Environ. Sci. Technol. 2005, 39, 1011-1017
Experimentally Derived Sticking Efficiencies of Microparticles Using Atomic Force Microscopy TRACY L. CAIL* AND MICHAEL F. HOCHELLA, JR. NanoGeoscience and Technology Laboratory, Department of Geosciences, Virginia Tech, Blacksburg, Virginia 24061-0420
The sticking efficiencies (R) of colloidal particles have been derived from the intersurface potential energy between 2 µm carboxylated polystyrene microspheres and a silica glass plate using the interaction force boundary layer model. The intersurface potential energies were calculated from force-distance data measured using atomic force microscopy (AFM) and from calculations based on Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. AFM forces were measured in aqueous solutions over a range of pH and ionic strength conditions, and DLVO calculations were performed on identical systems. In most conditions, sticking efficiencies that were calculated from AFM data are considerably larger than values calculated from DLVO predictions. Sticking efficiencies vary between 0 and 1 and are strongly dependent upon solution chemistry. AFM-derived sticking efficiencies are consistent with measured microsphere and collector ζ-potentials; sticking efficiencies are lower for more negatively charged surfaces. These results provide the first R estimates of a microparticle-collector system that are calculated directly from physically measured interfacial nanoforces. This study clearly demonstrates that significant differences exist between DLVO- and AFM-derived sticking efficiencies.
Introduction Biotic and abiotic colloidal particles are abundant in many natural environments (1-4), and the transport of contaminants by sorption to these colloids is widely reported. In groundwater systems, colloid-facilitated transport of radionuclides (5, 6), rare metals (7), heavy or toxic metals (8), organic material (9), and viruses (10) is heavily documented. Accurate predictions of colloidal transport through porous media would greatly benefit from the study of contaminant transport; unfortunately, there is rarely any agreement between predicted and measured transport of colloids in real systems (11). Colloidal particles that have a diameter between 0.1 and 2 µm are the most mobile of all colloids and are transported in solution by convective diffusion (3). The adhesion of these particles to a collector surface is an important removal mechanism and is a function of the interfacial forces between the colloid and the collector surfaces (12). These interfacial forces include electrostatic, hydrophobic, hydration, and van der Waals components. Israelachvili gives a thorough review * Corresponding author present address: Environmental Sciences Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6038; phone: (865)576-5134; fax: (865)576-8646; e-mail:
[email protected]. 10.1021/es0352000 CCC: $30.25 Published on Web 01/07/2005
2005 American Chemical Society
of the types of forces that act between particles of these dimensions (13). Since the 1940s, Derjaguin-LandauVerwey-Overbeek (DLVO) theory has been used to describe the interaction energies between surfaces as a function of separation distance. For like-charged surfaces, DLVO theory predicts a potential energy barrier that hinders adhesion, and the successful attachment of a colloid to a surface is limited to those particles with enough energy to overcome that barrier. In 1974, Spielman and Friedlander (14) derived the interaction force boundary layer (IFBL) model, which uses the magnitude of that energy barrier to predict the removal of colloidal particles from suspension by adhesion to a surface. In the IFBL model, sticking efficiency (R), which is defined as the probability that a colloidal particle colliding with a collector surface remains attached to that surface, is calculated as a function of the potential energy barrier between a particle and a collector. Accurate predictions of particle transport in the subsurface are important to several fields in the geosciences. While a complete description of particle transport requires detailed information about the physical and chemical components of the system, two parameters, sticking efficiency and collision efficiency (η), are commonly used to estimate particle removal from solution. Collision efficiency is a physical parameter that describes the probability that a particle approaching a collector will collide with that surface. Collision efficiency can be predicted using the Smoluchowski-Levich equation (15). It is the combined probability of sticking and collision efficiencies (R × η), called the collector efficiency, that is ultimately used to predict distances of particle transport through porous media (15):
C 3 L ln ) - (1 - f)Rη Co 2 d
