Langmuir 2004, 20, 7333-7338
7333
Overcharging of Nanoparticles in Electrolyte Solutions
Sathyajith Ravindran and Jianzhong Wu*
Department of Chemical and Environmental Engineering, University of California,
Riverside, California 92521
Received March 11, 2004. In Final Form: May 22, 2004
Monte Carlo simulations are performed to investigate the effects of salt concentration, valence and size
of small ions, surface charge density, and Bjerrum length on the overcharging of isolated spherical
nanoparticles within the framework of a primitive model. It is found that charge inversion is most probable
in solutions containing multivalent counterions at high salt concentrations. The maximum strength of
overcharging occurs near the nanoparticle surface where counterions and coions have identical local
concentrations. The simulation results also suggest that both counterion size and electrostatic correlations
play major roles for the occurrence of overcharging.
I. Introduction
Charge inversion or overcharging is an electrostatic
phenomenon frequently encountered in colloids or polyelectrolyte solutions in which a macroion adsorbs an excess
amount of counterions such that the overall charge of the
macroion along with its surrounding smaller ions has a
sign opposite to its bare charge. This nonintuitive phenomenon was first reported a long time ago by Bungenberg
de Jong1 and more extensively by Strauss et al.2 observing
the reversal of the electrophoretic mobilities of highly
charged macroscopic colloidal aggregates or polysoaps.
The charge inversion phenomenon is not supported by
the classical electrostatic theories based on the PoissonBoltzmann (PB) equation, which ignores counterion size
and the correlation of small ion distributions. According
to the PB theory, counterions can only partially neutralize
a macro charge by electrostatic screening. There had been
erroneous speculations that the reversal of charges was
probably due to some binding forces between counterions
and the macroion other than the Coulombic interactions.2
The phenomenon of charge inversion regains substantial
research interests in recent years for both scientific and
technical considerations. On the theoretical side, the
limitations of the traditional PB approaches for representing various electrostatic phenomena in colloids and
biological systems become ever-increasingly evident in
comparison to the results from molecular simulations and
experiments. More advanced statistical-mechanical theories are required for quantitative and sometimes even
qualitative interpretation of novel electrostatic behavior
such as attraction between similar charges and charge
inversion. Theoretical efforts to capture the overcharging
phenomenon beyond the PB-like approaches were recently
reviewed in two excellent accounts from different perspectives but both emphasizing the correlations among
counterions in the vicinity of the macroion surface.3,4 One
concludes that the correlation of counterions at the surface
* To whom correspondence should be addressed. E-mail: jwu@
engr.ucr.edu.
(1) Bungenberg de Jong, M. G. In Colloid Science; Kruyt, H. R., Ed.;
Elsevier: New York, 1949; p 335.
(2) Strauss, U. P.; Gershfeld, N. L.; Spiera, H. J. Am. Chem. Soc.
1954, 76, 5909-5911.
(3) Grosberg, A. Y.; Nguyen, T. T.; Shklovskii, B. I. Rev. Mod. Phys.
2002, 74, 329-345.
(4) Quesada-Perez, M.; Gonzalez-Tovar, E.; Martin-Molina, A.;
Lozada-Cassou, M.; Hidalgo-Alvarez, R. ChemPhysChem 2003, 4, 235248.
of a highly charged macroion, which resembles the cohesive
energy of a highly correlated two-dimensional fluid, leads
to the attraction of extra counterions to the surface.3
Conversely, the other approach, based on the integral
equation theory of electrolyte solutions, is focused on the
spatial correlations between small ions at the charged
surface and beyond.4 In recent years, charge inversion
was also extensively studied using computer simulations
at planar charged surfaces,5,6 cylindrical rods,7-9 and
spheres.10-13 Effects of macroion geometry on the overcharging were also investigated using a novel random
sampling technique.14 Simulation results indicate that
the correlation and the excluded volume effects of counterions are the prime reasons of charge inversion. On the
other hand, charge inversion is also of current interest
from a technical point of view because this phenomenon
has been widely utilized for the delivery of therapeutic
biomacromolecules,15,16 immobilization of biosensors,17,18
and layer-by-layer fabrication of materials.19 In addition,
there has been extensive experimental work on the
complexation of DNA with positively charged liposomes20,21
and with cationic protein aggregates22,23 and on the
(5) Denton, A. M.; Lowen, H. Thin Solid Films 1998, 330, 7-13.
