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Langmuir 2009, 25, 112-115
Correspondence between Curvature, Packing Parameter, and Hydrophilic-Lipophilic Deviation Scales around the Phase-Inversion Temperature Werner Kunz,*,† Fabienne Testard,‡ and Thomas Zemb§ Institute of Physical and Theoretical Chemistry, UniVersity of Regensburg, D-93040 Regensburg, Germany, CEA, IRAMIS, SerVice de Chimie Mole´culaire (SCM), LIONS, F-91191 Gif sur YVette Cedex, France, and Institut de Chimie Se´paratiVe de Marcoule-UMR 5257, CEA/CNRS/UMR/ENSCMBP 17171-F-30207 Bagnols-sur-Ce`ze Cedex, France ReceiVed September 3, 2008. ReVised Manuscript ReceiVed October 21, 2008 We show in this paper that three ways of characterizing “spontaneous” lateral packing of amphiphiles are equivalent: the spontaneous curvature, the molecular packing parameter, and the refined hydrophilic-lipophilic balance known as HLD (hydrophilic-lipophilic deviation). Recognition of this equivalence, with its underlying hypothesis of incompressible fluid with lowest surface energy, reinforces the single parameter bending energy expression implicit in the classical papers by Ninham and Israelachvili, as well as all the predictive models of solubilization developed as yet.
Introduction Today, amphiphilic molecules and their monolayers are usually described by three different concepts. The oldest one is based on the hydrophilic-lipophilic balance (HLB) of surfactant molecules and was initially proposed as a convenient tool to classify surfactants for practical use.1 In parallel and over the years, other practical concepts, such as the Winsor ratio2 and the phase inversion temperature (PIT) approach3 were developed. Recently, these descriptions were unified and generalized by Salager and co-workers in the so-called hydrophilic-lipophilic deViation (HLD) from optimum formulation concept.4 The second concept is a description of surface monolayers with a packing parameter, p, a dimensionless scalar that is the ratio of two areas and was introduced more than 30 years ago.5 In the meantime, this “packing” concept was cited and used more than 5000 times, unfortunately quite often with an inadequate definition of packing. Whereas the HLD concept is more a pragmatic description of phenomena, this geometrical concept is based on an evaluation of chemical potentials developed in geometrical variables.6 In 2003, Acosta et al.7,8 introduced a critical scaling model, called the net average curvature (NAC), to scale the curvature of micelles to the optimum formulation point using the HLD as the scaling variable. The two “curvatures” as defined in this paper do not correspond to quantities with easy * To whom correspondence should be addressed. Fax: (+) 49 941 943 45 32. E-mail:
[email protected]. † University of Regensburg. ‡ Service de Chimie Mole´culaire (SCM). § Institut de Chimie Se´parative de Marcoule.
(1) Griffin, W. C. J. Soc. Cosmet. Chem. 1949, 311, 1. (2) Winsor, P. A. SolVent Properties of Amphiphilic Compounds; Butterworth: London, 1954. (3) Shinoda, K.; Arai, H. J. Phys. Chem. 1964, 68, 3485. (4) Salager, J. L.; Marquez, N.; Graciaa, A.; Lachaise, J. Langmuir 2000, 16, 5534. (5) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc. Faraday Trans. 2 1976, 72, 1525. (6) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc. Faraday Trans. 2 1981, 77, 601. (7) Acosta, E.; Szekeres, E.; Sabatini, D. A.; Harwell, J. H. Langmuir 2003, 19, 186. (8) Acosta, E. J. Colloids Surf. A 2008, 320, 193–204.
interpretation using microstructural studies, hence this NAC model did not lead to the production of tables. The final concept of spontaneous curVature, H0, introduced by Helfrich,9 is based on the mechanics of solid thin shells and was developed in the 19th century. This approach has allowed phase diagrams to be described,10 as well as microstructures of microemulsions in a regime where fluctuations dominate due to the introduction of large quantities of cosurfactants.11 The spontaneous curvature and the associated rigidity constants can be qualitatively estimated from phase diagrams, surface tension, scattering, and conductivity experiments.12 However, this concept has implicit contradictions when used for fluid systems, for which average and Gaussian curvatures cannot be separated as in a solid, as pointed out by Fogden et al.13 In the present paper, we first recall the implicit approximations in these three concepts. We also analyze shortly a few immediate consequences of these approximations, before we show how the numerical scales inferred from these three concepts can be unified so that their values can be converted from one scale into the others for CiEj nonionic surfactants. The evidence of a correspondence between curvature, packing, and practical scales of lipophilic/hydrophilic balance of surfactants is a first step toward a better theoretical justification of semiempirical rules that are widely used in industry.
