dr khalid elhasnaoui 3

SOP TRANSACTIONS ON THEORETICAL PHYSICSISSN(Print): 2372-2487 ISSN(Online): 2372-2495

DOI: 10.15764/TPHY.2014.03008Volume 1, Number 3, September 2014

SOP TRANSACTIONS ON THEORETICAL PHYSICS

Phase Diagram of Colloids Immersed inBinary Liquid MixturesM. Badia1, K .Elhasnaoui3*, A. Maarouf3, T.El hafi3, M. Benhamou2,3

1 Royale Air School, Mechanical Dept, DFST, BEFRA, P.O.Box 40002, Menara, Marrakech2 ENSAM ,Moulay Ismail University P.O.Box 25290, Al Mansour, Meknes3 LPPPC, Sciences Faculty Ben M’sik, P.O.Box 7955, Casablanca, (Morocco)

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

Abstract:We consider an assembly of spherical colloids of radius R immersed in a A — B binary liquidmixture close to its consolute point Tc It is assumed that particles prefer to be surrounded byone of the two components (A). Fluctuations of composition imply a reversible flocculation ofthese particles in the non-preferred B-rich phase, which originates from a universal long-rangeattractive Casmir potential with a well established decay r�2x

y (the critical exponent 2xy

is about1.03). The aim is a quantitative study of the phase diagram shape of this colloid System,using the random-phase approximation with a hard-sphere reference. The phase diagramis drawn in the h � t plane, where h is the packing fraction and t = (T �Tc)/Tc the reducedshift of temperature T from the consolute point Tc. The problem is governed by three relevantparameters, which are the packing fraction h, the temperature shift t and an energy strengthu(attraction energy between two particles separated by a distance that is equal to the hard-sphere diameter s per kBT unit). This energy is given by an explicit universal function dependingonly on the ratio of the real particle diameter b = 2R to s . We first determine the coordinates(h⇤, t⇤) , of the critical point, where t⇤ is the temperature shift at the critical temperature t⇤. Toa given value of the r�2x

y energy strength u corresponds a value of the temperature shift t⇤.All possible points (t⇤,�u) constitute a continuous critical line, along which the colloid Systemundergoes a phase transition from liquid state to gas state. We show that this curve is describedby a universal equation, i.e. �u ⇠ (t⇤)gt where gt is the critical exponent characterizing thebehavior near the consolute point, of the compressibility of binary liquid mixture without colloids.Second, we determine the exact from of the spinodal curve in the (h/h⇤, t/t⇤)�plane. We showthat this curve is universal. Third, we determine exactly the state equation of colloid fluid and putit on a universal form. We finish by drawing the complete phase diagram involving coexistenceand spinodal curves in the (h/h⇤, t/t⇤)�plane.

Keywords:Colloids; Hard Sphere Casimir Potential; Phase Diagram

108

Phase Diagram of Colloids Immersed in Binary Liquid Mixtures

1. INTRODUCTION

We start by considering a binary mixture made of two incompatible liquids A and B. At some criticaltemperature (consolute point), the mixture phase separates. From a thermo-dynamic point of view,phase separation is a real phase transition, which is characterized by the appearance of macrodomainsalternatively rich in A and B liquids. The size of these domains is the thermal correlation length x ,which becomes divergent when the consolute point is reached, i.e. x ⇠ a |T �Tc|�nt . Here, a is atypical microscopic length and nt a critical exponent. Phase separation is thus accompanied by long-wavelength fluctuations extending over spatial distances of order of x . These fluctuations are responsiblefor phenomena investigated below.

The real System we consider here consists of an assembly of inert colloidal particles immersed ina binary liquid mixture near its consolute temperature Tc. Colloids are particles of mesoscopic size,which are the subject of numerous theoretical and experimental studies, because of their abundantindustrial applications. To simplify, colloids will be assumed to be spherical of the same diameter b = 2R(monodisperse System). Thus, we are in the presence of a critical binary liquid mixture in contact withspherical surfaces.

