Crystallization of Bidisperse Repulsive Colloids in Two-Dimensional

Jun 9, 2010 - Crystallization of Bidisperse Repulsive Colloids in Two-Dimensional Space: A Study of Model Systems Constructed at the Air−Water Inter...
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Crystallization of Bidisperse Repulsive Colloids in Two-Dimensional Space: A Study of Model Systems Constructed at the Air-Water Interface Jaehyun Hur, Nathan A. Mahynski, and You-Yeon Won* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907 Received April 2, 2010. Revised Manuscript Received May 25, 2010 We report the structural behavior for mixtures of two differently sized (“bidisperse”) silica microspheres at the air-water interface under three different size ratio conditions (R ( RS/RL) = 0.375, 0.500, and 0.579). These bidisperse silica monolayers were studied via measurement of the surface pressure-area isotherm and optical microscopy at various particle surface coverages (θ  θS þ θL) during compression. The silica colloids used in these trials were found to possess purely repulsive pair interactions at the air-water interface, which was confirmed by the pair correlation function calculated from the analysis of many optical images of the particles taken at dilute concentrations. The results revealed that, at certain mixture compositions (β  θL/θS), compression can lead to the formation of 2D binary crystal structures. Specifically, at a particle size ratio of R = 0.375, LS1 crystal domains were observed at a surface coverage of θ ≈ 0.619 when β = 7.00 and 3.50, although this LS1 structure was not observed at higher total particle densities (where the system became phase-separated). At a size ratio of R = 0.579, compression produced 2D LS2 binary crystals at particle surface coverages (θ) above 0.641 when β = 3.00, 1.50, or 1.00. However, at a size ratio of R = 0.500, compression triggered macroscopic phase separation, leading to the formation of two separate hexagonal-close-packed domains consisting purely of either large or small particles. In general, when the mixture composition (β) was too different from the stoichiometric ratio needed for the formation of LS1 or LS2 superlattices, the bidisperse monolayer was observed to remain in an amorphous state rather than evolving to an ordered phase under compression. These findings suggest that, in two dimensions, contrary to what has been speculated in the literature, (1) purely repulsive pair potentials can give rise to LS1 and LS2 binary crystals under compression and also (2) perfectly spherical particles can form LS2 crystals. This discrepancy between our results and the predictions of previous simulations might indicate that the capillary interaction and/or the many-body effects play a significant role in determining the structure of bidisperse colloids at the air-water interface.

I. Introduction Binary crystals occur frequently in interatomic compounds (e.g., NaCl, AlB2, and NaZn13). It is well-known that colloidal crystals isostructural with these metallic compounds, such as LS1, LS2 and LS13 superlattices, can also be formed in mixtures of two differently sized (i.e., “bidisperse”) colloids. Examples of bidisperse colloids in which ordered three-dimensional (3D) superlattices have been observed include the Brazilian opal, which is composed of 180 and 100 nm diameter SiO2 colloids dispersed in CaCO3,1 bidisperse poly(methyl methacrylate) (PMMA) latices,2 and binary mixtures of various metal nanoparticles.3 Also, in two dimensions (2D), bidisperse colloids have been shown to be able to form binary crystals; for instance, 2D LS1 (cubic) and LS2 (hexagonal) superlattices have been observed in monolayers of bidisperse/hybrid mixtures of Au and Ag nanoparticles.4 For 3D systems of bidisperse hard spheres, both theory5 and experiment2b-e have shown that the translational entropy of the *To whom correspondence should be addressed. E-mail: yywon@ecn. purdue.edu. (1) (a) Sanders, J. V. Philos. Mag. A 1980, 42(6), 705–720. (b) Sanders, J. V.; Murray, M. J. Nature 1978, 275(5677), 201–203. (2) (a) Bartlett, P.; Campbell, A. I. Phys. Rev. Lett. 2005, 95(12), 128302. (b) Bartlett, P.; Ottewill, R. H.; Pusey, P. N. Phys. Rev. Lett. 1992, 68(25), 3801–3804. (c) Hunt, N.; Jardine, R.; Bartlett, P. Phys. Rev. E 2000, 62(1), 900–913. (d) Schofield, A. B. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2001, 64(5 Pt 1), 051403. (e) Schofield, A. B.; Pusey, P. N.; Radcliffe, P. Phys. Rev. E 2005, 72(3), 031407. (3) Shevchenko, E. V.; Talapin, D. V.; Kotov, N. A.; O’Brien, S.; Murray, C. B. Nature 2006, 439(7072), 55–59. (4) (a) Kiely, C. J.; Fink, J.; Brust, M.; Bethell, D.; Schiffrin, D. J. Nature 1998, 396(6710), 444–446. (b) Kiely, C. J.; Fink, J.; Zheng, J. G.; Brust, M.; Bethell, D.; Schiffrin, D. J. Adv. Mater. 2000, 12(9), 640–643. (5) (a) Eldridge, M. D.; Madden, P. A.; Frenkel, D. Nature 1993, 365(6441), 35– 37. (b) Trizac, E.; Eldridge, M. D.; Madden, P. A. Mol. Phys. 1997, 90(4), 675–678.

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particles can give rise to thermodynamically stable binary superlattice structures with LS1, LS2, and LS13 symmetries. Incorporation of electrostatic attraction between the two component colloids (by using pairs of oppositely charged colloids) has been shown to increase not only the stability of the resultant superlattices but also the diversity of binary crystal structures that can be formed.2a,3,6 Interestingly, however, for 2D situations, random sequential adsorption (RSA) and melting simulations on bidisperse hard spheres suggest that translational entropy alone cannot produce the LS1 or LS2 structures observed experimentally.7 It has been shown by using the same simulation method that the addition of van der Waals, osmotic, and steric contributions to the pair interaction potentials can stabilize LS1 domains under compressive conditions, suggesting that the attractive interactions may play an important role in the stabilization of the LS1 morphology in two dimensions.8 On the other hand, the RSA/melting simulations predicted that even the attractive potential cannot stabilize the LS2 phase in 2D geometry, from which the authors of the study speculated that it is perhaps the faceted structure of Au/Ag nanoparticles that has enabled the LS2 structure observed in experiment.8 We also note that, in the experiments that demonstrated LS1 and LS2 superlattices in bidisperse metal nanoparticles, the monolayer samples were formed on solid surfaces by solvent evaporation, and thus the capillary interaction between particles is expected to have been strongly operative in the formation of the (6) Leunissen, M. E.; Christova, C. G.; Hynninen, A. P.; Royall, C. P.; Campbell, A. I.; Imhof, A.; Dijkstra, M.; van Roij, R.; van Blaaderen, A. Nature 2005, 437(7056), 235–240. (7) Doty, R. C.; Bonnecaze, R. T.; Korgel, B. A. Phys. Rev. E 2002, 65(6), 061503. (8) Rabideau, B. D.; Bonnecaze, R. T. Langmuir 2004, 20(21), 9408–9414.

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binary crystals; it is theoretically possible, but not necessarily easy in practice, to incorporate the capillary effect into a standard pairpotential model.9 Therefore, these previous findings raise a few interesting questions regarding the crystallization behavior of bidisperse colloids in two dimensions: one, is an attractive potential necessary for the formation of LS1 domains; two, does particle shape play a role in the stabilization of LS2 domains (would bidisperse mixtures of perfect spheres be able to form LS2 superlattices); and three, how important is capillary interaction in the stabilization of these structures? In an attempt to answer some of these questions, in particular the first two questions, we have conducted an experimental investigation of the structural behavior of bidisperse silica microspheres spread at the air-water interface. The present paper reports the results of this study. In this work, we used monodisperse silica microspheres of four different diameters (0.75, 1.50, 2.00, and 2.59 μm) to prepare bidisperse mixtures of these particles at three different size ratios, R ( RS/RL) = 0.375, 0.500, and 0.579; these R values represent the respective size ratio conditions under which the LS1, phase-separated hexagonal, and LS2 binary crystal structures have previously been observed in monolayers of bidisperse metal nanoparticles.4 For all combinations of pairs of particles, the 2D pair interaction potentials have been measured to be purely repulsive. The compressioninduced crystallization phenomena in these bidisperse silica monolayers have been studied by optical microscopy and surface pressurearea isotherm measurements at various overall dimensionless surface coverages (θ  θS þ θL) and various surface coverage ratios between small and large particles (β  θL/θS). The results are compared in detail with previous experiments and simulations.

