Letter pubs.acs.org/Langmuir
Self Assembly of Janus Ellipsoids Ya Liu,*,† Wei Li,† Toni Perez,† James D. Gunton,† and Genevieve Brett‡ †
Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, United States Department of Physics, Skidmore College, Saratoga Springs, New York 12866, United States
‡
ABSTRACT: We propose a primitive model of Janus ellipsoids that represents particles with an ellipsoidal core and two semisurfaces coded with dissimilar properties, for example, hydrophobicity and hydrophilicity, respectively. We investigate the effects of the aspect ratio on the self-assembly morphology and aggregation processes using Monte Carlo simulations. We also discuss certain differences between our results and those of earlier results for Janus spheres. In particular, we find that the size and structure of the aggregate can be controlled by the aspect ratio.
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INTRODUCTION Colloidal particles with anisotropic properties interact through an energy that depends not only on their spatial separation, but also on their relative orientations. This is a relatively new field that has been receiving considerable attention in recent literature.1−7 Various site-specific techniques such as template-assisted fabrication and physical vapor deposition have been developed to synthesize patchy colloidal particles of different shapes, patterns, and functionalities.8−10 The selfassembly of those building blocks into a desired mesoscopic structure and function are considered as a bottom-up strategy to obtain new bulk materials that have potential applications in broad fields including drug delivery, photonic crystals, biomaterials, and electronics. The effects of anisotropy have been classified by Glotzer and Solomon using the concept of an anisotropy dimension including patchiness, aspect ratio, and faceting.1,11 Patchiness on spheres with different number, size, and arrangement has been successfully fabricated in colloid experiments and reveal interesting properties and crystal structures.3−5,12 One particular example is the Janus sphere with two dissimilar semisurfaces, which has been extensively studied by experiments and theories.3,13−15 Examples of dissimilar, coded surfaces include hydrophobic/hydrophilic, charged/uncharged, and metallic/polymer surfaces. A variety of stable structures and unusual phase behaviors have been found under different chemical conditions of the solution by both experiments and computer simulations. Without losing the generality of the dissimilarity of two surfaces, a primitive twopatch Kern-Frenkel model has been used to model Janus spheres.13,16 In Monte Carlo simulations, this simplified model reproduces the main experimental features including selfassembly morphology and sheds light on potential applications in engineering and theoretical studies of reentrant phase diagrams.15,17 © 2011 American Chemical Society
As suggested by Glotzer and Solomon, it is natural to extend the study to explore the role of aspect ratio, in which patches are arranged on an anisotropic core such as spheroids. The anisotropy dimension is related to the aspect ratio, and recent studies by analysis of the energy landscape reveal many interesting structures such as shells, spirals, and helices for different sizes of clusters where ellipsoids interact via site−site potential.18−21 However, how the aspect ratio affects the selfassembly morphology is still not clear. In experiments, ellipsoidal colloids can be engineered with high monodispersity using techniques such as deforming the spherical silica by ion fluence,22,23 which makes the patched ellipsoidal surface possible if one can combine this with the template-assisted fabrication technique. Recently, unpublished work shows that Janus football-like ellipsoid has been fabricated.24 Suspensions of Janus ellipsoidal particles provide a good candidate to study the effect of aspect ratio on self-assembly and the new feature of colloidal phase transformation. In this report, we first propose a theoretical model of Janus ellipsoids with hard-core repulsion and quasi-square-well attraction. The properties of the model are also discussed. In the second part, we use Monte Carlo simulations to study the effect of aspect ratio of ellipsoids on the self-assembly morphology and dynamical properties. Although the Monte Carlo method does not provide an accurate dynamics of the complex system, it still reveals some useful information, and has been used for Janus spheres to make a comparison between the simulation and experimental observations.9 In the last section, we present a brief conclusion. Received: August 17, 2011 Revised: December 14, 2011 Published: December 15, 2011 3
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Letter
MODEL
In our simulation study, a primitive model is presented for Janus spheroidal particles, with the lengths of the principal axes denoted by a ≠ b = c. Depending on the aspect ratio, defined as ε = a/b, the ellipsoid is characterized as oblate if ε < 1 and prolate if ε > 1. Consider an ellipsoid centered at r0 = (x0, y0, z0) whose axial orientations are given by uT = (u1, u2, u3), vT = (v1, v2, v3), wT = (w1, w2, w3). The equation for this ellipsoid has the explicit form
[u · (r − r0)]2 a2
+
Figure 1. Plot of two interacting ellipsoids labeled as i and j, in which attractive and hardcore repulsive surfaces are coded by red and blue, respectively. rij is center-to-center displacement pointing from j to i. ui and uj is the patchy orientation. θi is the patch angle of ith ellipsoid.
