Journal of Contaminant Hydrology 181 (2015) 141–152

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Journal of Contaminant Hydrology
journal homepage: www.elsevier.com/locate/jconhyd

Equilibrium and kinetic models for colloid release undertransient solution chemistry conditions
Scott A. Bradford a,⁎, Saeed Torkzaban b, Feike Leij c, Jiri Simunek da

US Salinity Laboratory, USDA, ARS, Riverside, CA, United States
CSIRO Land and Water, Glen Osmond, SA 5064, AustraliacDepartment of Civil Engineering and Construction Engineering Management, California State University, Long Beach, CA 90840-5101, United StatesdDepartment of Environmental Sciences, University of California, Riverside, CA 92521, United Statesba r t i c l e

i n f oArticle history:
Received 3 December 2014
Received in revised form 26 March 2015
Accepted 8 April 2015
Available online 17 April 2015
Keywords:
Colloid
Microorganism
Model
Transients
Chemistrya b s t r a c t
We present continuum models to describe colloid release in the subsurface during transientphysicochemical conditions. Our modeling approach relates the amount of colloid release tochanges in the fraction of the solid surface area that contributes to retention. Equilibrium, kinetic,
equilibrium and kinetic, and two-site kinetic models were developed to describe various rates ofcolloid release. These models were subsequently applied to experimental colloid release datasets
to investigate the influence of variations in ionic strength (IS), pH, cation exchange, colloid size,
and water velocity on release. Various combinations of equilibrium and/or kinetic release models
were needed to describe the experimental data depending on the transient conditions and colloid
type. Release of Escherichia coli D21g was promoted by a decrease in solution IS and an increase inpH, similar to expected trends for a reduction in the secondary minimum and nanoscale chemical
heterogeneity. The retention and release of 20 nm carboxyl modified latex nanoparticles (NPs)were demonstrated to be more sensitive to the presence of Ca2+ than D21g. Specifically, retentionof NPs was greater than D21g in the presence of 2 mM CaCl2 solution, and release of NPs only
occurred after exchange of Ca2+ by Na+ and then a reduction in the solution IS. These findings
highlight the limitations of conventional interaction energy calculations to describe colloid
retention and release, and point to the need to consider other interactions (e.g., Born, steric, and/or hydration forces) and/or nanoscale heterogeneity. Temporal changes in the water velocity did
not have a large influence on the release of D21g for the examined conditions. This insensitivity
was likely due to factors that reduce the applied hydrodynamic torque and/or increase theresisting adhesive torque; e.g., macroscopic roughness and grain–grain contacts. Our analysis
and models improve our understanding and ability to describe the amounts and rates of colloid
release and indicate that episodic colloid transport is expected under transient physicochemical
conditions.
Published by Elsevier B.V.

1. Introduction
An understanding of and an ability to accurately predict therelease of colloids (including microorganisms, particulateorganic matter, clay, metal oxides, and nanoparticles) in porous
media are needed for many agricultural, environmental, and⁎ Corresponding author. Tel.:+1 951 369 4857.
E-mail address: Scott.Bradford@ars.usda.gov (S.A. Bradford).

http://dx.doi.org/10.1016/j.jconhyd.2015.04.003
0169-7722/Published by Elsevier B.V.

industrial applications. Under steady-state flow and solution
chemistry conditions low amounts of colloids are slowly
released from the solid phase when the kinetic energy ofdiffusing colloids exceeds the adhesive interaction energy(Ryan and Elimelech, 1996; Ryan and Gschwend, 1994; Shenet al., 2007). Conversely, significant amounts of colloid release
may rapidly occur during temporal changes (i.e., transients) in
solution chemistry (Grolimund et al., 2001; Roy and Dzombak,
1996) and/or water velocity (Bergendahl and Grasso, 1998,

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S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

2000). These observations indicate that different mechanisms
control colloid release during steady-state and transientconditions. Transient physicochemical conditions may occur
in the subsurface as a result of infiltration and recharge,
groundwater and surface water interactions, injection and/or
extraction wells, and contamination events.
Colloid immobilization and release at a particular locationon the collector surface depend on the strength of the resistingadhesive torque in comparison to the applied hydrodynamic
torque (Bergendahl and Grasso, 1998, 2000), and the adhesiveinteraction energy relative to the kinetic energy of diffusing
colloids (Shen et al., 2007; Simoni et al., 1998). Transients inwater velocity alter the hydrodynamic force that acts on
immobilized colloids (Bergendahl and Grasso, 1998, 2000),
and is spatially variable because of the complex geometry of thepore space (Bradford et al., 2011). Transient solution chemistry
conditions impact the adhesive force by altering the macroscopic zeta potentials and the electric double layer thickness(Grolimund and Borkovec, 2006; Ryan and Elimelech, 1996). In
addition, the dependence of the adhesive force on solution ionicstrength (IS) is a function of the colloid size and the nanoscale
heterogeneity (e.g., the distribution of charge and roughness)within the zone of electrostatic influence (Bendersky and Davis,
2011; Duffadar and Davis, 2007). Natural colloid and solid
surfaces always exhibit some degree of nanoscale heterogeneity due to surface roughness, mineral defects, isomorphic
substitution, protonation/deprotonation of hydroxyl groups,
and adsorption of different ions, organics, and clay particles(Suresh and Walz, 1996; Tufenkji and Elimelech, 2005;
Vaidyanathan and Tien, 1991). Consequently, spatial variabilityof the adhesive force is also expected. The above information
indicates that transient physicochemical conditions initiate
colloid release by altering the force and/or torque balance thatacts on immobilized colloids, and that only a fraction of theretained colloids may be released because of spatial variationsin the adhesive and hydrodynamic forces and torques.
Several continuum models have been developed to simulate colloid release during transients in solution chemistry(Bedrikovetsky et al., 2011; Bradford et al., 2012; Grolimundand Borkovec, 2006; Lenhart and Saiers, 2003; Tosco et al.,2009). In general, colloid transport equations are coupled to thesolution chemistry through explicit dependencies of retention
and/or release parameters on solute concentrations (e.g., IS, pH
and/or adsorbed divalent cations). The transport equations forsolution chemistry and colloids are subsequently solved andcolloid transport parameters are updated at each time step. Allof the continuum models employ different first-order kinetic
formulations for colloid retention and release; e.g., someconsidered population heterogeneity, blocking, or multiple
retention sites. In addition, the functional dependency ofretention and release parameters on solution chemistry varies
significantly among the models. Only some of these models canbe easily extended to describe colloid release with transients inwater velocity (Bedrikovetsky et al., 2011; Bradford et al.,2012).
One challenge in continuum modeling of colloid release isdetermining the amount of colloid release with the physicochemical perturbation. A finite number of colloids can beretained in porous media under steady-state conditions. The
maximum solid phase concentration of retained colloids (Smax)
is obtained when all retention sites are filled. Under steady-state

conditions, the value of Smax can be determined by direct
microscopic observations in simplified systems (Adamczyk
et al., 1992), inverse optimization of experimental breakthrough curve data that exhibit blocking behavior (Adamczyk
et al., 1994; Johnson and Elimelech, 1995), or theoreticallyestimated from torque balance calculations (Bradford et al.,2013). Experimental and theoretical results demonstrate that
Smax is sensitive to the solution chemistry and hydrodynamic
conditions (Adamczyk et al., 1995; Bradford et al., 2013;
Sasidharan et al., 2014). Such changes in Smax with physicochemical conditions have been related to the amount of colloid
release during transient conditions (Bedrikovetsky et al., 2011;
Bradford et al., 2012). However, the dependence of Smax on
physicochemical conditions is more complicated during colloid
release than retention because some of the retained colloids
cannot be released by a given perturbation (e.g., irreversible
retention) (Adamczyk et al., 1992). The functional dependenceof Smax on IS therefore exhibits hysteresis during retention andrelease phases (Bradford et al., 2012). The importance oftemporal changes in Smax with transient conditions on colloid
release is not widely appreciated, and has not yet been
incorporated into continuum models that account for the fullrange of equilibrium and/or kinetic release behavior.
A second challenge in continuum modeling of colloid
release with transient physicochemical conditions is describing
the dynamics of colloid release. The adhesive interaction at aparticular location on the collector surface may change with aphysicochemical perturbation as ions transfer and/or react in
the gap separating the colloid from the collector surface. Spatial
variations in the adhesive (and associated separation distances) and hydrodynamic forces produce a distribution ofmass transfer and reaction rates for ions on collector surfaces.
Indeed, various combinations of equilibrium or kinetic models
are needed to describe geochemical reactions of specific ionsin soils (Mayer et al., 2002; Šimůnek and Valocchi, 2002).
Similarly, colloid release with transients has sometimes been
observed to occur slowly and at other times very rapidly(Bedrikovetsky et al., 2011; Bradford et al., 2012; Grolimundand Borkovec, 2006; Khilar and Fogler, 1984; Lenhart andSaiers, 2003; Shen et al., 2012; Torkzaban et al., 2013; Tosco
et al., 2009). It is therefore logical to anticipate that equilibrium,
kinetic, combined equilibrium and kinetic, or even two-sitekinetic models may be needed to describe colloid release undertransient physicochemical conditions. However, continuum
models for colloid release have not yet been developed todescribe all of these possible conditions.
The overall objective of this research is to improve ourunderstanding of and ability to simulate colloid release undertransient physicochemical conditions. This aim is accomplishedby relating the amount of colloid release with transients tochanges in Smax, and then developing equilibrium, kinetic,
combined equilibrium and kinetic, and two-site kinetic models
to describe a wide variety of observed colloid release behavior.
These continuum models were subsequently applied toexperimental colloid release datasets under various transientconditions to investigate the influence of IS, pH, cationexchange, colloid size, and water velocity on release. Analysisof these data improves our knowledge of fundamental factors
influencing colloid release during transient physicochemical
conditions, and helps to identify the proper model formulations
to describe such behavior.

