Field-Dependent NMR Relaxation Study of Aggregation and

J. Phys. Chem. B 2000, 104, 1529-1538

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Field-Dependent NMR Relaxation Study of Aggregation and Dynamics in Dilute to Concentrated Micellar Decylammonium Chloride Solutions Maria To1 rnblom, Ruslan Sitnikov, and Ulf Henriksson* Physical Chemistry, Royal Institute of Technology, S-100 44 Stockholm, Sweden ReceiVed: July 27, 1999; In Final Form: NoVember 8, 1999

The surfactant aggregation behavior in aqueous decylammonium chloride solutions for concentrations between 5 and 29 wt % was studied by 2H nuclear magnetic relaxation dispersion in the 0.5-77-MHz frequency range. Below the second cmc the micelles can be described as polydisperse fairly small aggregates undergoing unhindered Brownian rotation. Above the second cmc the micelles are rod shaped and grow with increasing concentration, and as a consequence the tumbling of the micelles eventually becomes restricted by neighboring aggregates, which is demonstrated by the low-frequency relaxation data. The restricted reorientation of the micelles in the concentrated solution has been modeled as rotational diffusion in a hard-walled double cone potential of mean torque.

Introduction The study of self-assembled surfactant systems has for many years attracted enough interest to be regarded as a field of its own. Of the many experimental methods employed in the field, different NMR methods have proven to be among the most useful. Through a number of different techniques, NMR can provide information about order and dynamics on both the molecular and macroscopic level in the different phases of these systems.1,2 In the course of our continuing work in this field we have earlier3 interpreted 2H NMR relaxation data from micellar solutions in quantitative terms by employing a model for the dynamic behavior of the aggregated surfactant molecules. It was found that a model of free Brownian rotation of rodlike aggregates with simultaneous surface diffusion of the molecules4 could reproduce relaxation data in the accessible frequency range for all investigated samples. We have also employed this method in the investigation of micelles containing solubilized hydrocarbon molecules.5 Both these studies were, however, confined to a C12 and a C16 surfactant of concentrations up to 15 wt % and to temperatures close to room temperature. Under these experimental conditions the main part of the frequency dependence associated with aggregate end over end rotation (tumbling) fell below 2.0 MHz (the lower frequency limit of our equipment at that time) unless the aggregates had a very low axial ratio. This limited the possibility to detect deviation from the free rotation model even though the compliance of the model in the zero frequency limit was tested through the spin-spin relaxation rate, R2. In present study we apply our method to a surfactant with a shorter hydrocarbon chain and a smaller headgroup; decylammonium chloride (C10ACl), in a much wider range of temperatures (13-55 °C) and concentrations (5-29%). We have also rebuilt our variable frequency spectrometer so that it now permits measurements down to frequencies as low as 0.5 MHz.6 The shift of the relaxation rate curves to higher frequencies with smaller dimensions and higher temperatures, combined with the extension of the frequency range, has made it possible to follow * Corresponding author. E-mail: [email protected]. Fax: +46 8-7908207.

a large part of the relaxation dispersion caused by the slowest dynamic processes and thereby to study these in greater detail. Our interest in the decylammonium chloride/water system originates in the peculiar phase diagram where the usual order of liquid crystal phases7 is not followed but the hexagonal phase area forms an “island” surrounded on all sides by liquid, spectroscopically isotropic phases.8 (This is partly demonstrated in Figure 1.) Trying to interpret relaxation rates from a sample in the concentrated part of the micellar regime,9 we found a relaxation behavior not encountered in systems of lower concentration: Although the field-dependence of the spinlattice relaxation rate, R1, could be reproduced by the dynamic models used in our earlier investigations and also by the empirical three-step model,10 the simultaneous fit to R1 and R2 data required an additional term in the spectral density function, not accounted for by these models. Considering the high concentration of the sample it is tempting to interpret this deviation as the result of the breakdown of the model of free Brownian rotation due to interaggregate interactions and perhaps even the formation of domains with local order. To investigate this further, we have performed a broad study where we follow the development of the relaxation behavior with increasing concentration in the micellar phase in the C10ACl system at three different temperatures; see Figure 1. In this way it is possible to investigate the limits of the applicability of the free Brownian rotation model and gain information on the concentration and temperature dependence of the aggregation in this very interesting system. We discuss models for reorientation and other dynamic processes of molecules in concentrated solutions and how these may be applied to concentrated micellar solutions and investigate how they are able to reproduce data from the regime where the free Brownian rotation model fails. This study may also be seen as a further test of the applicability and usefulness of field-dependent NMR relaxation as a tool in experimental investigations of micellar solutions. Experimental Section To synthesize decylammonium chloride-R-d2 the C10-amide was reduced with LiAlD4 in ether solution to form decylamine-

10.1021/jp992601m CCC: $19.00 © 2000 American Chemical Society Published on Web 01/29/2000

1530 J. Phys. Chem. B, Vol. 104, No. 7, 2000

Figure 1. Partial phase diagram for the decylammonium chloride/ water system. The diagram shows the limits of the liquid crystalline phases. The two-phase regions surrounding these have been left out. The concentrations and temperatures where measurements were made are marked.