II. Experimental Section II.1. Preparation of Particle Suspensions. In this study, we used hydrophilic (hydroxy-functionalized) monodisperse silica microspheres of four different sizes. The diameters of these silica particles were 2.59 ( 0.12 μm (Bangs Laboratory, cat. # 5803), 2.00 ( 0.10 μm (Polysciences, cat. # 24328), 1.50 ( 0.08 μm (Polysciences, cat. # 24327), and 0.75 ( 0.09 μm (synthesized in our laboratory by the St€ ober method10), where the errors represent the standard error of the mean. Prior to use, the particles were purified by repeated centrifugation and redispersion with ethanol (Pharmco Products 200 proof, two times), isopropyl alcohol (Aldrich, three times), Milli-Q pure water (three times), and methanol (Aldrich, 99.9% purity, three times), followed by drying under vacuum overnight at 70 °C in a heated silicone oil bath. For preparation of a particle mixture, appropriate amounts of purified individual particles at a designated mixing ratio were added to a glass vial, and then a measured volume of methanol (Aldrich, 99.9% purity) was added to the dry particle mixture; these particle suspensions were prepared typically at a concentration of 0.01 g/mL. The mixture of silica particles dispersed in methanol was stirred using a magnetic stir bar for at least 24 h and then sonicated for at least 15 min prior to use to ensure homogeneous dispersion of the bidisperse particles. Typically, samples prepared through the thorough purification procedure described above did not aggregate when spread at the air-water interface.

II.2. Preparation of the Langmuir Monolayers and Measurement of the Surface Pressure-Area Isotherms of One- and Two-Component Silica Microspheres at the Air-Water Interface. Figure 1 depicts, schematically, the procedures used for the preparation and observation of the monolayers of monodisperse (9) Fernandez-Toledano, J. C.; Moncho-Jorda, A.; Martinez-Lopez, F.; HidalgoAlvarez, R. Theory for Interactions between Particles in Monolayers. In Colloidal Particles at Liquid Interfaces; Binks, B. P., Horozov, T. S., Eds.; Cambridge University Press: Cambridge, UK, 2006; pp 108-151. (10) (a) Stober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26(1), 62–&. (b) Bogush, G. H.; Tracy, M. A.; Zukoski, C. F. J. Non-Cryst. Solids 1988, 104(1), 95–106.

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Figure 1. Sketches describing the experimental procedures for the preparation and imaging of monolayers of monodisperse and bidisperse silica microspheres at the air-water interface using a Langmuir trough mounted on an inverted microscope. and bidisperse silica particles confined at the air-water interface. The commercial Langmuir miniature trough (Kibron Microtrough XS) was thoroughly cleaned before the preparation of the subphase by rinsing it with ethanol and warm water in an alternating fashion at least five times, and then drying it with filtered compressed air. Quickly afterward, fresh Milli-Q purified water (30 mL) was applied to the trough to form the subphase. The subphase surface was then repeatedly aspirated to remove any surface contaminants until the surface pressure became stable to within (0.05 mN/m during compression from 118 to 10 cm2. The Wilhelmy probe (metal alloy) used to take these surface pressures was cleaned by flaming it with a butane welding torch for approximately 30 s to burn off any organic contaminants. The surface pressure was measured by the Wilhelmy plate method; a microbalance attached to the plate measures the force applied on the plate due to wetting, which allows the surface tension of the film on the subphase to be calculated with the known wetted length of the Wilhelmy plate, and the contact angle (∼0°). For the preparation of a Langmuir monolayer of particles, the appropriate volume (∼0.400 mL) of the particle suspension was carefully dispensed dropwise onto the subphase surface using a 50 μL microsyringe (Hamilton 705N), and then the solvent was allowed to evaporate for 30 min; this time scale was found by measuring the time-dependent surface pressure after application, where the time scale for surface pressure stabilization was found to be approximately 30 min. Surface pressure-area isotherms were measured by recording the surface pressure over the course of the monolayer’s compression, with the barriers set at a compression speed of 1.0 mm/min controlled by the computer software (FilmWare v.3.5), at a constant subphase temperature of 25 °C (controlled by a circulating water bath). The trough area (A) available to the system was converted into the dimensionless number, 1/θ (the reciprocal value of the area fraction occupied by the particles, or the reciprocal dimensionless surface density); the relationship for a one-component system with N total particles in it is 1=θ ¼

A NπR2

ð1Þ

while for a two-component system 1=θ ¼

A NS πRS 2 þ NL πRL 2

ð2Þ

where NS, NL, RS, and RL denote the number of small particles, the number of large particles, the radius of small particles, and the radius of large particles, respectively. Because of the hydrophilic nature and the high density of the silica material, loss of a certain fraction of particles by sedimentation into the subphase was unavoidable during spreading. The sedimented amounts were Langmuir 2010, 26(14), 11737–11749

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Figure 2. (A) Representative optical microscopy image of 1.50 μm diameter silica particles at the air-water interface taken at an area fraction occupied by the particles of θ = 0.004. The size of the image is 155 μm  112 μm. (B) 2D pair correlation functions, g(r), of the particles as functions of the interparticle center-to-center distance, r, were estimated from the optical microscopy images of the particles taken at two different particle densities, θ = 0.004 (open circle) and θ = 0.006 (cross). The two g(r) curves coincide within 1% difference. (C) Pair interaction potentials U(r)/kBT, computed from the pair correlation functions shown in (B) using the Boltzmann relation, demonstrating that the pair interactions between the silica particles are purely repulsive. (D) Comparison of the pair interaction potential of the 1.50 μm diameter silica from experiment (circles) with the predictions of the theoretical model described in section III.1 (lines). The theoretical curves were obtained by fitting the experimental data to the model with either one (black curve) or three (red curve) adjustable parameters as described in the text. measured to be 6%, 7%, 8%, and 10% for the 0.75, 1.50, 2.00, and 2.59 μm diameter silica particles, respectively; this information was obtained by comparing the theoretical surface coverage of the particles calculated based on the stoichiometry of the starting material and the actual coverage value (θ) measured by optical microscopy. It was found that no additional sedimentation normally takes place during the monolayer compression. Therefore, it needs to be clarified that all the θ values used or reported in this paper represent the surface coverage values corrected for this initial sedimentation of the particles.

