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Monolayer Frustration Contributions to Surface and Interfacial Tensions: Explanation of Surfactant Superspreading Alexey Kabalnov Hewlett-Packard Company, 1000 NE Circle Boulevard, Corvallis, Oregon 97330 Received September 14, 1999. In Final Form: November 16, 1999 A simple model for calculation of the spreading coefficient of an aqueous surfactant solution on an apolar solid is proposed. The spreading coefficient is predicted to have two components: (i) the van der Waals component, which is similar to the spreading coefficient of the alkane, making up the surfactant tail; (ii) the monolayer frustration component, dependent on the bending moduli and the spontaneous curvature of the surfactant. The frustration term is minimized at a negative spontaneous curvature, above the surfactant cloud point. The maximum of the spreading coefficient as the function of spontaneous curvature H0 does not coincide with the minimum of the surface tension. The latter is predicted to occur approximately 2 times higher in absolute value negative H0. For a solution to spread, the van der Waals component of the spreading coefficient must be positive and larger than the monolayer frustration term. The spreading is facilitated by surfactants having very short and branched alkyl tails.
Introduction Spreading of aqueous surfactant solutions over lowpolarity solids, called superspreading, is a topic of considerable theoretical and practical interest.1-3 The most powerful empirical correlation for spreading of liquids on solids is the critical surface tension concept of Zisman.1 According to this approach, every solid is characterized by a specific critical surface tension value, σc. All the liquids that have a surface tension σ larger than σc, tend to contract into a lens on this solid, while the liquids with σ less than σc tend to spread. The correlation works well for one-component fluids. Experiments have shown, however, that it is less accurate for aqueous surfactant solutions where the low (16-25 mN/m) surface tension does not warrant superspreading.3,4 Alternatively, it has been suggested that superspreading is facilitated by a low spontaneous curvature of the surfactant,5,6 the presence of dispersed vesicles in water,7 specific “umbrellatype” or “hammer-type” architecture of the surfactant tail,4 and a low polarizability of the tail (e.g., the presence of siloxane groups in the tail). It has been shown that neither of these features is required, but all of them facilitate spreading to some degree.3 The mechanism underlying these effects remains obscure. Recently, a systematic study of surfactant-assisted spreading on apolar surfaces was conducted by Stoebe et al.8-11 The authors studied spreading of a wide variety of (1) Zisman, W. A. In Contact Angle: Wettability and Adhesion; Fowkes, F., Ed.; American Chemical Society: Washington, DC, 1964; Vol. 43, pp 1-51. (2) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827-863. (3) Hill, R. M. Curr. Opin. Colloid Interface Sci. 1998, 3, 247-254. (4) Ananthapadmanabhan, K. P.; Goddard, E. D.; Chandar, P. Colloid Surf. 1990, 44, 281-297. (5) Griffin, W. C. J. Soc. Cosmetic Chem. 1949, 1, 311-327. (6) Griffin, W. C. J. Soc. Cosmetic Chem. 1954, 5, 249-256. (7) Zhu, S.; Miller, W. G.; Scriven, L. E.; Davis, H. T. Colloids Surf. A 1994, 90, 63-78. (8) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1996, 12, 337-344. (9) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1996, 13, 7282-7286. (10) Stoebe, T.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7276-7281.
surfactants over the surface of gold, chemically modified by organosulfur monolayers. Chemical modification allowed the authors to continuously vary the surface energy of the substrate. The authors concluded that the spreading kinetics as a function of polarity of the solid follows a very general pattern. At low surface polarity, no spreading occurs. As the surface tension of the solid increases, the spreading rate steeply increases, reaches maximum, and then decays back to zero. This trend has been shown to apply to all classes of surfactants studied (ionic, nonionic, hydrocarbon, and siloxane tail). The presence of vesicles, hammer shape of surfactant molecules, and the low polarity of the surfactant tails shifted the onset of spreading to lower surface energies of the solid, but did not change the overall trend. Another interesting aspect of superspreading is its temperature dependence. Recently, Wagner et al. demonstrated that monodisperse ethoxylated siloxanes show a maximum of the spreading rate as a function of temperature, which is located at the two-phase region just above the single-phase L3 region.12 Again, the origin of this effect remains unclear. The objective of this paper has been to evaluate the spreading coefficient S of the surfactant solution on the solid. S is described by
S ) σsa - σsw - σaw
(1)
where σsa, σsw, and σaw are the interfacial tensions between the solid and air, solid and water, and air and water, respectively. For S > 0, the solution spreads on the solid, while for S < 0, it contracts into a lens with a finite contact angle. In the model, the solid is assumed to be “dry”; that is, no precursor film on the surface is allowed. Also, the surfactant is assumed to form micelles in water. By “micelles”, any equilibrium type of surfactant aggregates is assumed, including the lamellar phase. The surfactant concentration is therefore assumed to be larger than the (11) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7270-7275. (12) Wagner, R.; Wu, Y.; Czichocki, G.; Berlepsch, H. V.; Rexin, F.; Perepelittchenko, L. Appl. Organomet. Chem. 1999, 13, 201-208.
