Measurement and Interpretation of Particle−Particle and Particle−Wall

Jun 4, 2005 - Measurement and Interpretation of Particle−Particle and Particle−Wall Interactions in Levitated Colloidal Ensembles. Hung-Jen Wu,Tod...
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Langmuir 2005, 21, 9879-9888

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Measurement and Interpretation of Particle-Particle and Particle-Wall Interactions in Levitated Colloidal Ensembles† Hung-Jen Wu, Todd O. Pangburn, Richard E. Beckham, and Michael A. Bevan* Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122 Received March 11, 2005. In Final Form: April 29, 2005 This paper reports measurements of particle-wall and particle-particle interactions in levitated colloidal ensembles using integrated total internal reflection microscopy (TIRM) and video microscopy (VM) techniques. In levitated colloidal ensembles with area fractions of φA ) 0.03-0.25, ensemble TIRM measured height distribution functions are used to interpret particle-wall interactions, and VM measured pair distribution functions are used to interpret particle-particle interactions using inverse Ornstein-Zernike (OZ) and three-dimensional inverse Monte Carlo (MC) analyses. An inconsistent finding is the observation of an anomalous long-range particle-particle attraction and recovery of the expected Derjaguin-LandauVerwey-Overbeek (DLVO) particle-wall interactions for all concentrations examined. Because particlewall and particle-particle potentials are expected to be consistent in several respects, the analytical and experimental methods employed in this investigation are examined for possible sources of error. Comparison of inverse OZ and three-dimensional inverse MC analyses are used to address uncertainties related to dimensionality, effects of particle concentration, and assumptions of the OZ theory and closure relations. The possible influence of charge heterogeneity and particle size polydispersity on measured distribution functions is discussed with regard to inconsistent particle-wall and particle-particle potentials. Ultimately, achieving a consistent understanding of particle-wall and particle-particle interactions in interfacial and confined colloidal systems is essential to numerous complex fluid and advanced material technologies.

Introduction In this paper, we report measurements of particlewall and particle-particle interactions in interfacial colloidal systems. By combining total internal reflection microscopy (TIRM)1 and video microscopy (VM)2 methods (Figure 1), we track three-dimensional trajectories of colloids near surfaces with spatial resolution on the order of nanometers.3 From these measurements, we construct equilibrium particle-wall and particle-particle distribution functions that we interpret using Boltzmann’s equation,1 a two-dimensional inverse Ornstein-Zernike (OZ) analysis,4 and a three-dimensional inverse Monte Carlo (MC) method.5 Our measurements of particle-wall interactions are consistent with the DLVO theory while our measurements of particle-particle interactions indicate an anomalous long-range attraction. By employing threedimensional inverse MC methods to extract unique particle pair interactions from measured distribution functions, we are able to remove uncertainties related to concentration, dimensionality, and assumptions of the inverse OZ analysis. Inconsistent measurements of expected DLVO particle-wall interactions and anomalous long-range particle-particle interactions are discussed in terms of the influences of charge heterogeneity and particle size polydispersity on measured distribution functions and interpreted interactions. * To whom correspondence should be addressed. E-mail: [email protected]. † Part of the Bob Rowell Festschrift special issue. (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. (3) Wu, H. J.; Bevan, M. A. Langmuir 2005, 21, 1244. (4) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (5) Soper, A. K. Chem. Phys. 1996, 202, 295.

Figure 1. (a) Schematic illustration of ensemble TIRM-VM apparatus with HeNe laser, prism, flow cell, microscope, CCD camera, and data acquisition PC. Inset shows schematic representation of single levitated particle scattering evanescent wave with intensity, I(h), as a function of particle-wall surface separation, h. (b) Transmitted light CCD images of levitated colloidal ensembles measured in Figures 2 and 4 with interfacial concentrations expressed as projected area fractions φA ) 0.053, 0.11, 0.21, and 0.25 (left to right, top to bottom).

Understanding how particle-particle and particle-wall interactions control equilibrium self-assembly of colloidal structures on substrates is crucial to numerous complex fluid and advanced material technologies. While the present investigation focuses on the “inverse” problem of interpreting particle-particle and particle-wall interactions from measured distribution functions, the motivation is related to the “forward” problem of predicting how to tune particle and surface interactions to assemble desired equilibrium particle configurations. Understanding the links between particle interactions and configurations at interfaces will ultimately provide the ability to intelligently manipulate interactions that control self-assembly of complex three-dimensional structures on physically and chemically patterned substrates. Before theories and

10.1021/la050671g CCC: $30.25 © 2005 American Chemical Society Published on Web 06/04/2005

