J . Phys. Chem. 1989, 93, 2512-2519
2512
(A.12)
where Ak yields J
to.
Dividing eq A.9 by eq A.8 in order to eliminate
+ 0'+ l)CYkj(Rk_I/Rk)*j+' 1
- (Ykj(Rk-I/Rk)*j+'
]
j
=
.k+I[
+ 0' + l)ak+lj 1 - ak+lj
]
Equation A. 1 1 and eq A. 12 formulate an iterative procedure in which one replaces in each step the first k shells by a sphere with a homogeneous permittivity t k j and calculates the polarizability f f k + l j (eq A.12) of the sphere with k 1 shells. From eq A.11 one then obtains t k + l j and proceeds with the next step. In the case of a homogeneous particle with permittivity t l ( n = 1, c2 = to), one obtains the familiar result:
+
(A.lO)
(A.13)
If we introduce the shorthand notation
we may write eq A.10 in the form
The derivation of eq 12 follows from the above formulas by straightforward numerical evaluation of the multipole polarizabilities for the typical case considered. Registry No. AOT, 577-1 1-7; isooctane, 540-84-1.
Static and Dynamic Properties of Nonionic Amphlphile Micelles: Triton X-100 in Aqueous Solutlon Wyn Brown,* Roger Rymdik, Jan van Stam, Mats Almgren, and Goran Svensk Institute of Physical Chemistry, University of Uppsala, Box 532, 751 21 Uppsala, Sweden (Received: February 16, 1988; In Final Form: August 26, 1988)
Aqueous solutions of the nonionic surfactant Triton X-100 have been examined by using pulsed field gradient NMR (PFG-NMR), static and quasi-elastic light scattering (QELS), and time-resolved fluorescence quenching within a broad range of concentration (0.5-35%) and over the temperature span 10-45 O C . The infinite dilution values of self-diffusion coefficients from PFG-NMR correspond to a temperature-independent micellar radius of 31 A. Comparison of DNMRand DQELs indicate a significant size polydispersity in micellar size at least at low concentrations. The QELS time correlation functions were bimodal with components of narrow size distribution. The slow mode of low amplitude that is present in addition to the micellar mode is considered to derive from clusters of small micelles. The clusters become larger with increase in concentration and also appear to be temperature labile. Molecular weights from static light scattering gave a value of 43 500 at 10 OC (commensurate with a hard-sphere radius of about 25 A), rising to M = 96000 at 45 O C , and infer micellar growth with temperature. When the micellar mass is estimated from the ratio DQEs/DNMR, it is found to increase with concentration. The fluorescence measurements support the above conclusion that the micelles grow with both concentration and temperature increase.
Introduction Nonionic amphiphiles of the Triton X category ((alkylphenoxy)poly(oxyethylene) monoethers) have received considerable attention directed toward elucidation of the size and shape of their micelles formed within the confines of the isotropic L1 region of the phase diagrams. Some groups have maintained that the micelles are spherical, for example, ref 1-3, while more recent paperse7 concluded that an oblate form is the more likely. Another current question concerning nonionic micelles deals with the influence of temperature and concentration on the apparent growth of the micelles and possible shape changes. Aggregation and/or a dominant influence of critical concentration fluctuations have Kushner, L. M., Hubbard, W. D. J. Phys. Chem. 1954, 58, 1163. Razin, S . Biochim. Biophys. Acta 1972, 265, 241. Ribeiro, A. A.; Dennis, E. A. Biochemistry 1975, 14, 3746. Robson, R. J.; Dennis, E. A. J . Phys. Chem. 1977, 81, 1075. Tanford, C.; Nozaki, Y.; Rohde, M. F. J . Phys. Chem. 1977,81, 1555. Paradies, H. H. J. Phys. Chem. 1980, 84, 599. Mandal, A. B.; Ray, S.; Biswas, A. M.; Moulik, S. P., J . Phys. Chem. 1980, 84, 856. (8) Corti, M.; Degiorgio, V. Opt. Commun. 1975, 14, 358. (9) Corti, M.; Degiorgio, V. J . Phys. Chem. 1981, 85, 1442. (10) Corti, M.; Minero, C.; Degiorgio, V. J . Phys. Chem. 1984, 88. 309. (1 1) Cantu, L.; Corti, M.; Degiorgio, V.; Minero, C.; Piazzo, R. J. Colloid Interface Sci. 1985, 105, 628. (1) (2) (3) (4) (5) (6) (7)
been advanced as alternative explanations for the observed solution properties. In an attempt to shed more light on these aspects, we have applied several complementary techniques to the Triton X-100 system to give a broad approach to the size/shape problem. The techniques employed are pulsed field gradient N M R (PFGNMR), quasi-elastic light scattering, static light scattering, and time-resolved fluorescence quenching. The PFG-NMR self-diffusion measurements allow the frictional coefficient to be evaluated, while the light-scattering methods give access to thermodynamic quantities in addition to the mutual diffusion coefficient. The fluorescence quenching method provides an independent method for determining the average aggregation number of the micelles and is not subject to the same ambiguities in interpretation as scattering techniques. The concentration range of the measurements has been extended to 35%, which is close to the phase boundary, since the earlier investigations were restricted to dilute solutions (Le., less than about 2%, which is close to the critical concentration). It will be shown that the average molecular weight of the micelles (estimated from DQELSIDNMR at finite concentrations) increases with concentration, particularly at the higher temperatures but where the latter are still much below the cloud point. This apparent size increase has been noted in the earlier studies and will be shown below to be at least partly due to an increasing paucidisperse character of the particle sus-
0022-36~4/89/2093-2~12$01.50/0 0 1989 American Chemical Society
Nonionic Amphiphile Micelles pension at higher concentrations.