(6) Greberg, H.; Kjellander, R. J. Chem. Phys. 1998, 108, 29402953.
(7) Deserno, M.; Jimenez-Angeles, F.; Holm, C.; Lozada-Cassou, M.
J. Phys. Chem. B 2001, 105, 10983-10991.
(8) Montoro, J. C. G.; Abascal, J. L. F. J. Chem. Phys. 1995, 103,
8273-8284.
(9) Das, T.; Bratko, D.; Bhuiyan, L. B.; Outhwaite, C. W. J. Chem.
Phys. 1997, 107, 9197-9207.
(10) Messina, R.; Holm, C.; Kremer, K. Comput. Phys. Commun. 2002,
147, 282-285.
(11) Messina, R.; Gonzalez-Tovar, E.; Lozada-Cassou, M.; Holm, C.
Europhys. Lett. 2002, 60, 383-389.
(12) Tanaka, M.; Grosberg, A. Y. J. Chem. Phys. 2001, 115, 567574.
(13) Terao, T.; Nakayama, T. Phys. Rev. E 2001, 63, 041401.
(14) Mukherjee, A. K.; Schmitz, K. S.; Bhuiyan, L. B. Langmuir 2003,
19, 9600-9612.
(15) Trimaille, T.; Pichot, C.; Delair, T. Colloids Surf., A 2003, 221,
39-48.
(16) Dinsmore, A. D.; Hsu, M. F.; Nikolaides, M. G.; Marquez, M.;
Bausch, A. R.; Weitz, D. A. Science 2002, 298, 1006-1009.
(17) Nguyen, Q. T.; Ping, Z. H.; Nguyen, T.; Rigal, P. J. Membr. Sci.
2003, 213, 85-95.
(18) Rossi, S.; Lorenzo-Ferreira, C.; Battistoni, J.; Elaissari, A.; Pichot,
C.; Delair, T. Colloid Polym. Sci. 2004, 282, 215-222.
(19) Decher, G. Science 1997, 277, 1232-1237.
(20) Radler, J. O.; Koltover, I.; Salditt, T.; Safinya, C. R. Science
1997, 275, 810-814.
(21) Koltover, I.; Salditt, T.; Radler, J. O.; Safinya, C. R. Science
1998, 281, 78-81.
10.1021/la0493619 CCC: $27.50 © 2004 American Chemical Society
Published on Web 07/24/2004
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Langmuir, Vol. 20, No. 17, 2004
overscreening of clay surfaces.24 Charge inversion is also
frequently encountered in the adsorptions of multivalent
small ions and polyelectrolytes on oppositely charged
micelles, latex, and proteins.25-27 Although most practical
applications of charge inversion are concerned with the
adsorption of polyeletrolytes or highly charged biomacromolecules and the phenomena are necessarily more
complicated, the physics underlying the charge reversal
phenomena resembles that in relatively simple systems
involving structureless small ions and charged surfaces.28
All previous theoretical investigations on charge inversion have been primarily focused on macroions bearing a
uniform surface charge that is independent of solution
conditions. However, realistic colloidal particles and
biomacromolecules have discrete charges at the surface,
and the charge density is sensitive to solution pH and
local salt conditions. Using molecular dynamics simulations, Allahyarov et al. recently reported that the discrete
nature of macroion charges may lead to a striking
nonmonotonic variation of the osmotic second virial
coefficient with salt concentration. Such nonmonotonic
behavior is not captured within a smeared charge model.29
These simulation results provide a convincing explanation
to some long-standing experimental observations pertaining to the effect of salt concentration on the osmotic second
virial coefficients, solubility, and cloud point temperature
of protein solutions.
In this work, Monte Carlo simulations are applied to
investigating the effects of charge discreteness at the
surface of isolated nanoparticles on the distributions of
neutralizing small ions and charge inversion. In considering that the charge of each ionizable group at the
nanoparticle surface varies with the solution conditions
and it is influenced by the local environment of small ions,
we allow the discrete charges to migrate at the nanoparticle surface. The migrating discrete charges are strongly
correlated not only within the nanoparticle surface but
also with the neutralizing counterions. Such correlations
may result in a drastic effect on the small ion distributions
and, thereby, charge inversion.