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experiments can be adequately developed to understand more complex problems, it is essential to build a foundation by understanding the links between particle interactions and configurations in dilute interfacial colloidal ensembles, which has yet to be established. Literature measurements of electrostatic and van der Waals interactions between micron sized colloids and surfaces have consistently produced agreement with the DLVO theory for interactions on the order of kT. For example, equilibrium measurements of single charged particles electrostatically levitated above charged planar surfaces using TIRM have agreed with DLVO predictions of both electrostatic and van der Waals interactions.1,6-8 In addition, DLVO theory has been observed to successfully model particle-wall interactions in dilute ensembles levitated above a single surface3 or confined between parallel surfaces separated by several particle diameters.9 Nonequilibrium measurements of single particles sedimenting below a surface have also been shown to agree with DLVO theory,10 although dynamic simulations have shown that such measurements of multiple particles can produce apparently anomalous interactions if care is not taken with interpretation.11 In general, particle-wall interactions inferred from equilibrium and nonequilibrium measurements involving one surface or parallel confining walls have shown quantitative agreement with the DLVO theory. Despite the commonly observed agreement between particle-wall measurements and the DLVO theory, agreement with various particle-particle measurements has been mixed. A number of studies have used liquid structure theory to obtain particle pair interactions from pair distribution functions in interfacial colloidal ensembles. While a single investigation of this type near one surface has produced agreement with the DLVO theory,12 nearly all other similar studies involving single or confining parallel surfaces have reported an anomalous attraction between like charged particles near like charged surfaces.13-17 A different experimental approach has measured particle interactions from nonequilibrium trajectories of particle pairs periodically trapped and released from optical tweezers. While this type of measurement for particle pairs far from any boundaries produced agreement with DLVO theory,18 similar measurements between parallel confining walls suggest anomalous attractive and repulsive interactions.19 Anomalous attraction has also been inferred in nonequilibrium optical tweezer measurements near one surface,20 although this was due to misinterpretation of three-dimensional dissipative forces.11,21 In addition to indicating possible quantitative shortcomings of the DLVO theory, these various observations of attraction when repulsion is (6) Alexander, B. M.; Prieve, D. C. Langmuir 1987, 3, 788. (7) Prieve, D. C.; Luo, F.; Lanni, F. Faraday Discuss. 1987, 83, 297. (8) Bevan, M. A.; Prieve, D. C. Langmuir 1999, 15, 7925. (9) Kepler, G. M.; Fraden, S. Langmuir 1994, 10, 2501. (10) Behrens, S. H.; Plewa, J.; Grier, D. G. Eur. Phys. J. E 2003, 10, 115. (11) Anekal, S.; Bevan, M. A. J. Chem. Phys. 2005, 122, 034903. (12) Behrens, S. H.; Grier, D. G. Phys. Rev. E 2001, 64, R050401. (13) Kepler, G. M.; Fraden, S. Phys. Rev. Lett. 1994, 73, 356. (14) Carbajal-Tinoco, M. D.; Castro-Roman, F.; Arauza-Lara, J. L. Phys. Rev. E 1996, 53, 7345. (15) Han, Y.; Grier, D. G. Phys. Rev. Lett. 2003, 91, 038302. (16) Klein, R.; vonGrunberg, H. H.; Bechinger, C.; Brunner, M.; Lobaskin, V. J. Phys.: Condens. Matter 2002, 14, 7631-7648. (17) Royall, C. P.; Leunissen, M. E.; Blaaderen, A. v. J. Phys.: Condens. Matter 2003, 15, S3581-S3596. (18) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1994, 73, 352. (19) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1996, 77, 1897. (20) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (21) Squires, T. M.; Brenner, M. P. Phys. Rev. Lett. 2000, 85, 4976.

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predicted suggest serious qualitative limitations for describing the interactions of charged colloids near charged walls. Given numerous measurements of particle-wall interactions agreeing with the DLVO theory and an equally large number of particle-particle interactions that do not agree with DLVO theory, this investigation measures and interprets both interactions with the goal of examining expected consistencies between the two measurements. To the best of our knowledge, no previous study has attempted to resolve in a consistent manner whether these interactions agree with each other and the DLVO theory. In the following, we conduct a series of integrated TIRM and VM experiments to characterize particle-wall and particle-particle interactions as a function of interfacial particle concentration using inverse OZ and threedimensional inverse MC simulations. We discuss the influence of charge heterogeneity and particle size polydispersity in measured distribution functions that may account for the inconsistent observations of expected DLVO particle-wall interactions and anomalous longrange particle-particle attractive interactions. By working toward a quantitative understanding of the connection between particle pair potentials and equilibrium configurations in interfacial colloidal systems, our ultimate goal is to provide a foundation for measuring and manipulating colloidal interactions and structures in increasingly complex interfacial systems. Theory Colloidal and Surface Forces. Net particle-wall and particle-particle potentials can be calculated by adding potentials due to conservative surface and body forces acting on particles. For experiments presented in this paper, important interactions include electrostatic and gravitational forces. The net separation dependent potential is given by the sum of these interactions as

u(r) ) uedl(r) + ugrav(r)

(1)

where uedl(r) is the interaction between overlapping electrostatic double layers, and ugrav(r) is the gravitational potential energy due to the buoyant weight of a particle. For Debye lengths smaller than the surface separation (κ(r - 2a) > 1) and sphere radius (κa > 1), uedl(r) can be calculated using the nonlinear superposition and Derjaguin approximations. For the specific case of 1:1 electrolyte, the particle-particle interaction is given by22

uedl(r) ) B exp[-κ(r - 2a)] B ) 32πa

κ)

( ) kT e

(

2

( )

tanh2

)

2CNAe2 kT

eψp 4kT

(2) (3)

0.5

(4)

where a is particle radius,  is the dielectric permittivity of water, k is Boltzmann’s constant, T is absolute temperature, e is the elemental charge, ψp is the Stern potential of the particle, κ-1 is the Debye length, C is the bulk electrolyte concentration, and NA is Avogadro’s number. The particle-wall interaction is twice as strong (22) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989.