The physics of phase separation of binary liquid mixtures in contact with interacting surfaces is a veryrich problem. For example, near the consolute point, colloids have tendency to be surrounded by onecomponent (A) of the mixture [1]. Colloids clothed by A-liquid located in the B-rich sida reversiblyflocculate duo to the presence of long-wavelengtli fluctuations of composition. Quantitatively, thisflocculation results from a universal long-range attractive Casimir potential with a well established decayr�2x

y [2], where the critical exponent 2xy

is about 1.03 (see below). This problem was discussed in ashort note by Fisher and de Gennes [3]. Very recently, in a series of experiments [4, 5], one has consideredsilica beads with diameter about 0.1µm immersed in lutidine (A)-water (B) binary mixture. Near theconsolute point, it was found that the silica colloids exhibit a sharply defined reversible flocculation. Thosecolloids prepared using the St?ber method are known to adsorb on lutidine preferentially [4]. Aggregationtakes place in the water rich-side of the phase diagram.

The purpose of this work is the determination of the phase diagram shape of an assembly of sphericalcolloids of the same diameter b = 2R immersed in a binary liquid mixture close to the consolute point.

These colloidal particles attract each other by an attractive Casimir potential, whose form will beprecised below. Here, the effective medium (solvent) is the non-preferred phase (B). Use will be madeof the random-phase approximation (RPA) [6] with a hard-sphere reference. The phase diagram willbe drawn in the h � t plane, where h is the packing fraction and t = (T �Tc)/Tc the reduced shift oftemperature T from the consolute temperatureTc. As we will see below, three relevant parameters governthe physics of the problem, namely, the packing fraction h , the temperature shift t and an energy strengthu, which is the dimensionless attraction energy between two particles separated by a (minimal) distance athat is equal to the hard-sphere diameter. As shown below, the energy strength u is an explicit universalfunction depending only on the ratio of the real particle diameter b = 2R to s .

Our findings are the following. Within the framework of RPA, we first determine analytically thecoordinates (h ⇤ , t⇤) of the critical point, which is the top of the coexistence curve of colloid System.Here, t* is the temperature shift at the critical temperature T⇤ that should not be confused with consolutepoint Tc. To a given value of the energy strength u (or equivalently of the ratio b/s ) corresponds a valueof the temperature shift t*. All possible points (t⇤,�u) constitute a critical line (C), along which thecolloid system undergoes a phase transition from a liquid state to a gas state. We show that this curve is

109


described by a universal equation, that�u ⇠ (t⇤)gt , where gt is the critical exponent defining the behaviornear the consolute point of the compressibility relative to the binary mixture without colloids. Second,we determine the exact form of the spinodal curve in the (h /h*, t/t*)-plane. We show that this curve isuniversal. Finally, we determine the parametric equations defining the coexistence curve and draw thecomplete phase diagram in the (h/h⇤, t/t⇤)�plane.

The presentation of the paper is planned as follows. Sect. 2 is devoted to a succinct recall of theattractive Casimir potential expression. We present in Sect. 3 the RPA formulation of the problem. InSect. 4, we investigate the exact shape of the phase diagram within the framework of RPA. We draw ourconclusions in Sect. 5.

2. THE CASIMIR INTERACTION POTENTIAL

We first start by recalling some useful backgrounds. Consider a binary mixture made of two incompa-tible liquids A and B. We denote by fA and fB their respective compositions. For incompressible mixture,these compositions are related by the incompressibility condition, i.e. fA +fB = 1

To study the phase behavior near the consolute point, one introduces an order parameter y , which isthe composition fluctuation of one species, say A. It is defined as the distance between two points alongthe coexistence curve, that is

y =fA �fB

fA +fB· (1)

The critical behavior is usually investigated through the connected two-point correlation functiondefined as usual by [7–9]

G�r� r0

�=< y (r)y

�r0�>�< y (r)>< y

�r0�> · (2)

The notation < . > indicates the thermal average over ail possible configurations of the order parameter.The Fourier transform of the correlation function is directly proportional to the scattering intensity, whichcan be measured in a light of neutron scattering experiment.

