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Demixing of Mixed Micelles. Thermodynamics of Sodium Perfluorooctanoate-Sodium Dodecanoate Mixtures in Water R. De Lisi,*,† A. Inglese,‡ S. Milioto,† and A. Pellerito† Dipartimento di Chimica Fisica, Universita` di Palermo, via Archirafi 26, 90123 Palermo, Italy, and Dipartimento di Chimica, Universita` di Bari, via Orobona 4, 70126 Bari, Italy Received June 25, 1996. In Final Form: October 7, 1996X Conductivity, density, heat capacity, enthalpy of dilution, and osmotic coefficient measurements of water-sodium perfluorooctanoate (NaPFO)-sodium dodecanoate systems were carried out as functions of the surfactants’ total molality (mt) at different mole fractions (XNaPFO). From conductivity data, the critical micelle concentration (cmc) and the degree of ionization (β) of the micelles were derived. The cmc’s of the micelles are higher than those of the pure surfactants while β depends linearly on XNaPFO. At a given mole fraction, the apparent molar volume (VΦ) and heat capacity (CΦ) of the mixture increases and decreases monotonically with mt, respectively. From data in the premicellar region, the standard (infinite dilution) partial molar properties (Yo) of the mixtures were calculated for both volume and heat capacity. Yo depends linearly on XNaPFO according to the ideal behavior. From data in the postmicellar region, the excess properties (Yexc) for the mixed micelles formation from pure micelles were calculated. The values of the excess volumes and heat capacities are positive and negative, respectively, in the whole range of composition. The excess free energy (Gexc) is negative while both enthalpy and entropy are positive. On the basis of the pseudophase transition model and experimental evidences, the Gexc vs composition profile was derived. This profile indicates the presence of a critical point for 0.4 e XNaPFO e 0.6. According to this approach, Gexc was calculated for some mixtures. It turned out that sodium dodecanoate and sodium dodecyl sulfate micelles are miscible in the whole range of composition while ammonium perfluorononanoate (NH4PFN) and ammonium dodecyl sulfate micelles are partially miscible, and the coexistence of the two mixed micelles pseudophases occurs in the range 0.43 e XNH4PFN e 0.75. In the case of the sodium decyl sulfate-sodium perfluorooctanoate system, only a critical point is present.
Introduction Recently, we have undertaken in our laboratory thermodynamic studies dealing with hydrocarbon-hydrocarbon surfactant mixtures in water. Accordingly, the effect of the nature of the polar head and the alkyl chain length of the surfactant on the mixed micelles formation has been investigated through volume,1-3 heat capacity,1-3 free energy,3-5 enthalpy,3-5 and entropy.3-5 The increase in the alkyl chain tail of the surfactant, as well as the nature of the counterion binding, has been also analyzed in the literature6,7 through volume data. Also, calorimetric investigations on several mixtures of hydrogenated surfactant are reported.8,9 Mixed micelles containing fluorocarbon surfactant are scarcely studied from a thermodynamic point of view and as far we know, only reports on Neos Ftergent-sodium tetradecyl sulfate mixtures are available.10 Yet, these systems are very interesting from both practical and theoretical points of view; in fact, they drastically reduce * To whom all correspondence should be addressed. † Universita ` di Palermo. ‡ Universita ` di Bari. X Abstract published in Advance ACS Abstracts, December 15, 1996. (1) Bakshi, M. S.; Crisantino, R.; De Lisi, R.; Milioto, S. J. Phys. Chem. 1993, 97, 6914. (2) Milioto, S.; Crisantino, R.; De Lisi, R.; Inglese, A. J. Solution Chem. 1995, 24, 369. (3) De Lisi, R.; Inglese, A.; Milioto, S.; Pellerito, A. J. Colloid Interface Sci. 1996, 180, 174. (4) Crisantino, R.; De Lisi, R.; Milioto, S. J. Solution Chem. 1994, 23, 639. (5) De Lisi, R.; Inglese, A.; Milioto, S. Fluid Phase Equilib. 1996, 126, 273. (6) Yamanaka, M.; Kaneshina, S. J. Solution Chem. 1990, 19, 729. (7) Yamanaka, M.; Kaneshina, S. J. Solution Chem. 1991, 20, 1159. (8) Rathman, J. F.; Scamehorn, J. F. Langmuir 1988, 4, 474. (9) Hey, M. J.; MacTaggart, J. W. J. Chem. Soc., Faraday Trans. 1 1985, 81, 207. (10) Funasaki, N.; Hada, S. J. Phys. Chem. 1982, 86, 2504.
the surface tension, improve the wetting of several substances, form stable foams in drastic conditions, can dissolve large amount of oxygen, and so on. It seems that fluorinated and hydrogenated surfactants, depending on their nature, can form miscible or partially miscible mixed micelles. This problem has attracted the interest of many researchers, and the main conclusions have been collected in a recent book.11 Most of these studies are essentially based on the analysis of the cmc as a function of composition by using the regular solution theory. Notwithstanding, the problem is still unsolved, since often for a given system opposite conclusions are drawn.11 For instance, Mukerjee and Yang12 for the sodium dodecanoate (NaL)-sodium perfluorooctanoate (NaPFO) mixtures invoke the presence of two kinds of mixed micelles to explain cmc data while Shinoda and Nomura13 exclude it. Unfortunately, the energetics of these systems derived from experimental determinations is unknown and free energy changes for the mixed micelles formation, which is able to evidence the coexistence of two kinds of micelles, are calculated only theoretically.8 Therefore, in order to contribute to this topic, we thought it would be interesting to perform a thermodynamic study of the NaL-NaPFO mixtures. Namely, volume, heat capacity, enthalpy of dilution, and osmotic coefficient measurements were carried out as functions of the surfactants’ total molality at different mole fractions. The critical micelle concentrations and the degree of ionization of the mixed micelles were also determined. Experimental Section Materials. Sodium dodecanoate (NaL) (Sigma) was recrystallized from ethanol and then dried under vacuum at 333 K for (11) Ogino, K., Abe, M., Eds. Mixed Surfactant Systems; M. Dekker: New York, 1993. (12) Mukerjee, P.; Yang, A. Y. S. J. Phys. Chem. 1976, 80, 1388. (13) Shinoda, K.; Nomura, T. J. Phys. Chem. 1980, 84, 365.
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at least 48 h before use. Perfluorooctanoic acid (Fluka) was recrystallized from carbon tetrachloride and dried at room temperature. The sodium salt of perfluorooctanoic acid (NaPFO) was prepared by neutralizing the acid with a sodium hydroxide aqueous solution. The product was recrystallized twice from the ice-cold solution and dried in a vacuum oven at 333 K for at least 2 days before use. D-(+)-sucrose was dried in an oven at 333 K for at least 48 h before use. All solutions were prepared by weight using degassed conductivity water. The molality of the surfactant mixture (mt) was calculated as
mt ) mNaPFO + mNaL
(1)
Table 1. Critical Micelle Concentrations and Degrees of Dissociation of Sodium Perfluorooctanoate-Sodium Dodecanoate Mixtures at 298 K
a
XNaPFO
cmca
β
0 0.1022 0.2997 0.5014 0.7006 0.9006 1
28.1 29.8 33.9 37.3 35.9 31.5 30.0
0.46 0.47 0.51 0.48 0.56 0.58 0.56
Units are mmol kg-1 for cmc.
