Selective Determination of Surface Density of Bromide Ion through

methylammonium chloride (DTAC) mixture in the adsorbed film XC ... it was clearly shown that the criterion of an ideal mixing for the DTAB-DTAC system...
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Langmuir 2005, 21, 7398-7404

Selective Determination of Surface Density of Bromide Ion through XAFS and Its Application to Verification of a Criterion of an Ideal Mixing of Surfactant Mixture Makoto Aratono,* Kaoru Kashimoto, Takashi Matsuda, Souichiro Muroi, Youichi Takata, Norihiro Ikeda,† and Takanori Takiue Department of Chemistry, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan

Hajime Tanida‡ and Iwao Watanabe§ Experimental Facilities Division, Japan Synchrotron Radiation Research Institute, Hyogo 679-5198, Japan, and Faculty of Science, Osaka Prefecture University, Osaka 590-0035, Japan Received April 3, 2005. In Final Form: May 23, 2005 The mole fraction of chloride ion of dodecyltrimethylammonium bromide (DTAB) and dodecyltrimethylammonium chloride (DTAC) mixture in the adsorbed film XH C was estimated not only by the thermodynamic analysis of the surface tension data but also by analyzing the Br K-edge jump of the XAFS spectrum under the total reflection condition (TRXAFS method). The phase diagrams of adsorption (PDA) at several surface tensions from the two methods were in good agreement. On the basis of the PDA obtained, it was clearly shown that the criterion of an ideal mixing for the DTAB-DTAC system is not given by the 0 0 0 H linear relation between the total molality of surfactant mixture m and XH C , m ) mB + (mC - mB)XC , but H 0 2 0 2 0 2 H 2 2 by the one between m and XC , m ) (mB) + [(mC) - (mB) ]XC . Furthermore, it was demonstrated that the theoretical approach that provides the latter relation draws a distinction between the criteria for an ionic surfactant mixture without a common ion and that for an ionic surfactant mixture with a common ion.

Introduction A cationic surfactant usually has an inorganic anion as its counterion. When it is the bromide ion, it has been reported that its surface density can be determined by means of the total-reflection X-ray absorption fine structure (TRXAFS) technique by using the Br K-edge jump.1-3 This method is applicable even to the concentration range above the critical micelle concentration (cmc), where surface density cannot be calculated from the surface tension vs concentration curves through an adsorption equation. Furthermore the hydration structure of bromide ions in the interfacial region is elucidated by analyzing the EXAFS region of the spectrum.4,5 Another important feature of the method is its ion-selectivity; that is, we can selectively determine the surface density of bromide ions of bromide and chloride ion mixtures in the interfacial region. In this study, this important feature is utilized. The adsorption of surfactant mixtures has been extensively studied from the theoretical and experimental points of view because of not only scientific but also technological importance.6-8 One of the significant works in this field

is to elucidate the molecular interaction in the oriented and organized molecular state of adsorbed films by closely looking at deviation of physical quantities from a criterion of an ideal mixing constructed on the basis of thermodynamic consideration. Clint,9 Ingram,10 and Motomura et al.11 have proposed the criteria at a given surface tension for a nonionic [1]-nonionic [2] surfactant mixture as

m ) m01 + (m02 - m01)XH 2

(1)

1/m ) 1/m01 + (1/m02 - 1/m01)X2

(2)

and

Here m and X2 are the total concentration and the mole fraction of the second surfactant in the surfactant mixture and m01 and m02 are the concentrations of respective pure surfactants that give the γ value. XH 2 is the mole fraction of the second component in the adsorbed film defined in H terms of the respective surface densities, ΓH 1 and Γ2 , by H H H XH 2 ) Γ2 /(Γ1 + Γ2 )

(3)

* Corresponding Author. E-mail: [email protected]. Tel and Fax: +81 92 642 2577. † Faculty of Human Environmental Science, Fukuoka Women’s University, Fukuoka 813-8529, Japan. ‡ Japan Synchrotron Radiation Research Institute. § Osaka Prefecture University.

