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The CMC-decreasing effects of some added alcohols on the aqueous sodium dodecyl sulfate solutions
The CMC-Decreasing Effects of Some Added Alcohols on the Aqueous Sodium Dodecyl Sulfate Solutions K E I S H I R O S H I R A H A M A AXD T A D A S H I K A S H I W A B A R A Department of Chemistry, Faculty of Science and Engineering, Saga University, Saga, Japan
Received July 6, 1970; accepted December 1, 1970 The critical micelle concentration (CMC) of sodium dodeeyl sulfate in water at 25°C was measured as a function of the concentration of the added aleohols (C~-C4) including all isomers. On addition of alcohols, the CMC decreases. The CMC depression becomes more marked, the stronger the hydrophobieity of the added alcohol. As a measure of the hydrophobieity of the alcohols, the free energy of transfer of an alcohol molecule from pure liquid to the extremely diluted aqueous solution was employed. The logarithm of the CMC-decreasing effect vs. this free-energy plot gives a line which expresses the CMC-deereasing effect in terms of a limiting slope of a log (CMC) vs. mole fraction of the added alcohol plot. In order to explain the above phenomena, the factors that affect the CMC decrease may be considered as reduction of the free energy of a micelle due to the diluted surface charge density or~ a mieelle, the hydrophobic interaction to the hydrocarbon exposed to water, and the entropy of mixing of a mixed micelle on addition of alcohol, and they contribute in this order. INTRODUCTION
more essential factor that correlates the observed CMC decreases with the properties of the added alcohols.
Since the critical mice]le concentration of a surfactant solution is a characteristic parameter that is clearly related to the free energy of mieellization (1), it is important and informative to study factors that govern the C~IC. Thus the dependence of the C M C upon temperature or pressure has given deep insight into the mieellar picture. Addition of some third substance to a mieellar solution is also known to affect the C M C (2--4). For example, Shinoda was successful in describing a linear relation between the CMC-deereasing power and the number of carbon atoms in an alcohol molecule for the systems of some soaps and added alcohols (C2-C10) (5). Ooshika also derived a similar result b y means of statistieaI thermodynamics. However, their treatments could not distinguish a slight difference in the CMC-deereasing effect among isomeric alcohols (7). I n the present study, we looked for a
EXPERIMENTAL Materials. The sodium dodeeyl sulfate (SDS) was synthesized by sulfation of a fractionally distilled dodecanol with concentrated sulfuric acid, followed by neutralization with sodium hydroxide. The crude SDS was extracted by butanol, reerystalSzed from butanol three times and washed with ether in a Soxhlet's extractor for 40 hours, and finally recrystaUized from water three times. The purity of the SDS is indicated by the C M C (8.27 mM) and the limiting equivalent conductivity A0 -73.6 at 25°C, both of which are comparable with the values found by Mukerjee et al. (8) (8.1 mM, A0 = 71.70) and b y Parfitt and Smith (9) (8.1 mM, Ao = 72.64). I t was considered that this sample was sufficiently pure for the present purpose. The ethanol was refluxed on dried calcium Journal of ColZoidand InterfaceScience.VoI.36, No. 1, May 1971
65
66
SttlRAtIAMA AND KASHIWABARA
oxide for several hours and distilled. The other alcohols were simply distilled. Apparatus and Procedures. The measurement of electric conductivity of the surfactant solution was made with a universal bridge, Model 4255A of Yokogawa HewlettPackard Co., Ltd., equipped with a handmade variable capacitor (C = 0-0.9999 #F). The electric conductivity cell of dipping type, of Towa Dempa Co., Ltd., Model CG-201PL, was bright-platinized and immersed in a big test tube (3¢ × 20 cm) which was kept at 25°C in a thermostat. At first, 15 ml solvent (pure water or mixed solvent) was poured into the test tube; then the resistivity of the solvent was measured (r larger than 106 ohm.cm). The concentration of SDS was varied by adding portions of a concentrated surfactant solution (about 30 raM), and the resistivity of each solution was measured in the same manner as the solvent. RESULTS AND DISCUSSION The CMC is defined as a break point on a specific conductivity vs. surfactant concentration plot. Two examples of such a plot are shown in Fig. 1, where the slope below the CMC is the same whatever the composition of the solvent is, but the slope
5 o
v 4
~3
.~ 2
1
J
I
~
I 10
SDS concentration, (mM)
FIG. 1. Specific conductivity v s . SDS concentration plots. Open circle: pure water; full circle: 1.56 wt % n-butanol solution. Temperature: 25°C. Journal of Colloid and Interface Science, VoI. 36, No. 1, May 1971
