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Nov 27, 2003 - Environmental Engineering, The Johns Hopkins University,. 3400 North Charles Street, Baltimore, Maryland 21218. The deposition and ...
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Environ. Sci. Technol. 2004, 38, 210-220

Deposition and Reentrainment of Brownian Particles in Porous Media under Unfavorable Chemical Conditions: Some Concepts and Applications MELINDA W. HAHN† AND C H A R L E S R . O ’ M E L I A * ,‡ Environ Corporation, 708 Main, Suite 700, Houston, Texas 77002, and Department of Geography and Environmental Engineering, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218

The deposition and reentrainment of particles in porous media have been examined theoretically and experimentally. A Brownian Dynamics/Monte Carlo (MC/BD) model has been developed that simulates the movement of Brownian particles near a collector under “unfavorable” chemical conditions and allows deposition in primary and secondary minima. A simple Maxwell approach has been used to estimate particle attachment efficiency by assuming deposition in the secondary minimum and calculating the probability of reentrainment. The MC/BD simulations and the Maxwell calculations support an alternative view of the deposition and reentrainment of Brownian particles under unfavorable chemical conditions. These calculations indicate that deposition into and subsequent release from secondary minima can explain reported discrepancies between classic model predictions that assume irreversible deposition in a primary well and experimentally determined deposition efficiencies that are orders of magnitude larger than Interaction Force Boundary Layer (IFBL) predictions. The commonly used IFBL model, for example, is based on the notion of transport over an energy barrier into the primary well and does not address contributions of secondary minimum deposition. A simple Maxwell model based on deposition into and reentrainment from secondary minima is much more accurate in predicting deposition rates for column experiments at low ionic strengths. It also greatly reduces the substantial particle size effects inherent in IFBL models, wherein particle attachment rates are predicted to decrease significantly with increasing particle size. This view is consistent with recent work by others addressing the composition and structure of the first few nanometers at solid-water interfaces including research on modeling water at solid-liquid interfaces, surface speciation, interfacial force measurements, and the rheological properties of concentrated suspensions. It follows that deposition under these conditions will depend on the depth of the secondary minimum and that some transition between secondary and primary depositions should occur when the height of the energy barrier is on the * Corresponding author phone: (410)516-7102; fax: (410)516-8996; e-mail: [email protected]. † Environ Corporation. ‡ The Johns Hopkins University. 210

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order of several kT. When deposition in secondary minima predominates, observed deposition should increase with increasing ionic strength, particle size, and Hamaker constant. Since an equilibrium can develop between bound and bulk particles, the collision efficiency [R] can no longer be considered a constant for a given physical and chemical system. Rather, in many cases it can decrease over time until it eventually reaches zero as equilibrium is established.

Introduction This paper is written with four objectives: (1) to present the results of a study of the deposition of Brownian particles in porous media using a Monte Carlo/Brownian Dynamics simulation to track the motion of individual particles and thereby avoid averaging definitions in quantifying the attachment process, (2) to test this model experimentally, (3) using this approach, to examine the role of the secondary minimum in particle deposition and reentrainment, and (4) to speculate about some effects of reversible deposition in environmental systems. The deposition of a colloidal particle onto a surface can be considered to involve two sequential steps or processes: the transport process and the attachment process (1-4). The transport process brings a particle from the bulk liquid to very near the surface of the “collector” and the attachment process binds the two surfaces together in some way. In aquatic environments, convection, diffusion, and gravity dominate the transport step and colloidal interactions between the interacting surfaces usually control the attachment step. Electrostatic and van der Waals forces are frequently used to describe this attachment process. A comprehensive theory describing these colloidal interaction forces and their implications for colloidal stability was developed by Derjaguin and Landau (5) and Verwey and Overbeek (6) and is known as DLVO theory. The cornerstone of the DLVO theory is the assessment of the total interaction energy or force as a function of the separating distance between two charged surfaces. This interaction energy is comprised of contributions from Coulombic interactions, which can be attractive or repulsive, and van der Waals interactions that are attractive for aquasols. If the surfaces are similarly charged, the total interaction energy curve has a distinctive shape marked by certain features. Typical results are depicted in Figure 1. A deep primary attractive well is predicted at very small separating distances (contact), an energy barrier develops at larger separating distances (Φmax), and a smaller, secondary attractive region occurs at still larger separations. The secondary minimum is present because the Coulombic electrostatic repulsion decreases roughly exponentially with separating distance, whereas the van der Waals attraction decreases more slowly with increasing separation. The classic description of success in the attachment step is transport over an existing energy barrier to contact in the primary well. This type of attachment is considered to be irreversible due to the very large energy barrier to release from this primary energy minimum. Deposition rates, assuming contact in the primary well, depend on the value of Φmax. When there is a net repulsive interaction energy, the deposition rate is termed “unfavorable” or “slow” compared to the mass transport rate. Alternatively, when there is no net repulsive interaction, the deposition process is termed “favorable or “fast” and is set by the mass transport rate. 10.1021/es030416n CCC: $27.50