The miscibility, which implies a composition relation between the bulk solution and the adsorbed film and deviation from an ideal mixing, for several nonionic surfactant combinations has been investigated and often

(1) Watanabe, I.; Tanida, H. Anal. Sci. 1995, 11, 525. (2) Watanabe, I.; Tanida, H.; Kawauchi, S.; Harada, M.; Nomura, M.; Rev. Instrum. 1997, 68, 3307. (3) Takiue, T.: Kawagoe, Y.; Muroi, S.; Murakami, R.; Ikeda, N.; Aratono, M.; Tanida, H.; Sakane, H.; Harada, M.; Watanabe, I. Langmuir 2003, 19, 10803. (4) Harada, M.; Okada, T.; Watanabe, I. J. Phys. Chem. B 2003, 107, 2275. (5) Watanabe, I. J. Mol. Liq. 1995, 65/66, 245. (6) Abe, M.; Scamehorn, J. F. Mixed surfactant systems; Surfactant Science Series 124; Marcel Dekker: New York, 2005.

(7) Holland, P. M.; Rugingh, D. N. Mixed Surfactant Systems; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992. (8) Rosen M. J. Surfactants and Interfacial Phenomena; John Willey & Sons: New York, 1989. (9) Clint, J. H. J. Chem.. Soc. Faraday Trans. 1 1975, 71, 1327. (10) Ingram, B. T.; Luckhurst, Surface Active Agents; S.C.I. Symposium Proceedings, London, 1979; 89. (11) Todoroki, N.; Tanaka, F.; Ikeda, N.; Aratono, M.; Motomura, K. Bull. Chem. Soc. Jpn. 1993, 66, 351.

10.1021/la050876p CCC: $30.25 © 2005 American Chemical Society Published on Web 06/28/2005

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molecular interaction in the adsorbed films is expressed quantitatively by the activity coefficient in the adsorbed film.12-14 One parameter model proposed by Rosen15 provides the same criteria as eqs 1 and 2, but in this case, the dependence of the interaction parameter on the surface composition is not disclosed. Many researchers have adopted the above criteria and examined miscibilities in the adsorbed films even for the combinations of ionic-nonionic, ionic-ionic with a common ion, and ionic-ionic without a common ion. In the course of our studies on the surfactant mixtures, we have proposed different criteria that should be applied to the respective different types of surfactant combinations and investigated the miscibility on the basis of the criteria.12,16-18 For example,12 we have shown that the criterion for the dodecylammonium chloride (DAC)decylammonium chloride (DeAC) mixture is given by eqs 23 or 31 (see text and also eq 88 in ref 12), and on the other hand, the criterion for the DAC-decylammonium bromide (DeAB) is eq 43 (see text and also eq 54 in ref 12). Also, it was shown that a close examination of the criteria and the phase diagram of adsorption confirmed our criterion and eq 1 led to erroneous understandings. In this study, we apply the surface tension and TRXAFS techniques to the adsorption of the dodecyltrimethylammonium bromide (DTAB)-dodecyltrimethylammonium chloride mixture (DTAC). First the relation of composition of chloride ion between bulk solution XC and adsorbed film XH,γ C is evaluated from the surface tension. Then the surface density of bromide ion is determined selectively to evaluate the composition of chloride ions in the adsorbed film XH,x C . After showing two methods give almost the same values of the composition of the adsorbed film, we examine which one of the criteria between eqs 1 and 23 is plausible as the criterion of ideal mixing. Finally, the criterion for the ionic surfactant mixture without common ion is briefly summarized in a simple way. Experimental Section Materials. DTAB and DTAC (for ion-pair chromatography) were purchased from Tokyo Chemical Industry Co., Ltd. DTAB was purified by recrystallization three times from 1/5 (volume ratio) of ethanol/acetone mixture. DTAC powder was washed more than 10 times by hexane to remove the organic impurities and then recrystallized seven times from the ethanol and ethyl acetate mixture. The purity of surfactants was confirmed by that there was no minimum on the surface tension vs concentration curves around the critical micelle concentration (cmc). Water passed through ion-exchange resin was distilled three times. The second and third stages were performed from alkaline permanganate solution. Methods. Surface tension γ was measured by means of the drop volume apparatus19 at 298.15K under atmospheric pressure as a function of the total molality m of DTAB mB and DTAC mC mixture defined by

m ) mB + mC ) mDTA

(4)