II. Model and Simulation Method
We consider a negatively charged nanoparticle dispersed
in a primitive model electrolyte solution as shown
schematically in Figure 1. Both counterions and coions in
the solution are represented by hard spheres embedded
with fixed central charges, and the solvent is represented
by a dielectric continuum. The diameter of the charged
nanoparticle is assumed to be σN ) 20 Å in all simulations
performed in this work, and the charges are represented
by electronic units distributed on a spherical shell right
underneath the nanoparticle surface. The diameter of each
unit charge is assigned as σu ) 1 Å. The unit charges on
the nanaparticle surface are treated as mobile, and their
distribution is correlated with the local densities of small
ions. The electrostatic interactions between unit charges
(22) Grunstein, M. Nature 1997, 389, 349-352.
(23) Luger, K.; Mader, A. W.; Richmond, R. K.; Sargent, D. F.;
Richmond, T. J. Nature 1997, 389, 251-260.
(24) Kekicheff, P.; Marcelja, S.; Senden, T. J.; Shubin, V. E. J. Chem.
Phys. 1993, 99, 6098-6113.
(25) Radeva, T. Physical chemistry of polyelectrolytes; Marcel Dekker:
New York, 2001.
(26) Ladam, G.; Schaad, P.; Voegel, J. C.; Schaaf, P.; Decher, G.;
Cuisinier, F. Langmuir 2000, 16, 1249-1255.
(27) Paton-Morales, P.; Talens-Alesson, F. I. Langmuir 2002, 18,
8295-8301.
(28) Netz, R. R.; Andelman, D. Phys. Rep. 2003, 380, 1-95.
(29) Allahyarov, E.; Lowen, H.; Louis, A. A.; Hansen, J. P. Europhys.
Lett. 2002, 57, 731-737.
Ravindran and Wu
Figure 1. (a) Nanoparticle with migrating charges at the
surface. (b) Cubic simulation box with one nanoparticle fixed
at the center immersed in an electrolyte solution within the
primitive model.
are accounted for explicitly. Because these unit charges
are strongly repulsive to each other, a small variation of
σu will not significantly alter the results as long as the
size of the unit charge remains much smaller than the
nanoparticle itself, and the excluded volume effect does
not affect the charge distributions on the nanoparticle
surface. Table 1 lists the solution and model parameters
for various simulations performed in this work. In this
Table, C stands for the molar concentration of small ions;
N+ and N- are, respectively, the numbers of cations and
coions employed in the simulation; L is the length of the
cubic simulation box; σM is the diameter of microions (i.e.,
counterions and coions); ZN is the valence of the nanoparticle; and κ-1 is the Debye screening length. The
Bjerrum length, lB, represents the separation between
two unit charges where the pair electrostatic potential is
equal to the thermal energy kT.
All Monte Carlo simulations were performed in the NVT
ensemble where the number of particles, volume, and the
temperature of the system are constant. The cubic
simulation box (Figure 1) contains a single nanoparticle
fixed at the center and small counterions and coions. The
condition of electrostatic neutrality is satisfied for all
systems. To compensate the charge of the central nanoparticle, the overall charge density of cations is necessarily
larger than that of coions. Periodic boundary conditions
Overcharging of Nanoparticles
Langmuir, Vol. 20, No. 17, 2004 7335
Table 1. Parameters for Various Simulation Runs
Performed To Calculate the Radial Distributions of
Small Ions around a Nanoparticle with Migrating
Charges on the Surface
run salt
A
B
C
D
E
F
G
H
I
J
K
L
M
N
2:2
2:2
2:2
2:2
2:2
2:2
2:2
2:2
2:2
2:2
2:2
1:2
1:1
2:1
N+
N-
C (M) lB (Å) κ-1 (Å) L (Å) σM (Å)
740
740
740
740
220
220
220
220
220
220
220
220
220
220
730
730
730
730
215
210
205
210
210
210
210
100
200
420
2.4
7.14
0.71
7.14
1.23
7.14
0.154 7.14
0.71
7.14
0.71
7.14
0.71
7.14
0.71
7.14
0.71
7.14
0.71
5
0.71 10
0.71
7.14
0.71
7.14
0.71
7.14
0.98
1.81
1.37
3.88
1.81
1.81
1.81
1.81
1.81
2.16
1.53
2.08
3.61
2.08
80
120
100
200
80
80
80
80
80
80
80
80
80
80
4
4
4
4
4
4
4
2
6
4
4
4
4
4
ZN
-20
-20
-20
-20
-10
-20
-30
-20
-20
-20
-20
-20
-20
-20
are applied to each direction of the simulation cell. To
mimic the dispersion of an isolated nanoparticle in a bulk
solution, we choose the simulation box at least one order
of magnitude larger than the Debye screening length such
that the correlation between nanoparticles in different
image boxes is assumed negligible.
