2. 5 Elhasnaoui K et al.
J. Adv. Res. Appl. Phy. Appl. 2014; 1(2): 4- 11.
2
0
2
2 4
i
i
i
B
Zn
Tk
e
k
(2)
Here T is the absolute temperature,
Bk is the Boltzmann constant,
is the relative dielectric constant of the water,
and
in is the number density of particles
participating in the common Zi load screening.
So, this model involves several parameters
anZT ii ,,,, that control the interaction
force between particles. In fact, these parameters
do not have the same importance. If the
temperature T and dielectric constant of
aqueous media are readily determined, T the
product was not a relevant amount, because it
slowly decreases when T is increasing. The
charge of the spheres Z increases with the area
of the surface of the spheres, and varies
according to the preparation of these spheres.
Although the load is usually of the order of
1000, it is not measured accurately, but
determined by adjusting certain properties of the
measured system. In some cases, the smaller
particles have the largest effective charge. The
most critical parameters are the radius of the
spheres a and the number density in . If they are
not added to the electrolyte solution, the
solutions of ions are the only ones against the
hydrogen ions H+. To simplify the model, we
assumed that all areas are the same. Thus, the
only components of the system are the spheres
against the negatively charged ions and
positively charged, respectively density Sn and
cn number. The condition of electro neutrality
is:
0S Cn Z n (3)
Where Z is the charge of the spheres.
When this charge is very important 1Z the
screening parameter Debye-Huckel k will be
given by:
2
2 2
0
4
S
B
e
k n Z
k T
(4)
Increasing the density of the spheres or the
addition of an electrolyte (or salt) results in a
significant electrostatic shielding, and thus the
attractive van der Waals forces become dominant
in a large and intermediate interparticle distances.
These dispersion forces can play a major role in
many phenomena, such as irreversible
coagulation. Here, we used a new expression for
the potential of van der Waals proposed given by
Lu and Marlow, 3
and that takes into account the
effect of finite particle size. This potential has all
the characteristics of a semi-empirical potential
Van der Waals. Even if it was applied only to
ordinary molecules, it can be used universally in
the following form:
)()( 66
6
rf
r
C
rU LM
(5)
where 6C is a frequency integration of the
polarization density function and the so- called
nonretarded distance damping function 6 ( )f r is:4
(6)
Here 1a and 2a are two parameters that
characterize the atomic or molecular size in
the case of atoms or small molecules. In this
article, 1a and 2a are set equal since the
considered condensed bodies are always
composed of the same kind of molecules.
For simplicity, let a1 = a2 = a
(7)
2
2
1 1
1 1
3 2
( ) ,6 2
2 3
2 1 1
1 1
3 2 4
r
r r ae
a a
f r
r
r r r ae
a a a
2
2
6
1
2 3
2
1
1 1
, 1 1
3 2
2 1 1
1 1 .
3 2 4
j
j
r
a
j
j j j
r
a
j j j j
r r
f r a e
a a
r r r
e
a a a
3. Elhasnaoui K et al. 6
J. Adv. Res. Appl. Phy. Appl. 2014; 1(2): 4- 11.
When the distance r tends to infinity, the
function )(6 rf tends to 1 and )(rU LM
tends
asymptotically to 6
6
r
c
according to plan, but
6
6
72
)0(
a
C
rU LM
, which shows that
the potential is finite for any long distance, while
for extremely short distance, it is larger than the
electrostatic repulsion. The grouping together of
the two potentials of interaction leads to a
minimum adjacent at the point ar 2 .
Figure 1.The allure of potential Lu and Marlow depending on the distance renormalized x r
Figure 2.Correlation function with a Lu- Marlow potential using integral equation theory with
8 3
0.3795.10n nm
Variational method
The variational method of Mansoori et al. has
proven that it can be a fruitful way to estimate the
thermodynamic properties of fluids. 5, 6
Let us write
the Hamiltonian of the particles, H, as the sum of:
HHH R (8)
where RH is the Hamiltonian of a system of
reference, and H is the difference
RHHH .
4. 7 Elhasnaoui K et al.
J. Adv. Res. Appl. Phy. Appl. 2014; 1(2): 4- 11.
The Gibbs-Bogoliubov inequality associated with
the variational method, in terms of the Helmholtz
free energy F of the real system, is written as:
RRR HHFF (9)
Here, F is the actual free energy of colloidal
system under investigation,
RF is the free energy of a reference system, and
RRHH contribution accounts for the mean
interaction potential difference between the real
and the reference system.