where mNaPFO and mNaL are the number of moles of NaPFO and NaL per kilogram of water. The mixture’s composition and its molecular weight are expressed as
XNaPFO )
mNaPFO mt
XNaL )
mNaL mt
M ) MNaPFOXNaPFO + MNaLXNaL
(2) (3)
where MNaPFO and MNaL are the molecular weights of the pure surfactants. Equipment. Conductivity measurements were carried out by means of a digital Metrohm 660 conductimeter at a frequency of 2 kHz. The conductance measurements were performed at 298 K with a temperature control of (0.1 K. The densities of the solutions were determined with a vibrating tube flow densimeter (Sodev Model 03D) sensitive to 3 ppm or better. The densimeter was calibrated with water (d ) 0.997 047 g cm-3)14 and D-(+)-sucrose aqueous solutions whose density values are reported in the literature.15 The relative differences in heat capacities per unit volume ∆σ/σo ) (σ - σo)/σo were determined with a Picker flow microcalorimeter (Setaram) by setting a temperature increment of approximately 0.5 K. Measurements were carried out by taking water as a reference solvent (cpo ) 4.1792 J K-1 g-1).16 The specific heat capacities (cp) of solutions of density d are related to ∆σ/σo through the equation
cp ) cpo (1 + ∆σ/σo)do/d
(4)
where Cpo and do are the specific heat capacity and density of water, respectively. The temperature of both equipments was maintained constant at 298 K within 0.001 K by using closed loop temperature controllers (SodevModel CT-L). The enthalpies of dilution of the sufactants solutions with water were measured at 298 ( 0.01 K with a flow LKB 2107 microcalorimeter. The injection of the solutions and water into the microcalorimeter was made by means of a Gilson peristaltic pump (Minipuls 2). The flows of both the surfactant solution and water were determined by weight. The final concentrations were obtained as the product between the initial concentration (mt,i) and the dilution factor (fd), which is given by
fd )
φS φS + φw
(5)
where φw and φS are the flows of pure water and of water in the solution, respectively. The osmotic coefficients (Φ) were determined by using an Osmomat 070 (Gonotec) vapor pressure osmometer equipped with an automatic control unit. The following equation was used
Φ)
K∆R
∑νm
(6) t
where ∆R is the difference in the readings when water is replaced (14) Kell, G. S. J. Chem. Eng. Data Ser. 1967, 12, 66. (15) Garrod, J. E.; Herrington, T. M. J. Phys. Chem. 1970, 74, 363. (16) Stimson, M. F. Am. J. Phys. 1955, 23, 614.
Figure 1. Dependence of the critical micelle concentration of the sodium perfluorooctanoate-sodium dodecanoate mixtures on the mole fraction: b, present data; O, data from ref 12. by the solution in one of the two termistors, ν is the number of ions into which the surfactant dissociates (i.e., ν ) 2 for all the mixtures analyzed), while K is the calibration constant determined by measuring Φ of aqueous NaCl solutions at different osmolalities.17 The measurements were made at 310 K, the lowest operating temperature suggested for aqueous solutions. The Φ accuracy is less than 1%.
Results and Discussion Critical Micelle Concentration and Degree of Ionization of the Mixed Micelles. At a given surfactant mixture composition, the conductivity (χ) measurements were performed as a function of the total molality. For each mixture investigated, χ increases linearly with different slopes either above or below the cmc. Thus, the cmc was evaluated as the intersection point of the two straight lines while the degree of dissociation (β) of the micelles was obtained as a ratio between the slopes of these straight lines. The cmc and β values are collected in Table 1 and plotted in Figures 1 and 2, respectively, as functions of XNaPFO. The cmc’s of pure NaPFO and NaL agree well with the cmc’s reported in the literature.18,19 Cmc literature data12 for the NaL-NaPFO system are also shown in Figure 1. As can be seen, the two series of data agree well in the NaPFO rich region but not in the NaL rich region. The discrepancy can be ascribed to the fact that literature values were obtained in the presence of NaOH 1mM. As observed for mixtures of hydrocarbon and fluorocarbon surfactants having the same charge in the polar head, the cmc vs XNaPFO trend deviates positively from the ideal one calculated from the regular solution theory.20 As Figure 2 shows, β values are equal to those (17) Janz, G. J.; Gordon, A. R. J. Am. Chem. Soc. 1943, 65, 218. (18) Treiner, C.; Chattopadhyay, A. K. J. Colloid Interface Sci. 1984, 98, 447. (19) Brun, T. S.; Høiland, H.; Vikingstad, E. J. Colloid Interface Sci. 1978, 63, 89. (20) Clint, J. J. Chem. Soc. 1975, 71, 1327.
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Figure 2. Dependence of the degree of ionization of sodium perfluorooctanoate-sodium dodecanoate mixed micelles on the mole fraction.
Figure 3. Standard partial molar volumes and heat capacities of sodium perfluorooctanoate-sodium dodecanoate mixtures as functions of the mole fraction.
calculated by assuming a monotonic change between pure surfactants. The β value of 0.19 for the NaPFO-NaL mixture, at XNaPFO ) 0.67 and total concentration equal to 0.49 mol dm-3, was obtained from small-angle neutron scattering (SANS) measurements.21 This value, different from the present one, should reflect the concentration effect on β for mixed and pure surfactants.21,22 Accordingly, β decreases by increasing surfactant concentration. Volume and Heat Capacity. At a fixed mole fraction, the apparent molar volume (VΦ) and heat capacity (CΦ) of the surfactant mixture were calculated as
VΦ )
3 M 10 (d - do) d mtddo
CΦ ) Mcp +
103(cp - cpo) mt
(7)
(8)
The densities of the solutions with respect to that of water, the relative differences in the heat capacity per unit volume (∆σ/σo), the apparent molar volumes, and heat capacities of the surfactant mixtures are collected in Table 2. The partial molar properties (Y), reported in Table 2, were calculated as
Y)
∂(mtYΦ) ∂mt
(9)
by using analytical equations correlating YΦ to mt. At a given mole fraction, the standard (infinite dilution) partial molar property (Yo) and the interaction parameters for the unmicellized surfactant mixture (BY, CY, etc.) can be evaluated by applying eq 10 to experimental points in the premicellar region
YΦ ) Yo + AYmt1/2 + BYmt + CYmt3/2 + ...
(10)
where AY is the Debye-Hu¨ckel limiting slope. The values of 1.865 cm3 mol-3/2 kg1/2 and 28.95 J K-1 mol-3/2 kg1/2 were used23 for the volume and heat capacity, respectively, for each mixture since both the surfactants are 1:1 electrolytes. In the case of heat capacity, the pair (21) Caponetti, E.; Chillura Martino, D.; Floriano, M. A.; Triolo, R. Langmuir 1993, 9, 1193. (22) Berr, S. S.; Jones, R. M. J. Phys. Chem. 1989, 93, 2555. (23) De Lisi, R.; Ostiguy, C.; Perron, G.; Desnoyers, J. E. J. Colloid Interface Sci. 1979, 71, 147.
Figure 4. Apparent molar volumes, corrected for those at infinite dilution, for sodium perfluorooctanoate-sodium dodecanoate mixtures as functions of the reduced concentration: 4, XNaPFO ) 0.1; 2, XNaPFO ) 0.3; ], XNaPFO ) 0.5; b, XNaPFO ) 0.7; O, XNaPFO ) 0.9; solid line, XNaPFO ) 0; and broken line, XNaPFO ) 1.
