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SPECIAL SECTION: VADOSE ZONE MODELING

Colloid Transport and Retention in Unsaturated Porous Media: A Review of Interface-, Collector-, and Pore-Scale Processes and Models

^a USDA-ARS, U.S. Salinity Lab., 450 W. Big Springs Rd., Riverside, CA 92507
^b Dep. of Chemical and Environmental Engineering, Univ. of California, Riverside, CA
^* Corresponding author (sbradford{at}ussl.ars.usda.gov).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Received 15 May 2007.

	ABSTRACT

TOP ABSTRACT INTRODUCTION Interface Scale Collector Scale Pore Scale Conclusions REFERENCES

 
Our ability to accurately simulate the transport and retention of colloids in the vadose zone is currently limited by our lack of basic understanding of colloid retention processes that occur at the pore scale. This review discusses our current knowledge of physical and chemical mechanisms, factors, and models of colloid transport and retention at the interface, collector, and pore scales. The interface scale is well suited for studying the interaction energy and hydrodynamic forces and torques that act on colloids near interfaces. Solid surface roughness is reported to have a significant influence on both adhesive and applied hydrodynamic forces and torques, whereas non-Derjaguin–Landau–Verwey–Overbeek (DLVO) forces such as hydrophobic and capillary forces are likely to play a significant role in colloid interactions with the air–water interface. The flow field can be solved and mass transfer processes can be quantified at the collector scale. Here the potential for colloid attachment in the presence of hydrodynamic forces is determined from a balance of applied and adhesive torques. The fraction of the collector surface that contributes to attachment has been demonstrated to depend on both physical and chemical conditions. Processes of colloid mass transfer and retention can also be calculated at the pore scale. Differences in collector- and pore-scale studies occur as a result of the presence of small pore spaces that are associated with multiple interfaces and zones of relative flow stagnation. Here a variety of straining processes may occur in saturated and unsaturated systems, as well as colloid size exclusion. Our current knowledge of straining processes is still incomplete, but recent research indicates a strong coupling of hydrodynamics, solution chemistry, and colloid concentration on these processes, as well as a dependency on the size of the colloid, the solid grain, and the water content.

Abbreviations: AWI, air–water interface • DLVO, Derjaguin–Landau–Verwey–Overbeek • SWI, solid–water interface

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Interface Scale Collector Scale Pore Scale Conclusions REFERENCES

 
COLLOIDS ARE particles with effective diameters of around 10 nm to 10 µm. The lower limit of this size range corresponds to particles that are just larger than dissolved macromolecules and the upper limit to suspended particles that resist rapid settling in water (DeNovio et al., 2004). Colloids in natural subsurface environments include silicate clays, Fe and Al oxides, mineral precipitates, humic materials, and microorganisms (McCarthy and Zachara, 1989). These colloid particles can be released into the soil solution and groundwater through a variety of hydrologic, geochemical, and microbiological processes such as: translocation from the vadose zone (Nyhan et al., 1985), dissolution of minerals and surface coatings (Ryan and Gschwend, 1990), precipitation from solution (Gschwend and Reynolds, 1987), deflocculation of aggregates (McCarthy and Zachara, 1989), microbial-mediated solubilization of humic substances from kerogen and lignitic materials (Ouyang et al., 1996), and land application of raw and treated wastewater (Gerba and Smith, 2005). Consequently, colloid particles vary widely in concentration, composition, structure, and size depending on site-specific properties. Colloid concentrations have typically been reported to range from 10⁸ to 10¹⁷ particles per liter (Kim, 1991). DeNovio et al. (2004) has provided more specific information on colloid concentration ranges in the vadose zone.

An understanding and ability to characterize the transport and retention of colloids in subsurface environments is needed for a wide variety of purposes. For example, the migration of clay particles in porous media is an important process in soil genesis, erosion, and aquifer and petroleum reservoir production because it has a pronounced influence on the ability of porous media to transmit fluids and solutes (Khilar and Fogler, 1998; Mays and Hunt, 2005). Surface water and wastewater treatment processes such as groundwater recharge, riverbank filtration, infiltration ponds and galleries, and sand filtration rely on the efficient removal and inactivation of biocolloids (viruses, bacteria, and protozoan parasites) during passage through porous media (Schijven and Hassanizadeh, 2000; Tufenkji et al., 2002; Ray et al., 2002; Weiss et al., 2005). Many of these biocolloids pose a risk to public health and are therefore contaminants of concern in surface water and drinking water supplies and on agricultural produce (Gerba et al., 1996; Loge et al., 2002; Abbaszadegan et al., 2003). Efficient and cost-efficient design of bioremediation strategies (bioaugmentation and biostimulation) to clean up a variety of recalcitrant chemicals in the subsurface requires knowledge of the transport and fate of bacteria in these environments (Mishra et al., 2001; Vidali, 2001; Gargiulo et al., 2006). Furthermore, high-surface-area colloids that are mobile can facilitate the transport of many inorganic and organic contaminants that strongly adsorb to the solid phase (Grolimund et al., 1996; Kim et al., 2003; Chen et al., 2005; imnek et al., 2006). Hence, effective treatment processes for many colloids and contaminants relies on the optimization of colloid transport or retention in unsaturated or variably saturated porous media.

Considerable research has been devoted to the fate and transport of colloids in porous media (reviews have been given by Herzig et al., 1970; McDowell-Boyer et al., 1986; McCarthy and Zachara, 1989; Ryan and Elimelech, 1996; Khilar and Fogler, 1998; Schijven and Hassanizadeh, 2000; Harvey and Harms, 2002; Jin and Flury, 2002; Ginn et al., 2002; de Jonge et al., 2004; DeNovio et al., 2004; Rockhold et al., 2004; Sen and Khilar, 2006; Tufenkji et al., 2006). In spite of all of this research attention, the mechanisms of colloid transport and retention in porous media are still incompletely understood and quantified. For example, traditional colloid filtration theory assumes an exponential decrease in colloid retention with transport distance (e.g., Yao et al., 1971; Logan et al., 1995; Tufenkji and Elimelech, 2004). In contrast, under saturated conditions that are unfavorable for attachment (when repulsive electrostatic interactions exist between the colloids and the grain surfaces), retained colloids frequently exhibit a depth-dependent deposition rate that produces hyperexponential (a decreasing rate of deposition coefficient with distance) (Albinger et al., 1994; Baygents et al., 1998; Simoni et al., 1998; Bolster et al., 2000; DeFlaun et al., 1997; Zhang et al., 2001; Redman et al., 2001; Bradford et al., 2002; Li et al., 2004; Bradford and Bettahar, 2005) or nonmonotonic (a peak in retained colloids away from the injection source) (Tong et al., 2005; Bradford et al., 2006b) deposition profiles. Deviations between experimental observations and filtration theory predictions have been reported to increase for larger colloids and finer textured porous media (Bradford et al., 2003; Tufenkji and Elimelech, 2005a) and at larger transport distances (Bolster et al., 2000; Bradford and Bettahar, 2005).

