Environ. Sci. Technol. 2010, 44, 8996–9002
Development of a Trajectory Model for Predicting Attachment of Submicrometer Particles in Porous Media: Stabilized NZVI as a Case Study YU-TING WEI AND SHIAN-CHEE WU* Graduate Institute of Environmental Engineering, National Taiwan University, 71 Chou-Shan Road, Taipei 106, Taiwan, R. O. C.
Received June 29, 2010. Revised manuscript received October 22, 2010. Accepted October 26, 2010.
A new trajectory simulation algorithm was developed to describe the efficiency of a single collector (pore) to catch submicrometer particles moving through saturated porous media. A constricted-tube model incorporating the deterministic (interception, hydrodynamic retardation, van der Waals force and gravitational sedimentation), stochastic (Brownian diffusion), and thermodynamic (electrostatic and steric repulsion force) mechanisms was established to predict the transport and deposition of surface modified nanoscale zerovalent iron (NZVI) particles by applying Lagrangian trajectory analytical approach. The simulation results show good agreement with the results predicted by existing energy-barrier-free models except for the particle size less than 100 nm at low approach velocity. The number of realizations per start location could be decreased down to 100 with the simulations still exhibiting acceptable relative standard deviation for engineering purposes. With the consideration of energy barriers, the model successfully describes the breakthrough curve of polymermodified NZVI in a benchtop soil column as well. The novel simulation scheme can be a useful tool for predicting the behavior of the nanoscale colloidal particles moving through filter beds or saturated soil columns under conditions with repulsion and attraction forces among surfaces.
Introduction Application of nanoscale zerovalent iron (NZVI) in situ is a promising approach for the remediation of groundwater aquifers contaminated by chlorinated aliphatic hydrocarbons (CAHs) (1, 2) and heavy metals (3, 4). The characteristics of being stably dispersed and mobile ensure the effectiveness of in situ application of NZVI (5, 6) since NZVI with less mobility means lower efficiency and higher cost. NZVI synthesized with typical sol-gel method by reducing ferric ion with sodium borohydride tends to aggregate and poses quite a challenge to deliver NZVI in natural porous media or to form permeable reactive barriers for the remediation of dissolved CAHs. To overcome this limitation, a significant number of surface modification techniques have recently been developed to produce stabilized NZVI (SNZVI) (7-10) or to enhance the transport of SNZVI (11-13). Surface * Corresponding author phone: +886-2-2362-9435; fax: +886-22362-9435; e-mail:
[email protected]. 8996
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modifiers such as polymers and surfactants have been effectively applied to stabilize NZVI due to their surfaceactive properties, which can alter the electrostatic charges or provide steric hindrance to create the repulsion (energy barrier) between two adjacent particles. But, the influence of the applied modifier on the enhancement of SNZVI transport in the porous media cannot be determined without the results of column tests. To save the time for trial-anderror and the cost of manufacturing a mobile SNZVI, it is then beneficial to have a tool to promptly evaluate the mobility of engineered SNZVI prior to the column transport experiment or costly pilot test. However, no estimating tool based on mechanistic colloid transport has yet to exist in predicting the collection efficiency of anthropogenic nanoparticles, with repulsive and attractive forces among grain surfaces (14), let alone the newly engineered SNZVI. The theoretical framework commonly used to predict colloid transport and deposition through porous media is the colloid filtration theory (CFT). CFT was first presented by Yao et al. (15) in the 1970s to present the three mechanisms, interception, gravitational sedimentation, and Brownian diffusion. Since then, a number of expressions for the collection efficiency have been developed, with an equation developed by Rajagopalan and Tien (RT) being one of the most