 2004 American Chemical Society Published on Web 11/27/2003

FIGURE 1. Typical repulsive, attractive, and net interaction energy curves between a sphere and a flat plate based on DLVO theory using eqs 1-3. (dp ) 100 nm, I ) 0.1, ψ1 ) -30 mV ) ψ2, A ) 10-20 J, T ) 298 K, constant potential interaction). Classic studies of colloid deposition rates on surfaces were performed by Marshall and Kitchener (7) and by Hull and Kitchener (8) in which Brownian monodisperse latex and carbon black particles were deposited on rotating disk collectors made of various materials (glass, polystyrene, etc.). When the particles and the rotating disk had surface charges of opposite sign (considered favorable conditions for deposition), the observed deposition rates agreed with Levich’s solution (9) for mass transport rates for this case. When the particles and the disk collectors were both negatively charged (unfavorable deposition), deposition rates were hindered but not to the extent predicted using DLVO theory (5, 6). Significant deposition rates were observed even when DLVO theory predicted energy barriers of 250 kT (167 times the average three-dimensional thermal energy of a Brownian particle). It was suggested that the large discrepancy between theory and experiment for unfavorable deposition was due to roughness of the disk collectors. Gregory and Wishart (10) performed packed column experiments using 0.2 µm latex particles and 2 µm diameter alumina fiber collectors and also observed significant deposition under unfavorable chemical conditions, apparently despite large energy barriers. Deposited particles appeared to be irreversibly bound, but some particles were released upon washing with 10-3 M NaOH. Gregory and Wishart also pointed to surface roughness and added surface charge heterogeneity as a possible contributing factor because electron micrographs of the alumina fibers with particles deposited at high pH showed deposition only at certain sites where small clusters of particles were found. Studies of the deposition in packed columns have shown similar discrepancies between classical DLVO theory and experimentally observed deposition rates. Using nonBrownian latex particles and glass beads for media, Tobiason (11-13) observed particle deposition rates that were considerably larger than predicted using DLVO theory. In addition, while DLVO theory predicts a substantial particle size effect on deposition under unfavorable conditions because the height of the repulsive energy barrier scales directly with suspended particle size, experimental deposition rates did not show such a particle size effect. Elimelech (1416) observed similar discrepancies from DLVO theory using Brownian particles and glass beads. Similar discrepancies

