Monomer Exchange Kinetics, Radial Diffusion, and Hydrocarbon

The low-frequency term with concentration-dependent relaxation time, τ1, between 0.06 and 3.5 μs is discussed in terms of the surfactant monomer exc...
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J. Phys. Chem. B 2007, 111, 1625-1631

1625

Monomer Exchange Kinetics, Radial Diffusion, and Hydrocarbon Chain Isomerization of Sodium Dodecylsulfate Micelles in Water R. Polacek and U. Kaatze* Drittes Physikalisches Institut, Georg-August-UniVersita¨t, Friedrich-Hund-Platz 1, 37077 Go¨ttingen, Germany ReceiVed: October 24, 2006; In Final Form: December 20, 2006

Molecular relaxation properties of sodium dodecylsulfate in aqueous solutions with surfactant concentrations between 0.009 and 0.4 mol/L have been studied using broadband ultrasonic spectrometry in the frequency range 0.1-2000 MHz. Ultrasonic excess attenuation characteristics were found that could be well represented by a sum of three relaxation terms, each one characterized by a discrete relaxation time. The low-frequency term with concentration-dependent relaxation time, τ1, between 0.06 and 3.5 µs is discussed in terms of the surfactant monomer exchange. The noticeable effect from the incomplete dissociation of the surfactant counter ions and the variation of the monomer concentration following thereby is discussed. The second relaxation term (0.9 e τ2 e 2.5 ns) is assigned to the limited radial diffusion of monomers within the micelles, yielding a mean diffusion length of 0.5 nm, in correspondence to protrusions by four methyl groups. The high-frequency relaxation term (0.1 e τ3 e 0.2 ns) reflects the structural isomerizations of the hydrocarbon chains in the micellar cores, largely resembling those of liquid alkanes.

1. Introduction

and, thus, by a well-defined change

Association colloids formed by amphiphilic molecules play a significant role in the biosphere and also in a multitude of technological processes, such as pharmaceutical formulations, cosmetics fabrications, and paper production. In water, amphiphilic molecules spontaneously self-organize into liquidlike structures, of which the best understood seems to be the micelle.1-5 At concentrations above the critical micelle concentration, cmc, micellar systems constitute a dynamical equilibrium in which the amphiphilic molecules continuously exchange between the aggregates and the bulk phase. The exchange kinetics of ionic, zwitterionic, and nonionic micelle solutions has been intensively investigated in the past utilizing a variety of measurement methods. Proper micelle systems have been found well-represented by the Aniansson-Wall model6-10 of micelle formation. In this context, the insufficiently defined term “proper micelles” means almost globularly shaped structures with a mean aggregation number, m j , larger than ∼50, corresponding to a critical micelle concentration smaller than 0.01 mol/L in water. The Aniansson-Wall model proceeds from an isodesmic reaction scheme, kfi

A1 + Ai-1 y\rz Ai ki

(1)

with A1 denoting an amphiphile, and Ai, a multimer made of i ) 2, 3, ... monomers. Parameters kfi and kri are the forward and reverse rate constants, respectively, of the ith step in the coupled reaction scheme. Hence, each step is assumed to be characterized by a stability constant,

Ki ) kfi /kri

(2)

* Corresponding author. Fax: +49-551-39-7720. E-mail address: [email protected].

∆G0i ) RT ln(Ki)

(3)

in the free energy (i ) 2, 3, ...). In eq 3, R denotes the gas constant, and T, the temperature. It is presumed that the monomer concentration, [A1], is significantly larger than that of any aggregate, [Ai], so that direct association of oligomers Ai + Aj ) Ai + j; i, j > 1, can be neglected. Also empirically introduced is a symmetrically bell-shaped, nearly Gaussian-sized distribution of micelles. r [A h i+1] - kri [A h i])/[A h i] ) -(i - m j )kr/σ2 (ki+1

(4)

Here, [A h i] denotes the equilibrium concentration of species Ai, m j is the mean aggregation number, and σ2 is the variance of the size distribution. Parameter kr is the reverse rate constant for micellar sizes around the mean (i ≈ m j ). According to this model, the system is predicted to, after a small disturbance, approach thermal equilibrium following a function that is characterized by two relaxation times. These characteristic times can be identified in a suggestive way with particular modes in the reformation of the equilibrium size distribution. The fast relaxation process, with relaxation time τf on the order of nanoseconds and microseconds, is related to the monomer exchange at an almost constant number of micelles but slightly changing mean aggregation number m j . The slow process proceeding in parallel, with relaxation time τs in the range of milliseconds or even seconds, establishes the final equilibrium distribution by a readjustment of the number density of the micelles. The relaxation phenomena associated with the coupled isodesmic scheme of reactions can be favorably studied by ultrasonic spectrometry and its time domain analogues, particularly pressure jump methods.11 On the basis of the Aniansson-Wall model, the Teubner-Kahlweit theory yields the following relations for the amplitude Af and relaxation time