()
(1)
where Co and C are influent and effluent concentrations, f is porosity, L is length or distance of transport, and d is diameter of the collector media. Empirical methods that are used to predict particle transport in porous media generally under estimate transport distances as compared to field observations (16). Empirical methods are also constrained to very restricted physicochemical systems. Purely theoretical descriptions of particle transport typically predict unduly small sticking efficiencies and are rarely applicable to real, heterogeneous systems (11, 16). In this study a new method of predicting colloidal sticking efficiency has been developed, combining the IFBL model with interparticle force data measured by atomic force microscopy (AFM). The IFBL model is well-suited for utilizing data of this sort, even though it was developed before the AFM was invented. Experiments were conducted to determine the sticking efficiencies of abiotic colloid-sized particles onto a homogeneous silica collector surface. Sticking efficiencies were calculated from intersurface potential energies that were calculated from integrated force-distance data measured by AFM. The sticking efficiencies of 2 µm polystyrene spheres adhering to silica glass have been calculated in aqueous solutions of varying pH and ionic strength. This empirical method of determining sticking efficiency should lead to more accurate predictions of particle transport in porous media because it relies on measured interactions that control attachment at the nanoscale. To validate the results of this study and make a detailed comparison of theoretical versus empirical values, sticking efficiencies were also calculated VOL. 39, NO. 4, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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from DLVO predictions of intersurface potential energy using exactly the same solution chemistry and the IFBL model. The results of this study illustrate a promising new method of calculating sticking efficiency using AFM data. This project is the first to successfully combine the theoretical framework of Spielman and Friedlander (14) with direct force measurements, and further, the first to use atomic force microscopy to predict sticking efficiency.
Theory of the Interaction Force Boundary Layer Model The IFBL model has received recent attention in the literature as a method of predicting sticking efficiency (17-24). It was developed to describe colloidal particle adhesion to a collector surface, and it applies only to perfectly smooth, homogeneous surfaces in solutions without chemical gradients. A basic assumption of the IFBL model is that colloidal particles attach to surfaces at a primary energy minimum and that sticking efficiency is equal to the proportion of particles that cross a primary energy maximum (23). The model was derived by solving the convective diffusion equation including a term for intersurface forces between particles. The solution is specific to spherical particles and collectors and to particles that are small enough to experience Brownian motion. The IFBL model pertains only to surfaces that are free of any previously deposited particles. Also, sticking efficiencies can be calculated only for repulsive surfaces. Because no energy barrier is predicted between ideal attractive surfaces, sticking efficiency in this case is always equal to 1. In real systems the sticking efficiency of electrically attractive surfaces may be less than 1 due to the presence of deposited particles and/or compositional heterogeneity on the particle and collector surfaces. The theoretical sticking efficiency derived by Spielman and Friedlander (14) is
R)
(β +β 1)S(β)
(2)
where S(β) is a dimensionless function and describes the collection of Brownian particles onto a spherical collector. The values of S(β) have been solved analytically and are tabulated in Spielman and Friedlander (14). β is a dimensionless sticking parameter defined by
1 1 D β ) (2)1/3Γ As-1/3 3 3 Ur
( ) (k′rD)
()
1/3
(3)
where Γ is the mathematical gamma function, As is a dimensionless porosity-dependent flow model constant (As ) 60; 23), D is the diffusion coefficient (kT/6πµr), U is the undisturbed flow velocity, r is the particle radius, k is Boltzmann’s constant, T is temperature, and µ is fluid viscosity. k′ is the surface reaction rate coefficient describing adhesion and is equal to the ratio of the diffusive rate of transfer of particles to the collector surface to the overall rate of adhesion, which can be expressed as follows:
k′ )
D
∫ (e 0
φ/kT
∞
- 1) dx
(4)
where φ is the intersurface potential energy and x is the particle-collector separation distance. A modification of the solution by Dahneke (25) approximates the retarding effects of particle interactions on fluid velocity. The corrected reaction rate coefficient becomes
