J . Phys. Chem. 1989, 93, 2506-2512
2506
lOOO:1) of 60 Torr. Labile radioactivity was removed by hydrogen exchange with water followed by lyophilization. Nonlabile radioactivity was measured on a liquid scintillation counter. The relative standard deviation of measurements were 13%. The reaction products were examined by thin-layer radiochromatography on cellulose plates in tert-butyl alcohol-methyl ethyl ketone-ammonium hydroxide-water (40:30:20:10 mL). From 90 to 95% of the radioactivity accumulated in thymine and dihydrothymine.
Discussion of Experimental Results Figure 4a shows the obtained dependence of the total radioactivity of the products (Ci/mmol of initial thymine) on the reaction time. The radioactivity of the products easily shows that in all experiments less than 2%of the initial thymine entered the reaction. The regression of the data by eq 14 yields the following parameters: aRT a p = 0.228 m i d , b = 0.0597 Ci/mol (relative standard regression deviation (rsrd) is 13.2%). Figure 4a presents the graph of function 14 with these values of the parameters. The regression of the kinetic data of the radioactivity accumulation in dihydrothymine by (13) yields a p = 0.0316 min-' (rsrd =
+
14.9%). Hence, aRT= 0.196 min-' and kp/kRT = 0.161. Figure 4b shows the theoretical kinetic curve for [P]int/[RT]int calculated for these parameters. The experimental data on the kinetics of radioactivity ratio in thymine and dihydrothymine are presented in the same figure. Since the experiments were carried out not with tritium but with a Hz-T2 mixture (1000:1), the quantity of molecules containing several tritium atoms was negligible and the radioactivity ratio equals the concentration ratio. It follows from the Figure 4b that at the beginning of the reaction (approximately before 240 min) the experimental data agree well with the prediction of the diffusion model. But the remainder of the reaction shows a divergence that can be accounted for by a variety of reasons, particularly by the catalyst poisoning. The comparison of the experiment with the elaborated diffusion model allows the following conclusions to be drawn: the specific configuration of the [PIint/[RTIintkinetic curve predicted by the diffusion model is experimentally observed. A satisfactory quantitative agreement exists between the experimental kinetics and the curve predicted by this concrete variation of the diffusion model. Registry No. Thymine, 65-71-4.
Dielectric Study of Temperature-Dependent Aerosol OT/Water/Isooctane Microemulsion Structure M. A. v. Dijk,+ J. G. H. Joosten,*Y. K. Levine, Department of Molecular Biophysics, Physics Laboratory, University of Utrecht. Princetonplein 5, 3584 CC Utrecht, The Netherlands
and D. Bedeaux* Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, POB 9502, 2300 R A Leiden, The Netherlands (Received: January 19, 1988; In Final Form: August 30, 1988)
The low-frequency permittivity of an AOT/water/isooctane water-in-oilmicroemulsion has been measured over the temperature range 10-45 OC and as a function of the concentration of water droplets. The results are explained in terms of the polarizability of the system on introducing an extension of the Clausius-Mossotti equation to higher volume fractions. It is shown that the results indicate that the droplet shapes are temperature independent while individual droplets may form dimers. Binding enthalpies of this clustering process have been obtained.
Introduction A water in oil (W/O) microemulsion is a thermodynamically stable, isotropic, transparent liquid dispersion of small (- 10 nm) spherical water droplets, surrounded by a monomolecular layer of surfactant molecules, in a continuous oil phase. The molar water/surfactant ratio Woessentially determines the radius of the droplets while the molar oil/surfactant ratio Sodetermines their concentration in the oil phase.' Such systems of particles with high permittivity dispersed in an insulating low permittivity medium are amenable to studies with dielectric The usual drawback to such experiments is the lack of a detailed description of the observed macroscopic dielectric behavior in terms of the properties of the components of the mixture. Rather, one has to resort to effective medium theories, which furthermore cannot account for the temperature dependence of the dielectric properties. We shall show in this paper that an analysis of dielectric permittivity measurements in terms of cluster polarizabilities overcomes many 'Present address: Kon/Shell Laboratory, Separation Technology Department, POB 3003, 1003 AA Amsterdam, The Netherlands. 'Present address: DSM Research, Afdeling FA/GF, POB 18, 6160 M D Geleen, The Netherlands.
