Control of Randomness in Microsphere-Based Photonic Crystals

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Control of Randomness in Microsphere-Based Photonic Crystals Assembled by Langmuir-Blodgett Process Sarun Atiganyanun, Mi Zhou, Omar K. Abudayyeh, Sang Min Han, and Sang Eon Han Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03060 • Publication Date (Web): 08 Nov 2017 Downloaded from http://pubs.acs.org on November 20, 2017

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Control of Randomness in Microsphere-Based Photonic Crystals Assembled by Langmuir-Blodgett Process Sarun Atiganyanun†, Mi Zhou†, Omar K. Abudayyeh‡, Sang M. Han†,‡, and Sang Eon Han*,†,‡ †

Nanoscience and Microsystems Engineering, University of New Mexico, Albuquerque, NM, 87131, USA ‡ Department of Chemical and Biological Engineering, University of New Mexico, Albuquerque, NM, 87131, USA ABSTRACT. Photonic structures in biological systems typically exhibit an appreciable level of disorder within their periodic framework. However, how such disorder within the ordered framework renders unique optical properties has not been fully understood. Towards the goal of improving this understanding, we have investigated Langmuir-Blodgett assembly of microspheres to controllably introduce randomness to photonic structures. We theoretically modelled the assembly process and determined a condition for surface pressure and substrate pulling speed that results in maximum structural order. For each surface pressure, there is an optimum pulling speed, and vice versa. Along the trajectory defined by the optimum condition, however, the structural order decreases moderately as the pulling speed increases. This moderate decrease in structural order would be useful for controlled introduction of randomness into the periodic structures. Departing from the trajectory, our experiment reveals that a small change in pulling speed at a given surface pressure can significantly disrupt the structural order. For

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multilayer assembly, we find that, at a fixed pulling speed, the surface pressure should increase as the number of layers increases to achieve maximum structural order.

In totality, we

quantitatively present the optimum trajectories for nth layer assembly relating surface pressure and pulling speed.

1. INTRODUCTION Photonic crystals – artificial materials that are periodic on the optical wavelength scale – can strongly interact with light and exhibit unique optical effects, such as photonic bandgap1-2 and absorption enhancement/suppression.3-5 When light frequency is within a photonic bandgap, light can be localized in or guided through defects in photonic crystals.6-8 Outside the bandgap, however, non-periodic structural defects and disorders scatter light randomly. Random light scattering in photonic crystals is generally regarded as a problem and in many cases, a result of fabrication errors. Notwithstanding this view on structural defects and disorders, many photonic patterns found in animals and plants involve an appreciable degree of disorder in the periodic structures.9-11 For example, butterfly wings, humming birds, and blue Pollia condensata (African marble berry) generate iridescent colors by employing imperfect periodic structures.12-13 These biological structures possess a degree of disorder that is much greater than that in typical photonic crystals fabricated by various techniques, including optical lithography and micromachining.

These imperfections observed in nature suggest that, if the biological

structures have been optimized over millions of years of evolution, a certain degree of structural randomness may be not only favorable, but even required for the best optical performance to ensure their survival. The usefulness of structural randomness can also be found in artificial systems. For example, for solar photovoltaics, the control of spatial correlation in structures can improve light-trapping efficiency over both periodic and random structures.14-16


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Self-assembly of colloidal microspheres provides simple, scalable means to fabricate photonic crystals. When properly applied, the self-assembly technique can achieve a very low defect density (∼1% stacking fault, ~1 point defect per 103 unit cells) over centimeter-scale samples.17 The convenience and long-range periodicity resulting from self-assembly have led to significant advances in experimental research in photonic crystals.18 Popular self-assembly techniques include convective assembly on vertical substrates,17-19 slow sedimentation of colloidal microspheres,20-21 and Langmuir-Blodgett (LB) assembly.22-25

Among these

techniques, convective assembly has been highly successful in achieving long-range orders.17 However, if the desired goal is to introduce controlled randomness into periodic structures in a prescribed manner, these techniques must be carefully reevaluated. Here, we have investigated LB assembly to establish how LB process parameters can be manipulated to controllably introduce randomness. In previous studies on microsphere LB assembly, much attention has been given to the surface pressure – the decrease in surface tension by the presence of microspheres – in the liquid subphase.26-28 A wide range of surface pressures (0 and ~25 mN/m) has been claimed to achieve high degree of order.24-29 In this work, we demonstrate that the degree of randomness in photonic crystals can be controlled in LB assembly by manipulating the substrate pulling speed in addition to the surface pressure. We also modify the model by Dimitrov and Nagayama30 to describe the monolayer LB assembly and determine a condition that interrelates surface pressure and substrate pulling speed (two key experimental parameters) to achieve maximum structural order. Our experiment confirms that such a condition exists. Further, we find that increasing the pulling speed, while this condition is met, gradually increases randomness.

