pubs.acs.org/Langmuir © 2010 American Chemical Society
Importance of Surface Patterns for Defect Mitigation in Three-Dimensional Self-Assembly Jatinder S. Randhawa,† Levi N. Kanu,† Gursimranbir Singh,† and David H. Gracias*,†,‡ †
Department of Chemical and Biomolecular Engineering, ‡Department of Chemistry, The Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218 Received March 12, 2010. Revised Manuscript Received May 21, 2010
This article investigates the three-dimensional self-assembly of submillimeter scale polyhedra using surface forces. Using a combination of energy landscape calculations and experiments, we investigate the influence of patterns of hydrophobic surfaces on generating defect-free, closed-packed aggregates of polyhedra, with a focus on cubic units. Calculations show that surface patterning strongly affects the interaction between individual units as well as that of the unit with the growing assembly. As expected, an increase in the hydrophobic surface area on each face results in larger global minima. However, it is the distribution of hydrophobic surface area on each cubic face that is strongly correlated to the energetic parameters driving low-defect assembly. For patterns with the same overall area, minimizing the radius of gyration and maximizing the angular distribution leads to steep energy curves, with a lower propensity for entrapment in metastable states. Experimentally, 200-500 μm sized metallic polyhedra were fabricated using a self-folding process, and the exposed surfaces were coated with a hydrophobic polymer. Cubes with surface patterns were agitated to cause aggregative self-assembly. Experimental results were consistent with energy calculations and suggest that geometric patterns with large overall areas, low radii of gyration, and high angular distributions result in efficient and low-defect assembly.
Introduction The bottom-up three-dimensional (3D) assembly of functional metamaterials and devices from discrete units necessitates the utilization of units with preprogrammed interactions to increase the complexity and fidelity of the resulting assemblies. Although there have been numerous elegant 3D self-assembly demonstrations in various academic laboratories, the capabilities of this approach have yet to be realized in a mass producible industrial setting since the rules that enable low-defect and robust selfassembly are not well understood. Hence, there is an urgent need to develop strategies that reduce the magnitude and different types of defects in self-assembling systems. In looking toward biological assembly for inspiration, one observes that biological units such as cells and viruses have, among other attributes, elaborate nonrandom molecular surface patterns. Although patterns on units in biological assembly are not static, it is well-known that the patterns themselves play a critical role in unit attachment and aggregative assembly.1,2 In synthetic self-assembly, patterns on units can be used to code information, provide recognition, and alter self-assembly energy landscapes. However, previous experimental studies of selfassembly at submillimeter scales have been limited to components with geometric shapes such as spheres and rods with little to no surface patterning.3-7 In contrast, simulations have shown that *Corresponding author. E-mail:
[email protected]. Phone: (410) 516-5284. Fax: (410) 516-5510. (1) Vereb, G.; Szollosi, J.; Matko, J.; Nagy, P.; Farkas, T.; Vigh, L.; Matyus, L.; Waldmann, T. A.; Damjanovich, S. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 8053. (2) Rossmann, M. G.; Rueckert, R. R. Microbiol. Sci. 1987, 4, 206. (3) Velev, O. D.; Furusawa, K.; Nagayama, K. Langmuir 1996, 12, 2347. (4) van Blaaderen, A.; Ruel, R.; Wiltzius, P. Nature 1997, 385, 321. (5) Xia, Y.; Gates, B.; Yin, Y.; Lu, Y. Adv. Mater. 2000, 12, 693. (6) Manoharan, V. N.; Elsesser, M. T.; Pine, D. J. Science 2003, 301, 483. (7) Rycenga, M.; McLellan, J. M.; Xia, Y. Adv. Mater. 2008, 20, 2416. (8) Zhang, Z.; Glotzer, S. C. Nano Lett. 2004, 4, 1407. (9) Zhang, Z.; Keys, A. S.; Chen, T.; Glotzer, S. C. Langmuir 2005, 21, 11547.
