J . Phys. Chem. 1990, 94, 323-326
323
Dependence of Critical Micelle Concentrations on Intensive Variables: A Reduced Variables Analysis Camillo La Mesa Department of Chemistry, Calabria University, Arcavacata di Rende, CS, 87030, Italy (Received: September 15, 1988; In Final Form: May 8. 1989)
An analysis of the relations between critical micellar concentrations (cmc's) and temperature, or between cmc's and applied
pressure, indicates that experimental data fit on U-shaped curves whose minimum is cmc*Tand Tc,or and P*,respectively. Irrespective of alkyl chain length, polar head groups, and counterions, cmc versus temperature data can be fitted in reduced equations keeping the form cmc(T)/cmc*T- 1 = 11 - T/T*]Y, where y is an exponent. Similar fits account for the relations between cmc's and the applied pressure. Some considerations on the thermodynamics of micelle formation are discussed according to the above statements. In particular, the reduced variables approach allows to determine accurate values for the entropy of micelle formation.
Introduction The relations between critical micellar concentrations, cmc's, and the temperature observed in ionic surfactant solutions are quite unusual compared to those for nonionic ones. As a rule, the critical micelle concentrations for the latter compounds regularly decrease on increasing T up to the critical solution temperature.' Conversely, ionic surfactant solutions show a more complex trend and cmc versus temperature data fit on U-shaped curves: the trend is well-documented in the l i t e r a t ~ r e . ~ ~ ~ No information was deduced from these phenomena, even is Skinner4 and Desnoyerss*6suggested that the enthalpy of micelle formation, AH",, can be temperature dependent. In the present paper we show that the occurrence of minima in cmc versus temperature plots is common to most ionic surfactants and that further cmc's can be forecast in a wide range of experimental conditions by a reduced variables fit of cmcrd versus T d . We show that a similar behavior is observed in the dependence of cmc's on applied pressure. Finally, we discuss some questions concerning the thermodynamics of micelle formation according to the statements suggested by the present approach. Experimental Section Muteriais. Potassium perfluorooctanoate (PFOK), dodecyltrimethylammonium bromide (DTABr), cetyltrimethylammonium bromide (CTABr), sodium dodecylbenzenesulfonate(DBzSO,Na), and sodium dodecyl sulfate (DS0,Na) were of the highest available purity, from Fluka or BDH. Lithium perfluorononanoate (PFNLi) was as in previous studies.'** The surfactants were dissolved in hot anhydrous ethyl alcohol, filtered, cooled, precipitated upon addition of cold acetone, and vacuum-dried at 90 "C for 1 day. Their purity was inferred from the absence of minima in surface tension versus In (molality) plots. Anhydrous ethyl alcohol and acetone were of analytical purity. Water was bidistilled, deionized, and degassed. The mother surfactant solutions were prepared by weight and stored at room temperature until use. Methods. The apparatus for measuring the surface tension, u, was a Kraus unit, operating with a calibrated platinum Du Noiiy ~~~
( I ) Becker, P. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1967, Chapter XV. (2) Flockhart, B. D. J . Colloid Sci. 1961, 16, 484. (3) Stead, J. A.; Taylor, H. J. Colloid Interface Sci. 1969, 30, 482. (4) Pilcher, G.; Jones, M. N.; Espada, L.; Skinner, H. A. J . Chem. Thermodyn. 1969, I 38 1 ; 1970, 2, I . ( 5 ) De Lisi, R.; Ostiguy, C.; Perron, G . ; Desnoyers, J. E. J . Colloid Znterface Sci. 1979, 71, 147. (6) Desnoyers, J. E.;Caron, G.; De.Lisi, R.; Roberts, D.; Roux, A.; Perron, G . J . Phys. Chem. 1983, 87, 1397, and references therein. (7) La Mesa, C. Ann. Chim. (Rome) 1987, 77, 93. (8) La Mesa, C.; Sesta, B. J. Phys. Chem. 1987, 91, 1450.