2. EXPERIMENTAL


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Microsphere Solution Preparation. The surface of silica microspheres (~900 nm in diameter) was functionalized with allyltrimethoxysilane. This functionalization has been shown to provide a balance between hydrophobicity and hydrophilicity.31 The allyl groups rendered the microsphere surface hydrophobic, enough to prevent the aggregation of microspheres in solution and allow them to remain near the surface of the water.28 However, the microsphere surface was still hydrophilic enough, such that the microspheres were almost fully immersed just below the water surface in the LB trough (see Supporting Information). The microspheres purchased from Polysciences were delivered in an aqueous solution, and the polydispersity of this batch was 15 % based on the vendor-provided product information as well as our electron micrograph images. For the surface functionalization, 500 µl of 10 wt% microsphere solution was first centrifuged, so that the microspheres sedimented to the bottom. Then, the water was decanted, and the microspheres were dispersed in ethanol. This solution was sonicated for 45 minutes, and 10 µl of allyltrimethoxysilane was added to the solution. The solution was then further sonicated for 2 hours. After sonication, the microspheres are washed in ethanol 3 times to remove residual allyltrimethoxysilane from the solution. Lastly, the microspheres were dispersed in 500 µl of chloroform. Based on the microsphere density of 2.0 g/cm3 provided by the manufacturer, the total number of microspheres in the chloroform solution (50 µl) was estimated to be 6.9 × 1010. Substrate Preparation. Substrates (3 cm × 1 cm) diced from n-doped (100) silicon wafers with a resistivity > 100 Ω-cm were used for LB assembly. A substrate was first treated for 3 minutes in a Piranha solution, which consisted of 125 ml of 96 wt% sulfuric acid and 75 ml of 30 wt% hydrogen peroxide. The substrate was then cleaned for 2.5 minutes with a buffered oxide etchant, a 20:1 volume mixture of 40 wt% ammonium fluoride and 49 wt% hydrofluoric


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acid. Finally, the substrate was treated in the Piranha solution for 3 minutes, so that the silicon surface was oxidized and made hydrophilic. Langmuir-Blodgett Isotherm. The LB trough was first cleaned with deionized water and chloroform, after which a water subphase was added to the trough. The microsphere solution in chloroform was then carefully dispersed by a micropipette onto the subphase surface to form a monolayer of microspheres. The monolayer was compressed by two barriers of 20 cm in width. Each barrier moved toward the trough center line at 0.6325 cm/min, while the surface pressure was monitored and recorded. A filter paper was used as a Wilhelmy plate to measure the surface pressure. The effective projected area occupied by a single microsphere in the monolayer, which includes part of the spacing between spheres and is slightly greater than the true projected area per microsphere (see the inset in Figure 1), was calculated by dividing the area between the trough barriers by the number of microspheres. An LB isotherm was obtained from the effective area per microsphere and the surface pressure. Langmuir-Blodgett Assembly. The LB assembly was performed at ~22°C and ~20% relative humidity. The monolayer of microspheres was first compressed by the barriers, so that a target pressure was reached. A surface-treated silicon substrate was then pulled out of the LB trough in the vertical direction at a set velocity while the surface pressure was held constant. During substrate pull, a monolayer of microspheres was transferred onto the substrate in a closepacked arrangement. To transfer additional layers, the substrate with microsphere layers was first dipped into water while the monolayer on the water surface was decompressed. Then the cycle of pulling and dipping and of the sample was repeated. For each additional layer, we varied assembly parameters to determine the optimum pull speed and surface pressure. Once the


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assembly was complete, the samples were dried in air and characterized for their structural properties. Characterization. The microstructures of the LB-assembled samples were imaged by a scanning electron microscope (FEI Q3D FIB/SEM DualBeam system) at 10 keV electron beam energy.

