1. SOP TRANSACTIONS ON APPLIED PHYSICS
ISSN(Print): 2372-6229 ISSN(Online): 2372-6237
DOI: 10.15764/APHY.2014.04001
Volume 1, Number 4, December 2014
SOP TRANSACTIONS ON APPLIED PHYSICS
Structure and Thermodynamics of
Solutions of Colloids Interacting Through
Yukawa or Sogami Potentials
M. Badia1, A. Maarouf3*, K. ELhasnaoui3*, T. El hafi3, M. Benhamou2,3
1 Royale Air School, Mechanical Dept, DFST, BEFRA, P.O.Box 40002, Menara, Marrakech
2 ENSAM ,Moulay Ismail University P.O.Box 25290, Al Mansour, Meknes
3 LPPPC, Sciences Faculty Ben M’sik, P.O.Box 7955, Casablanca, (Morocco)
*Corresponding author: abdelwahad.maarouf@gmail.com; elhasnaouikhalid@gmail.com
Abstract:
The aim of the present work is the determination of the structure and thermodynamics of a
monodisperse colloidal solution. We assume that the interaction potential between colloids is
of Yukawa or Sogami types. The former is purely repulsive, while the second, it involves, in
addition to a repulsive part, a Van der Waals attractive tail. We compute the structure factor and
thermodynamics properties, using, the integral equation one with the hybridized mean spherical
approximation. We first compare the results relative to this theory, and with this obtained within
Monte Carlo simulation. We show that results from integral equation method with a Sogami
potential and those of simulation are in good quantitative agreement. Finally, our theoretical
results are compared to those of experiment by Tata and coworkers. We find that integral
equation theory with Sogami potential agrees well with experiment.
Keywords:
Colloids; Pair-potential; Structure; Thermodynamics; Monte Carlo Simulation; Integral Equation
1. INTRODUCTION
Colloids are particles of mesoscopic size, which are the subject of numerous theoretical and experi-
mental studies, because of their abundant industrial applications. Colloids immersed in a polar solvent
(water for instance) often carry an electric charge. This implies a strong Coulombian interaction between
colloidal particles. Actually, this interaction is screened out due to the presence of proper counterions and
co-ions coming from a salt or an electrolyte [1]. However, particles also experience a long-range Van der
Waals attractive interaction. The former is responsible for dispersion, while the second, for flocculation.
Dispersion and flocculation are the two crucial problems in colloid science.
From a theoretical point of view, colloids constitute special statistical systems. Thus, to study their
physical properties such as structure, thermodynamics and phase diagram, use is made of statistical
mechanics methods. Among these, we can quote variational and integral equation approaches.
c(r)g(r)The more reliable approach is the Ornstein-Zernike (OZ) [2] integral equation method [3]. The
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2. SOP TRANSACTIONS ON APPLIED PHYSICS
quantity solving this equation is the pair-correlation function , which is a crucial object for determining
most physical properties. But, this equation involves another unknown that is the direct correlation
function
. Thus, this necessitates a certain closure, that is, a supplementary relationship between these two
correlation functions. Integral equation has been intensively used in the modern liquid theory. It has been
solved using some techniques, which are based on the analytical or numerical computation. One has
used different closures, namely, the Percus-Yevick approximation [4], the hypernetted chain [5], the mean
spherical approximation and its modification that is the hybridized-mean spherical approximation [6]
(HMSA) we apply in this work.
The purpose of this paper is the determination of both structural and thermodynamic properties of a
dilute solution of spherical colloids of the same diameter (monodisperse system). We assume that particles
interact through Yukawa [7] or Sogami [8, 9] potentials. The former is purely repulsive, and then it favors
dispersion of colloids. Beside the repulsive contribution, Sogami potential involves a Van der Waals
attractive tail. In fact, this latter is responsible for condensation phenomenon of colloids. To investigate
the structure and thermodynamics of the system, we have used the integral equation with HMSA. First,
we have compared results obtained with Monte Carlo (MC) simulation results [10, 11]. We have shown
that results from integral equation method and those of MC are in good quantitative agreement. Finally,
our theoretical results are compared to those measured in experiment by Tata et al. [12], for the same
values of parameters of the problem. We have found that theory with a Sogami potential agrees with
experiment by the authors.
This paper is organized according to the following presentation. In Sec. II, we describe the theory of
integral equation with HMSA enabling us to compute the physical properties of interest. We present in
Sec. III the results and make discussion. Comparison between theory and experiment is the aim of Sec.
IV. We draw our conclusions in Sec. V.
2. THEORY
2.1 Pair-potential
In this paper, we choose separately two kinds of potentials that are of Yukawa [7] and Sogami types
[8, 9]. The aim of this section is to recall their forms.
