J. Phys. Chem. 1994,98, 4949-4954
4949
Evolution of the Diffusion Coefficient and Correlation Length of Aqueous Solutions of ClzEs T.M. Kole, C. J. Richards, and M. R. Fisch' Department of Physics, John Carroll University, University Heights, Ohio 441 18 Received: February 17, 1994'
We have used dynamic and static light scattering to study rodlike micelles of C & in HzO. Nine surfactant concentrations, C, in the range 3 < C < 510 mg/mL and in the temperature, T, range 15 < T < 50 OC were studied. The mean diffusion coefficient, determined using dynamic light scattering, shows a well-defined minima as a function of concentration along each isotherm. The concentration at which the minimum occurs is temperature dependent and to a very good approximation marks the threshold crossover concentration from dilute to semidilute solution behavior. From these data, using the recent theory of Carale and Blankschtein ( J . Phys. Chem. 1992, 96, 459), we determine the persistence length, a measure of micellar flexibility, as a function of temperature. It obtains a value of 15 nm near 20 OC and 6 nm near 50 OC. From static lightscattering measurements we determine the characteristic length which can be interpreted as a pore size in semidilute solutions. This quantity shows a maximum at concentrations near where the diffusion coefficient obtains a minimum. From this measurement the persistence length is also determined. By comparison of theoretical expressions with these measurements and the dynamic light-scattering measurements the persistence length was also deduced. This determination of the persistence length is essentially temperature independent with a value of approximately 2 nm.
Introduction
corresponding to the overlap threshold concentration and a concentration well in the semidilute regime is showp in Figure Micelles are self-assembledaggregates of surfactant molecules 1.23 The crossover from dilute to semidilute solution behavior in solution. The morphologies of the micelles in solution depend has also been observed in micellar solutions using a variety of in a complex manner on the intermolecular forces between the techniques.16 In the micellar case it is important to note that L, surfactant molecules, the solvent molecules, and the cross the length of a micelle, is a function of the nudber oflmonomers pair-solvent-surfactant molecules and on thermodynamic conin the micelle which, in turn, is a function of thermodynamic ditions such as temperature, concentration of surfactant and variables such as surfactant concentration, tempprature, conadditives to the solution, pressure, and the 1ike.l-3 The morcentration of salts and the solvent, and pressure. Thus the phologic behavior of the micelleswith temperature, concentration crossover concentration may well be a function of some of the of surfactant, and concentration of salts, urea, and other additives same thermodynamic variables. I is generally better known than the details of the above interThe study of crossover concentrations in micellar solutions is molecular forces.k7 It is known that very dilute surfactant of interest for several reasons. First, most light-scattering solutionsgenerally display roughly spherical "ungrown" shapes.* experiments on micellar systems assume that because the However, depending on the relative balance of the above molecular surfactant concentration is low (typically a few percent by weight) forces the micelles can grow (with changes of temperature, the solution is in the dilute solution regime. The experimental concentration,pressure, etc.) into either two-dimensional sheetlike (disk-shaped) micelles or one-dimensional,rodlike s t r u c t ~ r e s . l - ~ ~ ~data are then interpreted in terms of modelsvalid for such systems. As discussed earlier this may not be the case, and in fact, such Rodlike micelles are similar to flexible linear polymers (a major systems may be entangled. Thus, the interpretation of the data difference is that micelles are not covalently linked particles) should be changed from that characterizing a dilute system of flexible and, in fact, can be modeled as s u ~ h . ~ J o J sFurther, J~ roughly independent particles of hydrodynamic radius, Rh, to rodlike micelles are common, they have been observed in a large that of a transient network of characteristic size {.22.24 Further, number of surfactant systems.'7,9,10,15-21 From an experimental as discussed by Carale and Blankschtein,25entanglements may viewpointls.16 flexible, linear micelles may be treated as a have an impact on phase transitions and the theoretical interpolydisperse mixture of linear polymers each of total length, L pretation of these phase transitions. Last, the systematic study and persistence length, 5, where the persistence length is roughly of such systems will ultimately allow one to predict the effects the length over which the micelle may be regarded as rigid.22 of changing solute conditions on the flexibility of the micelles. The very dilute concentration regime of solutions of flexible This has practical significance because a semidilute solution of rodlike micelles is called the dilute solution regime. In this very flexible micelles will form in isotropic phase, while a solution concentration regime the solution consists of a solvent (including of similar concentration containing rigid micelles may form a monomers) containing very weakly interactingmicelles. However, nematic liquid crystalline phaseeZ4These two phases have very upon addition of more surfactant the micelles can become different physical properties.24 entangled and form transient networks of overlapping miIn this paper we will present the results of a static and dynamic celles.16.22.24The overlap threshold concentration, where the light-scattering study of aqueous solutions of n-dodecylhexakismicellesfirst begin to overlap, defines the onset of the semidilute (ethylene oxide) (C&). These results will be used to calculate concentration regime; the concentration at which this occurs is the persistence length of the micelles in two different ways: first, not a sharp point; nevertheless, a good estimate of this concentraby comparing the hydrodynamic radius and the radius of gyration tion, which may be rather low, on the order of a few weight obtained using light-scattering techniques to theoretical calculapercent, may be obtained experimentally.16 A schematic picture tions of the same parameters, and second by using the minima showing the dilute concentration regime, the concentration in the measured diffusion coefficient as input data in the recent calculation of Carale and Blankschtein.25 Finally, since the Abstract published in Aduance ACS Abstracts. April 1, 1994. 0022-3654/94/2098-4949504.50/0
0 1994 American Chemical Society
Kole et al.