k′ )
∫ [( 0
∞
D r φ/kT 1+ e - 1 dx x
)
]
(5)
Materials and Methods Particle and Collector Materials. The colloidal particles used in this study are fluorescent carboxylated polystyrene mi1012
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FIGURE 1. FE-SEM image of a 2 µm polystyrene bead attached to the end of a silicon nitride cantilever. The image was collected using a 2 kV electron beam at 20 k× magnification. crospheres measuring 2 µm in diameter (Molecular Probes, Eugene, OR). Carboxylated polystyrene microspheres are commonly used as model particles because of their strong negative charge, homogeneity, and simple morphology. Also, the fluorescent nature of the spheres makes optical imaging much easier (important for AFM work). Silica glass coverslips were used as model collectors in each experiment. The flat coverslip surface makes force measurements with the microparticles much easier and more reproducible and is still relevant to the IFBL model because it approximates a spherical collector with a large radius. Silica glass was selected because it has many similarities to quartz, which is the single most abundant mineral in nature and a very common mineral in most types of soils. Silica glass and quartz each have very low solubility in waters of near-neutral pH and similar points of zero charge (26). Preparation of Materials. To remove possible organic contamination, the silica glass and AFM cantilevers were cleaned in a 3:1 H2SO4/H2O2 mixture (known as piranha solution) for 30 min (note that combining H2SO4 and H2O2 is extremely exothermic and great care must be taken in the preparation of this solution), rinsed in sterile deionized water, and dried at 60 °C for 30 min prior to all experiments (27). The aqueous solutions were buffered using sodium acetate and acetic acid to final pH values of 4-7 with ionic strengths (i.s.) of 0.05 and 0.005 M. ζ-Potential Measurements. The ζ-potentials of the carboxylated microspheres and silica glass were each measured in buffered solution using a Malvern Instruments Zetasizer 3000HS. The piranha-cleaned silica glass was ground to a fine powder using an agate mortar and pestle and suspended in the buffered solutions. Ten ζ-potential measurements were collected for the particles and collectors in each buffered solution, and the results are presented as average values with standard deviations. Atomic Force Microscopy Experiments. AFM experiments were performed at room temperature using a Nanoscope IIIa Multimode SPM and tipless silicon nitride cantilevers (Veeco Metrology, Santa Barbara, CA). The cantilever spring constants were measured using the Cleveland method (28) and were found to be equal to the manufacturer’s reported value of 0.06 nN/nm. Individual polystyrene microspheres were attached to the ends of the AFM cantilevers using 5 minute epoxy (Uhu glue). Using the piezoactuator of the AFM, cantilevers were lowered onto a drop of wet epoxy resin until a minute amount of the epoxy coated the cantilever apex. The cantilevers were then lowered onto glass slides covered with microspheres until just one isolated sphere was attached to each cantilever. Cantilevers with attached microspheres were imaged using a LEO 1550 field-emission scanning electron microscope (FESEM) to determine the exact position of the bead and to ensure that no epoxy would contribute to force measurements (Figure 1). By using a low-voltage electron beam,
coating was not necessary, and cantilevers could be imaged without any modification before and after AFM experiments. Data were captured in contact-mode over a 300 nm ramp in the z direction at a cycle rate of 1 Hz. Changes in the cantilever velocity did not noticeably affect measured force curves. Force measurements were made at various collector sample locations. The AFM data were collected as photodiode voltage (v) versus piezo displacement (nm) and processed into force versus distance of separation curves using an Igor Pro routine and the method outlined by Kendall and Hochella (29). A complete description and interpretation of AFM force curves is provided by Cappella and Dietler (30). Tens to hundreds of force curves were collected in each experiment. AFM force measurements were also performed in aqueous solutions using naked cantilevers and cantilevers with a small amount of epoxy at the apex. Forces were collected against a cleaned glass collector under the same experimental conditions and in the same solutions as used in the bead experiments in each buffered solution. Naked cantilevers experienced strong repulsive forces (up to 0.5 nN) toward the silica glass collector at separation distances between 6 and 