10.1021/jp066974g CCC: $37.00 © 2007 American Chemical Society Published on Web 01/30/2007

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Polacek and Kaatze

τf of the fast relaxation term in the spectra:12,13

Af )

(

π(∆V)2cmc σ2 σ2 1 + X X m j m j κ∞RT s

r -2 + X/m j) τ-1 f ) k (σ

)

TABLE 1: Mass Fraction, Y, Molar Concentration, c, Density, G, Shear Viscosity, ηs, and Sound Velocity, cs, of Aqueous Solutions of Sodium Dodecylsulfate at 25 °C

-1

(5) 102Y

(6)

Herein, ∆V is the reaction volume, assumed to be identical for all steps in scheme (1); κ∞s is the adiabatic compressibility, extrapolated to frequencies well above the relaxation region; and

X ) (c - [A1])/[A1]

(7)

is the scaled concentration, with c ) ∑ii[Ai] denoting the total surfactant concentration. Recent broadband ultrasonic measurements of ionic and nonionic micelle systems14-17 with surfactant concentrations, c, near the cmc, feature characteristic differences to the predictions of the Teubner-Kahlweit model. The fast relaxation process displays a relaxation time distribution instead of a discrete relaxation time. The relaxation rate, τ-1 f , obtained from the principal relaxation time of the distribution function does not monotonically increase with X, and in addition, there exists an ultrafast relaxation term with relaxation frequency τuf on the order of 100 ps. These special characteristics in the ultrasonic relaxation spectra have been assigned to properties in the micelle size distribution of short-chain surfactant solutions, especially to the comparatively large concentration of oligomeric species. The agreement between an extended model of stepwise association18 and the experimental results for solutions of nonproper micelles from short-chain surfactants, however, is imperfect. It is, thus, interesting to first inspect ultrasonic relaxation data of a proper micelle system in terms of the existing models. We thus measured at various solute concentrations broadband ultrasonic attenuation spectra for the prominent sodium dodecylsulfate/water surfactant system. The results of the measurements are discussed to show the appropriateness and limitations of the Teubner-Kahlweit-Aniansson-Wall model of the micelle formation/decay kinetics. 2. Experimental 2.1. Surfactant Solutions, Auxiliary Measurements. Sodium dodecylsulfate (SDS, CH3(CH2)11SO4-Na+, Fluka, > 99%) was used as delivered by the manufacturer. Aqueous solutions were prepared by weighing appropriate amounts of the surfactant into suitable volumetric flasks and adding doubly distilled, deionized, and UV-sterilized water up to the measurement line. The density, F, of the samples was measured to within ∆F/F < 0.002 using a pycnometer (20 mL) that had been calibrated against deionized, distilled, and degassed water. The shear viscosity ηs of the liquids was determined with an error smaller than ∆ηs/ηs ) 0.02 utilizing a falling ball viscometer (B/BH, Haake, Karlsruhe, Germany). At some concentrations, an Ubbelohde capillary viscometer (Schott, Mainz, Germany) was additionally employed. The sound velocity, cs, of the liquid samples was obtained as a byproduct from the measurements of ultrasonic attenuation coefficients19 (see Section 2.2). At frequencies ν < 23 MHz, cs was derived from successive resonance peaks of the cavity resonator cells, carefully considering the nonequidistant distribution of the resonance frequencies. At higher frequencies, cs values have been calculated from the waviness in the transfer function of the sample cells at varying

0.260 0.289 0.347 0.405 0.722 1.155 1.443 1.731 2.162 3.738 5.166 7.157 11.40

c F 103ηs cs mol l-1 (0.2% g cm-3 (0.2% Pas (0.2% ms-1 (0.5% 0.009 0.010 0.012 0.014 0.025 0.040 0.050 0.060 0.075 0.130 0.180 0.250 0.400

0.9974 0.9969 0.9974 0.9978 0.9979 0.9986 0.9993 0.9994 1.0006 1.0029 1.0048 1.0074 1.0119

