Langmuir 2000, 16, 8917-8925
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A Simple Molecular Model for the Spontaneous Curvature and the Bending Constants of Nonionic Surfactant Monolayers at the Oil/Water Interface† Vesselin N. Paunov,‡ Stanley I. Sandler, and Eric W. Kaler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received March 13, 2000. In Final Form: July 13, 2000 A simple analytical model for the spontaneous curvature and the bending constants of nonionic surfactant monolayers at an oil/water interface is developed with the goal of allowing correlation and prediction of phase behavior. The surfactant molecules are treated as diblock copolymers grafted to the interface. The change in the free energy of the surfactant monolayer due to the bending of the interface is calculated as a sum of the contributions of the hydrophobic and the hydrophilic blocks. The equilibrium thickness of a spherical and cylindrical surfactant monolayer is found by minimizing the sum of the free energy of mixing and the stretching energy of the hydrophilic and the hydrophobic blocks. Comparison with the Helfrich model gives expressions for the spontaneous curvature H0, the bending elasticity κ, and the saddle splay modulus κj, of the surfactant monolayer. An analytical expression is obtained for the phase inversion temperature, which correlates well with experimental data for a wide series of nonionic surfactants CiEj and oils (n-alkanes). The model gives the temperature dependence of the spontaneous curvature and the bending parameters of nonionic surfactant monolayers. Phase boundaries in C12E5/octane/water microemulsions are calculated without use of any adjustable parameters and are in a good agreement with experimental data.
1. Introduction At a microscopic level microemulsions are structured into water and oil domains separated by an amphiphilic monolayer. It has long1 been recognized that the properties of the microscopic surfactant film control to a great extent the type of the microstructure formed. The phenomenological model of Helfrich2 for the curvature energy of amphiphilic structures has been successful in reducing the number of natural parameters used to describe the phase behavior and microstructural characteristics of complex systems such as microemulsions, vesicles, mixtures of surfactants and polymers, and biological membranes. Helfrich introduced the concepts of a natural radius of curvature and bending constants (mean and saddle splay moduli). The properties of microemulsions are largely determined by the values of the bending constants and the natural curvature of the amphiphilic monolayer together with entropic contributions. A variety of molecular models have been developed to relate the curvature and bending constants to the molecular properties of the surfactant molecules and oils used. A comprehensive review of curvature and elastic properties of monolayers and membranes is given by Petrov et al.3 Huh4 was the first to use the ideas proposed by Meier,5 Hesselink,6 and Dolan and Edwards7 together with * To whom correspondence may be addressed. Phone: 1 (302) 831 3553. Fax: 1 (302) 831 4466. E-mail:
[email protected]. † Part of the Special Issue “Colloid Science Matured, Four Colloid Scientists Turn 60 at the Millennum”. ‡ Present address: Surfactant & Colloid Group, Department of Chemistry, University of Hull, Hull, HU6 7RX, United Kingdom. E-mail:
[email protected]. (1) Winsor, P. A. Solvent Properties of Amphiphilic Compounds; Butterworth: London, 1954. (2) Helfrich, W. Z. Naturfrsch., C 1973, 28, 693. (3) Petrov, A. G.; Bivas, I. Prog. Surf. Sci. 1984, 16, 389. Petrov, A. G.; Mitov, M. D.; Derzhanski, A. Phys. Lett. A 1978, 65, 374. (4) Huh, C. J. Soc. Pet. Eng. 1983, 23, 829. (5) Meier, D. J. Phys. Chem. 1967, 71, 1861.