TABLE
I
T H E SLOPES ABOVE THE C]~/[C ON THE SPECIFIC CONDUCTIVITY VERSUS S D S CONCENTRATION PLOT
Alcohol concn (wt %)
Pure water n-Propanol 1.58
2.32 3.86 n-Butanol 0.80 1.56 2.27 Below the CMC
Slope (mho/mole)
0.026 0.031
0.034 0.043 0.037 0.044 0.052 0.066
above the CMC becomes steeper as the alcohol concentration increases, as seen also in Table I. It is clear that this effect is stronger for the alcohol with a larger a]kyl group. As the break on the specific conductivity vs. surfactant concentration plot is primarily attributable to the binding of the counterion onto the micelle surface, because of the high charge density, addition of alcohol may bring about a reduction of the surface charge density. The CFIC's thus obtained are shown in Table II and in Fig. 2, from which it is apparent that the more hydrophobic and the more concentrated the added alcohol, the more marked becomes the decrease in CMC. Ethanol and propanol have been known to raise the CMC at higher concentrations (2, 3, 7) than seen in Fig. 2, but here we are mainly interested in the concentration region where the CMC is depressed. To discuss this more quantitatively, the following model is proposed: Added alcohol molecules are partitioned between the bulk and the micellar phases, and a resultant entropy of mixing and a "diluted" surface charge density on the mixed micelle, and, in addition, the hydrophobic interaction between the solubilized alcohol and the hydrocarbon part of the micelle exposed to water cause a reduction of the free energy of the micelle, and the CMC is lowered. The chemical potentials of a singly dispersed surfactant and of a micellized
CMC OF SODIUM DODECYL SULFATE IN WATEI~ AT 25° C
67
TABLE II THE CMC's OF SDS IN WATERTO WHICHA SMALL AMOUNT OF ALCOHOLIS ADDED (TEMP 25°C) Alcoholcohen (wt %)
CMC (raM)
Pure water
8.27
Ethanol 2.48 3.83
6.88 6.76
5.94 n-Propanol
7
~G v
f~5
5.86
1.58
6.32
2.32
5.93
3.86 i-Propanol
4.79
1.51 2.95 4.32 n-Butanol 0.80
6.84 5.78 5.30
o
i
4
.3
i
7.14 5.88 5.30
2.77
5.15
4.44
4.84
s-Butanol 0.78 1.50 3.01 t-Butanol 0.72 1.46 2.22
I
I
5
6
panol; n, t, s, and i signify respective butyl alcohol isomers. Full circles were used merely to avoid confusion.
3.06
1.55
I
4
Alcohol weight percent
4.05
2.27 /-Butanol 0.33 0.79
I
3
FIG. 2. The CMC's of SDS in mixed solvents (25°C). ( / ) - - e t h a n o l , (2)--i-propanol, (3)--m-pro-
5.39
1.56
2
low the CMC, X~ is the mole fraction of the surfaetant in a mixed micelle, f~ is the activity coefficient, e is an electronic charge, and 40 is an electric potential of the micelle, taking the potential at infinity from the mieelle as zero. The other notations have the usual meanings. Equating Eqs. [1] and [2], after a slight rearrangement we have
6.40 5.34 4.63 6.92 5.78 4.65
in C~ = (½) (ln XJ~ + ®o +
surfactant are expressed as ~, = ~0 ÷ 2kT in C,
[1]
tt,~ = t~ ° + /cT In X~f, + e4~o,
[2]
and where v o and t~,~° are the standard chemical potentials of the singly dispersed, and of the mieellized, surfaetants, respectively, and C, is the concentration of the singly dispersed surfactant, which should be an activity, b u t since a contribution from the activity coefficient to the total effect was found to a m o u n t to only 1.6%, as shown in Appendix A, the concentration was used. Here the factor of 2 before t h e / c T t e r m in Eq. [1] was introduced because SDS can be regarded as a 1:1 strong electrolyte be-
,X~°/1~T)
[3]
where q)0 = eeoo/lcT a reduced electric potential and A~ ° = # 0 _ #o. Now C, is equal to the CMC. To obtain the mole fraction of the alcohol in the mieelle phase a partition eoeffeient K of the alcohol between the bulk and the mieelle phases is now introduced.
K
=
Xo/Y~
=
exp
(-AF°/RT).
[41
I n this equation Xa and Y~ are the mole fractions of the Meohol in the mixed mieelle and the bull; phases respectively and AF ° is the standard free-energy change for the partition process. Now Eq. [3] is rewritten in t e r m of YaK with the relation Xa -t- X, = 1 in mind: In C~ = (1~) [ln (1 - KYo)
+
lnf~ +
~0 +
Journal of Colloid and Interface Science,
a~O/kT].
[51
VoI. 36, No. 1, May 1971
68
SHIRAHAMA AND KASHIWABARA
Differentiation of the above equation with respect to Y~ gives din C,/d Y. = ( ~ ) [ - K / ( 1
- KYa)
+ d l n L / d Ya + d ¢o/d Y~]. [6] The din f~/d Ya in Eq. [6] is a term that describes the stabi]ization of micelle on solubilization of alcohol molecules. Among effects a hydrophobic interaction of the alcohol molecule with the hydrocarbon part of micelles exposed to water may be important. Now lnf~ is expanded around i n f , 0 , the logarithm of an activity coefficient at Xa = 0, lnf~ = In f,0 + 5X~.
fi/d Y~ = - ~ K .
