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Langmuir 2006, 22, 6826-6836
Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes Hung-Jen Wu, W. Neil Everett, Samartha G. Anekal, and Michael A. Bevan* Department of Chemical Engineering, Texas A&M UniVersity, College Station, Texas 77843-3122 ReceiVed February 21, 2006. In Final Form: May 11, 2006 We report a new method for mapping patterned surfaces based on monitoring the interactions of freely diffusing colloidal probes with pattern features to generate measured potential energy landscapes. Evanescent wave scattering and video microscopy are used to track 3D center positions of nominal 2 µm silica colloids as they diffuse over 5-20-nm-thick patterned gold films. An analysis of ensemble-averaged particle height histograms on different pattern features using Boltzmann’s equation produces local electrostatic and van der Waals potentials in good agreement with independent measurements and predictions. Absolute separation is obtained from theoretical fits to measured potentialenergy profiles and direct measurement by depositing silica colloids onto gold surfaces via electrophoretic deposition. As colloidal probe and pattern feature dimensions become comparable, potential energy profiles suffer some distortion due to the increased probability of probes sampling pattern feature edges. An analysis of interfacial colloidal probe diffusion in conjunction with potential energy measurements demonstrates a consistent interpretation of dissipative and conservative forces in these measurements. Future extensions of this work should produce similar approaches for interrogating physical, chemical, and biomolecular heterogeneous/patterned surfaces and structures with diffusing colloidal probes.
Introduction Physical and chemical surface heterogeneity is ubiquitous in synthetic materials, geochemical systems, and biological interfaces. Although homogeneous surfaces are an ideal model often assumed in theory and experiment, real surfaces have finite, if not extensive, surface heterogeneity. A unique type of surface heterogeneity encountered in many emerging technologies is the case where surfaces are intentionally patterned with regular chemical or physical features.1 Such surface patterns can serve as templates to initiate and orient colloidal assembly2,3 or as biomolecular arrays for use in high-throughput combinatorial measurements.4 Given the prevalence of problems involving either natural surface heterogeneity or microfabricated surface patterns, the measurement of physical and chemical heterogeneity over a broad range of length and energy scales is essential to numerous problems in science and technology. Spectroscopic techniques (e.g., surface plasmon resonance,5,6 total internal reflection fluorescence,7 etc.) provide one approach by nonintrusively monitoring equilibrium adsorption via changes in interfacial optical properties. Such measurements are sensitive to interactions between molecules and surfaces on the order of kT and can be used to infer surface heterogeneity via adsorption models. Because spectroscopic measurements generally measure large ensembles of molecules interacting with large surface areas, such measurements also have the benefit of being statistically significant. Scanning probe techniques (e.g., atomic force microscopy,8 chemical force microscopy,9 etc.) directly measure physical and * To whom correspondence should be addressed. E-mail: mabevan@ tamu.edu. (1) Xia, Y.; Whitesides, G. M. Angew. Chem., Int. Ed. 1998, 37, 550. (2) van Blaaderen, A.; Ruel, R.; Wiltzius, P. Nature 1997, 385, 321. (3) Lee, W.; Chan, A.; Bevan, M. A.; Lewis, J. A.; Braun, P. V. Langmuir 2004, 20, 5262. (4) Lee, K.-B.; Park, S.-J.; Mirkin, C. A.; Smith, J. C.; Mrksich, M. Science 2002, 295, 1702. (5) Smith, E. A.; Corn, M. R. Appl. Spectrosc. 2003, 57A, 320. (6) Shumaker-Parry, J. S.; Campbell, C. T. Anal. Chem. 2004, 76, 907. (7) Wertz, C. F.; Santore, M. M. Langmuir 2001, 17, 3006.
chemical surface heterogeneity via the mechanical deflection of cantilevers at different normal and lateral positions on surfaces. Such measurements are capable of measuring piconewton forces in aqueous media and are routinely used to measure physical and chemical features on inhomogeneous and patterned surfaces.10-12 Because scanning probe measurements directly interrogate surfaces with a single nanoscale tip, they are capable of probing highly localized surface properties in normal and lateral directions on a variety of different substrates. Although spectroscopic and scanning probe methods have been developed to measure atomic, molecular, and macromolecular interactions on heterogeneous surfaces, few studies have investigated interactions of colloids with heterogeneous and patterned surfaces in aqueous media. Many studies have probed the irreversible deposition of colloids on surface patterns,13 but such measurements provide only limited information on nonequilibrium particle-surface attractive interactions. A single study has investigated the interaction of a colloidal particle with hydrophobic and hydrophilic pattern features14 using the method of colloid attachment to an AFM cantilever tip.15 To our knowledge, no direct measurements have been reported for nanometer- and kT-scale interactions between colloids and physical or chemical patterns in aqueous media. In this article, we report measurements of numerous diffusing colloidal probes interacting with patterned surfaces as a novel means of interrogating interfacial potential energy landscapes on the order of kT. This novel method integrates and extends single-particle total internal reflection microscopy (TIRM)16 and (8) Binnig, G.; Quate, C. F.; Gerber, C. Phys. ReV. Lett. 1986, 56, 930. (9) Frisbie, C. D.; Rozsnyai, L. F.; Noy, A.; Wrighton, M. S.; Vezenov, D. V.; Lieber, C. M. Science 1994, 265, 2071. (10) Butt, H.-J. Biophys. J. 1992, 63, 578. (11) Heinz, W. F.; Hoh, J. H. Biophys. J. 1999, 76, 528. (12) Mazzola, L. T.; Frank, C. W.; Fodor, S. P. A.; Mosher, C.; Lartius, R.; Henderson, E. Biophys. J. 1999, 76, 2922. (13) Aizenberg, J.; Braun, P. V.; Wiltzius, P. Phys. ReV. Lett. 2000, 84, 2997. (14) Kokkoli, E.; Zukoski, C. F. Langmuir 2001, 17, 369. (15) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353), 239. (16) Prieve, D. C. AdV. Colloid Interface Sci. 1999, 82, 93.
10.1021/la060501j CCC: $33.50 © 2006 American Chemical Society Published on Web 07/07/2006
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multiparticle video microscopy (VM)17 methods to directly track 3D colloidal trajectories near surfaces with nanometer resolution of particle-surface separation and half-pixel resolution of lateral particle center coordinates. These new measurements and analyses of many particles on patterned surfaces build on our previous studies of ensemble-averaged particle-surface and particleparticle interactions near homogeneous surfaces.18,19 These measurements provide average equilibrium interactions of levitated colloidal particles with different pattern surface features. Our results demonstrate how measurements of the average interaction between numerous freely diffusing silica colloidal probes and thin patterned gold films are used to map interfacial potential energy landscapes. An analysis of particle height histograms on different pattern features produces particle-surface potentials in good agreement with independent measurements and predictions of electrostatic and van der Waals potentials for gold-coated glass substrates. Absolute particle-surface separation is directly measured via electrophoretic deposition of colloidal probes on gold patterns and indirectly inferred from theoretical fits to measured potential energy profiles. Although this work is primarily concerned with equilibrium particle-surface interactions, colloidal probe transport is quantified to avoid misconceptions concerning the role of hydrodynamic interactions in the reported equilibrium measurements.20,21 Finally, future extensions of this work are discussed in relation to “imaging” physical, chemical, and biomolecular surfaces and structures with diffusing colloidal probes.
