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Mass transfer model of nanoparticle-facilitated contaminanttransport in saturated porous media

Wan Lutfi Wan Johari, Peter J. Diamessis, Leonard W. Lion*

School of Civil & Environmental Engineering, Hollister Hall, Cornell University, Ithaca, NY 14853, USA

a r t i c l e i n f o

Article history:

Received 7 July 2008

Received in revised form

19 November 2008

Accepted 16 March 2009

Available online 1 April 2009

Keywords:

Facilitated contaminant transport

Gamma distribution

Nanoparticle

* Corresponding author. Tel.: þ1 607 255 75E-mail address: LWL3@cornell.edu (L.W.

0043-1354/$ – see front matter ª 2009 Elsevidoi:10.1016/j.watres.2009.03.033

a b s t r a c t

A one-dimensional model has been evaluated for transport of hydrophobic contaminants,

such as polycyclic aromatic hydrocarbon (PAH) compounds, facilitated by synthetic

amphiphilic polyurethane (APU) nanoparticles in porous media. APU particles synthesized

from poly(ethylene glycol)-modified urethane acrylate (PMUA) precursor chains have been

shown to enhance the desorption rate and mobility of phenanthrene (PHEN) in soil.

A reversible process governed by attachment and detachment rates was considered to

describe the PMUA binding in soil in addition to PMUA transport through advection and

dispersion. Ultimately, an irreversible second-order PMUA attachment rate in which the

fractional soil saturation capacity with PMUA was a rate control was found to be adequate

to describe the retention of PMUA particles. A g-distributed site model (GS) was used to

describe the spectrum of physical/chemical constraints for PHEN transfer from solid to

aqueous phases. Instantaneous equilibrium was assumed for PMUA–PHEN interactions.

The coupled model for PMUA and PHEN behavior successfully described the enhanced

elution profile of PHEN by PMUA. Sensitivity analysis was performed to analyze the

significance of model parameters on model predictions. The adjustable parameter a in the

g-distribution shapes the contaminant desorption distribution profile as well as elution

and breakthrough curves. Model simulations show the use of PMUA can be also expected to

improve the release rate of PHEN in soils with higher organic carbon content. The

percentage removal of PHEN mass over time is shown to be influenced by the concentra-

tion of PMUA added and this information can be used to optimize cost and time require to

accomplish a desired remediation goal.

ª 2009 Elsevier Ltd. All rights reserved.

1. Introduction organic contaminant such as a polynuclear aromatic hydro-

Modeling of facilitated transport in the context of soil reme-

diation requires appropriate description of the soil transport

of the agent (or carrier) introduced to enhance contaminant

mobility (e.g. attachment/sorption, re-entrainment/desorp-

tion, equlibria and rates for these processes, size exclusion

from pores, etc) and of the desorptive release of the contam-

inant from soil. In this paper, we develop a model designed to

consider the case where the pollutant is a hydrophobic

71; fax: þ1 607 255 9004.Lion).er Ltd. All rights reserved

carbon (PAH) and the remediation agent is a synthetic nano-

particle designed to mimic a surfactant micelle with

a hydrophobic core and hydrophilic exterior. Laboratory data

for the facilitated transport of a model PAH, phenanthrene

(PHEN) by amphiphilic nanoparticles have been reported

previously (Tungittiplakorn et al., 2005, 2004) and provide

a test case for model evaluation.

Remediation of soil containing hydrophobic contaminants

such as PAH compounds is hindered by their slow desorption

.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 7 1029

kinetics. Research has shown that sorption can take years to

reach equilibrium (Carroll et al., 1994; Coates and Elzerman,

1986). The persistence of PAHs in soil particles and their slow

desorption kinetics necessitate modeling approaches

amenable to simulation of long time scales. While several

models have been proposed to describe the mass transfer of

PAHs in porous media, we restrict consideration here to the

two-site model (TS) and g-distributed site model (GS). In a TS

model, the soil matrix is divided into two types of sorption

sites (van Genuchten and Wagenet, 1989). One site type is

assumed to be at instantaneous equilibrium with the aqueous

phase and the other site type is governed by a single mass

transfer coefficient. In the GS model, the rate constants for

mass transfer are assumed to follow a probability distribution

(Ahn et al., 1996; Connaughton et al., 1993; Culver et al., 1997;

Pedit and Miller, 1994) which is used to describe the spectrum

of physical (flow and diffusion) and chemical (site) constraints

on contaminant release from the stationary soil matrix to the

groundwater. For a contaminant with a known sorptive

distribution coefficient, both TS and GS models have two

adjustable parameters that can be changed to fit experimental

data. The flexibility of the GS model has allowed successful

prediction of the release rates of naphthalene from fresh and

aged contaminated soil samples (Ahn et al., 1996, 1999a),

predicted transport of volatile organic compounds in unsat-

urated soil (Lorden et al., 1998) and soil transport of transition

metals in the presence of a mobile metal-binding ligand

(Jensen-Spaulding et al., 2004b). Beyond its ability to fit

experimental results, the GS model has been shown to have

a predictive ability that the TS model lacked in comparative

tests (Ahn et al., 1996; Chen and Wagenet, 1995; Jensen-

Spaulding et al., 2004b; Lorden et al., 1998). Thus, the GS model

is selected for use in this evaluation for description of PAH

behavior in the presence and absence of a mobile amphiphilic

nanoparticle carrier.

Surfactant micelles have been used as mobile agents to

enhance the desorption rate of hydrophobic organic contam-

inants from soil particles and shorten the time required for

remediation (Abdul and Gibson, 1991; Ang and Abdul, 1994;

Deitsch and Smith, 1995; Grimberg et al., 1995, 1996; Rouse

et al., 1993). Micelles possess a hydrophobic core which serves

as sorbent matrix for the organic contaminants in soil and

a hydrophilic surface that facilitates the transport of surfac-

tant aggregate in porous media. However, micelles are not

stable and the surfactant molecules will also react with soil

surfaces (Abdul and Gibson, 1991; Rouse et al., 1993). The

structure of surfactant micelles can only be maintained when

the aqueous concentration of the surfactant monomers rea-

ches or exceeds the critical micelle concentration. Several

studies have been focused on the surfactant designs to mini-

mize losses due to breakage, to develop reuse and recycle

techniques and to eliminate toxicity effects on biodegradation

(Hasegawa et al., 1997; Lipe et al., 1996; McCray and Brusseau,

1998; Rouse et al., 1993).

