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Langmuir 1996, 12, 2454-2463
Thermodynamics of Vesicle Formation from a Mixture of Anionic and Cationic Surfactants Magnus Bergstro¨m† Department of Chemistry/Physical Chemistry, Royal Institute of Technology, S-100 44 Stockholm, Sweden Received October 16, 1995X Detailed model calculations on mixed sodium dodecyl sulfate/dodecylammonium chloride vesicles show that the work of bending a planar bilayer into a geometrically closed bilayer vesicle rises steeply as the molar ratio between aggregated anionic and cationic surfactant approaches unity. Likewise, the bending work increases as the mole fraction of either of the surfactants approaches unity, resulting in a bending free energy minimum on each side of the equimolar composition. In the vicinity of that composition the bending work is too large to permit vesicle formation to any appreciable extent, while at compositions where one of the surfactants is in excess, the bending free energy is much lower, thus enabling the formation of small unilamellar vesicles (R < 500 Å). These calculation results are in good qualitative agreement with recent experimental findings on mixed anionic/cationic vesicles. Moreover, the fluctuations in composition, chain packing density, shape, and size contribute to making the vesicle size distribution fairly polydisperse with a relative standard deviation of σR/Rmax equal to 0.283.
Introduction For a long time it was generally believed that only double-chained lipids can form bilayer aggregates in water such as large unilamellar vesicles and multilamellar liposomes, whereas single-chained surfactants normally generate small compact or elongated micelles. Later on, however, it was discovered that bilayer vesicles/liposomes do form when adding a nonionic cosurfactant, such as a long-chained alcohol, to a surfactant solution.1 More recently, several observations have been reported2-9 on the spontaneous formation of vesicles from mixtures of a surfactant and an additional amphiphile. It has been suggested that the presence of a charged surfactant might imply that intervesicular interactions of electrostatic origin can account for the stability of unilamellar vesicles. However, if interactions between the vesicles were crucial for their formation, the aggregates should be destabilized when the solution becomes sufficiently dilute and these interactions become negligible. In a recent paper Chiruvolu et al.10 investigate this idea by means of direct surface force measurements on vesicle forming bilayers composed of an anionic and a cationic surfactant and they conclude that vesicle-vesicle interactions cannot alone account for the spontaneous formation of vesicles. Attempts to account for vesicle formation in mixed systems by starting out from the intrinsic properties of the bilayer membranes have been made by several † Present address: Department of Solid State Physics, Risø National Laboratory, DK-4000 Roskilde, Denmark, e-mail
[email protected], telefax +45 42 37 01 15. X Abstract published in Advance ACS Abstracts, April 15, 1996.
(1) Hargreaves, W. R.; Deamer, D. W. Biochemistry 1978, 17, 3759. (2) Schurtenberger, P.; Mazer, N.; Ka¨nzig, W. J. Phys. Chem. 1985, 89, 1042. (3) Kaler, E. W.; Murthy, A. K.; Rodriguez, B. E.; Zasadzinski, J. A. N. Science 1989, 245, 1371. (4) Murthy, A. K.; Kaler, E. W.; Zasadzinski, J. A. N. J. Colloid Interface Sci. 1991, 145, 598. (5) Ambu¨hl, M.; Bangerter, F.; Pier, L. L.; Skrabal, P.; Watzke, H. J. Langmuir 1993, 9, 36. (6) Egelhaaf, S. U.; Schurtenberger, P. J. Phys. Chem. 1994, 98, 8560. (7) Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. J. Phys. Chem. 1995, 99, 1299. (8) Hoffmann, H.; Thunig, C.; Munket, U. Langmuir 1992, 8, 2629. (9) Hoffmann, H.; Munkert, U.; Thunig, C.; Valiente, M. J. Colloid Interface Sci. 1994, 163, 217. (10) Chiruvolu, S.; Israelachvili, J. N.; Naranjo, E.; Xu, Z.; Zasadzinski, J. A.; Kaler, E. W.; Herrington, K. L. Langmuir 1995, 11, 4256.
authors.11-14 Safran et al.11,12 argue that specific interactions between different surfactant headgroups account for the tendency of mixed surfactant systems to form vesicles. The bond distance for “1-2” surfactant pairs is supposed to be different from the average bond distance for “1-1” and “2-2” pairs, resulting in a release of curvature frustration upon forming vesicles. This theory suggests that the genesis of vesicle formation to a large extent is dependent upon the kind of surfactant headgroups involved, and hence, it cannot constitute a generally valid approach. Moreover, Safran and co-workers seem to overlook the fact that there is a rapid exchange of surfactant monomers between the vesicles and the surrounding bulk solution which results in an adjustment of the compositions in the inner and outer monolayers in order to minimize the vesicle free energy. According to our present and previous calculations the overall result is a mixed bilayer with a spontaneous curvature equal to zero (cf. Figure 3). In the theoretical treatment we are going to present in this paper, we invoke neither vesicle-vesicle interactions nor any specific headgroup interactions. In essence, the energetic and statistical-mechanical basis of this approach has been presented in two recent papers15,16 where we were able to account for the spontaneous formation of mixed surfactant/long-chained alcohol bilayer vesicles. Of particular interest is the recent observation, first reported by Kaler et al.17 and later on by others,18-21 that vesicles may form reversibly from a mixture of anionic and cationic single-chained aliphatic surfactants. These aggregates are only seen to arise when one of the two (11) Safran, S. A.; Pincus, P.; Andelman, D. Science 1990, 248, 354. (12) Safran, S. A.; Pincus, P.; Andelman, D.; Mackintosh, F. C. Phys. Rev. A 1991, 43, 1071. (13) Kumaran, V. J. Chem. Phys. 1993, 99, 5490. (14) Porte, G.; Ligoure, C. J. Chem. Phys. 1995, 102, 4290. (15) Bergstro¨m, M.; Eriksson, J. C. Langmuir 1996, 12, 624. (16) Bergstro¨m, M.; Eriksson, J. C. Submitted for publication to Langmuir. Bergstro¨m, M. Thesis, Royal Institute of Technology, Stockholm, 1995. (17) Kaler, E. W.; Herrington, K. L.; Murthy, A. K.; Zasadzinski, J. A. N. J. Phys. Chem. 1992, 96, 6698. (18) Watzke, H. J. Prog. Colloid Polym. Sci. 1993, 93, 15. (19) Marques, E.; Khan, A.; Miguel, M. G.; Lindman, B. J. Phys. Chem. 1993, 97, 4729. (20) Zhao, G. X.; Yu, W. L. J. Colloid Interface Sci. 1995, 173, 159. (21) Kondo, Y.; Uchiyama, H.; Yoshino, N.; Nishiyama, K.; Abe, M. Langmuir 1995, 11, 2380.