0.855 0.882 0.869 0.859 0.926 0.953 0.960 0.988 1.024 1.228 1.305 1.446 1.716

1498 1497 1499 1498 1499 1498 1497 1499 1498 1497 1498 1497 1496

sample thicknesses, resulting from multiple reflections of the acoustical signal at small transducer spacing. The error in the sound velocity data is ∆cs/cs ) 0.005 at 100 kHz < ν e 500 MHz and ∆cs/cs ) 0.01 at 500 MHz < ν e 2 GHz. A survey of the SDS solutions and their densities, shear viscosities, and sound velocities is given in Table 1. 2.2. Broadband Ultrasonic Attenuation Spectrometry. Between 0.1 and 2000 MHz, the ultrasonic attenuation coefficient R has been measured as a function of frequency, ν, applying two different methods of measurement. At frequencies below 23 MHz, a resonator method was adapted to the rather small R of the samples. With this method, the path length of interaction of the acoustical field with the sample liquid is significantly increased by multiple reflections. Five cylindrical cavity resonator cells, completely filled with the liquid, were available. The cells differed from one another mainly by their dimensions and by the shape of their end faces. Two cells with plane faces20 were used, one cell with one plane and one concave transducer,21 and two additional cells with both faces concavely shaped.22 A concave reflector shape leads to a reduction of diffraction effects in the low-frequency part of the measuring range of the cell, but entails also a resonator mode spectrum different from that of biplanar cells. Using the resonator method, the attenuation coefficient R of the sample is determined relative to a reference liquid with matched sound velocity and density. For the present SDS solutions, water was a suitable reference. At frequencies above 3 MHz, absolute R measurements were performed by transmitting pulse-modulated ultrasonic waves through a cell of variable sample length. Four cells have been employed, each one matched to a particular frequency range. The cells differed from one another by the sample volume, V, and by the type of piezoelectric transmitter and receiver transducer: V ) 130 cm3, X-cut quartz disks, 3 e ν e 63 MHz;23 V ) 10 cm3, Y-cut lithium niobate disks sticking to the back face of delay rods made of fused quartz, 30 e ν e 530 MHz;24 V ) 3 cm3, lithium niobate rods, operated according to the Bo¨mmel and Dransfeld25 surface excitation method, 0.5 e ν e 2 GHz;26 V ) 0.5 cm3, zinc oxide films sputtered on the back face of sapphire delay lines, 1.1 e ν e 2 GHz. The temperature, T, of the samples was controlled to within (0.03 K. Temperature gradients within the cells as well as differences in the temperatures of different cells affected the attenuation coefficient data by less than ∆R/R ) 0.001. Variations in the frequency of the synthezised signal generators were smaller than ∆ν/ν ) 0.0001 throughout. Typical errors in the attenuation coefficient data are presented in Table 2. Due to the overlaps of the frequency ranges of different methods,

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J. Phys. Chem. B, Vol. 111, No. 7, 2007 1627

TABLE 2: Experimental Errors in the Sonic Attenuation Coefficient Data ν, MHz 0.1-0.8 0.8-12 12-60 60-400 400-530 530-2000 ∆R/R 0.07 0.04 0.02 0.01 0.015 0.01

specimen cells, and electronic setups, systematic errors exceeding the values given in the Table are unlikely to remain unnoticed. 3. Results and Treatment of Spectra In Figure 1, the ultrasonic attenuation spectra of SDS solutions are displayed at two surfactant concentrations in the frequencynormalized format. Within the measuring range, the R/ν2 data of both spectra decrease with ν to asymptotically reach a frequency-independent value B′ ) limνf∞(R/ν2) at high frequencies. The negative slope (d(R/ν2)/dν < 0 ) in the frequency normalized attenuation coefficient spectra indicates relaxation behavior. Obviously, the total relaxation strength, limνf0(R/ν2) - B′, depends noticeably upon the surfactant concentration. The inset of Figure 1 shows one set of data additionally as an excessattenuation-per-wavelength spectrum. Here,

(Rλ)exc ) Rλ - Bν ) Rλ - csB′ν

(8)

where λ ) cs/ν denotes the wavelength of the ultrasonic field within the liquid. Also represented by dashed lines in that plot are the graphs of Debye-type relaxation terms that result when the measured attenuation-per-wavelength spectrum is represented by the function 3

RDDD(ν) )

∑ i)1

ADiωτi

1 + (ωτi)2

+ Bν

(9)

In this function, the ADi, i ) 1, 2, 3 are amplitudes, the τi are discrete relaxation times, and ω ) 2πν. Since the intermediate relaxation term, with relaxation frequency (2πτ2)-1 on the order of 80 MHz, becomes evident by a shoulder in the (Rλ)exc spectra only (Figures 1, 2), we have also attempted to represent the measured data by alternative spectral functions. Particular efforts have been made to apply the Hill relaxation term,27-29 corresponding to a continuous relaxation time distribution, and to describe the experimental spectra by the function

RDH(ν) )

ADωτD 1 + (ωτD)2

+

AH(ωτH)mH (1 + (ωτH)2sH)(mH+nH)/(2sH)

+ Bν (10)

In this equation D and H distinguish the Debye and the Hill term parameters, respectively. Parameters mH, nH, and sH ∈ (0, 1) control width and shape of the relaxation time distribution. Equation 10 did not allow for a satisfactory representation of the measured spectra within their limits of experimental error. Hence, the Hill term that had been successfully applied to spectra of surfactant solutions with high cmc14,15 turned out to be less suitable here. We finally used a nonlinear least-squares algorithm, minimizing the reduced variance,

χ2 )

1

N

∑

N - J - 1n)1

(

)

(Rλ)n - RDDD(νn, Pj) ∆(Rλ)n

Figure 1. Ultrasonic attenuation spectra in the format R/ν2 versus ν of SDS solutions in water at 25 °C and at two surfactant concentrations: O, 0.18 mol/L; b, 0.4 mol/L. In the inset, the latter spectrum is also shown as excess attenuation per wavelength. Dashed lines indicate the subdivision of the spectrum into three Debye relaxation terms. Full lines are graphs of the sum of these terms with the parameter values given in Table 3.