The average domain size of the microsphere crystals was determined from light

diffraction experiment. For light diffraction, a He-Ne laser beam at 632.8 nm was normally illuminated on the assembled sample. Diffraction patterns were projected onto a screen and recorded by a digital camera. The patterns approximately assumed circular symmetry because of the random distribution of the crystalline domain orientations and were fit to an intensity distribution given by I (r ) = I 0 e

−

3πG 2  r cos θ    4 λ2  R 

2

,

(1)

where I (r ) is the intensity at a distance r from the pattern center, I 0 is the intensity at the center, λ is the wavelength of the laser, θ is the diffraction polar angle, R is the distance between the illuminated spot on the sample and the primary peak on the screen, and G is the representative crystal domain size.

The derivation of eq 1 is found in Supporting Information.

The

reproducibility of the experiment was tested by examining the G values of 10 samples prepared under the same experimental conditions. When the substrate pulling speed was 1.5 mm/min, and the surface pressure was 4.8 mN/m, the error in G was within ± 10% of the average value.

3. MODELING We begin our discussion by presenting a model, modified from the work of Dimitrov and Nagayama,30 that describes the LB process for microsphere assembly. Our model relates the surface pressure and the substrate pulling speed that would induce the maximum structural order.


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Figure 1 schematically shows the LB assembly process where the liquid subphase is water. We assume that the top of the microspheres is close to the water surface. This happens when the microsphere surface is hydrophilic at the appropriate level. The assumption is approximately valid in our experiment as verified in Supporting Information. The submerged microspheres with a diameter d at the water surface are pushed laterally toward a vertical substrate by two barriers on either side of the substrate. Each barrier is moving horizontally with a speed of vb, and the substrate is vertically pulled out of the water with a speed of vc. As the substrate is pulled out, the microspheres on the liquid surface are transferred onto the substrate. Water evaporates from the meniscus over the vertical layer exposed to the ambient air at a volumetric flow rate per unit substrate width, Jevap (cm3/cm-sec).

Jevap is approximately a constant

determined by the humidity of the ambient air.30 The thickness of the water film at the lowest point of the assembled microsphere crystal is defined as hf. Multilayer photonic crystal structures can be obtained by repeating the transfer process. In this layer-by-layer approach, randomness is naturally introduced in each layer, and our goal is to control the overall degree of randomness. The thickness of each layer of the crystal is h, which is the same as d for a monolayer and 0.816d for hexagonally close-packed multilayers. At steady state, the upward volumetric flow rate of water through the area defined by hf and the substrate width (W) is equal to the volumetric water evaporation rate from the meniscus above the area, and hence j w h f = J evap

,

(2)

where jw is the volumetric flux of water in the vertical direction through the area (hf × W). jw consists of water flux from the surface layer (jw,s) flowing in together with microspheres pushed


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by the barrier and wafer flux from the bulk solution (jw,b) flowing in due to the upward substrate motion and Laplace pressure with a weight factor of α and 1 – α, respectively, so that j w = α j w ,s + (1 − α ) j w ,b

,

(3)

where we assume that α is a constant. We define the surface layer as the layer of thickness d that streams water and microspheres pushed by the barrier. The surfaces layer extends from the area where hf is defined to the edge of the barrier. The volumetric flux of microspheres in the surface layer is denoted by jp,s. The volumetric flow rate of microspheres in the surface layer is equal to that in the top assembled layer: j p , s d = vc h (1 − ε ) ,

(4)

where ε is the void fraction in the assembled layer. The ideal maximum order corresponds to a single crystal domain of hexagonally close-packed microspheres, where ε is calculated to be 0.395 for monolayers and 0.260 for multilayers. We assume that the ratio of volumetric flux of water to that of microspheres in the surface layer is equal to the ratio of volume fraction of water to that of microspheres in the layer:

j w, s 1 − ϕ = , j p ,s ϕ

(5)

where ϕ is the volume fraction of the microspheres in the surface layer. From eqs 2 – 5, we obtain the expression for the substrate pulling speed as

vc =

d ϕ  J evap − (1 − α ) j w,b h f (1 − ε ) hh f 1 − ϕ  α

 . 