interaction parameter BC was assumed to be zero since its uncertainty was of the same order of magnitude as its value, while in the case of volume the CV parameter was not considered. The fitting parameters are reported in Table 3. The standard partial molar properties as functions of XNaPFO are plotted in Figure 3, where the lines represent the ideal behavior, calculated as o o Yo ) XNaPFOYNaPFO + XNaLYNaL
(11)
o o and YNaL are the standard partial molar where YNaPFO properties of the two pure surfactants in water. The deviation of Yo values from the line represents the excess of the unmicellized surfactant mixture-water interactions with respect to those between water and pure surfactants. As can be seen, the NaPFO-NaL mixtures behave ideally as observed for other systems.6,7 At a given mole fraction, VΦ and CΦ slightly change with mt in the premicellar region while increasing and decreasing monotonically with the total molality, respectively, in the postmicellar region. By increasing XNaPFO, both the trends of VΦ and CΦ as functions of mt are shifted toward the higher values of those of pure NaPFO. Figures 4 and 5 show the plots of VΦ and CΦ, corrected for the corresponding properties at infinite dilution, against the
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Table 2. Apparent Molar Volumes and Heat Capacities of Sodium Perfluorooctanoate-Sodium Dodecanoate Mixtures in Water at 298 Ka mt
103∆d
VΦ
V
0.015 01 0.016 64 0.020 06 0.022 44 0.024 75 0.027 56 0.029 82 0.039 91 0.048 71
0.713 0.789 0.949 1.060 1.168 1.297 1.381 1.788 2.102
196.96 197.03 197.13 197.16 197.19 197.29 198.01 199.45 201.03
197.37 197.45 197.61 197.72 197.82 197.95 198.22 201.39 203.07
0.014 79 0.017 43 0.019 97 0.022 48 0.024 95 0.030 00 0.035 22 0.040 17 0.049 99
1.323 1.555 1.783 2.001 2.220 2.663 3.086 3.491 4.249
197.04 197.24 197.11 197.35 197.34 197.47 198.53 199.17 200.93
197.44 197.56 197.67 197.78 197.89 198.10 198.52 200.67 203.46
0.010 00 0.015 00 0.019 79 0.024 87 0.029 52 0.034 87 0.040 13 0.048 84 0.074 65
1.319 1.975 2.598 3.265 3.864 4.555 5.190 6.214 9.165
197.53 197.66 197.88 197.77 198.03 198.16 199.34 201.24 205.09
197.81 198.04 198.26 198.48 198.67 198.90 200.44 203.74 208.05
0.010 01 0.012 48 0.014 99 0.017 47 0.020 00 0.024 86 0.027 49 0.029 98 0.035 01 0.039 89 0.043 60 0.049 95
1.730 2.163 2.592 3.025 3.449 4.286 4.730 5.161 6.024 6.820 7.396 8.391
198.99 198.41 198.76 198.42 199.04 198.91 199.16 198.99 198.91 199.84 201.06 202.51
199.08 199.11 199.14 199.17 199.20 199.24 199.26 199.28 199.32 201.02 202.72 204.82
0.009 91 0.012 52 0.014 99 0.017 50 0.020 00 0.022 52 0.024 94 0.027 46 0.029 97 0.035 10 0.039 85 0.044 56
2.126 2.688 3.212 3.749 4.289 4.816 5.340 5.856 6.385 7.454 8.418 9.360
199.72 199.44 199.76 199.70 199.37 199.86 199.49 200.27 200.35 200.82 201.74 202.75
199.70 199.73 199.76 199.79 199.82 199.84 199.86 200.04 200.19 201.14 203.21 204.73
0.010 01 0.012 45 0.014 96 0.019 81 0.022 27 0.024 63 0.027 40 0.040 73 0.058 09 0.074 79 0.079 48
2.356 2.927 3.511 4.652 5.225 5.772 6.423 9.428 13.205 16.763 17.822
-103∆σ/σo
103∆d
VΦ
V
-103∆σ/σo
CΦ
Cp
- 0.33 0.41
XNaPFO ) 0.1022 955 957 0.074 34 942 958 0.100 1 945 958 0.125 2 960 959 0.150 3 956 959 0.175 1 957 959 0.190 0 0.247 4 866 784 0.299 7 803 704 0.342 9
2.996 3.885 4.750 5.602 6.411 6.879 8.699 10.224 11.567
203.71 205.04 205.72 206.22 206.70 207.01 207.68 208.41 208.52
205.65 206.91 207.62 208.10 208.43 208.59 209.03 209.27 209.42
2.88 5.31 7.71 10.41 12.79 14.32 20.10 25.10 29.27
685 628 593 561 545 535 508 497 487
592 543 516 498 486 481 465 457 452
-0.57
979
-0.79 -0.78 -0.85 -1.05 -1.28 -1.02 -0.37
XNaPFO ) 0.2997 975 0.074 22 0.099 92 988 976 0.125 2 967 977 0.150 2 965 977 0.171 4 970 978 0.200 1 981 961 0.249 2 936 880 0.299 7 868 774 0.398 7
6.030 7.901 9.709 11.498 12.994 15.023 18.273 21.595 27.933
204.33 206.11 207.25 207.87 208.30 208.62 209.69 210.25 210.91
206.84 208.49 209.41 210.00 210.36 210.72 211.13 211.41 211.75
1.47 4.27 6.86 9.61 11.69 14.64 19.41 24.05 33.04
767 677 629 590 573 551 531 520 504
644 579 543 520 506 491 475 463 450
-0.37 -0.51 -0.66 -0.81 -0.94
964 965 966 967 967
-1.06 1.40
XNaPFO ) 0.5014 964 0.099 73 965 0.124 9 966 0.150 3 967 0.173 7 967 0.203 4 0.249 8 0.300 0 930 842 0.349 8 775 664 0.400 3
11.980 14.771 17.553 20.098 23.274 28.189 33.275 38.555 43.428
207.15 208.38 209.32 209.87 210.48 211.00 211.84 211.43 212.10
209.79 210.75 211.36 211.76 212.12 212.50 212.77 212.96 213.10
4.03 6.36 9.22 11.41 14.30 18.83 23.36 28.08 32.33
691 650 608 590 571 548 537 521 518
598 563 541 527 514 501 492 485 480
-0.37 -0.42 -0.55 -0.59 -0.68 -0.87 -1.01 -1.05
979 969 983 969 973 975 985 976
XNaPFO ) 0.7006 980 0.059 51 981 0.075 17 981 0.100 2 982 0.124 9 982 0.149 0 983 0.174 9 983 0.211 3 984 0.247 5 0.298 2 0.350 0 971 908 0.394 5 923 826
9.872 12.294 16.115 19.832 23.484 27.227 32.512 37.716 44.889 52.028 57.946
204.30 206.13 208.04 209.32 209.72 210.84 211.54 211.86 212.22 212.57 213.05
206.88 208.87 210.55 211.46 212.03 212.46 212.86 213.14 213.41 213.60 213.72
0.00 1.28 3.84 6.30 8.80 11.38 14.85 17.74 22.87 26.81 30.50
851 787 703 656 619 597 575 568 544 542 538
747 672 609 575 555 539 524 514 505 498 493
-0.94 -0.98 -1.09 -1.25 -1.39 -1.14 -0.73
XNaPFO ) 0.9006 1008 0.049 19 1009 0.072 93 1009 0.100 3 0.125 1 0.149 7 1008 1011 0.176 0 996 1011 0.199 7 1001 1011 0.250 3 1010 1012 0.299 7 1004 983 0.349 6 961 892 0.395 3 914 829
10.279 14.933 20.265 24.999 29.640 34.579 38.895 48.214 56.977 65.684 73.524
203.63 206.88 208.47 209.69 210.51 210.95 211.45 211.95 212.53 212.91 213.15
205.89 209.24 210.97 211.83 212.40 212.81 213.09 213.50 213.76 213.95 214.08
-0.33 1.78 4.34 6.57 9.00 11.42 13.49 18.11 22.29 26.11 30.04
877 759 684 649 618 598 587 565 555 552 544
784 662 605 577 559 547 538 526 518 513 509
-0.41 -0.56 -0.71 -0.82 -0.97 -1.01 -1.19 -0.76 0.83 2.46 2.83
1006 1023 1034 1008 1017 1007 1017 923 798 729 716
XNaPFO ) 1 1016 0.090 07 1017 0.102 9 1017 0.131 7 1018 0.136 0 1018 0.156 5 1019 0.196 6 1019 0.233 2 829 0.259 6 695 0.300 7 640 0.360 5 630
20.123 22.843 28.915 29.867 34.122 42.380 49.811 55.124 63.284 74.826
3.76 5.36 7.51 7.80 9.14 13.35 16.93 19.09 22.03 29.71
691 650 632 629 626 586 565 559 559 515
611 589 571 568 558 545 537 533 528 521
- 0.48 - 0.48 - 0.59 - 0.74 - 0.79 - 0.88
-1.37 -0.94
-0.45 -0.53 -0.64
CΦ
Cp
mt
1023 1007 1011
a Units are mol kg-1 for m, g cm-3 for d, cm3 mol-1 for V and V and J K-1 mol-1 for C and C . Density data for pure NaPFO were Φ Φ p taken from ref 31.