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patterned spherical patchy particles8,9 and a range of polyhedra10-15 can assemble into a variety of ordered 3D geometric orientations. In the assembly of patchy particles, it has been noted that the patches can guide the geometry of the overall structure of the self-assembly as well as the relative unit position within the assembly. These assemblies have yet to be experimentally realized, mainly because it is challenging to fabricate the required patterned units. It is noteworthy that experimental investigations of the influence of unit surface patterning in guiding self-assembly have been explored to some extent using millimeter scale units fabricated by hand16,17 and in the assembly of isolated units on two-dimensional (2D) substrates.18-22 While these studies have provided a glimpse of the capabilities of self-assembly, design rules to yield lowdefect assemblies wherein surface patterning can be controlled are still not known. Here, we investigate the influence of different hydrophobic patterns in the closed-packed self-assembly of precisely patterned 200-500 μm scaled self-folded polyhedra with a focus on cube (10) Cohn, H.; Kumar, A. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 9570. (11) Torquato, S.; Jiao, Y. Nature 2009, 460, 876. (12) Xiong, X. R.; Liang, S. H.; Bohringer, K. F. IEEE Int. Conf. Robot. Autom. 2004, 2, 1141. (13) Zhang, X.; Zhang, Z. L.; Glotzer, S. C. J. Phys. Chem. C 2007, 111, 4132. (14) Chan, E. R.; Ho, L. C.; Glotzer, S. C. Int. J. Mod. Phys. C 2009, 20, 1443. (15) Haji-Akbari, A.; Engel, M.; Keys, A. S.; Zheng, X.; Petschek, R. G.; PalffyMuhoray, P.; Glotzer, S. C. Nature 2009, 462, 773. (16) Breen, T. L.; Tien, J.; Oliver, S. R. J.; Hadzic, T.; Whitesides, G. M. Science 1999, 284, 948. (17) Gracias, D. H.; Tien, J.; Breen, T. L.; Hsu, C.; Whitesides, G. M. Science 2000, 289, 1170. (18) Jacobs, H. O.; Tao, A. R.; Schwartz, A.; Gracias, D. H.; Whitesides, G. M. Science 2002, 296, 323. (19) Zheng, W.; Buhlmann, P.; Jacobs, H. O. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 12814. (20) Fang, J.; Bohringer, K. F. J. Micromech. Microeng. 2006, 16, 721. (21) Lin, C.; Tseng, F. G.; Chieng, C. C. J. Micromech. Microeng. 2009, 19, 115020. (22) Mastrangeli, M.; Abbasi, S.; Varel, C.; Van Hoof, C.; Celis, J. P.; Bohringer, K. F. J. Micromech. Microeng. 2009, 19, 083001.
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shaped units. We calculated the change in the overlap surface area to compute an energy landscape at different stages of aggregation. We investigated correlations between energetic landscape parameters and geometric parameters that defined the area distribution of the pattern on each face of the unit; striking correlations were observed. We validated our calculations by self-assembly experiments with polyhedral units that were fabricated using selffolding. Self-folding, a more deterministic form of self-assembly where panels are connected to each other prior to assembly, is a process that has been shown to create precisely patterned polyhedra with sizes ranging from 100 nm to 2 mm.23-26 The highlight of this process is that surface patterning on the faces of the units is defined using 2D lithography, which is extremely precise and enables considerable pattern variability.
Experimental Section
Aldrich) was mixed in at 90:9:1, respectively, by volume. The quantity of this adhesive was adjusted based on the total surface area of the polyhedra such that the final surface concentration of adhesive on the surface of the polyhedra was approximately 10-7 μL/μm2, an experimentally optimized value. The polymer adhesive was mixed by gentle agitation and approximately 1.5 mL of the solution was removed and deionized water was slowly added. The water was then removed until the polyhedra were barely submerged, and the vial was refilled with deionized water. This process was repeated several times to remove the remaining ethanol. The vial was then covered with aluminum foil to block ultraviolet light exposure, in order to prevent the curing of the adhesive during the process of selfassembly. Self-assembly was carried out by securing the vial in a vortex mixer and agitating the vial at approximately 1525 rpm for 12 h. The vial was then taken out and exposed under a mercury lamp (BLAK-RAY Long UV lamp, for 4 h) to cure the assemblies in place.
