Journal of Contaminant Hydrology 152 (2013) 60–69

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Journal of Contaminant Hydrology

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Analysis of the fate and transport of nC60 nanoparticles in thesubsurface using response surface methodology

Chunmei Bai a, Kent M. Eskridge b, Yusong Li a,⁎a Department of Civil Engineering, University of Nebraska, Lincoln, 362R Whittier Building, 2200 Vine Street, Lincoln, NE 68583, United Statesb Department of Statistics, University of Nebraska, Lincoln, 343D Hardin Hall North, Lincoln, NE 68583, United States

a r t i c l e i n f o

⁎ Corresponding author. Tel.: +1 402 472 5972; faxE-mail address: yli7@unl.edu (Y. Li).

0169-7722/$ – see front matter © 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jconhyd.2013.06.001

a b s t r a c t

Article history:Received 6 February 2013Received in revised form 11 June 2013Accepted 13 June 2013Available online 21 June 2013

Predicting the distribution of engineered nanomaterials (ENMs) in the environment willprovide critical information for risk assessment and policy development to regulate theseemerging contaminants. The fate and transport of ENMs in natural subsurface environmentsare complicated by various factors, such as hydraulic gradient, initial release concentration,nanoparticle size, and collision efficiency factor. Based on advanced statistical methodologies(i.e., response surface methodology (RSM)), we explore simple relationships between keyfactors that control ENM transport (collision efficiency factor, particle size, hydraulic gradient,and initial release concentration) and key parameters that describe the ENM concentrationdistribution in porous media (maximum standardized concentration, the mass percentage ofinjected nanoparticle attached in the aquifer, the x-centroid of aqueous phase nC60 plume, andthe x-centroid of attached phase nC60 distribution). Hypothetical scenarios for the release ofnanoparticles into an aquifer were simulated numerically with randomly generated permeabilityfields that were based on mildly and highly heterogeneous sites. RSM was used to developpolynomial regression equations based on a statistical experimental design. High R-squaredvalues (greater than 0.9) of the regression equations were obtained for all the models developedbased on themildly heterogeneous site. On the highly heterogeneous site, the R-squared value ofthe regression equation for the percentage of nanoparticles attached (by mass) was more than0.9. The ability to accurately estimate aqueous phase ENMconcentration distribution using simpleregression equations is particularly critical for risk assessment. Even though the correlationsdeveloped in this study were site and scenario specific, this work represents a first effort ofapplying RSM for predicting the distribution of engineered nanomaterials in porous media.

© 2013 Elsevier B.V. All rights reserved.

Keywords:nC60 nanoparticlesMobilitySubsurfaceResponse surface methodologyDesign of experiments

1. Introduction

The nanotechnology revolution has brought more than 1300nanotechnology-enabled consumer products into the market-place (LuxResearch, 2009; WWICS, 2011). Strong evidence(Benn and Westerhoff, 2008; Benn et al., 2010; Geranio et al.,2009; Kaegi et al., 2008; Zhang et al., 2008) has suggested thatnanoparticles may enter the subsurface environment fromwastewater reuse (Zhang et al., 2008), by directly leachingfrom underground waste sites (Benn et al., 2010), and from

: +1 402 472 8934.

ll rights reserved.

the agricultural use of biosolids that contain engineerednanomaterials (ENMs) (Benn and Westerhoff, 2008). Thepotential environmental impacts of these nanomaterialsonce they enter the environment are of concern. To provideeffective risk assessment and to support policy developmentthat regulates ENMs, it is important to predict ENM concen-trations and distributions in the subsurface environment.

In this work, nano-fullerenes (C60), which are one typeof widely used engineered nanomaterials, were chosenas model nanomaterials for investigation. The toxicities offullerene nanoparticles to microbes (Fortner et al., 2005), fish(Oberdorster, 2004), and human cells (Fortner et al., 2005),were previously reported. In aqueous systems, C60 is capable

61C. Bai et al. / Journal of Contaminant Hydrology 152 (2013) 60–69

of acquiring negative surface charges and forming stablenano-scale aggregates (nC60) (Brant et al., 2005a). These nC60aggregates may interact with natural organic matter andundergo surface transformations that result in the formationof stable aggregates that can travel very long distances, whichincreases the potential that these nanomaterials will reachdrinking water sources (Chen and Elimelech, 2007; Espinasseet al., 2007; L.L. Wang et al., 2012; Y. Wang et al., 2012).In addition, the high sorption capacity of C60 to organiccontaminants (Hu et al., 2008) (e.g., PAHs) indicates that C60

could serve as a carrier that facilitates the transport of othertoxic chemicals.