0022-3654/89/2093-2506$01.50/0
of these obstacles for the system under consideration. The polarizability of a microemulsion system containing a sufficiently low volume fraction, &, of particles can be obtained from the experimental results using the Clausius-Mossotti formula. In this regime the particles are considered to be independent. However, this assumption breaks down in the case of concentrated systems where many-body effects become important in the calculation of the average polarizability. These latter effects may be taken into account in the Clausius-Mossotti formalism by introducing a correction term, I(@p, T)@p2,where T is the absolute temperature, in an operational (1) Ekwall, P.; Mandell, L.; Fontell, K. J . Colloid Inferface Sci. 1970, 33, 215. (2) Peyrelasse, J.; Boned, C. J , Phys. Chem. 1985, 89, 370. (3) Eicke, H. F.; Shepherd, J. C. W. Helu. Chim. Acta 1974, 57, 1951. (4) van Dijk, M. A,; Boog, C. C.; Casteleijn, G.; Levine, Y . K. Chem. Phys. Lett. 1984, I l l , 571. (5) Sjoblom, J.; Jonsson, B.; Nylander, C.; Lundstrom, I . J . Colloid Interface Sci. 1983, 96,504. (6) van Dijk, M. A. Phys. Rev.Lett. 1985, 55, 1003. (7) Bhattacharya, S.; Stokes, J. P.; Kim, M. W.; Huang, J. S . Phys. Reo. Lett. 1985, 55, 1884. (8) van Dijk, M. A,; Casteleijn, G.;Joosten, J. G . H.; Levine, Y. K. J . Chem. Phys. 1986,85, 626.
0 1989 American Chemical Society
Aerosol OT/ Water/Isooctane Microemulsion Structure
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2507
definition. As such it can be extracted from the experimentally determined low-frequency polarizabilities of the microemulsion systemeg It will be shown below how the contribution to this term from particle pair correlations may be calculated; see also ref IO. We report here the experimental determination of I(C#Jp,T) in microemulsions of Aerosol OT/water/isooctane in the temperature range 10-45 "C. This function was found to exhibit a simple Arrhenius temperature dependence and to be independent of C#Jp over a large range of volume fractions. The findings may be rationalized on assuming that the microemulsion particles cluster and that the process involves a binding enthalpy AH per contact. It will be shown that a good description of the temperature and concentration dependences of the microemulsion permittivity is obtained in terms of only three parameters: the polarizability of a single microemulsion droplet, the binding enthalpy per contact, and a prefactor which is related to the average number of contacts per particle and the excess pair polarizability of a bound pair. The clustering hypothesis is supported by recent experiments using other and by the temperature dependence of the volume fraction where the percolation transition occurs.