Energy Landscape Calculations. The patterns were generated in MATLAB and the energy landscape was computed using the Surface Evolver27 software. The calculation involved computing the change in the overlap surface area as one pattern was translated or rotated over another pattern. We utilized a step size of 1 μm for translational calculations and a step size of 1 degree for angular calculations. The rotational analysis was done at the global energy minimum for 1-1 interactions. If we assume that the polymer adhesive layer has negligible thickness compared to the unit size, we can assume that the surface energy is directly proportional to the hydrophobic surface area on the face of the polyhedron.28 Hence, energy landscapes can be generated as a function of the type of pattern and the number of interacting polyhedra. Self-Folding of Tetrahedra and Cubes. We utilized the same fabrication methodology as described previously.24 Briefly, 2D nickel (Ni) templates with solder hinges were fabricated on a sacrificial layer and then subsequently released. Self-folding of the 2D templates to 3D polyhedra occurred on heating above the melting point of the solder. Surface Functionalization of Units. After self-folding, polyhedra were sorted and perfect units were selected for self-assembly experiments. In our experiments, we utilized 200 μm sized tetrahedra and 500 μm sized cubes. The inner and outer surfaces of the polyhedra were coated with a thin layer of gold (Au; approximately 2 μm thick) that was electroplated using a commercial plating solution (TechnicTG-25T-RTU). The units were then soaked in a 10 mM hexadecane thiol solution (TCI America) in ethanol for 12 h which rendered the gold surfaces hydrophobic as the thiols formed a self-assembled monolayer (SAM) on the gold surface.29 Self-Assembly. The aggregative self-assembly strategy using a hydrophobic adhesive and Au units was based on a methodology that has been described in detail elsewhere.30,31 Briefly, several polyhedra were transferred into a 2 mL vial and 1 mL of ethanol was added. Then, a photocurable adhesive composed of a monomer (lauryl methacrylate, Aldrich), a cross-linker (1,6-hexanediol diacrylate Aldrich), and a free radical initiator (benzoin isobutyl ether, (23) Gracias, D. H.; Kavthekar, V.; Love, J. C.; Paul, K. E.; Whitesides, G. M. Adv. Mater. 2002, 14, 235. (24) Leong, T. G.; Lester, P. A.; Koh, T. L.; Call, E. K.; Gracias, D. H. Langmuir 2007, 23, 8747. (25) Filipiak, D. J.; Azam, A.; Leong, T. G.; Gracias, D. H. J. Micromech. Microeng. 2009, 19, 075012. (26) Cho, J. H.; Gracias, D. H. Nano Lett. 2009, 9, 4049. (27) Brakke, K. A. Exp. Math. 1992, 1, 141. (28) Xiong, X.; Hanein, Y.; Fang, J.; Wang, Y.; Wang, W.; Schwartz, D. T.; Bohringer, K. F. J. Micromech. Microeng. 2003, 12, 117. (29) Bain, C. D.; Troughton, E. B.; Tao, Y. T.; Evall, J.; Whitesides, G. M.; Nuzzo, R. G. J. Am. Chem. Soc. 1989, 111, 321. (30) Tien, J.; Breen, T. L.; Whitesides, G. M. J. Am. Chem. Soc. 1998, 120, 12670. (31) Gu, Z.; Chen, Y.; Gracias, D. H. Langmuir 2004, 20, 11308.
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Results and Discussion We utilized energy landscape calculations and experiments to understand the influence of patterning on defect mitigation in closed-packed self-assembling systems. The self-folding process has been previously utilized to fabricate different polyhedral shapes such as tetrahedra, cubes, octahedra, and dodecahedra. Of these polyhedra, we focused on aggregation of cubes for a number of reasons. Initially, our studies involved an examination of the selfassembly of units composed of Platonic solid shapes that would theoretically aggregate with the lowest (tetrahedron; packing density (η) = 0.367) and highest (cube; η = 1) packing densities. The cube is also the only Platonic solid that can form closedpacked structures with a packing density of one. In contrast, it is impossible to tile space with regular tetrahedra because the assembly of five tetrahedra along a common edge results in an angular gap of 7.36° and the assembly of 20 tetrahedra about a shared vertex results in a gap of 1.54 steradians.32 We confirmed this theoretical fact in our experiments. Here, polyhedra were first self-folded from planar preconnected nets24 (Figure 1A) and were then chemically treated so that all solid surfaces were coated with a hydrophobic adhesive.33 Several units of patterned tetrahedra (Figure 1B) and cubes (Figure 1C) were fabricated using this approach. Prior to assembly, the surfaces of each polyhedron were composed of either hollow or hydrophobic regions with any desired pattern. The polyhedra self-assembled during agitation in water (Figure 1D) as a result of the minimization of the surface energy of exposed hydrophobic regions. Figure 1E,F shows optical microscopy images of the 3D selfassembly of tetrahedra and cubes, respectively, patterned with a hollow window pattern. Even though it is impossible to tile space with tetrahedra, the surface-coating adhesive that extended out from the surface filled the narrow angular gap between tetrahedra to create a pseudo closed-packed geometry (Figure 1E). However, for tetrahedral assemblies, it was challenging to isolate the influence of patterns on defects, since geometric angular gaps associated with pseudo closed-packing were also important. In contrast, the cubes formed closed-packed aggregates and over several trials we observed that the predominant defects in their self-assembly involved a planar shift between adjacent units. We observed that this dominant defect mode was dependent mainly on surface patterning and so we could study the influence of patterns in (32) Conway, J. H.; Torquato, S. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 10612. (33) Clark, T. D.; Tien, J.; Duffy, D. C.; Paul, K. E.; Whitesides, G. M. J. Am. Chem. Soc. 2001, 123, 7677.
