3293
J. Phys. Chem. 1991, 95. 3293-3300
the activation temperature; namely, higher melt index polymer tribution of sites that differ in the type of polymer produced, in is obtained, having a narrower molecular weight d i s t r i b u t i ~ n . ~ ~ their reactivity toward reducing agents, and possibly also in their polymerization activity. High activation temperature seems to Discussion narrow the distribution of sites, just as the polymer MW distriPoisoning experiments give an upper limit on the concentration bution is narrowed. of active centers. Therefore, they can only be conclusive when The MW of polymer from Cr(VI)/silica increases with reaction a very small amount of poison is successful in totally killing a time, or yield. This effect may also be due to a broad population reaction. For ethylene polymerization over Crfsilica under the of sites that are activated and deactivated at different times, the high pressure and temperature of commercial conditions, this was earlier sites having some tendency to produce lower MW polymer. not the case. Even by use of the technique of C O preadsorption However, this argument assumes nothing about the valence or on the divalent catalyst, the limit obtained would appear to be other characteristics of these sites. And it says nothing about a rather large fraction of the chromium on these 1% Cr catalyst, whether, or in what direction, the MW distribution should change somewhere between 6% and 30% of the total Cr. It is possible with time.22 Even at the lowest yields studied a broad M W that the active-center concentration actually is this high,' and not distribution is obtained, so even the earliest sites do not constitute the 1% or less that is sometimes postulated. Cutting the total a narrow population type. Thus as more sites develop, the MW amount of chromium present begins to inhibit activity a t levels distribution could broaden further, narrow, or even go through below about 0.5% Cr. These experiments are consistent with the a maximum breadth.2s active fraction being rather large, but do not constitute absolute Registry No. Cr, 7440-47-3; 02,7782-44-7; MeOH, 67-56-1; HC=proof. CH, 74-86-2; CO, 630-08-0; H2C=CH2, 74-85-1; polyethylene, 9002These experiments support our impressions from studying 88-4. polymers. Cr/silica catalysts apparently contain a broad dis~~
(24) Size exclusion chromatography indicates that the broader shear response that is characteristic of higher activation tempcratures is in reality not due to a broader MW distribution. Instead, it may reflect increased long-chain branching or other phenomenon not yet understood.
(25) As a simple example, consider a system having only two types of sites, A and B. At time t = 1, the MW composition might be 70% A/3046 B. But as more B sites develop, the composition becomes 50/50 by t = 2 and the MW distribution appears to be broadening. By t = 3, however, the composition might be 30/70; so between t = 2 and 3, the distribution seems to narrow.
Self-Diffusion in Nonionic Surfactant-Water Systems Mikael Jonstromer, Bengt Jonsson, and Bjorn Lindman* Physical Chemistry 1 , Chemical Center, University of Lund, P.O. Box 124, S-221 00 Lund, Sweden (Received: July 24, 1990)
The self-diffusion behavior of the micellar phase in some aqueous solutions of the nonionic dodecyloligoethylene oxide surfactants CIIESand ClZE8was analyzed according to the cell-diffusion model. Our calculations show that the hydration number per ethylene oxide (EO) group is independent of the surfactant concentration. The maximum extension of the headgroup shell for a C12E,,micelle was estimated to be around 18 A at both 5 and 66 OC, correspondingto an average mean water concentration of 8-9 water molecules per EO group. By comparing the water diffusion within the micellar headgroup shell with the water diffusion in poly(ethy1ene oxide)-water solutions of the same concentration expressed in water molecules per ethylene oxide group, our calculations show that the micellar water content decreases mainly from the innermost ethylene oxide groups at increasing temperatures. This approach further allows an analysis of the surfactant concentrated regime, where no 'free" water exists. The experimental data are then found to be consistent with a water continuous structure up to at least 80 wt % CI2E,,at 70 OC. We also present a model that accounts for the retardation of the micellar diffusion constant due to obstruction between prolate- and oblate-shaped aggregates. With this model, in combination with the results obtained from the cell-diffusion model, we find the observed surfactant diffusion data to be consistent with rodlike micelles with an axial ratio of 1:40 for CI2Esat 25 OC and an axial ratio of 1:lO for ClzE8at 70 OC over a large concentration interval.
