Monte Carlo Simulation on the Assembly of Nanorods with Anisotropic

Mar 8, 2011 - res.1-3 For example, gold nanorods show stronger absorption,4 enhanced ..... The solid lines are guides for eyes. Figure 8. Variation of...
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Monte Carlo Simulation on the Assembly of Nanorods with Anisotropic Interactions Jianhua Huang* and Guanfeng Liu Department of Chemistry, Zhejiang Sci-Tech University, Hangzhou 310018, China

bS Supporting Information ABSTRACT: The assembly of nanorods with anisotropic interactions in dilute solution is studied using lattice dynamic Monte Carlo simulation. Each nanorod consists of two different beads with one lattice length apart. The anisotropic interactions are of orientational dependence, and two kinds of nearest-neighbor interactions are considered: (a) side-to-side attraction εxy when two nanorods are parallel with head-head and tail-tail being nearest-neighbors and (b) end-to-end attraction εz when they are in end-to-head alignment. Three structures, wire, sheet, and pillar, are assembled by slowly cooling the system. Nanowires are assembled at large ratio r = εz/εxy, while sheets are assembled at small r. At intermediate values of r, three-dimensional (3D) pillars are formed. The aspect ratio A of the 3D pillar increases with r, which can be expressed as A = 0.238r2.13 ( 0.05. The exponent is different from that of theoretical calculation, which is probably caused by a kinetic effect.

1. INTRODUCTION Due to their anisotropic structure, nanorods present much more interesting optical properties as compared to nanospheres.1-3 For example, gold nanorods show stronger absorption,4 enhanced photoluminescence,5 and higher surface-enhanced Raman scattering activity.6 Much research has been focusing on the controlled assembly of nanorods into functional architectures.3,7,8 On account of their shape anisotropy, nanorods can be assembled via two orientational modes, i.e., end-to-end and sideto-side, which exhibit different optical properties.9 End-to-end 1D assembly of gold nanorods has been achieved by using the streptavidinbiotin linkage,10,11 thiolated-DNA hybridization schemes,12 and dithiol linkers.13,14 Also, the side-to-side assembly of gold nanorods was induced by the addition of sodium citrate in solution.9 Kotov and co-workers15 obtained V-shaped Te assemblies from L-cysteine stabilized CdTe nanoparticles. They thought that V-particles arose from collisions of individual nanorods followed by oriented attachment. Computer simulation can provide critical quantitative insight into nanoscale self-assembly, which is hard to achieve by other means. Simulations suggest that the “conventional” phase behavior can be recovered by stretching spheres into rods or adding attractive van der Waals interactions between particles.16 The Kotov group performed Brownian dynamic simulation to study the formation of V-shaped assemblies in nanorod suspension.17 With the assumption that close encounters between nanorod tips resulted in their merger into V-particles and the incidence of nanoscale V-shape formation was governed by the frequency of r 2011 American Chemical Society

interactions between nanorod tips, they clearly explained why V-particles were formed for short nanorods and absent for long nanorods, which was observed in their experiment.15 Meanwhile, the assembly of polymer-tethered nanorods has been extensively simulated.18-21 Various self-assembled structures, such as hexagonal cylinders and bilayer lamellae, were obtained by varying the tether length, tether location, and solvent property, etc. Endtethered nanorods are similar to the rod-coil block polymers or amphiphiles, so the similar phase behaviors can be predicted to a certain extent.22 However, the geometry of the tethered building block profoundly affects the geometry of the resulting phases.19 It is noticed that the anisotropic interaction of nanorod among themselves was not considered in these studies. Experiments have demonstrated that dipole-dipole interaction is one of important driving forces in nanoparticle selforganization.23-25 Recently, the Glotzer group took into account the anisotropic interaction and carried out MC simulation on the dipole-induced self-assembly of nanocubes at low concentration.16 They also pointed out that the strength of face-to-face attraction between nanocubes can significantly affect the selfassembled structures.16 Besides the dipole-dipole interaction, simulations showed that nanoparticle shape and directional attraction, coming from van der Waals interaction of surface Received: November 10, 2010 Revised: February 12, 2011 Published: March 08, 2011 5385

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Figure 2. Three basic movements: (1) and (2) are two translational movements, and (3) is the rotational movement. Arrows represent the movement directions. The nanorod in movement 2 is shifted left to clearly show the movement.

