WATER RESOURCES RESEARCH, VOL. 48, W09509, doi:10.1029/2012WR012468, 2012

Modeling colloid and microorganism transport and releasewith transients in solution ionic strength
Scott A. Bradford,1 Saeed Torkzaban,2 Hyunjung Kim,3 and Jiri Simunek4
Received 30 May 2012; revised 17 July 2012; accepted 20 July 2012; published 7 September 2012.

[1] The transport and fate of colloids, microorganisms, and nanoparticles in subsurface
environments is strongly influenced by transients in solution ionic strength (IS). A
sophisticated dual-permeability transport model was modified and a theory was developed
to mechanistically account for the transport, retention, and release of colloids withtransients in IS. In particular, colloid release in the model was directly related to the balanceof applied hydrodynamic and resisting adhesive torques that determined the fraction of thesolid surface area that contributed to colloid immobilization (Sf). The colloid sticking
efficiency (a) and Sf were explicit functions of IS that determined the rates of colloid
interaction with the solid, immobilization on the solid, colloid release from the solid andback into the bulk aqueous phase, and the maximum amount of colloid retention. The
developed model was used to analyze experimental transport and release data withtransients in IS for 1.1 and 0.11 mm latex microspheres, E. coli D21g, and coliphagefX174. Comparison of experimental values of Sf (IS) with predictions based on meaninteraction energies indicated that predictions needed to account for the influence ofphysical and/or chemical heterogeneity on colloid immobilization. This was especially true
for smaller colloids because they were more sensitive to microscopic heterogeneities
that produced mainly irreversible interaction in a primary minimum and greater
hysteresis in Sf (IS) with IS. Significant deviations between experimental and predictedvalues of a(IS) were observed for larger colloids when hydrodynamic forces were notaccounted for in the predictions. A sensitivity analysis indicated that colloid releasewith IS transients was not diffusion controlled, but rather occurred rapidly and with lowlevels of dispersion. The calibrated model provided a satisfactory description of theobserved release behavior for a range of colloid types and sizes and a general theoreticalfoundation to develop predictions for the influence of solution chemistry on thetransport, retention, and release of colloids.
Citation: Bradford, S. A., S. Torkzaban, H. Kim, and J. Simunek (2012), Modeling colloid and microorganism transport andrelease with transients in solution ionic strength, Water Resour. Res., 48, W09509, doi:10.1029/2012WR012468.

1. Introduction[2] Release of colloids from soils may occur as a resultof diffusion [Ryan and Gschwend, 1994; Simoni et al.,
1998; Dong et al., 2002; Shen et al., 2007], an increase inhydrodynamic forces [Hubbe, 1984; Sharma et al., 1992;
Torkzaban et al., 2007; Bradford et al., 2011a], and areduction in adhesive forces [Ryan and Gschwend, 1994;
Bergendahl and Grasso, 1999; Lenhart and Saiers, 2003;
Torkzaban et al., 2010]. The detachment coefficient is
1

U.S. Salinity Laboratory, ARS, USDA, Riverside, California, USA.
CSIRO Land and Water, Glen Osmond, South Australia, Australia.
3
Department of Mineral Resources and Energy Engineering, Chonbuk
National University, Jeonju, South Korea.
4
Department of Environmental Sciences, University of California,
Riverside, California, USA.
2

Corresponding author: S. A. Bradford, U.S. Salinity Laboratory,
ARS, USDA, 450 West Big Springs Rd., Riverside, CA 92507, USA.
(scott.bradford@ars.usda.gov)©2012. American Geophysical Union. All Rights Reserved.
0043-1397/12/2012WR012468

frequently considered to be a diffusion controlled processthat is a function of the diffusion coefficient and the boundary
layer thickness [Ryan and Gschwend, 1994; Ryan andElimelech, 1996]. However, this approach does not accountfor the strength of the adhesive interaction and the removal of
colloids due to hydrodynamic mechanisms such as rolling,
lifting, or sliding [Bergendahl and Grasso, 1998, 1999; Tsaiet al., 1991]. A constant value of the colloid detachment
coefficient is therefore unlikely to account for temporalchanges in colloid release that will commonly occur as aresult of transients in solution chemistry and flow rate during
infiltration and recharge events, freshwater intrusion, dilutionof contaminant plumes, and near injection and extraction
wells.
[3] It is well accepted that changes in solution chemistry(IS, pH, cation exchange, and concentration of organic matter)
will impact the electric double layer thickness and/or the zeta
potential and thereby alter electrostatic interactions [Ryan andElimelech, 1996; Grolimund and Borkovec, 2006]. Chemical
perturbations are often used to create interaction energies
between colloids and surfaces in order to induce particle
attachment [Goldshmid et al., 1973; Nocito-Gobel andW095091 of 10