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as the particle-particle interaction at the same separation as given by

B ) 64πa

( ) kT e

2

( ) ( )

tanh

eψp eψw tanh 4kT 4kT

(5)

where ψw is the Stern potential of the wall. The gravitational potential energy is the buoyant particle weight, G, multiplied by its height, h, above the wall. The gravitational potential energy is given by

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

(6)

where Fp and Ff are the particle and fluid densities and g is the acceleration due to gravity. Single and Ensemble Particle TIRM. In the TIRM experiment, the scattering intensity of levitated colloids in an evanescent wave (see Figure 1a) can be used to determine their instantaneous particle-wall separation, h, using23,24

I(h) ) I0 exp(-Rh)

(7)

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

R)

4π (n sin θ1)2 - n22 λx 1

(9)

where p(h) is the probability distribution of heights, A is a normalization constant related to the total number of height observations, β ) 1/kT, and u(h) is the total particle-wall potential given by eq 1. By measuring the number of times a particle samples each height during the course of an experiment, a particle height histogram, n(h), can be measured and substituted into eq 9 to give

β[u(h) - u(href)] ) ln

[ ] n(href) n(h)

∫

h(r) ) c(r) + F c(r′) h(|r - r′|) dr′

(10)

where href is a reference height often chosen as hm which is the most probable height in n(h). The time dependent height fluctuations of many particles levitated above a surface (see Figure 1b) can be averaged together to produce an ensemble average histogram, 〈n(h)〉, as3

h1(t), h2(t), ..., hi(t) f n1(h), n2(h), ..., ni(h) f 〈n(h)〉 (11) where hi(t) are the time dependent height fluctuations of each single particle, ni(h) are the time averaged histograms of each single particle, and 〈n(h)〉 is the time and ensemble averaged histogram of i particles. Both single particle and ensemble average histograms can be analyzed using eq 10. (23) Chew, H.; Wang, D. S.; Kerker, M. Appl. Opt. 1979, 18, 2679. (24) Prieve, D. C.; Walz, J. Y. Appl. Opt. 1993, 32, 1629.

(12)

where h(r) ) g(r) - 1 is the total correlation function, c(r) is the direct correlation function, and F is the average particle number density. Different closure relations relate u(r) and c(r) such as the hypernetted chain (HNC) and Percus-Yevick (PY) approximations given by4

c(r) ) -βu(r) + h(r) - ln g(r) (HNC)

(13)

c(r) ) [1 - exp(βu(r))][1 + h(r)] (PY)

(14)

By solving eq 12 using these closure relations, g(r) can be obtained for a given u(r) in the forward OZ (fOZ) problem or u(r) can be obtained for a given g(r) in the inverse OZ (iOZ) problem. Rapid solutions of this convolution type integral equation are obtained by applying a FourierBessel transformation to eq 12 to give25-27

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

p(h) ) A exp[-βu(h)]

Ornstein-Zernike and Monte Carlo Forward and Inverse Analyses. The time averaged, two-dimensional equilibrium configuration of particles levitated above a surface can be characterized in terms of a pair distribution function (PDF), g(r). The pair potential, u(r), and g(r) can be related using the Ornstein-Zernike (OZ) equation given by4

C(k) )

H(k) 1 + FH(k)

(15)

Monte Carlo (MC) simulations provide a numerical method to relate u(r) and g(r). In forward MC (fMC) simulations, g(r) is obtained for a given u(r) using standard algorithms.28 In inverse MC (iMC) simulations, u(r) is obtained from iterative fMC simulations that converge to a given g(r). In the iMC algorithm, an initial guess of u(r) is used in an fMC simulation to produce a simulated g(r). The potential of mean force, w(r), is computed from its definition as

w(r) ) -β-1 ln[g(r)]

(16)

for both the simulated PDF, gs(r), and the measured PDF, gm(r), to revise the estimate of ur(r) from its initial or previous value, ui(r), using5

ur(r) ) ui(r) + [wm(r) - ws(r)]

(17)

which can be rewritten using eq 16 as

ur(r) ) ui(r) + β-1 ln[gs(r)/gm(r)]

(18)

This procedure is repeated until gs(r) is obtained within some tolerance of gm(r) using5

χ)

[∑(

)]

gm(r) - gs(r)

r

(r)

2 1/2

(19)

where the error, (r), can be specified as a function of separation. With convergence of gs(r) to gm(r), a unique solution is obtained for a given g(r) provided that more (25) Lado, F. J. Chem. Phys. 1967, 47, 4828. (26) Lado, F. J. Chem. Phys. 1968, 49, 3092. (27) Lado, F. J. Comput. Phys. 1971, 8, 417. (28) Allen, M. P.; Tildesley, D. J. Computer simulation of liquids; Oxford Science: New York, 1987.