widely used expressions (16). In 2004, Tufenkji and Elimelech (TE) (17) reworked a numerical solution of the convective-diffusion equation (Eulerian approach) using the Happel sphere-in-cell model (18) and obtained a new correlation equation that, unlike the RT equation, considers the influence of hydrodynamic and van der Waals interactions on the deposition of particles dominated by Brownian diffusion. However, TE equation neglects the particle rotation caused by hydrodynamic interactions which was considered by the RT equation for the deterministic motion. With the Eulerian approach, a potential function is required to describe the interactive forces when energy barriers are present (19). As the potential function is generally unknown for newly engineered nanoscale particles or biological colloids, the application of Eulerian method is sometimes limited. Nonetheless, trajectory approach is able to solve these problems by directly providing the mechanistic description of transport based on Newton’s second law (20). By applying the Langevin equation in the trajectory approach, the effects of deterministic (interception, hydrodynamic retardation, the van der Waals force and gravitational sedimentation), stochastic (Brownian diffusion), and thermodynamic (electrostatic and steric repulsion forces) mechanisms on the transport of colloids could all be considered. However, the energy-barrier condition has not been computed by the trajectory approach in any known cases, only the energy-barrier-free cases have been prevalent, such as one in Happel sphere-in-cell (19) and another in hemispheres-in-cell models (14) to predict the deposition of bare particles at various sizes. Other than the mechanistic model, semiempirical models have also been implemented to model the transport of bare (21) or macromolecule coated particles (22) in porous media. This study was motivated to fulfill the need for an estimating tool based on mechanistic colloid transport. This research first developed a trajectory simulation algorithm with constricted-tube model, which would provide accurate predictions of the deposition of submicrometer particles moving through porous media in both the absence and presence of interaction forces with acceptable computation efforts. Next, the proposed algorithm was tested to demonstrate its performance by comparing the simulation results 10.1021/es102191b
2010 American Chemical Society
Published on Web 11/10/2010
Description of Flow Field. A two-dimensional flow field equation established by Chow and Soda (24) is adopted. Their analytical solution of the steady laminar flow of an impressible Newtonian fluid in an axisymmetric sinusoidal constricted tube is suitable for simulating the groundwater environment. The stream function (ψ) and perturbation solution up to the second-order term is provided in the SI with error correction for zero-order term of velocity component along the r direction. The zero-order stream function is used to calculate the single-collector efficiency of the constricted tube. The perturbation solution is used to compute the fluid velocity (v) components along the axial and radial directions. DeterminationofParticleTrajectory.UsingtheLagrangian approach is a straightforward methodology to track particle position over time, according to the principles of Newton’s second law. Mechanistically, particle trajectory simulation is based on a Langevin equation given as mp
FIGURE 1. Geometry of sinusoidal constricted tube. rc ) 0.35rg, rmax ) 0.865rg, h ) dg (grain diameter). When simulating the particle trajectory, the translation and rotation of coordinates are necessary to calculate the trajectory during each time step interval. Translation depends on the position of particle in r-z coordinate system. Rotation uses angle θ formed by z-axis and the tangent to the tube surface at point O′′. θ is negative when z < h and positive when z > h. The r-z coordinate system is translated first and then rotated by - π/2 + θ to become normal and tangential to tube surface coordinates. with the existing theoretically predictions under energybarrier-free condition and experimental data of SNZVI which is incorporated with the effect of steric repulsions.
