Transport Properties of Aqueous Solutions of Alkyltrimethylammonium Bromide Surfactants at 25°C

Journal of Colloid and Interface Science 239, 264–271 (2001) doi:10.1006/jcis.2001.7555, available online at http://www.idealibrary.com on

Transport Properties of Aqueous Solutions of Alkyltrimethylammonium Bromide Surfactants at 25◦ C Gerardino D’Errico,1 Ornella Ortona, Luigi Paduano, and Vincenzo Vitagliano Dipartimento di Chimica, Universit`a di Napoli “Federico II,” Complesso di Montesantangelo, via Cinzia, Naples, Italy Received September 28, 2000; accepted March 17, 2001

Intradiffusion coefficients, D, of n-alkyltrimethylammonium bromides [CH3 –(CH2 )n−1 –N(CH3 )3 Br, Cn TAB] (n = 6, 8, 10, 12) in mixtures with heavy water were measured by the PGSE–NMR technique at 25◦ C. The experimental data permitted evaluation of the influence of the alkyl chain length on the surfactant selfaggregation process. For all the surfactants considered, the D trend showed a slope change corresponding to the critical micellar composition (cmc). In the premicellar composition range, D decreased linearly with the square root of the surfactant molality. The D values extrapolated at infinite dilution were related to the limiting mutual diffusion coefficients, determined through the Taylor dispersion technique. In the micellar composition range, solubilized tetramethylsilane (TMS) molecules were used to determine the micelle intradiffusion coefficient, DM , from which the aggregate radii and the aggregation numbers were obtained. The decreasing trend of DM with increasing surfactant molality was interpreted in terms of interparticle electrostatic repulsion. DM values allowed evaluation of the Gouy–Chapman layer thickness. The solvent intradiffusion coefficient in the heavy water–Cn TAB mixtures, Dw , was also measured. It decreased with increasing surfactant molality. For n = 8, 10, 12 the Dw trend presented a slope change at the cmc, which could be ascribed to the strong decrease in hydration of surfactant molecules upon micellization. Because of its short hydrophobic tail, C6 TAB exhibited peculiar aggregation behavior. Its cmc, which is poorly marked, is lower than the value predicted by extrapolating the cmc values obtained for the other terms of the series. The C6 TAB aggregates do not solubilize TMS molecules; the estimated aggregation number is extremely low (∼3). Finally, no abrupt slope change in the solvent intradiffusion coefficient trend was detected. This evidence suggests that C6 TAB molecules do not micellize in aqueous solution, but form trimers in which the surfactant hydrophobic tails are not hidden from contact with water molecules. °C 2001 Academic Press Key Words: micelle(s); n-alkyltrimethylammonium bromides; intradiffusion coefficient; limiting mutual diffusion coefficient.

INTRODUCTION

The association of surfactant molecules into micellar aggregates in aqueous solution leads to reduction of the energetically unfavorable contact between water molecules and the surfactant 1 To whom correspondence should be addressed. E-mail: derrico@chemistry. unina.it.

0021-9797/01 $35.00

C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.

