J. Pbys. Chem. 1983, 8 7 , 5025-5032
have been traced via a study of corresponding singlechained surfactants to the much larger head-group area associated with the OH- ion, which, with ita strong affinity for water, sits much further from a charged micellar surface than the other anions. In the language of a current theo $ ~ of surfactant aggregation, for the dialkyl halides ulal, = 1 (bilayers favored), and for the dialkyl hydroxides ulal, i= lI2. Theoretically one expects peculiar behavior when such a high degree of curvature, opposed by chain packing, is imposed. It has been shown: although only within the pseudophase approximation for dilute solutions, that depending on chain stiffness, as head-group area is lowered by, e.g., increasing salt or counterion concentration, it is possible for the favored aggregates to be bilayers at u / a l = lI2, revert to vesicles as u l a l increases, and eventually revert again back to bilayers. The analysis of a situation which allows a distribution of aggregate sizes instead of the pseudophase approximation is more complicated. Clearly interaggregate forces, and entropy, and the physical limitations of close packing of large aggregates at high concentrations are all involved. It seems fairly clear that dialkyldimethylammonium hydroxides do form spontaneous stable vesicles whose size
5025
and distribution depend on concentration and salt type. Even microtubules can be constructed from these syst e m ~ . ~There ’ is some evidence that vesicles formed by partial titration have an asymmetric distribution of anions and with differing interior and exterior pH. This is important. For mooted pragmatic reasons and because of their evident biological interest the subject of what might be termed vesicle chemistry is an expanding phase. If the phenomena that we have reported hold up to intensive scrutiny, the possibility exists that some design features necessary for the making of vesicles for a particular task might be revealed. Acknowledgment. We thank Dr. Y. Talmon, Dr. B. Kacher, and Dr. J. K. Thomas for collaborative work on electron microscopy, VECPM and intramicellar excimer fluorescence studies, Dr. V. A. Parsegian for assistance, criticism, and advice, and R. G. Laughlin for much helpful criticism. D.F.E. acknowledges support of U.S. Army Contract DAA G29-81-K-0099. G.J.W. acknowledges support of Dr. V. Bloomfield through NSF grant PCM 81-18107. Registry No. DDAOH, 23381-53-5.
Ion Binding and the Hydrophobic Effect D. Fennel1 Evans*t and B. W. Nlnhamt*§ Department of Chemlcal Engineering and Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota 55455 (Received: Aprll25, 1983; In Flnal Form: July 28, 1983)
Two apparently opposing interpretations of ionic surfactant aggregation are reconciled. Phenomenological constants ascribed to ion association at micellar surfaces are reinterpreted and shown to emerge naturally from a treatment of electrostatic head-group interactions via an explicit approximate solution of the nonlinear Poisson-Boltzmann equation. Given critical micelle concentrations (cmc’s) and aggregation numbers, one can calculate the free energies, enthalpies, and entropies of micellization. Enthalpy-entropy compensation is studied with water as solvent, and earlier conclusions based on the phenomenological model are confirmed. Hydrazine exhibits very different behavior. By removing one parameter from consideration,the theory shows considerable light on the nature and importance of chain interactions in determining micellar structure, and on the validity of fluid-interior models for micelles.
Introduction In ionic micellar chemistry, no concept is more firmly entrenched than a belief in ion binding.’ Such a concept is not ab initio unreasonable. All observations can indeed be subsumed into a single phenomenological “constant” which characterizes a presumed degree of dissociation of counterions at the surface of a micelle. And this “constantn has been seen in the past as the one firm piece of information on which to build an understanding. On the other hand, extant theories2” make no appeal to ion binding and usually assume that micelles are fully dissociated. There is an apparent dichotomy here which it is our purpose to resolve. While the ion-binding model appears to account for many features of micellar chemistry quite satisfactorily, Department of Chemical Engineering. *Institute for Mathematics and Its Applications. $Permanent address: Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra ACT 2600, Australia.
its phenomenological base masks and obscures all information on the molecular interactions which determine the shape, size, and energetics of micelle formation. The estimated degree of counterion binding as determined from experiment varies widely (*20%) depending on the experimental technique employed6and is thus not even an operationally well-defined quantity. Since the determination of micellar thermodynamic quantities and the interpretation of almost all transport and equilibrium measurements is necessarily model dependent, the use of (1) Mukerjee, P. J. J. Phys. Chem. 1962, 66, 1375. (The thermodynamics of micelle formation was first described by an equation of this form in 1935 by G. S. Hartley in his well-known book. Given the then wide acceptance of the Bjerrum theory of ion pairing in electrolytes, it seemed sensible). (2) Tanford, C.“The Hydrophobic Effect”; Wiley: New York, 1973. (3) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (4) Mitchell, D. J.; Ninham, B. W. J. Chem. SOC.,Faraday Trans. 2 1981, 77, 601. (5) Wennerstrom, H.; Lindman, B. Phys. Rep. 1979, 52, 1. (6) Kresheck, G. C. In “Water-A Comprehensive Treatise”; Franks, F., Ed.; Plenum Press: New York, 1975; Vol. 4,Chapter 2.