2

(11)

to fit the three-Debye-term model function (eq 9) to the measured attenuation-per-wavelength spectra. In eq 11 νn,

Figure 2. Ultrasonic excess attenuation-per-wavelength spectra for aqueous solutions of SDS at 25 °C and at five concentrations: 2, 0.025; 3, 0.075; 1, 0.13; O,; and b, 0.4 mol/L. The lines are graphs of the relaxation spectral function RDDD(ν) defined by eq 9 with the parameter values from the nonlinear fitting procedure (Table 3).

n ) 1, ..., N, are the frequencies of measurements; (Rλ)n and ∆(Rλ)n are the data and their experimental errors, respectively, at these frequencies; and Pj, j ) 1, ..., J are the parameters of RDDD. The values of these parameters, as following from the regression analysis, are collected in Table 3. 4. Discussion 4.1. Monomer Exchange. As is obvious from the examples given in Figure 2, the amplitudes as well as the relaxation times of the low-frequency relaxation term in the spectra of the SDS solutions vary significantly with surfactant concentration. In contrast, in the high-frequency region of the spectra ( ν > 10 MHz) the amplitudes increase indeed with c; the relaxation times, however, seem to be largely independent of surfactant content. Corresponding to previous results for ionic surfactant systems,14,30-32 the low-frequency Debye relaxation term is assigned to the fast relaxation in the Teubner-KahlweitAniansson-Wall model, reflecting the exchange of surfactant molecules between micelles and the bulk phase. The amplitudes AD1 and relaxation rates τ-1 1 of the lowfrequency relaxation term (Figure 3) display a remarkable dependence upon the concentration difference, c - cmc , where cmc ) 8.3 × 10-3 mol/L.33 If, as normally done with nonionic surfactant systems, the monomer concentration, [A1], is identified with the critical micelle concentration, the amplitude of the monomer exchange relaxation in the Teubner-KahlweitAniansson-Wall model (eq 5) increases monotonically with c - cmc to asymptotically approach the concentration-independent value,

limAf )

cf∞

R(∆V)2cmc κ∞s RT

(12)

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Polacek and Kaatze

TABLE 3: Parameters of the RDDD(ν) Relaxation Spectral Function (eq 9) for SDS Solutions in Water at 25 °Ca c, mol l-1

103AD1

τ1, ns

103AD2

τ2, ns

0.009 0.010 0.012 0.014 0.025 0.040 0.050 0.060 0.075 0.13 0.18 0.25 0.40

0.013 ( 0.005 0.08 ( 0.1 0.34 ( 0.1 0.55 ( 0.1 0.97 ( 0.1 1.35 ( 0.1 1.44 ( 0.1 1.40 ( 0.3 1.30 ( 0.2 1.05 ( 0.05 0.73 ( 0.05 0.55 ( 0.01 0.38 ( 0.01

1206 ( 10 1382 ( 200 2250 ( 100 3472 ( 100 2087 ( 200 1603 ( 10 1413 ( 120 1138 ( 30 788 ( 20 482 ( 20 256 ( 10 149 ( 30 60 ( 10

0.07 ( 0.07 0.14 ( 0.1

0.9 ( 0.3 2.4 ( 0.2

0.04 ( 0.03

2.5 ( 1.4

0.7 ( 0.6

0.2 ( 0.2

0.18 ( 0.03 0.20 ( 0.05 0.30 ( 0.03 0.37 ( 0.03 0.52 ( 0.04 0.46 ( 0.04 0.60 ( 0.06

2.1 ( 0.1 1.8 ( 0.3 1.4 ( 0.2 1.5 ( 0.2 1.5 ( 0.1 1.9 ( 0.2 2.1 ( 0.4

1.1 ( 0.1

0.1 ( 0.1

1.2 ( 0.9 2.5 ( 0.3 4.5 ( 2.3 4.3 ( 0.7 7.7 ( 0.3

0.14 ( 0.1 0.16 ( 0.1 0.12 ( 0.1 0.19 ( 0.3 0.17 ( 0.1

a

103AD3

τ3, ns

B, ps 31.9 ( 0.2 31.9 ( 0.1 31.7 ( 0.2 32.9 ( 0.2 32.4 ( 0.3 32.6 ( 0.1 33.0 ( 0.1 34.4 ( 0.1 33.4 ( 0.1 34.6 ( 0.1 35.4 ( 0.1 37.5 ( 0.3 40.4 ( 0.2

No data are given for terms with too uncertain results from the fitting procedure.

Figure 3. Relaxation amplitude AD1 (b) and relaxation rate τ-1 1 (O) of the low-frequency relaxation term in the function for the SDS solutions in water at 25 °C , displayed as a function of c - cmc.