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than one u(r) does not satisfy eq 19 within the specified tolerance.29 Experimental Section 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. A custom water purification system using reverse osmosis, deionization, filters, and ultraviolet light treatments was used to produce deionized water with resistivities of 17.2-17.4 Ω/cm. The silica particle size and zeta potential in deionized water were measured using dynamic and phase analysis light scattering (ZetaPALS, Brookhaven Instrument Corporation, Holtsville, NY). Microscope cover glasses from Corning (Corning, NY) were used as surfaces in all experiments. Cover glass surfaces were initially washed for 30 min in Nochromix (Godax Laboratories, Takoma Park, MD) followed by rinsing with double deionized water (DDI) and drying with high purity nitrogen. In TIRM experiments, a polydimethyl siloxane (PDMS, DC 184, Dow Corning) spacer was clamped between a microscope slide and cover glass to produce a flow cell (Figure 1a) with inlets and outlets provided by two syringe needles. Optical Microscopy Methods. In ensemble TIRM experiments, an Axioplan 2 optical microscope (Zeiss, Germany) and CCD camera were used to dynamically track and monitor evanescent wave scattering from ensembles of levitated particles as described in a previous manuscript.3 A flow cell (Figure 1a) was optically coupled to a 68° dovetail prism (Reynard Corp., CA) using index matching oil (n ) 1.515). The flow cell and prism were then mounted on a three point leveling microscope stage. In particle-wall measurements, a 40× objective (NA ) 0.65) and 1.6× magnifying lens were used in conjunction with a 12 bit CCD camera (ORCA-ER, Hamamatsu, Japan) operated with 4× binning to produce a capture rate of 28 frames/s with 336 × 256 resolution (379.6 nm pixels, 128 × 97 µm2 image). The evanescent wave was generated using a 15 mW, 633 nm helium-neon laser (Melles Griot, Carlsbad, CA) with an incident angle of 62° to produce a decay length of 426 nm (ng ) 1.515, nw ) 1.333 in eq 8). In measurements of two-dimensional pair distribution functions, a 63× objective (NA ) 0.75) and 1.6× magnifying lens were used in conjunction with the 12 bit CCD camera using a image capture interval of 2 s/frame and 1344 × 1024 resolution (60.2 nm pixels, 81 × 62 µm2 image). Transmitted light was used to illuminate the levitated particles as shown in Figure 1b. In confocal scanning laser microscope (CSLM) measurements, a LMS 5 PASCAL scanner was used with an Axiovert 200M MAT inverted microscope stand (Zeiss, Germany) and 63× immersion oil objective (NA ) 1.4). Sedimentation cells were placed on a three point leveling stage and imaged from below in reflection mode using a 15 mW, 633 nm HeNe laser. Because levitated particles freely diffuse above cover glass surfaces, image analysis algorithms coded in FORTRAN were used to track lateral particle motion and integrate evanescent wave scattering from each particle. Standard video microscopy algorithms were used to locate and track centers of each particle.2 The evanescent wave scattering intensity from each particle was obtained by integrating all pixels within a specified radius of each scattering signal’s center pixel. All image analysis was performed using PCs and multi-page TIFF files containing either 20 000, 336 × 256 images in ensemble TIRM particle-wall measurements or 5000, 1344 × 1024 images in VM particleparticle measurements. Monte Carlo Simulation Methods . Allen and Tildesley’s NVT fMC code was used as a starting point for the fMC and iMC simulation codes used in this work.28 The starting configuration for all simulations consisted of 256-10 000 particles on a hexagonal lattice. Both fMC and iMC simulations were performed with an initial equilibration period consisting of 1-10 million (29) Gray, C. G.; Gubbins, K. E. Theory of molecular fluids; Oxford University Press: Oxford, 1984; Vol. I.

Wu et al. particle steps. After equilibration, 20-50 million particle steps were used to generate statistically significant distribution functions. To ensure efficient and robust convergence of iMC simulations, initial u(r) estimates were obtained from an iOZ analysis. For combined high density and low electrolyte conditions when the iOZ analysis did not converge, a DLVO repulsive potential was used for the initial u(r) estimate. All MC simulations employed a simulation box with charged confining walls on the top and bottom (normal to gravity) and periodic boundary conditions in the two dimensions parallel to the confining walls. The step size normal to the wall was dynamically changed to give an equal number of unaccepted MC steps both normal and parallel to the wall, which ensured proper sampling of all thermodynamically accessible configurations.

Results and Discussion TIRM-VM Measured Ensemble Particle-Wall Distribution Functions and Interactions. To consistently measure and interpret conservative particle-wall and particle-particle pair interactions in interfacial colloidal ensembles, three-dimensional particle coordinates are measured and analyzed as equilibrium distribution functions. Evanescent wave scattering is used to measure particle-wall separation with ∼1 nm resolution1 and video microscopy is used to measure lateral particle center-tocenter separations with ∼10 nm resolution (see Figure 1).2 With the ability to construct both particle-wall and particle-particle distribution functions, equilibrium statistical mechanical interpretations are used to obtain ensemble average particle-wall and particle-particle pair interactions. To extract potentials from measured distribution functions, Boltzmann’s equation (eq 9) is used to interpret particle-wall interactions since particles are dilute normal to the wall, while two-dimensional inverse Ornstein-Zernike (iOZ) (eqs 12-15) and three-dimensional inverse Monte Carlo (iMC) (eqs 16-19) analyses are used to interpret particle-particle interactions due to nondilute conditions parallel to the wall. Figure 2 shows ensemble average particle-wall distribution functions and potentials for 2.34 µm nominal sized silica colloids dispersed and levitated above a glass microscope slide in deionized water with no added electrolyte. A range of interfacial particle concentrations, expressed as area fractions from φA ) 0.03-0.25, are investigated from near infinite dilution up to relatively dense liquid configurations (Figure 1b). Figure 2a shows particle-wall distribution functions generated using eq 11 from time and ensemble averaging the height excursions for 20 000 frames collected over 12 min at each concentration. Figure 2b shows ensemble average particlewall potentials interpreted from the distribution functions in Figure 2a using Boltzmann’s equation. All distribution functions and potentials in Figure 2 are reported relative to the most probable separation, hm, since absolute particle-wall separation was not measured. On the relative separation scale, h-hm, the Debye length is determined from the exponential decay of the electrostatic repulsion (eq 2) in each profile, and particle size is determined from the buoyant weight given by the slope of the linear gravitational interaction (eq 6) in each profile (Table 1). The Debye lengths are found to vary between κ-1 ≈ 100-300 nm from the ensemble particle-wall interactions. Although these κ-1 values could not be confirmed via independent measurements, they correspond to ionic strengths between 10-5-10-6 M consistent with carbon dioxide saturated water with trace ionic contaminants. The average particle sizes determined from measured buoyant weights and the manufacturer reported density of 1.96 g/cm3 vary between a ) 1960-2283 nm. This size range agrees well with the size distribution