Algorithm of Trajectory Simulation Geometry of the Collector for Simulating the Porous Media. Differing from the commonly employed models using Happel sphere-in-cell (19) or Hemishperes-in-cell (14) geometry in trajectory simulation, constricted-tube geometry (23) has been chosen to account for the geometric nature of the pores where collectors (soil grains) are piled up and leave many interstitial pores with constricted passages. The sinusoidal constricted tube adopted in this study has wall geometry described as
[ (
rw ) rm
)
( )]
rmax - rc z 1+ cos 2π rmax + rc h
dup ) FH + FB + FG + FvdW + FEDL + FE dt
where mp is the particle mass, up is the particle velocity vector, and the terms on the right-hand side are force vectors acting on the particle including hydrodynamic force (FH), random Brownian (FB), gravity (FG), van del Waals (FvdW), electrostatic force (FEDL), and other external forces (FE), which is represented by only steric force in this study. The expressions of the deterministic forces (FH, FG, FvdW) and energy barriers (FEDL, FE) are listed in Table 1. Among the forces, FvdW, FEDL, and FE only have effects on particles in perpendicular direction relative to the collector surface. As for Brownian force, combining the continuous and molecular description of the fluid into a single equation is not justifiable, as pointed out by Chandrasekhar (28). Thus, Chandrasekhar’s derivation is adopted to solve the Brownian force calculation in eq 2. To obtain particle velocities, a time-interval approach was applied in integrating the forces acting on the particle. At each time interval, the integration was derived from normal and tangential directions relative to the collector’s surface, respectively. With the velocity and the initial position known at time, t ) τ, and the interval of time given, the position and the velocity at t ) τ + ∆t could be derived. The derived position and velocity then became the initial position and velocity of the next step. Thus the velocity and position of each step were calculated in this time-interval approach as follows
[
τ τ0 -β∆t ) up⊥ e + v⊥f1f2(1 - e-βcs∆t/f1) + up⊥
]
∆Fgsin θ f + Fpβcs 1
F f (1 - e-βcs∆t/f1) + Ru⊥ (3) mpβcs 1
(1)
where rw is the wall radius, rm() (rmax + rc)/2) is the average tube radius, rmax is the maximum radius of the tube, rc is the radius of the constricted throat, z is the Cartesian coordinate of axial direction, and h is the length of constricted tube. The two-dimensional coordinate represents a three-dimensional flow under the assumption of axisymmetric flow. For the simulation of filtration, it is assumed that the filter bed is composed of a number of unit bed elements (UBEs) connected in series. Each UBE contains a number of constricted tubes in uniformed size. The total number of the constricted tubes present in an UBE, Nc, is applied to calculate the average axial velocity () in the constricted tube so that it is comparable with the approach (superficial) velocity (U) of the filter bed in other models. The schematic diagram of particle passing through half of the constricted tube is shown in Figure 1. Details on the calculation of Nc and < U0 > are provided in the Supporting Information (SI) .
(2)
(
τ τ0 -β∆t ) up|| e + v||f3 + up||
)
∆Fgcos θ f (1 - e-βcs∆t/f4) + Ru|| Fpβcs 4 (4)
where F is the total external force, including gravity, van der Waals, electrostatic, and steric forces. θ () tan-1(drw/dz)) is as shown in Figure 1. Ru⊥ and Ru | | are two random deviates described in the latter section. Once the particle velocity vector is determined, the updated particle position could be resolved from first-order integration as τ τ0 τ0 xp⊥ ) xp⊥ + up⊥
[
(
v⊥f1f2 +
∆t -
(1 - e-βcs∆t/f1) f1 + βcs
)
∆Fgsin θ F f + f × Fpβcs 1 mpβcc 1
]
(1 - e-βcs∆t/f1) f1 + Rx⊥ βcs
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(5)
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TABLE 1. Expressions for Forces Applied in Eq 4a forces
expressions
drag
FH ) -β(up - v)/mp
gravitational
FG )
VvdW
(20)
4 πa 3(F - Ff)g 3 p p
-
dVvdW dH Aap 14H )1+ 6H ς
FvdW ) van der Waals (retarded)
references
{ }[
(25)
-1
]
dVEDL dH ) 4πεrε0apζ2ln(1 + e(-κH))
FEDL ) electrostatic
VEDL
FE )
steric
6πapkT 5s
2
[( H +2L2δ )
5/3
-1
(26)
]
(27)
a β() 6πµap/csmp) is the friction coefficient per unit mass (cs is Cunningham correction factor, set to 1 in this study); v is the fluid velocity vector; Fp and Ff are the density of particle and fluid, respectively; g is the gravitational acceleration vector; A is the Hamaker constant; H is the separation distance between particle and tube surface; ς is a characteristic wavelength of the dielectric, usually taken to be 100 nm; εrε0 is the permittivity of water; and ζ is the surface potential of particle and collector. In this study, zeta potential is used instead; κ is the reciprocal Debye length; k is Boltzman’s constant; T is temperature; s is the distance between the chain anchoring points; L is the polymer end-to-end distance; and δ is the compressed thickness.