0022-3654/90/2094-0323$02.50/0
ring. The measuring vessel was thermostated within to f0.05 "C by circulating water. Depending on T and/or concentration, the accuracy on individual surface tension readings ranges from 0.20 to 0.35 mN m-'. The apparatus for measuring the electrical conductance was a Leeds and Northrup resistance bridge, connected to a Daggett-Krauss cell of about 0.25-L capacity. The cell was located in an oil bath, whose temperature is controlled to f0.005 OC by a Muller resistance thermometer. The cell constant was determined by calibration with standard KC1 solutions. Electrical conductance data were determined by adding to the liquid in the cell known amounts of the mother surfactant solution under test. Accurate x values were determined even at concentrations as low as 7 X 10" mol kg-I. The apparatus for measuring the optical turbidity of the surfactant solutions at temperatures close to the Krafft point9 has beeen described elsewhere.I0
Results ( a ) cmc's. According to Swarbrick et al.," the experimental critical micellar concentrations were determined from first-derivative plots of surface tension, da/d In m(y.-+ y+), versus In m(y- + y+), where the concentration is expressed in molality. The cmc's were obtained also from axlam versus m plots for systems studied by electrical conductance: the accuracy on cmc values is within f1.5% in the former and f0.8% in the latter case. No spurious effects due to the presence of solid surfactant did occur: this was confirmed by measuring the Krafft points, Tko's, for the systems considered here. Tkovalues were obtained from Tk = Tk"[l A exp(m - cmc)/a] (1)
+
where Tk is the clearing temperature of the solution containing m moles of surfactant; A and a are constants which depend on the system. Tkovalues by eq 1 are accurate to f0.6 "C. In some cases the experiments were run at T < Tko. No anomalies in the conductivity plots were observed, and it was possible to determine cmc values even in undercooled, metastable micellar solutions. The experiments were performed to complete data for homologue surfactant series, to support data from one source with respect to others, when some discrepancies were observed or when available data were scarce and incomplete. Selected cmc values at different temperatures for some compounds are reported in Figure la-c. They are in excellent (9) Shinoda, K. In Colloidal Surfacranrs; Academic Press: New York, 1963; Chapter I. (10) La Mesa, C.; Ranieri, G. A,; Terenzi, M. Thermochim. Acta 1988, 137, 143. ( 1 1) Molyneux, P.; Rhodes, C. T.; Swarbrick, J. J. Chem. SOC.1965, 61, 1043.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. I , 1990
324
'4
TABLE I: Minimum cmc Values, and Corresponding Temperatures, T * , for Some Ionic Surfactants, Calculated by Eq 2 and 3" surfactant 103cmc*T,m P,K ref OS0,Na 129.9 303.5 2 DeS0,Na 32.1 301.9 2 DS0,Na 8.16 300.4 2, this work 1 .03b this work DBsS03Na DMePy Br 8.28 289.9 3 DMePyCl 12.2 299.3 3 DTABr 14.1 297.3 this work
A
A U
z
U
A
A
I
CTABr PFNLi PFOK
A
A A
A A A
8
4
I
1
I
z i3
T
La Mesa
513
b /
0
81
0.88 3.82 26.2
accuracy close to the experimental uncertainty, higher terms in the series were not considered. It is also possible to write cmc's in reduced form according to In cmc(T)/cmc*, = In Cred= CBilT - T*'I cos (ir) I=I
i=l
1
I I
T
31 3
35+
0 V
s
0
U
*d
I
t
I
298
Cred
I T
3;8
Figure 1. Critical micellar concentrations, cmc's, in mol kg-I, as a function of temperature, in kelvin. The arrows at the bottom of the figures indicate the Krafft temperature, Tko.Full symbols refer to data from surface tension. Data in (a) refer to DS04Na in (b) to DTABr, and (c) to PFOK. agreement with available literature data.2,'2.13 ( 6 ) Determination of cmc*Tand P.cmc versus temperature data can be expressed according to the polynomial equation In cmc(T) = A - BT
+C
p - DT3 + E p
+ ...