The fate and transport of nC60 in the subsurface are relativelywell studied in comparison with other ENMs. Multiple lab scaleone-dimensional column experiments (Espinasse et al., 2007;Wang et al., 2008; L.L. Wang et al., 2012; W. Zhang et al., 2012;L.L. Zhang et al., 2012; Y. Wang et al., 2012) were conducted toelucidate themechanisms that control themobility of nC60 in thesubsurface. One-dimensional mathematical models based onmechanistic understandings (Mauter and Elimelech, 2008; L.L.Zhang et al., 2012)were developed to simulate the experimentalcolumndata bymodifying the classic filtration theory (CFT) (Yaoet al., 1971). Recently, such models were incorporated intomulti-dimensional models to simulate field scale nC60 transportin the subsurface (Bai and Li, 2012; Cullen et al., 2010). In ourrecent work (Li et al., 2008), we developed amechanistic modelbased on column experiments (Mauter and Elimelech, 2008;Wang et al., 2008), which considers that nC60 nanoparticleattachment rate is linearly rated to the available soil surfaces andis subject to a maximum retention capacity. This model wasincorporated into the Modular three-Dimensional TransportModel for Multi-Species (MT3DMS) (Zheng and Wang, 1999).Various release scenarioswere investigated for the transport andretention of nC60 in a layered aquifer, including the entry of nC60from an injection well and its release from a waste site. Cullenet al. (2010) also incorporated the nC60 transport model into atwo-dimensional simulator and evaluated the nC60 mobility fora range of hydrological and geological conditions.

These studies demonstrated that the transport andretention of nC60 in the field are affected by various factors,such as the subsurface heterogeneity, hydraulic gradient,release concentration, aggregate size, dispersivity, collisionefficiency factor, maximum retention capacity, and releasescenarios. Clearly understanding the influences of thesefactors on the nC60 subsurface distribution could benefitrisk assessment analysis and remediation design. However,developing a three-dimensional numerical model that canreflect the specific site condition and appropriately handlingnatural uncertainty can be a challenge, particularly if theusers are not professional numerical modelers. Therefore, it isuseful to develop simple mathematical relationships that canbe easily used to predict nanoparticle transport and distribu-tion in the field based on some key factors. The goal of thiswork is to explore such relationships by applying responsesurface methodology (RSM).

RSM is a collection of statistical and mathematical methodsuseful for planning experiments, approximating complexquantitative relationships between factors and their responsesand for identifying response optimizing factor combination(Roebuck et al., 1995; Tsapatsaris and Kotzekidou, 2004). RSMis widely applied in the industrial, food, social, physical, and

engineering sciences (Myers et al., 2009). Here,we use the RSMto develop second-order polynomial regression equations thatapproximate the relationships between the controlling factorsand response variables describing the nC60 distribution inthe field. These polynomial regression models were developedbased on randomly generated aquifer field sites that representmildly and highly heterogeneous aquifers.

A description of the multiple-dimension numerical modelthat was used to simulate nC60 transport in the field is givenbelow. Hypothetical nC60 release scenarios and example simu-lation results are presented for mildly and highly heterogeneoussites. The subsequent section demonstrates the application ofthe RSM for developing polynomial regression equationsrelating key factors and response variables. In the Resultsand discussion section the derived equations are analyzed andvalidated. Following a brief summary we conclude by highlight-ing the viability of applying the RSM to analyze nanoparticletransport and distribution in the subsurface.

2. nC60 transport modeling

2.1. Mathematical model

The transport and retention of nC60 in saturated porousmedia were simulated using a modified MT3DMS model. TheMT3DMS model is a program that was originally developedby the U.S. Army Corps of Engineers to simulate advection,dispersion/diffusion, and chemical reactions of contaminantsin subsurface environments with various boundary condi-tions and external sources and sinks (Zheng and Wang,1999). In a previous study (Bai and Li, 2012), we incorporat-ed a nanoparticle transport module into the MT3DMS modelto simulate nC60 transport in the subsurface. Here, the nC60transport and attachment were simulated as follows (Mauterand Elimelech, 2008):

∂C∂t þ

ρb

n∂S∂t−∇ D ⋅∇C

� �þ V→

⋅∇C ¼ 0 ð1Þ

ρb∂S∂t ¼ nkatt 1− S

Smax

� �C ð2Þ

where C is the aqueous concentration of nC60 [M/L3], S is theattached (solid-phase) nC60 concentration [M/M], t is the time

[T], ρb is the soil bulk density [M/L3], n is the soil porosity, V→

isthe pore water velocity vector [L/T], D is the hydrodynamicdispersion tensor [L2/T], katt is the nC60 attachment ratecoefficient [1/T], and Smax [M/M] is a particle retention capacity.The katt is estimated as follows:

katt ¼3 1−nð Þv

2dcαη0 ð3Þ

where dc is the mean grain size [L], α is the collision efficiencyfactor [−], and η0 is the single collector efficiency [−]. Inaddition, α represents the fraction of nC60 that remains attachedafter collision and η0 represents the frequency of nC60 collisionswith the surfaces of the porous medium. The η0 is calculated byusing a correlation equation (Tufenkji and Elimelech, 2004) thatincorporates the integrated contributions of diffusion, intercep-tion and sedimentation processes. The Smax for nC60 can be