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Tobiason, 1996; Shiratori et al., 2007; Tosco et al., 2009]or detachment [Bales et al., 1991; McDowell-Boyer, 1992;
Ryan and Gschwend, 1994; Nocito-Gobel and Tobiason,
1996; Roy and Dzombak, 1996; Grolimund et al., 2001;
Lenhart and Saiers, 2003; Shiratori et al., 2007; Tosco et al.,2009; Bradford and Kim, 2010]. Ryan and Gschwend [1994]
reported that changes in solution chemistry must be great
enough to eliminate the energy barrier in order to successfully
achieve detachment. A critical salt concentration has beenreported for particle release [Khilar and Fogler, 1984;
Lenhart and Saiers, 2003]. Tosco et al. [2009] reported thatchemical perturbations that induced detachment were wellcorrelated with the disappearance of secondary minima underunfavorable attachment conditions.
[4] Hysteresis in the amount of colloid/microbe retention
and release has been observed with changes in the IS of theeluting solution [Torkzaban et al., 2010; Bradford and Kim,
2012]. Consideration of mean adhesive interactions cannot
explain this behavior [Bradford and Kim, 2012]. Bradford
and Kim [2012] found that hysteresis occurred because
retention and release were dependent on the relative size ofthe colloids/microbes to microscale chemical (metal oxides)
or physical (surface roughness) heterogeneity, the Debye
length, and spatial variations in the applied hydrodynamic
torque. A larger number of colloids interacted with theseheterogeneities at thinner (higher IS) than a thicker (lowerIS) double layer as they move near the solid surface and finda local minimum in the interaction energy. It is therefore
more difficult to remove colloids from local minima foundunder higher than lower IS conditions by changing the IS.
Colloids tended to be reversibly retained in sand with transients in IS when they were larger than the heterogeneity,
whereas they were mainly irreversibly retained when they
were similar in size to the heterogeneity. This occurs
because the zone of electrostatic influence and the appliedhydrodynamic torque are functions of the particle size
[Torkzaban et al., 2007; Duffadar and Davis, 2008].
[5] Several models have been proposed to simulate colloid release with transients in solution IS [Lenhart andSaiers, 2003; Grolimund and Borkovec, 2006; Tosco et al.,2009]. Lenhart and Saiers [2003] considered heterogeneity
in the interaction energy of the colloid populations bydividing attached colloids into a series of compartments,
each with a characteristic critical salt concentration for
release. Similarly, Grolimund and Borkovec [2006] divided
particles into discrete populations. However, in their model asingle attachment coefficient was employed for all populations, whereas separate detachment coefficients were used
for each population. Tosco et al. [2009] simulated the releaseof colloids during transient IS conditions using a two-sitekinetic attachment and detachment model. The models
of Lenhart and Saiers [2003] and Tosco et al. [2009] also
included terms for blocking, whereas the model of Grolimundand Borkovec [2006] included an empirical term in theattachment and detachment coefficients that accounted for thefraction of exchange sites occupied by divalent cations. All of
the above models employ attachment and detachment coefficients that were empirical functions of IS. Determination ofthese functions required extensive optimization to experimental data to achieve satisfactory descriptions, and true predictions were not possible.
[6] Bradford et al. [2011b] developed a mathematicalmodel to mechanistically account for colloid transport in theW09509bulk aqueous phase and adjacent to the solid surface, andrates of colloid collision, interaction, release and immobilization on the solid phase. This model is capable of simulating
exponential, hyperexponential, uniform, and nonmonotonic
retention profiles. The objective of this work is to improve
our ability to simulate colloid transport, retention, and releasewith transients in solution IS by extending this model.
Continuum-scale parameters were developed for the modelto account for the influence of diffusion, adhesion, andhydrodynamic forces and/or torques on colloid retention andrelease with variations in IS. This general modeling framework was subsequently used to analyze experimental transport and release data with transients in IS for two sized latex
microspheres, E. coli D21g, and coliphage 8X174. Thismodel and analyses improves our understanding of the hysteretic dependence of the adhesive interactions on solution IS
and velocity, and the rates of colloid release and dispersion.