Experimental Section The dielectric permittivity measurements were carried out in the frequency range I O kHz-13 MHz with a Hewlett-Packard HP4917A impedance analyzer with use of a thermostated cell, similar to the one described in ref 19. The temperature was stable to 0.1 OC. The permittivity spectra of AOT/water/oil microemulsions show a (temperature-dependent) dielectric relaxation in the megahertz range.2-8 Within our experimental frequency range we always observed a frequency independent portion of the real part e' of the permittivity at the low-frequency side of the spectrum. The corresponding value of e' we denote the low-frequency limiting permittivity elf. The Aerosol OT or AOT (sodium bis(2-ethylhexyl) sulfosuccinate) was obtained from Fluka AG (purum), and inorganic impurities were removed by following the method of ref 20. Isooctane (2,2,4-trimethylpentane)was obtained from Baker, and the water was deionized and quadruple distilled in an all-quartz still. Solutions were freshly prepared by dissolving appropriate weights of AOT, water, and isooctane. On calculating volume fractions, we assumed the bulk density values for water and isooctane and a value of 1.14 g/mL for the density of AOT.' Theoretical Section Microemulsion Model. A W / O microemulsion may be considered as a macrofluidI6 of small droplets suspended in an oil continuum. Each droplet consists of a spherical water core surrounded by a monomolecular layer of surfactant molecules.1~1~21 We shall here neglect the small polydispersity of about 20%123'3321 found experimentally and assume that all droplets are of identical size. Each water spherule, radius R,, is surrounded by n, AOT molecules, Figure 1, each occupying an average interfacial areal A = 65 AZ.The ionic SO< groups (molecular volumeZous ii: 50 A') will be assumed to be in the water phase and may donate their Na+ counterions to that phase. For a microemulsion system (9) van Dijk, M. A.; Broekman, E.; Joosten, J. G.H.; Bedeaux, D. J. Phys. 1986, 47, 727. (10) Bedeaux, D.; Wind, M. M.; van Dijk, M. A. Z.Phys. B 1987,68,343. (11) Lemaire, B.; Bothorel, P.; Roux, D. J. Phys. Chem. 1983, 87, 1023. (12) Kotlarchyk, M.; Chen, S. H.; Huang, J. S.; Kim, M. W. Phys. Reu. Lett. 1984, 53, 941. (13) Kotlarchyk, M.; Chen, S.H.; Huang, J. S.; Kim, M. W. Phys. Reu. A 1984, 29, 2054. (14) Ober, R.; Taupin, C. J. Phys. Chem. 1980, 84, 2418. (15) Guering, P.; Cazabat, A. M. J . Phys. Lett. 1983, 44, 601. (16) Hilfiker, R.; Eicke, H. F.; Gieger, S.; Furler, G.J . Colloid Interface Sei. 1985, 105, 378. (17) Eicke, H. F.; Hilfiker, R.; Thomas, H. Chem. Phys. Lett. 1985, 120, 212. ~-
(18) Chatenay, D.; Urbach, W.; Cazabat, A. M.; Langevin, D. Phys. Reo. Lett. 1985, 20, 2253. (19) van Beek, W. M.; Touw, F. v. d.; Mandell, M. J . Phys. E 1979, 9, 785
(20) Tavernier, S. M. F. Thesis, University of Antwerp, 1981. (21) Zulauf, M.; Eicke, H. F. J . Phys. Chem. 1979, 83, 480.
n
CH3
oil EO
El E
h
L'
L1
Figure 1. Schematic model of an AOT microemulsion particle. The aqueous core has a radius R,, L is the length of the AOT molecule ( L = 10.5 A), R, = R, L is the outer radius of the particle, and LI is the length of the apolar part of the AOT molecule (LI 7.5 A). The shaded area indicates the polar part. The permittivities of the water, the polar part, the apolar part, and the oil are denoted respectively e,, eh, el, and e,.