Theory Potentials Due to Conservative Colloidal, Surface, and Body Forces. The net separation-dependent potential energy profile for a single colloidal particle levitated above a wall can be calculated as the sum of surface and body forces acting on the particle as
u(h) ) uedl(h) + uvdw(h) + ugrav(h)
(1)
where h is the separation between particle and substrate surfaces, uedl(h) is the potential due to the interaction of electrostatic double layers, uvdw(h) is the van der Waals potential, and ugrav(h) is the gravitational potential due to the buoyant particle weight. The interaction of electrostatic double layers between micrometer-sized particles near planar surfaces, uedl(h), is accurately described using nonlinear superposition and the Derjaguin approximation as22
uedl(h) ) B exp(-κh)
() ( ) ( )
eψp eψs kT tanh tanh B ) 64πa e 4kT 4kT
[
]
∑ e2ziCiNA 1/2 kT
uvdw(h) ) -aAh-p
(5)
where A and p are determined for different Au film thicknesses as described in the Appendix. The gravitational potential energy is the buoyant particle weight, G, multiplied by its height, h, above the wall given by
ugrav(h) ) Gh ) mgh )
(34)πa (F - F )gh 3
p
f
(6)
where m is the buoyant particle mass, g is the acceleration due to gravity, and Fp and Ff are the particle and fluid densities, respectively. The sum of interactions relative to the minimum potential is obtained from eqs 1-6 as
u(x + hm) - u(hm) aB′ ) exp[-κ(x + hm)] + kT kT 3 u(hm) aA a G′ (x + hm) - (x + hm)-p (7) kT kT kT where B′ ) B/a and G′ ) G/a3 from eqs 3 and 6, x ) h - hm, and hm is the most probable height. Colloid-Surface Potentials on Homogeneous and Heterogeneous Surfaces. To analyze numerous levitated colloids interacting with heterogeneous and patterned surfaces, it is useful to first describe the analysis of a single levitated colloid above a homogeneous surface. The scattering intensity of a spherical colloid in an evanescent wave is related to the instantaneous particle-surface separation, h, by16
I(h) ) I0 exp(-βh)
(8)
where I is the scattering intensity, I0 is the intensity at particlesurface contact, h ) 0, and β-1 is the evanescent wave decay length given by
(2) β)
2
κ)
Ci is the bulk electrolyte concentration of species i, zi is the valance of species i, and NA is Avogadro’s number. The van der Waals interaction between a sphere and a coated half-space is accurately predicted using the Lifshitz theory23-25 and the Derjaguin approximation26 as described in the Appendix. For ease of use when fitting theoretical potentials to measured potential energy profiles, rigorous van der Waals predictions are well represented over the separation and energy ranges of interest by a convenient power law expression as
(4πλ)[(n sin θ ) - n
2 1/2
2
1
1
2
]
(9)
(3)
(4)
where a is the particle radius, is the dielectric permittivity of water, k is Boltzmann’s constant, T is the absolute temperature, e is the elemental charge, ψp and ψs are the Stern potentials of the particle and the wall, respectively, κ-1 is the Debye length, (17) Crocker, J. C.; Grier, D. G. J. Colloid. Interface Sci. 1996, 179, 298. (18) Wu, H. J.; Bevan, M. A. Langmuir 2005, 21, 1244. (19) Wu, H.-J.; Pangburn, T. O.; Beckham, R. E.; Bevan, M. A. Langmuir 2005, 21, 9879. (20) Anekal, S.; Bevan, M. A. J. Chem. Phys. 2005, 122, 034903. (21) Anekal, S.; Bevan, M. A. J. Chem. Phys., accepted for publication, 2006. (22) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989.
where n1 and n2 are the refractive indices of the incident and transmitted media, respectively, and θ1 is the angle of incidence. For a freely diffusing levitated particle, its time-dependent height fluctuations, h(t), measured using eq 8 can be used to construct a time-averaged height histogram, n(h), as
h(t) f n(h)
(10)
which is related to the average particle-surface potential using (23) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: New York, 1976. (24) Prieve, D. C.; Russel, W. B. J. Colloid Interface Sci. 1988, 125, 1. (25) Parsegian, V. A. Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists; Cambridge University Press: Cambridge, England, 2005. (26) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992; p 450.
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Wu et al.
Boltzmann’s equation as
[
]
u(h) kT
n(h) ) A exp -
(11)
near a planar surface, particle-surface hydrodynamic interactions hinder the particle’s diffusion for a given height above the wall in directions both parallel and normal to the surface as given by
where A depends on the total number of height measurements used to construct n(h). To obtain the particle-surface potential relative to a reference potential, eq 11 is inverted as
[ ]
n(href) u(h) - u(href) ) ln kT n(h)
(12)
where href is a reference height often chosen as either the most probable height, hm, in n(h) or h ) ∞ where the surface potentials decay to zero. The resulting single-particle potential, u(h), is averaged over time and many surface locations (x(t), y(t)). An analysis of many single particles diffusing laterally and normally over homogeneous surfaces can be used to measure ensemble-averaged particle-surface potential profiles.18,19 By assuming uniform particles, the time-dependent height fluctuations of all particles can be analyzed as an ensemble-average histogram as
h1(t), h2(t)...hP(t) f n1(h), n2(h)...nP(h) f 〈n(h)〉 (13) where hP(t) represents the time-dependent height fluctuations of each single particle, nP(h) represents the height histograms of each single particle averaged over all surface locations (xP(t), yP(t)) sampled in a given time period, and 〈n(h)〉 is the histogram averaged over all particles and surface locations that can be analyzed using eq 12. For many uniform particles freely diffusing over heterogeneous or patterned surfaces, the analysis is modified to construct particle height histograms for different surface locations as
h1,S(t), h2,S(t)...hP,S(t) f n1,S(h),n2,S(h),...nP,S(h) f 〈n(h)〉S (14) where hP,S(t) represents the time-dependent height fluctuations of each particle on a surface location, S, nP,S(h) represents the height histograms for each particle on a given surface location in a given time period, and 〈n(h)〉S is the histogram averaged over all particles that sample a surface location in an observation period. Surface locations can be defined by arbitrary grids of “pixels” or by independently determined “regions of interest”. The average particle-surface potential energy profile for each location S from the average particle height histogram on each location S is
〈
〉 [
u(h) - u(href) kT
) ln
S
]
〈n(href)〉S 〈n(h)〉S
(15)
