Bending Elasticity of Charged Surfactant Layers: The Effect of Mixing

6796

Langmuir 2006, 22, 6796-6813

Bending Elasticity of Charged Surfactant Layers: The Effect of Mixing L. Magnus Bergstro¨m* Department of Pharmacy, Pharmaceutical Physical Chemistry, Box 580, Uppsala UniVersity, SE-751 23 Uppsala, Sweden ReceiVed February 23, 2006. In Final Form: May 5, 2006 Expressions have been derived from which the spontaneous curvature (H0), bending rigidity (kc), and saddle-splay constant (khc) of mixed monolayers and bilayers may be calculated from molecular and solution properties as well as experimentally available quantities such as the macroscopic hydrophobic-hydrophilic interfacial tension. Three different cases of binary surfactant mixtures have been treated in detail: (i) mixtures of an ionic and a nonionic surfactant, (ii) mixtures of two oppositely charged surfactants, and (iii) mixtures of two ionic surfactants with identical headgroups but different tail volumes. It is demonstrated that kcH0, kc, and khc for mixtures of surfactants with flexible tails may be subdivided into one contribution that is due to bending properties of an infinitely thin surface as calculated from the Poisson-Boltzmann mean field theory and one contribution appearing as a result of the surfactant film having a finite thickness with the surface of charge located somewhat outside the hydrophobic-hydrophilic interface. As a matter of fact, the picture becomes completely different as finite layer thickness effects are taken into account, and as a result, the spontaneous curvature is extensively lowered whereas the bending rigidity is raised. Furthermore, an additional contribution to kc is present for surfactant mixtures but is absent for kcH0 and khc. This contribution appears as a consequence of the minimization of the free energy with respect to the composition of a surfactant layer that is open in the thermodynamic sense and must always be negative (i.e., kc is generally found to be brought down by the process of mixing two or more surfactants). The magnitude of the reduction of kc increases with increasing asymmetry between two surfactants with respect to headgroup charge number and tail volume. As a consequence, the bending rigidity assumes the lowest values for layers formed in mixtures of two oppositely charged surfactants, and kc is further reduced in anionic/cationic surfactant mixtures where the surfactant in excess has the smaller tail volume. Likewise, the reduction of kc is enhanced in mixtures of an ionic and a nonionic surfactant where the ionic surfactant has the smaller tail. The effective bilayer bending constant (kbi) is also found to be reduced by mixing, and as a result, kbi is seen to go through a minimum at some intermediate composition. The reduction of kbi is expected to be most pronounced in mixtures of two oppositely charged surfactants where the surfactant in excess has the smaller tail in agreement with experimental observations.

1. Introduction It is well known that the mixing of two or more surfactants may have a large impact on the behavior of surfactant systems. For example, the critical micelle concentration (cmc) or surface tension of a surfactant film may become significantly reduced in a nonideal fashion, usually referred to as synergistic effects.1-3 Surfactant mixing may also have a substantial influence on the structural behavior of surfactant aggregates. For instance, the promoted spontaneous formation of long threadlike micelles and geometrically closed bilayer vesicles has frequently been observed in surfactant mixtures, in particular, in mixtures of an ionic and a nonionic surfactant or in mixtures of two oppositely charged surfactants.4-6 From a theoretical point of view, it has become widely accepted during recent years that effects due to the curvature of selfassembled structures play a crucial role in determining the size and shape of surfactant aggregates. To explicitly account for the

curvature-dependent behavior of a surfactant film, it has been proven fruitful to employ the expansion to second order in curvature known as the Helfrich expression. Hence, we may write the free energy per unit area γ at a single point on the surface of a surfactant film, most conveniently defined at the interface that divides the hydrophobic and hydrophilic regions from each other, as a function of the mean and Gaussian curvatures, H ≡ (κ1 + κ2)/2 and K ≡ κ1κ2, respectively,

γ(H, K) ) γ0 + 2kc(H - H0)2 + khcK

where κ1 and κ2 are the two principal curvatures at a single point.7 In the present context, we have defined κ1 and κ2 to be positive for a film that appears convexly curved from a position in the hydrophilic phase, such as ordinary surfactant micelles in water. Three parameters, generally referred as to the bending elasticity constants, appear in eq 1, and they are formally defined as follows:

* E-mail: [email protected]. Tel: +46 18 471 43 69. Fax: +46 18 471 42 23. (1) Holland, P. M. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; Chapter 2, p 40. (2) Bergstro¨m, M. Langmuir 2001, 17, 993-998. (3) Rosen, M. J. Surfactants and Interfacial Phenomena, 3rd ed.; John Wiley and Sons: Hoboken, NJ, 2004. (4) Khan, A.; Marques, E. F. Curr. Opin. Colloid Interface Sci. 2000, 4, 402410. (5) Tondre, C.; Caillet, C. AdV. Colloid Interface Sci. 2001, 93, 115-134. (6) Gradzielski, M. J. Phys.: Condens. Matter 2003, 15, R655-R697.

(1)

H0 ) H at kc ≡

∂γ (∂H )

K

)0

( )

1 ∂2γ [H ) H0, K ) 0] 4 ∂H2 K

(7) Helfrich, W. Z. Naturforsch. 1973, 28C, 693-703.

10.1021/la060520t CCC: $33.50 © 2006 American Chemical Society Published on Web 06/27/2006

(2)

(3)

Bending Elasticity of Charged Surfactant Layers

khc ≡

Langmuir, Vol. 22, No. 16, 2006 6797

(∂K∂γ) [H ) K ) 0]

(4)

H

The bending elasticity constants are related to different aspects of the curvature of a surfactant film of given area:8,9 (i) The spontaneous curvature (H0) determines the sign and magnitude of the preferential curvature of the surfactant film. (ii) The bending rigidity or mean bending constant (kc) determines the stiffness or rigidity of the film (or its resistance to deviations from a uniform mean curvature of H0). This means that monodisperse rigid and geometrically homogeneous structures are favored by large values of kc whereas polydisperse, flexible, and geometrically heterogeneous aggregates are expected to form at low kc values. Most interestingly, kc > 0 is a criterion for a surfactant film to be stable because γ otherwise reaches a maximum at H ) H0. In the latter case, higher-order terms in the expansion of γ in curvature may give rise to more than one minimum, and a phase separation is expected. (iii) The saddle splay or Gaussian bending constant khc has the interesting property of determining the topology of the surfactant film. For instance, a collection of small geometrically closed aggregates (micelles or vesicles) are favored by negative values of khc whereas macroscopic bicontinuous structures with several handles or holes are favored by khc > 0). The remaining parameter γ0 may be interpreted as the stretching free energy per unit area (i.e., the surface tension) of a surfactant film with H ) H0 and K ) 0 and may be determined from a constraint of fixed overall surfactant concentration.10 In the present article as well as in previous papers,10-12 we are dealing with the bare bending elasticity constants (which means that neither thermal undulations nor dispersion effects have been incorporated into the definitions of H0, kc, and khc13,14). The free energy of a single surfactant monolayer of arbitrary size and shape may be calculated by simply integrating γ in eq 1 over the entire surface area A:

E≡

∫

γ dA ) γ0A + 2kc A

∫

(H - H0)2 dA + khc A

∫

K dA A (5)

As a consequence, a surfactant film or aggregate that is open in a thermodynamic sense will assume the size and shape that minimizes the overall free energy of a dispersed phase of selfassembled aggregates for a given set of values of H0, kc, and khc.10 The effect of mixing on the bending properties of surfactant monolayers and bilayers has previously been investigated by Safran et al.15,16 using a theory based on pairwise interactions between like and unlike surfactant molecules. It was demonstrated that the effective bilayer bending constant may be reduced as a result of mixing provided the compositions are different in the inner and outer monolayers of a vesicle bilayer, although any mechanism responsible for bringing about this difference in composition was not suggested. Later on, however, it was demonstrated that the additional degree of freedom present in a surfactant mixture may result in the lowering of the bending rigidity of equilibrated surfactant layers.14,17,18 As a result, the (8) Porte, G. J. Phys.: Condens. Matter 1992, 4, 8649-8670. (9) Hyde, S.; Andersson, S.; Larsson, K.; Blum, Z.; Landh, T.; Lidin, S.; Ninham, B. W. The Language of Shape; Elsevier: Amsterdam, 1997. (10) Bergstro¨m, L. M. J. Colloid Interface Sci. 2006, 293, 181-193. (11) Bergstro¨m, M. J. Chem. Phys. 2003, 118, 1440-1452. (12) Bergstro¨m, L. M. Langmuir 2006, 22, 3678-3691. (13) Morse, D. C. Curr. Opin. Colloid Interface Sci. 1997, 2, 365-372. (14) Safran, S. A. AdV. Phys. 1999, 48, 395-448. (15) Safran, S. A.; Pincus, P.; Andelman, D. Science 1990, 248, 354-356. (16) Safran, S. A.; Pincus, P.; Andelman, D.; Mackintosh, F. C. Phys. ReV. A 1991, 43, 1071. (17) Kozlov, M. M.; Helfrich, W. Langmuir 1992, 8, 2792-2797. (18) Porte, G.; Ligoure, C. J. Chem. Phys. 1995, 102, 4290-4298.

effective bilayer bending constant has been shown to become considerably reduced for thermodynamically open vesicles that are formed in surfactant mixtures, and as a consequence, the compositions in the two vesicle monolayers have been found to be different in accordance with the minimization of the free energy in each monolayer.19-21 Similarly, the bending rigidity of bilayers as calculated from a detailed model was found to be reduced as a result of mixing.22 In addition, the spontaneous curvature and bending rigidity for mixtures of surfactants with different tail lengths have been calculated by May and BenShaul for monolayers that are closed in a thermodynamic sense.23 In a recent paper, we have presented a detailed molecular theory from which the different bending elasticity constants for equilibrated, charged one-component surfactant monolayers and bilayers of finite thickness may be quantified.12 It was found that the calculated values of H0, kc, and khc largely depend on whether the surfactant consists of a flexible or rigid hydrophobic tail. The spontaneous curvature was found to become significantly reduced in the former case whereas the bending rigidity kc was increased. Most interestingly, it was demonstrated that the bending elasticity constants of films composed of surfactants with a flexible tail, and equilibrated with respect to layer thickness, are identical as for a film that is bent at constant layer thickness. In the present article, we have extended our recent model for pure surfactant layers so as to also include binary surfactant mixtures.