τ τ0 τ0 xp|| ) xp|| + up||
(
)
∆Fgcos θ (1 - e-βcs∆t/f4) f4 + v||f3 + f × βcs Fpβcs 4
[
∆t -
]
(1 - e-βcs∆t/f4) f4 + Rx|| (6) βcs
The particle position can then be obtained incrementally. The time interval, ∆t, is set sufficiently small so that the external forces, velocity, and position over ∆t can be treated as constants, but it is still sufficiently larger than the particle’s momentum relaxation time, τB (where τB ) mp/6πµap; µ is fluid viscosity, ap is particle radius), in order to neglect the particle inertia effect. ∆t ) 1 µs is adopted for all simulations in this study. f1, f2, f3, and f4 are universal hydrodynamic functions described in the SI. Ru and Rx are two random deviates that are bivariate Gaussian distributed, which Chandrasekhar (28) used to represent the Brownian force. The expressions of Ru and Rx described by Jin et al. (29) was adopted as shown in the SI. As the hydrodynamic retardation effect is direction dependent, normal (Ru⊥ and Rx⊥) and tangential (Ru | | and Rx | |) components of Ru and Rx are presented separately. Comparing to the similar terms used in ref 20, the expressions of Ru and Rx are found to better fit the proposed approach. Estimation of Collection Efficiency. The estimation method of collection efficiency is based on the limiting particle trajectory concept. According to Tien and Ramarao (20), the limiting particle trajectory is defined as that the trajectory which barely grazes the collector. Limiting particle trajectory occurs when the Brownian diffusion effect is insignificant. A particle entering the tube at a specific position could be collected before exiting the tube. Once this specific position is established, all particles entering at the limiting position, which defines the distance between a specific position and the wall, would also be collected before exiting 8998
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the tube. The collection efficiency could be the ratio of the amount of the fluid entering the tube within the limiting position to the total inflow. For the constricted-tube collector, the collection efficiency of limiting trajectory would be (30) ηlim )
ψ[0, rlim] - ψ[0, rmax] ψ[0, 0] - ψ[0, rmax]
(7)
where ψ[0,rlim] represents the value of zero-order stream function at z ) 0, r ) rlim. If the effect of Brownian diffusion is significant, eq 7 needs to be modified. In this paper, eq 8 was created to calculate the total collection efficiency by summarizing the probability of collection efficiency of limiting trajectory 100
∑ P[0, (1 - 0.01n)∆R] × {ψ[0, (1 - 0.01n)∆R] n)1
- ψ[0, (1 - 0.01(n - 1))∆R]} ψ[0, 0] - ψ[0, rmax] ψ[0, ∆R] - ψ[0, rmax] + ψ[0, 0] - ψ[0, rmax]
η)
(8) where P[0,∆R] represents the probability of collection, starting at position z ) 0, r ) ∆R. ∆R ) rmax - ap, and the stream function applied here is zero-order. To obtain the estimation of η, the initial coordinate of the trajectory is set at (0, rmax - ap). Then trajectory realization is performed for every 0.01(rmax - ap) interval in the r direction at z ) 0. A particle is determined as collected when the separation distance is less than 5 Å (19). Within this distance, the attractive force would ultimately draw the particles to the surface of collector and not allow the particles to return
TABLE 2. Summary of Parameters Used in Comparing the Simulated Collection Efficiency for Energy-Barrier-Free Cases parameter
value (range)
particle diameter, dp grain diameter, dg approach velocity, U porosity, ε particle density, Fp fluid density, Ff fluid viscosity, µ Hamaker constant, A fluid temperature, T
0.01 - 10 µm 0.327 mm 8 × 10-6 - 3.4735 × 10-4 m/s 0.372 1.07 g/cm3 0.997 g/cm3 8.9 × 10-4 kg/m/s 1 × 10-20 J 298 K
to the field with energy barrier. Each probability of collection was computed from the start location at 0.99 (rmax - ap) down to the origin of the center of the constricted tube accordingly. If the probability of collection is zero for the location, the probabilities of collection computed from the subsequent locations are assumed to be all zero so that the start locations required to be computed will be less than 100. The relative standard deviation of the prediction by the trajectory model simulation of start location is 0.5% for 100 realizations, which may be acceptable for preliminary engineering applications with reasonable computational effort. More accurate predictions require a higher number of realizations. For example, 2000 or more are necessary for a relative standard deviation less than 0.1%. Implementation of Computational Program. A Fortran program was implemented to compute the collection efficiency in accordance to the above-mentioned equations and the flowchart described in the SI. The Intel Fortran compiler (version 11.1.048) with IMSL 6.0 numerical library was applied, with the random number generator of IMSL function used to calculate Brownian motion with automatically seeding according to the computer system clock time. Newton’s method of approximation was used to calculate the minimum distance between particle and the tube surface. Table 2 summarizes the values of the parameters used in the simulation of energy-barrier-free filtrations.