Experimental Section Samples and Solutions. Triton X-100 (Merck, 99.5%; @( I , I ,3,3-tetramethylbutyI)phenyl)poly(oxyethylene)) was used without further purification. All solutions were prepared by weight and filtered through 0.22-pm Millipore filters. For the fluorescence measurements the Triton X-100 sample was checked with regard to absorbing and emitting impurities and found to be sufficiently pure. Solutions were prepared in distilled water essentially free from photoactive impurities. Pyrene (Aldrich) and benzophenone (Kebo, Stockholm), of analytical grade, were recrystallized twice from ethanol. Quasi-Elastic Light Scattering (QELS). The experimental arrangement has been described earlier.I2 The light source was a 488-nm Ar-ion laser, and the detector system consisted of an ITT FW 130 photomultiplier, the output of which was digitized by a Nuclear Enterprises amplifier/discriminator system. An ALV-Langen Co. (FRG) multibit, multi-7 autocorrelator was operated with 23 simultaneous sampling times, which covered, for example, delay times over the range 1 p s to 1 min. The sample cell (IO-mm precision-bore N M R tubes) was thermostated in an index-matching liquid (trans-decalin) controlled by an external thermostat. All QELS runs were routinely processed by using two- and three-parameter cumulant fits and also by discrete multiexponential fits. Since the time correlation functions at higher concentrations were found to deviate from a single exponential, distributions of relaxation times were also obtained by maximum entropy analysis (MAXENT). This is a new technique, the application of which to QELS data has been described in detail by Livesey et al.16317The power of the method to deal with single and multiple peaks as well as broad distributions was demonstrated by using both simulated data and experimental correlation curves for colloidal systems. The programs, based on the Cambridge maximum entropy suite of subroutines (MEMSYS), provide a unique solution which is robust to noise and shows features (peaks) only if demanded by the data. Pulsed Field Gradient NMR. These measurements were made on protons at 99.6 MHz on a standard JEOL FX-100 Fourier transform N M R spectrometer. An internal D 2 0 lock was used for field frequency stabilization as previously described.I2J3 The time between the 90 and 180' radio frequency pulses was 140 ms for all 6 values, where 6 is the duration of the gradient pulses. Static Light Scattering. These experiments were made by using a photon-counting apparatus from Hamamatsu to register the scattered signal as a function of angle. The light source was a He-Ne laser. A liquid-filled light guide connected the scattering cell to the photomultiplier. The instrument was calibrated with benzene, and this gave a flat angular dependence of the reduced intensity, after angle correction (sin e), over the range 45-135'. The value of dn/dc = 0.15 mL g-' was from ref 8. The inverse osmotic compressibility was evaluated as
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2513 of poly(ethy1ene oxide) with = 280000, obtained from Toya Soda Ltd., Tokyo. Time-Resolved Fluorescence Quenching. Fluorescence decay data were collected with the time-correlated single-photon-counting technique, using a mode-locked Ar-ion laser (Spectra Physics 17 1/ 12) to synchronously pump a cavity-dumped dye laser (Spectra Physics 375, 3448) for excitation. The dye used was DCM (4-dicyanomethylene-2-methyl-6-@-(dimethylamino)styryl)-4H-pyran) operating at 670 nm, frequency doubled in a KDP crystal to 335 nm so as to match the absorption of pyrene. The emission was measured at 395 nm, selected in a double monochromator (Jobin-Yvon H10) using a 4-nm bandwidth. For detection a channel plate multiplier (Hamamatsu 1564U) was employed. The multiplier output was conditioned by using a Tennelec CFD (TC455) and used as the stop pulse in the TAC (Tennelec TC861A). The start pulse was provided by a diode monitoring part of the excitation beam. The output of the TAC was connected via an ADC to a multichannel analyzer based on a P D P l l / 2 3 microcomputer. Data analysis was carried out on a VAX 780 computer, using software based mainly on the programs developed by Lofroth.22 Twelve different sets of solutions of Triton X-100 were prepared in the concentration range 0.5-35%. Each set was divided into three samples with benzophenone concentrations corresponding to a calculated amount of quenchers per micelle of 0, 0.5, and 0.75, respectively (assuming 100 monomers per micelle). The pyrene concentration was the same in all samples, Le., approximately M, low enough to prevent excimer formation. Emission measurements were made at four temperatures, 13, 25, 35, and 45 'C for all sets. The solutions were air-saturated so that the natural decay rate of pyrene includes the quenching by oxygen. The fluorescence decay of an excited probe, here pyrene, follows a single exponential in the absence of quenchers. With quenchers present the initial decay rate is faster but eventually an exponential decay is reached, characterized by a decay constant A2. If the quenchers are stationary in the micelles during the lifetime of the excited state, A2 will equal ko, the decay constant in the absence of quencher. The faster initial decay is due to micelles containing both excited probe and quencher, and the last, single-exponential, portion represents micelles without quencher at the excitation instant. From fluorescence decay data, it is possible to calculate the distribution of quenchers between the micelles, assuming a Poisson distribution, and thus the total quantity of micelles. This leads to the average aggregation number, N . Two different equations were fitted to the data sets. Equation 2 is the Infelta equation:23
F(t) = F(0) exp(-A2t
+ A,(exp(-A,t)
(12) Brown, W.; Johnsen, R. M.; Stilbs, P.; Lindman, B. J . Phys. Chem. 1983, 87, 4548.
