Supporting Information for “Particle Size Distribution, Concentration, and Magnetic Attraction

Affect Transport of Polymer-modified Fe0 Nanoparticles in Sand Columns” by Tanapon

Phenrat1, Hye-Jin Kim

1, Fritjof Fagerlund

2, Tissa Illangasekare

2,

Robert D. Tilton3,4

, Gregory V. Lowry1,3*

Center for Environmental Implications of Nanotechnology (CEINT) and 1 Department of Civil &

Environmental Engineering, 3

Department of Chemical Engineering, 4

Department of Biomedical

Engineering, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA.

2Center for Experimental Study of Subsurface Environmental Processes at Colorado School of Mines

Submitted to

Environmental Science and Technology

*Corresponding author: Greg Lowry (glowry@cmu.edu) 412-268-2948(ph) 412-268-7813 (fax)

Fe0 Content Determination via Acid Digestion and Atomic Absorption Spectrometry

The Fe0 content of RNIP can be determined (separately from total iron) by acid digestion in a

closed container with headspace as described in our previous study (1, 2). H2 produced from the

oxidation of Fe0 in RNIP by H

+ is used to quantify the Fe

0 content of the particles (Eq. S1) (3). The

RNIP concentration was calculated assuming a Fe0(core)/Fe3O4(shell) morphology, and using the

measured Fe0 content, α’ (Eqs. S2 and S3). The factor 1.38 in Eq. S3 is to convert the mass of iron (Fe)

to magnetite (Fe3O4).

↑+→+ ++2

20 2 HFeHFe (S1)

[ ] [ ] 043

][FeOFetotal FeFeFe += (S2)

totaltotal FeFeRNIP ]['])['1(38.1][ αα +−= (S3)

The dissolved iron concentration in the samples after acid digestion was quantified by atomic

absorption spectrometry (AA) (GBC 908AA, Avanta) using Fe lamp at the wavelength of 372 nm. The

calibration curve covered an iron concentration of 1 to 50 mg/L. The relative standard deviation (RSD)

of triplicate measurements was less than 2, while RSD of 4 is recommended for QA/QC purpose.

Applying Ohshima’s Soft Particle Analysis to Estimate Adsorbed Polyelectrolyte Layer

Properties on NZVI

The procedure for extracting the characteristics of the polyelectrolyte layer from electrophoretic

mobility (EM) data involves fitting Ohshima’s model, eqn. S4, with terms defined as in eqns. S5-S8 to

the experimental electrophoretic mobility vs. concentration of a symmetrical electrolyte (NaCl in this

study) to obtain the best fit N, λ, and d (1, 4-9). All other parameters in equations S5 to S8 are fixed for

a given salt concentration.

222

0

11.

4tanh.

8

11m

m

dd

B

B

m

DONm

e

mee

Tk

ze

ze

TkZeN

a

dfu

κλ

κλζηλε

ηλλκλψκψ

ηε κλ

−

−++

+

+=

−−

(S4)

where ε is the electric permittivity of the liquid medium, η is its viscosity, λ is a frictional parameter

given by (γ/η)1/2

, and κm is the effective Debye-Hückel parameter of the surface hydrogel layer, which

includes the contribution of the fixed charge ZeN (5). ζ is the apparent zeta potential of the bare

particles calculated from EM measurements using Smoluchowski’s formula. The function f(d/a) varies

between 1 for a thin adsorbed layer relative to radius of the core particle (a), to 2/3 for a thick layer.

Eqn. S4 is valid when λd and κd > 1 (4). The corresponding expressions for ψDON, ψ0, f(d/a), and κm are

given in eqn S5 to S8 (4, 5, 10, 11),

= −

zn

ZN

ze

TkB

DON2

sinh 1ψ (S5)

Tk

zee

ze

Tk

Tk

ze

ze

Tk

B

dB

B

DONBDON

m

4tanh.

4

2tanh0

ζψψψ κ−+

−= (S6)

( )

++=

3

12

11

3

2

ada

df (S7)

21

cosh

=

Tk

ze

B

DON

m

ψκκ (S8)

where kB is Boltzmann’s constant, T is absolute temperature, and κ is the Debye-Hückel parameter of

the solution. Use of the Ohshima method requires data for the electrophoretic mobility for both the

bare particles and for the polyelectrolyte-coated particles as a function of the bulk solution ionic

strength.