(2)
where the cmc's are in molality, Tis the experimental temperature in kelvin, and A, B, C, etc., are constants which depend on the s ~ r f a c t a n t . ' ~Since fourth-order equations fit the data with an ( 1 2) (a) Shincda, K.; Soda, T. J . Phys. Chem. 1963,67, 2072. (b) Shinoda, K.; Katsura, K. J . Phys. Chem. 1964, 68, 1568. (13) Mukerjee, P.; Mysels, K. J. Critical Micellar Concentrations of Aqueous Surfactant Systems. Natl. Stand. Ref: Dota Ser. (US., Natl. Bur. Stand.) 1971, NSRDS-NBS 36. (14) Note: I n the case of dodec I surfactants ty ical values for A, B, C, D, and E constants are 10'-102. IOg, 104-10-, and IO-'-lO-*, respec-
-
=
- TredlY
(4)
where Td is the ratio T I P and y is an exponent whose numerical value is 1.74 f 0.03. This value was derived from a logarithmic regression fit and is independent of the system. The accuracy on and T* values is reflected from the overlapping of experimental data on a single curve. The scattering of data due to the experimental uncertainty on cmc values is indicated in Figure 2b, where it is also reported the modification of the reduced variables (KV) fit consequent with a small variation on y values. According to eq 4, it is possible to draw a unified plot for all systems considered here, as reported in Figure 2a. An approach similar to the one developed above can be used to rationalize cmc versus pressure data. Experimental values relative to dodecylpyridinium bromide (DPyBr) and to sodium dodecyl sulfate (DS0,Na) were inferred from the l i t e r a t ~ r e . ~ ~ - ~ ~ In the cmc-P plane the RV fit can be expressed as lcred
P
(3)
where B;s have the same meaning as constants B, C, etc., in eq 2, the term cos ( i r ) accounts for the odd-even distribution of sign in the series, and T and T* are the experimental and minimum temperature, respectively. Minimum cmc values, hereafter termed cmc*,, were calculated first by use of eq 2: the values were refined by an iterative procedure until convergence of eq 3 to zero. The uncertainty on cmc*,values and on P ones obtained in this way are f0.5-1.0% and f O . l "C, respectively. Data relative to PFOK, PFNLi, DTABr, CTABr, DS04Na, and DBzS03Na, as well as literature data relative to sodium octyl sulfate (OS04Na), sodium decyl sulfate (DeS04Na),*sodium dodecyl sulfate (DS04Na),2potassium perfluorooctanoate (PFOK),I2vi3and dodecyl-4-methoxypyridiniumbromide (D M ~ P Y B ~ ) ~ and chloride ( D M ~ P Y C I were ) , ~ analyzed by combining eq 2 and 3. The minima for each function, having coordinates cmc*, and P, respectively, were determined by routine computing programs and are reported in Table I . ( c ) Reduced Variables Fit. We have assumed that the points having coordinates cmc*, and T* are reference values for the micellization process: accordingly, we can fit the data in the equation
0
I 28 3
this work this work 1 1 , this work
With the only exception of DBzS03Na, cmc's were determined at six or more temperatures. bThe range which has been studied is between 303 and 318 K.
= EBi(-l)i17l - T*'(
0
tively.
302.1 3 13.2 315.3
-
=
l1 -
(5)
where Pd is the ratio P I P * , Crd has the same meaning as before, (15) Tanaka, M.; Kaneshima, S . ; Tomida, T.; Ncda, K.: Aoki, K. J . Colloid Interface Sei. 1973, 44, 525. (16) Nishikido, N.; Yoshimura, N.; Tanaka, M. J . Phys. Chem. 1980, 84, 5 5_ R _. _ (17) Brun, T. S.; Hailand, H.: Vikingstad, E. J . Colloid Interface Sci. 1978, 63, 8 9 , 590; 1979, 72, 59.
Dependence of cmc’s on Intensive Variables
The Journal of Physical Chemistry, Vol. 94, No. 1. I990
325
0
a
*+
mv
00
vA
Figure 3. Reduced variables fit of ICr, - 11 versus II - P J , calculated by eq 5 , for DPyBr. Experimental data are from ref 16. Black symbols refer to values at pressures higher than P*.
