Langmuir 2001, 17, 7675-7686
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Molecular Interpretation of the Mean Bending Constant for a Thermodynamically Open Vesicle Bilayer Magnus Bergstro¨m* Department of Chemistry, Surface Chemistry, Royal Institute of Technology, SE-100 44 Stockholm, Sweden, and YKI, Institute for Surface Chemistry, Box 5607, SE-114 86 Stockholm, Sweden Received April 14, 2001. In Final Form: August 2, 2001 Expressions for the various molecular contributions to the mean bending constant kc of a thermodynamically open vesicle bilayer have been derived, from which kc may be calculated from an appropriate molecularthermodynamic model as a function of the structure of the surfactants making up the aggregates as well as the solution state. It is demonstrated that kc determines the shape of a vesicle bilayer insofar as spherical vesicles form when kc is large and positive whereas nonspherical vesicles predominate at values of kc close to or below zero. It is found that contributions due to electrostatics and residual headgroup effects, which are present for one-component as well as mixed aggregates, mainly give rise to a positive value of kc whereas geometrical packing constraints are less important. However, the mixing of two or more surfactants can significantly reduce kc to values where nonspherical vesicles may begin to form. The magnitude of the reduction of kc due to mixing increases with increasing asymmetry with respect to headgroup cross-section area, charge number, and hydrocarbon tail volume between two surfactants in a binary surfactant mixture. The asymmetry is most pronounced for a binary mixture where the surfactant that carries the charge has the larger headgroup and the smaller tail. The reduction of kc due to mixing is, however, expected to be less than the corresponding effect for the bilayer bending constant of a spherical vesicle as a result of an additional positive contribution in the expression for the compositional contribution to kc.
Introduction A fruitful way to theoretically evaluate the structure of surfactant aggregates is to investigate the free energy per unit area as a function of the local curvature at the aggregate interface. This approach was suggested more than 25 years ago by Helfrich1,2 who considered the free energy per unit area γ of a surfactant film, e.g., a vesicle monolayer, as an expansion to second order with respect to the mean and Gaussian curvature, H and K, respectively; that is,
γ(H,K) ) γ0 + 2kc(H - H0)2 + k h cK
(1)
The mean and Gaussian curvatures are defined as H ≡ (c1 + c2)/2 and K ≡ c1c2, respectively, where c1 and c2 are the two principal curvatures at a single point on the aggregate surface most conveniently defined at the hydrocarbon/water interface. The various parameters in the Helfrich expression are usually interpreted as mechanical elasticity constants for a surfactant film that is bent at constant aggregation number and monolayer thickness. However, in some recent works3-5 the Helfrich expression has been reinterpreted so as to apply for thermodynamically equilibrated systems; that is, all parameters involved in eq 1, such as H0, kc, and k h c, are defined at constant chemical potentials of free surfactants in the surrounding bulk solution. Accordingly, in a thermodynamic context the mean and Gaussian h c, respectively, as well as the bending constants kc and k spontaneous curvature H0 are constants depending on * Tel: +46 8 790 99 05. Fax: +46 8 20 89 98. E-mail:
[email protected]. (1) Helfrich, W. Naturforsch. 1973, 28c, 693. (2) Helfrich, W. Phys. Lett. 1973, 43A, 409. (3) Bergstro¨m, M.; Eriksson, J. C. Langmuir 1996, 12, 624. (4) Bergstro¨m, M. Langmuir 1996, 12, 2454. (5) Bergstro¨m, M. J. Colloid Interface Sci. 2001, 240, 294.
the structure of the surfactants making up the aggregates as well as the solution state, that is, aggregate composition, electrolyte concentration, temperature, and so forth. By defining the surface as located at the hydrocarbon/ water interface of a vesicle monolayer, it may be shown that the work of bending a bilayer into a spherical vesicle consisting of an outer and an inner monolayer is a constant independent of the vesicle size.1 In accordance, the free energy of forming a spherical vesicle with radius R may be written as a sum of the work of forming a planar bilayer and the work of bending the bilayer into a spherical shell, that is,
E(R) ) 4π(kbi + 2γpR2)
(2)
where γp is the free energy per unit area of a planar vesicle h c], and monolayer, kbi ) 2kcbi + kcbi ) 2[2kc(1 - 2ξpH0) + k ξp is the half thickness of a planar bilayer.3,5 The size distribution of spherical vesicles may be evaluated from a multiple equilibrium approach (see further below) to
φbil(R) ∝ RRe-E(R)/kT
(3)
where φbil(R) is the volume fraction density of vesicle bilayers the shape fluctuations of which are taken into account by the pre-exponential factor ∝ RR and the value of the exponent R depends on the particular structure of a vesicle bilayer deviating from spherical shape.6,7 The average vesicle radius may be derived from eqs 2 and 3 to
Γ[(R + 2)/2](2φbil)1/(R+1) 4πkbi/(R+1)kT e Γ[(R + 1)/2](R+2)/(R+1)
〈R〉 = ξ
(4)
where ξ is half the thickness of the vesicle bilayer and Γ(x) (6) Bergstro¨m, M.; Eriksson, J. C. Langmuir 1998, 14, 288. (7) Morse, D. C.; Milner, S. T. Phys. Rev. E 1995, 52, 5918.
10.1021/la0105432 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/24/2001
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Figure 1. The average vesicle radius 〈R〉 according to eq 4 plotted against the pre-exponential factor R. The bilayer bending constant kbi ) 2[2kc(1 - 2ξpH0) + k h c] equals 1 kT (solid line), 1.5 kT (dashed line), and 2 kT (dotted line). Half the bilayer thickness was set to ξ ) 10 Å, and the total volume fraction of vesicles φbil ) 0.01.
Figure 2. The relative standard deviation σR/〈R〉 plotted against the pre-exponential factor R in accordance with eq 6.
is the gamma function. The total volume fraction of vesicle bilayers
φbil ≡
∫0∞φbil(R) dR
(5)
is a conserved quantity for a given overall surfactant concentration () φbil + φfree) since φfree is determined by equilibrium conditions.8 Likewise, the following expression for the polydispersity in terms of the relative standard deviation of the distribution function (eq 3) may be derived:
σR 〈R〉
)
x
[
]
2(R + 1) Γ((R + 1)/2) Γ(R/2) R2
2
-1
(6)
where σR ≡ x〈R2〉-〈R〉2. The average size and polydispersity according to eqs 4 and 6, respectively, are plotted against the pre-exponential exponent R in Figures 1 and 2. Both the average vesicle radius 〈R〉 and the relative standard deviation σR/〈R〉 are seen to decrease with R. The behavior of eq 4 is dominated by the factor exp[4πkbi/ (R + 1)kT], and hence 〈R〉 decreases rapidly with R for low R-values. This may be rationalized by considering the preexponential factor as a logarithmic contribution to the (8) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991; Chapters 16 and 17.