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Figure 2. Model of a spherical, unilamellar bilayer vesicle. The two monolayers formally adhere at the radial distance R. τ is set equal to the length of the fully extended hydrocarbon chain. For a C12 chain, τ equals 16.7 Å. Figure 1. Drawing of the water-rich corner of a ternary phase diagram for an anionic surfactant/cationic surfactant/water system. Two one-phase lobes, one cation rich (V+) and one anion rich (V-), containing isotropic solutions of vesicles are seen. Outside the lobes there is a two-phase region where the additional phase (La+ or La-) is a lyotropic lamellar liquid crystal (LR). M denotes micellar regions.
surfactants is in excess, resulting in two one-phase lobes of unilamellar vesicles in the water-rich corner of the phase diagram, one in the cationic region and one in the anionic region. The two vesicle lobes are separated by a twophase region, located close to equimolar composition of the two surfactants, and where, in addition, a lamellar phase is also present (Figure 1). In the present paper we describe our calculations of the bending constant of a mixed sodium dodecyl sulfate (SDS)/ dodecylammonium chloride (DAC) bilayer, which follow the same lines as we have used previously for a mixed SDS/dodecanol bilayer.15 This choice of surfactants is mainly dictated by the fact that quantitative expressions for the packing-dependent conformational free energy are available for an aliphatic C12 chain and that similar model calculations have been made earlier on SDS micelles of different shapes22 and on DAC-laden water solution/air interfaces.23 However, our neglecting of any specific headgroup interactions suggests that our main conclusions in the present article should be valid for all equilibrated vesicles formed from a mixture of anionic and cationic surfactants. The main difference compared with our previous calculations on mixed vesicles is that here we consider two aggregated surfactant ions of opposite charge which form a pseudo-double-chained zwitterionic surfactant, and accordingly, the free energy of mixing the surfactants in the bilayer is modified as will be further discussed below. The charges of the two monovalent headgroups cancel and a pair of counterions are released to the surrounding bulk solution. The Free Energy of a Geometrically Closed Bilayer Vesicle We consider a spherical vesicle as being comprised of two monolayers which adhere to each other at a radial distance R from the center of the vesicle (Figure 2). The volume within the spherical surface located at R is filled up with Ni ) Ni+ + Ni- aggregated surfactant monomers forming the inner monolayer and Niw water molecules which constitute the water core enclosed by the hydrocarbon part of the bilayer. Outside R there is a spherical shell of fixed thickness τ containing Ne ) Ne+ + Ne(22) Eriksson, J. C.; Ljunggren, S. Langmuir 1990, 6, 895. (23) Eriksson, J. C.; Ljunggren, S. Colloid Surf. 1989, 38, 179.
monomers forming the outer monolayer and New water molecules which are loosely associated with the external hydrocarbon/water interface. The excess free energy corresponding to the work of forming an additional vesicle in a solution where equilibrium between vesicles and monomers already is established, we denote by . This is the free energy of the vesicle in the Ω-potential sense, implying that we merely consider free energy contributions which are in excess of the level set by the chemical potentials in the surrounding solution. We will assume that the molecular constituents of the inner and outer monolayer parts of the vesicle define a vesicular state. Hence, a particular vesicular state, denoted by ν, is fully defined by means of a set of independent variables, Ni+, Ni-, Niw, Ne+, Ne-, New. A symbolic “chemical reaction” can be written down for each state ν
N+DAC + N-SDS + NwH2O f Aν
(1)
Correspondingly, there is an aggregation equilibrium condition
ν + kT ln φν ) 0
(2)
one for each vesicle state. The second term given by eq 2 is the entropic contribution due to admixing of a small amount of vesicles in state ν to the solution where their volume fraction is φν. The total volume fraction of vesicles can now be obtained as a sum over all accessible vesicular states
φtot )
∑ν e- kT ) ∫ν e-(ν)/kT dν ) ∫0 ν
R
S(R)e-(R)/kT dR (3)
where S(R) is a statistical-mechanical factor that accounts for fluctuations in chain packing density, composition, and shape of the vesicular aggregate. We have previously derived expressions for both the factor S(R)16 and the (equilibrium) free energy function (R)15 in order to derive the vesicle size distribution, to which we subsequently shall return. Curvature Dependence of the Bilayer Tension Our model calculations yield comparatively large sizes of the vesicles (R > 100 Å) which is in good agreement with most experiments so far available (cf. for example ref 17). Hence, it is reasonable to expand the (monolayer) interfacial tensions referred to the water/hydrocarbon dividing surface (located somewhat underneath the polar headgroups) to second order in curvature
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γ ) γ∞ +
Bergstro¨ m
k1 k2 + 2 r r
(4)
where we take the curvature 1/r to be positive for a convex and negative for a concave surface. k1 and k2 stand for two (state-dependent) constants related to the change of tension when curving a single monolayer. For a spherical vesicle we obtain setting r ) -Ri and r ) Re for the inner and outer monolayer tensions, respectively
γi ) γ∞ -
k1 k2 + 2 R1 R
(5)
k1 k2 + Re R2
(6)
i
γe ) γ∞ +
e
where γ∞ is the equilibrium interfacial tension of the planar monolayer. The chain packing densities at the hydrocarbon/water contact interfaces, Γi ) Ni/Ai ) 1/ai and Γe ) Ne/Ae ) 1/ae, are given as functions of the radial distances to the hydrocarbon/water interface of the inner and outer monolayers Ri and Re, respectively, by the geometrical relations
Γi )
Γe )
R3 - R3i 3vsR2i R3e - R3 3vsR2e
( (
ξi ξi ξ2i ) 1+ + vs Ri 3R2 i
) )
ξe ξe ξ2e ) 1+ vs Re 3R2
(7)
(8)
e
where ξi ) R - Ri and ξe ) Re - R are the thicknesses of the hydrocarbon parts of the monolayers and ai and ae are the surface areas per aggregated monomer. We can likewise expand the monolayer thickness to second order in curvature, i.e.