0022-3654/83/2087-5025$01.50/00 1983 American Chemical Society
5026
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983
micelle ion-binding parameters and their derivatives is very limiting indeed. The same limitations, and lack of predictability, have been built into existing t h e ~ r i e s which ~-~ characterize the balance between attractive and repulsive molecular forces in terms of a head-group area per monomer. In the totality and hierarchy of aggregates formed by surfactants, micelles constitute only a small but important class. With their well-defined critical micelle concentrations and in many cases, near monodispersity, their characterization is simpler than the more elaborate vesicles, microemulsions, and biological membra ne^.^^'^^ For these more complex systems notions of specific ion binding and bridging of aggregates particularly by divalent ions is endemic, and often reduced to alchemy. A clarification of the specificity of cation and anion binding in micellar chemistry would appear to be a prerequisite for a more comprehensive understanding of all aggregation processes in aqueous solution. This approach had been anticipated in the pioneering work of ParsegianlO and in fact this has already been done in part in the important paper of Wennerstrom and his colleague^.^ They generated selfconsistent solutions to the Poisson-Boltzmann equation to compute electrostatic free energies within the framework of the Wigner-Seitz cell approximation. Comparison between the results of their theory and phenomenological ion-binding models shows that, were one to insist on the latter, then an “effective” degree of dissociation, weakly dependent on salt or surfactant concentration, would indeed emerge, and emerge correctly. Ion binding is then seen to be an unnecessary decoration. Ultimately it is important, especially in the context of membrane biology and macromolecules, to distinguish those situations where binding, e.g., a t carboxylate head groups, is a real issue. Our paper is concerned in part with this issue. The treatment of Wennerstrom et al. involves heavy computation, and it seems desirable to have explicit simple analytic formulas to make the physics of the problem clearer and explicit. A preliminary report indicating that this can be done has been given elsewhere.’l In this paper we use the theory to extract more convincing information concerning the temperature dependence of the hydrophobic effectlZfrom data presently available. Thermodynamics of Micellization The concept of ion binding to the surface of an ionic micelle is at first sight reasonable. Thus, a whole series of observations on an (apparently)wide class of surfactants show that the thermodynamics of micelle formation is well characterized by an equation of the form1v6J3
KCN = CINCZQ (1) where CN, C1,and Cz are the concentrations of micelles, monomers, and counterions. For univalent ions these are linked by the requirement of charge neutrality C = C, + NCN = C, + QCN- C3 (2) Here C and C3 are the concentrations of the surfactant and co-ions, respectively. It can be shown that, if Co is defined (7) Israelachvili,J. N.; Mitchell, D. J.; Ninham, B. W. Biochim. Biophys. Acta 1977, 470, 185. (8) Israelachvili,J. N.; Marcelja, S.; Horn,R. G. Q. Reo. Biophys. 1980, 13 121. ~.
(9) Gunarsson, G.; Jonson, B.; Wennerstrom, H.J.Phys. Chem. 1980, 84. - -,2114. - - - --
(10) Parsegian, V. A. Trans. Faraday S O ~1966, . 62, 848. (11) Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1983, 87, 2996. (12) Wightman, P. J.; Evans, D. F. J. Colloid Interface Sci. 1982,86,
515
(13) Evans, D. F.; Mukherjee, S.; Mitchell, D. J.; Ninham, B. W. J. Colloid Interface Sci. 1983, 93, 184.
Evans and Ninham
by the relation K = NCON-’(C3+ Co)Q,the system exhibits a critical micelle concentration for large N given by cmc i= Co. That is, as surfactant concentration increases, the concentration of micelles exhibits a step-function jump from essentially zero at Co. Thereafter CN/C remains nearly constant, with a slow decline until at a sufficiently large surfactant concentration an apparent second cmc is exhibited. As N decreases, this behavior, reflected in any measurement, will become a much more gradual phenomenon. In this phenomenological model q = Q / N represents the fraction of counterions “bound”, and 1- q the degree of “dissociation”. The cmc, expressed now in mole fraction units, will be defined in terms of kT In Xcmc = A ~ H-PkTq In (X,,, + X3) (3) where X,,, is the mole fraction of surfactant at the cmc and AgHp is a constant. On the other hand, in t h e ~ r y , ~ , ~ the equilibrium between monomer and micelles is given by the law of mass action as
X N = N X i N exP[N(M1’ - PNo)/kT] (4) where X N / N and X 1 are the concentrations of micelles and monomer in mole fraction units with a cmc given as kT In X,,, (5) PNO - plO = A ~ H P + g, There pNoand p: describe the chemical potentials of micelles of aggregation number N and monomer, respectively. (Polydispersity is an unimportant complication which we have no need to ~onsider.~ It can be shown quite g e n e r a l l ~ that ~ - ~ plo- p N o has a form given by eq 5 where AgHp is the (constant) hydrophobic free energy of transfer of hydrocarbon tails from water to an oil-like region which forms the micellar interior, and g, describes surface contributions. The predictions of eq 3 and 5 will be equivalent provided g,/kT = -4 In (X3 + XcmJ (6) The burden of theory is now to compute plo - pNo, a computation which is model dependent. Phenomenology vs. Models It is here that we enter on uncertain ground. One thing is certain. If we consider a dilute micellar solution and in particular a single ionic micelle of aggregation number N , the important quantity of interest, pNo, can be shown4 to have the form NpNo = FB(N)+ PoV,, - surface terms + packing terms (7)