Polymeric nanoparticles have been synthesized to produce

attributes similar to those of surfactant micelles and can be

used as an alternative carrier to enhance hydrophobic

contaminant transport in porous media. Amphiphilic poly-

urethane (APU) nano polymer particles were developed by

Kim et al. (2000) from polyurethane acrylate anionomer (UAA)

precursor chains. Unlike surfactants, the APU particle

precursor chains are cross-linked preventing the break up of

nanoparticle in soil. Thus, the stability of APU particles is

maintained regardless of precursor chain concentration.

The properties of APU particles can be controlled by using

alternative formulations and synthesis methods (Tung-

ittiplakorn et al., 2004). Nanoparticles with polar hydroxyl

surface functional groups are less susceptible than those with

carboxyl groups to coagulation by divalent cations (Tung-

ittiplakorn et al., 2004). Nanoparticles with an average size of

80� 15 nm synthesized from poly(ethylene glycol)-modified

urethane acrylate (PMUA) precursor chains have been shown

in laboratory experiments to improve the desorption rate of

PHEN from soil, facilitate the contaminant’s mobility in

porous media, and increase the bioavailability of PHEN to

microbial populations. In contrast, surfactant micelles can

inhibit PAH biodegradation rates (Chen et al., 2001; Kim and

Weber, 2003; Laha and Luthy, 1991).

The motivation of this study is to model the transport

behavior of PMUA nanoparticles in porous media and to

explore alternative ways to explain nanoparticle-facilitated

contaminant transport in the soil matrix. Below, a one-

dimensional model is developed by coupling use of the GS

model to describe the PHEN sorption and desorption as well as

alternative models for PMUA–soil interaction. Model param-

eters are fit to the experimental data of Tungittiplakorn et al.

(2004) and simulation results are compared to experimental

observations. Because of the long time scales involved in

PHEN desorption and rapid reaction of PHEN with PMUA

(Tungittiplakorn et al., 2004), instantaneous equilibrium is

assumed for PMUA–PHEN interactions. Coupling of a GS

contaminant sorption model with a colloid transport model

has not been exploited in prior attempts to model facilitated

transport. The resulting model is also utilized here to evaluate

the importance of selected parameters to PHEN and PMUA

behavior in porous media. Other researchers (Montas and

Shirmohammadi, 2004) have modeled the data presented by

Tungittiplakorn et al. (2004). However, the PHEN sorption

model employed in that case was a two region (TR) model,

which is mathematically equivalent to a TS model (van Gen-

uchten and Wagenet, 1989) and has been proven to be inad-

equate to describe the heterogeneity of sorption behavior of

PAH in the soil (Ahn et al., 1999b). The model of Montas and

Shirmohammadi also was unable to describe both PMUA BTCs

using the same constitutive equations with the final condition

for first application as the initial condition for the second

application. The limitation of this approach provided addi-

tional motivation for evaluation of an alternative model.

2. Methods

2.1. PMUA nanoparticle transport behavior

Four processes are considered in modeling the mass transfer

and transport of PMUA particles in porous media: advection,

dispersion, attachment (deposition) and detachment

(re-entrainment). The effluent concentration of PMUA parti-

cles, N [M/V], from a column of length L is modeled using

generalized colloid transport equation:

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 71030

vNvtþ r

nvNS

vt¼ v

vz

�DN

vNvz� uNN

�(1)

where NS is the concentration of PMUA particles attached to

the soil matrix [M/M], DN is the PMUA dispersion coefficient

[L2/T], uN is the average pore velocity of PMUA particles, r is

the bulk density of the porous media [M/V], n is soil porosity

[V/V], t is time [T] and z is length [L].

In the case where binding of nanoparticles to soil is

reversible, mass transfer of PMUA particles between the

aqueous and soil phases can be expressed by the difference

between the attachment and detachment rates:

vNS

vt¼ kanN� kdnNS (2)

where kan is the first-order PMUA particles attachment coef-

ficient [V/(MT)] and kdn is the first-order order detachment

coefficient [1/T] for soil-bound PMUA particles. kdn equals zero

if nanoparticle binding to soil is irreversible.

In many instances, a blocking phenomenon occurs when

the colloid concentration on the porous media approaches

maximum colloidal retention capacity. In these cases, the

attachment rate can be also described as dependent on

the fraction of porous medium available for deposition, J. The

attachment rate will approach to zero near the maximum

retention capacity and, at that point, colloidal particle trans-

port is no longer influenced by interaction with the soil phase.

The second-order kinetic equation that describes this

phenomenon is given by (Saiers et al., 1994):

vNS

vt¼ jkanN� kdnNS (3)

j ¼ 1� NS

NmaxS

(4)

where J is the dimensionless colloid retention function [�]

and NmaxS is the maximum solid-phase particle concentration

[M/M].

The TR model is used here as an alternative model to

describe the fate and transport of nanoparticles (van Gen-

uchten and Wagenet, 1989). The TR model is given by:

vNS

vt¼ fmKN

S

vNvtþ lm

rðN�N2Þ (5a)

lm

rðN�N2Þ ¼

h1þ

�1� fm

�rn

KNS

inr

vN2

vt(5b)

where fm is the fraction of mobile fluid, lm is the mass transfer

rate coefficient from mobile to immobile fluid [1/T], and N2 is

the PMUA concentration in the immobile region [M/M]. Poros-

ities in mobile and immobile regions are assumed to be equal.

2.2. PHEN transport behavior

PHEN transport in porous media can be described by one-

dimensional advection–dispersion–sorption equation (Ahn

et al., 1999b; Chen and Wagenet, 1995):

vCvtþ r

nvSvt¼ v

vz

�D

vCvz� uC

�(6)

where C is the PHEN concentration in aqueous phase [M/V], u is

the average pore-water velocity [L/T], D is the PHEN dispersion

coefficient [L2/T] and vS=vt is the PHEN mass transfer between

the aqueous phase and the solid matrix. In this analysis, a GS

model is used to describe vS=vt and the average pore velocity

for PHEN and PMUA are assumed to be equal.