reduced concentration (mt/cmc) at the different XNaPFO analyzed. It can be seen that, in the case of heat capacity, the curves superimpose on each other regardless of the
mixtures composition, while in the case of volume the curves are shifted toward higher values up to XNaPFO ) 0.5 beyond which they are moved toward lower values. This
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Table 3. Standard Partial Molar Volume and Heat Capacity and Interaction Parameters for Volume and Relative Enthalpy of the Sodium Perfluorooctanoate-Sodium Dodecanoate Mixtures in Water at 298 K XNaPFO
Vo (cm3 mol-1)
BV (cm3 mol-2 kg)
Cp° (J K-1 mol-1)
BL (kJ mol-2 kg)
CL (kJ mol-5/2 kg2)
0 0.1022 0.2997 0.5014 0.7006 0.9006 1
195.79 ( 0.08a 196.49 ( 0.04 196.60 ( 0.20 197.19 ( 0.16 198.80 ( 0.10 199.42 ( 0.07 200.00 ( 0.05a
17 ( 4a 18 ( 2 17 ( 9 17 ( 6
942 ( 4 952 ( 1 970 ( 4 960 + 3 976 ( 2 1004 ( 3 1012 ( 4
20.0 ( 2.8 -4.5 ( 2.6 11.2 ( 2.2 0.67 ( 0.94 -5.7 ( 1.7 8.8 ( 2.8
43.6 ( 15.3 152 ( 14 59.8 ( 11.4 109.9 ( 5.0 165.4 ( 8.7 97.3 ( 16.1
a
From ref 31.
Figure 5. Apparent molar heat capacities, corrected for those at infinite dilution, for sodium perfluorooctanoate-sodium dodecanoate mixtures as functions of the reduced concentration: 4, XNaPFO ) 0.1; 2, XNaPFO ) 0.3; ], XNaPFO ) 0.5; b, XNaPFO ) 0.7; O, XNaPFO ) 0.9; solid line, XNaPFO ) 0; and broken line, XNaPFO ) 1.
behavior essentially reflects the composition effect on the micellization properties. Therefore, it is expected that the heat capacity of micellization is independent of composition and that a maximum is present in the profile of the volume of micellization vs composition. Enthalpy and Osmotic Coefficient. The apparent molar relative enthalpies (LΦ) were calculated from the enthalpies of dilution (∆Hid) by means of the following equation:
∆Hid ) LΦ,f - LΦ,i
(12)
where LΦ,f and LΦ,i refer to the final and initial concentrations, respectively. The apparent molar relative enthalpies in the premicellar region were calculated by combining eq 12 and eq 10 applied to LΦ 1/2 ∆Hid - AL(f1/2 d - 1)mt,i
(fd - 1) mt,i
) BL + CL
Figure 6. Apparent molar relative enthalpies for sodium perfluorooctanoate-sodium dodecanoate mixtures as functions of the reduced concentration: 4, XNaPFO ) 0.1; 2, XNaPFO ) 0.3; ], XNaPFO ) 0.5; b, XNaPFO ) 0.7; O, XNaPFO ) 0.9; solid line, XNaPFO ) 0; and broken line, XNaPFO ) 1.
The enthalpies of dilution and the apparent and partial molar relative enthalpies at the initial and final concentrations are reported in Table 4. The dependence of LΦ on mt/cmc at the different mole fractions is shown in Figure 6. As can be seen, for a given mixture composition, LΦ increases slightly with concentration up to the cmc, beyond which it sharply increases to a constant value at high mt. Also, all the LΦ vs mt/cmc trends practically superimpose on each other in the premicellar region; in the postmicellar region they are very close in the dilute region and become different beyond mt/cmc ) 2. In particular, the trends are shifted toward larger values up to XNaPFO ) 0.7 while at higher compositions they are shifted toward lower values of those of pure NaPFO. The osmotic coefficients at 298 K (Φ298) were calculated from the experimental ones at 310 K (Φ310) by means of the following equation derived by assuming that the partial (Cp) and the apparent molar heat capacities do not depend on temperature
1/2 (f3/2 d - 1)mt,i
(fd - 1)
(13)
By taking 1973 J kg1/2 mol-3/2 for AL,23 the BL and CL values were calculated as intercept and slope of the plot of the quantity at the left-hand side of eq 13 against 1/2 [(f3/2 d - 1)mt,i /(fd - 1)], respectively; they are collected in Table 3. In the postmicellar region, for the dilution processes whose final concentration is smaller than the cmc, the LΦ,f and LΦ,i values were calculated by using eqs 10 and 12, respectively. For the dilution processes whose final concentration is larger than the cmc, LΦ,f was interpolated from the LΦ vs mt plot and LΦ,i was calculated by using eq 12. The partial molar relative enthalpies (L) were calculated by applying eq 9 to LΦ.
Φ298 ) Φ310 -
298 - 310 [(L - LΦ) - 298(Cp (νR)(298)(310) Cp - CΦ 298 (14) CΦ)} ln νR 310
( )
The activity coefficients were calculated by means of the following equation:
ln γ( ) (Φ - 1) +
∫om Φm-t 1 dmt t
(15)
For all the mixtures, the experimental points in the premicellar region were scattered around the line calculated through the Debye-Hu¨ckel limiting law. For this
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Table 4. Enthalpies of Dilution and Apparent and Partial Molar Relative Enthalpies of Sodium Perfluorooctanoate-Sodium Dodecanoate Mixtures in Water at 298 K mt,id
fdmt,ia
-∆Hidb
LΦ,ib
LΦ,fb
Lib
0.009 95 0.015 02 0.017 00 0.019 99 0.022 47 0.024 49 0.029 82 0.048 41 0.074 34
0.004 86 0.007 32 0.008 65 0.009 74 0.011 40 0.011 93 0.014 30 0.023 35 0.035 59
0.212 0.281 0.278 0.367 0.362 0.458 0.95 3.49 3.58
0.42 0.60 0.67 0.77 0.86 0.93 1.54 4.32 6.14
0.24 0.33 0.38 0.42 0.47 0.49 0.59 0.87 2.62
0.77 1.12 1.26 1.47 1.65 1.80 5.00 9.00 9.47
XNaPFO ) 0.1022 0.42 0.100 1 0.59 0.125 2 0.68 0.150 3 0.75 0.175 1 0.87 0.190 0 0.90 0.247 4 1.07 0.299 7 1.71 0.342 9 7.30