(
ξ ) ξp 1 +
)
kξ′ kξ′′ + 2 r r
(9)
where ξp denotes the thickness of a planar monolayer and kξ′ and kξ′′ are state-dependent constants related to the variation of monolayer thickness with curvature. ξp, kξ′ and kξ′ can be obtained from our model calculations. Comparing eqs 7 and 8 and eq 9 and expanding the chain packing density to second order in curvature we obtain
Γ ) Γp
(
)
ka′ ka′′ 1+ + 2 r r
(10)
where the constants involved can be identified as Γp ) 1/ap, ka′ ) kξ′ - ξp and ka′′ ) ξ2p/3 - 2kξ′ ξp + kξ′′. As we have demonstrated before15 eqs 5 and 6 yield a rather simple expression for the excess free energy of a spherically shaped bilayer, that is, for a spherical vesicle which is fully equilibrated with the surrounding solution
(R) ) 8π(ξpk1 + k2 + γ∞R2) ) 4π(kbi + 2γ∞R2)
(11)
where we have set the total bilayer thickness equal to twice the thickness of a planar monolayer, i.e. ξi + ξe ≈ 2ξp which, according to eq 9, is a valid approximation for comparatively large R. By neglecting higher order terms in eq 4, terms proportional to R-2, R-4, and so on in the expression for (R) will not appear. However, vesicle formation generally occurs for rather large R values (g100 Å), on the rising
branch of the (R) function far from the ascending branch at very small radii. Thus, for the present investigation, the free energy function given by eq 11 is sufficiently accurate. Furthermore, making use of eqs 5 and 6 we can show that
kbi ≡ 2(ξpk1 + k2) ) R(ξeγe - ξiγ1) + R2(γe + γi - 2γ∞) (12) i.e., kbi is the sum of the work (divided by 4π) it takes to change the bilayer tension from 2γ∞ to (γe + γi) at a constant hydrocarbon/water contact area equal to 8πR2 (second term), and the work of changing the surface areas to those of an outer and an inner monolayer (cf. ref 15). Contributions to the Excess Free Energy of a Mixed DAC/SDS Vesicle The gain in free energy upon bringing a hydrocarbon chain from water solution into an n-alkane bulk phase has been estimated by Tanford24 and, accordingly, its contribution to the free energy per aggregated monomer of forming either of the monolayers of a DAC/SDS vesicle becomes
tan ) -[19.960 + (1 - x-) ln xcat + x- ln xan] kT
(13)
where the mole fraction of aggregated SDS monomers in a vesicle monolayer is x- ) N-/N. A positive contribution to the free energy of a vesicle arises when the electrical charges from headgroups, counterions, and co-ions are concentrated in a restricted volume near the hydrocarbon/water interface. This contribution has been evaluated for moderately curved surfaces by Mitchell and Ninham25 and by Lekkerkerker,26 using the (nonlinear) Poisson-Boltzmann approximation and expanding to second order in 1/κRel, where Rel is the radial distance to the smeared-out surface charges. The inverse Debye length is given by
κ)
x
2(can + ccat + csalt)N2Ae2 0rRT
(14)
and NA denotes the Avogadro number. The free energy per unit charge for a spherical surface becomes
[
]
el xS2 + 1 - 1 ) 2 ln(S + xS2 + 1) kT S
[
(
)
1 + xS2 + 1 4 ln + κSRel 2
([
])]
2 2 1 + xS2 + 1 2 D 1 + ln 1 2 2 Sκ2Rel S2 S2xS2 + 1 (15) where the Debye function is defined by
D1(x) )
∫0x ett -dt1
(16)
and the dimensionless, reduced charge parameter, is given by (24) Tanford, C. In The Hydrophobic Effect; Wiley: New York, 1980. (25) Mitchell, D. J.; Ninham, B. W. Langmuir 1989, 5, 1121. (26) Lekkerkerker, H. N. W. Physica A 1989, 159, 319.
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Thermodynamics of Vesicle Formation
S)
σ
x8(can + ccat + csalt)0rRT
Langmuir, Vol. 12, No. 10, 1996 2457
(17)
σ ) e/ael is the surface charge density and e is the elementary charge. can and ccat are the concentrations of the two surfactants in solution and csalt is the concentration of monovalent salt released during the zwitterion formation plus, if any, added salt. The net charge of the surface is
Nch ) |N+ - N-|
(18)
and the surface area per charge 2 4πRel ael ) Nch
(19)
By convention Rel is positive for the outer convex surface and negative for the inner concave surface. Furthermore, Rel is assumed to be located 3 Å from the hydrocarbon/ water surface. A curvature-dependent hydrocarbon/water interfacial tension has some empirical support, and we have set this contribution equal to
γhc/w )
p γhc/w
(
2 1r
)
Nzi )
conf ) 3.12630 - 1.15534ξ + 0.18273ξ2 kT 1.40763 × 10-2ξ3 + 4.77423 × 10-4ξ4 (21) ξ as before denoting the thickness of the hydrocarbon layer. Bringing headgroups residing in solution to the vesicle surface gives an additional excess free energy contribution which accounts for a number of less known effects, above all (i) shielding of the hydrocarbon/water contact by the headgroups, (ii) different hydration of headgroups, counterions, and co-ions near the hydrocarbon bilayer, and (iii) repulsive correlation interactions between headgroups, counterions, and co-ions, which have not been accounted for in the PB approximation assuming a smeared out surface charge distribution. The details of these effects are complex and no model for them is available at present. However, comparison with surface tension data for DAC shows that they yield a constant contribution to the excess free energy per monomer23 and thus we have introduced headgroup parameters in our calculations in accordance with the relationships
(22)
p pg+ ) ppg+ + 2apg+γhc/w /r
(23)
(27) Gruen, D. W. R.; Lacey, E. H. B. In Surfactants in solution; Lindman, K. M. a. B., Ed.; Plenum: New York, 1984; Vol. I, p 279.
