1. Journal of Colloid and Interface Science 287 (2005) 114–120
www.elsevier.com/locate/jcis
Study on hydration layers near nanoscale silica dispersed in aqueous
solutions through viscosity measurement
S. Song a,b,∗
, C. Peng b
, M.A. Gonzalez-Olivares b
, A. Lopez-Valdivieso b
, T. Fort c
a Department of Resources Engineering, Wuhan University of Science and Technology, Av. Heping 947, 430081, Wuhan, China
b Instituto de Metalurgia, Universidad Autónoma de San Luis Potosí, Av. Sierra Leona 550, San Luis Potosí, C.P. 78210, Mexico
c Department of Chemical Engineering, Vanderbilt University, Nashville, TN 37235, USA
Received 13 September 2004; accepted 24 January 2005
Available online 9 March 2005
Abstract
On the basis of the Einstein theory of viscosity of dispersion, a parameter, termed as solvation factor, is presented to evaluate the solvation
degree of nanoscale particles dispersed in a liquid in this work. The value of the parameter is obtained through the measurements of relative
viscosity of the dispersions as a function of the volume fraction of dry particles. The solvation factor has been used to study the hydration
layers near nanoscale silica particles dispersed in water and aqueous electrolyte (NaCl and CaCl2) solutions in this work. The experimental
results have shown that a strong hydration indeed applied to the silica surfaces in aqueous solutions, leaving a large volume of hydration
layers on the surfaces. Also, it has been found that the hydration of the nanoscale silica particles could be greatly enhanced if they were
dispersed in aqueous NaCl or CaCl2 solutions, which might be attributed to that the hydrated cations (Na+ or Ca2+) bind onto the silica/
water interface and thus increase the volume of the hydration layers.
 2005 Elsevier Inc. All rights reserved.
Keywords: Hydration layers; Viscosity; Nanoscale silica; Electrolyte; Solvation factor
1. Introduction
The viscosity of a colloidal dispersion increases with the
increase of particle concentration in the system. If the col-
loidal particles are rigid spheres, the relationship between
the viscosity and the particle concentration can be expressed
by [1,2]
(1)
η
η0
= 1 + kφ + k1φ2
+ ···,
where η and η0 are the viscosities of the dispersion and the
pure liquid, respectively; φ is the volume fraction of the dis-
persed particles; k and k1 are constants. For a very dilute
dispersion where the spheres do not have significant inter-
action with each other, when flow velocity is low and there
is no slippage of fluid at the particle surfaces, Eq. (1) be-
* Corresponding author.
E-mail address: shaoxian@uaslp.mx (S. Song).
comes [1]
(2)
η
η0
= 1 + 2.5φ.
This equation is well known as the Einstein theory of vis-
cosity of dispersion. It is applicable to all monodispersions
of dilute rigid spheres if correct values of φ are chosen,
regardless of the size and the property of the spheres. If dis-
persion is dilute enough to neglect the interaction between
the spheres in the system, the Einstein equation is also ap-
plicable to polydispersion of rigid spheres.
As it is known, there are solvation layers coated on
lyophilic surfaces immersed in a liquid. In the layers, liquid
molecules may orient toward the surfaces in a good order
because of strong attractions between the molecules and the
surfaces. It is generally accepted that solvation layers have a
different molecular structure from bulk liquid [3], and higher
density and viscosity than the bulk liquid [4,5]. Obviously,
solvation layers would swell the volume of the “dry” parti-
0021-9797/$ – see front matter  2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcis.2005.01.066

2. S. Song et al. / Journal of Colloid and Interface Science 287 (2005) 114–120 115
Fig. 1. Diagram of solid sphere with and without solvation layers.
cles, increasing the viscosity of the dispersion, as shown in
Fig. 1. However, since solvation layers are not rigid, they
would not increase the viscosity of dispersions in the same
way as the solid rigid spheres described by the Einstein
equation. If we assume that the volume of solvation layers
contributes to increasing the viscosity of the dispersion in c
times as that of the rigid solid spheres does, Eq. (2) becomes
(3)
η
η0
= 1 + 2.5(φp + cφl),
where φp and φl are the volume fraction of the dry particles
and the solvation layers in the dispersion, respectively. They
can be expressed by
(4)φp =
Vp
V
,
(5)φl =
Vl
V
,
where Vp and Vl are the volumes of dry particles and solva-
tion layers, respectively; V is the volume of the dispersion.
