Supporting Information
for
Aggregation kinetics of CeO2 nanoparticles in KCl and CaCl2 electrolytes:
Measurements and modeling
Kungang Li, Wen Zhang, Ying Huang, and Yongsheng Chen*
School of Civil and Environmental Engineering,
Georgia Institute of Technology, Atlanta, Georgia 30332, United States
*Corresponding author: e-mail: yongsheng.chen@ce.gatech.edu; Phone: (+1) 404-894-3089
Submitted to
Journal of Nanoparticle Research
1
S1. Model derivations
von Smoluchowski’s population balance equation describes the irreversible aggregation
kinetics of particles (Smoluchowski 1917) and is expressed as
dnk 1
    ri , rj    ri , rj  ni n j  nk    ri , rk    ri , rk  ni
dt
2 i  j k
i 1
(S1)
where nk (or ni and nj) is the number concentration of aggregates comprised of k (or i and j)
primary particles (also called k-class or k-fold particles or aggregates),  (ri,rj) and  (ri,rj) are
the collision efficiency function and collision frequency function for class i and j particles, and ri
and rj are the radii of class i and j particles.
For the same class of particles,  (i,i) is equal to 8kT/3, where k is Boltzmann’s constant
(1.38×10-23 J/K), T is the absolute temperature (298 K), and  is the viscosity of the solution
(1×10-3 Pa·s). Taking into account the van der Waals forces and hydrodynamic interactions, the
collision frequency rate is then expressed as (Holthoff et al. 1996):
8kT
  i, i  
3
 
exp VA  h  kT  
du 
 2   u 
2
2  u
 0

1
(S2)
where VA (h) is the van der Waals attraction energy (kT); h is the surface-to-surface separation
distance between two particles (nm); r is the particle radius (nm); u=h/r; and  (u) is the
correction factor for the diffusion coefficient, which is related to the separation distance by the
equation (Honig et al. 1971):
 u  
6  u   13  u   2
2
(S3)
6 u   4 u 
2
The collision efficiency  is the reciprocal of the stability ratio, which is defined as (Chen
and Elimelech 2006; Mcgown and Parfitt 1967):
2

exp VA  h  kT    
exp VT  h  kT  
  i, i       u 
du       u 
du 
2
2
2  u
2  u
 0
  0

1
(S4)
where VT (h) is the total interaction energy between particles separated with a distance h. In
classical DLVO theory, VT (h) is the sum of the van der Waals attraction energy VA (h) and the
electrical repulsion energy VR (h). However, as discussed above, DLVO may not quantitatively
explain experimental observations. The EDLVO theory adopted includes an additional term,
Lewis acid-base interaction energy VAB (h), such that VT (h) = VA (h)+ VR (h)+ VAB (h).
The interaction energies between two identical particles in a 1-1 electrolyte solution are
expressed using equation (S5 a-e) (Abu-Lail and Camesano 2003; Schwarzer and Peukert 2005;
van Oss 2003; Gregory 1975):
VA  
Aa
12h 1  11.12h / c 
VR  h  
128 kBTn 1 2
2

(S5a)
r2
exp   h 
2r  h
(S5b)
 zi e Si 

 4kT 
 i  tanh 
 1 
(S5c)
 0 kBT
(S5d)
2 N A Ie2
 h0  h 
VAB  h    r GhAB
exp


0
  
(S5e)
where A is the Hamaker constant and for CeO2 a value of A of 5.57×10-20 J was obtained from
(Karimian and Babaluo 2007). a is the particle radius. h is the separation distance between the
interacting surfaces. c is the “characteristic wavelength” of the interaction, often assumed to be
100 nm (Abu-Lail and Camesano 2003). n is the concentration of electrolytes. kB is the
3
Boltzmann constant, 1.38×10-23 J/K; T is absolute temperature, 298 K. zi is the valency of the ith
ion. e is unit charge, 1.602×10-19 C. ψSi is the intrinsic constant surface potential (V) of the
interacting particles in an aqueous medium. κ-1 is the Debye length (nm). ε0 is the dielectric
permittivity of a vacuum, 8.854×10-12 CV-1m-1. ε is the relative dielectric constant of water, 78.5;
NA is Avogadro’s number, 6.02×10-23 mol-1. I is the ionic strength (M), I=0.5·ΣciZi2, where ci is
the molar concentration of one species of ions (i). λ is the correlation length, or decay length, of
the molecules of the liquid medium (for pure water, this value is estimated to be 1 nm (van Oss
2003)); Gh0AB is the polar or acid-base free energy of interaction between particles at the
distance h0 (Grasso et al. 2002), which is the minimum equilibrium distance due to Born
repulsion, 0.157 nm (van Oss 2003).
In 2-1 electrolyte solutions, equation (S5b) and equation (S5c) are replaced by equations
(S5f) and (S5g), respectively (Ohki and Ohshima 1999; Ohshima 2006):
VR  h  
384 kBTn 1 2
2