Figure 1. Twelve pairs of nearest-neighbor nanorods. Beads in different color represent different kinds of beads in nanorod.

atoms and hydrogen bonding between stabilizers, are important to determine the final self-assembled structures.26 Anisotropic surface energy can also drive nanoparticles into nanorods.27 The interfacial interaction strength is dependent on crystal face.17 These results indicated that chemical interaction is also an important anisotropic interaction when one considers the assembly of much smaller particles. In the present work, we study the assembly of nanorods with anisotropic interactions in dilute solution. For simplicity, we consider two pair interactions, named side-to-side attraction εxy and end-to-end attraction εz, analogous to the interactions between crystal faces. The effect of the competition between εxy and εz on their assembly behavior is investigated. Three structures, 1D nanowires, 2D sheet, and 3D pillar, are aggregated by annealing the system from high temperature to low temperature. The transition types and phase diagram are investigated.

2. MODEL AND SIMULATION METHOD We use a minimal model of nanorods on the simple cubic (SC) lattice. Each nanorod consists of two different beads located at one lattice length apart. Every bead occupies one lattice site. Excluded volume interaction is considered as each site cannot be occupied by two beads simultaneously. To describe the anisotropy of nanorods, the interactions adopted in our simulation are of orientational dependence. Only the nearest-neighbor interactions are considered for simplification. For a pair of nearestneighbor nanorods, they can be arranged in eight parallel ways and four perpendicular ways, as sketched in Figure 1. In order to simulate the effect of nanorod-nanorod interactions on the end-to-end and side-to-side growth of nanorod aggregate, two nearest-neighbor interactions are considered for simplification: (a) side-to-side attraction εxy, within a range of -0.05 to -3.8, if two nanorods arrange in mode 1 in Figure 1 , and (b) end-to-end attraction εz, in the range of -0.2 to -12, when they are arranged in mode 6 in Figure 1. Such selective interactions may be achieved via selective adsorption of different molecules on the ends and sides of nanorods.28 Other interactions between nanorods and those involving solvent molecules are set to zero. This allows us to treat the sites occupied by solvent molecules as vacant sites. Thus, the behavior of the system depends only on the value of the nanorod-nanorod interaction. We perform canonical ensemble MC simulations with the conventional Metropolis algorithm.29 In the simulation, we first

randomly choose a nanorod and then randomly choose an elemental movement from a total of 14 elemental movements. They include six translational movements of one lattice forward and backward along x, y, and z directions, and eight rotations of 90° rotation on the lattice around one of its beads. Each elemental movement is of equal probability in the simulation. Figure 2 shows two kinds of translational movements and one kind of rotational movement. The trial move is accepted with a probability P = min(1, exp(-ΔE/kBT)), where ΔE is the system energy shift for the trial move. The Boltzmann constant kB is set as unity in the simulation. One Monte Carlo step (MCS) contains N attempted moves; here N is the number of nanorods in the system. That means every nanorod attempts on average one movement in each MCS. Simulation starts from a sufficiently high temperature to ensure an initial disordered configuration. The cooling process is achieved by slowly decreasing the temperature with a step ΔT = 0.01. By monitoring the system energy, we can ensure that equilibrium is reached at every temperature. We also monitor the structure of the system using a 3D graphic software written by ourselves. The cooling process is completed when both the energy and structure do not change with temperature any more. The heating process is likewise simulated after the completion of the cooling process. At each temperature step, a typical simulation run of 5  106 MCS is carried out. A longer time simulation of 1  108 MCS is used near the critical temperature where the phase transition occurs. Long simulation time ensures that the configuration at the end of each temperature and the final structure at low temperature are stable. On the SC lattice, nanorods will grow in the xy direction (sideto-side) with four nearest-neighbors and in the z direction (endto-end) with two nearest-neighbors. However, the growth rate will be dependent on the interactions εxy and εz, so the final structure at low temperature will be dependent on the interactions. The effect of the ratio r = εz/εxy on the assembled structures of nanorods is investigated. The ratio r may depend on the properties of the solution, such as pH, ion concentration, and surfactant. Different structures are observed in the simulation.