estimates of the electrostatic contribution to the bending energy to derive an approximate expression for the interfacial bending stress of weakly charged surfactant monolayers. Different groups have proposed a variety of approaches to the same problem. The statistical mechanical approach of Ben-Shaul et al.8 was used by Szleifer et al.9,10 to develop a molecular theory of the curvature elasticity in surfactant films and monolayers. Barneveld et al.11,12 have applied the self-consistent field lattice approach of Scheutjens and Fleer13 to develop a numerical scheme for evaluating the spontaneous curvature and the bending elasticity parameters for both bilayers and monolayers of alkyl polyethylene glycols. For the case of ionic surfactants, the effects of the electric double layer of the curved interface have to be accounted for, and this was first done self-consistently by Winterhalter and Helfrich14 for slightly charged interfaces. Later, Mitchell and Ninham15 considered the nonlinear case of high surface potentials. Their results were rederived by Lekkerkerker16,17 from both energetic and mechanical points (6) Hesselink, F. Th. J. Phys. Chem. 1969, 73, 3488. (7) Dolan, A. K.; Edwards, S. F. Proc. R. Soc. London, Ser. A 1974, 337, 509. (8) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 4601. (9) Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Phys. Chem. 1990, 94, 5081. (10) Szleifer, I.; Kramer, D.; Ben-Shaul, A. J. Chem. Phys. 1990, 92, 6800. (11) Barneveld, P. A.; Scheutjens, J. M. H. M.; Lyklema, J. Langmuir 1992, 8, 3122. (12) Barneveld, P. A.; Hesselink, D. E.; Leermakers, F. A. M.; Lyklema, J.; Scheutjens, J. M. H. M. Langmuir 1994, 10, 1084. Leermakers, F. A. M.; van Noort, J.; Oversteegen, S. M.; Barneveld, P. A.; Lyklema, J. Faraday Discuss. 1996, 104, 317. (13) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 63, 1619. (14) Winterhalter, N.; Helfrich, W. J. Phys. Chem. 1988, 92, 6865. (15) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans 2 1981, 77, 610. (16) Lekkerkerker, H. N. W. Physica A 1989, 159, 329. (17) Lekkerkerker, H. N. W. Physica A 1990, 167, 384.
10.1021/la000367h CCC: $19.00 © 2000 American Chemical Society Published on Web 09/08/2000
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of view. Despite the development of a few well-accepted molecular models for the curvature elasticity of nonionic surfactant monolayers, their practical application for predictive calculations is not easy. The purpose of this study is to develop an analytical molecular model for the spontaneous curvature of monolayers of nonionic amphiphiles. Consider a monolayer of surfactant molecules adsorbed at the interface between oil and water. The nonionic surfactant molecules consist of hydrophilic and hydrophobic blocks, each in contact with the favorable solvent. Here only the case of strong amphiphiles is considered, so there is no adsorption (or desorption) of surfactant molecules from the oil/water interface due to bending of the monolayer. The mixing of the surfactant hydrocarbon tails with oil molecules and the mixing of hydrophilic blocks (e.g., -CH2CH2O- groups) with water are considered separately. Both the hydrophobic and the hydrophilic blocks are treated here as polymer chains, anchored to the interface. Even for a flat interface, there is some penetration (mixing) of the solvent molecules into the appropriate part of the surfactant monolayer. When the monolayer is bent toward the water, the hydrophobic part of the monolayer is “diluted” with oil, while the hydrophilic part is “concentrated”, thus expelling water molecules from the “headgroup” region. If the curvature is reversed, the opposite occurssoil molecules are expelled from the hydrophobic region of the surfactant monolayer and water molecules enter the hydrophilic region of the monolayer (see Figure 1). The effects of mixing (and the osmotic effect) on the surface free energy of the curved monolayer can be approximately accounted for by the Flory18 theory of polymer solutions. In addition, the conformational contribution to the free energy of the monolayer due to steric constraints and the stretching energy of the surfactant chains need to be taken into account when calculating the properties of the film. This study is organized as follows. Section 2 gives the physical background of the model and the basic equations for the curvature dependence of the interfacial free energy. In section 3 a comparison is made with the Helfrich mechanistic model to identify the expressions for the bending moduli and the spontaneous curvature. Section 4 illustrates the practical application of the molecular model for calculation of the phase boundaries of one-phase nonionic microemulsions. 2. Physical Background Here we first consider the mixing of a layer of polymeric molecules (say, the surfactant hydrophilic blocks, Ej) with the water molecules. The number of segments per polymeric chain is denoted by j. A similar treatment will be used for the hydrophobic part of the surfactant monolayer. The free energy of mixing, δFmix of the hydrophilic surfactant blocks with the water in the volume element δV at the interface is18
δFmix )
kBT {(1 - FjVj) ln(1 - FjVj) + Vw χjFjVj(1 - FjVj)}δV (2.1)
where Vw is the solvent (water) molecular volume, χj is the solvent-segment interaction parameter, kB is the Boltzmann constant, T is temperature, Fj is the local number density of surfactant hydrophilic segments (e.g., (18) Flory, P. J.; Krigbaum, R. 1950, 18, 1086. Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
Paunov et al.
