6086
J . Phys. Chem. 1988, 92, 6086-6094
Size and Shape of Nonionic Amphiphile Micelles: NMR Self-Diffusion and Static and Quasi-Elastic Light-Scattering Measurements on C12E5,C,*E,, and C12E8in Aqueous Solution Wyn Brown,* Zhou Pu,and Roger RymdCn Institute of Physical Chemistry, University of Uppsala, Box 532, 751 21, Uppsala, Sweden (Received: February 18, 1988; In Final Form: May 3, 1988)
Aqueous solutions of the homologous nonionic amphiphiles CI2E5,CIZE7,and Cl2E8have been studied by using pulsed field gradient NMR (PFG-NMR) and static and quasi-elastic light scattering (QELS) over a broad range of concentration (0.05-15%) and in a temperature interval such that a substantial part of the micellar phase region was examined in each case. Cl2E7 and C&8 are found to have closely similar properties: their micelles are small and spherical (RH= 30 A) at low temperatures and form a polydisperse suspension at the higher concentrations. Analysis of the QELS time correlation functions at these concentrations revealed a paucidisperse mixture of micelles and loose clusters. This is a probable source of the apparent micellar growth determined by static light scattering and the observed long-range concentration fluctuations. Cl2ESshows qualitatively different behavior. The micelles are always large (RH2 70 A) and polydisperse. Micellar growth appears likely with increasing temperature and micellar clusters are not observed with increasing concentration and temperature. At low temperatures (510 "C) the micelles are large and probably rodlike and dissociate into smaller entities at low concentration.
Introduction Micellar solutions of nonionic surfactants are currently the focus of considerable interest. Among various questions which have not yet been satisfactorily answered, one concerns possible micellar growth with changing temperature (and concentration) within the isotropic (LI) micellar region of the phase diagrams. Some laboratories have maintained that small spherical micelles persist as the main entity up to the cloud point where two isotropic phases with different amphiphile concentrations are formed. This has been the interpretation in particular of g r o u p ~ l -using ~ neutronscattering techniques, but the results are still e q u i ~ o c a l . Corti ~ et al.5-7 have postulated that the behavior observed in light scattering is due to critical concentration fluctuations and that the phase transition is due to interactions between small globular micelles. have interpreted their results to mean that substantial changes in size/shape occur with temperature change. Some of this ambiguity derives from the length scale probed by a given technique-short lengths in neutron scattering versus long length scales in light scattering. However, the literature dealing with this controversy is now extensive and these introductory comments are necessarily cursory. It is generally accepted that
(1) Cebula, D. J.; Ottewill, R. H. Colloid Polym. Sei. 1982, 260, 1 1 18. (2) Triolo, R.; Magid, L. J.; Johnson, J. S., Jr.; Child, H. R. J . Phys. Chem. 1982, 86, 3689. (3) Zulauf, M.; Weckstrom, K.; Hayter, J. B.; Degiorgio, V.; Corti, M. J . Phys. Chem. 1985.89, 3411. (4) Ravey, J. C. Colloid Interface Sei. 1982, 94, 289. (5) Corti, M.; Degiorgio, V. J . Phys. Chem. 1981, 85, 1442. (6) Corti, M.; Minero, C.; Degiorgio, V. J . Phys. Chem. 1984, 88, 309. (7) Cantu, L.; Corti, M.; Degiorgio, V.; Minero, C.; Piazza, R. J . Colloid Interface Sei. 1985, 105, 628. (8) Becher, P. J . Colloid Sei. 3961, 16, 49. (9) Corkill, J. M.; Goodman, J. F.; Ottewill, R. H. Trans. Faraday S o t . 1961, 57, 1627. (IO) Balmbra, R. R.; Clunie, J. S.; Corkill, J. M.; Goodman, J. F. Trans. Faraday S o t . 1962, 58, 1661; 1964, 60, 979. (11) Elworthy, P. H.; MacFarlane, C. B. J . Chem. SOC.1963, 907. (12) Elworthy, P. H.; McDonald, C. Kolloid, Z . Z . Polym. 1964, 16, 195. (13) Attwood, D. J . Phys. Chem. 1968, 72, 339. (14) Corkill, J. M.; Walker, T. J . Colloid Interface Sei. 1972, 39, 621. (15) Brown, W.; Johnsen, R. M.; Stilbs, P. J . Phys. Chem. 1983,87,4548. (16) Brown, W.; RymdBn, R. J . Phys. Chem. 1987, 91, 3565. (17) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. J . Chem. Phys. 1986, 85, 7268. (18) Zana, R.; Weill, C. J . Phys. (Paris) Lett. 1985, 46, L-953. (19) Kato, T.; Seimiya, T. J . Phys. Chem. 1986, 90, 3159.