vesicle free energy which becomes more favorable (i.e., its magnitude increases) with increasing R. However, the corresponding free energy contribution per aggregated surfactant (∼R ln R/R2 since N ∝ R2) decreases with increasing R and, consequently, small vesicles are increasingly favored with an increasing R. From eq 4, it is also seen that for a given value of R, 〈R〉 increases h c]. exponentially with increasing kbi ) 2[2kc(1 - 2ξpH0) + k The exponent in the pre-exponential factor has been determined by Morse and Milner7 to R ) 2/3 by considering shape undulations in vesicle bilayers with a persistence length much smaller than the size of a vesicle. This value was obtained as a result of finite-size effects of an undulating bilayer subjected to the constraint that it must be geometrically closed. In accordance, it is entropically unfavorable to form a vesicle from a geometrically open bilayer since the number of accessible undulation conformations of the bilayer must be restricted when the bilayer is geometrically closed. Because of the decrease in entropy associated with this process, the resulting (logarithmic) term to the free energy becomes positive and, hence, the contribution to R is negative, that is, -1/3 (the remaining contribution to R equal to unity comes from a variable shift in the distribution function, see further below). For the number-weighted distribution N(R) ∝ φbil(R)/ R2, the pre-exponential exponent obtained in ref 7 becomes negative!, that is, equal to R - 2 ) -4/3. As a consequence, the polydispersity is only prevented from becoming infinite by higher order bending terms proportional to 1/R2, 1/R4, and so forth which are no longer negligible in the expression for E(R) in eq 2. Hence, the number-weighted polydispersity according to the expression in ref 7 must be very large and is expected to increase indefinitely with increasing average vesicle size. Moreover, the number of smaller vesicles must be much larger than the number of larger vesicles within the very broad distribution so that virtually only small vesicles will appear in a distribution N(R) with a pre-exponential exponent equaling -4/3, despite the fact that the (volume-weighted) average size must be very large [cf. eq 4 and Figure 1]. The assumption of a bilayer persistence length much smaller than the size of a vesicle is clearly essential for the value of R predicted by Morse and Milner. It is, however, difficult to find support for the existence of highly undulating bilayer vesicles from the various experiments on spontaneously formed vesicles, for example, the enormous number of electron micrographs of vesicles published in recent years. On the contrary, vesicles deformed from spherical geometry appear to be approximately shaped as oblates or prolates or, occasionally, as long tubular bilayer structures.9-11 In accordance, we have below adopted a different approach than the one in ref 7 by only considering spheroidal (oblate or prolate) deviations from spherical shape. This assumption may be justified by considering vesicle bilayers with a persistence length of the order of magnitude of or larger than the size of the vesicles assuming undulations with a larger radius of curvature to be energetically suppressed by higher order bending terms. From this treatment, it follows that the stability of a spherical bilayer shell with respect to shape deformations is determined by the mean bending constant kc so that the (9) Kaler, E. W.; Herrington, K. L.; Murthy, A. K.; Zasadzinski, J. A. N. J. Phys. Chem. 1992, 96, 6698. (10) Yatcilla, M. T.; Herrington, K. L.; Brasher, L. L.; Kaler, E. W.; Chiruvolu, S.; Zasadzinski, J. A. N. J. Phys. Chem. 1996, 100, 5874. (11) Yaacob, I. I.; Bose, A. J. Colloid Interface Sci. 1996, 178, 638.
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vesicles are mainly spherical for rather large and positive values of kc whereas a transition to ellipsoidal or even tubular vesicles is expected when kc assumes values close to or below zero. Moreover, it has been proven that in the h bi lamellar regime large values of k c ) kbi - 4kc favor the formation of a bicontinuous L3 structure (sponge phase) h bi whereas small values (possibly negative) of k c favor 12 stacks of planar lamellar sheets (LR). The final outcome of calculated values of the curvature related constants kc and k h c depend on the constraints imposed upon the bending process, for example, if the bending occurs at constant bilayer thickness, constant surface charge density, and so forth. In analogy with recent calculations of the curvature free energy of spherical vesicle bilayers5 and generally (tablet-) shaped micelles,13 we have in the present paper treated fully equilibrated surfactant bilayers that are open in a thermodynamic sense; that is, the various constants in the Helfrich expression (eq 1) are interpreted and evaluated at constant chemical potentials of free surfactant monomers in the surrounding bulk solution. In accordance, we have below derived expressions for the different contributions to the mean bending constant kc from which it is possible to calculate kc from knowledge of the structure of the constituent surfactants as well as the solution state of a binary surfactant mixture. This work, together with the earlier similar treatments of ours, is different from other molecular-thermodynamic approaches14-16 insofar as we have employed the curvature free energy approach introduced by Helfrich in a treatment based on equilibrium thermodynamics. As a result, our conclusions to be presented below are not dependent on any particular detailed molecular model. On the contrary, the curvature free energy may be quantitatively calculated from any appropriate model using the expressions for the h c] derived various contributions to kbi ) 2[2kc(1 - 2ξpH0) + k in ref 5 and kc derived in the present paper. Size Distribution of Reversibly Formed Bilayer Vesicles The thermodynamics of a dispersed solution of surfactant aggregates may be described with a set of multiple equilibrium conditions.17,18 Hence, we may write
EN,Nw + kT ln φN,Nw ) 0
(7)
where EN,Nw is the free energy of forming a vesicle bilayer enclosing Nw water molecules out of N free surfactant monomers. To account for (small) oblate or prolate deviations from spherical shape of the vesicle bilayer, we have allowed the number of surfactant molecules N in the bilayer and the number of water molecules Nw in the interior core to be independent variables in the summation and assumed the surfactant and water molecules to be incompressible. In the second term in eq 7, which accounts for the free energy of mixing comparatively large vesicles with the surrounding water molecules, we have employed the volume fraction φN,Nw of vesicles in a state defined by (12) Porte, G. J. Phys.: Condens. Matter 1992, 4, 8649. (13) Bergstro¨m, M. J. Chem. Phys. 2000, 113, 5569. (14) Eriksson, J. C.; Ljunggren, S. Langmuir 1990, 6, 895. (15) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (16) Yuet, P. K.; Blankschtein, D. Langmuir 1996, 12, 3802. (17) Hill, T. L. Thermodynamics of Small Systems (Parts I and II); Dover: New York, 1963-64. (18) Bergstro¨m, M. Fluctuations Promoting the Spontaneous Formation of Surfactant Micelles and Vesicles; Royal Institute of Technology: Stockholm, Sweden, 1995.
Figure 3. Schematic representation of the cross-section of a spherically or cylindrically shaped unilamellar vesicle with radial distance R to the surface that divides the inner and the outer monolayers, the thicknesses of which are denoted ξi and ξe, respectively.
the independent variables [N, Nw] rather than mole fraction in accordance with the result of lattice statistical calculations by Guggenheim and others.19 The vesicle size distribution function in eq 3 may be derived from eq 7 by summing φN,Nw over all combinations of the independent variables N and Nw giving
φves )
∑ exp(-EN,N /kT)
(8)
w
N,Nw
where φves is the total volume fraction of vesicles including the water cores enclosed by each bilayer. The summations in eq 8 can be approximated with the following integration:
φves )
∫0N∫0N exp(-EN,N /kT) dN dNw ) ∫0R∫1χ|J| exp(-E(R,χ)/kT) dχ dR w
w
(9)
where in the second equality the variables N and Nw have been replaced by R and the shape factor χ ≡ Am/4πR2. Am is the area of the surface dividing the inner and outer monolayer at a radial distance R from the center of a spherical vesicle [cf. Figure 3] or, for χ * 1, the volume equivalent nonspherical vesicle. The Jacobian
J)
|
|
∂(N,Nw) ∂(R,χ)
)
32π2ξpR4 vpvw
(10)
may be derived from the geometrical relationships N ) 8πξpR2χ/vp and Nw ) 3πR3/3vw where vw is the volume per water molecule, vp is the (average) volume per surfactant tail, and ξp is half the thickness of the planar bilayer with identical chemical potentials (see further below) as the vesicle bilayer with R . ξp. Moreover, the free energy of a vesicle bilayer defined by the state (R,χ), that is, having a volume 4πR3/3 and an area Am ) 4πR2χ, may be written as a sum of the work of forming a spherically shaped bilayer E(R) and the work (19) Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, 1952.