[8]
As for the variation of the electrostatic contribution to the free energy, we have the relation, d ~0/d Yo
:-
(d ¢o/d ~)(d ~/d Yo).
AFO = #o _
[9]
The d ¢0/d ~ = 0.020 is obtainable in Appendix B. And if ~ = ~0(1 -- KYa) is assumed we have d ~/d Y~ = - ¢ 0 K where ~o = 106, which corresponds to a reduced surface potential ~o -- 4.4 (10). Substitution of Eqs. [7], [8], and [9] into [6] gives at the limit Y~ --+ 0
The value of ~ is difficult to evaluate, but tentatively assigned as - 7 0 0 cal/mole = --1.18RT, which is comparable with the standard free-energy change at 25°C of the hydrophobic bond between two methyl groups in a]anine as deduced by Nemethy and Scheraga (ii).
Then, Eq. [10] becomes dln C~/d Y~ = --2.15K.
[11]
Journal of Colloid and Interface Science, Vol. 36, No. 1, May 1971
[13]
Note in Eq. [13] that the sign is reversed, because the solubilization is considered as the reverse process. In Fig. 3, in ( - 2 . 3 0 J ) is plotted against AF~r/RT, where J is the slope of log (CMC) vs. mole fraction of an alcohol Ya at infinite dilution, and the AF°,.'s are quoted from Buffer's papers (12, 13). From Fig. 3, it is apparent that the CMCdecreasing effect is well correlated with the AF°r's of the alcohols including all isomers. A parallel movement of the experimental line by 0.74RT unit would bring it into agreement with the theoretical line. In order to do so, 5 in Eq. [10] should be - 5 . 8 5 £ T units, which is five times the value assigned above, or some five methylene groups of a
A 5
[10]
[12]
Kt~ = exp (AF~/RT).
din C,/d Yo = - K ( 1 - ~ + 2.12)/2 = - K ( 3 . 1 2 - 3)/2.
Fao = R T l n f O ,
where f J is the activity coefficient of the alcohol at infinite dilution, when the pure alcohol is taken as the standard state. This free-energy change is a little different from that which appeared in Eq. [4] but is tentatively used:
[7]
Here the first two terms were retained, because only fl remains after differentiation and making Xa ---+O. Equation [7] can be rewritten in term of KY~ as lnf~ = Inf,0 + ¢~(1 -- KY~). Then we have din
In order to examine the applicability of Eq. [11], K should be known. However, since this quantity is inacessible, we employed the difference between the standard free energy ~a° in the alcohol-water binary mixture, and the free energy Fa °, of the pure alcoho], namely,
O O
B
~o
3
2
I
0
l
2
3
4
5
£Ftr/RT
FIG. 3. Correlation between the CMO-decreasing effect and the free energy of transfer. (A)-experimental, (B)--theoretical.
CMC OF SODIUM DODECYL SULFATE IN WATER AT 25° C SDS molecule are exposed to the water phase; this is unlikely (14). Thus, it, may be concluded that a probable line must fall somewhere around the A line in Fig. 3. The solubilized alcohol molecules are inferred to lie in thermodynamically similar surroundings as in pure alcohol. FinMly, it is referred to the intentionally omitted surface free-energy term in Eq. [2]. A mieelle is considered as a monolayer entity whatever the micellar model, for example, the lamellar micelle is a bi-monolayer, and the spherical micelle is a monolayer with a curvature. Then the distinction between the surface part and the bulk part of a micelle becomes meaningless. So, the surface free energy, if any, may be considered to be included in t*~° in Eq. [2]. There is a question as to how the ~ 0 term varies on addition of alcohol. However, possible factors that significantly affect the CMC were explicitly expressed in the preceding discussions. Then, t,~ ° can be regarded as a constant.
69
1.8, corresponds to 1.6% of the total contribution. The use of concentration in Eq. [1] in place of activity may be allowable. (B), For a spherical colloid, the electric potential obeys the following differential e quation: 1
d
d~
where q) = e ¢ / k T , a reduced electric potential, r = KR, R is the distance from the center of the particle, ~ = ~/8-~72/DtcT, the Debye-Huckel parameter, n being the number of ions in 1 cm a, and D is the dielectric constant of the medium. Although Eq. [A4] cannot be analytically solved, numerical solutions have been obtained (15, 16). The values of ~ = - [ 7 / ( 1 + r) G0] (d¢/dr)~=~ in Hoskin's paper (15) were extrapolated to r = 0.5, where r = aK, a being a radius of the colloidal sphere; for SDS, the micellar radius a is estimated as 16.6 ~, (17), and 1/K at CMC is 33.0 ~, and is tabulated in Table III. APPENDIX The surface potential q)0 and the (dimen(A) The neglect of the activity coeffi- sionless) surface charge density ~ are recient term in Eq. [1] is justified in the following manner. TABLE III We are concerned with the variation of the activity coefficient term on addition of THt~ V.~LVJ~SOF v = --It/(1 + r)aSol(d~/dr)~=~ ±T = 0.5 alcohol. Then, we have dlnf/Y,~ = ( d l n f / d x / 5 ~ ) ( d v / ~ / d Y ~ ) ,
[A1]
where f is the neglected activity coefficient, which is approximated by the DebyeHuckel limiting law, that is, In,( = - 0 . 5 1 V / ~ / 2 . 3 0 at 25°C for a 1:1 strong electrolyte. Therefore, d l n f / d % / ~ -- - 0.22.