apolar groups, called tails, while the polar groups, called heads, are still exposed to the aqueous medium. The aggregation properties of ionic surfactants depend on the balance between the hydrophobic interactions among the tails and the electrostatic repulsion among the heads (1). The micellization process is not a first-order transition; in fact, it appears through a composition range whose extension depends on the length of the surfactant hydrophobic tail. Within a series of surfactants with the same polar heads, the terms with longer tails present a micellization process quite sharp so that the assumption that it occurs at a unique surfactant critical micellar composition, the cmc, is often taken. However, micellization of surfactants with shorter hydrophobic chains occurs through a large composition range; leading to the formation of loose and not well-defined micelles (2, 3). The extent to which the length of the hydrophobic chain influences the aggregation properties depends on the surfactant class considered. In the past we have extensively studied micellar systems formed by the anionic sodium sulfonate surfactants with a hydrophobic chain composed of a number of carbon atoms, n, varying from 5 to 11 (4, 5). It was found that the shorter terms of the series have aggregation behavior similar to that of the longer ones, and the micellization parameters, such as cmc value and aggregation number, vary regularly within the series. In this work, we present a similar investigation on a cationic surfactants class, the n-alkyltrimethylammonium bromides [CH3 –(CH2 )n−1 –N(CH3 )3 Br, Cn TAB]. Various studies are present in the literature concerning the micellar behavior of these surfactants (6–12); however, most of them deal with the longer terms of the series (n > 8). In this paper we report and comment on intradiffusion coefficients, determined through the PGSE–NMR method, and limiting interdiffusion coefficients, determined through the Taylor dispersion technique, for the Cn TAB (n = 6, 8, 10, 12) surfactants in aqueous solution. Our focus is on the similarities and differences of the aggregation behavior of the shorter term of the series, C6 TAB, in respect to that of the terms with longer hydrophobic tail. An ionic micelle can be represented as a spherical aggregate whose inner core region is formed by the surfactant hydrophobic tails. Because of micellar aggregation, the number of water molecules involved in the surfactant hydration shell is strongly reduced. The surfactant charged head groups are located on the

264

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DIFFUSION OF Cn TAB SURFACTANT SOLUTIONS

aggregate surface, in contact with the aqueous medium. A fraction of counterions are condensed on the micellar surface, forming the so-called Stern layer, in order to reduce the electrostatic repulsion among the head groups. The residual charge density on the micelle surface produces an electric potential difference with the medium surrounding the micelle. Because of this potential difference, in the aqueous medium close to the micelle surface there are excess unbound counterions. The counterion concentration decreases continuously from the micelle surface to the bulk solution. However, a simple approximate model is often assumed, in which the excess counterions are confined in a well-defined region, called the Gouy–Chapman layer, surrounding the micelle. In this work we show that surfactant intradiffusion measurements are a good tool for estimating the dimension of the micelles, as well as those of the Stern and Gouy–Chapman layers. We also monitor the reduction of surfactant hydration by measuring the solvent intradiffusion coefficient in all the mixtures. Our results show that, among the considered surfactants, C6 TAB is borderline. In fact, because of its short hydrophobic tail, it forms very small oligomers. Analysis of the solvent intradiffusion coefficient makes it possible to infer that in these oligomers the surfactant hydrophobic tails are still strongly hydrated by water molecules. EXPERIMENTAL SECTION AND RESULTS

Materials. Hexyltrimethylammonium bromide (C6 TAB, Mw = 224.18), octyltrimethyl ammonium bromide (C8 TAB, Mw = 252.24), decyltrimethylammonium bromide (C10 TAB, Mw = 280.29), and dodecyltrimethylammonium bromide (C12 TAB, Mw = 308.35) were Fluka products with stated purity >98%. All solutes were reagent grade and were used without further purification. For the density, viscosity, and Taylor dispersion measurements doubly distilled water was used as solvent; for the PGSE–NMR measurements D2 O obtained from Sigma (>99.96% isotopic purity) was used. All solutions were prepared by weight. As is discussed later, solubilized tetramethylsilane (TMS, Sigma, purity >99.9%) was used in the micellar composition range to measure the micelles’ intradiffusion coefficients. Intradiffusion measurements. The intradiffusion coefficients measurements were performed on a Varian FT80 NMR spectrometer using 1 H NMR at 80 MHz with an internal deuterium lock. The spectrometer was equipped with a pulsed magnetic field gradient unit, especially made by Stelar. All measurements were performed at 25.0 ± 0.1◦ C; a Varian Model 87 temperature controller (VTC87) was used to keep the sample temperature constant within 0.1◦ C. The pulsed-gradient spin– echo method described in Refs. (13, 14) was used. Individual signal amplitudes are described by the equation ¶¸ · µ δ , A = A0 exp −γ 2 g 2 Dδ 2 1 − 3