10.1021/la991215t CCC: $19.00 © 2000 American Chemical Society Published on Web 02/19/2000
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critical micelle concentration (CMC). Whenever a surface or interfacial tension of a surfactant solution is mentioned, the value above the CMC is implied. The spreading coefficient of surfactant solutions is studied as a function of the surfactant spontaneous curvature, H0, which is a parameter characterizing the surfactant shape. At positive H0, the surfactant head is larger than the tail and vice versa. The spontaneous curvature is allowed to vary over the whole spectrum, from positive to negative values. For aqueous surfactant systems, this means that the system is not in the singlephase state over the whole H0 range. Indeed, on one hand, a positive H0 is compatible with a single-phase O/W micellar solution, whereas zero spontaneous curvature in a binary surfactant-water system corresponds to a twophase coexistence of a dilute isotropic aqueous phase, L1, and a lamellar phase, LR, or an L1 and a sponge phase, L3. On the other hand, negative spontaneous curvatures correspond to a two-phase equilibrium of an inverse micellar solution, L2 and L1. This model does not require the surfactant solution to be in a single-phase state, and a multiphase state is allowed. Finally, the paper only addresses the question of whether the solution spreads on the surface of a given polarity. The spreading kinetics is beyond the scope of this paper. The paper has the following outline. In the first section, the surface and interfacial tensions are split into the van der Waals term and the monolayer frustration term. In the next section, the surface tension of aqueous surfactant solutions is considered in detail. Then, the theory is extended to the solid-water interfacial tension. Next, the spreading coefficient is calculated and its behavior is analyzed as a function of the monolayer spontaneous curvature and the solid polarity. The modeling is done on an example of aqueous solutions of ethoxylated alcohols, CiH2i+1(OCH2CH2)jOH ) CiEj, because the phase behavior and monolayer bending parameters of these systems have been relatively well established.13-17 Preliminary results of this study have been discussed elsewhere.18 Components of Spreading Coefficient In surfactant science, it is common to split the surface tension of surfactant solutions into two terms as follows,
σaw ) σ0aw - Πaw(µ)
(2)
where σ0aw is the surface tension of pure water and Πaw is the surfactant surface pressure, which is the function of the surfactant chemical potential in solution, µ. A similar equation can be written for the solid-water interfacial tension:
σsw ) σ0sw - Πsw(µ)
(3)
The spreading coefficient now becomes
S ) S0 + Ssurf(µ)
(4)
where S0 ) σsa - σ0aw - σ0sw is the “bare spreading coefficient”, that is, the spreading coefficient of the pure (13) Kahlweit, M.; Strey, R. Angew. Chem., Int. Ed. Engl. 1985, 24, 654. (14) Strey, R.; Schoma¨cker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86, 2253-2261. (15) Strey, R. Colloid Polym. Sci 1994, 272, 1005-1019. (16) Sottman, T.; Strey, R. J. Chem. Phys. 1997, 106, 8606-8615. (17) Sottman, T.; Strey, R.; Chen, S. H. J. Chem. Phys. 1997, 106, 6483-6491. (18) Kabalnov, A. Eur. Phys. J. 1999, submitted.
Figure 1. Pressure profiles in surfactant monolayers adsorbed at the air-water and solid-water interfaces. Here ds is the dividing surface of eq 6, and ns (z ) 0) is the neutral surface of eq 11.