0022-3654/88/2092-6086$01.50/0
surfactants in a homologous series may behave very differently according to structural factors influencing the delicate balance on which the process of micelle formation depends, viz., the relative lengths of hydrophobic (methylene chain) and hydrophilic (ethylene oxide chain) moieties. Thus, for example, Nilsson et al.'' using self-diffusion and N M R proton relaxation measurements over wide ranges of concentration and temperature, have demonstrated the considerable differences in the properties of micellar suspensions of the closely related amphiphiles C12E5and CI2Es. We have p r e v i ~ u s l yexamined ' ~ ~ ~ ~ the Cl2E6system in some detail using N M R self-diffusion and light-scattering techniques, both on dilute and concentrated suspensions. This approach is now extended to the neighboring members of the series, C12E5, CI2E7,and CI2E8,in order to elucidate trends in micellar size and shape as a function of the ethylene oxide segment length. The above combination of experimental techniques has proved fruitful. For example, inclusion of the nonscattering method, pulsed field gradient N M R (PFG-NMR), gives insight into the frictional properties of the micelle suspensions without an inherent coupling to the thermodynamic parameters of the system. This technique yields the well-defined self-diffusion coefficient and one obtains a direct measure of the particle size from the infinite dilution value. The method was first introduced by James and McDonald32 and has also been applied to a number of multicomponent systems, (20) Nilsson, P.-G.; Wennerstrom, H.; Lindman, B. J . Phys. Chem. 1983, 87, 1377.
(21) Lindman, B.; Soderman, 0.; Stilbs, P. In Surfactants in Solution; Mittal, K. L.; Ahluwalia, R., Eds.; Plenum: New York, in press. (22) Richtering, W. H.; Burchard, W.; Jahns, E.; Finkelmann, H. J . Phys. Chem., submitted for publication. (23). Gull, S . F.; Skilling, J. In Indirect Imaging; Roberts, J. A,, Ed.; Cambridge University Press: London, 1987. (24) Livesey, A. K.; Licinio, P.; Delaye, M. J . Chem. Phys. 1986.84, 5102. (25) Licinio, P.; Delaye, M.; Livesey, A. K.; Legtr, L. J . Phys. (Paris), 1987, 48, 1217. (26) Pike, E. R.; Pomeroy, W. R. M.; Vaughan, J. M. J . Chem. Phys. 1975, 6 2 , 3 1 8 8 .
(27) Neeson, P. G.; Jennings, B. R.; Tiddy, G. J. Faraday Discuss. Chem. Soc. 1984, 76, 353. (28) Degiorgio, V. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.: North Holland: Amsterdam, 1985.
(29) Carnahan, N. E.; Starling, K. E. J . Chem. Phys. 1969, 51, 635. (30) Okawauchi, M.; Shinozaki, M.; Ikawa, Y.;Tanaka, M. J . Phys. Chem. 1987, 91, 109. (31) Hill, T. L. Thermodynamics of Small Systems; W. A. Benjamin: New York, 1963. (32) James, T. L.; McDonald, G. G. J . Magn. Reson. 1973, 11, 5 8 .
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 6087
Size and Shape of Nonionic Amphiphile Micelles for example, the self-diffusion of solvents and small molecules in microernulsi~ns,~~ micellar suspension^,^^ gels,35 and polymer solutions.36 The light-scattering methods are, however, an essential feature in order to separately study the thermodynamic parameters which determine the associative phenomena (e.g., micellar growth/aggregation) as a phase boundary is approached. The measurements have been made over an extended concentration range (up to 15% w/v) in contrast to the earlier investigations which were restricted to dilute solutions. The temperature range typically covered the region in which (small) single micelles are observed up to the vicinity of the cloud point where more complex behavior is the rule. The results have substantiated the previous observationZothat the micellar size/shape pattern and the way in which it changes with temperature and concentration are critically determined by small differences in surfactant structure. There are thus serious pitfalls in extending general models of behavior to this series of compounds.