where γ is the gyromagnetic ratio of the nucleus, D is the intradiffusion coefficient of the species responsible for the NMR signal, g is the gradient strength, and δ and 1 are the length and spacing of the gradient pulses. Echo delays were kept fixed so that the effects of relaxation were constant and need not be taken into consideration. The gradient pulse lengths were varied over a range suitable for observing the decay of the spin-echo signal A. D was obtained by applying a nonlinear least-squares routine to the decay of A as a function of δ. Measurements to establish the values of g were performed on a reference sample with known intradiffusion coefficient. In this work we used heavy water with trace amounts of light water (DHDO = 1.872 × 10−9 m2 s−1 , Ref. 15). The intradiffusion coefficients of the alkyltrimethylammonium cations, D, were measured following the signal intensities of the CH3 groups’ protons adjacent to the nitrogen atom (chemical shift = 3.2). The solvent intradiffusion coefficient, D w , was determined following the signal intensities of the OH groups present as impurity in the heavy water (chemical shift = 4.8). The experimental errors on the intradiffusion coefficients were generally less than 2%. In the intradiffusion coefficient measurements, heavy water was used as solvent for NMR field/frequency lock purposes and in order to enhance the NMR signals of interest. The isotopic substitution of the solvent might result in an alteration of the structural properties of the micellar aggregates. Berr (16) showed that these differences are very small and become appreciable only for surfactants with long hydrophobic chains. Furthermore, the solubilization of guest molecules in micellar system is almost unaffected by the deuteriation of surfactant, solvent, or both (17). For these reasons we neglected the effect of solvent isotopic substitution. To evaluate the intradiffusion coefficients in light water from those measured in heavy water, the experimental data were multiplied by the ηD∗ 2 O /ηH∗ 2 O = 1.23 ratio (18), where ηD∗ 2 O and ηH∗ 2 O are the viscosities of pure heavy and light water, respectively. Surfactants and solvent intradiffusion data are collected in Table 1 and shown in Figs. 1 and 2. Taylor dispersion measurements. The Taylor dispersion method used for determining the mutual diffusion coefficient in liquid solutions, Dmd , has been widely described in the literature (19, 20). It is based on the combined effect of convective and diffusive flow generated by the slow running of a solution, called carrier, in a long capillary tube and the dispersion of a very small amount of a solution at a slightly different concentration injected into the carrier. The dispersion profile can be analyzed by fitting the following equation to the refractometer voltage, which is the instrumental output, µ v(t) = B0 + B1 t + vmax

[1]

tmax t

¶ 12

¸ · 12Dmd (t − tmax )2 , exp − r 2t [2]

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D’ERRICO ET AL.

TABLE 1 Intradiffusion Coefficients for the Systems D2 O–Cn TAB at 298.15 K m (mol kg−1 )

D × 109 (m2 s−1 )

0.0316 0.0532 0.0872 0.1020 0.1699 0.3427 0.6024 0.9302 1.0990 1.3580 1.5169 1.9832 2.1324 2.2739 2.6795 2.9271 3.0324 3.6654

0.603 0.585 0.578 0.563 0.552 0.513 0.477 0.382 0.348 0.310 0.295 0.261 0.250 0.248 0.220 0.204 0.203 0.165

0.0081 0.0136 0.0256 0.0311 0.0383 0.0539 0.0549 0.0663 0.0673 0.1082 0.1397 0.1577 0.1647 0.2094 0.2283 0.3345 0.4118 0.5618

0.496 0.486 0.473 0.472 0.463 0.461 0.454 0.391 0.374 0.274 0.207 0.193 0.187 0.160 0.145 0.112 0.0938 0.0817

D w × 109 (m2 s−1 )

DM × 109 (m2 s−1 )a

m (mol kg−1 )

D × 109 (m2 s−1 )

D w × 109 (m2 s−1 )

DM × 109 (m2 s−1 )

0.0095 0.0917 0.1910 0.3410 0.4826 0.0868 0.1300 0.1450 0.1912 0.2118 0.2200 0.2764 0.2810 0.4204 0.4848 0.4870 0.5740 0.7070 0.7340 0.7800 0.9740 1.1453 1.3273