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Figure 2. Schematic, experimental and equilibrium energy diagrams of 1-1 interactions of cubes with two different patterns, I and II. (A and B) Schematic and experimentally assembled dimers which aggregate as a result of 1-1 interactions. (C and D) Translational equilibrium energy landscape projected along the plane of interaction. (E and F) Inverted 3D energy landscape plotted as a function of planar translational and angular (inset) variations between two units.
Figure 1. (A) Schematic showing self-folding of tetrahedral and cubic units from 2D templates, and subsequent surface functionalization. (B and C) Optical images of as-fabricated tetrahedra and cubes, respectively. (D) Schematic showing self-assembly of hydrophobic units by fluidic agitation in a vial of water. (E and F) Optical images of self-assembled aggregates of tetrahedra and cubes, respectively. (G and H) Schematic showing that the dominant planar interactions of an approaching unit to the growing assembly are of the same type for tetrahedral assembly but vary in cubic assembly.
isolation. It should be noted that these defects were persistent and could not be mitigated by merely increasing the strength of the agitation, while maintaining the overall integrity of the assembly. We noted another important difference in this defect mode in the self-assembly of tetrahedral and cubic shapes (Figure 1G,H). During densely packed self-assembly of tetrahedra, the dominant planar interaction of an approaching unit to the growing assembly (referred to as a 1-X unit interaction) involves one-one (1-1) interactions because growing aggregates of proximal tetrahedra do not present an extended planar surface (greater than the individual face) (Figure 1G). On the other hand, during the closedpacked assembly of cubic building blocks (Figure 1H), an extended planar surface is generated which can be visualized in a 4-unit assembly with 1-1, 1-4, and 4-4 interactions. The generation of this extended plane can cause the approaching unit 12536 DOI: 10.1021/la101188z
to attach in many defective states with planar translational shifts, and patterns displayed on single units as well as this extended plane are both important. In contrast, only 1-1 interactions are important for planar shift defects in tetrahedral assembly. We varied the patterns of hydrophobic regions on the cubes and studied the interaction of isolated units to form dimers (Figure 2A-B). Energy landscape calculations with respect to translation along each axis reveal two important parameters, namely a translational global energy difference (denoted as |ΔEglobal|) and a force (-dE/dx calculated at the equilibrium point, denoted as F11). As expected, a 4-fold symmetry in the potential energy surface consistent with the symmetry of the overall face was observed (Figure 2C,D). We also observed that cubes with larger areas of hydrophobic regions generated greater global energy minima, as expected. Pattern I with a hydrophobic area of 0.22 mm2 had |ΔEglobal| = 22 au (arbitrary units) (Figure 2E), whereas pattern II with a hydrophobic area of 0.15 mm2 had |ΔEglobal| of 15 au (Figure 2F). We observed that patterns with a larger distribution of hydrophobic areas in the center of the face resulted in larger magnitudes of F11 (i.e., |F11| = 0.13 for pattern II as compared to |F11| = 0.067 for pattern I). To analyze the rotational stability we also performed energy landscape calculations by rotating one surface over the other at the equilibrium translarot | = 7.8 au (Figure 2F, tional location. Pattern II had a higher |ΔE11 rot inset) as compared to pattern I with |ΔE11 | = 4.7 au (Figure 2E, inset), which suggests a lower propensity for angular defects in pattern II. While the 1-1 planar interaction is an important design criterion to study the self-assembly of two units, it provides us Langmuir 2010, 26(15), 12534–12539
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Figure 3. Schematic, experimental and equilibrium energy diagrams of 1-4 interactions of cubes with two different patterns, I and II. (A and B) Schematic and experimental aggregates with dominant 1-4 interactions. The cube edges have been outlined with red lines to enable visualization of shift defects. (C and D) Translational equilibrium energy landscape projected along the plane of interaction. (E and F) Inverted 3D energy landscape plotted as a function of planar translational variations between two units composed of one cube and four cubes.