Experimental Section Samples. High-purity CI2Es,CI2E7,and C12E8were purchased in crystalline form from Nikko Chemicals, Tokyo, and used without further purification. Solutions were prepared by weight in either D 2 0 (99.8%) purchased from Stohler Isotope Chemicals, Switzerland, for the pulsed field gradient N M R measurements or in distilled water for light scattering. Light-scattering measurements (see below) were repeated in D 2 0 and gave identical results for molecular weights and virial coefficients in these solvents. The critical micelle concentrations (cmc) are all very much lower than the lowest concentration used in these measurements. (For example, cmc (C12E8)= 7.1 X M = 0.004% at 25 "C). The weighed-in concentrations are thus employed in all diagrams. All solutions were filtered by using 0.2-llm Millipore filters into precision-bore N M R tubes which were employed as scattering cells in light-scattering measurements. These tubes were immersed in a large-diameter bath of decalin for refractive index matching. Quasi-Elastic Light Scattering. The experimental arrangement has been described previously. The light source was either a 488-nm Ar ion laser or a 633-nm He-Ne laser. An ALV-Langen Co. multibit, multitau autocorrelator was operated with 23 simultaneous sampling times (covering, for example, delay times in the range 1 p s to 1 min) in the logarithmic mode and 191 channels. All measurements were made in the homodyne mode. The stability of the photon count at all angles indicated that the solutions were essentially dust-free. The data were routinely analyzed by the method of cumulants (two and three terms) and also by successively fitting to discrete exponential functions. In the case of the data for CI2E7and C&,, the distribution of relaxation times (see Figure 7, B and C) showed small amounts of a component with long relaxation times. Consequently, the main component was isolated by using the following expression for the normalized second-order correlation function ?(t)
- 1 = P(Af exp(-rft)
+ A , exp(-r,t) + B )
(1)
where the subscripts f and s refer to the fast and slow modes of relaxation, r is the relaxation rate, is an instrumental constant, and B is a base-line term. The separation between the modes is wide and the diffusion coefficient for the major component may be obtained free of contamination from the slower species. The data in the diagrams for C12E7 and CI2E8thus refer to the major component determined as above. Distributions of relaxation times were also obtained by using maximum entropy analysis (MAXENT), which is a new technique (33) Stilbs, P.; Moseley, M. E.; Lindman, B.J . Magn. Reson. 1980, 40, 401. (34) Stilbs, P. J . Colloid Interface Sci. 1980, 80, 608; 1982, 87, 385. (35) Nystrom, B.;Moseley, M. E.; Brown, W.; Roots, J. J . Appl. Polym. Sci. 1981, 26, 3385. (36) Moseley, M. E. Polymer 1980, 21, 1479. (37) (a) Altenberger, A. R.; Tirrell, M. J . Chem. Phys. 1984,80, 2208. (b) Altenberger, A. R.;Tirrell, M.; Dahler, J. S. J. Chem. Phys. 1986, 84, 5122.
for data analysis that has been successfully applied to a variety of problem areas involving image reconstruction. A review has been given by Gull and S k i l l i r ~ g .Its ~ ~application to QELS data has been described by Livesey et al.24325and the 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. It is particularly relevant that the method provides a unique solution which is robust to noise and only shows features (peaks) if demanded by the data. Static Light Scattering. Light-scattering-intensity measurements were made using a photon-counting apparatus supplied by Hamamatsu to register the scattered signal. The light source was a He-Ne laser. The optical constant for vertically polarized light is
K = 4mo2(dn/dc)*/NAX4 where n, is the solvent refractive index, dn/dc the refractive index increment (=0.134 mL-g-' at 25 'C), and X is the wavelength (633 nm). dn/dc values at other temperatures were interpolated from the data given by Balmbra.Io The reduced scattering intensity, K C / R t , was measured on the same solutions and at the same temperatures as used for the measurements described above. Here C is the concentration and R Bis the Rayleigh ratio obtained by calibration measurements using'benzene RW = 8.51 X The angle-corrected intensity for benzene was found to be constant over the angular range 45-135'. As an additional internal standard, intensity measurements were made on dilute solutions of an essentially monodisperse fraction of poly(ethy1ene oxide), M , = 280000, obtained from Toya Soda Ltd., Tokyo. It was found that K C / R t was angle-independent over the angular range 45-135' at all concentrations and temperatures used with the three surfactant systems. The data are thus given in the diagrams (Figure 10A-D) as K C / R W . All measurements were made in a concentration region where multiple scattering effects could be neglected. The inverse osmotic compressibility was evaluated from the absolute scattered intensity as (3)
Self Diffusion (Pulsed Field Gradient N M R ) . These measurements were made on protons at 99.6 MHz on a standard JEOL FX-100 Fourier transform spectrometer. An internal D 2 0 lock was used for field frequency stabilization as described in the previous communication^.'^^^^ The time between the 90' and 180' rf pulses was 140 ms for all 6-values, where 6 is the duration of the gradient pulses. The experimental uncertainty was ~ 1 . 5 % in D at a value of lo-" m2 s-I. Since the cmc is very much lower than the lowest concentration used, the values of D should be negligibly influenced by monomer diffusion even at the lowest concentration employed, and thus reliable determinations of the hydrodynamic radius are possible using the Stokes-Einstein equation in conjunction with D values at infinite dilution. However, as discussed in the Results and Discussion section, monomer transfer does occur between micelles at very high concentrations when micelle diffusion has become very slow and this leads to some complications in data interpretation.20 Data Representation. All curves shown in the diagrams are given as guides to the eye, except when otherwise stated.