II.3. Reversibility of the Pressure-Area Isotherms of OneComponent Silica Microspheres at the Air-Water Interface, and the Effect of the Compression Speed on the Isotherm Shape. Pressure (π)-area (1/θ) isotherms were also collected for a

system of monodisperse 1.50 μm diameter silica spheres in an effort to examine the dependence, if any, of the structures formed on the rate at which the trough’s barriers were compressed, and the reversibility of these structures. The monolayer was prepared in the same way as described in section II.2, and compressions were performed at 1.0, 3.0, and 5.0 mm/min. Trials at each of these speeds were then independently repeated several times to ensure reproducibility. Compressions of the monolayer at the different speeds do not yield different isotherms, in particular, in the compressible (fluidlike) and incompressible concentration regimes (see Supporting Information Figure S1); also, see section III.2 for the respective definitions of the “compressible” and “incompressible” states of the monolayer. All isotherm data presented in this Article (except for the data in Figure S1) were measured at a compression speed of 1.0 mm/min. Reversibility trials were conducted by preparing the monolayer as previously described, compressing at 1.0 mm/min, then pausing compression at representative points (congruent with those designated in Figure 4), and then allowing the barriers to relax at the same rate Langmuir 2010, 26(14), 11737–11749

of 1.0 mm/min. Each film was prepared independently for each trial to prevent any pre-existing condition induced by a prior compression or relaxation of the film to influence the results of an ensuing one. Any difference observed in the isotherm between the compression and relaxation runs would indicate an irreversible change in the properties of the monolayer. As discussed in section III.2, complete reversibility is observed (see Supporting Information Figure S2) until the formation of wrinkled structures (see Figure 4B(7)) causes the film to irreversibly deform, depositing some of the silica microspheres into the subphase. Prior to this limit, throughout the formation of hexagonally close-packed structures, complete reversibility is still maintained. II.4. Optical Microscopy. For optical microscopy imaging of the monodisperse/bidisperse silica monolayers under various particle concentration conditions, the Kibron Microtrough XS (23.0 cm  5.9 cm  0.1 cm) instrument was mounted on a stage of a Leica DMIRB inverted optical microscope, housed within a plexiglass chamber to prevent any disturbance or contamination of the monolayer sample from the environment during measurement. In order to minimize vibration noise, the microscope was placed on a vibration isolation device (TS-140 Table Stable). In order to take images, the monolayer was compressed in sequential steps to designated values of particle concentration. At each concentration condition, once the concentration is reached through compression, the monolayer was allowed to relax for 15 min before imaging. Images were recorded using a 40 objective lens and imaging software (QCapture Suite v.2.90.1.0). To ensure an image was statistically representative, many images were taken at multiple locations in the film at sufficient distances from one another.

II.5. Measurements of the 2D Pair Correlation Functions. The pair correlation functions, g(r), for the monodisperse and bidisperse silica microsphere monolayers (presented in Figures 2B and 3A,C,E) were computed from particle positions estimated by DOI: 10.1021/la101313r

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Figure 3. 2D pair correlation functions (g(r)) (A, C, E) and pair interaction potentials (U(r)) (B, D, F) between particles of type i and type j, as

functions of the normalized interparticle center-to-center distance (r/dij where dij  (di þ dj)/2), for bidisperse mixtures of silica microspheres at three different particle diameter ratios, i.e., R ( dS/dL) = 0.375 (A, B), 0.500 (C, D) and 0.579 (E, F). For each size ratio condition, the pair correlation functions have been estimated for both like (i.e., small-small (gSS) and large-large (gLL)) and unlike particle (small-large (gSL)) combinations. In all plots, we use red, blue and black symbols to denote, respectively, the SS, LL and SL pair combinations. For each size ratio, the particle pair correlations have been evaluated at two different total particle area fractions: e = 0.004 (circle) and 0.006 (cross).

analysis of the optical images using the following formula:29 N P

gðrÞ ¼ Fbulk

N P

i¼1

ni ðrÞ ð3Þ

½2πrdr - δAi ðrÞ

i¼1

where ni(r) denotes the number of particle centers within a ring of radius r and thickness dr centered at the ith particle, 2πrdr is the bin area, δAi(r) is the missing bin size due to the finite size of the image (“edge effect”) centered at the ith particle near the boundary of the image, and Fbulk (= N/Atotal) is the average particle density for the entire image. In the bidisperse mixture cases, the pair correlation functions between like-particles (small-small (gSS) and large-large (gLL)) and unlike-particles (small-large (gSL)) were both calculated.

III. Results and Discussion III.1. Confirmation of the Purely Repulsive Nature of the Interparticle Interactions of Silica Colloids at the AirWater Interface. We measured the pair interaction potentials, U(r), for silica microspheres confined at the air-water interface using the established procedures;11 for each particle system of interest, we digitally recorded optical images of the particles in 11740 DOI: 10.1021/la101313r

equilibrium at sufficiently low particle densities, analyzed the images to calculate the two-dimensional pair correlation function, g(r), of the particles as a function of the interparticle center-to-center separation, r (see section II.4 for procedures), and converted this g(r) information into a U(r) profile using the Boltzman relation which holds in the limit of infinite dilution of particles, U(r)/kBT ≈ -ln(g(r)).12 Figure 2A displays a representative image obtained from a system of 1.50 μm diameter silica spheres. Optical microscopy measurements were conducted at various particle surface coverages (i.e., at various particle area fraction (θ) values). At the lowest surface coverage examined (θ = 0.004), for instance, about a thousand images, taken at a fixed field of view at time intervals of 5 s, containing a total of more than 105 particles were collected to determine, with sufficient statistics, the ensemble-averaged pair correlation function, g(r); see Figure 2B. An increase of θ from 0.004 to 0.006 produces no change in the g(r) curve within an error of (1% (Figure 2B), which indicates that these surface coverage values are (11) (a) Behrens, S. H.; Grier, D. G. Phys. Rev. E 2001, 64(5), 050401. (b) Chen, W.; Tan, S. S.; Huang, Z. S.; Ng, T. K.; Ford, W. T.; Tong, P. Phys. Rev. E 2006, 74, 021406. (12) Hansen, J.-P.; McDonald, I. R. The Theory of Simple Liquids, 2nd ed.; Elsevier Academic Press: San Diego, CA, 1986.

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Figure 4. (A) Surface pressure (π) versus dimensionless area per particle (1/θ) isotherm for monodisperse 1.50 μm diameter silica spheres at the air-water interface. The filled circles denote the points at which the images shown in (B) were taken. (B) Representative optical microscopy images (73 μm  57 μm) of the monodisperse silica monolayer taken during the compression of the monolayer at the seven different dimensionless surface coverages of θ = (1) 0.225, (2) 0.355, (3) 0.571, (4) 0.651, (5) 0.709, (6) 0.820, and (7) 0.971. The notations θo and θHS denote, respectively, the surface coverages at the onset of the compressible-to-incompressible transition and at the onset of hard-sphere-like repulsion.

sufficiently within the dilute region where many-body correlations are effectively negligible, thus confirming that the aforementioned Boltzmann-distribution equation is applicable for the calculation of the pair potential. The resultant pair interaction potential (Figure 2C) indicates that the interaction between the 1.50 μm silica particles is purely repulsive and that this repulsion is significantly long-ranged relative to the screened Coulomb repulsion expected from the DLVO theory; for example, U(r)/kBT reaches a value of about 9 at any distance less than r/d = 3 (where d denotes the particle diameter), whereas the Debye screening length (κ-1) is estimated to be only about 10 nm, which is only one one-hundred-and-fiftieth of the diameter of the particle. To explore the origin of this longrange repulsion, we fit the experimental U(r) profile with the following theoretical equation (as suggested by Chen et al.13 in the no-added-salt limit where the in-plane contribution to the dipole-dipole interaction is negligible): UðrÞ ¼ Uvdw ðrÞ þ Uele ðrÞ þ Udip ðrÞ Uvdw ðrÞ = Uele ðrÞ =

Ad 24ðr - dÞ

ð5Þ

q2 expð- Kðr - dÞÞ 2

4πε0 εð1 þ Kd=2Þ r

Udip ðrÞ =

Pz 2 4πε0 εr3

ð4Þ

ð6Þ

ð7Þ

(13) Chen, W.; Tan, S. S.; Zhou, Y.; Ng, T. K.; Ford, W. T.; Tong, P. Phys. Rev. E 2009, 79(4), 041403.