Various hypotheses have been proposed in the literature to account for the observed deviations from filtration theory predictions. Chemical explanations include porous media charge variability (Johnson and Elimelech, 1995), heterogeneity in surface charge characteristics of colloids (Bolster et al., 1999; Li et al., 2004), deposition of colloids in the secondary energy minimum of the Derjaguin–Landau–Verwey–Overbeek (DLVO) interaction energy curve (Redman et al., 2004; Hahn et al., 2004; Tufenkji and Elimelech, 2005a), time-dependent attachment (Tan et al., 1994; Liu et al., 1995), and colloid detachment (Tufenkji et al., 2003). Other researchers have suggested that deposition may occur as a result of physical factors that are not included in filtration theory, such as straining (deposition of colloids in small pore spaces such as those formed at grain–grain contacts) (Cushing and Lawler, 1998; Bradford et al., 2002, 2003, 2004, 2005, 2006a,b; Li et al., 2004; Tufenkji et al., 2004; Bradford and Bettahar, 2005; Foppen et al., 2005), soil surface roughness (Kretzschmar et al., 1997; Redman et al., 2001), and hydrodynamic drag (Li et al., 2005).

Most of the above-cited research pertains to saturated media; less is known about colloid transport and retention in unsaturated systems (Wan and Wilson, 1994b; Choi and Corapcioglu, 1997; Wan and Tokunaga, 1997; Schafer et al., 1998a,b; Saiers et al., 2003; Saiers and Lenhart, 2003; Cherrey et al., 2003; de Jonge et al., 2004; DeNovio et al., 2004; Chen and Flury, 2005). Colloid retention mechanisms in the vadose zone are even more complicated than in the saturated zone, mainly due to the presence of air in the system. In unsaturated porous media, water flow is restricted by capillary forces to the smaller regions of the pore space and flow rates are relatively small. Colloid transport may be influenced by increased attachment to the solid–water interface (SWI) (Chu et al., 2001; Lance and Gerba, 1984; Torkzaban et al., 2006a), attachment to the air–water interface (AWI) (Wan and Wilson, 1994a,b; Schafer et al., 1998a; Cherrey et al., 2003; Torkzaban et al., 2006b), film straining in water films enveloping the solid phase (Wan and Tokunaga, 1997; Saiers and Lenhart, 2003), and retention at the solid–air–water triple point (Chen and Flury, 2005; Crist et al., 2004, 2005; Zevi et al., 2005; Steenhuis et al., 2006). Transients in water content during infiltration and drainage processes can also significantly influence these unsaturated colloid retention mechanisms (Saiers et al., 2003; Saiers and Lenhart, 2003; Torkzaban et al., 2006b).

The above literature indicates that many colloid retention processes are still poorly understood and quantified. To improve our knowledge of colloid fate in unsaturated porous media, this review focuses on physicochemical and hydrodynamic factors that will influence the transport and retention of colloids at the interface, collector, and pore scales. In this work, the interface scale is used to study colloid interactions near a single SWI or AWI that occur across the size range of several colloid diameters. The collector scale is used to study colloid transport and interactions on a single solid grain or air bubble collector, while the pore scale is used to study these processes in pore spaces that are defined by several collectors or multiple interfaces. The study of colloid retention at these small scales provides insight into different mechanisms and factors that influence the transport and fate of colloids at the larger scales that are typically considered in the laboratory and the field. Furthermore, diverse modeling approaches and experimental methodologies are needed to investigate colloid transport and retention processes at the small scale. The main objectives of this work are to: (i) review our current understanding of mechanisms, factors, and models of colloid transport and retention at the interface, collector, and pore scales; (ii) identify gaps in knowledge; and (iii) provide recommendations and illustrative examples of how to tackle these knowledge gaps at the small scale. Biological aspects of colloid retention and fate (growth, inactivation, and degradation) are not considered here. The interested reader is referred to recent reviews on this topic (Baveye et al., 1998; Schijven and Hassanizadeh, 2000; Harvey and Harms, 2002; Jin and Flury, 2002; Ginn et al., 2002; Rockhold et al., 2004; Tufenkji et al., 2006).

	Interface Scale

TOP ABSTRACT INTRODUCTION Interface Scale Collector Scale Pore Scale Conclusions REFERENCES

 
Derjaguin–Landau–Verwey–Overbeek Interactions
The interface scale is well suited for quantifying the interaction energy of a colloid as a function of separation distance from the SWI, the AWI, or other colloids. The interaction energy plays a critical role in determining the potential for colloid attachment to these interfaces, as well as in the stability of colloidal suspensions (Elimelech et al., 1998). Interaction energies have typically been calculated using DLVO theory as the sum of electrostatic and van der Waals interaction energies (Derjaguin and Landau, 1941; Verwey and Overbeek, 1948):

[1]

where _total [M L² T^–2], _el [M L² T^–2], and _vdW [M L² T^–2] are the total, electrostatic, and van der Waals interaction energies, respectively, and h [L] is the separation distance between the colloids and the interface of interest. Values of _total, _el, and _vdW are commonly made dimensionless by dividing by the product of the Boltzmann constant (k_B = 1.38 x 10^–23 J K^–1) and the absolute temperature (T_K).

Expressions for _el are available in the literature for different system geometries and assumptions (Elimelech et al., 1998). These expressions were derived from various approximations of the Poisson–Boltzmann equation that accounts for electrostatic interactions of charged bodies in ionic solutions as a result of the overlap of their diffuse double layers. The electrostatic double layer interactions can be determined using the constant surface potential interaction expression of Hogg et al. (1966) for a sphere–sphere interaction as

[2]

where (dimensionless) is the dielectric constant of the medium, _o [M^–1 L^–3 T⁴ A^–2, where A denotes ampere] is the permittivity in a vacuum, r_c [L] is the radius of a colloid, r_c2 [L] is the radius of the second sphere, ₁ [M L² T^–3 A^–1] is the surface potential of the colloid, ₂ [M L² T^–3 A^–1] is the surface potential of the second sphere, and [L^–1] is the Debye–Huckel parameter. The value of is inversely related to the thickness of the diffuse double layer thickness and is given as

[3]

where N_A is Avagadro's number (6.02 x 10²³ molecules mole^–1), M_i is the molar concentration of the electrolyte (mol L^–1), e is the charge of an electron (1.602 x 10^–19 C), and z (dimensionless) is the valence of the electrolyte. The colloid-collector system is frequently treated as a sphere–plate interaction, and a similar expression is obtained from Eq. [2] by taking the limit as r_c2 goes to infinity. In this case, the quantity (r_cr_c2)/(r_c + r_c2) is replaced by r_c.

Measured zeta potentials are frequently used in place of surface potentials in Eq. [2]. Zeta potentials for clean quartz and glass typically range from around –10 to –80 mV depending on the solution chemistry (Elimelech and O'Melia, 1990; Elimelech et al., 2000; Redman et al., 2004). The AWI has also been reported to be negatively charged (Ducker et al., 1994; Wan and Wilson, 1994a; Kelsall et al., 1996; Abdel-Fattah and El-Genk, 1998; Chen and Flury, 2005; Saiers and Lenhart, 2003; Lazouskaya et al., 2006), and reported measurements range from around –15 to –65 mV. In the calculations presented below, the zeta potentials for quartz and the AWI were assumed to be –20 and –50 mV, respectively.