O0
1
v
2
4
6
8
1.015 1 . 0 5 4 1 . 2 2 5 1 . 5 8 0 1.980 19.35 40.26 9 3 . 5 8 1 8 1 . 0 455.3 8 ~--
[A2]
6
5
On the one hand, we have from Fig. 2, for
~ 4
4
n-butanol,
[A3]
2
Then the value of Eq. [All is 1.8. On the other hand, Ktr for n-butanol is 52.9. Substituting these figures into Eq. [11], we have 113.7 as a total contribution to the CMC decrease, The value evaluated by Eq. [All, that is,
o
dV/~/dY~
= --8.2.
32 x
[ o
loo
200
300
4~o
6 FIG. 4. The reduced surface potentiM Ooand its derivative d,~o/do vs. the dimentionless surface charge density ~. Journal of Colloid and Interface Science, Vol. 36, No. 1, May 1971
70
SHIRAHAMA AND KASHIWABARA
lated by ¢ =
--(D/4~r) (d~o/dr)~=r -- - ( D / 4 v ) [ ( 1 + ~)/~1 " ~0, [A5]
w h e r e , d e s i g n a t e s t h e figure in T a b l e I I I . N o w , it is e a s y to o b t a i n a n e m p i r i c a l f o r m u l a t h a t connects ~0 w i t h ~. ~0 = 0.1462 X 1 0 - % 3 - - 0.1499 X 10-~z ' 4- 0.5553 X l O - l e -- 0.6682
X 10 -2.
[A6]
F r o m E q . [A6], (d ~0/d~) was c a l c u l a t e d a n d s h o w n in Fig. 4, where t h e surface p o t e n t i a l , ~0 was also seen. T h u s , n e c e s s a r y figures in E q . [9] can be r e a d f r o m Fig. 4. REFERENCES 1. MUKERJEE, P., Advan. Colloid Interface Sei., 1,241 (1967). 2. SHIRAHAMA,K., HAY.&SHI,M., AND MATUURA, R., Bull. Chem. Soe. Jap. 42, 1206 (1969); ibid. 42, 2123 (1969). 3. EMERSON, 1M:.F., AND I-IOLTZER, A., J. Phys. Chem. 71, 3320 (1967). 4. SHINODA,K., NAKAGAWA,T., TAMAMUSItI,]3., AND ISEMURA, W., "Colloidal Surfactants-Some Physicochemical Properties," p. 58. Academic Press, New York, 1963.
Journal of Colloidand InterfaceScience,Vol. 36, No. 1, May 1971
5. SHINODA-,K., Bull. Chem. Soc. Jap. 26, 101 (1953). 6. OOSHIKA,Y., J. Colloid Sei. 9,254 (1954). 7. FLOCKHART, B. D., AND UBBELOHDE, A. R., J. Colloid Sei. 8,428 (1953). 8. MUKERJEE, P., ~k/[YSELS,•. J., AND DULIN, C. I., J. Phys. Chem. 62, 1390 (1958). 9. PARFITT, G. D., ANE SMITH, A. L., Trans. Faraday Soc. 61, 2736 (1965). 10. MUKERJEE, P., Advan. Colloid Interface Sei. 1,271 (1967). 11. NEMETH¥, G., AND SCHERAGA,H. A., or. Phys. Chem. 66, 1773 (1962). 12. BUTLER, J. A. V., THOMSON, D. W., AND MACLENNAN, W. H., or. Chem. Soc. 1933, 674. 13. BUTLER, J. A. V., RAMCHANDANI,C. N., AND TI~OMPSON, D. W., J. Chem. Soc. 1935, 28. 14. MUEERJEE, P., Advan. Colloid Interface Sci. 1,268 (1967). 15. HosxlN, N. E., Trans. Faraday Soc. 49, 1471 (1953). 16. LOEB, A. L., OVERBEEK, J. Th. G., AND WIERSEMi, P. H., "The Electric Double Layer around a Spherical Colloid Particle." M. I. T. Press, Cambridge, Massachusetts, 1961. The book arrived at the author's laboratory just before sending in this manuscript. The calculated results therein are, of course, in agreement with those in reference 15 as well as with those in Table III. 17. STIGTER,D., J. Phys. Chem. 68, 3603 (1964).