3. RESULTS AND DISCUSSION 3.1. Assembly of Nanorods. The assembly of nanorods is investigated by annealing the system from high temperature to low temperature. Nanorods randomly disperse in the system at high temperature, while they assemble into aggregates at low temperature. The structures at low temperature are stable and depend on the interaction ratio r. Figure 3 shows five typical final structures assembled by nanorods at different r with εz = -1.8. Here the simulation is carried out in a 60  60  60 simulation 5386

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The Journal of Physical Chemistry C box containing 500 nanorods. That is, the nanorod concentration Cn = 0.0023 (the volume fraction of nanorods in the system). For a very weak side-to-side attraction, 1D nanowires are assembled through end-to-end growth. Figure 3a shows the nanowires formed at r = 36. Theoretically one single wire has the lowest energy. However, it is hard to obtain a single wire in simulation. One reason is the low entropy of a single wire, so it is difficult to obtain kinetically. Another reason is that our simulation is performed in a finite system where a single wire will be longer than the system size. With the increase of the side-to-side attraction, nanorods assemble into a 3D pillar structure with height Nz (defined as the layer number). The height decreases with the increase of |εxy| as shown in Figures 3b, c, and d. For instance, Nz decreases from 14 at r = 3 (Figure 3b) to 3 at r = 1 (Figure 3d). With a further increase in the side-to-side attraction, nanorods eventually assemble into a 2D sheet as shown in Figure 3e at r = 0.47. We have run more than five independent initial starting configurations with long simulation time and small temperature step. We find the final structures are roughly the same, indicating that the results are equilibrium. In other words, it is not a result of the system being kinetically entrapped in a metastable state.28 We have also simulated the assembly of nanorods in different system sizes and with different concentrations. We find the structures shown in Figure 3 are always observed.

Figure 3. Final structures assembled at various r = 36 (a), 3 (b), 2 (c), 1 (d), and 0.47 (e). The system size L = 60, and the nanorod concentration Cn = 0.0023.

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3.2. Transition and Phase Diagram. After we obtain the final structure by annealing, we then slowly heat the system to high temperature. Figure 4 shows the total system energy E at different temperatures for both cooling and heating processes at five values of r. For a very weak side-to-side attraction εxy = -0.05, i.e., r = 36, the plot shows continuous transition with a negligible hysteresis (Figure 4a), suggesting an “equilibrium polymerization” type of transition.16 At intermediate values of r, for example, r = 3, 2, and 1, the transition from disorder to 3D pillar structure takes place at a specific temperature accompanied with a sharp decrease of the energy E; see Figure 4b. An obvious hysteresis in E is observed, suggesting that the transition from randomly distributed nanorods to 3D pillar structure is a first-order phase transition. At small value of r, for instance, at r = 0.47, the hysteresis in E becomes small (Figure 4c). The effect of nanorod concentration Cn on the transition temperature has been studied at r = 1 where 3D pillar is formed. Here, Cn increases from 0.0005 to 0.0033, while the system size is kept at L = 60 in all three directions. They all possess the similar E-T curve as shown in Figure 4b (Supporting Information, Figure S1). The transition temperature Tc is identified by locating the maximum energy gradient in the E-T curve in the cooling process. It is expected that the transition temperature increases with the concentration of particles.16 Our simulation on anisotropic nanorods is in agreement with this conclusion as shown in Table 1. Clearly, the concentration dependence of transition temperature is a character of self-assembled systems. Therefore, we study the effect of system size on the aggregation behavior of nanorods at fixed nanorod concentration. Here we show the results for nanorod concentration Cn = 0.0023, while the system size is varied from 30 to 70. Three typical values of r (=36, 1, and 0.47) with εz = -1.8 are adopted, corresponding to three different structures. For the system with a large value of r = 36, nanorods aggregate into 1D nanowires through a continuous mechanism similarly to Figure 4a in various systems. All curves of the energy density E/V versus temperature T collapse (Supporting Information, Figure S2), indicating that the transition temperature Tc is roughly independent of the system size. For the system with a small value of r = 0.47, they aggregate into a single 2D sheet showing a small hysteresis in the energy density E/V-T curve (Supporting Information, Figure S3). Tc increases with the system size L and saturates at large L. Using the same method, the transition temperature Tc,h in the heating

Table 1. Transition Temperature Tc at Various Nanorod Concentrations Cn Cn Tc

0.0005 0.65

0.0010 0.72

0.0014 0.76

0.0023 0.82

0.0033 0.89

Figure 4. Plots of the system energy E vs temperature T for cooling and heating processes at various r = 36 (a), 3, 2, and 1 (b), and 0.47 (c). L = 60, Cn = 0.0023. 5387

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Figure 5. Dependence of the transition temperature Tc and ΔTc on the system size L at r = 0.47 (a) and 1 (b), respectively. Cn = 0.0023.

Figure 7. Phase diagram of nanorods at different -εxy/T and -εz/T. The solid lines are guides for eyes.