and the mole fraction of chloride ion XC in the aqueous solution (12) Aratono, M., Villeneuve, M., Takiue, T.; Ikeda, N.; Iyota, H. J. Colloid Interface Sci. 1998, 200, 161, and references cited therein. (13) Ohta, A.; Matsubara, H.; Ikeda, N.; Aratono, M. Colloid Surf. 2001, 183-185, 403. (14) Villeneuve, M.; Nomura, T.; Matsuki, H.; Kaneshina, S.; Aratono, M. J. Colloid Interface Sci. 2001, 234, 127. (15) Rosen, M. J.; Hua X. J. J. Colloid Interface Sci.1982, 86, 164. (16) Matsuki, H.; Aratono, M.; Kaneshina, S.; Motomura, K. J. Colloid Interface Sci. 1997, 191, 120. (17) Matsubara, H,; Ohta, A.; Kameda, M.; Ikeda, N.; Aratono, M. Langmuir 2000, 16, 7589. (18) Matsuda, T.; Asoh, Y.; Villeneuve, M.; Matsubara, H.; Takiue, T.; Aratono, M. Colloid Polym. Sci. 2004, 282, 324. (19) Matsubara, H.; Muroi, S.: Kameda, M.; Ikeda, N.; Ohta, A.; Aratono, M. Langmuir 2001, 17, 7752.

Figure 1. Surface tension versus total molality curves at constant composition: XC ) (1) 0, (2) 0.1938, (3) 0.3801, (4) 0.5000, (5) 0.6250, (6) 0.7500, (7) 0.8750, and (8) 1. (we call this the bulk composition in the following) by

XC ) mC/(mB + mC) ) mC/mDTA

(5)

The γ values were reproducible within (0.05mN m-1. TRXAFS measurements were achieved at BL-7C of the Photon Factory of the National Laboratory for High Energy Physics (Tsukuba, Japan). The principle of the TRXAFS, the procedures how to evaluate the Br K-edge absorption jump (J values) and estimate the surface densities of Br ions from the J values, and the schematic views of the XAFS setup are fully described in our previous papers.1-3 Here is the brief summary of them. The X-ray beam monochromatized by a double-crystal monochromator [Si(111)] strikes the solution surface of the trough on a floating boat under the total reflection condition at about 1 mrad. The incident beam strength I0 was measured by a gas ionization chamber filled with nitrogen gas, the signal intensity I was detected by the total-conversion Helium ion yield method, and thus the I/I0 vs energy plots (XAFS spectrum) was obtained. The jump was observed at around 13470 eV and the J values were estimated by the appropriate extrapolation processes. Since the J value is associated with the distribution of Br ions at the interface C(z), the intensity of the evanescent wave P(z), and the surface area over which the X-ray strikes, we may put J as

∫

J ) kS

∞

0

∫

C(z)P(z) dz ) kSP(0)

∞

0

C(z) exp(- z/λ) dz (6)

where P(0) is the intensity of the incident X-ray and λ the penetration depth. When the concentrations of both surfactant and counterions are higher enough compared to the bulk solution and the thickness of the adsorbed monolayer including its electrical double layer is thin enough compared to the penetration depth, J is assumed to be proportional to the surface excess concentration of bromide ion per unit area Γ as

J ) kSP(0)Γ

(7)

The proportionality constant is determined at a concentration by using the Γ value evaluated from the surface tension measurement.

Results The surface tension of the DTAB-DTAC system was measured and analyzed thermodynamically in our previous paper.20 The surface tension data are quoted here again for the convenience of the reader and the discussion in the following. The γ value is plotted against m at given XC in Figure 1. The shape of curves changes regularly with XC from one pure surfactant to another. The critical micelle concentrations are identified as the break points on the curves. Here we similarly define the total surface (20) Yamanaka, M.; Amano, M.; Ikeda, N.; Aratono, M.; Motomura, K. Colloid Polym. Sci. 1992, 270, 682.