4950 The Journal of Physical Chemistry, Vol. 98, No. 18, 19'94
Dilute
Crossover
Semidilute
Figure 1. Drawings showing schematically a dilute solution, a solution near the crossover concentration, and a solution near well into the
semidilute solution regime. concentration at which the minima in the diffusion coefficient occur is very similar to the critical concentration for phase separation we will discuss the relation between this experiment and other experiments on this system. The nonionicsurfactant, C&, was chosen for several reasons. First, the Carale and Blankschtein25calculation is for this material. Their calculation requires a detailed knowledge of the molecular thermodynamics of micellar growth in order to calculate the threshold crossover concentrationand hence the persistencelength of the micelles. Thus, other systems which are less well characterized are not yet amenable to such an analysis. Further, there is ample evidencethat the micelles in these solutions exhibit one-dimensional growth and are rodlike over a large range of temperatures and concentrations below the lower consolute point.29-3s Nonetheless, this has been a controversial point.363* A critical discussion of micellar growth in aqueous solutions of nonionic micelles which summarizes the various points of view and concludesthat there is in fact some growth into rodlike micelles has been recently published by Lindman and WennerstromU39 In light of this body of experimental evidence that the micelles grow into flexible rodlike micelles, we assume as a starting hypothesis that this is indeed the case. Our experimental results in the dilute solution regime are consistent with this hypothesis, and from these data we obtain a measure of the persistence length, f .
Experimental Materials, Measurements, and Methods The C & , was obtained from Nikko Chemicals, Tokyo (lot no. 1014) and used without further purification. The solutions were prepared using water obtained from a Millipore Milli-Q water purification system that produces H2O with a resistivity greater than 10 Mil cm. To ensure sample stability this water was deoxygenated by bubbling it with argon for approximately 1-2 h.38.39.42.43 Samples were prepared, filtered, and capped with this water under an argon atmosphere in a glovebag. The samples were freshly used or immediately refrigerated for short periods of time. The samples were clarified of dust by filtration through 0.22-pm filters into cylindrical glass scattering tubes and then centrifugation for periods of up to 1 h. The data were obtained as a function of increasing temperature. Except at the highest concentration, the data were taken from a temperature of 15 OC, below the start of the sphere-to-rod transition (which is about 18 OCI), to 50 OC in 5 O C steps. The highest temperature studied is a few degrees below the lower consolute point which occurs at a temperature of approximately 51 O C . Thus, for samples very near the critical concentration, the data are taken close to the critical temperature. However, at concentrations well removed from the critical concentration this temperature is severaldegrees below the coexistence curve for this system. Note that the temperature range from 15 to 50 OC is the only temperature window for these experiments. Below the lower temperature the
micelles are ungrown,I and above the highest temperature some of the samples will phase separate. Our light-scattering apparatus is a Malvern system 4700 laser light scattering system. This instrument allows us to perform both static and dynamic light-scatteringmeasurements. The light source is a Lexel Ar+ laser operating at a yavelength of 5145 A, and power levels between 0.05 and 0.4 W. The laser light level was reduced by means of attenuators rather than by operating the laser at very low currents. The signal analysis was performed using a 128-channel autocorrelator. The z-average diffusion