1998
J. Phys. Chem. 1992, 96, 1998-2006
of this paper is to present a new technique of observing this transition, a technique which is quicker, cheaper, and more direct than any other (e.g. light scattering). Also, we present the first evidence that this is not a sharp, single-point transition but one which occurs over a range of composition with a fairly distinct beginning and end (in some cases). Using this technique, we have been able to catalog the effects of a number of compounds on the transition. We briefly summarize four types of effects: counterions, co-ions, cosurfactants, and cosolvents (organic modifiers). While to some readers this may invite discussions of mechanisms and causal effects suggested by Ruckenstein, Ninham, and I~raelachvili,~*-~~ we are well aware (28) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1977, 60, 221. (29) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sei. 1979, 71, 580. (30) Israelachvili, J. N.; Mitchel, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525.
that these are observations on a single surfactant at a single concentration at a single temperature. Until we have convinced the scientific community of the validity of our observations, any discussion of mechanism would be speculation. Registry No. NaC1, 7647-14-5; NaBr, 7647-15-6;NaF, 7681-49-4; NaNO,, 7631-99-4; NaSCN, 540-72-7; KCI, 7447-40-7; LiC1, 744741-8; NH,Cl, 12125-02-9;sodium dodecyl sulfate, 151-21-3;1-pentanol, 71-41-0; 1-hexanol, 111-27-3; 3-methyl-I-butanol, 123-51-3;2-pentanol, 6032-29-7; 1-hexylamine,11 1-26-2;p-dioxane, 123-91-1;benzyl alcohol, 100-51-6;urea, 57-13-6. (31) Ruckenstein, E.; Chi, J. C. J. Chem. SOC.,Faraday Trans. 2 1975, 71, 1690. (32) Ruckenstein, E. Chem. Phys. Lett. 1978, 57, 517. (33) Ruckenstein, E. Chem. Phys. Lett. 1983, 98, 573. (34) Israelachvili,J. N. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Proceedings of the International School of Physics "Enrico Fermi", 1985; Degiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam,
1986.
Application of Tanford's Micellization Theory to Gel Filtration Chromatographic Data for Nonionic Surfactants Noriaki Funasaki,* HyangSook Shim, and Sakae Hada Kyoto Pharmaceutical University, Misasagi, Yamashina-ku, Kyoto 607, Japan (Received: March 19, 1991; In Final Form: September 25, 1991)
The micelle formation of hepta(ethy1eneglycol) decy1 ether ( C I A ) has been investigated by frontal gel filtration chromatographic (GFC) and surface tension methods at 25 'C and explained in the framework of the Tanford theory. From an analysis of the concentration dependence of the centroid volume of the GFC pattern, the monomer concentration and weight- (n,)and number- (n,) average aggregation numbers of CI0E7micelles are estimated as a function of the total CloE7concentration, The n, value increases rapidly above the critical micellization concentration (cmc) and levels off at higher concentrations. The ratio n,/nn has a maximum around the cmc and approaches unity at higher concentrations. The derivative GFC pattern suggests the formation of premicelles. The dimerization constant of CI0& is smaller than that of Cl&. These experimental results are quantitatively explained on the basis of many micellization models, originally considered by Tanford. These models are different in the configurations of the decyl and hepta(oxyethy1ene) chains, the penetration of water into the micelle core, the roughness of the micellar surface, and the micelle shape. A most probable CIOE7micelle of n, = 60, conforming to the present and literature data, seems to be an oblate ellipsoid whose minor semiaxis is the sum of the length of the fully extended decyl chain and that of the randomly coiled hepta(oxyethy1ene) chain. The micelle size distribution function is calculated on the basis of several micellization models.
Introduction It is well-known that a surfactant forms micelles above the critical micellization concentration (cmc) in aqueous solution, but little is yet known about the formation of premicelles (dimers and oligomers) and the relation of micelle size and shape to the chemical structure of the s~rfactant.I-~For the investigation of these problems, the Tanford theory is of most promise among a number of theories for micelle formation.'-3 Tanford related micelle size, the cmc, and the other micellar properties to a size-dependent free energy of micellization which is a function of the chemical structure of the surfactant. Then he used the definition of the cmc as the total concentration x at which the monomer concentration x, is 95% of x, although a better definition of the cmc would be x at which d3xl/dx3 = 0.4-7 Furthermore, (1) Wennerstrom, H.; Lindman, B. Phys. Rep. 1979, 52, 1. (2) Tanford, C. J. Phys. Chem. 1974, 78, 2469. (3) Tanford, C. The Hydrophobic Effect; John Wiley: New York, 1980; Chapters 2-8. (4) Ben-Naim, A,; Stillinger, F. H. J. Phys. Chem. 1980, 84, 2872. (5) Warr, G.G.;White, L. R. J. Chem. SOC.,Faraday Tram. 2 1985,81,
549.
0022-3654/92/2096-1998$03.00/0
he could reproduce weight-average micellar aggregation numbers
n,at the cmc on the basis of his theory. The agreement between theory and experiment for nonionic surfactants is poorer than that for ionic and zwitterionic surfactants. The values of n, and x , should both depend on x , but only those values at the cmc were compared with theoretical v a l ~ e s . ~He , ~considered that no premicelle forms below the cmc3 Recently we have developed a gel filtration chromatographic (GFC) method for the determination of n, and x l as a function of x.&Io For hexa(ethy1ene glycol) decyl ether (CloE6),we have shown that the concentration dependence of n, determined by GFC is in good agreement with that by static light ~cattering.~ For octa(ethy1ene glycol) decyl ether (CloE8),we have shown that premicelles (mainly dimers) form and that micelle size grows rapidly with increasing concentration above the cmc, as compared (6) Funasaki, N.; Shim, H.-S.; Hada, S. J. Chem. Soc., Faraday Trans. 1991, 87, 957. (7) Funasaki, N.; Hada, S . Bull. Chem. SOC.Jpn. 1991,64, 682. (8) Funasaki, N.; Hada, S.;Neya, S . J. Phys. Chem. 1988, 92, 7112. (9) Funasaki, N.; Hada, S.; Neya, S. J. Phys. Chem. 1990, 94, 8322. (IO) Funasaki, N.; Hada, S.; Paiement, J. J. Phys. Chem. 1991,95,4135, and references cited therein.