S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

2. Materials and methods
2.1. Electrolyte solutions and porous medium
Electrolyte solutions were prepared using deionized (DI)water (pH = 5.8), and NaCl (0, 1, 5, 25, 50, and 100 mM NaCl)or CaCl2 (2 mM CaCl2) salts. Sodium bicarbonate and carbonate
salts were added to the 100 mM NaCl solution to buffer the pH
at values of 7, 8, 9, and 10 (Delory and King, 1945; Waentig
et al., 2009). Ultrapure quartz sand (Iota Quartz, Unimin Corp.
NC) was employed as porous medium in the transport andrelease experiments discussed below. The sand was thoroughly
rinsed with DI water to eliminate background fines from thesand before use. The median grain size of this sand wasmeasured to be 238 μm (standard deviation of 124 μm) with alaser scattering particle size and distribution analyzer (Horiba
LA 930).
2.2. Colloids
Experiments discussed below employed pure cultures ofEscherichia coli D21g or carboxyl modified latex (CML) nanoparticles (NPs). D21g is a gram-negative, nonmotile bacterial
strain that produces minimal amounts of lipopolysaccharides
and extra-cellular polymeric substances. E. coli D21g was grown
24 h before initiating experiments. A single colony of E. coli D21g
was inoculated into 1.0 l of Luria-Bertani media containing
30 mg/L gentamycin and incubated with shaking at 37 °C
overnight. The bacterial suspension was centrifuged and rinsed
three times before diluting the concentrated suspension into thedesired electrolyte solution. Influent and effluent concentrationsof E. coli D21g were determined using a spectrophotometer(Unico UV-2000, United Products & Instruments, Dayton, NJ) ata wavelength of 600 nm and a calibration curve. The average
optical density at 600 nm for the influent cell suspension was
about 0.2, which corresponds to an input concentration (Co) of
approximately 107 CFU ml−1. The size of E. coli D21g in thevarious solution chemistries was measured using the laser
scattering particle size and distribution analyzer to be 1.46 μm.
The CML NPs (Molecular Probes, Eugene, OR) were
fluorescent with excitation at 505 nm and emission at
515 nm. The manufacturer provided values of the CML NP size
(20 nm in diameter), shape (spherical), density (1.05 g cm−3),
and hydrophobicity (hydrophilic). The stock CML NP solution
was diluted to achieve a value of Co = 5 ∗ 107 N ml−1 fortransport experiments. Average (three measurements) CML NP
concentrations reported herein were determined using a Turner
Quantech Fluorometer (Barnstead/Thermolyne, Dubuque, IA)
and reproducibility was typically within 1% of Co.
2.3. Interaction energy calculations
The zeta potential of E. coli D21g, CML NPs, and crushed
ultrapure quartz sand in the various solution chemistries wasdetermined from measured electrophoretic mobilities using a
ZetaPALs instrument (Brookhaven Instruments Corporation,
Holtsville, NY) and the Smoluchowski equation. The totalinteraction energy of E. coli D21 g and CML NP upon approachto the quartz sand under the various solution chemistries wascalculated using Derjaguin–Landau–Verwey–Overbeek (DLVO)
theory and a sphere–plate assumption (Derjaguin and Landau,

143

1941; Verwey and Overbeek, 1948). Electrostatic double layerinteractions were quantified using the expression of Hogg et al.
(1966). As is commonly assumed, zeta potentials were used inplace of surface potentials in these calculations (Elimelech
et al., 1995). The retarded London–van der Waals attractive
interaction force was determined from the expression ofGregory (1981). The Hamaker constant was 6.5 ∗ 10−21 J for
D21g (Rijnaarts et al., 1995) and 4.04 ∗ 10−21 J for CML NPs(Bergendahl and Grasso, 1999). Table 1 summarizes measured
values for zeta potentials for D21g, CML NPs, and sand, and thecalculated energy barrier height and depth of the secondary
minimum for the DLVO interaction energies in the various
solution chemistries.
It should be mentioned that DLVO theory is based on anumber of assumptions (in addition to those noted above) that
may be violated. For example, non-DLVO interaction energies
such as Born repulsion, steric interactions, hydrophobic
interactions, and/or hydration effects frequently need to be
included in interaction energy calculations to explain experimental observations (Elimelech et al., 1995). Furthermore,conventional DLVO calculations employ mean values of zeta
potentials and implicitly assume geometrically smooth andchemically homogeneous surfaces (Bendersky and Davis, 2011;
Duffadar and Davis, 2007; Shen et al., 2012). However, natural
colloid and solid surfaces always exhibit some degree ofnanoscale heterogeneity. Consequently, the above DLVO calculations should therefore be viewed as only a first approximationof expected changes in mean adhesion with solution chemistry.
2.4. Column experiments
Glass chromatography columns (15 cm long and 4.8 cm
inside diameter) were wet packed with the ultrapure quartz
sand. The saturated water content, bulk density, and length
for the packed column were determined to be about
0.44 cm3 cm−3, 1.48 g cm−3, and 12.8 cm (adjustable fittingat the column top), respectively. Colloid retention in the sandwas achieved by pumping six pore volumes (PVs) of a selectedsuspension upward through the vertically oriented columns at
a steady flow rate using a peristaltic pump (Phase 1), followedby continued flushing with colloid free solution having thesame solution chemistry for an additional 4 PVs (Phase 2).
Colloid release was then studied by systematically changing thechemistry or velocity of the eluting solution (Phase 3) for 4–6
PVs at each alteration step. Specific experiments and justification for the selected physicochemical sequence are provided
below. Effluent samples were continuously collected during thetransport experiment at selected intervals using a fraction
collector. The effluent samples were then analyzed for D21g or
CML NPs as described above. Experiments were typically
replicated and exhibited good reproducibility.
The influence of IS reduction on D21g release was studied
using the following solution chemistry sequence: 100 (Phases 1
and 2), 50 (Phase 3a), 25 (Phase 3b), 5 (Phase 3c), 1 (Phase 3d),and 0 (Phase 3e) mM NaCl when the solution pH = 5.8 and theDarcy water velocity (qw) equals 0.27 cm min−1. This solution
chemistry sequence was selected to encompass a range inmean adhesive interactions that ranged from favorable toprogressively more unfavorable for D21g attachment (Table 1).
The effects of pH increase in 100 mM NaCl solution on D21g
release was studied when qw = 0.27 cm min−1 using the144

S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

Table 1
Zeta potentials and DLVO interaction energy parameters. The standard error associated with the zeta potentials is given in the parenthesis.
Colloid

NaCl orCaCl2 (mM)

pH

Sand zeta
potential (mV)

Colloid zeta
potential (mV)

Energybarrier (kT)Secondary energy
minimum (kT)