1. Introduction Nonionic surfactants of the type oligooxyethylene alkyl ethers (C,,H,l(OCH2CH2)xOH) (here denoted C S x ) exhibit a complex phase pattern in aqueous solution.'-5 This pattern is not only a function of the chemical structure of the surfactant and of its concentration in the solution but also to a high degree dependent on the temperature. Of special interest is the existence of a lower consolute t~mperature,~.~ above which the micellar solution phase separates into one water-rich and one surfactant-rich isotropic phase. This critical phenomenon, called 'clouding", has previously been reported for simple polymer-water solutions of the type poly(ethy1ene oxide) (PEO)? Therefore, the interaction between solvent molecules and the surfactant ethylene oxide (EO) groups is believed to be responsible for not only the clouding phenomenon but also the phase behavior of nonionic surfactant systems in To whom correspondence should be addressed.
0022-3654/91/2095-3293$02.50/0
general.'.* The common picture is a disfavored solute-solvent interaction at an enhanced temperature, which leads to a reduced hydration of the surfactant E O shell and thus a decrease of the mean area occupied by each headgroup. From simple geometrical considerations,' this implies a change in the spontaneous curvature (1) Mitchell, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T.; McDonald, M. P.J . Chem. SOC.,Faraday Trans. 1 1981, 79, 915. (2) Adam, C. D.; Durrant, J. A.; Lowry, M. R.; Tiddy, G. J. T. J . Chem. SOC.,Faraday Trans. 1 1984,80, 789. (3) Degiorgio, V. In Physics of Amphiphiles: Micelles, Vesicles and
Microemulsions; Degiorgio, V., Corti, M., Eds.; North-Holland Amsterdam, 1985; p 303. (4) Lang, J. C.; Morgan, R. D. J . Chem. Phys. 1980, 73, 5849. (5) Strey, R.; SchomHcker, R.;Roux, D.; Nallet, F.;Olsson, U. J . Chem. SOC.,Faraday Trans. 1990,86, 2253. ( 6 ) Saeki, S.;Kuwahara, N.; Nakata, M.; Kaneko, M. Polymer 1976,17, 685.
(7) Lindman, B.;Karlstr6m, G. Z . Phys. Chem. (Munich) 1987,155,199. (8) Karlstrom, G. J . Phys. Chem. 1905.89.4962.
0 1991 American Chemical Society
3294 The Journal of Physical Chemistry, Vol. 95, No. 8, 1991
Jonstromer et al. 1 0.8
\
-
0.4
1 Oa2
t 0
40 60 wt% C12Eg
20
80
100
Figure 2. Water self-diffusion coefficient as a function of surfactant concentration for the system Cl2E8/D20. Do denotes the self-diffusion in pure D 2 0 at the experimental temperature.
Figure 1. Cross section of a C&g micelle, illustrating the extended headgroup region for this kind of surfactant aggregates. The hatched area represents the hydrocarbon region.
of the surfactant interface to become more curved toward the polar solvent at higher temperatures. As a consequence, the micelles are forced to grow, and the liquid crystalline phases change in the order hexagonal-lamellar-reverse hexagonal. For liquid crystalline phases, such transitions have been frequently reported, not only as a function of temperature but also when chaining the headgroup area by varying the number of EO groups.' The structure of the isotropic solution phase is not accessible in the same direct way as that of the liquid crystalline phases, but indirect observables such as the location of intensity peaks in scattering experiments9 and rheological quantitieslO may be analyzed in terms of both solution structure and micellar sizes. Another experimental observable, frequently used for structural investigations of micellar solutions, is the long-range molecular self-diffusion coefficient (D),11-15obtained with, for example, the NMR PGSE technique.16 This quantity expresses the molecular weighted mean displacement for each type of species in the solution. We will in this work analyze self-diffusion data within the so-called cell-diffusion mode1,17J8which in a quantitative way treats the diffusion behavior of molecules in a system of colloidal particles. As opposed to earlier theories, this model does not assume a "bound" state of water molecules interacting with the surfactant headgroups. Furthermore, it accounts for motional retardations, owing to both changes in the volume fraction and the shape of the obstructing aggregates. The model has also proven to be valid for high concentrations of particles, and hence it is well suited for investigations of the surfactant-rich regime. The aim of this report is to investigate the isotropic solution phase of the C12&/D20and C12E5/D20systems in order to gain (9) Kato, T.; Anzai, S.; Seimiya, T. J. Phys. Chem. 1987, 91, 4655. (10) Tanford, C.; Nozaki, Y.;Rohde, M. F. J. Phys. Chem. 1977, 81, 1555. (1 1) Nilsson, P. G.; Lindman, B. J. Phys. Chem. 1983, 87, 4756. (12) Brown, W.; Johnson, R.; Stilbs, P.; Lindman, B. J. Phys. Chem. 1983, 87, 4548. (13) Nilsson, P. G.; Wennerstriim, H.; Lindman, B. J. Phys. Chem. 1983, 87, 1377. (14) Kato, T.; Seimiya, T. J. Phys. Chem. 1986, 90, 3159. (15) Faucompre, B.; Lindman, B. J. Phys. Chem. 1987, 91, 383. (16) (a) Stilbs, P. Prog. Nucl. Magn. Spectrosc. 1987, 19, 1. (b) Callaghan, P. T. Ausr. J . Phys. 1984, 37, 359. (17) Jonsson, B.; Wennerstrom, H.; Nilsson, P. G.; Linse, P. Colloid Polym. Sci. 1986, 264, 77. (18) Jiinsson, B.; Jansson, M. Unpublished in: Jansson, M. Thesis, University of Uppsala, Uppsala, Sweden, 1988.
