Langmuir 2003, 19, 4923-4932
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Aggregation Process of the Mixed Ternary System Dodecylethyldimethylammonium Bromide/ Dodecylpyridinium Chloride/H2O: An Experimental and Theoretical Approach Elena Junquera, Francisco Ortega, and Emilio Aicart* Departamento de Quı´mica Fı´sica I, Facultad de Ciencias Quı´micas, Universidad Complutense de Madrid, 28040-Madrid, Spain Received February 7, 2003 The mixed aggregation process of two cationic surfactants with the same number of carbon atoms on the hydrophobic tail and different polar heads, dodecylethyldimethylammonium bromide (D12EDMAB) and dodecylpyridinium chloride (D12PyC), has been characterized in aqueous solution by conductivity, speed of sound, density, and static and dynamic light scattering measurements at 298.15 K. From these data, the monomeric and micellar phases of the mixed aggregates were fully analyzed through the determination of the total (cmc*) and partial (cmc1* and cmc2*) critical micellar concentrations, the dissociation degree of the mixed micelle (β), the total (N*) and partial (N1* and N2*) aggregation numbers, apparent molar volumes (Vφ), and isentropic compressibilities (Ks,φ) and hydration numbers (nh), as well as the corresponding changes in the latest properties due to the mixed aggregation process. The marked differences among the micelles of the pure surfactants studied in this work result in clear deviations from ideal behavior when the mixed aggregation process occurs. These experimental evidences have also been analyzed in terms of several theoretical models.
I. Introduction Mixed micellar systems, comprising a wide variety of surfactants, have become widely interesting due to their capacity to serve as good models for the study of molecular interactions on complex supramolecular aggregates, to mimic the behavior of biological systems and functions (ion transport, drug delivery, and so forth), and to provide better performance characteristics in their technological applications (pharmaceutical, food, detergency, and cosmetic industries, micellar solubilization, enhanced oil recovery, and so forth), than those consisting of only one type of surfactant.1-3 Most of the studies of mixed micelles found in the literature have been focused on the determination of the total critical micellar concentration, cmc*, and/or the total aggregation number, N*.1,3 However, there is a lack of information in the literature about the contribution of each surfactant to the monomeric phase of the mixed system through cmc1* values, as well as to the micellar phase, through the aggregation numbers N1*, from which the composition of the micellar phase, Ximlc, can be analyzed. The values usually reported for these quantities are those predicted by the theoretical models developed for mixed aggregated systems.1,3 Obviously, it would be interesting to have both experimental and theoretical information, for the sake of comparison and with the aim of fully characterized mixed micellar systems. One of the aspects widely analyzed in the studies reported in the literature is the influence of the hydrophobic chain length of surfactants on mixed micelle * To whom correspondence should be addressed. Fax: 34-13944135. E-mail:
[email protected]. (1) Holland, P. M.; Rubingh, D. N. Mixed Surfactant Systems; American Chemical Society: Washington, DC, 1992. (2) Christian, S. D.; Scamehorn, J. F. Solubilization in Surfactant Aggregates; Marcel Dekker: New York, 1995; Vol. 55. (3) Ogino, K.; Abe, M. Mixed Surfactant Systems; Marcel Dekker: New York, 1993.
properties.1 Nevertheless, data regarding the effect of the polar head are scarce, given that, since it is expected to be less than that of the chain length,4 only very accurate experimental techniques are able to carry on these studies with enough reliability. We have recently reported an experimental study of the mixed ternary system D12EDMAB/D12TAB (dodecyltrimethylammonium bromide)/H2O, based on conductometric and fluorescence measurements.4 Following a novel procedure fully described previously, both techniques allowed for the determination of the partial contribution of each surfactant to the mixed micellization process, the monomeric and micellar phases of the mixed surfactant system being, thus, completely analyzed.4 In this work, we present the study of the D12EDMAB + D12PyC mixed surfactant system in aqueous solution, by means of conductometric, ultrasonic, densitometric, and static and dynamic light-scattering measurements. These techniques allowed for the determination, in the whole composition range, of the mixed micellar critical concentration, cmc*, the dissociation degree, β, the aggregation number, N*, the hydration numbers, nh, isentropic compressibility, κs, and apparent and partial molar properties, Yφ and Y2, of the mixed micelles, as well as the change of the free Gibbs energy, ∆G°,mic, partial molar volume, ∆V2mic, partial molar isentropic compressibility, ∆Ks,2mic, and hydration number, ∆nhmic, due to the mixed micellization process. Furthermore, the partial contribution of each surfactant to the mixed aggregates, through their critical micellar concentrations, cmc1* and cmc2*, their aggregation numbers, N1* and N2*, and, from these values, the composition of the micellar phase, X1mic and X1mic, have been determined by using the experimental procedure designed by us. The behavior of the mixed system D12EDMAB/D12PyC/H2O has also been analyzed by using several descriptive and predictive theoretical (4) Junquera, E.; Aicart, E. Langmuir 2002, 18, 9250.