(24)
(
)
Nch + Nzi mix ) Nch ln(1 - xzi) + Nzi ln xzi + ) ln kT Nzi lnx2π(Nch + Nzi) (1 - xzi)xzi ) N [(1 - xzi) ln(1 - xzi) + xzi ln xzi] + (1 + xzi) ln
x
(1 - xzi)xzi (1 + xzi)
+ ln x2πN (25)
We can rewrite this equation considering that
xzi ≡
Nzi 1 - |2x- - 1| ) Nzi + Nch 1 + |2x- - 1|
(26)
as
mix ) N[|2x- - 1| ln |2x- - 1| - |x- ln x- kT (1 - x-) ln(1 - x-)|] + ln
x
|2x- - 1| (1 - |2x- - 1|) (1 + |2x- - 1|)
+ lnx2πN (27)
By considering the mixing of zwitterions and excess monomer in the bilayer rather than the mixing of anionic and cationic monomers, we have excluded all of those states where the monomer in deficit is not associated with a monomer of opposite charge. The total excess free energy of forming a single monolayer out of monomers in solution is now easily derived by summing up the different contributions
(r, a, x-, solution state) ) N[tan + conf + aγhc/w + (1 - x-)pg+ + x-pg- + |2x- - 1|el + |2x- - 1| ln |2x- - 1| - |x- ln x- - (1 - x-) ln(1 - x-)|] + ln
and
N - Nch N - |N+ - N-| ) 2 2
The free energy of mixing is now obtained from of the combinatorial expression
(20)
where the interfacial tension of the planar hydrocarbon/ water interface is assumed equal to its macroscopic value p ) 50.7 mJ/m2. at 25 °c, i.e., γhc/w The conformational flexibility of the hydrocarbon chains packed inside the bilayer core becomes restricted when the headgroups are spatially fixed at the vesicle surface and the free energy cost compared with chains in an n-alkane bulk phase has been estimated by Gruen and de Lacey27 by means of a single-chain, mean field model to
p /r pg- ) ppg- + 2apg-γhc/w
p p where pg) 0.5962 kT and pg+ ) 1.445 kT and apg- and apg+ are the cross sections of the anionic and cationic headgroups, respectively. Comparing with experimental data, these values were found to generate reasonable critical micelle concentration (cmc) values and essentially correct interfacial tension vs concentration plots when applied to solutions with and without added salt.22,23 We have to eqs 22 and 23 added curvature-dependent terms accounting for a reduction in magnitude of the curvaturedependent part of the hydrocarbon/water interfacial tension, given by eq 20, as the headgroups shield the hydrocarbon/water interface. Finally, we have the free energy of mixing the aggregated monomers in the vesicle. Due to the formation of zwitterions, the mixing that occurs is between the zwitterions and excess monomer. The number of zwitterions becomes
x
|2x- - 1|(1 - |2x- - 1|) (1 + |2x- - 1|)
+ ln x2πN (28)
For a given solution state and a fixed radius R, we minimize the free energies of the inner and outer monolayers according to eq 28, under the geometrical constraints given by eqs 7 and 8. Hence, we obtain the overall
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Bergstro¨ m
(
x- ) xp- 1 +
)
kx′ kx′′ + 2 r r
(31)
The electrostatic free energy, given by eq 15, has an explicit curvature dependence at a fixed surface charge density. However, for the process of forming a vesicle from a planar bilayer at constant chemical potentials the reduced charge parameter S has a curvature dependence as well, and hence, it is convenient to write
(
p el ) el 1+
)
kel′ kel′′ + 2 r r
(32)
Figure 3. Free energy functions /kT and */kT plotted against the square of the vesicle radius R2. It is evident that the latter function is better approximated by a straight line than the former. The bulk solution state is can ) 9.46 × 10-6 mM and ccat ) 10 mM with a released salt concentration of csalt ) 12.9 mM, due to the formation of zwitterions in the vesicle bilayers. The bilayer tension was γ∞ ) 5.1 × 10-3 mJ/m2 and the bending parameter kbi ) 3.28kT. The optimal area per hydrocarbon chain for a planar bilayer at the same solution state is ap ) 32.2 Å2 and the mole fraction of aggregated SDS monomer xp- ) 0.272.
The curvature dependence of the monolayer thickness, given by eq 9, invokes, according to eq 21, a curvature dependence of the hydrocarbon chain conformational free energy
free energy of a vesicle bilayer as the sum of the two monolayer free energies ) i + e. The last two terms in eq 28 which originate from the combinatorial free energy expression arise because the size of the aggregate is finite; i.e., they are not due to the bilayer curvature. For this reason, it is convenient to write the vesicle free energy in the form
Inserting eqs 10, 31, 32, and 33 in eq 30 gives the monolayer tension expressed as a power series with respect to curvature. Collecting all terms proportional to 1/r and 1/r2, we obtain explicit expressions for the two bending constants k1 and k2, respectively
) * +
k1 )
lnx2πNi|2xi- - 1| (1 - |2xi- - 1|)/(1 + |2xi- - 1|) +
lnx
2πNe|2xe-
- 1| (1 -
|2xe-
- 1|)/(1 +
|2xe-
- 1|) )
lnx2πNiNe/C (29)
where C ≈ (1 + |2xp- - 1|)/|2xp- - 1|(1 - |2xp- - 1|). To a very good approximation C is a constant as the curvaturedependent parts of the anion mole fraction in the inner and outer monolayers to a large extent cancel (cf. eq 31 below). It appears that the simple free energy expression kbi + 2γ∞R2 is a better approximation of */4π than of /4π (cf. Figure 3) and a more accurate size distribution function will result using *(R) ) 4π(kbi + 2γ∞R2).
