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Langmuir 2008, 24, 5382-5392

Two-Dimensional Colloid Crystals Templated by Polyelectrolyte Multilayer Patterns Jaehyun Hur† and You-Yeon Won* School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907 ReceiVed December 27, 2007. ReVised Manuscript ReceiVed February 16, 2008 In this paper, we demonstrate that two-dimensional (2D) periodic patterns of polyelectrolyte multilayers (PEMs) can be used as surface templates for assembling highly ordered 2D colloidal microarrays. We report detailed structural features of the 2D colloid crystals produced by depositing silica microspheres onto periodic micrometer-scale PEM patterns arrayed in a square or hexagonal lattice with a pattern pitch (approximately) twice the pattern diameter. Analysis of the images of these 2D colloid monolayers reveals that the distributions of the distances by which the adsorbed particles deviate from the corresponding PEM pattern centers are typically bell-shaped, with the majority of the deposited particles located within a relatively short distance from the respective pattern centers. We show that this behavior reflects the effect of the electrostatic focusing force that (occurs because of the finite size of the PEM pattern and) becomes effective when the depositing particle approaches the pattern site to a small distance. Also, in these 2D colloid crystals, the orientations of the off-center displacements of the deposited particles are strongly correlated spatially over the entire sample size. We present experimental evidence that this unusually long-ranged orientational correlation is due to the close spacing of the patterns, which causes an overlap of the excluded volumes between the neighboring deposited particles.

1. Introduction Developing a method for precise fabrication of threedimensional (3D) colloid crystals has been a subject of great research interest in recent years,1,2 driven in part by the goal of developing colloid-based photonic band-gap materials.3 In nearly all methods demonstrated in the literature, crystal fabrication processes require the presence of a substrate surface that can provide an appropriate boundary condition for guiding particles to a desired two-dimensional (2D) array arranged as in the first layer of the crystal surface. In particular, there are situations in which the precision and/or the complexity required of the crystal product necessitates the use of a patterned substrate, with the most common approach being the creation of holes on the surface in a symmetry that reflects a specific cross section of a desired 3D crystal structure. For instance, a perfect face-centered-cubic (fcc) crystal that is free of stacking faults can be constructed by sedimentation of colloidal particles onto a substrate containing a square pattern of holes that corresponds to the (100) plane of the fcc crystal.4,5 Also, it has been demonstrated that complex non-closed-packed structures such as a diamond-lattice crystal structure can be fabricated on the basis of this template-directed approach used in combination with nanorobotic single-particle manipulation and appropriate etching procedures.6 In most prior studies demonstrating the epitaxial growth of colloid crystals, photolithography has been predominantly used for creating the 2D substrate patterns.4–6 However, for submicronsized patterns (say, of about a few hundred nanometers as needed * To whom correspondence should be addressed. E-mail: yywon@ ecn.purdue.edu. † E-mail: [email protected].

(1) Weitz, D. A.; Russel, W. B. MRS Bull. 2004, 29, 82–84. (2) Glotzer, S. C.; Solomon, M. J.; Kotov, N. A. AIChE J. 2004, 50, 2978– 2985. (3) Xia, Y.; Gates, B.; Li, Z.-Y. AdV. Mater. 2001, 13, 409–413. (4) Blaaderen, A. V.; Ruel, R.; Wiltzius, P. Nature 1997, 385, 321–324. (5) Shall, P.; Cohen, I.; Weitz, D. A.; Spaepen, F. Science 2004, 305, 1944– 1948. (6) Garcia-Santamaria, F.; Miyazaki, H. T.; Urquia, A.; Ibisate, M.; Belmonte, M.; Shinya, N.; Meseguer, F.; Lopez, C. AdV. Mater. 2002, 14, 1144–1147.

for producing a diamond-structure colloid crystal with a band gap at the 1.55-µm optical communication wavelength), additional machineries are normally necessary in implementing the photolithography processes,7 and other techniques such as soft lithography8,9 or dip-pen nanolithography10 become reasonable alternatives in terms of fabrication cost and simplicity. In fact, it has recently been shown that soft-lithography printing can be used as the basic patterning method for the formation of a 2Dordered layer of colloids;11 key to this development was the use of the electrostatic attraction between the substrate pattern (coated with polyelectrolyte multilayers) and an oppositely charged colloid, which enabled the selective deposition of the colloidal particles into the patterns spaced by distances of many particle diameters. However, it remains unexplored whether a similar approach can be applied to the fabrication of a tightly spaced 2D colloid crystal (on which a 3D crystal structure can be built), and this is the question that we seek to explore in this paper. Specifically, the issue is whether the polyelectrolyte patterns generated based on soft lithography (which therefore do not contain any significant height differences between the regions inside and outside the pattern) can provide a high enough degree of regularity in the positions of the deposited particles so that the resultant 2D colloid arrays could be used as structural templates for growing 3D crystals thereon. A step toward addressing this question is taken in the present work by examining detailed structural features of 2D-ordered arrays of charged microspheres deposited to the oppositely charged polyelectrolyte multilayer (PEM) patterns. (7) Madou, M. J. Fundamentals of Microfabrication, 2nd ed.; CRC Press: Boca Raton, FL, 2002. (8) Kumar, A.; Biebuyck, H. A.; Whitesides, G. W. Langmuir 1994, 10, 1498– 1511. (9) Michel, B.; Bernard, A.; Bietsch, A.; Delamarche, E.; Geissler, M.; Juncker, D.; Kind, H.; Renault, J.-P.; Rothuizen, H.; Schmid, H.; Schmidt-Winkel, P.; Stutz, R.; Wolf, H. IBM J. Res. DeV. 2001, 45, 697–719. (10) Piner, R. D.; Zhu, J.; Xu, F.; Hong, S.; Mirkin, C. A. Science 1999, 283, 661–663. (11) Lee, I.; Zheng, H.; Rubner, M. F.; Hammond, P. T. AdV. Mater. 2002, 14, 572–576.

