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The surface of tension, the natural radius, and the interfacial tension in the thermodynamics of microemulsions
The Surface of Tension, the Natural Radius, and the Interfacial Tension in the Thermodynamics of Microemulsions E. RUCKENSTEIN 1 Institut fiir Physikalische Chemie 1 der Universitdt Bayreuth, Postfach 3008, D-8580 Bayreuth, West Germany Received October 28, 1985; accepted January 29, 1986 The possibility of the existence of a surface of tension for rnicroemulsions is examined. Its choice as the dividing surface has the advantage that a single characteristic o f the surface, namely the interfacial tension, can be used in the differential of the free energy. One concludes that such a surface may exist if the free energy Afdue to the entropy of dispersion of the globules in the continuous phase depends only on the combination 4J/r3 of the "actual " radius r and volume fraction ~ of the dispersed phase. The available expressions for Afdo not satisfy this requirement, even though the terms containing only this group provide the main contribution. However, a less constrained surface of tension could be identified whose radius is equated to the "natural radius." This is employed to relate the interfacial tension a between microemulsion and excess dispersed phase to the actual radius and volume fraction of the globules. © 1986AcademicPress,Inc. 1. INTRODUCTION
In order to formulate the theory of capillarity in its greatest simplicity, Gibbs has introduced the concept of dividing surfaces. The surface for which there is no explicit dependence of the differential of the free energy on the surface curvature was called surface of tension. Much of the traditional thermodynamics was formulated by using the surface of tension as the dividing surface. This has the advantage that a single characteristic of the surface, namely the interfacial tension, can be used in the expression of the differential of the free energy. In contrast, any other choice involves, in addition, two bending stresses associated with the two principal curvatures of the interface (see Eq. [4]). Gibbs and later Buff (1), Hill (2), and Kondo (3, 4) have been concerned with various dividing surfaces and with the corresponding Gibbs adsorption equation. A lucid presentation of the subject was provided recently by Rusanov (5). Kondo's pro-
cedure, which introduces the surface of tension by means of virtual displacements of the dividing surface, is particularly illuminating. He starts from the Helmholtz free energy F of a two phase system in equilibrium, with a spherical interface between them, F = "yS+ Z # i N i - p " V , -paVa,
[1]
where #i and Ni are the chemical potential and the number of molecules of species i, 3' is the interfacial tension, V, = 4a-r3, S = 4a-r2, r is the radius of phase a, and V, is the volume of phase/~. A virtual shift of the radius r, with the constraint that the physical characteristics of the system, F, ],ti, Ni, pa pt3, and V, + Va, should not change, leads to
t On leave from Department of Chemical Engineering, State University o f New York, Buffalo, New York 14260 as a Humboldt Award Winner.
p~ p~=23" 403" r Or"
[2]
The surface of tension corresponds to a Laplace-type relation between p" and p~, hence to that value of r for which 03"/Or = 0. The interfacial tension 3' has a minimum for this value of r. One can show that the two bending stresses (which for a spherical interface are equal) are in this case equal to zero.
173
Journalof Colloidand InterfaceScience, Vol. 114,No. 1, November1986
0021-9797/86 $3.00 Copyright© 1986by AcademicPress,lnc. All rightsof reproductionin any formreserved.
174
E. RUCKENSTEIN
Can one extend the surface of tension based thermodynamics of two phases separated by a specified interface to microemulsions? Such a possibility will have the advantages mentioned above in connection with the traditional thermodynamics. Obviously, in the thermodynamics of microemulsions additional complexities arise because the interface is not given but is itselfa result of the condition of thermodynamic equilibrium. In other words, assuming spherical globules, the radius of the globules is also provided by the condition of the minimum of the free energy. The surface of tension approach might be inappropriate for microemulsions, either because it is inconvenient (the radius corresponding to the surface of tension being, for instance, too different from the actual radius), or because it is incompatible with other constraints specific to microemulsions. It will be shown that the expressions which are employed for the entropy of dispersion of the globules in the continuous phase are incompatible (or only approximately compatible) with the existence of a surface of tension. However, a less constrained surface of tension provides an expression for the "natural radius" which is useful in calculating the energy associated with the bending of the interface. This expression is further employed to derive an equation which relates the interracial tension between microemulsion and the excess dispersed phase to the actual radius of the globules of the dispersed phase and their volume fraction. Because the present contribution is based on the two scale thermodynamics ofmicroemulsions (6, 7), the relevant equations are summarized in Section 2. Following this, the extension of the surface of tension approach to microemulsions is discussed by using the virtual shift method as well as the method based on the main differential equation for the variation of the free energy. Section 4 is concerned with the introduction of a less constrained surface of tension whose location provides an expression for the natural radius. The latter quantity is employed to establish an equation for the Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