r2
exp   h 
2r  h
(S5f)
1/2


 e 
2 exp  Si  / 3  1/ 3  1

3
 kT 

i  
1/2
2

 e Si 
 2 exp  kT  / 3  1/ 3  1




(S5g)
Going back to equation (S1), we can write the change rate of number concentrations of each
class of particles.
dn1
 1111n12  12 12 n1n2  13 13n1n3 
dt
dn2 1
 1111n12  12 12 n1n2   22  22 n2 2   23  23n2 n3 
dt 2
dn3
 12 12 n1n2  13 13n1n3   23  23n2 n3   33 33n32 
dt
4
(S6)
dn4 1
  22  22 n2 2  13 13 n1n3  14 14 n1n4   24  24 n2 n4   34 34 n3 n4 
dt 2
where ij and ij stand for  (i,j) and  (i,j), respectively.
If the particle size distribution is not broad, e.g., the NP sizes differs by a factor of
approximately two or less, it is safe to assume ij to be constant (ii) (Elimelech 1995). In the
collision efficiency function ij approximation, the collision efficiency between two primary
particles (11) is used as a substitute under the assumption that only the two involved primary
particles determine the interaction energy between aggregates (see Fig. S1 for more illustrations)
(Schwarzer and Peukert 2005).
Under above approximations, we summed the terms in equation (S6) and obtained a simple
equation (S7), which showed the rate of change of the total particle concentration.
dntot
1
  11 ii ntot 2
dt
2
(S7)
Replacing ii and 11 with equations (S2) and (S4), respectively, we obtained the equation
(S8).
1
dntot
4kT

dt
3
 
exp VT  h  kT  
du  ntot 2
 2   u 
2
2  u
 0

(S8)
We used a symbol “w” to represent the complex integration equation, and it is actually the
classical expression of inverse stability ratio:
 
exp VT  h  kT  
w   2   u 
du 
2
2  u
 0

1
(S9)
dntot
4kTw 2

ntot
dt
3
(S10)
5
Solving equation (S10) yields:
ntot 
n0
1  4kTwn0t / 3
(S11)
where ntot is the total number concentration of various classes of particles, and n0 is the initial
number concentration of primary particles.
The structures of aggregates have been recognized to be fractal and can be described as
nr-dFor n=cr-dF, where n is the number of aggregates, r is the radius of aggregates, dF is the
fractal dimension (Lin et al. 1989; Lin et al. 1990) and c is a constant. Thus, equation (S11) can
be rewritten as equations (S12 a-e):
c  r  dF 
r  dF 
c  a  dF
1  4kTwn0t / 3
(S12a)
a  dF
1  4kTwn0t / 3
(S12b)
log r dF  log adF  log 1  4kTwn0t / 3 
(S12c)
d F log r  d F log a  log 1  4kTwn0t / 3 
log r 
(S12d)
 4kT