(13) Brown, W.; Rymden, R. J . Phys. Chem. 1987, 91, 3565. (14) Almgren, M.; van Stam, J.; Swarup, S. Lofroth, J.-E. Langmuir 1986, 2, 432. (15) Richtering, W. H.; Burchard, W.; Jahns, E.; Finkelmann, H. J . Phys. Chem. 1988, 92, 6032. (16) Livesey, A. K.; Licinio, P. Delaye, M. J . Chem. Phys. 1986,84, 5102. (17) Licinio, P.; Delaye, M.; Livesey, A. K.; Ltger, L. J . Phys. (Paris) 1987, 48, 1217. (18) Nilsson, P.-G.; Wennerstrom, H.; Lindman, B. J . Phys. Chem. 1983, 87. 1377. (19) Carnahan, N . F.; Starling, K. E. J . Chem. Phys. 1969, 51, 635. (20) Degiorgio, V. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; North Holland: Amsterdam, 1985. (21) Vrij, A. J . Colloid Interface Sci. 1982, 90, 110.
(2)
The parameters in (2) are
A , = n(k,/(k,
A4 = k, As an additional internal standard, intensity measurements were made on dilute solutions of an essentially monodisperse fraction
- 1))
+ k-))2 + k-
where [Qlf is the concentration of quenchers in the intermicellar phase, n is the mean number of quenchers per micelle, Le., N = n[SImic/[Qlmic, k+ is the second-order entrance rate constant, kis the exit rate constant (first order), and kq is the quenching constant (first order). Equation 2 was used by Malliaris et al.24 The second equation (eq 3) is a sum of exponentials with decay F ( t ) = A,, exp(-kot)
+ x ( A i exp(-kit)) I
(3)
constants held fixed. ko was chosen to be the same as that obtained (22) Lofroth, J.-E. Eur. Biophys. J . 1985, 13, 45. (23) Infelta, P. P.; Gratzel, M.; Thomas, J. K. J . Phys. Chem. 1974, 78, 190. (24) Malliaris, A,; Le Moigne, J.; Sturm, J.; Zana, R. J . Phys. Chem. 1985,89, 2709. (25) Russel, J. C.; Wild, U . P.; Whitten, D. C. J . Phys. Chem. 1986, 90, 1319.
2514
The Journal of Physical Chemistry, Vol. 93, No. 6 , 1989
from the unquenched emission, whereas the constants ki have no physical meaning but are used only to describe the initial nonsingle-exponential part of the decay. The amplitudes, A , in eq 3 were determined to give the best fit to the emission data. The amplitude from the first exponential, Ao, is the extrapolated value of the fluorescence intensity at the excitation instant according to the final exponential decay as described earlier.26 In the same way the sum of amplitudes A,, = EiAigives the total fluorescence intensity at the excitation moment. The mean aggregation number is derived from eq 3 as
Brown et al.
03-
(9)1d5
A
I
NMR
C/gmi-l A
0
005
01
015
02
025
B
where [SI,,, is the concentration of surfactants in the micellar phase and [QlmIcis the concentration of quenchers in the micellar phase. For polydisperse micellar systems, the probability of finding quencher or probe is proportional to its size and hence the aggregation number. Application of eq 4 yields an apparent, Qaveraged aggregation number, NQ,which depends on the quencher concentration as shown in
N Q = Nw - ) / Z U , ~ 4- 1/67f,q2 - ...