Results and Discussion Pulsed Field Gradient N M R (PFG-NMR). Measurements using PFG-NMR on the C12E5, CI2E7,and CI2E8solutions are shown as a function of concentration at various temperatures in Figure 1, A-C. The self-diffusion coefficients have been normalized by using the ratio q / T , where 9 is the solvent viscosity at absolute temperature T. The systems CI2E7and CI2E8differ from Ci2E5 in several respects. At a given concentration (e.g., at C = 2% as illustrated in Figure 2) the reduced diffusion coefficient first shows a small increase with temperature and then decreases monotonically toward the cloud point. The maximum corresponds to the temperature at which the particle approximately conforms to the hard-sphere model (see Figure 9 later). Changes
Brown et al.
6088 The Journal of Physical Chemistry, Vol. 92, No. 21, 1988
10
5 \
I 0
5
10
15
Figure 2. Values of the reduced self-diffusion coefficients, D q / T , at a concentration of 2% as a function of temperature for the surfactants shown. Tc is the critical temperature.6
b
80
-
I
I
I
4 1 1 / / /A
60
I
&.--'2 2oL
5
10
c
15 10*/gml-'
170
1
13CL I 90
I0
I
30
I I TC
50
70
I
.-
-.-L
I
%E5
I 1 TC
-o-o-o-
'C
50 10
20
30
Figure 3. The hydrodynamic radius, RH.obtained from the diffusion coefficients at infinite dilution from self-diffusion (NMR) and dynamic light-scattering(QELS) data as a function of temperature. Filled points QELS; open points PFG-NMR.
Figure 1. Self-diffusion (PFG-NMR) coefficients, normalized with solvent viscosity and absolute temperature, as a function of concentration for CI2E5(A), CI2E7(B), and CI2Es(C) in aqueous solution at the
temperatures shown. in D v / T indicate size/shape changes or differences in the intermicellar interactions, possibly leading to aggregate formation. The initial increase may indicate desolvation and/or a progressive contraction of the oxyethylene chain or a change in the packing arrangement of the surfactant molecules. This behavior at the low temperature end (up to 33 O C for C12&and 25 "C for CITE7) corresponds to the region in which the micelles are small particles and/or the intermicellar interactions are insignificant. In the case of these two amphiphile species the micellar form is probably close to spherical and the size distribution is sharp (see the main peaks in Figure 7). A similar result was found for CI2E6at low temperatures in a previous investigation. The subsequent monotonic decrease in Dq/ T suggests that, at finite concentrations and starting at a rather well-defined temperature for a given system, the micelles become larger or more asymmetric and/or that there is a strong increase in the inter-
micellar interactions. This conclusion also applies to the CI2ES system. The curvature in the plots (Figure 1A-C)is pronounced at low concentrations. Since at low concentrations the dependence of the self-diffusion coefficient on concentration is a single-exponential function,37 linear plots of log (Dv/T ) versus concentration were made to extrapolate the values to infinite dilution. From the latter the hydrodynamic radii were evaluated by using the StokesEinstein equation. RH values are summarized in Figure 3. For each surfactant type, there is a plateau region in which the micellar radius is small and approximately constant (C&, 29 A; C12E7, 30 A; and C12E5, 68 A). Nilsson et aLZ0have reported an average value of RH = 31 A for C&8. The large apparent particle size for CIZESindicates substantial deviation from spherical form and a very different packing arrangement for the surfactant molecules in the micelles. The estimates of RH for C12E5are also less certain than the values for the other two surfactants owing to the distortion of the log (Dv/T ) curves which occurs with increasing concentration at the higher temperatures which has been ascribed to monomer transfer between micelles. Thus, for example, with Cl2E5 at 20 and 25 "C, D v / T passes through a minimum and increases at higher concentrations. This effect has been discussed by Nilsson et and was also established for aqueous solutions of CI2E6. The form of the D v / T curves at intermediate temperatures may also be changed in this way even though a minimum is not apparent; see, for example, the data for C12Esat 70.6 OC which cross the curves at 54 and 45 OC. The log ( D v / T ) plots are then no longer linear and the extrapolations become uncertain. In each of the systems CI2E7and C&, RH starts to increase at a considerable tem-
Size and Shape of Nonionic Amphiphile Micelles
The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 6089 L ‘C
15 I
5’
5
10
( D l / T ) . 10’7/NK-’ 5
20-
C12E5.
