Dr khalid elhasnaoui 2

SOP TRANSACTIONS ON APPLIED PHYSICSISSN(Print): 2372-6229 ISSN(Online): 2372-6237

DOI: 10.15764/APHY.2014.04002Volume 1, Number 4, December 2014

SOP TRANSACTIONS ON APPLIED PHYSICS

Numerical Study of the Structure andThermodynamics of Colloidal Suspensionsby Variational MethodA. Maarouf1*, K. Elhasnaoui1*, M. Badia2, M. Benhamou1,3

1 LPPPC, Sciences Faculty Ben M’sik, P.O.Box 7955, Casablanca, (Morocco)2 Royale Air School, Mechanical Dept, DFST, BEFRA, P.O.Box 40002, Menara, Marrakech3 ENSAM, Moulay Ismail University P.O.Box 25290, Al Mansour, Meknes

*Corresponding author: [email protected], [email protected]

Abstract:In this paper, we present the results of calculation of the structure and thermodynamics propertiesof the dispersions colloidal solution ended using the variational method with the scaling process.We assume that the interaction potential between colloids is of Yukawa or Sogami types. Theformer is purely repulsive, while the second, it involves, in addition to a repulsive part, a Van derWaals attractive tail. We compute the structure factor and thermodynamics properties, using,first, the variational method with the rescaling technique, and second, the integral equationone with the hybridized mean spherical approximation. We first compare the results relative tothese theories, and with those obtained within Monte Carlo simulation. We show that resultsfrom integral equation method with a Sogami potential and those of simulation are in goodquantitative agreement. Finally, our theoretical results are compared to those of experiment byTata and coworkers. We find that integral equation theory with Sogami potential agrees wellwith experiment.

Keywords:Colloids; Pair-potential; Hard Sphere (HS); Structure; Thermodynamics; Monte Carlo Simulation;Variational method; Integral Equation

1. INTRODUCTION

Colloids, are dispersions of mesoscopic particles size (of the order of 1-500 nm in diameters) suspendedin an atomistic fluid, that have been recently investigated by means of both experimental techniques [1, 2]and theorical and numerical studies [3, 4]. The great effort is well justified by the importance that thesesystems play in industrial, biological, and medical applications (paints, inks, pharmaceutical product,waxes, ferrofluids, etc.) [5]. Colloids immersed in a polar solvent (water for instance) often carry anelectric charge. This implies a strong Coulombian interaction between colloidal particles. Actually, thisinteraction is screened out due to the presence of proper counterions and co-ions coming from a salt or anelectrolyte [6]. However, particles also experience a long-range Van der Waals attractive interaction. Theformer is responsible for dispersion, while the second, for flocculation [7]. Dispersion and flocculation

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Numerical Study of the Structure and Thermodynamics of Colloidal Suspensions by Variational Method

are the two crucial problems in colloid science.

From a theoretical point of view, colloids constitute special statistical systems. Thus, to study theirphysical properties such as structure, thermodynamics and phase diagram, use is made of statisticalmechanics methods. Among these, we can quote variational and integral equation approaches.

The procedure variational is a familiar tool in many areas of physics, namely, in Quantum Mechanics[8], Quantum Field Theory [9] and Statistical Mechanics [10]. For instance, for liquid metals, this hasbeen used by several authors [11, 12]. To apply such a method, one needs a reference system. For fluidsand colloids, one often chooses the hard-sphere (HS) systems reference. For the HS model, particles areviewed as impenetrable spheres, which repel each other through a hard-core potential. The variationalparameter may be the hard-sphere diameterσ . In spite of its simplicity, the weakness of the variationalmethod is its mean-field character.

The more reliable approach is the Ornstein-Zernike (OZ) [13] integral equation method [14]. Thequantity solving this equation is the pair-correlation function g(r) which is a crucial object for determiningmost physical properties. Still, this equation involves another unknown which is the direct correlationfunction c(r).