10 nm at pH g5. The cantilever with epoxy and no microbead was very strongly attracted to the silica surface. The strong adhesion force between the epoxy and glass and the strong repulsive force between the cantilever and glass were useful in identifying the cantilevers that were not properly prepared. DLVO Modeling. Classical DLVO modeling was performed for a sphere and flat plate using electrostatic double-layer forces and the Derjaguin approximation for van der Waals interactions (31). The intersurface potential energy was calculated using a Hamaker value of 1.8 × 10-20 (32) and ζ-potentials measured in this study. Forces were derived from the DLVO energy versus distance of separation for a qualitative comparison with AFM data. Data Processing. More than 1000 force curves were collected for this study, and each curve was inspected individually. Force curves with obvious artifacts due to system or environment aberrations were removed. For each aqueous condition, tens to hundreds of force curves were pooled together, and the average curves from each data set were calculated. Only one variable, the intersurface potential energy (φ), is required to calculate R using the IFBL model. Intersurface potential energy is itself a function of the attractive and repulsive forces experienced between the particle and collector as the surfaces approach and make contact. This energy can be obtained by integrating the force (F) of the approach curve measured using AFM with respect to separation distance (x):
energy (φ) )
∫ F dx 0
50
(6)
The integration is performed from the first point of interaction detected by the AFM (at approximately 50 nm of separation) until the particle and collector make contact (separation distance ) 0 nm). In this study, there were no detectable interactions between the surfaces at particlecollector separation distances greater than 50 nm. Instrument and background noise, as well as optical interference from stray laser light and air bubbles, caused a minor amount of deviation from zero force at large distances of separation. DLVO calculations of intersurface potential energy and energy integrated from AFM measurements were used to calculate sticking efficiencies in solutions of eight different pH and ionic strength conditions. Sticking efficiencies were calculated from φ using the equations outlined above.
TABLE 1. Measured ζ-Potentials (mv) of Carboxylated Polystyrene Microspheres and Silica Glass ζ-potential carboxylated polystyrene
silica
pH
i.s. 0.05 M
i.s. 0.005 M
i.s. 0.05 M
i.s. 0.005 M
7 6 5 4
-34.6 ( 2.1 -29.8 ( 0.9 -24.2 ( 0.4 -23.1 ( 0.1
-59.5 ( 1.0 -57.0 ( 0.8 -54.3 ( 0.1 -51.1 ( 0.9
-31.6 ( 0.9 -26.7 ( 1.2 -23.6 ( 0.4 -23.7 ( 0.5
-46.3 ( 1.9 -40.9 ( 4.4 -36.5 ( 1.8 -29.9 ( 0.4
Results ζ-Potential of the Particle-Collector System. The measured ζ-potentials of the carboxylated polystyrene beads and silica glass collector are presented in Table 1. Both the carboxylated beads and glass are negatively charged at the pH range investigated, and the ζ-potentials of both materials become more positive with decreasing pH. The ζ-potentials of each surface also become more positive as ionic strength increases. At ionic strength conditions greater than 0.1 M, ζ-potential could not be measured due to particle flocculation. The carboxylated polystyrene beads do not reach zero charge at the isoelectric point of carboxyl groups (pI ) 4.25) due to residual sulfate groups on the polystyrene surface that are incorporated during the manufacturing process. AFM Measurements. The average force curves from each buffered solution are presented in Figure 2. Scatter in the AFM data is minimal, and results are reproducible within (0.1 nN in force and ( 5 nm in distance. The forces were measured upon approach to and retraction from the collector; however, only the interaction forces measured upon approach are plotted and discussed because it is the repulsive interaction (measured just prior to the jump to contact region; Figure 2) that is used to calculate sticking efficiency. These repulsive forces correspond to interfacial potential energy barriers that define sticking efficiency in the IFBL model. Due to the large adhesive forces in this system after contact between the surfaces is made, potential energy barriers are not detected as the surfaces are separated. Approach curves display both repulsive and attractive interactions before the two surfaces make contact. In this study, each approach curve has an attractive region with a constant slope that occurs before particle-collector contact is made (Figure 2). This region is called the jump to contact region, and it occurs when the gradient of the attractive force between the particle and the collector is greater than the cantilever force gradient. The