[v· (r − r0)]2 + [w· (r − r0)]2 b2
f (rij, u i , ⎧1 =⎨ ⎩0
(1)
=1
A12 A13 A14
u12
+
v12
+
if
⎪
In terms of a matrix representation, eq 1 can be rewritten as XTAX = 0, where XT = (x, y, z, 1) and A is a 4 × 4 symmetric matrix. The ten independent elements of A are listed as follows: 1)
A11 =
uj)
⎪
u i ·riĵ ≤ − cos δ , uj· riĵ ≥ cos δ
otherwise
(2)
as shown in Figure 1. Here, rij = ri − rj and r̂ij = rij/rij with rij the center-to-center distance between ellipsoids: rij = |rij|. δ = π/2 corresponds to Janus particle. An attractive interaction exists if two red patches face each other. The standard square-well potential has been applied for the study of Janus spheres;13,15 however, the determination of the accurate spatial relation between ellipsoids is computationally time-consuming. We thus introduce a quasi-square-well potential defined as
w12
a2 b2 c2 uu vv ww = 1 22 + 122 + 12 2 a b c u1u3 v1v3 w1w3 = 2 + 2 + 2 a b c = − x0 A11 − y0 A12 − z 0 A13
⎧ if particles overlap ⎪∞ U=⎨ ⎪ ⎩− U0H(σij + 0.5σ − rij) otherwise
(3)
where U0 is the well depth, H(x) denotes the Heaviside function, σ represents the length of the longer axis (max(2a,2b)).
2)
u2 v2 w2 A22 = 22 + 22 + 22 a b c u 2u3 v2v3 ww A23 = 2 + 2 + 22 3 a b c A24 = − x0 A21 − y0 A22 − z 0 A23
σij = 2b −1/2 ⎡ ⎛ (r · u + r ̂ · u )2 (riĵ · u i − riĵ · uj)2 ⎞⎤ ij j ⎥ ⎢1 − χ ⎜ iĵ i ⎟ + ⎢ 2 ⎜⎝ 1 + χu i · uj 1 − χu i · uj ⎟⎠⎥⎦ ⎣
(4)
where χ = (ε − 1)/(ε + 1), b, r̂ij, ui, and uj are defined in eqs 1 and 2.25 σij is introduced as an approximation to characterize the spatial relation, such that there is no overlapping interaction if rij ≥ σij, provided that the ellipsoids are represented by a Gaussian function: exp(−r · γ−1 · r) with range tensor γ = a2uu + b2(vv + ww).25 This approximation has been widely used to study anisotropic particles such as liquid crystals and granular material.26 We note that, in our study, this approximation is only applied to the potential but not to the geometric overlapping which will be determined using a precise method.27 Therefore, in our case H(σij + 0.5σ − rij) represents a quasisquare-well potential with width 0.5σ. Specific cases under the condition ui = uj are illustrated in Figure 2 for the aspect ratio ε = 0.1, 0.5, and 0.9 from left to right. Two particles will interact if their red shells touch each other. The hard-core repulsion is provided by the following geometric relation. Given two ellipsoids ( : XTAX = 0 and ) : XTBX = 0, one introduces the characteristic polynomial F(λ) = det (A − λB). A and B are normalized so that the interiors of ( and ) satisfy XTAX < 0 and XTBX < 0. The roots of the characteristic equation F(λ) = 0 have two positive real values and the rest characterizes the geometric relation between ellipsoids27,28 such 2
3)
u2 v2 w2 A33 = 32 + 32 + 23 a b c A34 = − x0 A13 − y0 A23 − z 0 A33 A 44 = − 1 − x0 A14 − y0 A24 − z 0 A34 Here, A is normalized so that a point in the interior of the ellipsoid X0 satisfies X0TA X0 < 0, and thus, det(A) < 0. This normalization will be used to determine the spatial relation between ellipsoids in the late section. Analogous to the patchy spherical model introduced by Kern and Frenkel,16 we choose the ellipsoids to interact through a pair-potential that depends on their separation and orientation: Uij = Uf(rij; ui; uj). As illustrated for two oblate ellipsoids in Figure 1, attractive patches are coded by red and the orientation is chosen to coincide with the principal axis u in the body-fixed frame of reference. The orientational interaction is defined as 4