10.1021/la034214g CCC: $25.00 © 2003 American Chemical Society Published on Web 05/06/2003
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models, previously reviewed.1,4 A comparison among them, as well as the concordance with the experimental results, is also discussed. II. Experimental Section A. Materials. D12EDMAB and D12PyC were from Aldrich Co. D12EDMAB has a purity of 99%, while D12PyC has 5.9% mass of water content, which was considered to calculate solute concentrations. Double-distilled water was deionized using a Super Q Millipore system and finally was also degassed with a vacuum pump, prior to the preparation of the solutions. B. Apparatus and Methods. Conductivity data were collected at 298.15 K ((0.5 mK) with a Hewlett-Packard 4263A LCR Meter, using an electrode with a cell constant of 0.8129 cm-1. Mixtures were prepared from a digital buret, whose cylinder was kept at the same constant temperature of the measuring cell. The conductometer and the buret were controlled via IEEE-488 Bus and RS-232C interfaces, respectively. The experimental procedure was widely described previously.5 The accuracy of the specific conductivity κ, obtained as an average of 2400 measurements for each concentration, is believed to be better than 0.03%. The conductivity measurements were made (i) as a function of total surfactant concentration, [S]tot ()[D12EDMAB] + [D12PyC]), at several constant values of their molar fraction, R1, and (ii) as a function of [D12EDMAB], for different constant values of premicellar D12PyC concentrations, with the range of [D12EDMAB] going from pre- to postmicellar regions. The speed of sound, u, and the density, d, were continuous, simultaneous, and automatically measured with a highly accurate experimental technique, recently developed by us.6 The digital elements of the equipment are fully computerized by a PC, via IEEE-488 bus and RS232C serial port interfaces. The software, a LabView program developed by us as well, controls the automatic measuring procedure. The speed of sound (u ) L/t), operating with broadband pulses at 2.25 MHz, is obtained for each concentration, from an average over 150 measurements of the time, t, between two consecutive echoes of a signal with 250 000 digitized points of 8 bits of resolution. Calibration of the distance, L, between the transducer and the reflector was made from the speed of sound in pure water at 298.15 K.7 The solution is recirculated, by using a peristaltic pump, from the ultrasonic measuring cell to the vibrating tube of a densimeter, where the density, d, is obtained from an average over 90 000 measurements of the period of oscillation, τ, of this vibrating tube using the equation
d ) d0 + C(τ - τ02)
(1)
where d0 and τ0 are the density and the period for the reference solvent (water), respectively. The constant C was determined by calibrating the densimeter with vacuum and water.8 Both speed of sound and density are extremely sensitive to the temperature, which is controlled up to (0.5 mK with a procedure described elsewhere.6 The reproducibilities of the speed of sound and density measurements are (2 × 10-2 m s-1 and (2 × 10-3 Kg m-3, respectively. Measurements of speed of sound and density of the aqueous solutions of D12EDMAB + D12PyC were performed, at 298.15 K, as a function of total surfactant concentration, by diluting an initial solution of around 65 mM. The light scattering (LS) experiments9 have been performed on a Malvern 4700 instrument, using the green line, λ0 ) 514.5 nm, of an Ar+ ion laser Coherent M-300. We used the Malvern K7032 correlator equipped with 256 channels working in parallel mode, that allows us to monitor simultaneously four to five time decades. The scattered intensity is obtained at each fixed angle through a 500 µm aperture, by collecting 10 times the total number of photocounts during 10 s periods and averaging. The sample cells, cylindrical quartz cells of 1 cm diameter, were placed (5) Junquera, E.; Aicart, E. Rev. Sci. Instrum. 1994, 65, 2672. (6) Junquera, E.; Ruiz, M.; Lopez, S.; Aicart, E. Rev. Sci. Instrum. 2002, 73, 416. (7) Kroebel, W.; Mahrt, K. H. Acoustica 1976, 35, 154. (8) Kell, G. S. J. Chem. Eng. Data 1967, 12, 66. (9) Martin, A.; Ortega, F.; Rubio, R. G. Phys. Rev. E 1996, 54, 5302.
Figure 1. Plot of specific conductivity, κ, as a function of total surfactant concentration, [S]tot at 298.15 K, at various fixed values of the molar fraction, R1, for the mixed system D12EDMAB (1) + D12PyC (2): 0, 0; 9, 0.204; O, 0.406; b, 0.506; 4, 0.592; 2, 0.798; ], 1. in an isorefractive index bath, thermostated by an external bath controlled by a Hart Scientific 2100 with a calibrated PT-100 probe. The long-term temperature stability was better than (5 mK. The stability of the instrument was verified using toluene (PRA grade), filtered through 0.2 µm membranes Millipore, in the angular range (30-145°) used for the static light-scattering measurements. Deviations were found to be smaller than 1%. Toluene was also used as the standard reference for static measurements, assuming a Rayleigh factor of Rv ) 3.05 × 10-5 cm-1. This factor was obtained from the data reported earlier,10 that correspond to measurements with a 488 nm laser source, after a correction to take into account our refraction index, n, and λ0 values. Samples were filtered with 0.2 µm Millipore membranes, and the absence of dust was checked first by viewing the sample with cross polarizers. A stringest test was done by collecting the light scattered at 30° for several hours; samples with short-time deviations greater than 5% were refiltered or rejected. LS measurements were made as a function of total surfactant concentration, [S]tot ()[D12EDMAB] + [D12PyC]), at several constant values of their molar fraction, R1. These R1 values were exactly those for the study (i) of conductivity, since the final solutions were used on these LS experiments.
III. Results and Discussion Figure 1 shows the experimental values of specific conductivity, κ, as a function of total surfactant concentration, [S]tot, at several constant values of molar fraction, R1, (study i) for the aqueous solutions of the mixed micellar system D12EDMAB (1) + D12PyC (2). From these data, the mixed critical micelle concentration, cmc*, of the system, as the concentration at which there is a break in the property, and the dissociation degree of the micelle, β, as the ratio between the slopes above (a2) and below (a1) the cmc*, have been determined and reported in Table 1 for all the molar fractions, R1, studied herein.11,12 Also resumed in Table 1 is the change in Gibbs free energy due to micellization, ∆G0,mic, which can be calculated by using (10) Bender, T. M.; Lewis, R. J.; Pecora, R. Macromolecules 1986, 19, 244. (11) Hoffmann, H.; Ulbright, W. Z. Phys. Chem., N. F. 1977, 106, 167. (12) Zana, R. J. Colloid Interface Sci. 1980, 78, 330.