2. Experimental Information[7] Bradford and Kim [2012] presented experimental
deposition and release data with transients in IS for fluorescent 1.1 and 0.11 mm carboxyl modified latex (CML)
microspheres, E. coli D21g, and coliphage 8X174. Experimental details are given in this publication, but will bebriefly highlighted below. Ottawa sand that is 360 mm insize was cleaned using a salt cleaning method [Bradford andKim, 2010] to remove trace amounts of clay. The sand wasthen wet packed to a porosity of 0.32 to 0.36 in 4.8 cm
diameter by 15 cm long columns that were equipped with an
adjustable fitting at the top so that different column lengths
were possible. Various NaCl solutions at selected IS (6, 31,
56, and 106 mM) and buffered to pH = 10 were prepared forthe resident, tracer, and eluting solutions. This pH = 10 was
selected to maximize the potential for colloid/microberelease with transients in solution IS. It should be mentionedthat colloid/microbe release is expected to be diminished atlower pH values, and will be completely eliminated underfavorable attachment conditions.
[8] The sand in the columns was equilibrated by flushing
the column with several pore volumes (PVs) of resident
solution at a given solution IS, leaving this solution in thecolumn over overnight, and then flushing several additional
PVs of resident solution through the column. A tracer solution of microspheres, E. coli D21g, or 8X174 was pumped
through the packed columns at a steady Darcy velocity of
0.09 to 0.12 cm min1 for several pore volumes (phase 1)
followed by continued flushing with the eluting solution at
the same velocity and IS (phase 2). The influent IS was subsequently lowered in several steps to study mobilization/
release (phase 3). After changing the IS in each step, thecolumn was flushed with the lowered solution IS until thechange in effluent concentration was minimal. The concentrations of fluorescent microspheres, E. coli D21g, and8X174 were determined in the column effluent using a fluorometer, a spectrophotometer, and the double agar overlay
method, respectively. A mass balance was conducted basedon measured concentrations in the influent and effluent.
[9] The zeta potentials for the CML microspheres, E. coli
D21g, 8X174, and Ottawa sand in the various solution
chemistries were measured and sphere-plate interactionenergies were subsequently calculated using DLVO theory[Derjaguin and Landau, 1941; Verwey and Overbeek,

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1948]. It should be mentioned that mean zeta potentials andcalculated interaction energies do not account for the influence of microscopic physical and chemical heterogeneity on
colloid adhesive parameters [Foppen and Schijven, 2006;
Duffadar and Davis, 2007, 2008; Kim et al., 2008; Metge etal., 2010]. Bradford and Kim [2012] reported that the salt
treated sand exhibited significant amounts of heterogeneity.
For example, the sand surface roughness was visualizedusing a scanning electron microscopy and found to have alarge influence on the measured surface area (a BET surface
area of 560 cm2 g1 compared to a geometric value of around
64 cm2 g1). To determine the influence of chemical heterogeneity the sand was stored in 12 N HCl acid overnightand 12.3 mg of Fe per gram of sand was measured in the acidrinse. It should be mentioned that E. coli D21 g is a mutant
with minimal amounts of lipopolysaccharides [Gmeiner andSchlecht, 1980; Walker et al., 2004]. The influence of surface
macromolecules on calculated interaction energies is therefore expected to be minimal for E. coli D21 g.

3. Mathematical Model[10] Our conceptual model for colloid transport, retention,and release in porous media assumes that the pore space isdivided into two mobile regions, a higher-velocity region 1 inthe bulk solution and a low-velocity region 2 adjacent to thesolid surface. The rate of colloid exchange between regions 1
and 2 is quantified using first-order kinetic expressions. The
fraction of the colloids in region 2 that interacts with the solid
phase at any given time is a function of IS, and this fraction is
subject to kinetic retention and release that also depends on
the solution chemistry. Immobilized colloids on the solid
phase may fill up retention locations over time or a fraction
may be released into region 2 with transients in solution IS orvelocity. Corresponding mathematical expressions for thisconceptual model are provided below.
[11] We assume that colloid transport and retention is afunction of IS that varies with distance and time. The IS of a
1:1 electrolyte can easily be determined from the solution of
the mass balance equation for a conservative tracer (e.g.,chloride ion) throughout the entire pore space (sum ofregions 1 and 2) as∂ðqwt Cis Þ
∂Jis
¼
∂z
∂t

ð1Þ

where qwt (dimensionless) is the volumetric water content,t [T; T denotes units of time] is time, and z [L; L denotes
units of length] is the depth, Cis [Mis L3; Mis denotes thetracer mass] is the tracer concentration (equal to the IS for a
1:1 electrolyte) in the aqueous phase, and Jis [Mis L2 T1]
is the tracer flux.
[12] The colloid mass balance equations for regions 1(bulk fluid), 2 (adjacent to the solid phase), and the solid are
given as∂ðqw1 C1 Þ∂J1
 qw1 ak12 C1 þ qw2 ð1  aÞk21 C2
¼
∂z
∂t

ð2Þ

W09509∂ðqw2 C2 Þ∂J2
¼þ qw1 ak12 C1  qw2 ð1  aÞk21 C2∂t∂z qw2 y 2s k2s C2 þ rb kd ðs  fc si Þ