coefficient, (D), was determined at four scattering angles from the forward direction (30,60,90,and 120O) by using the cumulants method to analyze correlation functions that were accumulated for 30 s. The results of 8-10 of these cumulant fits were averaged to obtain and calculate the mean diffusion coefficient, (D)and polydispersity parameters which were used in the subsequent analysis. The data obtained from several different groups of 8-10 runs at a given temperature and concentration showed no systematic variation; such variation was taken to indicate that thermal equilibrium had not been reached. The polydispersity parameter obtained from these fits was approximately 0.1 with an uncertainty nearly as large. Within this uncertainty the polydispersity parameter was independent of concentration. However, the polydispersity parameter did increase somewhat with increasing micellar size (or equivalently temperature). The smallness of the polydispersity parameter and its relative temperature independence of the polydispersity parameter disagrees with the simple theory of the sphere-to-rod transitionswhich predicts that the polydispersityparameter should increase from near zero for ungrown micelles to 0.7 or more for very large, grown micelles such as observed at temperatures not far from the consolute temperature. This is indicative of the need for further work on this problem. The reported diffusion coefficients are the values obtained by linearly extrapolating graphs of (D)versus k2 (kis the scattering vector defined below) to a scattering angle of Oo (or k = 0). This value of (D)is denoted simply by D. The extrapolation to zero scattering angle was done to eliminate the effects of internal polymer modes on the measured diffusion coefficient.b" In the dilute solution regime the mean hydrodynamic radius at zero scatteringvector was determined using the Stokes-Einstein relation:
(Rh(O))= k , T / h D
(1)
where k~ is the Boltzmann constant, Tis the absolutetemperature, and q is the shear viscosity of the solvent. The static light-scattering experiments were performed by measuring the time-averaged scattered light intensity as a function of scattering angle. The data were corrected for excess scattering due to the solvent and scattering due to the sample cell. The characteristic length, {,was found by fitting thedata to theGuinier and Fournet form:&,47
I ( e ) = z0 exp(-k2?) where k = k(0) = (4rn/X) sin(0/2) is the scattering vector, and 0 is the scattering angle measured from the forward direction, n is the index of refraction of the solution, X is the vacuum wavelength of the laser light, and IOthe (extrapolated) scattered light intensity at 0 = 0. In dilute solutions, where interparticle interference of the scattered light can be negle~ted,e.~'this form allows the determination of the z-average radius of gyration, (R,2),.45These two quantities are related by (R82)z= p / 3 . The z-average mean square radius of gyration, (R&[))z,where the explicitdependence of the z-average radius of gyration squared on persistence length has been shown, will be used in some of the later calculations.
The Journal of Physical Chemistry, Vol. 98, No. 18, I994 4951
Diffusion Coefficient and Correlation Length of C12E6
40
-
30 -
20
-
10 -
2
3
4 5 6
101
2
3
4
5 6
102
2
3
4 5 6
-
-
0
4
Figure 2. Graphs of isotherms of &(7')/q(reQ versus concentration of surfactant: (V) T = 50 O C ; (A) T = 45 O C ; (V) T = 40 ' c ; (A) T = 35 'C; (0) T = 30 O C ; (D) T = 25 OC; (0) 7'= 20 O C ; (0) T = 15 'C; curve such that &(T)/q(ref) a e."). (-a)
Generally this will be written as (Rg2),.This is defined through the following equation:
where n is the number of monomers in a micelle, X . is micellar size distribution as a function of n, Rg2(n,[)= (L&/3) - t2 + (2[3/L,) - (2[4/Ln2) [ 1 - exp(-L,/[)], [ is the persistence length of the micelle, and L, is length of a micelle composed of n monomers.