R-d2. The ammonium chloride was subsequently precipitated with HCl from an ethanol solution of the amine. The nondeuterated surfactant was prepared by precipitation from a solution of nondeuterated decylamine. Samples were prepared by weighing into 10-mm NMR tubes or 7-mm test tubes which were subsequently flame sealed. The 7-mm tubes were inserted into 10-mm NMR tubes for the NMR measurements. Samples of the concentrations marked in Figure 1 and of four low concentrations were prepared with the deuterated surfactant as was a sample of 0.6% (below reported cmc values11,12). In addition samples of 0.6% and of 2.0% were prepared with nondeuterated surfactant for 13C T1 measurements. 2H NMR relaxation measurements at 0.5-13.8 MHz were performed with a Bruker MSL 90 spectrometer equipped with a variable-field iron-core magnet rebuilt to operate down to 0.5 MHz.6 Measurements were also performed with the MSL or a DMX 200 equipped with a 4.7 T cryomagnet (at 30.7 MHz), with a Bruker AMX 300 spectrometer (at 46.07 MHz), and with a Bruker AM 400 spectrometer (at 61.4 MHz), and with a Bruker DMX 500 spectrometer (at 76.77 MHz). 13C T ’s for monomers and spherical micelles were deter1 mined under broad-band proton decoupling at 75.47 MHz on the AMX 300. The self-diffusion coefficients of the surfactant in the three most concentrated samples were determined at 35.1 ( 0.3°C with the stimulated echo extension of the PGSE method on the DMX 300 equipped with a gradient aggregate manufactured by Digital Specialities and a 10-mm gradient probe from Cryomagnet Systems. Reliable measurements could not be performed at 55 °C due to convection effects. T2 was determined at 30.7 MHz with the CPMG method using only every second echo. T1 was determined with the standard inversion recovery method, usually at 11-15 frequencies. Due to the broad signals of the most concentrated samples at 13 °C it was not possible to measure T1 at 0.5 MHz for all these at this temperature. The approximate criterion for if a measurement could be made was that it should be possible to obtain a signal/ noise ratio better than 9 within 32 000 scans. With this noise level the standard deviation in T1 given by Monte Carlo calculations was (6-10%. The standard deviations in the measured T1 below 2.0 MHz were otherwise generally (26% with higher uncertainty the broader the signal. At the higher

To¨rnblom et al. fields the S/N ratios were excellent and the standard deviation in Ti was usually below 1%. Relaxation measurements were performed at 13.2 °C, 35.1 °C, and 55.1 °C. The temperature was controlled by B-VT 2000 or B-VT 3000 units. The temperature at the sample position was always measured before and between measurements by an external thermometer with a Pt-100 sensor mounted in a 10mm NMR tube. With these procedures we estimate the standard variation in temperature between measurements to be (0.3 °C. The uncertainties in the relaxation rates caused by this variation were estimated with Arrhenius interpolation to be (0.5-1% in R1 at high and (1.5-3% at low frequencies. In R2 the uncertainties due to temperature variation range from (1% at low concentrations up to (2-2.5% for the most concentrated samples. The estimated total uncertainty in the relaxation rates is the sums of the uncertainties caused by noise and those caused by temperature variation. These errors are in good agreement with variation found for points where the measurement was repeated and with the average deviations of the data points from fitted curves as those presented in Figures 3 and 5. Since the relaxation measurements extended down to concentrations of the order of magnitude of the cmc, all relaxation rates were corrected for contributions from monomeric C10ACl10 at a concentration equal to the cmc. The cmc’s for the actual batch of surfactant were determined at four temperatures (the above mentioned and 25.1 °C) through linear extrapolation of the 2H R1 for 1.5 and 2.0% samples down to Rmon. The cmc’s thus obtained were in good accordance with one of the literature values.11 This procedure is rather crude, giving large uncertainties in cmc values and overestimating the monomer concentration at higher concentrations, but the final results of the fittings to the relaxation rates are quite insensitive to this. The relaxation data were evaluated by fittings of the theoretical relaxation rates for the dynamic models discussed below. The fittings were performed with programs based on the TWOSTEP routine.13 This routine minimizes the relative rootmean-square (RMS) deviation of the theoretically predicted relaxation rates from the experimental ones. The fittings performed in this manner included error analyses by the Monte Carlo method. The uncertainties in the parameters given in the tables correspond to 90% confidence levels and were obtained by 100 repetitions of the fittings. It must be pointed out, however, that the relative variation in the data points in these calculations are based on the RMS deviations in the fittings and set equal over the entire frequency range. They are not based on actual experimental uncertainties. In fittings where there are no data from frequencies below 2.0 MHz and where the average deviations are very small this ought not to present any problem, but in other cases it may lead to an underestimation of the uncertainties in the parameters mainly determined by the lowfrequency dispersion and overestimation of those in others. In the fittings the quadrupole coupling constant, χ, for the R-deuterons in the decylammonium ion was set to 170 kHz. This choice is based on the ratio of the 13C to the 2H T1’s for monomers. This ratio was ≈10, in good agreement with what has been found by Jansson et al. for the same ion in another system.14 The earlier employed value of χ ) 181 kHz originated in measurements on hexanoate ions giving a ratio of ≈13.15,16 If it is assumed that the coefficient of lateral diffusion, Dlat (see below), of the surfactant ion is independent of aggregate shape and surfactant concentration it can be determined for spherical micelles and kept constant in the fittings throughout the concentration interval.3 This assumption was invoked and

Aggregation and Dynamics in Aqueous C10ACl Solutions

J. Phys. Chem. B, Vol. 104, No. 7, 2000 1531

it was concluded from R2 data that the highest concentration where spherical micelles predominate is 2.0% at all temperatures except 13 °C were no certain estimate could be made. The 2H relaxation rates at several fields and 13C T1 at one high field were therefore measured for a 2% sample at 25.1, 35.1, and 55.1 °C and the data were evaluated within the two-step model (the 13C-1H dipolar coupling constant was set to 23.33 kHz16). The Dlat values were calculated from these results by means of the two-step model as described in ref 3 with the hydrocarbon core radius set to 14.2 Å17 and the micellar radius set to 16.7 Å.18 This yielded Dlat ) 140 ( 10 µm2 s-1 at 25.1 °C, 220 ( 30 µm2 s-1 at 35.1 °C, and 440 ( 100 µm2 s-1 at 55.1 °C. The error intervals were calculated from the errors in the correlation time τs, which appears as a parameter in the two-step model. They correspond to a 90% confidence interval and do not account for errors in the model or in the choice of a concentration where spherical micelles predominate. An Arrhenius plot of these values is linear within uncertainties with an activation energy of 31 kJ/mol giving Dlat ) 85 ( 10 µm2 s-1 at 13.2 °C. These lateral diffusion coefficients were determined with the hydrocarbon core interface defined as the surface of free diffusion. Results and Evaluation A partial phase diagram for the decylammonium chloride/ water system determined by 2H NMR spectra9 and light polarization is shown in Figure 1. The concentrations and temperatures at which the micellar solution (L1) has been investigated by field-dependent relaxation measurements are marked in the diagram. The concentration dependence of the R2’s (transverse relaxation rates) at three temperatures is shown in Figure 2 (note the different scales). The observed R2 values give a qualitative indication of the aggregate size; fast relaxation implies large aggregates. At all three temperatures we see an accelerating increase of R2 with concentration somewhere in the middle of the investigated concentration range. This is an indication of what is commonly called “second cmc behavior”, a sudden onset of aggregate growth, probably due to the formation of a new type of micelle that grows to large dimensions more easily than those prevailing at low concentrations.19,20 This change is not very sharp at any of the temperatures and it is therefore difficult to more precisely determine a concentration where it takes place. We can see in the figure that this concentration is higher at higher temperature, though. It is located somewhere between 10 and 15% at 13 °C, somewhere between 15 and 20% at 35 °C, and closer to 20% at 55 °C. The corresponding concentrations obtained from 1H line widths have been reported to be 0.75 M (≈13%) at 25 °C and 1.1 M (≈18%) at 40 °C.21 It remains to establish if the observed faster rise in R2 at high concentrations really reflects faster micellar growth with increasing concentration. This is one possible interpretation of the data in Figure 2 but there are, as discussed below, a number of factors apart from aggregate dimensions that may influence R2. Theoretically, there is a possibility that the observed increase in relaxation rate to a large extent is brought about by the restriction of the rotational motion with higher aggregate concentration. To study this and other aspects of aggregation behavior and dynamic processes, the R2 measurements were supplemented with measurements of the variation of R1 with the field, or the Larmor frequency, throughout the accessible frequency range. R2 was measured only at one frequency, since this is sufficient to obtain information about the zero frequency spectral density. The frequency dependence of R2 contains the