ð3Þ



∂ðrb sÞ
¼ qw2 y 2s k2s C2  rb kd s  fc si∂tð4Þwhere subscripts 1 and 2 denote the respective regions,
C [Nc L3; Nc denotes the number of colloids] is the colloid
concentrationintheaqueousphase,J[Nc L2 T1]isthecolloid
flux (sum of the advective and dispersive flux), s [Nc M1;
M denotes units of mass of soil] is the colloid concentration on the solid phase, si [Nc M1] is the value of s at theend of the deposition phase 1 and before a reduction in IS,
fc (dimensionless) is the fraction of colloids on the solid
phase that still remain immobilized after a reduction in IS,
kd [T1] is the colloid detachment rate coefficient from thesolid phase to region 2, k2s [T1] is the colloid immobilization rate coefficient from region 2 to the solid phase, k12[T1] is the mass transfer coefficient for colloids fromregion 1 to 2, k21 [T1] is the mass transfer coefficient forcolloids from region 2 to 1, a (dimensionless) is the colloid sticking efficiency, and rb [M L3] is the bulk density.
The parameter y 2s (dimensionless) accounts for time andconcentration dependent blocking using a Langmuirian
approach as [Adamczyk et al., 1994]
y 2s ¼ 1 

s
s max

ð5Þ

where smax [Nc M1] is the maximum solid phase concentration of retained colloids. The total water flux(qt; LT1), volumetric water content, and flux concentration ofcolloids (Ct, Nc L3) are given in the model as
qt ¼ q1 þ q2

ð6Þ

qwt ¼ qw1 þ qw2

ð7Þ

q1 C1 þ q2 C2
q1 þ q2

ð8Þ

Ct ¼

where q1 and q2 [LT1] are the Darcy velocities forregions 1 and 2, respectively.
[13] Equations (2)–(8) are similar to that presented by
Bradford et al. [2011b], but now colloid transport andretention parameters are functions of Cis and a detachment
term has been added from the solid phase to region 2. For agiven time step, equation (1) is solved first for the solution
IS, then the colloid transport parameters are updated, andfinally equations (2)–(5) are solved for the colloid concentrations. Determination of the model parameters and thecoupling with Cis will be described in detail in the next
section.
[14] The model outlined above has been implemented into
the COMSOL software package (COMSOL, Inc., Palo Alto,
California). For the simulations discussed below, a thirdtype boundary condition was used at the inlet, and a concentration gradient of zero was fixed at z equal to the outlet
depth. The simulation domain was selected to be consistent

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with the packed column lengths (Lc, L), and the initial concentration in the simulation domain was zero.

4. Model Parameters[15] Bradford et al. [2011b] described in detail methods toestimate the above model parameters. In brief, q2 = v2qw2
and qw2 = AsL2, where v2 (L T1) is the median watervelocity in region 2 which occurs at a distance of 0.5 L2 fromthe solid-water interface (SWI), L2 = 2rc + h is the boundary
thickness (L), rc (L) is the particle radius, h (L) is the separation distance, and As (L1) is the geometric surface area ofthe porous medium per unit volume. Pore-scale simulationsof water flow in sphere packs were used to determine thecumulative density function (CDF) of water velocities inregion 2 [Bradford et al., 2011a]. This information was
further extended by these authors using scaling and interpolation techniques to predict v2 for a range of grain sizes anddistributions, water velocities, and particle sizes. Information
on q2 and qw2 was used in conjunction with equations (6) and(7) to determine q1 and qw1, respectively. Values of thehydrodynamic dispersion coefficients in regions 1 and 2 wereset equal to the product of their respective pore water velocity
and a constant dispersivity (l, L). The value of k12 was predicted using filtration theory [Yao et al., 1971] and a published correlation for the collector efficiency [Tufenkji andElimelech, 2004]. The value of k21 was set equal to k12 tobe consistent with reported values in the literature [Gargiulo
et al., 2007, 2008; Wang et al., 2011] and linear equilibrium
sorption with the retardation coefficient (R) equal to R = 1 +a/(1  a).
[16] The model parameters a, smax, k2s, y 2s, fc, and kd are
all functions of the adhesive interaction energy between thecolloid and the solid surface (Fmin) which varies with Cis.
When mean zeta potentials are used in DLVO calculations[Derjaguin and Landau, 1941; Verwey and Overbeek, 1948]
the value of Fmin equals the depth of the secondary minimum under unfavorable attachment conditions. The relationship between these parameters and Fmin will first be
discussed before we address the determination of Fmin(Cis).
[17] The kinetic energy method [Simoni et al., 1998; Donget al., 2002; Shen et al., 2007] is used herein to provide apreliminary estimate for a as
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
a ¼ erf F min rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
4F minexpðF min Þpð9ÞThis approach assumes that the amount of colloids thatinteract with the solid surface in region 2 is only dependent
on the kinetic energy of diffusing colloids and the strength ofthe mean adhesive interaction energy.
[18] The value of smax for a given value of Cis can be
estimated as [Kim et al., 2009; Bradford et al., 2009]:
s max ¼

ð1  g ÞAs Sf
Ac rb

ð10Þ

where Ac [L2 N1] is the cross section area per colloid, As [L1]
is the solid surface area per unit volume, Sf (dimensionless) is
the fraction of the surface area that is favorable for colloid
retention, and g (dimensionless) is the porosity of a monolayer
packing of colloids on the solid surface. In this work we