5
6
7 8 1 0 1
1.5
2
3
4
5
6 7 8 1 0 2
Concentration in (mg/ml)
Conc'entrotion in mg/mi
Figure 3. Isothermsof (R,)versusconcentration. The symbols have the same meaning as in Figure 2, except there is no theoretical curve.
at the lowest concentrations (dilute solutions) the correlation length, { = Thus a t the lowest concentrations the correlation length may also be related to the z-average meansquare radius of gyration. The correlation length is too small to measure on our instrument a t weight concentrations above 52 mg/mL. At the highest concentration at which measurements were made the characteristic size ofthe pores (the correlation length), {determined using static light-scattering techniques is approximately 8 f 1.5 nm, independent of concentration. The same quantity determined using dynamic light scatteringobtains a value of 8.4 f 3 nm.
dz!8
Discussion Results There are two major light-scattering results from this study. The first is the measurement of the diffusion coefficient of the micelles as a function of temperature and concentration in the C12E6-H20 system. While light-scattering studies have been performed on this system in the past,29.30J8 the present study is unique in that it was undertaken to specifically verify the Carale and Blankschtein25 calculation and study the possible effects of increasing concentration on the critical phenomena in this system and hence covers a very broad range of both temperature and concentration. The second is the determination, under the same conditions, of the characteristic size { of the micelles. The first major result of our study is the determination of the mean diffusion coefficient as a function of temperature and concentration. A graph of these results is shown in Figure 2. Note that this is not merely a graph of D versus concentration for various temperatures, but rather the abscissa has been multiplied by v(T)/v( TREF) where v ( T ) is the shear viscosity of water at the temperature of the measurement and ~ ( T R E = F) 1.OO centipoise. A similar procedure has been used by Cates and Candau.16 The reason for multiplying by this dimensionless number is the following. First, &( T)/v(ref) is nearly temperature independent (over the range in temperature of the present study v varies by a factor of greater than 2). There is, however, a residual 10%temperaturedependence in &( T)/v(ref). Further, at low concentrations, below those at which the minima in these curves occur, the micelles are only weakly interacting and the micelle's hydrodynamic radius is inversely proportional to v( 7')D. In the intermediate region the diffusion coefficient yields no simple interpretation. Finally, well into the semidilute regime, the Stokes-Einstein relation may once more be employed only this time to determine the mean characteristic size, {. This parameter is also inversely proportional to the product of the viscosity times the mean diffusion coefficient. The second major result is the determination of the characteristic size, also called the correlation length of the micelles. The results of this part of the study are shown in Figure 3. Note that
The following discussion has four sections. The first discusses in general terms the concentration dependence of the isotherms of the diffusion coefficient. The second section discusses how the persistence length and its temperature dependence are determined from the results in Figures 2 and 3. The third section analyzes the present experiment in light of the work, studying critical phenomena in this system, of Wilcoxon et al.38 The last section discusses several caveats that suggest the need for further experimental work on the present problem. The general shape of the diffusion coefficient isotherms, shown in Figure 2 (except possibly near the lower consolute point where critical fluctuations are important) may be understood using the following models. Along an isotherm in the low-concentration regime, where the solution is dilute, the mean hydrodynamic radius, (Rh)should increase with concentration as there is micellar growth in this system. This means that D should decrease along these curves as shown in Figure 2. Similarly, along curves of constant concentration (Rh) should increase with increasing temperature as once more there is micellar growth. Hence, as observed D should decrease with increasing temperature, obtaining its highest value a t the lowest temperature. In the high concentration regime, D should increase once more. The reason for this is that in semidilute solutions { 0: Q,-0.77,22.24where Q, is thevolume fraction of surfactant. Sincelight scattering measures the cooperative diffusion coefficient, Dooopand DmP a $1 in this regime D 0: Q,0.77 and therefore D should increase with increasing concentration. On the assumption that the surfactant weight concentration, C, in g/mL of solvent a: CP, we might expect that in this regime D a 0 . 7 7 . The dotted line without data points in Figure 2 is a line of this functional form. The persistence length of the micelles can be determined from the threshold crossover concentration, C( T,[), using the CaraleBlankschteinz5 theory. The result of using this theory to analyze the data shown in Figure 2 is shown in Figure 4. The measured size and concentration of the micelles are consistent with the micelles just beginning to overlap a t the threshold concentration. In this figure one data point was obtained from each isotherm,
4952 The Journal of Physical Chemistry, Vol. 98, No. 18, 1994 180 160 h
?. 5 cn 01
g
140
120 100
01 +
80
1
40
20
25
30
35 Temperature (C" )
40
45
Figure 4. Persistence length, deduced from the crossover concentration, versus temperature. The solid line is a least-squares fit to the data.