jump to contact region is sensitive to the mechanics of the system and the slope of the jump to contact line and is equal to the cantilever spring constant (ksp) (Figure 2A). To assess the possible consequences of integrating over the jump to contact region, and to determined the influence of the cantilever spring constant, in the calculations of R, potential energies (φ) were calculated by integrating the force curve from 50 nm all the way to the point of particle-collector contact and from 50 nm just to the beginning of the jump to contact region. Because the denominator of the surface reaction rate equation (eq 4) quickly reaches -1 when the intersurface potential energy is negative, there are no appreciable differences between the sticking efficiencies calculated using the two integration methods. In a solution of high ionic strength (i.s. ) 0.05 M) and pH 4, AFM measures only attractive interactions between the microsphere and glass surface (Figure 2A). However, at solution pH g 5 the surfaces experience a weak repulsive force prior to the jump to contact at a separation distance starting between 30 and 8 nm. The maximum repulsive forces VOL. 39, NO. 4, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. Measured force-distance curves for 2 µm carboxylated polystyrene beads and silica glass in aqueous solutions at (A) high ionic strength (0.05 M) and (B) low ionic strength (0.005 M) and varying pH. The slope of the jump to contact region in each curve is constant and equal to the cantilever spring constant (ksp). Force sign convention: (+) repulsive; (-) attractive. measured are 0.05, 0.03, and 0.01 nN at pH 7, 6, and 5, respectively. At low solution ionic strength (i.s. ) 0.005 M) and over a pH range from 4 to 7, AFM measurements recorded similar trends compared to higher ionic strength (Figure 2). Repulsive forces are larger at higher pH and consistently decrease with lowering pH (0.25, 0.04, and 0.01 nN at pH 7, 6, and 5, respectively). When solution pH dropped below 5, there was no measurable repulsive force between the particle and collector. Overall, repulsive forces increase as solution ionic strength decreases. The magnitudes of the attractive forces do not appear to be sensitive to solution ionic strength but do increase as solution pH decreases. The slope of the force curve in the region of particlecollector contact (not shown in Figure 2) indicates the degree to which the particle and collector surface deform during contact. In all experiments performed in this study, the slope of the contact region is uniform indicating that the microspheres and glass are not being inelastically deformed. Images collected using FE-SEM after AFM experiments were completed also show no signs of particle deformation during data collection. DLVO Modeled Force Curves. Force versus distance of separation curves were derived from each interfacial potential energy curve calculated using classical DLVO theory. The resulting DLVO force curves are presented in Figure 3. 1014
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The magnitudes of the interaction forces that are predicted by classical DLVO theory are different than those measured for identical systems using AFM. DLVO-predicted repulsive forces are larger and occur at much smaller separation distances than those measured by AFM. In addition, DLVO predicts stronger attractive forces at very close range than those measured by AFM. Sticking Efficiency. The maximum potential energy values and corresponding sticking efficiencies calculated using AFMand DLVO-derived energies are presented in Table 2. The maximum peak of the energy barriers calculated from AFM force curves range from 4.4 to 1021 (φ/kT) at low ionic strength and from 1.7 to 112 (φ/kT) at higher ionic strength over a pH range of 4-7. Correspondingly, the maximum energy value predicted from DLVO calculations are between 1920 and 3630 (φ/kT) at lower ionic strength and between 134 and 722 (φ/kT) at higher ionic strength over the same pH range. Both sets of results show a trend of increasing R with decreasing pH, which is consistent with results from other studies (Table 3). Sticking efficiencies could not be calculated using either AFM or DLVO data at high ionic strength and pH 4 because no repulsive interactions were detected between the particle and collector surfaces. The same condition occurs at low i.s. and pH 4 in the AFM curves and at high i.s. and pH 5 in the DLVO-derived force curves. By
FIGURE 3. Classical DLVO force-distance predictions for 2 µm carboxylated polystyrene beads and silica glass in aqueous solutions at (A) high ionic strength (0.05 M) and (B) low ionic strength (0.005 M) and varying pH. Note that the scales on both axes of Figure 3 are not the same as Figure 2.