0 1 ' 0
'
20
(
?
I
40
"
60
"
80
" 100
wt% C12Eg
Figure 3. Relative C&g self-diffusion coefficient in the system CI2E8/D20as a function of the amphiphile concentration. Do is the C & self-diffusion coefficient at infinite dilution, which was estimated m 2 d at 70 O C (see text). m 2 d at 5 OC and 9.0 X to be 1.7 X
further insights into the motional state of water of hydration as well as into the micellar structure at different temperatures and surfactant concentrations. We will start with a presentation of the experimental results, which have all been reported in previous articles by Nilsson et al. and which we will refer to concerning the experimental conditions.' 1 9 1 3 We then present the theoretical model which in the last section is used to discuss the extension of the micellar EO layer as well as the shape of the aggregates at different temperatures. All equations used to evaluate the experimental results for nonspherical aggregates are finally summarized in the Appendix. 2. Experimental Observations A nonionic surfactant system, which previously has been shown to consist of fairly small micelles at temperatures around and below room temperature, is Cl2&/D20lF2' (see Figure 1). The solution phase exists for this system over a wide concentration and temperature range and, in particular, all from neat water to neat surfactant in the temperature interval 60-80 OC (clouding around 80 "C). For a complete phase diagram, see ref 1. In Figure 2, water self-diffusion coefficients from the isotropic L, phase are plotted as a function of surfactant concentration at 5 and 66 OC. In addition, the two most concentrated samples at 5 OC correspond to water diffusion within a discrete cubic crystalline liquid phase.22 To allow a comparison between data obtained at different temperatures, the relative diffusion coefficients are given (19) Zulaf, M.; Weckstriim, K.; Hayter, J. B.; Degiorgio, V.;Corti, M. J. J. Phys. Chem. 1985,89, 341 1. (20) Jennings, B. R.; Nash, M. E.; Tiddy, G. T. J. Colloid Interface Sci. 1989, 127, 537.
(21) Binana-Limbeli, W.; Zana, R. J. Colloid Interface Sci. 1988, 121, 81. (22) Jahns, E.; Finkelmann, H. Colloid Polym. Sci. 1987, 265, 304.
Self-Diffusion in Nonionic Surfactant-Water Systems
0.15
h
Po
0.1
-
0.05
-
. e a"
The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 3295
0
0
0 ' 0
3
"
10
"
"
20 30 wt% CizE,
u
'
40
'
I
50
Figure 4. Relative self-diffusion coefficient for the amphiphile in the systems CI2EJ/D20and CI2&/D20at 25 OC as a function of amphiphile concentration, Do represents the self-diffusion coefficients of the surfactant monomers in D 2 0 and is estimated to be 3.8 X 1O-Io m2& for m 2 d for CI2E8at this temperature. CI2EJand 3.5 X
in the figure, where Do is the value of neat D 2 0 obtained at the relevant temperature. The relative amphiphile self-diffusion coefficients for the same system, obtained at 5 and 70 OC, are given in Figure 3. Do denotes here the diffusion constant of CI2EImonomers in pure D,O and is estimated to be 3.5 X m2.s-I at 25 OC and assumed to vary with temperature according to q D / T = constant (eq 3) where q denotes the water viscosity." One example of a system where micellar growth has been reported, as a function of both concentration and temperature, is CI2E5/D2O.l3At temperatures just below clouding (-30 "C), the Ll phase is limited at relatively high concentrations by a lamellar liquid crystalline phase at around 55 wt 76 surfactant and at lower temperatures by a hexagonal phase. A recent, detailed phase diagram of the C12E5/H20system is given in ref 5. In order to display the large difference in the diffusion properties between CI2E5and CI2Ee,the amphiphile self-diffusion coefficients obtained a t 25 O C for the two systems are compared in Figure 4. Water diffusion data for the system C12E5/D20 obtained at 25 O C are presented further on (Figure 9). The water self-diffusion coefficient for PEO-D20 solutions of different concentrations at 5 and 66 "C was also obtained from ref 11. The experimental results were found to fit the following equation (see also Figure 5)
0
I/. 0
, 10
,
,
X
IOd exp
X
+ 0.02(T- 215.5)
where X is the number of water molecules per EO group and T the temperature in kelvin. The self-diffusion coefficient, D, is given in m 2 d .