+
composed of Wowater molecules (molecular volume u, per AOT molecule we thus have n,A = 4nRW2
i=
(area)
n,(Wou, + us) = Y37rRw3
(volume)
30 A3) (1)
(2)
where the volume of the Na+ ions has been neglected. Equations 1 and 2 can be solved for R, and n, to give R, = (3/A)(WoU,
+ v,) = (1.4W0 + 2.3) 8,
n, = 47rRW2/Ai= 0.19RW2/A2
(3) (4)
A typical microemulsion droplet in a system with Wo= 30 has a water core with a radius R, = 44 A and is surrounded by =370 AOT molecules, while the aqueous phase contains ~ 1 . X2 IO4 water molecules. Such a droplet is shown schematically in Figure 1. The volume fraction C#JP of the droplets is obtained from the volume fraction of water 4., The latter quantity can be calculated from the weighed-in amounts of AOT, water, and isooctane used to form the microemulsion on taking the density of AOT1,20as 1.14 g/mL and the bulk values for the density of water and isooctane. We now have30
where L ii: 10.5 8, is the length of an AOT molecule without the SO3-group.20 Dielectric Permittivity of a Dispersion of Spherical Particles. We shall here consider the dielectric permittivity.of a dispersion of small spherical water droplets surrounded by a layer of surfactant molecules in a continuous oil phase. The ClausiusMossotti function, or as it may be called the effective polarizability a due to the droplets per unit volume of the mixture, is defined by t cy=-
€
-
€,
+ 2€,
(6)
were E is the (measurable) permittivity of the mixture and e, the
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The Journal of Physical Chemistry, Vol. 93, No. 6,1989
permittivity of the continuous oil phase. Our definition of polarizabilities per unit of volume differs by a factor 3t0 from the more usual definition. The dipole polarizability cy, per unit of uolume of a homogeneous sphere of permittivity cp suspended in oil is given by a, =
-
tp ~
tp
d
€0
+ 2t0
(7)
where our definition differs again from the more usual one by a factor of 3t0. In our case it is more appropriate to describe the sphere as a composite sphere consisting of shells of different permittivities and conductivities. A calculation of the dipole polarizability cy, as well as the multipole polarizabilities of such a sphere is given in the Appendix. For a sufficiently dilute suspension of spheres the Clausius-Mossotti relation t -t
€o
+ 2t0
-
= ap4,
where I(4,,7‘) gives the volume fraction and temperature-dependent correction to Clausius-Mossotti, Neglecting excess polarizabilities due to clusters of three and more particles, we findlo
where g is the volume fraction and temperature-dependent pair correlation function for the distribution of the spheres. Furthermore, a ( R ) is the dipole polarizability per unit of uolume of two spheres at a distance R averaged over the orientations of the pair relative to the electric field. At large separation this polarizability reduces to the one-sphere result lim cy(R) = a,
R--
(11)
In ref 10 we discussed in detail how to calculate a ( R ) given the multipole polarizabilities found in the Appendix of the present paper. Using the fact that the inner shells of the droplets are either conductive or have a permittivity much larger than the permittivity of oil, one finds for the multipole polarizabilities, using the formulas given in the Appendix: a , = cy v+2)13 (12) P
A precise definition of the multipole polarizabilities is also given in the Appendix. As is clear from this equation, one has a, = a,. The above formula shows that for the purpose of calculating the dielectric constant one may replace the sphere with radius R , by a somewhat smaller hypothetical sphere with a radius R,
-
n v) I
ts
0.3 0.2
I
-I
0.1
0.0
i 1.2
1.4
1.6
S cy
is valid, where 4, is the volume fraction occupied by the particles. When the volume fraction increases, this relation is no longer correct. The reason for this is the fact that the polarizability per unit of uolume of a cluster of spheres with intersphere distances comparable to the diameter of the spheres may increase significantly above the value found for one sphere. In ref 10 we have given a systematic expansion in terms of polarizabilities of larger and larger clusters. It is convenient to write the more general expression in the form
J
0.4
v. Dijk et al.
= RP
P 113
(13)
containing a material which is either conductive or has an infinite permittivity. Outside these smaller droplets, and in particular also between R Eand R,, one may use the dielectric permittivity of the oil. The dipole polarizability of a pair of these conducting spheres has been calculated and is plotted in Figure 2 as a function of RIR,. It is clear from this figure that the spheres must approach each other to a distance such that the high permittivity (or conducting) cores almost touch each other to see a significant increase of the polarizability. As R , C R,, such an approach of two spheres implies an overlap of the apolar tails of the AOT molecules. This
Figure 2. Excess polarizability of a spherical particle with polarizability ap= 1 due to another identical particle at a distance R as a function of s = R/2R,, where R, is the particle radius.
