colloid transport in saturated porous media: elimination of attachment efficiency in a new colloid...

Colloid transport in saturated porous media: Elimination ofattachment efficiency in a new colloid transport model

Lee L. Landkamer,1 Ronald W. Harvey,2 Timothy D. Scheibe,3 and Joseph N. Ryan4

Received 9 October 2012; accepted 11 March 2013; published 30 May 2013.

[1] A colloid transport model is introduced that is conceptually simple yet captures theessential features of colloid transport and retention in saturated porous media when colloidretention is dominated by the secondary minimum because an electrostatic barrier inhibitssubstantial deposition in the primary minimum. This model is based on conventional colloidfiltration theory (CFT) but eliminates the empirical concept of attachment efficiency. Thecolloid deposition rate is computed directly from CFT by assuming all predictedinterceptions of colloids by collectors result in at least temporary deposition in thesecondary minimum. Also, a new paradigm for colloid re-entrainment based on colloidpopulation heterogeneity is introduced. To accomplish this, the initial colloid population isdivided into two fractions. One fraction, by virtue of physiochemical characteristics (e.g.,size and charge), will always be re-entrained after capture in a secondary minimum. Theremaining fraction of colloids, again as a result of physiochemical characteristics, will beretained ‘‘irreversibly’’ when captured by a secondary minimum. Assuming the dispersioncoefficient can be estimated from tracer behavior, this model has only two fittingparameters : (1) the fraction of the initial colloid population that will be retained‘‘irreversibly’’ upon interception by a secondary minimum, and (2) the rate at whichreversibly retained colloids leave the secondary minimum. These two parameters werecorrelated to the depth of the Derjaguin-Landau-Verwey-Overbeek (DLVO) secondaryenergy minimum and pore-water velocity, two physical forces that influence colloidtransport. Given this correlation, the model serves as a heuristic tool for exploring theinfluence of physical parameters such as surface potential and fluid velocity on colloidtransport.

Citation: Landkamer, L. L., R. W. Harvey, T. D. Scheibe, and J. N. Ryan (2013), Colloid transport in saturated porous media:Elimination of attachment efficiency in a new colloid transport model, Water Resour. Res., 49, 2952–2965, doi:10.1002/wrcr.20195.

1. Introduction

[2] Models that simulate colloid transport in saturated po-rous media are needed for a variety of applications, such astransport of pathogens through drinking water aquifers andtreatment systems or colloid-facilitated contaminant trans-port. Most of the widely used models derive from colloid fil-tration theory (CFT) originally developed by Yao et al.[1971] and Rajagopalan and Tien [1976]. When conditionsare favorable for colloid deposition, i.e., when colloidand collector surfaces are of opposite charge or solution ionicstrength is very high, CFT calculations closely match experi-

mental results [Tufenkji and Elimelech, 2004b]. However,when unfavorable deposition conditions exist due to electro-static repulsion between colloids and collector surfaces, CFToverpredicts colloid retention if all colloid-collector interac-tions are assumed to result in permanent retention. To addressthis issue, Yao et al. employ the concept of collision efficiencyor attachment efficiency, defined as the ratio of ‘‘collisions’’resulting in attachment to the total number of ‘‘collisions’’, toadjust predictions when unfavorable deposition conditionsexist. This practice is still widely used. However, the use ofattachment efficiency detracts from an understanding of thephysical processes affecting colloid transport and can result ina poor model fit to experimental data because it does not rep-resent mechanisms thought to control colloid transport.

[3] The model presented in this paper eliminates the em-pirical concept of attachment efficiency and uses a paradigmfor colloid re-entrainment based on colloid population het-erogeneity that captures the observed deposition behaviorwhen unfavorable deposition conditions exist. The fittingparameters in this model correlate with calculated values ofsecondary minimum interaction energy and fluid velocity,allowing the model to be used as a heuristic tool to investi-gate the effect of the secondary minimum depth and fluidvelocity on colloid transport.

1Department of Civil and Environmental Engineering, Colorado Schoolof Mines, Golden, Colorado, USA.

2U.S. Geological Survey, Boulder, Colorado, USA.3Pacific Northwest National Laboratory, Richland, Washington, USA.4Department of Civil, Environmental, and Architectural Engineering,

University of Colorado at Boulder, Boulder, Colorado, USA.

Corresponding author: L. L. Landkamer, Department of Civil and Envi-ronmental Engineering, Colorado School of Mines, 1500 Illinois St.,Golden, CO 80401, USA. ([email protected])

©2013. American Geophysical Union. All Rights Reserved.0043-1397/13/10.1002/wrcr.20195

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WATER RESOURCES RESEARCH, VOL. 49, 2952–2965, doi:10.1002/wrcr.20195, 2013

1.1. Terminology

[4] This paper uses terminology related to colloid depo-sition suggested by Johnson et al. [2009] and as defined inthe following paragraphs to avoid semantic confusion. Theterm ‘‘retention’’ is used for colloids trapped in the second-ary minimum, which does not involve physical contactbetween the two surfaces. The term ‘‘attachment’’ is re-served for colloids that are deposited in the primary mini-mum; however, ‘‘attachment’’ may be used in conjunctionwith previously defined legacy terms such as attachment ef-ficiency that do not specifically discriminate between thetwo modes of association. The term ‘‘irreversibly’’ retainedis used to differentiate colloids that remain trapped in thesecondary minimum from those that are retained temporarilyand subsequently re-entrained. However, the ‘‘irreversibly’’retained colloids are in fact, subject to re-entrainment ifphysiochemical conditions change in a way that affects thebalance of forces experienced in the secondary minimum(e.g., a decrease in ionic strength). For the purpose of thispaper, we assume physiochemical conditions remain con-stant; hence, colloids in the secondary minimum that are notsubject to re-entrainment are retained ‘‘irreversibly’’.

[5] The term ‘‘deposition rate’’ is used to quantify the rateat which colloids are delivered to the near-surface vicinity ofa collector, or more specifically, to the secondary minimumsurrounding a collector surface. Therefore, ‘‘deposition’’does not necessarily imply ‘‘attachment’’ to the physical sur-face of a collector but also includes delivery to the secondaryminimum. Lastly, rather than using collision (which impliesphysical contact) when describing interactions between acolloid and a collector, the term ‘‘interception’’ is used in ageneral sense for when a colloid is transported to the near-surface region of a collector via mechanisms of interception,sedimentation, and diffusion as described by Yao et al.[1971].