For uniform particles, the particle-surface potential energy profiles depend on the physicochemical properties at each location S. The resulting height-dependent particle-surface potential energy profiles for each lateral location, S, can be rendered as a 3D potential energy space from which 2D slices can be extracted to visualize interfacial potential energy landscapes. Colloidal Diffusion near Surfaces. The diffusion coefficient of an isolated colloidal particle undergoing 3D diffusion far from any boundaries is given by the Stokes-Einstein equation as
D0 )
kT 6πµa
D|(h) ) D0 f|(h)
(17)
D⊥(h) ) D0 f⊥(h)
(18)
where f|(h) and f⊥(h) are conveniently represented by fits to the exact solution (with less than (0.001 relative error) as21
f|(h) )
12 420R(h)2 + 5654R(h) + 100 12 420R(h)2 + 12 233R(h) + 431
f⊥(h) )
6R(h)2 + 2R(h) 6R(h)2 + 9R(h) + 2
(20)
where R(h) ) h/a. For levitated colloids diffusing parallel to planar surfaces and within potential energy wells normal to underlying surfaces, the average lateral and normal diffusion coefficients are given by21
〈D|〉 )
∫D|(h) n(h) dh ∫n(h) dh
(21)
〈D⊥〉 )
∫D⊥(h) n(h) dh ∫n(h) dh
(22)
where n(h) is the height histogram either measured using eq 10 or related to the particle-surface potential energy profile via eq 11. Mean square displacements (MSD) provide a measure of particle diffusion in a given direction, x, as a function of time, t, as
Wx(t) )
1
Np
[xi(t) - xi(0)]2 ) 2Dt ∑ N i)1
(23)
p
where D is the apparent diffusion coefficient. For isolated particles diffusing far from any boundaries or other particles, D ) D0 (eq 16) in all directions. For particles diffusing parallel to a surface but far from other particles, D ) 〈D|〉 (eq 21). Particle diffusion normal to a surface but far from other particles has D ) 〈D⊥〉 (eq 22) at short times and D ) 0 at long times because of confinement within the normal potential well. For elevated interfacial particle concentrations, eq 23 is a measure of selfdiffusion, and the interpretation requires the consideration of multi-body hydrodynamic interactions and many-body packing effects.21 For noninteracting colloidal particles diffusing parallel to surfaces that also experience migration in an external force field, the interpretation of MSDs must also consider the net velocity, V, superimposed on the diffusive motion given by27
Wx(t) ) 2〈D|〉t + [Vxt]2
(24)
where the velocity can be related to the external force acting parallel to the surface, F, and the average lateral diffusion coefficient (via the general relationship D ) kT/m) as
(16)
where µ is the surrounding medium viscosity. For a single colloid
(19)
F ) 〈m|〉Vx )
( )
kT V 〈D|〉 x
(25)
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Figure 1. (a) Transmitted light CCD image of 2.22 µm silica colloids electrostatically levitated in aqueous 1 mM NaCl above 75 µm × 75 µm × 10 nm (l × w × h) Au square films separated by 40 µm bare glass regions. Au films appear darker than uncoated glass. (b) Diffusion of levitated silica colloids on a patterned surface for 24 min from initial positions in part a. Colored regions of interest represent the average value of the particle-surface potential energy profile at h ) 77 nm (relative to zero potential energy at h ) ∞). The inset color scale shows an energy range from -2kT (red) to 4kT (violet).
For confined diffusion within lateral potential wells, the interpretation of MSDs must account for the particles’ limited ability to achieve displacements beyond a characteristic length scale of the well, L, using the following expression27
[
(
Wx(t) ) 〈L2〉 1 - A1 exp -
)]
A22〈D|〉t 〈L2〉
(26)
where A1 and A2 are constants that can be related to the well shape. Experimental Section Nominal 2.34 µm silica colloids (Bangs Labs., Fishers, IN) with a reported density of FSiO2 ) 1.96 g/mL were diluted in aqueous solutions to obtain bulk particle concentrations that produced the desired interfacial concentrations after sedimentation equilibrium was attained. The particle size distribution was measured using confocal microscopy and dynamic light scattering to have an average diameter of 2.22 µm and a standard deviation of 0.11 µm as described in a previous paper.19 The aqueous solution ionic strength and pH were controlled using analytical-grade NaCl, KNO3, KOH, and HNO3. Silica particle zeta potentials as a function of ionic strength and pH were measured using a ZetaPALS instrument (Brookhaven Instrument Corp., Holtsville, NY). Au films were patterned on glass microscope slides (Corning Inc., Corning, NY) by metal evaporation using TEM grids as masks. Glass slides were cleaned prior to metal deposition by soaking in piranha solution (3:1 H2SO4/H2O2) for 1 h and then rinsing with DI water. TEM grids (Gilder Grids, 300 and 2000 meshes, Ted Pella Inc., Redding, CA) were placed on top of glass slide surfaces in a 306 metal evaporation chamber (BOC Edwards, Wilmington, MA). A 2 to 3 nm Cr layer was initially deposited at 0.15 nm/s to improve Au film adhesion, and then 2-15 nm Au films were deposited at 0.15 nm/s. After washing patterned surfaces with DI water and drying with high-purity N2, 10 mm i.d. × 12 mm o.d. Viton O-rings were attached to surfaces using vacuum grease to produce batch sedimentation cells. To measure deposited particle intensities via electrophoretic deposition, electric fields were applied between an upper transparent ITO electrode (Delta Technologies Inc., Stillwater, MN) and conducting patterned Au films on the bottom surface. DC electric fields of 0.7-1.5 V were applied across the 1 mm batch cell height (27) Saxton, M. J.; Jacobson, K. Annu. ReV. Biophys. Biomol. Struct. 1997, 26, 373.
by connecting the Au electrodes to the negative and positive terminals of a power supply (Kepco, ATE55-10DM, Flushing, NY). Electric fields were maintained at less than 1.5 V to avoid metal film reduction. Particle trajectories were measured using a combined evanescent wave16 and video microscopy17 apparatus.18,19 The evanescent wave was generated using a dovetail prism and a 15 mW 632.8 nm heliumneon laser (Melles Griot, Carlsbad, CA). Images were obtained using either 40× (NA ) 0.65) or 63× (NA ) 0.75) objectives (Zeiss, Germany) in conjunction with a 12 bit CCD camera (ORCA-ER, Hamamatsu, Japan) operated with 4× binning. In addition, images were obtained at 27 frames/s with 336 × 256 resolution to produce either 607 nm pixels (204 × 155 µm2 image) with the 40× objective or 241 nm pixels (81 × 62 µm2 image) with the 63× objective and a 1.6× magnifying lens. Image analysis algorithms coded in Fortran were used to track the lateral motion and integrate the evanescent wave scattering intensity for each particle.
