Direct Measurement of Single and Ensemble Average Particle

Direct Measurements of kT-Scale Capsule–Substrate Interactions and Deposition ... Colloidal Energy and Diffusivity Landscapes in Macromolecular Solu...
1 downloads 0 Views 704KB Size
Download PDF
1244

Langmuir 2005, 21, 1244-1254

Direct Measurement of Single and Ensemble Average Particle-Surface Potential Energy Profiles Hung-Jen Wu and Michael A. Bevan* Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122 Received August 23, 2004. In Final Form: November 13, 2004 This work involves the development of a novel technique that integrates total internal reflection and video microscopy methods to simultaneously measure single particle and ensemble average particlesurface interactions. For the 2 µm silica colloids and glass coverslip used in this study, particle size polydispersity is found to be a dominant factor in determining the distribution of single particle profiles about ensemble average profiles. In conjunction with this observation, chemical and physical nonuniformity are not evident in any of our measurements even with sensitivity to interactions on the order of kT. One advantage of using ensemble averaging in conjunction with time averaging is the ability to dramatically decrease the time required to measure average particle-wall interactions which scales inversely with interfacial particle concentration. A number of experimental issues are addressed in the development of this technique including (1) combining single particle distribution functions, (2) statistical sampling of distribution functions using both time and ensemble averaging, and (3) correcting overlapping scattering signals between adjacent particles. The capabilities of the ensemble averaging technique are also demonstrated to provide unique measurements of particle-surface interactions in metastable systems by selecting only height excursions of levitated particles when calculating potentials. Ultimately, this new technique provides several important advantages over single particle measurements, which provides a foundation for measuring interactions in increasingly complex interfacial systems.

Introduction In this paper, we report results using a new technique that simultaneously measures single particle and manyparticle ensemble average potential energy profiles in interfacial colloidal systems. This method combines and builds on single particle total internal reflection microscopy (TIRM)1 and video microscopy (VM)2 methods to track three-dimensional colloidal trajectories near interfaces with nanometer resolution on particle-surface separation and half pixel resolution on particle excursions parallel to surfaces. Our results demonstrate several benefits of ensemble averaging including explicit measurement of chemical and physical nonuniformity, faster measurement rates, and an ability to measure interactions corresponding to metastable conditions. In the implementation of this new technique, a number of experimental issues are addressed including the importance of distinguishing evanescent wave scattering from adjacent particles. This new multiparticle technique preserves all the beneficial features of TIRM and VM but provides more comprehensive information to better understand interactions in interfacial colloidal systems. Single particle TIRM studies have previously demonstrated the sensitivity of this technique for quantifying particle-wall interactions on the order of kT without any external manipulation of the particle.1 While single particle measurements offer minimal complexity and maximum sensitivity, the statistical significance and generality of individual results can often be a concern. Reproducing single particle measurements a small number of times does not guarantee a representative result since selecting individual results can be subtly biased. * To whom correspondence should be addressed. E-mail: [email protected]. (1) Prieve, D. C. Adv. Colloid Interface Sci. 1999, 82, 93. (2) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298.

This concern generally applies to other methods including the surfaces forces apparatus3 and the atomic force microscope4 in which only one or two surfaces are measured in a single experiment. Because minimal statistical sampling is inherent in these techniques, it can be difficult to assess in any single measurement the role of nonidealities such as chemical and physical surface heterogeneities, size polydispersity, and generally nonuniformities. In contrast to two-body force measurement methods, scattering techniques can probe large particle ensembles to provide statistical parameters related to distributions in particle properties and interactions. One complicating factor associated with scattering methods is the uncertainty associated with measuring statistical properties without knowledge of the explicit distributions of properties. Although methods for interpreting scattering data are well established,5,6 another concern is the uncertainty associated with analyzing reciprocal space results, particularly when signal noise complicates matters. Because of limitations in both single particle and ensemble measurements, one approach to confidently and objectively characterizing colloidal systems is to conduct both types of experiments and compare results for consistency. Another approach to measuring colloidal interactions involves monitoring more than two particles using optical microscopy techniques. The most common of these approaches is the measurement of equilibrium distribution functions in pseudo two-dimensional colloidal systems.7-9 (3) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (4) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831. (5) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969. (6) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (7) Kepler, G. M.; Fraden, S. Phys. Rev. Lett. 1994, 73, 356. (8) Behrens, S. H.; Grier, D. G. Phys. Rev. E 2001, 64, R050401.