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Figure 2. Ensemble TIRM measurements for interfacial particle concentrations of φA ) 0.036 (O), 0.053 (0), 0.11 (4), 0.21 (3), and 0.25 (]). (a) Ensemble average particle-wall distribution functions. (b) Ensemble average particle-wall potentials determined from the distribution functions in (a) using Boltzmann’s equation (eq 9). Solid lines are the theoretical fits to each measured potential using eq 1. Table 1. Parameters Fit to Potentials in Figures 2b and 5 Using Equations 1-6 particle-wall

particle-particle

φA

2a, nm

κ-1, nm

κ-1, nm (iOZ)

κ-1, nm (iMC)

ψp, mV (iOZ)

ψp, mV (iMC)

0.036 0.053 0.109 0.21 0.25

1964 2189 2146 2283 2283

195 132 217 278 278

204 110 172 213 263

113 99 91 235 179

-5 -13 -4 -3 -3

-20 -18 -18 -3 -4

determined from a dried particle pair distribution function measured using confocal microscopy (Figure 3a) and dynamic light scattering measurements (Figure 3b), which give most probable sizes of 2a ) 2.22 µm and 2a ) 2.21 µm. Because a relatively small number of particles are sampled in ensemble TIRM experiments, which depends on interfacial concentration, it is reasonable to obtain ensemble average sizes occurring anywhere within the statistically significant distributions in Figure 3. It is not possible to determine the surface potential on the particles or wall in these experiments because absolute separation was not measured. There is no significant dependence of the measured particle-wall potentials on the interfacial particle concentration within the expected variations in particle size and ionic strength. These results in Figure 2 are not unexpected given numerous single particle TIRM measurements1 and recent ensemble TIRM measurements3 that have consistently agreed with the DLVO theory. Agreement with DLVO electrostatic repulsion has also previously been observed for particles between parallel confining walls.9 Confirmation of the expected DLVO repulsion in Figure 2 as a function of lateral particle concentration is an important control experiment before proceeding to measurements of lateral particle-particle interactions, which are known to display anomalous interactions for particle ensembles between parallel confining walls.13-15 The results in Figure 2 provide baseline measurements of particle-wall potentials in levitated particle ensembles, which indicate expected

Debye lengths and particle sizes with no obvious anomalous interactions. TIRM-VM Measured Ensemble Particle-Particle Distribution Functions and Interactions. In addition to the ensemble particle-wall measurements in Figure 2, the integrated TIRM and VM techniques were used to measure particle-particle pair distribution functions (PDF) for interpretation using iOZ and iMC analyses. PDFs were constructed in Figure 4 for a range of interfacial particle concentrations from φA ) 0.03-0.25 by time and ensemble averaging relative lateral particle excursions in the plane parallel to the underlying glass surface for 5000 frames collected over 166 min at each concentration. The particle-wall and particleparticle PDFs in Figures 2 and 4 are measured consecutively since particle-wall separations are measured from evanescent wave scattering and particle-particle separations are measured from transmitted light images. In addition to the measured PDFs in Figure 4 shown as points, fitted PDFs from iOZ and iMC analyses are shown as solid and short dashed lines. Long dashed lines are from forward MC (fMC) simulations using the DLVO fits to the iOZ pair interactions in Figure 5a. Figure 5a,b reports particle pair interactions interpreted from the measured PDFs in Figure 4 using the iOZ and iMC analyses. All pair potentials in Figure 5 are reported on the absolute particle center-to-center separation scale, r. Fitting DLVO electrostatic repulsion (eq 2) to the exponentially decaying portion of each potential in Figure 4 indicates Debye lengths between κ-1 ≈ 100-300 nm and average particle surface potentials from ψp ) -(320) mV as reported in Table 1. The insets of Figure 5a,b show semilog plots of these same fits in the purely exponential repulsive region for both the iOZ and iMC analyses. The fits in the inset of Figure 5a are significantly better than the fits in the inset of Figure 5b, which results from the lack of “smoothing” with the iMC method compared to the iOZ method, but Debye lengths are similar in both cases. The particle pair potentials in Figure 5 do not depend strongly on density or the inversion method but all display anomalous long-range attractions well beyond the ex-

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Figure 3. (a) Confocal scanning laser microscope measured pair distribution function for nominal 2.34 µm silica colloids dried on a microscope slide with first peak at r ) 2a ) 2.22 µm. Inset shows first peak magnified for images with pixel sizes of 190, 95, 48, 24, and 12 nm. (b) Dynamic light scattering measured log-normal distribution of nominal 2.34 µm silica colloids with most probable size of 2a ) 2.21 µm.

Figure 4. VM measured pair distribution functions for interfacial particle concentrations of φA ) 0.036 (O), 0.053 (0), 0.11 (4), 0.21 (3), and 0.25 (]). Solid lines are theoretical fits to each measured distribution function using the inverse OZ analysis (eqs 12-15) and short dashed lines are fits using the inverse MC analysis (eqs 16-19). Long dashed lines are from forward MC simulations using the iOZ pair interactions obtained in Figure 5a. Data are intentionally offset vertically for clarity.

pected range of van der Waals interactions.8 These potentials share some similarities with previous literature measurements of two-dimensional averaged PDFs for particle ensembles confined between two walls at lower area fractions (φA< 0.08).13,15 The PDFs obtained using the iOZ and iMC analyses fit the measured data well and are similar at each particle concentration. The particle pair potentials in Figure 5 display minor quantitative differences in their fitted Debye lengths and attractive