interfacial tension between microemulsion and the excess dispersed phase. 2. THERMODYNAMIC EQUATIONS
In what follows one assumes that the microemulsion contains spherical globules of uniform radius and one denotes by r the "actual" radius of the globules. The dispersion of these globules in the continuous phase is accompanied by an increase in the entropy of the system and Afdenotes the corresponding free energy change per unit volume of microemulsion. Expressions for Afare available (8, 9) and are used later in the paper (Eqs. [21] and [22]). Two length scales characterize a microemulsion. A microlength of the order of the size r of the globules and a macrolength, on the scale of the microemulsion, which is large compared to the size of a globule. To better clarify this point let us note that a single molecule in a gas does not sense the pressure and entropy of the gas, which constitute characteristics of a large number of molecules. Being a macroscopic body, a single globule in a microemulsion possesses its own pressure P2 and senses the pressure Pl in the continuous phase in the space between the neighboring globules. It does not feel, however, the macro (thermodynamic) pressure p which includes, in addition to pl, an osmotic contribution due to the entropy of dispersion of the globules in the continuous phase. Obviously, the latter effect is felt only on the macroscopic scale and involves a large number of globules. This suggests separating the free energy Afdue to the entropy of dispersion of the globules in the continuous phase by writing the free energy f per unit volume of microemulsion as f = f o + zXf
[3]
The micropressures P2 and Pl should be defined on the basis off0 and the thermodynamic (macro) pressure p on the basis o f f According to Gibbs, the variation at constant temperature T of the Helmholtz free energy J; is given by
THERMODYNAMICS
175
OF MICROEMULS1ONS
f=q/A+ ~ u i n i - p z 4 , - P ~ ( 1 - 4 , ) + A f
dfo = 7dA + C~dcj + C2dc2+ ~uidn~
[8]
Let us now consider an imaginary displacement of the dividing surface by virtual changes where 3, is the interfacial tension, A is the inof r and ~. Of course, the physical quantities terfacial area per unit volume of microemulf t~i, ni, Pl, P2, and Afshould not be affected sion, ~i is the chemical potential of species i, by these changes. Consequently, n~ is the number of molecules of species i per df = 3,dA + Ad3,-(p2-pDd4,=O, [9] unit volume of microemulsion, P2 is the pressure inside the globules, Pt is the pressure in where the continuous phase in the neighborhood of O3' Oy d3, =~rdr+-~d4~ [10a] a globule, 4, is the volume fraction of the globules, C~ and 6"2 are bending stresses and c~ and c2 are the principal curvatures. The radius r and, because A = 34#r, and the volume fraction 4~are the actual ones. [10b] dA = 3r d ~ - - ~34~- dr " For spherical globules, c1 = (?2 = 1/r and Cl = Cz = C/2, and Eqs. [3] and [4] lead to Here r and 4, are not independent quantities df = 3,dA + Cd( I/r) + ~ uidni because of the constraint that k f should be -p2d4,-pld(1-4))+dAf [51 invariant when the dividing surface is shifted. Equation [9] can be rewritten as The minimum of the free energy with respect =23, +~[1 34,dr ~ to r and 4, provides the equations (6, 7)
-p2d4,-p~d(1-O),
r2 (0 S /
[4]
P2 --171
r
t" \
r d~b]
c
3' = 3da\ Or ]~ 34,
+34, {03, dr
[6]
and ( 0 V / + r (OZ~
P2-P'=\-~],
C
g~-~-r )~ r4,"
[71
Because
--'~-F ]m = ~-'~'-r ]4a T \
O~ It'
_~(1 3chdr\
where m is the number of globules per unit volume, Eq. [7] can be rewritten, with the help of Eq. [6], in the more revealing, generalized Laplace form
23, C r [OAI~ p 2 - p , = r 34,r + ~-~/T~-r )m.
[7a]
Equation [Ta] in which the derivative of Af with respect to r is carried out at constant m will be useful in the next section. 3. T H E
SURFACE
The location of the surface of tension is provided by that value of r for which Laplace equation holds. Equation [11] acquires the Laplace form for those values of r and ~ for which
OF TENSION
Integrating Eq. [5] at constant 7, 1/r, #i, P2, and p~, one obtains
34,[03'dr
03,\
The first term in [ 12] is due to a change in the number of globules, while the second is a result of a change in curvature. Indeed, the number of globules m per unit volume of microemulsion is given by
m = ch/(47rr3), from where one obtains
dm
Omdr
am
3 (1
-d-~=-&-rd~+-O-~-=4--~3r3.
3q~dr] -Td-~]"
[13]
Since m should not change with the shift in the dividing surface, one has Journalof ColloidandInterfaceScience,VoL 114, No. 1, November 1986
176
E. RUCKENSTEIN m = const.
[ 14a]
In such cases, the equations
and Eq. [ 12] becomes
d3, _ O~ dr +O~ =0. d4) Or dd~ Oqb
C
'~'- ~
[20]
r t3
[151
More insight can be gained by employing Eq. [7a], instead of the equivalent, more formal expression [ 11 ]. It is clear that Eq. [7a] acquires the Laplace form for those values of ~b and r for which r /02xf~
3ekr ~-~--~t--~r ), :O.
