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Department of Chemistry, Faculty of Science, Kyushu University, Fukuoka 812, Japan, and XDepartment of Education, Yamaguchi University, Yamaguchi 753, Japan Received January 30, 1975; accepted April 29, 1975 Critical micelle concentrations (CMC) have been theoretically calculated for mixed micelles between ionic and nonionic detergents, i.e., sodium dodecyl sulfate-octaoxyethylene dodecyl ether, sodium dodecyl sulfate-tetraoxyethylene octyl ether, sodium dodecyl sulfate-dodecaoxyethylene octyl ether, sodium pentadecyl sulfate-hexaoxyethylene decyl ether, and sodium dodecyl sulfate-hexaoxyethylene octyl ether systems in aqueous solution and sodium dodecyl sulfate-octaoxyethylene dodecyl ether, sodium decyl sulfate-dodecaoxyethylene octyl ether, and sodium dodecyl sulfate-sodium decyl sulfate systems in electrolyte solution. The CMC of all of the above mixtures showed the vat form with variation of the composition, except the first and the last, which gave a monotonously decreasing CMC with an increase of the amount of the component with lower CMC value. The CMC from the theoretical calculation was in fair agreement with the experimental results. The variation of CMC with composition could be explained by the assumption that the state of the mixed micelle is a linear combination of the state of pure ionic detergent and that of one ionic detergent molecule surrounded by infinite number of nonionic detergent molecules. All CMC values were obtained from the table and figures of the paper by Lange (4). INTRODUCTION

The observed CMC of mixed micelles of homologous detergents has been reported to be in good agreement with theoretical results (1-5), whereas reports of nonhomologous detergent mixtures differing in both hydrophobic and hydrophilic parts are very scanty (6). Indeed, the CMC of ionic-nonionic detergent mixtures has also been reported (4, 7, 8), but the interpretation of the results remains qualitative. When the CMC of an ionic detergent is much higher than that of a nonionic one, the C M C of the detergent mixture decreases rapidly with an increase of nonionic detergent from that of a pure ionic one to a C M C approximately equal to that of pure nonionic detergent and then comes asymptotically close to the latter value (7, 8).

I n this case an anionic-nonionic detergent mixture has a larger rate of decrease of C M C than a cationic-nonionic mixture. The initial rapid decrease of C M C comes from a decrease of electrical repulsion due to the insertion of a nonionic detergent molecule into an ionic micelle, while the latter change seems to be due to the strong interaction between a hydrophilic part of an anionic detergent and the oxygen atom of the ether bond of the nonionic detergent (9). On the other hand, when the CMC of an ionic detergent is approximately equal to that of a nonionic one, the C M C of the mixture takes a v a t form against composition of the mixture (4). The decrease of the CMC from that of a pure nonionic detergent due to the addition of ionic detergent comes from a propensity of the ionic detergent to

356 Journal of Colloid and Interface Science, Vol. 52, No. 2, August 1975

Copyright ~ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.

CMC OF IONIC-NONIONIC DETERGENTS penetrate into the nonionic detergent micelle (4). Indeed, these interpretations must be true, but the extent of these contributions to the mixed micelle formation has not been made quantitatively clear. Accordingly, in this paper, the theoretical interpretation of the CMC of ionic-nonionic detergent mixture is examined. The following abbreviations are used : SDeS, SDS, and SPS for sodium decyl, sodium dodecyl, and sodium pentadecyl sulfate, respectively;

C~(EO)~[C~H2~,O (CH2CH~O)mH-] for polyoxyethylene alkyl ether. THEORETICAL CONSIDERATION The following theoretical equation on the CMC of pure sodium alkyl sulfate can be given in line with the preceding paper, the micellar surface being assumed to be plane

(10): (I+K~) in C~=In ( l O 0 0 / N m v o ) - - m W ' / k T + K g In (2000~ra~/DNkT)+K - 1 [1-]

357

As for the pure nonionic detergent, polyoxyethylene alkyl ether, a similar equation can be derived, neglecting an electric term: in C~ = In (lO00/Nv) - m ' W ' / k T + nW'/kr