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1999
Application of Tanford's Micellization Theory
TABLE I: Structural Models for Spherical Micelles Possessing Maximum Core Radius b,, and Aggregation Number i,, for Cl& (Models A-E), C,& (Model F), and C12E, (Models C and H) model n,' L",8, I,,? A b,,,, A i,,, rR, 8, A 9 12.885 10.18 10.18 16 13.0b B 9 12.885 10.18 12.89 33 13.6c C 10 14.150 10.90 10.90 18 12.3d D 10 14.150 10.90 14.15 40 12.3d E 10.90 10 14.150 14.15 40 19.Y F 10 14.150 10.90 14.15 40 13.q G 11 15.415 11.56 11.56 20 H 12 16.680 12.15 16.68 56
Figure 1. Hypothetical spherical micelle of CI0E7and definitions of a few distances.
with CloE6.6 Furthermore, these results have been explained on the basis of a modification of the Tanford theorye6 For poly(ethylene glycol) dodecyl ethers (CI2E,) at 25 'C, micelle size remains almost unchanged with x at m 2 7, but increases at m -< 6.11-13 In this work we determine the concentration dependence of xl and n, for hepta(ethy1ene glycol) decyl ether (ClOE7) by GFC and rigorously test the Tanford theory in comparison with these results and the literature data.6,'4
"Calculated from Table I of ref 2. bFor a randomly coiled CH2(OCH2CH2),0H group with a flexible alkyl chain. cFor a randomly group with a fully extended alkyl chain. coiled CH2(OCH2CH2)70H For a randomly coiled (OCH2CH2)70Hgroup. e For a helical (OCH2CH2)70Hgroup. /For a randomly coiled (OCH2CH2)80Hgroup.
i (Figure l), and the expression of the last term in eq 1 for the repulsive interaction may be correct for short-range interactions which occur for nonionic compounds. Since AHi and ARi depend on i, AGio changes with i. From eqs 1 and 2 we can calculate the equilibrium concentration xi (in mole fraction units) of i-mer: xi = x l i exp(-iAGio/RT) (3) The total stoichiometric concentration x of surfactant in the solution (including water as a component) is expressed as
Theoretical Background
(4)
Let us consider the Tanford the or^^,^ in light of recent advances. A general expression of the Tanford theory for i-mer formation of a nonionic surfactant possessing the number n, of carbon atoms is written as AGi' = -2100 - 700(n,' - 1)
and the weight- and number-average aggregation numbers of micelles as m
+ u ( A H-~21) + D + 7 / A R ?
iAl
Ai
(2)
The first four terms on the right-hand side of eq 1 express the hydrophobic free energy which is the driving force of micellization, and the last term expresses the repulsive interaction between the head groups at the micellar surface which opposes micellization. The value of n,' denotes the number of carbon atoms in the hydrocarbon core of the micelle and u (in cal mol-' per A2 molecule-') denotes the free energy of miwllization per the surface area of contact between hydrocarbon and water. In eq 1, Tanford employed values of u = 25 cal mol-' A-2 and n,' = n, - 1 . The value AHidenotes the area (in A2 molecule-I) in a micelle of size i at the closest approach (rH) of water molecules to the hydrocarbon core surface within which no water molecule penet r a t e ~ . ~ ~ ~Therefore, J ~ - l ~ the third term in eq 1 expresses the contribution due to incomplete shielding of a few methylene groups (mainly the a-methylene group) adjacent to the head group from water m0lecules.23~The value of D is an adjustable constant which is between 0 and 700 cal m01-I.~ The value of ARi(in A2 molecule-') denotes the available area per head group at the distance rR from the surface of the hydrocarbon core in a micelle size of (11) Tanford, C.; Nozaki, Y.;Rohde, M. F. J. Phys. Chem. 1977, 81, 1555. (12) Funasaki, N.; Hada, S.; Neya, S. Bull. Chem. Soc. Jpn. 1989, 61, 2485, and references cited therein. (13) Funasaki, N.; Hada, S.; Neya, S. J. Phys. Chem. 1988,92, 3488, and
references cited therein. (14) Imae, T. J. Colloid Interface Sei. 1989, 127, 256. (15) Hermann, R. B. J. Phys. Chem. 1972, 76, 2754. (16) Reynolds, J. A.; Gilbert, D. B.; Tanford, C. Proc. Nail. Acad. Sei. U.S.A. 1974, 71, 2925. (17) Hermann, R. B. Proc. Narl. Acad. Sei. U.S.A. 1977, 74, 4144. (18) Funasaki, N.; Hada, S.; Neya, S.; Machida, K. J. Colloid Interlace Sei. 1985, 106, 255.
m
m
n, =
(5)
i=2
i-2
(1)
Here the standard free energy iAGio (in cal mol-') is for the formation of i-mer ( A i )from i monomers (Al):
m
n, = E i 2 x i / E i x i
Eixi/Cxi
i-2
(6)
i=2
A hypothetical spherical micelle of CI0E7is shown in Figure 1. This micelle model is depicted by reference to theoretical and experimental studies of the molecular cod1 ration of alkyl chains in micelle^.'^-^^ The core volume V (in of the micelle with size i may be calculated from2 V = i(27.4 26.9n,') (7)
E)
+
This equation gives molar volumes close to our volumetric data.23 The maximum length l,,, (all-trans conformations) for an alkyl chain with n,' carbon atoms (in A) may be calculated f r 0 m ~ 3 ~ I,,, = 1.5 + 1.265~; (8) Some instances are shown in Table I. In fact, however, some of the CH2CH2-CH2CH2groups in the micelle are in the gauche (g) conformation, being in equilibrium with the trans (t) conformation.24 As a result, an average length la,of the alkyl chain could be the b,,, value (models A, C, and G in Table I) smaller than 1-. Tanford considered that the maximum spherical micelle has the radius of this 1," value, viz., b,,,, and calculated the maximum aggregation number of the spherical micelle from
,i
= (4~/3)(6,,,)~/(27.4
+ 26.9n,')
(9)
Recently it has been shown that though much hydrocarbon, including some chain ends, is exposed to water at the core interface, ~.~~ the core is virtually devoid of internal ~ a t e r . ~Furthermore, (19) Gruen, D. W. R. J. Colloid Interface Sei. 1981, 84, 281. (20) Istraelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1985; Chapters 15 and 16. (21) Dill, K. A,; Flory, P.J. Proc. Natl. Acad. Sei. U.S.A. 1981, 78,676. (22) Dill, K.A.; Koppel, D. E.; Cantor, R. S.; Dill, J. D.; Bendedouch, D.; Chen, S.-H. Nature 1984, 309, 42. (23) Funasaki, N.; Hada, S.; Neya, S. J. Phys. Chem. 1984,88, 1243; J . Phys. Chem. 1986, 90, 5469. (24) Flory, P. J. Statistical Mechanics of Chain Molecules; John Wiley: New York, 1969; Chapter 5.