The interaction potential between a pair of migrating
unit charges at the nanoparticle surface is given by
{
e2
r g σu
φuu(r) ) 4π0r
∞
otherwise
(1)
where r is the center-to-center distance, e ) -1.602 ×
10-19 C is the electronic charge, 0 ) 8.854 × 10-12 C2/(J
m) is the permittivity of free space, and the dielectric
constant is assumed the same as that of the solvent
(otherwise there will be image charges). A similar expression is applied for interactions between microions in the
solution. Because the discrete unit charges are free to
move on the nanoparticle surface, their distribution is
strongly correlated with the local density of small ions.
The total potential between the fixed nanoparticle and a
microion of charge qM and diameter σM is given by
φMN(r) )
{
eqM
1
∑
4π R r
∞
0
r g σMN ) (σM + σN)/2
Ri
(2)
otherwise
where r is the center-to-center distance between the
nanoparticle and a microion and rRi is the distance between
a unit charge R and a microion i. Because all charges are
explicitly considered, no screening is applied to the
interactions between microions or between a microion and
the charges located on the nanoparticle surface.
The Ewald sum method is used to take into account the
long-range electrostatic interactions introduced as a result
of the periodic boundary conditions. The system is allowed
to reach equilibrium by making trial moves of microions
and the unit charges at the nanoparticle surface following
the standard Metropolis algorithm. In the course of
simulations, a microion or a unit charge of the nanoparticle
is randomly chosen, and it is subjected to an infinitesimally
small displacement. In the case of a microion, a small
linear displacement is given along all three coordinates.
When a unit charge is chosen, it is displaced by a small
angle along the nanoparticle surface calculated using
Euler’s method.30 In this technique we generate three
angles R, β, and γ uniformly distributed over 0 to 2π using
a random number generator. Then the unit charge on the
nanoparticle is rotated with respect to the center of the
nanoparticle. The move is accepted or rejected on the basis
of the difference in energy of the original and the trial
microstates. If the energy of a trial configuration is lower
than that of the original configuration, the Monte Carlo
move is always accepted; otherwise, a random number is
generated and the move is accepted only if it is larger
than the ratio of the Boltzmann factors of the trial and
the original configurations. The number of microions in
each simulation varies from approximately 400 up to 1500
particles. By monitoring the total energy of the system,
we find that the system attains equilibrium after approximately 5 × 105 simulation steps per particle. To calculate
the radial distributions of the counterions and coions, we
divide the simulation box into approximately 250 concentric shells of equal thickness. The numbers of counterions and coions in each shell are counted during the
simulations, and they are averaged over 107 Monte Carlo
moves. The average densities of counterions and coions
are calculated by dividing these numbers with the shell
volume.
III. Results and Discussion
The radial distributions of small ions near each fixed
nanoparticle were calculated at various concentrations of
the electrolyte, valence and size of small ions, the surface
charge, and the Bjerrum length. The key quantity reflecting charge inversion is the integrated charge distribution
function defined as4
P(r) ) -ZN +
∑i ∫σ
r
MN
ziFi(r)4πr2 dr
(3)
where ZN is the number of unit charges of the nanoparticle
and zi stands for the valence of a counterion or coion. Even
though the system is overall electrically neutral, the local
charge depends on the densities of the counterions and
coions. At the surface of the fixed nanoparticle, P(r ) σMN)
is equal to the total number of unit charges for the particle,
and the absolute value of P(r) decreases as the surface
charge is neutralized by adsorbing counterions and
approaches 0 far from the surface (bulk). Charge inversion
occurs when P(r) changes the sign from negative to positive
at some intermediate separations.