and the solid line is the least-squares fit to the data described in the next paragraph. The data points were obtained by locating the minimum of the diffusion coefficient on a given isotherm. Then by knowing the temperature of the isotherm, the overall concentrationof surfactant, and the concentration at the diffusion coefficient minimum, C, the persistence length was found by varying [ in the theoretical expression for C ( T , t ) = C ( T , (Rl([))z) until there was good agreement in C and T. The major uncertainty in determining the persistence length is locating the minima. The relationship between the concentration of a given minimum and thecorresponding persistencelength is rather strong; hence the error bars shown are an estimate of the uncertainty. It is clear from Figure 4 that [ obtained from the CaraleBlankschtein calculation is temperature dependent. Since the persistence length depends primarily on short-range interactions which are in part temperature dependent, such behavior is not surprising. Two numerical expressions for [( T ) were studied. The theory of linear p o l y m e r ~ ~predicts ~ . ~ 4 that 5 = where t o is the persistence length at infinite temperature, and TOis the characteristic temperature that describes this process. On leastsquares fitting, these data yield & = 1.15 X 10-3A and TO= 3450 K. This fit is shown by the solid line in Figure 3. The smallness of 40 is surprising and may indicate the inappropriateness of applying this functional form to our data. Clearly, the temperature range of the present experiment is small and the excellent fit to this exponential form may be fortuitous. In fact, a fit to the linear form 5 = A - BT with A = 1152 A and B = -3.46A/K is almost as good as the exponential fit. Regardless of the fitting form used, it is clear that the persistence length decreases with increasing temperature. The persistence length was also determined in another way. In this analysis both the static and dynamic light-scattering results are used to independently determine the persistence length as a function of temperature. This is done by explicitly calculating (R82), via eq 3 and (l/Rb(O)) via the theory of Yamakawa and F u j i P using the data of Carale and Blanks~htein~~ to calculate X,. These theoretical calculations contain as a parameter and hence can easily be compared to experiment in dilute solutions where these calculations should be valid. The persistencelengths determined in this way are very small (approximately 2 nm) at all temperatures. This is clearly substantially smaller than those determined using the analysisdescribedin the previous paragraph. In light of this result the parameters that determine X, were varied over a range of values. It was found that for a given persistence length the calculated values of (R82)*and ( 1/Rh(O)) are very sensitive to the monomer concentration in solution. Further, by an appropriate selection of the parameters that determine X, (different from those in refs 1 and 25) agreement between the persistence lengths obtained from the diffusion coefficient minima and this technique could be obtained. Clearly further work is needed on this problem.49
Kole et al. The view that the temperature and concentration dependence of the diffusion coefficientmay be interpreted in terms of a model which describes the solution in terms of interacting grown flexible rodlike micelles is not universal.38 These analyses assume that there is negligible micellar growth and that light scattering and neutron scattering from this system can be interpreted in terms of simple binary fluid critical phenomena with a very large critical region (out to approximately 25 OC below the consolute point). The analysis is successful; the one exception to simple binary mixture behavior is that there is some excess scattering at intermediate scattering vectors near the critical temperature. These experiments measured the osmotic compressibility and the diffusion coefficient along isotherms between 25 and 45 OC at concentrations near the critical concentration. On a given isotherm the diffusion coefficient had a minimum and the osmotic compressibilityhad a maximum at concentrationsnear the critical concentration. These two results are expected.sO.51 The osmotic compressibility should diverge at the consolute point as should the critical correlation length, Fc (since D a &-I, D should obtain its minimum value near the consolute point). In fact, in critical solutions the maximum in the osmotic compressibility and the critical correlation length should occur at the same concentration. These experiments do not explain the large size of the critical region or how or why micellar growth apparently stops and critical behavior begins approximately 25 OC below the consolute point. This latter point is at variance with thermodynamic theories that successfully predict the behavior of nonionic surfactant solutions over a wide range of temperatures, concentrations,and surfactant identity.' The results of these earlier experiment~3~ may also be interpreted to indicate that micellar growth is occurring in this system. The single particle diffusion coefficient in micellar systems and particle in solvent solutions have'been successfully modeled by the generalized Stokes-Einstein relation.52 This equation defines the concentration dependence of the diffusion coefficient for weakly interacting particles (dilute solutions):