with limited insight for translational slip defects of a growing aggregate. For a cube, the 4-unit aggregate contains all possible pattern combinations for planar slip defects. Hence, we also studied the assembly of a single unit with the planar surface presented by a single 4-unit aggregate (Figure 3) as well as two 4-unit aggregates with each other (Figure 4). The key difference between 1-1 (Figure 2A,B) and 1-4 (Figure 3A,B) interactions is seen as the central region in the energy plots (Figure 3C,D). In addition to |ΔEglobal| and |F11| for the 1-4 interaction two more parameters need consideration: |ΔEdefect| = |ΔEglobal - ΔElocal| (Figure 3F), the energy difference for planar defects; and |F14| (Figure 3E) which is the slope of the energy curve when a single unit interacts with a unit composed of four polyhedra (here, the value of |F14| was evaluated at the same point for all patterns, away from the Eglobal point, along the x-axis and toward the center of the four units (Figure 3E)). The value of |ΔEdefect| for pattern I is 3.06 au as compared to 5.8 au for pattern II. The smaller magnitude of the energy gap (|ΔEdefect|) in the energy landscape for pattern I implies a higher propensity for defect formation. For a constant fluidic agitation energy, a higher value of |ΔEdefect| translates to a lower probability for planar defects. The magnitude of force (|F14|) for pattern II (0.107 au) is higher than that for pattern I (0.017 au) which indicates a higher restoring force to the global energy minimum and hence a lower propensity for defects. Langmuir 2010, 26(15), 12534–12539
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Figure 4. Schematic, experimental and equilibrium energy diagrams of 4-4 interactions of cubes with two different patterns I and II. (A-D) Schematic and experimental aggregates with dominant 4-4 interactions. The cube edges have been outlined with red lines to enable visualization of shift defects. (E and F) Translational equilibrium energy landscape projected along the plane of interaction. (G and H) Inverted 3D energy landscape plotted as a function of planar translational variations between two units composed of four cubes each.
For the 4-4 interactions (Figure 4A-D), we observed that, in pattern I, the Eglobal region (red) was diffuse and spread over ∼150 μm (Figure 4E), while in pattern II it was concentrated within ∼30 μm (Figure 4F). |F44| for pattern I (Figure 4G) was smaller (0.17 au) as compared to 0.38 au for pattern II (Figure 4H). Experimentally, we consistently observed more translational planar defects in the assembly of cubes with pattern I (Figure 4C) as compared to assembly with pattern II (Figure 4D). Over five trials, the average linear shift defect was 20 ( 5% and 6 ( 2% (of the cube side length) for pattern I and II, respectively. A key aim in this work is to develop predictive models to enable experimentalists to design patterns that enable self-assembly with low defect rates. Patterns I and II in Figures 2-4 represent only two cases. In order to generalize this observation to patterns with variable area and area distribution, we calculated energy landscapes while systematically varying both the total exposed hydrophobic region area (A) on each face (patterns 1, 2, and 3 in Figure 5A,B while keeping identical pattern symmetry) and the type of pattern (patterns 3, 4, 5, and 6 in Figure 5A,B while keeping an identical total exposed hydrophobic region area). It should be DOI: 10.1021/la101188z
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symmetry and internal features in these patterns were varied such that I kept decreasing from pattern 3 to 6 (Figure 5C). A and I are two independent design variables which define the physical quantities that play a role during the self-assembly process. It is intuitive and it was observed that |ΔEglobal| is directly proportional to the total exposed area (Figure 5D); in contrast, |ΔEdefect| depends more on the distribution of area. To capture the influence of the distribution of area on the pattern, we calculated the radius of gyration Rg of the area which is the defined as Rg = (I/A)1/2. We also defined a parameter (Rθ) which measures the angular distribution (details in Supporting Information). Intuitively, Rθ represents the distribution of area at different angles; it was defined such that any circular-shaped pattern has Rθ = 1, a semicircular pattern has Rθ = 0.5, and a thin line strip has Rθ close to zero. The magnitude of Rθ increases from pattern 3 to 6 (Figure 5E) while Rg decreases (Figure 5F). We observed that the important parameters that minimize defects in the self-assembly (i.e., |ΔEdefect|, |ΔErot 11 |, |F11|, |F14|, and |F44|) were directly correlated with Rθ and inversely correlated with Rg (Figure 5G). These values were calculated by evaluating the correlation matrix between geometric design (i.e., Rg and Rθ) and energy landscape (i.e., |ΔEdefect|, |ΔErot 11 |, |F11|, |F14|, and |F44|) parameters. Based on the design criteria above, pattern 6 was predicted to self-assemble with low defects; pattern II which is from this class of patterns was already experimentally shown to have much smaller shift defects in 2 2 2 assemblies as compared to pattern I (which resembles pattern 2 in Figure 5). The lower translational and rotational defects were also confirmed in larger experimental assemblies composed of 20-40 units (Figure 5J compared to Figure 1F).