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where A is the effective Hamaker constant at the air-water interface, assumed to be equal to (Asilica/air/silica þ Asilica/water/silica)/2 (that is, we assume that the immersed portion of one sphere only interacts with the immersed portion of another sphere, and likewise for the unimmersed hemispheres),9 q is the total charge per particle, ε0 is the vacuum permittivity, hε is the effective dielectric constant of the airwater interface (≈ (εair þ εwater)/2), and Pz is the dipole moment of the particle perpendicular to the air-water interface; it should be pointed out that the above assumption regarding the effective Hamaker constant does not influence the nature of our analysis, because the magnitude of Uvdw(r) is relatively very small. The Hamaker constant values are known in the literature: Asilica/air/silica = 6.5  10-20 J14 and Asilica/water/silica = 7.9  10-21 J.15 However, q, κ-1, and Pz are unknown parameters. Two different fitting procedures were used: in one approach, we fixed the value of κ-1 to 29 nm, which has been estimated from a measurement of the conductivity of a 0.01 g/mL suspension of 1.50 μm silica particles in (initially deionized) water (see our previous publications16 for details), the value of q was set to -4.6  103 e per particle (where e is the elementary charge), which was estimated from the above κ-1 value and the measured pH of the suspension (≈7.9) using the known correlation between these quantities,17 and Pz was used as the only adjustable parameter to fit the experimental U(r) profile with the theory. The best-fit value of the out-of-plane dipole moment was found to be Pz = 9.6  10-23 C 3 m (2.9  107 D); the fit result is shown as a black curve in Figure 2D. In the other procedure, we used all three parameters to fit the data; the best-fit values were found to be κ-1 = 10 nm, q = -3.4  103 e, and Pz = 8.3  10-23 C 3 m (2.5  107 D), which are not much different from the values obtained in the first method. The best-fit curve for the second method is also displayed as a red line in Figure 2D. Overall, it is obvious from Figure 2D that neither of the methods used is capable of precisely reproducing the experimental data. For instance, even the threeparameter fit significantly underestimates the repulsive potential at short separations, suggesting that other (repulsive) contributions to the pair-interaction energy, which have not been considered in the above model, may exist. One might suspect that a likely cause of this discrepancy is the capillary interaction between the partially immersed silica spheres (i.e., the interaction due to the overlap of the water menisci between the silica spheres); for the capillary interaction, a simple equation, such as those given above for other types of interactions, does not exist. However, a theory based on numerical integration of the Laplace equation predicts that, for hydrophilic colloids such as the silica particles used in our experiments (of which the water contact angles are expected to be less than 20° on the basis of literature data),18 the meniscus interaction will lead to the opposite effect, creating an attraction between particles.19 The hydrophilic character of the silica may produce a repulsive potential due to the structuring of water molecules near the particle’s surface. However, this effect is known to occur at significantly shorter ranges (typically, less than a few nanometers)20 than the repulsive potential seen in Figure 2D. At the present time, we speculate that the discrepancy between the model and the experiment might be due to a possible nonuniformity of the shape of the (14) Hough, D. B.; White, L. R. Adv. Colloid Interface Sci. 1980, 14(1), 3–41. (15) Ren, J.; Song, S.; Lopez-Valdivieso, A.; Shen, J.; Lu, S. J. Colloid Interface Sci. 2001, 238(2), 279–284. (16) (a) Hur, J.; Won, Y. Y. Soft Matter 2008, 4(6), 1261–1269. (b) Hur, J.; Won, Y. Y. Langmuir 2008, 24(10), 5382–5392. (17) Bolt, G. H. J. Phys. Chem. 1957, 61(9), 1166–1169. (18) Deak, A.; Hild, E.; Kovacs, A. L.; Horvolgyi, Z. Phys. Chem. Chem. Phys. 2007, 9(48), 6359–6370. (19) Kralchevsky, P. A.; Paunov, V. N.; Ivanov, I. B.; Nagayama, K. J. Colloid Interface Sci. 1992, 151(1), 79–94. (20) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992.

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three-phase contact line surrounding the particle, which has been suggested to be capable of producing a repulsive pair potential at separation distances of the order of the particle diameter;21 this effect might, in fact, be non-negligible given that the surface of a silica microsphere is laterally heterogeneous on the scale of tens of nanometers,13 which is predicted to be sufficient to produce the above-described effect.21 Precise determination of the nature of the interactions between silica microspheres at the air-water interface would be beyond the scope of this paper. Nonetheless, it is important to note (again) that, under the given experimental conditions, the silica pair interaction potential is purely repulsive, and a major contribution to this repulsion appears to result from the polarization of the charge distribution on the silica surface (e.g., Udip(r) is estimated to be about 4kBT at r/d ≈ 3, whereas Uele(r) becomes a negligibly small quantity (, kBT) at that separation distance). It is also worth noting that, even under lower ionic strength conditions, Chen et al.13 reported an attraction between 0.74 μm diameter silica particles at the air-water interface at a particle area fraction of θ = 0.010; we suspect the difference in θ values used in the respective studies to be the source of this discrepancy in interpretation, because at this higher area fraction we also observed a similar behavior, namely, the existence of a peak in the g(r) profile for our system (data not shown), which one might have interpreted as an indication of a net attractive pair interaction. Similar purely repulsive behaviors have also been confirmed for bidisperse mixtures of silica microspheres at the air-water interface. We used three different bidisperse systems having three different size ratios (R ( RS/RL) = 0.375, 0.500, and 0.579); these size ratio conditions were produced by using, respectively, a 1:1 (ratio by number, βN ( NL/NS) = 1) mixture of 0.75 and 2.00 μm diameter silica particles, a 1:1 (βN) mixture of 0.75 and 1.50 μm silica particles, and a 1:1 (βN) mixture of 1.50 and 2.59 μm silica particles. Figure 3 displays the two-dimensional pair correlation functions, gij(r), and the pair interaction potentials, Uij(r), between particles of type i and j under the three different size ratio conditions. For each size ratio, the pair correlation functions have been estimated for both like (i.e., small-small (gSS) and large-large (gLL)) and unlike particle (small-large (gSL)) combinations. For all bidisperse systems examined, the particle pair correlations have been evaluated at two different total particle area fractions: θ (= θS þ θL) = 0.004 and 0.006. The gij(r) profiles are found to be identical between the two surface coverage conditions, which justifies the use of the Boltzmann equation (i.e., Uij(r)/kBT ≈ -ln(gij(r))) for the calculation of the pair interaction properties, Uij(r). As shown in Figure 3, at all conditions studied, the pair potentials have been purely repulsive in nature (i.e., Uij(r) > 0 for all r). Also of note, the range of the interaction between an unlike particle pair, USL(r), is generally found to be exactly intermediate to those of the small-small and large-large particle combinations (i.e., USS(r) and ULL(r), respectively), which is consistent with the assumption used in the simulation study of ref 7. III.2. Structure and Surface Pressure Behavior of Monodisperse Silica Colloids at the Air-Water Interface. In this section, we will provide an explanation of (i) the general surface pressure-area isotherm behavior of a silica monolayer at the air-water interface and (ii) the relationship between the surface pressure property and the structure of the monolayer. As a model system for this investigation, we used a monolayer consisting of monodisperse 1.50 μm diameter silica spheres; the analysis of this (21) Stamou, D.; Duschl, C.; Johannsmann, D. Phys. Rev. E 2000, 62(4), 5263– 5272.