In the GS model, a g distribution of first-order sorption/

desorption rate constants is used to account for the influence of

soil matrix heterogeneity on the dynamic interaction between

the soil matrix and a contaminant. PHEN is used as represen-

tative PAH for purposes of comparison to available data.

The change of the sorbed PHEN concentration, vCS=vt, in

a continuum of soil ‘‘compartments’’ differentiated by their

mass transfer coefficients, ki, is described by the following

kinetic expression for PHEN desorption from soil:

vCS;i

vt¼ �ki

�CS;i � KC

S C�

(7)

where CS;i is the PHEN concentration which is attached onto

the ith soil compartment [M/M], C is the aqueous PHEN

concentration [M/V], KCS is the distribution coefficient of PHEN

to the soil [V/M], ki is the rate constant for the ith compart-

ment [1/T] and k is described by a g-distribution density

function, where b is a scale parameter [T] and a is a shape

parameter [�]:

fðkÞ ¼ baka�1expð�bkÞ=gðaÞ (8)

where: gðaÞ ¼RN

0 xa�1expð�xÞdx. The expected value of the

distribution [1/T], E(k) is given by a=b. The total amount of

PHEN per unit amount of soil, S [M/M] and its time derivative

are given by:

S ¼ZN

0

CSðkÞfðkÞdk (9)

vSvt¼ZN

0

vCSðkÞvt

fðkÞdk ¼ a

bKC

S C�ZN

0

kCSðkÞfðkÞdk (10)

As comparison, a TS model is used to describe the soil-

sorbed PHEN (van Genuchten and Wagenet, 1989). The TS

model is given by:

vSvt¼ fKC

S

vCvtþ vS2

vt(11a)

vS2

vt¼ l�ð1� fÞKC

S C� S2

�(11b)

where f is the fraction of equilibrium sorption sites at

instantaneous equilibrium, l is the mass transfer rate coeffi-

cient [1/T], and S2 is the PHEN concentration in the time

dependent site [M/M]. van Genuchten and Wagenet, (1989)

introduced variables in dimensionless forms of Eqs. (5a), (5b),

(11a), and (11b) that are mathematically equivalent and

conveniently interchangeable between TS and TR models.

2.3. PMUA-facilitated PHEN transport

A three-component model has been used to describe the

enhancement of contaminant transport by mobile carriers in

porous media (Jensen-Spaulding et al., 2004a; Liu et al., 2001;

Magee et al., 1991). This model is used here to represent the

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 7 1031

interactions between PMUA, PHEN particles and soil and is

illustrated in Fig. 1. The equilibrium isotherm of PHEN to

PMUA particles was assumed to be linear (in accord with the

observations of Tungittiplakorn et al. (2004)) with a distribu-

tion coefficient, KCN, and the same whether the PMUA particles

were in aqueous or solid phase.

The transport of nanoparticle-facilitated contaminant in

porous media can be described by a one-dimensional advec-

tion–dispersion equation (Johnson et al., 1998):

vCvtþ r

n

�vSvtþ vðCNNSÞ

vt

þ vðCNNÞ

vt¼ v

vz

�D

vCvz� uC

�

þ v

vz

�DN

vðCNNÞvz

� uNðCNNÞð12Þ

where: CN is the concentration of PHEN which is sorbed to

PMUA [M/M]. The left-hand side (LHS) of the equation sums

the rate of change of PHEN concentration in porous media and

the right-hand side (RHS) sums the advection–dispersion

terms for the aqueous PHEN and sorbed PHEN on mobile

PMUA. Assuming instantaneous equilibrium for contaminant

sorption to nanoparticles (i.e. CN ¼ KCNC), Eq. (12) can be

simplified to:

RvCvtþ r

nvSvt¼ DT

v2Cvz2� uT

vCvz

(13)

where: R ¼ 1þ KCNNþ r=nKC

NNS; DT ¼ Dþ DNKCNN; uT ¼ u

ð1þ KCNNÞ; and KC

N is the distribution coefficient of contaminant

and nanoparticle [V/M]. The detailed derivation of Eq. (13) is

given by (Wan Johari, 2007).

Fig. 1 – Schematic diagram for three-component model

describing the interactions between PHEN particles, PMUA

nanoparticles and soil particles. C and S are the PHEN

concentration in aqueous phase [M/V] and solid phase

[M/M], respectively; N and NS are the PMUA concentration

in aqueous phase [M/V] and solid phase [M/M],

respectively; KCS and KC

N are the distribution coefficients of

PHEN to soil and to PMUA, respectively, [V/M]; NmaxS is the

maximum saturation capacity for binding PMUA [M/M], kan

and kdn are the PMUA attachment [V/(MT)] and detachment

rate [1/T] coefficients, respectively; and g indicates that

PHEN sorption is governed by the GS model.

2.4. Boundary conditions, initial conditions, anddimensionless parameters

In our simulation of laboratory column experiments, the

initial condition of contaminant in the column is given by

Cðz; 0Þ ¼ Ci where Ci is the initial concentration of the solute

[M/V]. A Dirichlet inlet boundary condition is used, Cð0; tÞ ¼ C0

where C0 is the input concentration at the inlet [M/V]. For

a finite system of length L, a zero concentration gradient is

assumed at the outlet corresponding to a Neumann boundary

condition, vCðL; tÞ=vz ¼ 0.

The mass fraction of contaminant eluted from the column

experiments (Tungittiplakorn et al., 2004) was calculated from

the area under the PHEN breakthrough curves (BTCs). For

PHEN elution with PMUA the total concentration of PHEN

eluted in aqueous phase is the sum of the aqueous concen-

tration of PHEN, C and the concentration of PHEN sorbed to

mobile PMUA, CN and is given by CT ¼ Cð1þ KCNNÞ.

2.5. Numerical methods and parameter estimation

Effluent concentrations of both PMUA and PHEN are computed

with a 2nd order finite difference scheme in space and a higher

order (O(Dt3) – Dt is the timestep) temporal discretization. Eqs.

(3) and (4) or Eqs. (5a) and (5b) are numerically solved in

concurrence with Eqs. (7), (8), and (10) before being absorbed

into Eqs. (1) and (13), respectively. Eq. (1) is computed inde-

pendently at each timestep and the output is fed into Eq. (13). In

the analysis presented in this paper, a spatial resolution of 20

longitudinal grid points was employed. A model using twenty

grid points was found to produce the same results as one with

a spatial resolution of 200 grid points (Wan Johari, 2007). All

simulations in this paper focus on the low resolution case for

the purpose of fast run turnaround. 10-point Gauss–Legendre

quadrature was used to calculate the area under the g proba-

bility curve (Carnahan et al., 1969; Davis and Rabinowitz, 2007).