2
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Here, B2hs = 2/3πσ3 stands for the second viral coefficient for hard spheres with diameter equal to σ. The numerical error is less than 1%. The solid curve represents the theoretical prediction for a Janus sphere: B2/B2hs = 1 − 1/4(δ3 − 1)(eβU0 − 1) with the interaction range δ = 1.5σ in the study.16 As ε increases to 1, B2 approaches the value for the Janus sphere from above, which indicates that the effective interactions increase as the aspect ratio increases.
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Figure 2. Plot of quasi-square-well potential under the condition ui = uj for ε = 0.1, 0.5, and 0.9 from left to right. The red and blue surfaces represent the attraction range and hardcore repulsion associated with each ellipsoid, respectively. View angle of each figure is tuned for better visualization.
SIMULATION RESULTS Standard Monte Carlo (MC) simulation in the NVT ensemble has been applied to study the self-assembly pathway of these interacting Janus particles. We investigate a system in a 30 × 30 × 30 box with periodic boundary conditions. The particle aspect ratios range from 0.1 to 0.9 with number density ρ = 0.037; the attractive energy is set as βU0 = 3 with β = 1kBT, where kB is the Boltzmann constant. The system is initialized as a randomly distributed noninteracting gas of monomers. A random translation followed by a random rotation is carried out for each monomer. In particular, the rotation is performed using the method of quaternion parameters. More than ten independent runs for each aspect ratio have been carried out up to 5 × 106 Monte Carlo steps (MCS) and an ensemble average is taken by averaging over all runs. We monitor the evolution of −E/U0 in log−log and linear−linear plots (shown in Figure 4),
that (1) ( and ) are separate if and only if F(λ) = 0 has two distinct negative roots; (2) ( and ) touch each other externally if and only if F(λ) = 0 has a negative double root. (3) Otherwise, ( and ) overlap. Sturm sequence methods are applied to numerically decide if two roots are distinguishable. The primitive model we propose recovers the Kern-Frenkel Janus sphere model16 when ε = 1, since under this condition, σij = σ, and consequently, the attraction is simplified to H(1.5σ − rij). The introduced quasisquare-well potential has the advantage that there is no ambiguity when defining the connectivity of aggregates during the self-assembly process, and could be easily generalized to more realistic models. Our model is different from the Kern-Frenkel model when ε ≠ 1 since then the magnitude U is a function of both the separation and orientation due to the anisotropic ellipsoidal core. The aspect ratio ε affects the shape of the interacting potentials. To characterize the effective interaction between particles, we calculate the second viral coefficient B2:
B2 =
1 1 2V (4π)2
⎡
∫ ⎢⎣1 − e−βU
⎤
12⎥
⎦
dr1 dr2 du1 du2
(5)
where V is the system volume and U12 the pair interaction. The results of Monte Carlo integrations of B2/B2hs as a function of temperature for different ε are shown in Figure 3, where
Figure 4. Log−log plot of −E/U0 vs MCS. The curves from top to bottom correspond to the aspect ratio ε = 0.1 (green), 0.5 (red), and 0.9 (blue). Inset shows the enlarged section of the same curves from 4 million to 5 million MCS, in a linear−linear plot. A linear fit to each curve in the inset yields values for the slopes of 4.0 × 10−7, 6.4 × 10−8, and 2.4 × 10−8 per MCS, respectively.