From Eqs. (3), (4), and (5), we obtain
η
η0
= 1 + 2.5 φp + c
Vl
V
= 1 + 2.5 φp + c
Vl
Vp
φp
(6)= 1 + 2.5 1 + c
Vl
Vp
φp.
In nanoscale particle dispersions, the ratio of the volumes
of solvation layers and dry particles (Vl/Vp) could be in-
dependent of the volume fraction of the dry particles (φp) if
particle concentration is not negligible. Accordingly, a graph
of η/η0 against φp should yield a straight line for a given dis-
persion with the intercept of 1, and
(7)Slope = m = 2.5 1 + c
Vl
Vp
.
Now, we define
(8)f = c
Vl
Vp
,
where f is termed solvation factor. Obviously, f repre-
sents the solvation degree of solid spherical particles in a
liquid. For a given solid material that the Asp (specific sur-
face area) is constant, a stronger solvation of a liquid to the
solid particles would lead to a larger Vl, and thus a larger
solvation factor. Therefore, solvation factor can be used to
determine the degree of different liquids to solvate the same
colloidal solid material. The larger the solvation factor, the
more strongly the colloidal particles are solvated by the liq-
uid. From Eqs. (7) and (8), we obtain
(9)f = 0.4m − 1.
Accordingly, solvation factor can be calculated through the
measurement of relative viscosity of a colloidal dispersion
as a function of volume fraction of the dry particles.
As it is known, the thickness of solvation layers on solid
surfaces immersed in liquids is in the range of nanometers.
If solid particles are too large, say 1 µm in diameter, the vol-
ume of solvation layers is negligible compared with that of
the dry particles, leading to a very small solvation factor. It
indicates that the contribution of solvation layer to increas-
ing the viscosity of the dispersion can be neglected in the
case of large colloidal particles. In other words, solvation
factor cannot be used to study the solvation layers on large
colloidal particles dispersed in a liquid, as reported in our
previous paper [6]. However, it is significant for the disper-
sions of nanoparticles, say minus 200 nm.
The overlapping of solvation layers on particle surfaces
in a liquid leads to a non-DLVO-repulsive force between the
surfaces, which is called as solvation force [7,8]. Between
hydrophilic surfaces in water, the force (which is termed hy-
dration force) may be extremely strong, being an order of
magnitude larger than the electrical double-layer repulsive
force in a short-range separation, on the base of the exper-
imental measurements of forces between silica surfaces or
mica surfaces in water [9,10]. Obviously, such a strong sur-
face force certainly plays an important role in stabilizing
colloidal dispersions. This method is termed stabilization by
solvation force [11].
Recently, nanoscale materials have attracted considerable
attention in scientific and industrial fields, such as electronic
and optical devises and biological analysis [12,13]. A high
stability of particle dispersion is very important to the prepa-
rations and applications of nanoscale particles, which is a
great research interest in this subject. Obviously, it is sig-
nificant to obtain a greater understanding of solvation layers
on nanoscale particles dispersed in a liquid in order to sta-
bilize nanoscale particle dispersion through stabilization by
solvation force.
In this work, we attempted to study the hydration lay-
ers near nanoscale silica particles dispersed in aqueous so-
lutions through the determination of solvation factor from
the measurements of relative viscosity of the dispersion as
a function of volume fraction of dry particles. The objec-
tives are to obtain a greater understanding of hydration layers
near nanoscale silica dispersed in aqueous solutions from the
viewpoint of the volume of hydration layers, and to find out
the effect of electrolytes on the hydration layers. It is ex-
pected that a better medium (aqueous electrolyte solution)
will be found for attaining a high stability of nanoscale sil-
ica dispersion.

3. 116 S. Song et al. / Journal of Colloid and Interface Science 287 (2005) 114–120
]
Fig. 2. TEM images of the nanoscale silica. (a) N14; (b) N7.
2. Experimental
2.1. Materials
Two nanoscale silica samples with different size distri-
butions were used in this work, which were obtained from
Sigma-Aldrich Co. (St. Louis, MO). According to the man-
ufacturer, the mean particle sizes of the two samples were 7
and 14 nm, which were termed N7 and N14, respectively,
in this paper. They were fumed silica with a purity of
higher than 99%, and appeared in dry powder. Fig. 2 shows
the TEM (transmission electron microscope) images of the
nanoscale silica particles. It can be observed from the im-
ages that the particles were not of perfect spheres, but were
not very irregular. Some of the particles had a high ratio
of spheres, while others were elliptical. Also, the particles
distributed in a narrow size range. For the N14 sample, the
particles were close to 14 nm in size, while those were close
to 7 nm for the N7 sample.