The maximum k value is determined from the g-cumulative

distribution function (CDF) and set equal to the rate k with

a probability of occurrence less than 99.5%.

A fractional stepping scheme is used to integrate Eqs. (1)

and (13) in time (see Supplemental Information and Diamessis

et al. (2005) for more detail). Temporal derivatives are dis-

cretized using a third-order backward difference formula

(BDF3) (Hairer et al., 1993; Karniadakis et al., 1991). The

advection term is advanced explicitly in time using a third-

order Adams–Bashforth scheme (AB3) (Durran, 1991; Gear,

1971). The dispersion term is treated implicitly using a Crank–

Nicolson scheme (Carnahan et al., 1969; Smith, 1978). An

adaptive time-stepping scheme is implemented by

prescribing the coefficients of AB3 and BDF3 as a function of

the three previous timesteps (Hairer et al., 1993; Slinn and

Riley, 1998). Adaptive time-stepping enables the use of small

timestep values at early times, when the rates of change of

concentration are subject to rapid variations due to the

predominance of fast reaction rates, while allowing a seam-

less transition to larger timestep values at larger values when

slow reaction rates dominate. A twofold reduction in

computational cost with respect to a simulation using

a constant timestep is thus attained. Any variation in time-

step, Dt is restricted by the numerical stability conditions

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 71032

Cr< 1 and r ¼ D Dt=ðDzÞ2 < 0:5, where ½Cr ¼ uðDt=DzÞ� is the

Courant number and Dz is the spatial resolution (Smith, 1978).

The numerical model for the advective–dispersive transport

equation in the absence of a chemical reaction term has been

successfully validated against the corresponding analytical

solution (Cleary and Adrian, 1973; van Genuchten and Alves,

1982; Wan Johari, 2007).

Optimal values of unknown parameters were estimated

using a multiple nonlinear Marquardt method (Constantinides

and Mostoufi,1999; Marquardt, 1963).The method is flexible and

combines Gauss–Newton and steepest descent optimization

methods. The optimization was satisfied when the sum of

squared errors was minimized. Selected PMUA parameters,

such as DN; kan; kdn;NmaxS ; fm and lm, were optimized using the

experimental data (Tungittiplakorn, 2005). Three parameters

a;EðkÞ and KCS control the behavior of PHEN and these parame-

ters were optimized using the experimental data obtained over

thefirst10porevolumes(prior to introductionofPMUA) andalso

using all data, i.e., the combination of data obtained both prior

and subsequent to PMUA input in Fig. 7 of Tungittiplakorn et al.

(2004). The initial estimates for a;EðkÞ and KCS were determined

fromlaboratorybatch experiments (Tungittiplakorn etal.,2004;)

and a was constrained to values less than 1.

3. Results and discussion

3.1. Modeling and parameter estimation

The published experimental parameters used in optimization

and simulation are summarized in Table 1. Tungittiplakorn

et al. (2004) report PMUA BTCs for two sequential pulse

Table 1 – Reported experimental parameters used inoptimization and simulation (Tungittiplakorn, 2005;Tungittiplakorn et al., 2004).

Parameter Value

Porosity n 0.4 mL/mL

Porous medium

bulk density

r 1.3 g/mL

Average pore-water

velocitya

u 0.003 cm/s

Flow rateb Q 20 mL/h

Dispersionc D 0.00031 cm2/s

PHEN parameters

Influent concentration C0 0 mg/L

Initial concentration Ci 1 mg/L

PMUA parameters

Influent concentrationa N0 15 g/L

Influent concentrationb N0 12 g/L

Maximum saturation capacity NmaxS 1.1 mg/g

Distribution coefficients

PMUA-soil KNS 0.69 mL/g

PHEN-soil KCS 17.32 mL/g

PHEN-PMUA KCN 3263 mL/g

a PMUA breakthrough experiments.

b Enhanced PHEN elution with PMUA experiment.

c Based on measured breakthrough of nitrate.

applications. The mass recovery of the nanoparticles in the

first application was 38% and recovery was 100% in the second

application. Thus Jz0 for the second application and opti-

mizing the dispersion coefficient was sufficient to describe

PMUA behavior. The reported average pore velocity of 20 mL/h

was used and the PMUA dispersion coefficient was estimated

to be 0.00128 cm2/s, which is over four times greater than the

reported value determined from a nitrate BTC (0.00031 cm2/s).

The difference between these two values likely results from

the different size of PMUA nanoparticles vs. the nitrate anion.

The reported dispersion value for nitrate was retained in the

simulations as the PHEN dispersion coefficient.

The attachment rate coefficient, kan, and the maximum

saturation capacity, NmaxS , for PMUA particles on soil were

optimized using experimental data from both the first and

second applications simultaneously. The simulations for both

curves using Eqs. (1), (3), and (4) are shown with experimental

data in Fig. 2. The attachment rate coefficient, kan is estimated

to be 1.36 mL/(g h) and the optimal NmaxS was determined to be

1.79 mg/g, compared to 1.10 mg/g determined by the

experimenters.

A PMUA detachment coefficient, kdn, was also considered

in the optimization processes to test the efficiency of adding

another parameter on model performance; however, the

contribution of desorption rate coefficient in the model did not

markedly improve the fit of the data. Thus, the PMUA particle

transport through porous media under water-saturated

condition can be sufficiently described by irreversible attach-

ment subject to a maximum saturation capacity (Pieper et al.,

1997; Rajagopalan and Tien, 1976).

Other parameters determined through optimization are

summarized in Table 2. Fig. 2 shows that by neglecting J in

optimizing PMUA data, the model fails to describe both data

sets; therefore, the soil-sorbed PMUA concentration

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Pore volume (ut/L)

Co

ncen

tratio

n (C

/C

0)

Fig. 2 – PMUA parameters were estimated by fitting two

breakthrough curves simultaneously using multiple

nonlinear Marquardt method. (-) is the first set and (,) is

the second set of PMUA experimental data. Different lines

show different parameters being optimized: (d d) kan;

( ) kan and kdn; ( ) fm and lm (TR parameters);

( ) kan and NmaxS ; and ( )kan, kdn and Nmax

S .