Results and Discussion Potential Energy Landscape for Colloids on a Gold-Glass Pattern. Figure 1a shows a CCD image of 2.22 µm silica colloids levitated in aqueous 1 mM NaCl (126 µS/cm, pH 5.4) above a glass microscope slide surface patterned with 75 µm × 75 µm × 10 nm thick Au films separated laterally by 40 µm. The image in Figure 1a is obtained using transmitted light to illuminate the particles and pattern features with the Au films appearing darker than uncoated glass regions because of different light attenuation levels. The ability to visualize the Au pattern using transmitted light allows the coated and uncoated regions to be specified as distinct regions of interest for interrogation with diffusing colloidal probes. The Au-coated and uncoated regions of the patterned glass surface in Figure 1a are expected to have different surface charges and dielectric properties to produce different electrostatic and van der Waals interactions between levitated silica colloids and each surface region on the kT scale. By using a CCD camera to measure the evanescent wave scattering from levitated colloids as they freely diffuse over the patterned surface in Figure 1a, the particle-surface separation, h, is measured with nanometer resolution, and particle centers (x, y) are tracked in the plane parallel to the surface with ∼300 nm resolution (half of a CCD pixel).18,19 The Au film does not significantly scatter the evanescent wave at the patterned film edges. The intensity of the evanescent wave (I0 in eq 8) decreases
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because of differences in the reflection and absorption of the incident 632.8 nm laser light by the Au film compared to those of the uncoated glass surface. Differences in I0 values on different pattern regions are accounted for in analyses presented in the following sections. Absorbance of the 632.6 nm laser light (15 mW, 1 mm diameter) by the Au films does not produce any apparent heating effects as confirmed by the observation of expected equilibrium and dynamic results for levitated colloids. The value of the evanescent wave decay length is not known a priori because the ∼10 nm Au film optical properties are not well defined. Surface plasmons on patterned Au films should produce intensity decay lengths of βSP-1 ) 90 nm (using (λ/ 4π)[(Au + w)/-w2]0.5)28 for continuous films with bulk Au properties.29 However, the effective optical properties of sub20-nm Au films29 are not well established from either measurements30 or models,31 particularly because such films are often discontinuous with preparation-dependent properties. Electrical conductivities of the Au films in this work, which are also used as electrodes for electrophoretic deposition, suggest that the films are beyond their percolation threshold but do not provide information on the film microstructure. All measured potential energy profiles in this work are consistent with an evanescent wave decay length that is essentially unchanged on the Au films compared to the glass-water interface value of β-1 ) 114 nm. In particular, using β-1 ) 114 nm to compute potential energy profiles produces quantitative agreement between (1) κ-1 from electrostatic potentials and solution conductivities, (2) particle size distributions from gravitational potentials18 and light-scattering measurements,19 and (3) van der Waals potentials and theoretical predictions.23-25 For this initial work monitoring colloidal scattering in evanescent waves on sub-20-nm Au films, we proceed cautiously using β-1 ) 114 nm to obtain potential energy profiles consistent with independent measurements using no adjustable parameters. Future work will directly measure evanescent wave decay lengths on Au films from the scattering of particles deposited on MgF2 spacer films of known thickness in index-matched media.32 By assuming that Au films and bare glass surfaces are locally homogeneous on length scales probed by micrometer-sized colloids, measurement of the normal and lateral excursions of levitated silica colloids allows particle-surface height histograms to be constructed and analyzed for different surface regions as average potential energy profiles (eqs 14 and 15). This analysis assumes that the silica particles are uniform so that each particle’s height histogram is identical with an ensemble-averaged histogram for all particles sampling a given surface region. This assumption was previously demonstrated to be reasonably accurate in measurements of unpatterned surfaces using the same silica colloids used in this study.18,19 In particular, size polydispersity was easily accounted for in a3-dependent gravitational potentials, had a minimal effect on a1-dependent colloid-surface interactions, and was effectively averaged out in ensemble analyses of many single colloids. As a matter of terminology, the resulting “maps” of normal particle-surface potential energy profiles on different lateral surface locations are referred to as potential energy landscapes. This jargon is intended to be consistent with inverting particlesurface histograms using Boltzmann’s equation to produce pairwise potential energy profiles unaffected by any other nearby (28) Knoll, W. Annu. ReV. Phys. Chem. 1998, 49, 569. (29) Johnson, P. B.; Christy, R. W. Phys. ReV. B 1972, 6, 4370. (30) Dvoynenko, M. M.; Goncharenko, A. V.; Romaniuk, V. R.; Venger, E. F. Physica B 2001, 299, 88. (31) Kreibig, U.; Fragstein, C. V. Z. Phys. 1969, 224, 307. (32) Prieve, D. C.; Walz, J. Y. Appl. Opt. 1993, 32, 1629.
Wu et al.
particles. These definitions anticipate future experiments where high particle concentrations will produce multiparticle packing effects such that the inversion of histograms with Boltzmann’s equation will produce potentials of mean force that capture particle-surface interactions mediated by other nearby and/or intervening particles. Because potentials of mean force include energetic and entropic contributions, such surface-energy maps will be referred to as free-energy landscapes rather than potentialenergy landscapes. Figure 1b shows trajectories of silica colloids from their initial positions in Figure 1a as they diffuse laterally over the patterned surface to probe surface potentials. The colors of the two regions of interest in Figure 1b correspond to the average particlesurface interaction potentials at h ) 77 nm for each region. The potential energy scale from -2kT to +4kT relative to zero interaction at infinite particle-surface separations is represented by the color spectrum in the inset of Figure 1b. The value of the particle-surface potential at h ) 77 nm is obtained from the average particle-surface potential energy profile for each region reported in Figure 2a. The profiles in Figure 2a were obtained from an analysis of particle height histograms (〈n(h)〉SiO2 and 〈n(h)〉Au) averaged over all particles sampling each region of interest using eqs 14 and 15. The histograms for each region were obtained by monitoring the normal and lateral excursions of a total of 65 levitated colloids within the image window with 40 000 images acquired over a 24 min period. The total number of particles is greater than the average number (∼45) in the image at any given instant because particles enter and leave the imaging area via lateral diffusion during the course of the experiment. Several deposited particles were ignored in the analysis. Although deposition is not theoretically predicted for the 1 mM conditions in this experiment, it is well known in practice that reaction-limited deposition occurs at much higher rates probably because of surface heterogeneity on length scales much less than the particle size.33 Normal excursions of levitated particles were also ignored in the case of strong lateral interference of signals for particle centers within 7 µm (∼8a) of each other. The profiles in the inset of Figure 2a include the particles’ average gravitational potential energy whereas the main plot shows profiles with only electrostatic and van der Waals potentials between particles and surface regions. Solid lines in Figure 2 are theoretical fits to the measured potential energy profile for the bare glass and Au surfaces using eq 7, which are in good agreement with each other. When fitting eq 7 to the profiles in Figure 2, values of κ, G′, A, and p were fixed from conductivity measurements, the manufacturer-reported particle density, and van der Waals predictions. The values of A and p were obtained from the thickness of the Au films determined via calibration of the metal evaporator and Lifshitz predictions described in the Appendix. Initial values of a and B′ were provided by independent dynamic light scattering and zeta potential measurements and literature measurements of the pHionic strength-dependent Au surface potential.34 Values of all fixed and adjustable fit parameters are reported in Table 1. The fit value of the most probable separation, hm, for each profile is used to determine the absolute particle-surface separation scale reported in Figures 1b and 2a. The particle-surface interactions on the bare glass and Au regions produce distinct potential energy profiles that can be visualized as a potential energy landscape by plotting their interactions at a fixed height above each region in Figure 1b. (33) Velegol, D.; Thwar, P. K. Langmuir 2001, 17, 7687. (34) Barten, D.; Kleijn, J. M.; Duval, J.; von Leeuwen, H. P.; Lyklema, J.; Stuart, M. A. C. Langmuir 2003, 19, 1133.