W09509assume a value of g = 0.5 in all simulations based on information presented by Johnson and Elimelech [1995].
[19] Bradford et al. [2011a] presented a detailed approachto predict the value of Sf based on the balance of appliedhydrodynamic (Tapplied, ML2T2) and resisting adhesive(Tadhesion, ML2T2) torques. In summary, pore-scale simulations and scaling approaches were used to determine thelognormal CDF of Tapplied on packs of smooth, sphericalcollectors for different velocities, collector sizes and colloid
sizes. The CDF of Tapplied is subsequently evaluated atTadhesion to determine Sf as
Sf ¼

1 1
þ erf
2 2



lnðTadhesion Þ  m
pﬃﬃﬃ
s 2


ð11Þ

where m and s are the mean and variance of the lognormal
CDF of Tapplied. The value of m for the lognormal distribution is defined as
m ¼ lnðT50 Þ

ð12Þ

The value of T50 [ML2T2] corresponds to the medianvalue of Tapplied on the SWI.
[20] The value of Tadhesion in equation (11) is equal to the
product of a lever arm (lx, L) and the pull-off adhesive force(FA, M L T2) that is required to mobilize a colloid from
Fmin [Bergendahl and Grasso, 1998]. The value of FA may
be estimated using the Derjaguin and Langbein approximations as Fmin(Cis)/h [Israelachvili, 1992; Bergendahl andGrasso, 2000]. Only a portion of the colloid’s projection
on the SWI makes a meaningful contribution to FA andthis is referred to as the zone of electrostatic influence[Duffadar and Davis, 2008]. A value of lx occurs within
the zone of electrostatic influence due to colloid deformation[Bergendahl and Grasso, 1998] and/or friction arising fromsurface roughness and/or chemical heterogeneity [Duffadar
and Davis, 2008]. Some researchers have assumed anempirical value of lx to account for friction [Duffadar andDavis, 2008]. Alternatively, theory by Johnson, Kendall,
and Roberts (JKR) may be used to estimate lx as a result ofresistance due to deformation at separation [Johnson et al.,
1971; Maugis, 1992] for colloid attachment under favorable
or unfavorable conditions [Bergendahl and Grasso, 2000; Liet al., 2005; Bradford et al., 2007, 2011a; Torkzaban et al.,2007, 2008; Shen et al., 2010]. It should be mentioned that
Derjaguin-Muller-Toporov developed an alternative model(the DMT model) to determine lx as a result of deformation[Derjaguin et al., 1975]. In contrast to the JKR model, theDMT model predicts that the lx = 0 at separation. No colloid
immobilization occurs in the presence of fluid flow whenlx = 0 because FA is perpendicular to the fluid drag force. This
result is not consistent with many experimental observations
summarized by Torkzaban et al. [2009, 2010]; e.g., microscopic observations of colloid immobilization, significant
amounts of colloid retention under unfavorable attachment
conditions, and abrupt colloid release with changes in thesolution chemistry. We have therefore chosen to use lx at
separation from the JKR model.
[21] The parameter Sf is not only an important parameterfor determining smax and y 2s according to equations (10)and (5), respectively, it also is strongly related to k2s, fc,
and kd as will be discussed below. Bradford et al. [2011b]
estimated k2s from the median time it takes a colloid in

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W09509s > fcsi can occur when fc changes with Sf (which is a function of Fmin(Cis)). In the absence of additional information,
equation (16) assumes an upper limit for kd equal to k21.

5. Determining Sf (IS) and a(IS)Figure 1. Experimental and predicted values of Sf as afunction of IS for 0.11 and 1.1 mm CML microspheres.
Experimental values of Sf were obtained by optimization of
the simulated and measured breakthrough curves duringphases 1 and 2. The predicted values of Sf were obtainedusing equations (11) and (12) in conjunction with reported
values of Fmin [Bradford and Kim, 2012] and JKR theory[Johnson et al., 1971] to determine Tadhesion.
region 2 to move to a location where the torque balance isfavorable for retention:
k2s ¼

2Nf v2
pd50 1  Sf

ð13Þ

where Nf (dimensionless) is the number of favorable locations of equal size and distribution on the collector surface
and d50 [L] is the median grain size. On a smooth, chemically homogeneous collector the value of Nf may reflect the
average number of grain-grain contacts; e.g., Nf = 5.
Alternatively, if surface roughness or chemical heterogeneity is controlling the value of Sf then Nf is given as
Nf ¼

pd50 Sf
2df

ð14Þ

Here df [L] is equal to the average size of the heterogeneity
parallel to the collector surface.
[22] The amount of colloids that are released with transients in IS is directly related to changes in the balance of
Tapplied and Tadhesion. The value of fc is therefore directly
related to Sf as
fc ¼