2000 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992
Funasaki et al.
j3 depends on aliphatic or aromatic hydrocarbon^.'^ When a sphere a few studies showed that the maximum radius b,,, for the of radius rHrepresenting a water molecule slides over the surface spherical micelle is 1-, rather than I,. For instance, such current models predicted an i,,, value of 60 for the spherical micelle of of the solute molecule, the surface traced out by the center of the sodium dodecyl sulfate (SDS)20and 38 for that of dodecanoate sphere is the cavity surface area AH.’’ Two methods for estimation (n, = 9).22 The corresponding values for spherical micelles with of A H are available in the literature. Hermann used the Bondi smooth core surfaces are 56 (model H ) and 33 (model B). The radii38and a water molecular radius of rH = 1.5 A and took into minor discrepancies between these values may be ascribed to some account the effect of molecular conformations of hydrocarbon on roughness of the hydrocarbon core surface. Anyway, these values AH. The area AHcalculated by this method is denoted as AcoM. are much larger than those predicted by the Tanford model (e.g., He obtained a value of u = 33 cal mol-’ A-2.15In the other see model G for SDS). method, the number of the hydrogen atoms (rH= 1.0 A) of solvent With respect to the configuration of the hepta(oxyethy1ene) water molecules in contact with a hydrocarbon molecule conchain, a few models are suggested: e.g., random coil, helix, and structed with the CPK molecular model is counted and this meander configuration^.",^^,^^^^ In the crystalline state, the most number is converted to a molecular area.16J8,34This area will be denoted as ACpK. Reynolds et al. determined ACpK’sof aliphatic stable conformation of poly(ethy1ene glycol) (PEG) and its alkyl ethers is HO-(-t-CH2-g-CH2-t-O),-R.28~2g-31 This conformation hydrocarbons under the three assumptions of the fully extended forms a 7/2 helix having the length of 2.78 A/CH2CHz0group.29 conformations of the hydrocarbons, the direct proportionality The meander configuration is suggested to have a length of 2 between In x, and AH (viz.,j3 = 0 in eq 1l), and the proportionality A/CH2CH20 g r o ~ p . The ~ ~ ,PEG ~ ~ chain is flexible and takes between ACoM and A C ~and K estimated a range of u = 20-25 cal various conformations in the crystalline complexes with urea and mol-’ A-2.‘6 These assumptions and their conclusions were HgC12.29 In organic solvent32and aqueous ~ o l u t i o nthe , ~ ~ ~ criticized ~ ~ ~ ~ ~by Hermann” and us.18We determined both ACOM’S gauche conformation of the OCH2-CH20group is more favorable and ACpK’sof aliphatic hydrocarbons and estimated values of u than the trans conformation. From the Raman spectra, it was = 31 cal mol-’ A-2 and j3 = 1943 cal mol-’ for ACoMand values suggested that one of the most predominant conformational seof u = 41 cal mol-’ A-2 and fl = 829 cal mol-’ for ACpK.18These quences in aqueous micelles is 0-t-C-g-C-t-O-g-C-g-C-t-O-f-Ccavity surface area methods were successfully used for correlation g-C-t-0-.28 The rR value for this conformation is smaller than with hydrophobicity of alcohols36and ether^.^^,^^ The difference that for the helix conformation. On the other hand, the hydroin u between the two surface area approaches will be ascribed dynamic and thermodynamic properties are well explicable on the mainly to the rHvalue, viz., the position of the hydrocarbon-water basis of the random coil configuration of the PEG chain.2,3*26g33 interface. As rHincreases, the cavity area increases and conseThe length (in A) of the random coil (CH2CH20),H group may quently the u value decreases.39 On the basis of the studies by be calculated from3) Hermann15 and Reynolds et al. (their u values are probably questionable),I6 Tanford finally employed a value of u = 25 cal rR = 5.03(m - 1 ) ’ / 2 mol-’ k2 (10) in eq 1.3 Another problem is concemed with the equality of the u values in eqs 1 and 11. The aqueous solubilities of normal The rR value for model E is calculated on the basis of the helix alkanes change with n, as3 configuration, whereas all rR values except for this model are RT In x, = -884n, + constant (12) calculated from eq 10. Since the rRvalue for the meander configuration is between those for the other two configurations, it whereas the cmc values of aqueous nonionic surfactants possessing was not used for this work. Each of the values rR and la, shown linear alkyl chains change with n, as3 in Table I should be regarded as an average over all molecules RT In xcmc= -700 n, + constant (13) in the micelle. The rR values in Table I show the length of the The comparison between these coefficients suggests a slight hydrophilic group that is outside the hydrophobic core. In models difference in u between eqs 1 and 11. Therefore we could estimate A and B, the core consists of the nonyl group, and therefore, the a reasonable value of u = 31 X 700/884 = 24.5 cal mol-’ rRvalue denotes the length of the CH2(0CHzCH2)70Hgroup. since the dependence of cmc on n, for a homologous series can The difference in rR between models A and B stems from the be ascribed almost entirely to the second term in eq l.40 This difference in the configuration of the decyl group. value for u is close to the value estimated by Tanford, although The u value in eq 1 may be estimated from the solubilities x, the reasoning is different from ours. The interfacial tension (in mole fraction units) of hydrocarbons in water: between hydrocarbon and water is -72 cal mol-’ A-2. Although ~ ~ , ~studies ~ this value was employed for u in a few s t u d i e ~ ,recent RT In x, = -uAH + fl (11) have suggested that u is -26 cal mol-’ Here A H is the cavity surface area of the hydrocarbon defined When long-range repulsive interactions between ionic head by HermannIs and fl is a constant independent of the type (linear, groups play a predominant role, eq 1 may be rewritten as branched, or cyclic) of aliphatic hydrocarbon,’ though AGiO = -2100 - 700(n,‘ - 1) + u ( A H-~21) + D + a/ARi A-2.20942
5-17,34936,37
(14) (25) Meneer. F. M. Nature 1985.313.603. Dill. K. A. Nature 1985.313. 603. Caban; B:; Zemb, T. Nafure 1985; 314, 385.’ Chen, S.-H.; Dill, K.A: Nature 1985, 314, 385. (26) Bailey, F. E., Jr.; Koleske, J. V. In Nonionic Surfactanrs; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; Chapter 16. (27) Ribeiro, A. A.; Dennis, E. A. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; Chapter 17. (28) Matsuura, H.; Fukuhara, K. J . Mol. Sfruct. 1985, 126, 251. (29) Takahashi, Y.; Tadokoro, H. Macromolecules 1973, 6, 672. (30) Connor, T. M.; McLauchlan, K. A. J . Phys. Chem. 1965,69, 1888. (31) Dorset, D. L. J . Colloid Interface Sci. 1983, 96, 172. (32) Mark, J. E.; Flory, P. J. J . Am. Chem. Soc. 1%5,87, 1415; 1966.88, 3702. (33) Nagarajan, R. Adu. Colloid Inferface Sci. 1986, 26, 305. Equation IO may hold only for m L 8.33 (34) Funasaki, N.; Hada, S . ; Neya, S.; Machida, K. J . Phys. Chem. 1984, 88, 5786. (35) Rosch, M. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1967; Chapter 16. (36) Amidon, G . L.; Yalkowsky, S . H.; Anik, S. T.; Valvani, S . C. J . Phys. Chem. 1975, 79, 2239. (37) Funasaki, N . ; Hada, S.; Neya, S. J . Phys. Chem. 1985, 89, 3046.