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Langmuir, Vol. 22, No. 16, 2006 6831
Figure 2. (a) Average potential energy profiles for 2.22 µm silica colloids interacting with bare glass (O) and 10 nm Au films (0) in aqueous 1 mM NaCl (from Figure 1). The main plot shows particle-surface potentials without gravitational potentials (relative to zero potential energy at h ) ∞), and the inset includes gravity (relative to zero potential energy at h ) hm). Solid lines (-) are theoretical curve fits to eq 7. (b) Lateral mean squared displacements in the x (O) and y (0) directions (from Figure 1). Reference lines are shown for bulk diffusion (‚‚‚) from eq 16 and lateral surface diffusion on glass (- -) and 10 nm Au (-‚-) regions from eq 21. Solid lines (-) are theoretical curve fits to eq 24. Table 1. Average Particle-Surface Potential Energy Profile Fit Parameters in Figures 2a, 4a, and 6a eq δAu/nm κ-1/nm B′/J nm-1 -ψp/mV -ψs/mV Α/kT nmp - 1 p a/nm hm/nm hm/nm
4 7 3 5 5 7 7 8
Figure 2 0 9.58 5.53 59 56 2.10 2.15 1066 103
10 9.58 1.21 59 24 8.44 2.04 1074 77
Figure 4 9 9.64 1.77 59 30 8.29 2.04 1051 81 72
18 9.64 1.64 59 29 8.97 2.03 1144 79 75
Figure 6 5 10.0 0.53 59 16 7.20 2.06 1050 72 96
15 10.0 0.36 59 13 8.86 2.03 1050 61 105
This procedure produces an energy map of the surface that can be interpreted with theoretical fits in terms of varying surface properties including surface potentials and dielectric properties of each region. The theoretical fit parameters are consistent with other measurements of similar homogeneous glass and Au surfaces, particularly within the uncertainties associated with the origin of charge on each surface, the role of surface roughness, etc.34-36 In contrast to the different electrostatic and van der Waals interactions measured for the bare glass and Au regions, the gravitational potential and inferred radius of the levitated particles in the inset of Figure 2a are insensitive to the surface region as expected within the limits of particle size polydispersity.37,38 One drawback of the measurement reported in Figures 1 and 2 is that the absolute particle-surface separation was obtained via the fit to eq 7 rather than being measured directly. The separation is obtained in standard TIRM experiments by depositing particles on the wall surface at the conclusion of measuring levitated particles to determine the value of I0 in eq 8. This approach is not easily implemented in the experiment in Figures 1 and 2 because (1) each particle has a unique I0, even (35) Iler, R. K. The Chemistry of Silica; Wiley: New York, 1979; p 866. (36) Bevan, M. A.; Prieve, D. C. Langmuir 1999, 15, 7925. (37) Pangburn, T. O.; Bevan, M. A. J. Chem. Phys. 2005, 123, 174904. (38) Pangburn, T. O.; Bevan, M. A. J. Chem. Phys. 2006, 124, 054712.
for nearly monodisperse samples (which remains poorly understood) and (2) strong convection associated with adding salt solutions to cause particle deposition also causes colloidal probes to be “lost” via translation such that each particle’s specific I0 value cannot be measured. Despite each particle having a different I0 value, the ensemble-averaged profiles in Figure 2a are obtained by aligning single-particle profiles by their most probable intensity, Im, which is described in detail in our previous work.18,19 In a following section of this article, we describe a method to overcome problems with directly measuring absolute particlesurface separation by using electrophoretic deposition to deposit particles reversibly onto the patterned surface immediately at the conclusion of monitoring diffusing probe excursions. Colloidal Diffusion on a Gold-Glass Pattern. Before attempting to measure absolute particle-surface separation, we provide an analysis of lateral particle diffusion via MSDs for the measured particle trajectories shown in Figure 1b. Figure 2b shows MSDs computed separately for excursions in the x and y directions (eq 23), which are averaged over all particles and multiple time origins.21 The experimental MSD curves are fit to eq 24 to interpret the average lateral particle diffusion coefficient, 〈D|〉, rate of migration, V, and the magnitude of force field components parallel to the surface, F, (eq 25) in the x and y directions. As a side note, analysis of MSD data in this work does not require a consideration of the spatial gradient of 〈D|〉 that is required in the equation of motion for simulating Brownian particle trajectories.21,39 Fit parameters are reported in Table 2. Figure 2b also shows lateral diffusion coefficients predicted by eq 21 from each region’s average particle-surface potential and the exact result for particle-surface hydrodynamic interactions.21 A curve representing bulk diffusion (D0 in eqs 16 and 23) indicates the degree to which lateral diffusion is hindered by particlesurface hydrodynamic interactions.21 These results illustrate several interesting dynamic features in these experiments, but perhaps more importantly, they demonstrate a consistent interpretation of both conservative forces due (39) Ermak, D. L.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352.