-- 1

[-3-]

where v is the partial free volume of detergent in the micelle; m t is the carbon number in the detergent; n is the number of oxyethylene group in the detergent; and W" is the free energy difference per oxyethylene group between the micellar and solution states. Next, we consider a mixed micelle formation between ionic and nonionic detergents. The physicochemical state of the ionic head group of the ionic detergent must be different, depending on the composition in the micelle. On the other hand, that of the nonionic detergent can be assumed to be unchanged; if changed all, it certainly is not changed as much as the ionic one. Then, the following two equations can be obtained concerning Ka and K: K~ = K~°x,~ + K~'(1 -- x,4

F4]

g = g ° x ~ + K'(1 -- x~),

[5-]

and

and X = 1.40 + 0.025 (T -- 298)

[-2"]

where Kg is the effective coefficient of electrical energy; Cm is the CMC of the detergent in molarity with carbon number m; N is Avogadro's number; and v0 is the partial molar volume of methylene group in micelle. According to the preceding paper, Nvo is given as 2.72 cm 3, W I is the free energy difference per methylene group including the water effect around hydrocarbon chain between micellar and solution states, k is Boltzmann's constant, T is the absolute temperature, a is the charge density at the micellar surface, and D is the dielectric constant of the intermicellar solution. In this paper the following equation is used, D = 78 -- 0.26(T --298), and K is the contribution of water around the ionic head group to the free energy change of micellization.

where x,~ is the mole fraction of ionic detergent in the micelle; K~° and K ° are Kg and K in the case of ionic detergent only, respectively; and Kg' and K ~ are Kg and K in the case that one ionic detergent molecule is surrounded by infinite number of nonionic detergent molecules in the micelle, respectively. The initial mole fraction of ionic detergent to total detergent x is almost equal to that just above CMC. On the other hand, Kg is defined as the ratio of the real surface electrical potential to the completely ionized hypothetical surface electrical potential (10). Thus, Kg also can be rewritten as Kg = In (200(br~/DNkTCx)/ in {2000~r(~ox~)2/DNkTCx}

[-6]

where ~0 is the completely ionized hypothetical surface charge density. The value of a0 can be

Journal of Colloid and Interface Science, Vol. 52, No. 2, August 1975

358

MOROI ET AL.

calculated from the CMC and Kg of pure ionic detergent (10). As for the critical concentration for mixed micelle formation C, which will be called mixed micelle concentration, Eqs. [13 and [33 are transformed into the following two equations: In Cx = In (lO00/Nmvo) -- m W ' / k T + Kg In (20007w2/DNkTCx) -SK--

1-51nx,~

[7]

and In C(1 - x) = In (lO00/Nv) - m ' W ' / k T -5 n W " / k T

-- 1 -5 In (1 -- x~).

[-83

Thus, the mixed micelle concentration can be obtained as the solution of simultaneous Eqs. ['4~-[8~, Kg' and K ' being parameters. The procedure to obtain the solution will be described in detail in the next section. The

solution of the simultaneous equations was obtained by a computer F A C O M 360-60. RESULTS AND DISCUSSION I t is necessary to know the practical values of m a n y variables in Eqs. [-4~-[-83 to obtain the mixed micelle concentration. Some of these values are available from our previous papers (6, 10). Hereafter the following numerical values are employed: Nvo = 2.72 cmS; W t / k T = 1.033 and 1.038; K o ° = 0.638 and 0.549; K = 1.40 and 1.78; and D = 78 and 74.4 (esu) 2 erg -1 cm -z at 25 and 40°C, respectively. A surface charge density (~) of the ionic detergent can be obtained from Eq. [-13 b y introducing the above values and the CMC. As for SDS, ~ and ~0 are calculated to be 2.95 X 104 and 1.56 X 105 (esu/cm2), respectively,

TABLE I C M C ' s USED IN THIS REPORT OF POLYOXYETHYLENE ALKYL ETHER AND FREE ENERGY DIFFERENCE PER OXYETHYLE1NVE GROUP BETWEEN MICELLAR AND SOLUTION STATES ( W " / k T ) W't/kT

CMC (mmole/1)

C8(EO)3

7.5

C8(EO)4 C8(EO)6 Cs(EO)0 Cs(EO)12

9.8 13

9.0

C10(EO) ~

0.60

C10(EO)6 C10(EO)9

0.9 1.3

C12(EO)6 Cn(EO)7 C12(EO) s

0.08~

C12(EO)9

C12(EO)1~ SDeS SDS SPS

7.0 8.5

0.090 8.5 (0.25 N NaCt)

0.80b

0.129

0.08 0.98 0.10~ 0.14~

0.06 (0.5 N NaC1)