We recall that the two-point correlation function behaves at small distances compared to the thermalcorrelation length x as [7–9]

G�r� r0

�' B

|r� r0|2xy

,��r� r0

�� x

�· (3)

The characteristic length x becomes singular when the consolute point is approached, i.e.

x ⇠ a |t|�nt , t =T �Tc

Tc, (4)

where a is a typical microscopic length, t the reduced shift of temperature from the consolute point Tc

and nt a thermal critical exponent. In relationship (3), By

is some known amplitude [2] and xy

the scalingdimension of field y , which is related to the traditional thermal critical exponent ht by [7–9]

110


xy

=12(d �2+ht) (5)

where d is the space dimensionality. The three-dimensional best value of exponent xy

was found to bex

y

' 0.515 [2] . Of course, the correlation function vanishes (exponentially) for distances much largerthan x .

Now, consider an assembly of N spherical colloids immersed in a binary liquid mixture close to itsconsolute point. We assume that these colloids have the same radius R. Thus, the polydispersity effectsare neglected. Recently, it was established that fluctuations of composition imply a long-range attractiveCasimir force [2] between particles located in the non-preferred phase, say (B). Each colloid is surroundedby the preferred phase (A).

Intuitively, the Casimir interaction potential Uc (r� r0) between two particles separated by a distance|r � r0| must be proportional to the two-point correlation function. As a matter of fact, one has shown theresult [2]

Uc (r� r0)kBT

=�Ey

R2xy G

�r� r0

�(6)

where R is the colloid radius and Ey

some known universal amplitude [2].

Since the values of the two-point correlation function are appreciable only for distances much smallerthan the length x , thon, the expected Casimir potential reads

Uc

kBT'�D

y

✓br

◆2xy

, (r < x ) . (7)

Here, b = 2R is the colloid diameter and Dy

a universal amplitude. From its expansion to first order ine = 4�d(4 is the critical dimension of the System) extracted from Ref. [2], we find for this amplitudethe three-dimensional value D

y

' 6.64. The result (7) was established using the so-called small-sphereexpansion combined with conformal invariance [2].

The above expression for the Casimir effective potential call several remarks. First, this potential has auniversal character, independently on the chemical details of system. Second, the value of exponent x

y

,at d = 3 implies a decay r�1.03of the Casimir potential for large separations. In fact, this power law isconform with the decay r�1as found by de Gennes using an approximate theory [10]. We note that theabove form of the Casimir potential is the same for two widely separated colloidal particles immersed ina one-component fluid near the liquid-vapor critical point. Finally, this result was extended to heliumat lambda temperature [2], whose scaling dimension x

e

(of composite field y

2 is xe

= d � 1/nt . Atthree-dimensions, one has x

e

' 1.51 [2]. This corresponds to a slow decay r�3.02 of the Casimir potentialwith an amplitude of order of unity.

In principle, we have all ingredients to investigate the phase diagram of colloids immersed in binaryliquid mixtures close to their consolute point, using the established form (7) of the effective interactionpotential. This is precisely the aim of the following section.

111


3. THE RPA FORMULATION

To determine the shape of the phase diagram, use will be made of the RPA method [6, 11]. It is knownthat this approach is reliable only for low particle density. To use it, we will need a reference potential.We choose the simpler one, which is the hard sphere-potential

U0 (r) =

(•, r � s

0, r � s

(8)

where s is the hard-sphere diameter s � b. By reference System, we mean a colloid System at thesame temperature T and bulk density r but in which the interaction potential is U0 (r).

Then, the effective potential governing the physics of the colloid System is the sum

U (r) =U0 (r)+Uc (r) , (9)

where Uc (r) is the long-range Casimir potential (s < r < x ) defined by Eq. (7). Beyond the thermal

Figure 1. Reduced interaction potential U (r)/kBT versus the reduced distance r/s .

correlation length, that is for r � x , the interaction potential goes exponentially to zero. The abovedecomposition means that, there is a competition between the Casimir potential favoring the flocculationof colloids and the hard-sphere potential favoring rather their dispersion. The shape of potential U (r) isdepicted in Figure 1. The spirit of the RPA consists of writing the direct correlation function c(r) of amonodisperse colloidal dispersion as

c(r) = co (r)�bUc (r) (10)