10.1021/la047892r CCC: $30.25 © 2005 American Chemical Society Published on Web 01/20/2005

Particle-Surface Potential Energy Profiles

Langmuir, Vol. 21, No. 4, 2005 1245

These experiments provide ensemble average potentials, but an often complicating factor associated with these approaches is the questionable inference of anomalous attraction between like charged particles. Although it is possible to obtain single particle information from these optical microscopy experiments, to date only ensemble average distribution functions have been reported. These experiments have the potential to complement direct twobody force measurement and scattering techniques in terms of single particle and ensemble average information but have yet to be experimentally and analytically developed into reliable methods for measuring colloidal and surface interactions. In the following, we describe measurements of both single particle-wall and ensemble average potentials interpreted from fully three-dimensional particle trajectory data obtained using real space optical microscopy techniques. By combining TIRM and VM methods, we describe how the normal Brownian excursions of many levitated silica particles can be measured as they freely diffuse laterally over glass coverslip surfaces. By analyzing particle fluctuations both independently and collectively, we first show how single particle profiles compare with ensemble average profiles. Because numerous particles are considered in the averaging process and various positions are sampled on the underlying substrate due to free lateral particle diffusion, these measurements contain information concerning both particle and substrate nonuniformities. After illustrating general features of the experiment and analysis, we focus on several more subtle issues related to (1) combining single particle distribution functions, (2) statistical sampling of distribution functions using both time and ensemble averaging, and (3) correcting overlapping scattering signals between adjacent particles. Finally, we present results demonstrating the unique capabilities of the ensemble averaging approach for measuring particle interactions in metastable systems. Ultimately, our new technique provides a number of advantages over separate single particle and ensemble techniques to provide a foundation for measuring distribution functions and interactions in increasingly complex interfacial systems. Theory Colloidal and Surface Forces. The total potential energy profile for a single colloidal particle levitated above a wall can be calculated by summing the surface and body forces acting on the particle. For experiments presented in this paper, important interactions include electrostatic, van der Waals, and gravitational forces. The net separation dependent potential energy profile is given by the sum of these interactions as

φ(h) ) φedl(h) + φgrav(h) + φvdw(h)

(1)

where φedl(h) is the interaction between overlapping electrostatic double layers on the particle and wall, φgrav(h) is the gravitational potential energy due to the buoyant weight of the particle, and φvdw(h) is the long range, continuum van der Waals attraction between the particle and wall mediated by the aqueous electrolyte medium. For Debye lengths smaller than particle-wall surface separations (κh > 1) and much smaller than the sphere radius (κa > 1), the interaction of electrostatic double layers on adjacent particle and wall surfaces, φedl(h), is accurately described using nonlinear superposition and (9) Han, Y.; Grier, D. G. Phys. Rev. Lett. 2003, 91, 038302.

Derjaguin approximations. For the specific case of 1:1 electrolyte, the interaction is given as1

φedl(h) ) B exp(-κh) B ) 64πa

( ) kBT e

κ)

( ) ( )

2

tanh

(

eΨp eΨw tanh 4kT 4kT

)

2CNAe2 kBT

(2) (3)

0.5

(4)

where a is particle radius,  is the dielectric permittivity of water, kB is Boltzmann’s constant, T is absolute temperature, e is the elemental charge, ψp and ψw are the Stern potentials of the particle and the wall, κ-1 is the Debye length, C is the bulk electrolyte concentration, and NA is Avogadro’s number. The gravitational potential energy is the buoyant particle weight, G, multiplied by its height, h, above the wall. The buoyant particle weight is the product of gravitational acceleration, g, and the buoyant particle mass, m, which depends on particle volume and the density difference between the particle and medium. The gravitational potential energy is given by

φgrav(h) ) Gh ) mgh ) (4/3)πa3(Fp - Ff)gh

(5)

where Fp and Ff are the particle and fluid densities, and g is the acceleration due to gravity. For particle-wall separations smaller than the particle radius but large enough that retardation is important, the van der Waals interaction is accurately described using the Lifshitz theory and Derjaguin approximations as10,11

φvdw(h) ) -

∫h∞

a 6

A132(l) dl l2

(6)

where A132(l) is the retarded and screened interaction between half spaces, and l is the half space surface separation. Calculations of Lifshitz half space interactions and values of water and silica dielectric spectra in this work are the same as those reported by Bevan and Prieve.10 Single Particle TIRM. In the TIRM experiment, the scattering intensity of a single levitated colloid in an evanescent wave (see Figure 1a) can be used to determine the instantaneous particle-wall separation, h, above a wall using the following expression:1,12

I(h) ) I0 exp(-βh)

(7)

where I is the scattered intensity, I0 is the intensity at particle-wall contact, h ) 0, and β-1 is the evanescent wave decay length given by

β)

4π (n sin θ1)2 - n22 λx 1

(8)

where n1 and n2 are the refractive indices of the incident and transmitted media, and θ1 is the incident angle. Using eq 7, measurements of scattering intensity from single levitated particles can be used to monitor their height fluctuations due to Brownian excursions normal to the (10) Bevan, M. A.; Prieve, D. C. Langmuir 1999, 15, 7925. (11) Pailthorpe, B. A.; Russel, W. B. J. Colloid Interface Sci. 1982, 89, 563. (12) Chew, H.; Wang, D. S.; Kerker, M. Appl. Opt. 1979, 18, 2679.