0.16 6.1 (0.25 N NaC1) 8.0 1.0b

a At 23°C. b At 40°C. Journal of Colloid and Interface Science, Vol. 52, No. 2, August 197.5

3.7 (0.5 N NaC1) 0.41 (0.5 N NaC1)

0.115 0.113~

CMC OF IONIC-NONIONIC DETERGENTS at 25°C from the C M C value of 8.0 mmole/1. These values are almost equal to the previous results (10). As for SPS, o, and o0 are 2.05 X 104 and 1.44 X 105 (esu/cm~),jrespectively, at 40°C from the CMC value of 1.0 mmole/1. On the basis of these results the following calculation processes are applied to the mixed micelle concentration. W " / k T can be obtained from the plot of In CMC versus oxyethylene unit number of the nonionic detergent as expressed in Eq. [3-]. The C M C values used in this paper and those obtained from other reports (11, 12), together with W ' / k T values thus obtained, are shown in Table I.

(1) CMC of Detergant Mixtures with No Added Electrolytes. First of all, the details of the calculation will be shown when any other electrolyte is not added to an aqueous solution of ionic-nonionic detergent mixture. The method of the first C12(EO)s--SDS mixture can be applied for another four mixtures. With use of the above values and CMC values in Table I, Eqs. [4]-[-8] lead to the following equations at 25°C. K~ = 0.638xm + K~'(1 -- x,~) K = 1.40x~ 4- K ' (1 -- x~)

[-9] [-10"]

359

o x o.2 0,6 0,4

•

-5

-I0

0

0.5

1.0

Kg"

FIG. 1. Relationship between Kg' and K' satisfying the experimental CMC at each mole fraction of SDS (x) at 25°C. diverse values of K I against Ko I come to decrease at both sides of Kg' value. In fact this tendency can be seen for any other mixtures examined, in particular for Cs(EO)4-SDS and C s ( E O ) r S D S mixtures, as shown in Figs. 3 and 6. But it seems reasonable that the divergence of the K ~ value decreases with an increase of Kg ~, because Kg ~ is related to the dissociation of the ionic detergent in the micelle and the K~ ~ value goes to unity when the con-

K~ = In (3.25 × 10-~VCx)/

]n (3.25 × lO-%~xj/Cx) inCx=

Ell-]

-2

--9.98 4- Kg in (3.25 X 10-9o2/Cx)

+K+lnxm

[12-]

and in C(1 -- x) =

/

-3 - - 9.23 4 - ] n (1 - - x,~).

E13-]

I n these five equations there are nine variables, Kg, KJ, K, K ~, x, x,,, C, % and 00. Then four definite variables make the other five variables definite. As ~0 was obtained above, C M C can be obtained against each x value, K ~ and Ko t being parameters. In Fig. 1 are shown the relationships between K ~ and Kg / Satisfying the experimental results for a few mole fractions covering almost all contents. The

jo

-4

j O j

0 Mote

J° ,

~ °

0.5 Fraction of SDS

1.0

Fro. 2. Comparison of experimental with theoretical curve (K' = -- 10.7, KJ = 1.0) for C12(EO)s-SDS mixture at 25°C : (O) experimental value; ( ) theoretical curve.

Journal of Colloid and Interface Science, Vol. 52, No. 2, August 1975

360

MOROI E T AL.

K~ 0

0.5

1,0

0.5

1.0

0

10 -5 E

10

%

¥

-5 E

-10 5

:-10 i

00

0.5

1.0

-15

Mole Fraction of SDS

00

FIG. 3. Comparison of experimental with theoretical curve (K / = - - 9.8, Kg t = 1.0) for C8(EO)~-SDS mixture at 25°C and relationship between Kg ~ and K' satisfying the experimental CMC with divergency: (O) experimental value; ( ) theoretical curve.

1.0 Mole Fraction of SPS

FIG. 5. Comparison of experimental with theoretical curve (K' = - - 1 4 . 5 , Ko ~ = 1.0) for CI0(EO)rSPS mixture at 40°C and relationship between K J and K ~ satisfying the experimental CMC with divergency: (O) experimental value; ( ) theoretical curve.