suggests using a pair correlation function for sticking spheres. Such a pair correlation function has the form 1 ~ ~+( gRf ( R ) (14) g ( R ) = s ~ , R , ~ ~ ; ~ R-; R,) where n, is the average number of ‘bonds” per particle and R, is the distance between two bound spheres. Furthermore, gf is the more usual fluidlike contribution to the pair correlation function. If the volume fraction is not too large, most bound clusters will be dimers and the following form for n, is assumed: n, = AdP exp(-AH/kBT)
(15)
Here AH is the enthalpy of binding, kB is Boltzmann’s constant, and A is a constant. The main contribution to I(4,,T) is due to the bound spheres because g f ( R )will be zero for distances R 5 R , while cy(R) is already almost equal to its asymptotic large R value a, for these distances. Substituting eq 14 and 15 into eq 10, neglecting gf then gives I(4p7n= A [ a ( R , ) / a p- 11 exP(-AH/kBT) = 10 exp(-AH/kBT) (16) Substituting this expression into eq 9, we find
(’ =)+ 4,
= ap[l
2to
+ 4,I0 exp(-AH/kBT)]
(17)
The analysis of our data will be done using this expression. As we discuss in more detail in ref 10, eq 17 is not exact. It neglects contributions that will result in a deviation from the linear dependence on @, for larger volume fractions. It follows from the experimental results that while such a deviation indeed occurs, it is small until one is very close to the percolation transition. See also ref 9. Figure 3 shows the +p dependence of the function I , eq 10, calculated for a pair of conducting spheres, cyp = 1, using the Percus-Yevick radial pair distribution function22for hard spheres with a radius R . This behavior can be compared with I(4,) calculated from its operational definition, eq 9, for two widely used effective medium theories:23 Bruggeman’s symmetrical (SB) and unsymmetrical (UB) equations. For the case of conducting particles one has ~ ’ ~ ( 4 ,= ) c0/(1 which yield
-
3 4 ~ ~ )tUB(dP) ; = to/(l
- dPl3 (18)
(22) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976. (23) Landauer, R. In Electrical Transport and Optical Properties of Inhomogeneous Media--1977; Garland, J . C.; Tanner, D. B., Eds.; AIP Conference Proceedings 40; AIP: New York, 1978; p 2.
Aerosol OT/Water/Isooctane Microemulsion Structure
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2509 l ~ " ~ " ' " " ~ l " " l " " I ' ~ ' ' l ' ' " I
70 6
5
-4
e
v
-
3 2
20
,./-
1
lo
0
Figure 3. GP dependence of the function I for conducting particles (ep = a,cyp = 1) according to Bruggeman's symmetrical (SB) and unsymmetrical (UB) effective medium theories (eq 15) and according to eq 12 with a Percus-Yevick hard sphere distribution function (PY). The dashed line indicates the relation between the value of I(@,) and the critical volume fraction for the divergence of the permittivity (eq 16 with cy, = 1).
Figure 4. Low-frequency permittivity elf as a function of the particle volume fraction @, at four different temperatures of a microemulsion with
w,= I.