, +’
(5) 005
01
015
02
025
03
C
03
t
D 1015
02 a
(6)
This dimension is in approximate accord with the length of the component amphiphile molecules, using a length of 17 8, for the oxyethylene chain and I O 8, for the octylphenyl group.4 These estimates are, however, uncertain, as is also expression in terms of a spherical particle. As pointed out by Robson and D e n n i ~ , ~ if a sphere is assumed, from packing considerations some of the oxyethylene groups must be embedded in the core, extending its size. Moreover, the degree of chain extension of the ethylene oxide moiety is an open question, as is the extent of micellar hydration. We note here that the RH value is similar to that found for the closely related nonionic amphiphile micelles formed in the C& and CI2E7systems ( R H = 31 In the latter systems the micellar form has been taken to be close to spherical, in agreement with the conclusions of other investigations of the relatively well-studied C12E8system. The temperature-independence of R , over broad ranges is also similar to the behavior in these systems, where RH increases only in the vicinity of the phase boundary. (On the other hand, in the closely related CI2E6system, the micelles grow with increasing temperature over a similar range.) The concentration dependence of the self-diffusion coefficient becomes complicated at higher concentrations. As previously observed by Nilsson et a1.I8 in self-diffusion studies on nonionic surfactant systems, D N M R passes through a minimum, the probable reason for the subsequent increase in DNMR at high concentrations being an exchange of amphiphiles between the diffusing particles. (26) Almgren, M.; Alsins, J.; van Stam, J.: Mukhtar.
C/gml-l
NMR
Results and Discussion SelfDiffusion Coefficients ( P F G - N M R ) . Figure 1 shows the concentration dependences of self-diffusion coefficients, where the D values have been normalized by using the solvent viscosity (7) and the absolute temperature ( T ) at each measurement tempera ture. To obtain the infinite dilution values of ( D v / T ) , for calculation of the hydrodynamic radii, linear plots of log ( D q / T ) versus C were extrapolated to C = 0. These gave an approximately constant value of (@/T)NMR = 23 X lo-’’ at each of the four temperatures, which corresponds to an equivalent hydrodynamic radius (RH) of 3 1 A, by using the Stokes-Einstein equation:
f o l y m . Sci., in press.
/
Y
0
where N , is the weight-average aggregation number and uWand 7f3are the second and third cumulants of the weight distribution, sN,/CpN,; N, is the number of aggregates with s monomers. q = [Q]/[s] is the ratio of quenchers to monomers in the aggregates. The unequivocal determination of the Q-average aggregation number by eq 4 and eq 5 requires that the experimental data show a well-developed final exponential portion having a decay constant equal to that of the unquenched decay, Le., ko.
RH = k T / 6 ~ 7 D
03
E. f r o g . Colloid
Figure 1. Plots of the normalized diffusion coefficients from pulsed field NMR and quasi-elastic light scattering as a function of concentration, where T is absolute temperature and 7 the solvent viscosity. Triton X-100 in aqueous solution at (A) 10.3, (B) 25, (C) 34.2, and (D) 44.1 OC.
This trend is discernible with the present data at the highest concentrations. As will be shown below, an aggregate species (micelle clusters) is present at the higher concentrations, and it is possible that such amphiphile exchange would be facilitated within the cluster; thus the existence of the latter may serve to explain the observed augmentation in &MR at high concentrations. Mutual diffusion coefficients are also presented in Figure I , where the measurements were made on the same solutions and at the same temperatures as the PFG-NMR experiments. It is noted that the intercepts from the two data sets differ substantially and this difference increases with increasing temperature. In principle, the difference in intercepts could derive from the presence of a small amount of free monomer to which the N M R method is highly sensitive. However, the critical micelle concentration (cmc) is very low in this system (0.24 mM, corresponding to 0.015%), and thus the contribution from free monomer should be negligible. There could also be some hydrolysis products present, e.g., free ethylene glycol entities that could contribute to the N M R signal. Another and more general source of polydispersity could be a broad micellar size distribution since the averages pertaining to the two measurement techniques are different. DNM, is close to the number average (12), and D Q E L ~ equals the Z average. Thus the different intercepts may instead
Nonionic Amphiphile Micelles
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2515 10k
05
1^' 10
30
20
40
50
-10
10
IO'
,bl
1'04
1b5
1'06
'
- 11
RELAXATION TIME /IS
-12
c - 1:
102/9mr'
\
I
1
I
10
20
30
Figure 3. A. Concentration dependence of diffusion coefficients obtained from a bimodal fit to the time correlation function; data are at angle 60° and 34 OC. B. Temperature dependence of the relative amplitude for the slow mode obtained by using a bimodal fit. Figure 2. A. Time correlation function (&t) - 1 vs log t ) for Triton X-100 in aqueous solution (C = 10%)measured at an angle of 30' and 25 "C. B. Distribution of relaxation times obtained by using maximum entropy analysis.16J7
indicate a significant polydispersity of the micellar suspension itself rather than an effect coming into play only at very low concentrations. The Triton X- 100 molecules are themselves polydisperse, with an average length of the oxyethylene chain of about 9.5 ethylene oxide units. At low concentrations, amphiphile molecules having longer oxyethylene chains should participate less readily in micellization and/or the micelles formed may be smaller. However, similar differences between N M R data and QELS data were also found in the C12E8system with fresh samples of high purity. Further study of this aspect is in progress owing to the far-reaching consequences of polydispersity for the interpretation of the solution properties. The relative variance ( ~ ~ / iobtained = ' ~ ) in QELS measurements from a cumulants fit to the time correlation function also provides an index to the polydispersity. We find that the relative variance is approximately constant (r0.27 at 25 "C) over the concentration range (2-35%). The variance decreases, however, with increasing temperature to about 0.16 at 45 "C, which may stem from a reduction in the number/size of micellar aggregates, see below. The time correlation functions were observed to be nonexponential above a concentration of about 2%. This is illustrated in the upper part of Figure 2, which was obtained by using a multi-7 autocorrelator permitting monitoring of the spectrum of relaxation times over a wide span of delay time, extending over about 8 decades. Analysis using the maximum entropy method (MAXENT) illustrates the discrete and easily separable character of the modes showing that the dispersion is in fact paucidisperse. We not. here that the micellar solutions in the CI2Esand C12E7 systems pssess similar character. The relaxation rates of the two modes, derived by using a multiexponential fit, were found to be q2 dependent over the angular range 2O-12Oo, where q is the scattering vector ( ( q = (4s/X) sin 8/2), X is the wavelength of the light and 8 is the scattering angle). This shows that the components are diffusive in character. The concentration dependences of the fast and slow modes are shown in Figure 3A.