Figure 1. Sketch of the monolayer of nonionic surfactant (considered as a diblock copolymer) at the interface between oil and water. The nonionic surfactant molecule consists of a hydrophobic and hydrophilic block, like a CiEj surfactant. The free energy of mixing of both blocks of the surfactant molecule with the adjacent solvent depends on the curvature of the interface. The mean curvature is positive in (b) and negative in (c). Here R is the curvature of the monolayer, hi is the thickness of the hydrophobic block (Ci), and hj is the thickness of the hydrophilic block (Ej) of the surfactant monolayer. The area per surfactant molecule in the monolayer is A0.
-CH2CH2O- groups), and Vj is the molecular volume of the segment. The elastic contribution to the free energy due to the stretching of the hydrophilic surfactant blocks can be estimated as (see ref 19)
δFelast )
1 (j) 2 k z Pj(z)δV 2 s
(2.2)
Here 1/2ks(j)z2 is simply the elastic energy of stretching of the chain, z is the coordinate normal to the interface, Pj(z)δV is the probability of finding a free end of the polymeric chain in the volume element δV between z and z + dz, ks(j) ≈ kBT/(jlj2) is the “elastic constant” of the polymeric chain, where lj is the length of the E segment (-CH2CH2O- group) of the surfactant chain. An adequate approximation for Pj(z) is19,20 (19) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Addison-Wesley: New York, 1994. Safran, S. A. Adv. Phys. 1999, 48, 395. (20) Pincus, P. In Phase Transitions in Soft Condensed Matter; Riste, T., Sherrington, D., Eds.; Plenum: New York, 1989.
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Pj(z) ≈ Fj(z)/j
(2.3)
This approximation says that the probability of finding the free end of the chain between z and z + dz is simply proportional to the segment number density. This is likely true in the outer part of the monolayer, but it may be inaccurate in the monolayer core, especially when the surfactant blocks are highly stretched. However, the greater part of the free interfacial energy due to bending comes from the outer parts of the surfactant monolayer, which justifies the use of eq 2.3. (See the comment after eq 2.16.) The total free energy of the hydrophobic part of the monolayer is the sum of elastic and mixing free energies (eqs 2.1 and 2.2) integrated over the volume of the monolayer (j)
{
∫V
F ) kBT
}
2 1 z Fj dV (2.4) χjFjVj(1 - FjVj)] + 2 (jl )2 j
To find the segment density distribution, Fj(z), eq 2.4 should be minimized under the constraint that the volume integral of Fj(z) must give the number of segments, j, in the hydrophilic block of the surfactant molecule
∫V Fj dV ) j
(2.5)
Safran19 has considered a similar problem for a planar monolayer of grafted chain polymers; what follows is an analysis for spherical and cylindrical monolayer geometries. 2.1. Spherical Monolayer. The counterpart of eq 2.5 for a spherical monolayer is
A0
∫R
2
drr Fj(r) ) jR
2
(2.6)
Here A0 is the area per molecule, hj is the thickness of the hydrophilic part of the monolayer, R is the radius of curvature of the spherical interface (see Figure 1), and Fj(r) is the segment density distribution for a spherical monolayer. For the sake of simplicity, the integrand of eq 2.4 is expanded in power series with respect to the segment density up to the second virial coefficient. Thus for a spherical monolayer: 2 F(j) sphere ≈ 4πR (constant) +
1 k T4π 2 B
[
∫RR+h drr2 BjFj2(r) + j
]
(r - R)2Fj(r) (jlj)
Fj(r)R+hj) ) 0
(2.9)
The segment density Fj(r) must minimize the functional 2 F ˆ (j) sphere ≈ 4πR (constant) +
1 k T4π 2 B
∫R
R+hj
[
2
2
drr BjFj (r) +
(r - R)2Fj(r) (jlj)2
]
- µFj(r)
(2.10)
where µ is a Lagrange multiplier. The condition for equilibrium δF ˆ (j)/δFj(r) ) 0 gives
1 [(1 - FjVj) ln(1 - FjVj) + Vw
R+hj
are the respective second virial osmotic coefficients,18 which are assumed to be positive (Bi,j > 0). Here Vk is the volume of the oil molecule. On the other hand, we have the boundary condition that the segment density function vanishes at the end of the monolayer