62 C. Bai et al. / Journal of Contaminant Hydrology 152 (2013) 60–69

Fig. 1. An example simulated nC60 concentration distribution at the mildlyheterogeneous site after 1 h of release and 8 h of flushing with clean water isshown for the (A) aqueous-phase and (B) attached-phase concentrations.The following parameters were used: α = 0.001, dp = 100 nm, i = 0.1,and C0 = 1 mg/L.

estimated with the following equation (Mauter and Elimelech,2008):

Smax ¼ 19:6vdcDm

� �1=3 dcdM

� �� �−1:2

ð4Þ

whereDm is themolecular diffusion coefficient [L2/T] (which canbe estimated using the Einstein–Stokes equation) (Einstein,1956) and dM is the mean diameter of the medium (0.5 mm).This model (Eqs. (1)–(4)) provided a good fit for nC60 transportin a column thatwas packedwith Ottawa sand of between 0.125and 0.710 mm diameter at a flow velocity of between 1.03 and8.34 m/d (Li et al., 2008). The modified MT3DMS model wasvalidated by comparing these results with those from theHYDRUS-1D model (Simunek et al., 2009) for one-dimensionalproblems with various flow velocities, injection durations, andcollision efficiency factors (Bai and Li, 2012). Please note thatstraining, ripening, and detachment processes were not consid-ered in this model, which may be important for othernanomaterial transports in the subsurface.

2.2. nC60 release scenarios

The modified MT3DMS model was used to simulate nC60release scenarios in aquifers that are mildly and highlyheterogeneous. The mildly heterogeneous site was based on afield site in Oscoda, MI (Christ et al., 2005; Cullen et al., 2010;Lemke et al., 2004a, 2004b; Phelan et al., 2004; Ramsburg et al.,2004),whichwas extensively characterized during a surfactantenhanced remediation project (Ramsburg et al., 2004). Thegrain size distribution and hydraulic conductivity were ana-lyzed by using 167 homogenized core subsamples fromfourteen aquifer cores (Lemke et al., 2004b). As typically formost subsurface field sites, the amount of hydraulic conductivitymeasurements is significantly below the required resolution forsimulation of groundwater flow and contaminant transport.In modeling practice, the hydraulic conductivity distributionin a field site was thus often considered as a random field.Geostatistical approaches can be applied to generate a series ofrandom hydraulic conductivity fields based on the statistics andspatial correlation information from the availablemeasurements(Lemke et al., 2004a; Ramsburg et al., 2004). Simulations basedon these random field realizations will provide a range ofpossible flow and transport behaviors at the site. In thiswork, 15realizations of the non-uniformpermeability random fieldswereconstructed with conditional sequential Gaussian simulation(SGS). This number is similar to the number employed formultiple similar previous works on nanoparticle and DNAPLdistribution studies (e.g. Christ et al., 2005; Cullen et al., 2010;Kueper and Gerhard, 1995; Lemke et al., 2004a). The hydraulicconductivity estimates varied from 1 to 48 m/d, with a mean of16.8 m/d and a lnK variance of 0.29. The vertical and horizontalcorrelation lengths were 0.36 and 2.33 m, respectively (Lemkeand Abriola, 2003). Fig. 1 provides an example realization for themildly heterogeneous site.

Aquifer statistics of the highly heterogeneous sitewere generated based on a field site on the Columbus AirForce Base, MS (Barlebo et al., 2004; Bohling et al., 2012).As one of the most famous experimental sites for conductingmacrodispersion experiments, this site was fully characterizedin the last 25 years during three large-scale tracer tests

(Barlebo et al., 2004). In a similar approach for the mildlyheterogeneous site, fifteen realizations of the non-uniformpermeability random fields were constructed with SGS thatwere based on the site aquifer statistics. Hydraulic conductivityestimates varied from 0.035 to 1289 m/d, with a mean of4.77 m/d and a lnK variance of 4.5. Vertical and horizontalcorrelation lengths were 4.1 and 57 m, respectively (Egglestonand Rojstaczer, 1998). Fig. 2 provides an example realizationfor the highly heterogeneous site.

The simulation domain for both sites was 7.925 m ×9.754 m in the horizontal and vertical directions, respective-ly. The domains were uniformly discretized in the x and zdirections, with a grid size of 30.48 cm and 7.62 cm for themildly and highly heterogeneous sites, respectively. Longitu-dinal dispersivity values of 0.3 m and 7.62 cm were used forthe mildly and highly heterogeneous sites, respectively. Thetransverse dispersivity was estimated as one tenth of thelongitudinal dispersivity (Lemke et al., 2004b).