An explicit expression of a for ionic micelles is a ~ a i l a b l e . ~ Empirical theories, based on the Tanford theory, are available in the l i t e r a t ~ r e . ~ -For ~ , ~spherical ~ micelles with values of rH = rR = 0, we can obtain from eqs 1 and 7 iAGio = [-Ai Bi2I3 + Ci2]RT (15) Then, if eq 14 is used instead, we can 0 b t a i n ~ 3 ~ ~
+
(38) Bondi, A. J . Phys. Chem. 1964, 68, 441. (39) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J . Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (40) As eq 19 shows, the interactions between the head groups (terms B and C in eq 19) influence the cmc value. These effects on the cmc are almost constant for homologous surfactants possessing the same head group, as is seen from the agreement between the coefficients (700) of n, in eqs 1 and 3.’ (41) Nagarajan, R.; Ruckenstein, E. J . Colloid Inferface Sci. 1977, 60, 221. (42) Jonsson, B.; Wennerstrom, H. J . Colloid Interface Sci. 1981,80,482. (43) Nagarajan, R.; Ruckenstein, E. J . Colloid Interface Sci. 1983, 91, 500.
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 2001
Application of Tanford's Micellization Theory iAG:
= [-Ai
+ Bi2I3+ Ci4I3]RT
(16)
These coefficients in eqs 15 and 16 may be regarded as adjustable parameters. Since AG: should be zero at i = 1, Warr and White5 obtained from eq 16 iAG? = [-A(i - 1)
+ B(i2i3- 1) + C(i413-1)IRT
(17)
By using the same reasoning, we can obtain from eq 15 iAGio = [-A(i - 1)
+ B(i2I3- 1) + C(i2 - l ) ] R T
TABLE U Values of cmc and the Molecular Area at the Air-Water Interface at 25 O C
surfactant
cmc, mmol dm-'
CI0E6 CIOE7 CloEs
0.90 0.96 1.03
1.62 X 1.73 1.86
52.6 56.9 67.7
21.5 21.5 23.9
(18)
From some model calculations based on eq 17, empirical equations for x,,, and a rough value of i were In x,,, = -A
+ 1.87(BC)1/2+ 0.25
I = (B/G3I2
(19)
Equations 19 and 20 are useful for a rough estimation of A-C in eqs 15-1 8 and u, D,and y in eq 1. If i-mer only forms over all concentrations, we can expect x = x,
+ iKixli
(21)
When dimer and i-mer coexist with monomer, we can expect
+ 2Kzxi2+ iKixli
x = xl
(22)
In these equations the one-step aggregation constant Ki for eq 2 is defined as Ki = x i / x i i
(23)
Experimental Section Materials. A pure sample of CIOE7was obtained from Nikko Chemicals, Tokyo. This sample showed a single peak on the gas chromatogram and showed no minimum in the surface tension vs concentration plot. Blue dextran and Sephadex G-10 were purchased from Pharmacia Fine Chemicals, and poly(ethy1ene glycol) 6ooo was from Wako Chemicals, Osaka. Solute molecules possessing molecular weights larger than 700 are excluded from the G-10 gel The double-distilled water was degassed just before the GFC experiments. Methods. The surface tension of aqueous C10E7 solution was measured by the Wilhelmy method a t 25.0 'C, as already rep0rted.4~ GFC experiments were carried out with a column bed volume of 58.28 cm3 under a flow rate of -0.6 cm3 min-I. The column was jacketed for maintaining a constant temperature of 25.0 f 0.2 OC. A large amount of sample was applied so that the plateau region would appear on the elution curve (frontal GFC method). The elution process was monitored with a Shimadzu RID2A refractive index detector and recorded with a data processor. Analyses and simulations of chromatograms have been reported in detail.89 The best fit (optimization) was obtained when the sum, SS, of the squares of the differences in V, between theory and experiment was minimized:46 n
ss = i=C(Vc,calcd - Vc,obsd)Z 1
20.
(20)
(24)
Here n denotes the number of data; n = 18 for CIOE7and n = 17 for CloE8. To solve a polynomial equation such as eq 4, we used the Newton-Raphson method.47 Surface Tension Data. The surface tension u of an aqueous CIOE7 solution decreased with increasing concentration and remained almost constant above the cmc. By applying the Gibbs (44) Sephadex Gel Filtration: Theory and Practice; Pharmacia Fine Chemicals: Uppala, Sweden. (45) Funasaki, N.; Hada, S . Bull. Chem. SOC.Jpn. 1976,48, 2899. (46) Yamaoka, K.Analysis of Pharmacokinetics with Microcomputers; Nankodo: Tokyo, 1984. (47) Mortimer, R.G. Mathematics for Physical Chemistry; Macmillan: New York, 1981; p 249.
1510
12
14
16
nC
Figure 2. Plot of AAwm-'/*against n, for poly(ethy1ene glycol) alkyl ethers at 25 'C. Original data were taken from this work (O), ref 46 (0),and ref 47 (A).
adsorption equation to the data below the cmc, we can evaluate the surface excess r and molecular occupied area AAW (in Az molecule-') of C10E7 at the air-water interface from r = 10'6/AA~NA= -(da/d In x ) / R T (25) Since the surface pressure is the lowering of u caused by addition of surfactant, we can obtain the pressure-area curve of the surfactant at the air-water interface from the concentration dependence of u. Tanford used the pressure-area curve a t the oil-water interface to estimate the repulsive interaction term in eq 1.3 Since it is not easy to determine the pressure-area curve at the oil-water interface, the data at the airwater interface might be used for the same purpose, though no calculation based on this suggestion was actually attempted. Table I1 shows the observed values of cmc and AAW at the cmc for Cl0E6,CI0E7,and CloEs. These cmc (in mmol dmT3)values are well correlated with the oxyethylene unit m: log cmc = 0.029m - 0.219 (26) For the surfactants having the same alkyl group, the value of AAwm-'/' is likely to be independent of m?' In Figure 2, AAwm-'I2 is plotted against n,, together with the literature data A linear least-squares analysis gave the relationship at 25.0 0C.48*49 A AW m-lI2 = -1.265nC + 35.67 (27) Although we did not use these values of cmc and AAW for further calculations, these values are important parameters for theories of micellization. The samples of the oligo(ethy1ene glycol) decyl ethers used seem to be pure, since there is no minimum in the surface tension vs concentration plots and the cmc values are consistent with literature values.49 In a few theories of micellization, values much smaller than the AAW values shown in Table I1 are used for the cross-sectional areas of head groups of poly(ethylene glycol) alkyl ether^.^'^^^ CFC Data and Evaluation of Micellar Properties. The frontal GFC profile of C10E7 was determined at 18 concentrations of C = 0.1993-30.0219 mmol dm-3. The centroid volumes of this profile at the leading and trailing boundaries were determined and the average V, values were used for analysis (Figure 3), as already r e p ~ r t e d . ~Since , ~ , ~ Sephadex G-10 gel has small pores (48) Lange, H.; Jeschke, P. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; Chapter 1. (49) Meguro, K.;Ueno, M.; Esumi, K.In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; Chapter 3. (SO) Puwada, $3.; Blankschtein, E. J . Chem. Phys. 1990, 92, 3710.