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Figure 3. (a) Transmitted-light CCD image of 2.22 µm silica colloids electrostatically levitated in aqueous 1 mM KNO3/HNO3 above 75 µm × 75 µm × 18 nm (l × w × h) Au square films separated by 40 µm regions with 9 nm Au films (all on glass substrate). Au films (18 nm) appear darker than 9 nm Au films. (b) Diffusion of levitated silica colloids on a patterned surface for 24 min from initial positions in part a. Colored regions of interest represent the average value of the particle-surface potential energy profile at h ) 79 nm (relative to zero potential energy at h ) ∞). The inset color scale shows an energy range from -2kT (red) to 4kT (violet). Table 2. Particle Diffusion, Migration, and Confinement Parameters in Figures 2b, 4b, and 6b eq δAu/nm 〈D⊥〉/D0 〈D|〉/D0 〈D|,x〉/D0 〈D|,y〉/D0 Fx/fN Fy/fN A1,x, A2,x A1,y, A2,y Lx/a Ly/a
22 21 24 24 25 25 26 26 26 26
Figure 2
Figure 4
Figure 6
0 10 0.119 0.108 0.478 0.454 0.373 0.343 0.852 0.360
9 18 0.115 0.096 0.461 0.445 0.440 0.434 1.50 × 10-6 0.574
5 15 0.113 0.111 0.455 0.449 0.449 0.449 1.02, 1.36 1.01, 1.17 4.67 4.43
to particle-surface potentials and dissipative forces associated with the hydrodynamic interactions important to interfacial particle diffusion. Theoretical fits to the measured MSDs indicate 〈D|〉 values in good agreement with those predicted from particlesurface forces and hydrodynamic interactions using eq 21, with both values being significantly smaller than the isolated singleparticle diffusion coefficient (〈D|〉/D0 ≈ 0.4). Values of 〈D|〉 display small differences on the two surface regions due to small differences in van der Waals attractions (Figure 2a) but are not significantly different in the x and y directions because hydrodynamic interactions have no directional bias due to the intervening fluid properties or surface features. Theoretical MSD fits in Figure 2b indicate that lateral diffusion is superimposed on a finite lateral migration rate with a bias in the x direction. The lateral force corresponding to this migration rate is less than 1 fN in both the x and y directions and is most likely due to a minor misleveling of the sample cell (θ < 1°). Although flow fields due to a small leak or a sustained temperature gradient could also cause such migration over the ∼24 min experiment, rotating the cell indicated a preferred direction for migration consistent with misleveling. In any case, the source of migration is relatively unimportant once its rate is accounted for because it does not affect measured potential energy landscapes. There is no evidence of partial confinement of particles within the lower-energy Au features from the MSD data in Figure 2b. However, it is not expected that such confinement should be observable because the duration of the measurement in Figure 2b is not sufficiently long to allow MSDs
to occur on the length scales associated with pattern feature dimensions ((40 µm/a)2 and (75 µm/a)2). Particle diffusion normal to the underlying substrate was not extensively investigated in this work because it is relatively unimportant to how diffusing colloidal probes laterally sample potential energy landscapes determined by surface pattern features. Using eqs 21 and 22, the rate of lateral diffusion in this work is expected to be significantly greater than the rate of normal diffusion (〈D|〉/〈D⊥〉 ≈ 4) (〈D⊥〉/D0 reported in Table 2). The only significance of these relative diffusion rates to the present study is that normal excursions in successive frames can be expected to be smaller on average than lateral excursions. This ensures that particle height histograms used to construct potentialenergy profiles on each region (Figure 2a) do not contain too many points sampled at any one particular x, y position but produce statistically representative potential profiles for given surface regions. Colloidal Diffusion and Electrophoretic Deposition on a Gold-Gold Pattern. To obtain absolute particle-surface separations in experiments using diffusing colloidal probes to map patterned potential energy landscapes, we now investigate the elecrophoretic deposition of particles onto patterned Au films to directly measure each particle’s I0 value in eq 8. Figure 3a shows a CCD image of 2.22 µm silica colloids levitated in 1 mM ionic strength aqueous media (0.9 mM KNO3, 0.1 mM HNO3, 146 µS/cm, pH 4.2) above a continuous patterned Au film suitable for use as an electrode in electrophoretic deposition experiments. The Au film in Figure 3a was fabricated by initially depositing a laterally uniform 9 nm Au film and then a patterned 9 nm Au film to produce an electrically conductive surface with 75 µm × 75 µm × 18 nm thick Au films laterally separated by 40 µm × 9 nm Au films. As in Figure 1, the thicker 18 nm Au films are easily distinguished from the thinner 9 nm Au films by different transmitted light levels. Because the entire surface in Figure 3a is a Au film, the surface charge can be expected to be similar on all pattern features and therefore produce similar electrostatic interactions with diffusing colloidal probes. The conductive Au film’s surface potential can be controlled using externally applied electric fields and also via the medium ionic strength and pH.34 This allows the Au surface potential to be altered via solution conditions in levitated particle experiments and enables particles to be reversibly deposited on
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Figure 4. (a) Average potential energy profiles for 2.22 µm silica colloids interacting with 9 (O) and 18 nm (0) Au films in aqueous 1 mM KNO3/HNO3 (from Figure 3). The main plot shows particle-surface potentials without gravitational potentials (relative to zero potential energy at h ) ∞), and the inset includes gravity (relative to zero potential energy at h ) hm). Solid lines (-) are theoretical curve fits to eq 7. (b) Lateral mean squared displacements in the x (O) and y (0) directions (from Figure 3). Reference lines are shown for bulk diffusion (‚‚‚) from eq 16 and lateral surface diffusion on 9 (- -) and 18 nm (-‚-) regions from eq 21. Solid lines (-) are theoretical curve fits to eq 24.
Au film surfaces at the conclusion of experiments via electrophoretic (migration toward the surface in an electric field) and electrostatic (attractive interactions due to opposite charges on SiO2 particles and the Au surface) deposition. Differences in Au film thicknesses in different pattern regions in Figure 3a can be expected to produce different net van der Waals interactions. Figure 3b shows trajectories of levitated silica colloids on the patterned surface with the colors of the two regions corresponding to average particle-surface interaction potentials at h ) 79 nm for each region as determined from the potentials in Figure 4a. The histograms used to compute the profiles for each region in Figure 4a were obtained by monitoring 59 levitated colloids in the image window in 40 000 images over a 24 min period. The treatment of raw data and the analysis of potential energy profiles in Figure 4a are similar to those used in Figure 2a with values of all theoretical fit parameters reported in Table 1. The key difference between the results in Figures 4a and 2a is that absolute separation was measured directly in Figure 4a by depositing the particles onto the Au film to determine each particle’s I0 value in eq 8. I0 values differ slightly for the 9 and 18 nm Au film regions (for previously discussed reasons), and because many particles sample both regions, it is necessary to generate I0 values for each particle on each region. Because each particle’s I0 value is measured only on one of the two regions, the I0 value for the second region is estimated from the ratio of average deposited particle intensities in each region. In Figure 3, the ratio of the I0 values for the two regions (averaged over all particles) is (I0,18 nmAu/I0,9 nmAu) ) 1.47, which is either (1) multiplied by I0,9 nmAu to estimate I0,18 nmAu or (2) divided by I0,18 nmAu to estimate I0,9 nmAu. Values of the most probable separation, hm, which are used to determine the absolute separation, are reported in Table 1 from both direct I0 measurements and theoretical fits to eq 7. Excellent agreement between parameters obtained from theoretical fits to the potential energy profiles in Figure 4a and independent measurements suggest that the potential energy landscape rendered in Figure 3b is an accurate measurement of
diffusing colloidal probe interactions with the patterned Au film. As expected, electrostatic potentials are essentially independent of surface location because the Au surface is present everywhere on the pattern and should have a surface potential dependent on the aqueous medium ionic strength and pH.34 As discussed in relation to Figure 2a, the average gravitational potential and inferred size of the levitated particles is also insensitive to surface location. The van der Waals interaction is found to be stronger on the 18 nm Au film than on the 9 nm Au film, in good agreement with the Lifshitz predictions described in the Appendix. The potential energy profiles measured for each pattern region are in good agreement with independent measurements and theoretical predictions and are consistent with one another in terms of expected differences between Au films of different thicknesses. The experiment in Figures 3 and 4 demonstrates that the measurement of absolute separation via particle deposition is in excellent agreement with the estimate of absolute separation obtained via a theoretical fit (from a comparison of hm values in Table 1). This provides confidence in the absolute separation scale reported for the results in Figures 1 and 2 when particle deposition is not easily controlled. This also demonstrates a novel method for depositing particles onto surfaces to determine I0 values at the conclusion of experiments measuring evanescent wave scattering from levitated particles. An interesting feature of the electric-field-controlled deposition on the Au films in this work is its complete reversibility for all particles, which might allow for multiple measurements of diffusing probes on energy landscapes with direct measurements of particle-surface absolute separation. Figure 4b shows MSDs computed separately for x and y excursions with the same theoretical fits shown in Figure 2b and with fit parameters reported in Table 2. Fitted values of 〈D|〉 are in good agreement with those predicted from particle-surface interactions using eq 21, with both values being less than half of the isolated particle diffusion coefficient (〈D|〉/D0 ≈ 0.4). Similar to Figure 2b, values of 〈D|,x〉 and 〈D|,y〉 are not significantly different, although the average 〈D|〉 is smaller on the thicker Au
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Figure 5. (a) Transmitted-light CCD image of 2.22 µm silica colloids electrostatically levitated in aqueous 1 mM KNO3/HNO3 above 5 µm × 5 µm × 15 nm (l × w × h) Au square films separated by 6 µm regions with 5 nm Au films (all on glass substrates). Fifteen nanometer Au films appear darker than 5 nm Au films. (b) Diffusion of levitated silica colloids on a patterned surface for 24 min from initial positions in part a. Colored regions of interest represent the average value of the particle-surface interaction potential at h ) 61 nm (relative to zero potential energy at h ) ∞). The inset color scale shows the energy range from -2kT (red) to 4kT (violet).