0.561 0.554 0.549 0.533 0.534 0.525 0.511 0.510 0.481 0.482 0.469 0.396 0.383 0.349 0.305 0.327 0.285 0.251 0.240 0.234 0.182 0.148 0.113

1.86 1.83 1.80 1.70 1.72 1.64 1.51 1.52 1.42 1.32 1.34 1.27 1.25 1.19 1.21 1.18 1.13 1.05 1.10 1.05 1.02 0.911 0.875

0.0918 0.0796 0.0749 0.0735 0.0725 0.0700 0.0683

0.0049 0.0063 0.0097 0.0101 0.0114 0.0123 0.0131 0.0136 0.0162 0.0201 0.0256 0.0343 0.0596 0.0661 0.0780 0.0861 0.0976 0.1475 0.2289

C12 TAB 0.461 0.460 0.454 0.458 0.449 0.458 0.454 0.446 0.378 0.317 0.294 0.217 0.166 0.144 0.117 0.106 0.0994 0.0774 0.0659

1.86 1.85 1.84 1.83 1.83 1.83 1.83 1.83 1.82 1.82 1.82 1.81 1.81 1.80 1.80 1.79 1.79 1.78 1.72

0.0821 0.0809 0.0788 0.0768 0.0749 0.0703 0.0613

C6 TAB 1.76 1.80 1.68 1.70 1.61 1.50 1.27 0.982 0.906 0.688 0.693 0.518 0.415 0.389 0.274 0.217 0.187 0.110

C8 TAB

0.191 0.182 0.171 0.169 0.164 0.157 0.163 0.144 0.131 0.134 0.103

C10 TAB

a

1.83 1.85 1.80 1.79 1.79 1.72 1.74 1.68 1.70 1.66 1.63 1.67 1.64 1.62 1.63 1.58 1.53 1.53

0.101 0.0972 0.0950 0.0941 0.0877 0.0846 0.0737 0.0691 0.0610

Data computed through Eq. [8].

where B0 + B1 t is the baseline voltage, vmax the maximum peak height, tmax the time length after which the maximum is reached, and r the inner radius of the tube. The limiting intradiffusion coefficients of the cationic sur∞ , were measured by injecting factants under consideration, Dmd into a water stream an aqueous solution of the surfactant at molality largely below the cmc (21). Several experimental runs at decreasing molality were performed for each considered surfac∞ values were calculated by extrapolating tant. The limiting Dmd

√ the experimental Dmd data to m → 0, where m is the surfactant molality. The values obtained are collected in Table 2. Viscosity and density measurements. For all the systems considered, the viscosity at the cmc, ηcmc , was measured to evaluate, through the Stokes–Einstein relation, the hydrodynamic radii of the micellar aggregates from their intradiffusion coefficients. The viscosity measurements were carried out with an Ubbelohde viscometer with a relatively long water flow time

267

DIFFUSION OF Cn TAB SURFACTANT SOLUTIONS

FIG. 2. Cn TAB aqueous solutions: solvent intradiffusion coefficients. FIG. 1. Cn TAB aqueous solutions: (d) surfactant intradiffusion coefficients, (h) C6 TAB intradiffusion coefficients computed through Eq. [8], (j) TMS intradiffusion coefficients.

(203.00 ± 0.01 s) to minimize the kinetic energy correction and to have more accurate data for dilute solutions. All the measurements were carried out in a water bath (mgw Lauda CS) at 25.00 ± 0.01◦ C. At least three runs were performed on each solution with flow times differing no more than 0.05 s among them. The estimated error on measured relative viscosity data is less than 8 × 10−4 . For all the systems considered, the density at the cmc, d cmc , necessary to compute the dynamic viscosity from the kinematic one was measured with an Anton Paar DMA 60 vibrating tube densimeter operating at 25.00 ± 0.01◦ C, using distilled water and air (at measured pressure and humidity) for the calibration. Viscosity and density data are collected in Table 2. The experimental ηcmc values can be compared with those computed through the relation (22) ¢ ¡ η = ηH∗ 2 O 1 + Am 1/2 + Bm ,