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one-component system will be useful for discussions of the data for bidisperse systems in later sections. Figure 4A presents the surface pressure (π) versus dimensionless area per particle (1/θ) isotherm data for the monodisperse system of 1.50 μm silica particles, and Figures 4B(1)-(6) show representative optical microscopy images of the monolayer at seven different 1/θ values. It should be mentioned that the 1/θ values on the x-axis in all isotherm plots presented in this paper represent corrected 1/θ values for the loss of particles into the subphase which occurs during the initial spreading process (as explained in detail in the section II.2). As can be seen from Figure 4A, at low compression (e.g., at the points labeled (1), (2), and (3) in the figure), the silica monolayer is almost completely compressible; that is, the isotherm is nearly flat. We note that this behavior is different from that of the monolayers formed by molecular species such as lipids22 and polymers;23 in a Langmuir monolayer of lipids or polymers, the change in surface pressure follows the 2D ideal gas law at low concentrations of the material, and the isotherm curve normally starts deviating from the baseline at a very low surface coverage. We believe that this difference comes from the fact that in the silica particle case the Brownian motion of the silica microspheres is relatively very slow, and thus these particles do not create sufficient surface pressure until the surface coverage becomes sufficiently large. When the surface coverage passes a certain threshold (i.e., point (4) in Figure 4A), the monolayer undergoes a transition from the compressible state to an incompressible state; hereafter, we will use the notation θo to denote the surface coverage at the onset of this compressible-to-incompressible transition. In our system, the value of the onset coverage was determined to be θo = 0.65 ( 0.02; this onset point was defined as the point where the surface pressure deviates more than 0.1 mN/m from the baseline pressure. As shown in Figures 4B(1)-(4), the incompressible transition in the surface pressure behavior reflects the transformation of the monolayer structure from a fluidlike to a hexagonal close-packed arrangement of the particles. It should be noted that the measured θo value (= 0.65) is significantly lower than the critical surface coverage values predicted by simulation for the fluid-hexatic and hexatic-to-hexagonal phase transitions (θc = 0.70 and 0.735, respectively) in 2D hard-sphere systems,24 further supporting that the repulsion between the silica microspheres is longer-ranged than that of hard spheres (Figure 2C). However, the actual value of the interparticle center-to-center distance at the onset of the incompressible transition (r/d = 1.2) is much smaller than the range of repulsion (r/d ≈ 3) estimated from the pair potential data in Figure 2C, where the repulsion range is defined (somewhat arbitrarily but conservatively) to be the distance that gives a pair interaction energy U(r) of 10kBT; this comparison reveals that the many-body effect (i.e., the presence of other particles in the neighboring region) reduces the effective repulsive forces between two adjacent particles. In the incompressible region (e.g., at points (5) and (6) in Figure 4A), further compression of the monolayer induces a large mechanical resistance, as seen by the steep rise in the surface pressure. As shown in Figures 4B(5) and (6), in this regime, the monolayer film adopts a buckled shape in order to accommodate the reduced total area available to the particles. This buckling deformation is completely reversible; see Figures S2B-D of the (22) Kjaer, K.; Alsnielsen, J.; Helm, C. A.; Laxhuber, L. A.; Mohwald, H. Phys. Rev. Lett. 1987, 58(21), 2224–2227. (23) Witte, K. N.; Kewalramani, S.; Kuzmenko, I.; Sun, W.; Fukuto, M.; Won, Y. Y. Macromolecules 2010, 43(6), 2990-3003. (24) Gray, J. J.; Klein, D. H.; Bonnecaze, R. T.; Korgel, B. A. Phys. Rev. Lett. 2000, 85(21), 4430–4433.

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Supporting Information, which shows that as the monolayer that has initially been compressed to an incompressible state is relaxed back to the fluid state, the surface pressure decreases, retracing the exact same path of the compression isotherm. Therefore, in the incompressible regime, the deformation caused by the compressive stress is purely elastic and reversible. Following the analysis method of Kumaki,25 we constructed an extrapolation line from the linear portion of the isotherm curve encompassing the point of highest slope and determined the x-intercept of this line, which gave an estimation of the surface coverage at the onset of hard-sphere-like repulsion (θHS = 0.89 ( 0.02). The resulting θHS value agrees well with the theoretical value of the closest hexagonal-packing √ density of hard spheres in two dimensions, that is, θHS (= π/2 3) = 0.91. Compression of the monolayer beyond the elastic limit caused plastic deformation of the monolayer. For instance, at point (7) in Figure 4A, the particle monolayer can no longer elastically store the deformation energy and starts to yield to the compressive stress, as indicated by the sudden decrease in the magnitude of the slope of the isotherm curve. Optical microscopy images of the monolayer (e.g., the one shown in Figure 4B(7)) suggest that at this condition the monolayer film becomes heavily deformed (i.e., crumpled), possibly resulting in the formation of multilayer structures and/or the deposition of some of the silica microspheres into the subphase solution. In this case, the deformation was found to be irreversible and not recovered even when the original stress is removed (Supporting Information Figure S2(E)). It is also important to note that, as demonstrated in Figures 4B(6) and (7), under very high compression, the silica monolayer becomes deformed into a three-dimensional structure. On the other hand, the monolayer retains its two-dimensional-like nature at lower particle concentrations, that is, even when it is compressed beyond the incompressible transition point (θ J θo) up to a certain level. For instance, as demonstrated in Figure 4B(5), at θ = 0.709, the range of variation in vertical positions of the particles is well within the depth of field of the imaging system, which is estimated to be about 1.9 μm; this value is estimated using the equation, Z = nλ/NA2, where Z denotes the depth of field, n is the refractive index of the medium (n = 1 for air), λ is the wavelength of light in air (the highest, major-peak wavelength of a mercury arc lamp is 578 nm), and NA is the numerical aperture of the objective lens (= 0.55 in our case).26 It should be pointed out that, as will be discussed in later sections, the main conclusions of this paper are drawn mostly based on the data recorded in the compression range where the extent of the vertical deformation is much less than the diameter of the particles, and the arrangements of the particles can be considered to be effectively two-dimensional. III.3. Formation of the LS1 Structure by Bidisperse Silica Colloids at a Size Ratio of r = 0.375 at the AirWater Interface. Kiely and co-workers have reported observations of 2D binary cubic LS1 superlattice structures in 1:1 (by number) mixtures of gold and silver nanoparticles at size ratios (R  RS/RL) between 0.27 and 0.425.4 Random sequential adsorption (RSA) and melting simulations on hard spheres in 2D space suggested that the translational entropy alone cannot produce an LS1 structure.7 Later, similar simulation studies have demonstrated that addition of the van der Waals, osmotic, and steric contributions to the pair interaction potentials stabilizes the LS1 structure under compression in a bidisperse particle mixture at a size ratio of R = 0.375.8 In the present work, we examined the crystallization behavior of binary-size mixtures of 0.75 and 2.00 μm diameter silica (25) Kumaki, J. Macromolecules 1988, 21(3), 749–755. (26) Murphy, D. B. Fundamentals of Light Microscopy and Electronic Imaging; Wiley-Liss: New York, 2001.