For both sites, a steady state groundwater flow conditionwasapplied through a hydraulic gradient (ranged between 0.001and 0.1) along the xdirection. The left and right boundarieswere

63C. Bai et al. / Journal of Contaminant Hydrology 152 (2013) 60–69

Fig. 2. An example simulated nC60 concentration distribution at the highlyheterogeneous site after 1 h of release and8 hof flushingwith cleanwater for the(A) aqueous-phase and (B) attached-phase concentrations. The followingparameters were used: α = 0.001, dp = 100 nm, i = 0.1, and C0 = 1 mg/L.

subjected to Type I (Dirichlet) constant head boundary condi-tions and the top and bottomboundarieswere subjected to TypeII (Neumann) no flow boundary conditions. A constant nC60concentration was released for 1 h through a 1.2 m screen andwas followed by the release of clean background water for 8 hon the left boundary as shown in Eq. (5), whichwas subjected tothe Type I (Dirichlet) boundary condition. The numericalsimulation started with an initial time step of 10 s, which wasautomatically adjusted by the simulator based on mass balanceerror.

C 0; z; tð Þ ¼0; zb4:267m

C0; 4:267m≤z≤5:486m;0

>

t≤1hr0; z > 5:486m

:

8<: ð5Þ

2.3. Simulated nC60 transport and distribution example

Figs. 1 and 2 provide an example simulated nC60concentration distributions after 1 h of release and 8 hof flushing with clean water for the mildly heterogeneous

and highly heterogeneous sites, respectively. In the field,nanoparticles encounter various pore water velocities thatresult from spatially heterogeneous hydraulic conductivitiesand different soil grain sizes. The aqueous phase nanoparticleplume preferentially moved faster in the more permeablezones. Meanwhile, some nanoparticles attached to the aquifermaterials as the aqueous phase nanoparticle plume passedthrough. As shown in Figs. 1 and 2, the distributions of theaqueous and attached nanoparticles were less uniform at thehighly heterogeneous site. The highly nonuniformnanoparticledistribution originated from spatially variable hydraulic con-ductivities and grain sizes, which resulted in spatially variablepore water velocities, attachment rates katt, and maximumretention capacities Smax (Eqs. (3) and (4)).

As shown in Figs. 1 and 2, it is difficult to quantitativelydescribe the nC60 distribution. The four following responsevariables were selected to illustrate nanoparticle distributionfeatures based on concentration or plume shape, including:(1) the maximum standardized concentration C/C0 (maxNC),(2) the mass percentage of injected nanoparticles attachedin the aquifer (massAtt), (3) the x-centroid of aqueous phasenC60 plume (cCentroid), and (4) the x-centroid of attachedphase nC60 distribution (sCentroid). Here, maxNC and massAttrepresent the mass (or concentration) related feature of thedistribution and cCentroid and sCentroid represent the geo-metric shape of the nC60 distribution in the aquifer.

3. Response surface methodology

The ultimate goal of this work is to develop simplemathematical relationships between key input factors anddistribution of nC60 nanoparticles in the field using RSM. Inthe following paragraphs, the form of the RSM models andinput variables was first discussed, followed by the RSMexperimental plan designed to obtain data for fitting thesemodels. Finally, how the data were used to fit the models isbriefly described.

Here, the quantitative relationships between the indepen-dent input factors and the response variables were establishedby simulating responses and fitting a second-order polynomialregression Eq. (6) using the least squares method, as shownbelow:

Ym ¼ b0m þX4i¼1

bimZi þX4i¼1

biimZi2

þX4i¼1

X4i b j¼2

bijmZiZj þ εm m ¼ 1;2;3;4ð Þ ð6Þ

where Ym is the predicted response variables (i.e. maxNC,massAtt, cCentroid, and sCentroid), Zi is the log-transformedinput factor, b0 is the intercept, bi is the linear coefficient, bii isthe quadratic coefficient, bij is the cross-product coefficient andε is the stochastic error term.

Based on previous studies (Bai and Li, 2012; Cullen et al.,2010; Mauter and Elimelech, 2008), four input factors thatreflect the most important parameters for subsurface nano-particle transport were identified, including the collisionefficiency (α), nanoparticle aggregate size (dp), hydraulicgradient (i), and release concentration (C0) factors. A range ofbetween 0.0001 and 0.01 was used for α to reflect the wide

64 C. Bai et al. / Journal of Contaminant Hydrology 152 (2013) 60–69

range of possible conditions, especially for the transport of nC60through sand aquifer in the presence of natural organic matter(Y. Wang et al., 2012). dp values of between 10 nm and1000 nm were used based on previous nC60 formation studiesunder variousmixing and solution chemistry conditions (Brantet al., 2005b; Chen and Elimelech, 2006). Values between 0.001and 0.1 were used for i to reflect the realistic groundwater flowconditions (Bear and Cheng, 2010). Finally, initial releaseconcentration values (C0) of between 0.1 mg/L and 10 mg/Lwere used, which had an upper boundary that was within thesame order of magnitude that was used in other columnstudies (Petosa et al., 2010). Because the independent inputvariable values varied by several orders of magnitude, theirlogarithmic form was used (i.e., Zi in Eq. (6)).