the negative average potential scaled by the attraction strength, which characterizes the number of interacting neighbors of each particle. Since the initial system has no interaction, −E/U0 starts from 0. It grows quickly and the dynamics slows down while approaching equilibrium. We have performed linear fits to the data in Figure 4 in the region between 4 and 5 × 106 MCS and obtained slopes of 6.4 × 10−8 and 2.4 × 10−8 per MCS, respectively, for 0.9 and 0.5. It would appear from these results that the system for ε = 0.9 and 0.5 is close to equilibrium. For the case ε = 0.1, the slope is 4.0 × 10−7 per MCS. Although the energy is still slowly changing, it is approaching equilibrium. We would also note that, for this small aspect ratio, the ellipsoids take a much longer time to
Figure 3. Plot of B2/B2hs vs kBT/U0 for the aspect ratio ε = 0.1 (green down triangle), 0.5 (red triangle), and 0.9 (blue circle) from top to bottom. The solid curve is the theoretical prediction for a Janus sphere, i.e., ε = 1.
without specification, ε = 0.1, 0.5, and 0.9 are denoted by green down triangles, red triangles, and blue circles, respectively. 5
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the third one. As the oligomer grows, it starts forming a micelle (single layer) and a vesicle (double layer) as shown for clusters 13 and 39 (ε = 0.9) and 8 and 16 (ε = 0.5). For ε = 0.1, the relevant clusters (8 and 17) display two and three layers, respectively. Those structures have been found in the case of Janus spheres and have an effect on breaking the thermal correlations between particles.13 They thus affect the system phase behavior. Eventually, for ε = 0.9, clusters with different structures are formed, including two small oligomers joining together (46), as well as more complicated chains (72). For ε = 0.5, it is possible to form a triple-layer compact cluster (51) and a dumbbell cluster (62). For ε = 0.1, due to the relation between the attractive range and aspect ratio, multiple-layer structures have been observed in the simulations (30, 52). As is well-known, Monte Carlo simulation does not provide an accurate description of the kinetics of the system; however, it still reveals some useful features. We investigate the time evolution of number of clusters as illustrated in Figure 6. The number of clusters for larger ε drops faster until reaching about 106 MCS. The curves approach plateaus for ε = 0.5 and 0.9, indicating that the systems are close to equilibrium. For ε = 0.1, the number of cluster is slowly varying and the system is approaching equilibrium. These results are consistent with the evolution of the energy shown in Figure 4. At the early stage, the dynamics is dominated by monomer motion and the system at ε = 0.9 has a relatively stronger interaction which leads to faster cluster formation. Then, the dynamics is governed by cluster−cluster interaction so that for larger aspect ratio the number of clusters decreases more slowly than for smaller aspect ratio. Note that the cluster defined by energy interaction might in principle be inconsistent with the experimental observations, which could implicitly use the distance definition without considering orientations.9 The inset in Figure 6 shows that the difference between the energy definition (upper blue) and distance definition (lower red) for ε = 0.9 is smaller than the statistical error, which indicates that the energy-defined cluster is a good approximation as compared with experiments. We have calculated the distribution of cluster size, which illustrates the effect of aspect ratio on the system approaching equilibrium, as shown in Figure 7. For larger ε, the distribution has a broader range with a lower peak, which is consistent with the morphology shown in Figure 5. Due to the complex structure formation such as a chain, ε = 0.9 has extended configurations with larger size, in which the simulations show the largest cluster (a size 137 as shown in the inset of Figure 7). For ε = 0.1, the distribution is narrow and has a higher peak, which indicates that the clusters are more uniform. The largest cluster with size 55 has a similar shape as the oligomer (52) shown in Figure 5; the structure of this cluster is like a blob with several layers. The structure is more stable in terms of its energy and prevents further aggregation to form more complex forms. The case of an aspect ratio 0.5 (72) is intermediate between these two extremes of 0.1 and 0.9, and the resulting cluster reveals two vesicles forming together through a bridgelike structure. Next, in order to investigate in detail how ellipsoids organize in the cluster, we consider the correlation between patch orientations: ui · uj of two bonded ellipsoids (for which there exists a patchy attraction). The distribution P(ui · uj) is illustrated in Figure 8. For ε = 0.9, P(ui · uj) develops two peaks; typical configurations corresponding to this correlation are shown in