c .102/gml-’
,
15
10
I
c . 10/grnl-l 5
15
10
c 5
15
Figure 5. Reduced diffusion coefficients from NMR and QELS for
B.
5
10
10
20
10Z/gmi‘l 15
Figure 4. Reduced diffusion coefficients from quasi-elastic light scattering as a function of concentration for the surfactants as shown. Data for C12ESshown in A and B at low and high temperatures. Analogous data for C12E7 (C) and CIZEB(D).
perature interval from the cloud point; see Figure 3. This trend is clearly established since the PFG-NMR technique does not suffer from the possible complications from critical concentration fluctuations inherent in scattering experiments which may make data interpretation ambiguous even at a considerable distance from the cloud point. Referring to Figure 2, the temperature at which the monotonic decrease in D q / T commences is substantially lower than the corresponding point a t which RH starts to increase in Figure 3
for the infinite dilution data. This observation suggests that there is a significant micellar growth with increasing concentration. Quasi-Elastic Light Scattering (QELS). At low concentrations, the time correlation functions were approximately single exponentials and the relaxation rate was found to be directly proportional to the square of the scattering vector (q2) denoting a diffusive process. The relaxation rates were evaluated by using two- and three-parameter cumulant fits and also by using multiexponential fits as earlier described. Figure 4 depicts QELS data from measurements on the same solutions used for the PFG-NMR determinations summarized in Figure 1. With C12Es (Figure 4D) the data up to 32 OC are grouped together, with positive slopes and an approximately constant intercept. The data at 45 OC give a negative slope. The temperature at which the change in slope occurs corresponds to the position of the maximum for CI2& in Figure 2 ( z 3 3 “C) above which there is a strong decrease in D v / T , thus indicating a pronounced change in the size/shape and/or intermicellar interactions at this point. This conclusion is reinforced by the fit of the data to the curve predicted by the hard-sphere theory at this temperature (see Figure 9). The data at 54 O C are observed to turn upwards at low surfactant concentrations as well as having substantially lower values of D v / T . If the micellar size was independent of concentration, the initial decrease in Dv/ T would be attributed to a small or negative second virial coefficient in the dilute region. As the concentration is increased, higher terms in the virial expansion will become important and the slope becomes positive. However, the fact that there may be a significant growth of particle size with concentration raises some question as to such an interpretation. As will be seen below, aggregation with formation of a paucidisperse mixture of micelle sizes occurs at the higher temperatures. The data for C12E7 in Figure 4C at 25 “ C depart from the pattern prevailing at the lower temperatures. Here, again, this point coincides with the maximum in the data for this surfactant shown in Figure 2, denoting the point at which intermicellar interactions (and/or shape changes) occur. The data at 25 O C approximate the hard-sphere prediction. Analogous data are shown for C12E5 in Figure 4A,B. The low-temperature data (A) are found to be linear whereas those at higher temperatures (B) show a marked upswing at low surfactant concentrations which may indicate a breakdown of larger aggregates, although other interpretations are possible. The curves may be extrapolated at low concentration toward an approximate value of Do/ T corresponding to small micelles with RH = 24 A. The data at 8 OC approximately lit the hard-sphere prediction (see Figure 9). There is a close qualitative similarity in the trends in the PFG-NMR and QELS results for the three surfactants; thus the upward curvature at low C is present at higher temperatures in each system, although it is most pronounced in CI2E5. A comparison of the D q / T data obtained by using these two techniques indicates a substantial difference in the extrapolated intercepts at infinite dilution as shown in Figure 5 for the C12E5 system. This means that the suspensions are polydisperse. This aspect has been discussed p r e v i o ~ s l yand ~ ~is~a~consequence ~ of the different averages pertaining to the two techniques when the
6090 The Journal of Physical Chemistry, Vol. 92, No. 21, 1988
t
Brown et al. TABLE I: Micellar Dimensions and Second Virial Coefficients for Nonionic Amphiphiles typl A~ x 1051 C M, X nw Pa m6 k g 2 RIA"
Relative Variance
C12E6
I
I
I
I
0
20
60
60
Figure 6. Relative variance, p 2 / r 2 ,as a function of temperature for the various surfactants C I 2 E S( O ) , C12E7 (0),and C & (+) at a concentration of C = 1%.