Here, b = 1/kBT and co (r) is the direct correlation function of reference system. The Fourier transformof the function c(r) is defined by

c(q) =Z

dr c(r) exp(iq.r) (11)112


where q is the wave-vector. This transform is related to the structure factor S (q) through the standardOrnstein-Zernike relation

S(q) = 1�r c(q) (12)

Fourier transforming relationship (10) and according to the above equality, one finds the followingexpression for the structure factor within the framework of RPA

S(q) = S0(q)⇥1+brS0(q)Uc(q)

⇤�1 (13)

Here, eUc (q) is the Fourier transform of Casimir potential, which is directly proportional to the scatteringintensity from a binary liquid mixture without colloids. We will need the hard-sphere direct correlationfunction expression at vanishing wave-vector. It is a function of the packing fraction h = (p/6)rs

3

given by [6]

r c0 =�8h +2h

2

(1�h)4 (14)

(Here, the parameter h should not be confused with the thermal critical exponent ht defining the criticalbehavior of compressibility of the A — B mixture without colloids).

We are now interested in the quantitative nature of the spinodal decomposition, which drives the Systemfrom small colloid density state (gas) to high colloid density one (liquid). The desired phase diagram isgenerally drawn in the (h ,T )� plane.

4. THE PHASE DIAGRAM

4.1 The Critical Point

The top of the spinodal curve is the critical point where the System undergoes a phase separation. Itscoordinates (h⇤,T⇤) can be determined solving the coupled equations

1� c(0) = 0, (15)

∂

∂h

[1� c(0)] = 0. (16)

Explicitly, we find

1+4h +4h

2 �4h

3 +h

4

(1�h)4 +br

eUc (0) = 0 (17)

8+20h �4h

2

(1�h)5 +b

6ps

3eUc (0) = 0. (18)

113


By subtraction, we obtain the critical packing fraction

h⇤ ' 0.128. (19)

Incidentally, note that, within the framework of RPA, the critical packing fraction is independent on theform of interaction potential.

To get the critical temperature T*, we will need the expression of eUc (0) appearing in System (17)-(18).We first set

u =� kBT Dy

✓bs

◆2xy

, (20)

which is the contact energy between two particles, i.e. energy at minimum separation r = s Expressionof quantity eUc (0) can be obtained integrating the potential given by relation (7) over the r-variable runningfrom s to x , to find

b Uc (0)' 4p u(2�ht)s

3✓

x

s

◆ht�2

, (21)

with the notation u = u/kBT. We have used the scaling relations: d �2xy

= gt/nt and gt = nt (2�ht)

[7–9].

Near the consolute point Tc, we have the scaling law x ⇠ s t�nt , with an amplitude of order unity.This implies that the quantity b

eUc (0) defined by Eq. (21) scales as

b Uc (0)' 4p u(2�ht)s

3t�gt , (22)

where t is the temperature shift defined above. Therefore, the contribution of the Casimir interactionscales as the isotherm compressibility (or susceptibility).

We are know all ingredients to determine the critical temperature T*. We first set

t⇤ =T �Tc

Tc, (23)

which is the corresponding temperature shift. In fact, to each value of the dimensionless energy strengthu corresponds a value of the critical temperature parameter t*. The choice of u is equivalent to give theratio of the particle diameter b to the hard-sphere one s . We have then a critical line along which thecolloidal System undergoes a phase transition, which will be drawn in the (t, -u)-plane. The analyticalshape of this line can be deduced from Eqs. (17) and (22). We find

� u ' A⇤ (t⇤)�gt , (24)

with the amplitude

A⇤ ' 0.138. (25)114


The critical line is thus governed by the thermal critical exponent gt whose best three-dimensional valueis gt ' 1.2411 [8]. We have used the best value of the critical exponent ht at d = 3, which is ht ' 0.031[8]. The shape of critical line (C) of Eq. (24) is depicted in Figure 2.

Figure 2. Critical line (C) in the (t ,�u )-plane.

This curve separates two domains I and II. The former corresponds to high energy strength, wherecolloids have tendency to flocculate (liquid state). Low energy strength domain II corresponds rather to adispersion of colloids (gas state).