Table 2 – Parameters estimated for both PMUA applications.

Optimization condition J included in simulation Parameters R2

kan (mL/(g h)) kdn (h�1) NmaxS (mg/g) fm lm (h�1) First Second

3 Parameter optimization Yes 1.1 0.0466 1.79 – – 0.901 0.971

2 Parameter optimization Yes 1.36 – 1.79 – – 0.856 0.964

2 Parameter optimization No 0.441 0.237 – – – 0.801 0.368

1 Parameter optimization No 0.387 – – – – 0.711 0.334

TR model parameters – – – – 0.035 0.431 0.451 0.691

Table 3 – Enhanced PHEN elution optimization results.

Optimization condition Parameters R2 (all data)

a E(k) (h�1) b (h) KCS (mL/g) f l (10�4 h�1)

From Fig. 7 (Tungittiplakorn et al., 2004). Simulation results are shown in Fig. 3.

Fit all data 0.1968 0.00084 234.739 – – – 0.937

Fit all data 0.0901 0.0129 6.9832 58.82 – – 0.991

Fit first 10 pore volume 0.0117 0.0178 0.6567 – – – 0.024

Fit first 10 pore volume 0.0065 0.015 0.4379 11.18 – – 0.007

TS model parameters – – – – 0.0146 6.958 0.943

TS model parameters – – – 15.85 0.0024 5.349 0.924

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mass fractio

n rem

ain

in

g (M

/M

0)

Pore volume (ut/L)

α and E(k) are optimized

f and λ are optimized

α, E(k) and KSC are optimized

f, λ and KSC are optimized

Fig. 3 – Experimental data and simulation results for PHEN

mass fraction retained in a column. The first dashed

vertical line indicates the start of PMUA particle

introduction to the column and the second dashed line

indicates where the PMUA application was stopped.

( ) is the experimental data, ( ) is the simulation

where a, E(k) and KCS were optimized, and ( ) is the

simulation where a and E(k) were optimized. ( ) is the

simulation with TS model for PHEN desorption where f, l

and KCS were optimized, ( ) is the simulation where f

and l were optimized, and (-) or (C) indicate 24-h

stoppage of flow to allow passive PHEN desorption.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 7 1033

influences the particle deposition on soil. The TR model also

failed to describe the experimental data for the second

application of PMUA. The DN; kan and NmaxS parameters deter-

mined from PMUA BTCs were used in subsequent simulations

of the experimental data for PHEN removal by PMUA which

are discussed below.

The results of parameter optimization for PMUA-enhanced

PHEN elution curves are summarized in Table 3. Assuming that

the mass fraction of PHEN remaining in the column data with

no PMUA application would be sufficient to describe the g

distribution for PHEN desorption, a and E(k) values were first

estimated for the first 10 pore volume of experimental data in

Fig. 3 (i.e. data prior to introduction of PMUA). The reported

distribution coefficient for PHEN, KCS , was used in the calcula-

tion. The a value determined from this optimization was 0.0117

and the E(k) value was 0.0178 h�1. The same a and E(k) values

were then applied throughout the remaining simulation with

PMUA application using the published value of PMUA–PHEN

interactions in Table 1. Although the model successfully

simulated the initial data, it predicted less eluted PHEN mass

when PMUA was injected into the column (results not shown).

Thus, the experimental data from the initial 10 pore volumes

was not sufficient to describe the slow desorption of PHEN.

We also evaluated optimization of the entire data set (i.e.

data both without and with PMUA application) by fitting a,

E(k) and KCS (KC

S influences the driving force for desorption; i.e.,

CS;i � KCS C) to the PHEN elution data simultaneously and

compared this to a three-parameter fit using the TS model

(KCS , f and l were adjusted). Results of these simulations are

shown in Fig. 3. All of these optimizations included use of the

fitted values for PMUA transport. The GS model with three

adjustable parameters better described the data than the TS

model because the TS model prediction deviated significantly

from the data after the onset of the PMUA application at 10

pore volumes. The a value estimated from the GS optimiza-

tion was 0.0901, the E(k) value was 0.0129 h�1 and the KCS was

58.82 mL/g. While the fit of the data was excellent, the fitted

KCS value was higher than that measured in batch experi-

ments (17.32 mL/g). We deem the discrepancy between the

fitted and reported KCS values to be too great to reconcile with

experimental error.

a and E(k) values in the GS model were then estimated

using the entire data set using the reported distribution

coefficient, KCS , for PHEN. This was compared to a two-

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pore volume utL

Co

ncen

tratio

n (C

/C

0)

a

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Pore volume (ut/L)

Co

ncen

tratio

n (C

/C

i)

b

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mass fractio

n rem

ain

in

g (M

/M

0)

Pore volume (ut/L)

c

Fig. 4 – The effects of different a value on (a) PHEN breakthrough curves; (b) PHEN elution curves and (c) mass fraction

remaining for PHEN elution curves in (b). The curves are simulated E(k) value of 0.01 hL1 with different a values. a [ 0.05

( ); a [ 0.1 ( ); and a [ 0.2 ( ).

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 71034

parameter fit using the TS model ( f and l were adjusted and

the reported KCS was specified). For the GS model, the resulting

a value was 0.1968 and the E(k) value was 0.00084 h�1. The

resulting simulation over-estimated PHEN elution between 10

and 15 pore volumes, and underestimated PHEN elution

afterwards (Fig. 3, thin solid line). However, considering that

only two adustable parameters were used to describe the

entire data set, we deem this to be a reasonably satisfactory

agreement. The discrepancies in the simulation of PHEN

elution likely reflect the shortcoming of using a smooth

gamma distribution to represent the actual distribution of

mass transfer rates in the test soil. The fit of the data using the

two-parameter TS model agreed with the data between 15 and

30 pore volumes somewhat better but overpredicted elution of

PHEN after 30 pore volumes relative to the GS model (Fig. 3,

dotted line). The failure of the TS model with a single

desorptive rate constant was anticipated for simulation of

long term data. Johnson (2000) used a two-rate model to

describe facilitated elution of hydrophobic contaminants;

however, the model was unable to predict the free aqueous

contaminant concentration, which, in turn, affects the

desorption rate of the contaminant and hence the prediction

of contaminant transport Thus, the two-rate model was not

evaluated as part of this research.