D21g
D21g
D21g
D21g
D21g
D21g
D21g
D21g
D21g
NP
NP
D21g⁎
NP⁎

1
5
25
50
100
100
100
100
100
1
100
2
2

5.8
5.8
5.8
5.8
5.8
7.0
8.0
9.0
10.0
5.8
5.8
5.8
5.8

−66.8 (3.0)−63.3 (2.4)−57.2 (1.0)−41.4 (1.9)−17.3 (1.8)−21.0 (7.1)−37.0 (4.9)−38.9 (2.4)−37.2 (3.1)−66.8 (3.0)−17.3 (1.8)−28.6 (0.9)−28.6 (0.9)−57.2 (1.3)−54.6 (1.5)−41.4 (1.4)−32.3 (1.1)−17.5 (2.3)−16.2 (2.6)−16.2 (3.1)−26.4 (6.4)−18.9 (1.6)−34.1 (2.4)−28.1 (8.9)−24.8 (0.5)−14.9 (0.9)

3579.82892.21453.1
600.7
NB
NB
56.2
214.1
49.2
22.1
1.5
465.9
3.2

0.0
−0.4
−2.8
−6.7
–
–
−23.3
−19.6
−30.1
0.0
−0.1
−0.8
0.0

NP — Carboxyl modified latex nanoparticle (20 nm in diameter).
NB — no barrier to attachment.
kT — product of the Boltzmann's constant and the absolute temperature.
⁎ — CaCl2 solution.

following solution chemistry sequence: pH = 5.8 (Phases 1 and2), pH = 7 (Phase 3a), pH = 8 (Phase 3b), pH = 9 (Phase 3c),and pH = 10 (Phase 3d). Increases in solution pH can cause
deprotonation of surface hydroxyl groups that may decrease
the charge of nanoscale chemical heterogeneity sites (Tufenkji
and Elimelech, 2005; Vaidyanathan and Tien, 1991). The
solution IS also increased with pH because of the addition ofvarious amounts of buffering salts (sodium bicarbonate andcarbonate salts). The 100 mM NaCl solution and buffer produced
conditions that were mainly favorable for attachment basedon mean zeta potentials (Table 1), but that neglect the potential
effects of nanoscale chemical heterogeneity. This solution
chemistry sequence was therefore chosen to study the effects
of changes in nanoscale chemical heterogeneity on D21g release.
Another experiment examined the influence of qw on therelease of D21g. In this case, D21g was deposited in 100 mM
NaCl at pH 5.8 and qw = 0.27 cm min−1 (Phases 1 and 2), andthen the IS was reduced to 5 mM NaCl at pH = 5.8 (Phase 3).
The value of qw was then sequentially increased during Phase 3
using the following sequence: 0.27 (Phase 3a), 0.54 (Phase 3b),1.08 (Phase 3c), 2.16 (Phase 3d), and 5.4 (Phase 3e) cm min−1.
The solution IS was chosen to be initially high (100 mM NaCl)in order to achieve similar initial conditions to the IS and pH
release experiments discussed above. The solution was subsequently lowered to 5 mM NaCl in order to reduce the adhesiveinteraction (Table 1) and thereby facilitate release withchanges in water velocity. Increases in water velocity were
expected to initiate release by increasing the hydrodynamic
force on immobilized D21g.
The influence of cation type and IS (pH = 5.8) on the releaseof D21g and CML NPs was investigated using the following
solution chemistry sequence: 2 mM CaCl2 solution (Phases 1
and 2), DI water (Phase 3a), 100 mM NaCl (Phase 3b), and DI
water (Phase 3c) when the pH = 5.8 and qw = 0.27 cm min−1.
Adsorbed multivalent cations, such as Ca2+, can create
nanoscale chemical heterogeneity on the solid surface thatcan neutralize or reverse the surface charge at specific locations(Grosberg et al., 2002). The initial solution composition was
therefore selected to be 2 mM CaCl2 in order to study the effects
of a nanoscale chemical heterogeneity arising from theadsorption of Ca2+ on colloid retention and release. The meanadhesive interaction was dramatically reduced in the presenceof DI water during Phases 3a and 3c (Table 1). Cation exchange(Ca2+ is exchanged by Na+) may be initiated when 100 mM
NaCl is added during (Phase 3b), and this may alter the amountof nanoscale chemical heterogeneity arising for adsorbed Ca2+.
The second DI water pulse (Phase 3c) was conducted toobserve differences in release following alteration of thenanoscale heterogeneity by cation exchange. Two colloid sizes(20 nm CML colloids and 1.46 μm D21g) were considered
because the effects of nanoscale chemical heterogeneity on thelocal adhesive interaction are known to be a function of thecolloid size (Bendersky and Davis, 2011; Duffadar and Davis,
2007). Ideally, both colloids would have the same surface
properties. However, the literature indicates that even thesurface properties of CML colloids vary with size (Bradford andKim, 2012; Treumann et al., 2014). Trends for D21g retention,
release, and surface properties have been observed to begenerally consistent with 1 μm CML colloids (Bradford and Kim,
2012; Bradford et al., 2012). Consequently, D21g was used as
the larger colloid to be consistent with other release studies
discussed above.
3. Continuum model development
Continuum scale models for colloid transport, retention andrelease are described below. These models were implemented
into the Hydrus-1D (Šimůnek et al., 2008) software package.
Details of the numerical solution and iterative procedure fornonlinear flow and transport problems are already presented
in the literature (Šimůnek et al., 2008). Hydrus-1D includes
a nonlinear least squares routine for inverse parameter
optimization.
3.1. Steady-state colloid retention and release
Colloid transport in the aqueous phase was described using
the advection–dispersion equation that includes an exchange
term to/from the aqueous and the solid phases:


∂θw C
∂S
∂
∂C
∂q C
− w
þ ρb
¼
θw D
∂t
∂t ∂z
∂z
∂z

ð1Þ

S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

where C [NL−3; where N and L denote units of number andlength, respectively] is the colloid concentration in the aqueous
phase, D [L2T−1; where T denotes units of time] is thehydrodynamic dispersion coefficient for colloids, θw [L3L−3] isthe volumetric water content, ρb [M L−3; where M denotes
units of mass] is the soil bulk density, and S [N M−1] is the solid
phase colloid concentration. The first and second terms on theright hand side of Eq. (1) account for the dispersive andadvective fluxes of the colloids, respectively.
The solid phase mass balance equation during steady-statephysicochemical conditions (e.g., Phases 1 and 2) is given inthis work as:ρb∂S¼ θψksw C−ρb krs S∂tð2Þ

where ksw [T−1] is the retention rate coefficient, krs [T−1] is thesteady-state release rate coefficient, and ψ [–] is a dimensionless blocking function. The parameter ψ accounts for time andconcentration dependent blocking using a Langmuirian approach as (Adamczyk et al., 1994):
ψ ¼ 1−

S
Smax

ð3Þ

3.2. Amount of transient colloid release
A first step in modeling colloid release with transientphysicochemical conditions is determining the amount of
colloid release with a given perturbation. Fig. 1 presents anillustration of the hypothetical fraction of the solid surface areathat contributes to colloid immobilization (Af) for two sphericalcollectors after equilibrating with three IS conditions (10, 50,
and 100 mM). The value of Af increases with the IS because of
an increase in the depth of the secondary minimum and/or an
increasing influence of nanoscale heterogeneities (Bendersky
and Davis, 2011; Tufenkji and Elimelech, 2005). There is alinear relationship between Af and Smax that is given as(Bradford et al., 2009):
Smax ¼

ð1−γ ÞAs A f
A c ρbð4Þ

IS=10 mM

145

where Ac [L2 N−1] is the cross-sectional area per colloid, As[L−1] is the solid surface area per unit volume, and γ [–] is theporosity of a monolayer packing of colloids on the solid surface(a value of γ = 0.5 was selected based on information
presented by Johnson and Elimelech (1995)). Consequently,
the value of Smax also increases with IS in Fig. 1. It should be
mentioned that more complex, non-uniform distributions of Af
may occur at the pore-scale as a result of physical and/or
chemical heterogeneity that produces primary and/or secondary minimum interactions. Nevertheless, Af and Smax will still befunctions of IS.
Fig. 1 indicates that a reduction in IS may produce a
corresponding decrease in Af and Smax. This change in Af andSmax induces colloid release within the area defined by Aif-Af,
where Aif [–] is the initial value of Af before chemical alteration.
The new equilibrium value of S is given as:
ð5Þ