K6 0.5

K6 1.00

0

0.5

1.0

I

0

t0

-5

-5 E 'o

-5

'£ t~

\

-10

~o

O

O

-10

0

0.5

-15 0

0

0

0'.5

1.0

Mole Fraction of SDS

Mole Fraction of SOS

FIG. 4. Comparison of experimental with theoretical curve (K r = - - 1 3 . 5 , Ka ' = 1.0) for Cs(EO)a2-SDS mixture at 25°C and relationship between Kg' and K' satisfying the experimental CMC with divergency: (O) experimental value; ( ) theoretical curve.

FIG. 6. Comparison of experimental with theoretical curve (K ~ = - - 11.4, Ka' = 1.0) for Cs(EO)8-SDS mixture at 25°C and relationship between Ka' and K t satisfying the experimental CMC with divergency: (O) experimental value; ( ) theoretical curve.

.lournal of Colloid and Interface Science, V o l . 5 2 , N o . 2, A u g u s t 1975

CMC OF IONIC-NONIONIC DETERGENTS centration of ionic detergent goes to infinite dilution. Furthermore, the Kgt value must be almost equal to unity as defined by Eq. [-4]. On the other hand, it is theoretically possible to determine the Kg/ value from the plot of In CMC versus the logarithm of concentration of gegenion at a small mole fraction of ionic detergent. This method was examined to determine the KgI value. But it was found to be unsuccessful, partly because the CMC value of the nonionic detergent only decreases with an increase of added electrolyte (4) and partly because the accuracy of the CMC value is not high enough to permit separation of the CMC decrease into the contributions of nonionic and ionic detergent at such a small mole fraction of ionic detergent. The results obtained using the Kg ~ value equal to unity are shown in Fig. 2. Excellent agreement between the experimental and theoretical CMC's suggests the assumptions made above to be reasonable. Similar results can be obtained for the other four mixtures, Cs(EO)s-SDS, Cs(EO)z2-SDS, C10(EO)rSDS, andC8 (EO)6SDS. These results are shown in Figs. 3-6, in which the most probable relationship between Kg~ and K ~ is shown together with the divergence. It can be seen from these figures that the K t value decreases with an increase of Kg ~ value. This indicates that a free energy decrease due to K ~ is requested for a degree of dissociation of ionic head group to increase in the mixed micelle. The K' value means the free energy difference of ionic head group between a bulk and mixed micellar states. Then the extent of K / value from --10 to - - 1 5 in kT units shows a free energy decrease of --6 to --9 kcal/mole, which implies a fairly strong interaction such as a small ionic bonding between the ionic head group and hydrophilic group of nonionic detergent in the mixed micelle (9). Furthermore, the variation of the K ' value is due to the position of ionic head group in the mixed micellar phase which depends on the balance between the lengths of hydrophilic and hydrophobic groups of detergents combined.

361

(2) CMC of Detergent Mixtures with Added Electrolyte. Many papers have been published on the effect of added electrolytes on the CMC of a pure detergent in aqueous solution, but there are few for detergent mixtures, despite increasing interest in micellar catalysis in the presence of added electrolyte. The procedure for obtaining the CMC of mixed micelle with an added electrolyte is similar to that of no added electrolyte. The value of W~/kT is almost constant because a linear relationship between In CMC and the logarithm of concentration of gegenion is valid up to the concentration of 0.3/1 equivalents of gegenion (13). Thus, the CMC decrease due to an added electrolyte is due to a decrease of W"/kT value, which indicates that the added electrolyte makes the interaction between the oxyethylene group and water diminish. At any rate the right-hand side value of Eq. [-3"] is equal to the logarithm of CMC with added electrolyte. Then, Eq. [-8-] for Cz2(EO)s in 0.5 N NaC1 solution leads to lnC(1 - - x) = in (6.0 X 10 -5 ) + In (1 - xm).

[-14"]

On the other hand, the CMC value of SDS in 0.5 N NaC1 solution gives the values of and ~0. With use of these values, Eq. [-7] in Cx = - 9.98 + Ko2 In {3.40x~/(0.5 + cx)} + K + lnx~.