water on the solid, and Ssurf(µ) ) Πaw(µ) + Πsw(µ) is the surfactant contribution to the spreading coefficient. Because the surfactant precursor film at the air-solid interface is not allowed, the solid-air surface tension does not depend on the presence of a surfactant and is equal to the bare solid-air surface tension. In this paper, the surface and interfacial tensions are split into two terms in a less traditional way. Figure 1 shows the pressure profile at the air-water and solidwater interfaces covered with a nonionic surfactant monolayer. For the sake of simplicity, we start with the air-water interface. As one moves from the water phase to the air phase, one encounters regions of (i) steric repulsions between polar heads; (ii) van der Waals + polar attraction at the “oil-water” interface, that is, at the interface between the alkyl tails and polar heads; (iii) zone of steric repulsion between the alkyl tails; (iv) van der Waals attraction at the air-water interface. The surface tension of the interface σ, according to the Bakker equation, is the zeroth moment of the pressure profile p(z),
σaw )
∫-∞+∞[p(z) - p0] dz
(5)
where p0 is the pressure far away from the interface. In this paper, this integral is split in two: the one above and another below the dividing surface shown in Figure 1:
σaw ) σair-tails + σheads-tails
(6)
The first integral is identified with the van der Waals (vdW) interfacial tension between the surfactant tails and air. It is analogous to the surface tension of the respective ≡ σair-tails. The second alkane and is defined as σvdW aw integral is the interfacial tension between the heads and tails. It will be defined as the monolayer frustration contribution to the surface tension, σˇ aw ≡ σheads-tails. The surfactant tails are assumed to be completely apolar, so on one hand, the interactions between the tails and air are of purely dispersion origin, which justifies the use of the term “van der Waals component” throughout the paper. On the other hand, the interactions at the junction of the hydrophilic and hydrophobic surfactant blocks, which are incorporated into σˇ aw, include a strong polar component. After treating the interfacial tension between the apolar solid and water similarly, the spreading coefficient becomes the difference of two terms,
ˇ (µ) S ) SvdW - S
(7)
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vdW where SvdW ≡ σsa - σvdW sw - σaw is the spreading coefficient of the surfactant tails, called below the van der Waals component of the spreading coefficient, and S ˇ (µ) ≡ σˇ aw(µ) + σˇ sw(µ) is the monolayer frustration contribution to the spreading coefficient. It should be emphasized that the deconvolution of the spreading coefficient with eq 7 is completely different from that of eq 4. While the bare spreading coefficient of eq 4 is that of the pure water on the solid, the van der Waals component to the spreading coefficient in eq 7 is that of the “oil”, composed of the ˇ terms are not surfactant alkyl tails. Similarly, Ssurf and S equal to each other. While the Ssurf term measures the difference between the spreading of the surfactant solution and pure water, the S ˇ term evaluates the difference between the surfactant solution and the oil. On one hand, the separation of the terms, as described by eq 4, is appropriate at very low surfactant coverages. On the other hand, in the limiting case of nearly saturated monolayers considered in this study, eq 7 will be shown below to have more physical sense.19 The problem is reduced to evaluating σˇ aw(µ) and σˇ sw(µ) terms. The surfactant chemical potential is known to level off above the critical micelle concentration C ) cmc. One now needs to determine σˇ aw[µ(cmc)] and σˇ sw[µ(cmc)] values, utilizing the fact that the surfactant micelles and the monolayers at the solid-water and air-water interfaces are in thermodynamic equilibrium.
Surface Tension of Surfactant Solution above the Critical Micelle Concentration To evaluate σˇ aw, consider a surfactant micelle in equilibrium with the surfactant monolayer at the airwater interface. The chemical potential of the surfactant molecules in these assemblies has the following components: (1) stretching elastic energy; (2) bending elastic energy; (3) entropy of mixing. The entropy of mixing is neglected in this paper, by analogy with earlier theories of interfacial tension.21,22 The monolayer bending energy can be treated in terms of the Helfrich equation:23
∆G ) dA[2κ(H - H0)2 + κjK]
µˇ (a/w monolayer) ) 2κawHaw02As
Here, As is the area per surfactant molecule; the aw subscripts were introduced to indicate that the monolayer spontaneous curvature and bending modulus at the airwater interface are different from the ones at the oilwater interface. On the other hand, the frustration free energy of the monolayer in a micelle, neglecting the entropy contribution, is 24
µˇ (micelle) )
κjκH02 A* κ+ κj/2 s
(19) The term “van-de-Waals component of the spreading coefficient” can be misleading because it resembles the separation of the surface/ interfacial tensions of pure liquids into the dispersion and polar components, suggested by Fowkes.20 As has been already mentioned, the interactions of the surfactant tails with air are assumed to be of a purely dispersion origin, and no polar contribution is allowed. Therefore, S ˇ should not be interpreted as the term responsible for the “polar contribution” to the spreading coefficient. (20) Fowkes, F. M. Ind. Eng. Chem. 1964, 56, 40-52. (21) Robbins, M. L. In Micellization, Solubilization and Microemulsions; Mittal, K. L., Ed.; Plenum Press: New York, 1976; Vol. 2, pp 713-754. (22) de Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982, 86, 22942304. (23) Helfrich, W. Z. Naturforsch. 1973, 28c, 693-703.