have also been observed in aggregation experiments (1720). Other researchers have employed methods of direct observation such as ultramicroscopic examination and Total Internal Reflection Microscopy (TIRM) to study deposition and reentrainment in situ. The ultramicroscopic approach usually entails observing the attachment of individual particles on glass microscope slides. Dabros and van de Ven (21, 22) used this method to study the unfavorable deposition of 0.5 µm latex particles and observed attachment, detachment, and surface diffusion. They observed that the surface coverage was increasingly nonlinear with time and concluded that this was due to detachment and blocking effects. Sjollema and Busscher (23) and Meinders et al. (24) observed similar behavior with 0.82 µm latex particles on various surfaces (mica, poly(methyl methacrylate), and fluoroethenepropylene copolymer) and 0.736 µm latex particles on glass, respectively. This detachment suggests that the deposition process can, in fact, be reversible without changing the chemical or physical conditions of the system under study. With TIRM, separation distances between a deposited particle and a collector surface can be measured to within about 2 nm by monitoring Brownian fluctuations (25). Prieve et al. (26) investigated the unfavorable deposition of 10 µm (nonBrownian) latex spheres on a glass plate. They observed hindered diffusion as separation distance decreased and reversible deposition at low ionic strengths. At high ionic strengths, particles were apparently immobilized because no lateral Brownian motion was detected, but, as ionic strength decreased, particles were capable of lateral diffusion and of escape from the surface. Albery et al. (27) also observed this phenomenon with 0.46 µm carbon black particles on glass coated with tin-doped indium and tried to correlate it with DLVO theory. Where there was no barrier to deposition calculated, the separation distance was very small, but, when a large barrier was predicted, the particles seemed to find an equilibrium distance much farther away from the surface. This was interpreted to be the result of particles trapped in primary and secondary wells, respectively.

Modeling In this section the DLVO model for colloidal stability is briefly summarized. This is followed by a short description of deposition modeling in packed beds including the use of the Interaction Force Boundary Layer model (IFBL) for unfavorable deposition. The section is concluded with a summary of a Monte Carlo/Brownian Dynamics (MC/BD) model for particle deposition developed in this research. DLVO Modeling. Here we focus in the interaction energies using the classic DLVO approach. Hogg et al. (28) derived the following expression for the Coulombic interaction energy between a sphere and a flat plate when the surface potentials of each remain constant during the interaction as the two are brought together:

{

[

ΦCSP ) πorap 2ψ1ψ2ln

]

1 + exp(-κy) + 1 - exp(-κy)

(ψ12 + ψ22)ln[1 - exp(-2κy)]

}

(1)

Here ap is the particle size, o is the permittivity of free space, r is the relative permittivity, ψ is a surface potential, and y denotes the minimum surface-to-surface separating distance. Other assumptions include small surface potentials ( 5 (where κ is the inverse of the Debye length), a 1:1 symmetrical electrolyte, and point charges. Assuming pair wise additivity, the Hamaker expression for the retarded London-van der Waals VOL. 38, NO. 1, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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attractive interaction energy for the sphere-plate case is (29) as follows:

{ }[ ( )]

ΦVDW ) -

Aap 6y

1+

14y λ

-1

(2)

somewhat arbitrary]. For transport by convective diffusion, this single collector contact efficiency is denoted here by ηBr. For a single collector in an infinite medium, the following is obtained

ηBr ) 4.0Pe-2/3

(5)

Here A is the London-Hamaker constant and λ is a characteristic wavelength of the dielectric, usually taken to be 100 nm. Since the London-van der Waals interaction is due to induced dipole-induced dipole interactions, its magnitude is reduced at larger separating distances. This approach does not include the Keesom- and Debye-van der Waals interactions. These are small for most but not all solids in water. The total interaction energy is the sum of the Coulombic and van der Waals contributions:

in which Pe is the Peclet number (2Uac/D), ac is the radius of the collectors, and U is the superficial velocity of the packed bed. In a packed bed, each single collector is affected by neighboring collectors, altering the flow field around each. The Happel sphere-in-cell model (32-34) incorporates the effects of these neighboring collectors. The result, developed by Cookson (35) and used by Yao et al., is as follows:

ΦT ) ΦCSP + ΦVDW

ηBr ) 4.0As1/3Pe-2/3

(3)