0.009 94 0.014 79 0.019 96 0.024 67 0.029 96 0.035 22 0.040 17 0.049 99 0.074 22
0.005 00 0.007 44 0.010 03 0.012 41 0.015 05 0.017 69 0.020 02 0.024 91 0.036 91
0.130 0.212 0.325 0.398 0.542 1.08 1.81 3.00 3.90
0.30 0.45 0.62 0.79 0.99 1.62 2.43 3.80 6.17
0.17 0.24 0.31 0.38 0.46 0.54 0.62 0.80 2.27
0.58 0.91 1.31 1.72 2.22 4.80 6.70 9.00 10.74
0.010 00 0.015 00 0.019 79 0.024 87 0.029 52 0.034 87 0.040 13 0.048 84 0.074 65
0.005 08 0.007 64 0.009 99 0.012 58 0.014 88 0.017 57 0.019 77 0.024 05 0.036 73
0.151 0.222 0.303 0.359 0.448 0.572 1.20 2.47 4.50
0.38 0.53 0.67 0.83 0.97 1.18 1.87 3.27 5.92
0.23 0.31 0.38 0.46 0.53 0.61 0.67 0.80 1.42
0.010 01 0.012 48 0.014 99 0.017 47 0.020 00 0.022 24 0.024 86 0.027 49 0.029 98 0.039 89 0.043 60 0.049 95
0.004 92 0.006 11 0.007 32 0.008 53 0.009 56 0.010 67 0.011 97 0.013 25 0.014 95 0.019 88 0.021 71 0.024 71
0.133 0.172 0.225 0.272 0.306 0.346 0.383 0.443 0.478 1.30 1.95 2.83
0.32 0.38 0.46 0.53 0.61 0.67 0.76 0.85 0.93 1.90 2.60 3.58
0.009 91 0.012 52 0.014 99 0.017 50 0.020 00 0.022 52 0.024 94 0.027 46 0.029 97 0.035 10 0.039 85 0.044 56
0.004 92 0.006 20 0.007 42 0.008 66 0.009 90 0.011 61 0.012 83 0.014 11 0.015 39 0.017 99 0.019 39 0.021 69
0.153 0.178 0.229 0.284 0.325 0.373 0.429 0.481 0.568 1.26 2.11 2.82
0.010 07 0.014 79 0.014 98 0.017 77 0.019 56 0.020 28 0.024 48 0.033 12 0.038 00 0.039 93 0.043 00 0.048 09 0.049 97
0.005 10 0.007 55 0.007 60 0.009 00 0.009 90 0.010 67 0.013 24 0.016 30 0.018 70 0.020 97 0.021 15 0.023 65 0.023 97
0.161 0.240 0.259 0.306 0.330 0.324 0.409 1.28 2.24 2.48 3.03 3.62 3.87
a
Lfb
mt,id
fdmt,ia
-∆Hidb
LΦ,ib
LΦ,fb
Lib
Lfb
0.047 84 0.063 83 0.076 00 0.088 23 0.094 51 0.105 6 0.127 3 0.145 1
2.73 1.94 1.60 1.34 1.24 1.16 0.829 0.630
7.00 7.48 7.80 7.99 8.09 8.28 8.34 8.37
4.25 5.60 6.20 6.66 6.85 7.12 7.51 7.74
9.46 9.39 9.27 9.16 9.08 8.82 8.63 8.53
8.80 9.42 9.45 9.46 9.47 9.45 9.37 9.29
XNaPFO ) 0.2997 0.30 0.099 92 0.43 0.125 2 0.59 0.150 2 0.74 0.171 4 0.93 0.200 1 1.13 0.249 2 1.32 0.299 7 1.74 0.350 7 5.60 0.398 7
0.049 62 0.060 66 0.072 94 0.082 89 0.098 70 0.111 8 0.126 8 0.147 4 0.167 0
3.43 2.97 2.42 2.10 1.65 1.43 1.18 0.850 0.607
7.40 8.12 8.50 8.72 8.94 9.16 9.27 9.30 9.29
3.82 5.15 6.08 6.66 7.36 7.76 8.12 8.45 8.68
10.85 10.67 10.48 10.36 10.19 9.90 9.59 9.34 9.10
9.00 10.20 10.75 10.94 10.86 10.73 10.65 10.49 10.38
0.68 0.98 1.27 1.59 1.88 2.23 4.75 8.50 11.24
XNaPFO ) 0.5014 0.39 0.099 73 0.54 0.124 9 0.68 0.150 3 0.83 0.173 7 0.97 0.203 4 1.13 0.249 8 1.27 0.300 0 1.53 0.349 8 3.00 0.400 3
0.048 94 0.059 57 0.071 42 0.082 40 0.096 23 0.111 5 0.133 3 0.154 6 0.176 0
4.03 3.40 2.75 2.27 1.82 1.48 1.06 0.751 0.515
7.35 8.00 8.41 8.69 8.98 9.12 9.18 9.21 9.25
3.32 4.60 5.66 6.43 7.16 7.64 8.12 8.46 8.73
11.04 10.81 10.58 10.38 10.13 9.73 9.40 9.45 9.52
8.40 10.15 11.10 11.25 11.05 10.92 10.73 10.54 10.36
0.18 0.21 0.24 0.28 0.30 0.33 0.37 0.41 0.45 0.60 0.66 0.75
0.59 0.73 0.89 1.05 1.22 1.38 1.58 1.78 1.98 7.26 8.37 9.78
XNaPFO ) 0.7006 0.31 0.059 51 0.37 0.075 17 0.44 0.100 2 0.50 0.124 9 0.56 0.149 0 0.63 0.174 9 0.70 0.211 3 0.78 0.247 5 0.89 0.298 2 1.22 0.350 0 1.35 0.394 5 1.57
0.029 27 0.036 85 0.048 91 0.060 81 0.073 31 0.085 40 0.102 8 0.115 8 0.138 1 0.160 4 0.179 8
3.89 4.70 3.92 3.17 2.48 2.07 1.58 1.34 0.961 0.680 0.476
4.80 6.16 7.48 8.11 8.54 8.85 9.10 9.22 9.32 9.37 9.38
0.91 1.50 3.61 4.94 6.04 6.80 7.56 7.91 8.35 8.68 8.90
10.80 11.18 11.05 10.85 10.63 10.41 10.19 10.00 9.73 9.53 9.43
1.92 6.15 9.55 10.90 11.23 11.23 11.04 10.95 10.76 10.52 10.37
0.31 0.38 0.46 0.55 0.64 0.73 0.82 0.93 1.03 1.82 2.73 3.54
0.17 0.20 0.24 0.27 0.31 0.36 0.39 0.43 0.48 0.56 0.61 0.70
0.59 0.77 0.95 1.15 1.36 1.59 1.82 2.06 2.32 6.90 8.07 8.97
XNaPFO ) 0.9006 0.30 0.049 19 0.37 0.072 93 0.44 0.100 3 0.51 0.125 1 0.59 0.149 7 0.71 0.176 0 0.79 0.199 7 0.89 0.250 3 0.98 0.299 7 1.19 0.349 6 1.31 0.395 3 1.51
0.023 95 0.035 46 0.047 25 0.058 82 0.070 30 0.082 50 0.093 42 0.121 1 0.143 7 0.166 7 0.187 5
3.38 4.16 3.36 2.70 2.19 1.81 1.52 1.00 0.721 0.504 0.346
4.16 6.05 7.30 7.85 8.17 8.39 8.57 8.78 8.82 8.84 8.83
0.78 1.89 3.93 5.15 5.91 6.58 7.05 7.78 8.10 8.33 8.48
9.65 10.29 10.18 10.03 9.90 9.75 9.60 9.33 9.08 8.86 8.69
1.72 7.05 9.44 10.12 10.26 10.29 10.22 10.05 9.93 9.81 9.70
0.39 0.55 0.55 0.65 0.72 0.74 0.90 1.89 2.92 3.24 3.80 4.49 4.75
0.22 0.30 0.31 0.35 0.38 0.41 0.49 0.60 0.69 0.77 0.77 0.87 0.88
0.72 1.06 1.07 1.28 1.42 1.48 1.83 7.15 8.55 9.07 9.62 9.95 10.07
XNaPFO ) 1 0.39 0.052 08 0.55 0.058 14 0.55 0.070 31 0.65 0.099 44 0.71 0.120 2 0.76 0.145 4 0.95 0.158 9 1.17 0.161 1 1.36 0.200 0 1.54 0.261 5 1.55 0.297 7 1.76 0.350 2 1.78
0.025 53 0.028 45 0.039 50 0.055 59 0.067 02 0.080 91 0.074 69 0.082 37 0.093 74 0.121 7 0.138 1 0.167 4
4.00 4.36 3.07 2.13 1.66 1.27 1.64 1.32 1.15 0.671 0.496 0.243
4.94 5.51 6.27 7.39 7.75 8.03 8.13 8.14 8.34 8.44 8.47 8.44
0.94 1.15 3.20 5.26 6.09 6.76 6.50 6.82 7.19 7.77 7.97 8.19
10.15 10.19 10.12 9.75 9.53 9.36 9.24 9.24 8.96 8.61 8.55 8.47
3.61 5.20 8.92 10.20 10.16 10.02 10.10 10.00 9.84 9.53 9.40 9.20
Units for concentration are kg mol-1. b Units for the enthalpies kJ mol-1.