2
(2.7)
The terms linear in the segment density contribute to an additive constant in the energy. Here
Fj(r) )
[ ( )]
1 1 r-R µBj 2 jlj
2
(2.11)
Elimination of the Lagrange multiplier, µ using the boundary condition, eq 2.9, yields
Fj(r) )
1 [hj2 - (R - r)2] 2j2lj2Bj
(2.12)
and µ ) hj2/(2j2lj2). Substitution of eq 2.12 into the segment conservation relation, eq 2.6, gives a polynomial equation of fifth order with respect to hj. For relatively small curvatures (∼1/R), an asymptotic solution holds:
hj ) hj(0) +
hj(1) hj(2) + 2 + O(1/R3) R R
(2.13)
Combining eqs 2.13 and 2.12 with eq 2.6 and collecting the terms at equal powers of 1/R yields
( )
hj ≈ jlj
3Bj A0lj
1/3
-
2/3 2 2 2/3 4/3 3 1 3 lj Bj j 1 29 lj Bjj 1 + 4 R 80 A0 R2 A02/3 (2.14)
On the other hand, substituting eq 2.12 into eq 2.7 yields 2 F(j) sphere ≈ 4πR (constant) +
[
]
kBThj64πR2 1 1 hj 1 + 6j4lj4Bj 2 R 7 R2 (2.15)
Finally, combining eqs 2.14 and 2.15 and dividing by the area of the monolayer, 4πR2, for the free surface energy (j) (j) γsphere ) Fsphere /(4πR2) we obtain 2
γ(j) sphere ≈ γflat -
3 kBTBjj 1 + 8 A2 R 0
for the hydrophilic block 2
Vj Vw
Bj ) (1 - 2χj)
(2.8a)
for the hydrophobic block Vi2 Vk
Bi ) (1 - 2χi)
(2.8b)
1/3 4/3 3 2/3 219 kBT3 Bj j lj 1 (2.16) 1120 A 7/3 R2 0
Numerical calculations show that about one-third of the (j) due to bending comes contribution to the change of γsphere from the integration in eq 2.10 from r ) R to r ) R + 1/2hj and about two-thirds comes from the integration in eq 2.10 from r ) R + 1/2hj to r ) R + hj. Hence the outer part
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of the surfactant monolayer contributes more to the bending properties than the monolayer core, as assumed for eq 2.3. 2.2 Cylindrical Monolayer. The counterparts of eqs 2.6 and 2.7 for a cylindrical geometry of the monolayer are
A0
∫RR+h drrFj(r) ) jR j
(2.17)
F(j) cyl ≈ 2πLR(constant) +
∫R
1 k T2πL 2 B
R+hj
{
2
drr BjFj (r) +
}
(r - R)2Fj(r) (jlj)2
(2.18)
where R is the radius of curvature of the cylindrical interface and L is the length of the cylinder. Here Fj(r) and hj correspond to the cylindrical shape of the interface. The boundary condition at r ) R + hj for the cylindrical geometry is the same as eq 2.9. The functional
F ˆ (j) cyl ≈ 2πLR(constant) + 1 k T2πL 2 B
∫RR+h drr j
{
BjFj2(r) +
(r - R)2Fj(r) (jlj)2
- µFj(r)
}
(2.19)
is minimized with respect to Fj(r) and µ is found from eq 2.9. Fj(r) obeys an equation identical to eq 2.12 with R and hj corresponding to the cylindrical geometry of the monolayer. By analogy with the spherical case we obtain
( )
3Bj hj ≈ jlj A0lj
1/3
1 8
32/3lj4/3Bj2/3j2 A02/3
2
3
9 lj Bjj 1 1 + R 64 A0 R2 (2.20)
2
γ(i) sphere ≈ γflat +
0
1/3 4/3 3 2/3 219 kBT3 Bi i li 1 (2.23) 1120 A 7/3 R2 0
γ(i) cyl ≈ γflat +
kBThj62πRL 1 (2.21) ≈ 2πLR(constant) + 24j4lj4Bj R
Finally, combining eqs 2.20 and 2.21 and dividing by the surface area of the monolayer, 2πRL, we obtain an expression for the free energy per unit area γ(j) cyl ) F(j) cyl/(2πLR) of the cylindrical monolayer
γ(j) cyl
2 1/3 4/3 3 2/3 9 kBT3 Bj j lj 1 3 kBTBjj 1 + ≈ γflat 16 A 2 R 128 A07/3 R2 0 (2.22)
The area per surfactant molecule A0 is assumed to remain constant when bending the monolayer at the oil/water interface. The convention for the sign of the radius of curvature of the interface is that oil droplets (cylinders) in water have positive radius of curvature. The corresponding contributions from the hydrophobic part of the surfactant monolayer to the interfacial free energy can be formally obtained from eqs 2.16 and 2.22 by changing j with i, Bj with Bi, lj with li,and R with -R. The final results are
2 1/3 4/3 3 2/3 9 kBT3 Bi i li 1 3 kBTBii 1 + 16 A 2 R 128 A07/3 R2 0 (2.24)
The total interfacial free energy due to mixing and stretching is the sum of the contributions of the hydrophilic and hydrophobic parts of the monolayer (j) γsphere ) γ(i) sphere + γsphere,
(j) γcyl ) γ(i) cyl + γcyl
(2.25)