3. The Cell-Diffusion Model The so-called cell-diffusion model was first presented a few years ag0.l' This model gives a theoretical description of the molecular flow, or diffusion, in a system of colloidal sized particles. The model is based upon the widely used concept of dividing a macroscopic system into small subsystems, or cells, in such a way that they together may represent the macroscopic properties. The effective diffusion coefficient for a component i will in this model depend on both the diffusion of the cell and the diffusion within the cell. The equation for the total effective diffusion coefficient Dl of component i in a micellar system may according to ref 18 be written as (see also the Appendix) (2) Di = D;"[1 - Dmice~~e/D'/I+ Dmicelle where Dla is the effective self-diffusion coefficient in a cell centered around the micelle. Doi is the self-diffusion coefficient of component i in a bulk solution, which is the neat solvent or the amphiphile diffusion at infinite dilution, and Dmiallc the self-diffusion coefficient of the micelle.
,
12 ,
.
20 30 mol D20 / mol EO
.
40
50
Figure 5. Water self-diffusion coefficient in PEO-D20 solutions at 5 and 66 O C . The full lines correspond to eq 1.
For water diffusion in micellar solutions Orffis nearly the same as the effective total self-diffusion coefficient, since Dmiallc often is much smaller than Dowater.A rough estimate of Dmiallcis, therefore, often sufficient when water self-diffusion data is analyzed. The micellar self-diffusion constant is, however, important when analyzing amphiphile self-diffusion data, since the diffusion of the micelles often is nearly the same as the total self-diffusion coefficient of the amphiphile in the system. There is, of course, no problem in using eq 2 if experimental data for Dmialleare available, but this is often not the case. Therefore, we also discuss how the micellar self-diffusion coefficient in a concentrated solution may be determined from a theoretical model. Micelle Diffusion. Dmicclle of spheroidal aggregates at infinite dilution may be estimated from the Stokes-Einstein or Perrin equations23
kT
Domiccllc = -F(axial
ratio)
6aqb
(3)
where q is the viscosity of the surrounding medium, b is the length of the micellar short axis, and F(axia1 ratio) is a function depending only on the axial ratio (ar is the length of the long axis divided by the length of the short axis) of the aggregate. F is 1 for a spherical aggregate and may for a prolate ellipsoid be written as F(axial ratio) =
D = 3.605
,
In [ar + (ar2 - 1)'I2] (ar2 - 1)1/2
(4a)
For an oblate ellipsoid, F is given from arctan (ar2 F(axia1 ratio) =
1)1/2
(ar2 - 1)ll2
(4b)
At higher concentrations, the micellar excluded volume extends the diffusion paths and thus reduces the diffusion constant, socalled obstruction effects. The micelle-micelle obstruction in a system of hard spheres may at low concentrations be written as2' where rH is the hydrodynamic radius of the micelle and R the radius of the cell. An analogous expression has, to our knowledge, not been presented for a system with spheroidal aggregates. We will therefore as an approximation use eq 5 to estimate the micelle-micelle obstruction effect for spheroidal aggregates as well. By means of eq 3, ( r H / R ) 3may for a system of spheroidalshaped micelles be written as 0 ar'F(axia1 ratio)3
(6)
(23) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ai; Marcel Dekker: New York, 1986. (24) Ohtsuki, T.; Okano, K. J . Chem. Phys. 1982, 77, 1443.