(
p 1+ conf ) conf
(
)
Making a Taylor expansion of the curvature-dependent quantities in this expression enables us to calculate the bending constants k1 and k2. Mostly due to the curvature dependence of the surface area per headgroup, we will have a fine tuning of the monolayer composition upon curving a thermodynamically open vesicle giving rise to a curvature-dependent mole fraction of aggregated anionic surfactants
1 p [k ′′ + xp-kx′′′ + kconf′conf + ap a
and
k2 )
1 [k ′′′ + xp-(kx′′ + ka′kx′)′′ + ap a
p ]+ (kconf′′ + ka′kconf′)conf
xp-apg-)
+
xp-kx′
p 2γhc/w [ka′((1 - xp-)apg+ + ap
p el (apg- - apg+)] + [|2xp- - 1| × ap
(ka′kel′ + kel′′) ( 2xp-kx′kel′] +
Omitting the logarithmic terms originating from the contribution due to finite, size mixing, we obtain the monolayer tension from eq 27 as
*(r) 1 ) [-19.960kT + conf + A a (1 - x-) (pg+ - kT ln xcat) + x- (pg- - kT ln xan) + |2x- - 1|(el + kT ln|2x- - 1|) - kT|x- ln x- 2 p (1 - x-) ln(1 - x-)|] + γhc/w 1(30) r
(33)
p p [(1 - xp-)apg+ + xp-apg- - ap] + |2xp- - 1|kel′el ] 2γhc/w (34)
Contributions to the Bending Constants for a Mixed Spherical SDS/DAC Vesicle
γ*(r) ≡
)
kconf′ kconf′′ + 2 r r
xp-kx′2kT 2ap|2xp- - 1|(1 - xp-) (35)
where the last term in eq 35 comes from the curvature dependence of the (ordinary) free energy of mixing and is obtained by means of expanding the logarithmic expressions as shown in detail in the Appendix. The (+)-sign before the 2xp-kx′kel′ term refers to compositional states with xp- > 0.5 and the (-)-sign to states with xp- < 0.5. The first two terms in each of eqs 34 and 35 originate from the curvature-independent free energy contributions p ′ ) tan + conf + (1 - xp-)ppg+ + xp-ppg- + p |2xp- - 1|el + |2xp- - 1|kT ln|2xp- - 1| -
kT|xp- ln xp- - (1 - xp-) ln(1 - xp-)| (36) and
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Langmuir, Vol. 12, No. 10, 1996 2459
Table 1. Contributions to kbi, Surface Area per Aggregated Monomer (ap), the Vesicle Radius (Rmax), and the Constants Related to Chain Packing Density (Ka), Composition (Kx), and Shape Fluctuations (kbi c ) for Three Different Solution Statesa
ap/Å2 κa/nm-4 κx Rmax/Å ∆N kbi c /kT kgeom bi kcomp bi kel bi kconf bi khc/w bi kmix bi ∑ ) kbi
xp- ) 0.238
xp- ) 0.525
xp- ) 0.756
32.4 102 12.9 260 -62 0.983 19.80 -3.19 -9.66 -1.95 -6.06 3.53 2.47
31.1 134 106 -75 -1.17 23.01 -0.51 -0.45 -4.35 -5.97 0.67 12.40
32.3 102 13.1 541 -63 1.08 19.99 -3.23 -9.63 -2.07 -5.35 3.16 2.87
a The various contributions to the bilayer bending parameter k bi are given in kT units for the three cases. The change in aggregation number as a planar bilayer transforms into a spherical shell at a fixed hydrocarbon/water contact surface area is also included. See text for details.
( )
layer, we obtain a net composition change giving rise to the kbi contribution
xcat p ′′ ) kT ln ( 2el + (ppg- - ppg+) ( xan
[
kT ln
(2xp- - 1)2
xp-(1
-
xp-)
]
∆N′ ′ 2 4 R (ξi + ξe - 2ξp) - ξ3p (38) ) 4π vs 3
[
comp kbi )
(37)
where, as before, the (+)-sign refers to xp- > 0.5 and the (-)-sign to xp- < 0.5. Apparently, ′ is the free energy per monomer of a planar layer minus the contribution from the hydrocarbon/water interfacial tension. This is because we consider the work of bending the bilayer at a fixed overall hydrocarbon/water contact surface area. We may also note that if we write the overall free energy of a planar monolayer as ) N-- + N++ the free energy parameter defined in eq 37 is given by ′′ ) - - +. According to the expressions for k1 and k2 in eqs 34 and 35, on a molecular basis, we can subdivide the bending work parameter, kbi, into six different contributions. When a planar bilayer changes its shape into the spherical shell of a geometrically closed vesicle at fixed hydrocarbon/water contact area, there will be a change in volume of the hydrocarbon core of the bilayer. Since we assume the hydrocarbon chains to be incompressible, this volume change corresponds to a change in aggregation number ∆N, which in its turn gives rise to a contribution to the bilayer bending constant kbi geom ) kbi
Figure 4. Size distribution of mixed SDS/DAC vesicles where the fraction of aggregated SDS is 75.6 mol %. The bulk solution state is can ) 6.51 mM and ccat ) 1.0 × 10-5 mM, resulting in a salt concentration of csalt ) 11.6 mM, due to the formation of zwitterions. The overall volume fraction of vesicles is φtot ) 0.1 and the bilayer tension γ∞ ) 1.68 × 10-4 mJ/m2. The most probable radius Rmax is equal to 541 Å. The optimal value of the planar area per hydrocarbon chain is ap ) 32.3 Å2 and the bending parameters are kbi ) 2.87kT and kbi c ) 1.08kT. The fluctuation related parameters are κx ) 13.1 and κa ) 0.0102 Å4. τ was set equal to the fully extended hydrocarbon C12 chain, 16.7 Å.
]
∆N is generally negative, corresponding to a loss of aggregated monomers as the vesicle is formed, and since aggregation of monomers (keeping the hydrocarbon/water contact area fixed) is thermodynamically favored, the contribution captured by eq 38 is positive. This is also evident when considering that the free energy parameter ′ is always negative, which is seen as ′ ) ap (γ∞ - γhc/w) ≈ -apγhc/w. As a matter of fact this term turns out to be the main positive contribution to kbi, and hence, it plays a major role for the stability of large polydisperse bilayer vesicles15 (cf. Table 1). The composition differs in the two monolayers according to eq 31, and since the outer layer is larger than the inner
2xp-′′ (kξ′kx′ + kx′′) ap
(39)
Electrostatics gives rise to a large negative contribution el ) kbi
p 2el [|2xp- - 1| (kξ′kel′ + kel′′) ( 2xp-kx′kel′] ap
(40)
Together with the geometrical contribution it accounts for the main part to the overall bending work. The change of surface area per headgroup and composition when bending the bilayer lowers the electrostatic free energy in the outer monolayer and raises it in the inner layer resulting in a net negative contribution to kbi. The curvature dependence of γhc/w, which lowers the bending work and the shielding of the hydrocarbon/water contact area by the headgroups, gives hc/w ) kbi
p 4γhc/w [kξ′[(1 - xp-)apg+ + xp-apg-] + ap
xp-kx′(apg- - apg+) - vs] (41) The comparatively larger sulfate group of SDS shields the hydrocarbon/water interface more efficiently than the ammonium group of DAC and, hence, counteracting the curvature-dependent effect and gives a higher kbi for SDSrich vesicles (cf. Table 1) and, consequently, larger vesicles (cf. Figures 4 and 5). The conformational restrictions of the hydrocarbon chains upon bending the bilayer yields a net negative contribution for similar reasons as for the electrostatic contribution. conf kbi )
p 2conf (kξ′kconf′ + kconf′′) ap
(42)
Finally the free energy of mixing gives rise to the contribution mix kbi
xp-kx′2kT )
ap|2xp- - 1| (1 - xp-)
(43)
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Bergstro¨ m
Figure 5. Size distribution of mixed SDS/DAC vesicles where the fraction of aggregated DAC is 76.2 mol %. The bulk solution state is can ) 5.00 × 10-6 mM and ccat ) 12.2 mM, which gives a salt concentration of csalt ) 11.3 mM, due to the formation of zwitterions. The overall volume fraction of vesicles is φtot ) 0.1 and the bilayer tension γ∞ ) 7.27 × 10-4 mJ/m2. The most probable radius Rmax equals 260 Å. The optimal value of the planar area per hydrocarbon chain is ap ) 32.4 Å2 and the bending parameters are kbi ) 2.47kT and kbi c ) 0.983kT. The fluctuation related parameters are κx ) 12.9 and κa ) 0.0102 Å4. τ was set equal to the fully extended hydrocarbon C12 chain, 16.7 Å.