Figure 2. 2H transverse relaxation rate R2 as a function of the surfactant concentration at (a) 13.2 °C, (b) 35.1 °C, and (c) 55.2 °C.

same information as can be obtained with better accuracy from the R1s. The relaxation measurements generate data sets of the type shown in Figures 3-6. Figure 3 shows the influence of temperature and Figure 5 that of increasing concentration on the relaxation rates. In this investigation we are concerned with the relaxation of the 2H nuclei attached to the R-carbon of a selectively deuterated surfactant molecule in a micelle in a macroscopically isotropic solution. In this system the strong quadrupolar interaction, which in systems of nonvanishing order causes line splitting up to several kHz, is averaged out as the molecule reorient through all angles of space with equal probability. The strength of the interaction fluctuates with the orientation of the C-2H bond, assumed to coincide with the principal axis, zP, of the cylindri-

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To¨rnblom et al.

Figure 4. R1 (b) and R2 (O) from the 14.1% C10ACl solution at 13.2 °C. The lines represent the best five parameter (i.e., only the radius of the spherical micelles in the distribution fixed) fitting of the model of disks with semitoroidal rims with exponentially distributed aggregation numbers. The relative RMS deviation of the fitted from the experimental relaxation rates is 5.3%.

Figure 3. Frequency dependence of the measured 2H relaxation rates R1 (b) and R2 (O) from the 14.1% C10ACl solution at (a) 13.2 °C, (b) 35.1 °C, and (c) 55.2 °C. The solid and dashed lines are the theoretical relaxation rates given by the model of freely reorienting spherocylinders with exponentially distributed aggregation numbers with the parameters fitted as described in the text. The obtained parameter values are presented in Table 1.

cally symmetric electric field gradient tensor, relative to the direction of the magnetic field, zL. These fluctuations induce transitions between spin states at rates high enough for other relaxation mechanisms to be neglected. The relaxation rates depend explicitly on the dynamics of the reorientation processes through the spectral densities which by definition are FourierLaplace cosine transforms of the time correlation functions (TCFs), GLk , describing the time dependence of these inherently stochastic processes. In a truly isotropic system the lab-

Figure 5. Frequency-dependence of the measured 2H relaxation rates R1 (b) and R2 (O) at 55.2 °C from C10ACl solutions with concentrations of (a) 20.5%, (b) 23.8%, and (c) 28.8%. The solid and dashed lines are the theoretical relaxation rates given by the best fit of (a) the model of exponentially distributed spherocylinders undergoing free Brownian rotation and (b and c) monodisperse spherocylinders rotating in a hardwalled double cone potential of mean torque. The resulting parameters are presented in Tables 1 and 2.

frameTCFs are independent of the component index k and we only have to consider one of them:

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Figure 6. Fitting of the unrestricted Brownian rotation model for polydisperse spherocylinders to the experimental data from the 28% C10HACl solution at 55 °C (compare to Figure 5c). L GL(τ)) 〈VL/ 0 (ΩLP(0))V0 (ΩLP(τ))〉

R2(ω) ) T-1 2 (ω) )

3π2 2 χ [2J(ω) + 8J(2ω)] 4

(2)

3π2 2 χ [3J(0) + 5J(ω) + 2J(2ω)] (3) 4

The aggregation of surfactant molecules into micelles affects their reorientation processes in a characteristic manner. The purely molecular reorientation of the molecules within an aggregate consists of conformational changes and oscillations around an average molecular orientation or director, zM, which is assumed to coincide with the surface normal. These motions are complex but normally at least 2 orders of magnitude faster than the motional processes involving the aggregates.23-25 The spectral density may therefore, due to the time scale separation, be resolved into two terms:26

1 J(ω) ) S2MPJLM(ω) + (1 - S2MP) τMP 5

2

(1)

with VL0 the zero lab frame component of the normalized spherical interaction tensor, time dependent through the orientation ΩLP(τ). Hence there is only one lab frame spectral density function, J(ω), and the relaxation rates at Larmor frequency ω are22

R1(ω) ) T-1 1 (ω) )

interpreting these data in terms of different physical properties of the system. Dilute Micellar Solutions. In the model previously found to reproduce the relaxation behavior of micellar C16TABr and SDS solutions up to 15 wt % in the 2.0-61.4-MHz range,3 it was assumed that the further averaging of the quadrupolar interaction remaining after the fast motions proceeds in two steps. The lateral diffusion of the molecules over the aggregate surface modulates the angle between the molecular director zM and the aggregate symmetry axis zA. This results in further averaging to an extent described by the aggregate order parameter, SAM, which depends only on the geometry of the aggregate.4 The final averaging of the interactions to zero is brought about by the rotational diffusion of the entire aggregate. For the remaining interactions to be completely averaged out already at this stage these rotations have to be isotropic or unrestricted in the sense “probing all angles of space with equal probability”. If the aggregate is a symmetric top, the joint labframe TCF for these two dynamic processes is4