By using the BET method, the samples were determined
for the specific surface area. The results are given in Table 1.
Also, the volumetric mean diameters (dm) of the samples are
listed in the table, which is calculated by using the following
equation,
(10)dm =
6
ρAsp
,
Table 1
Specific surface area (Asp) and volumetric mean diameter (dm) of the silica
samples
Sample Asp (m2/g) Density (g/cm3) dm (nm)
N7 367.6779 2.550 6.40
N14 189.0357 2.550 12.45
where Asp is the specific surface area of solid particles, and
ρ is the density of the solid.
The sodium chloride (NaCl) obtained from J.T. Baker
(Xalostoc, Mexico), calcium chloride (CaCl2), and alu-
minum chloride (AlCl3) obtained from the Productos Quími-
cos Monterrey (Monterrey, Mexico) were used as elec-
trolytes, all of which were analytical purity. The water used
in this work was first distilled and then passed through resin
beds and a 0.2-µm filter. The residual conductivity of the
water was less than 1 µS/cm.
2.2. Measurements
A Physica RHEOLAB MC-1 concentric cylinder rheome-
ter (Physica Messtechnik GmbH, Ostifildern, Germany) was
used to determine the viscosity of nanoscale silica disper-
sions. The instrument is a rotational, shear stress and creep
rheometer. Samples are positioned in the measuring gap be-
tween the stationary measuring cup and the rotating bob
(Searle principle). The rheometer employs a highly dynamic
system consisting of a measuring drive and an optical en-
coder without gearing and a mechanical force transducer
with which torque is measured without deflection. The
torque resolution is 0.01 mN m. The temperature control
system permits viscosity measurements to be performed at a
fixed temperature with a precision of ±0.05 ◦C.
In this work, a given weight of dry silica powder was
first put into a 100-ml measuring flask. The flask was then
filled to the mark with water or aqueous electrolyte solu-
tions. Next, the suspension was mildly stirred with a mag-
netic agitator (Digital hot plate/stirrer 04644) at 150 rev/min
for 24 h. The pH of the suspension was controlled at 6.5.
Then, the suspension was transferred to the rheometer for
viscosity measurements. The measurements were carried out
at a shear rate of 1000 s−1 and temperature of 22 ◦C. Each
measurement was repeated 60 times. Results reported in this
paper were the arithmetic average of these 60 measurements.
The error estimation has been made for the 60 data. It was
found that the standard error of the reported values of mea-
sured viscosity was in the range of 0.0017 to 0.0029 mP s,
which depended on the value of measured viscosity (in this
work, the viscosities of nanoscale silica dispersions were
about 1 to 1.8 mP s).
The TEM images were obtained by using a Jeol JEM-
1200EX transmission electron microscope. The nanoscale
silica was first dispersed in alcohol with 0.05% solid con-
centration. Next, a 3-mm grill was immersed into the disper-
sion for a while, and then taken out for drying in air. The

4. S. Song et al. / Journal of Colloid and Interface Science 287 (2005) 114–120 117
nanoscale silica particles were adhered on the grills. After
that, the grill was moved to the TEM for an observation.
The specific surface area of the silica powder was de-
termined with a Micromeritics ASAP 2000 specific surface
area meter (Micromeritics Corporation, Norcross, GA). The
samples were previously dried at 120 ◦C for 6 h, and then
were degassed at 300 ◦C in a vacuum for another 6 h. The
adsorption tests were carried out at 77 K using nitrogen as
adsorbate. Specific surface areas of the samples were calcu-
lated by using the BET equation.