A MATLAB (the Mathworks, Novi, MI) code employing iterative least squares minimization

was used for this fitting the EM data. Ohshima’s model was used to fit the mean ue, mean ue+σ, and

mean ue-σ as a function of ionic strength to obtain three best-fit values of each fitting parameter (1/λ, N,

and d). The average and standard deviation of the fitting parameters determined for the mean ue, mean

ue+σ, and mean ue-σ was calculated and reported in Table S1. It should be noted that this procedure is

not meant to convey the goodness of fit of the data, rather it is used to bound the range of the

magnitude of each parameter (12-14).

Using a Constricted Tube Model to Estimate Apparent Shear Rate Acting on Aggregates in

Porous Media

Assuming a constricted tube model (Figure S6) as representative of the void space between

collectors in the porous media, the apparent shear rate (γs) acting on aggregates in a pore of a specific

diameter is given by Eq.S9

poreeff

sNd

Q3

32

πγ = (S9)

where Q is the total volumetric flow rate in the porous media, and Npore (Eq.S10) is the number of

pores in the column cross section. deff (Eq.S11) is the effective pore diameter.

2)4/( eff

column

pored

AN

πε

= (S10)

470.0

c

eff

dd = (S11)

dc (Eq.S12) is the equivalent diameter of the constriction, and Acolumn is the cross-sectional area of the

column.

5658.2

g

c

dd = (S12)

dg is the average diameter of collectors, which is 300 µm in this study.

Using a Constricted Tube Model to Estimate the Shear Experienced by a Retained

Particle/Agglomerate at the Pore Wall

Assuming a constricted tube model as representative of the void space between collectors in the porous

media, the shear experienced by a retained particle/agglomerate at the pore wall, ∂v/∂r, can be

expressed as Eq. (S13)(15):

22 )2/(

)2/(4

)4/(

/

z

z

z

pore

d

ad

d

NQ

r

v −=

∂∂

π (S13)

dz is the diameter of pore in the constricted tube model used to calculate νcolloid, the fluid velocity at the

location of the attached colloid (15) . Figure S6 illustrates dz as a function of z from the center of a

collector used in this study.

−

−+=

2

maxmax 5.022

42

2h

zdddd c

z (S14)

−−=

2

2 )2/(

)2/(1

)4/(

/2

z

colloidz

z

pore

colloidd

ad

d

NQ

πν (S15)

dmax (Eq.S16) is the maximum pore diameter.

cdd 141.2max = (S16)

Deposition of Agglomerates formed in Porous media.

This section summarizes a modified model that semi-quantitatively supports the effect of

hydrodynamic shear in porous media on the deposition of particles/agglomerates. The original model

was recently reported by Torkzaban and coworkers (16). When a particle or agglomerate collides with

a collector in a flow field an adhesive force (FA) between the particle and the collector promotes

attachment while a drag force (FD) due to the fluid flow promotes detachment. For PSS-RNIP the

adhesive force (FA) is estimated from the absolute value of the secondary minimum (Θmin) predicted by

extended DLVO theory which takes into account the electrosteric repulsions and the van der Waals

attraction between PSS-RNIP and sand (Figure S5 and Table S2 in Supporting Information). Using

eqn S17, the adhesive force is approximated as

min

min

hFA

Θ= (S17)

where hmin is the separation distance at the secondary minimum between the colloid and the collector

surface (Figure S2). The drag force (FD) acting on attached particles or aggregates can be calculated

using eqn S18 (16),

2)(205.10 ar

vFD ∂

∂= πµ (S18)

where µ is the fluid viscosity, ∂v/∂r is the shear experienced by a retained particle at the pore wall, and

a is the radius of the retained particle or the radius of gyration of the retained agglomerate. ∂v/∂r is

calculated assuming that a constricted tube model represents the void space between collectors in the

porous media (Figure S6, Supporting information) such that (15):

22 )2/(

)2/(4

)4/(

/

z

z

z

pore

d

ad

d

NQ

r

v −=

∂∂

π (S19)

where dz is the diameter of pore in the constricted tube model used to calculate νcolloid, the fluid velocity

experienced by the attached colloid (15) . Figure S6 illustrates dz as a function of z from the center of a

collector used in this study. Npore is the number of pores in the column cross section. The calculation of

Npore and dz assuming a constricted tube model is discussed previously.