where FB(N)is the bulk Helmholtz free energy for N amphiphiles at the (average) aggregate density and Po is the external pressure. If the aggregate is incompressible, the sum of the first two terms is the Gibbs free energy of N molecules at pressure Po,and N p N o will have an expression of the schematic form NPNO = NgHP i- N(7’0a gel -k gHS) (8) Here g, is the constant hydrophobic free energy of transfer. The remaining surface terms are extraordinarily difficult to quantify. At the lowest level of approxiation ’ 4 = Nroa where a is the surface they will include a term 7 area per surfactant molecule and yo, the surface tension at the hydrocarbon-water interface, is assumed constant. This will be opposed by repulsion head-group interactions. These involve as yet unquantified forces due to steric repulsion, hydration (gHs),and electrostatics (gel). There will be in addition contributions due to the packing and bending of hydrocarbon chains. Inclusion of all such terms would involve very much more detailed modeling. In
Ion Binding and the Hydrophobic Effect
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983 5027
earlier treatment^^-^ all of these effects have been subsumed into a form NpNo(surface)= Nyo(a u o 2 / a ) (9) A theory based on such a form essentially simply builds in the notion of opposing forces2at the micelle surface. It gives an excellent account of the data, when limitations imposed by chain packing are built in.3 We now extend this theory by focusing on a situation in which electrostatic forces are separated out, and explore a form NpNo(surface)= N ( y a + gel) (10) where gelis to be evaluated from double-layer theory. It must be remarked at once that by lumping all unknowns through the principle of compensating errors into a constant quantity y, an "effective" interfacial tension, we disguise a multitude of sins. This is indeed so, and a test for the theory will be in how well this assumption holds up. On the other hand, and important to our main thesis, if the model predicts temperature, salt, and solvent dependence of ionic micellar behavior correctly, one can refine it at leisure and overcome the limitations of the ion-binding approach. It must also be remarked that the Poisson-Boltzmann equation does have limitations to which we shall adhere. (1) Classical electrolyte theory gives a good account of 1-1 electrolyte activities and osmotic coefficients within the confines of the primitive model. It does not for 2-1 electrolytes where additional very strong forces due to solvent structure loom large. (2) For fully dissociated planar surfaces it gives a good account of the repulsive forces between mica and CTAB bilayers at low electrolyte con~entrations.l~-'~ (Very strong repulsive forces due to associated water structure dominate interactions below about 30-A separation between planar surface^.'^-'^ (3) At high surfactant and/or salt concentrations where the Debye length is of the order of head-group distance, discrete ion effects must render the continuum double-layer theory incorrect. (4) A t high surfactant (micelle) concentrations the meaning of the Debye length is quite unclear. While the Wigner-Seitz cell approach of Wennerstrom et al.9 is probably correct, we have no real feeling for its validity. Therefore, we restrict analysis to 1-1 surfactants and 1-1 salts. Double Layer at a Curved Interface We derive the appropriate quantities in the Appendix. For an isolated spherical micelle of radius R the doublelayer free energy per amphiphile of head-group area a is
+
gel= 2kT[[
In
(p +
)
[l + ( ~ / 2 ) ~ ] ' /+~
a
4 KRS
(1
)
+ [ l +2(s/2)211/2 + ...)
(11)
(14) Pashley, R. M. J . Colloid Interface Sci. 1981, 80, 153; 83, 531. (15) Pashley, R. M.;Israelachvili, J. N. J . Colloid Interface Sci., in press. (16) See aleo: Israelachvili, J. N.; Pashely, R. M.; Horn, R. G.;Ninham, B. W.; Parsegian, V. A. 'Proceedings of the IUTAM-IUPAC Conference, Canberra, 1981", Adu. Colloid Interface Sci. 1982. (17) Le Neveu, D.M.; Rand, R. P.; Parsegian, V. A. Nature (London) 1976, 259,601. (18)Cowley, A. C.; Fuller, N. L.; Rand, R. P.; Parsegian, V. A. Biochemistry 1978, 17, 3163. (19) Parsegian, V.A.; Fuller, N. L.; Rand, R. P. R o c . Natl. Acad. Sci. U.S.A. 1979, 76, 2750.
where s = 4re2/(t~akT)
= 8rnoe2/(tkT)
(12) and K - ~is the Debye length, t the dielectric constant of the solvent, e the unit charge, k Boltzmann's constant, T the absolute temperature, and no the electrolyte concentration. The first term here is the electrostatic free energy of a planar double layer, and the second an explicit correction due to curvature of the surface. In the limit of linear 0) this reduces to theory (s K~
-
(cf. eq 9) while in the nonlinear limit we have (In s - 1 - l / s ) - -((In 4 KRS
(s/4)
+ 2/s + ...)
Neither limiting form is of much quantitative use. The equilibrium micelle is determined from the condition
+
d(NpNO)/da = y (15) = yo - rei = 0 where aelis the electrostatic surface pressure. [Note that gel= US$, da and yel = -Sa dlClo, so that ya = US$^ da. Thus, y a gel g, eJlo.] We have the explicit form
+
. -_ =
7.1
-"( Fr[sinh2 (y0/4)+ -2 In cosh
(y0/4)
KR
r
I
+ ...