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γ′(R)∆Am of deforming the sphere into a spheroidal bilayer shell, that is,
E(R,χ) ) E(R) + γ′(R)∆Am ) E(R) + 4πR2γ′(R)(χ - 1) (11) where the magnitude of the coefficient for a Taylor expansion to first order with respect to ∆Am,
γ′(R) ≡
(
)
∂E(R,χ) ∂Am
(12)
R
determines the ability of the vesicle to resist deviations from spherical shape. Hence, by insertion of eqs 10 and 11 in eq 9 the vesicle size distribution φ(R) can be evaluated as follows: Figure 4. Example of a vesicle size distribution according to eq 15. The bilayer bending constant kbi was set equal to 2.47 -3 Å-5. The total volume kT, and 4πkT/(3kbi c apvw) ) 4.384 × 10 fraction of vesicles φves was set to 0.01 giving a planar bilayer tension γp ) 4.06 × 10-5 mJ/m2 and a most probable vesicle radius Rmax ) 899 Å.
32π2ξpR4 exp(-E(R)/kT) φ(R) ) vpvw
∫1∞exp(-4πR2γ′(R)(χ - 1)/kT) dχ ) 8πξpR2
exp(-E(R)/kT) (13)
vpvwγ′(R)
γ′(R) has been calculated by Milner and Safran20 using the Helfrich expression in eq 1 to be
γ′(R) )
2l(l + 1)kc R2
(14)
where, for an oblate or prolate surface, l ) 2. By insertion of eq 14 in eq 13, the following size distribution function for oblate/prolate fluctuating vesicles is obtained:
φ(R) )
4πξpR4kT
e-E(R)/kT ) l(l + 1)kcvpvw 2πξpkT 4 -4π(kbi+2γpR2)/kT Re (15) 3kcvpvw
The volume-weighted size distribution φ(R) ∝ R4 differs from what has previously been obtained (φ(R) ∝ R6)6 from a similar approach since in the latter work it was not taken into account that contributions from chain packing density fluctuations must be exactly canceled by finitesize effects.21 Moreover, since the volume fraction φves in eq 9 incorporates the total number of surfactants as well as the total number of molecules in the interior core of water enclosed by the vesicles, the pre-exponential factor in eq 15 corresponds to R ) 3 in the distribution φbil(R) in eq 3. An example of the size distribution in eq 15 is given in Figure 4. The corresponding relative standard deviation equals
σR 〈R〉
≡
x
〈R2〉 〈R〉
2
-1)
- 1 ≈ 0.323 x45π 128
(16)
As the size of the vesicles is approaching the same order of magnitude as the thickness of the bilayer terms proportional to 1/R2, 1/R4 must be included in the expression for E(R).5 Hence, the maximum value of σR/〈R〉, obtained in the limit R f ∞, is given by eq 16 (20) Milner, S. T.; Safran, S. A. Phys. Rev. A 1987, 36, 4371. (21) Bergstro¨m, M. J. Chem. Phys. 2000, 113, 5559.
whereas σR/〈R〉 is expected to decrease with decreasing vesicle size when R ∼ ξ provided E(R) is raised by the higher order terms that are significant at low R-values. It is customary to give the number density of vesicles rather than the volume-weighted size distribution as given in eq 15, and the corresponding distribution function N(R) ∝ R, with a relative standard deviation equaling 0.52, may be obtained by simply dividing eq 15 with the volume of a vesicle (V ∝ R3). This value of the polydispersity of equilibrated vesicles is in agreement with what has been observed for mixed cetyltrimethylammonium bromide (CTAB) or cetyltrimethylammonium tosylate (CTAT)/ dodecylbenzene sulfonic acid (HDBS) vesicles11 but somewhat larger than the observed values for mixed CTAB/ sodium octyl sulfate (SOS)22 and sodium dodecyl sulfate (SDS)/diddodecylammonium bromide (DDAB)23 vesicles. The lower values in the latter cases may be due to the influence of higher order terms in the expression for E(R) or by an additional electrostatic repulsion between overlapping double layers inside the vesicle core. The derivative of eq 15 equals zero at the position of the maximum of the distribution function in Figure 4 giving a most probable vesicle radius
Rmax )
x
kT 4πγp
(17)
Moreover, the total volume fraction
φves )
π3/2ξpkTR5max -4πkbi/kT e 16x2kcvpvw
(18)
is obtained by integrating eq 15 over all radii and eliminating γp using eq 17. From eq 18, the following relation between the bilayer bending constant kbi and Rmax is generated:
kbi/kT ) C +
5 ln Rmax 4π
(19)
where C ) ln(π3/2ξpkT/16x2kcvpvwφves)/4π is a constant for a given overall surfactant concentration. In Figure 5, kbi (22) Jung, T. H.; Coldren, B.; Zasadzinski, J. A.; Iampietro, D. J.; Kaler, E. W. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 1353. (23) Bergstro¨m, M.; Pedersen, J. S. J. Phys. Chem. B 2000, 104, 4155.
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as well as the corresponding relative standard deviation
σχ 〈χ〉
Figure 5. The bilayer bending constant kbi plotted against the most probable vesicle radius Rmax according to eq 19. The constant C is set equal to -0.328 corresponding to model calculations on C12 aliphatic chains.
is plotted against Rmax in accordance with eq 19 and it is seen that vesicles with R . ξ are obtained for values of kbi larger than about 2 kT. Accordingly, most probable vesicle radii in the range 10 nm < Rmax < 100 µm correspond to bilayer bending constants 1.5 kT < kbi < 5 kT and planar bilayer tensions 3 × 10-3 mJ/m2 > γp > 3 × 10-11 mJ/m2. The factor 5 before the ln Rmax term in eq 19 equals (R + 1); that is, the slope in the plot in Figure 5 increases with an increasing pre-exponential exponent R which means that the vesicle size decreases with increasing R for given values of kbi and φbil as illustrated in Figure 1. According to the size distribution in eq 15, vesicles with a finite average vesicle radius 〈R〉 much larger than ξ may form as the result of a balance between a large and positive kbi, tending to increase the size of the vesicles, and the entropy of self-assembling surfactant molecules, tending to decrease the size of the vesicles.5 In addition, oblate or prolate shape fluctuations of the vesicles favor smaller aggregates as the corresponding (favorable) free energy ∝ ln R/R2 decreases with increasing vesicle size. The described mechanism for the spontaneous formation of surfactant aggregates is analogous to that of rod-shaped micelles and is frequently referred to as an entropic stabilization in contrast to aggregates which are (energetically) stabilized by a minimum in the local free energy function E(R). Vesicles with an unfavorable curvature free energy (kbi > 0) may be stable with respect to flat bilayers (kbi ) 0) as a result of the elimination of the unfavorable rims of the latter aggregates when they close upon themselves to form vesicles. Curvature Free Energy of a Cylindrical Bilayer According to eqs 11 and 14, the size-independent free energy of deforming a spherical vesicle to oblate or prolate shape is ∆E(χ) ) 48π(χ - 1)kc. Hence, the corresponding Boltzmann-weighted frequency of different shape states can be obtained from eq 13, that is,
φ(χ) ) const × e-48πkc(χ-1)/kT
(20)
From the shape distribution (eq 20), the average shape factor
∫1 χe-48πk (χ-1)/kT kT )1+ 〈χ〉 ) ∞ -48πk (χ-1)/kT 48πk c ∫1 e ∞
c
c
(21)
≡
x
〈χ2〉 1 -1) 48πkc/kT + 1 〈χ〉
(22)
may be evaluated. According to eqs 20 and 21, large and positive values of the mean bending constant kc imply a large resistance for deviations from spherical shape, that is, φ(χ * 1) ≈ 0, whereas considerable shape fluctuations are expected when kc approaches zero. For values of kc close to or below zero, the expansion in eq 11 breaks down and the vesicles may obtain a nonspherical equilibrium shape, for example, a tubular shape. kc may be calculated by considering the free energy of forming a cylindrical bilayer. Inserting the appropriate values H ) 1/2r and K ) 0 in eq 1 gives the free energy per unit area of a cylindrical monolayer with a radial distance r to the hydrocarbon/water interface, that is,
γ(r) ) γp +
k1 k2 + 2 r r
(23)