films because particles sample positions closer to the surface more often, which produces more hydrodynamic hindrance. The theoretical fits in Figure 4b show no migration in the y direction and a finite migration rate in the x direction that is most likely due to minor misleveling (θ < 1°) of the batch cell. No evidence of lateral confinement within pattern features is observed or expected because the magnitude of MSDs (and the associated observation period) is not sufficient to probe the pattern feature length scales. Also similar to Figure 2b, the lateral particle diffusion rate is larger than the normal rate (〈D|〉/〈D⊥〉 ≈ 4), which ensures that particle height histograms are well averaged over lateral surface positions. Relative Colloidal Probe and Pattern Feature Sizes on a Gold-Gold Pattern. To provide an estimate of the lateral resolution of energy landscapes using diffusing colloidal probes, Figure 5 shows results for a patterned Au film with 5 µm pattern features that are much closer to the 2.22 µm SiO2 particle diameter than the 75 µm features in Figures 1-4. Figure 5a shows a CCD image of 2.22 µm SiO2 colloids levitated in 1 mM ionic strength aqueous media (0.9 mM KNO3, 0.1 mM HNO3, C ) 131 µs/cm, pH 6.0) over an electrically conductive surface with 5 µm × 15 nm Au films laterally separated by 6 µm × 5 nm Au films suitable for electrophoretic/electrostatic deposition. An overall 2.5× higher magnification was used in Figure 5 compared to that in Figures 1 and 3 to maximize the resolution of particle centers and scattering as well as smaller pattern features. Figure 5b shows trajectories of 24 levitated SiO2 colloids on the patterned surface from 40 000 images acquired over an ∼24 min period. The colors of the two regions in Figure 5b correspond to average particle-surface interaction potentials in Figure 6a at h ) 61 nm for each region. The treatment of raw data and the analysis of potential energy profiles in Figure 6a are similar to that used in Figures 2a and 4a with all theoretical fit parameters reported in Table 1. In addition to estimating the particle-surface absolute separation from curve fits to the potentials in Figure 6a, absolute separation was also measured directly using the electrophoretic deposition of particles on the Au films to determine each particle’s I0 as described for the experiments in Figures 3 and 4. The measured and fit potential profiles in Figure 6a display the expected variations in van der Waals attraction due to different
Au films thicknesses in different regions whereas the electrostatic and gravitational potentials are independent of the pattern region also as expected. The curve fits in Figure 6a do not fit the measured van der Waals and electrostatic potentials as well as the profiles in Figure 4a, although the fit to the gravitational potential is comparable. Because the electrostatic decay length (κ-1) and the van der Waals power law parameters (A, p) were fixed via independent measurements and predictions, the imperfect curve fits in Figure 6a still represent the best fit based on adjusting only the Au surface potential and absolute separation (B′ and hm in eq 7). Directly measured and theoretically fit values of hm disagreed with each other by up to ∼25% on the 6 µm × 5 nm Au film regions and by ∼40% on the 5 µm × 15 nm Au film regions. The imperfect curve fits and discrepancies in hm estimates in Figures 5 and 6 can be attributed to particles residing near pattern feature edges with a higher probability as particle and pattern dimensions become comparable. Differences in particle-surface interactions and particle evanescent wave scattering at pattern edges compared to homogeneous regions can both be expected to distort measured profiles and resulting curve fit parameters in Figure 6a. For particles at pattern feature edges, they should experience an average interaction with the two Au film thicknesses and produce an average evanescent wave scattering signal based on different I0 values. How distortion due to pattern edge effects propagates through measurements of ensemble and surface-averaged histograms and their interpretation is not obvious, and a complete analysis is beyond the scope of the present investigation. It is worth noting that averaging over particle polydispersity, multiple I0 values, and other particle/surface nonuniformities in ensemble TIRM experiments18,19,37,38 generally produces apparently weakened forces (F ) du/dh) and affects stronger forces to a greater extent. This is consistent with the strong electrostatic forces in Figure 6a suffering the greatest distortion, followed by the van der Waals forces, and finally the weakest gravitational forces being essentially unaffected. Weakening of strong forces near the potential energy minima in Figure 6a can also account for the 25-40% discrepancy between fit and measured values of hm. The directly measured hm values in Figure 6a are more reliable
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Langmuir, Vol. 22, No. 16, 2006 6835
Figure 6. (a) Average potential energy profiles for 2.22 µm silica colloids interacting with 5 (O) and 15 nm (0) Au films in aqueous 1 mM KNO3/HNO3 (from Figure 5). The main plot shows particle-surface potentials without gravitational potentials (relative to zero potential energy at h ) ∞), and the inset includes gravity (relative to zero potential energy at h ) hm). Solid lines (-) are theoretical curve fits to eq 7. (b) Lateral mean squared displacements in the x (O) and y (0) directions (from Figure 5). Reference lines are shown for bulk diffusion (‚‚‚) from eq 16 and lateral surface diffusion on 5 (- -) and 15 nm (-‚-) regions from eq 21. Solid lines (-) are theoretical curve fits to eqs 23 and 26.