(10)
where As* is the area per surfactant molecule in the micelle. Note that, from now on, the bending moduli and the spontaneous curvature parameters without subscripts refer to the monolayer, composing the micelle. In general, whenever the terms “spontaneous curvature” and “bending modulus” are mentioned below, the values of the micellar monolayer are implied, unless stated otherwise. One needs now to relate the κaw and Haw0 values to κ and H0. From the microscopic thermodynamics of surfactant monolayers, it follows that the spontaneous curvature is proportional to the first moment of the pressure profile:25,26
-2κawHaw0 )
∫-∞+∞(z - zj)p(z) dz
(11)
where zj is the location of the neutral surface, which does not change its cross-sectional area under the bend. Below, the origin of the z axis is placed at the neutral surface so that zj ) 0. The pressure profile within the surfactant monolayer at the air-water interface contains an additional van der Waals attraction region shown in Figure 1. It is assumed below that the bending modulus is not affected by the van der Waals attraction region: κaw ≈ κ, and the shift in the location of the neutral plane is neglected, zjaw ≈ zj ) 0; see ref 18 for arguments. However, there is a significant change in the monolayer spontaneous curvature. As becomes clear from Figure 1, one can expect that
(8)
where G is the free energy, A is the surface area of a monolayer patch, κ is the monolayer bending modulus, κj is the saddle splay modulus, H0, cm-1, is the spontaneous curvature, H ) (H1 + H2)/2 is the mean curvature of the monolayer, K ) H1H2 is the Gaussian curvature, H1 ) 1/R1, H2 ) 1/R2, and R1 and R2 are the principal radii of curvature. In this paper, the common sign convention is followed, according to which the positive curvature corresponds to oil-in-water drops. From eq 8, it follows that the bending free energy per molecule of the surfactant monolayer at the air-water interface, µˇ , is simply
(9)
Haw0 ) H0 +
δσvdW aw 2κ
(12)
where δ is the location of the air-tail vdW attraction with respect to the neutral plane. Here, the air-tail vdW pressure is modeled as a δ function; that is, the normal distribution of the vdW stress is neglected. The effect of z distribution of air-tail vdW stress is considered in the Appendix and is shown to be small. Similarly, the difference between the areas per surfactant molecule at the a/w monolayer and the micelle are neglected: As ≈ As* as a first approximation. From eqs 9, 10, and 12, one concludes that the curvature free energy difference between the micelles and the monolayer is
µˇ (a/w monolayer) - µˇ (micelle) ∆µˇ ) ) As* As* 2κ2 (H + ∆Haw)2 + σˇ °aw (13) κ + κj/2 0 where (24) Safran, S. A. Phys. Rev. A 1991, 43, 2903-2904. (25) Helfrich, W. In Physics of Defects; Balian, R., Ed.; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1981; pp 716-755. (26) de Gennes, P. G. J. Phys. Chem. 1990, 94, 8407-8413.
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∆Haw )
Kabalnov
δσvdW j/2) aw (κ + κ
(14)
2κ2
and
σˇ 0aw ) -
2 2 j (σvdW aw ) δ κ
(15)
4κ2
One can now make use of the fact that the macroscopic air-water monolayer and microscopic monolayer forming the micelle are in thermodynamic equilibrium. The curvature component of the surfactant chemical potential, µˇ , in the micelle is lower. Accordingly, the lateral pressure in the micelle must be higher to compensate for this difference. This means a lower surface pressure, Πs, at the macroscopic interface, so that
∆µˇ ) ∆µ(lateral pressure)
(16)
and
-∆ΠsAs* ) As*
(
2κ2 (H + ∆Haw)2 + σˇ 0aw κ + κj/2 0
)
(17)
One concludes that
σˇ aw ) -∆Πs ≈
2κ2 (H + ∆Haw)2 + σˇ 0aw (18) κ + κj/2 0
or, finally, assuming zero surface tension of the micelle in the Schulman limit
ˇ aw ) σvdW ˇ 0aw + σaw ) σvdW aw + σ aw + σ
2κ2 (H + ∆Haw)2 κ + κj/2 0 (19)
Equation 19 is based on the assumption that the areas per surfactant molecule in the micelle and in the a/w monolayer are the same. As the next approximation, this difference can be accounted for as follows:
(
2κ2 (H + ∆Haw)2 + σˇ 0aw - σvdW aw κ + κj/2 0 As ) As* 1 + λ
)
(20)
where λ is the monolayer compression modulus. The last equation reflects the fact that the monolayer at the airwater interface, in comparison with the micellar monolayer, is under the air-tails-extra vdW pressure, minus the frustration-related release in the pressure at the heads-tails interface. The effect of the difference between As and As* on the surface tension is expected to be small and will be ignored below, although it can be easily accounted for. One concludes, therefore, that the surface tension as a function of spontaneous curvature shows a parabolic dependence, just like the interfacial tension between the oil and water.21,22 The minimum in the surface tension is observed, however not in the balanced state at H0 ) 0, but at a negative spontaneous curvature, H0 ) -∆Haw. At this point, the surface tension does not vanish, but has a + σˇ 0aw. The area per surfactant finite value of σvdW aw molecule, As, has a minimum at H0 ) -∆Haw. The ∆Hair value is proportional to the distance of the vdW attraction versus the neutral plane, δ, the vdW surface tension, σvdW aw , and the reciprocal in the bending modulus, κ. One can verify eqs 19 and 20 by plotting the surface tension of surfactant solutions as a function of H0. The