The next step in RSM is to develop a statistical experimentaldesign that identifies the combinations of the four input factors(Zi) and the number of simulations needed to produce a goodfit between the model and data without wasting experimentalexpense. A Central Composite Inscribed Design (CCI) was usedin this study, which is a type of the Central Composite Design(CCD) (Myers et al., 2009). The CCI is particularly helpful forexcluding extreme conditions that are not realistic for thenatural subsurface environments. A CCD consists of three typesof points: factorial, axial, and central points. In a CCI design, theaxial points are located at the lower and upper bounds of theinput factor ranges and the factorial points are located insidethe inscribed design space. Fig. 3 shows the experimentalpoints for a two-factor design. When the factors are scaled sothe distance from the axial points to central points is ±1, thenthe distance from the coordinate points for each factorial pointon all axes to the central points is ±(1/λ). The value of λ ischosen to maintain the rotatability of the design as follows:

λ ¼ 2τ� 1=4; ð7Þ

Fig. 3. A Central Composite Inscribed Design (CCI) for two-factors. Herefactorial, axial, and central points are presented by circle, star, and crossrespectively.

,,

where τ is the number of independent input factors.Maintainingrotatability ensures that the points at equal distances from thecenter are well predicted (Acikalin et al., 2005). The number ofsimulation runs that were required for each of the 15 domainrealizations was 24 + 2*4 + 1 = 25. Table 1 shows the 25design points and input factors that were simulated allowing usto fit the polynomial regression Eq. (6).

Hypothetical nC60 release scenarios that were based on thedesign points in Table 1 were numerically simulated using themodified MT3DMS model for the mildly and highly heteroge-neous sites and for each randomly generated permeability fieldrealization. Specifically, 25 simulations were conductedfor each of 15 realizations for both sites. The four responsevariables (i.e., maxNC, massAtt, cCentroid, and sCentroid) thatwere chosen in the previous section (i.e. Section 2.3) werecalculated based on the aqueous phase and the attached phaseconcentration distributions from MT3DMS simulation (asillustrated in Figs. 1 and 2). Based on these 375 (25 × 15)simulated values for each response variable, the quantitativerelationships between the independent input factors and theresponse as described by Eq. (6) were estimated usingleast-squares regression methods (Myers et al., 2009). Testsof significance of the regression coefficients were conductedusing t-tests based on residual error and R2 was computed toassess model fit. SAS software (version 9.2) (SAS Institute,2002) was used to estimate polynomial regression equationsfor each variable of interest.

4. Results and discussion

The relationships between the input factors (i.e., α, dp, i,and C0) and the response variables (i.e., maxNC, massAtt,cCentroid, and sCentroid) are described through the regres-sion coefficients and R2 values for Eq. (6) and are summa-rized in Table 2 for the mildly and highly heterogeneous sites.

4.1. The mildly heterogeneous site

For the mildly heterogeneous site, the R-squared valuesfor the correlations between the input factors and theresponse variables maxNC, massAtt, cCentroid, and sCentroidwere 0.9513, 0.9897, 0.9106, and 0.9212, respectively. TheR-squared values that were greater than 0.9 indicated thatmore than 90% of the variation the response was explainedby the regression equation. The regression coefficients withp-values less than 5% are denoted by a star in Table 2. Instatistical hypothesis testing for regression coefficients, a lessthan 0.05 p-value indicates that the test is significant at the5% level. For themildly heterogeneous site, coefficients of b1 (α),b2 (dp), and b3 (i) were significant for all response variables,which indicated the importance of the linear effects of thesethree factors. However, b4 (C0) is only significant for massAtt.It is not surprising that C0 is not important for maxNCbecause this particular response variable was standardizedwith C0. However, C0 was expected to influence cCentroid andsCentroid, which are response variables that reflect how far thenC60 plume is transported in the aquifer and the mobility ofnC60 in the aquifer. Our previous results (Bai and Li, 2012)showed that a higher C0 facilitates a faster initial increasefor the attached phase concentration, which leads to a reducedeffective attachment rate and a higher mobility. This effect

65C. Bai et al. / Journal of Contaminant Hydrology 152 (2013) 60–69

Table 1Values of input factors obtained from a CCI design.