1246

Langmuir, Vol. 21, No. 4, 2005

Wu and Bevan

histogram, n(h), can be measured from time dependent height fluctuations as

h(t) f n(h)

(10)

With a large enough number of observations to ensure sufficient statistical sampling, n(h) can be considered a good approximation of the probability density of heights, p(h). The potential energy relative to a reference state can determined by measuring n(h), substituting for p(h) in eq 9, and then rearranging as

[ ]

φ(h) - φ(href) n(href) ) ln kBT n(h)

(11)

where href is often chosen as hm, which is the most probable height sampled in n(h). This relative separation scale is generally referenced to particle-wall contact at h ) 0, which is most commonly determined by measuring I0 in eq 7.1 Knowing the interaction relative to particle-wall contact is generally more important for understanding surface interactions, but the interaction relative to the most probable height is more easily measured. Ensemble TIRM. By assuming colloids and surfaces are chemically and physically uniform (monodisperse, homogeneous, etc.), the time dependent height fluctuations of many particles levitated above different positions on a surface (see Figure 1b) can be averaged together to produce an ensemble average histogram, 〈n(h)〉, as

h1(t), h2(t), ... hi(t) f n1(h), n2(h), ... ni(h) f 〈n(h)〉 (12) where hi(t) are the time dependent height fluctuations of each single particle, ni(t) are the time averaged histograms of each single particle, and 〈n(h)〉 is the time and ensemble averaged histogram of i particles monitored during the course of an experiment. By analyzing both single particle and average multiparticle histograms using eq 11, the ensemble TIRM method can simultaneously measure i single particles each interacting with specific surface locations and an ensemble particle-wall interaction averaged over all particles and surface positions. Experimental Section

Figure 1. (a) Schematic illustration of ensemble TIRM apparatus with HeNe laser, prism, batch cell, microscope, CCD camera, and data acquisition PC. The inset shows a schematic representation of levitated particle scattering evanescent wave with intensity, I(h), as a function of particle-wall surface separation, h. (b) CCD image from top view of levitated particles scattering evanescent wave (white spots) with transmitted light illuminating particles (dark rings).

wall. The probability of sampling each height above the surface is related to the potential energy of each height by Boltzmann’s equation,1

[

p(h) ) A exp -

]

φ(h) kBT

(9)

where p(h) is the probability density of heights sampled by a single particle, φ(h) is the particle-wall potential energy profile, and A is a normalization constant related to the total number of height observations. By measuring the number of times a particle samples each height during the course of an experiment, a particle height

Materials. Nominal 2.34 µm silica colloids were purchased from Bangs Laboratories (Fishers, IN) and used without further purification. The manufacturer reported particle density is FSiO2 ) 1.96 g/mL. In each experiment, particles were diluted in aqueous electrolyte solutions to obtain bulk particle concentrations that produced desired interfacial concentrations after sedimentation equilibrium was attained. Aqueous solution ionic strength and pH were controlled using analytical grade KOH (Fisher) and NaCl (Aldrich) without further purification. The silica particle zeta potential as a function of KOH and NaCl concentration was measured using dynamic light scattering and phase analysis light scattering measurements (ZetaPALS, Brookhaven Instrument Corp., Holtsville, NY). Microscope cover glasses from Corning (Corning, NY) were used as surfaces in all experiments reported in this work. Cover glass surfaces were initially washed for 30 min in Nochromix (Godax Laboratories, Takoma Park, MD). After washing with double deionized water (DDI) and drying with high purity nitrogen, a 10 mm ID × 12 mm OD Viton O-ring (McMaster Carr, Los Angeles, CA) was attached to the cover glass using polydimethyl siloxane (PDMS, Sylgard 184, Dow Corning) to produce a small batch sedimentation cell. Prior to use in each experiment, cells were washed for 30 min in 0.1 M KOH, rinsed with DDI water, and dried with nitrogen. Methods. Figure 1a shows a schematic representation of an optical microscope (Axioplan 2, Zeiss, Germany) and CCD camera setup for dynamically tracking and monitoring evanescent wave

Particle-Surface Potential Energy Profiles

Langmuir, Vol. 21, No. 4, 2005 1247

scattering from levitated particle ensembles as shown in Figure 1b. An O-ring/cover glass batch sedimentation cell is optically coupled to a 68° dovetail prism (Reynard Corp., CA) using index matching oil (n ) 1.515). The prism is mounted on a three point leveling stage. In each experiment, a 40× objective (NA ) 0.65) was used in conjunction with a 12 bit CCD camera (ORCA-ER, Hamamatsu, Japan) operated with 8× binning to produce a capture rate of 42 frames/s with 168 × 128 resolution (pixel ) 1214.6 nm, 204 × 155 µm2 image size). The evanescent wave was generated using a 15 mW, 632.8 nm helium-neon laser (Melles Griot, Carlsbad, CA) to produce an evanescent wave decay length of 113 nm (ng ) 1.515, nw ) 1.333 in eq 8). Because levitated particles freely diffuse above the cover glass surface, image analysis algorithms coded in FORTRAN were used to track the lateral motion and integrate the evanescent wave scattering intensity for each particle. Standard video microscopy algorithms were used to locate and track centers of the evanescent wave scattering signal on each particle.2 The total scattering intensity from each particle was obtained by integrating all pixels within a specified radius of the scattering signal center pixel. All image analysis was performed using PCs and multipage TIFF files containing up to 105 separate images in a single file.