where γp ) γ0 + 2kcH02 is the free energy per unit area of an infinitely large planar bilayer and k1 ) -2kcH0 and k2 ) kc/2 are two constants related to the (cylindrical) curvature of the vesicle monolayer. For an aggregate which is open in a thermodynamic sense, surfactant monomers are exchanged between the aggregates and the bulk solution so as to keep the chemical potentials µifree of the free surfactant monomers constant, that is,
k1 ≡
( ) ∂γ ∂r-1
[r-1 ) 0]
(24)
µifree
and
k2 ≡
( )
1 ∂2γ 2 ∂(r-1)2
[r-1 ) 0]
(25)
µifree
The free energy of a cylindrically shaped vesicle bilayer can thus be written as the sum of contributions from each vesicle monolayer, that is,
Ecyl(R) ) γiAi + γeAe
(26)
where γi ≡ γ(r ) -Ri) and γe ≡ γ(r ) Re) (the curvature is defined positive for the outer vesicle monolayer), Ai ≡ 2πRiL and Ae ≡ 2πReL are the areas of the inner and outer hydrocarbon/water interfaces located at the radial distances Ri ) R - ξi and Re ) R + ξe, respectively, and L is the length of the cylinder [cf. Figure 3]. As a result,
(
Ecyl(R) ) 2πL 2γpR +
)
kc R
(27)
The free energy per unit area of the bilayer may be obtained by dividing eq 27 with the midplane area Am ) 2πRL giving
γbi(R) ) 2γp +
kc R2
(28)
It may be interesting to compare eq 28 with the Helfrich expression applied to a vesicle bilayer rather than to a
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vesicle monolayer as in eq 1, that is,
free tan/kT ) x(υ1 - ln xfree 1 ) + (1 - x)(υ2 - ln x2 )
bi bi 2 γbi(H,K) ) γbi h bi 0 + 2kc (H - H0 ) + k c K
(29)
For a cylindrical geometry with H ) 1/2R and K ) 0, eq 29 equals bi γbi(R) ) γbi 0 + 2kc
(2R1 - H ) bi 0
2
(30)
By comparing eqs 26 and 28, we may identify kc ) kbi c /2 bi bi and γp ) γbi 0 /2 whereas kc H0 ) 0 since there are no terms proportional to 1/R in expression 28. Since, in general, bi kbi c * 0 (see further below), the spontaneous curvature H0 must equal zero for a thermodynamically open (pure as well as mixed) vesicle bilayer.5 Hence, by means of evaluating the free energy of a cylindrical bilayer the mean bending constant for a vesicle monolayer kc or, equivalently, for a vesicle bilayer kbi c ) 2kc may be calculated. Below, we will derive expressions for the different contributions to kc by means of investigating the various molecular-thermodynamic contributions to Ecyl(R). From these expressions, quantitative estimates of the different contributions to kc may be calculated from an appropriate detailed model such as those presented by Eriksson et al.,14 Nagarajan et al.,15 and Blankschtein and co-workers.16,24 As a result, we have below been able to explicitly reveal some interesting effects of surfactant aggregation into vesicles, for example, the promotion of nonspherical vesicles when mixing two surfactants that are asymmetric with respect to charge number, headgroup cross-section area, and tail length, and our conclusions are valid regardless of which specific model in refs 14-16 is utilized. Contributions to the Free Energy of Forming a Surfactant Aggregate The free energy per aggregated monomer ≡ E/N of a surfactant aggregate mixed by two surfactants (denoted with subscripts 1 and 2, respectively), minimized with respect to area per surfactant and composition, may be divided into the following five contributions: (i) the hydrophobic effect, (ii) electrostatics, (iii) residual headgroup effects, (iv) hydrocarbon chain conformational entropy, and (v) entropy of mixing two surfactants in a mixed surfactant aggregate;14,15,24,25 that is,
(r) ) x(r)1(r) + [1 - x(r)]2(r) ) tan + aγhc/w + el(r) + hg(r) + chain(r) + mix(r) (31) where the total aggregation number N ) N1 + N2 is the sum of the number of surfactant 1 and surfactant 2, respectively, and x ) N1/N is the mole fraction of aggregated surfactant 1. (i) The driving force for the process of self-assembling surfactant molecules in an aqueous solution is the reduction of the unfavorable interfacial area between hydrocarbon and water. According to Tanford,26 the free energy per aggregated monomer of bringing hydrocarbon and xfree from the surchains at concentrations xfree 1 2 rounding bulk solution into an n-alkane bulk phase may be written as (24) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618. (25) Ljunggren, S.; Eriksson, J. C. Prog. Colloid Polym. Sci. 1987, 74, 38. (26) Tanford, C. The hydrophobic effect; Wiley: New York, 1980; Chapter 7.
(32)
where υ1 and υ2 are constants that depend on the size and shape of the hydrocarbon moieties. The contribution to υi for an aliphatic hydrocarbon chain at 25 °C is about -1.5 for each methylene group (-CH2-) and -3.5 for a methyl group (CH3-). The free energy due to the interface of the hydrocarbon core that is exposed to the surrounding aqueous solvent with an area a per aggregated monomer may be calculated from the corresponding hydrocarbon/water interfacial tension γhc/w. The macroscopic curvature independent value of γhc/w ) 50 mJ/m2 at room temperature has previously been successfully employed when the free energy was minimized with respect to the radial distance ξ of the hydrocarbon core for spherical,27 cylindrical,28 and planar29 geometries. (ii) A large unfavorable contribution arises in ionic surfactant systems as a result of the free energy of forming a charged aggregate surface with an adjacent diffuse layer of counterions. The electrostatic free energy contribution favors highly curved structures, allowing for the formation of rather small aggregates at sufficiently large surface charge densities and low electrolyte concentrations. Hence, the curvature-dependent electrostatic free energy per aggregated charge of a cylindrical vesicle monolayer may be written as an expansion to second order in curvature at constant chemical potentials of the free surfactant monomers in the surrounding bulk solution, that is,
(
el ) pel 1 +
)
k′el k′′el + 2 r r
(33)
where pel is the electrostatic free energy per charge of a planar surface and r, as before, is the radius of curvature of a cylindrically shaped hydrocarbon/water interface. With the exception of solution states corresponding to very low surface charge densities and/or high electrolyte concentrations, that is, solution states where energetics rather than the entropy of mixing counterions and solvent is predominating,30 el is lowered for a positively curved aggregate surface which means that the first-order correction constant k′el usually assumes negative values. Explicit expressions that provide a rather accurate estimate of the electrostatic free energy to second order in curvature for a surfactant film consisting of monovalent ionic surfactants as functions of surface charge density and electrolyte concentration have been derived for a cylindrical surface by Evans and Ninham31 and for a spherical surface by Mitchell and Ninham32 and, independently, Lekkerkerker33 from the nonlinear PoissonBoltzmann theory. (iii) Several more or less known effects that contribute to the surfactant headgroup free energy other than electrostatics may be included in a single parameter hg characteristic for each aggregated surfactant headgroup. The most important curvature-dependent contribution to hg is obtained as a result of the entropy of mixing (27) Eriksson, J. C.; Ljunggren, S.; Henriksson, U. J. Chem. Soc., Faraday Trans. 2 1985, 81, 833. (28) Eriksson, J. C.; Ljunggren, S. J. Chem. Soc., Faraday Trans. 2 1985, 81, 1209. (29) Ljunggren, S.; Eriksson, J. C. J. Chem. Soc., Faraday Trans. 2 1986, 82, 913. (30) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain; VCH: New York, 1994; Chapter 3. (31) Evans, D. F.; Ninham, B. W. J. Phys. Chem. 1983, 87, 5025. (32) Mitchell, D. J.; Ninham, B. W. Langmuir 1989, 5, 1121. (33) Lekkerkerker, H. N. W. Physica A 1989, 159, 319.