TABLE 2. Sticking Efficiencies Calculated from AFM Data (rAFM) and DLVO Data (rDLVO) and Maximum Potential Energy Barriers in O/kT Calculated from AFM Forces and DLVO Theory for Polystyrene Particle/Silica Glass Collector System sticking efficiency (rAFM) pH i.s. ) 0.05 M i.s. ) 0.005 M
max potential energy AFM (O/kT) i.s. ) 0.05 M
i.s. ) 0.005 M
83 112 1.7 0
1021 36.5 4.4 0
7 6 5 4
1.9 × 10-26 5.4 × 10-48 0.66 1
pH
i.s. ) 0.05 M
i.s. ) 0.005 M
i.s. ) 0.05 M
i.s. ) 0.005 M
7 6 5 4
3.56× 10-142 4.22× 10-32 1 1
0 0 0 0
722 388 157 134
3630 3060 2570 1920
0 2.0 × 10-33 0.16 1
sticking efficiency (rDLVO)
max potential energy DLVO (O/kT)
IFBL definition, the sticking efficiency is equal to 1 in these instances. Sticking efficiencies calculated using DLVO data are typically much lower than AFM-derived values due to corresponding larger interfacial repulsive forces (Figures 2 and 3). At low ionic strength, DLVO calculations predict such large energy barriers to adhesion that sticking efficiency is
equal to zero at all pH conditions investigated. At higher ionic strength the sticking efficiencies calculated using the two methods are more similar.
Discussion The results of this study clearly illustrate that sticking efficiencies derived from theoretical data and those derived from AFM data are significantly different when calculated using the IFBL model. Other studies have reported sticking efficiencies of similar microparticle-collector systems using the IFBL model (Table 3). Dong et al. (23) calculated sticking efficiencies for a Comamonas cell-quartz collector system using the IFBL model and interfacial potential energies determined by DLVO theory. Their published theoretical sticking efficiency value (Rth) of the bacteria-quartz system at pH 6 and low ionic strength is 10-181 (23). The experimentally determined sticking efficiency (Rexp) from column experiments using the same bacteria-quartz system is 0.006 (33). Elimelech et al. (22) calculated sticking efficiencies of a silica colloid-silica collector system also using the IFBL model and DLVO theory. Their published Rth values are between 10-264 and 10-70 at pH ∼ 6 and low ionic strength. Column experiments predict sticking efficiencies of approximately 0.3 for similar silica systems (22). In earlier studies by Elimelech and O’Melia using polystyrene particles and glass collectors (17, 18), Rth were significantly lower than experimental results. VOL. 39, NO. 4, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 3. Theoretical and Experimental Sticking Efficiencies (rth and rexp) from Published Studies of Similar Systems sticking efficiency
particle
Rexp ) 0.006 Rexp ) 0.003-0.026
Comamonas sp.
collector glass beads porous sand
conditions
ref
pH 7, i.s. ) 0.003
33
Rth < 1 × 10-181
Comamonas sp.
