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Langmuir 2008, 24, 714-721
Electrostatically Confined Nanoparticle Interactions and Dynamics Shannon L. Eichmann, Samartha G. Anekal, and Michael A. Bevan* Department of Chemical Engineering, Texas A&M UniVersity, College Station, Texas 77843-3122 ReceiVed August 20, 2007. In Final Form: October 13, 2007 We report integrated evanescent wave and video microscopy measurements of three-dimensional trajectories of 50, 100, and 250 nm gold nanoparticles electrostatically confined between parallel planar glass surfaces separated by 350 and 600 nm silica colloid spacers. Equilibrium analyses of single and ensemble particle height distributions normal to the confining walls produce net electrostatic potentials in excellent agreement with theoretical predictions. Dynamic analyses indicate lateral particle diffusion coefficients ∼30-50% smaller than expected from predictions including the effects of the equilibrium particle distribution within the gap and multibody hydrodynamic interactions with the confining walls. Consistent analyses of equilibrium and dynamic information in each measurement do not indicate any roles for particle heating or hydrodynamic slip at the particle or wall surfaces, which would both increase diffusivities. Instead, lower than expected diffusivities are speculated to arise from electroviscous effects enhanced by the relative extent (κa ≈ 1-3) and overlap (κh ≈ 2-4) of electrostatic double layers on the particle and wall surfaces. These results demonstrate direct, quantitative measurements and a consistent interpretation of metal nanoparticle electrostatic interactions and dynamics in a confined geometry, which provides a basis for future similar measurements involving other colloidal forces and specific biomolecular interactions.
Introduction Interactions, dynamics, and transport of colloids and nanoparticles in confined geometries are important to numerous traditional and emerging technologies as well as biological and environmental systems that include, for example, chromatography, membrane separations,1 porous media,2 liquid film stabilization,3 Coulter counters,4,5 and microfluidic devices.6 In each of these applications, conservative colloidal and surfaces forces control equilibrium sampling of particles within confined environments7,8 while dynamic properties require additional consideration of dissipative hydrodynamic interactions.9 In cases where dissociated bulk (i.e., aqueous media) and interfacial charge (i.e., charged particles, boundaries) are present, coupled electrostatic and hydrodynamic interactions can lead to more complex electrokinetic phenomena.10-13 To design, control, and optimize technologies involving confined nanoparticles, it is necessary to understand the combined statistical mechanical and continuum fluid mechanical contributions that determine the equilibrium and dynamic behavior of such systems. Experimental observation is essential to understanding the interactions and dynamics of nanoparticles in confined geometries, but very little work has been reported in relation to direct * To whom correspondence should be addressed. E-mail: mabevan@ tamu.edu. (1) Deen, W. M. AIChE J. 1987, 33, 1409-1425. (2) Johnson, P. R.; Sun, N.; Elimelech, M. EnViron. Sci. Technol. 1996, 30, 3284-3293. (3) Sethumadhavan, G. N.; Nikolov, A. D.; Wasan, D. T. J. Colloid Interface Sci. 2001, 240, 105-112. (4) Anderson, J. L.; Quinn, J. A. ReV. Sci. Instrum. 1971, 42, 1257-1258. (5) Ito, T.; Sun, L.; Bevan, M. A.; Crooks, R. M. Langmuir 2004, 20, 69406945. (6) Squires, T. M.; Quake, S. R. ReV. Mod. Phys. 2005, 77, 977-1026. (7) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992. (8) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (9) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1965. (10) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417-439. (11) Anderson, J. L. Annu. ReV. Fluid Mech. 1989, 21, 61-99. (12) Lauga, E. Langmuir 2004, 20, 8924-8930. (13) Chen, P. Y.; Keh, H. J. J. Colloid Interface Sci. 2005, 286, 774-791.