3. Results and discussion
Fig. 3 shows the relative viscosity of nanoscale silica dis-
persions as a function of volume fraction of dry particles in
water, while the straight line from the Einstein equation is
also illustrated in the graph in the form of a dashed line. Er-
ror analysis for the results of relative viscosity presented in
the graph was made according to the standard error of the
viscosity measurements. It showed that the maximum er-
ror appeared at the result of η/η0 = 1.44, being ±0.0052,
which can be expressed by η/η0 = 1.44 ± 0.0052 at this
point. Since the error bars were too small to draw, we only
present the arithmetic average values of the relative viscos-
ity in the graph. As it can be seen from the graph, for the two
kinds of nanoscale silica samples, the measured relative vis-
cosities increased linearly with increasing volume fraction
of the dry particles. Linear regression of the experimental
data led to two equations as follows,
(11)
η
η0
= 1 + 56.5φp,
(12)
η
η0
= 1 + 71.4φp,
for the N14 silica and N7 silica dispersions, respectively. In-
deed, Eq. (6) was right for the two systems of nanoscale sil-
ica dispersed in water. There were a great difference between
Fig. 3. Relative viscosity of N7 and N14 silica dispersion as a function of
volume fraction of dry particle in water.
the “experimental” slope (56.5 and 71.4) and the “theoreti-
cal” slope (2.5). The smaller the particle size, the greater
the slope difference. This difference is certainly attributed to
the presence of hydration layers near the silica particles dis-
persed in water. However, the ellipticity of solid particles
also causes the slope to exceed the Einstein “theoretical”
value [14,15]. The deviation of the “experimental” slopes
from the Einstein “theoretical” value might not only be due
to the hydration layers, but also due to the ellipticity of the
particles. Therefore, the slope of the straight line (m) can be
expressed by
(13)m = 2.5 + ml + me,
where ml and me are the contributions of solvation layers
and particle ellipticity to the slope, respectively. In our case,
the nanoscale silica particles were not of perfect spheres, so
that me = 0. However, since the ellipticity of the silica par-
ticles was small (1 < a/b < 2) as shown in the TEM images
(Fig. 2), the me would be very small according to Lauffer’s
report [14]. Compared with the value of the slopes (56.5
and 71.4) as shown in Fig. 3, the me could be negligible.
Therefore, in order to simplify the calculation of solvation
factor, we neglect the contribution of particle ellipticity in
this study.
According to Eq. (9), the solvation factors (f ) for the N14
and N7 silica aqueous dispersions were calculated and found
to be 21.6 and 27.56, respectively. The larger solvation factor
of the N7 silica aqueous dispersion should be attributed to
the larger specific surface area of the N7 particles. At the
same volume fraction, there were much larger surface areas
for the formation of hydration layers near the N7 silica than
near the N14 silica. Accordingly, solvation factor cannot be
used to evaluate solvation degrees for the chemically similar
solid particles with different size distributions, but for the
same solid particles dispersed in different liquids.
In addition, we measured the solvation factors of the N14
silica particles dispersed in various electrolytic aqueous so-
lutions, in order to understand the effect of electrolyte on the
hydration layers near nanoscale silica surfaces. Fig. 4 illus-
trates the relative viscosity of N14 silica aqueous dispersion
as a function of volume fraction of dry particles at various
NaCl concentrations. Similarly, the experimental data were
almost located on the corresponding straight line. Linear re-
gression for the data led to following equations,
(14)
η
η0
= 1 + 82.4φp,
(15)
η
η0
= 1 + 106.8φp,
(16)
η
η0
= 1 + 120.7φp,
for the dispersions in 0.2, 0.6, and 1.0 mol/L NaCl aque-
ous solutions, respectively. Compared with the dispersion
with pure water, NaCl addition greatly increased the slope
of the straight line. The higher the NaCl concentration in the
aqueous solution, the larger the slope. As the contribution of

5. 118 S. Song et al. / Journal of Colloid and Interface Science 287 (2005) 114–120
Fig. 4. Relative viscosity of N14 silica dispersion as a function of volume
fraction of dry particle in aqueous NaCl solutions.
particle ellipticity was the same for the silica dispersion in
the presence and absence of NaCl, the increase of the slope
should be completely attributed to the increase of hydration
of the silica surfaces due to the addition of NaCl. Therefore,
the results suggest that the hydration of the nanoscale sil-
ica particles dispersed in aqueous solutions could be greatly
enhanced by adding NaCl.
Rabinovich et al. [9] reported the experimental measure-
ments of hydration force between two silica surfaces in
aqueous NaCl solutions. They found that the hydration re-
pulsive force increased with increasing NaCl concentration
in the aqueous solution. This phenomenon was also observed
in the force measurements between curved mica surfaces
in aqueous KNO3 or KCl solutions by Israelachvili [7] and
Pashley [10] and Israelachvili and Pashley [16]. As already
stated, hydration force between surfaces in aqueous solu-
tions closely correlates with the hydration degree of the
surface. It is clear that the results presented in Fig. 4 cor-
responded well with the previous reported results.
