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Ultrasonic Relaxation in Micellar Solutions of Nonionic Triblock Copolymers T. Thurn,† S. Couderc-Azouani,‡ D. M. Bloor,† J. F. Holzwarth,*,‡ and E. Wyn-Jones*,†,‡ Fritz-Haber Institut der Max-Planck Gesellschaft, Faradayweg 4-6, D-14195 Berlin-Dahlem, Germany, and School of Chemical Sciences, Science Research Institute, University of Salford, Salford, M5 4WT, U.K. Received December 30, 2002. In Final Form: February 24, 2003 During the last 6 years, chemical relaxation studies of the kinetics of micelle formation in triblock copolymers such as L64 (EO13PO30EO13) and B40 (EO13BO10EO13) which behave as nonionic surfactants have been studied. The micellar system is characterized by two well-defined relaxation times as predicted by Aniansson and Wall for the isodesmic model A1 + An-1 h An in which monomeric A1 and micellar, An, are the principal species. A closer examination of the published work, however, reveals that some issues have not been satisfactorily resolved. For example it is claimed that the fast relaxation time associated with the single step monomer/micelle exchange exists in two time domains, namely, 10-4-10-5 s (T-jump) and 10-7 s (ultrasonic) for the same triblock copolymers. In an attempt to resolve this predicament, we report here some ultrasonic relaxation measurements on a number of triblock copolymers as micellar systems and also as mixed micellar systems in the presence of hexadecyltrimethylammonium bromide using both H2O and D2O as solvents. We carried out a careful and systematic experimental study including light scattering and surface tension. Following the analysis of the ultrasonic absorption data, the resulting relaxation parameters were incorporated into well-documented relaxation equations to evaluate rate data and test thermodynamic predictions for the monomer/micelle exchange. After careful considerations we concluded that the ultrasonic relaxation observed in micellar solutions of water soluble triblock copolymers is not associated with the monomer/micelle exchange as defined in the Aniansson and Wall treatment. This conclusion is based on (a) the unreasonable values above the diffusion controlled limit of the resulting rate constants and (b) the amplitude of the relaxation process being incompatible with the known thermodynamic parameters for the micellization process. The most likely explanation for the origin of the ultrasonic relaxation in Pluronics is associated with critical phenomena involving concentration fluctuations in the micellar aggregates. We believe that this process is active in ultrasonic relaxation through a coupling with monomer concentration changes which accompany the fluctuations. The fast relaxation times measured in triblock copolymer systems with the iodine laser temperature jump by Holzwarth et al.12-15 therefore provide the correct rate constants for a monomer micelle/exchange process.
Introduction Triblock copolymers of the type poly(ethylene oxide)poly(propylene oxide)-poly(ethylene oxide) are high molecular mass nonionic surfactants which aggregate in water to form micelles above their critical micellar concentration (cmc) and temperature (CMT).1 Structural studies2-4 have shown that the micelles form a hydrophobic core consisting mainly of the propylene oxide (PO) blocks which are surrounded by an outer shell, the corona, of hydrated ethylene oxide (EO) blocks. The temperaturedependent difference in hydration of the EO and PO blocks causes a strongly temperature-dependent micellization process. This results in a decrease of the cmc of several orders of magnitude upon a small increase in temperature. The latter behavior has led to the widespread use of the CMT as a very useful and practical micellar parameter.1 This paper describes ultrasonic relaxation measurements on a number of triblock copolymers of the type EOnPOmEOn and also EO13BO10EO13 (BO is a butylene † ‡
University of Salford. Fritz-Haber Institut der Max-Planck Gesellschaft.
(1) Alexandridis, P.; Holzwarth, J. F.; Hatton, T. A. Macromolecules 1994, 27, 2414. (2) Mortensen, K.; Pederson, J. S. Macromolecules 1993, 26, 805. (3) Goldmints, I.; Yu, G E..; Booth, C.; Smith, K. A.; Hatton, T. A. Langmuir 1999, 15, 1651 and references therein. (4) Yang, L.; Alexandridis, P.; Steytler, D. C.; Kositza, M. J.; Holzwarth, J. F. Langmuir 2000, 16, 8555.
oxide group) with particular emphasis on the kinetics of the micellization process. When discussing the application of the law of mass action to micelle formation, Goodeve5 in 1935 pointed out that the simultaneous coming together of many molecules to produce a micelle is an improbable process and that the stepwise addition of monomer molecules to existing micelles is more likely. Accordingly the generally accepted mechanism for the formation of micelles occurs via a series of stepwise bimolecular equilibria following the aggregation scheme k+
A1 + An-1 y\ z An k
(1)
n ) 2, 3, 4, ... where A1 denotes the monomer and An is an aggregate made up of n monomers and the k values are rate constants. In this scheme only equilibria which involve an interaction between aggregates and monomers are considered to make a contribution. This means that the bimolecular equilibria involving only aggregates are neglected. In micellar solutions, it is generally accepted that monomers and micelles are present in substantial quantities which can be determined analytically. In comparison the concentrations of the intermediate species (5) Goodeve, C. F. Trans. Faraday Soc. 1935, 31, 197.