'C
I
10
20
30
I
LO
Figure 4. Hydrodynamic radius evaluated from QELS data using the Stokes-Einstein equation, ( 6 ) .
The diffusion coefficient of the slow mode decreases with increasing concentration, showing that the associated particle size increases. The data are not consistent with the presence of dust, partly owing to the narrow size distribution and also since dust would be most in evidence at the highest temperature where scattering from the micelles is lowest. Furthermore, the variation in the relative amplitude of the slow species with temperature as shown in Figure 3B indicates that the clusters are labile structures. An interpretation in terms of a mixture of small spheres and rodlike particles is a possibility, but the coexistence of two such disparate entities seems unlikely. As recently discussed by Richtering et al.,ls the slow mode most probably represents clusters of the smaller micelles and these in turn may form the precursor stage of the liquid-crystalline phase. A puzzling feature is, however, the apparently discrete nature of the slow component. This aspect will be the subject of a forthcoming contribution. The hydrodynamic radius evaluated for the fast micellar mode in QELS is larger than that determined from the self-diffusion coefficient and increases with temperature as shown in Figure 4. At 20 "C R13-50 A, a value similar to the estimates of other groups using scattering methods, e.g., 41.8 A in ref 6. The disparity in dimensions between NMR and scattering methods, which derives from significant polydispersity at infinite dilution coupled with the very different averages associated with these techniques,
2516
Brown et al.
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989
TABLE I: Micellar Dimensions for Triton X-100 in Aaueous Solution
temp/'C
(static LS)
n,
RIA"
(QELS)
RHIA
RHIA (NMR)
10.3 25 34.2 44.1
43.5 68.5 80 96.2
70 105 127 156
25 29 31 33
45 52 72 92
31 32 32 33
io-3Mw
"Calculated from M = ( ~ M V / ~ T N , ) ' / ~where , the partial specific volume = 0.91 mL g-1.8
3 010
Figure 5. Determination of the cloud point ( T c ) . Plot of the reciprocal total intensity from static light scattering as a function of temperature for Triton X-100 at a concentration of 3.02%; Tc = 68.5 O C .
I
3
c io3/& 10
10
30
LO
50
Figure 7. Static light scattering: the reduced scattering intensity ( K C / R g o )versus concentration at 10.3 OC.