well depths but remain qualitatively similar. The Debye lengths are expected to be similar in Figures 2 and 5 since particle-wall and particle-particle potentials are sensitive gauges of ionic strength. This appears to be the case within experimental error since Debye lengths interpreted from iOZ and iMC analyses in Figure 5 display similar variations to their differences with ensemble TIRM measured Debye lengths in Figure 2 (Table 1). Given the agreement between ensemble TIRM measurements of particle-wall interactions in Figure 2 and DLVO predictions, it is not obvious why DLVO particle pair interactions are not obtained in Figure 5. Although anomalous particle-particle attraction has been reported numerous times, particle-wall interactions have not been characterized in conjunction with these previous measurements. Kepler and Fraden observed DLVO particlewall interactions9 and anomalous particle-particle interactions13 for colloidal ensembles between parallel confining walls, but these results were not directly compared or considered together. Because the particlewall and particle-particle interactions measured in Figures 2 and 5 are not consistent with each other, it appears a unique explanation is required to reconcile these results. It is problematic that the assumptions of the DLVO theory for weakly overlapping electrostatic double layers appear to be satisfied in the particle-wall case but not the particle-particle case. Before further discussing the apparent discrepancy between the measured particle-wall and particle-particle potentials, there are a number of important differences in the iOZ and iMC analyses worth examining in closer detail. In particular, limitations of the iOZ theory related to dimensionality, particle concentration, liquid structure theory, closure relations, and so forth are addressed by the three-dimensional iMC method implemented in this investigation. It is important to establish confidence in the inverse analyses before further discussing experimental factors that might produce the inconsistent measurements in Figures 2 and 5. Comparison of Inverse Ornstein-Zernike and Inverse Monte Carlo Techniques. To consider the validity of the inversion methods used in Figures 4 and 5, forward MC simulations were performed for a range of

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Figure 5. Average particle-particle potentials obtained from inverse analyses of pair distribution functions in Figure 4. (a) Inverse OZ analysis for the following concentrations and κa values: (O) φA ) 0.036, κa ) 5.5; (0) φA ) 0.053, κa ) 10.1; (4) φA ) 0.11, κa ) 6.5; (3) φA ) 0.21, κa )5.2; and (]) φA ) 0.25, κa ) 4.2. (b) Inverse MC analysis for the following concentrations and κa values: (O) φA ) 0.036, κa ) 9.8; (0) φA ) 0.053, κa ) 11.2; (4) φA ) 0.11, κa ) 12.2; (3) φA ) 0.21, κa ) 4.7; and (]) φA ) 0.25, κa ) 6.2. Solid lines represent theoretical DLVO electrostatic repulsive potentials given by eqs 2-4. Data are intentionally offset by 1 kT for clarity. Insets show semilog plots of DLVO electrostatic repulsion fit to each data set in parts a and b. Inset data are intentionally offset horizontally for clarity.

parameters (φA ) 0.08, 0.16, 0.24, κ-1/nm ) 100, 200, 300) encompassing the experimental conditions in Figures 2, 4, and 5 and interpreted using the iOZ and iMC analyses. Figure 6 shows particle-particle PDFs and pair interactions interpreted using the iOZ analysis with PY and HNC closure relations and the three-dimensional iMC analysis. Particle-wall interactions were characterized in an identical fashion to the ensemble TIRM experiments in Figure 2 and produced expected DLVO potentials. The simulated particle-particle PDFs at higher κa’s in Figure 6 appear most similar to those measured in Figure 4 with experimental values of κa ≈ 4-12. The κa dependence observed in Figure 6 can be understood as a change in effective particle size and effective area fraction, which results in more order at lower ionic strengths and higher core particle area fractions. Although more structure might be expected for the PDFs at φA ) 0.21 and φA ) 0.25 in Figure 4 with Debye lengths of κ-1 ≈ 200-300 nm and κa ≈ 4-6, the polydispersity observed in Figure 3 could easily inhibit the formation of more order. For example, polydispersity can suppress crystallization in hard sphere systems.30 The particle pair potentials obtained from the threedimensional iMC analysis are identical with the input potential in all cases for the fMC simulations in Figure 6. This is a somewhat trivial case but demonstrates the reliable convergence of the iMC algorithm used to analyze the PDFs in Figure 4. It has been previously shown that iMC analyses yield unique pair potentials for given PDFs,29 and the convergence of all iMC analyses in Figure 6 to the correct input potentials suggests uniqueness is not a concern in our implementation of the iMC algorithm. Particle pair interactions obtained from the iOZ analysis in Figure 6 begin to deviate from input potentials and the (30) Phan, S.-E.; Russel, W. B.; Zhu, J.; Chaikin, P. M. J. Chem. Phys. 1998, 108, 9789.

iMC results by ∼0.1 kT for φA g 0.16 and for κa < 5.9 with the PY closure and for every case except φA ) 0.08 for κa )11.7 when using the HNC closure. As expected, the PY closure performs better than the HNC closure for the relatively short-range repulsive potentials investigated here that are not too dissimilar to effective hard spheres. Both closure relations produce correct potentials at infinite dilution in the iOZ analysis, but both fail for conditions involving high φA and low κa, or high effective area fractions. The mean spherical approximation (MSA) closure was also investigated but produced poor agreement with input potentials in all cases examined. Several immediate conclusions can be drawn with regard to the results in Figure 6. The fully threedimensional fMC simulations including both particlewall and particle-particle interactions remove uncertainties related to treating these experiments as pseudo two-dimensional systems. Although the particle-wall potentials in Figure 2 are vertically narrow and suggest negligible error in PDFs measured parallel to the wall, the results in Figure 6 clearly remove any remaining concerns with dimensionality. By using the two-dimensional iOZ and three-dimensional iMC analyses with three-dimensional fMC data, the results in Figure 6 demonstrate it is reasonable to assume that particles adopt two-dimensional equilibrium configurations as the result of lateral particle pair interactions only (at least for physically and chemically uniform wall surfaces). In addition, the numerical iMC simulations address uncertainties related to the iOZ analysis and use of approximate closure relations. Although the iOZ and iMC results in Figure 5 are similar to one another, the results in Figure 6 demonstrate the nature of the failure of the iOZ analysis at high effective area fractions. In none of the cases in Figure 6 does the iOZ analysis produce an anomalous attraction, but instead indicates only positive