2002 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 TABLE III: Geometric Parameters for Some Probable Mieellea of C&
micelle shape oblate prolate oblate prolate oblate prolate
model Ao
AP
Do
DP Fo
FP
i
60 60 60 60 46 46
b, A 10.18 10.18 14.15 14.15 14.15 14.15
Funasaki et al. (i = 60) and C I A (i = 46) at r H = 1.5 A a, A
AH,,A2
ARi. A’
molecule-I
molecule-I
Rb, A
20.42 40.97 17.32 21.20 15.17 16.26
71.3 84.1 66.0 67.3 72.8 73.0
179.2 209.9 170.8 173.2 216.1 216.5
29.2 32.0 28.6 28.8 28.1 28.2
C(mmol dm3)
Figure 3. Plot of centroid volumes against the total concentration of CIOE7at 25 OC. The solid line shows the theoretical values based on model Dol5y.
(from which compounds with molecular weights larger than 700 are excluded) relative to micelles of CIOE7including the dimer, V, can be written as9 Vc = [ClVI
-0.1
0
log(Cl(mmol dm-’1)
Figure 4. Plot of log (C - C,) against log C1. The solid line is drawn
through the experimental data points and the dashed line is calculated on the basis of model Dol5y.
+ (C- C,)V,,,l/C
(28) Here Vi denotes the elution volume of monomer and V,,,denotes that of the micelles. By extrapolation of V, to C = 0 and 1/C = 0, we estimated values of VI = 46.1 1 cm3and V,,, = 20.36 cm3, in the same way as reported.8 Once VI and V, have been estimated, we can determine C1as a function of C from
c, = (Vc - VM)C/(Vl - V m )
(29)
By using the concentration dependence of C,, we can evaluate the
n, value from39~5i~52 n, = d log (C - Cl)/d log C1
(30)
To calculate n, from eq 30, we must obtain the tangent at various points in Figure 4. The solid line in Figure 4 is drawn though the data points as closely as possible and the linearity is assumed at nine high concentrations (higher than 2.1943 mmol d m 9 . The n, values obtained from this solid line are shown by the circles in Figure 5. Furthermore, we can evaluate the n, values
Jniz
nn = (C - C,)/(S - Ci) S=
C d In C1
+ C*
Figure 5. Concentrationdependence of n, values evaluated from the solid line of Figure 4 by using eq 30.
(31)
’i
(32)
Here the total molarity S of the surfactant aggregates (summation of [Ai] over i = 1 to a)can be determined by graphical integration of eq 32, and CI denotes a very low concentration where no micelle is resent.^' The value of n, can be obtained as a function of C. The accuracy in n, is worse than that in n,. In Figure 6, the ratio of n, to nn is shown as a function of C by the circles. This result shows that the polydispersity in micelle size is maximal around the cmc. In Table 111, several geometric parameters of micelles of i = 60 and 46 are shown on the basis of some models described in Table I. These i values are the n, values obtained for C10E7 (Figure 5) and CloE2 by GFC under the assumption that n, remains constant at high concentrations. Since the i values in Table I11 are greater than any i, value in Table I, the minor (51)Funasaki, N.;Hada, S.; Neya, S. J . Phys. Chem. 1991, 95, 1846. (52)Corkill, J. M.;Goodman, J. F.; Walker, T.; Wycr, J. Proc. R. SOC. London 1969,A312, 243. Mukerjee, P.J. Phys. Chem. 1972,76,565.
4
Figure 6. Concentration dependence of n,/n,. The circles show the observed values and the dashed line is calculated on the basis of model
Dol5y.
semiaxis b of a prolate or oblate ellipsoid can be set to be b,,,. The major semiaxis a can be calculated from a combination of eq 7 with the mathematical formula for the volumes of these ellipsoids. The cavity micellar surface area AHiof the hydrocarbon
The Journal of Physical Chemistry, Vol. 96, No. 4 , 1992 2003
Application of Tanford's Micellization Theory
TABLE IV: Empirical Models for the MiceUizatioa of Cl& best-fit value model 1
B 17.09 20.72 27.17 20.21
estimated value C
SS,cm6
&'
0.022 1.201 1.590 0.027
0.0436 0.0434 0.0434 0.0435
56 58 56 55
K, 203 184 195 206
eq
A
2 3 4
15 16 17 18
16.26 20.57 23.65 17.28
model
eQ
K2
i
Ki
SS,cm6
nw'
K2
5 6
22 21
21 1
52
1.54 x 10239 1.07x 10277
0.0436 0.1510
52 60
0
best-fit value
estimated value
'Value at C = 30.0 mmol dm-).
0.08
-
?Ev
-
-E
0.07
5 %
0.06
E 'Oo0
0'05 40
45
50
40
45
t
50 500'
Figure 7. Derivative GFC patterns at the leading (solid lines) and trailing
(dashed lines) boundaries at C = 0.9043 (A and A) and 0.9986 mmol dm-3 (0and 0 ) : (a) experimental results, (b) simulations based on model 6 in Table IV, and (c) simulations based on model DslSy in Table V. For simulations, values of N = 65 and V, = 10 cm3 were employed as adjustable parameters.
core for the ellipsoid is calculable from the values of the major semiaxis ( a + r H )and minor semiaxis (b + r H ) . The AHivalue is slightly greater than the AAWvalue shown in Table 11. The surface area ARi and volume having the randomly coiled poly(oxyethylene) mantle were calculated similarly. The hydrodynamic radius Rh of the micelle was calculated from the Perrin equation under the assumption that the hydration of the micelle is independent of micelle shape.53 For CI0E7,Imae determined values of n, = 61 and Rh = 26.7 A, independent of C, by static and dynamic light ~cattering.'~Our observed value of n, = 60 at high concentrations is in excellent agreement with her n, value, and our calculated Rh values for i = 60 shown in Table 111 are close to her Rh value. Figure 7a shows some observed derivative elution patterns at the leading and trailing boundaries around the cmc. In general, whenever self-association occurs, the peak at the trailing boundary becomes broader than that at the leading bo~ndary.~'As Figure 7a shows, micellization occurs at C = 0.9043 mmol dm-3, a concentration lower than the cmc. Fitting of Miceilization Models to V, Data. Usually, the parameters of a micellization model are fitted by using the observed values of cmc and n, at the cmc by a trial-and-error search and are not ~ p t i m i z e d . ~In~this ~ *work ~ ~ ~we~ did not employ this usual procedure. To best fit the adjustable parameters by nonlinear least-squares methods, we generally need rough initial values for them, which may be chosen by a rough theoretical estimation or by a trial-and-error method. The former method was used, since it is better. The observed values of V, are the most appropriate quantities to be fitted, since the errors in V, are almost independent of C and the V,values are primary data in this work. To fit the parameters for AGio to the observed Vvalues, we used eqs 3, 4, (53)Perrin, F.J. Phys. Radium 1936, 7 , 1. Tanford, C. Physical Chemistry of Macromolecules; John Wiley: New York, 1961; Chapter 6.