Results and Discussion Comparison of Simulated Collection Efficiency for EnergyBarrier-Free Cases. RT and TE equations are the predictive models extensively used to predict the collection efficiency of colloidal particle in energy-barrier-free cases. The simulation results using our developed model in the absence of energy barriers were compared to those of RT and TE equations by using the same parameters shown in Table 2. It is necessary to note that the geometry of constricted tube is different from the one in Happel sphere-in-cell. The constricted (throat) and maximum radii, which determine the average radius for calculating the average flow velocity, were adjusted to fit the approaching velocity in simulation. The constricted radius was set as 0.35rg, according to Pendse and Tien (30). The maximum radius was set as 0.865rg based on the trial for dp ) 10 µm, rather than using the expression in ref 23. The geometric setting was applied for all simulations in this study. Collection efficiencies produced from simulations adopting the values of parameters in the range of those in Tufenkji and Elimelech (17) are shown in Figure 2. The three selected fluid velocities are frequently found in natural groundwater systems and are comparable to those used in other models and experiments. The results from the new simulation algorithm demonstrated good agreements with the results derived from RT and TE equations over a wide range of particle sizes except for particle diameter less than 100 nm. A trend was observed that the slower the approach velocity
FIGURE 2. Comparison of simulated collection efficiencies with the RT and TE predictions at three approach velocities: (a) 3.4375 × 10-4 m/s, also compared to the results from Nelson and Ginn, of which data are divided by a factor γ2 ) (1 - ε)2/3 to be comparable under the same definition of collection efficiency (19); (b) 4.63 × 10-5 m/s () 4 m/day); and (c) 8 × 10-6 m/s; the parameters differing from Table 2 are dg ) 0.4 mm, ε ) 0.36, Gp ) 1.05 g/cm3, T ) 288 K. is, the greater the difference between the simulation results and TE’s results at particle size less than 100 nm will be. Figure 2a also shows lower collection efficiencies for particle size less than 100 nm under Nelson and Ginn’s (19, 31) trajectory analysis. As explained in Nelson and Ginn (31), computational domain would better demonstrate the reason of the discrepancies for particle size less than 100 nm. The computational domain of our approach is based on the constricted tube model, which has varying dimensions of passages and is different from the Happel model used in TE or RT where the spherical collector is surrounded by a fluid envelope with uniform thickness. According to Song and Elimelech (32), the Happel model is valid only with moderate and larger Peclet numbers (Pe ) Udg/(kT/6πµap)). The Peclet number applied in the Happel model in Figure 2 is no doubt in the valid domain (Pe > 70 (33)), which also signifies that the diffusion boundary layer is confined within the uniform VOL. 44, NO. 23, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 3. Experimental Parameters Used in Calculating Net Force for Simulation FvdW
FEDL
A (J) case case case case
a
1 2b 3c 4d
8 8 6 9
× × × ×
-20
10 10-20 10-20 10-21
FE
ζ (mV)
K (nm-1)
s (nm)
L (nm)
δ (nm)
-47.8 -30 -1 -22.8
0.7888 0.3333 0.3333 0.9428
13.25 14.9 69 --
9.3 11 75 --
1.3 1.4 3.2 --
a b PAA-dispersed NZVI. PAA-dispersed NZVI. PSS-dispersed NZVI. d Nonionic surfactant-dispersed NZVI, please refer to the SI for more details on parameters.