(16) where yo is the scaled surface potential yo = t+,/kT to be determined from the equation s = 2 sinh y0/2 ( ~ / K R tanh ) (y0/4) ... (17)
+
+
The solutions embraced by the approximation eq 17 have been checked numericallylo and are accurate to 5% (maximum error) for KR 2 0.5. This covers the entire region of interest. [Rigorous bounds on the solutions have been determined by Weinberger (private communication) and show that the results are correct to order 1/(KRI2.] We remark that for typical applications the linear theory will be seen to give expressions which are grossly in error. Application to SDS and CllTAB Micelles We have already in an earlier publication1' tested these theoretical expressions for SDS micelles, and recapitulate. The equilibrium micelle is determined by eq 15, where we suppose y o is a constant. The assumption that y o is a constant, and its magnitude, can be tested by calculating yel using eq 16, and the experimentally determined aggregation number for sodium dodecyl, decyl, and octyl sulfates. The calculation proceeds as follows: Assume the micelle is a sphere and completely dissociated. The required quantities R, a are then determined from geometry by the relations 4rR2 = Nu (4r/3)R3 = Nu (18) For u we use the volume of the hydrocarbon core2v3given by u = 27.4 26.9n A3 where n is close to the number of carbon atoms per chain. The fully extended chain length is2y31, = 1.5 + 1.265n A. Once a is determined, so is s. Equation 17 can then be solved by iteration to give the surface potential yo, whence follows yep Table I lists computations for SDS and SOS. Consider data for SDS. First note that y is constant to within 2.5%. The value 15 dyn/cm is consistent with results expected from work on bilayer and monolayer films.8p20 (It is certainly not 50 dyn/cm. Nor is such a
+
5028
The Journal of Physical Chemistry, Vol. 87,
No. 24, 1983
Evans and Ninham
T = 25 Tu
TABLE I: Comparison of Theory and Experiment for Surfactants a t c,b
mol/L
+ cmc, mol/L
c
0 0.01 0.03 0.1 0.3
0.0081 0.0156 0.0331 0.1015 0.3007
0
0.134 0.151 0.202 0.369
0.03 0.1 0.3
N
R ,A
33.8 24.3 16.7 9.5 5.54
58 64 71 93 123
16.9 17.38 18.1 19.8 21.7
Sodium 62 60 58 53 48
8.30 7.82 6.76 5.00
24 25 29 31
11.16 11.31 11.88 12.15
Sodium Octyl 65.21 11.47 64.33 10.95 61.22 9.92 59.90 7.51
ti-l,
A
a, A ’
Y = -Ye19
y
S
Dodecyl 48.96 36.45 26.06 16.29 10.38
-AgHP, callmol
q
7.41 6.85 6.22 5.37 4.54
9745 9631 9608 9533 9492
0.83 0.84 0.84 0.85 0.87
4.38 4.31 4.19 3.70
6169 6189 6212 6157
0.72 0.70 0.66 0.55
dynlcm
gellkT
YalkT
g,lkT
15.8 15.47 14.87 14.87 14.63
5.03 4.604 4.119 3.458 2.821
2.38 2.25 2.09 1.91 1.72
11.42 11.40 11.58 10.76
2.65 2.53 2.46 2.13
1.81 1.78 1.72 1.56
Sulfate 7.44 6.88 6.21 5.37 4.53
Sulfate 4.43 4.36 4.22 3.73
Data for cmc’s and aggregation numbers after Huisman.21 Note that small differences in y and g J k T = ( g e l + 7 a ) l k T Salt reflect curvature approximations t o components gel and y a which become increasingly important for small tiR. concentration, TABLE 11: Comparison of Theory and Experiment for Surfactants a t T = 25 ‘CU c , ~ 102(cmc),
mol/L
0 0.0157 0.0237 0.460
M
K-’,A
N
R, A
1.31 1.04 0.925 0.722
26.6 18.8 16.8 13.2
56 93 101 142
16.7 19.8 20.3 22.8
a, A ’
y
S
Y = -Yelp
dynlcm
Dodecylammonium Chloride 63 38.0 6.94 15.0 53 31.9 6.69 16.6 52 29.2 6.53 16.7 46 25.7 6.35 18.0
ge,lkT
yalkT
gJkT
4.61 4.53 4.42 4.30
2.30 2.14 2.10 2.02
6.91 6.69 6.52 6.32
-AgHP 9 cal/mol
q
9040 9039 9013 9042
0.83 0.78 0.75 0.71
a Data for c m c and aggregation numbers after Debye.2’ (As for SDS, a t higher aggregation numbers, spherical approximation incorrectly estimates curvature effects.) Salt concentration,
TABLE 111: Tetradecyltrimethylammonium Bromide 1O2(cmc),
M
0.0038
ti-‘,
A
49.3
N
R, A
a, A’
70 64 60 55
18.5 17.9 17.5 17.1
61 63 64 66
s 72.4 7011 68.7 66.8
y 8.255 81172 8.117 8.037
dyn/cm
gellkT
YalkT
g,lkT
-AgHP, callmol
q
16.64 16.26 16.0 15.7
5.75 5.64 5.57 5.47
2.49 2.49 2.51 2.53
8.23 8.14 8.09 7.99
10551 10506 10467 10412
0.86 0.85 0.84 0.83
7 = -Yel,
value to be expected.) Its value is predicted on several assumptions. The first is that KR> 0.5 so that the analytic formulas are sufficiently accurate. This is well satisfied except at c = 0. The second is that K - ~>> mean distance between head groups so that a continuum description of surface charge is valid. This assumption breaks down at the highest salt concentration. A third is that the micelle packs into a sphere. Obviously it does not at the high salt concentrations. A fourth is that the sum of all terms which conspire to give a net attractive surface pressure contribution to cancel electrostatic pressures are proportional to area ya. A fifth assumption we shall defer till later. Next note that any errors in the term ya (2% variation) are much reduced in the total surface free energy g, = ya + geland even further reduced in the computed hydrophobic free energy (AgHP= -g, + kT In Xcmc).The value of this free energy is 9600 *50 cal/mol. [This is to be compared with the expected value2 (825 cal/mol for CH2 group, 2100 per CH3 group). At least one CH2 group is exposed to water at the surface, so that we expect A g H P 5 (10)(825) + 2100 = 10350.1 Consider now SOS. Here - y e l , while still roughly constant, *2.5%, has a different value = 11.5. The reasons for this seem very clear. [Continuum theory has already certainly broken down at the high salt concentrations (0.03, 0.1, 0.3).] Here the chains, being shorter, will have much less freedom to pack into the (assumed) oil-like interior. This reduces the net attractive term ya as observed. A higher fraction of CH2 groups at the surface must have exposure to water, so that we expect A g H p to be reduced
from the value estimated from solubility data. Again, if we assume that a single CH, group is immersed in water at the surface, AgHP5 (6)(825) + 2100 = 7050 cal/mol, whereas the computed value is 6400. The difference between A g H p for SDS and SOS is satisfactory. This difference is per CH2group (9631 - 6400)/4 = (3231 f 200)/4 = 810 f 50 (cf. 825/CH2 group). For sodium decyl sulfate y = 13 f ’ 2 % and AgHPhas intermediate expected values. In Table 11, the predicted values are given for dodecylammonium chloride.21 The free energies of chain transfer for this cationic surfactant agree quite well with those obtained for SDS. We conclude that the fluid oil-like interior model with double-layer theory must be a little suspect at short chain lengths, but is reasonable at higher chain lengths. If we reformulate our results in the language of the “ion-binding” model, computed values of q emerge in agreement with observation. Thus, the empirical approach embodied in the ion-binding model is replaced by a theory which invokes well-defined and determinable molecular parameters. Our remaining assumption concerns the radius, R , of the spherical micelle which defines the location of the surface charge. From the known volume of hydrocarbon tail and aggregation number the radius of the hydrocarbon region of the micelle Rhccan be deduced. The boundary defining the surface charge would be Rhc -tRhead roup = R. In the calculations presented we have assumed R = Rhc: In other words, we ought not to be too surprised if there are systematic differences between hydrophobic free en-
(20) Fendler, J. H. ‘Membrane Mimetic Chemistry”; Wiley-Interscience: New York, 1982. A large number of references are listed.