Conclusion
Figure 5. Results showing the variation of energy landscape parameters with varying patterns. (A) Patterns 1-3 have varying hydrophobic areas with the same type of pattern, whereas patterns 3-6 have the same hydrophobic areas with varying types of patterns. (B) Sketches of cubes with patterns 1 through 6 on all faces. (C) Plot of the total hydrophobic area (A) and the second moment of area (I) for patterns 1-6. (D) Plot of the magnitude of |ΔEglobal| and |ΔEdefect| for patterns 1-6. (E and F) Plot of Rθ and Rg for patterns 3 through 6, respectively (G) Plot of the correlation of Rθ and Rg with |ΔEdefect|, |ΔErot 11 |, |F11|, |F14|, and |F44| for patterns 3-6. (H-J) Optical images of individual, as-fabricated cubic units with type 6 patterns and their self-assembled aggregates of different sizes.
noted that pattern I used for cubic assemblies in Figures 1-4 is similar to the patterns 2 and 3 in Figure 5, whereas pattern II used in Figures 2-4 is similar to pattern 6 in Figure 5. The area distribution of different patterns were compared by the second moment of area (I), which is defined as I = Rcalculating 2 Ar dA, where r is the distance of the differential area element dA from the center of the face. All of the patterns had 4-fold 12538 DOI: 10.1021/la101188z
We have uncovered design rules that can be utilized for generating patterns which mitigate planar defects in 3D selfassembling systems. A common pitfall in designing such patterns is to focus on single unit interactions (1-1 interactions) and ignore the interactions with multicomponent units (1-4 and 4-4 interactions). Our study reveals that the patterns presented by a growing surface to an approaching unit are critical in enabling precise alignment and hence low defects. Additionally, easily computable variables Rθ and Rg capture the propensity for defects in the interaction of a single unit with itself (1-1) as well as a single unit with the surface of a multiunit component (1-4) and also that of multiunit components (4-4). We chose to use polyhedral building blocks in the hundred micrometer size range, since the self-folding process does not have a 100% yield and it was necessary to select “perfectly” self-folded polyhedra prior to 3D assembly to ensure that mis-folded polyhedra did not influence our results. At this size, sorting could be achieved with an optical microscope. However, we do believe that our results apply to smaller length scales provided that the magnitude of the energy defects |ΔEdefect| is comparable to the strength of the agitation. Our study also reveals the need to develop additionally geometric design rules to enable selfassembly as a viable and robust fabrication methodology. Apart from developing design rules to mitigate defects in 3D self-assembly, early studies of metamaterials composed of polyhedral unit arrays have revealed interesting optical properties.34 In the future, it would also be interesting to extend (34) Randhawa, J. S.; Gurbani, S. S.; Keung, M. D.; Demers, D. P.; Leahy-Hoppa, M. R.; Gracias, D. H. Appl. Phys. Lett. 2010, 96, 191108.
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these studies to include more complex polyhedral units such as dodecahedra. Acknowledgment. This material is based in part upon work supported by the National Science Foundation under Grant Number CMMI-0448816 and from the Dreyfus and the Beckman Foundations. Authors also wish to thank K. F. Bohringer for
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sharing a computation code, Matthew Jochmans for making changes in the code, and Michael D. Keung and Dilip Asthagiri for proof reading the manuscript. Supporting Information Available: Figure and formula defining the angular distribution function (Rθ). This material is available free of charge via the Internet at http://pubs.acs.org.
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