In this work, we have chosen a reference system
for which the Helmholtz free energy is easy to
determine which is close enough to the real
system, to provide a good estimate of .This free
energy system F is the system of hard spheres
(HS) with diameter . What constitutes a good
approximation for the thermodynamic and
structural properties. Under these conditions, the
above inequality takes the following explicit form:
drrrUr
HS
gn
HS
FF 2;2
Where rU is the sum of the repulsive
electrostatic rU DH
potential and the attractive
potential of van der Waals or Lu-Malow potential
rU LM
, previously defined in the inequality (9)
represents the correlation function of HS pair. The
above integral is calculable numerically, using the
classical algorithm by Throop and Bearman,
including corrections given by Verlet and Weis.
Moreover, the free energy of the system HS, has
an analytical expression: 7
2
1
2
2
3
1ln
TkF BHS (11)
Here, denotes the packing fraction, which is
related to the diameter of the HS and the
number n density by the relationship:
3
6
1
n (12)
Note that the number density n is that of
polystyrene spheres. However and are
parameters of the reference system HS, different
from the actual colloidal system, having the
effective volume fraction given by the
following equation as the parameter:
3
3
4
na (13)
The diameter of the HS is selected as a
variational parameter. The upper bound of the free
energy can be obtained by simply varying ,
to obtain the minimum of the right side of
inequality Gibbs-Bogolyubov, relation (9). The
kinetic energy and the entropy term of polystyrene
spheres are neglected in the expression for the free
energy as they are constant.
The variational method can be applied in such a
simple way for liquids. While for charged
colloidal suspensions, a significant change in
potential must be done to get the smallest upper
bound of the Helmholtz free energy. Different
approaches have been used to fix the potential. In
this work, we have adopted the process of
rescheduling. It is established that the structure
factors determined with the MSA (Mean Spherical
Approximation)8
: This is an amendment proposed
by Lebowitz and Percus closure. It treats the fluids
spherical particles acting on each other by a
potential which contains both a strongly repelling
portion short- range and low long- range attractive
part. MSA are in a good agreement with
experimental results for concentrated micellar
solutions, as they are not for systems of low
density, such as polystyrene spheres loaded.
However, to implement the MSA to colloidal
suspensions, Hayter and Hansen 9, 10
proposed a
rescheduling process, which is a generalization of
prescription introduced by Gillan for the plasma
component, based on the following argument:
Increased electrostatic repulsion at very short
distances between the particles is virtual,
since it never gets at closer distances.
The diameter is much larger than the
actual diameter of polystyrene spheres a2 ,
and the correlation function is zero for
r , so the pair remains substantially
small for the closest distance.
This means that the hard polystyrene spheres have
no significant role in determining the structure.
For this reason, Hansen and Hayter have suggested
increasing the diameter of the spheres from its real
value a2 to the effective diameter , keeping the
potential constant. The rescaled diameter increases
the effective, which is a complicated function
F
10
5. Elhasnaoui K et al. 8
J. Adv. Res. Appl. Phy. Appl. 2014; 1(2): 4- 11.
depending on an, et Z , and which is expressed as
volume fraction, and the MSA becomes more
accurate. The rescheduling process is
characterized by the potential parameters and with
a parameter, which is a complicated function
depending on an, et Z , and expressed as:
n
Z
s
1
(14)
Where and are two constants.
The useful expression of the Gibbs-Bogolyubov
inequality for colloidal systems thus becomes:
drrrUrgnFF T
HSHS
2
;;2
with
srUsrUsrU LMDH
;;;
The solution of this inequality is that the
minimization of the free energy with the respect to
the diameter is made numerically. Thus, one
gets all the thermodynamic and the structural
properties of the system in question.
Results and Discussion
Before presenting the model parameters and the
results, we briefly review the experimental results
of Grüner and Lehmann (GL).11
The experiments
were conducted at the room temperature
KT 298 , with polystyrene spheres of radius
nma 45 , disposed in the dielectric constant of
water . The values T and were
known with good accuracy, and the variation of
the radius of the spheres is estimated at 5%, but
the values of the density and the charge are not
accurate. The five densities for the sample to be
studied are known precisely. It must be treated
with caution, because the densities that are
measured by GL are higher than those announced
by the manufacturer by a factor of nearly 1.5. In
our work, we used those which were used by GL,
for which our results coincide with those of
experiments.
Concerning the charge, it was assessed to be
1501000 , using the pH , but it is well known
that the surface charge of the spheres must be
renormalized around 200 or 300. In our
calculations, we used a linear variation of the
effective charge Z varies 180 for the less dense
280 for denser suspensions, as has been suggested
by Grimson.12
This brief analysis of the
experimental conditions allowed us to see the role
of each parameter for characterizing the
electrostatic repulsion. Now to count the attractive
van der Waals interaction, we need to choose a
value for the parameter, 6C which is not easy to
determine as it depends on the properties of the
particles and the medium of dispersion. At this
stage of our study, we found for all densities
studied 67
6 105.1 JnmC
, so that the depth
of the potential well either 0.5 BK T . In table
1, we have presented the main characteristics of
structure factors calculated with the systems of
hard spheres as a reference system. We have found
that there is good agreement between the
experimental structure factors and those
calculated, in position maxq and the height
maxqS of the first peak, for different values of
density. But there is a disagreement 0S between
the experimental value and calculated value. To
solve this, we use the random phase approximation
of the Random Phase Approximation (RPA),
which is an approximation method in condensed
matter and nuclear physics. It was introduced by
David Bohm and David Pines as an important
outcome in a series of 1952 and 1953 founding.