3. Bending Constants and Spontaneous Curvature 3.1. Comparison with the Helfrich Model. The results above can be compared to the classical Helfrich model, assuming that there are no other contributions to the surface free energy of the monolayer. The Helfrich4 model gives
γ ) γflat - 4κH0H + 2κH2 + κjK
(3.1)
where H and K are the mean and the Gaussian curvatures of the interface, κ is the bending elasticity modulus, κj is the splay modulus, and H0 is the spontaneous curvature of the surfactant monolayer. For a spherical interface, H ) 1/R, K ) 1/R2, and eq 3.1 yields
γsphere ) γflat - 4κH0
1 1 + (2κ + κj) 2 R R
(3.2)
For a cylindrical interface, H ) 1/(2R), K ) 0, and eq 2.14 gives
γcyl ) γflat - 2κH0
On the other hand, substituting eqs 2.12 into eq 2.18 we obtain
F(j) cyl
3 kBTBii 1 + 8 A2 R
1 κ 1 + R 2 R2
(3.3)
The comparison of eqs 3.2 and 3.3 and their counterparts, eq 2.25 with eqs 2.16, 2.22, 2.23, and 2.24 yields expressions for both the spontaneous curvature, and the bending and splay moduli 1/3
κ)
9 3 kBT 4/3 3 2/3 (Bi i li + Bj4/3j3lj2/3) 64 A 7/3
(3.4)
0
κj ) -
1/3 3 3 kBT 4/3 3 2/3 (Bi i li + Bj4/3j3lj2/3) 35 A 7/3
(3.5)
0
4κH0 )
3 kBT (B j2 - Bii2) 8 A2 j
(3.6)
0
Combining eqs 3.4 and 3.6 for the spontaneous curvature gives
H0 )
2A01/3(Bjj2 - Bii2) 34/3(Bi4/3i3li2/3 + Bj4/3j3lj2/3)
(3.7)
For positive values of Bi and Bj, which correspond to a good solvent (χi < 1/2, χj < 1/2), the bending rigidity, κ, is
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always positive, κ > 0, while the saddle splay modulus can have only negative values, κj < 0. In addition, within these approximations κ and κj are proportional; i.e., κ/κj ) -105/64 ≈ -1.641. 3.2. Estimation of the Model Parameters. To determine the temperature dependence of the spontaneous curvature
H0 ) H0(T)
(3.8)
we consider that for certain temperature intervals, the Flory parameters are weak linear functions of the temperature
χi ≈ χi(0) + χi(1)T + ... χj ≈ χj(0) + χj(1)T + ...
(3.9)
The combination of eqs 2.8a, 3.7, and 3.9 predicts that the spontaneous curvature can change its sign at a given h ) ) 0. T h is known as the phase temperature, T h , i.e., H0(T inversion temperature (PIT). It is easily measured and plays an important role in interpreting microemulsion phase behavior and microstructure.21-24 For nonionic surfactants below the PIT the amphiphilic monolayer is preferentially curved toward oil (o/w microemulsion), while above the PIT it is curved toward the water (w/o microemulsion). From eq 3.7, T h can be estimated by solving the equation
h ) ) j2Bj(T h) i2Bi(T
(3.10)
Thus, eqs 2.8a and 3.8-3.10 yield an approximate analytical expression for the PIT
T h≈
VkVj2j2(1 - 2χj(0)) - VwVi2i2(1 - 2χi(0)) 2(χj(1) Vj2j2Vk - χi(1)Vi2i2Vw)
(3.11)
This is a principal result of this work, and it allows a direct correlation and prediction of phase behavior, as we will now illustrate. For the particular system of nonionic surfactant, CiEj/n-alkane/water, the Flory parameter for the hydrophobic block is very small (∼10-2) for the series of n-alkanes of similar length and can be neglected25
χi ≈ 0 Then, eq 3.11 reduces to
T h≈
1
[
2χj(1)
1 - 2χj(0) -
(3.12)
( )] VwVi2 i2 V h kVj2 j2k
(3.13)
where V h k ≈ Vk/k is the volume per oil segment. Note that h kVj2), which appears in eq 3.13, has a the ratio (VwVi2/V universal value of 0.230 for any ethoxylated alcohol (CiEj) and n-alkane oil (CkH2k+2). The parameters χj(0) and χj(1) can be estimated by fitting eq 3.13 to the data for PIT measured26 for a series of surfactant systems (see Figure (21) Safran, S. A.; Webman, I.; Grest, G. S. Phys. Rev. A 1985, 32, 506. (22) Aveyard, R.; Binks, B. P.; Lawless, T. A.; Mead, J. J. Chem. Soc., Faraday Trans. 1 1985, 81, 2155. (23) Kellay, H.; Binks, B. P.; Hendrikx Y.; Lee, L. T.; Meunier, J. Adv. Colloid Interface Sci. 1994, 49, 85. (24) Gradzielski, M.; Langevin, D.; Sottmann, T.; Strey, R. J. Chem. Phys. 1997, 106, 8232. (25) Barton, A. F. M. CRC Handbook of Solubility Parameters and Other Cohesion Parameters; CRC Press: Boca Raton, FL, 1983.