measurements of interaction potentials of submicron colloids with each other or confining surfaces. Scattering techniques can measure structure in dispersions containing colloids smaller than the optical diffraction limit,14,15 but such measurements are generally limited to unconfined (bulk) systems. In addition, interpretation of scattering data often requires inverse analyses based on liquid structure theory,16 which can be subject to complications from noise, polydispersity, and so forth. Microscopy-based techniques including total internal reflection (TIRM),17-19 video (VM),20-22 and reflectance interference contrast23 microscopies have been frequently applied to measure micron or greater sized colloids interacting with each other and single wall surfaces and, in one case, with parallel confining walls.24 More recently, differential electrophoresis and smallangle light scattering have been combined to interrogate the breakup of subdiffraction limit sized colloids,25 but this approach does not directly resolve separation-dependent pair interactions. In short, while scattering techniques can measure submicron sized particles, they are inherently indirect; in contrast, optical microscopy techniques directly measure micron sized colloids but, as a result, are often limited to studies of thin electrical double layers and near-field lubrication. A significant body of literature has been dedicated to measuring the transport of colloids near a single surface or within parallel confining walls. Dynamic evanescent wave scattering26-30 and (14) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969. (15) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (16) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (17) Prieve, D. C. AdV. Colloid Interface Sci. 1999, 82, 93-125. (18) Wu, H. J.; Bevan, M. A. Langmuir 2005, 21, 1244-1254. (19) Wu, H.-J.; Pangburn, T. O.; Beckham, R. E.; Bevan, M. A. Langmuir 2005, 21, 9879-9888. (20) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298-310. (21) Crocker, J. C.; Grier, D. G. Phys. ReV. Lett. 1996, 77, 1897-1900. (22) Biancaniello, P. L.; Crocker, J. C. ReV. Sci. Instrum. 2006, 77, 113702. (23) Clack, N. G.; Groves, J. T. Langmuir 2005, 21, 6430-6435. (24) Kepler, G. M.; Fraden, S. Langmuir 1994, 10, 2501. (25) Velegol, D.; Holtzer, G. L.; Radovic-Moreno, A. F.; Cuppett, J. D. Langmuir 2007, 23, 1275-1280. (26) Lan, K. H.; Ostrowsky, N.; Sornette, D. Phys. ReV. Lett. 1986, 57, 17.
10.1021/la702571z CCC: $40.75 © 2008 American Chemical Society Published on Web 01/05/2008
Electrostatically Confined Nanoparticle Interactions
VM studies31-37 of submicron and micron sized hard spheres between parallel walls have displayed good agreement with predicted hydrodynamic interactions except for some confusion over the best available theory. In electrostatically stabilized systems, dynamic TIRM measurements have displayed excellent agreement with exact theory38 for transport normal to wall surfaces.39-41 Dynamic evanescent wave scattering42 and VM measurements34 of electrostatically stabilized colloids between charged parallel confining walls have displayed anomalous drag at low ionic strengths beyond what is expected from predicted hydrodynamic interactions, but little explanation has been provided. Other measurements of single charged colloids near single charged walls in shear flows have revealed a normal force (“electrokinetic lift”) that has defied a satisfactory theoretical description.43-45 While the role of hydrodynamics in modifying the confined transport of hard sphere colloids appears to be fairly well understood, the interpretation of charged colloidal transport near charged boundaries has been more problematic. Based on comparisons with experiments, theories of colloidal transport within parallel confining walls are generally successful when electrostatic and hydrodynamic interactions can be treated separately, but are rather limited at describing electrokinetic phenomena. No exact theory is available to predict hydrodynamic interactions of single colloids with parallel confining walls; however, approximate expressions9,46,47 and more rigorous models48-51 display relatively minor quantitative differences ( 1.1 256 16 (10) For convenience, eqs 9 and 10 can also be represented using a simple rational expression as (with less than a few percent relative error)59,60 (57) Faxen, H. Ark. Mat., Astron. Fys. 1923, 17, No. 27. (58) Goldman, A. J.; Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22, 637651. (59) Anekal, S.; Bevan, M. A. J. Chem. Phys. 2005, 122, 034903. (60) Anekal, S.; Bevan, M. A. J. Chem. Phys. 2006, 125, 034906.