1.2. Background: CFT and the Use of AttachmentEfficiency

[6] CFT developed by Yao et al. [1971] and Rajagopa-lan and Tien [1976] and subsequently refined by Tufenkjiand Elimelech [2004a] has been useful for describingadvection and immobilization of colloids in well-definedporous media under favorable deposition conditions. Theseauthors use numerical modeling techniques that account forhydrodynamic, Brownian, gravitational and van der Waalsforces to predict colloid trajectories within Happel’s [1958]sphere-in-cell model. Electrostatic forces were not includedbased on the assumption that they are negligible relative tovan der Waals forces when favorable deposition conditionsexist (no electrostatic repulsion inhibiting deposition).These trajectory predictions can then be used to calculatethe theoretical ratio of the number of colloids intercepting acollector (a grain within the porous media) relative to thenumber approaching the collector by advection for a givenset of physical and chemical conditions. This ratio is calledthe single-collector contact efficiency (�o). These calcula-tions were performed over a range of relevant parametervalues such as colloid and collector diameter, fluid viscos-ity, density and velocity, media porosity and magnitude ofthe attractive van der Waals force. A regression analysis ofthese results was then performed to produce a closed-formexpression (correlation equation) that can be used to predict

�o for a given set of conditions without using numericalmodeling. The value of �o determined from the correlationequation can then be used to calculate the rate at which col-loids are deposited on collector surfaces (e.g., mineralgrains) in a packed bed when favorable deposition condi-tions exist :

kf ¼3 1� "ð ÞU�o

2dc; (1)

where kf is the deposition rate, " is the porosity of the me-dium, U is the approach or Darcy velocity of the water, anddc is the diameter of the collector. Equation (1) was derivedusing the assumption that each colloid-collector collision(interception) results in irreversible attachment, presumablyin the deep primary energy well predicted by Derjaguin-Landau-Verwey-Overbeek (DLVO) theory [Derjaguin andLandau, 1941; Verwey and Overbeek, 1948].

[7] Because of the close match between theoretical cal-culations and experimental results, the value of �o is rarelyquestioned for simple systems such as glass-bead filled col-umns when colloid and collector surfaces are of oppositecharge or solution ionic strength is very high (favorabledeposition conditions) [Tufenkji and Elimelech, 2004b].When collectors and colloids have the same charge and ionicstrength is low (unfavorable deposition conditions), a largeelectrostatic repulsive barrier inhibits deposition into the pri-mary energy minimum and application of CFT is consider-ably more problematic. When unfavorable conditions exist,a region of weakly attractive force often exists outside of therepulsive barrier. Called the secondary energy minimum,this region of net attractive force occurs between colloidsand collectors of the same charge-polarity at a separationdistance greater than the primary repulsive barrier becausethe repulsive electrostatic force decays with distance fasterthan the attractive van der Waals forces [Hahn and O’Melia,2004]. In such circumstances, calculations that include re-pulsive electrostatic forces predict colloids that would other-wise intersect the collector surface will follow a limitingtrajectory that is parallel to the collector surface and subse-quently exit the model cell because Happel’s sphere-in-cellmodel has no flow-stagnation zone [Payatakes et al., 1974].Calculations by Spielman and Cukor [1973] and Johnsonet al. [2007a] predict capture by the secondary minimumwhen a zone of flow stagnation at the rear of a spherical col-lector or other zones of low-fluid drag are present. Indeed,column experiments examining colloid transport underunfavorable deposition conditions result in colloid retentionbut less than that predicted by CFT when assuming favor-able deposition conditions.

[8] Because no reliable closed-form expressions exist forpredicting retention in the secondary minimum under unfav-orable deposition conditions, the existing correlation equa-tions developed for favorable deposition conditions are used,but the concept of attachment efficiency (�) is used to mod-ify the deposition rate:

kf ¼3 1� "ð ÞU�o�

2dc; (2)

where � is defined as the fraction of interceptions thatresult in retention. The attachment efficiency can vary

LANDKAMER ET AL.: ELIMINATION OF ATTACHMENT EFFICIENCY

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between 0 and 1 and may be thought of as the probabilitythat a given interception will result in retention. Attemptsto predict � have been largely unsuccessful [Ryan andElimelech, 1996]. For example, Ruckenstein and Prieve[1973] and Spielman and Friedlander [1974] independ-ently developed theoretical equations to predict the rate atwhich colloids would penetrate the electrostatic repulsivebarrier and become deposited in the primary minimum.Because these methods fail to account for retention in thesecondary minimum, they effectively under-estimate � bymany orders of magnitude. Others have attempted to de-velop empirical or semiempirical correlations predictingcollision efficiency, for example, Elimelech [1992] devel-oped a relationship between experimentally determinedcollision efficiencies and a dimensionless parameter basedon the Hamaker constant, inverse Debye length and themagnitude of the electrical double layer repulsive force.Although this relationship inadvertently captured the effectof retention in the secondary minimum over a limited rangeof experimental conditions it never gained widespread use.

[9] The phenomenon of temporary colloid retention in thesecondary minimum and subsequent release (re-entrainment),a common occurrence that results in tailing of breakthroughcurves [Nocito-Gobel and Tobiason, 1996; Li et al., 2005,Tosco et al., 2009], is poorly represented by attachment effi-ciency. Conceptually, attachment efficiency can conjureimages of colloids bouncing off collector surfaces immedi-ately upon impact rather than temporary retention in the sec-ondary minimum. Because the concept of � does not providea time-delayed release of colloids, it can impede a more-mechanistic understanding of how physiochemical processesgovern colloid transport.

[10] In order to capture the significant ‘‘tailing’’ or the con-tinued release of colloids after that predicted by advectionand dispersion, models must augment CFT with re-entrain-ment of colloids. Models that attempt to capture reversibleretention using both � and re-entrainment often becomeoverly complicated and difficult to parameterize due to pa-rameter correlation. More specifically, changes in � effec-tively change kf; however, decreasing the deposition rate orincreasing the re-entrainment rate both have the effect ofdecreasing colloid retention in a model. Additionally, modelsthat incorporate re-entrainment typically include a parameterthat defines the fraction of deposited colloids that are not sub-ject to re-entrainment. This fraction is dependent on the ratethat colloids are being deposited (kf), which would be affectedby �, if used. The shape of a colloid breakthrough curve doesnot contain enough information to uniquely determine allthree of these parameters due to their interdependence (pa-rameter correlation), resulting in an under-constrained model.

1.3. Prevalence of Secondary Minimum Retention

[11] Secondary minimum retention is important in thenatural environment because at circumneutral pH, mostmicroorganisms and many collector surfaces are negativelycharged resulting in repulsive electrostatic forces. Even aq-uifer surfaces that have a coating of metal oxy-hydroxides,which would normally be positively charged at circumneu-tral pH, can become coated with negatively charged naturalorganic matter [Davis, 1982].

[12] The existence of a secondary minimum is not only de-pendent on collector and colloid charge; it is also a function

of ionic strength, which influences the magnitude and decayof the electrostatic force. For instance, it is often acceptedthat electrostatic repulsion overwhelms attractive van derWaals forces at very low-ionic strengths so that no secondaryminimum exists. Conversely, high-ionic strengths can elimi-nate the repulsive barrier to deposition in the primary mini-mum resulting in no secondary minimum. However, forlarger microbes such as Cryptosporidium parvum oocysts(3–7 �m, diameter) there is evidence that a secondary mini-mum exists between biocolloids and clean quartz grains ationic strengths as low as 1�10�5 M [Abudalo, 2006]. Also,there is empirical evidence, e.g., tailing of breakthroughcurves [Chen and Zhu, 2004] that reversible retention canoccur at ionic strengths as high as 0.1 M where DLVO calcu-lations predict no electrostatic barrier to deposition in the pri-mary minimum. These lines of evidence suggest that DLVOcalculations for microorganisms are subject to error andretention in the secondary minimum may be more wide-spread than generally accepted. Sources of error in DLVOcalculations can range from uncertainties in surface potentialvalues to failure to include all relevant forces such as acid-base interaction forces suggested by Van Oss et al. [1986]and van Oss [1994]. The secondary minimum values pre-sented in this paper were calculated as outlined in the Meth-ods section using widely accepted assumptions; however, wedo not assume these values to be entirely accurate and inter-pretation of the results should consider this.