20.9 30 40 49.6 60 70.2
68 108 139 205 182 280
127 200 260 380 340 520
19.5 30 40 50 57 59.5
45 66 99 214 300 390
90 135 200 435 605 785
4 6 8 IO 14.4 20.4 25.2 29.8
1480' 2000' 22006 3500' 460 760 1100 1430
C,2ES 3800 5200 5700 9040 1190 1940 2850 3680
11.0 7.8 7.5 7.2 1.9 1.3
29 34 37 42 41 47
9.7 5.2 5.0 3.8 -0.8 -1 .o
25 29 33 43 48 52
C12E7
distribution of micellar sizes is not small: PFG-NMR (number average), QELS (Z average). A similar mismatch of the intercepts was found for the data for CI2E7and C&& The presence of even a small percentage of free monomer, for example, will contribute disproportionately to D N M R and may be responsible for the observed effect. A measure of polydispersity is provided by the relative variance obtained from the cumulants fit to the QELS data. With C&8 (and C12E7),the variance decreases with increasing temperature as shown in Figure 6 and may arise from the breakdown of micellar aggregates. In ClzE5the opposite trend is observed. The variance is almost independent of concentration above about 5% but increases strongly toward low concentration in all three systems. The origin of the increased variance at low Cis at present unclear; it could derive from an increasing proportion of some small micelles in this region. Amphiphile molecules should contribute insignificantly, however, because of their negligible scattering. With C12E5, there is clear evidence for a dissociation of larger structures at low concentrations at higher temperatures (see Figure 4B) and this will also contribute to the observed polydispersity. Hydrodynamic radii estimated from the intercepts from QELS experiments are included in Figure 3. The radii have values of approximately 34 8, (CI2E7)and 38 8, (C&) at low temperatures. For C12E7and C12E8the radii show a qualitatively similar trend with temperature to that found for the self-diffusion data. The linear data at high concentrations were extrapolated to C 0 in order to obtain the estimates of RH in the case of the higher temperature data in these systems. With ClzE5,however, only at temperatures below 8 OC can QELS data be extrapolated to infinite dilution. The corresponding radius obtained is approximately 120 8, for C12E5. At higher concentrations (above about
1.53 1.44 1.33 1 .o
84 93 131 153 57 67 76 83
a R = (3Mu/47rNA)'/' with u = 1.00 mL g-l. b M estimated at C = 1% using eq 2 with dn/dc from static light scattering.
10%) it was observed that the time correlation functions for ClZE6 and C12E7deviate increasingly from single exponentials. This characteristic is illustrated in Figure 7 (B and C) for these surfactants at an arbitrary concentration of 10%. The lower parts of the figures show the distributions of relaxation times obtained by analysis of the correlation function using MAXENT (see Experimental Section). In both systems, a satellite peak(s) is evident, and this is displaced from the main micelle peak by several decades in relaxation time. Moreover, the narrow widths of the component peaks would suggest that the suspensions are a paucidisperse mixture of micelles and micellar aggregates of narrow size distribution. The sharpness of the slower peak may be an artifact, however. The apparently discrete nature of the relaxational modes was an unexpected feature of the QELS experiments and this
-
C 1 2 E 7 (35-C)
I"'"1
~8
I '
5
I
'"P'I
''rn '''1.1 5
""'1 6
7
'm 8
log relaxatan tirne/ps
C
J
L
Figure 7. Time correlation functions (&t) - 1 vs log t ) for the three surfactants in solution at a concentration of 10%: (A) C12E, at 20 'C, (B) at 35 OC, and (C) CI2E8at 25 OC. The lower parts of the respective figures illustrate the distributions of relaxation times obtained by using maximum entropy analysis (MAXENT24'25).