S ¼ f nr Si

where Si [N M−1] is the initial value of S before chemicalalteration, and fnr [–] is the fraction of retained colloids that is
not released by the chemical alteration (fnr ≤ 1). Similarly, theamount of colloid release with the IS reduction is equal to(1-fnr)*Si. If the retained colloids are uniformly distributed
within Aif, then the value of fnr is related to Af and Smax as(Bradford et al., 2012):
f nr ¼

AfS¼ max
Aif Si max

ð6Þ

where Simax [N M−1] is the initial value of Smax. It should be
mentioned that the above analysis also applies to therepresentative elementary area or volume scales. In this case,values of Af and Smax are continuous functions of IS in Eq. (6),and Si and S may be functions of depth in Eq. (5).
Theoretical estimates of Af have been made for various
physicochemical conditions by conducting a torque balanceover an idealized porous medium surface (Bradford and
Torkzaban, 2012; Bradford et al., 2013). Variations in theamount, size, and charge of nanoscale chemical heterogeneity
on the solid and colloid surfaces produced localized primary
minimum interactions and corresponding Af values that

IS=50 mM

IS=100 mM

Af

Fig. 1. An illustration of the hypothetical fraction of the solid surface area that contributes to colloid immobilization (Af) for two spherical collectors at ionic strengths (IS)of 10, 50, and 100 mM. When IS is reduced from 100 to 50 and further to 10 mM, particles originally deposited on the “non Af” part (white) of the collector surface tend
to be released. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

146

S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

changed with the IS and colloid size (Bradford and Torkzaban,2012). In the presence of a secondary minimum, results
demonstrate that Af will depend on the IS, the zeta potentialof the colloid and porous medium, qw, the amount and size ofnanoscale heterogeneity on the solid surfaces, the colloid size,
and the surface roughness and pore–space geometry (Bradford
et al., 2013). A potential advantage of the theoretical approach is
independent model predictions for the amount of colloid
release over a wide range of conditions. Furthermore, theamount of colloid release from primary and secondary minimum interactions can be explicitly quantified for a variety ofsolid and colloid properties. Unfortunately, a priori informationon all the required nanoscale physical and chemical model
parameters for the theoretical approach is still not available, andfuture research is still needed to overcome these limitations.
Consequently, values of fnr (Eq. (6)) as a function of selected
physicochemical conditions were obtained directly using massbalance information (Bradford et al., 2012; Torkzaban et al.,2013) from transient release experiments discussed below.
Experimental data for fnr is expected to be of critical importancefor calibration and validation of theoretical predictions for Af.
3.3. Models for transient colloid release
The previous section demonstrates that the amount of
colloid release due to transient physicochemical conditions canbe related to changes in Af. Furthermore, this analysis holds forcolloids interacting in either a secondary or primary minimum.
A second critical step in modeling colloid release with transientphysicochemical conditions is accurately describing the rate ofrelease. This issue is addressed in this section.
Eq. (1) is again used to describe colloid transport duringtransient physicochemical conditions. In addition, similar transport equations with associated reactions are solved for geochemical species of interest (e.g., cations, anions, and protons),
and quantities such as the IS, pH, and the fraction of cationexchange sites occupied by multivalent cations (f++) can becalculated. The specific number of geochemical species, transport equations, and reactions types that are needed depend on
the solution chemistry and the soil properties (Mayer et al.,
2002; Šimůnek and Valocchi, 2002). The travel velocity ofchromatographic fronts (e.g., IS, pH, and cation exchange)
approaches that of a conservative tracer for sand with a very
low cation exchange capacity (Torkzaban et al., 2013).
An equilibrium expression for colloid release with achemical perturbation can be derived from Eqs. (5) and (6)by taking the partial derivative of S with respect to time andapplying the chain rule. The following equilibrium expressions
apply for colloid release with changes in IS shown in Fig. 1:
ρb

ρb

∂A f
∂S
S ∂A f¼ ρb iH −Aif ∂t o
∂t
∂t

ρb

∂S
S
¼ ρb iAif∂t¼ ρb Si

ρb

ρb

!
!
dA f ∂C pH∂C pH
HodC pH ∂t
∂t
!
!
∂C pH
df nr ∂C pH
Ho
dC pH ∂t
∂t


 

dA f ∂f þþ∂f∂SS
H o − þþ
¼ ρb iAif df þþ ∂t
∂t
∂t

 

∂f þþ
d f nr ∂f þþ
Ho −¼ ρb Si
df þþ ∂t
∂t

 

∂S
S dA f ∂qw∂qw
Ho¼ ρb iAif dqw ∂t
∂t
∂t

 

df nr ∂qw
∂qw
Ho
¼ ρb Sidqw ∂t
∂t

ð8Þ

ð9Þ

ð10Þ

where CpH is the aqueous phase pH (e.g., −log10(CH); and CH ismoles of hydrogen ions per liter). A composite expression may
also be developed for the total exchange during colloid releasewith changes in CIS, CpH, f++, and qw as:
ρb

!


 

∂S
S dA f ∂C IS∂C
H o − IS¼ ρb iAif dC IS ∂t
∂t
∂t

 

df nr ∂C IS
∂C IS
Ho −
¼ ρb Si
dC IS ∂t∂t

when − ∂C∂tIS ≥0 to turn release on when the IS decreases and thedepth of the secondary or primary minimum is reduced.
∂A
Eq. (7a) indicates that release only occurs when ∂tf b0. Notethat potential retention (e.g., reattachment) of released colloids
is not explicitly considered in Eq. (7a) and (7b). As a first
approximation we assume that subsequent retention of thereleased colloids during Phase 3 was negligible because
chemical conditions were altered in favor of colloid release,and it is only possible to experimentally determine the netchange in colloid release and retention. Significant amounts ofcolloid release occurred during transient physicochemical
conditions (Fig. 2), whereas negligible detachment occurredunder steady-state conditions (Table 2). Consequently, thevalue of krs was also set to zero during Phase 3.
In addition to IS, experimental information presented below
indicates that Af is also a function of pH and qw, with colloid
release initiated when the pH and qw increases. In addition,
theoretical information discussed above and experimental
results (Grolimund and Borkovec, 2006; Torkzaban et al.,2013) indicate that adsorption of multivalent cations mayalter the amount of nanoscale chemical heterogeneity, and that
Af may therefore be a function of f++. Similar equilibrium
expressions for colloid release may be derived for each of thesecases as:∂SS
¼ ρb iAif∂t!
!
dA f ∂C IS
dA f ∂C pH
dA f ∂f þþ dA f ∂qw
∂A f
þ
þ
þ
Ho −
dC IS ∂tdC pH ∂tdf þþ ∂tdqw ∂t
∂t

ð7aÞ

ð11Þ

ð7bÞ

Eq. (11) indicates that transient release is proportional to
the total derivative of Af with respect to time. This equation
may also be rewritten in terms of fnr in a similar manner toEqs. (7b)–(10). The determination of CIS, CpH, and f++ and their
partial derivatives with respect to time in Eqs. (7b)–(11)follows directly from the solution of the geochemistrytransport equations, whereas similar information for qw is
obtained from the solution of the water flow equation.

where CIS is the IS of the aqueous phase (e.g., moles/liter), and
Ho is a Heaviside function that is equal to 0 when − ∂C∂tIS b0 and 1

S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

147

release of latex colloids, microbes, and nanoparticles undertransient IS conditions and/or cation exchange.
A combined equilibrium and kinetic model for colloid
release during transient conditions can also be derived using
similar approaches. In this case, S is divided into equilibriumand kinetic sites, and Eq. (5) is rewritten to give the new
equilibrium value as:(a)

S ¼ Seq þ Sk1 ¼ F eq f nr Si þ F k1 f nr Si

where Feq [–] is the fraction of equilibrium sites, Fk1 [–] is thefraction of kinetic sites that is equal to Fk1 = 1-Feq, and Seq[N M−1] and Sk1 [N M−1] are the equilibrium and kinetic solid
phase concentrations of retained colloids, respectively. The
exchange terms for equilibrium and kinetic sites follow directlyfrom above as:

(b)

ρb

ρb

(c)

A kinetic expression for colloid release with transients insolution chemistry that is consistent with Eqs. (5) and (6) isgiven as (Bradford et al., 2012):∂S¼ −ρb k det1 ðS− f nr Si ÞH o ðS−f nr Si Þ
∂t

ð12Þ

where kdet1 [T−1] is the transient release rate coefficient. The
driving force for colloid release in this kinetic model is thedifference in the actual S and the equilibrium value of S given by
fnrSi in Eq. (5). Recall that Eq. (6) indicates that fnr = Af/Aif, and
Af is a function of CIS, CpH, f++, and qw. Bradford et al. (2012) andTorkzaban et al. (2013) employed Eq. (12) to simulate theTable 2Fitted model parameters to the colloid breakthrough curves during Phases 1
and 2. The standard error associated with the parameter ﬁt is given in theparenthesis.
Figures Transient

ksw
min−1

krs
min−1

Smax/Co
cm3g−1R2

2a, 3a2b, 3b2c4a4b0.087 (0.002)
0.090 (0.002)
0.076 (0.002)
0.057 (0.001)
0.195 (0.009)

0.002 (0.000)
0.001 (0.000)
0.002 (0.000)
0.002 (0.000)
0.002 (0.000)

2.11 (0.11)
5.36 (0.63)
4.27 (0.78)
3.35 (0.48)
5.03 (1.11)

0.98
0.98
0.96
0.98
0.75

IS
pH
qw
Exchange/IS
Exchange/IS

∂Seq
∂t

¼ −ρb F eq

∂A f
Si ∂A fH −Aif ∂t o
∂t

!