c
thickness of the fluid envelope. Unlike the Happel model, the constricted tube model does not have a uniform boundary but a boundary with varied thickness due to its wall geometry. With thicker boundary in some sections of the constricted tube allowing longer diffusion distance and lesser chance of being collected, the resulting collection efficiency would be lower than those derived from the Happel model for small particles. This explains the discrepancy shown in the prediction for particle size less than 100 nm, since the probability of being collected for small particles is dominated by diffusive movement. Comparison with Experimental Data of SNZVI. Simulations were conducted in the presence of energy-barrier and compared with experimental data of four cases from SNZVI breakthrough studies reported in the literature, two of which used poly(acrylic acid) (PAA) dispersed NZVI, one poly(styrene sulfonate) (PSS) dispersed NZVI, and the other one used a nonionic surfactant to disperse NZVI. Parameters applied in these four cases for calculating the net interactive forces are listed in Table 3. The breakthrough tests for case 1 and case 4 were conducted in our laboratory, with details explained in the SI. Cases 2 and 3 were research on polymermodified NZVI by Schrick et al. (34) and Phenrat et al. (35), respectively. The values of parameters used for model simulation were adopted from literatures, with only a limited number of them reasonably assumed, as explained in the SI. NZVI modified by surfactants or polymers incurs doublelayer and steric repulsions. Net force from these repulsions and van der Walls attraction is the key element in influencing SNZVI contact with porous media. The changes in net force with respect to the separation distance between particle surface and collector surface for these four cases are depicted in Figure 3. A simplified model is used to calculate the net force between modified particle surface and collector surface. The model with details explained in the SI can be envisioned as a particular case of Ohshima’s soft particle theory (36) with the assumption that the softness parameter is so small that the polymer and surfactant coated particle behaves like a rigid particle. Then, the slipping plane is positioned outward from bare particle surface to near the interface of bulk liquid and coated material. Based on the simplified model, the value of Hamaker constant applied to calculate van der Waals force was adopted from the value of polymer or surfactant in water. Measured zeta potential and ion strength were used to estimate doublelayer repulsion, which was considered to be significant in the nonionic surfactant case due to high ionic strength measured at 0.08 M. Steric repulsion considered in cases 1, 2, and 3 was estimated by referring to the parameters from literature under the similar condition; except for case 3, where an assumption of compressed thickness of polymer and a fitting value of distance between the polymer chain anchoring points were made due to the lack of reference. Figure 3 shows that although the same PAA with an average molecular weight of 2000 as used, the near-surface repulsion in case 2 is smaller than that in case 1 due to higher ionic 9000
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FIGURE 3. The energy barriers of four experimental cases. Case numbers are denoted in the upper cases. In cases 1, 2, and 3, particles were dispersed by polymer dispersants. In case 4, nonionic surfactant was used as dispersant.