Figure 6. Dependence of the transition temperature Tc on the interactions εxy (a) and εz (b).

process is obtained, and the hystersis size is defined as ΔTc = Tc,h - Tc. It is found that ΔTc is nearly independent of L (Figure 5a). For an intermediate value of r = 1, a single 3D pillar is aggregated. The transition tempareture Tc in the cooling process is roughly independent of the system size, and ΔTc saturates at large size L g 60; see Figure 5b. In short, our results show that the system size L = 60 is quite large to study the aggregation behavior of nanorods. For computational expediency, we focus on the simulations performed in a 60  60  60 system with Cn = 0.0023. The dependence of Tc to form pillar structure on the interactions εxy and εz is plotted in Figure 6. It is obvious that Tc is dependent on both the interactions εxy and εz, and it increases with an increase in |εxy| and |εz|. One can see that the difference of transition temperature Tc for different εxy (Figure 6a) or εz (Figure 6b) tends to be small with the increase of |εxy| or |εz|. Therefore the phase transition temperature is mostly dominated by εxy at |εxy| . |εz|, while it is dominated by εz at |εxy| , |εz|. Figure 7 presents the phase diagram of an interacting nanorod system. It is obtained by annealing the system and the subsequent heating process at various εxy and εz. Based on the equilibrated configuration, four phases are observed: a dispersed phase at high temperatures and three aggregated phases at low temperatures. The transition from 1D nanowires to 3D pillar locates at about εz/εxy = 22 and that from 2D sheet to 3D pillar locates at about εz/εxy = 0.55. We have examined how the system evolves from “pillar” to “sheet” and from “pillar” to “wire”. For the transition from pillar to sheet, the initial state is a single 3D pillar which is obtained by

Figure 8. Variation of the system energy E with the end-to-end attraction εz while keeping εxy = -2.5 and T = 1.

slowly decreasing temperature from T = 2 to T = 1 at εz = -6 and εxy = -2.5. We gradually increase εz from -6 to -0.01 and afterward decrease it to -6 again while keeping εxy = -2.5 and T = 1. In this circle, the ratio r is changed between 2.4 and 0.004. The results are presented in Figure 8. There is no transition when we ramp up εz; however, a phase transition is observed when we ramp down εz. The transition from sheet to pillar occurs at about r = 0.75. For the transition from pillar to wire, the initial state is a single 3D pillar which is obtained by slowly decreasing temperature from T = 2 to T = 1 at εz = -8.5 and εxy = -1.5. We gradually increase εxy from -1.5 to -0.05 and afterward decrease it to -1.5 again while keeping εz = -8.5 and T = 1. In this case, the ratio r is changed between 5.7 and 170. The results are presented in Figure 9. A transition from wire to pillar occurs at about r = 15 when we ramp down εxy. From this point of view, we think that there is a fundamental difference between sheet and pillar as well as that between wire and pillar. We would like to point out that the phase diagram presented in Figure 7 is obtained by annealing process from a random state at high temperature. From the phase diagram we will know the final 5388

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Figure 9. Variation of the system energy E with the side-to-side attraction εxy while keeping εz = -8.5 and T = 1.

structure of the system annealing from high temperature at given εxy and εz. Although we can observe transitions from “solid” phase to “solid” phase as presented in Figures 8 and 9, the transition points are not the same as that obtained by annealing. 3.3. Scaling of the Aspect Ratio of Pillar. In the simulation, the 3D pillar structure at low temperature can be represented by the aspect ratio A = Nz/(N)1/2, where Nz is the layer number of nanorods and Nxy the number of nanorods in one layer. Assuming each layer is a square with side length (N)1/2, the equilibrium surface energy of the pillar is pffiffiffiffi pffiffiffiffiffi ð1Þ σ ¼ - ð4εxy 3 N 3 Nz þ 2εz 3 N=Nz Þ Here the total number of nanorods N = NxyNz. To minimize the surface energy, we obtain Aµr

ð2Þ

by calculating ∂σ/∂Nz = 0. The equilibrium condition could be obtained by minimizing the free energy F = E - TS with E the system energy and S the entropy. However, in the present work, the pillar is stable at very low temperature, so the effect of entropy can thus be ignored. Figure 10 presents the simulation results of the aspect ratio A at different values of r. It is found that the dependence of A on the ratio r can be roughly expressed as A ¼ 0:238r 2:13 ( 0:05