10.1021/la704050t CCC: $40.75  2008 American Chemical Society Published on Web 04/24/2008

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Figure 1. (A) Schematic illustration of the procedures used for preparing 2D colloid crystals. These procedures involve multiple steps including (soft-lithography) microcontact printing of carboxylic acid-terminated alkanethiol, layer-by-layer deposition of alternating layers of cationic and anionic polyelectrolytes, and electrostatic particle deposition onto targeted pattern areas. Also shown are representative AFM images of the microcontactprinted thiol patterns of circles arranged in (B) square (Dp ) 1.15 µm) and (C) hexagonal (Dp ) 1.05 µm) arrays.

II. Methods and Procedures Materials. PEMs were fabricated using poly(diallyldimethylammonium chloride) (PDADMAC; 100 000–200 000 g mol-1, Sigma-Aldrich) and sulfonated polystyrene (SPS; 75 000 g mol-1, Sigma-Aldrich) as the polycation and polyanion species, respectively. For the microcontact printing of the negatively charged alkanethiol molecules (16-mercaptohexadecanoic acid, Sigma-Aldrich), we used a 2 mM solution of this compound in ethanol (200 proof) as the ink. The silica microspheres used in this study have diameters of 1.85 ( 0.10 µm (Polysciences, custom synthesized; the mean size and standard deviation values were provided by the vendor), 1.50 ( 0.08 µm (Polysciences, catalog no. 24329-15; the mean size and standard deviation values were provided by the vendor), and 0.75

( 0.09 µm (synthesized by the Stöber method;12,13 the mean size and standard deviation values were determined by scanning electron microscopy (SEM) image analysis (data not presented)). Preparation of PEM Patterns via Microcontact Printing and Deposition of Silica Microspheres onto the PEM Patterns. 2D periodic patterns of polyelectrolytes were prepared by using the procedures previously developed by Hammond and co-workers;11 a schematic summary of the procedures used is presented in Figure 1A. Briefly, a glass substrate was first cleaned by ultrasonication in isopropyl alcohol at room temperature for 20 min and then rinsed with a copious amount of Milli-Q pure water (18.2 MΩ cm-1). The (12) Bogush, G. H.; Tracy, M. A.; Zukoski, C. F. J. Non-Cryst. Solids 1988, 104(1), 95–106. (13) Stober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26(1), 62.

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Table 1. Summary of the Combinations of Values of the Diameter of Silica Particle (Ds), the Diameter of the Microcontact-Printed Pattern (Dp), and the Center-to-Center Distance between Adjacent Patterns (“Pitch”) and the Pattern Symmetry Used in This Studya diameter of the silica particle, Ds (µm)

pattern symmetry

pattern diameter, Dp (µm)

pattern pitch, ∆p (µm)

1.85 ( 0.10 1.85 ( 0.10 1.50 ( 0.08 1.50 ( 0.08 1.50 ( 0.08 1.50 ( 0.08 0.75 ( 0.09 0.75 ( 0.09

square square square square hexagonal hexagonal square hexagonal

1.15 ( 0.06 1.29 ( 0.08 1.25 ( 0.08 1.65 ( 0.11 1.05 ( 0.04 1.15 ( 0.05 1.03 ( 0.02 1.02 ( 0.02

2.00 2.00 2.00 2.00 2.45 2.45 2.00 2.45

a The information about the standard deviation of the particle size distribution was obtained either from the vendor (for the 1.85- and 1.50-µm silica cases) or by SEM measurement of the particle sizes (for the case of the 0.75-µm silica particles that were synthesized in our laboratory). The sizes and size distributions of the microcontact-printed (acid thiol) patterns were determined by AFM imaging, and representative AFM images of these patterns are presented in Figures 1 and S5 in the Supporting Information.