[ 191
[14b]
There is, however, another constraint on r and 4~which results from the condition of invariance of Af(r, 4). Because Afis a function of r and q~ (see Eqs. [21] and [22]) and rn is proportional to 4)/r 3 this condition is compatible with [ 14a] only if
Af(r, rb) = Af(ck/r3).
C(rt, q~t)= 0 and
[16a]
r 3
can, in principle, be used to obtain the values of rt and 4h that locate the surface of tension in terms of the actual radius r and volume fraction ~b. As a result, in such cases, the thermodynamics of microemulsions may be formulated as in the traditional surface of tension approach, using a fundamental equation of the same form as Eq. [5] without the term containing C. Several expressions have been proposed for the free energy Afdue to the entropy of dispersion of the globules in the continuous phase. One of them, derived on the basis of a lattice model, has the form (8)
34~kT[
-~ ln(1
-
141rr3 1
~b)+
O f course, one must have, in addition, m = const.
[21] [ 16b]
However, the constraint that Afshould be an invariant is compatible with Eq. [16b] only if Afhas the form of Eq. [15]. In such a case, Eq. [ 16a] reduces to the condition C=0,
[171
which constitutes the conventional condition for the location of the surface of tension. 2 Consequently, a surface of tension based on m = const, can exist if Af = Af(rk/r3).
[18]
2One can at least in principle avoid the condition m = const. Then, either Eq. [16a] or Eq. [17] could be associated with the condition of invariance of Afto choose a dividing surface. Even though these fictitioussystems, particularly the one based on Eq. [17], may simplifythe thermodynamic formalism (if solutions for the fictitious radius and volumefractionexistin the physicaldomain), they may, however,also obscure the physicalproblem. Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
whereas another, based on the CarnahanStarling approximation for hard spheres, has the form (9) 3~bkT[1
_
4-3~b
_ /47rr3\]
-~+ln~3-~)J'
[22]
where vc is the molecular volume of the continuous phase, k is the Boltzmann constant, and T is the absolute temperature. In neither of these expressions does Afdepend only on dp/r3, even though the main contribution to Af is provided by terms involving only that ratio. However, a somewhat different surface of tension could be identified. While it does not allow a thermodynamic formulation in which the variation of the free energy is free of the surface curvature, it provides an equation for the so-called "natural radius," which is useful in the calculation of the interfacial tension between a microemulsion and the excess dispersed phase. This problem is examined next.
THERMODYNAMICS OF MICROEMULSIONS 4. A LESS CONSTRAINED SURFACE OF TENSION APPROACH AND ITS USE IN THE CALCULATION OF THE INTERFACIAL TENSION As suggested previously (10), the interfacial tension e between microemulsion and excess dispersed phase can be related to the interracial tension 3" (which is defined at the "actual" surface of the globules) via the expression ~r = 3' - e,
[23]
where e is a free energy per unit area that accounts for the fact that the surfactant and cosurfactant are adsorbed on a curved interface. Equation [23] conveys that the interfacial tension a at the planar interface between the microemulsion and excess dispersed phase is equal to that at the curved interface minus the excess free energy generated by the bending of the interface. For spherical globules, e is evaluated using the expression proposed by Helfrich (12-14) e = 2K~-~0)
2,
[24]
where K is a constant (of the order of 0.1 eV) whose value is expected to depend upon the ratio of cosurfactant and surfactant, and R0 is the so called natural radius which accounts for the tendency of the interfacial layer to curve toward water or oil. It is important to note that Eq. [24] is approximate even for vesicles for which it was initially derived. The interfacial layer in a microemulsion is less rigid and as a result the elastic behavior involved in the derivation of Eq. [24] is open to question. It is expected the difference between the actual curvature 1/r and the curvature corresponding to a Laplace-type expression for P2 Pl to be a measure of the stresses generated by the bending of the interface. For this reason we suggest equating Ro to the radius R of the surface of tension, defined on the basis of Eq. [7a] as -
P2 --Pl -
177 23"(¢0,R) R
[25]
Here R and ¢0 are related via
c(R,¢o) R(o f I' 3¢0R = ~ o \'~F ]m'
[261