equilibrate due to the freedom of rotation. It would be desirable to perform even longer simulation runs (on the order of 107 or 108 MCS), based on Sciortino et al.’s study of Janus spheres (which are computationally less expensive to simulate than ellipsoids).13 However, this would require several months of computer time to carry out the simulation for each point we studied. It is important to note that the main focus of our study is the dependence of the cluster morphology on the aspect ratio. Since the morphology of the clusters is only slowly changing in this late time regime in all the cases we have studied, we believe that our simulations are of sufficiently long term to address this point. Also, for all three cases we believe that much of the variation in energy shown in the insert of Figure 4 is due to the thermal fluctuation of monomers in individual clusters, including orientational reordering. This different growth tendency for the different aspect ratios is due to the distinct aggregation mechanism that dominates at different times. As we will show in a later section, the aggregation at the early stage is dominated by monomer diffusion and interactions with small, formed oligomers. Depending on the aspect ratio, the system is composed of monomers, small oligomers, micelles, and vesicles. Afterward, the aggregation dynamics is mainly the diffusion and collision of those small clusters that have much smaller diffusion constants than monomers. The value of ε affects the dynamics, as shown in Figure 4, such that the aggregation process is relatively faster but reaches a less stable structure when ε is larger. To illustrate the difference in aggregation morphology, we take snapshots of the cluster growth as shown in Figure 5. The
Figure 5. Plot of cluster formation for ε = 0.9, 0.5, and 0.1 from left to right. The number next to each panel is the cluster size. The size and view angle of each figure have been tuned for better visualization.
cluster is defined such that two monomers are connected if they interact; there is no ambiguity with respect to this interaction in our model. From left to right in Figure 5, we show typical structures for ε = 0.1, 0.5, and 0.9 that are collected from all simulations. Small oligomers such as trimer, tetramer, pentamer, hexamer, and heptamer are similar and monomers form usual polygons, except that for ε = 0.1 there is more space enclosed by monomers and the heptamer has a different structure: a rough double-layer. When the aspect ratio is less than 0.5, a different feature shows up. Instead of building up polygons, two ellipsoids with the same orientation pack into two layers. This is due to the quasi-square-well attraction in our model of the attraction range 0.5σ. When ε < 0.5, two ellipsoids have the possibility to interact even when they are separated by 6
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Figure 6. Log−log plot of number of clusters vs MCS for ε = 0.9 (blue circle), 0.5 (red up triangle), and 0.1 (green down triangle). Error bars come from the statistical variance of independent runs. The inset shows the evolution of the number of clusters defined through energy (blue) and distance (red) for ε = 0.9 in log−linear plot.
Figure 8. Plot of distribution of ui · uj between two bonded ellipsoids for ε = 0.1 (green), 0.5 (red), and 0.9 (blue). Typical configurations corresponding to each peak are shown.
Figure 7. Plot of distribution of cluster size for ε = 0.1 (green down triangle), 0.5 (red triangle), and 0.9 (blue circle). Insets from top to bottom are configurations of the largest cluster found in the simulations for ε = 0.1, 0.5, and 0.9, respectively. The number associated with each configuration is the cluster size.
clusters at 5 × 106 MCS. For ε = 0.5, the configuration shows a somewhat different feature (middle panels of Figure 9). The system forms micelles and vesicles at 106 MCS and finally includes monomers, large numbers of micelles and vesicles, and their combined aggregates. For ε = 0.1, the configuration shows that clusters forms slowly and only includes small oligomers and a large number of micelles and vesicles at 5 × 106.