Results and Discussion Single and Ensemble Particle-Surface Potential Energy Profiles. Figure 2a shows potential energy profiles for 11 different 2.34 µm silica colloids levitated above a cover glass in aqueous media containing 0.1 mM KOH (pH ) 10) and 0.9 mM NaCl. The 11 individual potential energy profiles calculated using eqs 10 and 11 represent the average interaction between each particle and the portion of the cover glass surface sampled by each particle due to lateral diffusion (see Figure 1b). In addition to individual profiles, an ensemble average potential energy profile is also reported in Figure 2a, which was obtained by combining the 11 single particle histograms using eqs 11 and 12. Both the individual and ensemble averaged height histograms were constructed from 30 000 CCD images with 24 ms spacing for a total acquisition time of 12 min. In cases where scattering signals between adjacent particles overlap, these points were rejected so that several single particle histograms contain less than 30 000 total height observations. A correction for this signal overlap problem is described later in the discussion. The solid line in Figure 2a is a theoretical fit to the ensemble average profile using eq 1, and dashed lines represent the ensemble average profile refit to single profiles corresponding to the largest and smallest particles. Although absolute particle-wall separation can be directly determined by “sticking” each particle to the wall and measuring I0 in eq 7, initially we report all profiles on a separation scale relative to the most probable height, h - hm. Potential energy is reported on a scale relative to the minimum for each profile, φmin, which inherently occurs at the most probable separation, hm, as described by Boltzmann’s equation (eq 9). Because all single particle potential energy profiles essentially superimpose on these relative scales, it appears that particle and wall properties are sufficiently uniform to consider the particles as an “ensemble”. We use this observation to justify, at least initially, constructing ensemble average histograms by aligning single particle histograms on the same relative h - hm scales which assumes all particles have identical values of hm. Although directly measuring absolute separation for each profile is ultimately a more direct test of whether each profile is essentially identical, particularly in terms of surface potentials (ψ in eq 3), we first investigate obtaining absolute separation using theoretical curve fits to the van der Waals interaction in Figure 2a.

Figure 2. (a) Potential energy profiles for 11 different 2.34 µm silica colloids levitated above a cover glass in aqueous media containing 0.1 mM KOH (pH ) 10) and 0.9 mM NaCl. Gray open circles indicate 11 single particle profiles, and black open squares indicate the ensemble average profile. The solid line indicates the curve fit to the ensemble profile, and dashed lines indicate curve fits to maximum and minimum particle sizes (see text). (b) Potential profiles subtracted from the ensemble average profile.

The solid line in Figure 2a is a theoretical fit to the measured ensemble average potential energy profile. The theoretical function displaying the explicit particle size dependence for each force is obtained from eqs 1-6 to give

φ(x + hm) - φ(hm) B′ ) a exp[-κ(x + hm)] + kBT kBT φ(hm) A G′ 3 a (x + hm) + a(x + hm)p (13) kBT kBT kBT

1248

Langmuir, Vol. 21, No. 4, 2005

Wu and Bevan

Table 1. Parameters Fit to Potential Energy Profiles in Figure 2a Using Equation 13a κ-1/nm B′/J nm-1 -Ψ/mV a/nm hmin/nm R2 a

avg

max

min

9.59 17.4 ( 0.6 143 ( 8 1070 ( 2 114 ( 1 0.991

9.59 17.4 143 1140 ( 6 113 ( 1 0.947

9.59 17.4 143 995 ( 6 115 ( 1 0.915

The error range is only reported for adjustable parameters.

Table 2. Order of Magnitude Estimates of Number of Images Required to Obtain Statistically Well Sampled Height Distribution Functionsa area fraction (φA)

particles/window

total images

total time

0.0002 0.002 0.02 0.2 0.907

1 10 100 1000 4500

30000 3000 300 30 3

∼10 min ∼1 min ∼10 s ∼1 s ∼0.1 s

a Estimates are based on 25 ms sampling of 2.34 micron silica particles experiencing independent random walks on cover glass surfaces within a 204 × 155 µm2 window.