Sf
Sif

ð15Þ

where Sif (dimensionless) is the value of Sf before thereduction in IS. The value of kd is determined by the rate ofchange in adhesive force and/or torque with changes in IS.
This should theoretically be a very rapid response, as has
been confirmed by Lenhart and Saiers [2003]. In this work,
we initially estimate kd as
kd ¼ k21 Ho ðs  fc si Þð16Þ

where Ho(s  fcsi) is the Heaviside function that is equal to 1
when s > fcsi and 0 when s ≤ fcsi. Note that the value of[23] The above information indicates that colloid transportand retention will be very sensitive to Fmin(Cis) because of
its influence on Sf and a. Colloid transport and retention data
under various IS conditions [Bradford and Kim, 2012] will
be analyzed below to infer information on the dependency of
Sf and a on IS, and the macroscopic adhesive interaction.
Specifically, we examine whether observed experimental
trends are consistent with theoretical estimates based on themean value of Fmin(Cis) obtained from DLVO theory andmeasured zeta potentials [Bradford and Kim, 2012]. The
experimental values of Sf and a were obtained by simulating
the measured breakthrough curves during phases 1 and 2with the outlined model and simultaneously optimizing Sf
and a to the data. The Pearson’s correlation coefficient (R2)
between observed and simulated breakthrough curves duringphases 1 and 2 was typically >0.92. However, the R2 value
for the 0.11 mm CML colloids at an IS = 106 mM wassomewhat lower (0.57) because only 2.3% of the injectedmass broke through in the column effluent.
[24] Figure 1 presents experimental and predicted valuesof Sf as a function of IS for 0.11 and 1.1 mm CML colloids
during phases 1 and 2. The predicted values of Sf wereobtained using equations (11) and (12) in conjunction with
reported values of Fmin [Bradford and Kim, 2012] and JKR
theory [Johnson et al., 1971] to determine Tadhesion. Experimental values of Sf shown in Figure 1 tended to gradually
increase with IS, reflecting greater numbers of retention sites
when the double layer thickness was compressed and Fmin
increased. These values of Sf were always low, but Sf wasmuch higher for the 0.11 mm CML colloids (Sf < 0.075) than
the 1.1 mm CML colloids (Sf < 0.009) at a given IS. This
indicates that more retention sites were available for thesmaller colloids. These same trends were observed for predicted values of Sf, but the difference with size was muchlarger than experimentally observed. DLVO calculations
that were used to predict Sf reflect only the average adhesiveinteractions with the SWI (Fmin and Tadhesion). To better
understand the experimental and predicted values of Sf weneed to consider the potentially significant influence ofphysical (e.g., surface roughness) and/or chemical (e.g.,adsorbed multivalent cations) heterogeneity observed on thesalt-treated sand [Bradford and Kim, 2012]. Surface integration techniques have been developed and utilized to
calculate colloid interaction energies on idealized, microscopically heterogeneous solid surfaces [Hoek and Agarwal,
2006; Duffadar and Davis, 2007]. Surface integration calculations indicate that the strength of the interaction energy
will vary spatially on the SWI, and that local minima in theinteraction energy that are favorable for attachment may
occur near heterogeneities [Duffadar and Davis, 2008]. The
area of the zone of electrostatic influence is proportional tothe colloid radius size and inversely related to the DebyeHuckel parameter [Duffadar and Davis, 2008]. A greater
portion of the zone of electrostatic influence will therefore
be occupied by a given sized heterogeneity for smaller
colloids and higher IS. Consequently, smaller colloids
and higher IS are associated with more locations on a

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Figure 2. Hysteretic values of Sf for 0.11 and 1.1 mm CML
colloids when phases 1and 2 (arrow to the right) were conducted at IS = 106 mM and phase 3 (arrow to the left) consisted of step decreases in IS equal to 56, 31, 6, and 0 mM.
Experimental values of Sf were obtained by optimization of
the simulated and measured breakthrough curves duringphases 1 and 2, whereas equation (17) was used to predict
Sf during phase 3.

W09509[27] Figure 2 shows hysteretic values of Sf for 0.11 and1.1 mm CML colloids as a function of IS. Values of Sf duringphases 1 and 2 (arrow to the right) were taken from experimental values shown in Figure 1, whereas equation (17) wasused to predict Sf during phase 3 (arrow to the left) as the IS
was decreased from 106 to 56, 31, 6, and 0 mM. Hysteresis
in Sf occurred for both 0.11 and 1.1 mm CML colloids, but
was greater for the smaller colloids. DLVO calculations
indicate that the secondary minimum decreases and theheight of the energy barrier to the primary minimum increases as the IS decreases. Consequently, a reduction in solution
IS tends to induce release of colloids from a secondary
minimum, but not from a primary minimum. As mentioned
above, more primary minima interactions are expected forsmaller colloids on a heterogeneous SWI because a greater
portion of the zone of electrostatic influence is occupied by agiven sized heterogeneity [Duffadar and Davis, 2007, 2008].
Clearly, physical and/or chemical heterogeneity played asignificant role in determining Sf for 0.11 mm CML colloids