The Three Concepts and Their Approximations The Hydophilic-Lipophilic Deviation (HLD). A short overview of the history of HLD was given in a recent paper by Queste et al.14 As stated there, the HLD is based on the determination of the so-called optimum formulation with flexible interfacial films. At this optimum formulation composition, the surfactant affinity to the polar and apolar pseudophase of a (9) Helfrich, W. Z. Naturforsch. 1973, 28c, 693. (10) Gompper, G.; Zschocke, S. Phys. ReV. A 1992, 46, 4836. (11) Porte, G.; Gomati, R.; El Haitamy, O.; Appell, J.; Marignan, J. J. Phys. Chem. 1986, 90, 5746. (12) Gradzielski, M. Curr. Opin. Colloid Interface Sci. 1998, 3, 478. (13) Fogden, A.; Hyde, S. T.; Lundberg, G. J. Chem. Soc. Faraday Trans. 1991, 87, 949. (14) Queste, S.; Salager, J. L.; Strey, R.; Aubry, J. M. J. Colloid Interface Sci. 2007, 312, 98.
10.1021/la8028879 CCC: $40.75 2009 American Chemical Society Published on Web 12/10/2008
Characterizing Lateral Packing of Amphiphiles
microemulsion is equal, meaning that the free energy change of a surfactant molecule, when transferred from oil to water, is zero. For systems with flexible interfacial films, this point is usually found in a triphasic system (oil/microemulsion/water, i.e., Winsor III), when the interfacial tension of the system is a minimum. Under the conditions of this optimum formulation, the HLD value is set to zero; it is the reference state where the spontaneous curvature of the system is zero. Any changes in the systems, such as temperature, type of surfactant and oil, addition of salt or cosurfactants, will then lead to a departure from the reference state and to a positive or negative HLD value, depending on the then apparently more hydrophobic or hydrophilic behavior of the surfactant, respectively. For nonionic surfactants and given aqueous and oil phases, the HLD value of the whole system is defined from a practical approach as
HLD ) R - EON + b × salt - k × ACN + c(T - Tref) + aA (1) where R, EON, k, and c are characteristic parameters of the surfactants, “salt” is the weight percentage of salt in water, b is the sensitivity of the formulation balance to the presence of salt, ACN is the number of carbon atoms, in the case where a linear paraffin oil is used as oil (otherwise an effective alkyl carbon number EACN is defined), T is the actual temperature, Tref is an arbitrarily chosen reference temperature (usually 25 °C), and A is the mass percentage of possibly present cosurfactant (usually an alcohol) with a being the corresponding sensitivity parameter that depends both on the cosurfactant and the surfactant.15 For systems containing ionic surfactants, the corresponding expression is15
HLD ) σ + ln(salt) - k × ACN + c(T - Tref) + aA (2) where σ plays the role of R in eq 1. Both equations describe for example correctly the linear dependence of the surfactant hydrophilicitysand hence of the curvatureson temperature, with a much more pronounced dependence in the case of nonionic surfactants, of course. The characteristic sensitivity parameters can be experimentally determined for new and unknown surfactant systems by varying for example salinity and searching for the temperature shift that compensates the salinity shift. This fixes the ratio of two constants in eqs 1 and 2. A rigorous theory should in principle be able to predict these constants. This is achievable with ionic surfactants when electrostatics dominates, but not yet with the widespread nonionic “CiEj” surfactants, since the molecular mechanism of headgroup dehydration driving the oil-water solubility of head-groups is not known. The Packing Parameter Concept. The packing parameter is the ratio between a “sterical” area naturally linked to fluid incompressibility, as, and the actual area a of a surfactant molecule at the polar/apolar interface in a microemulsion (also defined as the neutral plane of the monolayer16). as ) V/l, the ratio of a volume to a length, and it characterizes a “sterical” area in a closely packed film of incompressible molecules. A quite widespread misuse of the concept is the use of the extended length, as determined by molecular modeling. The length l, which should be considered here, is the length averaged over all conformations:17,18 it is quite remarkable that for alkanes this (15) Salager, J. L.; Anton, R.; Anderez, J. M.; Aubry, J. M. Techniques de l’inge´nieur Volume ge´nie des proce´de´s; 2001; article J2157. (16) Ennis, J. J. Chem. Phys. 1992, 97, 663. (17) Tanford, C. The Hydrophobic Effect: The Formation of Micelles and Biological Membranes; Wiley: New York, 1980.