The isoelectric point (IEP) of the nanoscale silica in aque-
ous solution was determined to be pH 1.8 by using a Coul-
ter Delsa 440SX instrument (electrophoretic light-scattering
method) [17]. In the case of the experiments as shown in
Fig. 4 (suspension pH 6.5), the silica surfaces were charged
negatively. So, in the NaCl aqueous solution, cations (Na+)
adsorbed on silica/water interfaces to compress the electric
double layers on the silica surfaces. As it is known, cations
are strongly hydrated in aqueous solutions, forming a hydra-
tion shell in the vicinity of the ion [18]. The hydration shell
would swell the ions. For example, the radius of Na+ ion
is 0.102 nm, while that of hydrated Na+ ion is 0.237 nm [19].
As the O–H bond in water molecule is about 0.0991 nm in
length, hydrated Na+ ions would occupy a much larger space
than water molecules alone. Therefore, the participation of
hydrated Na+ ions in the hydration layers of silica surfaces
would certainly increase the volume of the hydration layers,
as shown in Fig. 5. In the hydration layers, the hydrated Na+
Fig. 5. Diagram of hydration layers near silica surfaces in pure water (a)
and in presence of hydrated Na+ ions (b).
ions together with water molecules orientate toward the sur-
face in a good order, forming a multilayer of water molecules
and hydrated Na+ ions near the surface. The molecules and
ions in the layers close to the surface are denser and more
orderly than those in the layers far from the surface (or close
to bulk water). Furthermore, in the presence of Na+ ions in
the dispersion, water molecules are not only attracted by sil-
ica surfaces, but also by the ions, leading to much stronger
hydration on the surfaces. This might be another mechanism
by which NaCl addition enhances the hydration of nanoscale
silica dispersed in aqueous solutions.
Fig. 6 shows the relative viscosity of the N14 silica aque-
ous dispersion as a function of volume fraction of dry parti-
cles at various CaCl2 concentrations. Similarly, we made a
linear regression for the experimental data and obtained fol-
lowing equations,
(17)
η
η0
= 1 + 86.6φp,
(18)
η
η0
= 1 + 110.5φp,

6. S. Song et al. / Journal of Colloid and Interface Science 287 (2005) 114–120 119
Fig. 6. Relative viscosity of N14 silica dispersion as a function of volume
fraction of dry particle in aqueous CaCl2 solutions.
(19)
η
η0
= 1 + 127.3φp,
for the dispersions in 0.2, 0.3, and 0.4 mol/L CaCl2 aque-
ous solutions, respectively. As it can be noted, the slope
increased with the increase of CaCl2 concentration, indicat-
ing that CaCl2 also enhanced the hydration of the nanoscale
silica dispersed in aqueous solutions.
From the slopes of the straight lines as shown in Figs. 4
and 6, the solvation factors of the N14 nanoscale silica dis-
persed in various aqueous solutions have been calculated
on the basis of Eq. (9). The calculated results were illus-
trated in the form of solvation factor as a function of elec-
trolyte concentration, as shown in Fig. 7. From this graph,
the effect of the two electrolytes on the hydration of the
nanoscale silica particles dispersed in aqueous solutions can
be clearly observed. With the increase of electrolyte concen-
tration, the solvation factor greatly increased. This increase
was much sharper for CaCl2 than for NaCl, indicating that
CaCl2 was stronger than NaCl to enhance the hydration of
the nanoscale silica particles dispersed in aqueous solutions.
In other words, Ca2+ ions increased the volume of the hy-
dration layers more greatly than Na+ ions, because cations
adsorbed on silica/water interfaces in this case.
The radius of Ca2+ ion is 0.10 nm, similar to that of Na+.
However, they have a different hydration number that is de-
fined for the number of water molecules in the vicinity of
the ion [7,18]. The calculations in the manner of model ra-
dius gave the hydration number of Na+ ion and Ca2+ ion
to be 3.5 and 7.2, respectively [20]. It means that the wa-
ter molecules in the hydration shell of Ca2+ ion are double
that of Na+ ion. There are two layers of water molecules
in the hydration shell of Ca2+ ions, while there is only one
layer in that of Na+ ions. Clearly, hydrated Ca2+ ions have
a larger radius and a stronger attraction to water molecules
than hydrated Na+ ions. Accordingly, Ca2+ ions would en-
hance the hydration of silica surfaces in aqueous solutions
more greatly than Na+ ions, leading to a larger volume of
hydration layers, as observed in Fig. 7.