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Langmuir, Vol. 17, No. 24, 2001 7681
headgroups and adjacent water molecules at the micellar surface.34 This contribution tends to favor a large aggregate curvature since the entropy of mixing increases upon diluting the headgroups. Hence, we may write
hg ) phg
(
)
k′hg k′′hg + 2 1+ r r
(
(
ξ(r) ) ξp 1 +
)
k′ξ k′′ξ + 2 r r
(37)
(34)
for a vesicle monolayer, where phg is the residual headgroup free energy per aggregated monomer of a planar bilayer. Since the headgroup contribution favors a positive curvature of the aggregate interface, the first-order correction constant phg k′hg usually assumes negative values the magnitude of which increases with increasing headgroup cross-section area.34 (iv) Since the end of each hydrocarbon chain that is attached to the surfactant headgroup is fixed to the micelle/solvent interface, the number of accessible chain conformations must be restricted in a micelle as compared with the corresponding n-alkane bulk phase. In analogy with the headgroup contributions, the corresponding curvature-dependent (unfavorable) free energy per aggregated surfactant in a vesicle monolayer may be written as
chain ) pchain 1 +
per unit area in eq 23, can be written as an expansion to second order in curvature, that is,
)
k′chain k′′chain + 2 r r
(35)
where, again, superscript p refers to a planar bilayer. The unfavorable chain conformational contribution decreases with geometry as sphere > cylinder > plane, and hence k′chain is in general positive, favoring less curved structures.27,29,35-38 (v) The ideal entropy per aggregated monomer of mixing two surfactants in a vesicle monolayer
mix ) x ln x + (1 - x) ln(1 - x)
(36)
is usually employed since an analytical expression of mix, in the derivation of which each microstate must be weighted with the appropriate Boltzmann factor, is for practical purposes not available. Contributions to the Mean Bending Constant To derive expressions for the various contributions to the mean bending constant kc, it is necessary to consider the curvature dependence of the thickness and composition of a thermodynamically open vesicle monolayer as well as curvature effects due to geometrical packing constraints. The thickness obtained by minimizing the aggregate free energy of a thermodynamically open bilayer leaflet may vary as a response to a change in curvature so as to keep the free monomer chemical potentials constant. Accordingly, ξ as a function of the curvature of a cylindrical hydrocarbon/water interface, analogous to the free energy (34) Bergstro¨m, M.; Eriksson, J. C. Langmuir 2000, 16, 7173. (35) Gruen, D. W. R.; Lacey, E. H. B. The Packing of Amphiphile Chains in Micelles and Bilayers. In Surfactants in solution; Mittal, K., Lindman, B., Eds.; Plenum: New York, 1984; Vol. I, p 279. (36) Gruen, D. W. R. J. Phys. Chem. 1985, 89, 153. (37) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. Amphiphile Chain Organization in Micelles of Different Geometries. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam, 1985; p 404. (38) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. J. Chem. Phys. 1985, 83, 3597.
Likewise, the (optimal) composition in a thermodynamically open cylindrical vesicle monolayer varies according to the following expansion to second order in curvature:
(
x(r) ) xp 1 +
)
k′x k′′x + 2 r r
(38)
The area per aggregated monomer at the hydrocarbon/ water interface of a vesicle monolayer is given by the geometrical relation
a-1(r) )
(
)
ξ(r) ξ(r) 12r v(r)
(39)
valid for a cylindrical surface located at a radial distance r. The volume per surfactant hydrocarbon chain v(r) ) v2 + ∆vx(r) is a function of the radius of curvature of a thermodynamically open vesicle monolayer consisting of surfactants with tails of different sizes, that is, ∆v ≡ v1 - v2 * 0. According to eq 39, a remains dependent on the monolayer curvature 1/r also for the special case where both ξ and v are constants. By means of employing the various curvature-dependent expressions, eqs 33-35 and 37-39, to calculate γ(r) ≡ (r)/a(r) from eqs 31 and 39 and collecting terms proportional to 1/r2, an expression for k2 ) kc/2 in terms of the various free energy contributions can be derived. The resulting expression for kc may be subdivided into the following contributions: hg chain + kcomp + kel + kmix (40) kc ) kgeom c c c + kc + kc c hg chain are entirely due to the explicit where kel c , kc , and kc curvature dependencies of the respective free energy contribution as given in eqs 33-35. The free energy of mixing two surfactants in a vesicle monolayer as given in eq 36 becomes dependent on curvature since the monolayer composition is a function of curvature x ≡ x(r) resulting in the contribution kmix c . However, the remaining contriand kcomp ) butions to the mean bending constant (kgeom c c are found to be independent of any curvature dependences of the different contributions to but are rather the result of geometrical packing constraints and a curvaturedependent monolayer thickness, respectively. Geometrical Packing Constraint Contribution to kc. As a result of the geometrical packing constraints implied in expression 39, an explicit contribution to kc is generated. This contribution may be expressed in terms of the change in number of aggregated surfactants ∆N ≡ Ae(1/ae - 1/ap) + Ai(1/ai - 1/ap) as a planar bilayer is curved into a cylindrically shaped shell at a fixed hydrocarbon/ water interfacial area and at constant free monomer chemical potentials. Ae and Ai are the areas of the outer and inner hydrocarbon/water interface, respectively, and ae, ai, and ap are the corresponding areas per aggregated monomer where subscript p denotes planar geometry. From eqs 37-39, we may derive the following expression
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Bergstro¨ m
Figure 6. The geometrical packing constraint contribution to the mean bending constant as given in eq 43 kgeom ) c -2γhc/wR2[(ξe + ξi)/2ξp - 1] plotted against the mole fraction of nonionic surfactant in a planar bilayer xp for a mixture of a monovalent ionic and a nonionic surfactant with identical C12 hydrocarbon chains (∆v ) 0). The monolayer thicknesses ξp, ξi, and ξe that determine kgeom were calculated from a detailed c model given in ref 3. The hydrocarbon/water interfacial tension is set to γhc/w ) 50 mJ/m2.
for ∆N per unit length of a cylindrically shaped tubular bilayer:
[(
)
4π 2 ξe + ξi ∆N ) R -1 L apR 2ξp ξp ∆v ∆v xp k′ξ - - xpk′x k′ + k′′x vp 2 vp x
((
)
)]
(41)
In eq 41, which is proportional to 1/R, we have neglected higher order terms, that is, terms ∝ 1/R2 and so forth. The geometrical contribution to the work of bending a bilayer into a cylindrical tube can then be obtained from the relation
2πLkgeom c ′∆N ) R
(42)
where ′ ) ap(γp - γhc/w) ≈ -apγhc/w (γhc/w . γp since γhc/w ≈ 50 mJ/m2 and γp ∼ kT/8πRmax2 , 1 mJ/m2) is the free energy per monomer of changing the volume of the hydrocarbon part of the bilayer at a fixed hydrocarbon/ water surface area. Hence,
[ (
)
ξe + ξi -1 + 2ξp ξp ∆v ∆v k′ξ - - xpk′x k′ + k′′x xp vp 2 vp x
) 2γhc/w -R2 kgeom c
((
)
)]
(43)
The first term in eq 43 () -2γhc/wR2[(ξe + ξi)/2ξp - 1]) due to a change in thickness is the contribution to kgeom c from 2ξp to ξe + ξi of a bilayer that may adjust its thickness in response to a change in curvature at constant free surfactant chemical potentials. The appearance of the remaining second term is a result of the difference in volume between the hydrocarbon parts of surfactants 1 and 2, and it vanishes for the special case of two surfactants as with identical hydrocarbon parts, that is, ∆v ) 0. kgeom c calculated from a detailed model is plotted in Figure 6 against the surfactant composition for a mixture of a monovalent ionic and a nonionic surfactant with identical C12 hydrocarbon parts (∆v ) 0)3 and in Figure 7 for a
Figure 7. The geometrical packing constraint contribution to the mean bending constant as given in eq 43 kgeom ) c -2γhc/wR2[(ξe + ξi)/2ξp - 1] plotted against the surfactant mole fraction in a planar bilayer for a mixture of a monovalent anionic and a monovalent cationic surfactant with identical C12 hydrocarbon chains (∆v ) 0). The monolayer thicknesses ξp, ξi, and ξe that determine kgeom were calculated from a detailed c model given in ref 4. The hydrocarbon/water interfacial tension is set to γhc/w ) 50 mJ/m2.