/ ;
I 2 rdA)
3
Figure 8. Dependence of u and D on r H ,calculated on the basis of models DslOy, DslSy, and Ds30y in Table V.
and 28 and a relationship between xi and x calculated from AGY. First, three adjustable parameters, A, B, and C, in eq 17 (which gives a theoretical relationship between x1and x) were fitted by using rough initial values for them which were estimated from a comparison of eqs 19 and 20 with the observed values of cmc and n,, The best-fit values for A-C in model 3 are shown in Table IV. These values for model 3 were used as initial values of A-C for models 1, 2, and 4. In model 6, only monodisperse micelles of i = 60 are assumed to be present over all concentrations, since such an assumption is often made. Judging from the value of SS, this model is worse than the other models shown in Table IV. As model 5 shows, the presence of dimer leads to the improvement of fit. The Kso value in eq 21 was best fitted to the observed V, values by using eqs 21 and 28. In model 5 , the values of K2, i, and Ki in eq 22 were regarded as adjustable parameters. Many micellization models based on the definition in Table I are represented in Table V. In this table, spherical micelles mean that all micelles including larger micelles than ,,i are spherical and oblate or prolate micelles mean that micelles smaller than i,,, are spherical and that those larger than i,,, are oblate or prolate ellipsoids. For instance, model A0307 means that n,' = 9, b = 10.18 A, i,,, = 16, rR = 13.0 8, (Table I), and rH = 3.0 All micelles less than i = 16 form spherical hydrocarbon cores composed of i nonyl groups, whereas the hydrocarbon cores of micelles greater than i = 17 form oblate ellipsoids of revolution with a minor semiaxis of b = 10.18 A. When the parameters of AGio in eq 1, together with eqs 3,4, and 28, are best fitted to the observed V, values, we kept the values of n;, b,,,, i,,,, rR,and rHat the above constants and minimized the SS value by adjusting the values of u, D,and 7 , which are shown in Table V. To choose initial values of u, D,and 7 , we used the values of A-C for model 2 shown in Table IV. The values of SS are almost independent of models A-E (Table V) and models 1-5 (Table IV). From the SS values only, therefore, we cannot decide which is the best model. As is evident in Table V, the main factor influencing the u value is the rHvalue. Figure 8a shows the dependence of u on rHfor
x.
2004 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992
Funasaki et al.
TABLE V Optimized Values of u, y , and D of Eq 1 to Experimental V , Values Using Fixed Values of D:, b,,, of SS. K,. and R , for C,& (Models A-E) and C,& (Model F)
rR,and rHand Estimated V d w s
~~
optimized value
micelle model Asl5-t As30; A0307 AP3b Bsl5y Col5y CPl5Y Ds 1Oy Dsl5y Dsl5a Dol57 DP15-Y Ds30y EslOy Esl5y FslOy Fsl5y Fol5y
shape
sohere sphere
oblate prolate sphere oblate prolate sphere sphere sphere oblate prolate sphere sphere sphere sphere sphere oblate
"Value at C = 30.0 mmol dm-).
fixed value rH,
A
1.5 3.0 3.0 3.0 1.5 1.5 1.5 1.o 1.5 1.5 1.5 1.5 3.0 1 .o 1.5 1 .o 1.5 1.5
a, cal
m o P A-2 24.06 14.81 14.93 15.27 24.01 23.15 23.54 27.85 22.96 25.30 22.95 22.84 14.28 27.63 22.86 26.43 21.74 21.75
7,
cal
D,cal
mol-' A" 14.1 X lo8 10.2 9.17 2.67 15.8 12.1 4.06 15.8 13.9 b 13.3 11.2 10.2 55.2 49.2 23.2 20.0 19.7
mol-' 179 464 426 344 184 792 718 676 83 1 185 838 866 1128 717 860 764 923 924
estimated value S S , cm6 K, 0.0432 0.043 1 0.0434 0.0479 0.0432 0.0433 0.0452 0.0432 0.0432 0.0436 0.0432 0.0432 0.0431 0.0433 0.0432 0.0140 0.0137 0.0138
184 168 164 140 187 182 167 191 189 165 186 188 170 197 188 413 408 403 ~~
La
5.5.
57 58 72 55 57 67 55 55 60 55 56 57 55 55 48 49 48
= 14.0 X IO4 cal mol" A-2 in eq 14.
models DslOy, DslSy, and Ds30y. An increase in rH causes a decrease in u. This change can be expected from eq 1. An increase in f H causes an increase in AHi. If the contribution of the third term of the right-hand side of eq 1 to AGio is a constant, u must decrease with an increase in rH. An increase in n,' causes a decrease in u (compare models A and B with models C-E). An increase in n,' increases AHiand, therefore, causes the same effect with an increase in r H . The effect of micelle shape on u is negligible. As expected from eq 1, the increase of n,' from 9 to 10 causes an increase of -700 cal mol-' in D (e.g., compare models AslSy and BslSy with DslSy and EslSy and also model As307 with model Ds30y). As Figure 8b shows, an increase in r H causes an increase in D. This increase can be expected from eq 1, since the value of D - 21 u in eq 1 would be kept a constant. The effect of micelle shape on D is small and rather complicated (e.g., compare models A307 and models DlSy), probably depending on rH. Our D values for models A fall within 0 and 700 cal mol-], as already predicted by T a n f ~ r d . ~The value of D markedly depends on micelle shape, when ,i is small, as is the case for models A and C (seeTable I). An increase in rR causes an increase in y (e.g., compare models E with models D). This increase is ascribed to increases in denominator and numerator of the last term in eq 1. The experimental results for C&8 have already been analyzed using eq 17.6 The values of u and D can be expected to be the same for both CIOE7and CloEs. This expectation is almost valid (e.g., compare model DolSy with FolSy and also DslOy with FslOy). The y value for CloE8is larger than that for C10E7 (compare models D with models F), as expected.2 Prediction of the Micellar Properties. Using the models shown in Tables IV and V, we can predict a number of micellar properties of Cl&7 which either can or cannot be determined experimentally. The solid line in Figure 3 shows the V, values calculated on the basis of model DolSy. That is, we used eqs 1 (with n,' = 10, b,,, = 10.90 A, i,,, = 40, r = 12.3 A, rH = 1.5 A, u = 22.95 cal mol-', and D = 838 cal mol-'), 3, 4, mol-' A-*, y = 13.3 cal and 28 (with V I = 46.1 1 cm3 and V, = 20.36 cm3). Very close values were obtained from models 1-5 (Table IV) and models A-E (Table V). Model 6 predicts slightly different values (data not shown). The dimerization constants K 2 can be calculated from K2 = exp(-2AG20/RT)
(33) The K 2 values estimated are shown in Tables IV and V. These K2 values are close to that for model 5 and the SS value for model 5 is as small as the other models except model 6 (Tables IV and V). These facts show that premicelles including dimer are formed. The same conclusion has also been reached from derivative elution GFC patterns. Figure 7c shows some derivative patterns simulated on the basis of model DslSy. The observed difference between
,
C(mmol dmi3
Figure 9. Concentration dependence of n, values calculated on the basis of models Dol57 (solid tine), A o 3 h (dashed line), and Ds15a (dot-dash line) in Table V.