[3]

where ηH∗ 2 O is the water viscosity, A and B are constants characteristic of each alkyltrimethylammonium bromide surfactant,

and m is the surfactant molality. The literature A and B values are reported in Table 2. ηcmc can by computed from Eq. [3], with m = cmc. The ηcmc values obtained in this way are in agreement with the experimental ones (see Table 2). DISCUSSION

cmc evaluation and premicellar composition range. For all the systems considered, the intradiffusion coefficient of the surfactant ion, D, plotted as a function of the surfactant molality, shows a change of slope at the onset of micellization, allowing the determination of the cmc (see Fig. 1). The change of slope becomes more evident with increasing surfactant tail length; it is extremely sharp in the case of C12 TAB, while it becomes very broad in the case of C6 TAB. The measured cmc values are collected in Table 2, where they are compared with literature values. In the case of ionic surfactants, the plot of log10 (cmc) vs the number of carbon atoms in the surfactant hydrocarbon chain, n, is usually found to show a linear trend with slope 0.29–0.30 (4). We found, for 8 ≤ n ≤ 12 (see Fig. 3), log10 (cmc) = 1.76(±0.11)–0.299(±0.011)n.

[4]

TABLE 2 Critical Micellar Composition and Physicochemical Properties for the Systems D2 O–Cn TAB at 298.15 K ∞ × 109 Dmd (m2 s−1 )

C6 TAB C8 TAB C10 TAB C12 TAB a

1.16 1.04 1.01 0.917

Experimental data. Data from Ref. (22). c Data computed through Eq. [3]. d Data from Ref. (12). e Interpolated data. b

a

d cmc (kg dm−3 )

ηcmc (cp)

1.013569 1.003016 0.998629 0.997415

1.16 1.02 0.920 0.898

A × 102 (mol−1/2 kg1/2 )

Bb (mol−1 kg)

0.75b 0.79e 0.83e 0.87b

0.48 0.65 0.79 0.91

c

ηcmc (cp)

1.15 1.02 0.936 0.903

cmc (mol kg−1 )

cmcd (mol kg−1 )

0.597 0.225 0.0618 0.0143

0.509 0.290 0.0661 0.0146

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D’ERRICO ET AL.

FIG. 3. Cn TAB aqueous solutions: logarithms of cmc values (in molality), log10 (cmc), as a function of the hydrocarbon chain length.

C6 TAB presents a cmc (0.60 mol kg−1 ) much lower than that predicted through Eq. [4] (0.92 mol kg−1 ). This evidence is in agreement with the results obtained by Mosquera et al. (12). These authors ascribed this deviation to the partial exposure of the C6 TAB hydrophobic tail to the solvent even when micellized. In the premicellar composition range the intradiffusion coefficients may be fitted as a function of the square root of the ionic strength (23), as is usual for electrolyte solutions: D=D

¡ ∞

1 − αI

¢ 1/2

.

[5]

The fitting parameters, D ∞ and α, are reported in Table 3. The intradiffusion coefficients extrapolated to infinite dilution, D ∞ , can be compared with those computed by the Nernst relationship D∞ =

RT ∞ λ , J- 2 +

[6]

where J- is the Faraday constant and λ∞ + is the limiting conductivity of the surfactant cation (24). The limiting intradiffusion coefficient of the surfactant cation, D ∞ , is related to the limiting mutual diffusion coefficient mea∞ sured through the Taylor dispersion technique, Dmd , which accounts for the diffusion of both the surfactant cation and the

FIG. 4. Cn TAB aqueous solutions: limiting surfactant intradiffusion coefficients, D ∞ , as a function of the hydrocarbon chain length. (d) Experimental values, (r) values computed through Eq. [6], (m) values computed through Eq. [7].

counterion, through the relation

D

∞

· =

2 1 ∞ − ∞ Dmd DBr −

¸−1

,

[7]