Design pointa α dp (nm) i C0 (mg/L)

1 0.001 100 0.01 12 0.0003 316.2 0.032 3.1623 0.0032 316.2 0.032 3.1624 0.0003 31.6 0.032 3.1625 0.0032 31.6 0.032 3.1626 0.0003 316.2 0.003 3.1627 0.0032 316.2 0.003 3.1628 0.0003 31.6 0.003 3.1629 0.0032 31.6 0.003 3.16210 0.0003 316.2 0.032 0.31611 0.0032 316.2 0.032 0.31612 0.0003 31.6 0.032 0.31613 0.0032 31.6 0.032 0.31614 0.0003 316.2 0.003 0.31615 0.0032 316.2 0.003 0.31616 0.0003 31.6 0.003 0.31617 0.001 31.6 0.003 0.31618 0.0001 100 0.01 119 0.01 100 0.01 120 0.001 10 0.01 121 0.001 1000 0.01 122 0.001 100 0.001 123 0.001 100 0.1 124 0.001 100 0.01 0.125 0.001 100 0.01 10

a Note Zi in Eq. (6) was the log transform of the input factors.

was not obvious here, potentially because a relatively low C0(0.1–10 mg/L) and short injection time periods (i.e., 1 h) wereconsidered. In most of the simulations, the attached nC60concentration was much lower than the Smax, which was notsufficient enough to significantly impact the effective attach-ment rate.

Table 2Estimated regression coefficients and R-squared values for the mildly and highly he

maxNC massAtt

Site IIb Site Ic Site II Site I

b0 −0.43a −0.26a 0.66a 0.50a

Linearb1 −0.14a −0.081a 0.19a 0.12a

b2 0.058a 0.039a −0.056a −0.049a

b3 −0.16a −0.095a 0.17a 0.10a

b4 −0.0035 −0.0022 0.0018 0.0078

Cross productb12 0.0057 0.00095 −0.0061a −0.0018b13 −0.030a −0.022a 0.025a 0.013a

b14 0.000029 −0.00070 0.0089 0.021a

b23 0.017a 0.017a −0.017a −0.028a

b24 0.0018 0.00058 −0.0014 −0.0029b34 −0.00012 0.00061 −0.0020 −0.0027

Quadraticb11 −0.0068 −0.0032a 0.014a 0.0075b22 0.0018 0.0016a −0.0032 −0.0085b33 −0.0098a −0.0020a 0.0059a −0.0059b44 0.0012 0.00075 −0.00091 −0.0008R2 0.6326 0.9513 0.9119 0.9897

The total degree of freedom of error is 360 for these equations.a Coefficients with p-values less than 0.05.b Site II refers to the highly heterogeneous site.c Site I refers to the mildly heterogeneous site.

In addition to the linear impact of the b1 (α), b2 (dp), andb3 (i) variables, in many cases, the quadratic effects (i.e., b11,b22, and b33) and the interactions of these three parameters(i.e., b12, b13, and b23) were also significant, as indicated inTable 2. The significance of these quadratic and interactioneffects between the variables highlighted the complexity ofthe system. All quadratic and interaction terms related to b4(C0) were insignificant except for massAtt for site I indicatingthat C0 did not have much of an influence on these responsevariables. However, the cCentroid was only significantlyaffected by b12 and b33 and the linear variables of b1 (α), b2(dp), and b3 (i). Among these factors, the hydraulic gradienthad the strongest impact on the cCentroid, as demonstratedby the large b3 and b33 values. The predominant hydraulicgradient effect was not surprising, because this responsevariable does not contain any mass related information andonly reflects a geometric feature of the aqueous phase nC60plume. In contrast, all of the coefficients (except for b44) weresignificant for massAtt. This observation further emphasizesthe fact that nanoparticle attachment is influenced by all ofthese factors and their interactive effects.

4.2. The highly heterogeneous site

As shown in Table 2, for the highly heterogeneous site, onlythe massAtt model had an R-squared value that was greaterthan 0.9. All other models had R-squared values from between0.44 and 0.64. Clearly, the higher variation in the highlyheterogeneous aquifer was not completely accounted for bythe polynomial regression equations. An R-squared value ofgreater than 0.6 indicates that themodel is able to explainmostof the variation. Therefore, models for cCentroid, and sCentroidare not expected to be useful for predictions and the maxNCmodel will only marginally predict the results. Here, the highly

terogeneous sites.

cCentroid sCentroid

Site II Site I Site II Site I

1.34a 2.92a −0.071a −0.48a

−0.79 −0.59a −0.57 −0.80a

1.05a 0.49a 0.41 0.58a

2.77a 3.23a 0.84a 0.84aa 0.093 0.056 0.039 0.043

a 0.16 0.069a 0.079 0.092a

0.00025 −0.013 −0.068 −0.11a

0.028 0.020 0.011 0.0130.12 0.032 0.038 0.092a

a −0.018 −0.010 −0.0078 −0.0095a −0.016 −0.013 −0.0066 −0.0092

a −0.073 −0.067 −0.038 −0.054aa −0.066 −0.048 −0.022 −0.021a 0.60a 0.66a 0.21a 0.27a

9 −0.052 −0.044 −0.014 −0.0210.5556 0.9106 0.4432 0.9212

66 C. Bai et al. / Journal of Contaminant Hydrology 152 (2013) 60–69

Table 3Summary of the empirical correlations between the response variables and factors.