the derivative patterns at the leading and trailing boundaries is reproduced by models A-E (Table V) and models 1-5 (Table IV), although not reproduced by model 6 (Figure 7b). Simulations of the derivative GFC patterns have been carried out on the basis of plate theory, as already reported.+l' The number N of plates and void volume V, were regarded as adjustable parameters. In particular, N was estimated by using the derivative patterns at concentrations much lower than the cmc, by a trial-and-error method. The values of N and Voare not very important parameters for this work. Figure 9 shows the concentration dependence of n, calculated on the basis of three models. The calculated n,value at 30.0 mmol dm-3 is shown in Tables IV and V. Models Dol57 and DplSy predict n, values very close to those predicted by model Dsl5y. This independence of n, from micelle shape is ascribed to a large i,,, value for model D (Table I). Since the ,,i value for model A (Table I) is small, the concentration dependence of n, is rather large (Figure 9) and the n, value at 30.00 mmol dm-3 markedly depends on micelle shape (Table V). The concentration dependence of n, for Dol57 is smaller than that for model Dsl5a. As i increases, ARidecreases and consequently the repulsion interaction is expected to increase. This effect is larger for model DolSy than for model DolSa, since the repulsive interaction for the former model is inversely proportional to the third power of ARi,instead of the first power for model Dsl5a. The ratio n,/n, is a measure of polydispersity in micelle size. This ratio, calculated on the basis of model DolSy, is shown by the dashed line in Figure 6 and has a maximum at C = 0.9986 mmol dm-3, beiig in good agreement with the observed value. The
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 2005
Application of Tanford's Micellization Theory
:
. 21
2 0 4 0 6 0 8 0 1 0 0 I
Figure IO. Micelle size distribution functions calculated on the basis of models A0307 (a), DslSy (b), DslSa (c), DolSy (d), and DplSy (e) at four CloE, concentrations (mmol dm-3), 0.6836 (0),0.9043 (A), 0.9986 (O), and 20.1046 (A),and model FolSy (f) at four C&8 conand 10.266 centrations (mmol dm-3), 0.826 (0),0.979 (A), 1.1 16 ),(.
(4. weight fraction wi (=ixi/x) of i-mer is shown as a function of i in Figure 10, where the calculations were carried out at four concentrations for six models. The concentration dependence of n, is closely related to the micelle size distribution function. That is, the broader the distribution function, the larger the concentration dependence of n, (compare Figure 9 with Figure 10). Below the cmc the population of dimer is greatest in the micelles. Above the cmc the distribution function becomes bimodal. As the total concentration is increased, the i value, iOpt,at the second peak gradually approaches n,. The effect of micelle shape on the distribution function appears at high concentrations. For instance, this effect appears at C = 20.1046 mmol dm-3 for models DslSy, DolSy, and Dpl5y. The order of width of the distribution is prolate > oblate > spherical micelles. Figure 10f shows the results calculated on the basis of model Fol5y for CloEs, which are comparable with those calculated from eq 17.6 Figure 11 shows the dependence of AGio, log Ki, and log ki on i, calculated on the basis of models DslSy, DolSy, and Dpl5y. The one-step aggregation constant Ki (eq 23) and stepwise aggregation constant ki can be calculated from
Ki = exp(-iAGio/RT)
(34)
ki = exp([(i - l)AGi-lo - iAGio]/Rg
(35)
The dependence of AGio on micelle shape is negligible a t small i values and becomes significant at i values greater than 75. The values of AGio have a shallow minimum at i = 63. The effect of micelle shape on log Ki appears a t i values greater than 90. Roughly speaking, log Ki changes linearly with i, but a slight deviation from linearity plays an important role in the determination of concentration dependence of n,. The value of log ki has a maximum at i = 31 for models DslSy and Dol57 and a t i = 34 for Dpl5y. The dependence of log ki on i a t large i values is small for model Dpl5y.
Discussion From an analysis of the concentration dependence of the centroid volumes, the concentration dependence of n, has been determined for CI0E7. These n, values increase rapidly with increasing concentration. This behavior is similar to that of C&8,6 but different from that of C10E6.9The most remarkable exper-
Figure 11. Dependence of AG?, log Ki, and log ki on aggregation number i, calculated on the basis of models DslSy (0),DolSy (A),and D P W (0).
imental result in this work is the formation of premicelles (Figures 5 and 7). Derivative GFC patterns can sensitively detect the appearance of some self-association (Figure 7). The ratio, h/n,,, has a maximum around C = 1.00 mmol dm-' (Figure 6). In general, when we investigate a self-associating system by transport methods such as chromatography and sedimentation velocity, the pressure or shear could disturb the equilibrium state of the system. Since the pressure difference between the inlet and outlet of the column was kept below -0.2 atm, this pressure effect will be negligible in this work. In fact our observed n, value (Figure 5 ) a t high concentrations is very close to a literature value of 61.14 Similar agreement between the n,values determined by GFC and the static light scattering method has been reported for some surfactant^^*^ and a drug.I0 Furthermore, the K2 value for the drug determined by GFC is in excellent agreement with the value obtained by spectrophotometry.I0 The present experimental results have been analyzed in the framework of the Tanford t h e ~ r y . The ~ , ~ strength of our approach stems from taking into account the concentration dependence of the micellar properties. The formation of premicelles has been quantitatively supported by a modification of the Tanford theory. The dimerization constant is 180 (3.2 dm3 mol-') for C10E7 and -400 (7.2 dm3 mol-') for CloEs. This difference in K2 has been ascribed mainly to the difference in rR, which affects the last term on the right-hand side of eq 1. The micelle size distribution function has two peaks, as shown in Figure 10. Such a bimodality has also been deduced from kinetic measurements of micellizat i ~ n . Whenever ~~ the size distribution function is bimodal, substantial amounts of premicelles must coexist together with monomer and micelles. In fact, we have shown that eq 22 (model 5 in Table IV) fits to the V, values (SSin Table IV) and to the derivative GFC patterns (Figure 7) better than eq 21 (model 6 in Table IV). Furthermore, model 5 differs apparently from the models shown in Table V. Nevertheless, it fits well to the V, and derivative GFC data, as do these complicated models. The K2 values for C1,,E7have been estimated by a number of micellization models shown in Tables IV and V. These K2 values are close to each other. By using the GFC and UV methods, we determined the K2 value for chlorpromazine hydrochloride; K2 = 0.124 (GFC) and 0.127 dm3 mmol-I (UV).Io This agreement in K2 between
-
(54) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielman, I.; Ulbricht, W.;Zana, R.; Lang, J.; Tondre, C. J . Phys. Chem. 1976, 80, 905.
2006 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992
Funasaki et al.