where the superscript prime indicates that r and ¢ should be replaced in the derivative by R and ¢0. Equation [26] is coupled with the condition m = const., which relates R and ¢0 to r and ¢ via ¢0_ ¢ R 3 r 3"
[27]
The surface of tension defined by Eqs. [26] and [27] is less constrained than that defined in the previous sections. The latter involved the additional constraint that Afshould be an invariant. The previous surface of tension (which exists only if A f = Af(¢/r3)) reduces the basic equation [5] to a form free of the term containing the bending stress C. The present surface of tension cannot produce such a simplification. It can, however, provide a meaningful radius which is useful in evaluating e. The interfacial tension 3, and the radius R can be expressed in terms of r and ¢ in the following way. The condition of thermodynamic equilibrium between microemulsion and the excess dispersed phase provides the equation (6, 7) 0A
;\-g-r] =0 t28]
Combining Eqs. [6], [22], and [28], one obtains (7)
kTr
3"=
/47rr3\
)
8 ¢ - 5 ¢ 2~_¢]
[29] 3
(1-¢)2
3Note that Eq. [29], which shows that 3' is almost inversely proportional to the square of the radius, with a proportionalityconstantweaklyr dependent,is verysimilar to an equation established by the author for the same quantity in 1981 (15). Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
178
E. RUCKENSTEIN with
and
3~bkT[ ,,,
ln/aTrr3 / C = 4~rr2 [ - z i n q b + 2 \3v¢}
~b0 _ q5 R3
-~ 6~b2 - 5q5 - ~ 3 ]
(1 - qS)2
" [301
Consequently, Eqs. [23], [24], and [29] lead to
kT [1 [4rrr3~ 8q~- 5~b2 + ~b]
nk377 ]
where R, obtained by combining Eqs. [26], [22], and [30], is given by 2
ln{47rR3] = 174~0- 24~b2 + 4~3 ~3--~o] (1 - ~bo)2 +
6~2(4 - 3q~o) (1 - ~bo)3
[321
TABLE I 107r (cm)
107R (cm)
3' (dyn/crn)
a (dyn/cm)
0,1
6 9 12 15 18
7.76 1 t.68 15.61 19.54 23.48
0.107 0.053 0.032 0.021 0.015
0.039 0.021 0.014 0.010 0,007
0.15
6 9 12 15 18
7.59 11.44 15.31 19.18 23.05
0.099 0.049 0.029 0.020 0.014
0,040 0,022 0.014 0,010 0.007
0.20
6 9 12 15 18
7.35 11.12 14.89 18.68 22.47
0.090 0.045 0.027 0.018 0.013
0.045 0.024 0.015 0.010 0.008
0.25
6 9 12 15 18
7.06 10.70 14.35 18.03 21.70
0.081 0.041 0.025 0.017 0.012
0.052 0.026 0.016 0.011 0.008
Journal of Colloid and Interface Science. Vol. 114,No. 1, November1986
[33]
r3 •
Because K is expected to be proportional to kT, Eq. [3 l] shows that ar2/kTis a weak function of r. 5. RESULTS
Equations [29], [31], and [32] have been used to compute 3', a, and R as a function of r and ~ for an oil in water microemulsion and the results are presented in Table I. The values obtained for a are of the fight order of magnitude, being in the range of 10-~ to 10-2 dyn/ cm which is provided by experiment (11). A detailed comparison with that experiment is not yet possible since data relating o- to r are not available. For reasons already mentioned, Eq. [24] is very approximate and probably provides only an upper bound for E. As a result, the calculated values probably give a lower bound for a. Of course, 3' provides an upper bound for a. At least two effects have been ignored in the above considerations: (1) the curvature contribution due to the double and hydrated layers, which can be relevant at relatively low ionic strength, and (2) the curvature contribution due to the strong organization of the water molecules in the vicinity of the internal interface (probably like in a liquid crystal) which probably occurs at high ionic strength because of the competition of the ions for water and interaction between the species involved (E. Ruckenstein, unpublished results). The curvature contribution arises because the above layers affect the interfacial tension of a planar and curved interface differently. Because the microemulsions of practical interest involve, in general, relatively large ionic strengths, the second effect may be significant and for the moment we can say that it probably increases the rigidity of the interfacial layer thus making Eq. [24] more plausible.
THERMODYNAMICS OF MICROEMULSIONS REFERENCES 1. 2. 3. 4.
Buff, F. P., J. Chem. Phys. 19, 1591 (1951). Hill, T. L., J. Phys. Chem. 56, 526 (1952). Kondo, S., J. Chem. Phys. 25, 662 (1956). Ono, S., and Kondo, S., in "Handbuch der Physik" (S. Fltigge, Ed.), Vol. 10. Springer, Berlin, 1960. 5. Rusanov, A. I., in "Modem Theory of Capillarity," p. 1. Akademie-Verlag, Berlin, 1981. 6. Ruckenstein, E., Chem. Phys. Lett. 98, 573 (1983); in "Macro- and Micro-emulsions," ACS Symposium Series 272 (D. O. Shah, Ed.), p. 21. Amer. Chem. Soc., Washington, D.C., 1985.