Accordingly, this necessitates a certain closure, that is, a supplementary relationship between these twocorrelation functions. Integral equation has been intensively used in the modern liquid theory. It has beensolved using some techniques, which are based on the analytical or numerical computation. One has useddifferent closures, namely, the Percus-Yevick approximation [15], the hypernetted chain [16], the meanspherical approximation and its modification that is the hybridized-mean spherical approximation [17](HMSA) we apply in this work.

The purpose of this paper is the determination of both structural and thermodynamic properties of adilute solution of spherical colloids of the same diameter (monodisperse system) [18]. We assume thatparticles interact through Yukawa [19] or Sogami [20, 21] potentials. The former is purely repulsive, andthen it favors dispersion of colloids. In addition to the repulsive contribution, Sogami potential involves aVan der Waals attractive tail. The latter, in fact, this latter is responsible for condensation phenomenonof colloids. To investigate the structure and thermodynamics of the system, we have used, first, thevariational method with rescaling techniques [22], second, the integral equation with HMSA. First, wehave compared results obtained within the framework of these two theories. Afterwards, we have made acomparison with Monte Carlo (MC) simulation results [22, 23]. We have shown that results from integralequation method and those of MC are in good quantitative agreement. Finally, our theoretical results arecompared to those measured in experiment by Tata et al [24], for the same values of parameters of theproblem. 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 ofintegral equation with HMSA enabling us to compute the physical properties of interest. We present inSec. III the results and make discussion. Comparison between theory and experiment is the aim of Sec.IV. The paper is closed with concluding remarks in Sec. V.

2. THEORY

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2.1 Pair-potential

In this contribution, we choose separately two kinds of potentials that are of Yukawa [19] and Sogamitypes [20, 21]. The aim of this section is to recall their forms.

Consider a monodisperse colloidal system made of polystyrene balls (polyballs) of spherical form. Wedenote by Ze the charge carried by one colloid [19], where c is the electron elementary charge. Because ofthe presence of counterions, and eventually, electrolyte or salt ions, Coulombian interactions are screenedout and colloids interact through a Yukawa pair-potential defined by

UY (r) =

∞, r <σ ,

σ

πεε0

( Ze2+κσ

)2 exp[−κσ

(r/σ−1)]

r/σ, r≥ σ ·

(1)

There, r is the interparticle center-to-center distance, σ the hard-sphere diameter, ε the relativepermittivity of solvent (water), ε0 the permittivity of free space, and κ the Debye-Hiickel inverse screeninglength. Parameter, κ is defined as usual by

κ2 =

4πe2

εε0kBT ∑i

niZ2i , (2)

Where ni stands for the number density of ion of type i and Zi, for their valency. the potential (1) takesthe form

UY (r)kBT

=

{∞, x < 1,Γ

exp(−kx)x , x≥ 1.

(3)

We have used the notations x = r/σ and k = κσ where κσ << 1, to mean respectively the renormalizedinterparticle distance and the renormalized electric screening parameter. There,

Γ =σ

πεε0

(Ze

2+κσ

)2

ek/

kBT (4)

is the coupling constant.

Another pair-potential used here is that derived by Sogami [20, 21], which describes the effectiveelectrostatic interactions between macrions of charge Ze , This potential involves a short-range Coulombrepulsion, whose origin is self-evident, in addition to a long-range exponential attractive tail. The latterwas derived using a self-consistent method [20, 21]. The Sogami potential has been used to describe thevapor-liquid transition and crystallization of charged colloids observed in experiments [25] and [26–35].Its expression is then [20, 21]

US(r) =(Ze)2

εε0σ

sinh2 (k/2)

k2

[2+kcoth

(k/

2)

x− k

]exp (−kx) ,x� 1, (5)

where x and k are those renormalized quantities defined above. The position of the potential minimumrmin is given as

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rmin ={

2+kcoth(k/

2)+[(

2+kcoth(k/

2))(

6+kcoth(k/

2))]1/2

}/2κ. (6)

rmin decreases monotonically, with increasing κσ , to the limiting value 2σ . The shape of such apotential is depicted in Figure 1.