The relaxation rate,

τ-1 f ) a(c - cmc) + b

(13)

is expected to linearly depend upon c - cmc, with slope a ) + 2 k+/(m j cmc) and with b ) τ-1 f (c ) cmc) ) k /(σ cmc). The concentration dependencies of the AD1 and τ-1 1 values, particularly the relative maximum in the relaxation amplitudes, indicate a significant effect from the ionic nature of the surfactant. In their study of the low-frequency relaxation of the micelle formation/decay kinetics in ionic surfactant solutions with low-weight salt added, Lessner et al.33 have explained consequences from the incomplete dissociation of the surfactant molecules. Proceeding from an effective degree of dissociation

Ri ) 1 - Ji/i

(14)

with Ji denoting the number of undissociated monomers per aggregate of class i , the law of mass action

[Ai] [Ai-1][A1][Ac](Ji-Ji-1)

) bi

(15)

relates the concentration of free counterions [Ac] to the concentrations [A1] and [Ai] of surfactant monomers and of class i aggregates, respectively. In eq 15, bi is the equilibrium constant. Assuming Ri for proper micelles independent of i, thus setting Ri ) Rm within the micelle region of the aggregate distribution, results in

[Am] m j

[A1] [Ac]mj (1 - Rm)

) bm

(16)

which, in turn, leads to an implicit relation for the monomer

Figure 4. Monomer concentration, [A1], of the SDS solutions as resulting from evaluation of the relaxation amplitudes, AD1, in terms of the extended Teubner-Kahlweit-Aniansson-Wall theory using eq 17. The dashed line shows [A1] on the assumption of completely dissociated surfactant molecules; the dotted line represents [A1] ) cmc. The inset shows experimental cmc values of SDS solutions, with NaClO4 added, as a function of low-weight-salt concentration.33

concentration [A1] as a function of the total surfactant concentration c.

[A1] ) Aγ(c/[A1] - 1)(1/mj (2-R))[1 + Rm(c/[A1] - 1)](1-R)/(2-R) (17) Here,

j bm)(1/mj (2-R)) Aγ ) (x2πσm

(18)

denotes the monomer concentration at the cmc. Deriving eq 17, m j -1≈m j , has been assumed. Using the monomer concentration from eq 17 in the scaled concentration X (eq 7), the experimental amplitudes AD1 and relaxation rates τ-1 1 at c > 0.05 mol/L are well-represented by the predictions of the Teubner-Kahlweit-Aniansson-Wall theory (Figure 3). Fitting eqs 5 and 6 to the experimental data, Rm ) 0.33 34 has been assumed, and relation 17 has been numerically solved in the regression analysis. In Figure 4, the monomer concentration following from the fitting procedures is displayed as a function of the total SDS concentration. Also shown is the [A1]-versus-c dependence at Rm ) 1, thus simulating the behavior of nonionic surfactants with an identical cmc. Whereas the latter feature a slight increase of [A1] with c, the monomer concentration of the ionic SDS system decreases substantially with surfactant concentration. This decrease in the monomer concentration corresponds to the decrease in the cmc of SDS solutions on addition of low-weight salts.33 The inset of Figure 4 shows the cmc values of SDS solutions with NaClO4 added, displayed versus low-weight electrolyte concentration. The experimental cmc data decrease in a quite similar manner as the [A1] values from eq 14 decrease with an increasing SDS concentration and thus with an

Sodium Dodecylsulfate Micelles in Water

J. Phys. Chem. B, Vol. 111, No. 7, 2007 1629

Figure 5. Relaxation rate, τ-1 1 , of the low-frequency relaxation term in the spectra of SDS solutions with surfactant concentrations close to the cmc plotted versus c - cmc. The dashed line indicates the lower limit of the frequency range of measurements.

increasing concentration of counterions. The tendency in the monomer concentration of ionic surfactants to decrease above the cmc is well-established (see, e.g., Figure 4.11 in ref 4). Using the mean aggregation number m j ) 64 from ref 34, the reaction volume ∆V ) 11 cm3/mol for the monomer exchange and the variance σ2 ) 20 in the Gaussian size distribution of aggregates follow from the theoretical description of the relaxation amplitude AD1. The representation of the relaxation rate, τ-1 1 , by the Teubner-Kahlweit-AnianssonWall model yields the rate constants at i ) m j as kb ) 4 × 106 s-1 and kf ) 1.8 × 108( mol/L)-1 s-1. Both constants are smaller than those for solutions of the smaller n-heptylammonium chloride14 (kb ) 1.3 × 109 s-1, kf ) 3.2 × 109(mol/L)-1 s-1), but are on the same order as the values obtained on the assumption of a surfactant-concentration-independent monomer concentration of the SDS system34 (kf ) 1 × 107 s-1 , kf ) 1.2 × 109 (mol/L)-1 s-1) . Close to the cmc, the relaxation rates, τ-1 1 , do not follow the predictions of the theory (Figure 5). Rather, the τ-1 1 values first decrease with surfactant concentration to increase according to the Teubner-Kahlweit-Aniansson-Wall model at c > 0.06 mol/L. The initial negative slope in the τ-1 1 -vs-c relation has been previously found for short-chain surfactant systems with large cmc’s14,15 and has been assigned to an effect from the comparatively high content of oligomeric species within the solutions.18 The large SDS molecules display larger relaxation times than the short-chain surfactants. For that reason, the minimum relaxation rates correspond to frequencies smaller than the lowest frequency of measurement. No definite conclusions can thus be drawn from the τ-1 1 values of the SDS solutions close to the cmc. 4.2. Fluctuating Protrusions of Micelle Molecules. The second term in the relaxation function RDDD(ν) representing the measured excess-attenuation-per-wavelength spectra (eq 9) reveals concentration-independent relaxation times, τ2 (Table 3). This result is a strong indication of the relaxation term to be due to a monomolecular reaction,11 which may be formally represented by the equilibrium