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Table 1. Values of Experimental Molar Fraction, r1, Mixed Micellar Critical Concentration, cmc*, Pre- (a1) and Postmicellar (a2) Slopes of Figure 1, Micellar Dissociation Degree, β, and Gibbs Free Energy of Micellization, ∆G°,mic, Obtained from Experimental Data of Study i, for the Mixed System D12EDMAB (1) + D12PyC (2)a R1
cmc* (mM)
104a1 (S cm-1 M-1)
104a2 (S cm-1 M-1)
β
∆G°,mic (kJ mol-1)
0 0.2044 0.4059 0.5058 0.5921 0.7980 1
16.77 15.06 14.29 14.08 13.92 13.75 13.97
865 897 909 920 920 919 917
387 364 332 319 310 277 246
0.447 0.405 0.366 0.347 0.337 0.299 0.268
-31.19 -32.46 -33.47 -33.92 -34.17 -35.00 -35.57
a Uncertainties on cmc*, a , a , and β are estimated to be less 1 2 than (2 × 10-5, (4 × 10-4, (3 × 10-4, (3 × 10-3, (2 × 10-5, and -5 (2 × 10 , respectively.
the following expression, based on the pseudophase separation model:13
∆G°,mic ) (2 - β)RT ln Xf
(2)
f
where X is the cmc* of the mixed system expressed in molar fraction. The surfactants studied herein are cationic surfactants of identical tail lengths (12 carbon atoms) and different polar heads. The cmc* values reported in Table 1 for the pure surfactants are in agreement with literature values, determined with a wide variety of experimental methods,14-19 and are also consistent with literature values for C12 surfactants with different polar heads.13,18,20 Table 1 also shows that the change in the polar head of an ethyldimethylammonium moiety by a pyridinium moiety increases the cmc around 20%. As can be seen in Table 1, premicellar slopes (a1) slightly increase as long as the molar fraction, R1, increases. This is consistent with the calculated values for the molar ionic conductivities at infinite dilution of D12Py+ and D12EDMA+ (λD12Py+° ) 25 S cm2 mol-1; λD12EDMA+° ) 26 S cm2 mol-1) and literature values for the counterions,21 thus indicating that the mobility of the ionic monomeric species D12Py+ is slightly lower than that of D12EDMA+. However, postmicellar slopes (a2) show a clear decrease as long as R1 increases, thus justifying the decrease found in the dissociation degree, β ()a2/a1). The fact that an aromatic ring stabilizes the charge on the polar head to a greater extent than an ethyldimethylammonium moiety does could be the explanation of these experimental evidences. Figure 2 shows a plot of specific conductivity, κ, as a function of [D12EDMAB], at several premicellar [D12PyC] values (study ii). The break in the experimental property is the [D12EDMAB] at which the mixed micellization process occurs for a given constant [D12PyC], these two concentration values informing about the contribution of each surfactant to the monomeric phase, referenced hereinafter as cmc1* and cmc2*, respectively.4 The sum of these magnitudes yields the cmc*, while the molar fraction (13) Attwood, D.; Florence, A. T. Surfactant Systems: Their Chemistry, Pharmacy and Biology; Chapman and Hall: London, 1983. (14) Maeda, T.; Satake, I. Bull. Chem. Soc. Jpn. 1984, 57, 2396. (15) Paluch, M.; Korchowiec, B. Colloids Surf., A: Physicochem. Eng. Aspects 1994, 82, 91. (16) Fujio, K.; Ikeda, S. Bull. Chem. Soc. Jpn. 1992, 65, 1406. (17) Liu, Y.; Han, B.-H.; Li, B.; Zhang, Y.-M.; Zhao, P.; Chen, Y.-T.; Wada, T.; Inoue, Y. J. Org. Chem. 1998, 63, 1444. (18) Junquera, E.; Pen˜a, L.; Aicart, E. Langmuir 1997, 13, 219. (19) Bakshi, M. S. Tenside, Surfactants, Deterg. 2001, 38, 103. (20) Junquera, E.; Pen˜a, L.; Aicart, E. Langmuir 1995, 11, 4685. (21) Tinner, U. Electrodes in Potentiometry; Metrohm: Herisau, 1989.
Figure 2. Plot of specific conductivity, κ, as a function of total [D12EDMAB] at 298.15 K, at various fixed values of the premicellar [D12PyC], for the mixed system D12EDMAB (1) + D12PyC (2): 9, 0.00 mM; O, 2.94 mM; 2, 6.08 mM; 3, 7.78 mM; [, 8.18 mM; ], 9.74; 0, 10.80 mM. Table 2. Values of Experimental cmc1* and cmc2*, Mixed Micellar Critical Concentration, cmc*, and Molar Fractions at the cmc*, r1cmc*, Obtained from Experimental Data of Study ii for the Mixed System D12EDMAB (1) + D12PyC (2)a cmc2*/mM
cmc1*/mM
cmc*/mM
R1cmc*
16.77 10.80 9.74 8.18 7.78 6.08 2.94 0
0 3.59 4.67 5.75 6.10 7.60 10.71 13.97
16.77 14.39 14.41 13.93 13.88 13.68 13.65 13.97
0.249 0.324 0.324 0.413 0.439 0.556 0.785 1
a Uncertainties on cmc *, cmc *, and cmc* are estimated to be 2 1 less than (2 × 10-5.
at the cmc*, R1cmc*(≡55.4X1f,cmc*), can be calculated as the ratio cmc1*/cmc*. Table 2 reports the values of all the above-mentioned magnitudes. A plot of cmc2* versus cmc1* is shown in Figure 3, together with the curve which best fits the experimental results. This curve, previously called by us the limit mixed micellization curve,4 divides the composition space into the mixed micelles region and the nonmicelles region, thus readily indicating the concentration range of both surfactants for which the mixed micellization for the D12EDMAB/D12PyC/H2O system may occur. Figure 4 includes a plot of the values of cmc1*, cmc2*, and cmc* from study ii as a function of R1cmc*, together with the lines which best fit the experimental data. By using these fits and the experimental cmc* values of study i, the cmc1* and cmc2* for this study have also been determined; they are collected as well in Figure 4 as a function of R1. It can be observed that cmc2* values vary almost linearly with R1, while cmc1* and cmc* show a clear negative curvature with respect to the ideal behavior (Clint’s model),22 described by
1 cmc*
)
Ri
∑i cmc
i
(3)
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Figure 3. Plot of cmc2* vs cmc1* (data from study ii). The line has been defined as the limit mixed micellization curve.