where B′ ) B/a and G′ ) G/a from eqs 3 and 5, x ) h hm, and φ(hm)/kBT corresponds to the first three terms in eq 13 evaluated at hm rather than x + hm. The theoretical van der Waals potential from eq 6 is accurately represented to within (1% for h > 20 nm and φvdw > -4 kBT by a convenient power law expression with A ) -2.095 kBT/nm and p ) -2.154. When fitting eq 13 in Figure 2a, values of κ, G′, A, and p were fixed from conductivity measurements, manufacturer reported particle density, and theoretical van der Waals calculations. Values of B′ and a were allowed as adjustable parameters with initial guesses provided from independent zeta potential and dynamic light scattering measurements. The surface potentials of the particle and wall are considered to be identical for the purpose of the curve fit. The value of hm was determined to be 114 nm from fitting eq 13 to the measured ensemble average profile in Figure 2a with fit parameters and their errors reported in Table 1. Excellent agreement is observed between the curve fit and data in Figure 2a, which illustrates a consistent treatment of relevant interactions and particle size to provide an estimate of absolute particle-wall separation. Single particle profiles in Figure 2a display a significant spread above and below the ensemble average potential energy profile. To consider the role of silica particle size polydispersity in Figure 2a, dashed lines were calculated by refitting eq 13 to single profiles displaying maximum and minimum weights (slopes of gravitational potential energy). The refitting procedure involved fixing all parameters to the ensemble average fit parameters in Table 2 except for the particle radius and the value of hm. As reported in Table 1, the relative deviations in the maximum and minimum particle sizes determined by the dashed lines in Figure 2 are within (0.07 of the average radius, which by coincidence both have absolute deviations of (74 nm from the mean size. By fitting a Gaussian curve to the particle sizes determined from each of the 11 profiles, a relative standard deviation of 0.04 was obtained. The values of hm changed by only (1 nm. The particle size distribution obtained in the fitting procedure is less than the manufacturer reported standard deviation of 11.5%, which is reasonable based on the small number of particles in Figure 2a. To better illustrate the distribution of individual profiles about the ensemble average profile in Figure 2a, deviations

of single particle profiles from the ensemble average curve fit are magnified in Figure 2b by reporting their absolute difference. The difference between the dashed lines in Figure 2a and the ensemble average curve fit are also shown as dashed lines in Figure 2b. The data in Figure 2b display a symmetric distribution about zero indicating that particle polydispersity consistently explains deviations in single particle profiles from the ensemble average profile. Because the spread is symmetric about zero on both the right and left of h - hm ) 0 where electrostatic and gravitational interactions dominate respectively, polydispersity appears to fully account for the data spread independent of the dominant interaction at any given separation. It does not appear necessary to independently consider differences in the gravitational interaction due to density variations, and likewise, electrostatic interactions do not appear to be separately influenced by chemical or physical surface heterogeneities on either the particles or cover glass. Because particles are free to diffuse laterally over the cover glass surface so that ensemble measurements represent an average particle-wall interaction, surface heterogeneities would certainly be important in the resulting average profiles if they were present. We find no evidence of significant surface heterogeneity for the data in Figure 2 on the kT energy scale. Based on the symmetric spread of data about zero that is contained within the dashed lines in Figure 2b, the distribution of single particle profiles about the ensemble average in Figure 2a can be fully accounted for by particle size polydispersity. Although rigorous “ensemble” averaging would require all particles to be identical (and the wall to be uniform), the size polydispersity in Figure 2 is sufficiently small so that combining the 11 single particle height histograms produces an average profile consistent with the average properties of the silica colloids tested. The mean and deviation of the parameters obtained for the fit in Figure 2a are consistent with other ensemble characterization techniques such as light scattering and zeta potential measurements. The novelty of the approach employed here is that by measuring both single particle and ensemble average interactions, average interactions and properties can be directly compared with the explicit distribution of interactions and properties compiled from single particle measurements. Because 11 particles comprise a small statistical set, a significant degree of time averaging was required in addition to ensemble averaging to generate the average particle-wall potential energy profile reported in Figure 2. With the successful application of the ensemble averaging method in Figure 2, we examine in the following section how statistical parameters associated with time and ensemble averaging influence measurements. Single Particle and Ensemble Profile Measurement Statistics. To understand how experimental sampling parameters influence single particle and ensemble measurements via distribution statistics, Figure 3 reports potential profiles interpreted from data sets with fixed sampling rates but different total numbers of points. In particular, Figure 3a shows profiles for one of the particles in Figure 2a calculated from height fluctuations sampled in 1000, 5000, 10 000, and 30 000 consecutive frames with 24 ms spacing. For comparison, Figure 3b shows ensemble average profiles constructed from the same 11 particles in Figure 2a also with height fluctuations sampled in 1000, 5000, 10 000, and 30 000 consecutive frames at 24 ms spacing. It is immediately obvious in both cases in Figure 3 that constructing histograms from more consecutive frames produces potential profiles displaying less noise. It is also clear that constructing ensemble averaged

Particle-Surface Potential Energy Profiles

Figure 3. Varying sampling statistics used to construct (a) single particle potentials and (b) ensemble average profiles for 11 particles (same as Figure 2). For both figures, 30 000, 10 000, 5000, and 1000 images are indicated by circles, squares, triangles, and inverted triangles.

histograms from 11 particles produces less noise compared to the same number of consecutive frames measured for a single particle at the same rate. In addition to simply obtaining a greater number of measurements when constructing the ensemble average potential in Figure 3b, the statistical independence of measured heights is also ensured. Because height measurements of single particles are not necessarily independent unless they are separated by a characteristic time interval, simply measuring more points does not guarantee a greater statistical significance of a measured distribution. The characteristic time between independent heights for the 2.34 µm particle in Figures 2 and 3 can be considered to range from ∼100 ms for complete decay of an auto-