Y h Y*

(19)

where Y* denotes another conformation of Y. On the basis of theoretical considerations,8 the second relaxation term in the ultrasonic spectra of ionic surfactant solutions30 has been assigned to the limited radial diffusion of monomers within micelles, leading to protrusions. If such a process, as sketched in the inset of Figure 6, is accepted as the cause of the relaxation term, then the mean diffusion time

〈tD〉 ) p2(exp(-z) - 1 - z)/Dp

(20)

Figure 6. Relaxation amplitude, AD2, of the intermediate relaxation term of the SDS solutions at 25 °C displayed as a function of surfactant concentration, c . The line is drawn just to guide the eyes. In the inset, a sketch of the idea of limited radial diffusion of molecules in the micelles is given.

follows from theory.8 In this relation, p is the mean radial diffusion length, z ) p/lcc , with lcc ) 0.125 nm35 denoting the length of a C-C bond in the hydrocarbon chain axis, and Dp is the diffusion coefficient of the radial surfactant motions within the micelles. With Dp ) 5 × 10-10 m2 s-1 for molecules with dodecyl chains, z ) 4 results from 〈tD〉 ) τ2 ) 1.8 ns, in nice agreement with z ) 4.4 for n-dodecyltrimethylammonium bromide micelles.30 The amplitude, AD2, of the ultrasonic relaxation term is related to the isentropic molar reaction volume, ∆VS, according to11,36

AD2 )

R∆VS2 κ∞S RT

Γ

(21)

where the stochiometric factor, Γ, is given by

Γ ) ([Y]-1 + [Y*]-1)-1 ) cpKp/(1 + Kp)2

(22)

Here, Kp is the equilibrium constant of the conformational change (eq 19), and cp ) c - [A1] . As illustrated by Figure 6, only at concentrations up to 0.1 mol/L does the relaxation amplitude increase proportional to c ≈ cp. At higher surfactant concentrations, the increase in the AD2 values is smaller than predicted by eqs 21 and 22. Probably, at higher micelle concentrations, activity corrections have to be taken into account. In the following estimation of the reaction volume, only the data of small surfactant content ( c e 0.1 mol/L) are thus considered. We do not know the distribution of species Y and Y* and, thus, the equilibrium constant of the radial diffusion equilibrium. Assuming equipartition [Y] ) [Y*], corresponding to Kp ) 1 , the maximum possible stochiometric factor Γmax ) 0.25cp follows so that the minimum volume change (∆VS)min can be derived from the experimental AD2 values using eq 21. (∆VS)min ) 2.1 cm3/mol results, in fair agreement with (∆VS)min ) 0.9 cm3/mol for dodecyltrimethylammonium bromide with surfactant concentration 0.5 mol/L.30 Partial molar volume data for aqueous solutions of alcohols37 may be taken to estimate a change of ∆V ) 1 cm3/mol in the molar volume of hydration water per additional methyl group of the solute. The minimum reaction volume of the limited radial diffusion of SDS molecules corresponds then to a minimum diffusion length pmin ) 2.1lcc ) 0.26 nm, which compares well to the mean diffusion length p ) 4 × lcc ) 0.5 nm from the relaxation times. We mention that the chain protrusions associated with surface roughness fluctuations have been intensively discussed in view of chain packing and hydration properties of

1630 J. Phys. Chem. B, Vol. 111, No. 7, 2007

Polacek and Kaatze core, thus affecting the volume change associated with the unimolecular reaction. 5. Conclusions

Figure 7. Relaxation amplitude, AD3, of the high-frequency relaxation term of the relaxation spectral function RDDD(ν) for the SDS solutions at 25 °C shown as a function of surfactant concentration, c.