(4)

The second term, representing the fast motions, is constant in the frequency range of 2H NMR and usually represented in terms of an effective correlation time, τMP, often denoted τf. The weight factor, SMP, is the second-rank order parameter of the reorientation of the C-2H bond relative to zM, a measure of how large a part of the interaction is averaged by the fast motions. It is often called the local order parameter, Sloc, but we adopt the above notation in order to maintain consistency with the notation used later on. JLM(ω) in the first term of eq 4 is the spectral density of the reorientation of the average molecular orientation in the lab frame and it is the transform of the corresponding time correlation function GLM(τ). This reorientation is brought about by a number of mechanisms all in some way related to the aggregation. If physical models for these are applied, theoretical spectral densities can be calculated and inserted into eqs 2 and 3, which can be fitted to experimental relaxation data thus

GLM(τ) ) S2AM gLA 0 (τ) +

AM (2 - δn0)gLA ∑ n (τ)gn (τ) n)0

(5)

With the rotational motion modeled as small step, i.e., Brownian, rotational diffusion27 the correlation functions, 28 with correlation times τLA which gLA n (τ), are exponential n depend on aggregate shape and dimensions, temperature, and the viscosity of the solvent,28 through the rotational diffusion coefficients parallel and perpendicular to the aggregate symmetry axis, R| and R⊥. Expressions for the rotational diffusion coefficients for spheroids are given by Perrin27 and for spherocylinders by Yoshizaki and Yamakawa.29 The effects of intermicellar interactions on the reorientation are taken into account in the model through their influence on the flow of the interaggregate medium and this is in practice accomplished by the application of a correction factor in the macroscopic viscosity of this medium.30 If the lateral diffusion over the aggregate surface is treated as a classical diffusion process with a constant diffusion coefficient, Dlat, it is possible to compute the diffusion correlation functions, gAM n (τ) that appear in eq 5, either through numerical methods or, in exponential approximations, as simple analytic expressions.4 In ref 3 the initial slope approximation was used for the spherocylindrical geometry which was found to give the best fit to experimental data. Within the exponential approximations each gAM n (τ) is characterized by a correlation depending on Dlat and the size and shape of the time τAM n aggregate as given in the Appendix of ref 3. The spectral density function derived from eq 5 is thus obtained as a sum of four Lorentzian terms:

J (ω) ) LM

[

1 5

S2AM

gLA 0 (0)

τLA 0

2 1 + (ωτLA 0 )

2

+

∑(2 - δn0)gAM n (0) × n)0

]

τLM n 2 1 + (ωτLM n )

(6)

with the joint rotation-diffusion correlation times given by

1 1 1 ) LA + AM LM τn τn τn

(7)

1534 J. Phys. Chem. B, Vol. 104, No. 7, 2000 The general expression for the spectral density JLM(ω) is of the same form but the last three terms are not necessarily Lorentzian. The first term in eq 6 is a pure rotation mode with its dispersion shifted to lower frequencies than the other three terms. The dispersions due to these three terms fall, under most of the commonly encountered experimental conditions, within the same frequency range and they are thus usually not resolved into separate steps but constitute one effective step in the experimentally observed dispersion. In combination with eq 4 this gives, in practice, a three-step spectral density. This explains why the empirical Lorentzian three-step model,10 or for aggregates close to spherical symmetry its two-step equivalent, often is able to reproduce relaxation data from micellar solutions.3,10,23 AM AM The correlation times τLA n and τn , the order parameter S , LA AM g0 (0), and gn (0) that appear as parameters in eq 6 are all, for a given solvent and temperature, functions of aggregate size and shape parameters and the lateral diffusion coefficient Dlat.3,4,28,29 For polydisperse samples spectral densities have to be summed over the contributions from aggregates with different aggregation numbers as was done in ref 3 where a simple exponential model, starting at spheres, for the distribution of molecules over different aggregates was employed. In this model the aggregate minor semiaxis or radius was allowed to decrease gradually with increasing aggregate length from the length of the surfactant ion with the chain in an all-trans configuration for spherical micelles to a smaller value, R∞, for the infinitely long micelles. Under these circumstances and with the lateral diffusion coefficient Dlat fixed to a predetermined value (see the Experimental section) there remain four parameters to be varied in the fittings of the free Brownian rotation model with the actual size distribution function: two parameters characterizing the fast motion, SMP and τMP, and two parameters related to the aggregate size and shape, e.g., the volume or aggregation number for the average aggregate and the radius of infinitely long aggregates. Up to a C10ACl concentration of 15% the experimental relaxation data at all three temperatures can be well described by the combination of the dynamic model and the size distribution model outlined above. For these samples, practically the entire dispersion in the 2H R1 falls within the measured frequency range, and it must be pointed out that the model fits astonishingly well within the whole frequency range despite the different character of the data sets. This is demonstrated in Figure 3, showing experimental and calculated relaxation rates for the 14.1% sample at three temperatures. The relative RMS deviation between experimental and calculated relaxation rates lies below 1.3% for all these three data sets. The free Brownian rotation model also fits for the 20.5% solution at 55 °C within 1.8% RMS deviation, which is mainly due to deviation at the lowest frequency, 0.5 MHz. A comparison with Figure 2 shows that the model applies for surfactant concentrations approximately up to the concentration where the faster increase in R2 sets in. The results of these fittings are given in Table 1. For concentrations above 14% at 13 and 35 °C and above 20% at 55 °C the model of rod-shaped micelles undergoing unhindered Brownian rotation is no longer sufficient to describe the experimental relaxation data. This shows up as systematic differences between the experimental relaxation rates and those predicted by the model, especially at the lowest frequency but also in the high-field data as demonstrated in Figure 6. These differences are systematic not only within but also between

To¨rnblom et al. TABLE 1: Parameter Values Obtained by Fitting the Motional Model for Polydisperse Spherocylindrical Micelles with Unhindered Brownian Rotation to Experimental Relaxation Data temp, concn, °C % wt 13.2d 13.2d 13.2d 35.1e 35.1e 35.1e 55.1f 55.1f 55.1f 55.1f