The van der Waals interactions also exist between colloids in suspension and charged surfaces due to the presence of intermolecular forces that occur as a result of polarization of molecules into dipoles. Various expressions for _vdW have been summarized by Elimelech et al. (1998). For sphere–sphere interactions, the retarded van der Waals interaction energy, _vdW, can be determined using the expression by Gregory (1981) as

[4]

where A₁₂₃ [M L² T^–2] is the Hamaker constant in this system, and [L] is the characteristic wavelength that is often taken as 100 nm (Gregory, 1981). When the colloid-collector system is treated as a sphere–plate interaction, the quantity (r_cr_c2)/(r_c + r_c2) in Eq. [4] is replaced by r_c.

The value of the Hamaker constant that is required in Eq. [4] is typically estimated from the following expression (Israelachvili, 1992):

[5]

where A₁₁ [M L² T^–2] is the Hamaker constant of the colloid, A₂₂ [M L² T^–2] is the Hamaker constant for the collector surface, and A₃₃ [M L² T^–2] is the Hamaker constant for the aqueous solution. The values of the Hamaker constant for polystyrene latex, glass, quartz, water, and air are reported to be 6.6 x 10^–20, 6.34 x 10^–20, 6.5 x 10^–20, 3.7 x 10^–20 J, and zero, respectively (Israelachvili, 1992). Hence, attractive van der Waals interaction occurs on glass and quartz surfaces and the value of A₁₂₃ is equal to 3.79 x 10^–21 J for polystyrene–water–glass systems, and 4.04 x 10^–21 J for polystyrene–water–quartz systems. In contrast, the value of A₁₂₃ for polystyrene–air–water systems is equal to –1.2 x 10^–20 J (Israelachvili, 1992), implying that the van der Waals interaction is repulsive for polystyrene colloids at the AWI.

The DLVO theory discussed above has proven to be a useful tool to explore the influence of solution and interface chemistry and colloid size on colloid attachment to various interfaces. Figure 1 presents plots of the calculated total DLVO interaction energy profiles for the 1- and 3-µm polystyrene latex microsphere colloids in 10 and 100 mmol L^–1 ionic strength solution on approach to a quartz surface (Fig. 1a) and the AWI (Fig. 1b). In these calculations, literature values for the zeta potential of 1- and 3-µm colloids were assumed to be –77 and –57 mV (Bradford et al., 2002), respectively. The DLVO calculations revealed the presence of a significant energy barrier to attachment in the primary minimum on quartz and the AWI at both ionic strengths of 10 and 100 mmol L^–1. Under these chemically unfavorable attachment conditions, the DVLO calculations predict that colloids can still interact with quartz due to the presence of a secondary energy minimum at separation distances greater than the location of the energy barrier (the depth of the secondary minimum for the 1- and 3-µm colloids was equal to –0.4 and –1.4 k_BT_K when the ionic strength was 10 mmol L^–1, and was –4.8 and –15.5 k_BT_K when the ionic strength was 100 mmol L^–1, respectively). The depth of the secondary energy minimum increased with colloid size and ionic strength due to an enhancement in attractive van der Waals interactions and compression in double layer thickness. In contrast, DLVO calculations predict that colloids at the AWI experience repulsive electrostatic and van der Waals interactions and therefore do not interact with the AWI. Indeed, Wan and Tokunaga (2002) demonstrated in bubble column experiments that only positively charged colloids attached to the negatively charged AWI.

View larger version (14K): [in this window] [in a new window]   FIG. 1. Plots of the calculated total interaction energy profiles when using standard Derjaguin–Landau–Verwey–Overbeek (DLVO) theory for the 1- and 3-µm polystyrene latex microsphere colloids in 10 and 100 mmol L–1 ionic strength solution on approach to (a) a quartz surface and (b) the air–water interface (AWI); (c) a similar plot when using extended DLVO theory that considers hydrophobic interactions (Eq. [2], [4], [6], [9], and [10]) for colloids on approach to the AWI when the suspension ionic strength was 10 mmol L–1 and the colloid contact angle was equal to 30 and 65°.

In unsaturated systems, hydrophobic interactions of colloids at the AWI may play a potentially significant role in colloid attachment (Schafer et al., 1998a,b; Lazouskaya et al., 2006; Johnson et al., 2006). Colloid stability and aggregation is also reported to be sensitive to the surface hydrophobicity (Crist et al., 2005; Breiner et al., 2006). The DLVO theory can be extended to include the potential influence of hydrophobic interactions on attachment as

[6]

where _Hyd [M L² T^–2] is the interaction energy due to hydrophobic effects.

Van Oss (1994) has proposed a mechanistic model for the calculation of hydrophobic interactions that is based on the Lewis acid–base free energy of adhesion. In brief, the Lewis acid–base interaction energy between a spherical colloid and a flat solid surface is given as (van Oss, 1994)

[7]

where _AB [L] is the water decay length for acid–base interactions that is typically accepted to be around 1 to 2 nm (Israelachvili, 1992), and d_o [L] is the distance of closest approach where physical contact occurs between the colloid and the solid surface and is assumed to be 0.158 nm (van Oss, 1994). The parameter _do^AB [M T^–2] is the free energy of adhesion at d_o and is given as (van Oss, 1994)

[8]

where _w⁺ [M T^–2], _s⁺ [M T^–2], _c⁺ [M T^–2], _w^– [M T^–2], _s^– [M T^–2], and _c^– [M T^–2] are the Lewis acid (superscript +) and base (superscript –) surface components of water (subscript w), solid (subscript s), and colloid (subscript c). Values of _w⁺ and _w^– are reported to be 25.5 and 25.5 mJ m^–2, respectively (van Oss, 1994). Bergendahl and Grasso (1999) reported that _c⁺ and _c^– for polystyrene latex was 0 and 5.9 mJ m^–2, respectively. These same researchers reported that _s⁺ and _s^– for glass was sensitive to the surface preparation, with _s⁺ ranging from 0.4 to 2.3 mJ m^–2 and _s^– from 26.2 to 62.2 mJ m^–2. Chen and Flury (2005) reported for various clays that _s⁺ ranged from 0.0 to 1.1 mJ m^–2 and _s^– from 27.5 to 44.5 mJ m^–2. It should be mentioned that values of _s⁺, _s^–, _c⁺ and _c^– are functions of their surface hydrophobicity, and that these parameters can be determined using measured contact angles and interfacial tension in several different fluids in conjunction with the Young–Dupre equation (e.g, Bergendahl and Grasso, 1999). To estimate Lewis acid–base interactions for colloids at the AWI, the values of _s⁺ and _s^– in Eq. [8] need to be replaced by _a⁺ and _a^– for the air phase. Values of _a⁺ and _a^– have both been reported to be equal to zero (van Oss, 2006).