Junquera et al.
Figure 5. Plot of speed of sound, u, as a function of total surfactant concentration, [S]tot, at 298.15 K, at various fixed values of the molar fraction, R1, for the mixed system D12EDMAB (1) + D12PyC (2): 9, 0; 0, 0.206; b, 0.399; O, 0.511; 2, 0.575; 4, 0.800; [, 1.
speed of sound and density by the Newton-Laplace equation:23
κs ) 1/du2
(4)
There can be found in the literature several methods to determine hydration numbers, that is, from ionic mobilities, entropy values, apparent molar properties at infinite dilution, ultrasonic measurements, and so forth. The latter allows us to obtain the primary hydration numbers nh, through the expression24
nh )
Figure 4. Plot of cmc1* (triangles), cmc2* (circles), and cmc* (squares) as a function of the molar fraction, R1, for the mixed system D12EDMAB (1) + D12PyC (2): open symbols, study i; solid symbols, study ii; dotted line, cmcid* (eq 3).
where cmci is the critical micellar concentration of the pure micelles constituted by the i surfactants. This marked deviation contrasts with the small deviation found in the D12EDMAB/D12TAB system,4 but it is comparable with the deviations from ideal behavior found when the changes take place on the surfactant tail1 instead of the polar heads. The good agreement between results from studies i and ii is remarkable, thus confirming the goodness of the method to analyze the occurrence of D12EDMAB/D12PyC mixed micellization. The thermodynamic study of the mixed ternary system has been carried on at 298.15 K, by measuring the speed of sound, u, and the density, d, of aqueous solutions of D12EDMAB + D12PyC. Given that ultrasonic absorption is negligible at the experimental conditions used herein, the isentropic compressibility, κs, can be determined from (22) Clint, J. H. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1327.
(
)
nw κs 1ns κs,0
(5)
where κs,0 is the isentropic compressibility of the pure water and nw and ns are the number of moles of water and solute, respectively. Figures 5-7 show the plots of u, κs, and nh, as a function of total surfactant concentration, [S]tot, at several constant values of molar fraction, R1, for the aqueous solutions of the D12EDMAB + D12PyC system. It can be seen that speed of sound, u, increases, while κs decreases as long as total surfactant concentration increases, which means that mixed micelles are more packed (structured) and less compressible than the corresponding monomeric phase, as can be expected. It is also worth mentioning that these properties behave similarly in the monomeric phase, while they show different values in the mixed micelles range, depending on the composition of the micelle, this trend being consistent with the conductivity data of Figure 1. Any one of the thermodynamic quantities is suitable to determine the cmc*, from the break in the property versus concentration. Both speed of sound and isentropic compressibility have been used, as an example, for that purpose, the results being reported in Table 3. It is (23) Le Neindre, B.; Vodar, B. Experimental Thermodynamics. Vol. 2, Experimental Thermodynamics of Non-Reacting Fluids; Butterworths: 1975. (24) Bockris, J. O. H.; Reddy, K. N. Modern Electrochemistry; Plenum: New York, 1977.
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Figure 6. Plot of isentropic compressibility, κs, as a function of total surfactant concentration, [S]tot, at 298.15 K, at various fixed values of the molar fraction, R1, for the mixed system D12EDMAB (1) + D12PyC (2): 9, 0; 0, 0.206; b, 0.399; O, 0.511; 2, 0.575; 4, 0.800; [, 1.
remarkable the very good agreement of the cmc* values obtained from both the conductometric and the thermodynamic studies. The derivatives of thermodynamic properties, such as apparent molar volumes, Vφ, and apparent molar isentropic compressibilities, Ks,φ, can be calculated from the density and isentropic compresibilities, using the following expressions:25
M 1000(d - d0) Vφ ) d mdd0 Ks,φ ) Vφκs +
1000(κs - κs,0) mdd0
(6)
(7)
where M is the molar mass of the solute, calculated as the averaged molar mass over those of the pure surfactants, considering R1 values, and m is the molality of the solution, respectively. Figures 8 and 9 show Vφ and Ks,φ values as a function of total surfactant concentration, [S]tot, at several constant values of molar fraction, R1, for the aqueous solutions of the D12EDMAB + D12PyC system. The partial molar quantities, Y2, have been obtained from the corresponding apparent molar quantities, Yφ, following the well-known expression:25
Y2 )
(
)
∆(Yφm) ∆m
T,p
(8)
As can be clearly seen in Figure 8, Vφ values show a small change with total surfactant concentration, ranging from about 3.5 to 6.5 cm3 mol-1; that is, it does not appear to be very much affected by the mixed micellization process. This feature contrasts with the one found in the case of pure ionic surfactants, where the micellization in water is accompanied by an increase of the partial molar volume from 7 to 20 cm3 mol-1.26-28 However, the effect of the (25) Zana, R. Surfactant Solutions: New Methods of Investigation; Marcel Dekker Inc.: New York, 1987. (26) Shinoda, K.; Soda, T. J. Phys. Chem. 1963, 67, 2072. (27) Sugihara, G.; Mukerjee, P. J. Phys. Chem. 1981, 85, 1612.