5.0 9.6 14.1 5.0 9.6 14.1 5.0 9.6 14.1 20.5

〈n〉a

〈F〉b

R∞,c Å

SMP

56 +5 -5 74 +6 -5 127 +7 -7 48 +2 -3 57 +2 -7 85 +4 -3 47 +4 -3 55 +0 -8 68 +0 -15 107 +3 -8

+0.07 1.23 -0.07 +0.05 1.53 -0.07 +0.05 2.73 -0.07 +0.05 1.11 -0.03 +0.03 1.24 -0.10 +0.04 1.63 -0.05 +0.05 1.10 -0.05 +0 1.21 -0.11 +0 1.40 -0.2 +0.10 1.94 -0.03

+1.6 15.1 -4.9 +0.6 16.1 -1.1 +0.4 15.4 -0.3 16.7

+0.009 0.209 -0.007 +0.006 0.217 -0.005 +0.003 0.218 -0.003 +0.008 0.212 -0.006 +0.015 0.222 -0.005 +0.007 0.217 -0.006 +0.01 0.20 -0.02 0.218 +0.030 -0 0.218 +0.030 -0 +0.006 0.218 -0.006

+0 16.7 -6.8 16.7 +0 16.7 -4.8 +0 16.7 -8.3 +0 16.7 -4.7 +0 16.7 -0.8

RMS τMP (ps) dev, % 30 +3 -4 30 +3 -3 33 +2 -2 12 +2 -2 11 +2 -5 14 +2 -3 9 +3 -3 7 +0 -7 7 +0 -7 8 +2 -2

1.6 1.2 0.7 0.9 0.9 1.3 0.6 0.8 0.9 1.7

a Average aggregation number. b Axial ratio of the average aggregate. R∞ was limited not to exceed 16.7 Å. d Dlat ) 85 µm2 s-1. e Dlat ) 220 µm2 s-1. f Dlat ) 440 µm2 s-1.

c

samples and temperatures and this has been the criterion on which models have been rejected rather than any definite limit for the RMS deviation. It was not possible to improve the quality of the fittings by modifications of the size distribution. Changing the shape of the micelles to disks increased the errors. We must therefore conclude that the failure of these models is ultimately a consequence of the high aggregate concentration that makes free Brownian rotation of the anisometric aggregates impossible. There is simply “one step too little” in this model to describe relaxation data as those in Figure 5b,c. The empirical Lorentzian three-step model also fails to describe these data. This is in contradiction to what was found by Nery et al.10 for a 2.0 M (≈28%) solution at 25 °C. Their experimental relaxation rates otherwise fall neatly between the results for our most concentrated sample at 13 and 35 °C. Both the fitting shown in Figure 1 in their paper and an analysis of their 2H data show, however, that the uncertainties in some data points have to be assumed to be very large if the three-step model is to fit those data. Concentrated Micellar Solutions. The most obvious explanation of the failure of the model presented above in the concentrated regime is that it neglects most of the possible effects of intermicellar interactions on the reorientation behavior. In the semidilute regime where aggregates start to overlap, it seems reasonable to apply a model where the Brownian rotation is slowed by collisions with neighboring aggregates, while still being isotropic on the rotational time scale.31-33 Within this model the time correlation function is still of the form of eq 5, although the rotational correlation times are not given by their hydrodynamic values. It is thus still of a three-step character and in the light of the results above it is not surprising that it cannot reproduce the experimental data from the concentrated regime. To introduce another slow step in the spectral density function it is not sufficient to introduce a restriction of the rates of the aggregate reorientation, there must be a restriction in the amplitude of the aggregate reorientation so that the interaction is not fully averaged on the rotational time scale. On this time scale we may view the aggregate as confined in a tube34,35 or cage36 formed by its neighbors which prevents it from rotating through all angles of space with equal probability. We can define a director, zI, coincident with the orientation of the tube, i.e., the average orientation of the aggregate. The degree of restriction of the rotation around this director is given by the rotational order parameter:

Aggregation and Dynamics in Aqueous C10ACl Solutions

SIA ) 〈d200(θIA)〉IA

J. Phys. Chem. B, Vol. 104, No. 7, 2000 1535

1 SIA ) cos β(1 + cos β) 2

(8)

where θIA(t) is the angle between the aggregate symmetry axis and zI. With the rotation restricted in space there must, since we observe spectra without quadrupole splittings, be other slower processes carrying out the final randomization of aggregate orientations. To proceed, the time correlation function GLM(τ), which is given for the unrestricted Brownian reorientation model in eq 5, has to be modified to describe restricted aggregate reorientation and to incorporate the slower processes. Restricted reorientation has generally been treated by a mean field approach where the influence of surrounding molecules is approximated by a nonspherical potential of mean torque (POMT), in which the studied object performs Brownian rotational diffusion.37,38 This POMT is for simplicity usually taken to be uniaxial. The model has its origin in work concerning nematic phases where there is nonvanishing order over larger domains. The extension to phases with no macroscopic order has been made by Freed and co-workers within the “slowly relaxing local structure” (SRLS) model.33,39 In this model a local director is defined and its orientation in the lab frame assumed to fluctuate over longer time periods and over space so that it on average becomes spherically distributed. The application of the SRLS model requires that there is a potential that restricts rotational diffusion, that this potential is independent of the orientation of the director it is related to (in this case zI), and that the reorientation of this director is statistically independent of the rotational diffusion,33,40 e.g., as a result of time scale separation. Applied in this way the assumptions above simply imply that the spatial diffusion of the aggregates is slow enough for the tube or cage formed by surrounding aggregates to have constant form and direction on the rotational time scale. Considering the slow processes responsible for the final averaging, the simplest, and in our case only possible, approach is to regard them as spherical, i.e., characterized by a single correlation function, gLI, independent of component index. Within the SRLS model and with the further assumption of a constant SIA the lab-frame to molecular-frame TCF, GLM(τ), can easily be derived from the equations of section II of ref 4 and we obtain

The exact evaluation of the gIA mn(τ) must be made by numerical methods43-45 but Moro and Nordio have also derived a single-exponential approximation,45 with the corresponding correlation times τIA mn, valid for arbitrary torques at high order, i.e., when SIA does not deviate too much from unity. (See also ref 4.) At moderate to low order, two or three exponential terms have to be retained in most of the gIA mn(τ) for better accuracy. Characteristic times and coefficients for such multiexponential approximations as polynomials in SIA have been presented for both the diffusion in a cone model42,46 and the harmonic potential.42 gLI(τ) is the time correlation function for the slow processes that are responsible for the final averaging of the interaction. From the available data it is impossible to draw any detailed conclusions about the mechanism of these slow processes. However, some possible combinations of solution structure and slow reorientation mechanisms which are plausible result in an exponential decay of correlation35,47,48 and we may use this form as an ansatz for gLI: LI 1 gLI(τ) ) e-τ/τ 5