10.1021/la020987d CCC: $25.00 © 2003 American Chemical Society Published on Web 04/15/2003
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are negligibly small and cannot be detected. On the other hand their presence is required in order for the above mechanism to operate. When micellar solutions are subjected to a rapid perturbation as in a relaxation experiment, the concentrations of monomers and of the various aggregate species shift from their equilibrium values following the perturbation. This leads to changes both in the monomer concentration and in the concentration of micelles. Although the overall process is complex and involves many steps, Aniansson and Wall6,7 clearly demonstrated in their theoretical considerations that micellar systems as described above are characterized by two discrete relaxation times which can be observed experimentally for solutions of a single surfactant above the cmc. The relaxations have been described as the fast time τ1 and the slow time τ2 and can be interpreted in terms of the stepwise association model of eq 1. Typically these times are separated by a factor of ∼103. The fast relaxation time τ1 can be attributed to a change in surfactant monomer concentration at constant concentration of micelles, and the slow relaxation time τ2 involves the redistribution of the number density of micelles and can be attributed to changes in the concentration of the micelles which remain effectively in equilibrium with the monomers throughout the slow process. In other words for well-defined micellar systems, τ1 is associated with the single step equilibrium between micelles and monomers, whereas τ2 is concerned with the dissolution/formation of micelles from monomers and vice versa. This second process takes place via all the intermediate steps in mechanism 1shence the so-called “slow” time description. The kinetics of micelle formation in Pluronics have been extensively studied using both joule heating8-11 and iodine laser temperature jump (ILTJ)12-16 techniques and also ultrasonic relaxation.9,10,17 The T-jump methods mainly used light scattering as the detection system. On the surface the impression given in these reports is that the kinetics of the micellar Pluronic systems behave according to the Aniansson and Wall predictions with fast and slow relaxation times being observed experimentally. This is a very good indication that the stepwise mechanism (1) also applies to the formation of Pluronic micelles. A critical examination of these reports suggests however that many details remain unclear. Although only one slow relaxation time, τ2, has been observed experimentally at any given condition, the interpretation of τ2 has been a controversial issue.11,14 It is clear that in order to explain the characteristics of the slow relaxation times that have been measured, it is necessary to invoke in Scheme 1 additional steps involving collisions between intermediate aggregates (6) (a) Aniansson, E. A. G.; Wall, S. N. J. Phys. Chem. 1974, 78, 1042. (b) Aniansson, E. A. G.; Wall, S. N. J. Phys. Chem. 1975, 79, 857. (7) Aniansson, E. A. G. Prog. Colloid Polym. Sci. 1985, 70, 2. (8) Hecht, E.; Hoffmann, H. Colloids Surf., A 1995, 96, 181. (9) Michels, B.; Waton, G.; Zana, R. Langmuir 1997, 13, 3111. (10) Waton, G.; Michels, B.; Zana, R. J. Colloid Interface Sci. 1999, 212, 593. (11) Waton, G.; Michels, B.; Zana, R. Macromolecules 2001, 34, 907. (12) Goldmints, I.; Holzwarth, J. F.; Smith, K. A.; Hatton, T. A. Langmuir 1997, 13, 6130. (13) Kositza, M. J.; Bohne, C.; Alexandridis, P.; Hatton, T. A.; Holzwarth, J. F. Langmuir 1999, 15, 322. (14) Kositza, M. J.; Bohne, C.; Alexandridis, P.; Hatton, T. A.; Holzwarth, J. F. Macromolecules 1999, 32, 5539. (15) (a) Kositza, M. J.; Bohne, C.; Hatton, T. A.; Holzwarth, J. F. Prog. Colloid Polym. Sci. 1999, 112, 146. (b) Dawson, A,; Gormally, J.; Wyn-Jones, E.; Holzwarth, J. F. J. Chem. Soc., Chem. Commun. 1981, 8, 386. (c) Marcandalli, B.; Winzek, C.; Holzwarth, J. F. Ber. BunsenGes. Phys. Chem. 1984, 88, 368. (16) Kositza, M. J.; Rees, G. D.; Holzwarth, A.; Holzwarth, J. F. Langmuir 2000, 16, 9035. (17) Marinov, G.; Michels, B.; Zana, R. Langmuir 1998, 14, 2639.
Thurn et al.
and micelles. In other words, the exact mechanism of the micelle dissolution/formation process can vary for the same Pluronic depending on the temperature range studied.12-15 In connection with the fast relaxation time, it has been reported by independent investigations that the relaxation of the monomer/micelle exchange process for some Pluronics exists in two apparently different time domains.12-17 In practice this is not possible since only one relaxation process can be attributed to the single step mechanism associated with the monomer/micelle exchange process. Clearly one of these reports cannot be correct. This paper describes ultrasonic relaxation measurements carried out on various aqueous solutions of triblock copolymers. One of the objectives of the work is to resolve the discrepancy in the literature concerning the interpretation and location of the fast relaxation time for the monomer/micelle exchange process. According to Holzwarth et al.12,14 the relaxation associated with the perturbation of the monomer/micelle process in the Pluronic EO13PO30EO13, code named L64, exists in the time domain of the ILTJ technique at ∼6 × 10-5 s for 1% L64 at 40 °C. On the other hand Zana et al.9,10 reported that this relaxation exists just outside the lower end of the ultrasonic frequency range corresponding to a relaxation time of approximately 10-7 s. In the ultrasonic work, it is not clear where the exact position of τ1 liesstwo apparently different times are quoted in the report for 5% w/v L64 at 40 °C of 1.6 and 3.3 × 10-7 s. It is however clear from these reports that the relaxation of the same chemical equilibrium is claimed to exist in two different time domains. The same scenario also seems to exist10 for other Pluronics although these have not been as extensively studied as L64. For practical purposes the concentration dependence of the fast relaxation time in micellar kinetics can be processed to obtain kinetic parameters by using two relaxation equations derived independently: (i) a twostate model,18 the Aniansson and Wall6,7 treatment, and also (ii) a phenomenological approach.19 For ionic surfactants, all these approaches give identical rate constants when reliable equilibrium data are available. Unfortunately in both above reports, the authors have not taken advantage of good quality relaxation data to carry out a rigorous kinetic analysissrather a qualitative semiquantitative approach has been used. At this stage we wish to point out that in the ILTJ method the iodine laser provides direct heating of the sample and the time dependence of the shift in equilibrium for the monomer/micelle process is monitored using light scattering or fluorescence changes.14 In principle, light scattering is a measure of the size and number of the L64 micelles and the fast relaxation time is essentially attributed to a change in surfactant monomer concentration at a constant concentration of micelles. This means that in such relaxation experiments changes in the surfactant monomer concentration following perturbation are indirectly monitored via changes in micelle size and/ or concentration. We also note that an unexpected angle dependence of the detected light scattering was observed in Joule heating9,10 T-jump experiments which could not be reproduced in the ILTJ studies, where there is no electric field change associated with the T-jump.14,15 In a basic ultrasonic experiment the sound absorption coefficient (R) of the solution is measured as a function of frequency (f), and relaxation occurs when R/f 2 decreases with increasing frequency. When used in a chemical (18) Sams, P. J.; Wyn-Jones, E.; Rassing, J. Chem. Phys. Lett. 1972, 13, 233. (19) Hall, D. G.; Gormally, J.; Wyn-Jones, E. J. Chem. Soc., Faraday Trans. 2 1983, 79, 645.