Figure 7. Plot of hydration number, nh, as a function of total surfactant concentration, [S]tot, at 298.15 K, at various fixed values of the molar fraction, R1, for the mixed system D12EDMAB (1) + D12PyC (2): 9, 0; 0, 0.206; b, 0.399; O, 0.511; 2, 0.575; 4, 0.800; [, 1. The inset at the bottom shows the smoothed curve of ∆(mnh)/∆m values vs m. Table 3. Values of Experimental Molar Fraction, r1, Mixed Micellar Critical Concentration from Speed of Sound, cmcu*, and Isentropic Compressibility Data, cmcKs*, and Changes of Partial Molar Isentropic Compressibility, ∆Ks,2mic and Hidration Numbers, ∆nhmic, upon Micellization, for the Mixed System D12EDMAB (1) + D12PyC (2)a R1
cmcu* (mM)
cmcκs* (mM)
1015∆Ks,2mic (m3 mol-1 Pa-1)
∆nhmic
0 0.20601 0.39934 0.51084 0.57467 0.80033 1
16.72 14.86 14.98 14.30 14.03 14.28 13.60
16.74 14.89 14.94 14.26 13.96 14.21 13.59
126 125 142 139 144 121 134
15 16 18 17 18 16 17
a Uncertainties on ∆K mic and ∆n mic are estimated to be less s,2 h than (5%.
composition of the system (R1) on Vφ results is evident, in both the pre- and postmicellar ranges: Vφ = 280 cm3 mol-1 for pure D12PyC (R1 ) 0) and Vφ = 305 cm3 mol-1 for pure D12EDMAB (R1 ) 1). The values of ∆(m.nh)/∆m and Ks,2 are calculated and plotted as a function of total molality in the insets of Figures 7 and 9, respectively. ∆Ksmic and ∆nhmic, which are obtained as the differences of the corresponding magnitudes extrapolated to the cmc*, inform about changes in the partial molar isentropic compressibility and in the number of hydration molecules surrounding surfactant molecules due to the micellization process. These values are reported, as well, in Table 3, for all the molar fractions, R1, studied herein. The abovementioned features indicate that, although V2, Ks,2, and ∆(mnh)/∆m are sensible for the composition of the mixed system, ∆V2mic, ∆Ks,2mic, and ∆nhmic values are comparable for both pure surfactants and do not show an appreciable variation with the composition of the mixed micelle, within the experimental uncertainties. Another thermodynamic quantity, also determined in this work, is ∆G°,mic. The (28) Tanaka, M.; Kaneshina, S.; Sin-no, K.; Okajima, T.; Tomida, T. J. Colloid Interface Sci. 1974, 46, 132.
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analyzed within the linear interaction theory if the ionic strength is increased by adding an inner electrolyte or if the cmc* of the system is sufficiently high. It is wellknown30,31 that when a mixed micelle of ionic surfactants is formed, one surfactant acts as an added salt for the other surfactant, conferring enough ionic strength to the micellar phase, even when the cmc* values are low. Furthermore, given that in the present case both pure surfactants and the mixtures present cmc* values higher than 10 mM (Tables 1-3), micelles are in a relatively high ionic strength environment. Within the linear interaction regime and when the micellar size is small in comparison with the wavelength of the incident light, we can write for the scattered intensity and the diffusion coefficient, Dm,25
H([S]tot - cmc*) 1 ) [1 + kt([S]tot - cmc*)] (9) Rθ M h w
Dm ) D0[1 + kD([S]tot - cmc*)] Figure 8. Plot of apparent molar volume, Vφ, as a function of total surfactant concentration, [S]tot, at 298.15 K, at various fixed values of the molar fraction, R1, for the mixed system D12EDMAB (1) + D12PyC (2): 9, 0; 0, 0.206; b, 0.399; O, 0.511; 2, 0.575; 4, 0.800; [, 1.
(10)
[S]tot is the total concentration of surfactant, kt and kD are the static and dynamic concentration coefficients, respectively, M h w is the weight average micellar molecular weight, D0 is the decay rate average micellar diffusion coefficient extrapolated to the cmc*, and Rθ is the excess Rayleigh ratio calculated from the scattered intensity of the solution, Ic, using25
Rθ )
Ic - Icmc* Rtol Itol
(11)
where Itol and Icmc* are the scattered intensity of a toluene reference and that of the micellar solution at the cmc*, respectively, Rtol is the toluene Rayleigh ratio, and H is an optical constant that, for vertically polarized light, is given by25
H)
4π2n02 ∂n λ 4N ∂c 0
Figure 9. Plot of apparent molar isentropic compressibility, Ks,φ, as a function of total surfactant concentration, [S]tot, at 298.15 K, at various fixed values of the molar fraction, R1, for the mixed system D12EDMAB (1) + D12PyC (2): 9, 0; 0, 0.206; b, 0.399; O, 0.511; 2, 0.575; 4, 0.800; [, 1. The inset at the top shows the smoothed curve of the partial molar isentropic compressibilities, Ks,2, vs m.
values reported in Table 1 show that the mixed micellization studied herein depends on the system composition, being slightly more favorable as long as the molar fraction of the system, R1, increases. Light-scattering experiments allow for the determination of aggregation numbers, N*, of the D12EDMAB/ D12PyC mixed micelles. For ionic surfactants, intermicellar electrostatic repulsions lead to strong dependences of the scattered intensity and the diffusion coefficients with the surfactant concentration.29 These dependences can be (29) Ortega, F.; Bacaloglu, R.; McKenzie, D. C.; Bunton, C. A.; Nicoli, D. F. J. Phys. Chem. 1990, 94, 501.