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Figure 5. (A) Surface pressure (π) versus dimensionless area per particle (1/θ) isotherms of the mixed monolayers of 0.75 and 2.00 μm diameter silica spheres (R ( RS/RL) = 0.375) at the air-water interface at five different monolayer compositions, β ( θL/θS) = 7.00, 3.50, 2.00, 1.00, and 0.50 (i.e., βN ( NL/NS) = 0.98, 0.49, 0.28, 0.14, and 0.07, respectively). The filled circles denote the conditions at which the optical microscopy images presented in Figures 6, S3, 7, S4, and S5 were taken. (B) Dimensionless surface coverage values at the onset of the incompressible transition (θo) estimated from the respective surface pressure-area isotherms shown in (A) for the five different surface coverage ratios (β).

microspheres (which gives a size ratio of R = 0.375) at different compositions, β ( θL/θS) = 7.00, 3.50, 2.00, 1.00, and 0.50 (i.e., βN ( NL/NS) = 0.98, 0.49, 0.28, 0.14, and 0.07, respectively). In section III.1, we have shown that for this particle mixture the net pair interaction potentials for the three different particle-pair combinations (i.e., the SS, LL, and SL combinations) are all purely repulsive (Figure 3F). We first examined the surface pressure-area isotherms of the mixed particle monolayers. As shown in Figure 5A, the general patterns of surface pressure behavior of the 0.75 and 2.00 μm silica mixtures at the five different compositions were found to be similar to that of the one-component 1.50 μm silica particle monolayer (Figure 4). However, careful comparison of the data across the various composition conditions revealed that there exist differences in the quantitative features of the surface pressure-area isotherms; as summarized in Figure 5B, when the number ratio (βN) becomes very asymmetric, for example, at βN=0.28, 0.14, or 0.07 (i.e., at β= 2.00, 1.00, or 0.50, respectively), the incompressible transition occurs at a relatively low overall surface coverage (i.e., θo = 0.57-0.58). At higher compositional ratios, for example, at βN=0.98 and 0.49 (i.e., at β=7.00 and 3.50, respectively), the onsets of the incompressible behavior were found to be at higher θ values (i.e., θo=0.62-0.63). It should also be noted that these θo values are still lower than the incompressible onset coverage measured from the monodisperse system, that is, θo = 0.65 (section III.2). These results suggest the following inferences; (1) at β < 3.50 (βN < 0.49), upon compression beyond the limit of compressibility, the particles become jammed in a less dense (likely amorphous) arrangement; (2) at β = 7.00 and 3.50 (i.e., at βN = 0.98 and 0.49, respectively), the incompressible transition involves compaction of the particles into a denser crystalline configuration; (3) in the latter situation, the symmetry of the crystal structure is expected to be non-closed-packed DOI: 10.1021/la101313r

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Figure 6. Representative optical microscopy images (155 μm  112 μm) of the mixed monolayer of 0.75 and 2.00 μm diameter silica spheres (R = 0.375) at a monolayer composition of β = 7.00 (βN = 0.98) taken during the compression of the monolayer at six different dimensionless surface coverages, θ ( θS þ θL) = (1) 0.071, (2) 0.258, (3) 0.425, (4) 0.619, (5) 0.752, and (6) 0.921. The insets in (4) and (5) show higher magnification images of the respective phase-separated domains.

Figure 7. Representative optical microscopy images (155 μm  112 μm) of the mixed monolayer of 0.75 and 2.00 μm diameter silica spheres (R = 0.375) at a monolayer composition of β = 2.00 (βN = 0.28) taken during the compression of the monolayer at six different dimensionless surface coverages, θ ( θS þ θL) = (1) 0.083, (2) 0.383, (3) 0.536, (4) 0.632, (5) 0.758, and (6) 0.883.

(i.e., nonhexagonal), because the hexagonal packing would give the greater density at the incompressibility onset point (θo =0.65). These interpretations were confirmed by optical microscopy images of the monolayers. As shown in Figure 6, at β=7.00 (βN= 0.98) the structure of the bidisperse monolayer evolves from a disordered state (Figures 6(1)-(3)) to an ordered state (Figure 6(4) and (5)) as the particle density is increased. This structural transition occurs near the onset of incompressible states (θo = 0.63). As shown in Figure 6(4) (θ ( θS þ θL)=0.619), the disorder-to-order transition accompanies phase separation of the system into a hexagonal phase composed purely of the larger particles and a small-particle-rich phase in which relatively small domains (∼900 μm2 per domain) of LS1 symmetry are randomly distributed. Upon further compression, for example, at θ = 0.752, the coexisting hexagonal and LS1 phases transform into two phaseseparated hexagonal closest-packed phases, with one phase consisting solely of the large particles and the other of the smaller particles (Figure 6(5)); the LS1 superlattice structure is no longer stable at this surface pressure. At the highest compression condition used (i.e., at θ=0.921), the phase-separated monolayer was found to become plastically deformed; see Figures 5A and 6(6). 11744 DOI: 10.1021/la101313r

Even when the coverage mixing ratio is reduced to β = 3.50 (βN = 0.49), the overall structural behavior was found to be identical to that seen in the β = 7.00 (βN = 0.98) case; representative optical images of the β=3.50 sample, taken at six different overall surface coverages, are presented in Figure S3 of the Supporting Information. Most importantly, the measurements confirmed the formation of the LS1 superlattice (again) only in the vicinity of the incompressible transition of the monolayer (Supporting Information Figure S3(4)). It is notable that the detection of the LS1 mor6 1) is in phology at this asymmetric composition (i.e., at βN ¼ contrast to the observation of Kiely and co-workers that a binary LS1 superlattice crystallizes only when βN = 1.4b At a further reduced coverage ratio of β = 2.00 (βN = 0.28), the monolayer of bidisperse 0.75 and 2.00 μm silica microspheres exhibits a random alloy structure throughout the entire range of surface coverages covering the compressible, incompressible, and irreversibly deformed states (Figure 7); the system is disordered at low densities, and upon compression the disordered structure becomes frozen in a glassy state at particle densities greater than the onset of incompressibility. This same behavior is observed for other values of β less than 2.00 (i.e., β = 1.00 and 0.50); see Supporting Information Figure S4 for images of the bidisperse Langmuir 2010, 26(14), 11737–11749

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Figure 8. Mixture composition (θS vs θL) structure diagram that summarizes the structural behavior of the bidisperse 0.75 and 2.00 μm silica monolayer system (R ( RS/RL) = 0.375) at the five different surface coverage ratios, β ( θL/θS) = 7.00, 3.50, 2.00, 1.00, and 0.50 (i.e., βN ( NL/NS) = 0.98, 0.49, 0.28, 0.14, and 0.07, respectively). The data points correspond to the composition conditions under which the optical images in Figures 6, S3, 7, S4, and S5 were obtained.

particle monolayer at β = 1.00 (βN = 0.14) and Figure S5 for β =0.50 (βN = 0.07). In Figure 8, we present a mixture composition (θS vs θL) structure diagram that summarizes all the abovedescribed results obtained from monolayers of bidisperse silica microspheres at the size ratio of R = 0.375. In closing this section, it should be noted that in our system an LS1 superlattice structure is observed even though there is no net attractive interaction between particles, and this result is discrepant from the finding of a simulation study that a stable LS1 structure is not formed when the particle pair interactions are purely repulsive.7 We suspect this discrepancy in fact points to the possibility that many body effects27 may play an important role in stabilizing the LS1 symmetry. Further study is needed to identify the exact cause of the LS1 structure. III.4. Formation of the LS2 Structure by Bidisperse Silica Colloids at a Size Ratio of r = 0.579 at the AirWater Interface. We also examined the structural behavior of bidisperse mixtures of 1.50 and 2.59-μm diameter silica microspheres at the air-water interface. In this case, the size ratio was therefore R ( RS/RL) = 0.579. This value for R is well within the range (i.e., 0.482 < R ( RS/RL) < 0.624) for which Kiely and coworkers have observed LS2 binary crystal structures in monolayers of bidisperse Au nanoparticles at a particle composition of βN ( NL/NS) = 0.5.4b Also, R = 0.58 is the condition under which LS2 superlattices have been observed in three-dimensional systems composed of bidisperse colloidal spheres; examples include systems of both natural1 and synthetic2b,28 origins. Interestingly, computer simulations of hard7 and sticky8 disks in two dimensions could not reproduce the LS2 superlattice structure observed experimentally in the monolayers of bidisperse metal nanoparticles, which led the authors of the simulation study to speculate that perhaps the nonspherical (faceted) structure of the nanoparticles used in the experiments contributes to the stabilization of the LS2 structure.8 It is of note that we confirmed by scanning electron microscopy (SEM) that the silica particles used in the present study are perfectly spherical (images not shown). The surface pressure behaviors of the mixed monolayers of 1.50 and 2.59 μm silica particles (R = 0.579) were examined at (27) Brunner, M.; Dobnikar, J.; von Grunberg, H. H.; Bechinger, C. Phys. Rev. Lett. 2004, 92, 7. (28) Yoshimura, S.; Hachisu, S. Prog. Colloid Polym. Sci. 1983, 68, 59–70. (29) Murray, M. J.; Sanders, J. V. Philos. Mag. A 1980, 42(6), 721–740.