3296 The Journal of Physical Chemistry, Vol. 95, No. 8,1991
Jonstromer et al.
co
when the radius is larger than b, and if Dc" and the mean concentration of the component in the region r < b are known D(r) C(r) = DICl r < b = D2C2 r>b (10)
g 0.1
then U(R) may be written as
0.15
I
h
e
U(R) = 0.05
1 -po 1
+ pCP/2
where 0
0
10 20 30 40 Volume% of obstructing particles
50
and
Figure 6. Relative micellar self-diffusion coefficient, calculated from eqs
3-6, for CI2EBmicelles of different prolate axial ratios (ar given in the figure) as a function of the micellar volume fraction.
where = (b/R)3arS is the micellar volume part of the cell (s = 1 for a prolate aggregate and s = 2 for an oblate aggregate). The micellar self-diffusion coefficient calculated from eqs 3-6 is in Figure 6 plotted as a function of the volume fraction CP for some prolate-shaped C12Esmicelles of different axial ratios. As in Figures 3 and 4, the diffusion data are given relative to the diffusion constant of the surfactant monomer in water. The micellar short axis was set to 30 A, corresponding to the radius of a spherical CI2E8micelle with the hydrocarbon chain fully extended and including a few water molecules per EO group.13 Self-Diffusion in a Fixed Cell. The cell-diffusion model presented in ref 17 is in the present work used to determine the effective diffusion coefficients, D:", for solvent and surfactant in a micellar system. The reader is referred to ref 17 for a complete description of the model. Moreover, to make it easier to follow the calculations, we only present the equations valid for systems that may be divided into spherical symmetric cells. All equations valid for systems of spheroidal symmetry (prolate and oblate ellipsoids) are summarized in the Appendix. To calculate the effective self-diffusion coefficient for a component i in a cell, the local variation of the product of the selfdiffusion coefficient and the equilibrium concentration (DC)must be known. The connection between the local DC profile and the effective self-diffusion coefficient may for a spherical symmetric system, where DC only depends on the radius ( r ) ,be written as (7) where U(R)is the value of the function U(r)at the cell boundary." The function U(r)is in the cell-diffusion model introduced in order to obtain a simple formalism for diffusion calculations. It is in ref 17 shown that U(r) C(r) D ( r ) is a continuous function that may be obtained from the following differential equation: dU d r- = (1 - U)(2 + U) - r U z ( l n (D(r) C ( r ) ) ) (8) dr The boundary condition a t r = 0 is
U(0)= 1
(9)
The differential equation (8) may be solved by numerical methods for any DC profile, a process which today is an easy operation on a personal computer. It is, therefore, straightforward to obtain the effective self-diffusion coefficient from eq 8 if the DC profile is known. DC may in some systems have one value in one region of the cell and another in the remaining part of the cell such as, for example, solvent molecules in a micellar system. The DCvalue may, however, also change smoothly over the whole cell, such as for the counterion concentration outside a charged micelle. A theoretical model, such as the Poisson-Boltzmann equation, may be used to obtain the DC profile in this case.I8 An important special case is when the DC value may be assumed to be constant in a region of the cell, since eq 8 has an analytical solution in that specific region. If DCin a cell is constant
Observe that eq 11 is also valid when the DC profile for radii less then b is not constant. The important aspect is that the mean concentration and the effective self-diffusion coefficient in this region must be known. The effective self-diffusion coefficient, a", for a species i in a micellar solution may now be simulated by dividing each cell (sized so they just include one surfactant aggregate) into regions of constant diffusion properties, each defined by the product DiCi together with the volume fraction it occupies. It should be noted that these regions always add up concentrically in the cell, fixing the location of the colloidal particle to the center. For the simplest case, with hard uniform particles dispersed in a solvent medium, the system is fully described by just dividing the cell into two parts: ,C one corresponding to the volume fraction of particles with, = 0 and the rest of the cell filled with pure solvent of bulk properties. Dcffwlvcnt calculated for this idealized case is found in and particle shape. Figure 4 of ref 17, as a function of both CPe For a real micellar system, this application of the model is oversimplified, since part of the solvent molecules interacts with the micellar headgroup shell and thus does not behave as in the bulk solution. In this case a third region, representing the micellar headgroup shell, has to be introduced, but this offers no problems in the effective diffusion calculations. The effective self-diffusion coefficient may be obtained by a successive use of eq 11 from the center of the cell, since the solutions from different regions may be matched by using the restriction that UDC must be continuous a t all points in the cell. Rather than a priori assuming a special type of DC profile, one may analyze diffusion data in another way if the DC product has a constant value outside the micelle and if the DC profile inside the micelle is independent of the micelle concentration. The effective self-diffusion coefficient in the micelle may then be obtained from a plot of (1 - U(R))/( 1 + U(R)/2) as a function of the micelle concentration, since 1 - U(R) = po 1 U(R)/2
+
U(R) is obtained from eq 7. This plot will give a straight line as long as the DC profile in the micelle is constant. The dimension of the micelle may be calculated from the value of 9 where the plot starts to deviate from a straight line and the effective selfdiffusion coefficient is obtained from the corresponding diffusion coefficient at that concentration. This way of evaluating diffusion data is especially suitable for solvent molecules, since the bulk value of the product D C may often be used as an approximation for C(R) D(R) and since the micellar diffusion may be neglected compared to the solvent diffusion as was mentioned earlier (cf. eq 2). It is important to point out that the above method of analyzing diffusion data is valid only for spherically symmetric systems. The U(R) correlation is for a system with spheroidal symmetry somewhat more complicated, as may be seen in the Appendix. The above method of analyzing diffusion data may therefore not in general be used for systems with spheroidal symmetry. The
Self-Diffusion in Nonionic Surfactant-Water Systems
...