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Figure 6. (left) Pair distribution functions for three-dimensional forward MC simulations of levitated colloidal ensembles. Simulations were performed for 2.34 µm silica colloids with area fractions of φA ) 0.08, 0.16, and 0.24 and Debye lengths of κ-1 ) 100, 200, and 300 nm to give values of κa ) 11.7, 5.9, and 3.9. (right) Pair interactions obtained from pair distribution functions on left using three-dimensional inverse MC analysis (solid lines) and inverse OZ analyses with PY (long dashed lines) and HNC (short dashed lines) closure relations. Data are intentionally offset vertically for clarity.

deviations from iMC results. This suggests the anomalous attraction in Figure 5 is not an artifact of the iOZ analysis. In fact, the positive deviation of the iOZ results in Figure 6 compared to the iMC results might explain the greater anomalous attraction observed in the iMC results in Figure 5b compared to the iOZ result in Figure 5a as a sort of compensating effect. It is also apparent from the threedimensional fMC simulated PDFs in Figure 4 using the “best” DLVO fitted electrostatic repulsion, that the measured PDFs deviate significantly from what is needed to obtain DLVO interactions in either inverse analysis. By comparing the iOZ and iMC analyses to each other in a series of three-dimensional fMC simulations, it is clear that only repulsive potentials should be re-

covered in Figures 2 and 5 if only DLVO electrostatic repulsive potentials were operating between the particles and wall. Possible Influences on Particle-Wall and Particle-Particle Measurements. Despite removing concerns related to concentration, dimensionality and inversion techniques via the iMC analysis, it remains that the DLVO ensemble average particle-wall potentials in Figure 2 are inconsistent with the anomalous attractive ensemble average particle-particle potentials in Figure 5. Given the same Debye lengths for the potentials in Figures 2 and 5 (Table 1), a unique resolution is required to produce the expected DLVO repulsion in one case and anomalous attraction in the other.

Interactions in Levitated Colloidal Ensembles

Novel charge distributions due to confinement have been suggested as one potential source of anomalous attraction in confined systems,31 although such a mechanism would probably be expected to affect particle-wall and particleparticle interactions in a similar manner. Electrostatic double layers experiencing anything other than pairwise overlaps, which may be the case for small κa systems (κa ≈ 1), have also been suggested as a source of anomalous attraction.16,17 This mechanism is also not expected to be significant for the range of κa ) 4-12 in Figures 4 and 5, which is consistent with the limited structure observed in the PDFs. It appears that additional factors should be considered to consistently interpret the DLVO particlewall interactions in Figure 2 and anomalous particleparticle interactions in Figure 5. The influence of particle and surface nonuniformities has yet to be discussed in relation to inferences of anomalous colloidal interactions. The probable presence of various forms of physical and chemical nonuniformities in the colloids and surfaces used in this study requires their consideration. From another perspective, effects of nonuniformities in Figures 2 and 5 should be understood before consideration of more complex mechanisms. The iOZ and iMC analyses assume that monodisperse, uniform particles adopt two-dimensional equilibrium configurations above a uniform wall as the result of only lateral particle pair interactions. Nonuniformities that might produce anomalous interactions because they are not considered in the inverse analyses include (1) charge heterogeneity on the wall, (2) charge heterogeneity on particles with the same average surface potential, (3) different average surface potentials on particles that are otherwise uniform, and (4) particle polydispersity. In the following, these nonuniformities are discussed in terms of their possible influences on the measurements in Figures 2 and 5. Charge heterogeneity on the wall surfaces alone could produce a potential energy landscape on which levitated colloids laterally sample some positions more often than others due to particle-wall interactions. For example, charge heterogeneity on microscope slides with characteristic length scales comparable to or greater than the 2.22 µm particle silica particle diameter could subtly perturb measured PDFs to yield deceptively long range anomalous interactions. In an extreme example of this effect, deliberately patterning charge heterogeneity on a wall surface could produce lateral particle configurations that produce seemingly anomalous interactions at any particle separation. Charge heterogeneity on the wall surface could produce unusual particle-particle PDFs and potentials but would not necessarily alter particle-wall interactions as they are characterized in Figure 2. Because particle-wall distribution functions and average potentials are reported on a separation scale relative to the energy minimum, the expo-linear dependence of the resulting particle-wall potential is preserved to produce expected Debye lengths and buoyant particle weight. We have not characterized charge heterogeneity on our microscope slide surfaces so it is only possible to speculate on its role with regard to the results in Figures 4 and 5. Surface charge heterogeneity on particles that otherwise have the same average surface potential could produce orientation and separation dependent PDFs that should not be interpreted using iOZ and iMC analyses considering only separation dependent interactions. In a somewhat (31) Grier, D. G.; Han, Y. J. Phys.: Condens. Matter 2004, 16, S4145S4157.