[-15"]

The theoretical CMC value can be obtained from the solution of simultaneous Eqs. [-9-], [10"], [-14"], and [-15"] against each x value with K' and KJ as parameters. In Fig. 7 the relationships between K' and Ko' satisfying the experimental results covering almost all contents are shown. The diverse values of K' against K J leads to a decrease with decrease of KJ value. This tendency can also be seen for the Cs(EO)i2-SDeS mixture in 0.25 N NaC1 solution. This indicates that the degree of dissociation of ionic detergent is very small in the nonionic detergent at high gegenion con-

Journal of Colloid and Interface Science, Vol. 52, No. 2, August 1975

362

MOROI ET AL. -2 _3 0

0.5

1.0

o/ o/

/o/ o/

×

,

0

0.5

0/0/0 ..0/

1.0

1.0

0 o

E Io

-2

=5

-4

(J

00

0.5

0.5

1.0

Mole Fraction of 5 1 ~ S

tent. The theoretical curve of the CMC is shown in Fig. 7, in which there is excellent agreement with experimental results. A similar result obtained for Cs(EO)12-SDeS mixtures between K J and K ' is shown together with the divergence (Fig. 8). The extent of decrease of

0.5

I

0

FIo. 7. Comparison of experimental with theoretical curve (K' = - 0.1, Kg' = 0) for C12(EO)s-SDS mixture in 0.5 N NaC1 solution and relationship between Kgt and K' satisfying the experimental CMC at each mole fraction of SDS(x) at 25°C: (©) experimental value; ( ) theoretical curve.

0

--4

a9 ~i - 2

Mole Fraction of SDS

10

o/°/

3

0 -1 -

o/

-3

FIG. 9. Comparison of experimental with theoretical curve for SDS-SDeS mixture in 0.5 N NaC1 solution at 25°C: (O) experimental value; (.... ) theoretical curve.

the K ' value is small compared to the results with no added electrolyte, which means a decrease of the interaction of ionic head group and hydrophilic group of nonionic detergent. This result seems to come partly from the decrease in ionization of the ionic head group and partly from the fact that the water structure around the oxyethylene group is destroyed in a bulk state on addition of electrolyte. The theoretical CMC values of SDS-SDeS mixtures in 0.5 N NaC1 solution are also calculated according to the procedure given in (5). The results are shown in Fig. 9, in which excellent agreement between the theoretical and observed values is obtained. We can conclude from these results that the theoretical equations in this paper properly indicate the mechanism of mixed micelle formation of ionic-nonionic detergent mixtures with or without added electrolyte. REFERENCES

1.0

Mote Fraction of 5DeS

FIG. 8. Comparison of experimental with theoretical curve (K' = 0.9, Kg' = 0) for Cs(EO)I~-SDeS mixture in 0.25 N NaC1 solution and relationship between K' and Kg' with divergency at 25°C: (C)) experimental value; ( ) theoretical curve.

1. LARGE,H., Kolloid-Z. 131, 96 (1953). 2. SHINODA, K., J. Phys. Chem. 54, 541 (1954). 3. TOK~WA,F., OHM, K., AWOKOKUSO,I., Bull. Chem. Soe. Japan 41, 2845 (1968). 4. LANOE,H., ANDBECK, K. H., Kolloid-Z.Z. Potym. 251, 424 (1973).

Journal of Colloid and Interface Science, Vol. 52, No. 2, August 1975

CMC OF IONIC-NONIONIC DETERGENTS 5. MOROI, Y., MOTOMIYRA~K., AND MATUURA, R., J. Colloid Interface S d . 46, 111 (1974). 6. MoRoI, Y., NISHIKIDO, N., AND MATUURA, R., J. Colloid Interface Sci. 50, 344 (1975). 7. SCmCK, M. J., AND MANZ~INO, D. I , J. Amer. Oil Chem. Soc. 43, 133 (1966). 8. ScHIcK, M. 1., J. Amer. Oil Chem. Soc. 43, 681 (1966). 9. SCHWUGER, M. 1", J. Colloid Interface Sci. 43, 491 (1973). i0. MOR01, Y., NISIIlKIDO, N., UEIIARA, H., AND

363

MATUURA, R., Y. Colloid Interface Sd. 50, 254 (1975). 11. LANGE, H., in "Proceedings of the International Congress on Surface Activity," 3rd ed., Vol. 1, p. 279, Universitittsdruckerei Mainz Gmblt, Cologne, 1960. 12. BECHER, P., in "Nonionic Surfactants," (M. J. Schick, Ed.), p. 478. Marcel Dekker Inc., New York, 1967. 13. SmNDDA, K., NAKAGAWA, T., TAMAMCSm, B., Am) ISEmJRA, T., "Colloidal Surfactants," p. 59, Academic Press, New York, 1963.

Journal of Colloid and Interface Science, Vol. 52, No. 2, August 1975

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