ð3Þ

The exponent in eq 3 is different from that of the theoretical estimation based on the equilibrium surface energy. In the assembly process, there exists a competition between the lateral and vertical growth of aggregate, which is dominated by the attractive interaction εxy and εz. Assuming one nanorod can be vertically or laterally assembled to an aggregate, the system energy will gain εz for a vertical growth. However, if it is assembled laterally, the system will gain εxy, 2εxy, or even 3εxy depending on the growing conditions; the latter two conditions are presented in Figure 11. It will then affect the growth rate in the lateral direction. Thus the growth in the lateral direction will be faster than that in vertical direction at r = 1. We find that(N)1/2 ≈ 12 is really larger than Nz = 3 for a system containing 500 nanorods at r = 1, as shown in Figure 3d. Instead, we have (N)1/2 ≈ Nz ≈ 8 at

Figure 10. Log-log plot of the aspect ratio A vs the interaction ratio r for systems with different interactions εxy and εz and different numbers of nanorods N. The solid line is a linear fitting with A = 0.238r2.13(0.05.

Figure 11. Sketch of two different growth sites. “A” has two nearest neighbors, and the decrease in E is 2  |εxy|; “B” has three nearest neighbors and the decrease in E is 3  |εxy|.

r = 2. The faster growth in the lateral direction results in a stronger dependence of A on r. However, the equilibrium surface energy (eq 1) does not consider the multiples of energy εxy. This may explain the difference between theoretical and simulation estimates. In other words, the difference is caused by a kinetic effect. 3.4. Equilibrium vs Metastability. We have studied the growth of 3D pillar structure. At T > Tc, nanorods randomly disperse in the system in the cooling process. Just below Tc, there is a steep decrease in energy E accompanied by the assembly of nanorods. As an example, we show the result of the system with r = 2 (εxy = -0.9, εz = -1.8). In the simulation, the final configuration at Tc = 0.5 is used as the initial configuration at temperature T = 0.49, and the simulation time is set as t = 0 at this moment. Then the system evolution with time is monitored. Figure 12 presents the evolution of energy E and four snapshots captured at different periods. Several small aggregates appear after a period of time. They dissolve and assemble randomly. However, one aggregate grows suddenly, and the energy drops sharply. The growth of the aggregate is very fast, and a large aggregate forms soon. Afterward it evolves slowly to reach an equilibrium state, and a relatively ordered structure is formed. Moreover, we can observe many sites with energy 2εxy in these snapshots, which favor the aggregate growth in the lateral direction. The same phenomenon is observed for r = 3 and 1. They all show that the assembly takes place suddenly with a sharp drop in energy. Metastable phases are observed if we decrease the temperature with a large step. For example, different phenomenon is observed when the temperature is suddenly decreased to T = 0.35 from T = 0.8 for the system with r = 2 (εxy = -0.9, εz = -1.8); i.e., a step ΔT = 0.45 is used. Nanorods immediately assemble into small 5389

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may be due to the kinetic effect resulting from more attraction pairs in the lateral direction in the assembly process.

’ ASSOCIATED CONTENT

bS Supporting Information. Figures showing the effects of nanorod concentration and system size on the phase transition. This material is available free of charge via the Internet at http:// pubs.acs.org. ’ AUTHOR INFORMATION Corresponding Author

*E-mail [email protected].

Figure 12. Evolution of energy E for a system with r = 2 at temperature T = 0.49. The initial configuration at t = 0 is the final configuration at T = 0.50.

’ ACKNOWLEDGMENT This research was financially supported by the National Natural Science Foundation of China (Nos. 20771092 and 20874088). ’ REFERENCES

Figure 13. Variation of the system energy E (left Y axis) and the number of aggregates Nagg (right Y axis) with time t. The inset is the final configuration after running for 25  106 MCS.

aggregates at T = 0.35. Figure 13 shows the time dependence of the system energy E and number of aggregates Nagg. It clearly shows that there is a rapid decrease in Nagg accompanied by a steep decrease in energy E at early stage. Afterward, large aggregates grow at the expense of smaller ones. It proceeds via a mechanism similar to Ostwald ripening, which is extensively reported in experiments27,30 and simulations.31 The size difference between aggregates becomes smaller and smaller with the time, so Ostwald ripening slows down. Thus multiple metastable aggregates are formed.

4. CONCLUSION In summary, the effect of anisotropic interaction on the assembly behavior of nanorods has been investigated. 1D nanowires and 2D sheet are assembled at a very small and large value of r = εz/εxy, respectively. At intermediate values of r, a first-order phase transition from disorder to 3D pillar takes place. The transition temperature Tc depends on both interactions εxy and εz. The aspect ratio A of 3D pillar structure is in power-law relation with the interaction ratio r as A = 0.238r2.13(0.05. The exponent is different from that of theoretical prediction, which

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