1
log 1 
wn0t   log a
dF
3


(S12e)
where a is the radius of the primary particles. The k, T, n0, ,and a are constants; and w can be
calculated using EDLVO theory.
The aggregation kinetics in equation (S12e) can be used to describe the growth of the
aggregate radius over time. However, this equation can be applied only in regimes where the
collision efficiency is relatively high or close to unity (i.e., in the DLA regime). In the RLA
regime and at other conditions with very low collision efficiencies, a rigorous expression does
6
not exist because the collision efficiency is determined by the aggregate structure in addition to
the interaction forces (Runkana et al. 2005; Ball et al. 1987). In such regimes, a large number of
collisions are required to achieve a successful aggregation, and the aggregates explore many
possible mutual configurations before they stick together firmly. The aggregation rate coefficient
in RLA (KRLA) is then directly proportional to the volume of the phase space Vc, over which the
center of one aggregate can be positioned to reach a bondable contact with another aggregate
(Ball et al. 1987). For two solid spheres with similar radii (r1  r2 and both are equal to r), Vc is
proportional to r2. Vc is expected to be larger for fractal aggregates with similar radii than for
solid spheres because the surfaces of the former are rough. In the RLA regime, it is proposed that
Vc  rdF (Ball et al. 1987).
Therefore, for two fractal aggregates with similar radii, the aggregation rate coefficient is
given by KRLA Vc rdF. Combining this expression with ntot  r-dF yields equation (S13):
KRLA= kRLAntot-1,
(S13)
where kRLA is the rate constant.
Equation (S13) is then substituted into the reduced von Smoluchowski’s population balance,
equation (S7), which yields
dntot
  k RLA ntot
dt
(S14)
Thus, the aggregation kinetics equation for RLA (r vs. t) is as follows:
ntot  n0 exp  kRLAt 
log r 
(S15)
2.303k RLA
t  log a
dF
(S16)
7
S2. The interaction energy between two large aggregates is determined by two involved
primary particles.
Fig. S1. Two primary particles (blue) determine the interaction energy between the two large aggregates (marked by
black dashed boxes).
8
S3. Interaction energy between CeO2 NPs under different electrolyte concentrations as
(a)
10
0
0
200
400
600
800
-10
0.001M
0.0025M
0.01M
0.025M
0.1M
-20
-30
-40
0.002M
0.005M
0.02M
0.05M
Separation distance between particles (Å)
5
(c)
0
0
100
200
300
400
500
600
700
-5
-10
-15
-20
-25
0.003M
0.005M
0.007M
0.01M
0.03M
0.004M
0.006M
0.008M
0.02M
0.05M
Separation distance between particles (Å)
800
Net energy between particles (kT)
20
Net energy between particles (kT)
Net energy between particles (kT)
Net energy between particles (kT)
calculated from DLVO and EDLVO theory.
(b)
20
10
0
-10
0
200
400
600
0.001M
0.0025M
0.01M
0.025M
0.1M
-20
-30
-40
800
0.002M
0.005M
0.02M
0.05M
Separation distance between particles (Å)
10
(d)
5
0
-5
0
-10
-15
-20
-25
100
200
300
400
0.003M
0.005M
0.007M
0.01M
0.03M
500
600
700
800
0.004M
0.006M
0.008M
0.02M
0.05M
Separation distance between particles (Å)
Figure S2. Interaction energy between CeO2 NPs under different KCl concentrations as calculated from (a) DLVO
and (b) EDLVO theory, and under different CaCl2 concentrations as calculated from (c) DLVO and (d) EDLVO
theory.
9
3.5
3
y = 0.8945x + 2.0985
R² = 0.8832
2.5
3
2.5
2
0.008M
1.5
y = 0.8936x + 2.1468
R² = 0.8107
1
2
0.007M
1.5
Linear (0.008M)
Linear (0.007M)
0.5
0
0.2
0.4
log (r) in 0.007M CaCl2
log (r) in 0.008M CaCl2
S4. Aggregation of CeO2 NPs in the intermediate aggregation regime.
1
0.6
log (1+4kTn0t/3w)
Figure S3. Aggregation kinetics model fitting the aggregation data for CeO 2 NPs in the intermediate aggregation
regime in 0.008M and 0.007M CaCl2 solution. The dashed lines and corresponding equations show linear fits to the
experimental data points.
10
S5. AFM images of CeO2 aggregates formed in the RLA and DLA regimes.
Figure S4. AFM images of CeO2 aggregates formed in the (a) RLA and (b) DLA regimes. The white bars are equal
to 50 nm. The aggregates in RLA have a more compact structure than those in DLA, indicating that the fractal
dimension of CeO2 aggregates is larger in the RLA regime.
11
3
3.5
(a)
3
2.5
2.5
2
0.025M, KCl
1.5
2
Model Prediction, KCl
CaCl2
0.009M, CaCl2
1
1.5
CaCl
Model Prediction,
CaCl22
KCl
log (r) in CaCl2 solution
log (r) in KCl solution
S6. The model predictions are consistent with experimental data.
1
0.5
0
0.5
1
KCl
1.5
log (1+4kTn0t/3w)
log (r) in KCl solution
3
(b)
2.5
2
1.5
2
0.012M, KCl
1
0.5
Model Prediction, KCl
1.5
0.004M, CaCl
CaCl22
1
CaCl
KCl
Model Prediction,
CaCl22
0
KCl
0
50
100
150
200
0.5
log (r) in CaCl2 solution
2.5
250
t (min)
Figure S5. Predicted and experimental aggregation kinetics of CeO 2 nanoparticles in the (a) DLA regime and (b)
RLA regime in KCl and CaCl2 solutions (solid lines are model predictions; symbols are experimental data).
12
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