(21) Huisman, H. F. Kon. Ned. Akad. Weten. h o c . Ser. B 1964, 67, 367, 376, 385, 407.
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983 5029
Ion Binding and the Hydrophobic Effect
TABLE IV: Comparison of Thermodynamics of Micelle Formation and Hydrocarbon Transfer in Water and Hydrazine surfactant H2
C,,TAB
0
H'lN2
C,,OSO,Na C ,,TAB
T , "C
N
35 135 35 45 35 45
70 35 60 55 60 55
-RT In Xcmc, cal/mol Present Theory
-Agnp, cal/mol
-gst
cal/mol
4440 4387 3750 3725
-AHHP, ASnp, kcalimol cal/(mol K )
10502 11155 8604 8430 7138 7012
4164 4043 3389 3287
3.0 15.6 14.0
25 -11 - 18
11.0
- 13
6.8 8.6 17.9 13.3 15.1
10 5 - 21 - 18 - 27
Equilibrium Expression
H20a
C,,TAB
HJZb
C ,,OSO,Na C ,,TAB
Reference 11.
25 40 135 35 35
9950 10160 9240
Reference 23.
TABLE V : Tetradecyltrimethylammonium Bromide in Water over the Temperature Range 25-166 "C Y = -Ye],
T , "C -105x,,,
N
R, A
25.2 40.3 54.2 76.3 95.5 114.0 134.9 166.0
70 65 60 55 50 35 30 20
18.9 18.4 17.9 17.4 16.9 15.0 14.2 12.4
6.865 7.604 8.901 12.059 17.837 24.432 38.523 70.900
a,
.A2
64 65.7 67.5 69.5 71.7 80.8 85.0 97.4
s
Y
69.21 64.76 58.19 50.33 40.70 31.31 24.23 16.17
8.159 8.016 7.785 7.503 7.068 6.467 5.941 5.063
dynicm g d k T 15.8 16.2 16.4 16.6 16.5 15.1 14.5 12.5
ergies of transfer as computed for widely differing surfactant head groups. This indeed is so. Tables I and I11 list a comparison between SDS and tetradecyltrimethylammonium bromide (C,,TAB). (The aggregation number for the latterz2is 70 > N > 64.) The difference between hydrophobic free energies of transfer should reflect only the difference associated with two CH2groups, Le., 1650 cal/mol. In fact, it is only 1000 cal/mol. The two results would come into line with expectation were we to choose the electrostatic radius of the C14TABmicelle to be the radius of the hydrocarbon core plus an additional several angstroms representing the distance of closest approach of head group and anion. The point is that we can expect significant difference and specificity due to this effect. By the same token one would expect specific counterion effects to become important. The behavior of surfactant with carboxylate or amine oxide head groups will show a strong dependence on pH. Effects of real association can be incorporated into the model. Temperature Dependence and the Hydrophobic Effect We wish to explore free energies, enthalpies, and entropies of hydrocarbon-water transfer as a function of temperature. Much of our intuition about hydrocarbonwater interactions comes from such transfer data, but with few exceptions these data are limited to temperature near 25 "C (0-50 "C). Over this temperature range, the unique properties of water are clearly evident; for example, the entropy of hydrocarbon transfer is positive. With increasing temperature, water becomes a more normal fluid and at high temperature normal solution thermodynamics should be obtained. In a previous paper,12 we reported the cmc's of tetradecyltrimethylammonium bromide from 25 to 166 "C as (22) Debye, P. Trans. N. Y . Acad. Sci. 1949, 51, 573.