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Figure 9. (A) Surface pressure (π) versus dimensionless area per particle (1/θ) isotherms of the mixed monolayers of 1.50 and 2.59 μm diameter silica spheres (R ( RS/RL) = 0.579) at the air-water interface at five different monolayer compositions, β ( θL/θS) = 3.00, 1.50, 1.00, 0.50, and 0.33 (i.e., βN ( NL/NS) = 1.01, 0.50, 0.34, 0.17, and 0.11, respectively). The filled circles denote the conditions at which the optical microscopy images presented in Figures 10, S6, S7, 11, and S8 were taken. (B) Dimensionless surface coverage values at the onset of the incompressible transition (θo) estimated from the respective surface pressure-area isotherms shown in (A) for the five different surface coverage ratios (β).

five different compositions, β ( θL/θS) = 3.00, 1.50, 1.00, 0.50, and 0.33 (i.e., βN ( NL/NS) = 1.01, 0.50, 0.34, 0.17, and 0.11, respectively). As shown in Figure 9A, for all the composition conditions tested, the overall patterns of the isotherm behaviors were consistent with the basic explanations given in section III.2; that is, at low particle densities, the particle-containing area is fully compressible, whereas the particle layer becomes significantly resistant to compression when it is compressed beyond a certain point (i.e., when 1/θ < 1/θo). Analysis of the surface pressure versus area data indicates that this incompressible transition occurs at a slightly higher particle density when β g 1.00 (βN g 0.34) than when β < 1.00 (βN < 0.34) (Figure 9B). As was seen for the R = 0.375 samples (section III.3), this trend was found to be related to whether the incompressible transition involves the ordering of the particles into a crystalline structure in which the particles are packed more efficiently than in the disordered state. For instance, at a surface coverage ratio of β = 3.00 (βN = 1.01), the onset of the compressible-to-incompressible transition was observed at a relatively high overall surface coverage of θo = 0.641. As shown in Figure 10, at this mixture composition, optical images of the bidisperse particle monolayer indicate that compression of the monolayer beyond the onset of incompressibility induces arrangement of the particles into an LS2 crystal structure (Figure 10(6)); prior to the incompressible onset point, the particles were found to be disordered at all stages of compression (Figures 10(1)-(5)). As shown in Figure 10(6), typically the LS2 binary crystalline phase was found to exist as small domains in the continuous random alloy phase. Notably, unlike the LS1 superlattice case wherein the LS1 structure is seen only near the onset of the incompressible transition and at higher particle densities the system undergoes further transition (i.e., densification) to phase-separated hexagonal states (section III.3), the LS2 DOI: 10.1021/la101313r

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Figure 10. Representative optical microscopy images (155 μm  112 μm) of the mixed monolayer of 1.50 and 2.59 μm diameter silica spheres

(R = 0.579) at a monolayer composition of β = 3.00 (βN = 1.01) taken during the compression of the monolayer at six different dimensionless surface coverages, θ ( θS þ θL) = (1) 0.067, (2) 0.256, (3) 0.387, (4) 0.459, (5) 0.602, and (6) 0.782. The inset in (6) shows a higher magnification image of an adjacent region.

Figure 11. Representative optical microscopy images (155 μm  112 μm) of the mixed monolayer of 1.50 and 2.59 μm diameter silica spheres (R = 0.579) at a monolayer composition of β = 0.50 (βN = 0.17) taken during the compression of the monolayer at six different dimensionless surface coverages, θ ( θS þ θL) = (1) 0.135, (2) 0.335, (3) 0.497, (4) 0.593, (5) 0.789, and (6) 0.835.

structure remained stable even at very high compression of the monolayer and no phase separation was observed. At slightly lower surface coverage ratios, that is, at β=1.50 and 1.00 (βN= 0.50 and 0.34, respectively), the overall structural behaviors were identical to that of the β = 3.00 case; see Supporting Information Figures S6 and S7 for representative images of the respective samples at these two lower composition ratios under various surface coverage conditions. However, when the particle number ratio is lowered significantly from the stoichiometric composition for the LS2 symmetry (i.e., βN =0.50, or β=1.50), for instance, at a coverage ratio of β = 0.50 (βN = 0.17), the random alloy configuration of the particles in the monolayer remains unchanged through the whole range of compression, and no crystalline morphology was observed even at particle densities past the incompressibility threshold (Figure 11). This result again confirms that a lower θo value is an indication that the particles become jammed in a random arrangement at the incompressible transition point. At a further lower coverage ratio, β = 0.33 (βN = 0.11), the overall behavior was identical to the β = 0.50 situation (Supporting Information Figure S8). 11746 DOI: 10.1021/la101313r

Figure 12. Mixture composition (θS vs θL) structure diagram that summarizes the structural behavior of the bidisperse 1.50 and 2.59 μm silica monolayer system (R ( RS/RL) = 0.579) at the five different surface coverage ratios, β ( θL/θS) = 3.00, 1.50, 1.00, 0.50, and 0.33 (i.e., βN ( NL/NS) = 1.01, 0.50, 0.34, 0.17, and 0.11, respectively). The data points correspond to the composition conditions under which the optical images in Figures 10, S6, S7, 11, and S8 were obtained. Langmuir 2010, 26(14), 11737–11749

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All the above-described results have been summarized in a mixture composition (θS vs θL) structure diagram presented in Figure 12. The results clearly demonstrate that bidisperse spherical particles with purely repulsive pair potentials can form LS2 crystals in two dimensions. We speculate, however, that theoretical modeling of these results will be challenging to achieve, because of the current lack of information regarding the roles that the capillary forces and the manybody effects play in determining the structure of bidisperse colloids at the air-water interface.

Figure 13. (A) Surface pressure (π) versus dimensionless area per particle (1/θ) isotherms of the mixed monolayers of 0.75 and 1.50 μm diameter silica spheres (R ( RS/RL) = 0.500) at the air-water interface at three different monolayer compositions, β ( θL/θS) = 2.00, 1.00, and 0.50 (i.e., βN ( NL/NS) = 0.50, 0.25, and 0.13, respectively). The filled circles denote the conditions at which the optical microscopy images presented in Figures 14, S9, and 15 were taken. (B) Dimensionless surface coverage values at the onset of the incompressible transition (θo) estimated from the respective surface pressurearea isotherms shown in (A) for the three different surface coverage ratios (β).