Disagreement about the origins of hydrophobic interactions, however, still prevails (Tsao et al., 1993, Yaminsky and Ninham, 1993; Rabinovich and Yoon, 1994; van Oss, 1994; Yoon and Ravishankar, 1996). Alternatively, asymmetric hydrophobic interactions between two surfaces can be calculated based on their contact angles (Yoon et al., 1997; Schafer et al., 1998a). The following empirical expression has been proposed to quantify this interaction as a function of separation distance for sphere–plate systems as (Schafer et al., 1998a)

[9]

where K₁₂₃ [M L² T^–2] is the force constant for the asymmetric interactions between macroscopic bodies 1 and 2 in medium 3. The value of K₁₂₃ has been determined as (Yoon et al., 1997; Schafer et al., 1998a)

[10]

where _c (°) is the contact angle on a colloid surface, and ₂ (°) is the contact angle on a second surface, and a and b (both dimensionless) are system-specific constants. For the illustrative examples presented below, we assumed that colloids had a value of _c equal to 30 and 65°, respectively. We also assumed that clean quartz or glass surfaces had ₂ equal to 0°, and the value of ₂ at the air surface is equal to 180° (Schafer et al., 1998a; van Oss, 2006). Values of a = –6 and b = –22 were taken from Crist et al. (2005).

Equation [10] indicates that hydrophobic interactions will be much greater on the AWI than on the quartz–water interface because of the pronounced difference in ₂ (0 vs. 180°). As an illustration of the potential significance of hydrophobic interactions on colloid attachment to the AWI, Fig. 1c presents a plot of the total extended DLVO interaction energy on approach of 1-µm polystyrene latex microspheres to the AWI when the suspension ionic strength was 10 mmol L^–1 and _c was equal to 30 and 65°. It can be observed that colloids with _c = 65° exhibit much greater affinity for the AWI than the _c = 30° colloids.

When the colloid enters into the AWI, a capillary force (F_Cap) will also act on the attached colloids. The vertical component of the capillary force that acts on a colloid at the AWI is given as (Zhang et al., 1996; Veerapaneni et al., 2000):

[11]

where [M T^–2] is the surface tension of water, x_c [L] is the horizontial distance measured from the axis of symmetry to the contact point of the AWI on the colloid surface, _c (°) is the angular inclination of the AWI interface to the horizontal at its line of contact with the colloid, and P_e [M L^–1 T^–2] is the excess pressure that acts on the colloid and is proportional to the height of the capillary rise on the colloid surface. For colloids with a hydrophobic surface, the capillary force will be dominated by the surface tension force (the first term on the right-hand side of Eq. [11]) (e.g., Johnson et al., 2006). The capillary force will only play a significant factor in attachment to the AWI once the colloids enter the interface, i.e., when the energy barrier to attachment has been overcome. In contrast to the SWI, the position of the AWI moves during wetting and drainage cycles. It has been postulated that movement of the AWI during water drainage could potentially capture colloids attached on the SWI by capillary forces (Saiers et al., 2003; Saiers and Lenhart, 2003; Torkzaban et al., 2006a).

Colloid Attachment and Detachment
For attachment to occur, the net adhesive force or torque acting on colloids in the vicinity of an interface must overcome the hydrodynamic forces and the applied torque. To obtain the adhesive force acting on colloids in the proximity of an interface in terms of the calculated interaction energy, the Derjaguin and Langbein approximations can be used (Israelachvili, 1992). Specifically, the value of the adhesive force (F_A) is estimated as _min/h, where _min [M L T^–2] is the absolute value of the secondary or primary minimum interaction energy. The adhesive or resisting torque (T_adhesive [M L² T^–2]) for colloids attached in either the secondary or primary minimum is represented by the net adhesive force (F_A) acting on a lever arm (l_x [L]) as

[12]

The value of F_A corresponds to the extended-DLVO force of adhesion, which must be overcome to detach the particle from the secondary or primary energy minimum. On smooth surfaces, the value of l_x is provided by the radius of the colloid–surface contact area that was estimated using the approach of Johnson et al. (1971). Since there is no direct physical contact between colloids attached in the secondary minimum and the interface, the corresponding contact radius is given as (Israelachvili, 1992)

[13]

where K [M L^–1 T^–2] is the composite Young's modulus (Johnson et al., 1971). Bergendahl and Grasso (2000) used a value of K = 4.014 x 10⁹ N m^–2 for glass bead collectors and a polystyrene colloid suspension.

Hydrodynamic forces also act on colloids that are in the vicinity of the SWI or AWI as a result of water flow. When the water flow is laminar, the lift force acting on the colloid perpendicular to the interface is negligible (Soltani and Ahmadi, 1994) and the drag force that acts on the colloid tangential to the interface is significant and can be determined using the following equation (Goldman et al., 1967; O'Neill, 1968):

[14]

where V/r [T^–1] is the hydrodynamic shear at a distance of r_c from the surface, and µ [M L^–1 T^–1] is the fluid dynamic viscosity.

A colloid that collides with an interface may not succeed in attachment or the previously attached colloids may detach from the interface. Lifting, sliding, and rolling are the hydrodynamic mechanisms that can cause colloid removal from an interface (Soltani and Ahmadi, 1994; Bergendahl and Grasso, 2000). Rolling has been reported to be the dominant mechanism of detachment from solid surfaces under laminar flow conditions (Tsai et al., 1991; Bergendahl and Grasso, 1998, 1999). Rolling occurs when the adhesive torque—the resistance to rolling—is overcome by the applied torque (T_applied [M L² T^–2]) from hydrodynamic forces (Johnson, 1985). The applied torque acting on the colloid in the vicinity of the solid interface due to the hydrodynamic shear force is given as (Goldman et al., 1967; O'Neill, 1968)

[15]

Due to the increase in velocity with distance from the interface, the drag force effectively acts on the attached particle at a height of 1.4r_c; thus, the drag force creates a torque by acting on a lever arm of 1.4r_c (Goldman et al., 1967; O'Neill, 1968).

In the above analysis of torques, a smooth interface and colloid were assumed. In this case, a single adhesion force reasonably describes the interaction (Burdick et al., 2005). The lever arms that act on the adhesive and applied torques, however, have been reported to be a strong function of the surface roughness of the interface and the colloid (Hubbe, 1984; Das et al., 1994; Burdick et al., 2005). Burdick et al. (2005) reported that the lever arm for the applied torque decreased with increasing size of surface roughness and was zero when the roughness was greater than the colloid radius. Conversely, the lever arm that acted on the adhesive torque was reported to increase with increasing size of the surface roughness. Hence, when the interface or the colloid was rough, a distribution of adhesion forces was obtained (Cooper et al., 2000a,b, 2001). Hoek and Agarwal (2006) reported that colloids in the immediate vicinity of multiple SWIs experience greater DLVO forces than colloids on a single SWI. All of these factors indicate that greater retention of colloids is expected on rough than on smooth interfaces.