spontaneous curvature can be controlled in many ways, including the change in alkyl chain architecture and length, polar head architecture and length, salting-out effect, and temperature.13,15,27 In general, H0 increases when the polar head gets more bulky and the alkyl tail gets more compact. For some of the CiEj surfactants, the experimental data on H0 has recently become available. In particular, it has been shown that the spontaneous curvature nearly linearly decreases with temperature because of the thermal dehydration of ethylene oxide groups,15
H0 ) R(T h - T)
(21)
where R is the expansion in series coefficient, and T h is the balanced temperature, at which the spontaneous curvature is equal to zero. Figure 2 provides a fit of the experimental surface tensions of C12E5 as a function of temperature. The selection of parameters is discussed in detail in the Appendix. The model predicts that the surface tension of C12E5 parabolically decreases with temperature and reaches minimum above 100 °C. The phase diagram of C12E5 in water is also shown in Figure 2. As the temperature is increased at a constant surfactant concentration (say, 3%), a number of phase transitions occur and the following sequence is seen: L1 - (L1′ + L1′′) LR - (LR + L3) - L3 - (L3 + L1) - (L1 + L2). Here, L1′ and L1′′ are the isotropic water-rich phases, L2 is the isotropic surfactant-rich phase, LR is the lamellar phase, and L3 is the sponge phase. Note that the minimum in the surface tension is predicted over the L1 + L2 twophase region, far above the balanced point (T h ∼ 60 °C) and the first cloud point (Tβ ) 32 °C). The balanced point in binary nonionic surfactant-water systems is normally associated with the midpoint temperature of the dilute lamellar-phase region. Figure 3 fits the experimental surface tensions of the homologous series of C12Ej surfactants. The surface tensions are predicted to decrease with the diminishing of the ethylene oxide chain length and increasing of the temperature, in a good qualitative agreement with experiment.28 The area per surfactant molecule, As, at the CMC of CiEj surfactants is known to increase with the ethylene oxide chain length and, less intuitively, to decrease with the alkyl tail length,31,32 see Figure 4. While the increase of As with j can be interpreted as an increase in the overall molecular size, the decrease with i can be attributed only to the spontaneous curvature effect, as described by eq 20. Figure 4 shows that, on one hand, the area per surfactant molecule at the air-water interface is smaller than As*, because of the presence of the excess air-tail vdW pressure at the air-water interface. On the other hand, the difference between As and As* increases with j and decreases with i because of the spontaneous curvature effect, as predicted by eq 20. (27) Shinoda, K.; Friberg, S. Emulsions and Solubilization; John Wiley & Sons: New York, 1986. (28) Some predictions of the model can be more readily seen on a homologous series of p-tert-octylphenoxypoly(ethoxyethanol)s, studied over a much wider temperature range.29,30 Here, the minima in the surface tension as a function of temperature are observed over twophase regions, more than 75 °C above the corresponding cloud points. (29) Crook, E. H.; Fordyce, D. B.; Trebbi, G. F. J. Phys. Chem. 1963, 67, 1987-1994. (30) Crook, E. H.; Trebbi, G. F.; Fordyce, D. B. J. Phys. Chem. 1964, 68, 3592. (31) Rosen, M. J.; Cohen, A. W.; Dahanayake, M.; Hua, X. J. Phys. Chem. 1982, 86, 541-545. (32) Rosen, M. Surfactants and Interfacial Phenomena; John Wiley: New York, 1989.
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Figure 2. (Left) Phase diagram of the C12E5-water system. Reproduced, with permission, from ref 37. Copyright Wiley & VCH. (Right) Surface tension above cmc versus temperature for C12E5 in water: theory (line) versus experiment (points). The experimental data are reproduced from ref 31.
Figure 3. Surface tensions of C12Ej surfactant solutions at 10-40 °C versus monolayer spontaneous curvature: theory (line) versus experiment (points). The experimental data are reproduced from refs 31, 32, and 38.