∂Sk1
¼ −ρb k det1 ðSk1 − F k1 f nr Si ÞH o ðSk −F k1 f nr Si Þ
∂t

ð14Þ

ð15Þ

A two-site kinetic release model can be developed in asimilar manner to Eqs. (13)–(15). In this case, S is divided
between the two kinetic sites to give the new equilibrium value
as:Fig. 2. Experimental values of fnr as a function of IS (Fig. 2a), pH (Fig. 2b), and qw(Fig. 2c) for E. coli D21g. The cells were initially deposited in 100 mM NaClsolution at pH = 5.8. The IS and pH were successively altered in a and b,
respectively. In c the IS was first decreased to 5 mM, and then qw was
successively increased. Values of fnr where determined from mass balance
information on cell release during Phase 3.

ρb

ð13Þ

S ¼ Sk1 þ Sk2 ¼ F k1 f nr Si þ F k2 f nr Si

ð16Þ

where the subscripts k1 and k2 on variables are used to denote
kinetic sites 1 and 2, respectively. Release from the first kinetic
site is again given by Eq. (15), whereas release for the second
kinetic site is given as:ρb∂Sk2¼ −ρb k det2 ðSk2 − F k2 f nr Si ÞH o ðSk2 − F k2 f nr Si Þ
∂t

ð17Þ

where kdet2 [T−1] is the release rate coefficient for kinetic sites
2 and Fk2 = 1-Fk1. Similarly, S can be divided between
equilibrium and two kinetic sites, and release may be simulatedusing Eqs. (14), (15), and (17).
The equilibrium (Eq. (11)), kinetic (Eq. (12), combined
equilibrium and kinetic (Eqs. (14)–(15)), and two-site kinetic(Eqs. (17)–(18)) models have 0, 1 (kdet1), 2 (Feq, kdet1), and 3(Fk1, kdet1, kdet2) fitting parameters, respectively. Simulations
can be quite different when using the various release model
formulations. In general, release occurs very rapidly whenusing the equilibrium model and this approach has limited
flexibility due to the lack of fitting parameters. The kinetic
model can reproduce a wide range of release behaviors
depending on the value of kdet1. When kdet1 is high the kinetic
model approaches equilibrium conditions, whereas when kdet1
is low the kinetic model predicts slow and prolonged release.
The greatest model flexibility occurs with the combined
equilibrium and kinetic or two-site kinetic models because of
the greater number of parameters. Variation of these parameters can produce different combinations of rapid (equilibrium)and slow kinetic release behavior. It should also be mentionedthat all of the above models are proportional to Si. Consequently, factors that promote smaller amounts of retention during

148

S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

Phases 1 and 2 (e.g., lower IS and influent colloid concentrations), will also produce smaller amounts of release during
Phase 3.
4. Results and discussion
4.1. Amount of transient colloid release
A critical issue in the application of the above models is thedetermination of the functional dependency of fnr (which isrelated to Af and Smax by Eq. (6)) on solution chemistry and/orhydrodynamic parameters. In this work, values of fnr weredirectly determined using mass balance information from thetransient release experiments. Fig. 2a presents experimental
values of fnr as a function of IS for E. coli D21g when the cells
were initially deposited in 100 mM NaCl solution at pH = 5.8. A
decrease in IS produces a nonlinear decrease in fnr, especially at
lower IS. This behavior reflects successive decreases in theadhesive interaction between D21g and the porous medium. In
particular, the calculated height of the energy barrier increases
and the depth of the secondary minimum decreases for
decreasing IS (Table 1). The final value of fnr = 0.47 in DI
water at pH = 5.8, which indicates that a significant amount of
D21g was still retained in the sand even when the secondary
minimum was eliminated. Most of the D21g can be recoveredfrom quartz sand after excavation of the sand into excess
solution of the same solution chemistry (Torkzaban et al.,2008). The nonlinearity of fnr at lower IS and continued D21g
retention in DI water likely reflect the effects of spatial
variations in the distribution of the applied hydrodynamic
and resisting adhesive torques that are discussed in Section 1.
It should also be mentioned that values of fnr shown in Fig. 2
can be converted to Af = fnrAif using Eq. (6). In this case, Eq. (4)can be used to calculate Aif from values of Smax (Table 2) thatwere obtained by fitting to the breakthrough curve during
Phases 1 and 2. Interestingly, the value of Aif was very small(b 0.8–1.9%) even though the solution chemistry (100 mM NaCland pH = 5.8) was predicted to be favorable for cell attachment(Table 1). Furthermore, DLVO theory predicts that the cells
would be irreversibly retained in a primary minimum becauseof its infinite depth. These discrepancies can be explained byfactors that are not considered in these DLVO calculations. For
example, the combined influence of Born repulsion andnanoscale surface roughness has been demonstrated to
produce a finite depth of the primary minimum that changes
with solution IS in a similar manner to the secondary minimum(Bradford and Torkzaban, 2013; Shen et al., 2012), and some of
the colloids in a primary minimum may therefore be susceptible to hydrodynamic removal (Treumann et al., 2014). Colloid
release from a primary minimum during physicochemical
perturbations has also been attributed to chemical heterogeneity and steric forces (Pazmino et al., 2014).
Fig. 2b presents experimental values of fnr as a function of pH
for E. coli D21g when the cells were initially deposited in 100 mM
NaCl solution at pH = 5.8. An increase in pH produces almost alinear decrease in fnr. Similar to the IS data, these changes
reflect successive decreases in the mean adhesive interaction.
An increase in solution pH tends to produce a decrease in thezeta potentials (more negative) for D21g and the sand thatmake conditions less favorable for attachment. However,strong primary or deep secondary minima (b−19.6 kT) werepredicted for all of these pH conditions because of the high IS
conditions (Table 1). Consequently, these DLVO calculations
cannot explain the observed release behavior. The DLVO
calculations employed macroscopic values of zeta potentials,
and therefore do not account for the influence of nanoscalechemical heterogeneity on interaction energies. Nanoscalechemical heterogeneity can have a large influence on themean and variance of the interaction energy (Bendersky andDavis, 2011; Bradford and Torkzaban, 2013). Consequently, theobserved sensitivity of fnr to pH implies that the amount or
charge of the chemical heterogeneity is decreasing with anincrease in solution pH due to deprotonation of positively or
neutrally charged sites. The presence of nanoscale chemical
heterogeneity may also explain some of the cell retention in DI(Fig. 2a). However, the final value of fnr = 0.58 in 100 mM NaClsolution at pH = 10, and this indicates that nanoscale chemical
heterogeneity cannot explain most of the D21g retention.
Additional experiments were conducted to examine theinfluence of qw on D21g release. In this case, D21g was
deposited in 100 mM NaCl at pH 5.8 (Phases 1 and 2), andthen the IS was reduced to 5 mM NaCl (Phase 3). The value ofqw was then sequentially increased from 0.27 to 5.4 cm min−1(Phase 3). Fig. 2c presents experimental values of fnr as afunction of qw for D21g when the IS = 5 mM NaCl. Little D21g
release was observed with increases in qw. Some researchers
have noted a similar insensitivity of colloid release to increasing
qw (Johnson et al., 2010), whereas others have shown a greater
dependency of colloid release on qw (Bergendahl and Grasso,
1998, 2000). These differences can likely be explained bydifferences in the adhesive interaction, the hydrodynamic
conditions, and the locations for retention. In particular, theeffects of qw are expected to increase when the resistingadhesive torque is smaller (e.g., lower IS and/or higher pH).
Furthermore, the applied hydrodynamic torque increases withthe cube of the colloid radius, and also increases in finer
textured media (Torkzaban et al., 2007). Larger scale roughness
locations and grain–grain contact points will dramaticallyincrease the resisting adhesive torque and decrease the appliedhydrodynamic torque at these locations in comparison tosmooth surfaces (Bradford et al., 2013; Burdick et al., 2005).
Both of these factors will tend to diminish the role of qw on
release.
4.2. Simulated colloid release
The developed models are used in this section to simulate
colloid release during transient solution chemistry conditions.
We did not attempt to simulate the colloid release behavior
with increases in qw (Fig. 2c) because only small amounts ofcolloid release occurred which were near the analytic detection
limit. The initial conditions for these simulations were
determined by fitting steady-state retention and releasemodel parameters (ksw, krs, and Smax/Co) to breakthroughcurve data for Phases 1 and 2. These model parameters areprovided in Table 2, along with the Pearson correlation
coefficient (R2) for the goodness of model fit. The dispersivity(λ = D/v; where v is the pore water velocity) was set equal to
0.1 cm for all experiments based on published tracer results(Wang et al., 2013), and preliminary optimization results
during Phases 1, 2, and 3. Fitted release model parameters
during Phase 3 are provided in Table 3, along with R2 and theS.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