( )
Av
T
2
(12)
where n0 is the solvent refraction index, λ0 is the vacuum wavelength of the incident light, and (∂n/∂c)T is the concentration refraction index increment at constant temperature. M h w can be easily related to the average micellar aggregation number, N h *, and from D0 the hydrodynamic radius, Rh, can be obtained using the Stokes-Einstein equation, D0 ()kT/6πηRh).24 From the D0 values we obtained micellar diameters of 2.5 nm for D12PyC and 2.8 nm for D12EDMAB, and intermediate values for the rest of the mixtures. As we were mainly concerned with the determination of the total aggregation number of both pure surfactants and the mixtures, we measure the scattered intensity for several concentrations between the cmc* and two times the cmc* for the D12EDMAB + D12PyC mixed system at different R1 values. The individual contribution of each surfactant to this micellar phase, that is, N1* (for D12EDMAB) and N2* (for D12PyC), and the molar fraction in the mixed micelles, Ximic, have been determined by using the (30) Sugihara, G.; Nakamura, A. A.; Nakashima, T.-H.; Araki, Y.-I.; Okano, T.; Fujiwara, M. Colloid Polym. Sci. 1997, 275, 790. (31) Nakano, T. Y.; Sugihara, G.; Nakashima, T.; Yu, S. C. Langmuir 2002, 18, 8777.
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Table 4. Values of Experimental Molar Fraction, r1, Total Surfactant Concentration, [S]tot, Molar Fraction in the Mixed Micelle, X1mic, Micellar Aggregation Number, N*, and Aggregation Number of D12EDMAB, N1*, and D12PyC, N2*, in the Mixed Micelle, for the Mixed System D12EDMAB (1) + D12PyC (2) R1
[S]tot (mM)
X1mic
N*
N1*
0 0.2044 0.4059 0.5058 0.5921 0.7980 1
34.68 33.09 31.93 30.88 31.04 28.88 37.68
0 0.2146 0.4065 0.4994 0.5771 0.7773 1
16 26 35 38 41 46 48
6 14 19 24 36 48
N2* 16 20 21 19 17 10
following expressions:
[S]totmic [S]1mic [S]2mic ) ) N* N1* N2* Ximic )
[S]imic [S]totmic
)
N1* N*
(13)
(14)
where [S]totmic, [S]1mic, and [S]2mic can be determined from the values of cmc1*, cmc2*, and cmc* obtained in the conductometric study (i), as follows:
[S]imic ) [S]i,tot - cmci*
(15)
N*, N1*, N2*, and Ximic results are reported in Table 4. It is remarkable the wide difference among the aggregation numbers of the pure surfactants. It is clear that the change of the polar head of a ethyldimethylammonium moiety by a pyridinium moiety has a marked effect on the aggregation number of the micelle, predominantly pointing to steric and electrostatic reasons as the main ones responsible for these experimental evidences.16,32 In fact, the effect is much higher than that observed in any of the other quantities measured and/or calculated in this work, that is, cmc*, β, ∆V2mic, ∆Ks,2mic, and ∆nhmic. These results are in very good agreement with literature values,16,18,33,34 which normally yield a much lower aggregation number for micelles constituted by surfactants with aromatic moieties on the polar head. Thus, the dodecylpyridinium micelles, with only 16 monomers, have a less packed structure, at least at the palisade layer, than the dodecylethyldimethylammonium micelles of 48 monomers.35 Figure 10 collects all the aggregation numbers as a function of molar fractions, R1. Also included in this figure is the line which best fits the experimental and calculated data, and that which represents the ideal behavior,36 described by the expression
1 N*
)
∑i
Ximic Ni
(16)
where Ni is the aggregation number of the pure micelles constituted by the i surfactants and Ximic is the molar (32) Eriksson, J. C.; Gillberg, G. Acta Chem. Scand. 1966, 20, 2019. (33) Ford, W. P. J.; Ottewill, R. H.; Parreira, H. C. J. Colloid Interface Sci. 1966, 21, 522. (34) Bales, B. L.; Zana, R. J. Phys. Chem. B 2002, 106, 1926. (35) Menger, M.; Jerkunica, J. M.; Johnston, J. C. J. Am. Chem. Soc. 1978, 100, 4676. (36) Bucci, S.; Fagotti, C.; de Giorgio, V.; Piazza, R. Langmuir 1991, 7, 824.
Figure 10. Plot of N1* (2), N2* (1), and N* (9) as a function of the molar fraction, R1, for the mixed system D12EDMAB (1) + D12PyC (2). Solid lines are the best fits to experimental values, and dotted line is Nid (eq 16).
fraction of the i surfactant in the micellar phase. It is worth mentioning the marked positive deviation of N* with respect to ideality, as well as the clear curvature of all the fit lines. Furthermore, a vertical line can be drawn on this figure for a given composition, R1, to obtain the aggregation numbers of each surfactant in the mixed micelle, N1* and N2*, and, from these values, the composition of the micellar phase, X1mic and X2mic. The aforementioned experimental results have been analyzed in terms of the most frequently used theoretical models for mixed micellar systems, previously reviewed by us.4 Rubingh’s model,37,38 based on the regular solution approach for the treatment of a nonideal mixing, considers the nonideality by introducing the activity coefficients, fi, in eq 3, as follows:
1
)
cmc*
Ri
∑i f cmc i
(17)
i
where fi is related with the β12 parameter, an energetic parameter that represents the excess Gibbs free energy of mixing, as follows:
ln β12 )
(
R1 cmc* mic
X1
) (
cmc1
(X2mic)2
ln
)
R2 cmc*
)
X2mic cmc2 (X1mic)2
(18)
The micellar composition Ximic and the activity coefficients fi can be obtained from β12 by using eqs 17 and 18. For most mixed micellar systems, β12 is negative, which implies that the critical mixed micellar concentration, cmc*, is lower than the averaged value of the critical micellar concentration of each surfactant, this effect being attributed to a nonideal behavior. Motomura et al.39 developed a thermodynamic method which determines the micellar composition from the experimental values of (37) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984. (38) Holland, P. M. Adv. Colloid Interface Sci. 1986, 26, 111. (39) Motomura, K.; Yamanaka, M.; Aratono, M. Colloid Polym. Sci. 1984, 262, 948.
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Junquera et al.