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relaxation experiment, such changes in the macroscopic behavior of the solution have to be interpreted on the microscopic level in terms of a mechanism. Thus both ILTJ and ultrasonic methods have their limitations for such experiments. As far as we are aware, however, these are the only two techniques which can be readily adapted to study this problem. This means that one of the basic prerequisites before embarking on the processing of chemical relaxation data to determine kinetic and thermodynamic parameters is an “a priori” knowledge of the nature of the chemical equilibrium being perturbed by a sound wave or temperature change. In some cases this exercise requires imaginative manipulation of the system by changing external parameters as well as a process of elimination of many possible mechanisms. When as many of these criteria as possible have been satisfied, the next step is to attempt to analyze the data to extract kinetic information. It is then a matter of applying conventional wisdom to determine whether the conclusions are reasonable in the context of other similar data which are available. To solve the above interpretation problem, the ultrasonic relaxation spectra of a number of aqueous solutions of triblock copolymers above their cmc’s have been measured in the frequency range 0.4-20 MHz. These include the triblock copolymers L64, EO13PO30EO13, and EO13BO10EO13, code named B40, at different concentrations at 40 °C and also 1% w/v of these polymers in the presence of various amounts of the cationic surfactant hexadecyltrimethylammonium bromide (CTAB) in both H2O or D2O as solvents. Finally we have also measured the ultrasonic relaxation at single concentrations of the Pluronics F108, P65, P104, and P85.
Figure 1. Typical ultrasonic relaxation data showing R/f 2 vs frequency (f) for the relaxation process observed in aqueous solutions of ([) L64 1% w/v and (]) L64 1% w/v + CTAB 1 mM at 40 °C (this work): points, experimental; line, calculated.
Experimental Section Materials. The Pluronics used in this work were poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) triblock copolymers. These nonionic surfactants were supplied by BASF, Mount Olive, NJ, and are codenamed with a general formula EOnPOmEOn as indicated below including the molecular mass (MM):
Figure 2. Typical ultrasonic relaxation data showing R/f 2 vs frequency (f) for the relaxation process observed in aqueous solutions of ([) B40 1% w/v and (]) B40 1% w/v + CTAB 1 mM at 40 °C (this work): points, experimental; line, calculated.
The error in MM is 5% and the MMs are average values. We used also a Butronic B40 whose formula is EO13BO10EO13 (BO denotes an oxybutylene group). Its molecular mass is 1900 g mol-1 and the reported cmc20a at 30 °C is 0.2 × 10-3 mol dm-3 . The low molecular mass helps to shift the ultrasonic relaxation into a faster time regime to avoid limitations of the technique in the slow time regime. CTAB was purchased from Sigma-Aldrich. The experimental methods used in this study were as follows. Ultrasonic Relaxation Studies. The ultrasonic sound absorption (R) and sound velocity (v) were measured in various aqueous Pluronic solutions using a modified version of the Eggers21,22 resonance technique which covers the frequency range
0.4-20 MHz. This modified version of the resonance method utilizes a Hewlett-Packard 4195A network/spectrum analyzer used in its spectrum analysis mode with the whole setup controlled by a microcomputer. Typical relaxation data which in all cases can only display the tail end of the relaxation are shown in Figures 1 and 2. Density Measurements. The densities of the L64 and B40 block copolymer solutions were measured at concentrations above and below the cmc at 40 °C using a commercial densitometer (DMA55 from A. Paar, Austria). Light Scattering (LS). Light scattering (LS) data were collected at 360 nm and 90° with a RF-5000 Shimadzu spectrofluorophotometer. A Haake F3-C bath was employed to control the sample temperature within (0.1 °C. Surface Tension (ST). The axis-symmetric drop shape analysis technique (ADSA)23 (constructed in our Salford laboratories) was used to determine the surface tension of liquids from the shape of a pendant drop. The basic principle is to capture the drop image and detect the edge of the drop profile to find the initial parameters of the Laplace’s equation which are then used in a minimization procedure24 to evaluate the surface tension. The temperature-controlled chamber allows experiments to be performed within an accuracy of (0.1 °C. The system was tested with an ethanol-water mixture standard, established by Bircumshaw25 and the surface tension results agreed very well in a range of (0.1 mN/m with the standard data.
(20) (a) Booth, C.; Attwood, D. Macromol. Rapid Commun. 2000, 21, 501. (b) Yu, G.-E.; Yang, Y.-W.; Yang, Z.; Attwood D.; Booth, C. Langmuir 1996, 12, 3404. (21) Eggers, F. Acustica 1967, 19, 323. (22) Eggers, F.; Funk, T.; Richmann, K. H. Rev. Sci. Instrum. 1976, 47, 361.