(11)

In the general case τLI has to be regarded as an effective correlation time but it still provides semiquantitative information about the slowest dynamic processes in the solution. If eq 11 and the single-exponential approximations for both the lateral diffusion and the restricted rotation TCFs are employed, the spectral density obtained by the transformation of eq 9 is a sum of Lorentzian terms:

J (ω) ) LM

5

2 2 SAM (2 m)0

∑

[

1

S2IA

-

S2AM

τLI

+

1 + (ωτLI)2

IA δm0)gm0 (0)

S2IA

τLA m

(2 - δm0)gLI× ∑ m)0

2

IA (τ) + (τ)gm0

2

∑ ∑(2 - δm0)(2 - δn0)gLI(τ)gIAmn(τ)gAM n (τ) m)0n)0

(9)

which reduces to eq 5 in the absence of restrictions. Fluctuations in SIA, as zI is defined in this work, do not lead to the final averaging of the interactions, whereas fluctuations in the spatial correlation are included in the slowest motional step. The lateral diffusion TCFs, gAM n (τ), in eq 9 are the same as in eq 5, and for spherocylinders the initial slope approximation may be applied. The nine independent restricted rotation TCFs, gIA mn(τ), can only be calculated if a model for the form of the POMT is applied. While a harmonic POMT is the physical choice for nematic or nematogenic structures it may, in analogy with a tube or cage model, be more natural to choose the simplest functional form of a uniaxial potential, which is a double cone with constant potential within the cone and infinitely hard walls at angles (β and ((π - β) relative to zI.36,41 In this model SIA has the simple form:42

+

2 1 + (ωτAM 0 )

2

+

2 1 + (ωτLA m )

(2 -

τAM 0

gAM 0 (0)

2

∑ ∑(2 - δm0) ×

m)0n)0

AM δn0)gIA mn(0)gn (0)

2

2 GLM(τ) ) S2IA S2AMgLI(τ) + S2IA gAM 0 (τ) + SAM

(10)

τLM mn

]

2 1 + (ωτLM mn )

(12)

with the correlation times given by

1 1 1 ) LI + IA LA τm τ τm0

(13)

1 1 1 1 ) LI + IA + AM LM τmn τ τmn τn

(14)

With the multiexponential approximation for the restricted 42,46 the number of Lorentzian terms with rotation TCFs gIA mn(τ), correlation times calculated in a similar fashion increases correspondingly but no new parameters are introduced. The restricted rotation or local structure model requires at least two additional parameters as compared to the free Brownian rotation model: a parameter characterizing the POMT such as the cone opening angle, β, and a correlation time for the slowest motion, τLI. Models for a polydisperse systems, where the rotational diffusion is restricted, must account for the variation of the POMT and/or the correlation time for the slowest motion with

1536 J. Phys. Chem. B, Vol. 104, No. 7, 2000

To¨rnblom et al.

TABLE 2: Parameter Values Obtained by Fitting the Motional Model for Monodisperse Spherocylinders Rotating in a Double Cone with Infinitely Hard Walls temp, °C

concn, % wt

SIAa

τLI, µs

nb

Fc

R,d Å

SMP

RMS dev, %

13.2e 13.2e 13.2e 35.1f 35.1f 35.1f 55.1g 55.1g

20.5 23.8 28.8 20.5 23.8 28.8 23.8 28.8

+0.03 0.57 -0.04 +0.09 0.64 -0.13 +0.07 0.56 -0.18 +0.03 0.60 -0.02 +0.03 0.64 -0.03 +0.07 0.67 -0.08 +0.09 0.55 -0.10 +0.03 0.65 -0.02

+0.1 0.9 -0.1 +0.8 1.1 -0.3 +3.2 2.3 -0.1 +0.01 0.12 -0.01 +0.02 0.21 -0.01 +0.10 0.35 -0.05 +0.02 0.07 -0.01 +0.01 0.10 -0.01

+5 139 -6 +22 151 -18 +22 162 -23 +6 107 -6 +15 127 -14 +35 184 -27 +26 110 -46 +20 158 -31

+0.1 3.7 -0.1 +0.6 4.1 -0.5 +0.5 4.1 -0.5 +0.1 2.6 -0.1 +0.1 3.1 -0.1 +0.2 4.0 -0.2 +0.1 2.3 -0.1 +0.1 2.9 -0.1

+0.2 14.3 -0.2 +0.2 14.1 -0.2 +0.3 14.5 -0.2 +0.3 14.9 -0.3 +0.6 14.8 -0.6 +0.7 15.1 -0.6 +1.0 15.7 -2.2 +0.5 16.1 -0.9

+0.002 0.224 -0.002 +0.003 0.228 -0.003 +0.003 0.227 -0.003 +0.003 0.225 -0.003 +0.005 0.228 -0.005 +0.003 0.226 -0.008 +0.03 0.22 -0.01 +0.011 0.219 -0.006

1.0 1.5 1.5 0.9 1.9 3.3 3.9 2.0

a Calculated from the cone-half opening angle β and eq 10. b Aggregation number. c Axial ratio of the aggregate. d R was limited not to exceed 16.7 Å. e Dlat ) 85 µm2 s-1, τMP ) 32 ps. f Dlat ) 220 µm2 s-1, τMP ) 13 ps. g Dlat ) 440 µm2 s-1, τMP ) 8 ps.