Received July 6, 1970; accepted December 1, 1970 The critical micelle concentration (CMC) of sodium dodeeyl sulfate in water at 25°C was measured as a function of the concentration of the added aleohols (C~-C4) including all isomers. On addition of alcohols, the CMC decreases. The CMC depression becomes more marked, the stronger the hydrophobieity of the added alcohol. As a measure of the hydrophobieity of the alcohols, the free energy of transfer of an alcohol molecule from pure liquid to the extremely diluted aqueous solution was employed. The logarithm of the CMC-decreasing effect vs. this free-energy plot gives a line which expresses the CMC-deereasing effect in terms of a limiting slope of a log (CMC) vs. mole fraction of the added alcohol plot. In order to explain the above phenomena, the factors that affect the CMC decrease may be considered as reduction of the free energy of a micelle due to the diluted surface charge density or~ a mieelle, the hydrophobic interaction to the hydrocarbon exposed to water, and the entropy of mixing of a mixed micelle on addition of alcohol, and they contribute in this order. INTRODUCTION
more essential factor that correlates the observed CMC decreases with the properties of the added alcohols.
Since the critical mice]le concentration of a surfactant solution is a characteristic parameter that is clearly related to the free energy of mieellization (1), it is important and informative to study factors that govern the C~IC. Thus the dependence of the C M C upon temperature or pressure has given deep insight into the mieellar picture. Addition of some third substance to a mieellar solution is also known to affect the C M C (2--4). For example, Shinoda was successful in describing a linear relation between the CMC-deereasing power and the number of carbon atoms in an alcohol molecule for the systems of some soaps and added alcohols (C2-C10) (5). Ooshika also derived a similar result b y means of statistieaI thermodynamics. However, their treatments could not distinguish a slight difference in the CMC-deereasing effect among isomeric alcohols (7). I n the present study, we looked for a
EXPERIMENTAL Materials. The sodium dodeeyl sulfate (SDS) was synthesized by sulfation of a fractionally distilled dodecanol with concentrated sulfuric acid, followed by neutralization with sodium hydroxide. The crude SDS was extracted by butanol, reerystalSzed from butanol three times and washed with ether in a Soxhlet's extractor for 40 hours, and finally recrystaUized from water three times. The purity of the SDS is indicated by the C M C (8.27 mM) and the limiting equivalent conductivity A0 -73.6 at 25°C, both of which are comparable with the values found by Mukerjee et al. (8) (8.1 mM, A0 = 71.70) and b y Parfitt and Smith (9) (8.1 mM, Ao = 72.64). I t was considered that this sample was sufficiently pure for the present purpose. The ethanol was refluxed on dried calcium Journal of ColZoidand InterfaceScience.VoI.36, No. 1, May 1971
65
66
SttlRAtIAMA AND KASHIWABARA
oxide for several hours and distilled. The other alcohols were simply distilled. Apparatus and Procedures. The measurement of electric conductivity of the surfactant solution was made with a universal bridge, Model 4255A of Yokogawa HewlettPackard Co., Ltd., equipped with a handmade variable capacitor (C = 0-0.9999 #F). The electric conductivity cell of dipping type, of Towa Dempa Co., Ltd., Model CG-201PL, was bright-platinized and immersed in a big test tube (3¢ × 20 cm) which was kept at 25°C in a thermostat. At first, 15 ml solvent (pure water or mixed solvent) was poured into the test tube; then the resistivity of the solvent was measured (r larger than 106 ohm.cm). The concentration of SDS was varied by adding portions of a concentrated surfactant solution (about 30 raM), and the resistivity of each solution was measured in the same manner as the solvent. RESULTS AND DISCUSSION The CMC is defined as a break point on a specific conductivity vs. surfactant concentration plot. Two examples of such a plot are shown in Fig. 1, where the slope below the CMC is the same whatever the composition of the solvent is, but the slope
5 o
v 4
~3
.~ 2
1
J
I
~
I 10
SDS concentration, (mM)
FIG. 1. Specific conductivity v s . SDS concentration plots. Open circle: pure water; full circle: 1.56 wt % n-butanol solution. Temperature: 25°C. Journal of Colloid and Interface Science, VoI. 36, No. 1, May 1971
TABLE
I
T H E SLOPES ABOVE THE C]~/[C ON THE SPECIFIC CONDUCTIVITY VERSUS S D S CONCENTRATION PLOT
Alcohol concn (wt %)