Fig. 7. Effect of electrolyte concentration on the solvation factor of N14
silica dispersed in aqueous solutions.
The study presented in this paper has shown that the sol-
vation factor is a useful parameter for evaluating the solva-
tion degree of a given nanoscale powder dispersed in various
kinds of liquids or solutions. The measurements of solvation
factor can guide the scientists and engineers in the field of
nanoscale materials to find out the best liquids or solutions
for the preparation and reservation of nanoparticles.
4. Conclusions
(1) In this paper, we have presented a parameter, solvation
factor, to evaluate the solvation degree of nanoscale particles
dispersed in a liquid, on the basis of the Einstein theory of
viscosity of dispersion. The value of the solvation factor can
be obtained through measurements of the relative viscosity
of nanoscale particle dispersion as a function of the volume
fraction of dry particles. The solvation factor might be useful
for guiding scientists and engineers to find out the best liquid
or solution for the preparation and reservation of nanoparti-
cles.
(2) The experimental results from this work have shown
that there was a large volume of hydration layers near the
surfaces of the nanoscale silica dispersed in water, suggest-
ing a strong hydration on the silica surfaces.
(3) The hydration of the nanoscale silica dispersed in
aqueous solutions could be enhanced by adding the elec-
trolytes of NaCl and CaCl2. The solvation factor of the
nanoscale silica was stronger if it is dispersed in aqueous
CaCl2 solution than in aqueous NaCl solution at the same
concentration.
Acknowledgments
Financial support for this work from the Consejo Na-
cional de Ciencia y Tecnología (CONACyT) of Mex-
ico under Grant 485100-5-38214-U and a scholarship for

7. 120 S. Song et al. / Journal of Colloid and Interface Science 287 (2005) 114–120
Gonzalez-Olivares from the Universidad Autónoma de San
Luis Potosí under Grant C01-FRC-12.17 are gratefully ac-
knowledged.
References
[1] P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface
Chemistry, third ed., Dekker, New York, 1997.
[2] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions,
Cambridge Univ. Press, Cambridge, 1989.
[3] B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces, Consul-
tants Bureau, New York, 1987.
[4] F. Merzel, J.C. Smith, Proc. Natl. Acad. Sci. USA 99 (2002) 5378.
[5] D. Zahn, O. Hochrein, Phys. Chem. Chem. Phys. 5 (2003) 4004.
[6] C. Peng, S. Song, Q. Gu, Chin. J. Chem., in press.
[7] J.N. Israelachvili, Intermolecular & Surface Forces, second ed., Acad-
emic Press, London, 1991.
[8] N.V. Churaev, J. Colloid Interface Sci. 172 (1995) 479.
[9] Y.I. Rabinovich, B.V. Derjaguin, N.V. Churaev, Adv. Colloid Interface
Sci. 16 (1982) 63–78.
[10] R.M. Pashley, J. Colloid Interface Sci. 80 (1981) 153.
[11] E. Kissa, Dispersions: Characterization, Testing, and Measurement,
Dekker, New York, 1999.
[12] A. Majetich, J.O. Artman, M.E. McHenry, N.T. Nuhfer, S.W. Staley,
Phys. Rev. B 48 (1993) 16845.
[13] J.J. Storhoff, R. Elghanian, R.C. Mucic, C.A. Mirkin, R.L. Letsinger,
J. Am. Chem. Soc. 120 (1998) 1959.
[14] M.A. Lauffer, J. Am. Chem. Soc. 66 (1944) 1188.
[15] P.C. Hiemenz, Polymer Chemistry: The Basic Concept, Dekker, New
York, 1984.
[16] J.N. Israelachvili, R.M. Pashley, in: F. Franks (Ed.), Biophysics of Wa-
ter, Wiley, New York, 1982, pp. 183–194.
[17] M.A. Gonzalez-Olivares, Master thesis, Universidad Autónoma de San
Luis Potosí (2004).
[18] Y. Marcus, Ion Properties, Dekker, New York, 1997.
[19] A.P. Lyubartsev, A. Laaksonen, J. Phys. Chem. 100 (1996) 16410.
[20] Y. Marcus, J. Chem. Soc. Faraday Trans. 87 (1991) 2995.