2. Free Energy of a Mixed Surfactant Layer Following the well-established approach previously adopted by Blankschtein et al.,24 Nagarajan,25 and others, we have written the free energy per aggregated molecule of a surfactant film of arbitrary curvature as a sum of several contributions12,26

 ) hb + chain + el + hg + mix

(6)

where the hydrophobic effect (hb) and chain conformational entropy (chain) are related to the surfactant tails whereas electrostatics (el) and residual headgroup repulsion (hg) are related to the surfactant headgroups. In addition, the free energy of mixing mix must be added for the case of a binary surfactant mixture. The various single free-energy contributions may be calculated from detailed expressions that are briefly discussed below. Hydrophobic Effect. The hydrophobic effect is the driving force for the self-assembly of surfactant molecules. The free energy of bringing hydrophobic tails at constant chemical potentials of free surfactant 1 and surfactant 2 with concentrations c1 and c2, respectively, in a hydrophilic bulk solution into the corresponding hydrophobic liquid bulk phase with a mole fraction x of surfactant 1 may be written as27

hb ) [x(ν1 - ln c1) + (1 - x)(ν2 - ln c2)]kT + aγhb

(7)

where ν1 and ν2 are two constants related to the size and structure of the surfactant tails. A contribution equal to aγhb must be included so as to account for the unfavorable interfacial contact between the hydrophobic film and the hydrophilic solvent, where (19) Bergstro¨m, M.; Eriksson, J. C. Langmuir 1996, 12, 624-634. (20) Bergstro¨m, M. Langmuir 1996, 12, 2454-2463. (21) Bergstro¨m, M. J. Colloid Interface Sci. 2001, 240, 294-306. (22) Bergstro¨m, M. Langmuir 2001, 17, 7675-7686. (23) May, S.; Ben-Shaul, A. J. Chem. Phys. 1995, 103, 3839-3848. (24) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567-5579. (25) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934-2969. (26) Bergstro¨m, M.; Jonsson, P.; Persson, M.; Eriksson, J. C. Langmuir 2003, 19, 10719-10725. (27) Tanford, C. The Hydrophobic Effect;; Wiley: New York, 1980; Chapter 7.

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Figure 1. Schematic representation of a curved surfactant monolayer. In our model, the surfactant film is considered to consist of a hydrophobic inner shell of thickness ξ with the charged headgroups located on a surface located a distance d outside the hydrophobichydrophilic interface. The area occupied by a single surfactant at the hydrophobic-hydrophilic interface is denoted a, whereas ael denotes the corresponding area at the surface of charge.

a denotes the area per surfactant at the hydrophobic-hydrophilic interface and γhb is the hydrophobic-hydrophilic interfacial tension. γhb is readily available from experimental measurements and equals about 50 mJ/m2 for a hydrocarbon-water interface at room temperature. It is important to note that hb becomes inherently dependent on the curvature of the film as a result of x and a both being curvature-dependent quantities. As a result, we may write 1/a and x as second-order expansions in curvature

1 1 2 ) (1 + k′aH + k′′′ aH + k′′′ a K) a ap

(8)

x ) xp(1 + k′xH + k′′xH2 + k′′′ x K)

(9)

and

where ap and xp are the area per surfactant and surfactant mole fraction for a planar layer. k′a, k′′a, k′′′ a and k′x, k′′ x, k′′′ x are constants with respect to H and K that are related to the curvature dependence of a and x, respectively.11 We have previously shown that k′a ) k′′a ) k′′′ a ) 0 for a thermodynamically closed film whereas for a thermodynamically open film k′a, k′′a, k′′′ a and k′x, k′′ x, k′′′ x usually assume some nonvanishing values that may be determined from the equilibrium conditions obtained by minimizing the overall free energy of the surfactant layer with respect to a and x, respectively.12 Chain Conformational Entropy. Restrictions in conformational entropy of aggregated aliphatic chains attached to surfactant headgroups as compared to the corresponding alkane bulk phase give rise to an additional free-energy contribution. It has been demonstrated that this contribution, for a planar layer formed by a single surfactant, may be written only as a function of monolayer thickness ξ (cf. Figure 1).28,29 For a binary mixture, we may write the chain conformational free-energy contribution per surfactant, to a first approximation, in the linear form

chain ) xchain1(ξ) + (1 - x)chain2(ξ)

(10)

where the contributions for surfactants 1 and 2, chain1 and chain2, respectively, are assumed to be independent of composition. Equation 10 is exact within the single-chain mean field approximation29 if the difference in chain length between the two surfactants is moderate. Nevertheless, significant nonlinear effects may arise as the monolayer thickness exceeds the length (28) Gruen, D. W. R.; Lacey, E. H. B. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984; Vol. I, pp 279-306. (29) Gruen, D. W. R. J. Phys. Chem. 1985, 89, 153-163.

of the fully stretched shorter chain.30,31 However, the chain conformational free energy is comparatively small in magnitude for an equilibrated monolayer (about a factor 10 or so lower than the electrostatic free energy or the hydrophobic effect),19,32 although it may become several orders of magnitude larger for thermodynamically closed films.30,31 Moreover, we have already shown in a previous paper that this contribution does not appear explicitly in the expressions for the different bending elasticity constants for a thermodynamically open pure surfactant monolayer.12 Instead, the chain conformational free energy of flexible chains was shown to give rise to huge indirect effects on the bending elasticity constants by making the monolayer behave as if it is bent at constant layer thickness.12 Below we will see that similar conditions also apply for binary mixtures. As a result, the linear approximation in eq 10 is expected to be fully satisfactory for our present purposes of calculating the various bending elasticity constants for thermodynamically open surfactant films. Moreover, for the sake of simplicity we have neglected any explicit curvature dependence that might contribute to chain.10,12 Nevertheless, chain in eq 10 must be implicitly curvaturedependent because both x (cf. eq 9) and ξ depends on H and K. In the latter case, we may write

ξ ) ξp(1 + k′ξH + k′′ξ H2 + k′′′ ξ K)

(11)

where ξp is the thickness of a planar layer and k′ξ, k′′ξ, and k′′′ ξ are constants related to the curvature dependence of ξ.11 Moreover, ξ and a are interrelated through the following condition of geometrical packing constraints

(

1 ξ ξ2 ) 1 - ξH + K a V 3

)

(12)

where V ) xV1 + (1 - x)V2 is the average volume per surfactant tail and the tail volumes of the two surfactants V1 and V2, respectively, are constants for an incompressible medium.11 As a matter of fact, eq 12 relates the three sets of constants defined in eqs 8, 9, and 11, and for the case of two surfactants with different tail volumes, we may write

k′a) k′ξ - ξp - yxpk′x

(13)

k′′a ) k′′ξ - 2ξpk′ξ - yxp(k′xk′a + k′′x)

(14)

k′′′ a )

ξp2 + k′′′ ξ - yxpk′′′ x 3

(15)

where

y≡

V1 - V2 ∆V ) Vp xpV1 + (1 - xp)V2

(16)

is the relative difference in tail volume.11 Terms including y in eqs 13-15 follow as a consequence of the average volume per surfactant tail V being a function of H and K, as a result of x being dependent on curvature in accordance with eq 9. Electrostatics. The electrostatic free energy per aggregated surfactant may be written as (30) Milner, S. T.; Witten, T. A. J. Phys. (Orsay, Fr.) 1988, 49, 1951-1962. (31) Szleifer, I.; Kramer, D.; Ben-Shaul, A.; Gelbart, W. M.; Safran, S. A. J. Chem. Phys. 1990, 92, 6800-6817. (32) Ljunggren, S.; Eriksson, J. C. J. Chem. Soc., Faraday Trans. 2 1986, 82, 913-928.

Bending Elasticity of Charged Surfactant Layers

Langmuir, Vol. 22, No. 16, 2006 6799

el ) |xz1 + (1 - x)z2|gel kT

(17)

for a mixture of two surfactants with charge numbers z1 and z2, respectively. k is Boltzmann’s constant, T is the absolute temperature, and gel denotes the electrostatic free energy per charge in the film. The electrostatic free energy of a charged surface, with an adjacent layer of pointlike counterions, immersed in a polar medium has been shown to be most accurately described by the (nonlinear) Poisson-Boltzmann (PB) equation.33 Within this theory, the electrostatic free energy shows a significant explicit curvature dependence, and as a result, gel may be written as a series expansion to second order in curvature

gel ) g0 + g1H + g2H2 + g3K

(18)

where the coefficients g0, g1, g2, and g3 have been derived for an infinitely thin surface by Lekkerkerker34 as well as Mitchell and Ninham.35 Recently, their results were modified so as to include the case where the surface of charge is located a distance d outside the hydrophobic-hydrophilic interface where H and K are defined (cf. Figure 1).12 As a result,

[

g0 ) 2 ln(S + xS2 + 1) g1 ) g2 )

(

( )

]

xS2 + 1 - 1 S

)

1 + xS2 + 1 4 ln κS 2

(

)

(

)]

2 2 2 8d 1 + xS2 + 1 ln 1 + + κS 2 κ 2S S2 S2xS2 + 1

g3 ) -

[ ((

))

1 + xS2 + 1 2 -1 κ D1 ln κS 2

1 + xS2 + 1 2

mix ) x ln x + (1 - x)ln(1 - x) kT

(26)

(19)

which is valid for an ideal surfactant mixture. mix becomes implicitly curvature-dependent because x, according to eq 9, depends on H and K.

(20)

3. Contributions to the Bending Elasticity Constants

(21)

+

2d ln

It may be noted that terms including d are present only in the expressions for g2 and g3, and by setting d ) 0, the expressions derived by Lekkerkerker and Mitchell and Ninham are recovered. Head Group Repulsion. Effects other than electrostatics as calculated from PB theory that are related to the surfactant headgroups may be incorporated into a single contribution hg. One obvious contribution to hg is due to the shielding of the hydrophobic-hydrophilic interface by an aggregated headgroup of finite size. However, this contribution is independent of curvature and consequently does not contribute to the various bending elasticity constants.10 The most important curvaturedependent contribution to hg is expected to be due to excluded volume repulsion and effects arising as a result of the counterions having a finite size.10 In the present article, we have neglected these contributions (i.e., we have always assumed hg to be a constant). Entropy of Mixing Contribution. Finally, the free energy of mixing two surfactants in a single monolayer may be approximately given by the expression

The free energy per unit area γ ) /a may be calculated from the various contributions mentioned in the preceding section, and by means of employing the definitions in eqs 2-4, we have recently been able to derive general expressions for the various bending elasticity constants in terms of molecular properties of the constituent surfactant as well as solution conditions and the hydrophobic-hydrophilic interfacial tension.11 Our results will be briefly summarized below. It was found that kc, khc, and kcH0 (but not H0 itself) may be written as sums of several contributions

(22) kcH0 ) (kcH0)hb + (kcH0)el + (kcH0)chain + (kcH0)comp (27)

where the Debye function in eq 22 is defined as

D1(x) )

∫0

x

t dt et - 1

2πlB aelκ

(24)

where ael is the area per charge at the surface of charge (cf. Figure 1). κ denotes the inverse Debye screening length, and the Bjerrum length is defined as

lB ≡

eel2 4π0rkT

(28)

khc ) khhb helc + khchain + khcomp c +k c c

(29)

(23)

The coefficients g0, g1, g2, and g3 are all functions of the reduced charge density, which is defined as

S≡

el chain kc ) khb + kcomp + kmix c + kc + kc c c

(25)

where eel is the elementary charge and 0 and r are the permittivity in a vacuum and the dielectric constant, respectively. lB ) 7.15 Å for an aqueous medium at 25 °C. (33) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain; Wiley-VCH: New York, 1994; Chapter 3. (34) Lekkerkerker, H. N. W. Physica A 1989, 159, 319-328. (35) Mitchell, D. J.; Ninham, B. W. Langmuir 1989, 5, 1121-1123.