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density ΓH of bromide and chloride ions to the total molality given by eq 4 by H H ΓH ) ΓH B + ΓC ) ΓDTA

(8)

and the mole fraction XH C of chloride ion in the mixture of chloride and bromide ions in the interfacial region (we call this the surface composition in the following) H H H H H XH C ) ΓC /(ΓB + ΓC ) ) ΓC /ΓDTA

(9)

Then the total differential of γ is expressed as a function of m and XC by

dγ ) - (2RTΓH/m) dm - (RTΓH/XBXC)(XH C - XC) dXC (10) at constant T and p by assuming that the surfactant solution is ideally dilute. Thus, ΓH and XH C are evaluated by applying the equations

ΓH ) -(m/2RT)(∂γ/∂m)T,p,XC

(11)

XH,γ C ) XC - 2(XBXC/m)(∂m/∂XC)T,p,γ

(12)

Figure 2. Total molality versus composition curves at constant surface tension: γ/mN m-1 ) (1) 60, (2) 55, (3) 50, (4) 45, and (5) 41; (-) m versus XC; (----) m versus XH,γ C .

and

to the surface tension curves given in Figure 1, where we put the superscript “γ” on XH C in eq 12 to discriminate it evaluated by eq 14 from TRXAFS measurefrom XH,x C ments. Here it should be mentioned that the assumption of an ideal dilute solution in eq 10 has no significant influence on the discussion and conclusion in this paper. From the thermodynamics of adsorption12 and the Debye-Hu¨ckel equations of activity coefficients of surfactants,21 the relation between the (XH,γ C - XC) value calculated using eq 12 and that corrected using activity coefficients, (XH,γ C - XC)corrected, is given by corrected /(XH,γ (XH,γ C - XC) C - XC) ) 1 + (∂ ln f(/∂ ln m) (13)

where f( is the mean activity coefficient of DTAB and DTAC and equal to each other for both surfactants within the Debye-Hu¨ckel approximation. Employing the expression for f(,21 the right-hand side is about 1.07 at 20 mM, the maximum concentration of this study. Taking account of that even the largest XH,γ C - XC is about 0.3, the uncertainty of the mole fraction from the assumption of ideal dilute solutions is estimated to be 0.3*0.07)0.021 at most. The XH,γ C values were calculated by adopting eq 12 to the m vs XC curves at some fixed γ values in Figure 2 and shown in the form of the phase diagram of adsorption (PDA) also in Figure 2. It should be noted that the PDAs have a cigar-shape and similar to the PDAs of the DACDeAC system, for which an ideal mixing is expected because DAC and DeAC are homologue and their carbon number is different from each other only by two. Therefore, the results in Figure 2 suggest that a criterion of an ideal mixing of this kind of ionic surfactant mixture with a common ion is not the linear relation between m and XH,γ C given by eq 1. (21) Rosen M. J. Surfactants and Interfacial Phenomena; John Willey & Sons: New York, 1989; p 65.

Figure 3. Total surface density versus total molality curves at constant composition: XC ) (1) 0, (2) 0.1938, (3) 0.3801, (4) 0.5000, (5) 0.6250, (6) 0.7500, (7) 0.8750, and (8) 1.

Now we utilize the TRXAFS method to make sure that the surface composition is calculated exactly by using eq 12 from the surface tension measurement. The TRXAFS was employed previously for the pure surfactant system mentioned above. The principle for the mixed surfactant system is simple: the total surface density ΓH is evaluated is estimated from the J values of from eq 11 and ΓH,x B TRXAFS spectrum, and then, the surface composition is calculated from H H H,x XH,x C ) (Γ - ΓB )/Γ

(14)

ΓH was evaluated at XC at which the surface tension was measured and plotted against m in Figure 3. The m value at the XC and given γ was read on the m vs XC curve in Figure 2, and the ΓH value at this combination of m, XC, and γ was read from Figure 3. Then the TRXAFS measurement was performed also at these combinations shown by open circles on ΓH vs m curves. The XH,x C values are given as open circles in Figure 4. Although there are some scatters in XH,x C at different runs due to an insufficient reproducibility of the TRXAFS

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examined for several mixtures.12,22 The equation applicable to the present mixture, i.e., the ionic surfactant mixture with a common ion, is briefly summarized here in a simple way. The mean chemical potential µC( of DTAC of the ideally dilute solution is given by

µC( ) µ/C( + (1/2)RT ln mCmDTA

(15)

and rewritten in terms of m and XC as

µC( ) µ/C( + RT ln xXCm

(16)

/ is the mean standard chemical potential at where µC( infinitely dilute solution. Furthermore, eq 16 is modified using the mean chemical potential of pure DTAC at the 0 to given surface tension µC(