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Table 5. Osmotic Coefficients at 298 and 310 K and Activity Coefficients, Nonideal Free Energies, Enthalpies, and Entropies of Sodium Perfluorooctanoate-Sodium Dodecanoate at 298 K mta
Φ310
Φ298
-ln γ(
-Gni b
Lb
0.016 85 0.024 68 0.029 92 0.034 92 0.050 04
0.933 0.965 0.905 0.758 0.582
0.934 0.964 0.904 0.768 0.587
0.15 0.18 0.20 0.46 0.75
0.75 0.91 1.00 2.26 3.72
1.28 1.76 5.00 7.35 9.10
0.014 94 0.019 94 0.022 76 0.029 17 0.033 34 0.039 24
0.947 0.963 0.923 0.879 0.806 0.734
0.950 0.966 0.926 0.881 0.811 0.741
0.14 0.17 0.18 0.26 0.35 0.46
0.71 0.82 0.88 1.28 1.73 2.25
0.020 39 0.040 26 0.050 08 0.074 65 0.075 29 0.099 73 0.125 1
0.944 0.753 0.667 0.526 0.499 0.413 0.332
0.949 0.756 0.672 0.534 0.507 0.419 0.338
0.11 0.42 0.57 0.86 0.89 1.13 1.35
0.030 20 0.049 60 0.058 32 0.072 82 0.100 9
0.916 0.732 0.651 0.539 0.407
0.925 0.741 0.660 0.547 0.413
0.039 79 0.048 80 0.075 67 0.099 88
0.766 0.673 0.462 0.340
0.010 00 0.018 93 0.034 87 0.039 06 0.047 65 0.071 50 0.098 60
0.947 0.927 0.868 0.822 0.728 0.556 0.449
a
TSni b
Φ310
Φ298
-ln γ(
-Gni b
Lb
TSni b
XNaPFO ) 0.1022 2.03 0.075 23 2.67 0.100 1 6.00 0.125 2 9.61 0.150 3 12.82 0.175 0
0.434 0.303 0.272 0.243 0.231
0.437 0.305 0.274 0.246 0.234
1.10 1.41 1.60 1.76 1.89
5.44 6.98 7.93 8.75 9.37
9.50 9.46 9.39 9.27 9.16
14.94 16.44 17.32 18.02 18.53
0.43 0.59 0.69 0.90 3.80 6.70
XNaPFO ) 0.2997 1.14 0.050 34 1.41 0.074 95 1.57 0.100 3 2.18 0.155 3 5.53 0.181 8 8.95 0.199 7
0.647 0.462 0.358 0.274 0.247 0.249
0.661 0.468 0.365 0.279 0.252 0.253
0.61 0.97 1.25 1.63 1.78 1.85
3.01 4.81 6.16 8.09 8.80 9.14
9.00 10.75 10.85 10.45 10.30 10.19
12.01 15.56 17.01 18.54 19.10 19.33
0.55 2.09 2.81 4.27 4.43 5.61 6.71
1.27 4.75 8.50 11.24 11.25 11.04 10.81
XNaPFO ) 0.5014 1.82 0.150 1 6.84 0.173 7 11.31 0.175 0 15.51 0.202 8 15.68 0.203 4 16.65 0.250 0 17.52
0.292 0.268 0.263 0.211 0.218 0.209
0.298 0.273 0.268 0.215 0.222 0.212
1.52 1.65 1.66 1.82 1.82 1.99
7.53 8.17 8.22 9.04 9.01 9.87
10.58 10.38 10.36 10.14 10.13 9.73
18.11 18.55 18.58 19.18 19.14 19.60
0.20 0.47 0.60 0.80 1.10
1.01 2.31 2.95 3.95 5.45
1.97 9.78 10.80 11.16 11.05
XNaPFO ) 0.7006 2.98 0.123 3 12.09 0.150 8 13.75 0.172 3 15.11 0.209 8 16.50
0.321 0.285 0.259 0.261
0.326 0.289 0.263 0.265
1.31 1.49 1.61 1.76
6.50 7.38 7.99 8.70
10.85 10.61 10.41 10.20
17.35 17.99 18.40 18.90
0.772 0.681 0.468 0.345
0.40 0.55 0.94 1.23
1.98 2.70 4.66 6.08
8.07 9.65 10.28 10.19
XNaPFO ) 0.9006 10.05 0.123 9 12.35 0.149 9 14.94 0.175 2 16.27
0.285 0.274 0.247
0.289 0.278 0.250
1.43 1.58 1.72
7.09 7.83 8.53
10.03 9.90 9.75
17.12 17.73 18.28
0.949 0.930 0.873 0.823 0.735 0.561 0.453
0.12 0.16 0.28 0.34 0.47 0.79 1.05
0.59 0.79 1.37 1.70 2.35 3.90 5.23
0.72 1.37 7.75 8.90 9.95 10.10 9.75
XNaPFO ) 1 1.31 0.124 9 2.16 0.150 3 9.12 0.173 7 10.60 0.203 4 12.30 0.250 0 14.00 0.299 4 14.98 0.349 8
0.354 0.306 0.268 0.218 0.209 0.200 0.146
0.357 0.309 0.271 0.221 0.210 0.199 0.145
1.30 1.47 1.61 1.78 1.96 2.11 2.29
6.42 7.29 7.99 8.83 9.69 10.45 11.36
9.53 9.30 9.14 8.92 8.66 8.53 8.45
15.95 16.59 17.13 17.75 18.35 18.98 19.81
mta
Units for mt are mol kg-1. b Units for Gni, L, and TSni are kJ mol-1.
reason, eq 15 can be written as
ln γ( ) (Φ - 1) + (2AΦ)xcmc +
m Φ - 1 dmt ∫cmc mt t
(16)
where AΦ()-0.39 kg1/2 mol-1/2 at 298 K) is the DebyeHu¨ckel limiting slope.23 The integral in eq 16 was solved graphically from the plot of (Φ - 1)/mt vs mt. The nonideal free energies (Gni) and entropies (TSni) were calculated as
Gni ) νRT ln γ(
TSni ) L - Gni
(17)
The osmotic coefficients at 310 and 298 K and the activity coefficients, the nonideal free energies, enthalpies, and entropies at 298 K are collected in Table 5. Figure 7 shows the dependence of Gni on ln(mt/cmc). The linear correlation observed in Figure 7 is predicted by eq 18, obtained on the basis of the phase transition model for the micellization process24
Gni ) (Gni)cmc - νRT ln(mt/cmc)
(18)
Figure 7. Partial molar nonideal free energies for sodium perfluorooctanoate-sodium dodecanoate mixtures as functions of the reduced concentration: 4, XNaPFO ) 0.1; 2, XNaPFO ) 0.3; ], XNaPFO ) 0.5; b, XNaPFO ) 0.7; O, XNaPFO ) 0.9; solid line, XNaPFO ) 0; and broken line, XNaPFO ) 1.
where (Gni)cmc is the nonideal free energy at the cmc.
Equation 18 was verified for several systems (ionic and nonionic,24 single and double chains,25 and pure and mixed surfactants3-5). However, the average experimental slope
(24) De Lisi, R.; Fisicaro, E.; Milioto, S.; Pelizzetti, E.; Savarino, P. J. Solution Chem. 1990, 19, 297.
(25) Milioto, S.; Bakshi, M. S.; Crisantino, R.; De Lisi, R. J. Colloid Interface Sci., 1993, 159, 354.