(6)

For the assembly of nth layer, in eq 6, we approximate hf and h as nd and d, respectively, so that

vc ≅

 ϕ  J evap − (1 − α ) jw,b d  .  (1 − ε ) dα 1 − ϕ  n  1

(7)


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Equation 7 approximately captures a condition for maximum order, where vc increases linearly with increasing ϕ /(1 − ϕ ) . As we will show later in Figure 3b, vc that yields maximum order increases approximately linearly with increasing ϕ /(1 − ϕ ) . Figure 3b further shows that the yintercept is nonzero. For this to be true, eq 7 must have a term independent of ϕ /(1 − ϕ ) that defines the y-intercept. The only variable in eq 7 that would contain the independent term is jw,b. Therefore, we split jw,b into two terms: one that becomes dependent on and the other independent of ϕ /(1 − ϕ ) , when multiplied by ϕ /(1 − ϕ ) :

α

  1−ϕ j0 + (1 − ε ) j1  ,  1−α  ϕ 

j w ,b =

(8)

where j0 and j1 are constants. Substituting eq 8 into eq 7, we obtain

vc ≅

 1 ϕ  J evap  − j0 d  − j1 . (1 − ε ) d 1 − ϕ  αn 

(9)

The microsphere volume fraction ϕ in the surface layer is given by ϕ=

NV p Ad

,

(10)

where A is the area between the substrate and the barrier, N is the number of microspheres within A, and Vp is the volume of a microsphere. Thus, from eqs 9 and 10, we have

vc ≅

 1 1  J evap  − j0 d  − j1 . (1 − ε )d Ad − 1  αn  NVp

(11)

Equation 11 dictates that as the area per microsphere A/N in the horizontal monolayer decreases, the substrate pulling speed must increase to keep the porosity in the vertical monolayer constant. To put this in physical terms, as one forces an increasing number of microspheres into the surface layer, the pulling speed must increase to achieve maximum order. We also note that the


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surface pressure increases as A/N decreases. Thus, as the surface pressure increases (i.e., as A/N decreases), the pulling speed must increase to achieve the maximum order. The efforts on LB process optimization in previous studies have been directed mostly to determine the optimum surface pressure irrespective of the pulling speed.26-28 Our modeling suggests that the two parameters, while independent parameters, must satisfy the relation in eq 11 to achieve maximum order.

4. RESULTS AND DISCUSSION Prior to validating our model, we have measured the LB isotherm for our system. Figure 2 shows our experimentally determined isotherm (black circles). The surface pressure increases sharply as the effective projected area per microsphere (Figure 1 inset) decreases and passes through ~1.2 µm2, which represents the monolayer collapsing point. Based on this area, the center-to-center average distance between two microspheres at the collapsing point is calculated to be ∼1.2 µm. When the microspheres of 0.9 µm in diameter form a hexagonally close-packed monolayer, the center-to-center distance would be 0.9 µm, and the effective projected area per microsphere would be 0.701 µm2. Thus, a monolayer in our LB system collapses when the average distance between the microspheres is close to but greater than their diameter. This observation is in agreement with previous studies.23, 28, and we speculate that the sphere-tosphere interaction is repulsive. The difference between the distance at the collapsing point and the microsphere diameter can be smaller29, 32 than the current value of ∼0.3 µm, depending on the system parameters, such as barrier speed, microsphere charging, degree of functionalization, polydispersity, etc. We fit the isotherm to the following 2-dimensional van der Waals equation of state:


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π=

k BT a − ,   A − A0  2  A − A0 − b     N   N 

(12)

where a is the microsphere interaction parameter, b = πd2/2 is the excluded area of a particle, T is the solution absolute temperature, kB is the Boltzmann constant, and A0/N is the difference between the effective projected area per microsphere for the monolayer collapsing point and that for the ideal close-packing. We set a and A0/N as fitting parameters and fit only the data for A/N ≥ 1.195 µm2. The model fit is shown as a red line in Figure 2 and is in good agreement with our experiment (a coefficient of determination of 0.984) for the data range greater than or equal to the area at the collapsing point. The best fit is obtained when a = –4.053 × 10-13 Nµm3 and A0/N = 1.191 µm2. The negative value of a indicates that the interaction between the microspheres is repulsive and is stronger than van der Waals attraction. We note that the 2-dimensional van der Waals equation of state provides only a semi-quantitative description of the microspheres in the LB trough. To test our model, we measured the average crystalline domain size, G, at various pulling speeds and surface pressures. Figure 3a shows the experimentally measured G values of a monolayer as a function of surface pressure (π) at different pulling speeds (vc). For the surface pressures of 4.6, 6.0, and 8.0 mN/m, the pulling speeds that correspond to the maximum G values are 1.5, 1.8, and 2.25 mm/min, respectively. Thus, for the maximum structural order, the pulling speed must increase as the surface pressure increases. The surface pressure is related to ϕ by eqs 10 and 12. Using the relation, we plot the vc at the maximum order for the three surface pressures as a function of ϕ / (1− ϕ). Figure 3b shows that vc increases approximately linearly as