Langmuir, Vol. 21, No. 4, 2005 1249

correlation function13 up to ∼1 min for diffusive time scales (a2/D(h)) evaluated using the expression of Goldman et al.14 In any case, it is clear that greater time intervals between successive frames result in a better chance of producing statistically independent height measurements and well sampled histograms for single particles. By measuring different particles, the statistical independence, or lack of redundancy, of measured heights ensures good sampling of the ensemble histogram. This may be an important factor in addition to the greater number of points that produce the smoother ensemble profiles in Figure 3b compared to the single particle profiles in Figure 3a. Because height fluctuations of different particles are uncorrelated in terms of their equilibrium sampling of potential energy positions above a wall, ensemble averaging can be used to construct statistically significant particle-wall histograms much faster than single particle measurements. To avoid potential confusion related to recent studies of multiple particles near surfaces, effects related to “hydrodynamic coupling”15,16 of particles near interfaces are inherently unimportant for the equilibrium measurements1 we investigate in this paper. Other factors to consider are multiparticle packing8 and many-body interactions17 that could invalidate our analysis for colloids at high interfacial concentrations or having long range interactions. Because we investigate relatively dilute particle concentrations and colloidal forces operating over distances small compared to particle radii in this work, it is reasonable to treat particles independently when constructing and analyzing ensemble histograms using eqs 9-12. The validity of this assumption is demonstrated by the agreement of each of our results in this paper with theoretical predictions. By combining time averaged histograms of many single particles, the ensemble averaging technique promises to provide significantly faster measurement times. Table 2 shows the estimated number of images required to obtain statistically significant height histograms using a combination of both time and ensemble averaging for the 2.34 µm silica particles in this study. If a 24 ms sampling is used, ∼12 min is required to measure 30 000 points for a single particle consistent with Figure 3a. Each time the number of particles in the ensemble is increased by an order of magnitude, the required time averaging period decreases in direct proportion by an order of magnitude. In the extreme case where an effective close-packed crystalline layer (4500 2.34 µm particles in the 204 × 155 µm2 window) with a lateral particle spacing on the order of κ-1 (10 nm for 1 mM 1:1 electrolyte) is levitated above a surface, a well sampled histogram and potential energy profile could be obtained in ∼0.1 s. This extreme case may invalidate the previously stated assumption concerning the unimportance of multiparticle packing effects. However, because a close-packed monolayer does not experience multiparticle packing effects normal to the wall, it is not obvious whether this experiment would or would not yield the correct ensemble particle-wall interaction using eqs 9-12. In any case, averaging over increasingly concentrated particle ensembles proportionally decreases the time required to obtain a statistically well sampled particle-wall histogram up to the point that multiparticle (13) Bevan, M. A.; Prieve, D. C. J. Chem. Phys. 2000, 113, 1228. (14) Goldman, A. J.; Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22, 637. (15) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (16) Squires, T. M.; Brenner, M. P. Phys. Rev. Lett. 2000, 85, 4976. (17) Klein, R.; vonGrunberg, H. H.; Bechinger, C.; Brunner, M.; Lobaskin, V. J. Phys.: Condens. Matter 2002, 14, 7631.

1250

Langmuir, Vol. 21, No. 4, 2005

Wu and Bevan

effects become important. In a later section, we describe another limitation that becomes significant before the onset of multiparticle packing effects with increasing concentration. Because no additional effort is required to measure particle ensembles compared to single particles in terms of experimental setup or data analysis, ensemble averaging produces significantly faster measurements of particle-surface interactions. Ultimately, ensemble averaging offsets the necessity for time averaging to produce rapid measurements of monodisperse colloids interacting with homogeneous surfaces. Constructing Ensemble Histograms from Single Particle Histograms. Although a diminishing number of single particle heights needs to be measured as the total particle number is increased with the ensemble averaging approach, in practice, an additional factor complicates the averaging process for large particle numbers and small acquisition times. Because latex colloids that appear uniform in every respect often scatter evanescent waves with different absolute intensities,18 it is not possible to directly construct an ensemble “intensity” histogram that can later be converted to a “height” histogram for use in eq 11. Although the origin of different single particle scattering intensities is unknown, it is not correlated with variations in density or dielectric properties since measured gravitational and van der Waals forces are found to be independent of any given particle’s absolute scattering intensity in this and previous work.10,18 As a result, our current strategy for generating ensemble average histograms is to convert single particle “intensity” histograms into single particle “height” histograms on the relative separation scale, h - hm, by specifying an arbitrary stuck intensity in eq 7. By using this relative separation scale, single particle height histograms can be reliably combined into an ensemble height histogram. Although this method works well as shown in Figures 2 and 3, poor statistical sampling of single particle excursions produces noisy histograms (see Figure 3a), which can confuse proper identification of hm and produce misalignment of single particle histograms on the h - hm scale. If unnoticed, this can inadvertently lead to broadened or multiple peaked ensemble histograms that erroneously indicate weaker forces or multiple attractive minima. Because it is beneficial to align histograms of numerous poorly sampled individual histograms to collectively provide sufficient statistics for an ensemble average histogram, here we describe a technique to improve estimates of the most probable height, hm, for sparsely sampled histograms. Of the methods we examined, the approach that we found to be the most general and robust in practice includes the following steps: (1) Initially, we identify hm,i as the most sampled bin in a noisy single particle height histogram. (2) We then calculate a weighted shift factor, ∆hm, using the following expression:

∆hm )

∑(hi - hm)Ni ∑Ni

(14)

where Ni is the number of samples for each bin location hi in a single particle histogram, and the limits of summation are set to include only bins sampled above a threshold frequency. (3) We adjust the initial (or previous) (18) Bevan, M. Ph.D. Dissertation, Carnegie Mellon University, Pittsburgh, PA, 1999.

value of hm,i by ∆hm to obtain a new value for the most probable height, hm,f, using

hm,f ) ∆hm + hm,i

(15)

Finally, (4) we repeat steps 2 and 3 using eqs 14 and 15 until hm,f converges to a constant value (∆hm becomes less than half of the histogram bin width). After this procedure is performed on each single particle height histogram to provide reliable estimates of hm, an ensemble average height histogram can be reliably constructed from all single particle histograms on the h - hm scale. This method was used to produce the ensemble average potential profiles in Figure 3b from sparsely sampled single particle histograms in Figure 3a. Excellent agreement is obtained for each of the ensemble profiles obtained in Figure 3b, even for cases with very little time averaging, which indicates the success of the weighting function approach in eqs 14 and 15. Because eq 14 uses a simple average weighting scheme, it is rigorously valid for improving estimates of hm only in symmetric histograms. Potential energy profiles generally have either expo-linear or expo-inverse power law dependences based on dominant contributions from either electrostatic and gravitational forces or electrostatic and van der Waals forces in the vicinity of their minima (see eq 1). In both of these cases, these potential energy profiles are asymmetric and also correspond to asymmetric histograms. To more rigorously consider the shapes of sparsely sampled histograms when improving estimates of hm, asymmetric weighting functions determined from preliminary fits to noisy histograms were explored as an alternative weighting scheme to the one used in eq 14. In simulated cases, estimates of hm were improved by ∼1 nm, which is comparable to experimental error and therefore relatively insignificant. The approximate symmetric weighting in eq 14 is reasonably successful at improving estimates of hm in noisy histograms because it only considers bins in the vicinity of the initial hm value. Because histograms corresponding to expo-linear and expo-inverse power law potential profiles are reasonably symmetric in the vicinity of their maximum frequency value, the simple weighting in eq 14 appears to be sufficient. Because eqs 14 and 15 are effective, simple, and do not require any assumptions concerning histogram shape, they are used to refine estimates of hm in single particle histograms to reliably produce ensemble average histograms for all experiments in this paper. An obvious next step is to increase the particle interfacial concentration, or the number of particles/image, to explore the limits of the ensemble averaging approach. As suggested by the scaling reported in Table 2, increasing the interfacial particle concentration can be expected to produce a proportional decrease in measurement time of ensemble profiles. Although we have resolved issues related to sparsely sampled single particle histograms, a separate issue becomes important at elevated particle concentrations: overlap of particle scattering signals from adjacent particles (see Figure 1b). Although this occurs even in the dilute systems reported in Figures 2 and 3, the relative frequency of overlaps in the ensemble average histogram is small. In many cases, effects of overlaps are not apparent, or specific observations known to suffer from signal overlap can be rejected based on particles approaching within some center-to-center cutoff distance. Because the relative frequency of signal overlap between adjacent particles increases with increasing interfacial particle concentration, simply rejecting these points significantly reduces the measurement rate and eventually