4.2 The Spinodal Curve

The spinodal curve is given by Eq. (17) emanating from the divergence condition of the compressibilityof colloid System. Without details, we give the equation defining this curve in the (h , t)� plane

t = t⇤ [ f (h)/ f (h)]�1/gt , (26)

With

f (h) =1+4h +4h

2 �4h

3 +h

4

h (1�h)4 . (27)

Spinodal curve of Eq. (26) calls the following important remark. First, note that this can be rewrittenon the following form, in terms of reduced variables h/h⇤ and t/t⇤

t�t⇤ = g(h/h⇤) (28)

Where

g�h

�h⇤

�= f (h)

�(h⇤). (29)

115


Second, the function g(x) is universal, that is independent on the chemical nature of the problem, on onehand, and the geometric form of colloids (through their radius R), on the other hand. The non-universalityis completely contained in critical temperature parameter t*. The spinodal curve of Eq. (28) is shown inFigure 3.

Figure 3. Spinodal curve in the (h /h*, t/t*)-plane.

Now, it remains us to determine the state equation and coexistence curve of colloid System.

4.3 The State Equation

Such an equation can be derived from the relationship giving the isotherm compressibility kT of colloidfluid, i.e.

rkT = ∂r

�∂ P (30)

where P is the pressure and r the particle density. Within the framework of RPA, the isothermcompressibility is given by

kBT r kT = [1�r c(0)]�1 (31)

where c(0) is the Fourier transform of the direct correlation function at vanishing wavevector. Combin-ing Eqs. (30) and (31) and after integration over the r� variable, we find the following equation for thepressure

P = P0 +r

2eUc (0) (32)

where Po is the hard-sphere pressure that is given by the following h-depending function [6]

P0

r kBT=

1+h +h

2

(1�h)3 . (33)

116


Relationship (32) tells us that the attractive Casimir interaction yields a reduction of the pressure.

Introduce now the dimensionless energy parameter

w =� 6p s

3 ·eUc (0)kBT

. (34)

At the critical point, we find from Eq. (18) that the critical value w* of this parameter is a pure number,i.e.

w⇤ =8+20h

⇤+8(h⇤)2

(1�h

⇤)5 ' 21.205 (35)

We have use the fact that h

⇤ ' 0.128. Now, set

ep ⌘ P�rkBT (36)

that is a dimensionless pressure. Combing Eqs. (22), (24) and (32) to (35) to obtain the universal stateequation

p= j (h , t) (37)

with the notation h ⌘ h/h⇤, t ⌘ t/t⇤. The two-factor universal scaling function j is found to be

j (h , t)=1+h

⇤.h +(h⇤)2 .h2

(1�h

⇤.h)3 � w⇤h

⇤

2· h · (t)�gt (38)

All non-universality is entirely contained in the critical temperature shift t* (throught). By universality,we mean that this state equation within the RPA is independent on the chemical nature of A and B-components and the radius R of colloids.

4.4 The Coexistence Curve

To determine the shape of such a curve, we will need the chemical potential expression. The startingpoint is the classical Gibbs-Duhem relation, i.e.

✓∂P∂r

◆

T= r

✓∂ µ

∂r

◆

T, (39)

where P is the pressure given by relationship (32). Integration of Eq. (39) over the r-variable yields

µ

kBT=

µ0

kBT�w⇤ ·h(

tt⇤)�gt (40)

with

117


µ0

kBT=

1+h +h

2

(1�h)5 +3

2(1�h)3 + ln(h

1�h

) (41)

which defines the hard-sphere chemical potential µ0. Universal relationship (40) shows a chemicalpotential reduction due to the attractive interaction between colloidal particles.

The analytical form of the coexistence curve can be determined through the equality of pressures andchemical potentials at coexistence, i.e.

P(hI) = P(hII) , (42)

µ (hI) = µ (hII) , (43)

where P is the pressure given by Eqs. (36) to (38) and µ the chemical potential defined by Eqs. (40)and (41). There hI and hII stand for the packing fractions of the two phases at coexistence. We show thatthe above equalities yield the relationships

h✓

hI

h⇤ ,hII

h⇤

◆= 0, (44)

tt⇤ = k

✓hI

h⇤ ,hII

h⇤

◆, (45)

where h (x, y) and k(x, y) are known two-factor universal scaling functions. These parametric equationsgive the coexistence curve.