Figure 6. Bilayer bending constant plotted against the mole fraction of aggregated SDS monomers in the planar bilayer, xp-. The large bending work near xp- ) 0.5 excludes the formation of vesicles in this region.
These different contributions are all listed for three different xp- in Table 1. Vesicle Size Distributions As we have demonstrated in a previous paper,15 fluctuations in composition, chain packing density, and shape give rise to a R-dependent preexponential factor of statistical-mechanical origin in the vesicle size distribution function which can be written as
φ(R) )
4Cπ3τkTR6 -4π(kbi+2γ∞R2)/kT e κxκav2wa3pkbi c
(44)
where κx and κa are “elasticity” constants related to the composition and chain packing density fluctuations, respectively. vw ) 30 Å3 is the van der Waal volume of a water molecule, ap is the area per hydrocarbon chain in an equilibrated planar bilayer, and the constant C is equal to (1 + |2xp- - 1|)/|2xp- - 1|(1 - |2xp- - 1|). kbi c is a bending constant related to the mean curvature of the bilayer and h bi it is defined as kbi ) 2kbi c + k c , where the saddle-splay bi constant k h c is related to the Gaussian curvature. Examples of size distributions for mixed anionic/cationic surfactant vesicles as obtained from our calculations are shown in Figures 4 and 5. From the size distribution, one can easily derive an expression for the most probable radius
Rmax )
x
3kT 8πγ∞
(45)
and the dispersion
σ2R ) 〈R2〉 - 〈R〉2 )
(175π - 512)kT 400π2γ∞
(46)
which implies that the relative width of the size distribution peak turns out to be a pure number:
σR ) Rmax
- 512 ≈ 0.283 x175π150π
(47)
Since the size distribution function is the same for pure
Figure 7. Contributions to the bilayer bending constant 2R(ξeγe - ξiγi) and R2(γe + γi - 2γ∞) plotted against the mole fraction of SDS in a planar mixed SDS/DAC bilayer, xp-. The sum of the two contributions equals kbi.
as well as mixed vesicles (only a constant equal to C/2κx differs), the expressions for Rmax and σR are actually of general validity for the equilibrium case. Integration of eq 44 yields the overall volume fraction of vesicles
φtot )
15x2Cτ(kT)9/2 e-4πkbi/kT 2 3 7/2 8192kbi c κxκaνwapγ∞
(48)
Discussion The bilayer bending constant kbi calculated according to the model described above is plotted in Figure 6. Distinct minima on each side of the equimolar composition are seen. The dependence of kbi on xp- can be rationalized by considering the two contributions to kbi given by eq 12 (cf. Figure 7). The first term, 2R(ξeγe - ξiγi), representing the work of relocating the spherical surfaces, is monotonously increasing in the interval 0 < xp- < 0.5, whereas the second term, R2 (γe + γi - 2γ∞), representing the change of monolayer tensions at R, is monotonously decreasing with the planar alcohol mole fraction xp-, in the same interval. The situation is switched in the region 0.5 < xp- < 1, and the final result is two minima for kbi as a function of xp-, one in the anion-rich region and one in the cation-rich region, and a distinct maximum at the equimolar composition xp- ) 0.5 (cf. Figure 6). As a consequence of the decrease of the electrostatic free energy of the headgroups when x- f 0.5, the monolayer thickness becomes larger, resulting in an increasing outer surface area and a decreasing inner surface area. Hence, we obtain an increasing γe and a decreasing γi, and consequently, 2R(ξeγe - ξiγi) increases as x- approaches equimolarity. Moreover, γi decreases
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Langmuir, Vol. 12, No. 10, 1996 2461
magnitude and would actually prevent vesicle formation. To yield a resulting kbi value which corresponds to reasonable vesicle sizes, there are a number of additional contributions to consider which turn out to be negative. The most important one is the electrostatical contribution, given by eq 40, which is also the contribution that largely accounts for the strong dependence of kbi on xp- as seen in Figure 6.
Figure 8. Bilayer bending constant kbi plotted against the most probable vesicle radius Rmax, according to eq 49. The following values were used for the different quantities and parameters: ap ) 32.4 Å2, xp- ) 0.238, κa ) 102.4 nm-4, κx ) 12.9, and kbi c ) 0.983kT, which corresponds to the same solution state as in Figure 5. The overall vesicle volume fraction was chosen equal to φtot ) 0.1 and τ was set to 16.7 Å.