Equation 3 is based on the DLVO approach. Many other interactions such as short-range steric and solvation forces are not addressed in this classic modeling effort. The results in Figure 1 were developed using eqs 1-3. Understanding DLVO interaction energies and how they can depend on relevant parameters (ionic strength, particle size, Hamaker constant) is important in this study. Increasing the ionic strength decreases the magnitude and range of the Coulombic repulsion between similarly charged particles. This probably does not materially alter the London-van der Waals attraction for the system so interactions become more favorable (deposition rates increase) as the height of the energy barrier is reduced and the depth of the secondary minimum is increased. Both the Coulombic repulsion (eq 1) and the London-van der Waals attraction (eq 2) depend linearly on particle size so that the total energy curve does also. Larger particles have both higher repulsive energy barriers and deeper attractive secondary minima than smaller ones. The van der Waals attractive energy depends linearly on the Hamaker constant, while Coulombic repulsion is independent of this parameter. A can vary over the range from about 3 × 10-21 to 10-19 J in water depending on the nature of the particles involved. Large Hamaker constants decrease the height of the energy barrier and increase the depth of the secondary minimum, thereby enhancing deposition rates. Modeling Particle Deposition in Packed Beds. The convective diffusion equation in an external force field is used to describe the interaction of suspended particles with media surfaces or stationary collectors in packed beds (30):

∂C DC +b v p‚∇C ) ∇‚ D‚∇C B F ∂t kT

(

)

(4)

Here C is the concentration of particles in suspension, b vp is the particle velocity vector arising from the bulk fluid flow, and D is the position-dependent diffusion coefficient for spherical particles. In an infinite medium, D∞ ) kT/6πµap, where µ is the fluid viscosity. B F is an external force vector that can be represented as the gradient of the total interaction energy, ΦT. The Smoluchowski-Levich approximation assumes that the increased drag on the particle due to increasing viscous forces as it approaches the collector is balanced by the attractive van der Waals forces (9, 31). The transport of Brownian particles is then due only to convective diffusion. Levich solved the diffusion equation analytically for this case for the rate of transport of particles to both a rotating disk and a spherical collector. Yao et al. (3) used the concept of a single collector contact efficiency, defined as the rate at which particles contact the collector to the rate at which they approach it [the description of this approach rate is 212

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(6)

As is given by

As )

2(1 - q5) 2 - 3q + 3q5 - 2q6

(7)

where q ) (1 - )1/3 and  is the porosity of the packed bed. For a porosity of 0.4, for example, As is 38; for this case the neighboring collectors increase particle transport to each collector by a factor of 381/3 or 3.36. For a packed bed of identical spherical media grains or collectors, the following expression has been derived from a number balance of particles across the length (L) of the bed (3)

()

CL 3 L ln ) - (1 - )RηT Co 4 ac

(8)

where CL and Co denote the effluent and influent particle concentrations, respectively, ηT is the total single collector contact efficiency for all three transport mechanisms (convective diffusion, interception, and gravity), and R is the sticking probability or attachment efficiency of the particles to the collectors. R is defined as follows:

R)

particle attachment rate particle collision rate

(9)

Negligible longitudinal dispersion is assumed in the development of eq 8; this is appropriate for most laboratory experiments. The single collector contact efficiency, ηT, has been modeled with considerable success (3, 36-37). It is primarily dependent on physical factors. The attachment efficiency or sticking probability, R, has been modeled with significantly less success. Often it is assessed experimentally by measuring the removal efficiency of a packed bed of given length, porosity, and media size, evaluating ηT from theoretical relationships such as eq 6, and then calculating an experimental R (Rexp) using eq 8. Attachment probabilities are primarily dependent on chemical factors. Two approaches to assessing them theoretically are provided here: (1) a brief summary of the Interaction Force Boundary Layer (IFBL) model developed by others and (2) new approaches using Monte Carlo and Brownian Dynamics (MC/BD) methods and the Maxwell distribution of kinetic energies of particles presented subsequently in this paper. The Interaction Force Boundary Layer model was developed by Spielman and Friedlander (38) to describe the deposition rates of Brownian particles under unfavorable chemical conditions. The convective diffusion equation is solved with an additional term for colloidal forces. The