Langmuir, Vol. 25, No. 1, 2009 113
length is to a good approximation 80% of the extended chain length, due to gauche/trans isomerism of linear alkanes. It should be noted that the thickening or thinning of films formed by surfactants with 10-20 methylene groups has a high cost in entropy and can be neglected, even in the case of solubilization of rigid molecules acting like a wedge in the surfactant monolayer.19 The molecular volume V to be used for determining as has to include not only the apolar part of the amphiphilic molecule but also the “penetrating oil”. This has been demonstrated by elegant experiments comparing phase boundaries in phase diagrams with homologue series of oils.20 It should be noted that volume and area used in defining as can be considered for the apolar part of the molecule, or as initially meant in the papers by Israelachvili et al., the “volume” may include the whole surfactant molecule, together with water of hydration of the headgroup. In this case, the area to consider is the area at the surfactant head-water interface instead of the headgroup-oil interface. Since V/(al) is a fractional quantity, the two methods are equivalent, provided consistent values of the volumes and area are considered. In the absence of “holes” and “tearing” of the surfactant film, the area per molecule a0 is the area per molecule, which minimizes the free energy. A numerical example of this is given in the calculation of cationic micellar mass variation upon variation of the chaotropic/cosmotropic nature of the counterion.21 The area per molecule in a real film a0 is imposed by an equilibrium between attractive and repulsive lateral interactions between surfactant molecules. Consideration of this point has allowed the development of predictive theories of micellar shapes as a function of salinity and the related ion-specific effects22,23 as well as their influence on equations of state describing two-dimensional surfactant and lipid films.24-26 With the definitions of as and a0, the spontaneous packing p0 at a given temperature, osmotic pressure (Π), salinity, etc. is given by the ratio of both quantities
p0 ) as/a0)V/(a0l)
(3)
where the Gibbs free energy G has its minimum (∂G/∂a)Π,T... ) 0. The simplest experimental derivation of spontaneous packing p0 is for “stiff” microemulsions, when packing constraints dominate over fluctuations: when more and more water is added to the microemulsion, the bicontinuous structure progressively transforms to monodisperse droplets. Any further added water is expelled from the swollen micelles due to the rigidity of the film. The result is the emulsification failure, where the excess “internal” phase is in equilibrium with swollen micelles. In the phase diagram, the tangent of this borderline between these two phases gives directly the spontaneous packing. At maximum swelling upon water, and when the conductivity measured is low, the microemulsions are in the dispersed spherical droplet (18) Ben-Shaul, A.; Gelbart, W. M. Annu. ReV. Phys. Chem. 1985, 36, 179. (19) Testard, F.; Zemb, Th. Langmuir 1998, 14, 3174. (20) Chen, S. J.; Evans, D. F.; Ninham, B. W.; Mitchell, D. J.; Blum, F. D.; Pickup, S. J. Phys. Chem. 1986, 90, 842. (21) Zemb, Th.; Belloni, L.; Dubois, M.; Aroti, A.; Leontidis, E. Curr. Opin. Colloid Interface Sci. 2003, 9, 74. (22) Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1984, 88, 6344. (23) Hayter, J. B. Langmuir 1992, 8, 2873. (24) Aroti, A.; Leontidis, E.; Dubois, M.; Zemb, Th.; Brezesinski, G. Colloids Surf. A 2007, 303, 144. (25) Aroti, A.; Leontidis, E.; Dubois, M.; Zemb, Th. Biophys. J. 2007, 93, 1580. (26) Leontidis, E.; Aroti, A.; Belloni, L.; Dubois, M.; Zemb, Th. Biophys. J. 2007, 93, 1591.
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Kunz et al.
regime. In oil-rich microemulsions, the maximum radius of the droplets obtained at the emulsification boundary is fixed by the internal volume (φ, volume fraction of water plus headgroup) to specific surface (Σ) ratio
Rmax)1/H0 ) 3φ/Σ
(4)
and the packing, identical to spontaneous packing at emulsification failure, when the rigid film cannot be easily deformed
p0 ) 1 + l/Rmax + 1/3(l/Rmax)
2
(5)
Table 1. Example of Given Correspondence between p0 and HLD for a Particular Surfactant Systema system C12E5/water/n-octane C12E5/water/n-octane C12E5/water/n-octane C12E5/water/n-octane C12E5/water/n-octane
at at at at at
20 K above PITb 20 K below PIT the PIT 10 K above PIT 10 K below PIT
p0
HLD
1.732 0.268 1 1.366 0.634
-1.2 1.2 0 -0.600 0.600
a p0 is obtained from eq 13 using ref 34 and HLD is obtained independently from eq 14. b Phase inversion temperature.