The drag force results in an applied torque (Tapplied, eqn S20) and the adhesive force results in

an adhesive torque (Tadhesive, eqn S21) (16).

Dapplied aFT 4.1= (S20)

3/1

4

==

K

aFFlFT A

AxAadhesive (S21)

lx represents the radius of the colloid-surface contact area and is estimated using the theory of Johnson,

Kendall, and Roberts (16). K is the composite Young’s modulus. For deposition of PSS-RNIP on silica

sand in this study we assume a value of K= 4.014 x 109 N m

-2 reported for glass bead collectors and

polystyrene particles (17). Particle/aggregate deposition is only favorable and irreversible when

Tadhesive>Tapplied. The magnitude of Tadhesive and Tapplied as a function of particle/aggregate sizes for PSS-

modified nanoparticles with the measured adsorbed layer properties (Table 1) under the hydrodynamic

and geochemical conditions used in this study were calculated (Figure S7). For a fixed pore water

velocity and geochemistry, a greater particle (or aggregate) size increases the probability for

irreversible attachment (i.e. Tadhesive >> Tapplied). Tadhesive increases with increasing radius of the retained

particle/agglomerate to the 1.67th

power (see Supporting Information). In contrast, Tapplied decreases

with increasing size of the retained particle/agglomerate for the range of particle/aggregate sizes used

because ∂v/∂r is inversely proportional to a (Supporting Information Figure S7 and eqn S19).

Scaling Argument that Tadhesive increases with increasing radius of the retained

particle/agglomerate to the 1.67th

power

From Eq.S21

3/1

4

==

K

aFFlFT A

AxAadhesive (S22)

Therefore,

3/13/4

aFT Aadhesive ∝ (S23)

From Eq.S17

min

min

hFA

Θ= (S24)

Θmin is governed by the magnitude of van der Waals attraction between a particle and a

collector. Vvdw for sphere-wall interaction is

s

HaVvdW

6

−= (S25)

Where H is the Hamaker constant, and s is the separation distance between particle and wall.

Therefore,

aFA ∝ (S26)

For this reason, from Eqs. S23 and S26

3/5aTAdhesive ∝ (S27)

Arguments to Support the Effect of Particle Concentrations on Time to Attain a Steady

State Agglomerate Size and to Support the Assumption that the Steady State

Agglomerate Size is Reached Relatively Quickly Compared to the Retention Time in

Column Experiments

Particle concentration can affect the time lag (τSS) for attainment of the steady state based on

the study by Spicer et al (1996). τ is a dimensionless time which can be expressed as:

(S28)

Where G is a shear rate (s-1

); for porous media, G can be calculated from Eq. S9 (in Supporting

Information). φ is the volume fraction of particle in solution (i.e. a function of particle concentration),

and t is time in second. The steady state time lag (τSS) for the number (τnSS) and for the volume (τvSS)

distributions are a function of Coagulation-Fragmentation Group (CF) (when 0.003<CF<0.07 ):

(S29)

When CF is

(S30)

tGφτ =

69.20log35.22 −−= CFnSSτ

496.0log60.5 −−= CFvSSτ

0

00

φβVS

CF =

Where V0 is the volume of particles at time zero (before agglomeration). S0 is the fragmentation rate

for initial particles (V0) and β0 is the collision frequency for initial particles (V0).

(S31)

(S32)

According to S29 to S32, the higher the particle concentration, the smaller the time (t; in second)

required to reach the steady state. Based on the pore water velocity and the porous media used in this

study, G can be calculated from Eq. S9 (Supporting Information). CF values calculated based on

Eq.R3 for PSS-RNIP-F1 particles of 1 to 6 g/L range from 2.7x1036

to 4.7 x1035

, indicating that τSS

should be on the order of seconds once subjected to the flow conditions in the column. This is much

smaller than the retention time of particles in the column (~ 25 minutes). Therefore, the assumption

that agglomeration reaches the steady during the transport in the columns is reasonable.