2. Conceptual Model

2.1. Model Overview

[13] The postulated mechanisms and variables control-ling colloid transport in saturated porous media when con-ditions of unfavorable attachment exist are numerous, somemore carefully quantified than others. In addition to reten-tion in the secondary minimum in a general sense, theyinclude wedging [Johnson et al., 2007a; Johnson et al.,2010], straining, colloid and collector shape, deposition inthe primary minimum due to surface heterogeneities [Bhat-tacharjee et al., 1998] or by penetrating the repulsive bar-rier, steric interactions arising from surface biomoleculeson microbial surfaces [Kuznar and Elimelech, 2005], mi-crobial cell motility, blocking and ripening [Camesanoet al., 1999], and translation of colloids trapped in a sec-ondary minimum to a region of the collector where flowstagnation exists [Johnson et al., 2007b; Kuznar and Eli-melech, 2007] that reduces the likelihood of re-entrain-ment. While not comprehensive, this list illustrates thedifficulty that would be encountered when including allmechanisms affecting transport in a comprehensive model.The strategy adopted for this paper is to use a simple non-mechanistic rate-based model that only considers retentionin the secondary minimum; the model does not attempt tosimulate any other of the above-mentioned mechanisms.The data used in this paper were collected under unfavora-ble attachment conditions using homogeneous sphericalcollector grains (e.g., no collector surface charge heteroge-neity). ‘‘Clean-bed’’ conditions also existed, minimizingthe probability of retained colloids impacting the ability ofthe secondary minimum to capture additional colloids.

[14] In this paper, we develop a one-dimensional colloidtransport model that eliminates the empirical attachment


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efficiency term (�) by assuming that all colloids that areintercepted by the secondary minimum surrounding a col-lector are retained at least temporarily. A new paradigm forthe behavior of colloids that are re-entrained after a reten-tion event is also incorporated. In this paradigm, the initialcolloid population is split into two fractions. One fraction,by virtue of physiochemical characteristics (e.g., size andcharge) will always be re-entrained after capture in a sec-ondary minimum. The remaining fraction of colloids, againas a result of physiochemical characteristics, will beretained ‘‘irreversibly’’ when captured by a secondary min-imum. The basis for such behavior is discussed in the fol-lowing section. Assuming the dispersion coefficient can beestimated from tracer behavior, this model has only two fit-ting parameters. One parameter is the fraction of the initialcolloid population that will be retained ‘‘irreversibly’’ uponinterception by a secondary minimum. The other parameteris the rate at which reversibly retained colloids leave thesecondary minimum.

[15] Although the model is simpler than most existingmodels, it captures the majority of tailing commonlyobserved in breakthrough curves and the colloid depositionprofile observed in many column experiments conductedwith unfavorable deposition conditions. We use publisheddata to validate the model and to illustrate the utility of themodel in elucidating physical processes controlling colloidtransport.

2.2. Deposition Rate

[16] For negatively charged colloids approaching a ho-mogeneous, negatively charged collector at relatively lowionic strength, a repulsive electrostatic barrier inhibits dep-osition in the primary minimum; however, an attractivesecondary energy minimum often surrounds the collectoroutside the repulsive barrier. For this paper and model, weare assuming that if a colloid is intercepted by a secondaryminimum, the colloid will become at least temporarilytrapped in the secondary minimum, although the residencetime may be very short. Using the assumption that all inter-ceptions result in retention, � is unity and the need for theconcept of attachment efficiency disappears. Indeed, thedeposition rate (kf) can now be quantified by the single col-lector contact efficiency (�o) and equation (1), just as whenfavorable deposition conditions prevail.

[17] We use �o, calculated using the physical diametersof the colloid and collector and the correlation equationdeveloped by Tufenkji and Elimelech [2004a], to quantifythe delivery of colloids to the near-surface region of thecollectors (as defined by the region where a secondary min-imum exists between the colloid and a collector surface).Due to the finite distance between the collector and the sec-ondary minimum, which is on the order of a few nano-meters to 10’s of nanometers, using the physical diameterof the collector could potentially introduce errors in �o.One source of error would stem from increasing the effec-tive diameter of the collector by including the distancefrom the particle surface to the secondary minimum. For acolloid diameter of 1 mm and a collector diameter of 0.4mm and a secondary minimum ‘‘standoff’’ of 40 nm, thedifference in �o is on the order of 0.01%. The error result-ing from the secondary minimum standoff distance is incon-sequential relative to colloid and collector measurement

errors and is not included in the CFT computations in thispaper. A potentially larger error may result because thehydrodynamic retardation acting on a colloid approaching acollector is less at the distance of the secondary minimumrelative to that at the physical surface. However, refinementof the �o correlation equation for this effect is beyond thescope of this paper and left to other authors.

[18] The assumption that all interceptions result in atleast temporary retention is central to the model, constrain-ing the calculation of kf as mentioned, and thereforedeserves additional discussion. Even in unfavorable condi-tions with no secondary minimum, colloids intercepted bya collector briefly interact as the trajectory of the colloid ischanged due to the repulsive electrostatic forces experi-enced between the colloid and the collector. Due to the fi-nite velocity of the colloids, this interaction may be viewedas a very brief retention followed by re-entrainment. Asnoted above, this eliminates the need for the parameter �and allows the calculation of kf directly from CFT so that kf

is no longer a fitting parameter.

2.3. Colloid Re-Entrainment

[19] Once a colloid is in the secondary minimum it willeventually escape and continue its journey through the po-rous medium if it possesses kinetic energy (Ek) in excess ofthe depth of the secondary energy minimum (�2min) itexperiences (Figure 1). If the colloid does not possess suffi-cient kinetic energy, it will remain trapped in the secondaryminimum until it either gains the required kinetic energyfrom an external source (hydrodynamic, thermal, vibra-tional, etc.) or the depth of the secondary minimum isreduced by changing chemical conditions. To reiterate, if acolloid’s kinetic energy is less than the secondary mini-mum it experiences, it will be retained in the secondaryminimum irreversibly, but if its energy is greater than the

Figure 1. Illustration of ‘‘irreversible’’ and temporarysecondary minimum retention. The rate at which colloidsare intercepted by the secondary minimum of a collector isdefined by the first order deposition rate, kf. A fraction ofthe colloids that interact with the secondary minimum willbe ‘‘irreversibly’’ retained, but the rest, by virtue of pos-sessing sufficient kinetic energy, will become re-entrainedin the pore water.