µC( ) µ0C( + RT ln (xXCm/m0C) Figure 4. Total molality versus composition curves at constant surface tension: γ/mN m-1 ) (1) 60, (2) 55, (3) 50, (4) 45, and H,x (5) 41; (-) m versus XC; (- -) m versus XH,γ C ; (O) m versus XC . H,x measurement, the coincidence between XH,γ C and XC is good at higher surface tension and not bad even at low surface tension, except very near the cmc (the curve at γ ) 41mN m - 1). The large gap near the cmc between the surface densities evaluated by two methods is not an experimental error but has a possibility that affords some useful information on the distribution of ions and/or micelles near the interface.3 This will be examined in a separate paper. Summarizing the results given above, we reasonably rate that the two methods gave credible values of the surface composition, and thus, the PDA was constructed rightly. Furthermore, it is reasonably expected that DTAB and DTAC are mixed ideally in the adsorbed film judging from the fact that the surfactant ion is common to them and both bromide and chloride ions are halide. Since the degree of binding to the surfactant ion is different for bromide and chloride, an ideal mixing is not necessary guaranteed. However, judging from the fact that the PDA of both DTAB-DTAC and DAC-DeAC systems follow the same criterion of ideal mixing and the latter exhibits almost ideal mixing similar to the nonionic homologous surfactant mixture having carbon number being different only by two.

Discussion Now let us examine a relation that expresses the criterion of ideal mixing of the ionic surfactant mixture on the basis of the PDA of the DTAB-DTAC system under the reasonable assumption that it shows an ideal mixing in the adsorbed film except around the cmc. Equation 1 has been most frequently cited as the criterion of an ideal mixing in the adsorbed film at a given surface tension. However, it is serious that this equation has been derived for a nonionic surfactant mixture in principle, and thus, neither dissociation of ionic surfactant into ions in the bulk and interface nor whether there is a common ion in the mixture is taken into account. That is, the criterion is the same irrespective of whether a surfactant mixture contains ionic surfactant and the two surfactants have common ions. This means that it is indispensable for us to explore whether eq 1 is applicable to an ionic surfactant mixture. On the other hand, the criterion in which the dissociation is explicitly taken into account has been reported and

(17)

In the previous paper,12 we have introduced an average electrical potential in the interfacial region φ h H and shown that the electrochemical potential of chloride ion µ˜ H C can be expressed as H,* + RT ln XH hH µ˜ H C ) µC C - Fφ

(18)

Introducing a similar equation for DTA cation and the electroneutrality condition for the surface densities, the mean chemical potential of DTAC in the interfacial region H µC( is given by H,* H H µH C( ) µC( + (1/2)RT ln XC XDTA

(19)

and rewritten in terms of the surface composition of chloride ion as H,* H µH C( ) µC( + RT lnxXC

(20)

H,* is the standard mean chemical potential. where µC( According to the definition of XH C given by eq 9, we have H,0 H,* µC( ) µC( and then

H,0 H µH C( ) µC( + RT ln xXC

(21)

H,0 is the mean chemical potential of pure DTAC where µC( in the interfacial region at the given surface tension. Thus for the equilibrium between the surface and H aqueous solution at the given γ, equating µC( to µC( and 0 H,0 using µC( ) µC( for the pure system yields

xXHC ) (m/m0C)xXC

(22)

m2 ) (m0B)2 + [(m0C)2 - (m0B)2]XH C

(23)

This leads to

It should be noted that the square of the total molality m2, not m itself, is a linear function of the surface composition XH C . The squares and square roots are originated from the dissociation of the surfactant into two ions. This forms a striking contrast to eq 1 that expresses a linear relation between m and XH C. (22) Lopez-Fontan, J. L.; Suarez, M. J.; Mosquera, V.; Sarmiento, F. J. Colloid Interface Sci. 2000, 223, 185.

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Figure 5. (a) Square of total molality versus surface composition evaluated from surface tension data; (b) total molality versus surface composition evaluated from surface tension data: γ/mN m-1 ) (1) 60, (2) 55, (3) 50, and (4) 45; (-) ideal mixing.