glass surface was then coated first with titanium (to a thickness of 10 nm) and then with gold (100 nm thickness) via e-beam evaporation in a high-vacuum environment. This gold substrate was cleaned in a piranha solution [3:1 (v/v) H2SO4/H2O2]. On the clean gold surface, an initial 2D template with patterned charge groups of 16mercaptohexadecanoic acid was created using the microcontactprinting (i.e., soft-lithography) technique. In this study, we used two different poly(dimethylsiloxane) (PDMS) stamps for the microcontact printing of the thiol molecules: one with a square array of 1.00µm-diameter circles on a 2.00-µm pitch (center-to-center distance) and the other with a hexagonal array of 1.00-µm-diameter circles on a 2.45-µm pitch. Using a Q-tip, the PDMA stamp was painted with the ink (2 mM mercaptohexadecanoic acid (HS(CH2)15COOH) in ethanol). On top of the PDMS stamp where the ink was applied, the gold substrate was gently placed using tweezers. Typically, a waiting time of 3 min was allowed for the transfer of the ink solution from the stamp to the gold surface. The substrate was then carefully detached from the stamp, washed with ethanol and deionized water (multiple times), and then finally dried with filtered compressed air. The size of the printed pattern is typically slightly larger than that of the original pattern on the PDMS stamp (1.00 µm) because of the diffusion of the alkanethiol molecules chemisorbed on the substrate during the microcontact-printing process; this effect is influenced by various factors, such as printing time, solvent evaporation rate, and temperature.9 The formation of the desired pattern on the gold substrate was confirmed by atomic force microscopy (AFM; Nanoscope IV, Digital Instruments) using the lateral force mode (i.e., by measuring the torsional deformation of the cantilever during contact scanning). Representative AFM images of the alkanethiol patterns used in this study are presented in Figures 1 and S5 in the Supporting Information. The sizes of these patterns were determined from these (and other) AFM images, and the measured mean diameter values are listed in Table 1. The remaining regions of the gold surface were coated with an 11-mercaptoundecanoic triethylene glycol (Prochimia) blocking agent.14 Subsequently, multiple alternating molecular layers of the two oppositely charged polyelectrolyte materials (i.e., PDADMAC and SPS) were constructed selectively on top of the charged pattern sites by layerby-layer electrostatic self-assembly processing.15 In all of the cases studied in this work, the PEMs were grown to the total 10.5 number of PDADMAC/SPS bilayers, as has been previously suggested.11 These polyelectrolyte-patterned substrates (in which the PDADMAC is predominantly expressed at the outermost surface of the PEM coating) were placed horizontally at the bottom of a glass vial (2.6(14) Ulman, A. An Introduction to Ultrathin Organic Films: From Langmuir-Blodgett to Self-Assembly; Academic Press: San Diego, CA, 1991. (15) Decher, G.; Hong, J.-D. Ber. Bunsenges. Phys. Chem. 1991, 46, 321–327.

cm inner diameter) and were immersed in a well-dispersed suspension (10 mL) of negatively charged silica microspheres at a silica particle concentration of 1.0 g mL-1 in water. In a typical experiment, a thin stack of silica particles on the PEM-patterned substrate was initially produced by gravitational sedimentation of the particles; different sedimentation times were applied for different sized particles (i.e., 12, 18, and 24 h for the 1.85, 1.50, and 0.75-µm-diameter silica particles, respectively). After careful rinsing (to remove all excess colloids), only the particles in the first layer that are in contact with the PEM-pattern surface remain attached at the initial positions; as discussed in Section III.1, the binding between the particles and the polyelectrolyte-coated surface is irreversible and sufficiently strong that the positions of the deposited particles are expected to be unaffected by the washing process. The colloid arrays produced as described above were imaged in water using a Nikon Eclipse L150 microscope (under Köhler illumination conditions), and these images were analyzed using the IDL image analysis software (version 5.6); the procedures and results of these analyses are presented in Section III.2. Measurement of the Adhesion Force between a Silica Particle and the PEM Surface via AFM. The AFM cantilever deflection versus sample position data shown in Figure 2A were obtained using “colloid” probes, which were prepared by epoxy-gluing a silica particle of 1.85 µm diameter to the end of a commercial AFM cantilever (Novascan Technologies, Inc.). The normal spring constants (k’s) of these colloidal cantilevers were determined by the thermal tune method.16 In this method, the frequency spectrum of the cantilever’s thermal vibration is measured, and this power spectral density profile of the cantilever is fitted to a Lorentzian curve. By integration of the Lorentzian line shape, the average fluctuation energy of the cantilever is computed, and a comparison between this energy and the thermal energy leads to an estimate of the cantilever’s spring constant (k). All of these calculations were performed within the Nanoscope 6.11r1 software. The data presented in Figure 2A were obtained from five independent measurements. In each measurement, the cantilever deflection was recorded as a function of the relative position of the PEM surface during the first tip approach and retraction cycle. For each measurement, a fresh colloid probe was used because the tip becomes contaminated by the polyelectrolyte molecules after the initial contact. The spring constants for the five different colloidal cantilevers used were measured separately, and their values were found to be in a narrow range between 0.051 and 0.056 N m-1. This spring constant information was used to convert the approach portions of the cantilever deflection versus surface position data (Figure 2A) to an average force–distance profile presented in Figure 2B using established procedures.17