2). The best-fit values are
χj(0) ≈ -0.2086,
χj(1) ) 2.061 × 10-3(1/K) (3.14)
and the quality of the fit is shown in Figure 2. The resulting values of χj ≈ χj(0) + χj(1)T lie between 0.36 and 0.48 for temperatures between 0 and 60 °C. The model is able to fit the experimental values within 1-2% (in °C) for a wide variety of alkane oils and different surfactants. The PIT of the surfactant monolayer is predicted to increase with increasing size of the oil molecule or the number of oxyethylene groups of the surfactant, and the PIT drops with increasing length of the alkyl group of the surfactant. 3.3. Numerical Results and Discussion. The calculation of the spontaneous curvature and the bending moduli requires the knowledge of the area per surfactant molecule, A0. According to Sottmann et al.,27 A0 is sensitive neither to the oil used nor to the alkyl group (Ci) of the surfactant, but depends linearly on the number (j) of oxyethylene groups in the surfactant molecule
A0 ≈ 29.3 + 6.20j (Å2)
(3.15)
In eqs 3.4-3.7 we use li ) 0.39 nm and lj ) 0.49 nm, as estimated from mass density data, to predict phase boundaries. Figure 3 shows the temperature dependence of the spontaneous curvature, H0(T), for different oils (a), different hydrophobic groups (b), and different hydrophilic groups (c) of the surfactant molecule. The spontaneous curvature changes more strongly with temperature for larger oil molecules (Figure 3a). The same trend occurs with respect to the increase of the length of the hydrophobic group (Figure 3b) or the hydrophilic group (Figure 3c). Figure 4 shows the dependence of the bending elasticity h , and the saddle splay modulus, κj(T h )/kBT h modulus, κ(T h )/kBT on the type of oil (Figure 4a), the length of the alkyl group of the surfactant (Figure 4b), and the length of the hydrophilic group (Figure 4c). The bending moduli are calculated at the respective PITs. Increasing the length of the hydrophobic group increases both κ and κj in magnitude, while increasing of the length of the hydrophilic group (Ej) has the opposite effect. Both κ and κj are less than kBT, which is consistent with experimental estimates.28 κ and κj are complex functions of the temperature (cf. eqs 3.4 and 3.5). Our prediction is that κ(T)/kBT and κj(T)/kBT decrease in magnitude with increasing temperature, shown for two different systems in Figure 5. These are predictions for the bare bending moduli, the values of which could be altered by interfacial fluctuations.29 Given the temperature dependence of H0, κ, and κj, the position of phase boundaries can be calculated as illustrated in the next section. 4. Temperature Induced Phase Transitions in Nonionic Microemulsions Consider the phase transitions in a microemulsion of water/n-octane/C12E5 (pentaethylene glycol n-dodecyl ether) caused by temperature and compositional changes. The system can be in one of the following states: oil-in-water (o/w) µE (droplets) coexisting with oil (denoted as 2); a single phase (droplets) o/w µE (L1); the lamellar phase (LR); a single phase water-in-oil (w/o) µE (droplets); or a (26) Sottmann, T.; Strey, R. J. Chem. Phys. 1997, 106, 8606. (27) Sottmann, T.; Strey, R.; Chen, S.-H. J. Chem. Phys. 1997, 106, 6483. (28) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (29) Golubovic, L.; Lubensky, T. C. Phys. Rev. A 1990, 41, 4343.