Figure 2 shows the integrated charge distribution
function P(r) for run A. In this case, the Debye screening
length is κ-1 ) 0.98 Å, much smaller than the size of
microions (σM ) 4 Å). According to a classical electrostatic
theory, the electrostatic interactions should be insignificant for this case. However, that is not true as shown by
Monte Carlo simulations. While P(r) approaches 0 when
r f ∞ as expected, it overshoots the neutralization line
(or isoelectric line) at r/σM ≈ 3.25 and reaches a maximum
at r/σM ≈ 3.7. At larger separations, the integrated charge
profile exhibits an oscillatory behavior indicating the
existence of alternating counterion- and coion-dominated
adsorption layers. In particular, the overall charge within
the shaded region is opposite to the bare charge of the
nanoparticle, indicating a charge inversion. The charge
inversion does not occur exactly at the nanoparticle surface
as speculated before. Instead, it shows a maximum
reversal charge at a position where the counterions and
coions have the same number density. This is most clearly
depicted in Figure 3, which presents the radial density
(30) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids;
Oxford University Press: New York, 1989.
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Langmuir, Vol. 20, No. 17, 2004
Ravindran and Wu
Figure 4. Effect of salt concentration on charge inversion.
The inset represents Pmax as a function of salt concentration
(runs A-D).
Figure 2. Integrated charge distribution for run A. The shaded
area indicates that the overall charge of the nanoparticle along
with the neutralizing counterions has a sign opposite to the
bare charge of the nanoparticle.
Figure 5. Overcharging in electrolyte solutions of different
valences (runs E, L, M, and N).
Figure 3. Density distributions of the counterions (solid line)
and coions (dotted line) for run A. Here, the density profiles are
in the units of 1/lB3.
distributions of the counterion (squares) and the coion
(triangles). At the position where the counterions and
coions have the same number density, the overall charge
of the nanoparticle and its surrounding small ions is
opposite to the nanoparticle bare charge. As the distance
is further increased, the additional adsorption of coions
diminishes the charge inversion. Figure 3 also indicates
that near the nanoparticle surface, the distributions of
small ions are significantly asymmetric. Different from
the predictions of the PB theory, the distributions of both
counterions and coions clearly exhibit nonmonotonic
behavior. The two density profiles cross at r/σM ≈ 3.7,
corresponding to the position of maximum charge reversal
as shown in Figure 2. At larger separations, the local
concentration of coions becomes larger than that for the
counterion. As a result, the integrated charge distribution
declines.
Figure 4 shows the effect of electrolyte concentration
on the integrated charge distributions (runs A-D). The
integrated charge densities exhibit oscillatory behavior
at high salt concentrations but become monotonic when
the concentration is below 0.154 M. In this case, no charge
inversion is observed. Figure 4 indicates that charge
inversion is more probable at high salt concentrations.
The inset of Figure 4 presents the maximum value of the
integrated charge distribution as a function of salt
concentration. It shows that the magnitude of charge
inversion increases monotonically with the concentration
of the electrolyte. However, the layer thickness decreases
with the solution concentration; that is, the first peak of
the integrated charge distribution becomes sharper as
the salt concentration increases.
Previous investigations point toward electrostatic correlations and the excluded volume effects as the main
causes of charge inversion.6 To investigate these two
different aspects separately, we simulate the charge
distributions near the nanoparticle at different valences
and sizes of small ions and at different Bjerrum lengths.
Figure 5 presents the integrated charge distribution
function P(r) for a nanoparticle immersed in 1:1, 1:2, 2:1,
and 2:2 electrolytes (runs E, L, M, and N). All these
electrolyte solutions have the same concentration. We find
that there is essentially no charge inversion in the 1:1
and the 1:2 electrolytes where the counterions are
monovalent. However, in solutions containing divalent
counterions as in the 2:1 and the 2:2 electrolytes,
significant charge inversion appears. Figure 5 suggests
that charge inversion is directly related to the strong
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Langmuir, Vol. 20, No. 17, 2004 7337
Figure 7. Integrated charge distributions for different microion
sizes (runs E, H, I).