FIGURE 4. Comparison of the collection efficiency of experimental data with the predictions by trajectory simulation. The inset shows the comparison of attachment efficiency. Case numbers are denoted in the upper cases. strength and consequently different values of parameters like the length of polymer. Steric repulsion seems dominant in case 3 with a thicker dispersant layer due to larger polymer and a higher dosage on the surface. Also, the double layer effect is negligible according to the estimation from default values of parameters. Therefore, a low repulsive force is extended to a longer distance from the surface in case 3, as compared to other cases. As the end result, case 4, with the use of nonionic surfactant, exhibited the highest energy barrier. The same surfactant applied in the field study (37) also demonstrated good SNZVI mobility, with the same trend predicted in case 4. Simulation and experimental results of the collection efficiency from the four data sets are compared in Figure 4. Of which, ηexp were estimated by using eq 9 ηexp ) -
( )
dg 2 C ln 3 (1 - ε)L0 C0
(9)
where L0 is the packed bed length, and C/C0 is the normalized particle concentration in column outlet at the initial stage of the particle breakthrough curve. Experimental attachment efficiency (Rexp) and simulated attachment efficiency (R) are respectively derived from
Rexp ) R)
ηexp η0
η ηsim
(10)
(11)
where η0 is the collection efficiency of a single-collector calculated by TE correlation equation, η is the overall simulated collection efficiency, and ηsim is the simulated collection efficiency under the energy-barrier-free condition. As Figure 4 indicates, the simulated results for predicted collection efficiencies of polymer-modified NZVI in three data sets correlate with experimental results. But the collection efficiency predicted from NZVI dispersed by nonionic surfactant is overestimated comparing to the experimental results. The inset of Figure 4 shows that the attachment efficiencies of the four data sets are comparable, except for a slight underestimation for the PAA-dispersed experimental set in case 2. In the nonionic-surfactant case, the surfactant concentration of 5000 mg/L, exceeding the critical micelle concentration, would create micelles in the suspension, as was observed in the results of our particle size analyses shown in the SI. Due to the forming of nonionic surfactant micelles, there is another external force called oscillatory structural force, which is also known as hydration force that is suspected to increase the repulsive force (38). Under such circumstance, the van der Waals and double-layer forces are not the only forces acting on particle surface. The lack of consideration for other external forces may result in the discrepancies between our simulation results and the results of the experiments with nonionic surfactants. Environmental Implications of the New Trajectory Simulation Algorithm. Since the simulation results demonstrated a good agreement with the SNZVI transport experiments, it would be beneficial if the new trajectory simulation algorithm could be applied to the prediction of the transport and deposition of other anthropogenic nanoparticles in the porous media to better understand the impact on the actual environment. Lecoanet et al. (39) have presented a well-defined system to assess the mobility of particles with size from 1.2 to 303 nm in porous media. One of their results showed that the monodispersed silica with size around 57 nm gain breakthrough ratio up to 97% through a 9.25 cm column at 0.04 cm/s linear velocity. Comparing to the experimental results of Lecoanet et al., the simulated collection efficiency, 0.00011, is close to the experimental result of 0.00013, with the experimental setting of particle size at 57 nm, zeta potential at -29.8 mV and Hamaker constant at 1.7 × 10-20 J (40) for silica in water. With the capability of further predicting the collection efficiency for particles smaller than 100 nm, the proposed approach is ready to be applied to predict the transport of biological colloids (such as bacteria and viruses) in the porous media to lower the impact on water environment. The new approach could also help in evaluating the fate of newly or in-progress engineered nanoparticles to adhere to green design standards. As for engineering applications, the proposed method is available to assist in the design of the filtration system for treating nanoscale particles. Some modifications can be taken for future applications. First, variable distributions of particle and collector size can be accepted. Also, the effects of agglomeration/aggregation (41, 42) should be notably considered during particle transport in porous media. Lastly, particle entrapment, which affects the size and shape of collector, should be taken into account for practical interest.
Acknowledgments The authors would like to thank the National Science Council (NSC), Taiwan, ROC for the financial support under Grant
no. NSC 97-2221-E-002-068-MY2. We are also grateful to Computer and Information Networking Center, National Taiwan University for the support of high-performance computing facilities.
Supporting Information Available Flow field of constricted tube applied to the simulation, the universal functions for hydrodynamic retardation correction of the drag and Brownian forces, the expressions of two random deviates accounted for the effect of Brownian force, the computational algorithm with the step-by-step description of calculating process, the number of trajectory realizations to show the convergence of collection efficiency as the number of realizations is increased, the parameter values used in experimental data simulation, the breakthrough column test of poly(acrylic acid) (Case 1) and a nonionic surfactant (Case 4) modified NZVI, and the modeling approach for calculating the total energy between polymer coated particle surface and collector surface. This material is available free of charge via the Internet at http:// pubs.acs.org.
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