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Langmuir, Vol. 13, No. 2, 1997 199
for 1:1 surfactants is -4.5 ( 0.2 kJ mol-1 while the theoretical one is -4.95 kJ mol-1. For the present system, we obtained the average slope value of -4.4 ( 0.1 kJ mol-1. At a given mole fraction, the nonideal entropy as a function of mt slightly changes in the premicellar region while increasing monotonically toward a constant value in the postmicellar region. The TSni vs mt trends for the mixtures essentially superimpose on each other, such as occurs for the two pure surfactants, and are shifted toward higher values than those of pure NaL and NaPFO. These trends reflect positive values for the entropy of micellization and the presence of a maximum in the plot of this property as a function of composition. Excess Property for Mixed Micelles Formation. A thermodynamic approach was reported elsewhere4 which permits us to calculate a given excess property (Yexc) for the mixing of two pure aqueous micelles to form aqueous mixed micelles. By the use of the two components j and k, Yexc is given by
Yexc ) YΦ - XM,jYΦ,j - XM,kYΦ,k
(19)
Figure 8. Excess volumes for the mixing of (O) sodium dodecyl sulfate and sodium dodecanoate micelles, (b) sodium perfluorooctanoate and sodium dodecanoate micelles, and (4) perfluorohexane and hexane pure liquids as functions of the second component mole fraction.
where YΦ refers to a given total concentration of the surfactant mixture in the micellized form (mt,M), YΦ,j and YΦ,k refer to pure surfactants in water, and XM,j and XM,k are the mole fraction of the j and k components in the mixed micelles, respectively. The apparent molar properties of the pure surfactants are interpolated from the plots of YΦ,j and YΦ,k vs stoichiometric concentration at
mj ) mt,MXM,j + cmcj
mk ) mt,MXM,k + cmck
(20)
In the case of free energy, the partial molar nonideal property is available. Therefore, the nonideal contribution to the apparent molar free energy (Gni Φ ) was calculated through the following equation:4 ni Gni Φ ) G - νRT(Φ - 1)
(21)
By combining eqs 19-21, we obtain the following equation for the excess free energy:4 ni ni Gexc ) (Gmix - XM,jGni j - XM,kGk ) + RT[νjXM,j(Φj Φmix) + νkXM,k(Φk - Φmix)] (22)
where the subscript “mix” refers to the surfactant mixture. From the Gexc and Lexc values, the excess entropy (TSexc) is calculated as
TSexc ) Lexc - Gexc
(23)
The excess properties were calculated at mt,M ) 0.2 mol kg-1, since starting from this concentration the stoichiometric mole fractions are close to mixed micelles compositions theoretically evaluated by using, for example, the Rubingh approach.26 Figure 8 shows that the excess volume is positive in the whole range of composition as already observed for the sodium dodecyl sulfate (NaDS)-NaL mixtures3 in water. For the latter system, Vexc should reflect the difference in the hydrophilic interactions between the two polar heads in the mixed micelles with respect to the pure micelles since the hydrophobic contribution can be assumed null as expected in the mixing of hydrocarbons. Although the change in the degree of ionization of the mixed micelles with the composition could play a more or less important (26) Rubingh, D. N. In Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum Press: New York, 1979.
Figure 9. Excess heat capacities for the mixing of (O) sodium dodecanoate and sodium dodecyl sulfate micelles and (b) sodium dodecanoate and sodium perfluorooctanoate micelles as functions of sodium dodecanoate mole fraction.
role in the hydrophilic interactions contribution to the excess property, the positive Vexc values certainly reflect the lack of affinity between the hydrogenated and fluorinated chains as the data of hexane-perfluorohexane mixtures27 indicate (Figure 8). To the best of our knowledge, excess volumes for hydrogenated-fluorinated surfactant mixtures are available only for the Neos Ftergent (NF)-sodium tetradecyl sulfate (NaTS) system.10 In this case, Vexc is still positive and the presence of two kinds of mixed micelles was invoked in a certain range of composition. On this topic we will return later. As far as the heat capacity is concerned, Figure 9 shows values are negative. As far we know, no that Cexc p literature data dealing with hydrogenated-fluorinated surfactant mixtures are available so that no comparison data for anionic-anionic can be made. The only Cexc p surfactants are for the NaDS-NaL mixtures;3 it is interesting to note that the data are equal to those for the present system. Therefore, no conclusions can be drawn since the two systems differ for hydrophilic and hydrophobic moieties. However, as a general feature, these results are consistent with the volume data since, as far as pure surfactant solutions are concerned, it is usually observed25,28 that an increase in the volume corresponds to a decrease in the heat capacity. (27) Dunlap, R. D.; Scott, R. L. J. Phys. Chem. 1962, 66, 631.
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Figure 10. Excess free energy, enthalpy, and entropy for the mixing of sodium dodecanoate and sodium perfluorooctanoate micelles as functions of the sodium perfluorooctanoate mole fraction. The line for the excess free energy was calculated according to eqs 18 and 22 (see text).
Figure 10 shows the excess free energy, enthalpy, and entropy for the NaL-NaPFO mixed micelle formation as functions of the surfactant mixture composition. As can be seen, the excess free energy is negative while both enthalpy and entropy are positive in the whole range of composition. Similar studies were performed on NaL-NaDS,3 dodecyldimethylamine oxide (DDAO)-NaDS4, and DDAOsodium decyl sulfate (NaDeS)5 mixtures. As a general feature, Gexc is always negative while enthalpy and entropy seem to be system specific. Therefore, excess enthalpy is positive for the present mixture and negative for the NaLNaDS system. In the case of DDAO-NaDS and DDAONaDeS systems, Lexc is positive or negative depending on the mixture composition. Demixing of Mixed Micelles. An important topic concerning hydrogenated and fluorinated surfactant mixtures deals with the existence of the equilibrium between two pseudophases formed by two kinds of mixed micelles, one rich in the hydrocarbon surfactant and another rich in the fluorocarbon surfactant. Depending on the technique and/or the approach used, controversial results are generally obtained. This was pointed out by Funasaki in a recent review,29 where for most of the systems taken into account there is no conclusive information on the micelles demixing. Free energy of mixing (∆Gmix) for the mixed micelle formation from pure micelles should clarify this topic. In fact, ∆Gmix vs composition profile should display an upward concave curve for completely miscible mixed micelles, while for those partially miscible a concave downward curve should be present in the demixing region. Since
∆Gmix ) Gexc + νRT(Xj ln Xj + Xk ln Xk)
(24)
if the total concentration is sufficiently high, as mentioned above, it is possible to assume that the stoichiometric mole fraction is equal to that of mixed micelles; therefore, ∆Gmix can be easily calculated provided that Gexc is known. Unfortunately, apart from the above mentioned systems, no experimental Gexc data are reported in the literature. Generally, this property is theoretically evaluated.8,26 Therefore, the problem cannot be solved. For example, (28) Desnoyers, J. E.; Perron, G. In Surfactant Solutions New Methods of Investigation; Zana, R., Ed.; M. Dekker: New York, 1987. (29) Funasaki, N. In Mixed Surfactant Systems; Ogino, K., Abe, M., Eds.; M. Dekker: New York, 1993.