179
7. Ruckenstein, E., Fluid Phase Equilib. 20, 189 (1985). 8. Ruckenstein, E., and Chi, J. C., J. Chem. Soc., Faraday Trans. 2 71, 1690 (1975). 9. Overbeek, J. Th. G., Faraday Discuss. Chem. Soc. 65, 7 (1978). 10. Ruckenstein, E., Chem. Phys. Lett. 118, 435 (1985). 11. Pouchelon, A., Meunier, J., Langevin, D., Chatenay, D., and Cazabat, A. M., Chem. Phys. Lett. 76, 277 (1980). 12. Helfrich, W., Z. Naturforsch. C 28, 693 (1973). 13. Safran, S. A., Turkevick, L. A., and Pincus, P. A., J. Phys. (Parisj 45, L69 (1984). 14. Safran, S. A., J. Chem. Phys. 78, 2073 (1983). 15. Ruckenstein, E., Soc. Pet. Eng. J., 593 (1981).
Journal of Colloid and Interface Science, Vol, 114, No. 1, November 1986
In order to formulate the theory of capillarity in its greatest simplicity, Gibbs has introduced the concept of dividing surfaces. The surface for which there is no explicit dependence of the differential of the free energy on the surface curvature was called surface of tension. Much of the traditional thermodynamics was formulated by using the surface of tension as the dividing surface. This has the advantage that a single characteristic of the surface, namely the interfacial tension, can be used in the expression of the differential of the free energy. In contrast, any other choice involves, in addition, two bending stresses associated with the two principal curvatures of the interface (see Eq. [4]). Gibbs and later Buff (1), Hill (2), and Kondo (3, 4) have been concerned with various dividing surfaces and with the corresponding Gibbs adsorption equation. A lucid presentation of the subject was provided recently by Rusanov (5). Kondo's pro-
cedure, which introduces the surface of tension by means of virtual displacements of the dividing surface, is particularly illuminating. He starts from the Helmholtz free energy F of a two phase system in equilibrium, with a spherical interface between them, F = "yS+ Z # i N i - p " V , -paVa,
[1]
where #i and Ni are the chemical potential and the number of molecules of species i, 3' is the interfacial tension, V, = 4a-r3, S = 4a-r2, r is the radius of phase a, and V, is the volume of phase/~. A virtual shift of the radius r, with the constraint that the physical characteristics of the system, F, ],ti, Ni, pa pt3, and V, + Va, should not change, leads to
t On leave from Department of Chemical Engineering, State University o f New York, Buffalo, New York 14260 as a Humboldt Award Winner.
p~ p~=23" 403" r Or"
[2]
The surface of tension corresponds to a Laplace-type relation between p" and p~, hence to that value of r for which 03"/Or = 0. The interfacial tension 3' has a minimum for this value of r. One can show that the two bending stresses (which for a spherical interface are equal) are in this case equal to zero.
173
Journalof Colloidand InterfaceScience, Vol. 114,No. 1, November1986
0021-9797/86 $3.00 Copyright© 1986by AcademicPress,lnc. All rightsof reproductionin any formreserved.
174
E. RUCKENSTEIN
Can one extend the surface of tension based thermodynamics of two phases separated by a specified interface to microemulsions? Such a possibility will have the advantages mentioned above in connection with the traditional thermodynamics. Obviously, in the thermodynamics of microemulsions additional complexities arise because the interface is not given but is itselfa result of the condition of thermodynamic equilibrium. In other words, assuming spherical globules, the radius of the globules is also provided by the condition of the minimum of the free energy. The surface of tension approach might be inappropriate for microemulsions, either because it is inconvenient (the radius corresponding to the surface of tension being, for instance, too different from the actual radius), or because it is incompatible with other constraints specific to microemulsions. It will be shown that the expressions which are employed for the entropy of dispersion of the globules in the continuous phase are incompatible (or only approximately compatible) with the existence of a surface of tension. However, a less constrained surface of tension provides an expression for the "natural radius" which is useful in calculating the energy associated with the bending of the interface. This expression is further employed to derive an equation which relates the interracial tension between microemulsion and the excess dispersed phase to the actual radius of the globules of the dispersed phase and their volume fraction. Because the present contribution is based on the two scale thermodynamics ofmicroemulsions (6, 7), the relevant equations are summarized in Section 2. Following this, the extension of the surface of tension approach to microemulsions is discussed by using the virtual shift method as well as the method based on the main differential equation for the variation of the free energy. Section 4 is concerned with the introduction of a less constrained surface of tension whose location provides an expression for the natural radius. The latter quantity is employed to establish an equation for the Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