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Figure 3. Predictions of the spontaneous (natural) curvature of a nonionic surfactant monolayer as a function of the temperature: (a) for different oils; (b) for different alkyl groups (Ci) of the surfactant; (c) for different hydrophilic groups (Ej) of the surfactant. Figure 2. (a) The phase inversion temperature (PIT), T h , of a monolayer of C12E5 at the water/n-alkane interface as a function of the number of oil carbon atoms. (b) The phase inversion temperature of a monolayer of CiE4 at water/n-octane interface as a function of the number of n-alkane carbon atoms. (c) The phase inversion temperature of a monolayer of C12Ej at water/ n-octane interface as a function of the number of oxyethylene groups of the surfactant. In all panels, the experimental data ([) are taken from Sottmann et al.,26 and the connected open symbols (0) represent the best fit of the model with χj(0) ≈ -0.2086 and χj(1) ≈ 2.061 × 10-3(1/K).
w/o µE (droplets) coexisting with water (2) and a single phase (droplets) w/o µE (L2). The phase structure of minimum free energy is formed at a given composition and temperature. The free energy of formation of a microemulsion of a particular structure is a sum of bending and entropic contributions (see, e.g., refs 21, 23, 24, 28, 30, and 31)
F ) γbA - T∆Smix
(4.1)
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Figure 5. Bending modulus κ/kBT and saddle splay modulus κj/kBT of a monolayer of nonionic surfactant (CiEj) at the water/ n-alkane interface vs temperature for two different systems: C10E4/n-decane/water and C12E5/n-octane/water.
how the entropic term affects the equilibrium conditions of the microemulsion. However, given the lack of a quantitative description of the entropic term, we present here the phase patterns predicted by considering only the bending energy.28 4.1. Phase boundaries 2/L1 and L2/2. Here we assume that the structure of the microemulsion does not change upon the transition from a two-phase to a singlephase microemulsion. When a microemulsion of spherical droplets coexists with a pure disperse phase, the droplet size should be the maximum possible at a given temperature. The latter is determined by minimizing the free energy, eq 4.1, with respect to the droplet radius, so
1 1 γbsphere ) γflat - 4κ|H0(T)| + (2κ + κj) 2 (4.2) R R and
∂γbsphere ∂(1/R)
) -4κ|H0(T)| + 2(2κ + κj)
1 )0 Rmax
(4.3)
On the other hand, the surface area-to-volume ratio of a spherical droplet microemulsion is given by the equation
A 3φ φcA0 φc ) ) ) V R νc lc R ) 3φlc/φc
Figure 4. Calculated bending modulus and saddle splay modulus (at T ) T h ) of a monolayer of nonionic surfactant (CiEj) at the water/n-alkane interface as a function of (a) the number of n-alkane carbon atoms, (b) the number of carbon atoms in the alkyl group of the surfactant molecule, and (c) the number of oxyethylene groups of the surfactant.
with γb given by eq 3.1, A the total area of the surfactant monolayer, and ∆Smix the entropy of mixing of the microemulsion structures with the continuous phase. There have been numerous discussions21,23,24,28,30,31 about (30) Talmon, Y.; Prager, S. J. Chem. Phys. 1978, 67, 2984.
(4.4)
Here φ is the droplet volume fraction, φc and νc are the surfactant volume fraction and molecular volume, and lc ≈ νc/A0 is the effective thickness of the surfactant monolayer. Other studies33 show that the thickness of a C12E5 monolayer at the octane/water interface is lc ) 1.3 ( 0.2 nm. The combination of eqs 4.3 and 4.4 finally gives
1+
φ κj ) 3lc|H0(T)| 2κ φc
(4.5)
where the temperature dependence of the spontaneous (31) Ruckenstein, E. Fluid Phase Equilib. 1985, 20, 189; J. Colloid Interface Sci. 1978, 66, 369; J. Colloid Interface Sci. 1998, 204, 143. (32) Strey, R.; Glatter, O.; Schubert, K. V.; Kaler, E. W. J. Chem. Phys. 1996, 105, 1175. (33) Vollmer, A.; Vollmer, D.; Strey, R. Phys. Rev. E 1996, 54, 3028.