ϕ / (1− ϕ) increases. This result is consistent with our model in eq 9. Note that, while the range


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of ϕ / (1− ϕ) is relatively small in Figure 3b, this represents a large corresponding surface pressure range (4.6 – 8.0 mN/m). A key observation in Figure 3a is that the G values can be highly sensitive to the pulling speed at a given surface pressure, abruptly changing over a narrow range of vc. For example, at

π = 6.0 mN/m, G dramatically decreases from >8.8 µm to 2.8 µm when the pulling speed decreases by only 0.1 mm/min from vc = 1.8 mm/min [Point (d) in Figure 3a]. The high sensitivity suggests that the introduction of randomness in the LB assembly should be performed carefully at precise values of surface pressure and pulling speed.

When the degree of

randomness should be increased slightly, one should not blindly attempt to make a small change in pulling speed from the optimum value, expecting a gradual change in G. A small change in vc may lead to an abrupt disorder. To visualize the dependence of G on the experimental conditions, we took scanning electron micrograph (SEM) images of the samples corresponding to (c)-(f) points in Figure 3a, as shown in Figure 3c-f. For the ease of visualization, we highlight the disordered regions by red and the crystalline domains by blue and green. Small point defects in crystalline domains are ignored in the coloring.

The shades of color between blue and green represent different

orientations of the crystalline domains. The orientation is characterized by the angle between a lattice vector and the horizontal line in the figures. Because of the 6-fold rotational symmetry of the lattice, the angle is between ±30°. The total area of red regions (disorder) in Figure 3f is much greater than that in Figure 3c. This increase in disordered area is consistent with the decrease in G for π = 4.6 mN/m in Figure 3a. Note that the G values are obtained from a large area of ~5 mm2 by laser diffraction, while the SEM images are taken over a much smaller area of 2.5×10-3 mm2. Thus, the G value is a


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much better representation of the structural order/disorder than the SEM images. In Figure 3c, the disordered regions form irregular thin lines between large single crystalline domains. In comparison, in Figure 3f, the disordered regions take a large space between the small crystalline domains. From this comparison, we can speculate a possible mechanism for how the structural order is established during the LB assembly. Initially, microspheres are randomly positioned or are gathered to form small crystalline domains on the subphase surface.33 The small crystalline domains are randomly oriented on the liquid surface. As the microspheres are transferred onto a substrate at the optimum pulling speed and surface pressure, these small crystalline domains appear to grow to form larger domains. When the assembly is not conducted under the optimum condition, the microspheres are not closely packed, and the crystalline domains do not significantly grow. As a result, the LB film is more disordered. We surmise that the growth of crystallographic domains has an associated characteristic time scale. Since enough time must be provided for the domains to grow, the structural order is likely to improve with decreasing pulling speed, provided that the pulling speed and the surface pressure are maintained at the optimum condition described by eq 11. Our speculation agrees with the results shown in Figure 3a, where the maximum G value increases from 5.0 to 8.8 to 11.0 µm as vc decreases from 2.25 to 1.8 to 1.5 mm/min. The visual comparison between Figures 3c-e (see Supporting Information) further supports our speculation that the crystalline domain orientation is random, and the domain size increases as the pulling speed decreases while meeting the optimum conditions by eq 11. When the optimum condition by eq 11 is not met at the same pulling speed (1.8 mm/min), going from Figure 3d to 3f, the average domain size becomes smaller, while the orientation distribution appears similarly random.

This


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understanding of structural ordering provides a simple solution to introducing a small degree of randomness into the structure: we increase the pulling speed while adjusting the surface pressure to be at the optimum condition. We also characterized the level of structural order in multiple layers, where the number of layers varies from 2 to 8. To determine how the optimum surface pressure depends on the number of layers (n) in the LB assembly, we fixed the pulling speed at 1.5 mm/min and determined the optimum surface pressure that maximized G for each layer in the multilayer structure. Figure 4a shows this optimum surface pressure as a function of n. The optimum surface pressure must increase as one increases the number of layers in the assembly. For the first three layers, the optimum surface pressure increases rather steeply as the number of layers increases. When the number of layers is greater than 3, the optimum surface pressure increases more gradually for the maximum order. This dependence of the optimum surface pressure on n further supports our model in eq 11. From eq 11, we express A/N in terms of n as

J evap  A Vp  1 j0 = ⋅ +1−  . N d  (1 − ε )αd (vc + j1 ) n (1 − ε )(vc + j1 )