The information presented above indicates that colloid attachment and detachment will be dependent on the hydrodynamic and adhesive forces. Published literature (Ryan and Elimelech, 1996) also suggests that the diffusion force will play a role in these processes. Brownian motion of colloids in suspension (diffusion) occurs as a result of fluctuations in the number of collisions between the fluid molecules and the colloids. The Brownian diffusion force (F_B) has been modeled as a Gaussian white noise process as (Gupta and Peters, 1985; Ahmadi and Chen, 1998; Kim and Zydney, 2004)

[16]

where U(t) (dimensionless) is a function that generates random numbers between –0.5 and 0.5, and t [T] denotes time. In the limit as time goes to infinity, the distribution of energies that are associated with diffusing colloids in suspension will approach a Maxwellian distribution (Chandrasekhar, 1943). In this case, the fraction of diffusing colloid particles (f₁) that possess energy less than a given dimensionless energy (divided by k_BT_K) of ₁ is given as (e.g., Simoni et al., 1998)

[17]

where E [M L² T^–2] is the kinetic energy of diffusing colloids, _i is the incomplete gamma function, and is the gamma function. Simoni et al. (1998) and Dong et al. (2002) have used Eq. [17] under unfavorable attachment conditions to estimate the fraction of colloids colliding with the solid surface that can be attached by setting ₁ to the absolute magnitude of the depth of the secondary energy minimum. Conversely, this analysis implies that the complementary fraction of colloids that collide with the solid surface, 1 – f₁, would detach from the solid surface via diffusion. This analysis, however, neglects the potential influence of hydrodynamic forces on colloid detachment.

	Collector Scale

TOP ABSTRACT INTRODUCTION Interface Scale Collector Scale Pore Scale Conclusions REFERENCES

 
At the collector scale, the aqueous flow field can be solved and the rate of mass transfer to a simple collector surface can be calculated. The water flow field around a solid grain or an air bubble spherical collector can be found from the solution of the Navier–Stokes equations:

[18]

where [M L^–3] is the fluid density, v [L T^–1] is the velocity vector, p [M L^–1 T^–2] is pressure, and g [L T^–2] is the acceleration due to gravity vector. In filtration theory, a simplified version of Eq. [18] was solved analytically (Yao et al., 1971; Rajagopalan and Tien, 1976). In this work, we solved the Navier–Stokes equation under laminar flow conditions in an axisymmetrical coordinate system using the COMSOL commercial software package (COMSOL, Palo Alto, CA); i.e., under steady-state laminar flow conditions the left-hand side of Eq. [18] is zero. The normal velocity and tangential stress at the side boundaries of the cell around the collector were set equal to zero. Normal pressure differences between the inlet and outlet of the cell were assumed to achieve a range of pore water velocities. A no-slip boundary condition was imposed along the collector surface for a solid grain collector. Since the viscosity of air is much less than that of water, the tangential component of the viscous force at the AWI vanishes (Bird et al., 2002) and there is no momentum transfer (a perfect slip boundary condition):

[19]

where t is the tangential unit vector at the AWI. It should be mentioned that other boundary conditions have been applied to the AWI (Lazouskaya et al., 2006). Partial-slip boundary conditions are likely to be more physically realistic when surface-active impurities accumulate at the AWI.

Figure 2 presents a plot of the velocity distribution at a distance of 0.5 µm from the surface of 400-µm spherical grain and air bubble collectors when the average pore water velocity was 1 m d^–1. Using Eq. [14], this velocity information can be used to calculate the distribution of drag forces that acts on 1-µm colloids that are attached to these collector surfaces. Notice that higher velocities occur on the sides of the collectors that are parallel to the flowing water, and zones of relative flow stagnation occur at the top (front stagnation point occurs at L/L_max = 0) and bottom (rear stagnation point occurs at L/L_max = 1) of the collectors that are perpendicular to flowing water. For a given average pore water velocity, the solid collector has lower velocities near the collector surface than the air bubble. This occurs because of the no-slip boundary condition on the solid grain collector surface (i.e., the velocity at the SWI is zero).

View larger version (21K): [in this window] [in a new window]   FIG. 2. The calculated distribution of water velocity at a distance of 0.5 µm from the surface of 400-µm spherical solid and air bubble collectors when the average pore water velocity was 1 m d–1. The distribution of water velocity along the collector surface is plotted vs. normalized distance (L/Lmax), which is defined as the distance from the front toward the rear stagnation point (L) divided by the distance between the front and rear stagnation points (Lmax = times the radius of the collector).

[20]

Here v_avg [L T^–1] is the average pore water velocity, d₅₀ [L] is the median grain diameter, (dimensionless) is the volumetric water content, (dimensionless) is the collector efficiency, and (dimensionless) is the collision or sticking efficiency. It should be mentioned that filtration theory was originally developed for favorable attachment conditions and in this case = 1 in Eq. [20].

The parameter in Eq. [20] accounts for the mass flux of colloids to the collector surface via diffusion, interception, and sedimentation and is defined as the ratio of the integral of the colloid flux that strikes the collector to the rate at which particles flow toward the collector (Yao et al., 1971). The parameter has been extensively studied for ideal systems, composed of a spherical collector with a smooth surface. Assuming a perfect sink at the collector boundary, the advection–diffusion equation is used to quantify mass transfer to the collector surface in the sphere-in-cell model as (e.g., Ryan and Elimelech, 1996)

[21]

where C [N_c L^–3, where N_c denotes the number of colloids] is the aqueous colloid concentration, D [L² T^–1] is the colloid diffusion tensor, and F [M L T^–2] is the external force vector. The first, second, and third terms on the right-hand side of Eq. [21] account for the colloid flux due to diffusion, advection, and external forces (e.g., gravity and adhesive forces), respectively. Correlation equations to predict as a function of system variables have been developed from simulation results (Rajagopalan and Tien, 1976; Tufenkji and Elimelech, 2004). More recently, the sensitivity of to variations in collector shape and roughness was found to be significant (Saiers and Ryan, 2005). It should be mentioned, however, that these correlations are only explicitly valid for saturated systems.

Colloid filtration theory originally assumed that colloids were irreversibly retained in the primary minimum of the DLVO interaction energy distribution (Ryan and Elimelech, 1996). In this case, physicochemical forces between colloids and collectors will determine the probability of the success in colloid attachment once they collide with the collector surface (Ryan and Elimelech, 1996), i.e., the value of in Eq. [20]. Differences in mineralogy and the presence of coatings of metal oxides or organic matter are expected to produce variations in surface charge (Davis, 1982; Tipping and Cooke, 1982; Song and Elimelech, 1993, 1994). In chemically heterogeneous porous media, it is possible to have localized regions that are favorable for attachment and the value of is therefore proportional to the fraction of the solid surface area that is "favorable" for attachment (Elimelech et al., 2000; Abudalo et al., 2005). Johnson and Li (2005) demonstrated that porous media charge variability and the influence of the DLVO secondary energy minimum should theoretically be consistent with an exponential deposition profile.