III. Results and Discussion III.1. Discussions and Comparisons of the Magnitudes of the Various Forces Operative during the Particle Deposition and Washing Processes. In order to discuss the detailed structural properties of 2D colloid crystals constructed using PEM patterns as surface templates (Section III.2), it is necessary to understand (i) how stable the binding between the colloid and PEM is against the disturbance caused by the rinsing of the sample after the particle deposition and (ii) how much of a role the repulsive electrostatic interactions between the colloids play in the determination of the positions of the deposited particles. In this section, we provide discussions on these issues. Estimation of the Shear Stress Caused by the Fluid Flow during the Washing Process. Typical rinsing procedures applied involve a movement of the colloid-coated surface in a back-and-forth motion normal to the surface at an estimated speed of ∼4 cm s-1. In a forward rinsing stroke, for example, the water flow is nearly stagnant at the geometric center of the surface, and the tangential velocity of water increases as a function of the radial distance from the center; the shear stress exerted on the deposited colloids is greatest at the edges of the substrate. For this flow

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Figure 2. (A) AFM cantilever deflection versus sample position data obtained from five independent measurements of the distance-dependent forces between a 1.85-µm-diameter silica particle (attached to an AFM cantilever) and a patternless, 10.5 PDADMAC/SPS bilayer-coated surface (the topmost layer is formed of PDADMAC) under a water environment. Each set of the deflection versus position data was acquired using a fresh colloid probe and a newly prepared PEM sample during the first cycle of the tip–sample approach and retraction processes. (B) Data in part A converted to an average force–distance plot (discrete points with error bars). The solid curve represents a fit of these data to the DLVO model using the PEM surface charge density and the Hamaker constant between the silica and the PEM-coated substrate in the water medium as two adjustable parameters; all other parameters are known (as explained in the text).

geometry, the stream function of the flow field in the boundary layer obeys the momentum equation18 ^ ^ ^ ^ ^ ∂2ψ ∂ψ ∂2ψ du ˜ ∂3ψ ∂ψ ˜ u ) ∂yˆ ∂x˜ ∂ yˆ ∂x˜ ∂yˆ2 dx˜ ∂yˆ3

(1)

ˆ denotes the stream function, u˜ is the tangential velocity where ψ outside the boundary layer, x˜ is the tangential coordinate, and ˆ (x˜,y˜) is yˆ is the normal coordinate, all in dimensionless units; ψ related to the tangential and normal velocities inside the boundary ˆ /∂yˆ and Vˆ y ) -∂ψ ˆ /∂x˜, respectively. Equation layer by V˜ x ) ∂ψ 1 can be reduced to a one-dimensional form by using a modified ˆ (x˜,yˆ)/[u˜(x˜) g(x˜)], which is assumed to stream function, f(η) ) ψ be a function only of a similarity variable, defined as η ) yˆ/g(x˜), where g(x˜) is the scale factor proportional to the boundary layer thickness. By numerically solving the similarity-transformed equation, the tangential velocity profile, V˜ x(x˜,yˆ), can be obtained. From this analysis, we estimate the maximum shear stress at the outer surface of the particle-coating layer (assuming that fluid flow is negligible within the colloid layer) to be τd ≈ 10-1 N m-1 at the edge of the plate (we used a 1 × 1 cm2 glass plate to grow a 3 × 3 mm2 2D colloid crystal thereon). This can be translated into an estimate for the maximum viscous drag force of Fd ≈ 10-3 nN that a particle experiences during the rinsing process, assuming Fd ≈ 2πR2τd to a reasonable approximation. Measurements of the Strength of the Particle-PEM Bond. For comparison with the above estimate for the washing-induced stress, we performed experiments to measure the pull-off force (Fp) for the adhesive contact between the silica particle of 1.85µm diameter and the PEM surface (10.5 bilayers of PDADMAC and SPS prepared on a patternless surface) using the AFM colloid probe technique.19–21 Figure 2 displays the cantilever deflection versus sample position profiles obtained from five independent force measurements under a liquid environment; each measurement was made using a fresh tip because, upon first contact with the PEM surface, the colloid probe normally becomes contaminated by adsorption of the polyelectrolyte. With a known spring constant of the cantilever, k (separately measured for each cantilever as described in Section II), the magnitude of the pulloff force was determined from the measured maximum displace(16) Hutter, J. L.; Bechhoefer, J. ReV. Sci. Instrum. 1993, 64, 1868–1873. (17) Cappela, B.; Dietler, G. Surf. Sci. Rep. 1999, 34(1–3), 1–104. (18) Deen, W. M. Analysis of Transport Phenomena; Oxford University Press: New York, 2003. (19) Ducker, W.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831–1836. (20) Clark, S. C.; Walz, J. Y.; Ducker, W. A. Langmuir 2004, 20, 7616–7622. (21) Kappl, M.; Butt, H.-J. Part. Part. Syst. Char. 2002, 19, 129–143.