(23) Susnar, S. S.; Neumann, A. W. Trans. Can. Soc. Mech. Eng. 2000, 24, 215. (24) del Rio, O. I.; Neumann, A. W. J. Colloid Interface Sci. 1997, 196, 136. (25) Bircumshaw, L. L. J. Chem. Soc., Faraday Trans. 1922, 121, 887.
-1
L64,
m ) 30,
n ) 13,
MM ) 2900 g mol
F108,
m ) 50,
n ) 132,
MM ) 14600 g mol-1
P65,
m ) 29,
n ) 18,
MM ) 3400 g mol-1
P104,
m ) 61,
n ) 27,
MM ) 5900 g mol-1
P85,
m ) 40,
n ) 26,
MM ) 4600 g mol-1
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Results and Discussion (i) Analysis of Ultrasonic Data. Our ultrasonic data for L64 were initially analyzed using eq 2
A R ) +B 2 f 1 + (f/fc)2
(2)
where R is the measured sound absorption at frequency f, A is an amplitude relaxation parameter, fc is the relaxation frequency (fc ) 1/(2πτ) with τ being the relaxation time), and B represents the background absorption which may contain contributions to relaxation processes at frequencies much higher than fc. We found that several sets of the parameters A, B, and fc emerge as solutions for the same experimentally measured R/f 2 values during the fitting process. The exact values of these parameters depend on the type of criteria set for minimization and also the experimental error. In all cases the fits between the measured and calculated values of R/f 2 were within (5%. Typical fits for L64 and B40 at 40 °C and 1% w/v are shown in Figures 1 and 2. This uncertainty arises because only the tail end of the relaxation could be measured (experimental restrictions) and the main conclusion here is that no unique set of relaxation parameters can be extracted from the data. This is an inherent problem in ultrasonic relaxation, and the same conclusion although not exactly spelled out also emerges if we critically examine the experimental data of Zana et al.9,10 For example, in this work two different relaxation frequencies (0.48 and 1 MHz) are mentioned for 5% L64 at 40 °C using a three parameter fit procedure and also an alternative approach from the maximum in the plot of R/f 2 against temperature at a fixed frequency. The only positive and noteworthy feature to emerge from the present fitting exercise is that for each set of A, B, and fc which satisfies the minimization criteria applied for fitting eq 2 there appears to be a self-compensating effect between the maximum absorption per wavelength µm ()1/2 Avfc) and τ ()1/(2πfc)) in such a way that µm/τ remains fairly constant for each set of A, B, and fc. We will therefore explore the possibility of analyzing the R/f 2 versus f data in an alternative fashion by rearranging eq 2 as follows:
(
)
R -B f2
-1
)
1 1 2 f + A Af 2
(3)
c
Here a plot of the left-hand side (LHS) against f 2 should be a straight line with intercept 1/A and slope of 1/Afc2. Since we are dealing with the tail end of a relaxation process with an apparent high amplitude, the plot should almost go through the origin (1/A ≈ O). The slope of this plot contains the product Afc2 which means that the compensating effect discussed above is a consequence which we can use as a discriminative handle to obtain more reliable estimates of Afc2, being related to µm/τ as follows
µm/τ ) πvAfc2
(4)
As shown in Figures 3 and 4 the straight line plots which emerge from the above exercise are encouraging. Therefore, despite the fact that τ and fc cannot be independently estimated from the ultrasonic data, the parameter µm/τ can be estimated with some confidence albeit with some generous estimates for the error. It is however reliable enough to allow us to pursue our objective to test various mechanisms as the origin of the observed
Figure 3. Plots of 1/(R/f 2 - B) vs f 2 (see eq 3 in the text) for aqueous solutions of ([) L64 1% w/v and (]) L64 1% w/v + CTAB 1 mM at 40 °C.
Figure 4. Plots of 1/(R/f 2 - B) vs f 2 (see eq 3 in the text) for aqueous solutions of ([) B40 1% w/v and (]) B40 1% w/v + CTAB 1 mM at 40 °C.
ultrasonic relaxation. At this stage we would also like to point out that the ratio µm/τ is the key relaxation parameter which enables the kinetics of fast processes, e.g., in micellization or mixed micelles, to be studied directly via a phenomenological treatment.19,26-30 The µm/τ values for various L64 solutions are listed in Table 1 together with values calculated from Zana’s data.9,10 In both sets of data, there is a good correspondence between the R/f 2 and the µm/τ values. If we take into account the fact that the L64 samples were of different origin and different purification was applied as well as the errors in the µm/τ values, the agreement is excellent. In Table 2 the corresponding µm/τ values determined from the slope analysis are listed for 1% w/v L64 containing various amounts of CTAB in both H2O and D2O as solvent. Finally the values of this relaxation product are also listed in Table 4 for other Pluronics studied and the Butronic B40. (ii) Relaxation Equation for Monomer/Micelle Exchange Process. For the monomer/micelle equilibrium represented by the general step (eq 5)
A1 + An-1 h An
(5)
where A1 is the monomer and n is the average aggregation (26) (a) Jobe, D. J. Verrall, R. E.; Junquera, E.; Aicart, E. J. Colloid Interface Sci. 1997, 189, 294. (b) Gormally, J.; Sztuba, B.; Wyn-Jones, E.; Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1985, 81, 395. (27) Wan-Badhi, W. A.; Palepu R.; Bloor D. M.; Hall D. G.; WynJones E. J. Phys. Chem. 1991, 95, 6642. (28) Gharibi, H.; Takisawa, N.; Brown, P.; Thomason, M.; Painter, D. M.; Bloor, D. M.; Hall D. G.; Wyn-Jones, E. J. Chem. Soc., Faraday Trans. 1991, 87, 707. (29) Takisawa, N.; Thomason, M.; Bloor, D. M.; Wyn-Jones, E. J. Colloid Interface Sci. 1993, 157, 77. (30) Wan-Badhi, W. A.; Lukas T, Bloor, D. M.; Wyn-Jones, E. J. Colloid Interface Sci. 1995, 169, 462.