As already mentioned, in the present article we have neglected residual headgroup repulsion effects, and as a result, this contribution has been omitted in eqs 27-29. Hydrophobic Effect. The hydrophobic contribution is the result of an unfavorable interfacial tension γhb for the hydrophobic-hydrophilic interface together with a curvature-dependent area per surfactant a. As a result, this contribution to the bending elasticity constants depends on γhb as well as on the curvaturerelated constants defined in eq 8:11

1 (kcH0)hb ) γhbk′a 4

(30)

1 khb c ) - γhbk′′ a 2

(31)

hkhb c ) -γhbk′′′ a

(32)

Electrostatics. The electrostatic free energy is explicitly

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Bergstro¨m

curvature-dependent in accordance with eq 18 and gives rise to the following contributions11

(kcH0)el ) -

kelc )

( )

∂gel 1 |z x + (1 - xp)z2| 4ap 1 p ∂H

[

( )

|z1xp + (1 - xp)z2| ∂gel k′a 2ap ∂H

+

hkelc )

K

K

(33)

[H ) K ) 0]

( )

2 1 ∂ gel + 2 ∂H2

( ) ( )

xpk′x ∂gel (z1 - z2) 2ap ∂H

[H ) K ) 0]

K

[H ) K ) 0]

K

[H ) K ) 0]

∂el 1 |z1xp + (1 - xp)z2| ap ∂K

H

]

(34)

[H ) K ) 0]

(35)

We have recently employed the Poisson-Boltzmann theory together with eqs 33-35 to derive expressions for the electrostatic contribution to the various bending elasticity constants for pure ionic surfactant layers.12 Further below, we will adopt the same approach in order to quantify kcH0, kc, and khc for charged layers formed in some different cases of binary surfactant mixtures. Chain Conformational Entropy. The corresponding contribution due to the free energy related to the more or less flexible hydrocarbon chains of a surfactant is given by similar expressions. For the case where any explicit curvature dependence of chain ≡ chain(ξ) has been neglected ,we may write11,12

(kcH0)chain ) -

[

]

ξpk′ξ dchain1 dchain2 + (1 - xp) xp 4ap dξ dξ

(36)

) xpkchain1 + (1 - xp)kchain2 kchain c c c xpk′x dchain1 dchain2 + ξpk′ξ (37) 2ap dξ dξ

(

) hkchain c

) ]

[

ξpk′′′ dchain1 dchain1 ξ xp + (1 - xp) ap dξ dξ

[

]

(39)

Composition Effects. All of the contributions mentioned above are also present for the case of one-component surfactant layers.12 However, for the case of a mixed monolayer it has been found that an additional contribution equal to

′′xpk′x 4ap

(40)

′′xp (k′ k′ + k′′x) 2ap a x

(41)

′′xpk′′′ x ap

(42)

(kcH0)comp ) ) kcomp c

) hkcomp c

surfactant between surfactants 1 and 2 aggregated in a planar layer, and the following expression

′′ ) -[2kTq(1 - xpy) + apγhby]

(43)

which is derived in detail in Appendix A, is based on the equilibrium condition for a planar monolayer with respect to surfactant composition. For the sake of simplicity, we have introduced the parameter12

q≡

Sp

xSp2 + 1 + 1

(44)

which assumes values between 0 and 1 as the reduced charge density of a planar layer Sp assumes values between 0 and ∞. Entropy of Mixing. An additional contribution appears as a result of the entropic penalty of having different compositions in differently curved parts of a geometrically heterogeneous aggregate (e.g., the inner and outer monolayers of a bilayer vesicle) (cf. Figure 2). We have previously derived the following expression11 for the always positive contribution to kc from the ideal entropy of mixing as given in eq 26

(38)

for a binary surfactant mixture where the bending rigidity for a single chain is given by

ξp dchain12 d2chain12 2 ) 2(k′ k′ + k′′ ) k′ kchain12 + ξ c ξ a ξ p ξ 4ap dξ dξ2

Figure 2. Schematic representation of a monolayer and a bilayer formed in mixtures of two surfactants with different spontaneous curvatures. The surfactant with the higher spontaneous curvature (white head) is enriched in more curved parts of the layer, causing the bending rigidity kc as well as the effective bilayer constant kbi to become significantly reduced in surfactant mixtures.

arises as a consequence of a curvature-dependent surfactant composition (cf. eq 9).11 ′′ is the difference in free energy per

kmix c )

xpk′x2kT 4(1 - xp)ap

(45)

This contribution is closely related to the effect of mixing on the bending rigidity to be discussed at length further below, and like all mixing effects, the entropy of mixing does not contribute at all to either kcH0 or khc.11

4. Bending Elasticity Constants for a Monolayer Formed by an Ionic and a Nonionic Surfactant From the various contributions accounted for in the preceding section, we may now derive expressions for the bending elasticity constants for charged layers formed in some different cases of binary surfactant mixtures in a similar manner as was recently done for one-component surfactants films.12 We begin with the case of mixtures between a monovalent ionic and a nonionic surfactant. Spontaneous Curvature. The electrostatic contribution to kcH0 for a mixture of a monovalent ionic and a nonionic surfactant may be obtained by setting z1 ) 1 and z2 ) 0 in eq 33. As a result,

( )

xp ∂gel (kcH0)el )kT 4ap ∂H

K

[H ) K ) 0]

(46)

Bending Elasticity of Charged Surfactant Layers

Langmuir, Vol. 22, No. 16, 2006 6801

Figure 3. Bending rigidity times the spontaneous curvature (kcH0) for a mixture of a monovalent ionic and a nonionic surfactant plotted against the mole fraction of ionic surfactant in a planar layer (xp) in accordance with eq 49 (solid line). The contribution to kcH0 of bending an infinitely thin surface (dashed line) and the contribution resulting from effects of the monolayer having a finite thickness ξp ) 12 Å with the surface of charge located a distance d ) 3 Å outside the hydrophobic-hydrophilic interface (dotted line) are also given. The hydrophobic interfacial tension was set equal to γhb ) 50.7 mJ/m2, the Bjerrum length lB ) 7.15 Å, which is valid for a temperature of T ) 298 K, and the Debye screening length was set to κ-1 ) 40 Å. The area per aggregated surfactant in a planar monolayer ap was set to 30 Å2.

ξ outside which the surface of charge is located at a distance d (cf. Figure 1). We see that kcH0 is always positive for an infinitely thin charged surface whereas it is significantly reduced by finite thickness effects. Furthermore, we note that the sharp increase in kcH0 with ionic surfactant mole fraction is primarily attributed to finite thickness effects, the magnitude of which also increases with ξp and d. As a matter of fact, eq 49 is identical to the expression obtained for a surfactant layer that is bent at constant layer thickness. Recently, we arrived at the very same conclusion for a one-component surfactant film, and we were able to show that this result appears to be an indirect effect of the chain conformational free energy of flexible surfactant tails being present.12 Setting xp ) 1 in eq 49 gives the same expression as was previously obtained for a monolayer formed by a single ionic surfactant with a flexible tail.12 Bending Rigidity. Setting z1 ) 1 and z2 ) 0 in eq 34 gives the following relation for kelc of a monolayer formed by a monovalent ionic and a nonionic surfactant

[

()

(

)

and as a result, we obtain the following rather simple expression

(

)

ξpγhb qxp xpπlBκ-1 kcH0 1 ln (ξ + 2d) ) + kT 2πlB qap 4kT 2ap p

(

]

[H ) K ) 0] (50)

K

)

xp [p(k′a + k′x - 2d)2 + 2q(k′′a + k′′x + k′ak′x)] 2ap

(51)

may be derived using PB theory, where we have introduced the parameter

p≡

Sp

(52)

xSp2 + 1

for which 0 e p < 1 as 0 e Sp < ∞. All contributions to kc in eqs 31, 37, 41, 45, and 51 may now be summed up, and by means of employing the two equilibrium conditions for a planar film in eqs 43 and 48 as well as the geometrical relation in eq 13, we finally arrive at the following rather tedious expression

(

)

kc 2(p - q)2 κ-1 ) 1kT 2πlB pq

+ (49)

Most interestingly, it is found that all terms proportional to k′x as well as k′ξ have been canceled as a result of the employment of the two equilibrium conditions in a similar manner as was previously found for terms including k′ξ in the case of a onecomponent surfactant film.12 Moreover, kcH0 may be subdivided into two contributions (cf. Figure 3). The first term in eq 49, which strongly depends on the Debye screening length, may be interpreted as kcH0 for an infinitely thin charged surface and is identical to what has previously been derived.34,35 The second and third terms in eq 49 appear as a result of the surfactant layer having a finite thickness

( )

2 1 ∂ gel 2 ∂H2

kelc 2(p - q)2 κ-1 pq ) 1(k′ + k′x - 2d) kT 2πlB pq πlB a

(47)

ξp dchain2 dchain1 γhb 2qxp xp )0 + (1 - xp) kT ap apkT dξ dξ (48)

[H ) K ) 0]

In Appendix B, it is demonstrated that the following expression

+

The total expression for kcH0 can now be obtained by summing up the differrent contributions given in eqs 30, 36, 40, and 47. Furthermore, because we are dealing with the case of athermodynamically open layer we may employ the equilibrium condition for a planar monolayer with respect to composition given in eq 43 as well as the corresponding condition obtained by minimizing the free energy in eq 6 with respect to layer thickness at H ) K ) 0. This is done in detail in Appendix A, giving

K

+

and in Appendix B we show in detail that the following expression is obtained using PB theory as given in eqs 18-22

(kcH0)el Sp qxp 1 ) ln (k′ + k′x - 2d) kT 2πlB 2q 2ap a

( )

kelc xp ∂gel ) (k′ + k′x) kT 2ap a ∂H

+ +

(

pq (ξ + 2d - k′ξ - k′x(1 - yxp)) πlB p

pxp xpk′x2 (ξp + 2d - k′ξ - k′x(1 - yxp))2 + 2ap 4(1 - xp)ap

)

γhb 2xpq k′ξ (ξp + k′ξ - yxpk′x) kT ap 2 ξpxpk′ξk′x dchain1 dchain2 + 2kTap dξ dξ +

[

(

]

ξp2k′ξ2 d2chain1 d2chain2 xp + (1 x ) p 4kTap dξ2 dξ2

)

(53)

6802 Langmuir, Vol. 22, No. 16, 2006

Bergstro¨m

Figure 4. k′ξ according to eq 54 plotted against mole fraction of ionic surfactant xp for a mixture of a monovalent ionic and a nonionic surfactant with identical tails (y ) 0). λ was set to 0.05 whereas γhb, lB, κ-1, ap, ξp, and d assume the same values as given in Figure 3.