Table 5. Values of Experimental Molar Fraction, r1, Experimental Mixed Micellar Critical Concentration, cmc*, Experimental and Theoretical Molar Fractions in the Micelle, X1mic,a Activity Coefficients, fi, and Rubingh’s Interaction Parameter, β12, for the Mixed System D12EDMAB (1) + D12PyC (2) R1
cmcexp* (M)
X1,expmic
X1,Rmic
X1,Mmic
X1,Gmic
f1
f2
β12
0 0.2044 0.4059 0.5058 0.5921 0.7980 1
16.77 15.06 14.29 14.08 13.92 13.75 13.97
0 0.2146 0.4065 0.4994 0.5771 0.7773 1
0 0.2671 0.4575 0.5443 0.6179 0.7962 1
0 0.3280 0.5279 0.5970 0.6499 0.7865 1
0 0.1870 0.3821 0.4824 0.5706 0.7856 1
0.8251 0.9074 0.9365 0.9548 0.9865
0.9748 0.9332 0.9106 0.8861 0.8126
-0.358 -0.330 -0.316 -0.317 -0.327
a
R, Rubingh’s model; G, Georgiev’s model; M, Motomura’s model.
the monomeric composition and the cmc*, by the expression
X2mic ) R2 -
( )( ) [ R1R2 ∂ cmc*
cmc*
∂R2
1-
T,P
δcdν1,cν2,d
]
ν1,cν2R1 + ν2,dν1R2
(19)
for a mixed system constituted by two surfactants, 1 and 2, which dissociate into ν1 (ν1,a and ν1,c) and ν2 (ν2,b and ν2,d) ions, respectively, where
Ri )
X2mic )
νiRi ΣνiRi
(20)
ν2X2mic
(21)
ΣνiXimic
cmc* ) cmc*ΣνiRi δcd
(22)
is 1 for c ) d and 0 for c * d
Georgiev’s model40 allows us to obtain the micellar composition, X1mic, from the values of Ri and two parameters, g1 and g2: mic
X1
) mic
X2
R1 g1R1 + R2 R2 g2R2 + R1
(23)
where the parameters gi, which represent the ratio between the equilibrium constants for the formation of micelles constituted by the same type of surfactants (Kii) and different types of surfactants (Kij), can be determined from any one of the linearization expressions of eq 23:
X1mic/X2mic X1mic/X2mic - 1 ) g1 g2 R1/R2 (R /R )2 1
(24)
2
There is a relationship between the gi parameters of Georgiev’s model and the interaction parameter of the regular solutions, β12:
ln g1g2 ) -β12
(25)
Theoretical calculations of the parameters used by the above-mentioned models to analyze the mixed micellization process studied herein are reported in Table 5. The micellar composition, X1mic, predicted by the above resumed descriptive models is plotted against the molar fraction, R1, of the system in Figure 11, together with the experimental values and those for the ideal behavior given
Figure 11. Plot of experimental and theoretical micellar molar fractions, X1mic, versus total molar fraction, R1.
by the next equation, which is valid only at the cmc.22
X1mic )
R1 cmc2 cmc1 + R1(cmc2 - cmc1)
(26)
It can be noticed that, among these theoretical models, that of Georgiev reproduces quite well the experimental results, slightly subestimating them. The models of Rubingh and Motomura overestimate the experimental results, although the deviations are much more accentuated with the predictions of Motomura’s model. A similar trend was found in a previous mixed micellar system studied by us,4 although in that case the differences among theoretical and experimental results were much smaller than that in the present system for the three theoretical models analyzed in this figure. The energetic interaction parameters β12 of Rubingh’s model (Table 5) yield an averaged value of β12 ) -0.33 among the whole composition range, confirming the clear negative deviation with respect to ideality, previously observed in Figure 4. This feature is also consistent with the values obtained for the gi parameters of Georgiev’s model (eq 24), g1 ) 0.94 and g2 ) 1.13, which result through eq 25 in a β12 value of -0.06, in consistency with that obtained with Rubingh’s model. It is well-known that when the decrease of the cmc* with the system composition is so accentuated that it is lower than that for any of the pure surfactants comprising the system, a synergistic effect occurs.1 This effect is of particular interest for surfactant technological applications, since they normally look for micelles to form at as low a surfactant concentration is possible. The cmc*
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Table 6. Values of Molar Fraction in the Monomeric Phase of the i Surfactant, Xif, Molar Fraction in the Micelle of the i Surfactant, X1mic, Aggregation Number, N*, Mixed Critical Micellar Concentration, cmc*, and Energetic Contributions, ∆µi°/kT, to the Overall Free Energy of Micellization, ∆µ˜ °/kT, Predicted by the Nagarajan Model for the Mixed System D12EDMAB (1) + D12PyC (2), in the Whole Composition Range X1f 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X1mic
N*
∆µtransfer°/kT
∆µdeform°/kT
∆µinterfac°/kT
∆µsteric°/kT
∆µionic°/kT
0.299 0.383 0.442 0.491 0.535 0.579 0.626 0.680 0.756
16 23 26 27 29 30 32 33 35 38 48
-19.82 -19.82 -19.82 -19.82 -19.82 -19.82 -19.82 -19.82 -19.82 -19.82 -19.82
0.586 0.751 0.801 0.837 0.868 0.890 0.924 0.955 0.991 1.043 1.382
9.352 7.950 7.614 7.390 7.212 7.057 6.907 6.751 6.579 6.347 5.166
0.262 0.425 0.482 0.526 0.564 0.601 0.640 0.685 0.740 0.825 1.234
1.609 2.036 2.170 2.269 2.354 2.433 2.515 2.605 2.712 2.871 3.816
values obtained from conductivity measurements (Table 1 and Figure 4) point, in the limit of the experimental uncertainties, to a possible synergism, which is not readily detected from the results reported in Table 3 from thermodynamic data. Several theoretical treatments have been developed to predict the appearance of synergism in a mixed surfactant system.41-45 The most used, proposed by Rosen,45 is based on Rubingh’s model and states two conditions for a system to present synergism: (i) the β12 interaction parameter must be negative, and (ii) |β12| > |ln cmc1*/cmc2*|. Both conditions are followed by the cmc1*, cmc2*, and β12 obtained in this work from conductivity experiments for the mixed ternary D12EDMAB/D12PyC/ H2O system. However, it is worth mentioning that condition ii is satisfied in the limit of experimental uncertainties due to the low value of β12. Nagarajan’s model1,46,47 is able to predict, from the value of the change in the Gibbs energy for the micellization process, previously minimized, the features of mixed micellar systems in the monomeric phase (cmc*), as well as in the micellar phase (N*, Ximic), using molecular information of the pure surfactants. The expression for the micellar composition of aggregation number N*, in terms of molar fraction XN*, is *