5.661 5.525 5.306 5.083 4.711 4.138 3.713 3.000
ralkT
g,
2.475 2.467 2.453 2.394 2.330 2.290 2.185 2.010
8.136 7.992 7.759 7.477 7.040 6.428 5.897 5.006
- Agnp, calimol 10502 10886 11112 11456 11481 11342 11155 10694
q
RT In x,,,
0.85 0.84 0.83 0.82 0.82 0.77 0.75 0.69
-5682 -5909 -6065 -6265 -6320 -6397 -6373 -6327
g, -4820 -4977 - 5046 - 5191 -5155 - 4944 - 47 81 -4367
determined by conductance measurements. The data were analyzed with the equilibrium model and typical results are summarized in Table IV. In this analysis all changes in the micellar surface free energy with temperature were attributed to changes in counterion binding and aggregation numbers were assumed constant. For comparison corresponding computations from the present theory are also listed in Table IV. The theory described in this paper gives an explicit expression for g, and thus permits the dependence of g, with N and T to be determined. Appropriate computations for C14TABare summarized in Table V. A t 25 "C, 64 < N < 70. At higher temperatures, breaks reflecting cmc become more gradual,12J3suggesting decreasing aggregation numbers. Micellar diffusion data23on CllTAB using a water-insoluble dye at 95 and 135 "C are consistent with N = 50 and N = 35. We have carried out calculations for a wide range of assumed aggregation numbers at each temperature. As shown in Tables I11 and V, g, varies only slowly with N . The results for AgHP,g,, and RT In X,,, are plotted in Figure 1. The continuous curve for AgHP passes through values for aggregation numbers deduced from diffusion measurements. The change in A g H P with temperature shows a maximum near 95 "C and results from very different dependence of RT In X,,, and g, on temperature. Above and below 95 "C,the transfer free energy decreases and reflects an increasing solubility of hydrocarbon in water. The high-temperature behavior typifies normal (23) Evans, D. F.; Mukherjee, S., unpublished data.
(24) Ramadan, M.; Evans, D. F.; Lumry, R. J . Phys. Chern. 1983,87, 4538. (25) Evans, D. F.; Ninham, B. W.; Wei, J. J.Phys. Chern., preceding article in this issue. (26) See also: Talmon, Y.; Evans, D. F.; Ninham, B. W. Science 1983, 221, 1047. (27) Lianos, P.; Zana, R. J. Phys. Chen. 1983,87, 1289. (28) Hashimoto, S.; Thomas, J. K.; Evans, D. F.; Mukherjee, S.; Ninham, B. W. J. Colloid Interface Sci., in press.
5030
The Journal of Physical Chemistty, Vol. 87, No. 24, 1983
Evans and Ninham
TABLE VI: Hydroxide vs. Bromide Surfactants
C,,TAB cmc = 0.0038 N 70
R
a
K,A - 1
18.55
61.2
49
s 72.36
Y
-Ye1
8.255
16.64
ge,lkT 5.755
TalkT 2.478
gslkT 8.233
-AgHP 10560
4 0.86
C,,TAOH cmc = 0.007 Res
aes
19.5 20.5 21.5 22.5 23.5 24.5
68.26 75.44 82.98 90.88 99.14 107.76
15.32 16.32 17.32 18.32 19.32 20.32
7 3.81 83.67 94.24 105.44 117.26 129.72
s
- Ye1
ralkT
gslkT
-AgHP
N ( 7 0 ) , R h , (18.5), K = 2.7535 X 36.3 A - ' 7.41 14.22 4.97 2.36 7.20 12.77 4.73 2.34 11.51 4.49 2.32 6.99 6.79 10.42 4.27 2.30 6.60 9.46 4.05 2.28 6.42 8.61 3.84 2.25
7.33 7.07 6.81 6.57 6.33 6.10
9660 9500 9350 9210 9070 8933
0.82 0.78 0.76 0.73 0.71 0.68
Assume: N ( 4 0 ) , Rhc (15.32), ahc (73.81), K = 2.7535 X lo6= 36.3 A - l 44.24 7.12 13.73 4.61 2.46 7.08 39.03 6.79 6.85 11.96 4.36 2.43 34.65 6.58 6.52 10.46 4.13 2.39 30.97 6.33 6.27 9.20 3.91 2.35 27.85 6.09 6.01 8.13 3.70 2.31 25.17 5.86 7.21 2.27 5.79 3.51
9510 9340 9180 9030 8880 8750
0.78 0.75 0.72 0.69 0.67 0.64
Assume: 47.84 43.29 39.35 35.93 3 2.94 30.30
Y
solution behavior. Below 95 "C, the increased solubility must be a consequence of the unique structural properties of water. This increasing solubility stabilizes surfactant monomers and accounts for the minimum in cmc's vs. temperature for many surfactants like SDS (T- N 20 "C). Similar conclusions were reached by S h i n ~ d afrom ~~ changes in the solubility of alcohols and aromatic hydrocarbons with temperature. The entropy and enthalpy of transfer can be estimated from the change of Agm with temperature and are given in Table IV for 35 and 135 "C. A t high temperatures, the negative enthalpies reflect the aggregation and orientation of hydrocarbon tails upon transfer to the micelle. The favorable free energy of this process at high temperature is a consequence of the energy change. At lower temperatures the transfer data are more difficult to interpret. The measured positive entropy change must be the composite result of two processes. (i) A large negative entropy change accompanies the transfer of hydrocarbon chain out of water and into the micelle similar to what is observed at high temperatures. (ii) A positive entropy change accompanies the removal of water from around the hydrocarbon chain. The measured enthalpy change in low-temperature water must also be the sum of two contributions of opposite sign. The removal of water is endothermic and the transfer into the micelle is exothermic. The very large changes in AS and AH over the entire temperature range (25-166 "C) suggest large, but nearly compensating, contributions to the free energy of transfer. The solubility of hydrocarbons in low-temperature water is increased, not decreased, by the unique structure of water around nonpolar groups.29 Micelles i n Hydrazine These conclusions regarding water are confirmed by a comparison of the micelle data for hydrazine and water (Table IV). Hydrazine and water24are both extensively hydrogen-bonded liquids with many similar physical properties (melting and boiling points, enthalpies and entropies of vaporization, viscosities, and surface tension). Unlike water, hydrazine has no volume decrease on melting, no temperature of maximum density, and no compressibility maximum. The heat capacity per atom of hydrazine at low temperatures is about one-half that of water. The similarities and differences between the two (29) Shinoda, K.J . Phys. Chem. 1977,81, 1300.