Figure 6. Effects of Bjerrum length on charge inversion. lB )
7.14 Å corresponds to an aqueous system at an ambient
condition (runs E, J, and K).
interactions between the counterions as shown for the
electrostatic attraction between two similarly charged
nanoparticles.31-33 In addition to stronger electrostatic
interactions, the entropic penalty associated with localization of multivalent counterions is much smaller than
that for monovalent counterions. This is because the
number of multivalent ions required to neutralize the
charged particle is smaller than that for the monovalent
counterions while the entropy loss for the immobilization
of each ion depends only on density. Even though Figure
5 shows no sign of overcharging when the solution contains
only monovalent counterions, a recent density functional
theory calculation predicts the appearance of overcharging
even in monovalent electrolyte solutions as long as the
salt concentration is sufficiently high.34 The integral
equation and molecular dynamics study of Deserno et al.
also indicated the possibility of overcharging caused by
large monovalent ions.7
We use the Bjerrum length, defined by lB ) e2/(4π0kT),
as an input in Monte Carlo simulations to control the
strength of electrostatic interactions. In real experiments,
this parameter can be controlled by changing the temperature or the dielectric constant of the solvent. The
change of Bjerrum length investigated in this work
corresponds to that due to the latter approach. A larger
Bjerrum length (or a smaller electrostatic constant) means
stronger electrostatic interactions among nanoparticles
and surrounding small ions. Figure 6 shows the charge
inversion for systems with three different Bjerrum lengths
(runs E, J, and K), approximately corresponding to those
for the ionic species dispersed in a solvent of N-propylpropanamide (lB ) 4.74 Å), water (lB ) 7.14 Å), and
dimethyl sulfate (lB ) 10.19 Å) at ambient conditions.
Figure 6 clearly indicates that a higher Bjerrum length
introduces stronger charge inversion. Both Figures 5 and
6 suggest that charge inversion is most likely in systems
with strong electrostatic interactions.
(31) Wu, J. Z.; Bratko, D.; Prausnitz, J. M. Proc. Natl. Acad. Sci.
U.S.A. 1998, 95, 15169-15172.
(32) Wu, J. Z.; Bratko, D.; Blanch, H. W.; Prausnitz, J. M. J. Chem.
Phys. 1999, 111, 7084-7094.
(33) Linse, P.; Lobaskin, V. Phys. Rev. Lett. 1999, 83, 4208-4211.
(34) Yu, Y. X.; Wu, J. Z.; Gao, G. H. J. Chem. Phys. 2004, 120, 72237233.
Figure 8. Effect of surface charge density of nanoparticles on
charge inversion (runs E-G).
The effect of excluded volume on overcharging may be
studied by changing the size of small ions while keeping
all other conditions fixed. Figure 7 shows the integrated
charge distributions in solutions containing microions of
different sizes (runs E, H, and I). We observed that charge
inversion is more significant for microions of larger size.
This implies that charge inversion at higher salt concentrations as shown in Figure 4 is likely attributed to the
excluded volume effects. Figure 7 also indicates that larger
counterions are significantly more efficient in shielding
the charge effects of nanoparticles. Qualitatively, the two
major mechanisms behind the overcharging, that is, the
electrostatic correlations and the excluded volume effect
as shown in Figures 5-7, are consistent with a previous
investigation by Greberg and Kjellander for the charge
inversion in a planar electric double layer.6
Finally, Figure 8 presents charge inversion for nanoparticles with different surface charge densities (runs
E-G). Here, the charge densities are expressed in terms
of ZN/σN2. It shows that the charge inversion becomes more
significant as the surface charge of the nanoparticle
increases. This can be explained by the fact that an
increase in nanoparticle charge attracts more counterions
near the surface, introducing stronger electrostatic and
excluded-volume correlations.
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Langmuir, Vol. 20, No. 17, 2004
IV. Conclusions
Monte Carlo simulations have been applied to studying
the ionic density profiles and integrated charge distribution functions near nanoparticles bearing mobile unit
charges at a variety of solution conditions. In general, the
results are qualitatively similar to those corresponding to
a smeared charged model or discrete but fixed charged
model. We find that charge inversion is most likely in
electrolyte solutions of high ionic strength. Upon the
increase of salt concentration, the magnitude of inversed
charges almost linearly increases while the thickness of
the first adsorption layer declines. The charge inversion
phenomenon can be attributed to two main factors,
namely, the strength of electrostatic potential and the
Ravindran and Wu
size of counterions. At similar solutions conditions, charge
inversion occurs in solutions containing multivalent
counterions and the magnitude of reversal charge increases with the counterion size. As for the interactions
between similarly charged nanoparticles, the counterions
play a dominant role in overcharging. Charge inversion
becomes more likely as the Bjerrum length or the charge
density of the nanoparticle surface increases.
Acknowledgment. This project was financially supported by a grant from the University of California Energy
Institute.
LA0493619