∞ −9 m2 s−1 is the Br− limiting inwhere DBr − = 2.081 × 10 tradiffusion coefficient (25). The D ∞ values computed through Eqs. [6] and [7] are collected in Table 3 and shown in Fig. 4. The experimental and the computed D ∞ values are in good agreement. The α values in Eq. [5] increase with the hydrophobic chain length; this evidence indicates that the hydrophobic interaction among the tails also enhances the surfactant–surfactant interaction in the premicellar composition range. Micellar composition range. Given the rapid exchange between free and micellized surfactant molecules, in the micellar composition range the experimental surfactant intradiffusion coefficient is a mean value between that of free monomers, DF , and that of the micellized molecules, DM (26). Thus

D = pF DF + (1 − pF )DM =

mF (1 − m F ) DF + DM , [8] m m

TABLE 3 Interaction Parameters and Limiting Diffusion Coefficients for the Systems D2 O–Cn TAB at 298.15 K D ∞ × 109 (m2 s−1 ) C6 TAB C8 TAB C10 TAB C12 TAB a b

0.781 0.721 0.636 0.588

Computed through Eq. [6]. Computed through Eq. [7].

D ∞ × 109 (m2 s−1 )

D ∞ × 109 (m2 s−1 )

cmc × 109 DM (m2 s−1 )

0.788 0.706 0.650 0.602

0.806 0.689 0.667 0.588

0.245 0.151 0.134 0.110

a

α

(mol−1/2

kg1/2 )

0.323 0.384 0.51 0.5

b

AM (mol−1 kg)

BM (mol−2 kg2 )

−0.143 −0.99 −1.51 −2.1

– 0.54 1.1 2.9

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DIFFUSION OF Cn TAB SURFACTANT SOLUTIONS

where pF is the fraction of amphiphile in the monomeric state and m F is the free monomer molality. DM can be estimated experimentally by the addition of tetramethylsilane to the system. In fact, for a compound which is entirely confined to the micelles and has a negligible solubility in the intermicellar solution, the observed intradiffusion coefficient will be the same as the intradiffusion coefficient of the micelles (27). With this purpose, we added TMS in trace amounts to our solutions. TMS is a strongly hydrophobic molecule; in the premicellar composition range it diffuses to the air–water interface in the absence of solubilization sites in solution (28), while in the micellar composition range it is solubilized in the micellar core. To be sure that the TMS insertion does not change the shape and dimension of micelles, two measurements were performed for each solution, before and after the TMS addition, checking that the surfactant intradiffusion coefficients were the same. In ionic micellar solutions, electrostatic repulsion should prevent intimate micelle–micelle contacts, barring collisional transfer of solubilized molecules that could lead to higher TMS intradiffusion values. For these reasons following the TMS NMR signal one can obtain a reliable estimation of the micelle intradiffusion coefficient, DM . We followed this approach for the Cn TAB surfactants with n ≥ 8. Concerning C6 TAB, we did not succeed in solubilizing TMS into the aggregates even in very concentrated solutions. This evidence suggests that C6 TAB forms aggregates whose inner core is not hydrophobic enough to act as a solubilization site for TMS molecules. The alternative hypothesis of no aggregate formation is denied by the presence of a cmc in the intradiffusion trend. Since the C6 TAB aggregates are not able to solubilize TMS molecules, in this case a different approach was followed to estimate the DM value. We assumed that above the cmc the monomers’ intradiffusion coefficient, DF , is constant and equal to the surfactant intradiffusion coefficient measured at the cmc, D cmc = 0.6 × 10−9 m2 s−1 . In this case DM can be estimated through Eq. [8], assuming m F = cmc in the whole micellar composition range. The assumptions made do not take into consideration the obstruction effect exerted by the micelles on the monomers’ intradiffusion coefficient, which should lead to a decreasing DF trend. As a consequence DM could be underestimated. However, this error increases with increasing surfactant

molality, so that the value extrapolated at the cmc should be unaffected. On the other hand, the slope of the computed DM trend with surfactant concentration is unreliable. The micelle intradiffusion coefficients, DM , are collected in Table 1. Inspection of Fig. 1 shows that DM → D as the surfactant molality increases and the monomeric contribution becomes negligible. This is a further confirmation that the addition of the solubilizate does not perturb the micelles’ shape and size. In the pseudophase transition model the surfactant molecules start to micellize at the cmc, which can be considered as infinite dilution for micelles. For this reason, the concentration dependence of DM can be expanded as a polynomial in (m − cmc), cmc b1 + AM (m − cmc) + BM (m − cmc2 ) + · · ·c. [9] D M = DM