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secondary minimum it will be retained reversibly and willeventually be re-entrained.

[20] The concept that some colloids have sufficientenergy to escape the secondary minimum, whereas the restdo not, arises from several physical phenomena. One lies inthe distribution of velocities possessed by colloids in a fluidat thermal equilibrium [Hahn and O’Melia, 2004]. Max-well’s kinetic theory states that whereas a colloid popula-tion possesses on average 1=2 kBT (kB¼Boltzmann’sconstant and T¼ absolute temperature) of kinetic energyfor each degree of translational freedom due to thermalenergy, a distribution of energies (velocities) is observedamong a population that can be approximated by the Max-well distribution. If the depth of the secondary minimum iswithin the range of energies possessed by a ‘‘homogeneouspopulation’’ of colloids, a defined portion of the colloidswill have sufficient energy to escape the secondary mini-mum. To be precise, only the colloidal kinetic energy thatacts normal to the collector surface will effect an escapefrom the secondary minimum, kinetic energy acting paral-lel to the colloid surface will simply result in translation ofthe colloid along the collector surface. When referring tocolloid kinetic energy, we mean the portion of kineticenergy that acts normal to the collector surface. However,applying this concept is problematic because the Maxwelldistribution is an instantaneous description and little isknown about how quickly the kinetic energy of individualcolloids changes with time. Therefore, tracking individualcolloids using this paradigm would not be practical.

[21] Another explanation involves the heterogeneity ofsurface potential and/or diameter of the colloid population.The secondary minimum experienced by a colloid is a

function of both colloid surface potential and colloid diam-eter. Heterogeneities in either of these parameters willresult in a distribution of secondary minimum depths expe-rienced by a population of colloids. Because all of the col-loids, regardless of diameter, have the same averagekinetic energy due to Brownian motion, those that experi-ence a secondary minimum depth that is less than Ek,ave

will not be retained permanently by a collector surface.Therefore, given a homogeneous population of collectorsand a heterogeneous colloid population with respect tocharge and/or diameter, the colloid population can be splitinto two groups; those with a kinetic energy greater than thesecondary minimum they experience (reversible retention)and those whose kinetic energy is less than the secondaryminimum they experience (‘‘irreversible’’ retention). In thispaper, we use the term ‘‘two-population paradigm’’ todescribe this type of behavior.

[22] To illustrate this concept graphically, consider a hy-pothetical, heterogeneous population of colloids interactingwith like-charged, homogeneous granular media. If the col-loid population has a distribution of surface potentials(individual colloids are homogeneous with respect to sur-face charge) as illustrated in Figure 2a, a distribution ofsecondary minimum depths will result because the depth ofthe secondary minimum experienced between an individualcolloid and a collector is dependent on the surface potentialof that colloid (Figure 2b). If a zeta potential of �15 mVcorresponds to a secondary minimum depth that is equal tothe average kinetic energy of the colloids acting normal tothe collector surface (e.g., 0.5 kBT), the colloids with a zetapotential more negative than �15 mV will be retainedreversibly (�2min< 0.5 kBT) and the rest of the colloids will

Figure 2. Zeta-potential distribution of (a) a hypothetical colloid population and (b) the correspondingdistribution of secondary minimum depths that result when the population of colloids interacts with anegatively charged collector surface. If a zeta potential of �15 mV (vertical bar in Figure 2a)) corre-sponds to a secondary minimum depth that is equal to the average kinetic energy of the colloids, colloidswith a zeta potential more negative than �15 mV will be retained reversibly and the rest of the colloidswill be retained irreversibly. The vertical bar in Figure 2b represents the same dividing line, butexpressed as the average kinetic energy of the colloids acting normal to the collector surface. Figure 2bis intended for illustration purposes only, the actual shape of the curve and the values of the secondaryminimum would depend on numerous factors, including the ionic strength and the surface potential ofthe collectors.


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be retained irreversibly (�2min> 0.5 kBT). This concept isdisplayed more directly in Figure 2b, where the vertical baris the average kinetic energy of the colloids (Ek,ave). Col-loids experiencing a �2min greater than Ek,ave will beretained irreversibly while the rest will be able to escapethe secondary minimum.

[23] Colloid surface potential and size heterogeneity arelargely unrecognized because most techniques that measurethese parameters report average values. However, bothBayer and Sloyer [1990] and Dong [2002] found monoclo-nal bacterial populations to have a distribution of electro-phoretic mobilities (a measure of surface potential). Evenmanufactured microspheres are not perfectly homogeneousin terms of charge and size [Dong, 2002; Wu et al., 2005].

[24] As an example of colloid population heterogeneityaffecting transport behavior, Simoni et al. [1998] observedthat the average electrophoretic mobility increased and thesize of Pseudomonas sp. decreased upon passage through acolumn packed with clean sand. This makes sense in thecontext of the proposed model because larger cells and oneswith a less negative surface potential would experience alarger secondary minimum and would be more likely tobecome permanently trapped in the secondary minimum.Cells that passed through the first column were removed at amuch lower rate in an identical second column, confirmingthat the first column fractionated the cell population.

[25] To define the two-population paradigm mathemati-cally, it is necessary to split the colloid population into twogroups: (1) those which, by virtue of their physical andchemical properties, will be retained ‘‘irreversibly’’ when acolloid is intercepted by a secondary minimum, and (2)those which will always be re-entrained after capture by asecondary minimum. This is accomplished by the variablefir, which is defined as the fraction of the total colloid popu-lation that will be retained irreversibly when captured by asecondary minimum. The variable fir is not the fraction ofcolloids that may be retained irreversibly at any given time,which would change with distance as irreversibly retainedcolloids are removed from the aqueous population. Instead,under the two-population paradigm, it is the fraction of theoverall colloid population that experiences a secondaryminimum that is greater than the kinetic energy of the colloidand will therefore be retained irreversibly should they beintercepted by a secondary minimum. Cir is the time and dis-tant dependent concentration of colloids in solution that willbe retained ‘‘irreversibly’’ when intercepted by a secondaryminimum surrounding a collector and Cr is the concentrationof colloids that will be retained temporarily (reversibly)when interception occurs. The variable fir defines the initialconcentration of these two fractions as follows:

Cr;0 ¼ 1� firð ÞC0 (3)

and

Cir;0 ¼ firC0; (4)

where C0 is the total initial concentration of colloids. Thetime-dependent equation for Cr is :

@Cr

@t¼ �v

@Cr

@xþ D

@2Cr

@x2� kf Cr þ

�b

"krSr; (5)

where t is time, x is distance, v is the average pore velocity,D is the longitudinal dispersion coefficient of the colloids,�b is the porous media bulk density, kr is the rate at whichthe reversibly retained colloids are re-entrained, and Sr isthe concentration of colloids reversibly retained by the po-rous media. Similarly,

@Cir

@t¼ �v

@Cir

@xþ D

@2Cir

@x2� kf Cir: (6)