FIGURE 2. Experimentally derived values of the stability ratio (rexp) compared to theoretical predictions (rIFBL) from the Interaction Force Boundary Layer Approximation. Adapted from ref 14. interfacial region is subdivided into two parts: an inner region, or the interaction force boundary layer, where colloidal forces dominate, and an outer region, where convective diffusion is dominant. DLVO modeling of Coulombic and van der Waals interactions is used for the colloidal interactions in the inner region. The distances over which these forces are significant range from several nanometers (for an ionic strength of 0.1) to several tens of nanometers (for ionic strengths of 10-3) when a 1:1 electrolyte is considered. The thickness of the outer region is on the order of the thickness of the diffusion boundary layer, ac(D/Uac)1/3 (38). The convective diffusion equations are solved for the two regions, and the solutions are matched at some intermediate distance. Particle fluxes to the surface and hence rates of attachment can then be calculated and used to determine a theoretical value for the attachment probability, Rtheor. Elimelech and O’Melia (14-16) used this approach to model attachment efficiencies (Rtheor) and compared these predictions with experimentally calculated values (Rexp). Some results are presented in Figure 2. The comparisons are not good; experimentally determined attachment probabilities are many orders of magnitude larger than theoretically calculated ones under unfavorable chemical conditions. In addition, the size dependence predicted with the DLVO model is not observed in the experimental results. Several possibilities for discrepancies between theory and experimental observations have been suggested: chemical heterogeneities on the surfaces (chemical patches), random distribution of local charges, physical irregularities (bumps), solvation, hydration, or structural forces, deposition in secondary minima, and others. There are several difficulties in the IFBL model that may contribute to the poor agreement observed by these authors and others. Implicit in the IFBL approximation is the definition of a successful attachment as involving transport over a repulsive energy barrier to a deep primary minimum at zero separation distance. This, together with the steadystate assumption, negates the possibility of describing deposition in the secondary minimum although the depth of this well can be significant (several tens of kT for large Brownian particles). Another problem in the IFBL approximation is that the particles are treated as point particles, and a concentration distribution of particles is computed over spatial scales similar to the Debye length, which is usually smaller that the particle size of interest. This results in an unphysical concentration distribution because the concentration cannot change over spatial scales smaller than the

particle. Due to these limitations, it seems prudent to examine a model for colloidal stability that does not use the convective diffusion equation to calculate fluxes to the collector in order to predict the collision efficiency. This approach is developed in the next section of this paper. Monte Carlo/Brownian Dynamics (MC/BD) Modeling. Let us consider a model system comprised of a single flat surface or collector (flat compared to the suspended particles) and 1000 identical Brownian particles able to move normally to the surface by thermal diffusion. The region of interest is bounded by the surface (the location of the primary well) and extends 200 nm into the bulk solution. Beyond this region, colloidal interactions are considered negligible. Particle trajectories are simulated in one dimension (normal to the surface), and particle-particle interactions are neglected. A combination of Monte Carlo and Brownian Dynamics methods are used to simulate the particle trajectories. Extensive additional information is presented by Hahn (40). In this hybrid model, the displacement of a particle by Brownian motion (∆y) is determined by

∆y ) R(∆t)

(10)

where R(∆t) has a Gaussian distribution with the following properties (39, 40):

〈R(∆t)〉 ) 0

(11)

〈R(∆t)2〉 ) 2Do(H)∆t

(12)

Here H is the dimensionless separation distance defined as y/ap. The superscript “o” indicates that the diffusion coefficient (D(H)) is evaluated at the beginning of a time step according to the following relationship:

D(H) ) D∞

(1 +H H)

(13)

Hydrodynamic retardation, also termed the lubrication effect, is incorporated here by a correction of the diffusion coefficient in the bulk suspension (D∞) whereby the coefficient decreases with decreasing separation distance until it vanishes at zero separation (42). The process described above has been accomplished numerically by choosing a random deviate (Xn) from a normal distribution with mean zero and unit variance using IMSL subroutine DRNNOF. The displacement then becomes

∆y ) Xnx2D(H)∆t

(14)

Following the Metropolis walk method (43), the new position, yf ) yi + ∆y, is accepted with probability

(

)