The fitting parameters of Eq. [9] are collected in Table 3. It is possible to relate the micelle intradiffusion coefficients cmc , to the hydrodynamic size of the extrapolated at the cmc, DM aggregates by using the Stokes–Einstein equation (29) to calculate the apparent radius, r , r=

kB T cmc , 6π ηcmc DM

[10]

where ηcmc is the viscosity of surfactants solution at the cmc, as determined experimentally. The r values obtained from Eq. [10] are shown in Table 4. They can be compared with the alkyl chain lengths computed according to the Tanford relation (1) l = 0.15 + 0.1265n

[11]

where l is approximately the hydrophobic core radius (in nm). The difference δS = r − l is related to the surfactant hydrophilic heads, the bound counterions, and their hydration water; it accounts for the so-called Stern layer surrounding the micellar hydrophobic core. As can be seen in Table 4, δS increases with the length of the hydrophobic chain; this evidence could be ascribed to the increased fraction of bound counterions, β, from 0.65 for C8 TAB (30) to 0.79 for C12 TAB (31). In the case of C6 TAB, δS is negative. For surfactants with very short tails the assumption that the radius of the aggregate hydrophobic core

TABLE 4 Aggregation Number and Structural Parameters of Cn TAB Micelles

C6 TAB C8 TAB C10 TAB C12 TAB a b

r (nm)

l (nm)

δS (nm)

V¯ Cn TAB,M (cm3 mol−1 )

0.773 1.41 1.77 2.21

0.909 1.16 1.41 1.67

−0.135 0.25 0.35 0.54

191.79a 228.7b 262.2b 296.2b

Data from Ref. (12). Data from Ref. (32).

s

sa

r0 (nm)

δGC (nm)

κ −1 (nm)

3 17 31 57

3 20 40 55

0.468 1.58 2.22 3.00

−0.305 0.165 0.455 0.795

0.393 0.640 1.222 2.541

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coincides with the length of the hydrophobic tail is not realistic, so the value obtained through Eq. [11] could be overestimated. The aggregation number, s, of the various surfactants can be estimated from the micelle hydrodynamic radii, r . Assuming a spherical shape, the following relation holds, s=

4 πr 3 3

V¯ Cn TAB,M + hVD∗2 O

,

[12]

where V¯ Cn TAB,M is the partial molar volume of the micellized surfactant (12, 32) and VD∗2 O is the molar solvent volume. h is the hydration number of the hydrophilic heads, which is reported in the literature (33) to be ∼10. The s values obtained through Eq. [12] are collected in Table 4, where they are also compared with some literature values. Intermicellar interactions. The dependence of DM on the surfactant molality, as described by the parameters AM and BM in Eq. [9], is generally ascribed to the combination of two effects: change of micelle size and intermicellar interactions. In dilute micellar solution the latter factor is predominant. In a previous work (5), we interpreted the micellar intradiffusion trend for sodium sulfonate surfactants in terms of the obstruction effect due to the micelles themselves. This effect is amplified by the electrostatic repulsion among the aggregates. In dilute micellar solution it is appropriate to express the micelle intradiffusion coefficient as a power series of the obstructing particles’ volume fraction (34), φ: cmc (1 − 1.73φ + 0.88φ 2 + · · ·). DM = DM

[13]