and consistent with results in Figure 4a by indicating higher Au surface potentials at the higher pH in Figure 6a.34 The results in Figure 6a suggest that laterally resolving potential energy landscapes on patterned surfaces with diffusing colloidal probes depends on the relative particle and pattern sizes. As a result, future work will investigate the use of smaller colloids including metal nanoparticles that have large scattering intensities despite their small size due to high refractive indices. Experimental challenges can be expected as particle and pattern dimensions approach and decrease below the optical diffraction limit. The ultimate lateral resolution limit via geometric considerations (based on the Derjaguin approximation26) of the sphere-wall interaction region can be expected to scale as (aδ)0.5, where δ is the range of colloid-surface interactions being investigated. A systematic study of nano- to micrometer-sized particles having near hard wall to long-range electrostatic potentials should help elucidate the role of the sphere-wall interaction region as a limiting factor in determining lateral resolution. The optimal choice of relative particle and pattern feature dimensions might also depend on dynamic factors that determine experimental measurement times. In any case, future work is planned to investigate the importance of absolute particle and pattern feature dimensions when using diffusing colloidal probes to map surface patterns. Finally, Figure 6b shows MSDs computed for the measurement in Figures 5 and 6a with theoretical fit parameters reported in Table 2. In contrast to the MSDs in Figures 2b and 4b, the MSDs in Figure 6b show the effects of lateral particle confinement on surface pattern features. The partial confinement on the 5 µm × 15 nm Au features is expected because of the ∼1kT deeper van der Waals minimum compared to the intervening 5 nm Au regions and is observable because the measurement period allows for MSDs greater than pattern feature dimensions (shown by dashed lines at 〈r2〉/a2 ) 19 ) (5 µm/1.15 µm)2 and 〈r2〉/a2 ) 32 ) (6.5 µm/1.15 µm)2). Fits to the measured MSDs indicate expected lateral diffusion coefficients at short times, partial lateral confinement within the 5 µm features at intermediate times
(evident in the partial MSD plateau), and long-time diffusion coefficients somewhat smaller than the short-time value. Differences in the MSD curves suggest some lateral particle migration in the x direction similar to the measurements in Figures 1-4, although the rate of migration is not easily separated from the partial confinement in Figure 6. The short-time lateral particle diffusion rate is large relative to the normal diffusion rate (〈D|〉/ 〈D⊥〉 ≈ 4), similar to that in the experiments presented in Figures 1-4.
Summary and Conclusions Results in this article demonstrate the ability to map patterned potential energy landscapes by monitoring interactions between diffusing colloidal probes with pattern features. Because diffusing colloids are dilute both normal and parallel to the underlying substrate, particle-surface histograms measured on different regions can be inverted in a straightforward manner using Boltzmann’s equation to yield potential energy landscapes. The patterned Au films investigated in this work modulate the particle-surface potential energy profiles by producing varying electrostatic surface potentials and van der Waals interactions based on the Au film surface and dielectric properties. For continuous Au films that can operate as electrodes, both electrophoretic and electrostatic deposition of colloidal probes are viable methods of measuring absolute separation at the conclusion of experiments monitoring colloidal probe diffusion. With reliable measurements of potential energy profiles, theoretical fits yield absolute separations in good agreement with absolute separations measured directly via electrophoretic deposition. As the diffusing colloidal probe size approaches the pattern feature dimensions, distortion to measured potential energy profiles result in less accurate estimates of absolute separation from theoretical curve fits. An analysis of colloidal probe diffusion on surfaces demonstrates the consistent interpretation of both conservative forces due to particle-surface potentials and dissipative forces due to
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particle-surface hydrodynamic interactions. Measurements of lateral diffusion via analysis of MSDs display excellent agreement with rigorous theoretical predictions that include surface, body, and hydrodynamic forces in each experiment. Lateral migration in external fields and confinement within energetic pattern features were also observed in MSDs, but as expected, they do not interfere with the equilibrium sampling of the surface necessary for constructing potential energy landscapes. Our results suggest a number of future directions for the refinement and extension of the new paradigm for surface imaging presented in this work. Using smaller nanoparticle probes might provide one route to obtaining higher-resolution potential energy landscapes. Employing higher colloidal probe concentrations might provide more rapid measurements of energy landscapes, although statistical mechanical analyses will become more complicated as a result of many-body effects. Monitoring diffusing colloidal probes can also be used to sensitively probe energy landscapes arising from nonspecific interactions between colloidal probes with synthetic chemical templates or specific interactions with biomolecular arrays. Although physically patterned surfaces are not generally compatible with evanescent waves without index matching or fluorescence, video or confocal microscopy of particle excursions might provide an alternative means to interrogate physicochemical energy landscapes with diffusing colloidal probes. Ultimately, complex 3D synthetic, geological, or biological structures could be probed energetically with diffusing colloids and any multidimensional particle-tracking method. Acknowledgment. We acknowledge financial support provided by NSF Career and PECASE awards, the ACS Petroleum Research Fund, the Robert A. Welch Foundation, and DARPA.
Appendix For ease of use when fitting eq 7 to experimental data, the van der Waals potential is specified using the power law expression in eq 5. To obtain these curve fits to the rigorous Lifshitz theory,23-25 the Hamaker “function”, A132(l), is computed for two silica half-spaces (materials 1 and 2) separated by water (material 3) as ∞
3 A132(l) ) - kT ′ 2 n)0
∑ ∫r
Figure 7. Power law parameters in the van der Waals particlesurface potential (eq 5) fit to Lifshitz half-space predictions (eq 27) with the Derjaguin approximation geometric correction (eq 29) for silica colloids interacting with Au films of thickness δ on the silica half-space. Power laws were fit to theory with (0.01 error for a silica colloid with 2a ) 2200 nm interacting with Au-coated surfaces in 1 mM ionic strength media for separations from 30 to 500 nm.
and the remainder of the terms in eqs 27 and 28 are described in extensive detail elsewhere.23-25,36 Water and silica dielectric spectra used in this work are the same as those reported by Bevan and Prieve,36 and Au dielectric spectra are from Parsegian and Weiss40 (constructed from the optical data of Johnson and Christy29). The thin chromium film (∼2 nm) was considered to be part of the Au film in calculations using eq 28. A 1 mM ionic strength medium was specified to account for screening of the zero frequency contribution in eq 27. To compute the van der Waals interaction between a 2.2 µm silica sphere and a Aucoated silica half-space, the half-space predictions from eq 27 were used in the Derjaguin approximation as26
uvdw(h) ) ∞
n
-x
x{ln[1 - ∆13∆23e ] + -x
h 23e ]} dx (27) ln[1 - ∆ h 13∆ where a uniform Au film (material 4) of thickness δ is included in eq 27 by replacing ∆j3 with
∆j3 f ∆j43 )
∆43 + ∆j4 exp(-δs4/l) 1 + ∆43∆j4 exp(-δs4/l)
A(l)
∫h∞ l2
a 6
dl
(29)
Values of A and p in eq 5 were obtained from curve fits to results generated using eqs 27-29 for a range of Au film thicknesses between 5 and 25 nm. In all cases, curve fits were obtained with less than (0.01 relative error for silica colloid-Au film surface separations in the range of 30-500 nm. Figure 7 shows values of A and p as a function of Au film thickness. LA060501J
(28) (40) Parsegian, V. A.; Weiss, G. H. J. Colloid Interface Sci. 1981, 81, 285.