Interfacial Tension between an Apolar Solid and an Aqueous Surfactant Solution The formalism of the previous section is directly applicable here, with the difference that the air phase is now replaced by a solid. That is, the van der Waals vdW pressure, σvdW aw ≡ σair-tails, needs to be replaced by σsw ≡ σsolid-tails, while all the other terms in the equations remain the same. The solid is assumed to be apolar. Therefore, the interactions between the surfactant tails and the solid are of a purely dispersion origin, which can be approximated by the classical Girifalco-Good equation:33 vdW vdW 2 σvdW sw ) (xσaw - xσsa )
Figure 4. Areas per surfactant molecule at the air-water interface at the CMC As for CiEj surfactants at 25 °C.32 For comparison, the area per surfactant molecule in surfactant micelles As* is shown.17 (a) C12Ej series; (b) CiE6 series.
interfacial tension between the solid and aqueous surfactant solution becomes equal to
σas ) σvdW ˇ sw ) σvdW ˇ 0sw + sw + σ sw + σ where
(22)
Recall that, in our notation, the σvdW aw term is the surface tension of the surfactant alkyl tails, not the water. The (33) Good, R. J. In Contact Angle: Wettability and Adhesion; American Chemical Society: Washington, DC, 1964; Vol. 43, pp 74-87.
2κ2 (H + ∆Hsw)2 κ + κj/2 0 (23)
∆Hsw )
δσvdW j/2) sw (κ + κ 2κ2
(24)
and
σˇ 0sw ) -
2 2 (σvdW j sw ) δ κ
4κ2
(25)
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One concludes that the interfacial tension between the surfactant solution and a solid shows the same parabolic dependence on the spontaneous curvature as the surface tension. The negative shift of the minimum with respect to the balanced point, ∆Hsw, increases with the surface tension of the solid. This has important implications for the spreading behavior, as discussed in the following section. Spreading Coefficient By combining eqs 19 and 23, one concludes that the spreading coefficient for a surfactant solution on a solid is determined by eq 7, where vdW vdW vdW vdW vdW SvdW ≡ σvdW sa - σsw - σaw ) 2(xσsa σaw - σaw ) (26)
S ˇ ) ∆H ˇ )
4κ2 (H + ∆H ˇ )2 + S ˇ0 κ + κj/2 0
(27)
∆Haw + ∆Hsw δ(κ + κj/2) vdW ) (σaw + σvdW sw ) (28) 2 2 4κ
and the S ˇ 0 value, called below the frustration minimum, is equal to
S ˇ0 ) -
vdW 2 δ2(σvdW j aw + σsw ) κ
8κ2
+
vdW 2 δ2(σvdW aw - σsw ) 4κ
(29)
Despite some bulkiness of the formulas, the results are quite transparent: the van der Waals spreading term is the classical spreading coefficient of an apolar liquid (in this case, hydrocarbon tails) over the solid, constructed with the Girifalco-Good theory. Within this theory, SvdW > 0 whenever the tails have a lower surface tension than the solid because the penalty of creating the solid-tail and the air-tail interfaces is always overcome by the free energy decrease related to the reduction of the air-solid ˇ interface. The SvdW value is counterbalanced by the S frustration term, which is positive (excluding from consideration some unrealistic cases, involving positive κj values) and, therefore, always opposes spreading. One can state that the frustration term incorporates into itself the free energy contributions that make the water drop covered by the surfactant monolayer spread differently from the oil drop. This difference comes not from the finite thickness of the alkyl tail layer, but rather from the monolayer frustration effect. (The effect of the finite thickness of the surfactant monolayer is discussed in more detail in the Appendix). The frustration term quadratically depends on the spontaneous curvature and has a minimum ˇ , which at the negative spontaneous curvature, H0 ) -∆H is situated at the midpoint between the σaw and σsw minima. The frustration term does not vanish at this point, but is equal to S ˇ 0. Accordingly, a surfactant solution spreads on a solid when the following conditions are met: (i) The van der Waals spreading coefficient, SvdW, is not only positive but also larger than the frustration minimum value, S ˇ 0. (ii) The spontaneous curvature is negative and located close to the point where the frustration term is minimized. Figure 5a shows the values of the surface and interfacial tensions and the spreading coefficient of C12Ej solutions on a solid with the surface tension of 40 mN/m, as a function of the monolayer spontaneous curvature at 25 °C. One can see that both the surface and interfacial tensions parabolically depend on the spontaneous cur-