The a, E(k) and KCS values determined above were also used

in simulating PHEN elution by PMUA in a different experiment

(Tungittiplakorn, 2005). Results and their discussion are

provided in the supplemental information for this publication.

In brief, the model predicted more PHEN mass eluted from the

column in the second experiment after two pore volumes.

Using the same expected value of the gamma distribution

(i.e. EðkÞ) and the same KCS , the a value was then optimized and

the value estimated was 0.0480 which was 1.8 times less than

the value previously estimated. Possible reasons for this

change in a include experimental differences in contaminant

aging and are discussed in the supplemental information.

3.2. Sensitivity analysis

Sensitivity analysis was used to explore the significance of the

a parameter in the GS model, varying soil organic carbon

content or PMUA concentration, and the presence or absence

of flow cessation. The expected value E(k) was held constant

and the value of kmax was determined from the CDF as

described above. Eqs. (6) through (10) were used to analyze the

effect of a values on simulations of the elution of a contami-

nant from a contaminated column and also for simulation of

contaminant BTC for a clean column. Simulations for a soil

with a higher carbon content were made by assuming KCS to be

proportional to the fraction of organic carbon (Karickhoff

et al., 1979). The model was also evaluated to observe the

effect of varying PMUA concentration on PHEN elution with

fixed a and E(k) values. Simulations of PHEN elution with and

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Co

ncen

tratio

n (C

T/C

eq)

100Pore volume ut/L

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

Co

ncen

tratio

n (C

T/C

0)

Pore volume (ut/L)

0 10 20 30 40 50 60 70 80 90 1000.5

0.550.6

0.650.7

0.750.8

0.850.9

0.951

Mass fractio

n rem

ain

in

g (M

/M

0)

Pore volume (ut/L)

010 20 30 40 50 60 70 80 90 1000

0.10.20.30.40.50.60.70.80.9

1

Mass fractio

n rem

ain

in

g (M

/M

0)

Pore volume (ut/L)

a b

c d

Fig. 5 – (a) Enhanced PHEN elution curves; (b) PHEN breakthrough curves and (c) Mass fraction remaining corresponding to

PHEN elution curves in (a). The curves are simulated with a value of 0.05 and E(k) value of 0.01 hL1 with carbon-type soil

( 0.5 g/L PMUA; 0 g/L PMUA) and sandy aquifer ( 0.5 g/L PMUA; 0 g/L PMUA). (d). PHEN mass fraction

remaining with 0 g/L ( ), 0.5 g/L ( ), 5 g/L ( ) and 10 g/L ( ) of PMUA in low carbon soil. The initial mass for

PHEN elution curves simulation is fixed at 0.0173 mg/g CT[Cð1DKCNNÞ.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 7 1035

without PMUA application were compared, and the effect of

a 24-h cessation in flow was also considered.

g distributions for PAH desorption from soil typically have

a value less than 1 and values ranging from 0.1 to 0.001 have

been reported (Ahn et al., 1999a; Bonas, 1996; Liu et al., 2001).

Therefore, it is informative to analyze the effect of different GS

distributions on model predictions. Model predictions were

found to be sensitive to parameters that describe the g-

distribution. To explain this, sensitivity values of a equal to

0.05, 0.1 and 0.2 were used to simulate elution from

a contaminated column and breakthrough of a continuous

input to a clean column. The simulations of breakthrough and

elution curves in Fig. 4(a and b) illustrate the effect of the

different a values on simulations of PHEN behavior. PAH

transport and removal is enhanced for distribution with

higher a value. Fig. 4(c) depicts the mass fraction remaining in

the column corresponding to the elution curves in Fig. 4(b).

Although E(k) was held constant, a decrease in a resulted in

slower desorption of the contaminant and lower mobility. In

distributions with smaller a values, the ki values in f(k)

compartmentalization were concentrated in smaller value

region which effected the sorption rate term in Eq. (9).

Different types of soils exhibit different affinities for

hydrophobic contaminants. Soils with higher organic carbon

contents typically have greater sorption distribution coeffi-

cients which would influence the efficacy of pump and treat

remediation performed with polymeric nanoparticles. Fig. 5(a)

through (c) show the effect of soil type on the fate of PHEN.

Simulations are presented for a ‘‘low-carbon’’ soil ( foc¼ 0.57%)

and a soil with foc¼ 3.95%. It was assumed that the contami-

nant sorptive distribution coefficient, KCS varied in proportion

to foc. While it is possible that the nanoparticle attachment

coefficient, kan, and the soil saturation coefficient, J, may also

vary, no data are currently available to predict how these

parameters will change as a function of foc; therefore, nano-

particle-related soil binding parameters were held constant.

The elution and breakthrough curves of PHEN are improved in

all cases with the use of PMUA as a carrier.

When present, PMUA acts to decrease the free dissolved

PHEN concentration and, therefore, acts to increase the

driving force (i.e. CS � KCS C) for PHEN desorption (and, there-

fore, increases the desorption rate, i.e. kðCS � KCS CÞ) from soil to

aqueous phase by lowering C (Liu et al., 2001). The peak in

Fig. 5(a) indicates that the PMUA application helps the elution

initially by enhancing desorption of PHEN from soil

‘‘compartments’’ that have a relatively fast time scale.

Implementing a no-flow condition a short amount of time

after PMUA release can passively improve the PHEN desorp-

tion and lower the cost of remediation.

Attachment of PMUA particles can also act to increase the

effective retardation of a contaminant in low carbon soils via

the contribution of trapped PMUA to contaminant sorption.

However, application of PMUA beyond NmaxS still results in an

overall increase in contaminant removal relative to a case

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 71036

where PMUA is not applied. This phenomenon as illustrated in

Fig. 5(b) by initial retardation of PHEN breakthrough for the

first 20 pore volume in the presence of PMUA followed by an

increase in the PHEN concentration relative to the simulation

where PMUA is not present.