For decades, physicists have tried to incorporate
the effect of the microscopic quantum mechanical
interactions between electrons in the theory of
material, of which the results are shown in table 2.
The results with the RPA are in good agreement
with the experimental results, for low values of the
wave vector transfer.
38
/10 m
n
nm maxS q
HS
maxS q
Exp
15
10
cm
qHS
15
10
cm
qEXP
HS
S 0 EXP
S 0
0.3795 0.328 548.6 1.63 1.62 1.17 1.16 0.07 0.30
0.7590 0.374 454.9 1.87 1.87 1.45 1.37 0.05 0.23
1.1385 0.398 405.7 2.03 2.03 1.63 1.63 0.04 0.14
1.5180 0.430 378.2 2.30 2.30 1.78 1.73 0.03 0.11
1.8975 0.481 364.5 2.92 2.9 1.89 1.91 0.02 0.07
Table 1.Characteristics of structure factors with HS for five densities compared with experimental results
54.78
6. 9 Elhasnaoui K et al.
J. Adv. Res. Appl. Phy. Appl. 2014; 1(2): 4- 11.
Table 2.Characteristics of structure factors with the RPA for the five densities compared with
experimental results
The comparison between theoretical and
experimental data of the structure factors for the
five densities studied is shown in fig. [3-7].Avec
HS we played well the first peak qS , in
position and height, and this for all densities
studied. But for small values of the wave vector
transfer q , the results obtained with the RPA are
close to the experience as those of HS.
Figure 3.Comparison between the experimental structure factors (OOO) those determined with HS (-)
and with RPA (- - -), for the density:
8 3
0.3795 10n nm
Figure 4.Comparison between the experimental structure factors (OOO) and those determined with HS
(-) and with RPA (- - -), for the density:
8 3
0.7590 10n nm
38
/10 m
n
nm maxS q
RPA
ExpS q
Max 15
10
RPA
cm
q
15
10
EXP
cm
q
RPA
S 0 EXP
S 0
0.3795 0.263 509.4 1.57 1.62 1.21 1.16 0.30 0.30
0.7590 0.320 431.7 2.11 1.87 1.46 1.37 0.23 0.23
1.1385 0.346 387.2 2.60 2.03 1.63 1.63 0.14 0.14
1.5180 0.375 361.3 3.34 2.30 1.78 1.73 0.11 0.11
1.8975 0.378 364.5 3.62 2.9 1.92 1.91 0.06 0.07
7. Elhasnaoui K et al. 10
J. Adv. Res. Appl. Phy. Appl. 2014; 1(2): 4- 11.
Figure 5.Comparison between the experimental structure factors (OOO) and those determined with HS
(-) and with RPA (- - -), for the density:
8 3
0.7590 10n nm
Figure 6.Comparison between the experimental structure factors (OOO) and those determined with HS
(-) and with RPA (- - -), for the density:
8 3
1.1385 10n nm
Figure 7.Comparison between the experimental structure factors (OOO) and those determined with HS
(-) and with RPA (- - -), for the density:
8 3
1.1385 10n nm
8. 11 Elhasnaoui K et al.
J. Adv. Res. Appl. Phy. Appl. 2014; 1(2): 4- 11.
Conclusions
In this work, we are interested in studying the
stability of a colloidal solution. The stability of
such a solution results from the balance between
attractive and repulsive interactions exerted on the
particles by preventing the aggregation of particles
of the dispersed phase. The two main mechanisms
of stabilization are steric stabilization and
electrostatic stabilization in the case that we had in
hand. In this study, we have used a new expression
for the potential of Van der Waals described by Lu
and Marlow, which takes into account the finite
size of the particles. To test this potential, we have
calculated the structure factor. As a method, we
have used the variational method based on the
Gibbs-Bogoliubov inequality. The resulting
theoretical structure factors obtained are found to
be in a good agreement with the experimental data
of Grüner and Lehmann, which justified the
interest in introducing such potential.
Acknowledgment
We are much indebted to Prof. J.-L. Bretonnet,
Prof. J.-M. Bomont and Prof. N.Jakse for the
helpful discussions. We would like to thank the
Laboratory of Condensed Matter Theory (Metz
University) for their kind hospitality during the
regular visits.
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