vature with the minima at -5 × 106 and - 7 × 105 cm-1, respectively. The frustration term is minimized in the midpoint, at - 3.1 × 106 cm-1. The spreading coefficient is positive over the range of spontaneous curvatures, from 5.1 × 106 to - 1.1 × 107 cm-1, with the maximum value of 12 mN/m at the midpoint. More specifically, the model predicts that surfactants with 2 < j < 7 will spread on the solid, while the surfactants with j g 7 will not. This contradicts the original Zisman critical surface tension concept because the surfactants with j ) 7-11 have the surface tension below 40 mN/m and therefore spread. The effect stems from the monolayer frustration term, not accounted for in the Zisman’s model. In other words, the maximum of the spreading coefficient is offset from the minimum of the surface tension, and the low surface tension may not warrant superspreading. This point is further illustrated in Figure 5b,c where the spreading coefficient of an aqueous solution of C12E5 is plotted as a function of temperature. One can see that a closed-loop temperature dependence of the spreading coefficient is predicted, with no spreading above and below specific temperature end points. For a lower energy solid with the surface tension of 25 mN/m, the spreading range becomes very narrow, only a few degrees centigrade. The maximum of the spreading coefficient is predicted to occur close to the L2-L1 two-phase coexistence line, both for the higher and lower energy solids. This location is substantially offset from the surface tension minimum, which is expected at much higher temperatures, although over the same L1 + L2 two-phase region. Another important aspect to consider is the dependence of the spreading coefficient on the surface tension of the solid at a constant value of the surfactant spontaneous curvature. This dependence is shown in Figure 6 where the C12E5 surfactant spreading is simulated at 25 °C. One finds that the positive spreading coefficients are observed in the range 34 < σsa < 200 mN/m, with no spreading outside this range. The reason for this behavior is the following. At very low surface tensions of the solid, the spreading does not occur because the van der Waals component of the spreading coefficient is negative. As the surface tension of the solid increases, the SvdW increases and initially overweighs the frustration term, and the spreading becomes favorable. However, as the frustration minimum shifts more and more toward negative spontaneous curvatures, the frustration term also increases. Eventually, the frustration term becomes larger than the van der Waals term and the spreading coefficient becomes negative. The theory is able therefore to model the “closedloop” wetting behavior by surfactants observed experimentally.8-11 Let us now discuss in more detail the requirements posed on surfactants to show superspreading behavior on solids with low surface energy, ∼ 9, which apparently means that beyond this range the spreading coefficient is negative. The theory of this paper provides an explanation for the statics of spreading. Although the dynamics of spreading is, strictly speaking, beyond the scope of this paper, it is interesting to note that the maximum of the spreading rate as a function of T is observed over the two-phase region, just above the single-phase L3 region. On a qualitative level, the location of the maximum found by Wagner et al. correlates well with the maximum of the spreading coefficient, as predicted by this model, and is shown in Figure 5a-c. One can hypothesize therefore that for the spreading of aqueous solutions on low-energy surfaces, the precursor film formation may not occur, and the spreading rate can be controlled by the value of the spreading coefficient. Indeed, very recently it has been shown that the spreading of aqueous solutions of siloxane surfactants on hydrocarbons occurs without the stage of precursor film formation.36 Several aspects of spreading have not been covered in this paper. Polar interactions, that is, the interactions of a non-van der Waals nature are traditionally invoked to explain the difference between the experimental contact angles of liquids on solids and the Girifalco-Good theoretical estimates.20 While this approach is permissible for the spreading of pure liquids on solids, it is less so for aqueous surfactant solutions, where the monolayer frustration term becomes another factor to consider. In other words, the polar interaction and the monolayer frustration contributions must be carefully discriminated. In this paper, on one hand, the interactions of the surfactant tails with apolar solids (such as, e.g., polyethylene) are assumed completely apolar. On the other hand, the interactions at the surfactant head-tail (oil-water) interface contain a substantial polar component. However, these interactions drop out from the final equations because they are present both in the micelles and in the adsorbed layers. Of course, there is some interaction between the solid-tails and head-tails planes, but it is expected to be rather small if the hydrocarbon tail is long enough, as discussed in the Appendix. Another aspect ignored in this paper is the entropy of mixing term of the micelles. This term is traditionally neglected in the treatment of the oil-water interfacial tension,21,22 so that the entropy of mixing contribution ends up being incorporated into the bending moduli. While this may sound permissible for the oil-water interfacial tension, it is less so for the surface tension, where the entropic term can skew the surface tension parabola. However, the entropy treatment for the L2 phase is not available at present. Appendix A1. Selection of Parameters. In the simulations shown in Figures 2, 3, 5, and 6, the following parameters (36) Chauhan, A. J.; Svitova, T. V.; Radke, C. J. Talk at 73rd Colloid and Surface Symposium, June 13-16; MIT: Boston, MA, 1999.