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Table 1. Relaxation Kinetic Data (assuming monomer/micelle exchange) from Ultrasonic Measurements for the Pluronic L64 at Different Concentrations and at 40 °Ca this work 2
R( (107 mol m-3 s-1)
ref 9 2
/τb
L64 concn (mol dm-3)
1/Afc (m)
µm (s-1)
1/Afc (m)
µm/τ (s-1)
this work
0.00345 0.0069 0.0103 0.0138 0.0172
3.80 0.520 0.490 0.150 0.120
1240 9060 9620 31400 39300
0.960 0.370 0.210 0.140
4890 12900 22300 33200
0.32 2.4 2.5 8.1 10.3
this work
ref 9
ref 9
k-/n (106 s-1)
k+/n (109 dm3 mol-1 s-1)
k-/n (106 s-1)
k+/n (109 dm3 mol-1)
1.3 3.3 5.7 8.5
1.2 3.8 2.6 6.2 6.1
1.4 4.4 2.9 7.1 7.0
2.1 3.5 4.4 5.2
2.4 3.9 5.0 5.9
a ∆V -6 m3 mol-1 from densitometry, ∆H ) 212 kJ/ mol. cmc of L64 equals to 0.89 mol m-3 at 40 °C. b µ /τ ) π × 1500 × L64 ) 3 × 10 m Afc2.
Table 2. Ultrasonic Relaxation Data for Different Block Copolymers from This Work block copolymer
concn % w/v
concn (mol dm-3)
T (°C)
1/Afc2 (m)
µm/τ (s-1)
R( (105 m3 mol-1)
m1c (mol m-3)
k-/nd (104 s-1 )
P65 P65 F108 P104 P85 P85 L64 B40 B40
1 2.5 1 1 1 2.5 1 1 0.654
0.00294 0.00735 0.000684 0.00169 0.00217 0.00543 0.00345 0.00526 0.00344
40 40 40 34 40 40 37 40 40
120 1.4 e 8.4 5.8 1.7 15.6 3.7 11.7
41 3300
1.1a 85a
1.03 1.03 0.027 0.013 0.108 0.108 2.12 0.076 0.076
5.5 140
570 820 2760 310 1300 410
15a 21a 71a 8a 590b 19b
87 100 130 58 1140 560
k+/n (107 dm3 mol-1 s-1) 5.4 130 6720 950 1230 27 15000 7400
a R( calculated with ∆V ) 3 × 10-6 m3 mol-1 (from densitometry, this work) and ∆H ) 212 kJ/mol for L64. b R( calculated with ∆V ) 7 × 10-6 m3 mol-1 (from densitometry, this work) and ∆H ) 118 kJ/mol (obtained from cmc values in ref 20b) for B40. c m1 corresponds to the cmc of the block copolymer in ref 1. d k-/n calculated according to eq 11. e No relaxation.
Table 3. Ultrasonic Data for Aqueous Solutions of L64 1% w/v with Increasing CTAB Concentrations at 40 °Ca
Table 4. Ultrasonic Data for Aqueous Solutions of B40 1% w/v with Increasing CTAB Concentrations at 40 °C
[CTAB] (mol dm-3)
1/(Afc2) in H2O (m)
µm/τ H2O (s-1)
1/(Afc2) in D2O (m)
µm/τ D2O (s-1)
[CTAB] (mol dm-3)
1/(Afc2) in H2O (m)
µm/τ H2O (s-1)
0.0001 0.0005 0.001 0.002 0.003 0.004 0.005
5 2.84 2.84 1.85 1.61 1.72 2.12
940 1660 1660 2550 2930 2740 2220
4.13 3.05 2.95 2.83 2.69 2.15
1140 1550 1600 1665 1750 2190
0.00005 0.0001 0.0005 0.001 0.002 0.003 0.004
11.8 10.1 11.5 10.9 9.7 9.5 9.3
400 470 410 430 490 500 510
a
Measured sound speed v ) 1500 m/s.
number of the micelle, the following equations6,7,19,26,27,31 relate various relaxation parameters to kinetic and thermodynamic constants and parameters associated with the above exchange process. These equations have been used successfully19,26,28 to investigate the fast kinetics of several anionic, cationic, and also some nonionic surfactants.
[
1 k- ka ) 2 + τ1 n σ
(
µm ) and
]
)
( )
R∆H 2 2 Fv FCp m1 2RT
π ∆V -
[(
µm τ
2Fv2 R∆H π ∆V FCp
)
]
2
σ2 a n σ2 1+ a n
) R(
(6)
(7)
(8)
Here ∆H is the enthalpy change, a ) [(C - m1)/m1] with C as the total surfactant concentration and m1 ) cmc; σ is the micellar distribution width, n the micellar aggrega(31) Teubner, M. J. Phys. Chem. 1979, 83, 2917.
tion number, Cp is the specific heat at constant pressure, R the coefficient of thermal expansion of water, v the sound velocity, F the sample density, k- the rate constant for surfactant monomers dissociating from the micelle, and R( the equilibrium forward ()backward) rate of the monomer/micelle equilibrium. Equation 6 is generally known as the Aniansson and Wall fast relaxation equation and relates the measured relaxation time with the surfactant concentration. Under favorable conditions, the backward rate constant and micellar distribution width σ can be evaluated from the plot of 1/τ against (C - m1)/ m1. This equation has been used successfully to analyze the measured relaxation times associated with the perturbation of the monomer/micelle exchange process in many systems.7,27-30 Equation 7 relates the concentration dependence of the ultrasonic maximum absorption per wavelength to a thermodynamic term (∆V - R∆H/[FCp]) associated with the micellization process. Finally eq 8 is the phenomenological rate equation which can be used for ultrasonic relaxation data when µm/τ is available. This equation has been successfully used to determine rate constants for a number of ionic micellar27-30 and also mixed micellar26 systems, and in all cases the data are consistent with those from eq 6. In favorable circumstances k- the backward rate constant can also be evaluated from this equation. A careful examination of the phenomenological equation shows that the quantity µm/τ is a direct measure of the backward ()forward) rate
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Figure 5. Plot of the light scattering intensity ([) and surface tension (O) in milli Newton per meter of aqueous mixtures of 1% w/v L64 in the presence of increasing CTAB concentration at 40 °C.