where C is the concentration (weight/volume), au/aC is the osmotic compressibility,f(C) is the hydrodynamic friction factor (proportional to the mean hydrodynamic radius of the micelles, and (1 - v C ) is the "reference frame correctio11".~~.53To the extent that micellar systems may be treated (1) as a hard object systems, (2) the virial expansions found in hard monodisperse disperse particulate systems are valid, and (3) this expression can be applied to polydisperse micellar systems with the result D( C) = kT/6uqRh to a very good approximation.52 The extension from hard object systems to micellar systems outlined in the previous paragraph, with the inclusion of thermodynamics to predict the micellar weight M,has been a paradigm that has led to no apparent serious errors in systems that are not t m close to a consolute point or do not exhibit critical behavior.I-z1 The extent to which this may be applied to systems such as the present remains to be determined and has been c o n t r o v e r ~ i a l , although ~ ~ - ~ ~ there is general agreement that there is some growth. The starting hypothesis of the ptesent analysis are (1) that both micellar growth and critical phenomena occur; however, (2) critical phenomena is important only within 5 OC or so of the consolute temperature, Tc. This latter hypothesis is supported by most of the experimental work;SO#s*we will present further evidence for its validity below. Thus, except in this temperature range near Tc diffusion in micellar solutions can be described by the Stokes-Einstein equation. The Stokes-Einstein equation indicates that a maximum in (ax/aC)-1 will produce a minimum in D(C) at the same concentration as the maximum unless there is a significant variation inf(C) which isindicativeofgrowth. To further examine
Diffusion Coefficient and Correlation Length of ClzEa 005
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,
The Journal of Physical Chemistry, Vol. 98, No. 18, 1994 4953
, . ,
i
0
~
3 0 1 ~ ' " ' ' 2 5 27 2 9
'
31
" " ' 33 35
37
' , ' , " ' I
39
41
43
45
TemperatLre (C)
Figure 5. Concentration at which minima in D(C) occur (R) present work and (0)Wilcoxon et al. and the concentration at which the maxi-
mum in ( d r / d C ) - l occur).(
Wilcoxon et al. versus temperature.
this point, a graph of the concentration a t which Wilcoxon et al. measured a maximum in ( d ~ / d C ) and - ~ a minimum in D(C), and the concentration a t which we measured the minimum in D(C) are graphed versus temperature in Figure 5 . There are only four temperatures at which the two sets of data overlap. Nevertheless, several conclusions may be drawn from this graph. First, there is a systematicdifference in the concentration at which the minima in the measured diffusion coefficient occurred in the two experiments. The two curves are nearly parallel, and the present study's results are approximately 10 mg/mL higher than the those of Wilcoxon et al. This may be due to different sample purities. More importantly, on an isotherm, the concentration at which the minimum in D(C)occurs is not the same as the concentration at which the maximum in (a~/aC)-1occurs. The concentrations a t which the minima in D(C) occur are higher than those at which the maximum in (&r/aC)-l occurs except at the temperature closest to the consolute temperature, where they are equal. These latter two curves are also not parallel. This indicates that critical phenomena is not the whole story. Rather, in light of the present work, as the temperature is raised at constant concentration both micellar growth and interactions occur. At lower temperatures these two effects cause the solution to evolve from a dilute to a semidilute solution. At higher temperatures (within about 5 OC or so of Tc)critical behavior dominates dilute or semidilute solution behavior and the solutions properties are indicative of binary fluid critical phenomena. Several aspects of the present study will require further experimental work. For example, the Carale and Blankschtein25 theory assumes 8 solvents and wormlike micelles. It is known that water is a good solvent (in the polymer sense) for C12E6 for temperatures less than about 20 O C . This is near the bottom of the temperature range studied and indicates, along with the existence of the lower consolute point, that water is not a good solvent over much of the temperature range of the present study. In the temperature range T > 8 a wormlike micelle is lessexpanded than a t thee temperature. Thismay helpexplain theexperimental observation that at the highest concentrations Dq( T)/q(ref) is not independent of temperature. In light of this observation, further experimental work on the analogy between polymers and surfactant-water solutions and the effects of non-8 solutions on the predicted persistence lengths may be in order. Further work is also needed in order to understand why the thermodynamic parameters that explain so much of the bulk thermodynamics of this system fail to adequately explain the mean radius of gyration and hydrodynamic radius of the micelles.