Pure water n-Propanol 1.58
2.32 3.86 n-Butanol 0.80 1.56 2.27 Below the CMC
Slope (mho/mole)
0.026 0.031
0.034 0.043 0.037 0.044 0.052 0.066
above the CMC becomes steeper as the alcohol concentration increases, as seen also in Table I. It is clear that this effect is stronger for the alcohol with a larger a]kyl group. As the break on the specific conductivity vs. surfactant concentration plot is primarily attributable to the binding of the counterion onto the micelle surface, because of the high charge density, addition of alcohol may bring about a reduction of the surface charge density. The CFIC's thus obtained are shown in Table II and in Fig. 2, from which it is apparent that the more hydrophobic and the more concentrated the added alcohol, the more marked becomes the decrease in CMC. Ethanol and propanol have been known to raise the CMC at higher concentrations (2, 3, 7) than seen in Fig. 2, but here we are mainly interested in the concentration region where the CMC is depressed. To discuss this more quantitatively, the following model is proposed: Added alcohol molecules are partitioned between the bulk and the micellar phases, and a resultant entropy of mixing and a "diluted" surface charge density on the mixed micelle, and, in addition, the hydrophobic interaction between the solubilized alcohol and the hydrocarbon part of the micelle exposed to water cause a reduction of the free energy of the micelle, and the CMC is lowered. The chemical potentials of a singly dispersed surfactant and of a micellized
CMC OF SODIUM DODECYL SULFATE IN WATEI~ AT 25° C
67
TABLE II THE CMC's OF SDS IN WATERTO WHICHA SMALL AMOUNT OF ALCOHOLIS ADDED (TEMP 25°C) Alcoholcohen (wt %)
CMC (raM)
Pure water
8.27
Ethanol 2.48 3.83
6.88 6.76
5.94 n-Propanol
7
~G v
f~5
5.86
1.58
6.32
2.32
5.93
3.86 i-Propanol
4.79
1.51 2.95 4.32 n-Butanol 0.80
6.84 5.78 5.30
o
i
4
.3
i
7.14 5.88 5.30
2.77
5.15
4.44
4.84
s-Butanol 0.78 1.50 3.01 t-Butanol 0.72 1.46 2.22
I
I
5
6
panol; n, t, s, and i signify respective butyl alcohol isomers. Full circles were used merely to avoid confusion.
3.06
1.55
I
4
Alcohol weight percent
4.05
2.27 /-Butanol 0.33 0.79
I
3
FIG. 2. The CMC's of SDS in mixed solvents (25°C). ( / ) - - e t h a n o l , (2)--i-propanol, (3)--m-pro-
5.39
1.56
2
low the CMC, X~ is the mole fraction of the surfaetant in a mixed micelle, f~ is the activity coefficient, e is an electronic charge, and 40 is an electric potential of the micelle, taking the potential at infinity from the mieelle as zero. The other notations have the usual meanings. Equating Eqs. [1] and [2], after a slight rearrangement we have
6.40 5.34 4.63 6.92 5.78 4.65
in C~ = (½) (ln XJ~ + ®o +
surfactant are expressed as ~, = ~0 ÷ 2kT in C,
[1]
tt,~ = t~ ° + /cT In X~f, + e4~o,
[2]
and where v o and t~,~° are the standard chemical potentials of the singly dispersed, and of the mieellized, surfaetants, respectively, and C, is the concentration of the singly dispersed surfactant, which should be an activity, b u t since a contribution from the activity coefficient to the total effect was found to a m o u n t to only 1.6%, as shown in Appendix A, the concentration was used. Here the factor of 2 before t h e / c T t e r m in Eq. [1] was introduced because SDS can be regarded as a 1:1 strong electrolyte be-
,X~°/1~T)
[3]
where q)0 = eeoo/lcT a reduced electric potential and A~ ° = # 0 _ #o. Now C, is equal to the CMC. To obtain the mole fraction of the alcohol in the mieelle phase a partition eoeffeient K of the alcohol between the bulk and the mieelle phases is now introduced.
K
=
Xo/Y~
=
exp
(-AF°/RT).
[41
I n this equation Xa and Y~ are the mole fractions of the Meohol in the mixed mieelle and the bull; phases respectively and AF ° is the standard free-energy change for the partition process. Now Eq. [3] is rewritten in t e r m of YaK with the relation Xa -t- X, = 1 in mind: In C~ = (1~) [ln (1 - KYo)
+
lnf~ +
~0 +
Journal of Colloid and Interface Science,
a~O/kT].
[51
VoI. 36, No. 1, May 1971
68
SHIRAHAMA AND KASHIWABARA
Differentiation of the above equation with respect to Y~ gives din C,/d Y. = ( ~ ) [ - K / ( 1
- KYa)
+ d l n L / d Ya + d ¢o/d Y~]. [6] The din f~/d Ya in Eq. [6] is a term that describes the stabi]ization of micelle on solubilization of alcohol molecules. Among effects a hydrophobic interaction of the alcohol molecule with the hydrocarbon part of micelles exposed to water may be important. Now lnf~ is expanded around i n f , 0 , the logarithm of an activity coefficient at Xa = 0, lnf~ = In f,0 + 5X~.
fi/d Y~ = - ~ K .
