the A blocks or the B blocks, respectively. 2D structure factors correspond to patterns obtained in small-angle X-ray scattering experiments. For a cylindrical

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Figure 5. Final structures formed at the end of simulation time t ) 5000 for a fixed nanoparticle concentration Cn ) 0.06 in the presence of different concentrations of DHBC: Cp ) 0.02 (a), 0.03 (b), 0.04 (c), 0.1 (d), 0.2 (e), 0.5 (f), and 0.9 (g).

Figure 6. Final structure formed with a concentration of nanoparticles Cn ) 0.06 and concentration of DHBC Cp ) 0.7.

Figure 7. Calculated structure factors for cylindrical structure at Cp ) 0.5 (a) and lamellae structure at Cp ) 0.9 (b). The concentration of nanoparticles Cn ) 0.06. The unit of wave vector k is 2π/rc (rc ) 1).

structure, B k is chosen to lie on the surface that is roughly perpendicular to the direction of the cylinders, while for the lamellae structure, the sheet usually does not parallel any surfaces of the simulation box as shown in Figure 5g. However, we find that one of the Cartesian surfaces (xy, yz or xz) will have the most stripes, indicating that the lamellae are mostly perpendicular to this surface. Therefore B k is chosen to lie on this surface. The calculated structure factors S(k B) are presented in Figure 7 for both cylindrical (Figure 5f) and lamellae structures (Figure 5g). Figure 7a shows that the cylindrical structure is hexagonally packed while Figure 7b conforms to an ordered lamellae structure. From the vector length of the first peak in S(k B), we determine the lattice spacing (one periodic distance of the

ordered lattice) between cylinders is about 6 and that between lamellae is also about 6. For the lamellae structure, the layer thickness of the A block is about 2.5 estimated from Figure 5g. Therefore, the nanoparticles are roughly separated by two interpenetrating A layers. To ensure that the formation of such patterns does not suffer from the finite size effect of the simulation box, we have run simulations in a large box of size 30 × 30 × 30 for Cp ) 0.5 and 0.9, respectively, at Cn ) 0.06. The same structures are found as those in a smaller simulation box. This indicates that the finite size effect is negligible at low nanoparticle concentration Cn ) 0.06. With an increase of the nanoparticle concentration, such as Cn ) 0.10, cylindrical and lamellae structures are still observed. However, for Cn exceeding 0.1, the length of the cylinder becomes comparable to that of the simulation box size. Therefore, the cylindrical structure formed begins to suffer from finite size effect at Cn > 0.1. The lamellae structure, however, does not suffer from obvious finite size effect even at Cn as high as 0.25. With increasing Cn, lamellae structure appears at a low concentration of DHBC Cp. For example, when Cn ) 0.25, a lamellae structure is observed when Cp ) 0.5, whereas at Cn ) 0.06, the lamellae structure is observed when Cp ) 0.9. Also at high Cn, the thickness of the lamellae structure is larger than that formed at low Cn, although the thickness of A blocks remains almost constant. At high Cn, there are two interpenetrating layers of A blocks of DHBC chains between sheets of nanoparticles. 3.4. Kinetics of Aggregation in the Presence of DHBC. Here we discuss the kinetics of aggregation in the presence of DHBC. At a low concentration of nanoparticles Cn and DHBC Cp, the kinetics of aggregation can be monitored by monitoring the total number of aggregates and the maximum size of aggregates formed. Figure 8 presents such data at Cn ) 0.06 with different Cp up to 0.1. With the addition of the DHBC, the maximum aggregate size decreases with the increase of Cp. The number of aggregates also decreases rather rapidly initially and seems to have little dependence on Cp except when Cp ) 0.1. With a further increase in DHBC concentration, we can no longer use the number of aggregates to monitor the aggregation process, since now the nanoparticles no longer form aggregates.

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Figure 8. Change of the number of nanoparticle aggregates Nagg with simulation time t in the presence DHBC. The inset shows the change of the maximum size of the aggregate formed by the nanoparticles. The concentration of nanoparticles Cn ) 0.06 and the concentration of DHBC Cp is shown in the legend.

Huang and Wang

Figure 10. Dependence of the average value of the structure factor near the six reciprocal vectors S1 on simulation time t for structures shown in Figure 9.