A growing body of evidence suggests that attachment in the secondary minimum can significantly contribute to the retention of colloids in saturated porous media (Franchi and O'Melia, 2003; Redman et al., 2004; Hahn and O'Melia 2004; Hahn et al., 2004; Tufenkji and Elimelech, 2005a). Colloids that are attached in the secondary minimum are only weakly associated with the solid phase. In this case, the value of has been related to the energy of the diffusing colloids and the depth of the secondary minimum according to Eq. [17] when = f₁ (Simoni et al., 1998; Dong et al., 2002). Recent experimental (Tong et al., 2005; Li and Johnson, 2005; Johnson et al., 2007a) and theoretical (Torkzaban et al., 2007) evidence also demonstrates that the value of decreases with increasing water velocity under unfavorable attachment conditions. Furthermore, it has been observed that colloids captured in the secondary energy minimum can be translated along the collector surface via hydrodynamic forces (Kuznar and Elimelech, 2007).

Torkzaban et al. (2007) examined the influence of hydrodynamic and adhesive forces and torques, discussed above, on colloid attachment to glass spheroidal (spheres and ellipsoids) collectors. Figure 3 presents a plot of the fraction of the grain collector surface area (S_f) where attachment may occur (T_adhesion > T_applied) for 1-µm colloids as a function of F_A at several pore water velocities. Notice that on a glass collector, S_f is close to zero at the lowest values of F_A because T_adhesion < T_applied across the vast majority of the collector surface. As F_A increases, the value of S_f increases to values between 0 and 1. In this case, partially favorable attachment conditions occur. This implies that attachment occurs on regions adjacent to the front and rear stagnation point (i.e., T_adhesion > T_applied), but that conditions are unfavorable for attachment near the collector center because T_adhesion < T_applied. Eventually a value of F_A occurs when S_f is equal to 1 because T_adhesion > T_applied across the entire collector surface. Increasing the pore water velocity tends to shift these curves to the right because of the higher value of T_applied. Torkzaban et al. (2007) also found that the value of S_f was a function of collector size and shape. For the same pore water velocity, smaller collectors exhibited smaller values of S_f than larger collectors due to the presence of greater values of T_applied near the collector surface. The collector shape also affected the distribution of T_applied around the collector surface and therefore S_f.

View larger version (18K): [in this window] [in a new window]   FIG. 3. Plots of the fraction of the grain collector surface area that is favorable for attachment (Sf) for 1-µm colloids as a function of the adhesive force (FA) at several pore water velocities (0.5, 1.0, and 10.0 m d–1).

Under unfavorable attachment conditions, the hypothesis of colloid charge variability has frequently been invoked to explain experimental deposition profiles for a variety of colloids, including microorganisms (Simoni et al., 1998) and latex microspheres (Li et al., 2004; Tufenkji and Elimelech, 2005b; Tong and Johnson, 2007). Figure 3 can also be used to study the influence of surface charge heterogeneity of colloids or porous media on colloid attachment to a single collector. Surface charge heterogeneity of the colloid and the collector will both influence F_A of colloids that collide with the collector surface. If the zeta potential distributions of either the collector or the colloid is known or assumed, it is possible to determine the fraction of the surface area that is accessible for attachment for a given solution chemistry when the charge heterogeneity (colloid or collector) is uniformly distributed across the collector surface into N categories. In this case, the value of S_f can be determined as

[22]

where i is the category index, S_i (dimensionless) is the value of S_f for the ith category, and f_i (dimensionless) is the charge heterogeneity fraction of the ith category. When N = 2, this approach is similar to that applied in chemically heterogeneous porous media to determine an effective value of (Elimelech et al., 2000; Abudalo et al., 2005). It should be mentioned, however, that when colloid surface charge variability is considered, the variance in the zeta potential distribution will change with transport distance and result in a decreasing attachment rate coefficient (Bradford and Toride, 2007). In this case, the value of f_i will also change with transport distance. In contrast, heterogeneity in the collector surface charge will not produce any change in the average attachment rate coefficient with transport distance (Johnson and Li, 2005).

The coupled roles of hydrodynamic and adhesive forces on colloid attachment to a spherical air-bubble collector may also be studied by considering the forces and torques that act on the attached colloids. For negatively charged colloids, hydrophobic and capillary forces are expected to play the dominant role in colloid attachment to an air bubble because both electrostatic and van der Waals interactions are repulsive (Fig. 1b). An additional complication arises at the AWI compared with the SWI due to the difference in boundary conditions. Figure 2 indicates that much higher velocities are possible at the AWI than the SWI. The type of boundary condition at the AWI (zero tangential momentum transfer at the interface) dictates that attached colloids are likely to experience a uniform drag force with distance from the interface, and therefore no applied torque due to hydrodynamic shear. In this case, sliding of colloids at the AWI is possible when (Bergendahl and Grasso, 1998)

[23]

where F_L [M L T^–2] is the lift force and µ_f (dimensionless) is the coefficient of sliding friction. Saffman (1965) provided a formula for calculating the lift force near the SWI, but this expression may not be applicable near the AWI due to the different boundary condition. When colloids are attached to the AWI, the value of µ_f is likely to be very small. We are not aware of any published values for µ_f adjacent to the AWI. If a value of µ_f equal to zero is assumed, then Eq. [23] predicts that colloids attached to the AWI will slide along the interface as a result of fluid drag, regardless of the magnitude of F_A. Additional research is warranted to study sliding of attached colloids at the AWI and to determine µ_f.

	Pore Scale

TOP ABSTRACT INTRODUCTION Interface Scale Collector Scale Pore Scale Conclusions REFERENCES

 
The pore scale consists of an ensemble of collectors and differs from the collector scale due to the presence of multiple SWI or AWI and contact points (grain–grain contacts and solid–water–air triple points) that makeup the pore space geometry. The aqueous flow field, mass transfer rate, and forces and torques that act on colloids can also be determined at the pore scale. Differences in pore- and collector-scale variables occur as a result of the pore space geometry. The pore scale is well suited for determining mechanisms of colloid retention in porous media. Indeed, studies have recently examined colloid transport and deposition processes at the pore scale using a variety of techniques (Ochiai et al., 2006). This research has provided valuable insight on colloid retention at the AWI and solid–water–air triple point (Wan and Wilson, 1994a; Sirivithayapakorn and Keller, 2003b; Crist et al., 2004, 2005; Wan and Tokunaga, 2005; Steenhuis et al., 2005), colloid size exclusion (Sirivithayapakorn and Keller, 2003a), colloid dispersion (Auset and Keller, 2004), and attachment and straining processes of colloid retention (Bradford et al., 2005, 2006a,b; Xu et al., 2006; Li et al., 2006; Yoon et al., 2006; Kuznar and Elimelech, 2007).

The geometry presented in Fig. 4 allows a first approximation to mechanistically study the influence of pore structure on colloid transport and retention. Similar geometries have been used by other researchers (Lenormand et al., 1983; Mason and Morrow, 1984, 1991; Li and Wardlaw, 1986; Tuller et al., 1999) to study unsaturated flow. The shape of the AWI at a given capillary pressure can be determined using the Young–Laplace equation:

[24]

where R₁ [L] and R₂ [L] are the principal radii of curvature of the interface, [L] is the matric potential head, and g [L T^–2] is the constant of gravitational acceleration. For a spherical interface, R₁ = R₂, whereas when R₂ is large, the second term on the right-hand side of Eq. [24] approaches zero. Tuller et al. (1999) has provided relationships to determine the saturation that corresponds to a given interface curvature.