c 10ygfnr-1 0
50
109
150
200
2M
300
Figure 6. Total intensity from static light scattering as a function of concentration at different temperatures. The curves have been nor-
malized to the same maximum intensity to allow comparisons. The broken line is according to the equation for hard spheres from Carnahan and Starling.lg underlines the necessity of using independent methods to establish particle conformation. The picture emerging from the above data is thus one in which approximately spherical micelles coexist with aggregates of loose clusters of such particles forming at higher concentrations. The clusters increase in size with increasing concentration at a given temperature. As the temperature is raised, the clusters decrease in size and/or become less numerous, giving a smaller contribution to the total scattering. Static Light Scattering. Static light-scattering measurements were also made at the same concentrations and temperature as the previous experiments. The cloud point was estimated by extrapolating the total intensity to an infinite value as shown in Figure 5 and give the value Tc = 68.5 OC. This value is somewhat larger than the 64 OC given by Corti et a1.,8 but this may be due to the differing source of the Triton X-100. Figure 5 shows the total intensity pattern as a function of concentration at four temperatures. The data are given in arbitrary units, normalized to the same maximum intensity to facilitate comparison with each other and with the theoretical curve for the hard-sphere model calculated from the Carnahan-Starling equation.lg At low concentration, at each temperature, the intensity increases linearly with concentration in agreement with scattering theory for independent particles. The displacement of the curves from the sphere model toward low concentration indicates a size/shape increase of the micelles and/or a progressive change in the intermicellar interactions with increasing temperature. It will be shown below (see, for example, Figure 9) that the nonideality also increases with temperature. Also, the average molecular weight
of the particles increases (Table I), although the decreasing relative variance from the QELS determinations did not indicate that the size distribution becomes broader. These results could be interpreted to mean that growth of the micelles occurs. The reduced scattering intensity (KCIR,) is shown in Figure 7 as a function of concentration for the data taken at 10 "C. The The intercepts at C - 0 give the average molecular weight (A?,,). latter are summarized in Table I at the four temperatures. The value of M at 25 OC (66000, aggregation number 105) corresponds to a radius of the nonhydrated particle of 29 A (assuming a sphere and with a value of the partial specific volume 8 = 0.91 mL g-l). This figure agrees quite well with the values for the hydrodynamic radius obtained from the N M R measurements (32 A), and this agreement is significant with regard to the viewpoint that critical concentration fluctuations in light scattering may lead to a misleading picture of the micellar dimensions of the individual micelle.M A comparison of the derived aggregation numbers shown in Table I with those from the fluorescence quenching determinations (Table 11) will be discussed below. We note that the value of M = 6.6 X lo4 at 25 "C is lower than the 9 X lo4 reported in ref 8, although the latter value was determined at low angle (20O) and may thus be artifactually high if aggregates are present. Reference 4 reports a summary of older data with a spectrum of molecular weights ranging from 6.3 X lo4 to 1.53 X 10'. Triton X-100 preparations are known to be polydisperse. There may also be significant differences in M between materials having differing molecular weight distributions obtained from different sources. The average molecular weight shows a modest increase with temperature in comparison with, for example, the nonionic amphiphile system CI2E6in which M increased by a factor of about 10 over a comparable temperature interval. This trend was also demonstrated in ref 8. All of the nonionic surfactant systems examined in a parallel studyz9 (e.g., CI2ES,CI2E7,and CI2Es) similarly increased in molecular weight with temperature change. The reduced scattered intensity increases strongly with increasing concentration as shown in Figure 8 using the derived inverse osmotic compressibility:
aT/ac= ( K C / R ~ ) ~ , ~ R T
(7) The latter quantity reflects the nonideality in the system as embodied in the virial coefficients: &/dC = ( R T / M ) ( 1 + 2 4 M C + ...). dn/dC shows a monotonic increase with concentration and finally decreases rapidly as the phase boundary is approached in the vicinity of C = 37%. Since the virial coefficients are related to the interplay between the excluded volume and the interparticle
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2517
Nonionic Amphiphile Micelles TABLE 11: Triton X-lOO/Water Data" T = 13OC 0.54 1.03 2.05 3.00 4.13 5.04 10.5 15.0 19.8 22.8 29.9 34.5
57.2 69.1 85.4 74.4 83.2 76.1 98.4 107.6 108.8 124.5 90.6 125.4
49.2 56.5 70.3 58.0 70.5 62.6 67.8 67.7 61.6 74.0 55.4 58.7
67.0 85.5 111.1 98.4 114.4 105.5 133.8 153.6 134.2 177.4 139.1 165.7
" ( a ) F obtained from sum of four exponentials with fixed i values.
55.8 66.0 81.3 71.7 82.0 72.2 78.6 88.0 71.9 86.3 75.5 69.9
101.4 102.9 156.0 127.3 151.2 131.7 171.1 188.7 192.9 194.8 194.0 201.4
T = 44 OC 103.2 125.9 193.6 156.4 185.8 162.4 194.3 244.3 243.5 237.6 255.1 279.3
71.9 78.1 99.2 78.4 88.8 78.7 91.1 105.4 94.2 90.1 89.4 89.6
72.4 86.3 106.6 97.7 108.5 94.3 112.1 131.6 130.0 98.0 103.4 92.1
( a ) I obtained from generalized Infelta model. b T =~Triton X-100. 50 1
i
25t
20
T = 35 OC
T = 25 OC
A
30
t
HARD-SPHERE
/ 1HEORY
c/gmr' 0
005
01
015
02
025
03
035
Figure 9. Plot of the reduced osmotic compressibility (where M and a?r/aC are from static light scattering experiments) as a function of concentration for Triton X-100 solutions at different temperatures: 10.3 (O), 25 (0),34.2 ( O ) , and 44.1 OC (A).