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Paunov et al.
curvature H0(T) is calculated from the molecular model, eq 3.7. For a surfactant dilution line, i.e., for the condition that the ratio of oil-to-water volume fractions is constant,
φb ) constant φa
(4.6)
1 - φc , φb ) λbaφa 1 + λba
(4.7)
λba ) and we find
φa )
φ ) φb + xbφc (for 2/L1 boundary) φ ) φa + xaφc (for L2/2 boundary) Here xa ) jVj/(iVi + jVj) ≈ 0.55 is the portion of the surfactant molecule “immersed” in the water phase, φa and φb are the volume fractions of water and oil, respectively. The combination of eqs 4.5, 4.6, and 4.7 gives
for 2/L1 boundary 1+
for L2/2 boundary 1+
(
)
(
)
λba(1 - φc) κj ) 3lcH0(T) + xb 2κ φc(1 + λba)
(1 - φc) κj + xa ) 3lcH0(T) 2κ φc(1 + λba)
(4.8)
The latter equation can be solved to find the temperature of the respective emulsification failure boundary (2/L1 or L2/2) at fixed composition of the microemulsion. 4.2. Phase Boundaries L1/Lr and Lr/L2. The position of these phase transitions is determined by the solution of the following energetic balances
γbL1(T) ) γbLR(T)
(for L1/LR boundary)
γbLR(T) ) γbL2(T)
(for LR/L2 boundary)
(4.9)
Since γbLR(T) ) γflat, the solutions of eqs 4.9 have the form
2|H0|R ) 1 +
κj 2κ
(for both L1/LR and LR/L2) (4.10)
Combining eqs 4.4 and 4.7 with eq 4.10 we obtain
for L1/LR boundary 1+
for LR/L2 boundary 1+
(
)
(
)
Figure 6. Phase boundaries for one-phase microemulsions of C12E5/octane/water. The symbols represent experimental data, taken from Vollmer et al.33 for surfactant dilution lines. The solid and dashed lines give the model predictions. The three graphs, (a), (b), and (c), correspond to three different ratios of the oil and water volume fractions. No adjustable parameters are used to calculate the theoretical curves.
λba(1 - φc) κj ) 6lcH0(T) + xb 2κ φc(1 + λba)
(1 - φc) κj + xa ) -6lcH0(T) 2κ φc(1 + λba)
(4.11)
for the surfactant dilution line. Note that the equations for the lamellar/droplet transitions and their counterparts for the emulsification failure boundaries differ only by a coefficient of 2 (cf., e.g., eqs 4.8 and 4.11). Figure 6 compares experimental data33 and predictions for the phase boundaries in one-phase microemulsions of
C12E5/n-octane/water. The three different plots correspond to the surfactant dilution line for three different values of Rv ) φb/(φa + φb). The solid curves are the calculated emulsification failure boundaries, while the dashed curves are calculated boundaries of lamellar and globular microemulsion phases. The agreement between the theoretical model and the experiment is good, without use of adjustable parameters. As pointed out by Vollmer et al.,33
Nonionic Surfactant Monolayers
the positions of both phase boundaries are insensitive to the values of the bending moduli, κ and κj, and depend only on their ratio and the geometry of the microstructure. The agreement between experimental data and theoretical model in Figure 6 (Rv ) 0.26 and Rv ) 0.85) is good even through the structure of the microemulsion is not one of spherical droplets but rather is bicontinuous.34 The latter means that it is not expensive in free energy to form necks that interconnect the microemulsion droplets, at least for this range of parameters. Similarly, in this parameter range, there is also no significant difference between the free energy of cylindrical and spherical structures.33
Langmuir, Vol. 16, No. 23, 2000 8925
expressions for the spontaneous curvature, H0, the bending elasticity, κ, and the saddle splay modulus, κj, of the surfactant monolayer. The model is based on minimization of the free energy of mixing and the energy of stretching of the hydrophilic and the hydrophobic blocks of the surfactant monolayer. This model also yields a simple expression for the phase inversion temperature (PIT) for a variety of combinations of nonionic surfactants CiEj and n-alkanes (oils). Calculated phase boundaries of one phase microemulsions of C12E5/octane/water agree well with experiment without the use of adjustable parameters.
5. Conclusions A molecular model for the bending properties of nonionic surfactant monolayers (CiEj) yields simple analytical
Acknowledgment. The authors appreciate the financial support of this study from the US Department of Energy (Grant No. DE-FG02-85ER13436). The authors are also indebted to a reviewer for valuable comments and suggestions.
(34) Anderson, D.; Wennerstro¨m, H.; Olsson, U. J. Phys. Chem. 1989, 93, 4243.
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