(13)

j1 is determined to be positive from the linear fit in Figure 3b, so that A/N decreases as n increases according to eq 13. Because A/N decreases as the surface pressure increases, this means that one must increase the surface pressure as n increases, which agrees with our experiment in Figure 4a. In fact, the least square fit to eq 13 with two fitting parameters agrees well with the experimental data shown in Figure 4a (a coefficient of determination of 0.995). The two non-dimensional fitting parameters are

J evap

(1 − ε )αd (vc + j1 )

and 1 −

j0 . (1 − ε )(vc + j1 )


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To clearly state the implication of eq 11 that governs maximum order in LB assembly, we present a graph (Figure 4b) of surface pressure as a function of pulling speed for the nth layer, using the two fitting parameters. Each line represents an optimum condition for maximum possible order. The experimentally determined optimum points found in Figures 3a and 4a are also displayed in Figure 4b. The model curve agrees well with the experimental data. According to the model calculations, one must increase the surface pressure as the pulling speed or the number of layers increases. The optimum surface pressure increases rapidly as the pulling speed increases when the pulling is high, especially when n is large. In prior studies, the pulling speed ranged from 1 to 10 mm/min.27-28, 32, 34 If one is to use a high pulling speed, the optimum surface pressure should be controlled carefully as the dependence between the two parameters is rather strong at a high pulling speed.

For controlled introduction of randomness in multilayer

assembly, one should start from the optimum pulling speed and pressure for the given nth layer and gradually increase the pulling speed while adjusting the surface pressure to satisfy eq. 11.

5. CONCLUSIONS We have determined how structural randomness can be controllably introduced into periodic structures during Langmuir-Blodgett assembly of microspheres. Specifically, we have investigated the relation between two process parameters: surface pressure and substrate pulling speed. To maximize the structural order and thus for minimal randomness, the two parameters must satisfy a relation based on material balance and Langmuir isotherm. We discovered that a small deviation from this relation can introduce a large degree of randomness, as in the case for varying the substrate pulling speed while holding the surface pressure constant. To controllably introduce randomness, we find that one should start the assembly from optimum pulling speed and surface pressure that satisfy their interrelation (eq 11) for a given nth layer, but gradually


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increase the pulling speed. During this increase, one will have to adjust the surface pressure to satisfy eq 11. Overall, we expect that our findings will prove useful in mimicking biological photonic structures, as they display appreciable randomness in periodic patterns to render unique optical properties.

In our future work, we will elucidate how the randomness in ordered

structures influences optical properties such as angle-dependent reflectance and emissivity. Our modeling is based on material balance alone; however, an investigation accounting for detailed energetics of 2D colloidal crystallization (e.g., wetting characteristics of substrate, line tension, latent heat of evaporation of water, etc.) would elucidate the Langmuir-Blodgett assembly process at a more fundamental physics level. In particular, energetics approach would be able to explain the role of surface energy and thermal energy in the crystallization process.

ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge via the Internet at http://pubs.acs.org. Derivation of eq 1; determination of the degree of sphere immersion; original SEM images of Figure 3c–f.

AUTHOR INFORMATION Corresponding Author *Email [email protected].

ACKNOWLEDGMENTS


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SEH and SMH acknowledge the generous financial support from the National Science Foundation (NSF) CAREER Award (DMR-1555290) and NSF Sustainable Energy Pathways (NSF-SEP) Program (CHE-1231046).


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Table of Contents Graphic


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Figure 1. Schematic illustration of the LB assembly of microspheres and the definition of the assembly parameters. Inset illustrates the true projected area and the effective projected area. 258x212mm (96 x 96 DPI)


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Figure 2. Experimentally determined LB isotherm (black circles) and a fitting to the 2D van der Waals equation of state (red curve). 88x67mm (300 x 300 DPI)


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Figure 3. (a) Crystalline domain sizes determined at various values of surface pressure and pulling speed for a monolayer assembly. (b) vc as a function of φ / (1–φ) for the (c)-(e) points in (a). (c)-(f) Representative SEM images corresponding to the monolayer assembly conditions indicated as (c)-(f) in (a). For the convenience of view, the crystalline domains are colored depending on the orientation from blue to green and the disordered regions are painted red. 477x356mm (96 x 96 DPI)


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Figure 4. Dependence of optimum surface pressure on (a) the number of layers and (b) the pulling speed for various numbers of layers for maximum order in LB microsphere assembly. 88x133mm (300 x 300 DPI)


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Control of Randomness in Microsphere-Based Photonic Crystals

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