The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 6091
Size and Shape of Nonionic Amphiphile Micelles
5-
1I
4 q2E5 (5°C)
3 21-
I
I
1
5
10
15
20
log relaxation tirne/p
E
I 30
5
-
k
10
C
20
15
/1 L 5' 92'
C12E5
Figure 8. Time correlation function and distribution of relaxation times using MAXENT for Cl2ESat low temperature ( 5 "C) where the data were collected on a solution with concentration C = 10%.
aspect will be treated in more detail elsewhere. Surprisingly, the solutions of C12Esare strictly single-exponential from about 15 OC up to close to the cloud point at about 31 OC (see Figure 7A). Since the micellar mass of the Ci2ESmicelles increases substantially with temperature, see Table I, the absence of micellar aggregates in this system presumably means that the CI2ESmicelles themselves grow with temperature above 15 O C and the micelles retain their narrow size distribution. Thus there is a qualitative difference in behavior between this surfactant and C12E8and CI2E-,. This conclusion also finds support in comparison with previous investigations on a surfactant with shorter ethylene oxide chain (CI2ES)where it was also concluded that the micelles grow with increasing temperature. At low temperatures, the Ci2E5system shows evidence for an increasing polydispersity. Figure 8 shows the presence of aggregates of narrow size distribution at a concentration of 10%and 5 OC. This finding is consistent with the broadening of the methylene signal observed in proton N M R with increasing concentration in the low-temperature range." The ratio DQEF/DNMRmay be usedlg to extract some information on the micellar conformation DQELS/DNMR
= M/RT(&/aC)Ty = (1
+ 2AzMC + ...)
(4)
where M/RT(a?r/ac)T,p is the reduced osmotic compressibility, R the gas constant, M the molecular weight, and A2 the second ) virial coefficient. The expressions for the mutual ( D Q E ~and self-diffusion coefficients ( D N M R ) are given, for example, by Y a m a k a ~ a .By ~ ~incorporating the value of (a?r/aC)T, obtained from light-scattering-intensity measurements, one can, in principle, obtain M at a given temperature and concentration. In the present systems this approach fails. There is the implicit assumption that the difference between D Q E and ~ I&R lies only in the nonideality terms. However, it is found in the CI2Esand Ci2E7systems that the apparent value of M derived from eq 2 decreases strongly with increasing concentration. This means that the micelles grow with increasing concentration and that the suspension also becomes significantly more polydisperse giving progressively smaller D Q E m / D N M R ratios owing to the different averages involved. In the case of Cl,Es, the apparent molecular weight increases strongly with temperature in the range below 10 O C (see data in Table I). (These estimates of M are given here since the static light(38) Yamakawa, H. Modern Theory of Polymer Solutions; Harper and Row: New York, 1971.
Figure 9. The ratio between DQEm/&MR as a function of concentration at different temperatures. The ratio is given by eq 3. In each diagram the theoretical curve for the hard-spheremodel is represented by the line (-
.-.-).
scattering data at low concentrations are nonlinear (see Figure lOD).) However, at about 15 OC, M is much smaller but subsequently grows again as the temperature is increased toward the cloud point at -31 "C. The marked discontinuity is possibly related to a decreased order in the aggregates; thus Nilsson et aLzo refer to large fluctuations in aggregate sizelshape in this temperature region. By comparison with the theoretically predicted concentration dependence of the reduced osmotic compressibility for a given particle shape (e.g., for a hard-sphere, using the Carnahan-Starling equation29),one may also draw conclusions regarding the particle form and the way in which it varies with temperature. Diagrams showing DQEu/DNMRas a function of concentration are given in Figure 9. The hard-sphere prediction is apparently approximated at the following temperatures Cl2E8, 32 "C
(e
= 0.58);
Tc = 77 OC
C12E7, 25 "C
(t
= 0.62);
Tc = 65 "C
= 0.74);
Tc = 31 OC
C12E5, 8 "C
(t
where t = Tc - T/Tc and Tc is the cloud point temperature taken from the summary in ref 6. While a spherical model may be reasonable for Ci2E7and CI2E8at low temperatures, the equivalent hydrodynamic radius is too large (Figure 3) for this to be plausible with CI2ES. Not only will polydispersity lead to values of DQEm/f&MR which are too smal1,lgbut both the size distribution and the average size will increase with increasing concentration as shown by the data in Figure 10D (lower). Thus conclusions as to micellar form cannot be meaningfully drawn from the data in Figure 9C. Referring to Figure 9, one notes that, a t a given concentration and as the temperature is increased, DQELS/DNMR first falls to a minimum value and thereafter increases, indicating micellar growth. With ClzE7and ClzESsome data points at the
6092 The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 .20 ' 9
2
05
10
15
B
/ t/o;gm,-l
51'
y=-p 5
,
54.