The complete phase diagram involving coexistence and spinodal curves is shown in Figure 4.

Figure 4. Complete phase diagram shape in the (h /h*, t/t*)-plane.

118


5. CONCLUSIONS

We considered an assembly of spherical colloids of the same radius R immersed in a binary liquidmixture close to its consolute point Tc. We assumed that particles prefer to be surrounded only by onecomponent (preferred phase). Long-wavelength fluctuations of composition imply a reversible flocculationof colloids in the other side of the phase diagram (non-preferred phase). This flocculation originates froma universal long-range attractive Casimir potential with a well established decay r�1.03. The aim was aquantitative study of the phase diagram structure of these colloids, within the framework of RPA with ahard-sphere reference. The phase diagram is drawn in the h — t plane, where h is the packing fractionand t = (T — Tc) /Tc the reduced shift of temperature T from the consolute point Tc. Three relevantparameters govern the physics of the problem, which are the packing fraction h , the temperature shift tand a dimensionless energy strength u, which is the attraction energy between two particles separated bya (minimal) distance that is equal to the hard-sphere diameter s , per kBT unit. This energy is an explicituniversal function depending only on the ratio of the real particle diameter b = 2R to s . We have firstdetermined the coordinates (h*, t*) of the critical point, where h* is the critical packing fraction and t*the temperature shift at the critical temperature T*. To a given value of the energy strength u correspondsa value of the temperature shift t*. All possible points (t*,�u) constitute a critical line (C), along whichthe colloid System undergoes a phase transition from a liquid state to a gas state. We have shown that,such a curve can be described by a universal equation, i.e. �u ⇠ (t⇤)gt , where gt is the critical exponentdefining the behavior near the consolute point of the compressibility of the binary liquid mixture withoutcolloids. Second, we have determined the exact form of the spinodal curve in the (h/h*, t/t*)-plane.We have shown that this curve is universal. Also, we determined the parametric equations defining thecoexistence curve and drawn the complete phase diagram in the (h /h*, t/t*)-plane.

We emphasize that the present analysis can be extended, in a straightforward way, to helium at lamda-point, substituting the critical exponent x

y

' 0.515 by xe

' 1.51 and the amplitude Dy

' 6.64 appearingin Eq. (7) by D

e

' 1.

Extension of the present studies to charged colloidal particles immersed in critical binary liquid mixturesis in progress.

ACKNOWLEDGMENTS

We are much indebted to Professors M. Daoud and J.-L. Bretonnet, for helpful discussions.

References

[1] R. R. Netz, “Colloidal flocculation in near-critical binary mixtures,” Physical Review Letters, vol. 76,no. 19, p. 3646, 1996.

[2] T. W. Burkhardt and E. Eisenriegler, “Casimir interaction of spheres in a fluid at the critical point,”Physical Review Letters, vol. 74, no. 16, p. 3189, 1995.

[3] M. Fishor and P. De Gennos C. R. Acad. Sci, Paris B, vol. 287, p. 207, 1978.[4] D. Beysens and D. Esteve, “Adsorption phenomena at the surface of silica spheres in a binary liquid

mixture,” Physical Review Letters, vol. 54, no. 19, p. 2123, 1985.

119


[5] M. L. Broide, Y. Garrabos, and D. Beysens, “Nonfractal colloidal aggregation,” Physical Review E,vol. 47, no. 5, p. 3768, 1993.

[6] J. Hansen and I. McDonald, “Theory of Simple Liquids,” 1976.[7] J. Zinn-Justin, “Quantum Field Theory and Critical Phenomena,” 1989.[8] C. Itzykson and J.-M. Drouffe, Statistical Field Theory: 1 and 2. Cambridge University Press, 1989.[9] D. Amit, Field Theory, Renormalization Group and Critical Phenomena. McGraw-Hill, New York,

1978.[10] P. De Gennes C. R. Acad. Sci., Paris II, vol. 292, p. 701, 1981.[11] P. Grimson, “This method was successfully applied to charged colloids,” J. Chem. Soc.Faraday

Trans., vol. 279, p. 817, 1983.

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