149

Table 3Model parameters for the colloid release curve during Phase 3. The standard error is given in the parenthesis.
Fig.

Transient

Model

kdet1
min−1

kdet2
min−1

Feq

Fk1

Fk2

R2

AIC

4a
4a
4a
4a
4b
4b
4b
4b
5a
5b

IS
IS
IS
IS
pH
pH
pH
pH
Ex./IS
Ex./IS

E
K
EK
2K
E
K
EK
2K
K
K

NA
0.395 (0.064)
0.420 (0.115)
0.063 (1.450)
NA
0.027 (0.002)
0.023 (0.002)
0.019 (0.003)
1.000 (0.275)
0.192 (0.008)

NA
NA
NA
0.391 (0.144)
NANA
NA
0.392 (0.118)
NA
NA

1.0
NA
0.01 (0.19)
NA
1.0
NA
0.10 (0.01)
NA
NA
NA

NA
1.00
0.99
0.01 (0.30)
NA
1.00
0.90
0.84 (0.03)
1.00
1.00

NA
NA
0.00
0.99
NA
NA
NA
0.16
NA
NA

0.54
0.78
0.78
0.78
0.25
0.47
0.60
0.65
0.94
0.95

NA
−631
−628
−626
NA
−963
−1007
−1017
−429
−552

NA — denotes not applicable.
Ex. — denotes cation exchange.
E — denotes equilibrium model.
K — denotes kinetic model.
EK — denotes equilibrium and kinetic model.
2K — denotes 2 site kinetic model.

2.5 Phases 1 & 2

Phase 3

2.0

Data
Model
IS

(a)

1.5

120
100
80
60

1.0

40

0.5

IS (mM)

where n [–] is the number of observations, σ2 [–] is the residual
variance estimated as the sum of squared residuals divided byn, and k [–] is the number of estimated parameters. The valueof R2 tends to increase with the number of model fitting
parameters. Conversely, the AIC is a measure of the goodness of
fit that penalizes for adding fitting parameters. The model that
produces that best description of the data with the smallest
number of fitting parameters (the lowest AIC) is preferred.
Fig. 3 presents observed D21g breakthrough and release
curves with changes in IS (Fig. 3a) and pH (Fig. 3b). Simulatedchanges in the solution composition (IS or pH) are also shownin these figures. The final reduction in solution IS from 1 mM to
DI water is difficult to see in Fig. 3a, but it occurs after around
420 min. The release curves were simulated using the various
model formulations and the model result with the lowest AIC
is shown in the figure. Release model parameters and AIC
information provide valuable information on the rates ofcolloid release and the most appropriate model formulation
under the considered physicochemical conditions, respectively.
Values of AIC indicate that the one-site kinetic model waspreferred for D21g release with changes in solution IS, and thatthe agreement between the model and data was reasonable(Table 3). The fitted value of kdet1 was high (kdet1 =0.395 min−1). This implies that colloid release and changes in
the mean adhesive interaction (primary or secondary minimum) with IS reduction (Table 1) occurred very rapidly andapproached equilibrium conditions. In contrast, AIC values
indicated that the two-site kinetic model was best to describe
release with changes in solution pH and that it provided areasonable characterization of the data (Table 3). The firstkinetic site exhibited a slow rate of colloid release (kdet1 =0.019 min−1), whereas the second kinetic site produced amuch more rapid rate of release (kdet2 = 0.392 min−1). The
rapid colloid release behavior likely reflects changes in themean adhesive interaction (Table 1) in a manner similar to IS

20

0.0

0

0

100

200

300

400

500

Time (min)Phase 3

1.0 Phases 1 & 2

(b)

0.8

12
10
8

0.6

Data
Model
pH

0.4
0.2

6

pH

ð18Þ

C/Co

 
2kðk þ 1Þ
2
AIC ¼ n log σ þ 2k þ
n−k−1

reduction (Fig. 3a). Conversely, the slow colloid release likely is
associated with alteration of nanoscale chemical heterogeneity
by deprotonation of hydroxyl sites as the pH increases. Most of
the initial colloid mass was associated with this slow rate ofcolloid release (Fk1 = 0.84).

C/Co

Akaike information criterion (AIC) (Akaike, 1974) for the
goodness of model fit. The AIC value was determined as:

4
2

0.0

0

0

100

200

300

400

500

600

Time (min)Fig. 3. Observed and simulated breakthrough and release curves for D21g withchanges of IS (a) and pH (b) using experimental values of fnr shown in Fig. 2.
Cells were initially deposited in 100 mM NaCl at pH 5.8 (Phases 1 and 2), thenthe IS (100, 50, 25, 5, 1, and 0 mM) or pH (5.8, 7, 8, 9, and 10) was sequentiallychanged during Phase 3. The value of qw was 0.276 cm min−1. The 1 site kinetic
model was used in (a), and the 2 site kinetic model was used in (b). Modelparameters are given in Tables 2 and 3.

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S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

Fig. 4 presents observed and simulated values of D21g
(Fig. 4a) and CML NP (Fig. 4b) breakthrough and release curves
when employing the following solution chemistry sequence:
deposition in 2 mM CaCl2 solution (Phase 1); eluting in 2 mM
CaCl2 solution (Phase 2); flushing with DI water (Phase 3a);flushing with 100 mM NaCl solution (Phase 3b); and flushingwith DI water (Phase 3c). Note that there are drastic differencesin the amount of D21g and CML NP retention and release
for this solution chemistry sequence. In particular, greater
amounts of retention occurred for CML NPs than D21g in 2 mM
CaCl2 solution. Traditional DLVO calculations cannot explain
this difference because the secondary minimum is greater for
D21g than CML NPs and the energy barrier is sufficiently highto eliminate most primary minima interactions (Table 1).
Release of D21g mainly occurred in the first DI water pulse(Phase 3a). Conversely, release of CML NPs mainly occurredwith the second DI water pulse (Phase 3c).
Nanoscale chemical heterogeneity provides one plausible
explanation for the differences in D21g and CML NP retention
and release shown in Fig. 4. In particular, surface integration
calculations demonstrate that nanoscale chemical heterogeneity can produce strong primary minima interactions for smaller
colloids, whereas larger colloids will continue to mainlyinteract in a weak secondary minimum (Bendersky and Davis,

6.0

Phases 1 & 2

5.0

(a)

Phase 3

80

Data
Model
IS

3.0
2.0

60
40

1.0

IS (mM)

100

4.0

C/Co

120

20

0.0

0

0

50

100 150 200 250 300 350

Time (min)Phases 1 & 2

(b)

0.8

C/Co

Phase 3

120
100
80

0.6

Data
Model
IS

0.4
0.2

60
40

IS (mM)

1.0

20

0.0

0

0

50

100 150 200 250 300 350

Time (min)Fig. 4. Observed and simulated breakthrough and release curves for D21g (a)and 20 nm CML NPs (b) when employing the following solution chemistry
sequence: deposition in 2 mM CaCl2 solution (Phase 1); eluting in 2 mM CaCl2
solution (Phase 2); flushing with DI water (Phase 3a); flushing with 100 mM
NaCl solution (Phase 3b); and flushing with DI water (Phase 3c). The
simulations employed values of fnr that were directly obtained from thisexperimental balance information and by fitted release parameters given in
Table 3.