[8]
As for the variation of the electrostatic contribution to the free energy, we have the relation, d ~0/d Yo
:-
(d ¢o/d ~)(d ~/d Yo).
AFO = #o _
[9]
The d ¢0/d ~ = 0.020 is obtainable in Appendix B. And if ~ = ~0(1 -- KYa) is assumed we have d ~/d Y~ = - ¢ 0 K where ~o = 106, which corresponds to a reduced surface potential ~o -- 4.4 (10). Substitution of Eqs. [7], [8], and [9] into [6] gives at the limit Y~ --+ 0
The value of ~ is difficult to evaluate, but tentatively assigned as - 7 0 0 cal/mole = --1.18RT, which is comparable with the standard free-energy change at 25°C of the hydrophobic bond between two methyl groups in a]anine as deduced by Nemethy and Scheraga (ii).
Then, Eq. [10] becomes dln C~/d Y~ = --2.15K.
[11]
Journal of Colloid and Interface Science, Vol. 36, No. 1, May 1971
[13]
Note in Eq. [13] that the sign is reversed, because the solubilization is considered as the reverse process. In Fig. 3, in ( - 2 . 3 0 J ) is plotted against AF~r/RT, where J is the slope of log (CMC) vs. mole fraction of an alcohol Ya at infinite dilution, and the AF°,.'s are quoted from Buffer's papers (12, 13). From Fig. 3, it is apparent that the CMCdecreasing effect is well correlated with the AF°r's of the alcohols including all isomers. A parallel movement of the experimental line by 0.74RT unit would bring it into agreement with the theoretical line. In order to do so, 5 in Eq. [10] should be - 5 . 8 5 £ T units, which is five times the value assigned above, or some five methylene groups of a
A 5
[10]
[12]
Kt~ = exp (AF~/RT).
din C,/d Yo = - K ( 1 - ~ + 2.12)/2 = - K ( 3 . 1 2 - 3)/2.
Fao = R T l n f O ,
where f J is the activity coefficient of the alcohol at infinite dilution, when the pure alcohol is taken as the standard state. This free-energy change is a little different from that which appeared in Eq. [4] but is tentatively used:
[7]
Here the first two terms were retained, because only fl remains after differentiation and making Xa ---+O. Equation [7] can be rewritten in term of KY~ as lnf~ = Inf,0 + ¢~(1 -- KY~). Then we have din
In order to examine the applicability of Eq. [11], K should be known. However, since this quantity is inacessible, we employed the difference between the standard free energy ~a° in the alcohol-water binary mixture, and the free energy Fa °, of the pure alcoho], namely,
O O
B
~o
3
2
I
0
l
2
3
4
5
£Ftr/RT
FIG. 3. Correlation between the CMO-decreasing effect and the free energy of transfer. (A)-experimental, (B)--theoretical.
CMC OF SODIUM DODECYL SULFATE IN WATER AT 25° C SDS molecule are exposed to the water phase; this is unlikely (14). Thus, it, may be concluded that a probable line must fall somewhere around the A line in Fig. 3. The solubilized alcohol molecules are inferred to lie in thermodynamically similar surroundings as in pure alcohol. FinMly, it is referred to the intentionally omitted surface free-energy term in Eq. [2]. A mieelle is considered as a monolayer entity whatever the micellar model, for example, the lamellar micelle is a bi-monolayer, and the spherical micelle is a monolayer with a curvature. Then the distinction between the surface part and the bulk part of a micelle becomes meaningless. So, the surface free energy, if any, may be considered to be included in t*~° in Eq. [2]. There is a question as to how the ~ 0 term varies on addition of alcohol. However, possible factors that significantly affect the CMC were explicitly expressed in the preceding discussions. Then, t,~ ° can be regarded as a constant.
69
1.8, corresponds to 1.6% of the total contribution. The use of concentration in Eq. [1] in place of activity may be allowable. (B), For a spherical colloid, the electric potential obeys the following differential e quation: 1
d
d~
where q) = e ¢ / k T , a reduced electric potential, r = KR, R is the distance from the center of the particle, ~ = ~/8-~72/DtcT, the Debye-Huckel parameter, n being the number of ions in 1 cm a, and D is the dielectric constant of the medium. Although Eq. [A4] cannot be analytically solved, numerical solutions have been obtained (15, 16). The values of ~ = - [ 7 / ( 1 + r) G0] (d¢/dr)~=~ in Hoskin's paper (15) were extrapolated to r = 0.5, where r = aK, a being a radius of the colloidal sphere; for SDS, the micellar radius a is estimated as 16.6 ~, (17), and 1/K at CMC is 33.0 ~, and is tabulated in Table III. APPENDIX The surface potential q)0 and the (dimen(A) The neglect of the activity coeffi- sionless) surface charge density ~ are recient term in Eq. [1] is justified in the following manner. TABLE III We are concerned with the variation of the activity coefficient term on addition of THt~ V.~LVJ~SOF v = --It/(1 + r)aSol(d~/dr)~=~ ±T = 0.5 alcohol. Then, we have dlnf/Y,~ = ( d l n f / d x / 5 ~ ) ( d v / ~ / d Y ~ ) ,
[A1]
where f is the neglected activity coefficient, which is approximated by the DebyeHuckel limiting law, that is, In,( = - 0 . 5 1 V / ~ / 2 . 3 0 at 25°C for a 1:1 strong electrolyte. Therefore, d l n f / d % / ~ -- - 0.22.