mixture of an anionic and a cationic surfactant with identical C12 hydrocarbon parts (∆v ) 0)4. It is seen that kgeom decreases from positive to negative values with c decreasing surface charge density since the bilayer thickness decreases with increasing curvature at high surface charge densities whereas it increases upon curving the bilayer at low surface charge densities. One contribuin eq 43 is proportional to -(xpk′x∆v/vp) and tion to kgeom c is always negative. This term is partly cancelled by another important term depending on ∆v in eq 43 if the surfactant with the larger tail is concentrated in the inner vesicle layer. kgeom vanishes for the case of surfactants with identical c hydrocarbon parts where the bilayer thickness is kept constant during the process of bending a planar bilayer to a cylindrical shell. This is in contrast to the geometrical contribution to the curvature free energy of a spherical where kgeom ) 2kbi,geom +k h bi,geom ) for vesicle () 4πkgeom bi bi c c geom which a large positive contribution to kbi proportional is generally to ξ2p is always present.3-5 As a result, kgeom bi geom bi,geom ) ( k k h )/2 implying that much larger than kgeom c bi c k h bi,geom usually assumes fairly large positive values. c Compositional Contribution to kc. As a result of the curvature dependence of the composition as expressed in eq 38, an explicit contribution to the curvature free energy is obtained, that is,
kcomp ) c
((
)
)
2ξp ξp ∆v x ′′ k′ξ - - xpk′x k′ + k′′x vp p 2 vp x
(44)
where ′′ ≡ p1 - p2 is the difference in free energy per monomer of surfactants 1 and 2, respectively, aggregated in a planar bilayer; that is, p1 and p2 are related to the free energy of a planar bilayer as p ≡ xpp1 + (1 - xp)p2. The following explicit expression may be evaluated from eqs 31-36 with r ) 0:
′′ ) υ1 - υ2 + kT ln
( ) xfree 2
xfree 1
+ z1p1el1 - z2p2el2 + phg1 -
p p phg2 + chain1 - chain2 + kT ln
( )
xp (45) 1 - xp
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Langmuir, Vol. 17, No. 24, 2001 7683
Figure 8. The compositional contribution to the mean bending constant kcomp as given in eq 44 plotted against the mole c fraction of nonionic surfactant in a planar bilayer xp for a mixture of a monovalent ionic and a nonionic surfactant with identical C12 hydrocarbon chains (∆v ) 0). The various parameters in eq 44 that determine kcomp were calculated from a detailed model c given in ref 3.
Figure 9. The compositional contribution to the mean bending constant kcomp as given in eq 44 plotted against the surfactant c mole fraction in a planar bilayer xp for a mixture of a monovalent anionic and a monovalent cationic surfactant with identical C12 hydrocarbon chains (∆v ) 0). The various parameters in eq 44 that determine kcomp were calculated from a detailed model c given in ref 4.
For a mixture of surfactants with identical hydrocarbon is mainly determined by a term parts (∆v ) 0), kcomp bi proportional to ξp/2 in eq 44 () -ξ2pxp′′k′x/vp) which is negative if ′′ and k′x have the same sign. The sign of ′′ may be further investigated by taking into account the equilibrium relation
at xp ) 0, 0.5, and 1 [cf. Figure 9]. kcomp as given in Figures c 8 and 9 was calculated from detailed models given in refs 3 and 4, respectively. This result may be compared with what has been obtained for the case of spherical vesicles5 for which a term proportional to ξp/2 (present in eq 44) is ) 2kcomp,bi +k h comp,bi and, absent in the expression for kcomp bi c c comp is mainly determined by a second-order hence, kbi constant (corresponding to k′′x) and is generally negative for mixtures of an ionic and a nonionic surfactant3 as well as for mixtures of an anionic and a cationic surfactant.4 As a result, the reduction of kc due to surfactant mixing [cf. further below] is expected to be less than for kbi ) h c]. 2[2kc(1 - 2ξpH0) + k For mixtures of surfactants with hydrocarbon sizes of unequal size (∆v * 0), an additional term () in -2ξp′′(xpk′x)2∆v/vp2) appears in the expression for kcomp c eq 43. This term is negative if p is lowered by the addition of the surfactant with the larger hydrocarbon part and otherwise positive. Electrostatic Contribution to kc. From eqs 33 and 37-40, we may derive the following expression for the electrostatic contribution to kc:
(
)
dp dp2 dp1 dp2 ) + ′′ + xp )0 dxp dxp dxp dxp
(46)
which minimizes the free energy per monomer in a planar bilayer with respect to bilayer composition. Equation 46 may be rearranged so as to give an explicit expression for ′′,
dp2 dp1 - (1 - xp) ′′ ) -xp dxp dxp
(47)
which vanishes if p1 and p2 are linear functions with respect to xp, that is, if p1 and p2 are constants, implying vanishes for this particular case. For mixtures that kcomp bi containing at least one ionic surfactant, the electrostatic free energy contribution usually dominates the dependence of p1 and p2 with composition, and for a mixture of an ionic and a nonionic surfactant only the free energy per monomer of the ionic surfactant contributes to p el electrostatics. As a result, ′′ ) -xel p (del/dxp ) which is p negative if el is raised by the addition of the ionic surfactant. equaling -ξ2pxp′′k′x/vp Hence, the contribution to kcomp c p becomes positive if el is raised by the addition of the surfactant that is enriched in the outer vesicle monolayer and otherwise negative. For a mixture of an ionic and a nonionic surfactant, pel is raised by the addition of the ionic surfactant which is enriched in the outer layer and, is positive in the entire compositional regime hence, kcomp c 0 < xp < 1 [cf. Figure 8]. Similarly, for a mixture of an anionic and a cationic surfactant, pel is raised by the addition of the surfactant in excess which is enriched in also the outer vesicle monolayer. As a result, kcomp c becomes positive for the catanionic surfactant case in the regimes 0 < xp < 0.5 and 0.5 < xp < 1 whereas it vanishes
kel c ) 2ξp p z + (z1 - z2)xp vp el 2
[|
|((
k′ξ -
)
)
ξp ∆v - xpk′x k′ + k′′el ( 2 vp el
]
(z1 - z2)xpk′xk′el (48) For the pure surfactant case, kel c is mainly determined by the usually positive term ) -ξ2p|z2 + (z1 - z2)xp|pelk′el/vp (k′el is usually negative, see above). In addition, terms proportional to the second-order constant in electrostatic free energy k′′el and the first-order constant in monolayer thickness k′ξ contribute to kel c . This is in contrast with the curvature free energy of a spherical vesicle for which a term proportional to ξp/2 is absent and only terms of the latter kind contribute to kel bi which is usually negative mainly as a result of the term including a secondorder constant with respect to the electrostatic free energy.5
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Bergstro¨ m
Figure 10. The electrostatic contribution kel c to the bilayer bending constant as given in eq 49 (solid line) plotted against the mole fraction of nonionic surfactant in a planar bilayer xp for a mixture of a monovalent ionic and a nonionic surfactant with identical C12 hydrocarbon chains (∆v ) 0). The contribution to kel c common for pure as well as mixed vesicles () -2ξppel(1 - xp)[(k′ξ - ξp/2)k′el + k′′el]/vp, dashed line) as well as the contribution only appearing for mixed vesicles () -2ξppelxpk′xk′el/vp, dotted line) are also given. A minimum at a given value of xp for the latter contribution appears as the compositions in the monolayers are able to adjust in response to a change in bilayer curvature. The various parameters in eq 49 were calculated from a detailed model given in ref 3.