g,llkT
-12000
,
I
I
I
I
4
I
-11500 -11000
-1050055 N
T
$ -6500 t
4
-I
-5500k
25
50
75
100
125
150
175
T, C '
Flgure 1. Free energy contributions to micelle formation for tetradecyltrimethylammonium bromide in water as a function of temperature. Upper curve: Free, gHP,of transfer of hydrocarbon tails from water to a micelle. Bars indicate extreme upper and lower bounds on gHP,computed at each temperature for various assumed aggregation numbers. Curve passes through (circles) values computed for measured aggregation numbers and as given in Table V. Middle curve: Measured free energy RT In X,. Lower curve: contribution, gs,from surface electrostatic and surface tension terms calculated for aggregatlon number of Table V.
solvents suggest that hydrazine differs from water in just those features which are generally attributed to the high degree of three-dimensional order available to water. A comparison of "hydrophobic" processes in the two solvents should provide a basis for distinguishing the unique feature of water. At low temperatures the enthalpies and entropies for the solvents are dramatically different. In fact, the results in hydrazine at 35 "C are very similar to those observed in water at 135 and 160 "C and reflect compensated behavior in aqueous solutions. Hydroxide Surfactants We have already alluded to differences which can emerge when electrostatic radii differ considerably from hydrocarbon core radii due to differences in ionic hydration. A class of surfactants which exhibit quite dramatically different behavior as a result of a related effect are those in
Ion Binding and the Hydrophobic Effect
The Journal of Physical Chemistty, Vol. 87, No. 24, 1983
which the counterion is hydroxide instead of a halide. We shall report on their properties in detail elsehwhere but remark that the dialkyldimethylammonium compounds form spontaneous vesicle^^^.^ whereas the corresponding bromides are completely insoluble, forming liquid crystals. This suggests that the hydroxide anion, being strongly associated with water, sits on average significantly further out from the surface than does a more innocuous anion. We consider only the single-chained surfactanb C1,TAOH ? ~ mol/L (cf. C14TAB,cmc = for which the C ~ C ~ is' 0.007 0.0038). The measured value of q is here about 0.5 rather than 0.8 as for C14TAB. Computation (Table VI) shows that this situation can come about within the confines of our model if aggregation numbers are considerably reduced, and if the counterion is at a distance of -4 A from the micellar surface. The calculations are carried out for different assumed aggregation numbers (which fix the radius of the hydrocarbon core Rh,). The computations are then performed assuming the OH- counterion sits at a further distance out from this radius, so that the double layer begins at the boundary R,. The required reduction in surface electrostatic free energy is accomplished by reduction in the number of amphiphiles per micelle, with compensating exposure of considerably more of the hydrocarbon tail region to water (cf. Agm, Table VI). To go beyond this qualitative calculation, some assumption would have to be made about the nature of the charge-free layer. The behavior of these surfactants is rather different (compare Table VI). Thus, addition of hydroxide causes micelles to increase only slowly in size ( N = 45 to N = 70 at 1 M NaOH2*), whereas addition of NaBr causes a transition to cylindrical micelles and eventually to a gel.25 The unusual features of hydroxide counterions are dramatically illustrated by the behavior of the double-chained surfactant^^^^^^ which form spontaneous vesicles. On addition of excess hydroxide the solution remains clear up to 0.5 M. With addition of excess salt, the vesicles grow and flocculate. This can be readily understood at a qualitative level only: addition of hydroxide forces the OHions to closer proximity to the charged micellar surface. This presumably induces a change in local water structure near the cationic head group. (Effectively the local dielectric constant is lowered.) There is therefore an additional strong repulsion between the cationic head groups which inhibits micellar growth, albeit at the expense of further exposure of hydrocarbon to water.
Conclusion We have laid to rest some ghosts, and simultaneously raised the specter of others. These things are important if one is ever to understand the specificity and nature of Ca2+and Mg2+in biology. Acknowledgment. D.F.E. acknowledges support of U.S. Army Contract No. DAA G29-81-K-0099. We are grateful to Prof. J. Th. G. Overbeek and Dr. V. A. Parsegian for their comments and advice on this problem. Appendix We derive here the formulas used and follow closely ref 10. Consider a spherical micelle of radius R immersed in a 1-1 electrolyte. The nonlinear Poisson-Boltzmann equation which describes the distribution of ions about the sphere is v2+
d2$ 2 dJ, 4rp(r) 8anoe = - + - - =-=e sinh &2 r dr e
($) (A.1)
5031
where no is the bulk electrolyte concentration, e the magnitude of unit charge, e the dielectric constant of the solvent, and T the temperature. The monomers are assumed to be completely dissociated and univalent. The boundary conditions are $(r) 0 as r m
- -
d+ = --4au atr=R
(A.2) dr e Introduce dimensionless variables through the substitution y = eJ,/kT x = Kr K~ = 8anoe2/(ekr) (A.3) where K - ~is the inverse Debye screening length. Then eq A.l and A.2 become
, I
.\
-
where u = e/a, and a is the area per surfactant molecule head group. In the limit R a,the term in (2/x)(dy/dx) which describes curvature correction to the planar problem drops out. The planar limit can be recovered by writing X = KR,x = X + [, and, taking the limit x 0 3 , then eq A.4 reduces to d2y/dx2 = sinh y
-
where the first integral is
yo = 2 In (s/2
(A.5)
+ [(s/2I2 + 1]'/2)
where yo is the scaled surface potential. If we substitute this expression into the curvature term of eq A.4, we have
xYosinhy dy - 2
;( z)
1yOldY
dy (A.6)