The validity of a micellization model is usually tested by a the two methods demonstrated the validity of our GFC method. comparison with the observed values of cmc and n, at the A K2 value of 250 dm3 mol-’ for SDS was estimated from an analysis of equivalent electric conductance data.55 However, a cmc.2,3,33,49 One of the problems is the theoretical definition of cmc. A few definitions were used: cmc satisfies the equation of later study based on the same technique and a different analysis suggested that no premicelle of SDS is formedSS6 There is still x I / x = 0.95,2 d2AGio/di2= 0,33 or d3xl/dx3 = 0.4-7 It was demonstrated that the cmc value significantly depends on these a matter of controvery about the formation of premicelles of definitions.‘+’ This results in a difficulty in rigorous comparison surfactants.’ This fact suggests that the K2 values are small for between theory and experiment. Among the above definitions most surfactants. We roughly assumed that the hydrocarbon core of cmc, the last definition will be b e ~ t . ~The - ~ second problem of the dimers of CIOE7and CIoEsis spherical. Recent theoretical studies showed that the ratio of gauche butane to trans butane is concerned with a comparison of n, at the cmc. The experimental in water is larger than that in the liquid and in organic ~ o l v e n t . ~ ~ ~ ~n,* value at the cmc is determined by extrapolation of experimental data at high concentrations to cmc. As Warr and White pointed,5 This prediction favors our spherical model for the dimers of C10E7 and CIOEB.Vold has proposed a different model for the structure the same extrapolation should be done to obtain a theoretical n, of the dimer of SDS and explained the K2 value of 250 dm3 mol-’ value at cmc. To our knowledge, there is no report on such a direct for SDS.59 comparison between theory and experiment. The last problem is the fitting procedure. The trial-and-error method is insufficient Recently Imae reported that CIOE7forms spherical monodisto test theory against experiment rigorously. In this work we have perse micelles (though this expression should not be accepted determined the values of C1and n, as a function of C. These data, literally) of n, = 61 and Rh = 26.7 A, independent of the total concentrati~n.’~ This conclusion contradicts the model considered however, are not used to fit theory, since they are secondary data above, since the maximum aggregation number of spherical Cl&7 derived from the V, values. All observed V, data are used to fit micelles should be less than 40 (Table I). To reconcile this the parameter values of the micellization models by nonlinear least-squares methods. The validity of the models may be judged discrepancy, though it is not seriously large, we must consider other from the SS values and the appropriateness of the parameter spherical micelle models or reinterpret her data. We have convalues. Judging from the SS values only, all models except model sidered micelles having smooth surfaces only. If micellar surfaces are rough by one ethylene oxide group, CI0E7could form a 6 shown in Tables IV and V will be good. For the reasons despherical micelle of i = 61 (e.g., see model H in Table I). A way scribed under Theoretical Background, we believe that model D for CIOE7and model F for CloEsare the most probable models to take into account the roughness of micellar surfaces is to set rH to be greater than 1.5 Ae2s3When the micellar surface of a in Table I. With respect to micelle shape, eq 1 predicts that oblate ellipsoids are more stable than prolate ellipsoids, at least in dilute poly(ethy1ene glycol) alkyl ether is rough, the OCH2CH2group solutions? As Table 111shows, the calculated Rhvalue for model adjacent to the a-CH2 group will contact (be embedded in) the Do is closer to the observed Rh value (26.7 A)I4 than those for hydrocarbon core. When CH30(CH2CH20)3His dissolved in dodecane, the excess molar volume is positive.23 This result the other models in Table I11 for C1&. Furthermore, we believe suggests that the contact of the ethylene oxide group with hythat eq 1 is more appropriate for poly(ethy1ene glycol) alkyl ethers drocarbon is thermodynamically unfavorable. Imae estimated than eq 14 and that rH = 1.5 A is best. Finally, model Dol57 the partial specific volume u of the C10E7 micelle from the Rh for CI0E7and model Fol5y for C&, would be the best models in Table V. As expected, the values of u and D for model Dol5y value.I4 According to this definition, u is estimated to be 1.686 are close to those for model Fol5y and the y value for model cm3 g-I. On the other hand, from volumetric data on related compounds,23we can estimate it to be 0.971 cm3 g-I. The former Dol57 is larger than that for model Fol5y. The u values for value may be termed an effective specific volume, which includes models Dol 5y and Fol5y are smaller than values suggested under the volume of hydrated (entrapped) water. This may lead to Theoretical Background. The reason for this minor difference reinterpretation of her data. Neither of these two possibilities will be that we have taken into consideration the formation of is excluded at the present. premicelles. Although models Dol 5y and Fol5y seem to be best for CIOE7and CloEs,the model for CI0E6,expected from these Tanford applied eq 1 to several oligo(ethy1ene glycol) dodec 1 models, is not well fitted to our experimental results for CloE6.6 ethers, employing values of n: = n, - 1, rH = 3.0 A, rR = 3.0 The Tanford theory may be modified for surfactants which form and u = 25 cal mol-’ A-2, but the calculated values of the cmc and n, were far from the observed values.2 By fixing values of rather large micelles. Several modifications of the Tanford theory have been proposed.5-7J3,39,41,42,50.60,61 A few theories take into n; = 9, i,,, = 16, r H = 3.0 A, rR = 13.0 A (model Ao30y), and consideration the intermicellar interaction^^^-^',^^ and have been u = 25 cal mol-‘ A-2, we obtained best-fit values of D = -524 cal mol-’ and y = 2.64 X lo8 cal mol-’ A4 and estimated value applied for the clouding p h e n o m e n ~ n .These ~ ~ ~theories ~ ~ ~ ~may ~ of SS = 0.0924 cm6, K2 = 8.00 X and n, (at C = 30.0 “01 be as useful as the Tanford theory. Studies on the hydration and conformations of poly(oxyethy1ene) chains provide important dm-3) = 5 1. Furthermore, the derivative GFC patterns simulated results which may be taken into account in such thermodynamic on the basis of this model are close to those shown in Figure 7b. Judging from the SS values and derivative patterns, this model theories.634s is worse than any model for Cl0& shown in Table V. As is evident Registry NO. CIOE7, 39840-09-0. in Figure 8a and Table V, a value of rH = 3.0 A gives too small u values. The effect of the roughness of the micellar surface, if (60) Kjellander, R. J . Chem. SOC.,Faraday Trans. 2 1982, 78, 2025. it is present, may be included in the D term of eq 1, although it (61) Gelbart, W. M.; Ben-Shaul, A.; McMullen, W. E.; Masters, A. J. is uncertain whether this is an adequate expression. Phys. Chem. 1984, 88, 861.
i,
(55) 1390. (56) (57) (58) (59)
Mukerjee, P.; Mysels, K. J.; Dubin, C. I . J . Phys. Chem. 1958, 62, Parfitt, G. D.; Smith, A. L. J . Phys. Chem. 1962, 66, 942. Pratt, L. R.; Chandler, D. J . Chem. Phys. 1977, 67, 3683. Jorgensen, W. L. J . Chem. Phys. 1982, 77, 5757. Vold, M . J. J . Colloid Interface Sci. 1987, 116, 129.
(62) Claesson, P. M.; Kjellander, R.; Stenius, P.; Christenson, H. K. J . Chem. SOC.,Faraday Trans. 1 1986,82, 2735. (63) Karlstrom, G. J . Phys. Chem. 1985,89, 4962. (64) Lindman, B.; Karlstrom, G. Z . Phys. Chem. (Munich) 1987, 155, 199. Lindman, B.; Soderman, 0.; Wennerstrom, H. In Surfacranr Soluzions; Zana, R., Ed.;Marcel Dekker: New York, 1987; Chapter 6. (65) Carlstrom, G.; Halle, B. J . Chem. SOC.,Faraday Trans. 1 1989,85, 1049.