the figure. These two peaks correspond to two ellipsoids facing toward (ui · uj ∼ −1) and opposite (ui · uj ∼ 1) each other; similar behavior has been found for Janus spheres.13 As ε decreases, one peak (ui · uj ∼ −1) vanishes, since ellipsoids selfassemble into a micelle-like structure (for example, panels 8 and 16 for ε = 0.5 and panels 8 and 17 for ε = 0.1 in Figure 5). Due to the geometric constraint, there is empty space inside the ellipsoids forming the cluster. This property could be useful for the encapsulation of different types of particles (e.g., as in drug delivery). The other peak shifts toward zero; thus, the curvature is increasing so that the structure is more compact and energetically favorable. This structure prevents further aggregation, as the thermal fluctuation is suppressed, which prevents long-range correlation. More details could be explored from the snapshots of configurations. As illustrated in Figure 9, the system for ε = 0.9 forms large clusters at 106 MCS and is composed of a large number of monomers, micelles, vesicles, and large complex
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CONCLUSION In the article, we propose a primitive model to study the selfassembly of Janus ellipsoids. The interactions between ellipsoids include a hard-core repulsion and a quasi-squarewell attraction, where the latter exists when the patchy surfaces of two interacting ellipsoids orient in designated directions. The anisotropy in our model comes from two aspects: the patch interaction and the anisotropic core, which are controlled by a patchy angle δ and an aspect ratio ε, respectively. We particularly focus on the Janus ellipsoids in which δ = π/2, number density ρ = 0.037, and ε ranges from 0.1 to 0.9, and address the effects of aspect ratio on the self-assembly morphology. We note that we have also briefly investigated 7
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Figure 9. Snapshots of the system for ε = 0.9, 0.5, and 0.1 from left to right. The upper and lower panels correspond to t = 106 and 5 × 106 MCS.
of the aggregates can be controlled by the aspect ratio, which should be an interesting result from a design viewpoint.
the self-assembly of ellipsoids when the aspect ratio is greater than one (prolate ellipsoids) and found that the morphologies for the cases we present here are more interesting. An important case for future study is that of the behavior of these systems at high density, where one, for example, could encounter gelation. Our model could be easily extended to consider more realistic systems. For example, the typical interaction between colloids is in the range 0.05−0.2σ, which, however, is computationally expensive to simulate. Janus ellipsoids with the interacting range comparable to 0.5σ can be realized by nanoparticles using a technique such as induced phase separation.29 In our study, the phase diagram is not yet known, but our initial choice of parameters (such as density and temperature) puts us in the gas regime, with stable clusters such as micelles, and vesicles, similar to the case for a model of spherical Janus particles.13 In that case, the authors found an anomalous behavior in which the density of the gas phase (coexisting with the liquid phase) increased with decreasing temperature. Our results show that, for larger aspect ratio, the self-assembly process is relatively faster and the morphology is more complicated, including chain structures as well as micelles and vesicles, which is consistent with our calculations of B2, since the larger magnitude of B2 corresponds to the larger aspect ratio and indicates a relatively stronger interaction. The structures for smaller aspect ratio are more uniform and multiple-layer vesicles dominate. Our simple model raises many potential avenues for investigation; for example, the possible extension of B2 scaling in the calculation of the phase diagram. This B2 scaling has been successfully applied, for example, for patchy spheres.30,31 In addition, it is important to see how our results depend on the range of the interaction, e.g., 0.2σ, as is the case for colloidal interactions, as well as the shape of potential. Recent studies show that the model we use might bring in a driving force for a large orientational ordering.19,32 We also find for small aspect ratio that ellipsoids tend to form vesicles, which suggests a potential application for particle encapsulation. Perhaps the most important result of the study for materials engineering is the fact that the size and structure
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AUTHOR INFORMATION
Corresponding Author
*
[email protected].
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ACKNOWLEDGMENTS This work was supported by grants from the Mathers Foundation and the National Science Foundation (Grant DMR-0702890). One of us (G.B.) was supported by the NSF REU Site Grant in Physics at Lehigh University. Simulation work was supported in part by the National Science Foundation through TeraGrid resources provided by Pittsburgh Supercomputing Center. We thank Wenping Wang at University of Hong Kong for providing the ellipsoid code.