The main contribution to the second integral of the right side comes from y = yo, near which x = X = KR. To a first approximationwe can remove this fador from the intergral to give N
2 sinh2y0/2 -
= 2 sinh2y0/2
+ -(cosh a KR
X
y0/2 - 1)
(A.7)
We have used the planar approximation eq A.5 to carry out the last integral. Hence
{ (:
2 sinh y0/2 1 + E
2 sinh y0/2
cash yo/2 - 1
sinh2yo/2
4 +tanh (y0/4) + ... KR
))
'/'
(A.8)
This analytic approximation can be compared with the numerical solution of eq A.4 and can be shown to be quite accurate. The worst error for KR 2 0.5 is 5% at KR E 0.5. For larger values of KR the error diminishes rapidly. The
5032
J. Phys. Chem. 1983, 87, 5032-5040
electrostatic free energy can now be evaluated explicitly as gel= aJ‘ILo da = a(
y)2(
z ) J s y o ds (A.9)
which gives eq 11of the main text. To find the equilibrium micelle, we must satisfy3 a(NpNo(surface))/aa = y o + yel = y o - rel= 0
(A.lO)
where re]is the electrostatic contribution to surface pressure. Instead of differentiating eq 10 directly, it is simpler to use the exact result
and to use eq (A.8). This gives eq 16 of the main text. For cylindrical or ellipsoid micelles, the analysis can be carried through in the same way. Thus, for cylindrical micelles the operator V2 becomes d2/dr2+ (l/r)(d/dr) instead of d2/dP + 2/r for spheres. Hence, the curvature term will be reduced by a factor of 2. Similarly for other shapes. The analysis can also be carried through for a situation where the surface of a micelle has dissociatedz6 groups, and/or for a mixture of 2:l and 1:l electrolyte. At high surfactant concentrations (>>cmc)the approximations of the Debye-Huckel theory or Poisson-Boltzmann equation break down. Micelles other than the one under consideration cannot be considered as point ions and intermicellar interactions must be built into the formalism. These could be included, either by the Wigner-Seitz method or through a perturbation expansion built by using the above approximate analytic forms. Registry No. H4N2, 302-01-2.
Influence of the Properties of the Oil and the Surfactant on the Phase Behavior of Systems of the Type H,O-Oil-Nonionic Surfactant M. Kahlwelt,’ E. Lessner, and
R. Strey
Mex-Plenck-Instltut fuer blophyslkellsche Chemle, P 3 4 0 0 Goettlngen, West Germany (Received: March 1, 1983; In Flnal Form: July 20, 1983)
In our preceding publications on this subject we have suggested studying “simple” systems of the type H20oil-nonionic surfactant in order to find qualitative rules for the phase behavior of such systems. In this paper we report on the influence of the chemical nature of the oils and of the surfactants on the phase behavior, As model oils we have used n-dodecane,n-octane, cyclohexane, and toluene, and as nonionic surfactantalkylpolyglycol ethers of the type CiE,.We found that the maximum mutual solubility of the three components is obtained at the ‘waist” which extends around the three-phase interval. The positions of the three-phase intervals on the temperature scale were determined as functions of the properties of the oils and the surfactants. The diagrams permit one to construct a scale for the hydrophobicity of any oil and may help one to choose the appropriate surfactant for obtaining a three-phase system for a given oil at a given temperature. Furthermore, the results are briefly discussed with respect to the existence of tricritical points in such ternary systems. The results may also be helpful for further discussion of the properties of so-called microemulsions. In a forthcoming second part we shall report on the corresponding results for quaternary systems with an electrolyte as fourth component.
I. Introduction Quaternary systems of the type H20-oil-nonionic surfactant-electrolyte have attracted increasing attention for both practical and theoretical reasons. The practical reasons result from the fact that with appropriate surfactants one can prepare so-called “microemulsions”,i.e., stable homogeneous solutions with high fractions of HzO and oi1.l Closely related herewith is the separation of the system into three phases with very low interfacial tensions between them, a property which appears to be of interest for tertiary oil recovery.2 The theoretical reasons result from the growing interest in critical phenomena in such systems, in particular, in the properties of multicritical point^.^^^ A necessary basis for the discussion of the properties of such systems is the understanding of their phase behavior, (1) Hoar, T. P.; Schulman, J. H. Nature (London) 1943, 152, 102. (2) See, e.g.: Morgan, J. C.; Schechter, R. S.;Wade, W. H. In “Solution Chemistry of Surfactants”;Mittal, K. L., Ed.;Plenum Press: New York, 1979; Vol. 11, p 749. (3) Widom, B. J. Phys. Chem. 1973, 77, 2196. (4) Griffiths, R. B. J. Chem. Phys. 1974,60, 195. 0022-365418312007-5032801 .SO10
i.e., the change of their phase diagrams with temperature and pressure, as well as the influence of the nature of the oil and the surfactant on the phase behavior. In this first part we shall present the results of our studies on the phase behavior of ternary systems of the type H 2 0 (A)-oil(B)-nonionic surfactant (C) restricting ourselves to the temperature dependence between 5 and 90 “C at atmospheric pressure. Accordingly, we shall represent the phase behavior of the ternary system in an upright prism with Gibbs triangle A-B-C as basis and temperature as ordinate. In a forthcoming second paper we shall then extend the studies to quaternary systems including an electrolyte. The fact that the addition of a third component may either increase or decrease the mutual solubility between two partially miscible substances has been well-known for about 100 years? Equally well-known is the fact that, with three substances which are only partially miscible with each other, the interplay between the three corresponding miscibility gaps may lead to the formation of a three-phase triangle (3PT) within the Gibbp triangle.6 All we have ( 5 ) Bancroft, W. D.
J. Phys. Chem. 1897, 1 , 414, 647.
0 1983 American Chemical Society
