Khalid elhasnaoui DR Version final (groupe LPPPC)

Journal of Advanced Research in Applied
Physics and Applications
J. Adv. Res. Appl. Phy. Appl. 2014; 1(2): 4- 11.
© ADR Journals 2014. All Rights Reserved
Structure and thermodynamics of
solutions of colloids interacting
through Yukawa or Lu-Marlow
potentials
F. Benzouine, K.Elhasnaoui, A.Maarouf, A. Derouiche
LPPPC, Sciences Faculty Ben M’sik, P.O.Box 7955, Casablanca, (Morocco)
Abstract
In this article, we used an expression 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 use 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 a potential.
Keywords : Computer simulations, colloids, pair potential , structure factor,
Structure, Thermodynamics variational method.

Introduction
A common and basic representation of a
suspension of colloid that is charged
polystyrene latex spheres immersed in an
aqueous medium[1]
. Many studies have been
devoted to the study of the structure of these
colloids. The associated structure factor is
similar to those of the simple liquids, with a
pronounced peak first, which is becoming
increasingly important and moves to a larger
values of the wave vector transfer, when the
concentration increases. This similarity with
atomic systems allows the physical colloidal
suspensions to be treated in terms of effective
pair potential. In very dilute suspensions
colloidal, particle interactions are absent,
whereas in more concentrated solutions;
interaction effects become significant and
therefore can be treated using the
approximation of a modified Debye-Hückel[2]
.
In order to study the structure, we used a
traditional repulsive potential Verwey and an
Overbeek, which come from the mutual
interaction of electrical double layers
surrounding each particle, and a new form of
the attractive van der Waals potential
described by Lu and Marlow[3]
. The main
advantage of this attractive potential is that it
is proportional uA the inverse sixth power of
the distance, for large separations, and in
addition, it involves the size of the particles.
Which justifies its importance.
I- Interparticle potential.
Before considering the interparticle
potential, we first define the system to be
studied. These polystyrene spheres immersed
in an aqueous medium, the sulfonic acid
groups are ionized by contact with water and
then produce negative charges which are
located on the surface of the spheres. The
interactions between these spheres are foiled
against by the presence of positively charged
ions in the solution. This is identical to the
shielding of impurities in an electron gas
processed in the Thomas-Fermi
approximation. If the solution is very dilute,
colloids interactions can be treated by the
Debye-Huckel approximation. By contrast,
the suspensions of polystyrene spheres of
finite size, the electrostatic repulsion between
the electrical double layers around these areas
are so high that the linear approximation of
the Debye-Hückel becomes inadequate and
must be changed. Thus, the potential for these
electrostatic interaction colloidal systems can
be described by the following expression
r
e
ka
eeZ
rU
krka
DH









2
0
22
1
)(

(1)
Where r is the distance between the centers of
two spheres interaction (common load Ze ,e
is the elementary charge of an electron),
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 in the number density. If they
are not added to the electrolyte solution, the
solution 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[4]
, 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 a the ordinary molecules,
it can be used universally in the 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 functio 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)
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
 ,this
shows that the potential is finite for any
distance. While extremely low long
distance, it is larger than the electrostatic
repulsion. The grouping together of the
two potentials of interaction leads to a
minimum adjacent the point ar 2 .
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




   
             
     
                  



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
 


II. variational method
The variational method of Mansoori [5,6]
et al has proven that it can be a fruitful way
to estimate the thermodynamic properties of
fluids. Let us write the Hamiltonian of the
particles, H, as the sum of:
HHH R  (8)
With RH is the Hamiltonian of a
system of reference and H is the
difference RHHH  . The Gibbs-
Bogoliubov inequality associated to the
variational method, in terms of the
Helmholtz free energy F of the real
system, is written:
RRR HHFF  (9)
Here, F is the actual free energy of
colloidal system under investigation.
There, RF is the free energy of a reference
system. The 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) in
diameter  . What constitutes a good
approximation for the thermodynamic and
structural properties. Under these
conditions, the above inequality takes the
explicit form:
      drrrUrHSgnHSFF 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
Throop and Bearman, including corrections
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)
 10

Note that the number density n is that of
polystyrene spheres. However  , and
 are a parameters of the reference system
HS, different from the actual colloidal
system, having as parameter the effective
volume fraction  given by:
3
3
4
na  (13)
The diameter  of the HS is selected as a
variational parameter. The upper bound of
the free energy F 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, which is based on the
following argument:
* Increased electrostatic repulsion at
very short distances between the particles is
virtual, since it never gets closer distances.
* The  diameter is much larger than
the actual diameter polystyrene spheres a2 ,
and the correlation function is zero for
r , that pair remains substantially small
for  the closest distance.
This means that the hard polystyrene
spheres has no significant role in
determining the structure. For this reason,
Hansen and Hayter have suggested
increasing the diameter of the spheres of its
real value a2 to the effective diameter ,
and the potential remains constant. The
rescaled diameter increases the effective,

which is a complicated function depends
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 depends an, et Z , and which
is 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.
III. Results and Discussion
Before presenting the model parameters
and the results, we briefly review the
experimental results of Grüner and
Lehmann)[11]
(GL). The experiments were
conducted at the room’s temperature
 KT 298 , with polystyrene spheres of
radius nma 45 , disposed in the dielectric
constant of water 54.78 . 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
measured by GL are larger announces the
manufacturer by a factor of nearly 1.5. In
our work, we used those which used by GL
for which our results coincide with those of
experiments.
Concerning the charge it was assessed
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 µ

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  0S is a
disagreement between the experimental and
calculated. To solve this, we use the
random phase approximation of the
"Random Phase Approximation (RPA)" 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
articles, 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 grouped. 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 III.1: Characteristics of structure factors with HS for five densities are studying
compares with experimental results.

Table III.2: Characteristics of structure factors with the RPA for the five densities are
studying 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  .
 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

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  .
0.7590 10n nm  .

1.1385 10n nm  .
1.1385 10n nm  .

IV. 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
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 case before us that we have at 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 use 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
potentil.
ACKNOWLEDGMENT
We are much indebted to Professors J.-L. Bretonnet, J.-M. Bomont and N.Jakse for helpful
discussions. Three of us (M.B., F.B. and A.D.) would like to thank the Laboratory of
Condensed Matter Theory (Metz University) for their kind hospitality during their regular
visits.

Page
1919
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