Particle-Surface Potential Energy Profiles

becomes impractical at very high concentrations. In the following section, we describe how the signal overlap problem can be reliably corrected to increase the number of particles/window to approach the scaling suggested in Table 2. Correcting Signal Overlap between Adjacent Particles. In this section, we describe a method to reliably extract the independent scattering intensities of two adjacent particles when their signals overlap. The goal is to further extend the ensemble method to measure higher interfacial particle concentrations, which will minimize the necessity of time averaging in favor of ensemble averaging to achieve the maximum measurement rate. A complication with increased interfacial particle concentrations and lateral collision frequency is the increased likelihood of signal overlap between adjacent particles. Because our image analysis algorithm integrates scattering signal in radial sweeps about a center position, signal in the overlap region between two particles is counted twice to make both particles appear brighter, which distorts both single particle and ensemble average histograms. Figure 4a shows representative potential energy profiles from two adjacent particles with significant scattering signal overlap. The square symbols in Figure 4a correspond to the profiles of two particles whose signals overlap for ∼15% of the 12 min measurement period. Because the radial integration distance is 4 pixels in Figure 4a, overlap is defined to occur when scattering signal centers approach within 8 pixels. Figure 4b shows overlapping scattering patterns for the two particles measured in Figure 4a. Potential energy profiles reported as circles in Figure 4a were constructed by simply rejecting intensity measurements when signal overlap was detected. Because the profiles in which overlapping signal was rejected are expected to be accurate, the theoretical in Figure 4a was obtained by fitting eq 13 to these data. The curve fits the data well and produces a consistent estimate of particle size and other parameters in good agreement with independent measurements and theoretical predictions. The two profiles where effects of signal overlap are ignored (squares) clearly deviate from the profiles where overlapping signal is rejected (circles) which are in excellent agreement with theory. Differences between the profiles are observed only for separations where electrostatic interactions are important, which is apparent from the departure from exponential decay behavior. Signal overlap increases at small particle-wall separations because particles scatter more intensely over a proportionally larger radial distance. As a result, scattering signals of particles at a given center-to-center distance may not overlap when they are far from the wall but can overlap considerably at decreased particle-wall separation. Spreading of the signal at higher intensities is illustrated by the single particle scattering profiles with different absolute intensities shown in Figure 5a. The net effect is that the profiles in Figure 4a deviate at a critical point on the electrostatic portion of the profile when the scattering distribution on each particle becomes spatially large enough in the lateral dimension to fall within the integration region of the adjacent particle. A number of methods were explored for correcting the signal overlap issue in ensemble experiments at higher interfacial concentrations. The simplest approach investigated was simply reducing the integration region on each particle. This was unsuccessful because the range could not be sufficiently reduced to avoid signal overlap in all cases, particularly at very small particle-wall separations. Although simply ignoring signal overlaps, which was

Langmuir, Vol. 21, No. 4, 2005 1251

Figure 4. (a) Potential energy profiles of two adjacent particles with overlapping scattering signals. Squares were obtained from analysis with no correction, circles were obtained by rejecting only overlap points, and triangles were obtained by correcting signal interference using eq 16. Dashed lines indicate the curve fit to triangles. (b) Intensity distribution overlap of two adjacent particles. The insets show the cross-sectional view and processed image.

adopted in Figures 2-4, is expected to produce accurate profiles, this is undesirable from the standpoint that rejecting data points reduces statistical sampling, which diminishes the advantage of the ensemble average method for rapidly measuring profiles at higher particle concentrations. In extreme cases such as a levitated crystalline monolayer of charge stabilized colloids, it is possible that all signal would be rejected, rendering the ensemble method useless. Rejecting measurements of particle intensities, and corresponding heights, is also unaccept-

1252

Langmuir, Vol. 21, No. 4, 2005

Wu and Bevan

the individual particle scattering intensity distributions, which are modeled as surfaces of rotation about central intensity maxima, numerous particles with a broad range of intensities were angularly averaged and fit using

I(r,Im) ) Ib(Im) +

Figure 5. (a) 12-bit intensity distributions of single particles relative to center positions in terms of CCD camera pixels. Open squares, triangles, inverted triangles, diamonds, and circles represent different local maximum intensities of Im ) 3799, 2999, 1799, 1399, and 999. Solid lines indicate curve fits using eq 16. (b) Linear curve fits to parameters determined in panel a.

able for planned experiments requiring uninterrupted monitoring of three-dimensional particle trajectories. To continuously monitor each particle’s scattering intensity for rapid ensemble averaging, we developed a method to successfully decouple overlapping signals. The method involves calibrating the scattering intensity surface for individual particles which can then be used to separate overlapping signals into discrete contributions. Figure 4b shows an intensity surface taken from an image of the overlapping particles used in Figure 4a, and the inset of Figure 4b shows a cross sectional view of this surface normal to the XY plane and in the plane connecting the particle centers. The cross sectional view in Figure 4b demonstrates the net signal is essentially a superposition of the isolated particle intensity profiles. To calibrate

Im - Ib(Im) 1 + [r/σ(Im)]

(16)

where r is the radial distance from the center of the scattering signal, Im is the maximum intensity at the particle center, Ib is the background intensity that depends linearly on Im, and σ is the distribution width that also depends linearly on Im. Figure 5a reports average data measured with different maximum intensities and curve fits using eq 16. Figure 5b reports the linear dependence of Ib and σ on Im. The functional form of eq 16 was chosen on purely empirical grounds based on its ability to accurately capture the scattering intensity distributions of particles as a function of their maximum intensity, Im. The relatively low resolution CCD images obtained in these experiments are thought to result in the dependence of σ and Ib on Im, as well as the smooth profile in eq 16 without interference fringes commonly observed for particles scattering an evanescent wave. It is possible using eq 16 to specify the intensity distribution for each particle based on its central intensity maximum, which can be determined even for overlap conditions. By correcting interfering signals between adjacent particles using eq 16, potential energy profiles reported as triangles in Figure 4a were constructed without rejecting any intensity measurements. Potential profiles in Figure 4a corrected by either ignoring overlap signal (circles) or the interference correction in eq 16 (triangles) produce nearly identical profiles. The corrected profiles both display excellent agreement with the theoretical fit using eq 13. Using this method, we are able to obtain ensemble average profiles similar to the results in Figure 2 for