The concentration of PMUA added also effects the

percentage removal of the mass of PHEN remaining as

shown in Fig. 5(d). A total 5 g/L of PMUA was able to remove

65% of the contaminant after 100 pore volumes and 10 g/L of

PMUA eluted 85%. Remediation designs in which PMUA is

used as an agent for facilitating transport must optimize the

trade off between cost of the carrier and the time required

to achieve a desired level of remediation. Implementing

a no-flow condition after PMUA introduction into a soil

can passively improve the PHEN desorption and would

presumably lower the cost of remediation. The cost can

potentially be ameliorated by recycling of injected particles

through capture in a withdrawal well, and removal of PMUA-

bound contaminants by a process such as treatment in a

bioreactor (Tungittiplakorn et al., 2005). Tungittiplakorn et al.

(2005) note that the feasibility of remediation using nano-

particles will depend upon assessment of their human

health effects.

4. Conclusion

In summary, a model for nanoparticle transport that assumes

irreversible particle attachment to soil subject to the satura-

tion of attached nanoparticles successfully described avail-

able data sets for breakthrough of PMUA nanoparticles. The

nanoparticle transport model coupled to a GS model for PHEN

desorption fit available data for PMUA-enhanced desorption

of PHEN. The model permits exploration of alternative PMUA

concentrations in meeting treatment goals. We anticipate the

use of a GS model for contaminant desorption will facilitate

predicting future data sets obtained over longer time intervals

and will better describe field scenarios where remediation

may occur on the time scale of years.

Acknowledgment

W.L.W.J is a recipient of a fellowship from Ministry of Higher

Education, Malaysia.

Appendix. Supplementary data

Supplementary data associated with this article can be found

in the online version at doi:10.1016/j.watres.2009.03.033.

r e f e r e n c e s

Abdul, A.S., Gibson, T.L., 1991. Laboratory studies of surfactant-enhanced washing of polychlorinated biphenyl from sandymaterial. Environmental Science & Technology 25 (4),665–671.

Ahn, I.-S., Lion,L.W., Shuler,M.L., 1996.Microscale-based modelingofpolynuclear aromatic hydrocarbon transport and biodegradationin soil. Biotechnology and Bioengineering 51 (1), 1–14.

Ahn, I.-S., Lion, L.W., Shuler, M.L., 1999a. Validation of a hybrid‘‘two-site gamma’’ model for naphthalene desorption kinetics.Environmental Science & Technology 33 (18), 3241–3248.

Ahn, I.-S., Ghiorse, W.C., Lion, L.W., Shuler, M.L., 1999b.Independent prediction of naphthalene transport andbiodegradation in soil with a mathematical model.Biotechnology and Bioengineering 65 (1), 65–75.

Ang, C.C., Abdul, A.S., 1994. Evaluation of an ultrafiltrationmethod for surfactant recovery and reuse during in situwashing of contaminated sites: laboratory and field studies.Ground Water Monitoring & Remediation 14 (3), 160–171.

Bonas, A. R., 1996. Desorption kinetics of native and freshly addednaphthalene from sediments. M.S. Thesis, Cornell University,Ithaca, NY.

Carnahan, B., Luther, H.A., Wilkes, J.O., 1969. Applied NumericalMethods. John Wiley & Sons, Inc.

Carroll, K.M., Harkness, M.R., Bracco, A.A., Balcarcel, R.R., 1994.Application of a permeant/polymer diffusional model to thedesorption of polychlorinated biphenyls from Hudson Riversediments. Environmental Science & Technology 28 (2), 253–258.

Chen, G., Strevett, K.A., Vanegas, B.A., 2001. Naphthalene,phenanthrene and surfactant biodegradation. Biodegradation12 (6), 433–442.

Chen, W., Wagenet, R.J., 1995. Solute transport in porous mediawith sorption-site heterogeneity. Environmental Science &Technology 29 (11), 2725–2734.

Cleary, R.W., Adrian, D.D., 1973. Analytical solution of theconvective–dispersive equation for cation adsorption in soils.Soil Science Society of America Proceedings 37, 197–199.

Coates, J.T., Elzerman, A.W., 1986. Desorption kinetics forselected PCB congeners from river sediments. Journal ofContaminant Hydrology 1 (1), 191–210.

Connaughton, D.F., Stedinger, J.R., Lion, L.W., Shuler, M.L., 1993.Description of time-varying desorption kinetics: release ofnaphthalene from contaminated soils. Environmental Science& Technology 27 (12), 2397–2403.

Constantinides, A., Mostoufi, N., 1999. Numerical Methods forChemical Engineers with MATLAB Applications. Prentice-HallInc., Upper Saddle River, NJ.

Culver, T.B., Hallisey, S.P., Sahoo, D., Deitsch, J.J., Smith, J.A., 1997.Modeling the desorption of organic contaminants from long-term contaminated soil using distributed mass transfer rates.Environmental Science & Technology 31 (6), 1581–1588.

Davis, P., Rabinowitz, P., 2007. Methods of Numerical Integration.Dover Publications.

Deitsch, J.J., Smith, J.A., 1995. Effect of Triton X-100 on the rate oftrichloroethene desorption from soil to water. EnvironmentalScience & Technology 29 (4), 1069–1080.

Diamessis, P.J., Domaradzki, J.A., Hesthaven, J.S., 2005. A spectralmultidomain penalty method model for the simulation of highReynolds number localized incompressible stratifiedturbulence. Journal of Computational Physics 202 (1), 298–322.

Durran, D.R., 1991. The third-order Adams–Bashforth method: anattractive alternative to leapfrog time differencing. MonthlyWeather Review, 702–720.

Gear, W.C., 1971. Numerical Initial Value Problems in OrdinaryDifferential Equations. Prentice-Hall, Inc.

Grimberg, S.J., Nagel, J., Aitken, M.D., 1995. Kinetics ofphenanthrene dissolution into water in the presence ofnonionic surfactants. Environmental Science & Technology 29(6), 1480–1487.

Grimberg, S.J., Miller, C.T., Aitken, M.D., 1996. Surfactant-enhanced dissolution of phenanthrene into water for laminarflow conditions. Environmental Science & Technology 30 (10),

2967–2974.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 1 0 2 8 – 1 0 3 7 1037

van Genuchten, M.T., Alves, W.J., 1982. Analytical Solutions of theOne-Dimensional Convective–Dispersive Solute TransportEquation. U.S. Salinity Laboratory, Reverside, CA.

van Genuchten, M.T., Wagenet, R.J., 1989. Two-site/Two-regionmodels for pesticide transport and degradation: theoreticaldevelopment and analytical solutions. Soil Science Society ofAmerica Journal 53 (5), 1303–1310.