(11)
For a single colloid confined between two parallel planar surfaces with separation, δ, the hydrodynamic hindrance to lateral diffusion can be described using a form similar to eq 8 as
D2w|(z,a,δ) ) D0f2w|(z,a,δ)
(12)
where a number of approximate analytical solutions exist for f2w|(z,a,δ). The simplest of these is the linear superposition approximation (LSA) (suggested by Oseen),9 which includes hydrodynamic hindrance of each wall from eq 9 using LSA (z,a,δ) ) [f1w|(z,a)-1 + f1w|(δ - z,a)-1 - 1]-1 f2w|
(13)
An alternative expression, originally developed for perpendicular motion within a confining gap,46 and sometimes referred to as the coherent superposition approximation (CSA),36 is given for parallel motion as ∞
CSA (z,a,δ) ) [1 + f2w| ∞
∑
(f1w||(z + nδ,a)-1 - 1) + ∑ n)0
(f1w||(nδ - z,a)-1 - 1) - 2
n)1
∞
∑(f1w||(nδ,a)-1 - 1)]-1
(14)
n)1
Another representation obtained via the matched asymptotic expansion (MAE) technique is given as47 MAE (z,a,δ) ) f1w| - g(z,δ)(1 - y)-1(2a/δ) + f2w|
h(z,δ)(1 - y)-3(2a/δ)3 (15) g(z,δ) ) 0.106195(1 - y)2 + 0.659475(1 - y)3 0.329357(1 - y)4 (16) h(z,δ) ) 0.622201(1 - y)4 - 0.798834(1 - y)5 0.495588(1 - y)6 (17) MAE where y ) (δ - 2a)/δ, and eq 15 is evaluated as f2w| (z,a,δ) for MAE z e δ/2 and f2w| (δ - z,a,δ) for z > δ/2. The average lateral diffusion coefficient, 〈D2w|〉, can be predicted as an average over the equilibrium distribution of heights sampled by confined particles as given by41
(8)
where f1w|(z,a) is given conveniently in terms of ω ) z/a from a combination of asymptotic solutions57,58 as47
f1w|(z,a) ) 1 - (9/16)ω-1 + (1/8)ω-3 - f(ω)
f1w|(z,a) )
〈D2w|〉 )
∫D2w|(z)n(z - a)dz ∫n(z - a)dz
(18)
where n(h) ) n(z - a) is the height distribution, which is connected via eqs 5 and 6 to the equilibrium potential energy profile in eq 1. The average lateral diffusion coefficient can be measured via mean squared displacements (MSD) of many noninteracting tracer particles in a given direction, x, as a function of time, t, using60
〈x 〉 ) 2
1 Np
Np
[xi(t) - xi(0)]2 ) 2〈D2w|〉t + ∆2 ∑ i)1
(19)
where Np is the number of particles, and ∆2 is related to the square of the uncertainty in the particle center location due to limited resolution in VM measurements.35,61 Materials and Methods Glass coverslips (Corning, Inc., Corning, NY) were cleaned by soaking in Nochromix (Godax Laboratories, Takoma Park, MD) for 1 h, sonicating in 1 mM KOH for 30 min, rinsing with deionized (61) Savin, T.; Doyle, P. S. Biophys. J. 2005, 88, 623-638.
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Table 1. Fit Parameters for Potential Energy Profiles in Figure 2 2a/nma C/mMb κ-1/nmc δ/nma ψ/mV hm/nm G/pN Fmax/pNd κa κhm
50 nm
100 nm
250 nm
57 ( 9 0.1 31.8 342 ( 36 -60 143 0.017 1215 0.893 4.49
123 ( 20 0.1 29.9 342 ( 36 -27 109 0.176 1328 2.06 3.65
213 ( 29 0.1 31.1 619 ( 68 -55 203 0.911 1025 3.43 6.53
a Particle averages and standard deviations from DLS. b Nominal electrolyte concentrations. c Debye lengths from conductivity measurements. d Maximum electrostatic force evaluated at 6kT.