more rapidly than γe increases since the inner monolayer is more curved, resulting in a monotonously decreasing second term of eq 12; i.e., R2 (γe + γi - 2γ∞), decreases as x- f 0.5. Combining the φtot expression (48) with eq 45 gives, for a given total vesicle volume fraction, the following relation between kbi and the most probable vesicle size Rmax
kbi )
7kT ln Rmax + const 4π
(49)
which is plotted in Figure 8. Very low values of the bilayer tension, 5 × 10-11 < γ∞ < 5 × 10-3 mJ/m2, are consistent with vesicle radii in the range 100 Å < Rmax < 100 µm, which approximately corresponds to what has been observed experimentally. According to our calculations of the bilayer bending constant plotted in Figure 6, small unilamellar vesicles (Rmax < 500 Å) can form in the regions 0.1 < xp- < 0.3 and 0.7 < xp- < 0.9. Comparison with our previous calculations15 shows that in these regions kbi assumes substantially lower values than is ever obtained for a SDS/dodecanol bilayer. As xp- approaches 0.5, the aggregates become larger and, if a limit of the upper vesicle size is set to Rmax ) 100 µm, no vesicles form when 0.4 < xp- < 0.6 because of the large bending free energy, and one expects large planar membranes to be the most stable bilayer form. Our calculation results are in excellent qualitative agreement with experiments performed by Kaler et al.17 on mixed cetyltrimethylammonium tosylate (CTAT)/ sodium dodecylbenzenesulfonate (SDBS) vesicles. Figure 1 shows the water-rich region of a ternary phase diagram for a mixture of anionic and cationic surfactants. Two lobes representing one-phase regions of isotropic vesicle solutions, one in the anion-rich and one in the cation-rich region, appear. Both of them have a maximal vesicle concentration at a certain ratio between the total concentrations of the two surfactants at approximately equal distance from the equimolar line. At higher surfactant concentrations, when the intervesicular interaction becomes significant, a lamellar phase (LR) begins to form. A precipitate was observed at the equimolar line in the phase diagram. In order to balance the negative dispersion (mixing) free energy contribution and satisfy eq 2, the bilayer bending constant has to assume a positive value. The main contribution determining the sign of kbi is the geometrical one. However, kgeom alone is too large in bi
Interestingly, the zwitterion formation influences kel bi in two different ways: (i) substituting an anion with a cation (or vice versa) in a monolayer changes the charge with two units, and (ii) a pair of counterions are released when a zwitterion is formed. The former results in a factor 2 in the last (always negative) 2xp-kx′kel′ term of eq 40 which is absent in the corresponding expression for a SDS/ dodecanol vesicle (cf. eq 64 in ref 15). This term brings down the electrostatic contribution to large negative values when xp- is removed from 0, 0.5, and 1, giving a lower kel bi for anion/cation surfactant vesicles than for the corresponding surfactant/long-chained alcohol vesicles. The increase of kbi with salt concentration is a result of two effects, both of which act in the same direction: (i) The increasing entropy of mixing the counter- and co-ions as a planar monolayer is convexly curved at constant surface charge density gives rise to the curvaturedependent terms in eq 15, and since the outer vesicle layer always dominates over the inner layer (the number of aggregated monomers in the outer layer is larger), there is a negative contribution to kbi, the magnitude of which becomes smaller as the concentration of salt increases. (ii) As the surface charge density decreases in the outer monolayer and increases in the inner layer upon bending the vesicle bilayer, there is a net decrease of kbi which becomes less significant as the concentration of electrolyte increases. Hence, the very large values of kbi near xp- ) 0.5 are mainly due to the increased ionic strength in the bulk solution consequent to the release of salt as the zwitterions are formed. In order to determine the electrolyte concentration in the bulk solution, we have, in our calculations, assumed the total amount of surfactant to be generally 50 mM. The overall picture as to the bilayer bending constant is seen in Figure 6. The smallest value is lower and the largest value is larger than for the SDS/dodecanol case (cf. Figure 14 in ref 15). In the present case of anion/ cation vesicles the electrostatic contribution ranges between -19kT < kel bi < 0, and for a SDS/dodecanol vesicle < -5kT. -14kT < kel bi The modified free energy of mixing expression due to zwitterion formation is largely included in the expression for kmix bi given by eq 43, but it has only a minor influence on the appearance of Figure 6. The basic model employed in the calculations includes an explicit curvature dependence of the hydrocarbon/water interfacial tension γhc/w resulting in a contribution to kbi of about -5kT; i.e., it brings down the bending free energy substantially, toward magnitudes which correspond to reasonable vesicle sizes. The headgroup parameters are influenced by the curvature dependence of γhc/w, and hence, they vary indirectly with curvature. However, it would not be unreasonable to assume a direct curvature dependence of the headgroup parameters, although it would result in a somewhat more complicated model. A repulsive headgroup interaction would imply a negative contribution to kbi, and hence, it might bring down the bilayer bending
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Langmuir, Vol. 12, No. 10, 1996
constant to vesicle-promoting values without any need to invoke an explicit curvature dependence for γhc/w. In Figures 4 and 5, size distributions are plotted for two mole fractions of aggregated SDS equal to 0.238 and 0.756, respectively. The maximum of the size distribution of SDS-rich vesicles (Figure 4) occurs at a larger radius than that of DAC-rich vesicles (Figure 5), mainly because the sulfate headgroups are more space demanding than the ammonium headgroups yielding a lower khc/w bi in the latter case (cf. Table 1). Moreover, the smaller difference between the two headgroup cross sections for the SDS/ assumes a lower value than DAC case implies that khc/w bi for the SDS/dodecanol case, and together with the electrostatic effect discussed above, this accounts for the formation of substantially smaller mixed SDS/DAC vesicles compared with the SDS/dodecanol case. Our conclusion in this and previous papers15,16 that kbi increases when salt is added which, eventually, destabilizes a vesicle solution, agrees with what Kaler et al.17 observed for SDBS-rich vesicles but not with their experiments on CTAT-rich vesicles. However, a rapid decrease of the region of vesicle stability upon adding electrolyte has been reported by Hoffmann et al.9 for mixed tetradecyldimethylammonium oxide/tetradecyltrimethylammonium bromide vesicles. From their experiments Kaler et al. presented vesicle size distributions as vesicle number density as functions of the vesicle radius. Since our size distribution in eq 44 represents the volume fraction density, we predict a number density size distribution function with a preexponential factor proportional to R3 resulting in a peak width of σR/Rmax ) 0.39, which is equivalent to σR/〈R〉 ) 0.37. This is in good agreement with σR/〈R〉 ) 0.36 reported by Kaler et al. for SDBS-rich vesicles but less so with their result on CTAT-rich vesicles, σR/〈R〉 ) 0.57. It should be stressed here that it is often enough experimentally difficult to assess whether a solution of vesicles has attained equilibrium or not.
Bergstro¨ m
Acknowledgment. I am most grateful to Professor Jan Christer Eriksson for giving valuable comments on this paper.