Maximuma C/C0maxNC siteIð Þ ¼ −0:26−0:082 Z1 þ 0:038 Z2−0:097 Z3−0:0037 Z12

þ0:0011 Z22−0:022 Z3Z1 þ 0:017 Z3Z2−0:0025 Z32

R2 ¼ 0:9506 �

maxNC siteIIð Þ ¼ −0:35−0:088 Z1 þ 0:049 Z2−0:15 Z3−0:031 Z3Z1 þ 0:018 Z3Z2−0:0087 Z23

R2 ¼ 0:6211 �

Percentage of the injected nanoparticle mass that becomesattached

massAtt siteIð Þ ¼ 0:051þ 0:013 Z1−0:0051 Z2 þ 0:011 Z3 þ 0:00077 Z4 þ 0:00080 Z12

−0:00018 Z2Z1−0:00079 Z22 þ 0:0013 Z3Z1−0:0028 Z3Z2−0:00053 Z32

þ0:00021 Z4Z1−0:00029 Z4Z2−0:00026 Z4Z3

R2 ¼ 0:9897 �

massAtt siteIIð Þ ¼ 0:070þ 0:020 Z1−0:0069 Z2 þ 0:017 Z3 þ 0:0016 Z12−0:00065 Z2Z1

þ0:0025 Z3Z1−0:0017 Z3Z2 þ 0:00075 Z32

R2 ¼ 0:9113 �

x-Direction centroid of the aqueous phase nC60 plume cCentroid siteIð Þ ¼ 4:06−0:16 Z1 þ 0:22 Z2 þ 3:52 Z3 þ 0:068 Z2Z1 þ 0:70 Z23

R2 ¼ 0:9094 �

cCentroid siteIIð Þ ¼ 3:96þ 0:058 Z2 þ 3:22 Z3 þ 0:65 Z23

R2 ¼ 0:5466 �

x-Direction centroid of the attached phase nC60 sCentroid siteIð Þ ¼ −0:20−0:69 Z1 þ 0:49 Z2 þ 0:89 Z3−0:037 Z21 þ 0:088 Z2Z1

−0:12 Z3Z1 þ 0:095 Z3Z2 þ 0:28 Z23

R2 ¼ 0:9204 �

sCentroid siteIIð Þ ¼ 1:61þ 1:20 Z3 þ 0:23 Z23

R2 ¼ 0:4216 �

a siteI and siteII denote mildly heterogeneous and highly heterogeneous sites, respectively.

heterogeneous site possesses a lnK variance of 4.5, which is15.5 times higher than that of the mildly heterogeneous site.The substantially lower R-squared values that were observedformost of the response variables for the highly heterogeneoussite highlight the dominant effect of subsurface heterogeneityon nanoparticle transport.

Nonetheless, the dependency of these response variableson the factors is similar between the two sites. For example,the single parameter coefficients b1 (α), b2 (dp), and b3 (i) aresignificant with R-square values that are greater than 0.6 (i.e.,maxNC and massAtt). In addition, the cross products of thesethree factors and their quadratic terms are significant formultiple occurrences, which indicate that these interactiveeffects are important. b4 (C0), its quadratic term b44, and allcross product coefficients that contained b4 (i.e., b14, b24, b34,and b44) were not significantly related to any responsevariable. The same explanation regarding the insignificanceof C0 for the mildly heterogeneous site can be applied here.

Finally, the R-squared value formassAtt was higher than 0.9.Here, massAtt is influenced by factors b1, b2, b3, their crossproducts and their quadratic terms, which underscores thecomplicated and interactive influences of these factors. The highR-squared value for massAtt occurred because this responsevariable mainly reflects the mass feature of the plume and doesnot account for the geometric distribution of the nanoparticles.In contrast, the variation introduced by the heterogeneity of thepermeability field mainly influences the transport pathway ofnanoparticles by producing various flow velocities and prefer-ential flow pathways. Therefore, the percentage of massattached is not very sensitive to the heterogeneous permeabilityfield. Although the percentage of mass attached does notprovide direct information regarding the distribution of nC60in the field, it can still serve as a good indicator for nanoparticlemobility. When the mass percentage of nanoparticles attachedis higher, the nanoparticles are less mobile.

4.3. Model validation

To evaluate the predictive ability of the RSM approach, finalpolynomial regression models were re-fitted only includingterms if their coefficients had statistically significant p-values(p b 0.05). Table 3 provides a summary of the final regressionequations. Please note that the coefficients and R-squaredvalues in Table 3 are slightly different from those in Table 2,because only statistically significant terms were included inthe re-fitting. The ability of these equations to predict thecorresponding response variables was evaluated by comparingthe predicted values with those observed from the modifiedMT3DMS simulations for a series of scenarios for the mildlyheterogeneous and highly heterogeneous sites.