20 r
I
-VW 0
r
10
Figure 2. (a) Reduced variables fit of C, - 1 versus II - Trdl for the following systems: O S 0 4 N a (O), DeS0,Na ( 0 ) ,D S 0 4 N a (O), PFOK (A),PFNLi (A), DTABr ( O ) , CTABr (m), DMePyCl (V),and DMePyBr (v).To avoid the signal overlapping, the scattering of data has been slightly exaggerated. (b) Reduced variables fit of C, - 1 versus (1 - Tdl for D S 0 4 N a . The full line in the figure indicates the trend expected if the y exponent is 1.74; the two dotted lines indicate the upper and lower limits of the fit, calculated by assuming the uncertainty on the y value to be i2%. The bar on the point indicates the uncertainty on Crd.
and 6 is an exponent whose value is close to 2.8. A fit of cmc’s versus reduced pressure, calculated by use of eq 5, is reported in Figure 3.
100 0C.18319 Similar assumptions are valid for the relations between cmc’s and the applied pressure. According to the values reported in Table 1, both cmc*T and the related temperatures are dependent on surfactant hydrophilic-lipophilic balance: for instance, T* values for the alkyl sulfates are approximately a linear function of the alkyl chain length. In the data analysis we choose, when possible, homologous surface active agents, such as sodium alkyl sulfates and alkyltrimethylammonium bromides, and/or surfactants having different counterions (Table I). In this way any specificity due to the alkyl chain, polar groups, or counterion could be detected by , C versus Tred.plots. Perhaps, as indicated in Figure 2a, the RV fit does not indicate any peculiar effect and the data fit on the same curve. This means that, even if the role played by polar groups and counterions noticeably changes from system to system, the compensation between electrostatic effects and the hydrophobic contribution to micelle formation is nearly the same for all systems considered here. The present results refer to surfactants that are strong electrolytes: we do not know, at the present stage, whether the RV fit can be applied even to soaps or whether the temperature dependence of hydrolytic equilibria modifies their cmc versus temperature curves. The RV fit does not allow to distinguish between stable and metastable micellar solutions,” but this is exactly the same trend which has been experimentally observed (Figure la-c). Some thermodynamic aspects inherent with the present formal approach deserve further considerations. We shall discuss them by assuming cmc’s are saturation concentrations and the validity of the phase separation model (PSM) for micelle formation.2’,22 There is no reason, however, to reject other approaches: for instance, it is possible to account for the “charged phase separation” model for micelle f0rmation,2~provided an independent
*
Discussion The reduced variables approach obtained by eq 4 and 5 allows to forecast cmc values in a wide range of experimental conditions. We point out, however, that the accurate determination of either cmc*T and P ,or cmc*p and P*,is a prerequisite to obtain the overlapping of data on a single curve. Incidentally, we remark give rise to a pronounced that variations as low as 1% on splitting of the curve in two branches. The RV approach predicts reliable cmc’s for undercooled, metastable, micellar solutions and for temperatures higher than
(18) (a) Wightman, P. J.; Evans, D. F. J . Colloid Interface Sci. 1982, 86, 515. (b) Evans, D. F.; Ninham, B. W. J . Phys. Chem. 1983, 87, 5025. (19) Shinoda, K.; Kobayashi, M.; Yamaguchi, N. J . Phys. Chem. 1987, 91, 5292. (20) (a) Mazer, N. A.; Benedek, G. 8.; Carey, M. C. J . Phys. Chem. 1976, 80, 1075. (b) Franses, E. I.; Davis, H. T.; Miller, W. G.;Scriven, L. E. J . Phys. Chem. 1980, 84, 2413. (21) Shinoda, K.; Hutchinson, E. J . Phys. Chem. 1962, 66, 577. (22) Wennerstrom, H.; Lindman, B. Phys. Rep. 1979, 52, 1 . (23) Hall, D. G.J . Chem. SOC.,Faraday Trans. 2 1972, 68, 668.