Appendix The (macroscopic) expression for ideal free energy of mixing per aggregated monomer due to zwitterion formation, id mix ) |2x- - 1| ln|2x- - 1| kT |(1 - x-) ln(1 - x-) - x- ln x-| (A1)
is part of the monolayer tension expression in eq 30 and contributes to the bending free energy constants k1, k2 and, hence, to the bilayer bending constant kbi. For convenience we treat each logarithmic term in eq A1 separately starting with the last one
(
(i) x- ln x- ) x- ln xp- + x- ln 1 +
)
kx′ kx′′ + 2 r r
(A2)
where we have used the anion curvature dependence given by eq 31. The first term is included in the expression for ′ and ′ given in eqs 36 and 37. Hence, from now on we concentrate on the last term of eq A2. Expanding the logarithm of this term yields
(
x- ln 1 +
) [
)]
(
kx kx′′ 1 kx′ kx′′ kx′ kx′′ + 2 ) x+ 2 + 2 r r 2 r r r r
2
(A3)
Conclusions To summarize, we have found the following: (i) For the catanionic surfactant system the (macroscopic) free energy of forming a vesicle out of monomers in solution can to a very good approximation be written as *(R) ) 4π(kbi + 2γ∞R2), where 8πR2γ∞ is the work of forming a planar bilayer (without edges) and 4πkbi is the work of bending the bilayer into a geometrically closed bilayer vesicle. (ii) The bilayer bending constant kbi ≡ 2(ξpk1 + k2), as calculated from a conventional, additive model assuming a set of separate contributions to the overall free energy of a mixed anion/cation bilayer, increases sharply as the bilayer composition tends to 0.5 and, correspondingly, the vesicle size increases. Close to equimolarity, the bilayer bending constant is very large and vesicle formation does not occur. In the regions 0.1 < xp- < 0.3 and 0.7 < xp- < 0.9, small unilamellar vesicles (Rmax < 500 Å) can form. This is in agreement with recent experimental findings indicating vesicle formation in phase diagram regions where one of the two surfactants is in excess. (iii) The vesicle size distribution is governed by a volume fraction expression of the form φ(R) ∝ R6 exp[-4π(kbi + 2γ∞R2)]. The vesicles are fairly polydisperse with a relative standard deviation σR/Rmax generally equal to 0.283.
Again inserting the expression of x- given by eq 31, eq A3 becomes
(
) ( )[ )] ( ) ( )
kx′ kx′′ kx′ kx′′ kx′ kx′′ + 2 ) xp- 1 + + 2 + 2 r r r r r r p 2 2 k ′ ′′ k ′ ′′ k ′ ′′ x k k k 1 x x x x x x ) xp(A4) + 2 + 2 + + 2 2 r r 2 r r r r
x- ln 1 +
(
where we have neglected higher order terms since they give no contribution to kbi. (ii) Essentially the same procedure for the second term in eq A1 gives
(1 - x-) ln(1 - x-) ) (1 - x-) ln(1 - xp-) +
[
(1 - x-) ln 1 -
xp(1 -
xp-)
(
)]
kx′ kx′′ + 2 r r
(A5)
Expansion of the second term in eq A5 and omitting terms of higher order results in
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Thermodynamics of Vesicle Formation
[
(1 - x-) ln 1 -
(
)]
kx′ kx′′ + 2 p (1 - x-) r r xp-
[
(
Langmuir, Vol. 12, No. 10, 1996 2463
and
≈
)
xp2 2(1 - xp-)2
[ (
1 - xp- 1 +
( )][ (
kx′ kx′′ + 2 r r xp2 -
2(1 - xp-)2
-
xp-
(
)
)]
kx′ kx′′ + 2 r r -
)
(
)
kx′ kx′′ + 2 p (1 - x-) r r
)]
kx′ kx′′ + 2 r r
2
≈
(
)
kx′ kx′′ kx′ kx′′ xp2 + 2 + + 2 p r r 2(1 - x-) r r
2
(A6)
(2x- - 1) ln(2x- - 1) ) (2x- - 1) ln(2xp- - 1) + 2x- - 1 (2x- - 1)ln p (A7) 2x- - 1
[
]
2x- - 1
2xp- - 1
[ (
2xp-
) ][
kx′ kx′ + 2 -1 r r 4xp2 -
2(2xp- - 1)2 2xp-
(
1-
2xp-
kx′ kx′′ 1+ + 2 r r
-
(
(
)
≈
)
kx′ kx′′ + 2 1 - 2xp- r r 2xp-
)]
kx′ kx′′ + 2 p 2 r 2(1 - 2x-) r 4xp2 -
)] (
kx′ kx′′ + 2 r r
2
)
(
)
kx′ kx′ kx′ kx′′ 2xp2 + 2 + + 2 p r r (1 - 2x-) r r
2
(A10)
We can now evaluate eq A1, which for xp- < 0.5 becomes id mix ) (1 - 2x-) ln(1 - 2x-) - (1 - x-) ln(1 - x-) + kT x- ln x- (A11)
Employing eqs A4, A6, and A10, eq A11 can be written as id mix ) (1 - 2x-) ln(1 - 2xp-) - (1 - x-) ln(1 - xp-) + kT 2 xp- (kx′/r + kx′′/r2) p x- ln x- + (A12) 2(1 - 2xp-) (1 - xp-)
)
(
)]
kx′ kx′′ (2x- - 1)ln 1 + p + 2 2x- - 1 r r
2xp- 1 +
2xp-
Apparently, there are no terms proportional to 1/r, and hence, we do not obtain any contribution to k1. Dividing eq A12 with the curvature-dependent expression for a, given by eq 10, gives the coefficient before the 1/r2 term for each monolayer. Hence, we obtain the mixing free energy contribution to kbi as twice this value, i.e.
the second term of which becomes
[ ] [
1-
2xp-
) (1 - 2x-)
1 - 2xp-
-2xp-
(iii) When evaluating the first term of eq A1, we have to differ between anion-rich and cation-rich compositional states. For xp- > 0.5 this term equals
(2x- - 1) ln
1 - 2x-
ln 1 -
2
xp-
[ ] ( [ )][ [ (
(1 - 2x-) ln
kx′ kx′′ (1 - x-) + 2 p (1 - x-) r r xp-
)
(
(
≈
)
kx′ kx′′ + 2 -1 r r
2xp-
2xp-
mix ) kbi
)]
kx′ kx′′ + 2 r r
xp- kx′ 2 kT ap (1 - 2xp-) (1 - xp-)
(A13)
An analogous procedure for composition states xp- > 0.5 gives
2
)
(
)
kx′ kx′′ kx′ kx′′ 2xp2 + 2 + + 2 p r r (2x- - 1) r r
mix ) kbi
2
(A8)
[
]
ap (2xp- - 1) (1 - xp-)
(A14)
and our result can thus be summarized with the following expression valid for all composition states
(iv) For composition states xp- < 0.5 the first logarithmic term of eq A1 becomes
(1 - 2x-) ln(1 - 2x-) ) (1 - 2x-) ln(1 - 2xp-) + 1 - 2x(1 - 2x-) ln (A9) 1 - 2xp-
xp- kx′2 kT
mix kbi
xp- kx′2 kT )
ap|2xp- - 1| (1 - xp-)
(A15)
which is identical to the expression for kmix bi in eq 43 and twice the last term in the overall expression for k2 given by eq 35. LA950873K