15
10
2
e
6
't 0
10
10
10
Figure 10. Reduced scattered intensity from static light scattering (KC/R90)as a function of concentration for the different surfactants: (A) CIzEsat low C. Insert shows the temperature dependence of the second virial coefficient. (B) CIZE7at low C. ( C ) C12E3at high C. (D) Shows the low concentration data for CiZE3. (E) Illustrates the temperature dependence of (KC/&) for CI2E5at a concentration of 5%.
highest concentrations and at the highest temperature are discrepant. This is in part due to the influence of monomer-transfer effects. These will be facilitated within the micelle clusters formed at high concentrations and also have the effect of making D N M ~ artifactually large. The interpretation placed on the large values of DQELS/DNMR at the highest temperatures measured on each
Brown et al. system is that the intermicellar interactions are predominantly repulsive and/or reflect the formation of larger particles (aggregates). Static Light Scattering. Light-scattering-intensity measurements were made on the same systems and at the same concentrations and temperatures as the N M R and QELS experiments in order to obtain information on (a) molecular weights and (b) the intermicellar interactions as embodied in the virial terms expressing the concentration dependence of the reduced scattering function KC/R9,. The measurements were initially made as a function of angle over the range 45"-135" and it was established that there was no significant angular dependence of the scattered light in the concentration regions and temperature ranges examined. Thereafter, a measurement angle of 90' was used. Light-scattering data obtained on dilute solutions of C,,Es are included in plots of KC/R90as a function of concentration at different temperatures in Figure 10. The molecular weights, obtained from the inverse intercepts, and the second virial coefficients (A,) are summarized in Table I. In each system, the apparent molecular weight increases with temperature. We also recall the increase in polydispersity with increase in temperature deduced from the differing N M R and QELS intercepts; see, for example, Figure 5 (whereas there will be a common intercept at infinite dilution for a monodisperse solute). The increase in molecular weight also follows the trends in the hydrodynamic radii, RH, illustrated in Figure 3. As pointed out by Richtering et a1.,2z the micellar radius may also be estimated from the Az values, using the expression relating this quantity to the excluded volume. It was found that these radii are in approximate agreement with the hydrodynamic radii derived from the N M R measurements. Radii have been also estimated from the molecular weights, assuming the micelles to be hard spheres. These values, included in Table I, are broadly in agreement with RH determined from self-diffusion. While CI2E8and ClZE7behave in the expected manner with regard to the change in the concentration dependence with temperature (Le., A, decreases monotonically toward the cloud point), Cl2E5 differs in this respect as illustrated by comparing Figure 1OC with Figure lOA,B. At low temperatures (Le., below 10 "C) the KC/RgOvalues are low and there is a very small dependence on concentration, except for the upturn at low concentrations which is shown enlarged in Figure 10D and which precludes evaluation of the virial coefficient. The low values of KC/Rw at all but the lowest concentrations show that the frictional coefficient is large and increases with concentration, which suggests a rcdlike particle form in this temperature region. The latter conclusion agrees with the results of N M R measurements of the methylene group line widthz0from which it was also concluded that at low temperatures there is growth of the micelles into long rods. At the lowest concentrations there is dissociation into smaller particle sizes. At higher temperatures (>14 "C), however, there is a pronounced dependence of KC/RgOon concentration. Figure 1OE exemplifies this change, which is not found with the other amphiphiles, for data on CI2E5selected at an arbitrary concentration of 5%. The values suggest that above approximately 10 "C the micellar size/shape and/or intermicellar interactions in ClzE5 change rather abruptly as a function of temperature at a given concentration. This observation also is in agreement with the N M R measurements of Nilsson et aL20 Thus the line width of the methylene group (also measured at 5% C12E5) passes through a maximum and it was suggested that there is growth of the micellar aggregates up to about 10-15 "C (in general one anticipates a greater line width with increasing temperature owing to enhanced molecular motions). The explanation for the decreased line width above 15 "C was attributed to large fluctuations in aggregate size/shape as the cloud point is approached. In this higher temperature range, we find behavior more akin to that of C12E7 and CI2Es, i.e., KC/RgOdecreases in magnitude and the second virial coefficient also decreases toward the cloud point at about 3 1 "C. Thus in this latter region the trends for CI2E5are similar to those observed for C12E8and C12E7. Okawauchi et aL30 have recently applied small-system thermodynamic^^^ to the analysis of static light-
Size and Shape of Nonionic Amphiphile Micelles I
The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 6093 The virial expansion describing the concentration dependence in static light scattering provides a sensitive index of the solutesolvent interactions in a system. In the present case the virial terms may potentially be complicated by size/shape changes of the micelles with change in concentration. However, in dilute solutions (