for the general case of two surfactants with different tail volumes. Most interestingly, it may be noted that all terms including k′′ξ and k′′x have been canceled in eq 53 as a result of the employment of the two equilibrium conditions. However, the two parameters k′ξ and k′x are still present in eq 53 and may be determined for a thermodynamically open film by simply minimizing eq 53 with respect to k′ξ and k′x.12 This may be done for the for the special case of two surfactants with identical tails (chain1 ) chain2 and y ) 0), giving k′ξ ) pqap/πlB - ξp(2p(1 - xp) + 1)(apγhb/2kT - qxp) + pxp(ξp + 2d) (2p(1 - xp) + 1)(apγhb/kT - 2qxp + λξp2) + pxp

(54) and

k′x )

2p(1 - xp)

[

]

qap + ξp + 2d - k′ξ 2p(1 - xp) + 1 xpπlB

(56)

for the chain conformational free-energy contribution implying that 2 1 d chain ) 2λ kT dξ2

xp approaches zero. As a matter of fact, nonionic amphiphilic molecules with rather small headgroups (e.g., long-chained alcohols) have been found to phase separate rather than form stable aggregates in an aqueous solvent. For nonionic surfactants with a voluminous headgroup, however, residual headgroup repulsion effects that have been neglected in the present work are expected to become important so as to yield kc > 0 in the regime where xp ≈ 0. Hence, we expect the case of bending a surfactant film at constant layer thickness (i.e., setting k′ξ ) 0) to be a good approximation, and eq 53 may thus be simplified so as to give

(

(57)

Comparing eq 56 with detailed statistical mechanical model calculations by Gruen, the parameter λ may be estimated to be approximately equal to 0.05.12,29 Equation 54 is plotted in Figure 4, and it is seen that k′ξ usually is rather close to zero. In Figure 5, we have plotted the bending rigidity for the case of two surfactants with identical tails, and it is seen that kc can be very well approximated by simply setting k′ξ ) 0, except in the limit of very low ionic surfactant mole fractions. We may note that a surfactant film is no longer stable as kc assumes negative values. Hence, we predict the surfactant film to become destabilized and a phase separation to occur as

)

kc 2(p - q)2 pq κ-1 1+ (ξ + 2d - k′x(1 - yxp)) ) kT 2πlB pq πlB p +

(55)

In eq 54, we have also employed the approximate parabolic function12

min chain chain 2 ) λ(ξ - ξmin) + kT kT

Figure 5. Bending rigidity (kc) for a mixture of a monovalent ionic and a nonionic surfactant with identical tails (y ) 0) plotted against the mole fraction of ionic surfactant in a planar layer (xp) in accordance with eqs 53-55 (solid line) as well as in accordance with eq 59 (k′ξ ) 0, dashed line). γhb, lB, κ-1, ap, ξp, and d assume the same values as given in Figure 3.

pxp xpk′x2 (ξp + 2d - k′x(1 - yxp))2 + 2ap 4(1 - xp)ap

(58)

However, the parameter k′x related to the curvature dependence of the composition is seen to have a large influence on the bending rigidity of surfactant films. xpk′x is plotted in Figure 6 against the surfactant mole fraction in accordance with eq 55, and it is seen to assume large positive values as well as go through a maximum at some intermediate composition, whereas it vanishes for the case of a pure surfactant layer (xp ) 0 and 1). The sign of k′x is determined by the minimization process of the free energy in eq 6, and positive values imply that the fraction of ionic surfactant increases as a response to an increased positive curvature of the surfactant monolayer. Inserting eq 55 with k′ξ ) 0 into eq 58 gives after some rearrangement

(

)

kc 2(p - q)2 κ-1 1) kT 2πlB pq

+

-

p2xp(1 - xp)(1 - xpy)2

[

]

pxp 2qap (ξp + 2d) + ξp + 2d 2ap xpπlB

[

qap + ξp + 2d 2 x (2p(1 - xp)(1 - xpy) + 1)ap pπlB

]

2

(59)

It is seen that eq 59 may be subdivided into three different

Bending Elasticity of Charged Surfactant Layers

Figure 6. xpk′x according to eq 55 plotted against mole fraction of ionic surfactant xp for a mixture of a monovalent ionic and a nonionic surfactant with identical tails (y ) 0). k′ξ was calculated from eq 54 (solid line) or set equal to zero (dashed line). λ ) 0.05 in the former case whereas γhb, lB, κ-1, ap, ξp, and d assume the same values as given in Figure 3.

Figure 7. Bending rigidity (kc) for a mixture of a monovalent ionic and a nonionic surfactant with identical tails (y ) 0) plotted against the mole fraction of ionic surfactant in a planar layer (xp) in accordance with eq 59 (solid line). The three contributions to kc due to the bending of an infinitely thin surface (dashed line), finite thickness effects (dotted line), and mixing effects (dash-dotted line) are also given. γhb, lB, κ-1, ap, ξp, and d assume the same values as given in Figure 3.

contributions that are all shown in Figure 7. The first term represents the rigidity of an infinitely thin charged surface as previously derived by Lekkerkerker34 and Mitchell and Ninham.35 The dependence of kc on the Debye screening length is mainly attributed to this contribution, and it is seen to be fairly constant with respect to surfactant composition except in the limit xp ≈ 0. The second term in eq 59 appears as a result of the surfactant layer having a finite thickness with the surface of charge being located outside the hydrophobic-hydrophilic interface. It is found to depend largely on the surfactant mole fraction as well as on ξp and d. These two contributions also appear for a one-component monolayer formed by an ionic surfactant (xp ) 1), as was recently demonstrated.12 However, for the mixed case there also appears a third contribution that is absent for a pure film and from the expression in eq 59, it is evident that this term is always negative. In other words, kc decreases per se as a result of the effect of mixing two or more surfactants. This may be rationalized as a

Langmuir, Vol. 22, No. 16, 2006 6803

Figure 8. Bending rigidity (kc) for a mixture of a single-chain ionic and a single-chain nonionic surfactant with identical tails (y ) 0, upper solid line), mixture of a single-chain ionic and a double-chain nonionic surfactant (y < 0, upper dashed line), and mixture of a double-chain ionic and a single-chain nonionic surfactant (y > 0, upper dotted line) plotted against the mole fraction of ionic surfactant in a planar layer (xp) in accordance with eq 59. The contributions to kc due to mixing effects are also given for the three cases (lower solid, dashed, and dotted lines, respectively). The area per surfactant in mixtures of surfactants with different tails was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å. γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

consequence of the lowering of the minimized free energy of an equilibrated surfactant layer that results from the presence of an additional degree of freedom (composition) in a binary mixture.14,17,18 Otherwise expressed, the bending rigidity of mixed surfactant films must always be softened as a consequence of the fact that the composition may be different in differently curved parts of a geometrically heterogeneous film (cf. Figure 2). We also see that the quantity y ≡ ∆V/Vp appears in the last term in eq 59, implying that the bending rigidity, because of geometrical reasons, is influenced by the difference in tail volume between the two surfactants. Hence, kc shows nonlinear behavior with respect to the difference in tail volume, although a linear expression for the chain conformational free-energy contribution has been employed in our derivations (cf. eq 10). The appearance of y in eq 59 is the result of a coupling effect between electrostatics and geometrical packing constraints, and its influence on kc is expected to be much larger than any nonlinear effects of the rather small chain conformational free-energy contribution. In Figure 8, we have plotted kc, together with its contribution due to mixing, for mixtures of two surfactants with different tail volumes, and it is seen that the reduction of the bending rigidity is enhanced for a mixture where the ionic surfactant has the smaller tail (y < 0) whereas kc is raised for an ionic/nonionic surfactant mixture where the nonionic surfactant has the smaller tail (y > 0). We may summarize our results with the general statement that the magnitude of the reduction of kc upon mixing increases with increasing asymmetry between two surfactants in a binary mixture.11 The effect of mixing on the bending properties of surfactant layers will be further discussed below in the treatment of bilayers. Saddle-Splay Constant. From eq 35, it follows that the electrostatic contribution to the saddle-splay constant for a mixture of a monovalent ionic and a nonionic surfactant equals

( )

xp ∂gel hkelc ) kT ap ∂K

H

[H ) K ) 0]

(60)

6804 Langmuir, Vol. 22, No. 16, 2006

Bergstro¨m

mixing has no explicit influence on khc, in agreement with what has previously been concluded by Porte and Ligoure.18

5. Bending Elasticity Constants for a Monolayer Formed by Two Oppositely Charged Surfactants Spontaneous Curvature. We may derive expressions for the bending elasticity constants for monolayers mixed by two oppositely charged surfactants in an analogous manner as was done above for the ionic/nonionic surfactant mixture. Setting z1 ) 1 and z2 ) -1 in eq 33 gives the electrostatic contribution to kcH0

( )

(kcH0)el |2xp - 1| ∂gel )kT 4ap ∂H

Figure 9. Saddle-splay constant (khc) for a mixture of a monovalent ionic and a nonionic surfactant plotted against the mole fraction of ionic surfactant in a planar layer (xp) in accordance with eq 62 (solid line). The contributions to khc of bending an infinitely thin surface (dashed line) as well as resulting from effects of the monolayer having a finite thickness with the surface of charge located outside the hydrophobic-hydrophilic interface (dotted line) are also given. γhb, lB, κ-1, ap, ξp, and d assume the same values as given in Figure 3.

[ ( ( ))

( )]

Sp 2q 2qxp 2 + (k′′′ + k′′′ x - d ) (61) ap a

+ 2d ln

[H ) K ) 0]

(63)

which may be further evaluated from PB theory given in eqs 18-22 so as to yield

()

Sp (kcH0)el 1 ln ) kT 2πlB 2q q|2xp - 1| 2xp k′a - 2d + k′ (64) 2ap |2xp - 1| x

(

)

where

2πlB κap

and in Appendix B it is demonstrated that, according to PB theory, we may write the electrostatic contribution to khc as follows:

hkelc Sp 1 -1 κ D1 ln )kT πlB 2q

K

Sp ) |2xp - 1|

(65)

for a mixture of a monovalent anionic and a monovalent cationic surfactant. It is straightforward to show that the two equilibrium conditions with respect to composition and layer thickness for a planar monolayer formed in mixtures of two oppositely charged surfactants equal

′′ ) -[4kTq(1 - xpy) + apγhby]

Summing up all contributions to khc in eqs 32, 38, 42, and 61, as well as taking into account the two equilibrium conditions in eqs 43 and 48, we finally arrive at the following expression

and

xpπlBκ-1 hkc κ-1 D ln )kT πlB 1 qap

ξp dchain2 γhb 2|2xp - 1|q dchain1 + (1 - xp) xp )0 kT ap apkT dξ dξ (67)

((

))