aggregate size. The perhaps simplest geometrical model is given by the extension of the tube model for rigid polymers34,35 to polydisperse systems in such a way that the tube radius is supposed constant throughout the solution and varied in the fittings and the cone opening angle for each aggregate is determined by the tube radius and the aggregate dimensions. This POMT can be combined with a slow reorientation time τLI, which can be assumed constant over the size distribution or varying with aggregate size as in the tube model. These models can be combined with any model for the size distribution. However, with the exponential distribution starting with spherical micelles, it was not possible to obtain good fittings to the experimental high concentration relaxation data. If the smaller aggregates were removed from the distribution by use of an exponential distribution starting at aggregates with higher aggregation number, the quality of the fittings increased. This procedure, which introduces one new parameter in the fittings, resulted, somewhat astonishingly, in rather narrow size distributions for these concentrated solutions, and we found that it is a practical and reasonable approximation to treat these solutions as monodisperse which, in fact, does not increase the RMS deviations in the fittings. The results from such fittings are presented in Table 2 and two of them also in Figure 5b,c. In these fittings the fast correlation time, τMP, has also been fixed since the high-frequency limit does not allow unambiguous determination of this parameter in complicated models such as these at the two higher temperatures. The value was determined from lower concentration samples. In our previous investigation3 τMP was constant within uncertainties over a wide concentration range. The rotational order parameters SIA have been calculated by eq 10 from the half cone angle, β, obtained from the fittings. In a comparison with Table 1 the RMS deviations may seem large for some of the entries but this is a result of large deviations in one or a few points below 2.0 MHz and, in all cases but one (see below), a consequence of the larger uncertainties in the relaxation rates measured at these low frequencies. A harmonic POMT yielded similar results for the values of the fitted parameters as well as for the RMS deviations. From this we may conclude that the theoretical relaxation dispersions are determined by the magnitude of SIA rather than the details of the models. Discussion The aggregation behavior found in the C10ACl/water system over the entire concentration interval in terms of the axial ratio of the average aggregate is presented in Figure 7 where results from fittings of both the free Brownian and the restricted reorientation models discussed above have been combined. At the three studied temperatures this has most features in common

Figure 7. Average axial ratios of the aggregates obtained from fittings of the model of exponentially distributed, freely rotating spherocylinders (filled symbols) or monodisperse spherocylinders rotating in a hardwalled double cone potential of mean torque (empty symbols). These results are also presented in Tables 1 and 2.

with what was earlier found in aqueous solutions of SDS/NaCl and C16TABr:3 Nonspherical micelles predominate already at quite low concentrations and as the concentration increases a point is reached where there is, as is seen in the figure, a sudden onset of faster aggregate growth; a “second cmc”. This increase in aggregate growth has already set in when the free rotation model fails to describe relaxation data. A comparison with Figure 2 shows that a large part of the increase in R2 above this point can be attributed to restriction of the rotational motion of the aggregates rather than to aggregate growth, and therefore an analysis of R2 alone leads to an overestimation of the second cmc. Determined from Figure 7 the second cmc is ≈10% at 13 °C, ≈12% at 35 °C, and ≈15% at 55 °C. The aggregation numbers found, which are presented in the tables, are slightly lower than those found by Fletcher and Gilbert11 with fluorescence quenching at 25 °C. It is noteworthy that it is only the data from the most dilute sample (5.0%) at 13 °C and the two most dilute samples (5.0% and 9.6%) at 35 and 55 °C that can be reproduced by a twostep Lorentzian spectral density. At higher concentration this model gives large and systematic deviations from the experimental data. A comparison with the results in Figure 7 and in Table 1 shows that the limit for detection of deviation from spherical shape actually is an axial ratio around 1.4 (aggregation number around 70) if the uncertainties in the measured relaxation rates can be kept at the levels obtained in this work. Another question, also discussed in the previous study,3 is if it is possible to distinguish between rodlike and disklike aggregates with the method used here. In the present study this question was further investigated, and just as in the previous

Aggregation and Dynamics in Aqueous C10ACl Solutions study we find that for the samples where the Lorentzian twostep model fits (see above) a model of exponentially distributed disk-shaped micelles with semitoroidal rims undergoing free Brownian rotation can be fitted equally well compared to the spherocylinder model using the same value for the lateral diffusion coefficient. For samples where the spherocylinder model gives axial ratios between 1.4 and 2.0 an approximately 50% faster lateral diffusion is required in order to fit the disk model to the experimental data. When the average aggregate is larger it is not possible to fit the model of freely reorienting disks at all. This is illustrated in Figure 4, showing the best fit, including polydispersity, to the data from the 14.1% solution at 13 °C. Even if all parameters in the model except the radius of the spherical micelles are varied, the relative RMS deviation of the best fit is above 5%. The spherocylindrical model gives an average aggregation number of 150 and, as seen in Figure 3a, fits excellently to these data using the same lateral diffusion coefficient as for the spherical micelles, i.e., with fewer parameters varied in the fitting procedure. Generally it is more problematic to differentiate between different micellar shapes in the restricted reorientation model. We found all data sets which cannot be reproduced by the model of freely reorienting disks, also the one shown in Figure 4, to be reproducible with a model of monodisperse disks rotating in a potential of mean torque. However, this required an at least 2-fold increase in Dlat compared to the values discussed above and this is not reconcilable with the notion that this coefficient is controlled mainly by headgroup interactions.49,50 Another observation to be made from Figure 7 is that the growth is linear within uncertainties at 35 and 55 °C, while at 13 °C it seems to decrease as the concentration increases. This is interesting since no hexagonal phase exists at this temperature but the first liquid crystalline phase formed is a discotic nematic formed at 47-48%.8 One might therefore expect a transition from rodlike to disklike, perhaps via biaxial aggregates, somewhere below this concentration. There is, however, nothing in our results, not even an increase in the radius of the micelles, that indicates that this transformation should have begun already at 29%. We show in this work that, at low concentrations, a model of free Brownian rotation of exponentially distributed spherocylinders describes the experimental variation of relaxation rates with frequency very well also when the entire low-frequency dispersion falls within the accessible frequency range, something that was not really put to the test in the former study.3 It is also clear from the results presented here that in order to differentiate between free and restricted Brownian rotation the slowest of the purely rotational modes must be “visible” within the studied frequency range. The experimental data for the two most concentrated of the previously investigated C16TABr solutions do not fulfill this requirement, and it cannot be ruled out that micellar tumbling is restricted in these solutions and axial ratios therefore overestimated. The relaxation dispersion curves in Figures 3c and 5 show the strong impact of increases in concentration on the relaxation dispersion in the low-frequency region, which reflects the slowest dynamic processes in the system. The data in these figures were recorded at 55 °C, but the picture is similar at least at 35 °C though smaller parts of the dispersion falls within the accessible frequency range. From the results in the tables it is seen that within the models presented here this is interpreted as a sudden occurrence of a rather high rotational order parameter, SIA, which for higher concentrations is practically constant, while the effective correlation time for the slowest