Kabalnov
were used. The bending and saddle splay moduli are assumed to be equal to κ ) 1 kT and κj ) -0.36 kT. These values refer to the C12E5 monolayer16 and are used here as estimates for the whole C12Ej homologous series. The σvdW aw value is assumed to be equal to 22 mN/m, which is the critical surface tension value for CH3 groups.1 For comparison, the surface tension of dodecane at 25 °C is equal to 24 mN/m.39 The critical surface tension value has been used instead of the surface tension of dodecane because the monolayer, as opposed to the liquid alkane, is covered by methyl groups which have a lower molecular polarizability than methylene groups.1 The spontaneous curvature of C12E5 is estimated with eq 21, assuming R h ) 60 °C; see ref 15. This value ) 1 × 105 cm-1 K-1, and T of R has been used as an approximation for all the members of the C12Ej homologous series. It has been also assumed that the spontaneous curvature increases by ∼1 × 106 cm-1 per each -CH2CH2O- group added, which corresponds to a 10 °C shift in the balanced point per each ethylene oxide group.37 In fact, the experimental dependence of T h on j is nonlinear; however, this effect is ignored. The δ parameter, which is the offset distance of the vdW air-tail pressure with respect to the neutral surface, has been used as an adjustable parameter of the model. In simulations of Figures 2, 3, 5, and 6, the value δ ) 0.25 nm is used, which is equal to the length of two carboncarbon bonds and cannot be associated with the full length of the hydrocarbon tail. The value increases to ≈0.415 nm, after one takes into account the normal distribution of the vdW air-tail stress; see section A3. A2. Other Effects of the Apolar Phase on Spontaneous Curvature. When the apolar phase can solubilize the surfactant tails, another effect, known as the oil penetration effect into the surfactant monolayer, becomes important.40,41 Thus, for alkane-water-C12E5 systems, the balanced temperatures are equal to 33 °C (octane), 39 °C (decane), and 49 °C (tetradecane), which is lower than the balanced point of C12E5 in water without the oils present (≈60 °C).16 This indicates a lower spontaneous curvature of C12E5 in the presence of alkanes. In this study, the solid and the surfactant tails are assumed to be insoluble in each other and this effect is irrelevant. A3. Distribution of vdW Stress in Normal Direc. tion: Effect on Haw0 and σvdW aw In this paper, the airtail stress is modeled as a δ function; that is, it is considered to be localized in a point. In reality, there is a distribution of the vdW stress in the z direction. This distribution can affect H0 and σaw. First, because of the distribution, the positive increment of H0 can be overestimated. Second, the air-tail surface tension can be somewhat different from the surface tension of the respective alkane because the alkyl tail region has a finite thickness. The estimates of these effects are given below within the classical macroscopic approach to van der Waals forces.42 The distribution of vdW molecular pressure at the interface between the surfactant monolayer and air is shown in Figure 7. Only the stress originating from the air-tail interface is shown. The interfacial layer is modeled as a sequence of a gas phase, called below “air”, hydrocarbon tail phase, called “oil”, and water phase, called (37) Strey, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 182-189. (38) Corkill, J. M.; Goodman, J. F.; Ottewill, R. H. Trans. Faraday Soc. 1961, 57, 1627-1636. (39) Korosi, G.; Kovats, E. J. Chem. Eng. Data 1981, 26, 323-332. (40) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601-629. (41) Aveyard, R.; Binks, B. P.; Fletcher, P. D. I.; MacNab, J. R. Langmuir 1995, 11, 2515-2524. (42) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: San Diego, CA, 1992.
Explanation of Surfactant Superspreading
Langmuir, Vol. 16, No. 6, 2000 2603
this distance is ≈0.165 nm for a wide variety of molecular liquids.42 In our more complicated case of three phases back-to-back, the surface tension becomes equal to
σ)
A
A
∫a∞p(z) dz ) 24πa1 2 - 24π(a 1+ h)2 + 0
0
0
A2 24π(a0 + h)2 Figure 7. Schematic van der Waals pressure distribution at the surfactant monolayer adsorbed at the air-water interface. Only the vdW pressure originating from the air-tail interface is depicted, and the decay length is artificially exaggerated. The vdW and polar pressures at the head-tail (oil-water) interface are not shown.
“water”. According to the classical Hamaker equation, the local pressure at the planar interface of a an apolar uniform phase can be estimated as
p(z) )
A 12πz3
σ)
A ∫a∞p(z) dz ) 24πa 2 0
Here, A1 and A2 are the Hamaker constants of hydrocarbon tails and water, respectively; h is the thickness of the hydrocarbon tail region. Assuming h ) 1.7 nm, A1 ) 5.0 × 10-20 J, and A2 ) 3.7 × 10-20 J, one concludes that the difference between the surface tension of the macroscopic oil phase and the surface tension of dodecyl alkyl tails is merely 0.05 mN/m and can be neglected. Similarly, the correction for the spontaneous curvature reads as follows:
∆2κHaw0 )
(31)
where A is the Hamaker constant. Accordingly, for the single-phase system, the surface tension can be evaluated as
Here, one needs to introduce the integration cutoff distance a0 of the order of molecular size. Israelachvili argues that
∫a∞p(z)(δ - z + a0) dz 0
(34)
After performing integration and dropping negligibly small terms, one concludes that
∆2κHaw0 ≈ σ(δ - a0)
(32)
0
(33)
(35)
This corresponds to a 0.165-nm correction to the value of δ. LA991215T