-

(

)

xpπlBκ-1 2d ln πlB qap

ξp2γhb 2qxp 2 (ξ - 3d2) (62) + 3 3ap p

where the planar reduced charge density has been set to Sp ≡ 2πlBxp/apκ. By analogy with to the cases of kcH0 and kc, it is found that all terms including k′′′ ξ and k′′′ x have been canceled a consequence of the employment of the two equilibrium conditions. As a result, eq 62 is the exact result as obtained for a surfactant monolayer that is bent at constant layer thickness, in a similar manner, as was recently demonstrated to be the case for pure surfactant layers.12 The saddle-splay constant according to eq 62 is plotted against xp in Figure 9. As for the other bending elasticity constants, khc may be subdivided into one contribution that is due to the bending of an infinitely thin surface and one contribution that appears as a result of the layer having a finite thickness. The former contribution is mainly dependent on κ-1 whereas the magnitude of the finite thickness effects increases with ξp and d. We may also note that the infinitely thin surface contribution decreases with ionic surfactant mole fraction whereas the finite layer thickness contribution increases with xp and, as a result, khc is seen to go through a minimum in Figure 9. Most interestingly, in contrast to the bending rigidity we find that the effect of

(

(66)

)

Finally, by summing up the various contributions in eqs 30, 36, 40, and 64, as well as taking into account eqs 66 and 67, the following expression

(

)

ξpγhb |2xp - 1|πlBκ-1 k cH 0 1 ) ln kT 2πlB qap 4kT q|2xp - 1| (ξp + 2d) (68) + 2ap may be deduced. Hence, as for a pure ionic surfactant layer12 as well as for an ionic/nonionic surfactant mixture all terms including k′ξ and k′x have been canceled in eq 68, and we arrive at the very same result as for a monolayer that is bent at constant layer thickness. kcH0 for an anionic/cationic surfactant mixture is plotted in Figure 10 together with its contributions. It is seen that kcH0 for an infinitely thin layer assumes positive values and vanishes at xp ) 0.5. The contribution due to finite thickness effects, however, decreases monotonically to large negative values with decreasing surface charge density and reaches a minimum at equimolar composition. We may note that kcH0 and its contributions as a function of xp shown in Figure 10 are also differentiable in the rather sharp minima always found at xp ) 0.5, and the

Bending Elasticity of Charged Surfactant Layers

Langmuir, Vol. 22, No. 16, 2006 6805

same holds true for kc and khc in Figures 11-13. As a result, the free energy Ε of a surfactant film is an analytical function with respect to xp over the entire range of composition also for the case of oppositely charged surfactants. Bending Rigidity. The electrostatic contribution to the bending rigidity of a monolayer formed in mixtures of two monovalent and oppositely charged surfactants may be obtained by setting z1 ) 1 and z2 ) -1 in eq 34

[(

)

kelc ∂gel |2xp - 1| ) k′a kT 2ap ∂H

K

[H ) K ) 0] + +

( )

2 1 ∂ gel 2 ∂H2

( )

xp ∂gel k′ ap x ∂H

K

[H ) K ) 0]

K

]

[H ) K ) 0] (69)

which may be further elaborated from PB theory so as to give

( [(

) [

2xp kelc 2(p - q)2 pq κ-1 1k′a - 2d + ) k′ kT 2πlB pq πlB |2xp - 1| x

)

2 |2xp - 1| 2xp k′x + p k′a - 2d + 2ap |2xp - 1| 2xp (k′ k′ + k′′x) + 2q k′′a + |2xp - 1| a x

(

)]

]

(70)

Figure 10. Bending rigidity times the spontaneous curvature (kcH0) for a mixture of two oppositely charged monovalent ionic surfactants plotted against the surfactant mole fraction in a planar layer (xp) in accordance with eq 68 (solid line). The contribution to kcH0 of bending an infinitely thin surface (dashed line) as well as the contribution resulting from effects of the monolayer having a finite thickness with the surface of charge located a distance outside the hydrophobichydrophilic interface (dotted line) are also given. γhb, lB, κ-1, ap, ξp, and d assume the same values as given in Figure 3.

(

The sum of the different contributions in eqs 31, 37, 41, 45, and 70 now gives

(

)

kc 2(p - q)2 κ-1 ) 1kT 2πlB pq

+ +

(

(

( ))

2xp pq - yxp ξp + 2d - k′x πlB 2xp - 1

(

(2xp - 1) 2xp - yxp p ξp + 2d - k′x 2ap 2xp - 1

+

))

2

xpk′x2 4(1 - xp)ap

(71)

for the case where k′ξ has been approximated to zero. In the derivation of eq 71, we have also employed the two equilibrium conditions in eqs 66 and 67 as well as the geometrical relation in eq 13. Once again, we note that all terms including k′′ξ and k′′x have been canceled. Moreover, the following expression

k′x )

2p(1 - xp)|2xp - 1|(2 - |2xp - 1|y) 2pxp(1 - xp)(2 - |2xp - 1|y)2 + |2xp - 1| qap + ξp + 2d (72) × |2xp - 1|πlB

[

]

is obtained for the case of an equilibrated monolayer by simply minimizing eq 71 with respect to k′x. Inserting eq 72 in eq 71 finally gives

)

kc 2(p - q)2 κ-1 ) 1kT 2πlB pq p|2xp - 1| 2qap + (ξp + 2d) + ξp + 2d 2ap |2xp - 1|πlB

[

]

p2xp(1 - xp)|2xp - 1|(2 - |2xp - 1|y)2 (2pxp(1 - xp)(2 - |2xp - 1|y)2 + |2xp - 1|)ap qap × + ξp + 2d |2xp - 1|πlB

[

]

2

(73)

As for the ionic/nonionic surfactant mixture, kc may be written as a sum of three contributions that are all given in Figure 11 for the case of two surfactants with identical tails (y ) 0). It is seen that the bending rigidity decreases from large positive values for the cases of pure surfactant and vanishes for an uncharged layer at equimolar composition, mainly as a result of finite thickness effects. Again, we note that kc vanishes (or even becomes negative if k′ξ is not approximated to zero), preventing stable surfactant aggregates to form, as surface charge density approaches zero and the amphiphilic character of the molecules is lost. In anionic/cationic surfactant mixtures, precipitates are frequently observed close to equimolar composition. Moreover, it is clear that the reduction of kc is enhanced in an anionic/ cationic surfactant mixture as compared to that in the corresponding ionic/nonionic mixture as a result of the larger magnitude of the always negative mixing term in eq 73 (compare Figures 7 and 11). This fact may be rationalized as a result of the larger asymmetry between two oppositely charged surfactants in terms of the difference in charge number |z1 - z2|. This quantity equals 2 for a mixture of a monovalent anionic and a monovalent cationic surfactant whereas it is equal to unity in the corresponding ionic/ nonionic surfactant mixture. Moreover, kc may be further reduced in a mixture of two surfactants with different tail volumes provided the surfactant in excess has the smaller tail (cf. Figure 12). The magnitude of the mixing contribution is reduced in the less asymmetrical case where the surfactant in excess has the larger tail. However, kc is also found to be significantly reduced in double-chain-rich

6806 Langmuir, Vol. 22, No. 16, 2006

Bergstro¨m

Figure 11. Bending rigidity (kc) for a mixture of two oppositely charged monovalent ionic surfactants with identical tails (y ) 0) plotted against the surfactant mole fraction in a planar layer (xp) in accordance with eq 73 (solid line). The three contributions to kc due to the bending of an infinitely thin surface (dashed line), finite thickness effects (dotted line), and mixing effects (dash-dotted line) are also given. γhb, lB, κ-1, ap, ξp, and d assume the same values as given in Figure 3.

Figure 13. Saddle-spaly constant (khc) for a mixture of two oppositely charged monovalent ionic surfactants plotted against the surfactant mole fraction in a planar layer (xp) in accordance with eq 76 (solid line). The contribution to khc of bending an infinitely thin surface (dashed line) and the contribution resulting from effects of the monolayer having a finite thickness with the surface of charge located a certain distance outside the hydrophobic-hydrophilic interface (dotted line) are also given. γhb, lB, κ-1, ap, ξp, and d assume the same values as given in Figure 3.

[ ( ( ))

hkelc Sp 1 -1 )κ D1 ln kT πlB 2q +

+ 2d ln

(

( )] Sp 2q

)

2q|2xp - 1| 2xp 2 k′′′ k′′′ a-d + ap |2xp - 1| x

(75)

Summing up all contributions to khc in eqs 32, 38, 42, and 75 and taking into account the two equilibrium conditions in eqs 66 and 67 gives the following expression for the saddle-splay constant

((

))

|2xp - 1|πlBκ-1 hkc κ-1 )D1 ln kT πlB qap

Figure 12. Bending rigidity (kc) for a mixture of an ionic singlechain surfactant and an oppositely charged double-chain surfactant (solid line) plotted against the mole fraction of single-chain surfactant in a planar layer (xp) in accordance with eq 73. The three contributions to kc due to the bending of an infinitely thin surface (dashed line), finite thickness effects (dotted line), and mixing effects (dashdotted line) are also given and so is kc for a mixture of two oppositely charged surfactants with identical tails (y ) 0, dash-dot-dot line). The area per surfactant was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å whereas γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

mixtures because the finite thickness contribution is lower for a double-chain surfactant.12 Saddle-Splay Constant. The electrostatic contribution to khc for a mixture of two oppositely charged surfactants equals

( )

hkelc |2xp - 1| ∂gel ) kT ap ∂K

H

[H ) K ) 0]

which may be further evaluated so as to give

(74)

-

(

)

|2xp - 1|πlBκ-1 2d ln πlB qap

ξp2γhb 2q|2xp - 1| 2 (ξp - 3d2) + 3 3ap

(76)

Again we note that all terms including k′′′ ξ and k′′′ x have been canceled and that eq 76 is the exact result as obtained for the procedure of bending the surfactant film at constant layer thickness. The saddle-splay constant according to eq 76, as well as its two components (infinitely thin film and finite thickness effects, respectively), are plotted in Figure 13. It is seen that both contributions generally assume negative values. However, whereas the magnitude of the infinitely thin surface contribution increases with increasing surface charge density and is zero for the uncharged equimolar composition, the finite size contribution shows the opposite trend (i.e., its magnitude decreases with increasing surface charge density). As a result, a minimum appears on either side of the equimolar composition in the khc versus xp plot shown in Figure 13.