Consider a monodisperse colloidal system made of polystyrene balls (polyballs) of spherical form. We
denote by Ze the charge carried by one colloid,, where c is the electron elementary charge. Because of the
presence of counterions, and eventually, electrolyte or salt ions, Coulombian interactions are screened out
and colloids interact through a Yukawa pair-potential defined by
UY (r) =



∞, r σ
πεε0ψ2
0
exp[−κ (r −σ)]
r
, r ≥ σ
(1)
There, r is the interparticle center-to-center distance, σ the hard-sphere diameter, Ψ0 the surface
potential, ε the relative permittivity of solvent (water), ε0 the permittivity of free space, and κ the
Debye-Hiickel inverse screening length. Parameter, κ is defined as usual by
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3. Structure and Thermodynamics of Solutions of Colloids Interacting Through Yukawa or Sogami Potentials
κ2
=
4πe2
εε0kBT ∑
i
niZ2
i (2)
where ni stands for the number density of ions of type i and Zi, for their valency. On the other hand, the
surface potential Ψ0 my be related to macroion valency Z by [7], in the limit of weak screening κσ 1
[13].
ψ0 =
Ze
πεε0 (2+κσ)
. (3)
Choosing σ as a distance unit, the considered potential takes the form
UY (r)
kBT
=



∞, x 1,
Γ
exp(−kx)
x
, x ≥ 1.
(4)
We have used the notations x = r/σ and k = κσ , to mean respectively the renormalized interparticle
distance and the renormalized electric screening parameter. There,
Γ = πεε0σψ2
0 ek
kBT (5)
is the coupling constant.
Another pair-potential used here is that derived by Sogami [8, 9], which describes the effective
electrostatic interactions between macrions of charge Ze, This potential involves a short-range Coulomb
repulsion, whose origin is self-evident, in addition to a long-range exponential attractive tail. This latter
was derived using a self-consistent method [8, 9]. The Sogami potential has been used to describe the
vapor-liquid transition and crystallization of charged colloids observed in experiments [14]. Its expression
is then [8, 9]
US
(r) =
(Ze)2
εε0σ
sinh2
k 2
k2
2+kcoth k 2
x
−k exp (−kx), x 1 (6)
where x and k are those renormalized quantities defined above. The shape of such a potential is depicted
in Figure 1.
Sogami potential is canceled in both cases, either r = 2 2+kcoth k 2 κ = 2A κ, or when r → ∞ .
The first derivative of the expression of the potential of Sogami he presented as follows
∂
∂r
US
(r) = −
(Ze)2
εε0σ
sinh2
k 2
k2
exp (−kx)
x2
−k2
x2
+Akx+A (7)
The position of the potential minimum Rm is given as
Rm = A+[A(A+4)]1/2
2κ = 2+kcoth k 2 + 2+kcoth k 2 6+kcoth k 2
1/2
2κ
(8)
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4. SOP TRANSACTIONS ON APPLIED PHYSICS
Figure 1. Reduced Yukawa and Sogami potentials U (r)/kBT versus the renormalized interparticle distance r/σ,
using the parameters the Tata et al [12].
Rm decreases monotonically, with increasing k = κσ , to the limiting value 2σ. The depth of the pair
potential grows rapidly in the interval 0 κσ 1 , reaches its maximum around κσ ≈ 1.19 , and then
decreases gradually to zero. Since Rm behaves as Rm ≈ 2 1+
√
2 κ for small κσ, the interparticle
distance can take a very large value in the suspension with small latex particle concentration so far as the
potential minimum keeps up a sufficient depth [8, 15].
Table 1. Comparison of the calculated distance Rm and the observed interparticle distance Rexp in dilute colloidal
suspensions of charged (charge number Z = 4 × 103) and spherical (radius σ = 1705) polymer particles
for different particle concentrations [8].
Concentration vol (%) κσ US (Rm) Rm 103 ˚A Rexp 103 ˚A
0.4 0.48 -0.49 18.0 18.0
0.55 0.56 -0.55 15.0 15.0
1.5 0.92 -0.73 11.0 10.0
4 1.50 -0.33 7.2 8.0
The following step consists in recalling the essential of the integral equation method used in this work.
2.2 Method of Equations Integrals (MEI)
Several approaches exist to study the structural property and thermodynamic a fluid from its interactions.