For the special case of a mixture of a monovalent ionic and a nonionic surfactant, eq 48 equals
kel c )
[
((
)
)
2ξp p ξp ∆v (1 - xp) k′ξ - - xpk′x k′ + k′′el vp el 2 vp el
]
xpk′xk′el (49) and for a mixture of two monovalent and oppositely charged surfactants
kel c )
[|
2ξp p 2xp - 1 vp el
|((
k′ξ -
)
) ]
ξp ∆v - xpk′x k′ + k′′el ( 2 vp el 2xpk′xk′el (50)
Equations 49 and 50, as calculated from a detailed model3,4 for surfactants with identical C12 hydrocarbon chains (∆v ) 0), are plotted in Figures 10 and 11, respectively. An additional always negative term proportional to k′x appears in eqs 49 () -2ξppel(1 xp)xpk′xk′el/vp) and 50 () (2ξppel|2xp - 1|xpk′xk′el/vp) for mixed vesicles. This term brings down kel c for surfactant mixtures as compared with one-component surfactant solutions, and a minimum in kel c is obtained at some value 0 < xp < 1 for the ionic/nonionic surfactant case [cf. Figure 10]. For the catanionic case, two minima on either side of equimolar composition are obtained [cf. Figure 11]. Moreover, a factor of 2 appears in the mixing term for the latter case but is absent in the former case, since the difference in charge number between two oppositely charged monovalent surfactants is 2 whereas it equals unity for a mixture of a monovalent ionic and a nonionic surfactant. In other words, the larger asymmetry between the anionic and the cationic surfactant enhances the reduction of kel c.
Figure 11. The electrostatic contribution kel c to the bilayer bending constant as given in eq 50 (solid line) plotted against the surfactant mole fraction in a planar bilayer xp for a mixture of a monovalent anionic and a monovalent cationic surfactant with identical C12 hydrocarbon chains (∆v ) 0). The contribution to kel c common for pure as well as mixed vesicles () 2ξppel|2xp - 1|[(k′ξ - ξp/2)k′el + k′′el]/vp, dashed line) as well as the contribution only appearing for mixed vesicles () (4ξppelxpk′xk′el/vp, dotted line) are also given. A maximum at equimolar composition, with a minimum at either side of it, appears for the latter contribution as a result of the fact that the compositions in the two vesicle monolayers are able to adjust in response to a change in bilayer curvature. Note that the mixing term (including k′x) contains a factor of 2 as compared with the corresponding term in the expression plotted in Figure 10 for the ionic/nonionic mixture. As a result, kel c is further brought down in the catanionic mixture where the difference in charge number between the two surfactants is larger. The effect is, however, counteracted by an increasing electrolyte concentration in the catanionic case as a consequence of a release of a pair of counterions to the bulk solution when two oppositely charged surfactants are aggregated. The various parameters in eq 50 were calculated from a detailed model given in ref 4.
If the two surfactants have different hydrocarbon sizes (∆v * 0), an additional contribution appears in expression 48 for kel c equaling
kel c ∆v ) -
2pelξp∆v |z2 + (z1 - z2)xp|xpk′xk′el vp2
(51)
which is negative if k′x and ∆v have opposite signs (k′el is negative), that is, if the surfactant with the smaller hydrocarbon part is enriched in the outer layer and otherwise positive. Hence, in the former case a large negative contribution from kel c ∆v may bring down kc to values close to, or below, zero where nonspherical vesicles are able to form. In an ionic/nonionic surfactant mixture (kel c ∆v ) -2ξp(1 - xp)xpk′xpelk′el∆v/v2p), it is usually the ionic surfactant that is concentrated in the outer layer and, hence, the term is negative if the nonionic surfactant has the larger hydrocarbon part. Similarly, for the catanionic case the term p 2 (kel c ∆v ) -2ξp|2xp - 1|xpk′xelk′el∆v/vp) must be negative if the surfactant in excess (which carries the charge and is enriched in the outer layer) has the smaller hydrocarbon part. This prediction may be compared with observations of mixtures of an anionic surfactant with a C8 aliphatic hydrocarbon chain and a cationic surfactant with a C16 hydrocarbon chain where the anionic-rich vesicles were observed to deviate appreciably from a spherical shape whereas vesicles rich in the long-chain cationic surfactant were found to be spherically shaped.10 Hence, these
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Langmuir, Vol. 17, No. 24, 2001 7685
observations are consistent with a reduction of kc to values close to zero in the anionic-rich case (the surfactant with the smaller hydrocarbon part carries the charge) where the contribution from kel c ∆v is negative in contrast to the cationic-rich case for which kel c ∆v is positive. Residual Headgroup Contribution to kc. From eqs 34 and 37-40, the contribution to kc due to surfactant headgroup free energy other than electrostatics may be derived as follows:
[ (( ((
) )
) )]
2ξp p ξp ∆v hg1xp k′ξ - - xpk′x k′ + k′′hg1 + vp 2 vp hg1 ξp ∆v phg2(1 - xp) k′ξ - - xpk′x k′ + k′′hg2 + 2 vp hg2 2ξp x k′ (p k′ - phg2k′hg2) (52) vp p x hg1 hg1
khg c )
where hg is usually brought down if a surfactant film is positively curved, that is, phgk′hg < 0. Hence, analogous to the electrostatic case the main contribution to khg c for onecomponent surfactant systems (k′x ) 0) is a term proportional to ξp/2 () -ξ2p phgxpk′hg/vp) which is positive. Also in analogy with the electrostatic case, terms including ∆v bring down khg c for a binary mixture where the surfactant with the smaller tail is concentrated in the outer layer but are positive if the surfactant with the larger tail is concentrated in the outer layer. Hydrocarbon Chain Contribution to kc. From eqs 35 and 37-40, we may derive the following expression:
) kchain c ξp 2ξp p ∆v chain1xp k′ξ - - xpk′x k′ + k′′chain1 + vp 2 vp chain1 ξp 2ξp p ∆v chain2(1 - xp) k′ξ - - xpk′x k′ + vp 2 vp chain2 2ξp p k′′chain2 + x k′ (p k′ - chain2 k′chain2) (53) vp p x chain1 chain1
[
[
((
)]
((
)
)
)]
for the contribution to kc as a result of the hydrocarbon p and k′chain chain conformational entropy. Both chain are usually positive, and hence the main contribution for one-component surfactant systems to kchain c p xpk′chain/vp) must be negative in contrast to () -ξp2 chain the headgroup contributions to kc. The last term in eq 53 p p k′chain1 - chain2 k′chain2)/vp) only appears for () 2ξpxpk′x(chain1 mixtures of surfactants with different hydrocarbon chains and is positive if the surfactant with the larger chain (for p is larger) is concentrated in the outer layer. which chain Terms including ∆v, on the other hand, are negative if the surfactant with the larger tail is concentrated in the outer layer and, hence, these mixing terms that both depend on the asymmetry with respect to surfactant chain length partly cancel each other. As a matter of fact, it is p k′chain has a linear straightforward to show that if chain dependence on the surfactant tail volume, which is expected to be approximately true at least for small differences in tail volume between two surfactants, the two terms exactly cancel. As a result, the contribution to kchain due to mixing is expected to be very small in c contrast to the corresponding effects due to the different headgroup contributions. The same holds true for the influence of mixing on the spherical bilayer bending .5 constant kchain bi
Entropy of Mixing Contribution to kc. The (ideal) free energy of mixing aggregated surfactants given in eq 36 gives rise to a contribution to kc equal to
kmix c
)
ξpxpk′x2kT vp(1 - xp)
(54)