Figure 6. (a) Square of total molality versus surface composition evaluated from TRXAFS measurement; (b) total molality versus surface composition evaluated from TRXAFS measurement: γ/mN m-1 ) (1) 60, (2) 55, (3) 50, and (4) 45; (-) ideal mixing.

Table 1. Standard Deviation from the Ideal Mixing Criterion (102σ)

m vs XH,x C plots as given in Figure 6 and Table 1. Taking into account that an ideal mixing is expected for the DTAB-DTAC mixture as mentioned above, the criterion given by eq 23 is much more plausible compared to that given by eq 1. Furthermore, eq 22 provides another form being identical to the criterion of eq. 23 that

m2 and m versus XH,γ C

m2 and m versus XH,x C

γ/mN m-1

Figure 5a

Figure 5b

Figure 6a

Figure 6b

60 55 50 45

0.8 1.8 0.5 2.2

7.9 9.8 9.2 11.0

5.5 4.1 4.0 7.8

11.9 8.2 9.1 13.1

m2 and m are plotted against the composition XH,γ C evaluated from the surface tension data by using eq 12 and compared with the criteria given by eqs 23 and 1 in Figure 5, panels a and b, respectively. The standard deviation from the straight lines H,id 2 ∑ (XH,R C - XC ) /n

σ)x

(24)

H,x H,id presents XH,γ is XH where XH,R C C or XC and XC C values calculated from eq 23 or eq 1, is given in Table 1. It is evident that m2 values follow the straight line given by eq 23 better than m values do the straight line given by eq 1. The situation is the same with respect to the m2 and

1/m2 ) 1/(m0B)2 + [1/(m0C)2 - 1/(m0B)2]XC

(25)

and the corresponding one to eq 1 is

1/m ) 1/m0B + (1/m0C - 1/m0B)XC

(26)

In Figure 7, 1/m2 and 1/m are plotted against XC at some fixed surface tensions. It is evident that 1/m2 follows the straight line much more than 1/m, and therefore, again the criterion given by eq 23 is much more plausible compared to that given by eq 1. Here it is valuable to inspect whether the approach given above, which derives eq 23, can discriminate a criterion for an ionic surfactant mixture without a common ion such as DTAB-decyltrimethylammonium chloride

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we arrive at the criterion of ideal mixing in terms of the newly introduced variables given by

m ˆ 2 ) (m ˆ 0B)2 + [(m ˆ 0C)2 - (m ˆ 0B)2]X ˆH C

(31)

1/m ˆ 2 ) 1/(m ˆ 0B)2 + [1/(m ˆ 0C)2 - 1/(m ˆ 0B)2]X ˆ2

(32)

and

eqs 31 and 32 are converted to eqs 23 and 25, respectively. For the DTAB-DTAC system, the surface densities of the respective ions are evaluated separately, and therefore, introducing Γˆ H and X ˆH C is not necessary and actually eqs 23 and 25 are essentially identical to eqs 31 and 32. However, for the DTAB-DeTAC system, the surface density of the respective ions cannot be evaluated even when the surface tension of the mixture is measured as a function of the total molality and the composition in the bulk solution.12,23,24 This means that, for example, the surface composition of chloride ion defined by H H H H H X ˆH C ) ΓC /(ΓB + ΓDTA + ΓC + ΓDeTA)

(33)

cannot be evaluated. The surface densities that can be H H H evaluated are ΓH B + ΓDTA and ΓC + ΓDeTA. Thus, it is essential to introduce the total surface density of ions Γˆ H and the surface composition X ˆH C defined by H H H Γˆ H ) ΓH B + ΓDTA + ΓC + ΓDeTA

(34)

H H ˆH X ˆH C ) (ΓC + ΓDeTA)/Γ

(35)

and

instead of eq 33, repectively. The corresponding total molality of ions m ˆ and the bulk composition X ˆ C are defined by Figure 7. (a) Reciprocal of square of total molality versus solution composition curves; (b) reciprocal of total molality versus solution composition curves: γ/mN m-1 ) (1) 60, (2) 55, (3) 50, and (4) 45; (-) ideal mixing.