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related case, different average surface potentials on otherwise uniform particles with some characteristic distribution could also distort measured PDFs and resulting interactions. To suggest the possible effects of such charge nonuniformities, either enhanced or diminished repulsion relative to uniformly charged particles could be obtained for (1) two particles interacting with either low or high charge density patches oriented towards one another or (2) interacting pairs of particle having different surface potentials either smaller or greater than the average. Results in Figure 5 show diminished repulsion as evident from the low surface potentials of ψs ) -(320) mV that are significantly less than measured zeta potentials of -60 mV. Effects of either particle surface charge heterogeneity or charge distributions on measured PDFs are not obvious. For particles with surface charge heterogeneity, separation and orientation dependent degrees of freedom are probably not independent; if high charge patches approach each other, low charge patches can rotate into position. For particle ensembles with a surface charge distribution, the distribution shape could have an important role in “weighting” the distortion to measured PDFs. In practice, particle surface charge heterogeneity or charge distributions over particle ensembles probably occur together. As demonstrated in Figure 4 by the simulated PDFs using DLVO potentials and the measured PDFs, anything producing enhanced sampling of the first peak could be manifested as an apparent attraction. As a result, a complete statistical mechanical treatment is required to fully understand effects of charge heterogeneity and distributions on measured PDFs and interpreted pair interactions. Particle surface charge heterogeneity or distributions were not characterized in this work, but literature studies have reported evidence of these nonuniformities in interacting pairs of silica and latex colloids.32-34 Similar to charge heterogeneity on the wall surface, the ensemble particle-wall potential measured in Figure 2 is not necessarily affected by particle charge heterogeneities because of the particular way it was constructed on a relative separation scale. Particle size polydispersity could also distort measured PDFs. The polydisperity of the nominal 2.34 µm silica colloids used in this investigation was characterized in the dried particle PDF and dynamic light scattering data in Figure 3. The most probable diameter in both measurements is ∼2.2 µm with an approximately log-normal distribution between ∼1.8-2.8 µm, which is consistent with TIRM measured sizes (Table 1). This particle size distribution could distort the PDFs in Figure 4 as it has in the dried PDF in Figure 3a by producing a ∼1 µm wide peak rather than a narrow peak. The significant polydispersity in Figure 3 is also consistent with the limited structure observed in the measured PDFs in Figure 4 compared to the fMC simulated PDFs in Figure 6, which is an expected effect of polydispersity.30 One problem with polydispersity is that apparent particle overlaps can occur for PDF separations comparable to the width of a particle size distribution. For example, two particles significantly smaller than the average particle size can interact at a distance but have center-to-center separations less than twice the average particle radius; this would deceptively appear as an (32) Velegol, D.; Thwar, P. K. Langmuir 2001, 17, 7687. (33) Feick, J. D.; Velegol, D. Langmuir 2002, 18, 3454. (34) Feick, J. D.; Chukwumah, N.; Noel, A. E.; Velegol, D. Langmuir 2004, 20, 3090.

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overlap of two averaged size particles. Particle overlaps appear to be observed in confined experiments for repulsive potentials truncated at 2-4 kT and apparent particle contact (r ) 2aavg).15,31 The lack of any van der Waals attraction in these measurements suggests that particles may not actually be in contact. Although particle overlaps are not observed in Figures 4-6, the low interpreted surface potentials from ψs ) -(3-20) mV relative to a measured zeta potential of -60 mV suggests a shorter range repulsion possibly related to polydispersity. The degree to which the first PDF peak is distorted by polydispersity will also be influenced by the particle size dependence of DLVO electrostatic repulsion. DLVO theory suggests that electrostatic repulsion may be affected by polydispersity via the hard core particle size (r - 2a in exponential of eq 2) and size scaling via the Derjaguin approximation (a in prefactor of eq 2, shown in B in eq 3). The particle size dependence of DLVO electrostatic repulsion in combination with multiparticle packing effects makes the effects of polydispersity on PDFs nonintuitive. As with other nonuniformities, polydispersity does not alter the ensemble particle-wall potentials in Figure 2. Polydispersity in ensemble TIRM measurements appears as many single particle-wall profiles distributed about the ensemble average particle-wall profile, which is explained in further detail in a previous paper focused on this issue.3 Due to the probable presence of finite charge heterogeneity and polydispersity in colloidal and interfacial systems, a concern is whether such nonuniformities can significantly perturb measured PDFs to produce anomalous interactions. Given the common observation of anomalous attractive wells of ∼0.2 kT in colloidal ensembles near one or two walls, it is possible that relatively small changes to PDFs due to the independent or collective influences of such nonuniformities could produce seemingly anomalous interactions. It may be important to consider additional factors including (1) separations at which different length scale surface heterogeneities are significant, (2) suppression of polydisperse “hard” core effects by “soft” electrostatic repulsion, and (3) diminished

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repulsion due to charge regulation35 in confined geometries. Given the complex nature of considering all these nonuniformities together, future characterization and modeling work are planned to identify any significant contributions that produce anomalous interactions. Conclusions In seeking the simplest explanation of measured anomalous particle-particle attractions and DLVO particle-wall interactions in levitated colloidal ensembles, an attempt has been made to systematically address several analytical and experimental concerns rather than considering novel conservative forces. In terms of analytical problems, inverse Ornstein-Zernike and threedimensional Monte Carlo analyses were implemented to remove uncertainties related to dimensionality, particle surface concentration, and assumptions of the liquid structure theory and closure relations when interpreting interactions from measured distribution functions. Because charge heterogeneity and particle size polydispersity are common in colloidal systems, the role of these nonuniformities was considered in relation to distorting particle pair distribution functions and interpreted pair potentials. A future investigation is planned to understand the role of nonuniformities in possibly reconciling inconsistent particle-wall and particle-particle interactions. Successfully understanding the role of nonuniformities could allow them to be specified as nonadjustable parameters in inverse Monte Carlo analyses to better understand colloidal interactions in interfacial and confined geometries. Acknowledgment. We are grateful to Prof. David M. Ford for valuable insights and helpful conversations related to implementing the inverse analyses. We acknowledge financial support for this work by the National Science Foundation (CTS-0346473), the donors of the ACS Petroleum Research Fund (41289-G5), and the Robert A. Welch Foundation (A-1567). LA050671G (35) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: London, 2001.