is proportional to the square of k′x We may note that kmix c and must therefore always be positive. The reason for this is that it is unfavorable from the point of view of free energy of mixing to have different compositions in the two vesicle monolayers. Influence of the Asymmetry between Two Surfactants on the Mean Bending Constant The net effect of surfactant mixing must be to bring down kc, that is, the sum of all terms including k′x must be negative, since the compositions in the two vesicle monolayers are determined by minimizing the free energy of respective monolayers. Nevertheless, the different mixing terms for each contribution to the curvature free energy may enhance or counteract each other depending on the structure of the two surfactants making up the aggregates. In accordance, we have previously shown that the bending constant kbi for a spherical bilayer shell is reduced by surfactant mixing and that the reduction of kbi may be described by a simple rule of thumb in terms of the asymmetry between two surfactants in a binary surfactant mixture with respect to the size of headgroup and hydrocarbon tail as well as the charge number of the headgroup of an ionic surfactant.5 From the expressions derived in the preceding section, it is found that the same holds true for the mean bending constant kc. Hence, additional terms (including k′x) in expressions 48 and 51 appear for vesicle bilayers mixed by two surfactants with identical hydrocarbon parts (∆v ) 0). Analogous to the case of kbi for a spherical vesicle, the magnitude of the mixing term for each of these contributions is expected to increase with increasing asymmetry between the two aggregated surfactants with respect to headgroup cross-section area and charge number. For mixtures of surfactants with tails of different sizes, there is an additional effect which is due to terms depending on both k′x and ∆v in eqs 48 and 52. These hg terms further bring down kc (kel c and/or kc ) if the surfactant with the larger tail is concentrated in the inner vesicle monolayer. in eq 53, one term proporIn the expression for kchain c tional to k′x but excluding ∆v is partly canceled by another term proportional to k′x but including ∆v. Moreover, terms in eq 43 partly including ∆v in the expression for kgeom c cancel each other if the surfactant with the larger tail is concentrated in the inner layer and, hence, the corresponding net contribution is small. As a result, kc is furthest brought down for a mixture as illustrated in Figure 12 where the surfactant that carries the charge has the larger headgroup size as well as the smaller tail. The magnitude of the reduction of kc increases with increasing difference with respect to these three quantities. However, since the compositional contribution to kc in eq 44 is usually positive [cf. Figures 8 and 9], in contrast to the corresponding contribution to the bilayer bending constant for spherical vesicles kbi ) 2[2kc(1 - 2ξpH0) + k h c],5 the effect of surfactant mixing to bring down kc is expected to be less than for kbi.
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Langmuir, Vol. 17, No. 24, 2001
Figure 12. The mean bending constant of a vesicle bilayer kc is, in general, brought down when two surfactants are mixed since the compositions in the two vesicle monolayers generally assume different values. If the magnitude of this reduction of kc, which depends on the difference in charge number, headgroup cross-section area, and tail volume between the two surfactants, is sufficiently large, it may promote the formation of nonspherical vesicles. The reduction of kc due to mixing is most pronounced for a binary surfactant mixture as illustrated in the figure; that is, the surfactant carrying the charge has the larger headgroup and the smaller tail.
Since large oblate/prolate shape fluctuations are expected to occur at low values of kc and a transformation from spherical to tubular vesicles is expected at kc ≈ 0, we may conclude that the reduction of kc by mixing at least two asymmetric surfactants promotes the formation of nonspherical vesicles. Conclusions The shape of a vesicle bilayer is mainly determined by the mean bending constant kc so that vesicles are approximately spherical at large and positive values of kc whereas significant deviations from spherical shape may occur at values of kc close to or below zero. In the present paper, we have derived expressions for kc in terms of the various contributions to the free energy of forming a surfactant aggregate. Hence, from these expressions kc may be calculated from the structure of the surfactants making up the vesicles (hydrocarbon chain length, crosssection area, charge number of the headgroup, etc.) as well as the solution state (electrolyte concentration, surfactant composition, etc.) using an appropriate detailed molecular-thermodynamic model. kc may be divided into several different contributions: (i) geometrical packing constraints, (ii) surfactant composition, (iii) electrostatics, (iv) residual headgroup effects, (v) chain conformational entropy effects, and (vi) entropy of mixing.
Bergstro¨ m
We have previously, in a similar manner, derived expressions for the bilayer bending constant kbi ) h c] of thermodynamically open vesicles,5 2[2kc(1 - 2ξpH0) + k and it may be suitable to compare the different contributions to kbi and kc. The main contribution to kbi is due to geometrical packing constraints subjected to the surfactant tails aggregated in a vesicle bilayer.3-5 For the case of surfactants with identical tails, the geometrical contribution may be divided into two terms: one is due to the bending of a planar bilayer to a spherical shell with the same thickness and is always large and positive, whereas the other is the result of a change in bilayer thickness upon bending a bilayer at constant free surfactant chemical potentials. The former term is, however, absent in the corresponding expression for kc which, as a result, is usually rather small and may be positive or negative depending on the solution state. A term proportional to -ξ/2 appears in each of the expressions for the compositional, electrostatic, headgroup, and hydrocarbon chain contributions to kc in eqs 44, 48, 52, and 53, respectively, which is absent in each of the corresponding expressions for kbi. Since these terms also are proportional to the first-order bending constant times the planar free energy for each contribution (′′k′x, pelk′el, etc.), they must be rather large and positive for the compositional contribution as well as for the contributions favoring a positive aggregate curvature, that is, the free energy contributions associated with the surfactant headgroups. As a result, these terms usually raise the value of kc to rather large and positive values, thus promoting a spherical shape of the bilayer vesicles. Analogous to the case of kbi, terms including k′x (which only appears for mixed vesicles) tend to bring down kc thus promoting deviations from spherical shape of mixed vesicle bilayers. The magnitude of these mixing terms increases with increasing asymmetry between two aggregated surfactants where the asymmetry may be with respect to charge number and cross-section area of the headgroups as well as volume of the hydrocarbon tail. The different asymmetries may enhance or counteract one another. For example, the larger asymmetry between a surfactant that carries charge (e.g., the surfactant in excess in an anionic/cationic surfactant mixture) with a smaller tail and a surfactant that does not carry charge (e.g., the surfactant in deficit in an anionic/cationic surfactant mixture) with a larger tail than vice versa is expected to generate a larger deviation from spherical shape in the former case. As a matter of fact, this prediction is consistent with observations of mixtures of an anionic surfactant with a C8 chain and a cationic surfactant with a C16 chain.10 The reduction of kc due to mixing is, however, expected to be less pronounced than the corresponding effect for the spherical bilayer bending constant kbi ) 2[2kc(1 h c] as a result of the positive term proportional 2ξpH0) + k to ξp/2 for kcomp in eq 44 the corresponding term of which c is absent in the latter case.5 LA0105432