The validation scenarios, as listed in Table 4, were selectedto cover a wide range for each parameter and to ensure that noscenario was used in the experimental design to develop theregression equations. The nC60 release scenarios were simulat-ed for three randomly chosen K-field realizations for each site.The observed values of the four response variables (maxNC,massAtt, cCentroid, sCentroid) were calculated based on thesimulated MT3DMS results while equations listed in Table 3were used to calculate the predicted response variable values.Fig. 4 provides a comparison between the predicted andobserved results with close agreement resulting when the allpredicted values fall near the straight line with a 1:1 slope.

As shown in Fig. 4, themodels for themildly heterogeneoussites predicted the observed values reasonably well which wasconsistent with the high R-square values that were observedfor these equations. Corresponding to a single release scenario,the observed values for three different random K-field re-alizations are typically very close to each other, reflected by thelargely overlapping symbols in the Figure. This indicated thatmild heterogeneity of the site did not significantly influence thefinal results of the response variables. For the highly

67C. Bai et al. / Journal of Contaminant Hydrology 152 (2013) 60–69

Table 4nC60 release scenarios simulated for empirical correlation validation.

Test α dp(nm)

i C0

(mg/L)

#1 0.001 31.6 0.01 1#2 0.0005 31.6 0.01 1#3 0.005 31.6 0.01 1#4 0.001 200 0.01 1#5 0.001 500 0.01 1#6 0.001 31.6 0.005 1#7 0.001 31.6 0.05 1#8 0.001 31.6 0.01 0.5#9 0.001 31.6 0.01 5

heterogeneous field, only the massAtt was adequately predict-ed for the validation scenarios. Predictions based on all of theother equations for the highly heterogeneous site wereconsiderably scattered around the 1:1 line, which wasconsistent with their low R-squared values. This result strongly

Fig. 4. Comparison between the predicted and observed (i.e., MT3DMS simulated) re(C) aqueous plume x-centroid, and (D) attached phase x-centroid distribution. HK-fields for the mildly heterogeneous field. The three filled circles represent the thr

indicates that the variability among the random K realizationsleads to large differences between the observed and responsevariables, which indicates a low predictive capacity for thehighly heterogeneous site.

5. Summary and conclusions

In this work, we sought to develop simple quantitativerelationships between key factors (i.e., dp, α, i, and C0)and response variables that represent the distribution ofnanoparticles (i.e., maxNC, massAtt, cCentroid, sCentroid).The mathematical model that was used in this study wasbased on an experimentally validated model for nC60 nanopar-ticle transport in porous media (Li et al., 2008) and waspreviously incorporated into the MT3DMS model (Bai and Li,2012). Hypothetical scenarios of nC60 nanoparticle releasesinto subsurface aquiferswere simulated in randomly generatedpermeability fields based on mildly and highly heterogeneoussites. A total of 25 simulations that were based on a CCI

sponse variable values for (A) maximum C/C0, (B) percentage mass attached,ere, the unfilled squares, diamonds, and triangles represent three randomee random K-fields for the highly heterogeneous field.

68 C. Bai et al. / Journal of Contaminant Hydrology 152 (2013) 60–69

statistical experimental design were conducted for each of 15random fields in the mildly and highly heterogeneous sites.RSM was used to develop the final predictive equations basedon the response variable summaries from each numericalsimulation.

The results showed that the regression equations developedfor the mildly heterogeneous sites had high R-squared values,which indicated that uncertainty from mild heterogeneity canbe easily accounted for. However, all of the regressions for thehighly heterogeneous sites (except for the correlation of theattached mass percentage) have R-squared values that areless than 0.7. The poor prediction performance for the highlyheterogeneous site highlights the importance of heterogeneityon nC60 nanoparticle transport. Validations were conducted bycomparing the predicted values that were calculated from theregressionmodels with the observed values that were obtainedfrom the numerical simulation results of several scenarios butwere not used when developing the correlations. All of themodels with high R-squared values were able to accuratelypredict the corresponding response variables in the validationscenarios.

This work represents a first effort for applying RSM forpredicting the distribution of engineered nanomaterials inporous media. Theoretically, the developed correlations can beused to predict the response variables (e.g. maxNC, massAtt,cCentroid, sCentroid), if a similar nanoparticle release scenariooccurs in an aquifer with similar permeability field statistics.However, since the regression equations were developed basedon synthetic data with no field validation possible at this stage,direct application of these regress equations needs to be donevery carefully. When estimating the nanoparticle distributionfor a nanoparticle release accidentwith limited or nomonitoringdata, we would rather suggest using these regression equationsby comparing the input factors with a scenario that has actualfield observation data. Finally, because the regression equationspublished here are based on a numerical model developedspecifically for nC60, they may not be necessarily suitable forother engineered nanomaterials.

Acknowledgments

We thank Drs. Linda M. Abriola and Kurt D. Pennell at TuftsUniversity for their helpful suggestions and discussions. Thisresearch was supported by the National Science FoundationAward No. CBET-0854136.

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