0
20 30 40 50 Volume% of obstructing particles
10
60
Figure 7. Three theoretically calculated examples of the dependence of ( 1 - v)/( 1 + U/2)on concentration for solvent molecules outside an
obstructing spheroidal particle.
10.4
--
3
0.2 0 0
20
40
60
80
Volume% C12Eg
Figure 8. Concentration dependence of (1 - U(R))/(l+ U(R)/2) for D20 molecules in the system C1&,-D2Or at 5 and 66 OC. U(R)has been calculated from eq 7, assuming bulk water properties of C(R)and D(R). The dotted line corresponds to a situation where the surfactant interacts with the water only as hard obstructing spheres.
numerical difference for the effective solvent diffusion coefficient around a prolate micelle and around a spherical micelle is, however, rather small, and the above-mentioned way of analyzing solvent diffusion data may, therefore, also be ased for prolate aggregates as a good approximation. It is, as may be seen from Figure 7, only around oblate aggregates with large axial ratios that a significant difference in the effective solvent diffusion is obtained. 4. Results and Discussion In this part we will analyze the diffusion data obtained for the two systems CI2ES/D20and Cl2EB/D20, presented in section 2. First, water diffusion data are analyzed, and quantitative considerations about both the dynamic nature of water of hydration and the solution structure are presented. The observed self-diffusion coefficients for the amphiphiles are then in the second part used to determine the shape of the micellar aggregates. Water SelfDgfuusion. We start our analysis with the C12&/D20 system, which, due to the expected small aggregates, permits us to apply the spherical approximation of the model without further consideration (cf. Figure 7). In Figure 8, the hydration properties of the micellar EO layer are analyzed by plotting (1 - U ( R ) ) / ( l U ( R ) / 2 ) as a function of the total amphiphile concentration at 5 and 70 O C . Because of the low cmc of ClzEBin water, the total amphiphile concentration is proportional to the micellar concentration. The plotted values follow at both temperatures a linear relation for concentrations up to around 3 5 vol % of surfactant, as may be seen in Figure 8. This clearly indicates that the hydration properties are not dependent on the surfactant concentration in this region. The different slopes of these lines compared with the slope of the dotted line, which represents an ideal C12E8system where the surfactant only interacts with the water as obstructing hard spheres, indicates that the EO layer contains a large amount
+
The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 3297
of water, in particular at 5 "C. If the deviation from a straight line of the experimental data at around 35 ~019%C& is assumed to be a consequence of overlapping EO groups from different micelles, the thickness of the EO layer will be 18 A. This thickness, corresponding to around 8 waters per EO group, may be considered as a maximum extension of the EO layer that may be observed from our experimental data. The effective water diffusion coefficient in the spherical shell containing a C12EB micelle with an extension of the headgroup layer of 18 A is now obtained from the experimental diffusion constant at 35 vol % of surfactant. Assuming that no water penetrates into the hydrocarbon part of the micelle, the water diffusion within the EO layer may be calculated from the above-deduced value of the diffusion in the micelle. The hydrocarbon part acts here as an obstruction of the water diffusion in the micelle, and the relations in eqs 11-13 may be used to estimate the water diffusion in the EO layer. The value we obtain for the water diffusion in the EO layer is DIDo = 0.60 at 5 OC and D/Do = 0.65 at 70 "C. This is as far the analysis can be carried without any further knowledge about the physical nature of water of hydration. In a recent 170NMR relaxation report,2s the dynamic state of water in several CI2Ex/D20systems was investigated and compared with PEO/D20 solutions of the same concentration expressed in water molecules per EO group. These data display a similar relaxation behavior, and it was concluded that the dynamic perturbation of water is the same irrespective of whether the EO groups are attached at one end (C12E,) or not (PEO). The authors also explicitly applied the cell-diffusion model on the same set of C,,E8/D20 diffusion data as used in this paper, which gave a close