2
L
6
Figure 8. A. Inverse osmotic compressibility (a?r/aC) versus concentration at various temperatures. B. a?r/aC versus C in the low concen-
tration range of Figure 8A. enthalpic interactions, one can deduce that the interactions are predominantly repulsive over the range of concentration studied. The extent of the change of the average micellar size with change in concentration is not known, of course, and thus more quantitative information cannot be derived from the inverse osmotic compressibility term at least at the high concentrations. At a given concentration, d.n/aC decreases with increasing temperature, meaning that the intermicellar interactions become less repulsive at higher temperatures. In Figure 8B the dilutesolution values of &/aC are shown as a function of C. At 45 OC the initial slope, which is related to the second virial coefficient, becomes approximately zero. In polymeric systems this temperature would correspond to the so-called 8 temperature, where the excluded volume is exactly counterbalanced by the attractive pair potential between particles and denotes incipient phase separation. The analogy is not complete since the aggregation number presumably changes with concentration. However, this is probably not a large factor at the low concentrations where the initial slope describing A2 may be determined. Thus one may
surmise that there will be a predominantly attractive pair potential operative in the temperature interval 45-68.5 "C that exerts a driving force for either micellar growth and/or aggregation. Since cluster formation was found to be a phenomenon, appearing only at higher concentrations (see Figure 2), we conclude that significant micellar growth occurs as reflected in the molecular weights given in Table I. In another system recently studied (CI2E6),in which micellar growth was establi~hed,'~.'~ it was found that A2 = 0 at a temperature interval of about 30 OC below the phase boundary. On the other hand, in systems exhibiting negligible growth (ClZE8and CI2E7)A2 = 0 only in the vicinity of the cloud point.29 These differences make it unlikely that the seat of the micellar growth lies only in the changes in interaction between the ethylene oxide segments. Rather it is the interplay between the packing requirements of the respective segment types and the sensitive balance between the hydrophobic and hydrophilic interactions that determine the growth/aggregation pattern. Figure 9 shows the dimensionless parameter (M/RT)(an/dC), where both M and a.n/dC have been obtained from static light scattering, as a function of concentration at four temperatures. The deviation from ideality is large and independent of the molecular size characterizing the micelle at each temperature. The theoretical curve for the hard-sphere model is included, and this serves as a good approximation at concentrations below 5%. One may thus suggest that this represents the concentration at which concentration-driven growth starts to become significant: This ratio thus in principle allows an independent estimation of as a function the nonideality terms. Figure 10 shows DQELS/hMR of concentration. Although expected to be similar, Figures 9 and 10 differ, however, mainly because polydispersity influences the measured D values very differently with the two techniques (see, for example, Figure 1). If, as appears to be the case, the micelles grow and the suspension becomes more polydisperse with increasing concentration, then the ratio D O E U / D N M R will be arti-
2518
Brown et al.
The Journal of Physical Chemistry, Voi.93, No. 6, 1989
V
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Figure 10. DQELS/DNMR = (V/RT)(dlr/dC),taken from QELS and NMR data. The broken line is from hard-sphere theory.I9
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Figure 11. A. Average molecular weight (evaluated from D Q E ~ / D N M R and with an/aC from static light scattering, eq 8) as a function of concentration at different temperatures. B. Apparent molecular weight as a function of temperature for a Triton X-100 concentration of 10.2%.
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factually too low. The ratio is also temperature-dependent since a I the average molecular weight changes as the temperature is al200 tered. However, in spite of these effects, it is still possible to obtain a useful semiquantitative estimation of the trends in the molecular weight as a function of T,C from D Q E u / h M R by utilizing aa/aC 150 from static light scattering. Figure 11A shows these estimates at different temperatures. At 34 and 44 OC, M increases with 100 concentration. This contrasts with C12E6,for which M was established to be a function of temperature but not c~ncentration.'~~'~ Since the polydispersity will increase with concentration as cluster formation proceeds, the concentration dependences shown in Figure 11A will be underestimates. Data beyond a concentration I I I , 1 I j of 10% have not been included since the contribution of monomer 5 10 15 20 25 30 35 transfer to D N M R becomes pronounced and seriously affects the X TRITON X-100 estimates of M . This will also lead to an underestimate of the Figure 13. Mean aggregation number dependence on concentration concentration dependence of M . Figure 11B shows that at an (percent) of Triton X-100. The unfilled squares correspond to results arbitrary concentration of lo%, the average molecular weight from eq 2 and the filled squares from eq 3. Temperatures are (top) 13 increases strongly with temperature. (The larger values in comand (bottom) 44 OC. The dashed lines are inserted as a visual guide. parison with those in Table I are a result of the differing polyregion, which extends for a considerable temperature interval from dispersity influences on the two D values.) the cloud point, can be explained by critical phenomena without Figure 1 2 illustrates the power law dependence of air/dC on postulating a large micellar growth as indicated by the apparent the reduced critical temperature ( e = (Tc - T ) / T c ) . These data molecular weights. However, this general thermodynamic repcan be described by resentation is quite compatible with the data presented here. While ( a . r r / a O T , P = cy (9) it is certain that the solutions become highly nonideal close to the cloud point ( Tc), the experimental data, for example, those given with C = 6.3 X lo4 m2 s2 and y = 0.93. The data were obtained on a solution with C = 3.02%, which is close to the critical conin Figure 2, show that the solution structure, at least at the higher centration for Triton X-100. Such a power law apparently affords concentrations, is more complex than would necessarily be inferred from a description such as that in Figure 12. Alternative sources a general representation of critical micellar solutions, and the of the long correlation lengths measured beyond the immediate parameter values are similar to those obtained by Corti and vicinity of Tc must also be considered. Degiorgio.*-" We also refer to the review of Degiorgio20of data Fluorescence Quenching Results. Representative decay curves on the nonionic ethylene oxide amphiphiles. These authors have suggested that the properties of such solutions in the power law are shown in Figure 13. The fitting of eq 2 and 3 to such data
f
Nonionic Amphiphile Micelles
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The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2519
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Figure 14. Fluorescence emission decay curves with pyrene as the emitting probe. The concentration of Triton X-100 is 1.03%, and the temperature is 13 'C. The concentration of benzophenone is (a) 0, (b) 0.203, and (c) 0.270 mM.