ment of the cantilever (d) (after which the tip breaks free of the adhesive forces and returns to the free position during the sample retraction); i.e., Fp ≈ kd. From the five different measurements (Figure 2A), we obtained an average value of Fp ≈ 18.3 ( 0.2 nN for the particle-PEM adhesion force. This result, however, cannot be directly compared with the estimated shear force caused by the agitation during the rinsing operation because any shearinduced rearrangement of the strongly bound colloids is most likely through the rolling of the particles.22 On the basis of the predicted relation between the pull-off force (Fp) and the rollingfriction force (Fr) for soft adhesive contacts, Fr ≈ 2Fpξ/R, where ξ is the critical rolling displacement and R is the reduced radius of curvature of the two surfaces,22 and using an arbitrary minimum value of ξ ≈ 0.2 nm, we obtain an absolute minimum for Fr on the order of 10-2 nN, which still exceeds the upper bound for Fd by 1 order of magnitude. These results (i.e., Fr » Fd) suggest that the washing likely has a negligible impact on the particle positions. Additional evidence further supporting this conclusion is provided in Section S1 of the Supporting Information. For the purpose of obtaining detailed quantitative information about the interactions between the 1.85-µm silica particle and the PEM surface at finite separation distances, the approach (“precontact”) portions of the AFM deflection versus piezo position data shown in Figure 2A were converted to an average force–distance profile using the established procedures;17 the resulting force versus distance plot is presented in Figure 2B. Further, using this force–distance data, we obtained a particlesurface interaction potential profile (not shown in this paper), and this experimental potential profile was fitted to the DerjaguinLandau-Verwey-Overbeek (DLVO) theory23 using the effective charge density of the PEM surface (σp) and the Hamaker constant for the silica particle interacting with the PEM-coated substrate across the aqueous medium (A132) as two fitting parameters; in this analysis, we used a value of 0.009 e-/nm2 for the surface charge density of the 1.85-µm silica and a value of 4.2 × 10-5 M for the small ion concentrations (estimated as explained later in this section). From the fitting, we obtained very reasonable estimates of σp ≈ 3.3 N+ nm-2 and A132 ≈ 1.56 kBT for the above parameters. Force versus distance data reconstructed on the basis of these fitting results are also shown in the solid curve in Figure 2B. The above σp and A132 values will be used for evaluations of the interaction potentials between the PEM surface and a (22) Heim, L. O.; Blum, J. Phys. ReV. Lett. 1999, 83, 3328–3331. (23) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997.

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Figure 3. Theoretical energy profiles (W(D)) of the repulsive interaction between two silica particles of the same diameter (solid curve) and the attractive interaction between the silica sphere and a flat surface coated with 10.5 PDADAMAC/SPS bilayers (dotted curve), calculated from the DLVO theory as a function of the surface-to-surface separation distance (D) for various silica particle sizes: i.e., Ds ) (A) 1.85, (B) 1.50, and (C) 0.75 µm. Also shown in the insets of the respective figures are the length scales at which the silica-silica repulsive and silica-PEM attractive interactions become comparable to the thermal energy, kBT.

silica particle of a different size (e.g., 1.50 or 0.75 µm; Section III.2). Role of the RepulsiVe Forces between the Like-Charged Silica Particles in the Structure of the Adsorbed Particle Monolayer. In all particle adsorption experiments discussed in this paper, the silica particles were deposited onto the PEM surface from suspensions prepared by diluting the as-purchased 0.1 g mL-1 solutions of silica microspheres with deionized water to a final particle concentration of 0.01 g mL-1. In these suspensions, the conductivities (σ) were measured to be 5.47 ( 0.06, 14.38 ( 0.05, and 11.50 ( 0.05 µS cm-1 for the 1.85-, 1.50-, and 0.75µm silica cases, respectively (the experimental procedures are presented in Section S3 of the Supporting Information), which give the number concentrations of small ions (nion) of approximately 4.2 × 10-5, 1.1 × 10-4, and 9.0 × 10-5 M for the above respective systems due to the relation σ ) e2nion/6πηah, where e, η, and ah respectively denote the elementary charge, the viscosity of the medium, and the hydrodynamic radius of the ions; here we assume for simplicity that all ions are monovalent and have the same radius equal to the hydrodynamic radius of water (≈1.6 Å).24 We also measured the pH values of the suspensions; the pH values were 7.1, 7.9, and 7.0 for the 1.85-, 1.50-, and 0.75-µm silica suspensions, respectively (the experimental procedures are presented in Section S4 of the Supporting Information). From these pH and salt concentration values and using the established correlation between the surface charge density of silica and the pH and salt concentration of the medium,25 we estimate the charge densities of the 1.85-, 1.50-, and 0.75-µm silica spheres to be 0.009, 0.013, and 0.007 e-/nm2, respectively, under the particle deposition conditions. Using these chargedensity values and the above estimates for the ion concentrations and on the basis of the DLVO theory with the literature value of the Hamaker constant for silica in an aqueous medium of 1.95 kBT,19 we calculated the interaction energy between two silica microspheres of identical diameter (1.85, 1.50, or 0.75 µm) (24) Venable, R. M.; Pastor, R. W. Biopolymers 1988, 27(6), 1001–1014. (25) Bolt, G. G. J. Phys. Chem. 1957, 61, 1166–1169.