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REFERENCES
(1) Glotzer, S. C.; Solomon, M. J. Nat. Mater. 2007, 6, 557−562. (2) Kegel, W. K.; Lekkerkerker, H. N. W. Nat. Mater. 2011, 10, 5−6. (3) Chen, Q.; Whitmer, J. K.; Jiang, S.; Bae, S. C.; Luijten, E.; Granick, S. Science 2011, 331, 199−202. (4) Pawar, A. B.; Kretzschmar, I. Langmuir 2009, 25, 9057−9063. (5) Chen, Q.; Bae, S. C.; Granick, S Nature 2011, 469, 381−384. (6) Romano, F.; Sciortino, F. Nat. Mater. 2011, 10, 171−173. (7) Vácha, R.; Frenkel, D. Biophys. J. 2011, 101, 1432−1439. (8) Pawar, A. B.; Kretzschmar, I. Macromol. Rapid Commun. 2010, 32, 150−168. (9) Jiang, S.; Chen, Q.; Tripathy, M.; Luijten, E.; Schweizer, K. S.; Granick, S. Adv. Mater. 2010, 22, 1060−1071. (10) Elsesser, V. N., M. T.; Pine, D. J. Science 2003, 301, 483−487. (11) Zhang, Z; Glotzer, S. C. Nano Lett. 2004, 4, 1407−1413. (12) Hong, L.; Jiang, S.; Granick, S. Langmuir 2006, 22, 9495−9499. (13) Sciortino, F.; Giacometti, A.; Pastore, G. Phys. Rev. Lett. 2009, 103, 237801−237804. (14) Miller, W. L.; Cacciuto, A. Phys. Rev. E 2009, 80, 021404. (15) Sciortino, F.; Giacometti, A.; Pastore, G. Phys. Chem. Chem. Phys. 2010, 12, 11869−11877. (16) Kern, N.; Frenkel, D. J. Chem. Phys. 2003, 118, 9882−9888. 8
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(17) Reinhardt, A.; Williamson, A. J.; Doye, J. P. K; Carrete, J.; Varela, L. M.; Louis, A. A. J. Chem. Phys. 2011, 134, 104905. (18) Fejer, S. N.; Wales, D. J. Phys. Rev. Lett. 2007, 99, 086106. (19) Fejer, S. N.; Chakrabraty, D.; Wales, D. J. Soft Matter 2011, 7, 3553−3564. (20) Fejer, S. N.; Chakrabraty, D.; Wales, D. J. ACS Nano 2010, 4, 219−228. (21) Chakrabraty, D.; Fejer, S. N.; Wales, D. J. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 20164−20167. (22) Dillen, T.; van; Blaaderen, A.; van; Polman, A. Mater. Today 2004, 7, 40−46. (23) Ho, C. C.; Keller, A.; Odell, J. A.; Ottewill, R. H. Colloid Polym. Sci. 1993, 271, 469−479. (24) DeConinck, A. J. M.A. Thesis, University of Illinois at UrbanaChampaign, 2010. (25) Berne, B. J.; Pechukas, P. J. Chem. Phys. 1972, 56, 4213−4216. (26) Care, C. M.; Cleaver, D. J. Rep. Prog. Phys. 2005, 68, 2665− 2700. (27) Wang, W.; Wang, J.; Kim, M. S. Computer Aided Geometric Design 2001, 18, 531−539. (28) Choi, Y. K.; Chang, J. W.; Wang, W.; Kim, M. S.; Elber, G. IEEE Trans. Visualization Comput. Graphics 2009, 15, 311−324. (29) Shah, R. K.; Kim, J. W.; Weitz, D. A. Adv. Mater. 2009, 21, 1949−1953. (30) Foffi, G.; Sciortino, F. J. Phys. Chem. B 2007, 111 (33), 9702− 9705. (31) Liu, H.; Kumar, S. K.; Sciortino, F.; Evans, G. T. J. Chem. Phys. 2009, 130, 044902. (32) Paramonov, L.; Yaliraki, S. N. J. Chem. Phys. 2005, 123, 194111.
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