microscopically heterogeneous SWI that have primary minimum interactions that contribute to Sf.
[25] The predicted value of Sf will also depend on the local
balance of Tapplied and Tadhesion for unfavorable locations
with a secondary minimum (equations (11) and (12)). Both
Tapplied and Tadhesion are functions of rc, but Tapplied decreases
more rapidly (proportional to r3c ) than Tadhesion with rc.
Hence, Sf was predicted to be larger for smaller rc (Figure 1)
even though Fmin was also smaller. It should be mentionedthat this same result will hold for unfavorable locations on aheterogeneous SWI. However, in this case the local value ofFmin on unfavorable locations is expected to be much lower
as a result of surface roughness [Hoek and Agarwal, 2006;
Bendersky and Davis, 2011] and because the zeta potential ismore negative than in favorable locations. Both of thesefactors will produce lower values of Sf on a heterogeneousthan a comparable homogeneous, smooth SWI that was
assumed in our DLVO calculations. Errors in the predictedvalues of Sf (especially for smaller colloids) therefore reflect
the importance of microscopic heterogeneities on colloid
immobilization, and the inability of mean zeta potentials toaccount for these observations.
[26] Recall that smax and Sf are linearly related throughequation (10) when retention locations are filled. Similarly,
it is logical to expect a linear relationship between theequilibrium value of s (seq) at a new IS and Sf when retention
locations are incompletely filled as (cf. equations (3), (4),and (15)):
seq ¼

Sf
si ¼ f c si
Sif

ð17Þ

This equation was subsequently used to estimate values of Sfduring phase 3. Values of si (model output) and Sif (Figure 1)
used in this calculation were determined after completion ofphases 1 and 2, and a mass balance was conducted for eachstep reduction in IS of phase 3 to determine seq.

Figure 3. Experimental and predicted values of a as afunction of IS for (a) 1.1 and (b) 0.11 mm CML microspheres. Experimental values of a were obtained by optimization of the simulated and measured breakthrough curves
during phases 1 and 2. The predicted values of a were initially obtained from Fmin [Bradford and Kim, 2012] using
equation (9). This prediction was subsequently corrected
for the influence of velocity by multiplying equation (9) by
Sf (obtained from predictions shown in Figure 1).

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W09509Figure 4. Observed and simulated transport and release behavior for 1.1 mm CML colloids whenphases 1 and 2 were conducted at IS = 106 mM and phase 3 consisted of step decreases in IS equal to56, 31, 6, and 0 mM. The values of (a) k21, (b) kd, and (c) l are systematically varied. Other model
parameters are given in Table 1.
and the interaction was therefore largely irreversible (primary
minimum). In contrast, 1.1 mm CML colloids experienced alargely reversible interaction with less hysteresis, suggestingthat the heterogeneity was sufficiently small relative to thezone of electrostatic influence to create a secondary minimum interaction.
[28] Figure 3 presents experimental and predicted valuesof a as a function of IS for 0.11 (Figure 3a) and 1.1(Figure 3b) mm CML colloids. Initial estimates of a at different IS were obtained from Fmin using equation (9).
Experimental and predicted (equation (9)) values of a bothincreased with IS, reflecting greater amounts of retention
when the double layer thickness was compressed andFmin increased. Furthermore, experimental and predicted(equation (9)) values of a were similar in magnitude at agiven IS for the 0.11 mm CML colloids (Figure 3a). In
contrast, predicted values of a were much larger thanexperimental values for the 1.1 mm CML colloids(Figure 3b). One potential explanation is due to hydrodynamic forces. Experimental values of a have been reported
to decrease with increasing water velocity [Johnson et al.,2007; Shen et al., 2010]. Shen et al. [2010] accounted forthis velocity dependency by multiplying a (equation (9)) by
Sf (equations (11) and (12)). Predicted values of aSf shownin Figure 3b were in closer agreement with experimental
values of a then predictions based only on equation (9).
However, this approach does not account for solid phase masstransfer from unfavorable to favorable regions. Consequently,
the predicted velocity dependency of a is expected to beoverestimated, especially for larger colloids. Indeed, predictedvalues of aSf underestimated the experimental value of a forthe 1.1 mm CML colloids at higher IS (Figure 3b).
[29] Piecewise cubic interpolation functions for Sf (IS)and a(IS) were used in subsequent simulations based on

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W09509

Table 1. Model Parameters Used in Simulations Shown in
Figures 4 and 5
Parameter

Values Used in Simulations

qwt
qt
Lc
l
d50
qw1, qw2, q1, q2
k12
k21
k2s
Nf

0.32 to 0.36
0.09 to 0.12 cm min1
11 to 13 cm(0.018 to 0.0025)*Lc, except for Figure 4c
360 mmBradford et al. [2011b]
Yao et al. [1971] and Tufenkji and Elimelech [2004]
k12, except for Figure 4aEquation (13)Nf = 5 for Figures 4, 5a, 5c, and 5d; equation (14)was used in Figure 5b with df = 1 mm
Equation (16), except Figure 4b
Interpolation function of fitted values of a duringphases 1 and 2
Equation (17) using measured values of seq andfitted Sif during phases 1 and 2
Model output after completion of phases 1 and 2
Equation (15)Equation (10)0.11 and 1.1 mm for CML, 0.5 mm for E. coli
D21g,a and 0.027 mm for 8X174
160 min for CML, 75 min for E. coli D21g, and70 min for 8X174

kd
a(IS)
Sf (IS)si
fc
smax
Colloid diameter
Pulse durationa

Selected to be consistent with the width of E. coli D21g.