of the equilibrium being perturbed by the sound wave. This approach allows the kinetics to be treated in a conventional way rather than having to derive relaxation equations such as (2) from mechanistic considerations (e.g., eq 1). (iii) Origin of Observed Ultrasonic and ILTJ Relaxation Process. We first consider the ultrasonic data where it has been established that the relaxation is associated with L64 micellessno relaxation was observed below the cmc or CMT. The ultrasonic relaxation seems to be consistent with a bimolecular mechanism such as eq 1 since the equilibrium rate as measured by µm/τ increases with increasing L64 concentration (see Table 1). Finally we have also chosen the L64/CTAB and B40/ CTAB systems to carry out further ultrasonic studies in an attempt to confirm or eliminate the monomer/micelle exchange as the mechanism being perturbed by the sound wave. The measurements for L64/CTAB were also carried out in H2O and D2O as solvent. It is well-known that after the formation of triblock copolymer micelles there is still a high hydration of the EO chain and to a lower extent of the PO interior. Therefore in micellar solutions an equilibrium exists between “bound” and “free” water molecules, and this process could make a contribution to the ultrasonic relaxation. In these mixed micellar systems the exchange process being monitored by the ultrasonic wave is exclusive to the triblock copolymer since the CTAB exchange process is frozen out on the ultrasonic time scale, so that it cannot be monitored in ultrasonics.26 When CTAB is added to L64 or B40 micelles, mixed triblock copolymer/ CTAB micelles are formed which are systematically broken down to smaller mixed aggregates as the concentration of CTAB increases. This is illustrated in Figures 5 and 6. This effect has already been discussed for F127 and ionic surfactants.32,34,35 The surface tension data of 1% w/v L64 and B40 as a function of increasing CTAB concentration are also shown in Figures 5 and 6. In the region where the L64 mixed micelles start breaking down there is a decrease in surface tension. Since most of the added CTAB is taken up in binding to the mixed micelles, the increased surface activity must be due to the release of the (32) Li, Y.; Xu, R.; Bloor, D. M.; Holzwarth, J. F.; Wyn-Jones, E. Langmuir 2000, 16, 10515. (33) Li, Y.; Xu, R.; Couderc, S.; Bloor, D. M.; Holzwarth, J. F.; WynJones, E Langmuir 2001, 17, 183. (34) Li, Y.; Xu, R.; Couderc, S.; Bloor, D. M.; Holzwarth, J. F.; WynJones, E. Langmuir 2001, 17, 5742. (35) Thurn, T.; Couderc, S.; Sidhu, J.; Bloor, D. M.; Penfold, J.; Holzwarth, J. F.; Wyn-Jones, E. Langmuir 2002, 18, 9267.
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Figure 6. Plot of the light scattering intensity ([) and surface tension (O) in milli Newton per meter of aqueous mixtures of 1% w/v B40 in the presence of increasing CTAB concentration at 40 °C.
hydrophobic L64 monomers resulting from the breakdown of the mixed micelles. At ∼10-3 mol dm-3 added CTAB, the surface tension starts to increase again when CTAB micelles start binding to monomer L64 to form monomeric L64/micellar CTAB complexes which render the polymer hydrophilic.35 Therefore during the systematic addition of CTAB to both triblock copolymer micelles there is a simultaneous increase in the number of mixed micelles and also the monomer concentration of the copolymer. As a result the equilibrium rate in mechanism 5 should display a corresponding increase. This means that the µm/τ is expected to respond to this increase. The µm/τ values are listed in Table 2. When CTAB is added to L64 and B40 micelles, the µm/τ values increase as the CTAB concentration increases in both H2O and D2O. On the basis of a monomer/micelle mechanism, the rate would be expected to change and in fact increases presumably because the concentration of micelles present also increases due to a reduction of aggregation numbers. Therefore the relaxation data are consistent with mechanism 5. If we compare the corresponding rates for L64 in H2O and D2O, then on the whole there is a slight reduction for the D2O rates. This would be expected for a monomer/ micelle exchange36 but not for an alternative mechanism involving the exchange of water between bulk and micellar state. In the latter case the isotopic effect should be manifest in a significant rate decrease in D2O compared to H2O. In addition, the large amplitude observed for the ultrasonic relaxation would also be inconsistent with a water exchange process. Therefore any involvement of the solvent as an important contributory factor to the ultrasonic relaxation is unlikely. In the above considerations of the behavior of the ultrasonic relaxation data all the evidence we have available is consistent with the monomer/micelle exchange process. This statement is also true for the behavior of the relaxation times measured using the ILTJ experiments. In such work 1/τ increases with increasing surfactant concentration and also as the anionic sodium dodecyl sulfate is added to L64 micelles.16 On the basis of the arguments discussed above, the T-jump data are also consistent with a monomer/micelle exchange. At this stage, we have no reasons to eliminate either the T-jump or ultrasonic data as a cause of the relaxation being dominated by the monomer/micelle exchange rate in L64. (36) Gettins, J.; Jobling P. L.; Walsh, M. F.; Wyn-Jones, E. J. Chem. Soc., Faraday Trans. 2 1980, 76, 794.