[26] The same deposition rate coefficient applies to bothsubpopulations but there is no re-entrainment term for theirreversibly retained subpopulation. The corresponding rela-tionships for the concentrations of reversibly and irreversi-bly retained colloids are:

@Sr

@t¼ kf Cr �

�b

"krSr (7)

and

@Sir

@t¼ kf Cir: (8)

[27] In simpler systems such as packed columns, kr and firare the only unknowns because kf is defined by �o, and D istypically estimated from conservative tracer transport behav-ior. For systems dominated by secondary minimum deposi-tion, the parameter fir should be directly related to the depthof the secondary minimum (�2min) and inversely related tothe average kinetic energy of the colloids (Ek,ave) or statedmathematically, fir / �2min/ Ek,ave. Conversely, kr could beexpected to behave in an opposite manner, i.e., kr / Ek,ave /�2min. This hypothesis assumes that deposition in the pri-mary minimum by ‘‘wedging’’ [Johnson et al., 2007a] orsurface heterogeneity is not significant relative to retentionin the secondary minimum. However, alternate explanationsto increasing retention with increasing secondary minimumare possible. As the secondary minimum increases withincreasing ionic strength, the barrier to deposition in the pri-mary minimum concurrently decreases, increasing the possi-bility of deposition in the primary minimum. The model asoutlined above cannot distinguish between irreversible reten-tion in the secondary minimum and deposition in the pri-mary minimum. In fact, with regard to fitting the model toexperimental data, it does not matter whether irreversiblyretained colloids are trapped in the secondary minimum orthe primary minimum, both mechanisms would be capturedby the parameter fir.

[28] This framework also assumes that colloids trapped inthe secondary minimum are not free to jump from collectorto collector via a continuous secondary minimum and even-tually exit the column to an appreciable extent. Althoughthe translation of colloids from collector to collector via thesecondary minimum has not been directly observed, Kuznarand Elimelech [2007] used a flow-cell packed with a singlelayer of glass beads to visually observe colloids, presumablytrapped in the secondary energy minimum, translating alongthe surface of individual glass beads and accumulating inthe rear flow-stagnation zone of the collectors.

[29] At this point, it is instructive to compare the re-entrainment behavior of the two-population paradigm with


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that used in other models that include re-entrainment. Anymodel that incorporates re-entrainment of previouslyretained colloids must define the behavior of re-entrainedcolloids with respect to subsequent interceptions. Mostmodels incorporate a definition of colloid-surface interac-tion that is based on collector heterogeneity, sometimesreferred to as the two-site paradigm. In a two-site para-digm, when a negatively charged colloid approaches a het-erogeneous collector (e.g., a quartz sand particle covered inpatches of ferric oxyhydroxide) it has a specified probabil-ity (fir) of hitting a positively charged site and attachingpermanently. If the colloid interacts with a negativelycharged site, it will not be retained permanently, in whichcase it will be re-entrained and continue its journey. In atwo-site paradigm, the detaching colloid will have the sameprobability as before of attaching permanently during asubsequent collision event. This behavior would also resultfrom heterogeneous colloids (in the sense of individual col-loids having positively and negatively charged portions)that must strike a homogeneous collector with a certain ori-entation to attach. It could also occur when both the col-loids and collectors are of the same charge, but themagnitude of the charge of the collectors varies such thatsome collectors have a very deep secondary minimum,from which, colloids are unlikely to escape while other col-lectors have a relatively shallow secondary minimum thatwill not retain colloids. The commonality in each of theseexamples is the subsequent deposition behavior of colloidsthat have been re-entrained. Examples of authors utilizing aparticle-tracking model that uses the two-site paradigminclude Scheibe and Wood [2003] and Li et al. [2004,2005]. All rate based nonparticle-tracking colloid transportmodels simulate the two-site paradigm because it is notpossible to track individual colloids unless they incorporatetwo colloid populations, one that attaches permanently andone that does not. The two-site and two-population para-digms are contrasted in Figure 3. The two-population para-digm is used in this paper unless otherwise noted.

[30] To re-iterate, this conceptual model has two newaspects. The first is that all interceptions result in at leasttemporary retention due to interaction of the colloid witheither the primary or the secondary minimum. Even inunfavorable conditions with no secondary minimum, col-loids briefly interact with the surface due to their finite ve-locity. This eliminates the need for the parameter � andallows the calculation of kf directly from CFT so that kf

is no longer a fitting parameter. The second is that thetwo-population paradigm is used to characterize colloidre-entrainment rather than the commonly used two-siteparadigm.

3. Methods

3.1. Data Set Selection

[31] To test the model, the results of published colloidtransport experiments performed in a well defined and con-trolled manner were simulated. High-quality data sets acquiredunder unfavorable deposition conditions were selected. Addi-tionally, information on colloid and collector zeta potentialswas required so �2min could be calculated. Lastly, data setswere selected that had variations in parameters likely to affectcolloid transport, such as ionic strength and fluid velocity so

that the correlation between fitting and physical parameterscould be observed.

3.2. DLVO Calculations

[32] Measured zeta potentials were used to calculate�2min. DLVO calculations assumed constant potential andsphere-sphere geometry [Hogg et al., 1966] along withGregory’s [1981] assumption of retarded van der Waalsforces. The Hamaker constant was set to 6.5 � 10�21 J formicrobe-water-quartz systems [Simoni et al., 1998; Trues-dail et al., 1998] and 1 � 10�20 J for latex microsphere-water-glass/quartz systems [Spielman and Fitzpatrick,1973]. Bacterial diameter was assumed to be 1mm unlessotherwise noted.

3.3. Single-Collector Contact Efficiency (go)Calculations

[33] The correlation equation developed by Tufenkji andElimelech [2004a] was used to calculate �o using the rele-vant published column parameters (e.g., media diameter,pore-water velocity, porosity). A density of 1070 kg m�3

[Wan et al., 1995] was used for bacteria and 1055 kg m�3

[Harvey et al., 2008] for latex microspheres. The fluid den-sity, viscosity, and temperature were set to 1000 kg m�3,0.001 kg m�1 s�1 and 293 K, respectively.

3.4. Computer Modeling

[34] A transport code called FlowTrack [Scheibe, 2005],that was modified in accordance with equations (3)–(8),was used to simulate the data. FlowTrack is a one-dimen-sional discrete particle-tracking model that simulatesadvection and dispersion using a conventional random-walk method [Scheibe and Wood, 2003]. The forward dep-osition rate, kf, was calculated using equation (1) for eachmodel run. FlowTrack was coupled with UCODE [Poeteret al., 2005], which adjusted fir, kr, and D to fit the model

Figure 3. Illustration of the difference in re-entrainmentbehavior between the two-site and the two-population para-digms. In the two-site paradigm, when a re-entrained col-loid interacts with a subsequent collector, it has a 1�firprobability of being re-entrained. Colloids that have beenre-entrained in the two-population paradigm will never beretained permanently by virtue of their surface properties.F.R. is ‘‘fraction re-entrained’’.