Figure 1. Reduced Yukawa and Sogami potentials βU (r/σ) versus the renormalized interparticle distance r/σ ,using parameters of Tata et al [24].

2.2 Variational Approach

The variational method is based on the so-called Gibbs-Bogoliubov (GB) inequality [36]

F ≤ F0 + 〈U−U0〉0 (7)

Here, F is the actual free energy of colloidal system under investigation. There, F0 is the free energy ofa reference system. The contribution 〈U−U0〉0 accounts for the mean interaction potential differencebetween the real and the reference system. In this work, we decide to choose the hard-sphere (HS)reference system having the same density n as the real system. The HS interaction potential is

UHS (r) =

{∞, r < σ ,

0, r ≥ σ .(8)

where σ is the hard-sphere diameter, which is not equal to the real diameter of particle 2a (σ > 2a).

Now, it is possible to express the mean value in the right-hand side of inequality (7) in term, of the HSpair-correlation function gHS (r) through

〈U (r)−UHS (r)〉HS =n2

∫ +∞

0gHS (r) [U (r)−UHS (r)]4πr2dr, (9)

The HS correlation function gHS (r) is a known quantity [14]. In particular, it vanishes, i.e. gHS (r) = 0,for interparticle, distances smaller than σ .

15


With these considerations, the GB inequality (7) can be rewritten as

F ≤ FHS (η)+2πn∫ +∞

σ

gHS (r)U (r)r2dr, (10)

where FHSstands for the HS free energy (per site), whose expression is known [14]

FHS (η) = kBT

[− ln(1−η)+

32

(1− 1

(1−η)2

)]. (11)

Here, η is the packing fraction, defined by

η =π

6nσ

3. (12)

This packing fraction will be the actual variational parameter rather than σ .

The approximative expression for the free energy of the real system within the framework of thevariational method is then

Fmin = FHS (ηmin)+2πn∫ +∞

σmin


where ηmin is the value of the packing fraction realizing the minimum of the GB free energy, i.e.,

∂

[FHS + 〈U−UHS〉HS

]∂η

∣∣ηmin = 0 . (14)

With the help of the GB free energy (11), we will compute the structure factor. Its computation withinthe framework of variational method should be done in conjugation with the rescaling procedure proposedmany years ago by Hansen and Hayter [19]. Determination of the structure factor will be the aim of Sec.IV.

The following step consists of recalling the essential of the integral equation method used in this work.

2.3 Method of Equations Integrals (MEIs)

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 afluid in a thermodynamic state given, characterized by its density n and its temperature T , for a potentialpair of u(r) which mobilizes the interactions between the particles. The calculation of the structure,represented by the function of radial distribution g(r), is an own approach to the theory. In fact, the factthat 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 onthe one hand, the Fourier transform of g(r) is the factor of structure

S (q) = 1+n∫

(g(r)−1)exp(iqr)dr (15)16


That is measured by the experiences of diffraction of X-ray or neutron in function of the vector transferq. There exist various routes through which thermodynamic properties of the liquid can be related tointegrals involving g(r)and u(r) the as the internal energy per particle. In the following, we review thethree most common ones. The energy equation

E = 〈Ekin〉+⟨U(rN)⟩ = 3

2NkBT +

12

nN∫

dr4πrg(r)u(r), (16)

expresses the internal energy E, of a one-component N− particle system with pair-wise additive U(rN)

in terms of u(r) and g(r). The internal energy is the sum of a kinetic ideal gas part,(3/

2)