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TEMPERATURE
Figure IS. Increase of aggregation number with temperature. Squares correspond to 1.03%, diamonds to 5.04%, and octagons 29.9% Triton
x-100.
gave values for the aggregation numbers (Q-average) as listed in Table I1 and displayed, in part, in Figure 14. It is not possible to decide from the decay rate data alone if quencher migration occurs. From experience with the pyrene-benzophenone-SDS system24 and CTACZS(SDS = sodium dodecyl sulfate; CTAC = cetyltrimethylammonium chloride), migration is expected to be insignificant on the microsecond time scale at all but the lowest surfactant concentrations. Other exchange mechanisms with hydrophobic solutes between micelles may operate at higher concentration^.^^^^^ In their analysis of pyrene excimer data in assumed that migration Triton X-100 solutions, Malliaris et could occur and used the equivalence to eq 2. According to this interpretation, their results suggested a rapid migration of the very hydrophobic pyrene molecules between very small micelles. The size grew somewhat with temperatiire (10 at 26 "C to 42 at 32.8 "C) but not with concentration. Even as interpreted by using eq 2, the present results suggest much larger micelles, which grow with increasing temperature (Figure 15) if it is assumed that no migration occurs but polydispersity effects are important, the aggregation numbers are even larger and show a clear growth with concentration at all temperatures. On the whole, the fluorescence quenching data suggest aggregation numbers intermediate between those estimated by using QELS and PFG-NMR self-diffusion. This also suggests that the system is polydisperse even when the average size is small, and this aspect warrants further study. As discussed above, the QELS time correlation functions (see Figure 2) suggest that extensive clusters of small micelles are formed at high surfactant concentrations. One would imagine that the migration of the probes and quenchers would be particularly fast (27) Zana, R.; Weill, C. J . Phys. (Paris), Lett. 1985, 46, 953. (28) Almgren, M.; Alsins, J . Prog. Colloid Polym. Sci. 1987, 7 4 , 5 5 . (29) Brown, W.; Zhou, P.; R y m d h , R. J . Phys. Chem. 1988, 92, 6086.
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Figure 16. A. Temperature dependence of the intrinsic viscosity for Triton X-100 in aqueous solution. B. Concentration dependence of the reduced viscosity (q,,/C) for Triton X-1 00 solutions.
in such clusters. However, only a small fraction of the surfactant and thus fluorescent probe is present in the clusters, and thus only a small part of the fluorescence will emanate from them. Viscosity measurements were made on the Triton X-100 solutions, and these data are collected in Figure 16. There is a marked dependence of the intrinsic viscosity (Figure 16A) on temperature, which may be due to dehydration and/or a more contracted conformation of the oxyethylene chains; it may also be related to the lability of the clusters at the higher temperatures. The concentration dependence of the reduced viscosity (Figure 16B) provides evidence in support of significant micellar growth, particularly at the higher temperatures. According to viscosity theory qsp/c = Vv+ KPC + ... (10) where Vis the specific volume of the solute and v and K are shape parameters. Assuming the specific volume either to be constant or most likely to decrease with increasing temperature as implied by Figure 16A, the data demonstrate a pronounced change in the asymmetry of the micelles with increasing concentration.
Conclusions The overall picture emerging from the present measiirements is that of a possibly polydisperse mixture of small micelles. These then grow as well as forming aggregates with increasing concentration. The change with increasing temperature is more ambiguous partly because the changes in M are modest but also because D,,, and DQEUgive conflicting trends, possibly as a result of the form of the size distribution. The presence of a slowly relaxing component is probably the result of the formation of loose micelle clusters that grow with concentration at a given temperature. In this way it is possible to reconcile the apparently conflicting finding of small micelles by using self-diffusion with the increase in the overall molecular weight determined by static light scattering with temperature. Acknowledgment. We are grateful to the Swedish Natural Science Research Council for financial support and to Peter Stilbs for assistance with maintenance costs for the JEOL instrument. Registry No. Triton X-100, 9002-93-1; benzophenone, 119-61-9; pyrene, 129-00-0.