interfacial tension between microemulsion and the excess dispersed phase. 2. THERMODYNAMIC EQUATIONS
In what follows one assumes that the microemulsion contains spherical globules of uniform radius and one denotes by r the "actual" radius of the globules. The dispersion of these globules in the continuous phase is accompanied by an increase in the entropy of the system and Afdenotes the corresponding free energy change per unit volume of microemulsion. Expressions for Afare available (8, 9) and are used later in the paper (Eqs. [21] and [22]). Two length scales characterize a microemulsion. A microlength of the order of the size r of the globules and a macrolength, on the scale of the microemulsion, which is large compared to the size of a globule. To better clarify this point let us note that a single molecule in a gas does not sense the pressure and entropy of the gas, which constitute characteristics of a large number of molecules. Being a macroscopic body, a single globule in a microemulsion possesses its own pressure P2 and senses the pressure Pl in the continuous phase in the space between the neighboring globules. It does not feel, however, the macro (thermodynamic) pressure p which includes, in addition to pl, an osmotic contribution due to the entropy of dispersion of the globules in the continuous phase. Obviously, the latter effect is felt only on the macroscopic scale and involves a large number of globules. This suggests separating the free energy Afdue to the entropy of dispersion of the globules in the continuous phase by writing the free energy f per unit volume of microemulsion as f = f o + zXf
[3]
The micropressures P2 and Pl should be defined on the basis off0 and the thermodynamic (macro) pressure p on the basis o f f According to Gibbs, the variation at constant temperature T of the Helmholtz free energy J; is given by
THERMODYNAMICS
175
OF MICROEMULS1ONS
f=q/A+ ~ u i n i - p z 4 , - P ~ ( 1 - 4 , ) + A f
dfo = 7dA + C~dcj + C2dc2+ ~uidn~
[8]
Let us now consider an imaginary displacement of the dividing surface by virtual changes where 3, is the interfacial tension, A is the inof r and ~. Of course, the physical quantities terfacial area per unit volume of microemulf t~i, ni, Pl, P2, and Afshould not be affected sion, ~i is the chemical potential of species i, by these changes. Consequently, n~ is the number of molecules of species i per df = 3,dA + Ad3,-(p2-pDd4,=O, [9] unit volume of microemulsion, P2 is the pressure inside the globules, Pt is the pressure in where the continuous phase in the neighborhood of O3' Oy d3, =~rdr+-~d4~ [10a] a globule, 4, is the volume fraction of the globules, C~ and 6"2 are bending stresses and c~ and c2 are the principal curvatures. The radius r and, because A = 34#r, and the volume fraction 4~are the actual ones. [10b] dA = 3r d ~ - - ~34~- dr " For spherical globules, c1 = (?2 = 1/r and Cl = Cz = C/2, and Eqs. [3] and [4] lead to Here r and 4, are not independent quantities df = 3,dA + Cd( I/r) + ~ uidni because of the constraint that k f should be -p2d4,-pld(1-4))+dAf [51 invariant when the dividing surface is shifted. Equation [9] can be rewritten as The minimum of the free energy with respect =23, +~[1 34,dr ~ to r and 4, provides the equations (6, 7)
-p2d4,-p~d(1-O),
r2 (0 S /
[4]
P2 --171
r
t" \
r d~b]
c
3' = 3da\ Or ]~ 34,
+34, {03, dr
[6]
and ( 0 V / + r (OZ~
P2-P'=\-~],
C
g~-~-r )~ r4,"
[71
Because
--'~-F ]m = ~-'~'-r ]4a T \
O~ It'
_~(1 3chdr\
where m is the number of globules per unit volume, Eq. [7] can be rewritten, with the help of Eq. [6], in the more revealing, generalized Laplace form
23, C r [OAI~ p 2 - p , = r 34,r + ~-~/T~-r )m.
[7a]
Equation [Ta] in which the derivative of Af with respect to r is carried out at constant m will be useful in the next section. 3. T H E
SURFACE
The location of the surface of tension is provided by that value of r for which Laplace equation holds. Equation [11] acquires the Laplace form for those values of r and ~ for which
OF TENSION
Integrating Eq. [5] at constant 7, 1/r, #i, P2, and p~, one obtains
34,[03'dr
03,\
The first term in [ 12] is due to a change in the number of globules, while the second is a result of a change in curvature. Indeed, the number of globules m per unit volume of microemulsion is given by
m = ch/(47rr3), from where one obtains
dm
Omdr
am
3 (1
-d-~=-&-rd~+-O-~-=4--~3r3.
3q~dr] -Td-~]"
[13]
Since m should not change with the shift in the dividing surface, one has Journalof ColloidandInterfaceScience,VoL 114, No. 1, November 1986
176
E. RUCKENSTEIN m = const.
[ 14a]
In such cases, the equations
and Eq. [ 12] becomes
d3, _ O~ dr +O~ =0. d4) Or dd~ Oqb
C
'~'- ~
[20]
r t3
[151
More insight can be gained by employing Eq. [7a], instead of the equivalent, more formal expression [ 11 ]. It is clear that Eq. [7a] acquires the Laplace form for those values of ~b and r for which r /02xf~
3ekr ~-~--~t--~r ), :O.
[ 191
[14b]
There is, however, another constraint on r and 4~which results from the condition of invariance of Af(r, 4). Because Afis a function of r and q~ (see Eqs. [21] and [22]) and rn is proportional to 4)/r 3 this condition is compatible with [ 14a] only if
Af(r, rb) = Af(ck/r3).