NkBT , and aninteraction or excess part,

⟨U(rN)⟩

. The latter can be understood on physical grounds as follows: Foreach particle out of N, there are 4πr2ng(r) neighbors in a spherical shell of radius r and thickness dr, andthe interaction energy between the central particles and these neighbors is u(r). Integration from 0 to ∞

gives the interaction energy part of E , with the factor 1/

2 included to avoid double counting of particlepairs, kB, is the constant of Boltzmann. The pressure equation,

P = nkBT − 2π

3n∫ +∞

0r3g(r)

du(r)dr

dr, (17)

relates the thermodynamic pressure, P, of a liquid with pairwise additive U to an integral over g(r) andthe derivative, d (u(r))

/dr, of the pair potential. Here, nkBT is the kinetic pressure of a classical ideal

gas, for the second term of the equation (11) is the excess pressure contribution due to particle interactionsthat can be derived along the same lines as the energy equation using, e.g., the classical virial equationFor repulsive pair potential where d (u(r))

/dr ≺ 0, nkBT is the positive pressure contribution originating

from the enhanced thermal bombardment of container walls by the mutually repelling particles.

The compressibility equation,

χ−1T = ρ

∣∣∣∣∂P∂n

∣∣∣∣T= nkBT −

(4πn2

3

)∫r(

du(r)dr

){g(r)+

(n2

)(∂g(r)

∂n

)}r2dr, (18)

χT , is the isothermal compressibility. The compressibility equation can be derived in the grandcanonical ensemble representing an open system, at constant V andT , which allows for fluctuations in theparticle number. This is perfectly appropriate since S (q) is related to the intensity of quasi-elasticallyscattered radiation. It shares the intermediaries of a study of fluctuations in the number of particles in thewhole grand canonical

S (q = 0) = nkBT.χT = 1+4πn∫

(g(r)−1)r2dr (19)

We can note that the isothermal compressibilityχT deducted from the pressure of viriel is equal to thatcalculated from the angle limit the diffusion of the zero factor structure.

2.4 Integral Equation Theories (IETs)

The methods used in this contribution to investigate the structural and thermodynamic properties ofa liquid are based on the Ornstein–Zernike (OZ) equation [13], which relates the direct and the totalcorrelations functions c(r) and h(r) of a system via [37, 38], is given by

17


h(r) = c(r)+n∫

c(∣∣r− r′

∣∣)h(r′)

dr′, (20)

Here, n is the number density of the system; moreover, the relation g(r) = h(r)+1 defines the radialdistribution function g(r). The static structure factor S (q) is defined through the relation

S (q) = 1+n∫

exp(−iqr)h(r)dr (21)

To solve of the equation (14), one needs a closure relation between these two quantities c(r) and h(r) .In this paper, we decide to choose the HMSA [36], and write

gHMSA(r) = exp [−βU1 (r)] ×{

1+exp [ f (r) {γ (r)−βU2 (r)}−1]

f (r)

}, (22)

Here β ≡ (kBT )−1, kB is Boltzmann’s constant, and T the absolute temperature, where the interactionpotential is divided into short-range part U1 (r) and long-range attractive tail U2 (r) as prescripted byWeeks et al [39]. There, the function γ (r) = h(r)− c(r) . is simply the difference between the total anddirect correlation functions, i.e., γ (r) Quantity f (r) is the mixing function [17], whose a new form wasproposed by Bretonnet and Jakse [22]. The virtue of such a form is that, it ensures the thermodynamicconsistency in calculating the internal compressibility by two different ways. The form proposed by theauthors is [40]

f (r) = f0 +(1− f0)exp(−1/

r), (23)

Here, f0account for the interpolation constant, This an adjustable parameter such that 0≤ f0 ≤ 1. Thisconstant that serves to eliminate the incoherence thermodynamic, can be fixed equating the compressibilitydeduced from virial pressure to that calculated from the zero-scattering angle limit of the structure factor,i.e.,

S (0) = nkBT χT . (24)

Now, it remains the presentation and discussion of our results, and their comparison with those relativeof MC simulation [22] and experiment [24].