N* ∆µ˜ ° kT
)
(27)
∑i Ximic ln Xif
(28)
(
XN* ) (Xf)N exp where
∆µ˜ ° kT
)
∆µ° kT
-
where ∆µ° is the sum of different energetic contributions due to the transfer of the surfactant tail from water to the micellar core, the deformation of the surfactant tail inside the micellar core due to packing constraints, the interfacial interaction between the core of the aggregate and water, the steric contribution from the headgroups in the micelle, the dipole interaction term between the headgroups, the ionic interaction between the headgroups, and the mixing of the surfactant tails. The calculation of the different contributions is widely explained by the author in successive papers, and previously reviewed by us.1,4,46,47 From the value of the total free energy of micellization, ∆µ˜ °, the cmc* and X1mic values have been calculated (40) Georgiev, G. S. Colloid Polym. Sci. 1996, 274, 49. (41) Bergstrom, M.; Eriksson, J. C. Langmuir 2000, 16, 7173. (42) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 90, 212. (43) Shiloach, A.; Blankschtein, D. Langmuir 1997, 13, 3968. (44) Zhu, B. Y.; Rosen, M. J. J. Colloid Interface Sci. 1984, 99, 442. (45) Rosen, M. J. Prog. Colloid Polym. Sci. 1994, 95, 39. (46) Nagarajan, R. Langmuir 1985, 1, 331. (47) Nagarajan, R. Adv. Colloid Interface Sci. 1986, 26, 205.
∆µmissing°/kT
∆µ˜ °/kT
cmc* (mM)
-0.610 -0.666 -0.686 -0.693 -0.691 -0.681 -0.661 -0.627 -0.556
-8.012 -8.505 -8.664 -8.753 -8.805 -8.831 -8.833 -8.812 -8.759 -8.649 -8.222
18.37 11.21 9.57 8.75 8.31 8.10 8.08 8.25 8.70 9.71 14.05
Figure 12. Plot of experimental and theoretical micellar molar fractions, X1mic, and cmc* versus monomer molar fraction, X1f.
following Nagarajan’s model (Table 6), and plotted against the molar fraction in the monomeric phase, X1f, in Figure 12, together with experimental values. As can be seen in Table 6, the energetic contributions (∆µ°)transfer and (∆µ°)mixing to the overall free energy of micellization stabilize the mixed micelles. It is worth mentioning that the former is independent of the micelle composition, given that both surfactants have identical hydrophobic tails. Thus, the positive contributions (deformation, interfacial, steric, and ionic) are responsible for the micellar composition behavior. It is clear from Figure 12 that Nagarajan’s model can reproduce at least the trend of the dependence of X1mic with monomeric molar fraction, although the predictions are somewhat poor. However, it cannot predict experimental cmc* values, although theoretical results show a remarkable synergistic effect. The predicted aggregation numbers, also reported in Table 6, are in poor agreement as well with experimental values. This poor theoretical analysis of Nagarajan’s model for the mixed system reported herein contrasts with that previously reported4 for the D12EDMAB/D12TAB/H2O ternary system. Given that both surfactants have identical hydrophobic tails, the marked differences found in the pure micelles must be due to the different polar heads. These differences are probably responsible for the clear deviations shown by the mixed micellization process studied herein, with respect to ideality. It seems that the model is not able to adjust the ∆µ˜ ° values which have to do with the polar heads, to predict the behavior of the monomeric phase,
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but mainly those of the micellar phase of the D12EDMAB/ D12PyC mixed micellar system. IV. Conclusions This work reports an experimental conductometric method to determine the individual critical micellar concentration of components 1 and 2 in a mixed system, cmc1* and cmc2*. These results, combined with N* data from dynamic and static light scattering, allow as well for the determination of the individual contribution of each surfactant, N1* and N2*, to the aggregation number of the system, the magnitudes of which have been analyzed to obtain the micellar compositions, X1mic and X2mic. Another important contribution of this work is a thermodynamic analysis of the mixed micellization process through measurements of speed of sound and density of the solutions, from which apparent molar volumes (Vφ) and isentropic compressibilities (Ks,φ), hydration numbers (nh), and the corresponding changes in the latest properties due to the mixed micelles formation have been reported. The marked differences among the micelles of the pure surfactants studied in this work result in clear deviations
Junquera et al.
from ideal behavior when the mixed aggregation process occurs, which could be attributed to a combination of steric effects and electrostatic interactions, including those between the pyridinium headgroup and the counterions. This behavior contrasts with the small deviation from ideality found in the D12EDMAB/D12TAB system, but is comparable with that one found when the changes take place on the surfactant tail instead of the polar heads. These experimental results have been analyzed in terms of several theoretical models, widely used in the field. Among them, those models based on either the regular solution theory (Rubingh) or a thermodynamic approach (Motomura) or molecular considerations (Nagarajan) do not predict well the mixed micellization process studied herein, while another model based on the Markov chain approach (Georgiev) reproduces quite well the experimental results. Acknowledgment. The authors thank the MCYT of Spain, Project No. BQU2002-00586, for supporting this work. We also thank the CAI de Espectroscopia of UCM. LA034214G