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III.5. Behavior of Bidisperse Silica Colloids at a Size Ratio of r=0.500 at the Air-Water Interface. At an intermediate size ratio of R ( RS/RL)=0.47, bidisperse particle mixtures in two dimensions have been observed to undergo phase separation and form two separate hexagonal phases of small and large particles.4a At this same size ratio, similar behavior has been suggested also for 3D bidisperse colloids on the basis of a simple geometric argument.29 Interestingly, at a slightly higher R value (R = 0.50), a 3D bidisperse particle system is expected to form a stable LS2 superlattice.2b,5a,29 In the present study, we have also examined, in two dimensions, the structures of bidisperse silica mixtures at this size ratio. For this study, we used 0.75 and 1.50 μm diameter silica colloids. The surface pressure-area isotherms of 2D mixtures of these particles have been measured at three different surface coverage ratios, β ( θL/θS) = 2.00, 1.00, and 0.50 (or βN ( NL/NS) = 0.50, 0.25, and 0.13, respectively) (Figure 13A). As displayed in Figure 13B, at β = 2.00, the incompressible transition occurs at a slightly lower θo value (θo = 0.598) than at the other two compositions examined (β = 1.00 and 0.50). Optical microscopy results indicate that the bidisperse monolayer at this β = 2.00 condition remains amorphous over the entire range of particle density and does not undergo phase separation (see images in Figure 14), which explains why the particles are less densely packed at the onset of incompressibility. The structural behaviors at β = 1.00 and 0.50 were found to be qualitatively different. As shown in Figure 15, for example, at a mixed particle composition of β = 0.50 (βN = 0.13), the large particles (darker dots) were seen to be expelled from the regions of the small particles (lighter dots) when the monolayer was compressed close to the incompressible transition point (Figure 15(3)), and further compression through the transition to the incompressible region (θo = 0.625) resulted in complete phase separation of the system into two hexagonal closepacked phases of the small and large particles (Figures 15(4) and (5)). Also, it is interesting to note that, in the plastically deformed regime (e.g., see Figure 15(6)), the wrinkled (crumpled) structures were observed predominantly in the small particle domains; similar behavior has also been observed with the R = 0.375 system (Figures 6(6) and S3(6)). These results are consistent with the previous finding in the literature that the contact adhesion force between two solid spheres scales

Figure 14. Representative optical microscopy images (155 μm  112 μm) of the mixed monolayer of 0.75 and 1.50 μm diameter silica spheres (R = 0.500) at a monolayer composition of β = 2.00 (βN = 0.50) taken during the compression of the monolayer at six different dimensionless surface coverages, θ ( θS þ θL) = (1) 0.125, (2) 0.267, (3) 0.423, (4) 0.565, (5) 0.611, and (6) 0.887. Langmuir 2010, 26(14), 11737–11749

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Figure 15. Representative optical microscopy images (155 μm  112 μm) of the mixed monolayer of 0.75 and 1.50 μm diameter silica spheres

(R = 0.500) at a monolayer composition of β = 0.50 (βN = 0.13) taken during the compression of the monolayer at six different dimensionless surface coverages, θ ( θS þ θL) = (1) 0.178, (2) 0.275, (3) 0.497, (4) 0.657, (5) 0.715, and (6) 0.898. The insets in (4) show higher magnification images of the respective phase-separated domains.

during the drying of the particle suspension onto the mercury surface.31 It is also of note that similar size-dependent segregation phenomena have also been observed in other 2D mixed particle systems such as tridisperse gold nanoparticles32 and tetradisperse polystyrene microgel spheres.33 The results of the present investigation suggest, however, that the compression-induced separation of different sized particles can occur only within a finite range of particle compositions; for instance, in our case when the concentration of the smaller-sized particles is relatively lean (e.g., β = 2.00 or βN = 0.50), neither the small nor large particles are able to form sufficiently large clusters of their own before they become jammed together at the point of the incompressible transition during the compression event. Figure 16. Mixture composition (θS vs θL) structure diagram that summarizes the structural behavior of the bidisperse 0.75 and 1.50 μm silica monolayer system (R ( RS/RL) = 0.500) at the three different surface coverage ratios, β ( θL/θS) = 2.00, 1.00, and 0.50 (i.e., βN ( NL/NS) = 0.50, 0.25, and 0.13, respectively). The data points correspond to the composition conditions under which the optical images in Figures 14, S9, and 15 were obtained.

IV. Conclusion

linearly with the harmonic mean radius of the two particles;30 this relationship gives an explanation as to why, in the phaseseparated bidisperse monolayer, the interparticle adhesion is weaker within the small-particle domain than within the largerparticle domain or at the boundary between the two types of domains. At a slightly higher surface coverage ratio of β = 1.00 (βN = 0.25), the overall structural behavior was found to be identical to that observed at β = 0.50; see Supporting Information Figure S9 for representative images of the sample at this mixture composition (β = 1.00) under various densities of particles. Again, the combined results for this size ratio (R = 0.50) have been plotted in a mixture composition (θS vs θL) structure diagram presented in Figure 16. These results appear to suggest that at R = 0.50 the behavior in two dimensions is discrepant from that in 3D systems. There has been another report which supports this point; bidisperse polystyrene lattices on a surface of mercury, at this same size ratio, have been shown to form phase-separated hexagonal structures

Two-dimensional bidisperse colloids with purely repulsive pair potentials have been prepared using mixtures of pairs of monodisperse silica microspheres at the air-water interface. The structural behaviors of these bidisperse silica monolayers have been examined at three different particle size ratios, R ( RS/RL) = 0.375, 0.500, and 0.579, under various particle surface coverage (θ  θS þ θL) and composition (β  θL/θS) conditions. For the R = 0.375 condition, the overall behavior can be summarized as follows; at the two highest β values examined (β = 7.00 and 3.50), LS1 domains were observed when the bidisperse monolayer was compressed to an overall surface coverage of θ ≈ 0.62 where the monolayer became no longer laterally compressible, and further compression resulted in phase separation of the system into two hexagonal phases; at other surface coverage ratios tested (β = 2.00, 1.00 and 0.50), the particles in the monolayer became jammed in a random alloy configuration at the incompressible transition point, and no crystallization was seen even at surface coverages beyond the incompressibility threshold; a mixture composition (θS vs θL) structure diagram has been presented in Figure 8 that summarizes the results obtained at R = 0.375. At the size ratio of R = 0.579, the structural behavior of the bidisperse particle monolayer can also be summarized as follows; at surface coverage ratios of β = 3.00, 1.50, and 1.00, compression of the monolayer beyond the onset of

(30) Heim, L. O.; Blum, J.; Preuss, M.; Butt, H. J. Phys. Rev. Lett. 1999, 83(16), 3328–3331. (31) Yamaki, M.; Higo, J.; Nagayama, K. Langmuir 1995, 11(8), 2975–2978.

(32) Ohara, P. C.; Leff, D. V.; Heath, J. R.; Gelbart, W. M. Phys. Rev. Lett. 1995, 75(19), 3466–3469. (33) Antonietti, M.; Hartmann, J.; Neese, M.; Seifert, U. Langmuir 2000, 16(20), 7634–7639.

11748 DOI: 10.1021/la101313r

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incompressibility produced LS2 binary crystals, and this LS2 structure remained stable even at very high compression of the monolayer; at β = 0.50 and 0.33, the monolayer was seen to exhibit a random alloy structure throughout the entire range of surface coverages; see Figure 12 for a summary of the results at R = 0.579. When R = 0.500, as also summarized in Figure 16, the bidisperse monolayer remained amorphous over the entire range of particle density at β=2.00, while the bidisperse system with β=1.00 or 0.50 underwent phase separation under high compression conditions. From these observations, we were able to make the following conclusions: contrary to what has been speculated in the literature, (1) purely repulsive pair potentials can give rise to LS1 and LS2 binary crystals under compression, and (2) spherical particles can form LS2 crystals. This discrepancy between our results and

Langmuir 2010, 26(14), 11737–11749

Article

the predictions of previous RSA/melting simulations, along with the observations described in section III.2, lead us to suspect that the capillary interaction and/or the many-body effects play important roles in determining the structure of bidisperse colloids at the air-water interface. This problem certainly requires more detailed study. Acknowledgment. The authors are grateful for financial support of this research from the Purdue Research Foundation and the 3M Company (Nontenured Faculty Award). The authors would also like to thank Polysciences, Inc. (Warrington, PA) who generously provided the 1.50 μm diameter silica samples. Supporting Information Available: This material is available free of charge via the Internet at http://pubs.acs.org.

DOI: 10.1021/la101313r

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