3. RESULTS AND DISCUSSION

In this paper, we have used those parameters values reported in experiment by Tata et al [24]. These are

σ = 1090◦A (particle diameter), T = 298 K (absolute temperature), ε = 78 (relative permittivity of water),

Z = 600 (colloid valency), ni = 1.75 1021m−3 (impurity ion concentration) and np = 1.33 x 1018m−3

(polyball number density). With these parameters values, we have Γ = 2537 and κσ = 0.558.

Our purpose is a quantitative investigation of thermodynamic and structural properties of a dilutesolution of polyballs (in water), using, first, the variational method, and second, the integral equationsone.

18


3.1 Variational Method Results

Within the framework of such an approach, the structure factor is given by the standard formula [14]

SRPA (q) =SHS (q)

1−nSHS (q)βU (q), (25)

Where U (q) is the Fourier transform of the interaction potential U (r) of Yukawa or Sogami types,relations (3) or (5). There, SHS (q) is the HS structure factor and β = 1/kBT .

First, we have made a comparison between the computed structure factor SRPA (q) with a Yukawapotential and the HS one SHS (q). We have found that these coincide in all q-range (Figure 2), for thespecial value ηmin = 0.37726 of the packing fraction minimizing the free energy, relation (14).

Figure 2. Structure factor with a Yukawa potential calculated by a variational approach, compared to the HS one.

In the case of a Sogomi potential, we found that the two structure factors coincide for the valueηmin = 0.380449 (Figure 3).

Figure 3. Structure factor with a Sogami potential calculated by a variational approach, compared to the HS one.

However, there is some deviation just around the principal peak. As a matter of fact, this is natural,because of the presence of a well-potential, which is the attractive part of Sogomi potential.

In Figure 4, we superpose the RPA structure factors with Yukawa and Sogami potentials. We note thatthe Sogami curve remains all the time above that of Yukawa. Indeed, this can be attributed to the fact thatthe repulsive part of Sogami potential is more important than the purely repulsive Yukawa potential.

19


Figure 4. Structure factors with a Yukawa and Sogami potentials calculated by a Variational Method, compared tothe HS one.

Having determining the structural properties of the colloidal solution of polyballs, the next step consistsin investigating the thermodynamic properties, namely, internal energy, pressure and compressibility.We start with the pressure, for the hard sphere fluid (HS), the atoms (or molecules) are describedby hard spheres of diameter σ , and its equation of state (EOS) is given with a high accuracy by theCarnahan-Starling equation (Carnahan and Starling, 1969) (CS) formula [41]

pCS

nkBT=

1+η +η2−η3

(1−η)3 , (26)

Whereη is the packing fraction, relation (12). From the above formula, we deduce the CS compress-ibility formula [14]

nkBT χCS =(1−η)4

1+2η +η3 (η−4). (27)

The internal energy E can be computed using its standard expression

E−E0

nkBT=

η (2−η)

(1−η)2 +2πn∫ +∞

σ


where the actual pair-correlation function g(r) was replaced by the HS one gHS (r). There, we have, E0 = 3kBT

/2, which is the energy of one particle of an ideal gas. Figure 5 shows the internal energy

shift ∆E = E−E0 versus the packing fraction η , with Yukawa and Sogami potentials. This energy shiftexhibits a minimum-point at ηmin.

Table 1 summarizes the values of pressure, compressibility and internal energy at minimum ηminwithYukawa and Sogami potentials, respectively.

Table 1. Thermodynamic proprieties for Yukawa and Sogami potentials, within the variational method.

Types of potential ηminPCS/nkBT nkBT χCS ∆E

/nkBT

Yukawa Potential 0.37726 6.0699 0.05219 2.56692

Sogami Potential 0.380449 6.18191 0.05078 2.566326

20


Figure 5. Reduced internal energy shift β∆E = β (E−E0) versus the packing fraction η , with Yukawa and SogamiPotentials.