6. Bending Elasticity Constants for a Monolayer Formed by Two Ionic Surfactants with Identical Head Groups but Different Tail Volumes Spontaneous Curvature. The electrostatic contribution to kcH0 for a mixture of two monovalent ionic surfactants with

Bending Elasticity of Charged Surfactant Layers

Langmuir, Vol. 22, No. 16, 2006 6807

considerably reduce kcH0. The latter contribution also shows a much stronger dependence on surfactant mole fraction. Bending Rigidity. It is straightforward to show that the electrostatic contribution to kc for a mixture of two ionic surfactants with identical headgroups equals

kelc )

(

)

2(p - q) κ-1 pq 1(k′ - 2d) 2πlB pq πlB a p q + (k′ - 2d)2 + k′′a (82) 2ap a ap 2

whereas the overall bending rigidity may be written as

(

)

kc 2(p - q)2 κ-1 pq ) 1+ (ξ + 2d + xpk′xy) kT 2πlB pq πlB p Figure 14. Bending rigidity times the spontaneous curvature (kcH0) for a mixture of a single-chain and a double-chain monovalent ionic surfactant with identical headgroups plotted against the mole fraction of a single-chain surfactant in a planar layer (xp) in accordance with eq 81 (solid line). The contribution to kcH0 of bending an infinitely thin surface (dashed line) and the contribution resulting from effects of the monolayer having a finite thickness with the surface of charge located a certain distance outside the hydrophobic-hydrophilic interface are also given (dotted line). The area per surfactant was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å whereas γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

identical headgroups is obtained by setting z1 ) z2 ) 1 in eq 33, giving

(kcH0)el ) -

( )

1 ∂gel 4ap ∂H

K

[H ) K ) 0]

(77)

xpk′x2 p 2 (ξ + 2d + xpk′xy) + + 2ap p 4(1 - xp)ap

where k′a ) -ξp - yxpk′x for the case with k′ξ ) 0 (cf. eq 13). Minimizing eq 83 with respect to k′x gives

k′x ) -

()

(kcH0)el Sp 1 q ) ln (k′ - 2d) kT 2πlB 2q 2ap a

(78)

The equilibrium conditions with respect to composition and layer thickness for a planar layer may be obtained in a similar manner as was done for the ionic/nonionic and anionic/cationic surfactant mixtures, giving

′′ ) -[4kTqxp - apγhb]y

(79)

and

(

)

ξp dchain2 dchain1 γhb 2q xp )0 - + (1 - xp) kT ap apkT dξ dξ

( )

qap + ξp + 2d πl 2p(1 - xp)xpy + 1 B 2

(

)

[

[

p2xp(1 - xp)y2

qap + ξp + 2d (2pxp(1 - xp)y + 1)ap πlB 2

+ 2d ln

( )] Sp 2q

]

]

2

(85)

2 + (k′′′ a - d )q

(80)

for an equilibrated monolayer, and we once again note that all terms including k′ξ and k′x have been canceled. As a matter of fact, eq 81 is identical to the corresponding expression recently derived for a monolayer formed by a single monovalent ionic surfactant.12 kcH0 in accordance with eq 81 is given in Figure 14, and as found for the other cases, we see that the infinitely thin surface contribution is always positive whereas finite thickness effects

(84)

The first term in eq 85 corresponds to an infinitely thin surface and is identical to what has been obtained for the previously treated cases whereas kc is significantly increased as a result of finite thickness effects. The third term in eq 85, which appears as a result of mixing two surfactants with different tail volumes expressed in terms of the parameter y ≡ ∆V/Vp, is always negative, except in the limit of pure surfactant where it vanishes, and its magnitude increases with increasing y (cf. Figure 15). Saddle-Splay Constant. The expressions for the electrostatic contribution to the saddle-splay or Gaussian bending constant

[ ( ( ))

(81)

]

2qap kc 2(p - q)2 κ-1 p ) 1+ (ξp + 2d) + ξp + 2d kT 2πlB pq 2ap πηlB

Sp hkelc 1 -1 κ D1 ln )kT πlB 2q

It may be noted that ′′ in eq 79 vanishes as y ) 0 and the two surfactants become identical. The sum of all contributions in eqs 30, 36, 40, and 78 equals

ξpγhb q(ξp + 2d) πlBκ-1 kcH0 1 ) + ln kT 2πlB qap 4kT 2ap

[

2p(1 - xp)y

which may be inserted into eq 83 so as to yield

and in accordance with PB theory we may thus write

(83)

as well as the sum of all contributions

[ ( ( ))

hkc πlBκ-1 1 -1 )κ D1 ln kT πlB qap

-

+ 2d ln

(86)

( )] πlBκ-1 qap

ξp2γhb 2q 2 + (ξ - 3d2) (87) 3 3ap p

are both identical as for a monolayer formed by a single monovalent ionic surfactant in the same way as was kcH0.12 Equation 87 is given in Figure 16, and it is seen that both contributions always assume negative values. The finite thickness contribution shows a stronger dependence on surfactant composition and is responsible for the fact that khc is lower for a double-chain than for a single-chain ionic surfactant.

6808 Langmuir, Vol. 22, No. 16, 2006

Bergstro¨m

hbi effective bilayer bending constant is defined as kbi ≡ 2kbi c + k c ) 2(2kc + khc - 4ξpkcH0). It has been demonstrated that kbi is directly related to the average equilibrium vesicle size as a result of a balance between a positive bending energy that tends to increase the vesicle size and dispersion effects that tends to decrease the vesicle size.10,21 As a result, the size of the vesicles is expected to increase with increasing values of kbi. By analogy to the bending rigidity, the effective bilayer bending constant may be written as a sum of three different contributions

kbi ) k0bi + kξbi + kmix bi

(88)

where

[

( ( ))]

k0bi 2κ-1 Sp 2(p - q)2 ) - D1 ln 1kT πlB pq 2q Figure 15. Bending rigidity (kc) for monolayers formed in mixtures of a single-chain and a double-chain monovalent ionic surfactant with identical headgroups plotted against the mole fraction of a single-chain surfactant in a planar layer (xp) in accordance with eq 85 (solid line). The three contributions to kc due to the bending of an infinitely thin surface (dashed line), finite thickness effects (dotted line), and mixing effects (dash-dotted line) are also given. The area per surfactant was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å whereas γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

(89)

is the bilayer bending constant for an infinitely thin surface. kξbi is the contribution to the effective bilayer bending constant resulting from the effects of the bilayer having a finite thickness with the surface of charge located somewhat outside the hydrophobic-hydrophilic interfaces, and for a mixture of a monovalent ionic and a nonionic surfactant, we obtain

kξbi ) kT

(

)

[

(

( ))

2qxp 2 Sp 4 4 γhb ξp + (ξp + d) pq - ln 3 ap πlB 2q +

2xp [p(ξp + 2d)2 - 2q(2ξp + d)d] ap

+ pqd

]

(90)

for the approximate case where k′ξ ) 0. The third alwaysnegative contribution to the bilayer bending constant equals

[

qap 4p2xp(1 - xp)(1 - xpy)2 kmix bi )+ ξp + 2d 2 kT πl (2p(1 - x )(1 - x y) + 1)a Bxp p

p

p

]

2

(91)

Figure 16. Saddle-splay constant (khc) for monolayers formed in mixtures of a single-chain and a double-chain monovalent ionic surfactant with identical headgroups plotted against the mole fraction of single-chain surfactant in a planar layer (xp) in accordance with eq 87 (solid line). The contribution to khc of bending an infinitely thin surface (dashed line) and the contribution resulting from effects of the monolayer having a finite thickness with the surface of charge located a certain distance outside the hydrophobic-hydrophilic interface (dotted line) are also given. The area per surfactant was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å whereas γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

7. Bending Properties of Mixed Surfactant Bilayers It has previously been shown that the bending elasticity constants for a bilayer may be expressed in terms of the corresponding constants for a single monolayer (i.e., Hbi 0 ) 0, bi 31,36,37 Moreover, the bending kbi ) 2k and h k ) 2k h 8ξ k H ). c c p c 0 c c energy of a geometrically closed bilayer vesicle has been found to be equal to 4πkbi, independent of vesicle radius, where the (36) Porte, G.; Appell, J.; Bassereau, P.; Marignan, J. J. Phys. (Orsay, Fr.) 1989, 50, 1335. (37) Ljunggren, S.; Eriksson, J. C. Langmuir 1992, 8, 1300.

for an ionic/nonionic surfactant mixture, and thus kbi is generally found to decrease upon mixing two surfactants. The mixing contribution in eq 91 originates from and is exactly four times larger in magnitude than the corresponding contribution seen in the expressions for the bending rigidity of a monolayer (cf. eq 59). The effective bilayer constant is plotted against surfactant mole fraction in Figure 17, and it is seen that kbi assumes large positive values as a result of finite thickness effects whereas the infinitely thin surface contribution assumes values close to zero. Moreover, it is evident from Figure 17 that kbi goes through a minimum at some intermediate surfactant mole fraction. This behavior has been previously predicted from detailed model calculations of bilayers formed by an ionic and a nonionic surfactant with identical aliphatic chains.19 The main difference between the expressions in eqs 88-91 and the result in ref 19 is that an explicit curvature dependence of γhb was introduced in the latter case that caused kbi to decrease by a certain constant amount. In addition, setting k′ξ ≈ 0 may have a significant influence on kbi at low values of xp. The kbi expression for a mixture of an ionic and a nonionic surfactant following PoissonBoltzmann theory has previously been given by Porte and Ligoure18 for the special case of an infinitely thin surface, and their result is recovered if kξbi is neglected in eq 88 and ξp and d are both set equal to zero in eq 91.

Bending Elasticity of Charged Surfactant Layers

Langmuir, Vol. 22, No. 16, 2006 6809

(

)

[

(

( ))

kξbi 4 γhb 2q 2 Sp 4 ξp + (ξp + d) pq - ln ) kT 3 kT ap πlB 2q +

+ pqd

2 [p(ξp + 2d)2 - 2q(2ξp + d)d] ap

]

(94)

and

[

]

4p2xp(1 - xp)y2 qap kex bi )+ ξp + 2d 2 kT πl (2pxp(1 - xp)y + 1)ap B

Figure 17. Effective bilayer bending constant (kbi) for a mixture of a monovalent ionic and a nonionic surfactant with identical tails (y ) 0, solid line) plotted against the mole fraction of ionic surfactant in a planar layer (xp) in accordance with eqs 88-91 together with the three contributions due to the bending of an infinitely thin surface (long dashed line), finite thickness effects (long dotted line), and mixing effects (long dash-dotted line). kbi for a mixture of a singlechain ionic and a double-chain nonionic surfactant is also given (y < 0, dash-dot-dotted line), together with the three contributions due to the bending of an infinitely thin surface (short dashed line), finite thickness effects (short dotted line), and mixing effects (short dash-dotted line). The area per surfactant in the latter case was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å whereas γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

In Figure 17, we have also plotted kbi for a mixture of a singlechain ionic surfactant and a double-chain nonionic surfactant. It is noted that kbi decreases further in the latter, more asymmetrical case as compared with the corresponding mixture of two surfactants with identical tails. The influence on kbi that arises as a direct consequence of the surfactants having different tail volumes (y * 0) can be completely attributed to the behavior of kmix bi . The infinitely thin surface contribution to kbi according to eq 89 is valid for a mixture of two oppositely charged surfactants as well whereas the finite thickness contribution may be written as

+

(

[

(

( ))

kξbi Sp 4 (ξ + d) pq - ln ) kT πlB p 2q

)

+ pqd

]