(Φ(yf)) - (Φ(yi)) kT

pv ) exp -

(15)

if the final DLVO interaction energy is greater than the initial value. Otherwise, the new position is accepted with a probability of 1. The decision is made by comparing pv to a uniformly distributed random number on the interval (0,1) chosen with IMSL subroutine DRNUNF. If pv is greater than the random deviate, then the new position is accepted; if not, it is rejected. This method includes the requirement that ∆t . τB (the Brownian relaxation time, mp/6πµap, in which mp is the mass of the particle) because it models Brownian motion as a series of random displacements, but it avoids the requirement ∆t , τF (the time over which the external force changes) because the probability of motion from one position to another is independent of the number of steps between them. Using the two models in combination solves problems inherent in VOL. 38, NO. 1, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 3. Simulation results for 300 nm particles over time (I ) 10-1.5, ψ1 ) -30 mV ) ψ2, A ) 10-20 J, T ) 298 K, ∆t ) 10-8 s): (a) t ) 0.000 s, (b) t ) 0.001 s, (c) t ) 0.004 s, and (d) t ) 0.016 s. each. The time scale difficulty of the Brownian Dynamics model is avoided, and the problem of relating sampling time and real time (44) of the Monte Carlo method is also avoided. In this way the kinetics of Brownian deposition in real time can be examined. The simulations were run on the RS/6000 cluster at the Cornell Theory Center. For each physical and chemical case examined, the model was applied to an ensemble of 1000 negatively charged particles moving in one dimension normal to a similarly charged collector surface. The initial state consisted of the particles distributed randomly over a distance of several nanometers from the surface, and the particles were allowed to diffuse in the electrical interfacial region following the relationships presented above. The difference in the DLVO interaction energies from the old position to a new position determined the probability that the new position is accepted. If a particle was transported out of the control area, it was replaced by another at some random position, preserving conservation of mass within the system.

Results MC/BD Simulations. Three decades ago Long et al. (45) suggested that deposition into secondary minima and subsequent escape of deposited particles would lead to an equilibrium between captured particles and particles in the bulk suspension. To examine this idea, a simulation was made using the MC/BD model to follow the distribution of separating distances over time and thereby monitor the evolution of the particle distribution. Results are presented in Figure 3. Here the symbols represent individual particles located at a distance, y, the minimum surface-to-surface separating distance for each particle and the flat surface. Note that the symbols are much smaller than the actual size of the particles due to considerations of scale. Note also that the vertical location of the particles in the figure is not relevant; it is chosen to enable presentation of the calculated separating distances along the x-axis. The total interaction energy, ΦT, calculated using eqs 1-3 is superimposed on the particle location distribution to permit their comparison. For the conditions indicated in the figure legend, the height 214

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FIGURE 4. Simulation results for 300 nm particles at various ionic strengths (ψ1 ) -30 mV ) ψ2, A ) 10-20 J, T ) 298 K, ∆t ) 10-8 s, and 800 000 time steps): (a) I ) 10-1.0, (b) I ) 10-1.5, (c) I ) 10-2.0, and (d) I ) 10-2.5. of the energy barrier is about 65 kT and the depth of the secondary minimum is approximately 2 kT. These results indicate that an equilibrium is developed over time where the number of particles residing in the secondary minimum is constant. For the conditions assumed here, none of the particles overcomes the energy barrier and makes contact with the surface; Rtheor ) RIFBL ) 0. The effects of ionic strength on particle locations at an interface are presented in Figure 4. Ionic strength was varied from 10-2.5 to 10-1. At I ) 10-2.5, the lowest ionic strength considered, particle locations are distributed over considerable distances. As the ionic strength is increased, more and more particles are captured in the secondary minimum. For the highest ionic strength considered, I ) 10-1, 928 of the 1000 particles are captured in the secondary minimum; RMC/BD ) 0.928. None of the particles crossed the 27 kT energy barrier to the primary well; RIFBL ) 0. These results indicate that the secondary minimum can dominate particle deposition in some cases. Simulations of the effects of suspended particle size (dp) on the distribution of separating distances are presented in Figure 5. Ionic strength is set at 10-1, and suspended particle size is varied from 30 to 1000 nanometers, a size range covering most clays and also biological particles from viruses to bacteria. For the smallest particle size (30 nm), the height of the energy barrier is only about 3 kT, and the energy difference between the bottom of the secondary minimum and the top of the energy barrier is about 4 kT. Most of these particles are able to diffuse across this energy difference, be deposited in the primary well, and attach to the surface; RIFBL f 1. For the 100 nm particles, the energy difference is more significant, about (9 + 3) or 12 kT. Only about six of the particles reach the primary well and make contact with the surface. About half of the remainder is captured in the secondary minimum. RIFBL ) 0.006; RMC/BD ≈ 0.5. For the larger particles (300 and 1000 nm), deposition in the primary well is not observed in the simulations, and most of the particles are captured in the secondary minimum. For these sizes, RIFBL ) 0 and RMC/BD f 1. This series of simulations