A typical example is water and alkane oils mixed with ionic double chain surfactants, such as the well studied didodecyldimethylammonium bromide.27 However, it should be noted that the packing parameter concept is not limited to rigid interfacial films. In the case of flexible films, the actual packing parameter p is no longer equal to p0, because the energy cost per unit area to bend the film, F, is small
F ) 1/2k*(p - p0)2
(6)
28,13
where k* is a generalized bending constant. As a consequence, it is difficult to determine p0 in a system with flexible interfacial films. The Concept of Spontaneous Curvature.9,29-32 The concept of a local spontaneous packing parameter p0 (and a packing parameter p depending on sterical constaints, due to composition) can be linked to the macroscopic concept of spontaneous and nonspontaneous curvatures. Here, H0 and K0 are the spontaneous average and Gaussian curvatures, adopted by the surfactant film in the absence of nonlocal constraints, and H and K are the corresponding values imposed by the composition of the microemulsion. Within this approach, Gaussian and average curvatures are independent variables, and the bending energy per unit area F is given by
F ) 2kc(〈H 〉 -H0)2 + k′(〈K 〉 -K0)
(7)
where, on a macroscopic scale, 〈H〉 and 〈K〉 are the average and Gaussian curvatures averaged over the whole sample, respectively. However, usually the spontaneous curvature K0 is neglected and the elastic constant k′ is unknown. The problem can be simplified by assuming that the surfactant film cannot disrupt. Then the concept of curvature can be transferred to the concept of the packing parameter with the help of the following so-called coverage relations33
p ) 1 + 〈H 〉 l + 1/2〈K 〉 l2
(8)
p0)1 + 〈H0〉 l + 1/2〈K0 〉 l
(9)
2
with the convention of a positive curvature toward water. The advantage using eq 6 instead of eq 7 as the basis of the free energy of any surfactant film dispersion is to relate the energy of deformation of surfactant film to the deviation of the effective surfactant parameter from its preferred value. The coverage relation (eq 8) implies that film tearing does not occur. Instead of two ill-defined bending constants, namely the average and the (27) Samseth, J.; Chen, S. H.; Litster, J. D.; Huang, J. S. J. Appl. Crystallogr. 1988, 21, 835. (28) Zemb, Th. Colloids Surf. A 1997, 129, 435. (29) Safran, S. A. Phys. ReV. Lett. 1983, 50, 1930. (30) Cates, M. E.; Andelman, D.; Safran, S. A.; Roux, D. Langmuir 1988, 4, 802. (31) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces and Membranes; Addison-Wesley: Reading, MA, 1994. (32) Hunter, R. J. Introduction to Modern Colloid Science; Oxford University Press: Oxford, 1993. (33) Hyde, S.; Andersson, S.; Larsson, K.; Blum, Z.; Landh, T.; Lidin, S.; Ninham, B. W. The Language of Shape; Elsevier: Amsterdam, 1997.
Figure 1. Correlation between the spontaneous packing parameter p0 and the HLD value for the water/n-octane/C12E5 system. The values are taken from Table 1. ∆T ) T0 - T.
Gaussian parameters, just one constant is needed in the firstorder, namely, the generalized bending constant associated with the packing parameter. Typically, this constant is much less than kT in all cases measured with linear nonionics and much higher than kT for all double chain surfactants or microemulsions containing long chain fatty alcohols as surfactants, gemini surfactants, or microemulsions formed with ionic liquids.
Unifying the Packing Parameter and the HLD Whereas the relations between the packing parameter and the curvature concept are known from the coverage relation, we will show here, how the third approach, the HLD concept, can be linked with them. To this purpose, we consider a system that was explicitly examined within the frameworks of both the packing parameter and the HLD concept. This system is the ternary mixture water/ n-octane/C12E5 (n-dodecylpentaethylene glycol ether).34 For this system, the spontaneous curvature was found to have the following temperature dependence:34 -3 -1 〈H0 〉 ) -1.22 × 10 (T° - T) Å
(10)
where T°, the temperature of zero spontaneous curvature, is equal to 32.6 °C for this system. This temperature is also defined as the PIT and corresponds to the optimal temperature used in the HLD scale. A number of experimental studies performed on CiEj have demonstrated that this temperature dependence of the curvature 〈H0〉 is valid over a significant temperature range covering (20 °C around the temperature of zero spontaneous curvature. The proportionality constant is always around 10-3 K-1 nm-1 for CiEj surfactants.35,36 Now we use the coverage relation (eq 9) and make the reasonable assumption that the average spontaneous (34) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (35) Olsson, U.; Wennerstroem, H. AdV. Colloid Interface Sci. 1994, 49, 113. (36) Le, T. D.; Olsson, U.; Wennerstroem, H.; Schurtenberger, P. Phys. ReV. E 1999, 60, 4300.