immersed in water under the above experimental conditions as a function of the surface-to-surface separation distance; the results are shown in Figure 3. As can be seen from the figure, the shortest approach distances beyond which the net electrostatic repulsion between the like-charged colloids becomes on the order of kBT are estimated to be 385, 225, and 185 nm for the 1.85-, 1.50-, and 0.75-µm silica spheres, respectively; this thermal energy value (≈kBT) was chosen arbitrarily for the purpose of making a comparison between the respective length scales associated with the particle–particle and particle-surface interactions (see below for the estimations of the latter quantities). For instance, we note that for the 1.85-µm silica case the estimated range of the repulsive electrostatic interaction between the like-charged silica particles (≈385 nm) is significantly smaller than the range of attraction between the PEM surface and the 1.85-µm silica particle (525 nm) estimated from the AFM force–distance data shown in Figure 2B, i.e., from the average distance between the point at which the elastic energy stored in the deflected cantilever becomes equal to kBT during the approach of the PEM surface toward the colloid probe and the point at which the cantilever suddenly springs in contact with the PEM surface upon further approach of the surface to the tip. Using the values of the PEM surface charge density and the Hamaker constant between the silica and the PEM-coated substrate in the aqueous medium obtained from the fitting of the data in Figure 2B with the DLVO theory (as discussed earlier in this section) and also using the above estimates for the surface charge density of the 1.50-µm silica sphere and the small ion concentration for the 1.50-µm silica suspension, we estimate the range of attraction between the PEM surface and the 1.50-µm silica particle to be about 321 nm, which is again significantly larger than the estimated range of the repulsive interaction between two 1.50-µm silica particles (≈225 nm). These analyses suggest that, although the repulsive interparticle interaction is sufficient to keep the particles dispersed as isolated objects in the suspension, this interaction plays a relatively weak role in determining the structure of the monolayer of the deposited particles. To the contrary, in the 0.75-µm silica case, the range of the interparticle repulsion (≈185 nm) is

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Figure 4. Representative optical micrographs demonstrating the 2Dordered (A) square (Ds ) 1.85 µm; Dp ) 1.15 µm) and (B) hexagonal (Ds ) 1.50 µm; Dp ) 1.05 µm) arrays of silica microspheres produced using the procedures outlined in Figure 1A. (C and D) Computerreconstructed images of the same areas visualized in (A and B, respectively), which display the calculated particle locations (discrete points) and pattern coordinates (meshed grids).

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Figure 5. Distributions of the off-center displacements of the silica particles deposited to the (A) square (Ds ) 1.85 µm; Dp ) 1.15 µm) and (B) hexagonal (Ds ) 1.50 µm; Dp ) 1.05 µm) polyelectrolyte patterns. These data are replotted in parts C and D to display the azimuthally averaged radial particle density (F(r)/2πr) profiles for the 2D square and hexagonal arrays, respectively.

estimated to be slightly larger than that of the attractive interaction between the silica and the PEM surface (≈177 nm); see Figure 3C. As will be discussed in the next section, this difference between this and the larger silica cases has a significant impact on the structures and qualities of the 2D colloid crystals produced with these particles using the PEM template-guided deposition method. III.2. 2D Colloid Crystals Templated by PEM Patterns. Detailed Structural Features of the 2D Colloid Crystals. Following the procedures described in Section II, we were able to produce 2D-ordered arrays of silica microspheres on the patterned PEM surfaces; see parts A and B of Figure 4 for representative images of the resultant 2D colloid crystals (lower magnification images of the same samples are presented in parts A and B of Figure S5 in the Supporting Information). In these examples, we used 1.85- and 1.50-µm-diameter silica particles, respectively, for preparing the square and hexagonal colloid crystals. The PEM pattern diameters used were 1.15 ( 0.06 and 1.05 ( 0.04 µm for the square and hexagonal cases, respectively. The resultant 2D square and hexagonal crystals of colloids respectively have average point defect (cavity) densities of about 10% and 2% over the 3 × 3 mm2 pattern area. The detailed structural properties of these 2D colloid crystals were examined in the water-immersed condition under an optical microscope. All of the photomicrograph data that are discussed in this section were taken under Köhler illumination conditions,26 and we confirmed the absence of optical distortion in the images27 by carefully examining images of a square-patterned silicon master (used in this work for making the PDMS stamp) taken under the exact same set of conditions as those in the imaging measurements of the 2D colloid crystal samples (see Section S5 of the Supporting Information for details); therefore, no image correction was necessary prior to analysis of the images.