(DI) water, and drying with nitrogen. Coverslips were assembled into confined cells immediately after cleaning and just prior to experiments. Nominal 50, 100, and 250 nm gold colloids (Ted Pella, Inc., Redding, CA) were dispersed in 0.1 mM sodium dodecyl sulfate (SDS) solutions using DI water filtered through 0.02 µm sterile filters (Whatman International Ltd., Maidstone, England). Silica spacer particles with average diameters of 342 nm and 619 nm were synthesized via the Stober method.62 Gold nanoparticle and silica spacer particle average diameters and their standard deviation were measured using dynamic light scattering (DLS) (ZetaPALS, Brookhaven Instrument Corp., Holtsville, NY) with values reported in Table 1. Prior to assembling confined cells, gold nanoparticles and silica colloid spacers were dispersed in aqueous electrolyte media. Upon confinement, lateral gold nanoparticle number densities greatly exceeded spacer particle densities but remained sufficiently low so that lateral interactions were unimportant (5-10 gold nanoparticles in 408 µm × 310 µm image windows correspond to area fractions of 10-5 - 10-6). An ∼10 µL drop of the nanoparticle/spacer colloid mixture was placed in the middle of a large coverslip (24 × 50 mm2), and then a smaller coverslip (18 × 18 mm2) was placed on top of the drop. Excess solution was wicked from the coverslip edges, and the cell was sealed with fast drying epoxy. The confined cell was optically coupled with an index matching oil to a dovetail prism (Red Optronics, Mountain View, CA) placed on a three-point leveling stage. Particle trajectories were measured using evanescent wave microscopy and VM (Axioplan 2, Zeiss, Germany), which is described in detail elsewhere.18,19 A 15 mW 632.8 nm helium-neon laser (Melles Griot, Carlsbad, CA) was used to generate an evanescent wave decay length of β-1 ) 113.7 nm (68° incident angle) at the bottom coverslip/solution interface. Images were obtained using a 40× (Achroplan, NA ) 0.65) objective (Zeiss, Germany) in conjunction with a 12-bit charge-coupled device (CCD) camera (ORCA-ER, Hamamatsu, Japan) operated at 27-42 frames/s in either 4-binning (336 × 256 pixels, 204 × 155 µm2, 607 nm/pixel) or 8-binning (168 × 128 pixels, 408 × 310 µm2, 1214 nm/pixel) modes. Instantaneous particle heights, h, were obtained via their exponential dependence on evanescent wave scattering intensity, I(h) ) I0 exp(-βh).63,64 Image analysis algorithms coded in FORTRAN were used to track lateral particle coordinates and integrate the evanescent wave scattering intensity for each particle.
Results and Discussion TIRM and VM. Figure 1a provides a schematic depiction of the experimental configuration for evanescent wave scattering and VM measurements of nominal 50, 100, and 250 nm gold colloids confined within nominal 350 and 600 nm gaps between parallel glass microscope slides. The schematic is accurately (62) Stober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62-69. (63) Chew, H.; Wang, D. S.; Kerker, M. Appl. Opt. 1979, 18, 2679. (64) Prieve, D. C.; Walz, J. Y. Appl. Opt. 1993, 32, 1629-1641.