The method of integral equations is one of these techniques which allows to determine the structure of a
fluid in a thermodynamic state given, characterized by its density ρ and its temperature T, for a potential
pair of u(r) which mobilize the interactions between the particles. The calculation of the structure,
represented by the function of radial distribution g(r), is a own approach to the theory. In fact, the fact
that in a liquid the particles are partially disordered implies his ignorance apriority. The function g(r),
which describes the arrangement medium of particles as a function of distance from an origin theory on
the one hand, the Fourier transform of g(r) is the factor of structure
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5. Structure and Thermodynamics of Solutions of Colloids Interacting Through Yukawa or Sogami Potentials
S(q) = 1+ρ (g(r)−1)exp(iqr)dr (9)
That is measured by the experiences of diffraction of X-ray or neutron in function of the vector transfer
q . On the other hand, the thermodynamic quantities of the fluids are functions of g(r) and the u(r) as the
internal energy per particle
E/ N = (3/2)kBT +2π u(r)g(r)rdr (10)
kB is the constant of Boltzmann, the pressure of the viriel
P = ρkBT −2
πρ
3
rdu(r)
dr
g(r)rdr (11)
Or the isothermal compressibility χT . This last can be obtained by two independent see, either by
deriving the pressure (6) by report to the density:
χ−1
T = ρ
∂P
∂ρ T
= ρkBT −
4πρ2
3
r
du(r)
dr
g(r)+
ρ
2
∂g(r)
∂ρ
r2
dr (12)
Either share the intermediare of a study of fluctuations in the number of particles in the whole grand
canonical
S(q = 0) = ρkBT.χT = 1+4πρ (g(r)−1)r2
dr (13)
We can note that the isothermal compressibility χT deducted from the pressure of viriel is equal to that
calculated from the angle limit the diffusion of the zero factor structure.
2.3 Integral Equation Approach
The starting point of such a method is the Ornstein-Zernike (OZ) integral equation satisfied by the
total correlation function h(r) = g(r)−1. The OZ integral equation that involves the so-called direct
correlation function c(r) [16, 17], is given by
h(r) = c(r)+n c r −r h r dr (14)
where n is the number density of macroions. This equation, however, contains two unknown quantities
h(r) and c(r).To solve it, one need a closure relation between these two quantities. In this paper, we
decide to choose the HMSA, and write
gHMSA
(r) = exp[−βU1 (r)] × 1+
exp [f (r) {γ (r)−βU2 (r)}−1]
f (r)
(15)
where the interaction potential is divided into short-range part U1 (r) and long-range attractive tail
U2 (r) as prescripted by Weeks et al [18]. There, the function γ (r) is simply the difference between the
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6. SOP TRANSACTIONS ON APPLIED PHYSICS
total and direct correlation functions, i.e., γ (r) = h(r)−c(r). Quantity f (r) is the mixing function [6],
whose a new form was proposed by Bretonnet and Jakse [19]. The virtue of such a form is that, it ensures
the thermodynamic consistency in calculating the internal compressibility by two different ways. The
form proposed by the authors is [19]
f (r) = f0 +(1−f0)exp −1 r (16)
where f0 the is the interpolation constant. This an adjustable parameter such that 0 ≤ f0 ≤ 1. This
constant that serves to eliminate the incoherence thermodynamic, can be fixed equating the compressibility
deduced from virial pressure to that calculated from the zero-scattering angle limit of the structure factor,
i.e.,
S(0) = nkBT χT (17)
Now, it remains the presentation and discussion of our results, and their comparison with those relative
of MC simulation [11] and experiment [12].
3. RESULTS AND DISCUSSION
In this paper, we have used those parameters values reported in experiment by Tata et al [12]. These
are σ = 1090 ˚A (particle diameter), T = 298K (absolute temperature), ε = 78 (relative permittivity of
water), Z = 600 (colloid valency), ni = 1.751021m−3 (impurity ion concentration) and np = 1.33x1018m−3
(polyball number density). With these parameters values, we have Γ = 2537 and k = κσ = 0.558.
Our purpose is a quantitative investigation of thermodynamic and structural properties of a dilute
solution of polyballs (in water), using the integral equations method.
3.1 Integral Equation Method Results
The HMSA integral equation is applied here for accomplishing an alternative computation of structural
and thermodynamics properties of the colloidal solution under investigation. Potentials used here are of
Yukawa or Sogami types, and the choosing mixing function f (r) is that pointed out in [19].
First, we have computed the main object that is the pair-correlation function g(r) versus the renormal-
ized interparticle distance r/σ. Figure 2 shows a superposition of two curves that are relative to Yukawa
and Sogami potentials. The important remark is that, the height of the peak of g(r) is more pronounced
for Sogami potential than of Yukawa type. Indeed, this can be understood by the fact that the colloid
system with Sogami interaction is more dense than that governed by Yukawa potential.
Second, we have reported in Figure 3 the pair-correlation function for a Sogami potential, together with
that computed using the MC simulation [20]. In fact, the two curves are in good quantitative agreement.
In Figure 4, we have reported the structure factors versus the renomalized wave-vector qσ computed
using integral equations and MC methods. Remark, first, that the results obtained within the integral
equations HMSA are in good agreement as in the case of the pair-correlation function discussed above.