C(rt, q~t)= 0 and
[16a]
r 3
can, in principle, be used to obtain the values of rt and 4h that locate the surface of tension in terms of the actual radius r and volume fraction ~b. As a result, in such cases, the thermodynamics of microemulsions may be formulated as in the traditional surface of tension approach, using a fundamental equation of the same form as Eq. [5] without the term containing C. Several expressions have been proposed for the free energy Afdue to the entropy of dispersion of the globules in the continuous phase. One of them, derived on the basis of a lattice model, has the form (8)
34~kT[
-~ ln(1
-
141rr3 1
~b)+
O f course, one must have, in addition, m = const.
[21] [ 16b]
However, the constraint that Afshould be an invariant is compatible with Eq. [16b] only if Afhas the form of Eq. [15]. In such a case, Eq. [ 16a] reduces to the condition C=0,
[171
which constitutes the conventional condition for the location of the surface of tension. 2 Consequently, a surface of tension based on m = const, can exist if Af = Af(rk/r3).
[18]
2One can at least in principle avoid the condition m = const. Then, either Eq. [16a] or Eq. [17] could be associated with the condition of invariance of Afto choose a dividing surface. Even though these fictitioussystems, particularly the one based on Eq. [17], may simplifythe thermodynamic formalism (if solutions for the fictitious radius and volumefractionexistin the physicaldomain), they may, however,also obscure the physicalproblem. Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
whereas another, based on the CarnahanStarling approximation for hard spheres, has the form (9) 3~bkT[1
_
4-3~b
_ /47rr3\]
-~+ln~3-~)J'
[22]
where vc is the molecular volume of the continuous phase, k is the Boltzmann constant, and T is the absolute temperature. In neither of these expressions does Afdepend only on dp/r3, even though the main contribution to Af is provided by terms involving only that ratio. However, a somewhat different surface of tension could be identified. While it does not allow a thermodynamic formulation in which the variation of the free energy is free of the surface curvature, it provides an equation for the so-called "natural radius," which is useful in the calculation of the interfacial tension between a microemulsion and the excess dispersed phase. This problem is examined next.
THERMODYNAMICS OF MICROEMULSIONS 4. A LESS CONSTRAINED SURFACE OF TENSION APPROACH AND ITS USE IN THE CALCULATION OF THE INTERFACIAL TENSION As suggested previously (10), the interfacial tension e between microemulsion and excess dispersed phase can be related to the interracial tension 3" (which is defined at the "actual" surface of the globules) via the expression ~r = 3' - e,
[23]
where e is a free energy per unit area that accounts for the fact that the surfactant and cosurfactant are adsorbed on a curved interface. Equation [23] conveys that the interfacial tension a at the planar interface between the microemulsion and excess dispersed phase is equal to that at the curved interface minus the excess free energy generated by the bending of the interface. For spherical globules, e is evaluated using the expression proposed by Helfrich (12-14) e = 2K~-~0)
2,
[24]
where K is a constant (of the order of 0.1 eV) whose value is expected to depend upon the ratio of cosurfactant and surfactant, and R0 is the so called natural radius which accounts for the tendency of the interfacial layer to curve toward water or oil. It is important to note that Eq. [24] is approximate even for vesicles for which it was initially derived. The interfacial layer in a microemulsion is less rigid and as a result the elastic behavior involved in the derivation of Eq. [24] is open to question. It is expected the difference between the actual curvature 1/r and the curvature corresponding to a Laplace-type expression for P2 Pl to be a measure of the stresses generated by the bending of the interface. For this reason we suggest equating Ro to the radius R of the surface of tension, defined on the basis of Eq. [7a] as -
P2 --Pl -
177 23"(¢0,R) R
[25]
Here R and ¢0 are related via
c(R,¢o) R(o f I' 3¢0R = ~ o \'~F ]m'
[261
where the superscript prime indicates that r and ¢ should be replaced in the derivative by R and ¢0. Equation [26] is coupled with the condition m = const., which relates R and ¢0 to r and ¢ via ¢0_ ¢ R 3 r 3"
[27]