J. Phys. Chem. B, Vol. 104, No. 7, 2000 1537 TABLE 3: Surfactant Self-Diffusion Coefficients and Characteristic Lengths in the Concentrated C10ACl Solutions at 35 °C concn, % wt

D, µm2 s-1

τLI,a µs

20.5 23.8 28.8

+0.5 51.9 -0.5 +0.7 36.8 -0.7 +0.2 27.2 -0.2

+0.01 0.12 -0.01 +0.02 0.21 -0.01 +0.10 0.35 -0.05

a

(6DτLI)1/2, nm +0.3 6.2 -0.2 +0.4 6.9 -0.3 +1.0 7.5 -0.6

From Table 2.

processes, τLI, increases considerably with concentration. This behavior of SIA is certainly not what would be expected. If the samples presented in Table 1 are classified into concentration regimes as suggested by Doi and Edwards34,35 for rigid polymers, it is seen that the model of free Brownian diffusion is able to reproduce data from samples with concentrations well into the semidilute region where crossover between aggregates has started to occur. Fittings of the restricted reorientation model to these data sets gave order parameters with low nonzero values when τLI was fixed at values corresponding to those found for the higher concentrations. From this we may conclude that the spectral density is not very sensitive to small deviations from free Brownian rotation under the conditions applied here. The micellar radii and axial ratios yielded by these fittings were not substantially different from those presented in Table 1, though. A further comparison of the results from the restricted reorientation model with the models of Doi and Edwards shows that the SIA values for the sample of highest concentration agree reasonably well with those given by the simple geometrical estimates of the tube model,34,35 while those at the other concentrations probably are too high. It is possible that this is an artifact brought about by the application of an erroneous form for the slow spectral density term. The Lorentzian ansatz for this term, eq 11, is not entirely consistent with experimental data as is shown by the 0.5 and 1.0 MHz points in Figure 5b, which is the only data set where a low-frequency dispersion with a high amplitude is almost entirely contained within the accessible frequency range. We can conclude that a large portion of the low-frequency dispersion has a Lorentzian form, though, since the model does not deviate in this systematic way from any other of the data sets. Though the correlation times τLI obtained with the Lorentzian model, in the light of what was said above, may seem to be an unreliable quantitative measure, they still show that the rates of the slowest dynamic processes decrease with increasing concentration. This observation rules out exchange of surfactant molecules between neighboring micelles with different orientations in a perfectly disordered structure as the dominating mechanism of the slowest processes. We also note that these processes are surprisingly fast. For comparison we can calculate the RMS displacement of a surfactant molecule during the time τLI from self-diffusion coefficients measured by the PGSE technique at 35 °C as is done in Table 3. These distances are of the order of one aggregate length and any mechanism based on the existence of domains with local order considerably larger than the aggregate dimensions is not likely to dominate and neither is any kind of collective fluctuations involving a large number of aggregates. The most plausible mechanisms for this process is the reorientation of the zI directors as the aggregates move relative to one another.34,35 In systems with local order this mechanism leads to the formation/breakup of the local structures and it seems this is the most important reorientation mechanism also in these systems.51 At the concentrations studied in this work there is always the question of whether the micelles are long and flexible,

1538 J. Phys. Chem. B, Vol. 104, No. 7, 2000 wormlike. We do not find this likely in this case. The samples we have worked with are visibly fluid even at the lowest temperature and certainly not viscous enough to entrap air bubbles as can be expected from wormlike micelles at these concentrations. The concentration dependence of the slowest dynamic processes also speak against wormlike micelles. These solutions are, as the results show, still in the concentration region where the correlation times of the slowest reorientational processes increase with concentration instead of being constant as in solutions of wormlike micelles52 and where the selfdiffusion coefficient, as can be seen in Table 3, still decreases with concentration.53 Acknowledgment. We thank Ann-Charlotte Hellgren for synthesizing the deuterium-labeled surfactant. This work was financed by the Swedish Natural Sciences Research Council. References and Notes (1) Halle, B.; Quist P.-O.; Furo´, I. Liquid Cryst. 1993, 14, 227. (2) So¨derman, O.; Stilbs, P. Prog. Nucl. Magn. Reson. 1994, 26, 445. (3) To¨rnblom, M.; Henriksson, U.; Ginley, M. J. Phys. Chem. 1994, 98, 7041. Erratum J. Phys. Chem. 1997, 101, 3901. (4) Halle, B. J. Chem. Phys. 1991, 94, 3150. (5) To¨rnblom, M.; Henriksson, U. J. Phys. Chem. B 1997, 101, 6028. (6) Sitnikov, R.; Furo´, I.; To´th, F.; Henriksson, U. ReV. Sci. Instrum., in press. (7) Bleasdale, T.; Tiddy, G. In The Structure, Dynamics and Equilibrium Properties of Colloidal Systems; Bloor, D. M., Wyn-Jones, E., Eds.; Kluwer Academic Publishers: Dordrecht, 1990; p 397. (8) Rizzatti, M. R.; Gault, J. J. Colloid Interface Sci. 1986, 110, 258. (9) Ginley, M. Unpublished result. (10) Nery, H.; So¨derman, O.; Canet, D.; Walderhaug, H.; Lindman, B. J. Phys. Chem. 1986, 90, 5802. (11) Fletcher, P.; Gilbert, P. J. J. Chem. Soc., Faraday Trans. 1 1989, 85, 147. (12) Mukerjee, P.; Mysels, K. Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. 1971, No. 36. (13) Walderhaug, H.; So¨derman, O.; Stilbs, P. J. Phys. Chem. 1984, 88, 1655. (14) Jansson, M.; Li, P.; Henriksson, U.; Stilbs, P. J. Phys. Chem. 1989, 93, 1448. (15) So¨derman, O. J. Magn. Reson. 1986, 68, 296. (16) So¨derman, O.; Henriksson, U. J. Chem. Soc., Faraday Trans. 1 1987, 83, 1515. (17) Tanford, C. Hydrophobic Effect; John Wiley & Sons: New York, 1980; Chapter 6.

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