Comparing Eqs. [9] and [13] one obtains AM = −1.73

sφ NA = −1.73VM0 = −2.31π 24 (r 0 )3 , m − cmc 10 [14]

where VM0 is the excluded volume due to the presence of micelles, r 0 is the corresponding radius (in nm), and NA is the Avogadro number. The r 0 values computed through Eq. [14] for the micellar aggregates formed by the alkyltrimethylammonium surfactants are shown in Table 4. Because of the electrostatic interaction, the radius of the spherical excluded volume due to the presence of a micelle, r 0 , is larger than the micelle hydrodynamic radius, r . The difference δGC = r 0 − r can be seen as the thickness of the layer surrounding the micelle, where other micelles cannot enter because of the repulsion between the two charged surfaces; it must be proportional to the dimension of the Gouy–Chapman layer. The thickness of this layer is usually identified with the Debye length, κ −1 (5). As shown in Fig. 5, κ −1 and δGC show the same increasing trend when reported as a function of I −1/2 , where the ionic strength was assumed to be equal to the cmc. This evidence confirms that δGC permits a realistic view of the electrostatic intermicellar repulsion.

FIG. 5. Cn TAB aqueous solutions: Gouy–Chapman layer thickness. (j) δGC , (r) κ −1 .

Solvent intradiffusion coefficient. For the longer terms of the series (Cn TAB, n = 8, 10, 12), the solvent intradiffusion coefficient trend shows a marked slope change at the cmc. This evidence can be ascribed to a change in the surfactant molecules’ hydration due to their micellization (35). The evaluation from our data of the surfactant hydration number in the premicellar and micellar composition range would require many assumptions concerning, for example, the counterions’ intradiffusion coefficients, the counterion hydration number, and the change of this value due to the micellization; these assumptions would make the results questionable. For this reason we prefer to limit ourselves to a qualitative discussion. In the premicellar composition range the solvent intradiffusion coefficient, D w , decreases with increasing surfactant molality mainly because a fraction of the solvent molecules are involved in the surfactant hydration shell. The micellization process reduces the number of hydrating solvent molecules, leading to a less decreasing D w trend (see Fig. 2). In the case of C6 TAB aqueous solutions, no marked slope change was evidenced (see Fig. 2), but D w shows a smoothly decreasing trend whose slope becomes less negative with increasing surfactant molality; this evidence could be ascribed to a gradual decrease of the hydration number, without an abrupt change in the organization of the surfactant molecules’ hydration. CONCLUSIONS

This paper has provided intradiffusion coefficients for an important class of ionic surfactants, the n-alkyltrimethylammonium bromides (Cn TAB, n = 6, 8, 10, 12). The cmc values and the surfactants aggregation numbers were determined. The micelles mobility is affected by the obstruction caused by their own excluded volume. This effect is enhanced by the intermicellar electrostatic repulsion. The measured hydrodynamic dimensions permit the determination of the thickness of the Stern and Gouy–Chapman layers related to the counterions condensed at

DIFFUSION OF Cn TAB SURFACTANT SOLUTIONS

the micelle external surface. The trend of the solvent intradiffusion coefficient shows the decrease of the surfactant molecules’ hydration caused by the micellization process. By comparing the results obtained for the various considered surfactants, it is evident that the self-aggregation behavior becomes less marked and cooperative with decreasing the number of carbon atoms in the tail. C6 TAB is a borderline case in which the self-aggregation process occurs through a large composition range, leading to the formation of very small aggregates, in which the C6 TAB molecules are still exposed to the aqueous medium. Furthermore, for C6 TAB there is no experimental evidence of counterions’ condensation at the aggregates’ external surface. Our findings, together with others reported in the literature (12), call into question whether C6 TAB aggregates should be considered micelles. We think preferable to discuss the C6 TAB association in terms of a gradual change from solvent–mediate to direct solute–solute interactions (36) with increasing surfactant concentration. The prevalence of direct hydrophobic interaction among surfactant tails at high C6 TAB concentration is favored by the increased ionic strength of the solution, which weakens the ionic repulsions among the hydrophilic heads. ACKNOWLEDGMENT This research was carried out with the financial support of the Italian MURST.

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