quartz
pH 6, i.s. ) 0.003
23
Rexp ) 0.35 Rexp ) 0.01 Rth < 1 × 10-4
polystyrene
glass beads
pH 6.7, i.s. ) 0.06 pH 6.7, i.s. ) 0.003 pH 6.7, i.s. ) 0.06
18
Rexp ) 0.01 Rexp ) 0.1 Rth < 1 × 10-4
polystyrene
glass beads
pH 6.7, i.s. ) 0.001 pH 6.7, i.s. ) 0.03 pH 6.7, i.s. ) 0.06
17
Rexp ) 0.3 Rth ) 1 × 10-265 Rth ) 1 × 10-70
silica colloids
quartz grains
pH 5.7, i.s. ) 0.0001 pH 5.7, i.s. ) 0.0001 pH 5.7, i.s. ) 0.01
22
Rexp ) 0.004 Rth ) 1 × 10-7.5 Rexp ) 0.01 Rth ) 0.3
silica colloids
iron oxide coated sand
pH 6, i.s. ) 0.0055
21
Although collector and particle materials, as well as experimental conditions, are not identical, the results of this study at pH 6 and pH 7 are similar to several previous studies providing theoretical sticking efficiencies that are dramatically lower than ones derived from column experiments (Tables 2 and 3). However, in solutions at pH 4 and 5, the sticking efficiencies calculated in this study are more comparable to empirical data than to theoretical predictions of sticking efficiency (17, 18, 22, 23, 33). In particular, at low ionic strength, the AFM derived sticking efficiencies are much more realistic than those predicted by DLVO calculations. In general, however, sticking efficiencies calculated by the IFBL model are somewhat lower than empirical values from comparable systems. There are several potential reasons why Rth values seem unduly low and do not agree with experimental results. As previously mentioned, it is plausible that DLVO theory does not provide an accurate estimate of intersurface potential energy at extremely small distances of separation. The differences in surface proximity of the energy barriers predicted by DLVO and those measured by AFM, which are on the order of several to over 20 nm, may be significant sources of error in calculating Rth. Forces measured using AFM regularly contradict theory and repulsive and attractive interactions are often detected at separation distances greater than 20 nm (34-39). Particle and collector surface roughness, morphology, and chemical/compositional heterogeneity may not be accurately represented in models that calculate potential energy barriers. Certainly, perfectly smooth, spherical, and compositionally homogeneous particles do not represent most inorganic colloids that exist in natural systems. Also, it has been suggested that fluid dynamics plays a significant role in the nanoscale interactions between surfaces in solution (19). Hydrodynamic drag, which is not considered in DLVO theory, may significantly alter the forces that exist between static particles (40); however, measured interaction forces in this study were not altered by changing the approach velocity of the cantilever. It is also possible that R measured in field and column studies may not correspond well with theoretical sticking efficiencies because of difficulties with measuring the former. Sticking efficiencies derived from field data and columns are only as accurate as the estimate of collision efficiency and the tools used to determine colloidal particle concentration. An underestimate of experimental collision efficiency would yield an overestimate of sticking efficiency (eq 1). A final assumption of the IFBL model is that particles adhere to surfaces by attaching at a primary energy minimum. 1016
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pH 5.5, i.s. ) 0.002
Although it has been suggested that particle deposition in secondary minima is negligible in some systems (17-20), any amount of deposition in secondary minima would cause Rth to underestimate sticking efficiency. There is a good possibility that sticking efficiencies determined from field and column studies include particles that are attached at secondary energy minima (17, 20, 23). Particles that attach to collector surfaces at secondary energy minima are more likely to be remobilized by changing solution chemistry and hydrodynamics (19). Estimates of R that include secondary deposition may overpredict long-term sticking efficiency values in natural systems. This study presents a new method of estimating the sticking efficiencies of colloid-sized particles and provides encouragement for researchers interested in predicting particle transport in natural systems. Although the IFBL model employed in this study is based on an ideal system and several assumptions about the particle and collector surfaces are made, the AFM technique used to determine the interfacial potential energy relies on no simplifying assumptions about the system. Nano- and even pico-Newton forces of interaction between the particle and collector are measured in situ, and any surface irregularities or chemical heterogeneities in the system are captured in the measured data. For this reason, the IFBL model coupled with AFM may provide an improved method of calculating sticking efficiency as compared to purely theoretical methods. Preliminary comparisons to DLVO models and previously published data suggest that certain sticking efficiencies calculated in this study are comparable with sticking efficiencies in laboratory and natural systems. Recent advances in AFM methods (41), which allow the measurement of continuous force-separation curves using tapping mode AFM, may improve sticking efficiency estimates by obliterating the jump to contact and jump from contact regions.
Acknowledgments This work benefited from discussions with D. Rimstidt, M. Schrieber, C. Tadanier, S. Lower, T. Kendall, A. Ritter, and P. Haskell and from the detailed comments and suggestions of three anonymous reviewers. S. McCartney helped with collecting FE-SEM images. The authors are grateful for the support of the National Science Foundation through Grant EAR 01-03053 and the Department of Energy through Grant DE-FG02-02ER15326.
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Received for review October 28, 2003. Revised manuscript received October 1, 2004. Accepted October 7, 2004. ES0352000
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