The surface of tension defined by Eqs. [26] and [27] is less constrained than that defined in the previous sections. The latter involved the additional constraint that Afshould be an invariant. The previous surface of tension (which exists only if A f = Af(¢/r3)) reduces the basic equation [5] to a form free of the term containing the bending stress C. The present surface of tension cannot produce such a simplification. It can, however, provide a meaningful radius which is useful in evaluating e. The interfacial tension 3, and the radius R can be expressed in terms of r and ¢ in the following way. The condition of thermodynamic equilibrium between microemulsion and the excess dispersed phase provides the equation (6, 7) 0A
;\-g-r] =0 t28]
Combining Eqs. [6], [22], and [28], one obtains (7)
kTr
3"=
/47rr3\
)
8 ¢ - 5 ¢ 2~_¢]
[29] 3
(1-¢)2
3Note that Eq. [29], which shows that 3' is almost inversely proportional to the square of the radius, with a proportionalityconstantweaklyr dependent,is verysimilar to an equation established by the author for the same quantity in 1981 (15). Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
178
E. RUCKENSTEIN with
and
3~bkT[ ,,,
ln/aTrr3 / C = 4~rr2 [ - z i n q b + 2 \3v¢}
~b0 _ q5 R3
-~ 6~b2 - 5q5 - ~ 3 ]
(1 - qS)2
" [301
Consequently, Eqs. [23], [24], and [29] lead to
kT [1 [4rrr3~ 8q~- 5~b2 + ~b]
nk377 ]
where R, obtained by combining Eqs. [26], [22], and [30], is given by 2
ln{47rR3] = 174~0- 24~b2 + 4~3 ~3--~o] (1 - ~bo)2 +
6~2(4 - 3q~o) (1 - ~bo)3
[321
TABLE I 107r (cm)
107R (cm)
3' (dyn/crn)
a (dyn/cm)
0,1
6 9 12 15 18
7.76 1 t.68 15.61 19.54 23.48
0.107 0.053 0.032 0.021 0.015
0.039 0.021 0.014 0.010 0,007
0.15
6 9 12 15 18
7.59 11.44 15.31 19.18 23.05
0.099 0.049 0.029 0.020 0.014
0,040 0,022 0.014 0,010 0.007
0.20
6 9 12 15 18
7.35 11.12 14.89 18.68 22.47
0.090 0.045 0.027 0.018 0.013
0.045 0.024 0.015 0.010 0.008
0.25
6 9 12 15 18
7.06 10.70 14.35 18.03 21.70
0.081 0.041 0.025 0.017 0.012
0.052 0.026 0.016 0.011 0.008
Journal of Colloid and Interface Science. Vol. 114,No. 1, November1986
[33]
r3 •
Because K is expected to be proportional to kT, Eq. [3 l] shows that ar2/kTis a weak function of r. 5. RESULTS
Equations [29], [31], and [32] have been used to compute 3', a, and R as a function of r and ~ for an oil in water microemulsion and the results are presented in Table I. The values obtained for a are of the fight order of magnitude, being in the range of 10-~ to 10-2 dyn/ cm which is provided by experiment (11). A detailed comparison with that experiment is not yet possible since data relating o- to r are not available. For reasons already mentioned, Eq. [24] is very approximate and probably provides only an upper bound for E. As a result, the calculated values probably give a lower bound for a. Of course, 3' provides an upper bound for a. At least two effects have been ignored in the above considerations: (1) the curvature contribution due to the double and hydrated layers, which can be relevant at relatively low ionic strength, and (2) the curvature contribution due to the strong organization of the water molecules in the vicinity of the internal interface (probably like in a liquid crystal) which probably occurs at high ionic strength because of the competition of the ions for water and interaction between the species involved (E. Ruckenstein, unpublished results). The curvature contribution arises because the above layers affect the interfacial tension of a planar and curved interface differently. Because the microemulsions of practical interest involve, in general, relatively large ionic strengths, the second effect may be significant and for the moment we can say that it probably increases the rigidity of the interfacial layer thus making Eq. [24] more plausible.
THERMODYNAMICS OF MICROEMULSIONS REFERENCES 1. 2. 3. 4.
Buff, F. P., J. Chem. Phys. 19, 1591 (1951). Hill, T. L., J. Phys. Chem. 56, 526 (1952). Kondo, S., J. Chem. Phys. 25, 662 (1956). Ono, S., and Kondo, S., in "Handbuch der Physik" (S. Fltigge, Ed.), Vol. 10. Springer, Berlin, 1960. 5. Rusanov, A. I., in "Modem Theory of Capillarity," p. 1. Akademie-Verlag, Berlin, 1981. 6. Ruckenstein, E., Chem. Phys. Lett. 98, 573 (1983); in "Macro- and Micro-emulsions," ACS Symposium Series 272 (D. O. Shah, Ed.), p. 21. Amer. Chem. Soc., Washington, D.C., 1985.
179
7. Ruckenstein, E., Fluid Phase Equilib. 20, 189 (1985). 8. Ruckenstein, E., and Chi, J. C., J. Chem. Soc., Faraday Trans. 2 71, 1690 (1975). 9. Overbeek, J. Th. G., Faraday Discuss. Chem. Soc. 65, 7 (1978). 10. Ruckenstein, E., Chem. Phys. Lett. 118, 435 (1985). 11. Pouchelon, A., Meunier, J., Langevin, D., Chatenay, D., and Cazabat, A. M., Chem. Phys. Lett. 76, 277 (1980). 12. Helfrich, W., Z. Naturforsch. C 28, 693 (1973). 13. Safran, S. A., Turkevick, L. A., and Pincus, P. A., J. Phys. (Parisj 45, L69 (1984). 14. Safran, S. A., J. Chem. Phys. 78, 2073 (1983). 15. Ruckenstein, E., Soc. Pet. Eng. J., 593 (1981).
Journal of Colloid and Interface Science, Vol, 114, No. 1, November 1986