similarity between the diffusion properties of water in a micellar headgroup region and in a polymer solution. Adopting these results, we are now in the pasition to estimate the water distribution in the micellar headgroup shell, since the diffusion of water as a function of EO concentration may be experimentally determined from results for the polymer/water system. The water content in the EO layer varies, of course, continuously from a low value near the hydrocarbon part of the micelle to a higher value further out, but we will here approximate the water distribution with one constant concentration in an inner region with 50% of the EO groups and another concentration in the outer region. The water concentrations in the two parts may then be evaluated from the earlier obtained effective diffusion coefficient in the two layers matched to eq 1. From this model we calculate the number of water molecules in the inner layer to be 2.8 per EO group at 5 "C, but only 1 per EO group at 70 OC. The outer layer was from this approach found to contain about 14 water molecules per EO group a t both temperatures. The lower number of water molecules per EO group at higher temperatures is a consequence of the strong temperature dependence in the EO-water interactionB and is a well-known phenomenon. That the decrease in the hydration seems to start in the inner region near the hydrophobic surface of the micelle is an observation that we, however, have not seen reported previously. All the results discussed so far have been obtained from diffusion data in the concentration range 0-35% surfactant. For the system C12EB/D20a t 70 OC, the isotropic solution range spans all the way from pure water to pure surfactant, and hence it is of interest to test the proposed model in this concentration range as well. To increase the applicability to concentrations where all water is interacting with the surfactant headgroups, a model of how water is distributed in the EO layer must be used. We thus assume that the water concentration in the inner part of the EO layer is constant until the concentration in the outer layer has decreased to the same value. At still higher micelle concentrations, the water concentration is assumed to be the same in all parts of the E O layer. The water diffusion constant as a function of the E O concentration was as previously obtained from eq 1. ( 2 5 ) CarlstrBm, G.; Halle. B. J . Chem. SOC., Faraday Trans. 1 1989,85, 1049.
3298 The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 I
Jonstrbmer et al.
9
0.1
I
i-
0.08
z .
30I
-3
t
h
no
2
2
n
0.04
20
0.02
IO 0' 0
0.06
. .,
I
I
'
20
40 wt % C12E,
60
0 0
80
Figure 9. Comparison of the experimentally obtained D20 self-diffusion coefficients with the values obtained from our theoretical model (solid curves) for the Ci2E8/D20system at 5 and 66 OC as well as for the C12Es/D20system at 25 OC.
Water self-diffusion values, calculated by use of this model, are in Figure 9 plotted together with the experimentally observed self-diffusion values for the CI2EE/D2Osystem at 5 and 66 OC as well as for the C12E5/D20 system at 25 OC. It is, as may be seen in the figure, possible to describe the experimental data rather well by the proposed model. Surfactant Self-Diffusion. In contrast to the analysis performed in the last section, the analysis of surfactant diffusion data is quite complex since the observed value is a function of several diffusion mechanisms weighted in a more involved way. Besides the translational motion of the individual micelles, the surfactant molecules are free to move within the micellar surface, a motion characterized by the lateral diffusion constant (Dl,,). The long experimental time scale (typically 10-100 ms) implies that the root-mean-squared (rms) displacement of surfactant molecules far exceeds the typical structural length scale. Thus, the lateral diffusion may not alone contribute to the observed diffusion constant. D,,, in combination with molecular diffusion between the micelles may, however, become an important transport mechanism for the surfactant molecules especially when the distance between the micelles is rather short, Le., at high micelle concentrations and/or large axial ratios. Let us, however, start our analysis with the low amphiphile concentrations, where the micelle diffusion gives the dominating contribution to the observed amphiphile diffusion. (At very low surfactant concentrations,