2|2xp - 1|q 2 4 γhb ξp 3 ap 2|2xp - 1| + [p(ξp + 2d)2 - 2q(2ξp + d)d] (92) ap

2

(95)

whereas k0bi is given by eq 89. The effective bilayer bending constant for the anionic/cationic surfactant mixture is plotted in Figure 18 for the case of two surfactants with identical single chains as well as for a mixture of a single-chain and a double-chain surfactant. Evident minima on either side of the equimolar composition, and with a maximum at xp ) 0.5, are clearly seen for both cases. The symmetrical plot for the case of identical tails much resembles what we previously have obtained from detailed model calculations.20 The reduction of kbi when either of the surfactants is in excess agrees with a large number of experimental observations where rather small vesicles have been found to form spontaneously in mixtures of an ionic and a nonionic surfactant.4,5,38 Moreover, kbi is seen to be further decreased in catanionic mixtures where the surfactant in excess has the smaller tail volume. As a matter of fact, unusually small vesicles with radii of less than 10 nm have been observed to form in mixtures of the anionic single-chain surfactant sodium dodecyl sulfate (SDS) and the double-chain cationic surfactant didodecyldimethylammonium bromide (DDAB) where [SDS]/ [DDAB] > 10 using small-angle neutron scattering (SANS).39 Likewise, vesicles formed in excess SDS were found to be considerably smaller than vesicles formed in excess DDAB according to cryo-TEM micrographs for the same system.40,41 The reduction of kbi upon mixing is a direct consequence of the reduction of kc as observed in eq 73. Hence, we also expect kbi c ) 2kc to decrease further in an anionic/cationic surfactant mixture where the surfactant with the smaller tail is in excess. Indeed, considerably nonspherical vesicles have been observed in anionicrich mixtures of an anionic surfactant with a C8 chain and a cationic surfactant with a C16 chain, indicating low bilayer bending rigidities.42 As expected, the vesicles in this system were found to be appreciably smaller in regions where the surfactant with the shorter chain is in excess. The effective bilayer bending constant for a mixture of a singlechain and a double-chain ionic surfactant with identical headgroups is plotted in Figure 19, and we see that kbi is also reduced upon mixing for this case. The promotion of the spontaneous formation of vesicles has been observed in mixtures of two anionic surfactants43 as well as two cationic surfactants44 with different tail volumes.

and the mixing contribution equals

4p2xp(1 - xp)(2 - |2xp - 1|y)2|2xp - 1| kex bi )kT (2pxp(1 - xp)(2 - |2xp - 1|y)2 + |2xp - 1|)ap 2 qap × + ξp + 2d (93) |2xp - 1|πlB

[

]

For a mixture of two ionic surfactants with identical headgroups but different tail volumes, we may write

(38) Kaler, E. W.; Herrington, K. L.; Murthy, A. K.; Zasadzinski, J. A. N. J. Phys. Chem. 1992, 96, 6698-6707. (39) Bergstro¨m, M.; Pedersen, J. S. J. Phys. Chem. B 2000, 104, 4155-4163. (40) Marques, E. F.; Regev, O.; Khan, A.; da Grac¸ a Miguel, M.; Lindman, B. J. Phys. Chem. B 1998, 102, 6746-6758. (41) Marques, E. F.; Regev, O.; Khan, A.; da Grac¸ a Miguel, M.; Lindman, B. J. Phys. Chem. B 1999, 103, 8353-8363. (42) Yatcilla, M. T.; Herrington, K. L.; Brasher, L. L.; Kaler, E. W.; Chiruvolu, S.; Zasadzinski, J. A. N. J. Phys. Chem. 1996, 100, 5874-5879. (43) Viseu, M. I.; Edwards, K.; Campos, C. S.; Costa, S. M. B. Langmuir 2000, 16, 2105-2114. (44) Boschkova, K. Adsorption and Frictional Properties of Surfactant Assemblies at Solid Surfaces. Doctoral Thesis, Royal Institute of Technology, Stockholm, 2002.

6810 Langmuir, Vol. 22, No. 16, 2006

Figure 18. Effective bilayer bending constant (kbi) for a mixture of two oppositely charged monovalent ionic surfactants with identical tails (y ) 0, solid line) plotted against the surfactant mole fraction in a planar layer (xp) in accordance with eqs 88, 89, 92, and 93 together with the three contributions due to the bending of an infinitely thin surface (long dashed line), finite thickness effects (long dotted line), and mixing effects (long dash-dotted line). kbi for a mixture of an ionic single-chain surfactant and an oppositely charged doublechain surfactant is also given (dash-dot-dotted line), together with the three contributions due to the bending of an infinitely thin surface (short dashed line), finite thickness effects (short dotted line), and mixing effects (short dash-dotted line). The area per surfactant in the latter case was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å whereas γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

Bergstro¨m

Figure 20. Contribution to the effective bilayer bending constant kbi that appears as a result of mixing plotted against surfactant mole fraction in a planar layer (xp) for a mixture of an ionic and a nonionic surfactant with identical tails (solid line), a mixture of two oppositely charged surfactants with identical tails (dashed line), and a mixture of a single-chain and a double-chain ionic surfactant with identical headgroups (dotted line) according to eqs 91, 93, and 95, respectively. The area per surfactant in the latter case was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å whereas γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

be further analyzed by writing eq 91 as follows

[

kmix bi A qap + ξp + 2d )kT ap xpπlB

]

2

(96)

where

A)

Figure 19. Effective bilayer bending constant (kbi) for a mixture of a single-chain and a double-chain monovalent ionic surfactant with identical headgroups plotted against the mole fraction of singlechain surfactant in a planar layer (xp) in accordance with eqs 88, 89, 94, and 95 (solid line). The three contributions due to the bending of an infinitely thin surface (dashed line), finite thickness effects (dotted line), and mixing effects (dash-dotted line) are also included. The area per surfactant was set to ap ) xpa1 + (1 - xp)a2 where a1 ) 30 Å and a2 ) 60 Å whereas γhb, lB, κ-1, ξp, and d assume the same values as given in Figure 3.

The contribution to the effective bilayer bending constant due to mixing kmix bi is plotted in Figure 20 for the three cases of surfactant mixtures given in eqs 91, 93, and 95. It is clearly seen that kmix bi is largest in magnitude for mixtures of oppositely charged surfactants because of the larger asymmetry between the two surfactants whereas kbi is least affected in mixtures of two surfactants with identical headgroups. The mixing contribution for an ionic/nonionic mixture may

4p2xp(1 - xp)

(97)

2p(1 - xp) + 1

is valid for two surfactants with identical tails. It is straightforward to show that A will reach its maximum value Amax ) 2(2 - x3) ≈ 0.54 at xp ) (3 - x3)/2 ≈ 0.63. In Figure 20, it is seen that the location of the minimum of kmix bi virtually corresponds to this value, which may be rationalized as a result of the finite thickness term ξp + 2d being appreciably larger than qap/xpπlB for xp values close to the minimum. It is the appearance of p in the expression for A in eq 97 that causes the reductions of kc and kbi to be most pronounced in excess ionic surfactant. The mixing term in eq 96 has previously been investigated by Porte and Ligoure18 for the case of an infinitely thin surface (ξp ) d ) 0) at low ionic surfactant mole fractions (xp ≈ 0). It may be demonstrated that the quantity qap/xpπlB equals the Debye screening length κ-1 at xp ≈ 0 whereas it is equal to ap/πlB for xp . κap/2πlB corresponding to Sp . 1. The latter quantity becomes as small as κap/2πlB ) 0.02 with κ-1 ) 40 Å and ap ) 30 Å, which means that kmix bi usually becomes rather small (i.e., less than kT) if finite thickness effects are neglected as done in ref 18. For the case of two oppositely charged surfactants with identical tail volumes, we may likewise write

[

qap kex bi A + ξp + 2d )kT ap |2xp - 1|πlB where

]

2

(98)

Bending Elasticity of Charged Surfactant Layers

A)

16p2|2xp - 1|xp(1 - xp) 8pxp(1 - xp) + |2xp - 1|

Langmuir, Vol. 22, No. 16, 2006 6811

(99)

It is clear from eq 99 that A vanishes at surfactant mole fractions equal to 0, 0.5, and 1, and it may be demonstrated that A reaches two maximum values equal to Amax ≈ 0.83 at xp ≈ 0.16 and 0.84, closely corresponding to the two minima appearing in Figure 20. We note that the numerator in the expression for A in eq 99 is 4 times larger than the corresponding quantity for the ionic/ nonionic surfactant mixture in eq 97 as a result of A being proportional to (z1 - z2)2. Hence, the larger asymmetry between two oppositely charged surfactants as compared to the ionic/ nonionic surfactant mixture implies a larger magnitude of the maximum value of A and is responsible for the enhanced reduction of kc and kbi in anionic/cationic mixtures. As in the situation for the ionic/nonionic surfactant mixture, the maximum of A is seen to be significantly shifted toward the pure surfactant cases (xp ) 0 and 1). For mixtures of a single-chain and a double-chain surfactant with identical charged headgroups, the mixing contribution to the effective bilayer constant may be written as

[

kex bi A qap )+ ξp + 2d kT a1 πlB

]

2

(100)

where

A) 4p2xp(1 - xp)(1 - r)2 (2pxp(1 - xp)(1 - r)2 + (r + xp(1 - r))2)(r + xp(1 - r)) (101) Here we have expressed the difference in tail volume with the ratio r ) V2/V1 rather than y ) ∆V/Vp as was done in the corresponding expression for the bending rigidity in eq 85. For the case where each chain in the double-chain surfactant is identical to the tail in the single-chain surfactant, we may set r ) 2 as well as assume ξp to be constant. As a result, we may write ap ) xpa1 + (1 - xp)a2, where a1 and a2 ) 2a1 are the areas per surfactant in a pure monolayer of either a single-chain or double-chain surfactant. From eq 101, it may be demonstrated that Amax ≈ 0.31 when xp ≈ 0.75, which exactly corresponds to the location of the minimum in Figure 20. Amax is smaller in magnitude as compared with the ionic/nonionic and the anionic/ cationic surfactant mixtures because the two surfactants are asymmetrical only with respect to tail volume whereas the headgroup charge numbers are identical.