in the case of hydrogenated-fluorinated charged surfactant mixtures, the cmc deviates positively from the ideal behavior, and according to the regular solution theory (RST), which is most commonly used, it follows that the ω parameter and Gexc (Gexc ) ωRTXjXk) are positive in the whole range of composition while the present results show opposite signs for ω and Gexc. Since the ideal contribution to ∆Gmix is a symmetrical upward curve, the ∆Gmix vs mixture composition profile can display a concave downward curve region if (i) Gexc is positive, but this is not the case for the mixed micelles formation and (ii) Gexc presents a concave downward curve region. Therefore, the study of Gexc vs composition can be informative on the coexistence of the two pseudophases. For this reason, we will next focus our attention on the excess free energy. The plot of Gexc vs XNaPFO, shown in Figure 10, indicates the completely solubility between NaPFO and NaL micelles. This is supported by SANS studies21 at XNaPFO ) 0.67 at different total concentrations. It was found that NaPFO and NaL micelles are mutually soluble and that the composition of mixed micelles is equal to the experimental one. The lack of a demixing region also agrees with Shinoda and Nomura13 but disagrees with the Mukerjie and Yang12 findings. In both cases, equations correlating the cmc values to the mixture composition were derived from the RST. The cmc values calculated according to the Mukerjie and Yang equation are quite close to the experimental ones indicating the coexistence of two kinds of micelles; the best agreement between the experimental cmc values and those calculated, according to the Shinoda and Nomura equation,13 is obtained by taking ω ) 1.6 ( 0.1. Since this value is lower than the theoretical critical one (ω ) 2), demixing for the present system is not expected. Indeed, our experimental Gexc values (Figure 10) do not clarify the above controversial point. In fact, the demixing region could be sufficiently narrow so that the study of a few mixtures could mask the existence of the two pseudophases. In order to clarify this point, we decided to calculate Gexc in narrow intervals of mixture composition as a function of XNaPFO by using the following procedure: (i) at a given XNaPFO, the cmc value was interpolated from the experimental cmc vs XNaPFO plot; (ii) the osmotic coefficient of the mixtures at mt,M ) 0.2 mol kg-1 was assumed to be 0.21 according to the values obtained for the mixtures experimentally analyzed; and (iii) Gni for the mixtures was calculated by using eq 18. As far as (Gni)cmc is concerned, it is to be stressed that the theoretically calculated (3Aφcmc1/2) values differ by 0.1-0.3 kJ mol-1 from those obtained as the intercept of the Gni vs ln (mt/cmc) plot. Therefore, the calculated Gexc values can be affected by an uncertainty of 0.1-0.3 kJ mol-1. Therefore, by using the procedure mentioned above (eqs 21 and 23), Gexc values at regular narrow intervals of the mixture composition were calculated. The profile of Gexc against XNaPFO is shown in Figure 10. As can be seen, within the uncertainty of the method used, the calculated data agree with the experimental. A concave downward curve is present in a narrow composition region but its amplitude being smaller than the uncertainty is inconclusive on the existence of the demixing process and indicates that only a critical point is likely present for 0.4 e XNaPFO e 0.6. The inconclusive results dealing with the present system suggested to us that we analyze other systems, in particular, the NaDS-NaL system, whose micelles are miscible, the ammonium perfluorononanoate (NH4PFN)ammonium dodecyl sulfate (NH4DS) system which, ac-
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Figure 11. Osmotic coefficients as functions of the reduced concentration for b, sodium perfluorooctanoate; 2, sodium dodecanoate (ref 3); and O, sodium decyl sulfate (ref 5).
Figure 12. Excess free energy for the mixed micelle formation of sodium dodecanoate-sodium dodecyl sulfate (‚‚‚), ammonium perfluorononanoate-ammonium dodecyl sulfate (- -), and sodium perfluorooctanoate-sodium decyl sulfate (s) as functions of the first component mole fraction.
cording to Shinoda and Nomura,13 presents a demixing region, and the NaPFO-NaDeS system for which controversial conclusions12,13 are reported. The excess free energies of these mixtures were evaluated by using the above procedure. Also, to obtain welldefined profiles, Gexc values were calculated at arbitrary mole fractions. The corresponding cmc were interpolated from the cmc vs composition plots reported in the literature.13,30 As far as the osmotic coefficients are concerned, the value of 0.15 was used for the NaDS-NaL mixtures according to the experimental data; for the NaPFO-NaDeS system, Gexc values were evaluated by using the cmc values reported by Aratono et al.30 Also, Φ ) 0.22 was assumed for all mixtures at mt,M ) 0.2 mol kg-1 according to the average value for the two pure surfactants.5 For the NH4PFN-NH4DS system, only the cmc’s are available.13 Since the osmotic coefficient contribution to Gexc is not very important, it was estimated by assuming that Φ for all mixtures, as well as for pure surfactant, is equal to that for NaDS,4 whose cmc is comparable to the cmc’s of the two surfactants. Accordingly, the osmotic coefficient vs concentration curves for NaL,3 NaPFO, and NaDeS,5 which have comparable cmc’s, essentially superimpose on each other (Figure 11). (30) Aratono, M.; Ikeguchi, M.; Takiue, T.; Ikeda, N.; Motomura, K. J. Colloid Interface Sci. 1995, 174, 156. (31) Milioto, S.; Crisantino, R.; De Lisi, R.; Inglese, A. Langmuir 1995, 11, 718.
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Figure 13. Free energy of mixing of pure ionic micelles to form mixed micelles calculated according to eqs 18 and 22 (see text). The cmc’s of the two hypothetical surfactants are 0.01 mol kg-1, and those of the mixtures were calculated according to ref 13 at ω values of -1.0, 0, 1.0, 1.5, and 2.0. When ω is increased, the curves are shifted toward less negative values.
Figure 14. Free energy of mixing of pure ionic micelles to form mixed micelles calculated according to eqs 18 and 22 (see text). The cmc’s of the two hypothetical surfactants are 0.001 mol kg-1 and those of the mixtures were calculated according to ref 13 at ω values of -1.0, 0, 1.0, 1.5, and 2.0. When ω is increased, the curves are shifted toward less negative values.
The profiles of Gexc vs mixture composition are shown in Figure 12. As can be seen for the NaL-NaDS mixture, a regular concave curve is obtained, confirming that the two surfactant mix in their micelles completely. In the case of NH4PFN-NH4DS, since the presence of two minima imply two conjugate phases, a demixing region is clearly present and the coexistence of the two pseudophases occurs for 0.43 e XNH4PFN e 0.75, which is close to that reported in the literature.13 For the NaPFONaDeS system, the Gexc vs composition profile does not clearly indicate the presence of a demixing region but rather that of a critical point, such as observed for the NaPFO-NaL mixture. This result disagrees not only with the Aratono et al.30 and Mukerjie and Yang12 findings (existence of a demixing region) but also with the Shinoda and Nomura13 (complete miscibility) findings. The latter findings were based on the fact that the best ω value fitting the cmc vs composition plot is 1.8 and therefore, is smaller than the critical value (ω ) 2). However, our and Shinoda and Nomura conclusions could apparently be different if we consider that the uncertainty of ω is 0.1. On the other hand, it is well-known that the RST is strictly valid for pure hydrocarbon-liquid mixtures; accordingly, ω should be temperature independent while experimental evidence9 shows the contrary. This means that for real systems,
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Figure 15. Free energy of mixing of pure ionic micelles to form mixed micelles calculated according to eqs 18 and 22 (see text). The cmc’s of the two hypothetical surfactants are 0.001 and 0.01 mol kg-1 while those of the mixtures were calculated according to ref 13 at ω values of -1.0, 0, 1.0, 1.5, and 2.0. When ω is increased, the curves are shifted toward less negative values. The abscissa scale indicates the composition of the surfactant having a lower cmc value.
like surfactant mixtures, the ω critical value could be different than the theoretical one. To clarify this point, ∆Gmix was calculated at mt,M ) 0.2 mol kg-1 for three hypothetical systems formed by two 1:1 ionic surfactants. In the first two cases it was assumed that the cmc value for the two pure surfactants is the same for each case, 0.001 and 0.01 mol kg-1, respectively; in the third case, the cmc’s of the pure surfactants differ by 1 order of magnitude being 0.001 and 0.01 mol kg-1, respectively.
De Lisi et al.
The procedure used to calculate Gexc values was the same as above, but the cmc for the mixtures was calculated by following the Shinoda and Nomura13 equations at different ω values by always using 0.5 for Kg. For the osmotic coefficient contribution to Gexc, because of the low cmc values, it was assumed to be equal to that used for the NH4PFN-NH4DS system. The ∆Gmix vs composition profiles are shown in Figures 13-15. As can be seen, the higher the cmc of pure surfactants, the less negative the ∆Gmix. In addition, comparable cmc values (independently of the magnitude) for pure surfactants lead to excess free energies much smaller than those obtained when the cmc values of the two surfactants differ. A close inspection of Figures 13-15 gives the following information: (i) negative ω values, i.e., negative deviations in the cmc vs composition plot from the ideal behavior, always predict complete miscibility between pure surfactant micelles; (ii) the critical point seems to be independent of the cmc values of pure surfactants, and it occurs for ω ≈ 1.8; (iii) the location of the demixing region depends on the difference of the cmc’s of the two pure surfactants; and (iv) if the cmc’s of the two surfactants differ by 1 order of magnitude (Figure 15), at least one of the two pseudophases is formed by micelles of the pure surfactant having the higher cmc value. Acknowledgment. The authors are grateful to the National Research Council of Italy (CNR) and to the Ministry of University and of Scientific and Technological Research (MURST) for financial support. LA960636X