Summary This paper has presented the results of a light scattering study of C12E6-HzO solutions over a large range of temperatures and surfactant concentrations. The results show that along a given isotherm, in the temperature region in which there is micellar growth, a minimum in the diffusion coefficient and a maximum
in the characteristic size determined by static light scattering occur a t some concentration. The concentration at which these quantities obtain extremum was identified as the threshold crossover concentration for the temperature of the isotherm. From the temperature dependence of the crossover concentration and the theoretical calculation of Carale-Blanks~htein~~ the persistence length and its temperature dependence were determined. The persistence length was also deduced from static and dynamic light-scattering measurements. The two sets of measurements are in very poor agreement, indicating that further work is needed on this problem. The interpretation of present experiment is consistent with a theoretical d e v e l ~ p m e n tthat ~ . ~makes ~ predictions covering the range of concentrations from below the critical micelle concentration (cmc) to semidilute, and temperatures from below the critical micelle temperature (cmt) to the critical region. Finally, this type of experiment and analysis can and should be applied to other systems.
Acknowledgment. This work was supported by NSF Grant DMR90-9007442. We also wish to thank Professor Daniel Blankschtein and Ms. Teresa Carale for their calculation of the persistence length, their interest in this project, and their many helpful suggestions. We also wish to thank the referees for their helpful comments. References and Notes (1) Puwada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710 and references therein, especially refs 1-8. (2) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J . Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (3) Israelachvili, J. N. Intermolecular and Surface Forces; Academic: New York, 1985. (4) Young, C. Y.; Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1978,82, 1375. ( 5 ) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J . Phys. Chem. 1980,84, 1044. (6) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J . Phys. Chem. 1983,87, 1264. (7) Porte, G.; Appell, J.; Poggi, Y. J . Phys. Chem. 1980, 84, 3105. (8) Tanford, C. The HydrophobicEffect;Wiley: New York, 1980. Also see: Mittal, K.L., Ed. Micelliration, Solubilization, and Micrvemulsions; Plenem: New York, 1977; Vols. 1 and 2. (9) Van De Sande, W.; Persoons, A. J. Phvs. Chem. 1985.89. 304. (10) Imae, T.; Ikeda, S.J . Phys. Chem. 1985,90,5216. Imae, T:; Ikeda, S. Colloid Polym. Sci. 1987, 265, 1090. (1 1) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1977, 60, 221; 1979, 72, 580. (12) Missel, P. J.; Mazer, N. A.; Carey, M. C.; Benedek, G. B. J . Phys. Chem. 1989, 93, 8354. (13) Mukerjee, P. J . Phys. Chem. 1972, 76, 565. (14) Ben-Shaul, A.; Gelbart, W. M. J . Phys. Chem. 1982,86,316. Also see: Gelbart, W.; Ben-Shaul, A.; McMullen, W. E.; Masters, A. J. Phys. Chem. 1984,88, 861. (15) Mishic, J. R.; Fisch, M. R. J. Chem. Phys. 1990, 92, 3222. (16) Cates, M. E.; Candau,S. J.J. Phys.: Condens. Matter 1990,2,6869, and references therein. Also see: Candau, S.J.; Hirsch, E.; Zana, R. In Physics of Complex and Supermolecular Fluids; Safran, S.A., Clark, N. A., Eds.; Wiley: New York, 1987; p 569. (17) Briggs, J.; Nicolli, D. F.; Ciccolello, R. Chem. Phys. Lett. 1980, 73, 149. (18) Eriksson, J. C.; Ljunggren, S.J . Chem. SOC.,Faraday Trans.2 1985, 81, 1209. (19) Candau, S. J.; Hirsch, E.; Zana, R.; Delsanti, M. Lungmuir 1989, 5, 1225. (20) Kato, T.; Anzai, S.;Seimiya, T. J. Phys. Chem. 1990, 94,7255. (21) Schurtenberger, P.; Magid, L. J.; Penfold, J.; Heenan, R. Lungmuir 1990,6, 1800. (22) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (23) After Figure 1 in ref 16. (24) Edwards, S.F.; Doi, M. The Theory OfPolymerDynamics; Clarendon, Oxford University Press: New York, 1986. (25) Carale, T. R.; Blankschtein, D. J . Phys. Chem. 1992, 96, 459. (26) Imae, T. Colloid Polym. Sci. 1989, 267, 707. (27) Imae, T. J . Phys. Chem. 1989, 93,6720. (28) Zhou, A.; Chu, B. J. Colloid Interface Sci. 1989, 133, 348. (29) Brown,W.; Rymden, R. J. Phys. Chem. 1987,91,3565 and references therein.
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