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Langmuir, Vol. 25, No. 1, 2009 115
Gaussian curvature is only a little temperature dependent (Gauss-Bonnet theorem).13 As long as the film does not break, we have
(p0(T) - 1)/l ) 〈H0 〉 ) - 1.22 × 10-3(T° - T) Å-1 (11) For this nonionic system, freeze fracture, electric conductivity, and scattering were studied simultaneously.34 The average length l of the surfactant molecules in the interfacial film was inferred from neutron scattering and found to be l ) 30 Å;37 consequently,
p0(T) - 1 ) - 0.0366(T° - T)
(12)
It should be noted here that a similar definition, but considering only the average length of the tail of surfactant, can be used if the same reference is taken to define the packing parameter. On the other hand, the factor c in eq 1, expressing the temperature dependence of HLD values of nonionic surfactants, is about 0.06 K-1, roughly independent of the type of nonionic surfactant considered.15 Equation 1, applied at the temperature of zero spontaneous curvature T° (where HLD ) 0) allows us to scale the HLD when only the temperature varies, all other quantities remaining constant (eq 13):
HLD ≈ 0.060(T - T°)
(13)
This linear equation approximates the experimental data over a significant temperature range sufficiently well.14 Combining the last two equations finally leads to the desired relation:
p0 - 1 ≈ 0.610 × HLD ≈ 〈H0 〉 × l
(14)
Equation 14 gives the fundamental link that allows now for the conversion of the three concepts discussed above. It should be noted that these conversions do not imply any assumption about the flexibility or rigidity of the surfactant films. As eqs 10 and 13 depend only slightly on the nature of the CiEj surfactant (cf. refs 34 and 35 for eq 10 and ref 15 for eq 13), the conversion between HLD, packing parameter, and curvature can be extended to the whole CiEj surfactant family. Table 1 and the corresponding Figure 1 illustrate the link between the packing parameter p0 (and hence the curvature) and (37) Strey, R.; Glatter, O.; Schubert, K. V.; Kaler, E. W. J. Chem. Phys. 1996, 105, 1175.
the HLD on the abscissa for the studied system. Beyond the correspondence between p0 and HLD, Figure 1 also illustrates the expected relation for nonzero spontaneous curvature, provided that the temperature is not too far from the PIT. It would now be interesting to follow the same strategy to obtain a correspondence between p0, H0, and HLD for other nonionic as well as zwitterionic and ionic surfactants. We can expect a strong dependence of the relation on the nature of the polar surfactant headgroup. A generalization of our approach could help to better explain the empirical hydrophilic-hydrophobic balancing rules that have been used in formulation chemistry for many decades now. At the end, we note that there is an inherent problem in the definition of the HLD scale that makes consistency with other scales difficult: the salt sensitivity parameter b in eq 1 and the arbitrary logarithmic expression of salt concentration dependence in eq 2 do not take into account any ion specificities. Consequently, cases where counterions are more or less “bound” or “complexed” to headgroups are not properly taken into account.24 The effect that the area per molecule depends on ion equilibrium and the spontaneous packing parameter is coupled to nonelectrostatic ion adsorption to the interface is not foreseen in the present formulation of the HLD equations. As long as only NaCl is considered, this is not a problem (because this effect is the only one that is included). However, other ions (very chaotropic or very cosmotropic ones) may negate the relations, depending on the type of surfactant.
Conclusion Although the models of curvature, global packing, and hydrophilic-lipophilic deviations are based on different concepts and were developed for different film flexibilities, they can be linked. The key is that all three concepts are based on a common reference point, which is given at the “optimum formulation”, where the spontaneous curvature is 0, the packing parameter is 1, and the HLD value is 0. Then, from the temperature dependence of curvature and HLD, the correspondence between all three parameters and concepts can be derived. LA8028879