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output to the experimental breakthrough curves. UCODEadjusts parameters using a modified Gauss-Newton methodthat minimizes the least-squares error between model out-put and experimental data with respect to parameter values.The model output curves are not smooth due to the statisti-cal nature of the program; a limited number of particlesand finite time steps must be used due to processing powerlimitations.

4. Results

[35] Chen and Zhu [2004] performed a series of bacteriatransport experiments by injecting Esherichia coli andPseudomonas fluorescens cells into columns packed with0.2–0.5 mm diameter quartz sand over a range of ionicstrengths. They modeled these data using an ‘‘equilibrium-kinetic, two-region’’ model that contained five variables ;four were allowed to vary for each set of conditions, thefifth variable was adjusted for the bacteria type but not forionic strength. We simulated the Pseudomonas fluorescensdata of Chen and Zhu [2004] using our much simplermodel (Figure 4a). The first step of the modeling procedurewas to calculate the deposition rate coefficient using con-ventional CFT, assuming all interceptions result in at leasttemporary retention. A value of �o¼ 0.021 was calculatedusing the relevant column parameters. The correspondingdeposition rate coefficient (kf) of 5.05 h�1 was calculatedusing equation (1) and was held constant for all ionicstrengths. The fraction of colloids that are retained irrever-sibly (fir) and the first-order re-entrainment rate (kr) werethen adjusted to fit the four breakthrough curves (C/Co, oreffluent colloid concentration (C) normalized by the influ-ent colloid concentration, versus time) using UCODE. Avalue of 16.4 cm2 h�1 for D (dispersivity¼ 2 cm, v¼ 8.2cm h�1, or 1.97 m d�1) gave the best overall fit as deter-mined by UCODE.

[36] The fitted values of fir and kr exhibit a correlationwith the calculated depth of the secondary minimum (Fig-ure 4b and Table 1). We hypothesize that the fraction of

P. fluorescens cells that are retained irreversibly (fir)increases as the secondary minimum increases becausefewer cells have the required escape energy as the secondaryminimum depth increases. Similarly, the first-order re-entrainment rate (kr) is larger at smaller secondary minimumdepths because the ratio Ek,ave /�2min is greater (i.e., kr /Ek,ave /�2min). Alternatively, the positive correlation betweenfir and ionic strength could also be explained by increasingdeposition in the primary minimum as the energy barrier todeposition in the primary minimum decreases. However,this would not explain the observed correlation between kr

and the depth of the secondary minimum.[37] There are only three data points in Figure 4b

because, at an ionic strength of 0.1 M, DLVO calculationspredict no barrier to primary minimum deposition (henceno secondary minimum). However, the substantial ‘‘tail-ing’’ in the breakthrough curve of the 0.1 M experiment(Figure 4a) indicates that reversible deposition was occur-ring, implying the presence of a secondary minimum. Thetrends observed between fir, kr, and �2min were also notedbetween fir, kr, and the zeta potential of the cells for all fourionic strengths (Figure 5). The correlation between fir, kr,and �2min are dependent on the calculated values of �2min

and would change if the �2min values were in error. How-ever, similar trends would be expected even if improvedcalculations changed the relative values of �2min.

[38] The correlation coefficient calculated by UCODEbetween the parameters fir and kr (at any one ionic strength)

Figure 4. (a) Effluent breakthrough curves of P. fluorescens at four ionic strengths [Chen and Zhu,2004] and model fits and (b) resulting model parameters fir (�, the fraction of colloids that are retainedirreversibly) and kr (�, first-order re-entrainment rate) plotted against the calculated depth of the second-ary minimum. The arrows in Figure 4b indicate the proper axis for each curve. There are only threepoints because at an ionic strength of 0.1 M, DLVO calculations predict no secondary minimum. Thepoints in Figure 4b correspond to ionic strengths of 0.001, 0.005, and 0.01 M (from left to right).

Table 1. Measured Zeta Potential of the P. fluorescens Cells as aFunction of Ionic Strength [Chen and Zhu, 2004] and Fitting Pa-rameters Used to Generate Model Fits in Figure 4

Ionic Strength (M) Zeta Potential (mV) fir 6 � kr (h�1) 6�

0.001 �62.6 0.494 6 0.003 15.4 6 0.670.005 �42.3 0.731 6 0.002 7.54 6 0.200.01 �29.6 0.856 6 0.001 5.06 6 0.140.1 �11.7 0.937 6 0.002 2.05 6 0.10


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for the Chen and Zhu’s data was low (0.25) indicating thatthe two parameters are likely unique and can be determinedindependently. Correlation coefficients greater than 0.95indicate that there may not be enough information in theexperimental observations to estimate the parameters indi-vidually [Poeter et al., 2005]. However, the parameters kr

and fir appear to be inversely and linearly related as theionic strength changes (Figure 6). This implies that kr isrelated to fir, which could further simplify the model.

4.1. Forward Modeling

[39] The E. coli data from Chen and Zhu [2004] werechosen to test the predictive (forward-modeling) capabilitiesof the model because the physical conditions (e.g., colloidand collector size, fluid velocity, etc.) were the same asthose in the P. fluorescens experiments. Whereas the physi-ochemical conditions were the same, the zeta potentials and

breakthrough curves of the E. coli were different from theP. fluorescens. The correlation between cell zeta potentialand fir and kr observed in the P. fluorescens data was usedto predict the fir and kr values of the E. coli cells based onthe measured zeta-potential values. The deposition rate, kf,was held constant for both species of bacteria as the diame-ter, density, and Hamaker constants of the cells wereassumed to be equal in the absence of specific informationto the contrary (for the given conditions, a 50% increase incell diameter would result in a 12% decrease in �o). Whilethe match between the model and experimental break-through curves for the E. coli is quite good for a predictivemodel (Figure 7), it is not expected that the correlationwould hold for colloids that are not as similar as these twospecies of Gram-negative bacteria or for different collectormedia, solution chemistry or flow velocities. However,these results suggest that the model has potential predictivecapabilities based on colloid zeta-potential measurements.

4.2. Retention Profile Modeling

[40] Tufenkji et al. [2003] and Johnson et al. [2003]demonstrated that the pattern of colloid deposition alongthe length of a column could be used to place additionalconstraints on a transport model. Subsequently, Tufenkjiand Elimelech [2005] examined the transport of 3 mmmicrospheres through columns packed with glass beadsover a range of ionic strengths to investigate the ‘‘mecha-nisms and causes of deviation from the classical CFT in thepresence of repulsive Derjaguin-Landau-Verwey-Overbeek(DLVO) forces’’ and used the pattern of colloid depositionto constrain their modeling efforts. Tufenkji and Elimelechdeveloped a ‘‘Dual Deposition Model’’ (DDM) to simulatethe data. The DDM model consists of a bimodal distributionof deposition rates that are described by kfast and kslow, themean ‘‘fast’’ and ‘‘slow’’ deposition rates; �fast and �slow, thecorresponding standard deviations that describe the Gaussiandistribution associated with each deposition rate; and ffast

and fslow, the fraction of the colloid population associated

Figure 5. Variables fir (�) and kr (�) resulting frommodel fitting of breakthrough curves plotted as a functionof cell zeta potential for P. fluorescens at all four ionicstrengths (0.001, 0.005, 0.01, and 0.1 M, from left to right).