Micelle Formation in Triblock Copolymers
Langmuir, Vol. 19, No. 10, 2003 4369 Table 5. Observed and Calculated Value of µm on the Basis of Section iv in the Text for L64 Solutions at 40 °C (this work) L64 concn % w/v
µm/τ ultrasonic (s-1)
τTjump (10-4 s)
µma (measd)
µm (calcd from eq 7)
1 2 3 4 5
1250 9200 9720 31600 40370
0.67 0.283 0.141 0.112 0.0850
0.0838 0.260 0.136 0.354 0.343
0.000308 0.000322 0.000327 0.000329 0.000330
a µ was calculated via eq 7 by using the cmc equal to 0.89 mol m m-3 and σ2/n ) 4.
Figure 7. Typical ultrasonic relaxation data showing R/f 2 vs frequency (f) for the relaxation process observed in aqueous solutions of L64 2% w/v at 40 °C. The calculated curves correspond to a typical best three-parameter minimization fitting procedure (- - -) and to a two-parameter minimization fitting procedure using the relaxation times from T-jump experiments (s).
(iv) Are the Ultrasonic and T-jump Techniques Measuring the Same Process? The next obvious step to consider is whether the partial relaxation measured in ultrasonics is also the tail end of the faster T-jump relaxation. One way to test this hypothesis is to reduce the fitting procedure for the ultrasonic data using eq 2 to a two-parameter fit for A and B by assuming that
fc ) 1/2πτTJ
(9)
where τTJ is the relaxation time quoted by Holzwarth et al.13-16 This type of fitting procedure essentially forces the data to fit eq 2. For all the systems investigated using both ultrasonic and T-jump data, acceptable fits were obtained in all cases and typical results are shown in Figure 7. On the other hand if we take into account our initial experiences with the three-parameter fitting procedures and the fact that in this exercise the data have been forced to fit eq 2, then the result is not surprising. Having achieved good correspondence between the τ values, we proceed with further considerations in this direction, by estimating the ultrasonic µm values from the µm/τ values in Table 1 using the τ values measured in the T-jump experiment. At the same time, we can also calculate the µm values from the RHS (right-hand side) of eq 7 from a knowledge of ∆V and ∆H and also by putting in reasonable estimates for the σ2/n values. Here ∆V is estimated from density measurements37 carried out over concentrations spanning the cmc using eq 10
∆V ) -[(M1 + C2*)/(F*)2][(∂F/∂C2)mon - (∂F/∂C2)mic] (10) where M1 is the molecular mass of the solvent, C2 ) N2/ N1, with N2 the number of moles of surfactant and N1 the number of moles of solvent, and M2 the molecular mass of the surfactant with the asterisk referring to measurements taken at the cmc, (∂F/∂C2) with the appropriate subscripts being the respective slopes of the plots of density vs C2 in the monomer and micellar regions. From density measurements carried out over the cmc at 40 °C, a ∆V value of 3 × 10-6 m3 mol-1 was obtained for L64. The ∆H value of 212 kJ/mol was taken from calorimetric measurements. If we assume values of σ2/n (see section vi), then we can calculate µm from eq 7 at different L64 (37) Benjamin, L. J. Phys. Chem. 1966, 70, 3790.
concentrations. The measured and calculated µm values based on the assumption that the relaxation of the monomer/micelle exchange process is centered at the ILTJ time domain of ∼2 × 10-5 s and that the tail end of this process is responsible for the observed ultrasonic relaxation in the megahertz range are given in Table 5. The experimental values of µm are in all cases a factor of some 103 higher than those calculated on a basis of a monomer /micelle exchange. This clearly shows that the starting point of assuming that the monomer/micelle relaxation spans over the ultrasonic and T-jump time domains is unreasonable. Since both the available ultrasonic and T-jump relaxation data are apparently consistent with such a mechanism, we will now independently analyze both sets of relaxation data to determine the kinetic parameters for the monomer/micelle exchange. Clearly both of these approaches cannot be correct, and we will make our choice by examining whether the numbers that emerge for the kinetic parameters are reasonable. (v) Analysis of Ultrasonic Relaxation Data for Kinetic Parameters Based on Monomer/Micelle Exchange. As we have mentioned previously, the uncertainty in the analysis prevents an independent estimation of the relaxation frequency and µm. Hence we are unable to use the Aniansson and Wall relaxation equation (6) or the Teubner equation (7)31 under such circumstances. We can however use the phenomenological rate equation (8) to evaluate the backward ()forward) rates of equilibrium 5 at all concentrations. µm/τ is known from the analysis of the relaxation data, and the bracketed thermodynamic term can be calculated as described above. The R( values are listed in Tables 1 and 2. The next step is to consider the backward rate process in equilibrium 5, i.e., the dissociation of the surfactant monomers from the micelles. This is generally accepted to be a first-order rate process which is proportional to the concentration of micelles as follows:
R- )
( )
k(C - m1) n
(11)
From the estimated rates R- and (C - m1), the values of k-/n are listed in Tables 1 and 2. (vi) Analysis of ILTJ Relaxation Data for Kinetic Parameters Based on a Monomer/Micelle Exchange. If we assume that the fast relaxation observed in the T-jump experiments is associated with the monomer/ micelle exchange process, the relaxation data can be analyzed graphically using eq 6 by plotting 1/τ measured from the ILTJ experiments against (C - m1)/m1. In the original reports on their T-jump measurements, the authors14 choose a similar approach. The plots of 1/τ against (C - m1)/m1 (see Figure 8) are reasonably linear and values of k-/n and k-/σ2 follow directly from the plots. As far as we are aware, all the ILTJ data are consistent
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Figure 8. Aniansson and Wall plot of the reciprocal relaxation times from T-jump experiments (1/τTJ) versus (C - m1)/m1 for aqueous L64 at 40 °C.