TABLE 4: Relaxation Time, τ3, of the High-Frequency Relaxation Term for Some n-Alkanes at 25 °C45 alkane

τ3, ns

n-decane n-dodecane n-tetradecane n-pentadecane n-hexadecane

0.08 0.12 0.19 0.25 0.29

amphiphilic aggregates.38,39 More recently, lipid-molecule protrusions modes have been considered to contribute to the repulsion forces between bilayers40,41 and, among other factors, play a role in the activation of enzymes at membrane surfaces.42 4.3. Structural Isomerization of Alkyl Chains. The relaxation times, τ3, of the high-frequency relaxation term in the spectral function RDDD(ν) likewise do not depend upon the surfactant concentration, c , whereas the relaxation amplitude AD3 increases linearly with c (Figure 7). Hence, this term also reflects a unimolecular reaction of the type represented by eq 19. In previous investigations of surfactant solution ultrasonic spectra, the high-frequency term has been assumed to be due to the isomerization of the alkyl chains in the cores of proper micelles. This assignment is supported by two findings. First, the high-frequency relaxation term with relaxation time on the order of some tenths of a nanosecond has been found with anionic,30 cationic,30 zwitterionic,43 and nonionic17,43 surfactant systems and also with suspensions of phospholipid bilayer versicles.44 Hence, the relaxation time does not depend significantly on either the nature of the hydrophilic head group of the surfactant or whether the amphiphiles are aggregated in micelles or in the fluid phase of bilayer structures. For an n-dodecyltrimethylammonium bromide solution, τ3 ) 0.15 ns has been found at c ) 0.5 mol/L and 25 °C,30 in nice agreement with the τ3 values ( 0.1 ns e τ3 e 2.1 ns, Table 3) of the SDS solutions. In addition, the relaxation times of the surfactant systems agree well with those for pure n-alkanes17,45 at the same temperature (Table 4). These results confirm the assignment of the high-frequency relaxation term to the structural isomerization of the alkyl chains, with this respect resembling a droplet of liquid alkane. The increment in the relaxation amplitudes of the highfrequency relaxation term of SDS solutions (AD3/c ) 19 × 10-3 (mol/L)-1 ) is somewhat different from the n-dodecyltrimethylammonium bromide solution30 (AD3/c ) 13 × 10-3 (mol/ L)-1), but is noticeably larger than that of n-dodecane45 (AD3/c ) 4 × 10-3 (mol/L)-1). These differences in the AD3/c values may be taken to show that the motions of the surfactant head groups are coupled to the chain isomerization within the micellar

At surfactant concentrations above the critical micelle concentration, broadband ultrasonic excess attenuation spectra of aqueous solutions of sodium dodecylsulfate can be wellrepresented by a superposition of three Debye relaxation terms. There does not exist a continuous distribution of relaxation times as characteristic of the low-frequency term of short-chain surfactant solutions near the cmc. This relaxation term, with relaxation times between 60 ns ( 0.4 mol/L, SDS, 25 °C) and 3.5 µs ( 0.014 mol/L), reflects the monomer exchange equilibrium in the Aniansson-Wall isodesmic reaction scheme of the micelle formation/decay process. The relative maximum in the dependence of the relaxation amplitudes upon surfactant concentration can be well-represented by theoretical models considering the incomplete dissociation of the surfactant counterions. Taking this feature into account, a significant reduction in the monomer concentration [A1] with increasing surfactant concentration, c, follows for the micellar solutions, in contrast to nonionic surfactant systems for which [A1] ≈ cmc at c > cmc. The intermediate relaxation term, the relaxation time of which ( 0.9 ns e τs e 2.5 ns) does not reveal a noticeable dependence upon surfactant concentration, is assumed to be due to the limited radial diffusion of surfactant molecules within the micelles. On this assumption, mean protrusion by z ) 4 methyl groups results for the dodecyl chains of the SDS system, in nice agreement with n-dodecyltrimethylammonium bromide micelles. Alternatively, the intermediate relaxation term may be taken to reflect a mode of hydrocarbon chain isomerization. The high-frequency relaxation term of the SDS solution spectra exhibits a relaxation time between 0.1 and 0.2 ns, in nice agreement with the relaxation time of n-dodecane (0.12 ns, 25 °C). This term is thus assigned to the structural isomerization of the hydrocarbon chains. It shows that, with respect to these isomerizations, the core of the micelles resembles liquid alkanes. Acknowledgment. Financial support by the Deutsche Forschungsgemeinschaft, Bonn, Germany, is gratefully acknowledged. References and Notes (1) Belocq, A. M. Micelles, Membranes, Microemulsions and Monolayers; Springer: Berlin, 1994. (2) Shah, D. O., Ed. Micelles, Microemulsions, and Monolayers; CRC Press: Boca Raton, FL, 1998. (3) Mittal, K. L.; Lindman B. Surfactants in Solution; Plenum: New York, 1999. (4) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain. Where Physics, Chemistry, and Biology Meet; Wiley-VCH: New York, 1999. (5) Zana, R., Ed. Dynamics of Surfactant Self-Assemblies: Micelles, Microemulsions, Vesicles and Lyotropic Phases; CRC Press: Boca Raton, FL, 2005. (6) Aniansson, E.; Wall, S. J. Phys. Chem. 1974, 78, 1024. (7) Aniansson, E.; Wall, S. J. Phys. Chem. 1975, 79, 857. (8) Aniansson, E. J. Phys. Chem. 1978, 82, 2805. (9) Aniansson, E. Ber. Bunsen-Ges. Phys. Chem. 1978, 82, 981. (10) Aniansson, E. Progr. Colloid Polym. Sci. 1985, 70, 2. (11) Strehlow, H. Rapid Reactions in Solution; VCH: Weinheim, 1992. (12) Teubner, M. J. Phys. Chem. 1979, 83, 2917. (13) Kahlweit, M.; Teubner, M. AdV. Colloid Interface Sci. 1980, 13, 1. (14) Telgmann, T.; Kaatze, U. J. Phys. Chem. B 1997, 101, 7758. (15) Telgmann, T.; Kaatze, U. J. Phys. Chem. A 2000, 104, 1085. (16) Telgmann, T.; Kaatze, U. Langmuir 2002, 18, 3068. (17) Hanke, E.; Telgmann, T.; Kaatze, U. Tenside, Surfactants, Deterg. 2005, 42, 23. (18) Telgmann, T.; Kaatze, U. J. Phys. Chem. B 1997, 101, 7766.