FIGURE 5. Simulation results for various particle sizes at I ) 10-1.0 (ψ1 ) - 30 mV ) ψ2, A ) 10-20 J, T ) 298 K, and 800 000 time steps): (a) dp ) 30 nm and ∆t ) 10-9 s, (b) dp ) 100 nm and ∆t ) 3.4 × 10-9 s, (c) dp ) 300 nm and ∆t ) 10-8 s, and (d) dp ) 1000 nm and ∆t ) 3.4 × 10-8 s.

FIGURE 6. Simulation results for 300 nm particles at various Hamaker constants (I ) 0.02, ψ1 ) - 30 mV ) ψ2, T ) 298 K, ∆t ) 10-8 s for 800 000 time steps): (a) A ) 3 × 10-21 J, (b) A ) 10-20 J, (c) A ) 3 × 10-20 J, and (d) A ) 10-19 J.

demonstrates the transition from deposition in the primary well to deposition in the secondary minimum that occurs when the effective energy barrier is about 10 kT. It also shows that deposition in the secondary well increases significantly as particle size increases due to the strong dependence of the depth of the well on suspended particle size. The last series of simulations presented here considers effects of the Hamaker “constant” on the distribution of particle locations. In these simulations the ionic strength is set at 0.02, and the Hamaker constant is varied from 3 × 10-21 to 10-19 J, covering a range from some organic substances to metal oxides and silver halides (46). Results are presented in Figure 6. For the smallest value of the Hamaker constant, 3 × 10-21 J, the energy barrier is large (130 kT), and the secondary minimum is shallow, 5 and LiNO3 g 10-2 M. At LiNO3 concentrations of 0.1 and 1 M, a jump into a secondary minimum from separations of about 10 nm was observed, followed by a steep repulsive force as the surfaces were forced closer together. The authors suggest several possible origins of this short-range repulsion including hydration, solvation or structural forces and also charge regulation at close separating distances as the surfaces transition from a constant potential interaction to act as constant charge surfaces. Chapel (79) used a surface force apparatus to measure forces between two pyrogenic silica sheets immersed in a series of monovalent electrolytes with different cations (LiCl, NaCl, KCl, and CsCl). In experiments at pH ∼6.3 and at salt concentrations from 10-4 to 10-1 M, a short-range repulsive force was detected in all cases: no adhesion was ever observed. Li+ showed the weakest short-range repulsion and Cs+ the strongest one; the others followed in the lyotropic series. Considering hydrated ion sizes, the largest cation (Li+) produced the weakest repulsion, and the others followed in the series Li+ < Na+ < K+ < Cs+. Chapel proposed that repulsion in excess of DLVO forces that is observed between silica at short separating distances is due to the structuring of water at the interface by surface silanol groups. Adsorbed cations weaken this hydrogen bonding network with the most hydrated cation, Li+, inducing the most weakening. This is contrasted with the hydration interaction proposed by others (83) for mica, where adsorbed cations are suggested as the origin of the non-DLVO repulsion at close separation with Cs+ producing the strongest repulsive force and Li+ the weakest. It is appropriate to note that the surfaces of silicas have been a subject of controversy for decades. For example, Healy (84) has considered that the surface of silica is comprised of steric barriers of polysilicate and bound cations. In either case (bound water interactions with adsorbed cations or polysilicates reacting with cations), the classic DLVO approach does not adequately describe the first few nanometers of these surfaces. The formation of primary minima can be hindered in subtle ways that are not quantified by DLVO theory. Rheological Properties of Suspensions. Leong et al. (85) examined the effects of short-range forces on the dispersion of concentrated suspensions using a series of anionic adsorbates in zirconia suspensions. Anions studied were sulfate, phosphate, pyrophosphate, and two polyphosphates as well as the organic anions lactate, malate, benzene-1,2,3tricarboxylate, and citrate. Static yield stress measurements were used to study particle interactions. Maximum attractive interactions as indicated by maximum static yield stresses were observed at the isoelectric pH of the zirconia particles, about 7.0, in the absence of adsorbed anions. Addition of adsorbing anions lowered both the iep and the maximum static yield stresses of the suspensions. Reductions in these