Figure 6. The first-order release rate coefficient kr plottedas a function of fir for P. fluorescens at four different ionicstrengths (0.001, 0.005, 0.01, and 0.1 M, from left to right).

Figure 7. Effluent breakthrough curves of E. coli at fourionic strengths [Chen and Zhu, 2004] and model predictionsgenerated by predicting kr and fir from measured zeta poten-tials of the cells and correlations observed in Figure 5.


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with each deposition mode. The DDM model as imple-mented by Tufenkji and Elimelech did not have a re-entrain-ment term and therefore could not simulate the tailingobserved in the breakthrough curves.

[41] The additional constraint of the measured depositionprofile contained in this data set provided an important vali-dation of the two-population paradigm over the two-siteparadigm under unfavorable deposition conditions. Wemodeled this data by calculating a value of 17.7 h�1 for kf

based on system parameters noted by Tufenkji and Elime-lech. The fir and kr values were determined by fitting onlythe breakthrough curves using UCODE and FlowTrack,which resulted in the model fits displayed in Figure 8a; theretention profiles (Figure 8b) resulted directly with no fur-ther fitting. The two-population paradigm was able to cap-ture the retention profile of the colloids even though theprofile was not used in the fitting procedure. Model param-eters are given in Table 2. The units for the normalizedretained microsphere concentration are the same as used by

Tufenkji and Elimelich [2005], which were derived bydividing the retained microsphere concentration (number ofmicrospheres/g of beads) by the influent concentration ofmicrospheres (number of microspheres/cm3).

[42] Also shown in Figure 8b is the fit to the 3 mM datausing the two-site paradigm, which fails to capture theretained profiles even though it can simulate the break-through curves (data not shown). The two-site paradigmgives the same slope for the retention profile as conven-tional CFT utilizing � and no re-entrainment as describedin Tufenkji and Elimelech [2004b]. Both Tufenkji and Eli-melech [2004b] and Li et al. [2004] observed that conven-tional CFT could not match measured retention profileswhen deposition conditions are unfavorable. The two-popu-lation paradigm produces a retained profile with a steeperslope than the two-site model because when a colloid is re-entrained by virtue of the properties of the colloid, all sub-sequent interception events involving this specific colloidwill result in only temporary retention. Therefore, thedownstream concentration of attached colloids will be lessrelative to the two-site paradigm where re-entrained col-loids have a chance of being retained permanently. For the3 mM data, the two-site paradigm yielded a value offir¼ 0.022, which is only about a third as large as that pro-duced by the two-population paradigm (fir¼ 0.062) but pro-duced similar values for kr.

4.3. Effect of Fluid Velocity

[43] Li et al. [2005] investigated the effect of hydrody-namic drag on colloid deposition and re-entrainment. Theyperformed experiments with 1.1 mm carboxylated latexmicrospheres and clean quartz sand at an ionic strength of0.006 M with pore-water fluid velocities of 2, 4, and 8 md�1. They modeled the data using the traditional two-siteparadigm and adjusting kf (adjustable �), kr, and fir. Li et al.concluded that hydrodynamic drag affected all three modelparameters; decreasing kf and fir and increasing kr withincreasing fluid velocity (although fir decreased only slightlybetween 2 and 4 m d�1 and was relatively unchanged as thevelocity increased from 4 to 8 m d�1).

[44] Rather then allowing the deposition rate (kf) to be afitting parameter, we assumed that fluid velocity does notaffect the deposition rate in any manner other than pre-dicted by CFT. We modeled Li et al.’s data [2005] by cal-culating kf using CFT for each fluid velocity; otherwise,the data were fit as described in the Methods section. Fig-ure 9a shows the model fits for the three fluid velocities andTable 3 contains the model variables.

[45] The model fits the experimental data reasonablywell from 0 to 4.5 pore volumes, capturing roughly 99.9%of the eluted microspheres. However, in Li et al. [2005],

Figure 8. (a) Model fit to breakthrough curves and (b)retention profiles that resulted from no further fitting (i.e.,the retention profiles were not considered when fitting themodel to the data) from Tufenkji and Elimelech’s [2005]microsphere transport experiments using the two-popula-tion paradigm. The dashed line in Figure 8b is the two-siteparadigm for the 3 mM (ionic strength) data.

Table 2. Fitting Parameters Used to Generate Model Curves inFigure 8 (Microsphere Transport Experiment [Tufenkji and Elime-lech 2005])

Ionic Strength (M) fir 6� kr (h�1) 6� D (cm)

0.003 0.062 6 0.003 158 6 4.8 0.050.030 0.364 6 0.003 166 6 3.4 0.050.100 0.628 6 0.002 154 6 4.6 0.050.300 0.910 6 0.001 159 6 5.3 0.05


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logarithmic plots reveal extended tailing at C/Co levelsbetween 0.001 and 0.0001 but our model quickly goes tozero and does not capture the extended tailing at very lowconcentrations.

[46] Figure 9b contains the model parameters fir and kr

plotted as a function of fluid velocity. While not obvious inFigure 9a, the leading edge of the breakthrough at 2 m d�1

has a sharp peak, after which the concentration declineswith time. Because kr is mostly a function of the slope ofthe leading and following edge of the breakthrough, the fit-ting routine resulted in an abnormally high value for kr at 2m d�1 relative to the other velocities. Because of this, kr

does not have the expected trend of decreasing value with

decreasing fluid velocity for the 2 m d�1 point. However,fir, which is primarily a function of the amount of break-through, has a strong inverse relationship with fluid velocityas expected. One interpretation of this inverse correlationbetween fir and pore velocity is that the hydrodynamic forceon colloids retained in the secondary minimum increases(thus increasing the kinetic energy of the colloids) as fluidvelocity increases. If higher fluid velocities result inincreased kinetic energy (Ek,ave) of the colloids retained inthe secondary minimum, more colloids would possess suffi-cient energy to escape the secondary minimum, resulting inthe observed correlation between fir and fluid velocity.Because the ionic strength was kept constant for theseexperiments, �2min did not change and was removed as avariable.

[47] Our modeling of the Li et al. [2005] data supports thehypothesis that fluid velocity affects the quantity of colloidsretained irreversibly in a consistent manner. Comparing ourmodeling results of the other two data sets examined in thispaper also gives insight into the effect of fluid velocity. Fora given depth of the secondary minimum the microspheresof Tufenkji and Elimelech [2005] had substantially lowervalues of fir and higher values for kr than the P. fluorescensused by Chen and Zhu [2004] (Figure 10).

Figure 9. (a) Microsphere breakthrough curves andmodel fits for three different fluid velocities and (b) result-ing model parameters fir (�, fraction of colloids that areirreversibly retained) and kr (�, first-order re-entrainmentrate) for the Li et al. [2005] data plotted as a function ofpore-water velocity. The deposition rate (kf) was recalcu-lated at each velocity using CFT (see Table 3 for values).