W09509experimental information similar to that shown in Figures 1
and 3, respectively.

6. Sensitivity Analysis[30] In this section we investigate the sensitivity of colloid
release with transient in IS to the parameters k21, kd, and l.
Figure 4 presents observed and simulated transport andrelease behavior for 1.1 mm CML colloids when phases 1
and 2 were conducted at IS = 106 mM, and phase 3 consisted of step decreases in IS equal to 56, 31, 6, and 0 mM.
All concentrations are normalized by the maximum input
concentration (Ci). The values of k21, kd, and l are systematically varied, as indicated in the legends of Figures 4a, 4b,
and 4c, respectively. Other model parameters are given in
Table 1. Notice that release during phase 3 is sensitive to thevalues of k21, kd, and l. In particular, higher values of k21
and kd are required to accurately capture observed release
behavior. High values of k21 and kd imply that release duringphase 3 is not consistent with a slow diffusion controlled
processes as is commonly assumed, but is rather controlled
by system hydrodynamics and changes in the torque balancewith IS. However, it should be noted that colloid release ismuch lower in phase 2 than 3 due to a lower value of kd = 0
and C2, and a higher value of a that yields lower (1  a)k21.
Results from solute transport studies in columns indicate thatthe value of l is typically around 0.1*Lc [Vanderborght andFigure 5. Observed and simulated transport and release for 1.1 and 0.11 mm CML microspheres, E. coli
D21g, and 8X174. Simulation parameters are summarized in Table 1.
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Vereecken, 2007]. However, considerably lower values of lwere required to capture the observed trends for colloid
release during phase 3. One plausible explanation is due toexplicitly accounting for differences in the velocity inregions 1 and 2 of our dual-permeability model formulation.

7. Simulation of Experimental Data
[31] Figure 5 presents representative observed and simulated transport and release for 1.1 and 0.11 mm CML
microspheres, E. coli D21g, and 8X174. Simulation parameters were predicted from measured data (a(IS) andSf (IS)), or selected based on the results of the sensitivity
analyses discussed above. Table 1 provides a summary of
the simulation parameters. The agreement between observed
and simulated behavior is quite good (the mean square error<0.008 and the R2 ranged from 0.80 to 0.98), indicating thatthe model formulation provided a satisfactory description ofa variety of colloid release data with transients in IS. It
should be mentioned that the above model formulation
effectively lumps first-order microbial death and inactivation
into the retention term for simplicity. Death and inactivation
terms could have been explicitly added to the model formulation if desired, but were not needed because little death/inactivation occurred during these relatively short duration
experiments [Bradford and Kim, 2012]. Bradford and Kim[2012] discuss in detail the differences in the release
behavior for 1.1 and 0.11 mm CML microspheres, E. coli
D21g, and 8X174. In brief, these differences are mainly due
to the hysteretic dependency of Sf on IS shown in Figure 2.
In particular, the value of Sf determines the amount of releasewith transients in IS by directly influencing fc according toequation (15).

8. Conclusions[32] A sophisticated dual-permeability model was extended
to simulate colloid transport and release with transients insolution IS by inclusion of a detachment term that is dependent
on torque balance information (equations (3), (4), and(15)–(17)), and by making a and Sf functions of IS. Thismodel was subsequently used to analyze experimental
transport and release data with transients in IS for two
sized latex microspheres, E. coli D21g, and coliphage
8X174. Comparison of experimental and predicted valuesof a suggested that hydrodynamic forces likely influenced
a, especially for larger colloids. Experimental values of Sf
with IS were found to be consistent with expected trends
on microscopically heterogeneous surfaces, and exhibited
hysteresis with transients in IS. Greater hysteresis in Sf
with IS occurred for smaller colloids likely because they
are similar in size to microscale heterogeneities andexhibited more irreversible primary minimum interactions.
Release of colloids during transient IS conditions wereconsistent with the modeled balance of applied hydrodynamic and resisting adhesive torques under unfavorable
attachment conditions. Colloid release rates were shown tobe very high (not diffusion controlled) once torque balance criteria for release were satisfied. All of this information improves our understanding and ability to simulate
colloid release with transients in IS, and highlights theneed to account for hysteretic adhesive interactions on
microscopically heterogeneous surfaces. Results strongly

W09509indicate that additional research is needed to better predicta and Sf as a function of IS on heterogeneous surfaces
and at different velocities.
[33] Acknowledgments. We would like to thank Lorena Altamirano
for her help in conducting the transport experiments. This research was supported by the ARS, USDA, NP 214. The USDA is an equal opportunity
provider and employer.

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