Specifically, for accurate determination of the positions of the individual particles in the 2D crystal with respect to the lattice points defined by the positions of the PEM-patterned circles, the surfacebound particles were digitally imaged with a resolution of about 75 nm per pixel over an area of 120 × 90 µm2, and from these images, the coordinates of the particle centers were first determined as singlepixel local intensity maxima using the established image processing and analysis algorithms.28 As shown in parts C and D of Figure 4, the simulated particle images constructed on the basis of the calculated particle locations precisely reproduce the particle arrangements in the original images. However, the positions of the polyelectrolyte-patterned circles could not be determined directly from the optical imaging measurements, and to obtain estimates of the substrate pattern coordinates, we used the following procedures; using constant values for the unit cell dimensions of the PEM pattern (taken from the corresponding variables in the PDMS stamp structure), three parameters [i.e., the orientation and translational shifts (in both x and y directions) of the grid points of the polyelectrolyte pattern] were adjusted to minimize the sum of the distances between the substrate pattern centers and the measured points for the corresponding particle centers, and an optimal set of values for these fitting parameters that minimize the discrepancy between the particle and pattern positions was determined and used to define the best estimates for the substrate pattern coordinates. The reasonableness of these estimates is supported a posteriori by the consistency between the average size of the circles in the pattern estimated on the basis of the best-fit pattern coordinates (as will be discussed with reference to parts C and D of Figure 5) and its measured value (parts B and C of Figure 1). Once the estimates of the coordinates of the pattern circles were obtained by the method described above, we applied these to the examination of the statistical variations of the off-center distances of the attached particles. As shown in parts A and B of Figure 5 (generated from analyses of over 2000 particles for

(26) Heath, O. V. S. Nature 1954, 174, 506–507. (27) Ray, S. F. Applied Photographic Optics, 3rd ed.; Focal Press: Woburn, MA, 2002.

(28) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298–310. (29) Feder, J. J. Theor. Biol. 1980, 87, 237–254. (30) Onoda, G. Y.; Liniger, E. G. Phys. ReV. A 1986, 33, 715–716.

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Figure 6. Reconstructed images showing larger area views of the 2D (A) square (Ds ) 1.85 µm; Dp ) 1.15 µm) and (B) hexagonal (Ds ) 1.50 µm; Dp ) 1.05 µm) arrays of silica particles, with a color scale indicating the angle of orientation of the displacement of a particle (θi) relative to an arbitrary reference orientation. Pair correlations in the displacement vectors defining the off-center positions of the particles (〈ri · rj〉 ) 〈|ri||rj| cos |θi - θj|〉) as a function of the separation distance between the corresponding pattern centers (Dij) for the (C) square and (D) hexagonal cases.

Figure 7. Representative optical micrographs demonstrating the 2Dordered (A) square (Ds ) 0.75 µm; Dp ) 1.03 µm) and (B) hexagonal (Ds ) 0.75 µm; Dp ) 1.02 µm) arrays of silica microspheres. (C and D) Computer-reconstructed images of the same areas visualized in (A and B, respectively).

each of the two pattern types), the angular distributions of the particle locations relative to the corresponding pattern coordinates are found to be largely isotropic, which allows the radial distance (r) to be used as the single variable to characterize the statistical distribution of the off-center positions of the particles. Parts C and D of Figure 5 display the azimuthal averages of the radial particle density distributions for the 2D square and hexagonal arrays, respectively; in the figures, F(r)/2πr denotes the 2D number density of the particles where F(r)dr equals the number of adsorbed particles at a radial distance between r and r + dr from the

Figure 8. Distributions of the off-center displacements of the silica particles deposited to the (A) square (Ds ) 0.75 µm; Dp ) 1.03 µm) and (B) hexagonal (Ds ) 0.75 µm, Dp ) 1.02 µm) polyelectrolyte patterns. These data are replotted in parts C and D to display the azimuthally averaged radial particle density (F(r)/2πr) profiles for the 2D square and hexagonal arrays, respectively.

pattern center. A common trend observed in these results is that the probability density for locating the center of a particle at a distance r shows a plateau as a function of r at small r, whereas the probability decays with distance at larger r. The latter observation is understandable considering that the particle density at large off-center distances becomes limited by the screened electrostatic repulsion caused by the presence of neighboring particles that have adsorbed on the adjacent lattice points; the effect of this interparticle electrostatic repulsion, though relatively short-ranged (as can be seen from Figure 3), is operative in our experiments because of the small spacing between two adjacent adsorption sites. Another important factor that contributes to the

Two-Dimensional Colloid Crystals

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Figure 9. Reconstructed images showing larger area views of the 2D (A) square (Ds ) 0.75 µm; Dp ) 1.03 µm) and (B) hexagonal (Ds ) 0.75 µm; Dp ) 1.02 µm) arrays of silica particles, with a color scale indicating the angle of orientation of the displacement of a particle (θi) relative to an arbitrary reference orientation. Pair correlations in the displacement vectors defining the off-center positions of the particles (〈ri · rj〉 ) 〈|ri||rj| cos |θi - θj|〉) as a function of the separation distance between the corresponding pattern centers (Dij) for the (C) square and (D) hexagonal cases.