8. Summary In a recent paper, we demonstrated that the bending properties of monolayers formed by a single ionic surfactant largely depend on whether the surfactant tail is rigid or flexible.12 In the latter case, we found that the spontaneous curvature is substantially lowered and the bending rigidity is raised whereas the Gaussian bending constant becomes increasingly negative with increasing layer thickness. A similar effect arises as a result of the surface of charge being located a certain distance outside the hydrophobic-hydrophilic interface. In the present article, we have extended our theory so as to account for binary mixtures of surfactants with flexible tails. We have treated three different cases of thermodynamically open charged monolayers and bilayers in detail: (i) mixtures of a monovalent ionic and a nonionic surfactant, (ii) mixtures of two

oppositely charged monovalent ionic surfactants, and (iii) mixtures of two ionic surfactants with identical headgroups but different tail volumes. It is straightforward to extend our theory so as to cover any ionic amphiphilic molecules using the PoissonBoltzmann mean-field model provided that all counterions are univalent. It is demonstrated that kcH0, kc, and khc may be subdivided into one contribution that is due to the bending properties of an infinitely thin surface and one contribution resulting from the effects of the monolayer having a finite thickness with the surface of charge separated from the hydrophobic-hydrophilic interface. The latter contribution is found to have a huge impact on the bending properties in the sense that H0 is extensively lowered and kc is considerably raised. Moreover, finite thickness effects are mainly responsible for giving the various bending elasticity constants a strong dependence on surfactant mole fraction. As previously shown for pure surfactant layers, the final expressions for kcH0 and khc are identical to those for a monolayer that is bent at constant layer thickness whereas this is approximately true for kc. In addition, a third contribution appears in the expressions for kc as a result of mixing effects but is absent for kcH0 and khc. This contribution must always be negative because it appears to be a consequence of the minimization of the free energy of the surfactant layer with respect to composition. In other words, the additional degree of freedom present in a surfactant mixture enables the surfactant with the larger spontaneous curvature to be enriched in parts of the surfactant film with a higher curvature. As a result, the comparatively high bending rigidities expected for monolayers and bilayers formed by a single ionic surfactant with a flexible tail may become considerably reduced upon mixing, and the magnitude of the reduction is found to increase with increasing asymmetry between two surfactants in a binary mixture. For instance, the magnitude of the mixing contribution to kc increases with the square of the difference in surfactant charge number. As a result, kc generally decreases in a mixture of two oppositely charged surfactants as compared with the value for a corresponding mixture of an ionic and a nonionic surfactant. Moreover, the mixing contribution to the bending rigidity may be further influenced if the two surfactants have different tail volumes. As a consequence, we find that the reduction of kc is enhanced for ionic/nonionic surfactant mixtures where the ionic surfactant has the smaller tail as well as in anionic/cationic mixtures where the surfactant in excess has the smaller tail. Likewise, the bending rigidity is also found to be decreased in mixtures of two ionic surfactants with identical headgroups but different tail volumes, and the magnitude of the reduction is found to increase with increasing difference in tail volume. Our results suggest that mixing per se promotes the formation of structures that are favored by low kc values (i.e., geometrically heterogeneous aggregates with high flexibility and polydispersity) but does not influence the tendency of the monolayer to curve (kcH0) or to change topology (khc). The effective bilayer bending constant is to a large extent determined by the bending rigidity of a single surfactant monolayer, hence kbi ) 2(2kc + khc - 4ξpkcH0) is also found to be lowered by mixing. As a result, kbi is seen to go through a minimum when plotted against the surfactant mole fraction, and the reduction of kbi upon mixing is found to be most pronounced for the case of two oppositely charged surfactants. As for the bending rigidity, kbi is found to be further decreased in mixtures of surfactants with different tail volumes provided the shorter tail is attached to the surfactant that carries the higher charge.

6812 Langmuir, Vol. 22, No. 16, 2006

Bergstro¨m

Appendix A: Equilibrium Conditions for Monolayers Formed in Mixtures of an Ionic and a Nonionic Surfactant The overall free energy of a surfactant monolayer may be obtained by summing up the various expressions in eqs 7, 10, 17, and 26. As a result, we obtain

aγhb  ) xν1 + (1 - x)ν2 + - x ln c1 kT kT - (1 - x)ln c2 +

xgel kT

Appendix B: Derivation of the Electrostatic Contribution to the Bending Elasticity Constants for Monolayers Formed in Mixtures of an Ionic and a Nonionic Surfactant The following expression may be derived from eq 18

( ) ∂gel ∂H

K

[H ) K ) 0] )

dg0 d (x ) xp) ) ′′ + apγhby + xpkT (x ) xp) ) 0 dx dx

(A2)

where

[H ) K ) 0] + g1 (B1)

K

where the reduced charge density for a mixture of an ionic and a nonionic surfactant is defined as

+ xchain1 + (1 - x)chain2 + x ln x + (1 - x)ln(1 - x) (A1) for a mixture of a monovalent ionic and a nonionic surfactant (z1 ) 1 and z2 ) 0). By minimizing eq A1 with respect to surfactant mole fraction x at constant layer thickness ξ, for a planar film with H ) K ) 0 the following equilibrium condition is obtained

( )

dg0 ∂S dS ∂H

2πlBx 2πlB a ) x κael κa ael

S(H, K) )

(B2)

The ratio between the areas per surfactant at the surface of charge and at the hydrophobic-hydrophilic interface equals

ael ) 1 + 2dH + d2K a

(B3)

if the surface of charge is located a distance d outside the hydrophobic-hydrophilic interface (cf. Figure 1). Combining eqs 8, 9, B2, and B3 enables us to write

S ) Sp(1 + k′SH + k′′S H2 + k′′′ S K)

()

γhb∆V c2 ′′ + ) ν1 - ν2 + ln + g0 kT c1 ξpkT

(B4)

where Sp ≡ 2πlBxp/apκ and

+ chain1 - chain2 + ln xp - ln(1 - xp)

k′S ) k′a + k′x - 2d

(B5)

k′′S ) k′ak′x + k′′a + k′′x - 2dk′S

(B6)

2 k′′′ S ) k′′′ a + k′′′ x-d

(B7)

(A3)

In eq A2, we have taken into account that

V2 + xp∆V ap ) ξp

(A4) From eq B4, we may now deduce that

(∂H∂S )

where ∆V ≡ V1 - V2 and y ) ∆V/Vp. From eqs 19, 24, and A4, it follows that

K

[H ) K ) 0] ) Spk′S

(B8)

which together with eqs B1, 19, and 20 gives

dg0 2q (x ) xp) ) (1 - xpy) dx xp

( )

(A5)

∂gel ∂H

K

[H ) K ) 0] ) 2qk′S -

()

Sp 4 ln κSp 2q

(B9)

where q is defined in eq 44. As a result, we may rewrite the equilibrium condition with respect to composition in eq A2 in the following manner

By inserting eq B9 in eq 46, the following expression for the electrostatic contribution to kcH0 is obtained:

′′ ) -[2kTq(1 - xpy) + apγhby]

x pq Sp (kcH0)el 1 ) ln (k′ + k′x - 2d) kT 2πlB 2q 2ap a

(43)

A second equilibrium condition is obtained as a result of the minimization of the free energy in eq A1 with respect to ξ at constant x

(

()

The second derivative of gel with respect to H may be evaluated from eq 18, giving

( ) ∂2gel

)

γhb dgel da dchain1 dchain2 d(/kT) ) -x +x + (1 - x) )0 dξ kT da dξ dξ dξ (A6) For a planar layer with da/dξ ) -ap/ξp and dgel/da ) 2q/ap, we may rewrite eq A6 so as to become

(

)

ξp dchain2 γhb 2qxp dchain1 + (1 - xp) xp )0 kT ap apkT dξ dξ (48)

(47)

∂H2

)

K

( )

d2g0 ∂S dS2 ∂H

)

2

+

K

()

d2gel ∂S dS2 ∂H

2 K

( )

dg0 ∂2S dS ∂H2

+

K

( )

dgel ∂2S dS ∂H2

+2

K

( )

dg1 ∂S dS ∂H

K

+ 2g2 (B10)

where, according to eq B4,

( ) ∂ 2S ∂H2

K

[H ) K ) 0] ) 2Spk′′S

(B11)

Bending Elasticity of Charged Surfactant Layers

Langmuir, Vol. 22, No. 16, 2006 6813

From eqs 19 and 20, we may deduce that

Appendix C: List of Symbols a

d2g0 2(p - 2q) [S ) Sp] ) 2 dS Sp2

(B12)

()

dg1 Sp 4 4pq ln [S ) Sp] ) 2 dS 2q κSp κSp2

(B13)

where p and q are defined in eqs 52 and 44, respectively. As a result,

( ) ∂2gel ∂H2

+

[H ) K ) 0] ) 2pk′S2 + 4q(k′′S - k′S2)

K

( () )

(

)

()

Sp Sp 4 8 8pq 2pq 16d ln k′S + 2 1 - 2 + ln κSp 2q κSp κSp 2q κ Sp Sp (B14)

We may now insert eqs B9 and B14 in eq 50, giving

(

)

kelc 2(p - q) pq κ-1 1(k′ + k′x - 2d) ) kT 2πlB pq πlB a +

xp [p(k′a + k′x - 2d)2 + 2q(k′′a + k′′x + k′ak′x)] 2ap

(51)

where k′S, k′′S, and k′′′ have been eliminated by means of S employing eqs B5-B7. From eq 18, it follows that

( ) ∂gel ∂K

)

H

( )

dg0 ∂S dS ∂K

H

+ g3

(B15)

where

(∂K∂S )

H

[H ) K ) 0] ) Spk′′′ S

(B16)

Combining eqs 19, 22, B15, and B16 gives

( ) ∂gel ∂K

H

[H ) K ) 0] ) 2qk′′′ S -

[ ( ( ))

Sp 2 -1 κ D1 ln κSp 2q

+ 2d ln

( )] Sp 2q

(B17)

which may be inserted in eq 60 so as to yield

[ ( ( ))

Sp hkelc 1 -1 κ D1 ln )kT πlB 2q

( )]

Sp 2q 2qxp 2 + (k′′′+ k′′′ x - d ) (61) ap a

+ 2d ln

area per aggregated surfactant at the hydrophobichydrophilic interface area per aggregated surfactant at the surface of charge ael ap area per aggregated surfactant for a planar monolayer d Distance between the hydrophobic-hydrophilic interface and the surface of charge D1(x) Debye function as defined in eq 23 gel electrostatic free energy per charge H mean curvature at the hydrophobic-hydrophilic interface spontaneous curvature H0 k Boltzmann’s constant K Gaussian curvature at the hydrophobic-hydrophilic interface effective bilayer bending constant kbi kc bending rigidity of a monolayer bending rigidity of a bilayer kbi c khc Gaussian bending constant of a monolayer Gaussian bending constant of a bilayer khbi c k′a, k′′a, constants related to the curvature dependence of a as and k′′′ defined in eq 8 a k′x, k′′x, constants related to the curvature dependence of x as and k′′′ defined in eq 9 x constants related to the curvature dependence of ξ as k′ξ, k′′ξ, and k′′′ defined in eq 11 ξ lB Bjerrum length as defined in eq 25 p parameter related to Sp and defined in eq 52 q parameter related to Sp and defined in eq 44 r ratio between the tail volumes of the two surfactants in a binary mixture S reduced surface charge density as defined in eq 24 reduced surface charge density for a planar monolayer Sp T absolute temperature V volume of the surfactant hydrophobic tail x mole fraction of aggregated surfactant 1 xp mole fraction of aggregated surfactant 1 in a planar monolayer y relative difference in tail volumes between two surfactants in a binary mixture as defined in eq 16 z charge number hydrophobic-hydrophilic interfacial tension γhb  free energy per surfactant aggregated in a monolayer ′′ difference in free energy between surfactants 1 and 2 as defined in eq A3 for an ionic/nonionic surfactant mixture chain conformational contribution to  chain el electrostatic contribution to  hydrophobic contribution to  hb hg residual headgroup contribution to  mix free energy of mixing per surfactant aggregated in a monolayer κ inverse of the Debye screening length λ parameter related to the quadratic behavior of chain as defined in eq 56 ξ monolayer thickness ξp thickness of a planar monolayer LA060520T