View larger version (43K): [in this window] [in a new window]   FIG. 4. The velocity distribution in (a) saturated and (b) unsaturated triangular capillary tubes when the average pore water velocity was 0.02 m d–1.

[25]

where w [L] is the thickness of the water films, and A_saw [M L² T^–2] is the Hamaker constant for the solid–air–water system. It should be mentioned that even under relatively moist conditions ( equal to –5 to –10 cm), the calculated thickness of the water film is around 20 nm. This thickness is much smaller than the size of colloids that are typically used in transport experiments. Under variably saturated flow conditions, equilibrium conditions may not be reached instantaneously and Zevi et al. (2005) reported that Eq. [25] did not provide a good prediction of their observed water film thickness of between 5 and 25 µm.

When steady-state and unit hydraulic gradient water flow occurs in a nonspherical capillary, the configuration of the water along the angular capillary is determined using Eq. [24] and [25]. The water velocity distribution was obtained by numerically solving the Navier–Stokes equations (Eq. [18]) for steady-state laminar flow conditions using the COMSOL software package. A no-slip boundary condition was imposed at all SWIs, and Eq. [19] was used as the boundary condition at the AWIs. For the case of a fully saturated capillary, normal pressure differences between the inlet and outlet of the cell can be selected to achieve various average water velocities.

Figure 4 presents a plot of the velocity distribution in the saturated (Fig. 4a) and unsaturated (Fig. 4b) pore space when the average pore water velocity is 0.02 m d^–1. Under saturated conditions, zones of relative flow stagnation occur in the smallest regions of the pore space where SWIs intersect (grain–grain contact points). In addition to these locations, unsaturated flow conditions also produce low-velocity regions near the solid–water–air triple point and on thin water films adjacent to solid surfaces. It should be mentioned that porous media may be represented as a bundle of tortuous capillaries of various geometries (e.g., Tuller et al., 1999). When a given hydraulic gradient is applied to a bundle of capillaries, lower flow rates occur in smaller capillaries and in the smaller regions (corners) of unsaturated pore spaces (Fig. 4b). Figure 3 suggests that variations in the pore-scale fluid distribution will also influence colloid retention. This finding is supported by recent simulation results from a stochastic stream tube model for colloid transport and deposition (Bradford and Toride, 2007).

The rate of mass transfer to the SWIs and AWIs can be determined at the pore scale by solving the advection–diffusion equation (Eq. [21]) and assuming a perfect sink for the colloids at these interfaces. To date, no correlation equations for the colloid mass transfer to the SWI or AWI have been developed for unsaturated systems; however, Torkzaban et al. (2006a) suggested that the limited virus movement under unsaturated conditions was due to increased virus mass transfer to the SWI as a result of the reduced diffusive length in unsaturated compared with saturated systems.

In contrast to the collector scale, only a fraction of the pore space may be physically accessible to colloids as a result of their size in unsaturated porous media. This size exclusion affects the mobility of colloids by constraining them to more conductive flow domains and larger pore spaces that are hydraulically accessible. Hence, colloids may be transported faster than a conservative solute tracer (Reimus, 1995; Cumbie and McKay, 1999; Harter et al., 2000; Bradford et al., 2003; Ryan and Elimelech, 1996; Ginn, 2002). The colloid-accessible fraction of the pore space will decrease with increasing colloid size, decreasing collector size, and decreasing water saturation (Bradford et al., 2006a). For example, pore-size distribution information for average characteristics of sand, silt, and clay soils indicates that under saturated conditions, 10.5, 36.1, and 83.3% of the pore space, respectively, will be smaller than 1-µm colloids.

The forces and torques that act on colloids that are attached to the SWI and the AWI can also be determined at the pore scale. As mentioned above, the flow field is strongly influenced by the pore geometry, and low values of F_D occur at the junction of multiple SWIs or AWIs (Fig. 4). In analogy to surface roughness (Hubbe, 1984; Das et al., 1994; Burdick et al., 2005; Hoek and Agarwal, 2006), the lever arms that act on the adhesive and applied torques is also likely to depend on the pore space geometry near contact points. Steenhuis et al. (2006) reported that the vertical component of the capillary force acts to pin colloids at the solid–water–air triple point. All of these factors indicate that greater retention of colloids is expected in the smallest regions of the pore space formed near multiple interfaces than compared with smooth collector surfaces. Furthermore, the unsaturated water conductivity also rapidly decreases with decreasing water saturation (van Genuchten et al., 1991). Hence, for a given hydraulic gradient, the hydrodynamic forces that act on attached colloids in unsaturated systems are expected to be much lower than under saturated systems.

The above discussion indicates that colloids that are retained near multiple interfaces (SWI–SWI, SWI–AWI, AWI–AWI, SWI–colloid interface, AWI–colloid interface, and colloid–colloid interface) experience different forces and torques than those on a single interface. Hence, colloid retention at the pore scale may occur by processes other than attachment on a single interface. Various terms for colloid retention at the pore scale have been applied in the literature and there is not yet a consensus on this terminology. Figure 5 presents a schematic of the various pore-scale colloid retention processes that have been proposed in the literature. In saturated systems, colloid attachment may occur on the SWI (Location 1). Retention of colloids at two bounding SWIs (Location 3) has been referred to as wedging (Herzig et al., 1970; Johnson et al., 2007b) or straining (Hill, 1957; Cushing and Lawler, 1998; Bradford et al., 2006a). When multiple colloids collide and are retained in a pore constriction (Location 4), this process has been referred to as bridging (Ramachandran and Fogler, 1999) or straining (Herzig et al., 1970; Bradford et al., 2002). When all of the pore spaces in a porous medium are smaller than the colloid diameter, then complete retention of these colloids occurs via mechanical filtration (McDowell-Boyer et al., 1986). In addition to these saturated retention processes, other related colloid-retention mechanisms may occur in unsaturated systems. Colloid attachment can occur at the AWI (Location 2). Film straining refers to retention of colloids in thin water films that are smaller than the colloid diameter (Location 6) (Wan and Tokunaga, 1997), and colloids may also be retained at the solid–water–air triple point (Location 5) (Crist et al., 2004, 2005; Chen and Flury, 2005) in much that same way as wedging at grain-to-grain contact points. It should be noted that colloids that are retained at the triple point may experience DLVO and hydrophobic forces that are associated with the SWI and AWI, as well as capillary forces if they penetrate the AWI (Chen and Flury, 2005). In this work, we refer to retention of colloids on a single interface (SWI or AWI) as attachment. Retention of colloids at multiple interfaces (wedging, bridging, film straining, and retention at the triple point) share many similarities (low-velocity regions), and can all be encompassed by the more general definition of straining as colloid retention in the smallest regions of the pore space (Locations 7) (McDowell-Boyer et al., 1986; Bradford et al., 2002, 2003, 2004, 2005, 2006a,b; Tufenkji et al., 2004; Foppen et al., 2005; Xu et al., 2006; Yoon et al., 2006).