Hairer, E., Nørsett, S.P., Wanner, G., 1993. Solving OrdinaryDifferential Equations I. Non-Stiff Problems. Springer Verlag,Berlin.

Hasegawa, M.A., Sabatini, D.A., Harwell, J.H., 1997. Liquid–liquidextraction for surfactant-contaminant separation andsurfactant reuse. Journal of Environmental Engineering 123(7), 691–697.

Jensen-Spaulding, A., Shuler, M.L., Lion, L.W., 2004a. Mobilizationof adsorbed copper and lead from naturally aged soil bybacterial extracellular polymers. Water Research 38 (5),1121–1128.

Jensen-Spaulding, A., Cabral, K., Shuler, M.L., Lion, L.W., 2004b.Predicting the rate and extent of cadmium and copperdesorption from soils in the presence of bacterialextracellular. Water Research 38 (9), 2230–2239.

Johnson, W.P., 2000. Sediment control of facilitated transport andenhanced desorption. Journal of Environmental Engineering126 (1), 47–50.

Johnson, W.P., Cabral, K., Lion, L.W., Corapcioglu, Y.M., 1998.Reconciliation of expressions for the modified retardationfactor and incorporation of non-linear contaminant sorptionto the stationary phase. Journal of Contaminant Hydrology 32(3–4), 247–266.

Karickhoff, S.W., Brown, D.S., Scott, T.A., 1979. Sorption ofhydrophobic pollutants on natural sediments. Water Research13 (3), 241–248.

Karniadakis, G.E., Israeli, M., Orszag, S.A., 1991. High-ordersplitting methods for the incompressible Navier–Stokesequations. Journal of Computational Physics 97 (2),414–443.

Kim, H.S., Weber, W.J., 2003. Preferential surfactant utilization bya PAH-degrading strain: effects on micellar solubilizationphenomena. Environmental Science & Technology 37 (16),3574–3580.

Kim, J.-Y., Cohen, C., Shuler, M.L., Lion, L.W., 2000. Use ofamphiphilic polymer particles for in situ extraction of sorbedphenanthrene from a contaminated aquifer material.Environmental Science & Technology 34 (19), 4133–4139.

Laha, S., Luthy, R.G., 1991. Inhibition of phenanthrenemineralization by nonionic surfactants in soil-water systems.Environmental Science & Technology 25 (11), 1920–1930.

Lipe, K.M., Sabatini, D.A., Hasegawa, M.A., Harwell, J.H., 1996.Micellar-enhanced ultrafiltration and air stripping forsurfactant-contaminant separation and surfactant reuse.Ground Water Monitoring & Remediation 16 (1), 85–92.

Liu, A., Ahn, I.-S., Mansfield, C., Lion, L.W., Shuler, M.L.,Ghiorse, W.C., 2001. Phenanthrene desorption from soil in thepresence of bacterial extracellular polymer: observations andmodel predictions of dynamic behavior. Water Research 35 (3),835–843.

Lorden, S.W., Chen, W., Lion, L.W., 1998. Experiments andmodeling of the transport of trichloroethene vapor inunsaturated aquifer material. Environmental Science &Technology 32 (13), 2009–2017.

Magee, B.R., Lion, L.W., Lemley, A.T., 1991. Transport of dissolvedorganic macromolecules and their effect on the transport ofphenanthrene in porous media. Environmental Science &Technology 25 (2), 323–331.

Marquardt, D.W., 1963. An algorithm for least-squares estimationof nonlinear parameters. Journal of the Society for Industrialand Applied Mathematics 11 (2), 431–441.

McCray, J.E., Brusseau, M.L., 1998. Cyclodextrin-enhanced in situflushing of multiple-component immiscible organic liquidcontamination at the field scale: mass removal effectiveness.Environmental Science & Technology 32 (9), 1285–1293.

Montas, H.J., Shirmohammadi, A., 2004. Modeling of soilremediation by nanoscale particles. Paper number 043053presented at the 2004 ASAE/CSAE Annual InternationalMeeting.

Pedit, J.A., Miller, C.T., 1994. Heterogeneous sorption processes insubsurface systems. 1. Model formulations and applications.Environmental Science & Technology 28 (12), 2094–2104.

Pieper, A.P., Ryan, J.N., Harvey, R.W., Amy, G.L., Illangasekare, T.H., Metge, D.W., 1997. Transport and recovery ofbacteriophage PRD1 in a sand and gravel aquifer: effect ofsewage-derived organic matter. Environmental Science &Technology 31 (4), 1163–1170.

Rajagopalan, R., Tien, C., 1976. Trajectory analysis of deep-bedfiltration with the sphere-in-cell porous media model. AIChEJournal 22 (3), 523–533.

Rouse, J.D., Sabatini, D.A., Harwell, J.H., 1993. Minimizingsurfactant losses using twin-head anionic surfactants insubsurface remediation. Environmental Science & Technology27 (10), 2072–2078.

Saiers, J.E., Hornberger, G.M., Liang, L., 1994. First- and second-order kinetics approaches for modeling the transport ofcolloidal particles in porous media. Water Resources Research30 (9), 2499–2506.

Slinn, D.N., Riley, J.J., 1998. A model for the simulation ofturbulent boundary layers in an incompressible stratified flow.Journal of Computational Physics 144 (2), 550–602.

Smith, G.D., 1978. Numerical Solution of Partial DifferentialEquations. Clarendon Press, Oxford.

Tungittiplakorn, W., 2005. Engineered polymericnanoparticles for remediation of polycyclic aromatichydrocarbon contaminated soils. Dissertation, CornellUniversity, Ithaca, NY.

Tungittiplakorn, W., Cohen, C., Lion, L.W., 2005. Engineeredpolymeric nanoparticles for bioremediation of hydrophobiccontaminants. Environmental Science & Technology 39 (5),1354–1358.

Tungittiplakorn, W., Lion, L.W., Cohen, C., Kim, J.-Y., 2004.Engineered polymeric nanoparticles for soil remediation.Environmental Science & Technology 38 (5), 1605–1610.

Wan Johari, W. L., 2007. Mass transfer model of nanoparticle-facilitated contaminant transport in saturated porous media.MS Thesis, Cornell University, Ithaca, NY.