l ~ ~ " l ~ " ~ l ' " " ' ' ' ' l ' ~ ' " l ' ~ '
50
The symmetric Bruggeman equation predicts a percolation transition from dielectric to conducting behavior for C#J~= 1/3 and is therefore in better qualitative agreement with the experimental results than the unsymmetric Bruggeman equation. The result found using Percus-Yevick also predicts a percolation transition for & = 0.58, a value smaller than the random close packing fraction 0.637. This follows from eq 9, where an infinite value of e on the left-hand side gives the general relation for the percolation threshold:
This relation between I and & is given in Figure 3 for the special case of a conducting sphere, a p = 1. The main reason that both the effective medium theories as well as the PY results are unsatisfactory is the fact that they predict a temperature-independent dielectric constant whereas the experimental results are strongly dependent on the temperature. We further note that the PY distribution function for hard spheres with radius R , peaks at contact just like the two-sphere polarizability. This gives values for Ipywhich are large compared to the value found if a more realistic gf,cf. eq 14, which is small at that distance, is substituted in eq 10. This explains once again why we neglect this contribution in our derivation of eq 17. In a paper by Eicke et al.29an interesting suggestion was made to overcome the lack of temperature dependence of the unsymmetric Bruggeman approximation. They replace 4pin eq 18 for euB by 4p times a temperature-dependent swelling factor that describes the formation of clusters which become increasingly "ramified" as the temperature approaches the percolation threshold. Our eq 17 for e may in fact be obtained in the same way by replacing 4pin Clausius-Mossotti by 4p(1 +doexp(-AH/kBT)). The factor between brackets may in this context be interpreted as a swelling factor. The temperature dependence we find is in fact similar to the temperature dependence found in ref 29 from their data. In our analysis the swelling factor is interpreted as an enhancement of the polarizability due to the clustering, and on the basis of this we are able to give an explicit expression. As for dimensional reasons the volume fraction and the polarizability always appear as products, the difference is to
+
(24) Sjoblom, J.; Jonson, B.; Nijlander, C.; Lundstrom, I. J . Colloid In-
ferface Sei. 1984, 100, 21. ( 2 5 ) Eicke, H. F.; Shepherd, J. C . W.; Steinemann, A. J . Colloid Inferface Sci. 1976, 56, 168. ( 2 6 ) Laqiies, M. J . Phys. Lett. 1979, 40, 331. ( 2 7 ) Lagourette, B.; Peyrelasse, J.; Boned, C . ;Clausse, M. Nafure 1979, 281, 60. (28) Eicke, H. F.; Hilfiker, R.; Hammerich, H. Helu. Chim. Acfa, in press.
(29) Eicke, H. F.; Geiger, S.;Sauer, F. A,; Thomas, H. Eer. Bunsen-Ces. Phys. Chem. 1986, 90, 872. ( 3 0 ) Notice the fact that the formula used in ref 9 implicitly used v, = 0, which is incorrect.
o
T=lOT
o
: T=15T
Figure 5. Low-frequency permittivity til as a function of the particle volume fraction bP at six different temperatures of a microemulsion with W,, = 25.
t
/
Figure 6. Low-frequency permittivity elf as a function of the particle volume fraction 4, at four different temperatures of a microemulsion with = 35.
w,
some extend merely a matter of names.
Experimental Results The low-frequency permittivity elf was measured as a function of the droplet volume fraction +p, eq 5, for three values of Wo in the temperature range 10-45 OC. The results for W, = 7 are shown in Figure 4,for Wo = 25 in Figure 5, and for Wo= 35 in Figure 6 . It can b e seen that under all conditions the value of e l f approaches that of the permittivity of isooctane (1.94) in the limit 4p 0. The value of elf increases monotonically with 4p until a leveling off is observed at a value much larger than e,,. In the systems with Wo = 25 and Wo = 35, e l f increases markedly with increasing temperature (Figures 5 and 6) over the whole range of volume fractions studied. On the other hand, for the system with Wo = 7, e l f exhibits a significant temperature dependence only for c $ ~ > 0.5, where it decreases with increasing temperature. -+
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The Journal of Physical Chemistry, Vol. 93, No. 6,1989
7 25 35
12.1 37.3 51.3
0.40 f 0.06 0.74 f 0.02 0.84 f 0.02
22.6 47.8 61.8
v. Dijk et al.
16.7 f 0.8 43.2 f 0.5 58.3 i 0.5
5.9 f 0.8 4.6 f 0.5 3.5 f 0.5
I
1.39 f 0.15 30.0 f 0.3 47.9 f 0.5