View larger version (14K): [in this window] [in a new window]   FIG. 5. A schematic of the various pore-scale colloid retention processes. Colloid retention at Locations 1 and 2 occurs via attachment to the solid–water and air–water interfaces, respectively. Colloid retention at Locations 3, 4, 5, and 7 occurs via various straining mechanisms, namely: 3, wedging; 4, bridging; 5, retention at the solid–water–air triple point; and 6, film straining. Locations 3, 4, 5, and 7 correspond to the more general definition of straining as colloid retention in the smallest regions of the pore space.

Straining processes have traditionally been assumed to be purely physical phenomena (Herzig et al., 1970; McDowell-Boyer et al., 1986) and therefore only determined by geometry considerations. Recent experimental evidence, however, has indicated a strong coupling of straining processes on solution chemistry and hydrodynamics (Bradford et al., 2006a, 2007; Torkzaban et al., 2008). Insight into the roles of solution chemistry and hydrodynamics on straining processes can be obtained from Fig. 3. This figure indicates that hydrodynamic and adhesive forces will have a big impact on the fraction of the collector surface where retention occurs, and that colloids that collide with a collector may roll along the surface until they come to a region that is chemically and hydrodynamically favorable for deposition. Figure 4 indicates that these favorable retention locations occur where multiple interfaces intersect (zones of relative flow stagnation). Increasing the adhesive force or decreasing the pore-water velocity will direct and retain a larger number of colloids in a given porous medium (Bradford et al., 2007). It is interesting to note that a reduction of the adhesive force will only liberate a fraction of the colloids retained in a given straining location, and that this fraction will depend on the relative size of the colloid and median grain diameter (Bradford et al., 2007).

Many research issues with regard to straining processes in saturated and unsaturated porous media still need to be addressed and quantified. For example, bridging is expected to be a function of solution chemistry, hydrodynamics, and colloid concentration, but this dependency is likely to be different than colloid retention at the triple point and wedging locations due to differences in the hydrodynamics and pore space configurations. It has been reported that bridging increases with increasing hydrodynamic forces and colloid concentration (Ramachandran and Fogler, 1999), whereas Fig. 3 suggests that retention at locations of grain–grain contacts will decrease with increasing hydrodynamic forces and colloid concentration (S_f fills more rapidly at a higher concentration). It is also possible that colloid aggregation may play a role in colloid retention at all of these straining locations (Bradford et al., 2006b), and this process is expected to be a function of hydrodynamics and the chemistry of the colloids and the solution. Additional research is also needed to quantify and to model the influence of temporal changes in solution chemistry and hydrodynamics on colloid retention and release.

Time- and concentration-dependent retention of the colloids in straining locations is to be expected due to filling of these locations (Foppen et al., 2005; Bradford and Bettahar, 2006). If the colloid size is known, then the number of colloids that is required to fill a given volume of the porous medium can be calculated (Foppen et al., 2005). The rate of filling of these sites is theoretically dependent on the concentration of the colloids in suspension (e.g., higher colloid concentrations fill straining sites more rapidly than low concentrations). Large numbers of colloids will be required to fill even small straining fractions of the pore space. As accessible straining sites become filled, water and colloids may be diverted from these regions and less colloid retention will occur with increasing time. Alternatively, straining processes may also produce clogging of pores that will lead to permeability reductions in the porous media. Reviews of colloid-induced clogging of porous media have recently been given by Baveye et al. (1998) and Mays and Hunt (2005).

	Conclusions

TOP ABSTRACT INTRODUCTION Interface Scale Collector Scale Pore Scale Conclusions REFERENCES

 
Our ability to accurately simulate colloid transport and retention in aquifers, and especially in the vadose zone, is currently limited by our lack of basic understanding of the governing processes that control colloid retention at the pore scale. This review discussed our current understanding of physical and chemical mechanisms, factors, and models of colloid transport and retention at the interface, collector, and pore scales. We have identified gaps in knowledge, and provided recommendations and illustrative examples of how to tackle these challenges at the pore scale.

The interface scale is well suited for studying the interaction energy and hydrodynamic forces and torques of colloids near solid–water, air–water, and colloid–colloid interfaces. The DLVO theory provides a useful approach to predict these interactions for various system conditions (zeta potentials of colloids and interfaces, solution ionic strengthen, and colloid size). The DLVO theory, however, does not account for non-DLVO forces such as hydrophobic and capillary forces that may also play a significant role in colloid attachment to the AWI. At present, non-DLVO interactions are incompletely understood and quantitative theory has not been developed or is not generally accepted. Surface roughness is reported to have a significant influence on both adhesive and applied hydrodynamic torques that act on colloids near solid interfaces.

At the single-collector scale, the aqueous flow field can be solved and the rate of mass transfer to a collector surface (solid grain or air bubble) can be calculated. Lower velocities occur adjacent to a spherical solid grain collector compared with an air bubble of similar size due to different boundary conditions at the interface (no slip compared with Eq. [19]). A balance of adhesive and applied hydrodynamic torques (rolling) that act on colloids that collide with a solid collector indicates that only a fraction of the collector surface may contribute to attachment due to variations in the flow field around the collector. The fraction of the collector surface that contributes to attachment will depend on both physical (water velocity and collector shape and size) and chemical (pH, ionic strength, zeta potentials of colloids and collectors, and surface charge heterogeneity) conditions. In contrast to the solid collector, colloids that collide with an air bubble collector will probably slide along this surface due to the presence of a relatively uniform water flow field adjacent to this interface and a low coefficient of sliding friction.

Similar to the collector scale, the flow field can be solved and the rate of mass transfer to an interface can be determined at the pore scale. Differences in flow, mass transfer, and colloid retention processes occur, however, due to the presence of multiple solid–water and air–water interfaces and contact points (grain–grain contacts and the solid–water–air triple point). Specifically, low-velocity regions occur near these contact points. These differences in the flow field and smaller diffusion path lengths that occur in unsaturated systems will influence the colloid mass flux to a particular interface. At present, no mass transfer correlations have been developed for truly pore-scale geometries or for unsaturated systems. Colloid retention mechanisms will also be influenced by the pore space geometry. At the pore scale, a variety of straining processes may occur in saturated (wedging and bridging) and unsaturated (wedging, bridging, film straining, and retention at the triple point) systems, as well as colloid size exclusion. Current knowledge of straining processes is still incomplete but recent research indicates a strong coupling of hydrodynamics, solution chemistry, and colloid concentration on these processes, as well as a dependency on the colloid and grain sizes and the water content.

	ACKNOWLEDGMENTS

 
This research was supported by the 206 Manure and Byproduct Utilization Project of the USDA-ARS and by a grant from NRI (NRI no. 2006-02541). Mention of trade names and company names in this manuscript does not imply any endorsement or preferential treatment by the USDA.

	REFERENCES

TOP ABSTRACT INTRODUCTION Interface Scale Collector Scale Pore Scale Conclusions REFERENCES

 

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