2011; Duffadar and Davis, 2007). Greater retention of CML NPs
than D21g is therefore expected during Phases 1 and 2 due tonanoscale chemical heterogeneity created by adsorbed Ca2+
(Grosberg et al., 2002). The greatest amount of D21g releaseoccurred with the first DI water pulse (Phase 3a) when thesecondary minimum was eliminated. Conversely, the greatest
amount of CML NP release occurred with the second DI water
pulse (Phase 3c). This difference in release with DI water for
D21g and CML NPs is also due to differences in the sensitivity of
D21g and CML NPs to chemical heterogeneity. Addition of
100 mM NaCl (Phase 3b) initiates cation exchange (data notshown) that likely converts some of the primary to secondary
minima interactions for CML NPs but not for D21g. Values of fnr
for CML NPs were therefore lower for the second than the firstDI water pulse.
Others have similarly observed that the release of clay,
coliphage, and nanoparticles was sensitive to the processesof cation exchange (Grolimund and Borkovec, 2006; Sadeghi
et al., 2013) or the sequence of cation exchange and IS
reduction (Bradford and Kim, 2010; Torkzaban et al., 2013).
Fig. 4 indicates that D21g and CML NP release was not directly
initiated by decreasing f++ (e.g., cation exchange during Phase3b). This implies that Af was mainly a function of IS, but that thisdependency on IS was altered by a decrease in the amount ofadsorbed Ca2+ (Phase 3b), especially for the smaller CML NPs.
Similar to release with a reduction of IS shown in Fig. 3a, therelease behavior of D21g and CML NPs occurred very rapidly
and approached equilibrium conditions that were reasonably
described by the one-site kinetic model (Table 3). Additional
research is needed to better resolve the dependence of Af on
f++ and IS.
5. Summary and conclusions
Two fundamental issues must be resolved in order tosimulate colloid release with transient physicochemical conditions. One challenge is determining the amount of colloid
release with a given perturbation. In this work, we relate theamount of transient colloid release to changes in the fraction of
the solid surface area that contributes to colloid retention (Af).
Functional relations between Af and selected physicochemical
conditions may be determined from experimental mass balance
information or estimated theoretically. A second challenge ismodeling the rates of colloid release. We develop equilibrium,
kinetic, and combined equilibrium and kinetic models to
describe various colloid release behavior with changes in Af.
Our approach is quite general and is applicable to colloid release
initiated by alteration of mean and/or local (nanoscale
heterogeneity) adhesive interactions (secondary or primary
minima), as well as hydrodynamic forces.
Column transport and release experiments were conductedto study the release behavior of E. coli D21g and CML NPs under
different physicochemical conditions, to determine Af relations,and to test the ability of the developed models to describe thisdata. Release of E. coli D21g from a primary minimum waspromoted by a decrease in solution IS, similar to expected
trends for a secondary minimum. However, a large fraction of
the D21g was still retained in the sand even when thesecondary minimum was eliminated, suggesting the importantrole of macroscopic surface roughness and grain–grain contacts
on cell retention. Cell release was also promoted by increases

S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

in solution pH even though strong primary or secondary
minimum were predicted. This observation suggested that a
reduction in nanoscale chemical heterogeneity with an increase in pH also contributed to cell release. In contrast to D21g,
release of CML NPs that were initially deposited in the presenceof 2 mM CaCl2 solution occurred only after exchange of Ca2+ by
Na+ and then a reduction in the solution IS. These observations
suggest that reversible primary minimum interactions canoccur for colloids on solid surfaces due to factors such as nonDLVO forces and nanoscale physical and chemical heterogeneity. Temporal changes in the water velocity did not have a large
influence on the release of D21g. This insensitivity was likely
due to factors that reduce the applied hydrodynamic torqueand/or increase the resisting adhesive torque; e.g., large-scaleroughness locations and grain–grain contacts.
The developed models provided a reasonable description of
the colloid release when parameters were optimized to theexperimental data. However, the required complexity of therelease model depended on the specific transient condition. For
example, D21g release with changes in IS was well describedwith a one-site kinetic model that approached equilibrium
conditions. This implies that changes in the mean adhesiveinteraction (primary or secondary minimum) with IS reduction
occurred very rapidly. In contrast, D21g release with changes in
pH was better quantified with a two-site kinetic model, withmost of the colloid release occurring slowly and only a small
fraction rapidly. The rapid colloid release likely reflects changes
in the mean adhesive interaction in a similar manner to the IS
data. The slow colloid release presumably occurred due toalteration of nanoscale chemical heterogeneity by deprotonation of hydroxyl sites as the pH increases.

Acknowledgments
This research was supported by the 214 Manure andByproduct Utilization Project of the USDA-ARS. Mention of
trade names and company names in this manuscript does not
imply any endorsement or preferential treatment by the USDA.
We would also like to acknowledge the help of Dr. Brendan
Headd in conducting these transport and release experiments.

References
Adamczyk, Z., Siwek, B., Zembala, M., 1992. Reversible and irreversible
adsorption of particles on homogeneous surfaces. Colloids Surf. 62, 119–130.
Adamczyk, Z., Siwek, B., Zembala, M., Belouschek, P., 1994. Kinetics of localized
adsorption of colloid particles. Adv. Colloid Interf. Sci. 48, 151–280.
Adamczyk, Z., Siwek, B., Szyk, L., 1995. Flow-induced surface blocking effects inadsorption of colloid particles. J. Colloid Interface Sci. 174, 130–141.
Akaike, H., 1974. A new look at statistical model identification. IEEE Trans.
Autom. Control 19, 716–722.
Bedrikovetsky, P., Siqueira, F.D., Furtado, C.A., Souza, A.L.S., 2011. Modified
particle detachment model for colloidal transport in porous media. Transp.
Porous Media 86, 353–383.
Bendersky, M., Davis, J.M., 2011. DLVO interaction of colloidal particles with
topographically and chemically heterogeneous surfaces. J. Colloid Interface
Sci. 353, 87–97.
Bergendahl, J., Grasso, D., 1998. Colloid generation during batch leaching tests:
mechanics of disaggregation. Colloids Surf. A Physicochem. Eng. Asp. 135,
193–205.
Bergendahl, J., Grasso, D., 1999. Prediction of colloid detachment in a model
porous media: thermodynamics. AICHE J. 45, 475–484.
Bergendahl, J., Grasso, D., 2000. Prediction of colloid detachment in a model
porous media: hydrodynamics. Chem. Eng. Sci. 55, 1523–1532.