Table 3. Calculated Values of kf and Fitting Parameters Used toGenerate the Model Curves for the Li et al. [2005] Data (Figure 9)

Pore Velocity(m d�1) kf (h�1) fir 6� kr 6 � (h�1) D (cm)

2 3.58 0.942 6 0.005 647 6 365 0.064 4.32 0.613 6 0.008 408 6 71 0.068 5.29 0.341 6 0.001 905 6 349 0.06

Figure 10. Model parameters (a) fir and (b) kr for P. fluo-rescens and 3 mm microspheres plotted against the calcu-lated secondary minimum depth. The pore water velocitywas 1.97 m d�1 for the P. fluorescens experiments and 19.4m d�1 for the microsphere experiments.


2962

[48] We believe the lower fir values and higher kr valuesfor the microspheres relative to the P. fluorescens are due tothe substantially higher fluid velocity used in the microsphereexperiments, our reasoning is as follows. In the absence ofan energy input such as vibration or cell motility, the kineticenergy of the cells and the microspheres would be equalbecause all of the experiments were done at room tempera-ture and the thermal energy of a colloid is independent ofcolloid diameter or mass (Ek ¼ 1=2 kBT). The depths of thesecondary minimum experienced by the microspheres weredeeper than those of the P. fluorescens because the Hamakerconstant of the latex microspheres is thought to be about50% larger than that of bacteria cells and the diameter of themicrospheres was three times that of the P. fluorescens cells(�2min / dc). The fact that the microspheres experience adeeper secondary minimum would imply that the micro-spheres should exhibit higher fir values and lower kr valuesin the absence of mitigating factors. However, the pore-watervelocity in the microsphere experiments was approximately10 times higher than in the P. fluorescens experiments (19.4versus 1.97 m d�1) and the diameter of the microspheres waslarger (3 versus 1 mm). Both of these factors would result inhigher colloid kinetic energy before an interception eventand larger hydrodynamic forces on the trapped microspheres(imparting more kinetic energy to the microspheres) relativeto the bacteria cells, accounting for the higher rate of escapeand lower retention of the microspheres for a given depth ofthe secondary minimum. Although this analysis is compli-cated by the fact that P. fluorescens are typically motile, itappears that the increased hydrodynamic forces on themicrospheres are a bigger factor than any motility of the P.fluorescens.

5. Conclusions

[49] The results of these modeling exercises show thatthe elimination of attachment efficiency (�) and introduc-tion of a re-entrainment term based on the two-populationparadigm produces a simple model that captures the trans-port behavior of colloids when unfavorable attachment con-ditions exist. The model also provides a heuristic tool forunderstanding the physical processes controlling colloidtransport. In all cases a good fit to the data was obtainedand the resulting model parameters displayed correlationswith physical parameters thought to control transport. Forexample, the fraction of colloids retained irreversibly (fir)increased as the depth of the secondary minimum increasedand as the pore velocity decreased. The rate at which col-loids were re-entrained (kr) increased as the secondary min-imum depth decreased.

[50] Using conventional CFT coupled with the two-siteparadigm, which assumes re-entrained colloids may subse-quently be retained irreversibly, produces a retention profilewith a slope that is lower than that observed experimentallywhen conditions of unfavorable attachment exist. For thefirst time, retention profiles and the associated breakthroughcurves of experiments conducted under unfavorable attach-ment conditions have been successfully simulated using asimple model with only two fitting parameters. This resultsupports the assumption that the two-population paradigmis representative of colloid re-entrainment behavior when

conditions of secondary minimum retention dominate col-loid transport.

[51] Although the idea of attachment efficiency is appeal-ing as a simple way to quantify the retention of colloidsunder unfavorable attachment conditions, � is an empiricalconcept that merely serves to confound the determination ofkf, kr, and fir, variables that represent physical processes.When � is not used, kf becomes a calculated parameter,which decouples kf from kr and fir during modeling exer-cises resulting in better estimates of these critical parame-ters. The use of � has inhibited development of simplerelationships between kr and fir and quantifiable physical pa-rameters such as zeta potential and flow velocity.

[52] Viewing the entrapment of colloids in a secondaryminimum as a balance between the attractive force of thesecondary minimum and the kinetic energy of the colloidsproduces additional benefits. It acknowledges that colloidsretained in the secondary minimum (fir) are not truly irre-versibly attached. External inputs of energy to a system,such as increasing flow velocity, vibration, or heat, mayresult in additional releases of colloids by increasing theirkinetic energy. Abudalo [2006] showed that Cryptospori-dium parvum oocysts deposited in a column tightly packedwith clean sand under unfavorable deposition conditionscould be released by mechanically vibrating the column.Abudalo also performed multiple experiments to show thatthe oocysts deposited in the column were trapped in thesecondary minimum of the sand surfaces and were notstrained. Minor changes in solution chemistry, e.g., ionicstrength, surfactant concentration, or NOM concentrationcan also result in the release of trapped colloids, presum-ably by changing the DLVO forces that determine thedepth of the secondary minimum [Chen and Zhu, 2004;Franchi and O’Melia, 2003; Hahn et al., 2004; Lenhartand Saiers, 2003; Redman et al., 2004; Tosco et al., 2009].If the released colloids are pathogens trapped in a riverbankfiltration system or a rapid sand filter, contamination ofdrinking water may result. Complications such as cell mo-tility, which increases the kinetic energy of a cell andpotentially its ability to escape a secondary minimum,could be accommodated by simply changing fir and kr.

[53] A system that would not be well characterized bythe presented model is one with heterogeneity between col-lectors. For example, a system could be envisioned that hastwo types of unfavorable interactions due to heterogeneitybetween collectors in which: (1) a substantial secondaryminimum exists on some collectors, and (2) other collectorshave a repulsive barrier to deposition in the primary mini-mum but no substantial secondary minimum. In such a sys-tem, the re-entrainment rate would likely be different forthe two types of interactions, requiring a more sophisticatedmodel. In all of the preceding examples, the elimination of� is still valid (fixed kf), but the re-entrainment behaviorwould be more complicated requiring aspects of both thetwo-population and two-site paradigms.

[54] In conclusion, this model is conceptually simplerand has fewer fitting parameters than most models thatattempt to simulate complicated retention/re-entrainmentbehavior and was able to simulate a diverse data set. Byeliminating �, the deposition rate can be calculated, leavingonly three variables in the model. The dispersion coeffi-cient can be estimated from tracer behavior reducing the


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number of fitting parameters to two. These two parameters,fir and kr, appear to be related to physical processes control-ling colloid transport. Lastly, the incorporation of the two-population paradigm into the model resulted in accuratesimulations of colloid retention profiles and presumably,better estimates of fir. The resulting model serves as a heu-ristic tool in understanding more fully the physics of col-loid retention in porous media.

[55] Acknowledgments. This material is based upon work supportedby a National Research Council Post Doctorial Fellowship. Any opinions,findings, and conclusions or recommendations expressed in this materialare those of the author(s) and do not necessarily reflect the views of theNational Research Council. The authors would also like to thank theauthors that shared their data that was the basis for the modeling in thispaper.

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