scaled for the case of a 213 nm gold nanoparticle in a 619 nm gap in aqueous 0.1 mM SDS (κ-1 ) 30 nm) media and for an evanescent wave decay length of β-1 ) 113.7 nm, which corresponds to one of the cases investigated in this work. To the knowledge of the authors, this work is the first TIRM investigation of metal nanoparticles having dimensions comparable to the evanescent wave penetration depth. Controlled submicron gaps between cover slips are obtained using monodisperse Stober silica colloids as spacers. The cell is oriented so that gravity is in the direction normal to the parallel slides. The gold nanoparticles are confined within potential energy wells resulting from electrostatic repulsive interactions with the two confining walls that arise from negative charges and potentials on the particle and wall surfaces. In all experiments, confinement did not allow gold nanoparticles to escape the evanescent wave, which allowed for continuous monitoring of normal excursions with nanometer resolution17 and lateral tracking of nanoparticle centers to within half a pixel35 (∼300-600 nm for the camera and microscope settings in the present work). Although the evanescent wave decay length and the gap dimensions are comparable in this work (βδ ) 3-5), there was no obvious increase in background intensity resulting from scattering of the evanescent wave from the top solution/glass interface. Figure 1b-d shows CCD images viewed from above in an upright microscope of three different sized gold nanoparticles irreversibly deposited onto the lower glass surface and scattering the evanescent wave. The laser and CCD settings were identical in each case so that the intensity differences in Figure 1b-d accurately represent the relative scattering intensity for the three gold nanoparticle sizes investigated in this work. The lateral extent of the scattering pattern in each case is on the order of ∼1-2 µm (scale bar is 2 µm) as a result of the combined effects of the optical diffraction limit, the evanescent wave spatial scattering pattern, and the microscope/camera settings. Owing to the relatively high refractive index of the gold nanoparticles, their scattering in the evanescent wave is sufficiently intense, despite their small size, to allow for quantitative intensity measurements using a standard CCD camera and optics. Confined Nanoparticle Potential Energy Profiles. Figure 2 shows potential energy profiles for electrostatic interactions of nanoparticles with confining walls with relative nanoparticle/ gap dimensions of (from DLS measurements in Table 1) (a) 57 nm/342 nm ) 0.17, (b) 123 nm/342 nm ) 0.36, and (c) 213 nm/619 nm ) 0.34. In each plot, the ordinate is potential energy in units of kT relative to the potential energy profile minimum, u(hm,), and the abscissa is the separation between the bottom surface of the nanoparticle and the bottom slide surface relative to the most probable height, hm, on the bottom axis and relative to particle-surface contact on the top axis. The black data points correspond to single particle profiles,17 and the red points are ensemble average profiles constructed by averaging all single particle profiles.18 The solid blue lines are fits to the net potential profile in eq 1, and the green dashed lines are fits to the electrostatic interaction (eq 2) between the nanoparticles with each surface. To fit eqs 1-4 to of each of the potential energy profiles in Figure 2, Debye lengths were fixed via solution conductivity measurements, and nanoparticle and gap dimensions were fixed to their average DLS measured values. The only parameters adjusted to minimize the error between the measured and predicted potential energy profiles were the particle and surface potentials. Although the surface potentials cannot be measured independently, the fit values display excellent correspondence with independent zeta potential measurements and literature re-
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Figure 2. Potential energy profiles for gold nanoparticles interacting with parallel confining walls. (left to right) (a) 56.8 nm particle in 342.1 nm gap, (b) 123.1 nm particle in 342.1 nm gap, (c) 213.2 nm particle in 619 nm gap. Potential energy profiles are shown for single particles (black circles), ensemble averages (red circles), theoretical curve fits (solid blue lines) (eq 1), and theoretical electrostatic interactions between particles with each wall (green dashed lines) (eq 2). Insets show scaled schematics for each case.
sults.18,19,65 Values of all fixed and adjustable parameters are reported in Table 1. Each of the potential energy profiles in Figure 2 has a symmetric nearly harmonic well shape. Although the gravitational potential energy is included in the net potential in eq 1, the buoyant nanoparticle weights are 3-5 orders of magnitude weaker than the greatest electrostatic forces measured in Figure 2 (see G vs Fmax in Table 1) using bulk material densities (FAu ) 19320 kg/m3, FH2O ) 1040 kg/m3). As a result, the contributions of gravity to the net potentials are imperceptible in terms of either skewing the harmonic well shape or shifting the most probable height from hm ) δ/2 - a. The ranges of electrostatic interactions in all cases in Figure 2 are significantly greater than the ranges of retarded van der Waals attraction. For example, the separations where van der Waals attraction decays to