151

Bradford, S.A., Kim, H., 2010. Implications of cation exchange on clay releaseand colloid-facilitated transport in porous media. J. Environ. Qual. 39,
2040–2046.
Bradford, S.A., Kim, H., 2012. Causes and implications of colloid andmicroorganism retention hysteresis. J. Contam. Hydrol. 138–139, 83–92.
Bradford, S.A., Torkzaban, S., 2012. Colloid adhesive parameters for chemicallyheterogeneous porous media. Langmuir 28, 13643–13651.
Bradford, S.A., Torkzaban, S., 2013. Colloid interaction energies for physicallyand chemically heterogeneous porous media. Langmuir 29, 3668–3676.
Bradford, S.A., Kim, H., Haznedaroglu, B.Z., Torkzaban, S., Walker, S.L., 2009.
Coupled factors influencing concentration-dependent colloid transport andretention in saturated porous media. Environ. Sci. Technol. 43, 6996–7002.
Bradford, S.A., Torkzaban, S., Wiegmann, A., 2011. Pore-scale simulations todetermine the applied hydrodynamic torque and colloid immobilization.
Vadose Zone J. 10, 252–261.
Bradford, S.A., Torkzaban, S., Kim, H., Simunek, J., 2012. Modeling colloid andmicroorganism transport and release with transients in solution ionicstrength. Water Resour. Res. 48, W09509. http://dx.doi.org/10.1029/
2012WR012468.
Bradford, S.A., Torkzaban, S., Shapiro, A., 2013. A theoretical analysis of colloid
attachment and straining in chemically heterogeneous porous media.
Langmuir 29, 6944-2952.
Burdick, G.M., Berman, N.S., Beaudoin, S.P., 2005. Hydrodynamic particle
removal from surfaces. Thin Solid Films 488, 116–123.
Delory, G.E., King, E.J., 1945. A sodium carbonate–bicarbonate buffer for alkaline
phosphatases. Biochem. J. 39, 245-245.
Derjaguin, B.V., Landau, L.D., 1941. Theory of the stability of strongly charged
lyophobic sols and of the adhesion of strongly charged particles in solutionsof electrolytes. Acta Physicochim. U.S.S.R. 14, 733–762.
Duffadar, R.D., Davis, J.M., 2007. Interaction of micrometer-scale particles withnanotextured surfaces in shear flow. J. Colloid Interface Sci. 308, 20–29.
Elimelech, M., Gregory, J., Jia, X., Williams, R.A., 1995. Particle Deposition andAggregation: Measurement, Modeling, and Simulation. ButterworthHeinemann, Oxford, England.
Gregory, J., 1981. Approximate expression for retarded van der Waals
interaction. J. Colloid Interface Sci. 83, 138–145.
Grolimund, D., Borkovec, M., 2006. Release of colloidal particles in natural
porous media by monovalent and divalent cations. J. Contam. Hydrol. 87,
155–175.
Grolimund, D., Barmettler, K., Borkovec, M., 2001. Release and transport ofcolloidal particles in natural porous media 2. Experimental results andeffects of ligands. Water Resour. Res. 37, 571–582.
Grosberg, A.Y., Nguyen, T.T., Shklovskii, B.I., 2002. Colloquium: the physics ofcharge inversion in chemical and biological systems. Rev. Mod. Phys. 74, 329.
Hogg, R., Healy, T.W., Fuerstenau, D.W., 1966. Mutual coagulation of colloidal
dispersions. Trans. Faraday Soc. 62, 1638–1651.
Johnson, P.R., Elimelech, M., 1995. Dynamics of colloid deposition in porous
media: blocking based on random sequential adsorption. Langmuir 11,
801–812.
Johnson, W.P., Pazmino, E., Ma, H., 2010. Direct observations of colloid retention
in granular media in the presence of energy barriers, and implications
for inferred mechanisms from indirect observations. Water Res. 44,
1158–1169.
Khilar, K.C., Fogler, H.S., 1984. The existence of a critical salt concentration forparticle release. J. Colloid Interface Sci. 101, 214–224.
Lenhart, J.J., Saiers, J.E., 2003. Colloid mobilization in water-saturated porous
media under transient chemical conditions. Environ. Sci. Technol. 37,
2780–2787.
Mayer, K.U., Frind, E.O., Blowes, D.W., 2002. Multicomponent reactive transportmodeling in variably saturated porous media using a generalized
formulation for kinetically controlled reactions. Water Resour. Res. 38,
1174. http://dx.doi.org/10.1029/2001WR000862.
Pazmino, E., Trauscht, J., Johnson, W.P., 2014. Release of colloids from primary
minimum contact under unfavorable conditions by perturbations in ionicstrength and flow rate. Environ. Sci. Technol. 48, 9227–9235.
Rijnaarts, H.H.M., Norde, W., Bouwer, E.J., Lyklema, J., Zehnder, A.J.B., 1995.
Reversibility and mechanism of bacterial adhesion. Colloids Surf. B 4, 5–22.
Roy, S.B., Dzombak, D.A., 1996. Colloid release and transport processes innatural and model porous media. Colloids Surf. A Physicochem. Eng. Asp.
107, 245–262.
Ryan, J.N., Elimelech, M., 1996. Colloid mobilization and transport ingroundwater. Colloids Surf. A Physicochem. Eng. Asp. 107, 1–56.
Ryan, J.N., Gschwend, P.M., 1994. Effects of ionic-strength and flow-rate on
colloid release — relating kinetics to intersurface potential-energy. J. Colloid
Interface Sci. 164, 21–34.
Sadeghi, G., Behrends, T., Schijven, J.F., Hassanizadeh, S.M., 2013. Effect ofdissolved calcium on the removal of bacteriophage PRD1 during soil
passage: The role of double-layer interactions. J. Contam. Hydrol. 144, 78–87.
Sasidharan, S., Torkzaban, S., Bradford, S.A., Dillon, P.J., Cook, P.G., 2014. Coupledeffects of hydrodynamic and solution chemistry on long-term nanoparticle

152

S.A. Bradford et al. / Journal of Contaminant Hydrology 181 (2015) 141–152

transport and deposition in saturated porous media. Colloids Surf. A
Physicochem. Eng. Asp. 457, 169–179.
Shen, C., Li, B., Huang, Y., Jin, Y., 2007. Kinetics of coupled primary- andsecondary-minimum deposition of colloids under unfavorable chemical
conditions. Environ. Sci. Technol. 41, 6976–6982.
Shen, C., Lazouskaya, V., Zhang, H., Wang, F., Li, B., Jin, Y., Huang, Y., 2012.
Theoretical and experimental investigation of detachment of colloids fromrough collector surfaces. Colloids Surf. A Physicochem. Eng. Asp. 410,
98–110.
Simoni, S.F., Harms, H., Bosma, T.N.P., Zehnder, A.J.B., 1998. Population
heterogeneity affects transport of bacteria through sand columns at low
flow rates. Environ. Sci. Technol. 32, 2100–2105.
Šimůnek, J., Valocchi, A.J., 2002. Geochemical Transport. In: Dane, J.H., Topp, G.C.
(Eds.), Methods of Soil Analysis, Part 4, Physical Methods, Third edition
SSSA, Madison, WI, pp. 1511–1536 (Chapter 6.9).
Šimůnek, J., van Genuchten, M.Th., Šejna, M., 2008. Development andapplications of the HYDRUS and STANMOD software packages and related
codes. Vadose Zone J. 7, 587–600. http://dx.doi.org/10.2136/VZJ2007.0077(Special Issue “Vadose Zone Modeling”).
Suresh, L., Walz, J.Y., 1996. Effect of surface roughness on the interaction energy
between a colloidal sphere and a flat plate. J. Colloid Interface Sci. 183,
199–213.
Torkzaban, S., Bradford, S.A., Walker, S.L., 2007. Resolving the coupled effects ofhydrodynamics and DLVO forces on colloid attachment to porous media.
Langmuir 23, 9652–9660.
Torkzaban, S., Tazehkand, S.S., Walker, S.L., Bradford, S.A., 2008. Transport andfate of bacteria in porous media: coupled effects of chemical conditions andpore space geometry. Water Resour. Res. 44, W04403. http://dx.doi.org/10.
1029/2007WR006541.
Torkzaban, S., Bradford, S.A., Wan, J., Tukunaga, T., Masoudih, A., 2013. Releaseof quantum dot nanoparticles in porous media: role of cation exchange andaging time. Environ. Sci. Technol. 47, 11528–11536.
Tosco, T., Tiraferri, A., Sethi, R., 2009. Ionic strength dependent transport of
microparticles in saturated porous media: modeling mobilization andimmobilization phenomena under transient chemical conditions. Environ.
Sci. Technol. 43, 4425–4431.
Treumann, S., Torkzaban, S., Bradford, S.A., Visalakshan, R.M., Page, D., 2014. An
explanation for differences in the process of colloid adsorption in batch andcolumn studies. J. Contam. Hydrol. 164, 219–229.
Tufenkji, N., Elimelech, M., 2005. Breakdown of colloid filtration theory: roleof the secondary energy minimum and surface charge heterogeneities.
Langmuir 21, 841–852.
Vaidyanathan, R., Tien, C., 1991. Hydrosol deposition in granular media underunfavorable surface conditions. Chem. Eng. Sci. 46, 967–983.
Verwey, E.J.W., Overbeek, J.Th.G., 1948. Theory of the Stability of Lyophobic
Colloids. Elsevier, Amsterdam.
Waentig, L., Roos, P.H., Jakubowski, N., 2009. Labelling of antibodies anddetection by laser ablation inductively coupled plasma mass spectrometry.
Part III. Optimization of antibody labelling for application in a Western blot
procedure. J. Anal. At. Spectrom. 24, 924–933.
Wang, Y., Bradford, S.A., Simunek, J., 2013. Transport and fate of microorganisms in soils with preferential flow under different solution chemistry
conditions. Water Resour. Res. 49, 2424–2436.