O0
1
v
2
4
6
8
1.015 1 . 0 5 4 1 . 2 2 5 1 . 5 8 0 1.980 19.35 40.26 9 3 . 5 8 1 8 1 . 0 455.3 8 ~--
[A2]
6
5
On the one hand, we have from Fig. 2, for
~ 4
4
n-butanol,
[A3]
2
Then the value of Eq. [All is 1.8. On the other hand, Ktr for n-butanol is 52.9. Substituting these figures into Eq. [11], we have 113.7 as a total contribution to the CMC decrease, The value evaluated by Eq. [All, that is,
o
dV/~/dY~
= --8.2.
32 x
[ o
loo
200
300
4~o
6 FIG. 4. The reduced surface potentiM Ooand its derivative d,~o/do vs. the dimentionless surface charge density ~. Journal of Colloid and Interface Science, Vol. 36, No. 1, May 1971
70
SHIRAHAMA AND KASHIWABARA
lated by ¢ =
--(D/4~r) (d~o/dr)~=r -- - ( D / 4 v ) [ ( 1 + ~)/~1 " ~0, [A5]
w h e r e , d e s i g n a t e s t h e figure in T a b l e I I I . N o w , it is e a s y to o b t a i n a n e m p i r i c a l f o r m u l a t h a t connects ~0 w i t h ~. ~0 = 0.1462 X 1 0 - % 3 - - 0.1499 X 10-~z ' 4- 0.5553 X l O - l e -- 0.6682
X 10 -2.
[A6]
F r o m E q . [A6], (d ~0/d~) was c a l c u l a t e d a n d s h o w n in Fig. 4, where t h e surface p o t e n t i a l , ~0 was also seen. T h u s , n e c e s s a r y figures in E q . [9] can be r e a d f r o m Fig. 4. REFERENCES 1. MUKERJEE, P., Advan. Colloid Interface Sei., 1,241 (1967). 2. SHIRAHAMA,K., HAY.&SHI,M., AND MATUURA, R., Bull. Chem. Soe. Jap. 42, 1206 (1969); ibid. 42, 2123 (1969). 3. EMERSON, 1M:.F., AND I-IOLTZER, A., J. Phys. Chem. 71, 3320 (1967). 4. SHINODA,K., NAKAGAWA,T., TAMAMUSItI,]3., AND ISEMURA, W., "Colloidal Surfactants-Some Physicochemical Properties," p. 58. Academic Press, New York, 1963.
Journal of Colloidand InterfaceScience,Vol. 36, No. 1, May 1971
5. SHINODA-,K., Bull. Chem. Soc. Jap. 26, 101 (1953). 6. OOSHIKA,Y., J. Colloid Sei. 9,254 (1954). 7. FLOCKHART, B. D., AND UBBELOHDE, A. R., J. Colloid Sei. 8,428 (1953). 8. MUKERJEE, P., ~k/[YSELS,•. J., AND DULIN, C. I., J. Phys. Chem. 62, 1390 (1958). 9. PARFITT, G. D., ANE SMITH, A. L., Trans. Faraday Soc. 61, 2736 (1965). 10. MUKERJEE, P., Advan. Colloid Interface Sei. 1,271 (1967). 11. NEMETH¥, G., AND SCHERAGA,H. A., or. Phys. Chem. 66, 1773 (1962). 12. BUTLER, J. A. V., THOMSON, D. W., AND MACLENNAN, W. H., or. Chem. Soc. 1933, 674. 13. BUTLER, J. A. V., RAMCHANDANI,C. N., AND TI~OMPSON, D. W., J. Chem. Soc. 1935, 28. 14. MUEERJEE, P., Advan. Colloid Interface Sci. 1,268 (1967). 15. HosxlN, N. E., Trans. Faraday Soc. 49, 1471 (1953). 16. LOEB, A. L., OVERBEEK, J. Th. G., AND WIERSEMi, P. H., "The Electric Double Layer around a Spherical Colloid Particle." M. I. T. Press, Cambridge, Massachusetts, 1961. The book arrived at the author's laboratory just before sending in this manuscript. The calculated results therein are, of course, in agreement with those in reference 15 as well as with those in Table III. 17. STIGTER,D., J. Phys. Chem. 68, 3603 (1964).