Figure 10. Representative optical micrographs demonstrating the 2Dordered (A) square (Ds ) 1.85 µm; Dp ) 1.29 µm) and (B) hexagonal (Ds ) 1.50 µm; Dp ) 1.15 µm) arrays of silica microspheres. (C and D) Computer-reconstructed images of the same areas visualized in (A and B, respectively).

decaying F(r)/2πr profile observed in the large r regime is that the portions of the surface outside the PEM patterns are noncharged, and therefore a particle approaching toward the PEM surface, especially near the boundary of the PEM-patterned circle, will be attracted toward the center of the patterned region. To assess whether this “electrostatic focusing” effect indeed has a substantial influence on the motions of the depositing particles, we calculated the electrostatic energy (W(D,r)) at near contact between the silica microsphere and the PEM pattern (D ≈ 0) as a function of the particle off-center distance r, to an approximation, by taking the linear summation of the preaveraged, screened

Figure 11. Distributions of the off-center displacements of the silica particles deposited to the (A) square (Ds ) 1.85 µm; Dp ) 1.29 µm) and (B) hexagonal (Ds ) 1.50 µm; Dp ) 1.15 µm) polyelectrolyte patterns. These data are replotted in parts C and D to display the azimuthally averaged radial particle density (F(r)/2πr) profiles for the 2D square and hexagonal arrays, respectively.

Coulombic pair potentials between all possible point charge-point charge combinations at an arbitrarily small value for the particle-surface separation distance (D ) 1 nm); see Section S6 and Figure S7 of the Supporting Information for details. It should be noted that the results are largely insensitive to the value of D in this short distance limit; for instance, using D ) 1 Å causes less than a few percent change in the W(D,r) versus r profile over the whole range of r. Also of note in the present analysis, we used the value for the (effective) surface charge density of the (PDADMAC/SPS)10.5 layer (σp ) 3.3 N+ nm-2) obtained from the fitting of the AFM force–distance profile (Figure 2B) with the DLVO theory (Section III.1). From the results of these

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Figure 12. Reconstructed images showing larger area views of the 2D (A) square (Ds ) 1.85 µm; Dp ) 1.29 µm) and (B) hexagonal (Ds ) 1.50 µm; Dp ) 1.15 µm) arrays of silica particles, with a color scale indicating the angle of orientation of the displacement of a particle (θi) relative to an arbitrary reference orientation. Pair correlations in the displacement vectors defining the off-center positions of the particles (〈ri · rj〉 ) 〈|ri||rj| cos |θi - θj|〉) as a function of the separation distance between the corresponding pattern centers (Dij) for the (C) square and (D) hexagonal cases.

W(D≈0,r) calculations, we estimated the driving forces for the horizontal movement of the depositing particle toward the pattern center, for instance, near the PEM boundary, to be Ff ≡ (∂W/ ∂r)D≈0 ≈ 3.2 × 10-6 nN for the 1.85-µm silica particle on a 1.15-µm-diameter PEM pattern and 5.4 × 10-6 nN for the 1.50µm silica on a 1.05-µm PEM pattern. These values are (3-10 times) less than the sedimentation forces (Fs ≡ (∆d)gV where ∆d is the density difference between the particle and the medium, g is the gravitational acceleration, and V is the particle volume) estimated for these particles (i.e., 3.25 × 10-5 nN for the 1.85µm silica microsphere, and 1.73 × 10-5 nN for the 1.50-µm silica); this gives an explanation for the tailing of the F(r)/2πr profile around the pattern boundary commonly observed in parts C and D of Figure 5. We note, however, that the estimated Ff values are significantly lower than the pull-off or rolling forces estimated for the 1.85-µm silica in adhesive contact with the PEM surface (Section III.1)(in the periphery of the PEM pattern, this particle adhesion force is expected to be smaller but on the same order of magnitude as that for the previously discussed case in which the surface was uniformly coated with the PEM), and therefore it is unlikely that, upon binding of the particles, this gradient of the electrostatic potential causes any further rearrangement of the surface-bound particles even near the pattern boundary. In an interpretation of the behavior of the F(r)/2πr profile observed at small r, it is useful to consider the following two possible cases; during the approach to the PEM pattern, the particles are allowed to rearrange their positions to reach the lowest electrostatic energy state, and therefore F(r)/2πr is a monotonically decreasing function of r (as was the case for larger r); on the other hand, if the magnitude of the focusing effect is sufficiently small relative to the sedimentation and adhesion forces (so that the particles make contact with the pattern surface at random locations and afterward become irreversibly bound at the positions of initial contact), the density distribution is expected to be nearly uniform with r. The experimental data in parts C and D of Figure 5 suggest that the process of particle deposition near the pattern center follows the latter scenario. This picture

is also supported by the calculation of the driving force for particle centering, which shows that Ff becomes vanishingly small in the small r region; for instance, Ff becomes less than a hundredth of Fs when r < 28% of the PEM pattern radius in the 1.85-µm silica case (see Figure S7 in the Supporting Information for plots of Ff (and Fs) versus r for the different combinations of particle and pattern sizes used in this study). Lastly, it is of note that, in parts C and D of Figure 5, the maximum off-center distances above which the particle density effectively drops to zero (say,