with the type of analysis which suggests that k-/n is reasonably constant over the concentration range studied. Discussion and Conclusion If we take the reported aggregation number for L64 at 40 °C (n ) 40)3 and with a cmc of 0.89 × 10-3 mol dm-3, the following rate constants are obtained:
ultrasonic data k+ ∼ 2.4 × 1011 mol-1 dm3 s-1,
k- ∼ 2 × 108 s-1
ILTJ k+ ∼ 2.4 × 108 mol-1 dm3 s-1,
k- ∼ 2 × 105 s-1
The ultrasonic and T-jump14 values of k+ and k- are respectively very similar for all the Pluronics studied12-15 but differ by a factor of 103 between the two techniques.. At this stage, we are only interested in the order of magnitude of these rate constants. When considering whether the above k+ and k- values are acceptable for equilibrium 5, the only reference available is similar work on ionic surfactants. Following the success of the Aniansson and Wall model, many investigators studied a diverse range of ionic surfactants which produced a conventional wisdom about k+ values in the sense that the association of a surfactant monomer to a micelle is an almost diffusion controlled38a,b process with k+ values in the range 1081010 mol-1 dm3 s-1 evaluated from eq 6. When due allowance is made for the monomer ionic surfactant concentration in the micellar range, the highest rate constant that we are aware of is ∼1010 dm3 mol-1 s-1 for a short C6 chain ionic or nonionic surfactant.27-30 The ultrasonic value of k+ for the Pluronic of 1011 dm3 mol-1 s-1 is at least 1 order of magnitude larger than any other similar rate constants measured for other surfactants and also greater than the rate expected for diffusion-controlled reactions38b,c (109-1010 dm3 mol-1 s-1). In addition, the molecular mass and chain length of the Pluronics far exceed those of C6-C8 conventional surfactants. In conclusion, the measured ultrasonic k+ and k- values are inconsistent and unreasonable and cannot be accepted in the context of our present knowledge. Finally, the ultrasonic µm values are also incompatible with those predicted for monomer/micelle exchange. If we take the (38) (a) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905. (b) Holzwarth, J. F.; Ju¨rgensen, H. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 526. (c) Holzwarth, J. F.; Meyer, F.; Pickard, M.; Dunford, H. B. Biochemistry 1988, 27, 6628.
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µm/τ values quoted in Tables 1 and 2 and estimate τ from anywhere in the range between the ultrasonic and the iodine laser T-jump time domain, the resulting values of µm are always at least an order of magnitude higher than those predicted using eq 7 and following the line of the discussion in section iv. On the other hand, the above rate constants evaluated from the T-jump experiments14 are perfectly reasonable within the context of the above arguments. Therefore we are forced to conclude that the ultrasonic relaxation in the block copolymer systems is associated with a different molecular mechanism involving the L64 micelles. As far as we are aware, the following mechanisms are worthy of consideration: (a) An ultrasonic relaxation associated with various conformational changes has been observed in aqueous solutions of a nonassociating nonionic polymer.39 However in the present systems, we would expect this mechanism to be active in both monomeric and micellar copolymers. In practice, a relaxation is only observed above the CMT and cmc; therefore it can be excluded. (b) In short chain, ionic, and nonionic surfactants, a weak relaxation, which is about 1 order of magnitude faster than the monomer/micelle process, has been attributed to premicellar aggregation.40-42 This relaxation is normally of very low amplitude and of the order of ∼10-4 times the amplitude of the relaxation process estimated in this work; therefore it can also be excluded. (c) Recently, Kaatze et al.43 have published a series of papers on the ultrasonic relaxation measured in conventional ionic and nonionic surfactants. These measurements have been carried out over an extensive ultrasonic frequency range from 0.1 MHz to 2 GHz. They have found ultrasonic relaxations which are much faster than the monomer/micelle exchange process in most of the micellar solutions. These fast relaxations are attributed to fluctuations in local aggregate concentrations near critical points, which means in our case near the cmc or CMT of the Pluronics. Although no detailed molecular mechanism has been proposed, it is believed that these fluctuations cover a large range of micellar sizes which results in a broad relaxation distribution. The relaxation of these fluctuations in micellar aggregates will also produce corresponding changes in monomer surfactant concentration. As a result, the coupling to the sound wave occurs via a monomer exchange mechanism with these fluctuating micellar aggregates. We believe that such fluctuations are the most likely explanation for the ultrasonic relaxation observed in the present work In the future, more ultrasonic measurements covering a more extensive frequency range are required together with significant progress in formulating a model which describes these concentration fluctuations on a molecular level. LA020987D (39) Pereira, M. C.; Jobling, P. L.; Wyn-Jones, E.; Morris, E. R.; Pethrick, R. A. J. Chem. Soc., Faraday Trans. 2 1983, 79, 977. (40) Reinsborough, V. C. Chemical and biological Applications of Relaxation Spectrometry; Wyn-Jones, E., Ed.; D. Reidel Publishing Co.: Dordrecht, 1975; p 159. Sams, P. J.; Rassing, J. E.; Wyn-Jones, E. Chemical and biological Applications of Relaxation Spectrometry; WynJones, E., Ed.; D. Reidel Publishing Co.: Dordrecht, 1975; p 163. (41) Hall, D. G.; Wyn-Jones,E. J. Mol. Liq. 1986, 32, 63. (42) (a) Telgmann, T.; Kaatze, U. J. Phys. Chem. A 2000, 104, 1085. (b) Groll, R.; Bottcher, A.; Jager, J.; Holzwarth, J. F. Biophys. Chem. 1996, 58, 53. (c) Genz, A.; Holzwarth, J. F. Colloid Polym. Sci. 1985, 263, 484. (d) Holzwarth, J.; Rys, F. Prog. Colloid Polym. Sci. 1984, 69, 109. (43) Telgmann, T.; Kaatze, U. Langmuir 2002, 18, 3068.