yield stresses denoted increased repulsive interactions among zirconia particles. The sequence observed was sulfate < lactate < benzene-1,2,3-tricarboxylate ≈ malate < phosphate < citrate. These anions were considered to exert a steric barrier at a separation of twice their molecular size. Effects such as these can be expected to be common in aquatic systems. Classic DLVO theory does not address such effects, does not accurately describe interaction energies in the first few nanometers of separation, and is not a reliable approach to quantifying primary minima in such systems. Franks and co-workers (86-89) determined zeta potentials and yield stresses of submicron silica and alumina in concentrated monovalent electrolyte solutions (up to 1.04.0 M). The salts investigated included LiCl, NaCl, KCl, CsCl, NaIO3, NaBrO3, and NaClO4. Zeta potential measurements indicated that well hydrated ions preferentially adsorbed to alumina (considered a well hydrated surface) and the poorly hydrated ions adsorbed to silica (considered a poorly hydrated surface). The most strongly adsorbed ions (Cs+ and K+ for silica and Li+ and IO3- for alumina) shifted the iep several units at concentrations of about 1.0 M. Maxima in the shear yield stresses shifted to pH values that corresponded roughly to the iep of the solid with its adsorbed ions. For silica the highest yield stresses occurred for Cs+- and K+containing suspensions, while for alumina the highest yield stresses occurred for Li+- and IO3- containing suspensions. Yield stresses measured with high concentrations of the strongest adsorbing ions were greater than those measured at the iep observed when no salt was added. The authors suggest that there is an additional attraction when surfaces are covered with strongly adsorbing ions and suggested further that this additional attraction may be due to ion-ion correlation forces. Here again, classic DLVO theory does not address such effects and hence is not a reliable approach to describing primary minima in such systems. Summary. Considering the results of these diverse approaches to solid-water interfaces, it is clear that interactions between solid surfaces in water can be described well by the DLVO model for most systems at separating distances of about 10 nm and larger. For separations of a few nanometers or less, non-DLVO forces will often produce strong interaction forces that can prevent the formation of a primary well when the surfaces have charges with the same sign. This supports the view that irreversible attachment or deposition in primary minima are not ubiquitous in aquatic systems. It is concluded here that reversible aggregation or deposition in secondary minima is common in many natural aquatic environments and in some technological aquatic systems.

Acknowledgments The authors wish to acknowledge the generous support provided by NSF Grant BCS9112766, an NSF doctoral fellowship to M.W.H., a grant from the supercomputer resources of the Cornell Theory Center, and the hospitality of the Particulate Fluids Processing Centre and the Department of Chemical and Biomolecular Engineering at the University of Melbourne to C.R.O.’M. Discussions with Tom Healy and Peter Scales of the University of Melbourne, George Franks of the University of Newcastle (Australia), and Bill Johnson of the University of Utah have been very helpful.

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Received for review April 2, 2003. Revised manuscript received August 25, 2003. Accepted August 28, 2003. ES030416N