(DeTAC) from that for the DTAB-DTAC system. For this purpose, let us newly introduce the total molality of ions m ˆ and the bulk composition of the second surfactant X ˆ C. These are defined for the DTAB-DTAC system by

m ˆ ) mB + mDTA,1 + mC + mDTA,2 ) 2mB + 2mC (27) and

ˆ ) mC/m X ˆ C ) (mC + mDTA,2)/m

(28)

where mDTA,1 and mDTA,2 are the molalities of dodecyltrimethylammonium ion that originated from DTAB and DTAC, respectively. It should be noted that the relations m ˆ ) 2m and X ˆ C ) XC hold. Similary, the total surface density Γˆ H and X ˆH C are defined H

Γˆ )

ΓH B

+

ΓH DTA,1

+

ΓH C

+

ΓH DTA,2

H

) 2Γ

(36)

ˆ X ˆ C ) (mC + mDeTA)/m

(37)

and

respectively. It is worth noting that, when these quantities are employed, the total differential of γ is given by the equation having the completely same formula as the dγ of a nonionic surfactant mixture

ˆ ) dm ˆ - (RTΓˆ H/X ˆ BX ˆ C)(X ˆH ˆ C) dX ˆC dγ ) - (RTΓˆ H/m C - X (38) The mean chemical potential of DeTAC in the bulk solution corresponding to eq 15 is expressed

µC( ) µ/C( + (1/2)RT ln mCmDeTA ˆ Cm ˆ /2) ) µ/C( + RT ln (X

(39) (40)

(29) and rewritten in terms of the mean chemical potential of 0 as the pure second component µC(

and H H ˆ H ) XH X ˆH C ) (ΓC + ΓDTA,2)/Γ C

m ˆ ) mB + mDTA + mC + mDeTA

(30)

repectively. Following the same procedure given above,

(23) Okuda H.; Ozeki, S.; Ikeda, S. J. Colloid Interface Sci. 1987, 115, 155. (24) Motomura, K.; Ando, N.; Matsuki, H.; Aratono, M. J. Colloid Interface Sci. 1990, 139, 188.

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µC( ) µ0C( + RT ln (X ˆ Cm ˆ /m ˆ 0C)

Aratono et al.

(41)

Furthermore, by making reference to eqs 18 and 19, the mean chemical potential in the adsorbed film is given by H,0 µH ˆH C( ) µC( + RT ln X C

(42)

H ) µC( gives Thus the equilibrium condition µC(

ˆ 0C - m ˆ 0B)X ˆH m ˆ )m ˆ 0B + (m C

(43)

instead of eq 31, and furthermore

1/m ˆ ) 1/m ˆ 0B + (1/m ˆ 0C - 1/m ˆ 0B)X ˆC

(44)

instead of eq 32 as the criterion of ideal mixing. Again we note that the criterion expressed in terms of m ˆ,X ˆ C, and X ˆH apparently has the same form as that for a nonionic C surfactant mixture given by eq 1. However, it should be noted that the quantities m ˆ and X ˆ C are different among each other for the different kinds of surfactant combinations; for example, when the first surfactant dissociates into ν1,a a ions and ν1,c c ions and the second into ν2,b b ions

and ν2,d d ions, putting ν1 ) ν1,a + ν1,c and ν2 ) ν2,b + ν2,d, we have m ˆ ) ν1m1 + ν2m2 ) (ν1X1 + ν2X2)m and X ˆ2 ) ν2X2/(ν1X1 + ν2X2), respectively. Similarly, the relations H between Γˆ H and ΓH and that between X ˆH 2 and X2 depend on the surfactant combinations. Thus, it was demonstrated above that the theoretical approach that provides eq 23 can discriminate a criterion for an ionic surfactant mixture without a common ion from that for an ionic surfactant mixture with a common ion. It should be mentioned that the eqs 43 and 44 are applicable also to a nonionic-ionic surfactant mixture. On the other hand, from the experimental viewpoint, it was clearly demonstrated that the PDA of the mixed system of DTAB and DTAC, in which an ideal mixing is reasonably expected, obeys definitely eqs 23 and 25 (or equivalent eqs 31 and 32). Thus, the criterion of an ideal mixing of a system such as DTAB-DTAC is not eq 1 but eq 23. Acknowledgment. This work was supported in part by Grant-in-Aid for Scientific Research (B) (No.16350075) from the Japan Society for the Promotion of Science. This work has been performed under the approval of the Photon Factory Advisory Committee (Proposal No. 2002G106) LA050876P