3/1

0

6.1

0 0047.0 VGS =

33/1

00 )2(31.0 VG=β

Table S1 Characteristics of the adsorbed polyelectrolyte layers at pH 8.0±0.1 as

estimated by Ohshima’s soft particle analysis.

ZN/NA a d

a 1/λ

a |ψDON|

ab |ψ0|

ab

(mole/m3) (nm) (nm) (mV) (mV) Particle Fraction

F1

2.0±0.2

63±15

13±0.6

2.4±0.2

1.7±0.1

F2

2.9±0.3

75±24

11±0.6

3.3±0.3

1.7±0.2

PSS70K-

RNIP

F3

1.7±0.9

75±12

15±0.5

2.0±1.0

1.0±0.4

F1

1.0±0.0

41±2

18±0.0

1.2±0.0

0.6±0.0

PSS70K-

Hematite

F2

1.0±0.0

46±3

18±0.6

1.2±0.0

0.6±0.0

a Errors are ± 1 standard deviation as described in the experimental section.

b The sign of ψDON and ψ0 is negative, and the data is for 10 mM Na

+.

Table S2 Θmin and hmin between PSS-RNIP particles or agglomerates and a flat plate (i.e. a

representative of a collector in porous media) using extended DLVO calculation by taking into

account electrosteric repulsions and van der Waals attractions.

Particle/agglomerate

size (nm)

Θmin

(10-20

N.m)

hmin

(nm)

25 1.0x10-5

190

45 4.1x10-4

170

367 1.5x10-1

170

1000 1.0 170

2000 2.9 170

5000 9.4 170

PSS-RNIP F1

F2

F3

PSS-hematite

F1

F2

(a) (b)

Figure S1 Volume-averaged particle size distributions of different fractions of (a) PSS-

modified RNIP and (b) PSS-modified hematite. Number averaged particles size

distribution of different fractions of (c) PSS-modified RNIP and (d) PSS-modified

hematite. The symbols for (c) and (d) are similar to (a) and (b), respectively

(c) (d)

PSS-RNIP PSS-hematite

(a)

(b)

Figure S2 Sedimentation curves of (a) PSS-RNIP-F2 and –F3 and (b) PSS-hematite-F1

and-F2 at the initial particle concentration of 30 mg/L and 1 g/L.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150

S (nm)

V/(

kBT

)

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 50 100 150 200 250

S (nm)

V(K

BT

)

(a)

(b)

VM

VvdW

Vosm

VT

Velas

VES

PSS-RNIP-F2

(a=25 nm)

PSS-hematite-F2

(a=66 nm)

VT and Vosm

Velas

VES

VvdW

Figure S3. Potential energy of particle-particle interaction calculated by extended DLVO

for (a) PSS-RNIP-F2 and (b) PSS-hematite-F2 using the physical properties according to

the 1st peak in Table 1.

(a) (b)

Figure S4 Normalized breakthrough curves for (a) PSS-modified RNIP and hematite

at particle concentration of 30 mg/L and (b) PSS-modified hematite at particle

concentrations from 1 to 6 g/L.

-5

-3

-1

1

3

5

0 100 200 300 400 500

S (nm)

V(K

T)

PSS-RNIP-F1

(Cluster of a=328)

VT

Velas

VES

VvdW

Vosm

Figure S5. A representative potential energy of particle-collector interaction calculated by

extended DLVO for a cluster with a =328 representing in PSS-RNIP-F1 dispersion.

Secondary minimum

z

dz

FD

Tapplied Tadhesive

vz Collector

Pore

Figure S6 Diameter of pore (dz) as a function of distance from the center

of a collector (z) used in this study assuming the constricted tube model as

a representation of the void space between collectors in the porous media.

RH= 25 nm

Tadhesive

RH= 45 nm

RH= 155 nm

RH= 367nm

RH= 1 µm

RH= 2 µm

RH= 5 µm

Tapplied

RH= 25 nm

RH= 45 nm

RH= 155 nm

RH= 367nm

RH= 1 µm

RH= 2 µm

RH= 5 µm

Figure S7 The calculated Tapplied and Tadhesive for different sizes of PSS-modified

RNIP (as individual particles and aggregates) for the transport conditions used in

this study assuming the constricted tube model (Figure S4) as a representation of the

void space between collectors in the porous media.

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