326 The Journal of Physical Chemistry, Vol. 94, No. I , 1990
11 5
I
AS", = -R In cmc*? (8) where cmc*Tvalues were obtained by eq 2 and 3. The entropy of micelle formation for homologous surfactants, calculated by use of eq 8, is reported in Figure 4. The trend indicates that the hydrophobic contribution to the entropy of micelle formation is additive. The accuracy on AS", values calculated by eq 8 is extremely high (linear correlation function equal to 0.999 998). In cmc versus pressure studies the thermodynamic variable of interest is the volume change associated with micelle formation, AVO,. Since AVO, = RT(d In cmc/dP), (9)
1
1'5
NC
Figure 4. Dependence of In cmc*ron alkyl chain length, N,, for sodium alkyl sulfates. Black symbols refer to data calculated by eq 2 and 3.
determination of counterion binding is available at each temperat~re.~~ Within the limits set up by the PSM, the free energy of micelle formation is AGO, = R T In cmc = AH', - T A P ,
(6)
where the cmc is expressed in mole fraction units. It is possible to get either AH', and AS", by combining eq 6 with the Gibbs-Duhem equation for the enthalpy of micelle formation
AH', = [d(AG'm/T)/a(l/T)I,
La Mesa
(7)
According to eq 7, a plot of In cmc versus 1 / T will be a straight line of slope AH',/R if, and only if, AH", is independent of the temperature. Experimental evidence contradicts the above hypothesis and indicates that AH", can be greater than or less than zero, depending on T.25 This led some authors to discuss the energetics of micelle formation in terms of a temperature-dependent compensation between enthalpic and entropic contributions.26 Experimental studies show that the temperature at which AH', is zero is practically the same as It is reasonable, on these grounds, to consider the cmc*? a value at which the free energy of micelle formation is only due to entropic contributions. According to the above statements, it can written T*.4-6325-27
(24) Zana, R. J . Colloid Interface Sci. 1980, 78, 330. (25) Muller, N. In Micellization, Solubilization and Microemulsions;
Mittal, K. L., Ed.; Plenum Press: New York, 1977; Vol. I, p 229. (26) Jolicoeur, C . ; Phillips, P. Can. J . Chem. 1974, 52, 1834. (27) De Lisi, R.; Perron, G.: Desnoyers, J. E. Can. J . Chem. 1980.58.959.
if cmc versus pressure data fit on U-shaped curves, there must be a pressure at which AVO, becomes zero. Either eq 9 or the equation In [cmc(P)/cmc*,] = In C,, = ELIJcos (j I)TIPJ - P*J~
+
J= I
(10)
formally equivalent to eq 3, allows us to determine P* and the corresponding cmc*p. For DS04Na the pressure at which AVO, becomes zero, calculated by eq 9 and 10, is 1130 atm: literature data are close to 1200,28in good agreement with the computed value. Not much information is available on the pressure dependence of cmc's for homologous surfactants or amphiphiles having different counterions. However, values reported by Mukerjee et aLZ9 and those by Tanaka and co-workers in a monograph30 are in appreciably good agreement with the present statements.
Conclusions The reduced variables approach allows us to predict the dependence of cmc's on intensive parameters with appreciable accuracy. The results presented here indicate that information on some thermodynamic quantities for micelle formation can be obtained. Acknowledgment. M.P.I. is acknowledged for financial support. Registry No. PFOK, 2395-00-8; DTABr, I 1 19-94-4; CTABr, 5709-0; DBzSO,Na, 251 55-30-0 DSO,Na, 151-21-3; PFNLi, 6087 1-92-3. (28) (a) Kaneshina, S.; Tanaka, M.; Tomida, T.; Matuura, R. J . Colloid Interface Sci. 1974, 48, 450. (b) Tanaka, M.; Kaneshina, S.; Shin-no, K.; Okajima, T.; Tomida, T. J . Colloid Interface Sei.1974, 48, 132. (29) Sugihara, G.; Mukerjee, P. J . Phys. Chem. 1981, 85, 1612. (30) Tanaka, M.; Kaneshina, S.; Sugihara, G.; Nishikido, N.; Murata, Y. In Solution Behavior of Surfactants; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. I, p 41.