Sodium Dodecylsulfate Micelles in Water (19) Eggers, F.; Kaatze, U. Meas. Sci. Technol. 1996, 7, 1. (20) Kaatze, U.; Wehrmann, B.; Pottel, R. J. Phys. E: Sci. Instrum. 1987, 20, 1025. (21) Eggers, F.; Kaatze, U.; Richmann, K. H.; Telgmann, T. Meas. Sci. Technol. 1994, 5, 1131. (22) Polacek, R.; Kaatze, U. Meas. Sci. Technol. 2003, 14, 1068. (23) Kaatze, U.; Ku¨hnel, V.; Menzel, K.; Schwerdtfeger, S. Meas. Sci. Technol. 1993, 4, 1257. (24) Kaatze, U.; Lautscham, K.; Brai, M. J. Phys. E: Sci. Instrum. 1988, 21, 98. (25) Bo¨mmel, H. E.; Dransfeld, K. Phys. ReV. Lett. 1958, 1, 234. (26) Kaatze, U.; Behrends, R.; Lautscham, K. Ultrasonics 2001, 39, 393. (27) Hill, R. M. Nature 1978, 275, 96. (28) Hill, R. M. Phys. Status Solidi 1981, 103, 319. (29) Menzel, K.; Rupprecht, A.; Kaatze, U. J. Acoust. Soc. Am. 1998, 104, 2741. (30) Kaatze, U.; Lautscham, K.; Berger, W. Z. Phys. Chem. (Munich) 1988, 159, 161. (31) Frindi, M.; Michels, B.; Zana, R. J. Phys. Chem. 1994, 98, 6607. (32) Verrall, R. E. Chem. Soc. ReV. 1995, 24, 135. (33) Lessner, E.; Teubner, M.; Kahlweit, M. J. Phys. Chem. 1981, 85, 1529. (34) Hall, D. G.; Wyn-Jones, E. J. Mol. Liq. 1986, 32, 63.

J. Phys. Chem. B, Vol. 111, No. 7, 2007 1631 (35) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley: New York, 1973. (36) Eigen, M.; De Maeyer, L. In Techniques of Chemistry, InVestigation of Rates and Mechanisms of Reactions; Hammes, G. G., Ed.; Wiley: New York, 1974; Vol. VI, Part II, p 351. (37) Franks, F. Water, a ComprehensiVe Treatise; Plenum: New York, 1973, Vol. 2, Chapter 1, p 1. (38) Lindman, B.; Wennerstro¨m, H.; Gustavsson, H.; Kamenka, N.; Brun, B. Pure Appl. Chem. 1980, 52, 1307. (39) Ben-Shaul, A.; Gelbart, W. M. Annu. ReV. Phys. Chem. 1985, 36, 179. (40) Israelachvili, J. N.; Wennerstro¨m, H. Langmuir 1990, 6, 873. (41) Israelachvili, J. N.; Wennerstro¨m, H. J. Phys. Chem. 1992, 96, 520. (42) Mouritsen, O. G.; Andresen, T. L.; Halperin, A.; Hansen, P. L.; Jakobsen, A. F.; Bernchou Jensen, U.; Jensen, M. Ø.; Jørgensen, K.; Kaasgard, T.; Leidy, C.; Cohen Simonsen, A.; Peters, G. H.; Weiss, M. J. Phys.: Condens. Matter 2006, 18, S1293. (43) Kaatze, U.; Berger, W.; Lautscham, K. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 872. (44) Schrader, W.; Halstenberg, S.; Behrends, R.; Kaatze, U. J. Phys. Chem. B 2003, 107, 14457. (45) Behrends, B.; Kaatze, U. J. Phys. Chem. A 2000, 104, 3268.