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7. Structure and Thermodynamics of Solutions of Colloids Interacting Through Yukawa or Sogami Potentials
Figure 2. Comparaison of the correlation function with Yukawa and Sogami potentials computed by HMSA integral
equation.
Figure 3. Correlation function with a Sogami potential using HMSA integral equation theory and MC simulation.
In the case of Yukawa potential Figure 5, we have reported the structure factors versus the renormalized
wave-vector computed within the integral equations method described above.
In fact, the same conclusions can be drawn. The only difference is that, the height of the peak is strongly
deviated. This is natural, because of the precised character of the integral equation and MC methods.
With potentials of Yukawa or Sogami types, we summarize in Table 2 thermodynamic properties, for
two values of the interpolation constant f0 .
For a small increase in density as they should and the energy and pressure increase, whereas the
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8. SOP TRANSACTIONS ON APPLIED PHYSICS
Figure 4. Structure factors with a Sogami potential, obtained within HMSA integral equation theory and MC simu-
lation.
Figure 5. Structure factors with a Yukawa potential, obtained within HMSA integral equation theory and MC simu-
lation.
compressibility decreases. We also note that the internal energy is negative in the case of potential Sogami,
then what is positive in the case of Yukawa potential that is due the attraction has the potential to Sogami,
the pressure is very important in the case the Yukawa potential which is purely repulsive character.
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9. Structure and Thermodynamics of Solutions of Colloids Interacting Through Yukawa or Sogami Potentials
Table 2. Thermodynamic proprieties for Yukawa and Sogami potentials, within HMSA integral equation method.
Sogami Potential Yukawa Potential
fo ρkBT χ E/nkBT P/ρkBT fo ρkBT χ E/nkBT P/ρkBT
0.465 0.08420 -5.3442 1.4115 0.470 0.03780 4.0803 10.2251
0.465 0.08280 -5.3566 1.4574 0.470 0.03750 4.1142 10.2965
0.465 0.0810 -5.3729 1.5191 0.475 0.03715 4.1629 10.3972
0.465 0.07931 -5.3887 1.5818 0.475 0.03681 4.2084 10.4924
0.465 0.07761 -5.4044 1.6451 0.475 0.03643 4.2543 10.4924
0.465 0.07603 -5.4197 1.7092 0.475 0.03642 4.2543 10.5882
0.465 0.07443 -5.4346 1.7743 0.480 0.03611 4.3039 10.6904
0.465 0.07292 -5.4348 1.7743 0.480 0.03572 4.3502 10.7869
0.465 0.07281 -5.4495 1.8300 0.480 0.03541 4.3967 10.8838
0.470 0.07123 -5.4609 1.9141 0.480 0.03501 4.4472 10.9874
0.470 0.06984 -5.4750 1.9818 0.480 0.03471 4.4942 11.0852
4. COMPARISON WITH EXPERIMENT
For our calculations, we have considered the same values of parameters as in experiment by Tata et
al. [12], which are σ = 1090 ˚A (particle diameter), T = 298K (absolute temperature), ε = 78 (relative
permittivity of water), Z = 600 (colloid valency), ni = 1.751021m−3 (impurity ion concentration) and
np = 1.33x1018m−3 (polyball number density).
We find that the structure factor computed within the framework of the HMSA integral equation agrees
with the measured one, in all q-range Figure 6.
Figure 6. Comparison between experimental structure factor with the calculated one within HMSA integral equation
method and Sogami potential.
However, this is not true for the structure factor relative to a Yukawa potential. As a matter of fact,
there is some inconsistency around the peak. This inconsistency originates from the fad that this potential
ignores the attractive interaction that exists within the sample Figure 7.
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10. SOP TRANSACTIONS ON APPLIED PHYSICS
Figure 7. Comparison between experimental structure factor with the calculated one within HMSA integral equation
method and Yukawa potential.
5. CONCLUSIONS
We recall that the purpose of this paper is the determination of the structure and thermodynamics
of a monodisperse colloidal solution. We assumed that the interaction potential between colloids is of
Yukawa or Sogami types. The difference between these two kinds of potential is that, the former is purely
repulsive, while the second is the sum of two contributions: a repulsive part and a van der Waals attractive
tail. We have computed the structure and thermodynamics, using, the integral equation one with HMSA.
We compared the results relative to this theory to the obtained within MC.
We have shown that results from integral equation method and those of MC are in good quantitative
agreement. Finally, our theoretical results are compared with recent experiment. We found that integral
equation theory agree with this experiment.
Further developments such as the studies of the phase behavior and density effects are in progress.
ACKNOWLEDGMENTS
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 kinds of hospitality during their regular visits.
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