3.2 Integral Equation Method Results

The HMSA integral equation is applied here for accomplishing an alternative computation of structuraland thermodynamics properties of the colloidal solution under investigation. Potentials used here are ofYukawa or Sogami types, and the choice of mixing function f (r) is that pointed out in [40].

First, we have computed the main object that is the pair-correlation function g(r) versus the renormal-ized interparticle distance r/σ . Figure 6 shows a superposition of two curves that are relative to Yukawaand Sogami potentials. The important remark is that, the height of the peak of g(r) is more pronouncedfor Sogami potential than of Yukawa type. Indeed, this can be understood by the fact that the colloidsystem with Sogami interaction is denser than that governed by Yukawa potential.

Figure 6. Comparaison of the correlation function with Yukawa and Sogami potentials computed by HMSA integralequation [42].

Second, we have reported in Figure 7 the pair-correlation function for a Sogami potential, together withthat computed using the MC simulation [43]. In fact, the two curves are in good quantitative agreement.

In Figure 8, we have reported the structure factors versus the renomalized wave-vector qσ computedusing variational, integral equations and MC methods. Notice, first, that the results obtained withinthe second and the third methods are in good agreement as in the case of the pair-correlation functiondiscussed above. But, the result derived within the framework of the former is slightly different, due to its

21


Figure 7. Correlation function with a Sogami potential using HMSA integral equation theory and MC simulation[42].

mean-field character.

Figure 8. Structure factors with a Sogami potential, obtained within variational and HMSA integral equation theo-ries, and MC simulation.

In the case of Yukawa potential (Figure 9), we have reported the structure factors versus the renormal-ized wave-vector computed within the three methods described above.

In fact, the same conclusions can be drawn. The only difference is that, the height of the peak is stronglydeviated. 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, fortwo values of the interpolation constant f0.

Table 2. Thermodynamic proprieties for Yukawa and Sogami potentials, within HMSA integral equation method.

Sogami Potential Yukawa Potential

fo nkBT χ E/nkBT P/nkBT fo nkBT χ E/nkBT P/nkBT

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

0.470 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

22


Figure 9. Structure factors with a Yukawa potential, obtained within variational and HMSA integral equation theories,and MC simulation.

For a small increase in density as they should and the energy and pressure increase, whereas thecompressibility 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 to the attraction has the potential to Sogami,the pressure is very important in the case the Yukawa potential which is a purely repulsive character.

4. COMPARISON WITH EXPERIMENT

In this section we present the our calculations, we have considered the same values of parameters as inexperiment by Tata et al. [24] .

Table 3. The details of the simulated colloidal systems.

σ (A) T(K) ε Z ni/1021 (m−3) np/1018 (m−3) Γ κσ

1090 298 78 600 1.75 1.33 2537 0.558

We find that the structure factor computed within the framework of the HMSA integral equation agreeswith the measured one, in all q-range Figure 10.

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 fact that this potentialignores the attractive interaction that exists within the sample Figure 11.

5. CONCLUSIONS

We recall that the purpose of this paper is the determination of the structure and thermodynamics of amonodisperse colloidal solution. We assumed that the interaction potential between colloids is of Yukawaor 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, first, the variational method with rescalingtechnique, and second, the integral equation one with HMSA. We first compared the results relative to

23


Figure 10. Comparison between experimental structure factor within variational and HMSA integral equation theo-ries, with a Sogami potential.

Figure 11. Comparison between experimental structure factor within variational and HMSA integral equation theo-ries, with a Yukawa potential.

these theories, and afterwards to those obtained within MC. We have shown that results from integralequation method and those of MC are in good quantitative agreement. Finally, our theoretical resultsare compared to those of a recent experiment. We found that integral equation theory agrees with thisexperiment.

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 (MetzUniversity) for their kind hospitality during their regular visits.

24


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