Figure 11. Snapshots of the time evolution at different times: t ) 100 (a), 500 (b), 1000 (c), and 1600 (d). The concentration of nanoparticle Cn ) 0.06 and that of DHBC Cp ) 0.9.

Figure 9. Snapshots of structure evolutions (left) and 2D structure factors (right) at several times: t ) 500 (a), 1500 (b), and 4000 (c). The concentration of nanoparticle Cn ) 0.06 and that of DHBC Cp ) 0.5. The scale of wave vector k is the same as that in Figure 7.

Instead, we use the structure factor to monitor the kinetics of aggregation formation. Figure 9 presents several snapshots of structures at different times for the system with Cn ) 0.06 and Cp ) 0.5 with the corresponding 2D structure factors. A transformation from orientationally disordered small cylinders to hexagonally packed long cylinders is observed. At time t ) 4000, a hexagonal structure, with six bright spots at the first six reciprocal vectors in the structure factor, is found. A bright spot means its structure factor S(k B) is large. At an early stage of evolution (t ) 500), there is no well-defined bright spot and

the structure is disordered. At a medium stage (t ) 1500), there are spots near the first six reciprocal vectors but they are much weaker than those at t ) 4000. During the formation of the hexagonally packed cylinders, the average value of the structure factor at the six reciprocal vectors, S1 ) 1/6∑(kh), is calculated for a disordered structure at short time and for a well-defined hexagonal structure at long time. Here S(kh) values are the structure factors at the six reciprocal vectors of a hexagonal structure. For a disordered structure and an ill-defined hexagonal structure, we counted six maximum values of S(kh) near the six reciprocal vectors. The dependence of S1 on the simulation time is presented in Figure 10. The value S1 increases smoothly from roughly zero at the beginning of simulation time to about 0.4 at a time around 4000. Snapshots for evolution of lamellae structure at Cn ) 0.06 and Cp ) 0.9 are shown in Figure 11. Although the segregation of A and B blocks occurs at a very early time (t ) 100), these blocks, however, are randomly distributed over the long-range length scale. The long-range ordering of the lamellae structure occurs at a much later time. At time t ) 1000, a bent lamellae structure appears. At time t ) 1600, an ordered lamellae structure is formed and the structure ceased to evolve any further in simulation. The lamellae structure can be characterized by

Double-Hydrophilic Block Copolymer-Directed Mineralization

Figure 12. Dependence of the average value of the structure factor near the first two reciprocal vectors S1 on simulation time t for structures shown in Figure 11.

two bright spots in the 2D structure factor as shown in Figure 7b. The two bright spots correspond to two reciprocal vectors of the lamellae structure. The average value of the structure factor of nanoparticles at the two reciprocal vectors, S1, is also calculated at different simulation times and the results are presented in Figure 12. Values of S1 increase rapidly from 0.1 to 0.4 at time t ≈ 1200, indicating a rather sharp transition from a disordered lamellae structure to an ordered lamellae structure. At t ≈ 1600, lamellae orient parallel to each other. Figure 12 also shows time evolution of the average values of the structure factor S1 calculated for the A blocks. Similar evolvement in the structure factor of the A blocks is observed here. This indicates that the long-range ordering of the lamellae structure occurs simultaneously for the block copolymers and the nanoparticles. 4. Conclusion We present a first dissipative particle dynamics simulation of polymer-directed mineralization of inorganic particles. The mineralization of inorganic particles is simulated via aggregation of hydrophobic nanoparticles from the solution. The polymers examined are double hydrophilic block copolymers (DHBC) where both blocks are hydrophilic in the solvent but repel against each other. One of the blocks is more attractive to the nanoparticles. In the absence of DHBC, the nanoparticles form one large aggregate. With the addition of DHBC, a variety of structures are formed, dictated by competing forces between aggregation formation of nanoparticles, interaction of DHBC with nanoparticles, and self-assembly of DHBC in solution. At low concentration of DHBC, nanoparticles form dispersed nearly spherical aggregates with DHBC adsorbed onto the surface of aggregates. The size of aggregates formed decreases with an increase in the concentration of DHBC. When the size of the maximum aggregates formed decreases to a certain value, nanoparticles are seen to form hexagonally packed cylinders. With a further increase in the concentration of DHBC, an ordered lamellae structure, dictated by the self-assembly be-

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