Langmuir 1994,10, 1387-1392
1387
Articles Viscosity of Droplet-Phase Water/AOT/Isooctane Microemulsions: Solid Sphere Behavior and Aggregation J. Smeets,* G. J. M. Koper, J. P. M. van der Ploeg, and D. Bedeaux Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, P.O.Box 9502, 2300 RA Leiden, The Netherlands Received November 22,1993. In Final Form: February 16,2994" We have measured the viscosity of water/AOT/isooctanemicroemulsions as a function of the droplet volume fraction at different temperatures for two values of the molar water to surfactant ratio WO: 25 and 35. From the data we conclude that the microemulsion droplets behave like solid spheres, since (1) the low volume fraction dependence of the relative viscosity follows Einstein's relation and (2) at sufficiently low temperatures the coefficients of the second- and third-order terms in the volume fraction expansion of the relative viscosity equal those found for latex and silica spheres. Furthermore, we find that the droplets aggregate reversibly at the higher temperatures. The enthalpies of binding of the observed clustering process were found to be 89 and 121 kJ/mol for wo = 25 and 35, respectively, which is in good agreement with dielectric permittivity measurements on the same system. 1. Introduction
Dispersions of water in oil (or oil in water) become thermodynamically stable after adding a sufficient amount of an appropriate surfactant. Depending on the relative concentrations of these constituents and on the temperature, so-called microemulsions are formed. Here we consider a homogeneous microemulsion where the water is subdivided into small spherical droplets that are coated with a monomolecularlayer of surfactant and are dispersed in the oil, the continuous phase. The surfactant used in this work is AOT (sodiumbis(2-ethylhexyl)sulfosuccinate), while isooctane (2,2,4-trimethylpentane) is used as the oil. The mean droplet size is determined by the number, wo,of water molecules per surfactant molecule.1*2Smallangle neutron scattering (SANS) studies performed by Kotlarchyk et al. clearly demonstrate that the droplet size is almost independent of volume f r a ~ t i o n .Moreover, ~ the dielectric permittivity measurements of van Dijk et al. strongly indicate that the shape of the microemulsion droplets is essentially independent of t e m p e r a t ~ r e The .~ size polydispersity of the droplets is approximately 10%.5 A dramatic increase in the physical properties of AOT microemulsions such as the dielectric permittivity6-8 and the conductivit~~Q as a function of the volume fraction cp has been reported. From the conductivity measurements in particular one concludes that the system undergoes a percolation transition.8 This transition occurs at a
temperature-dependent volume fraction. The viscosity of droplet-phase microemulsions has been measured before, and a similar temperature dependence as for the dielectric permittivity has been found.lOJ1 However, the authors could not distinguish droplet clustering from other explanations for the increase in viscosity, e.g. electroviscous effects. In this paper we report low shear viscosity measurements on droplet-phase AOT microemulsions. By studying in particular the temperature dependence of the viscosity in the low volume fraction regime, we demonstrate quantitatively the same behavior as has been reported for dielectric permittivity measurements performed by van Dijket al.,thereby supporting the picture of an aggregation process underlying the percolation behavior in these systems.4J2 It is established that hydrodynamically the droplets behave like solid spheres with stick boundary conditions. Regarding their statistical thermodynamic properties, the droplets are found to behave like sticky hardspheresrather than like hard spheres. Due to a short-range (compared to the radius) attraction extensive aggregation occurs, ultimately leading to percolation.8
2. Theoretical Section For low particle volume fractions cp the viscosity 7 of a suspension of solid particles is given by Einstein's f ~ m u l a ~ ~
Abstract published in Advance ACS Abstracts, April 1, 1994. (1)Hilfiker, R.; Eicke, H.-F.; Hammerich, H. Helu. Chim.Acta 1987, 70, 1531. (2) Eicke, H.-F.; Rehak, J. Helu. Chim.Acta 1976, 59, 2883. (3) Kotlarchyk, M.; Chen, S.-H.; Huang, J. S.;Kim, M. W. Phys. Rev. A 1984, 29, 2054. ( 4 ) van Dijk, M. A.; Joosten, J. G. H.; Levine, Y. K.; Bedeaux, D. J. Phys. Chem. 1989, 93,2506. (5) RiEka, J.; Borkovec, M.; Hofmeier, U. J. Chem. Phys. 1991, 94, 8503. (6) Bhattacharya, S.; Stokes, J. P.; Kim, M. W.; Huang, J. S. Phys. Reu. Lett. 1985, 55, 1884. (7) van Dijk, M. A,; Broekman, E.; Joosten, J. G. H.; Bedeaux, D. J. Phys. (Paris) 1986, 47, 727. (8)van Dijk, M. A.; Casteleijn, G.; Joosten, J. G. H.; Levine, Y. K. J. Chem. Phys. 1986,85,626. (9)Eicke, H.-F.; Hilfiker, R.; Holz, M. Helu. Chim.Acta 1984,67,361.
where qo is the viscosity of the solvent. The form factor f is equal to 1 for a solid sphere with stick boundary conditions, and it differs from 1 for instance in the case of solid spheroidal parti~1es.l~ In the case of microemulsion (10) Eicke, H.-F.; Kubik, R.; Hammerich, H. J. ColloidInterface Sci. 1982,90, 27. (11)Berg, R. F.; Moldover, M. R.; Huang, J. S. J. Chem. Phys. 1987, 87, 3687. (12) Robertus, C.; Joosten, J. G. H.; Levine, Y. K. Phys. Rev. A 1990, 42, 4820. (13) Einstein, A. Ann. Phys. 1906, 19, 289. (14) Hunter, R. J. Foundotiow of colloid science; Oxford University Press: Oxford, Great Britain, 1987; vol. I, p 546.
0743-7463/94/2410-1387$04.5~/0 0 1994 American Chemical Society
Smeets et al.
1388 Langmuir, Vol. 10, No. 5, 1994 droplets f is more appropriately interpreted as the ratio of an effective solid sphere volume and the volume calculated from the amount of dispersed matter. For larger volume fractions one must on the one hand account for hydrodynamic interactions between the spheres and on the other hand for direct interactions between the particles that are, e.g., of a thermodynamic origin. For uncorrelated spheres the hydrodynamic interactions can be accounted for by writing
..
- 1
with qr 7/70the relative viscosity of the suspension. This formula was first derived by Sait6,15 and it is analogous to the well-known Clausius-Mossotti expression for the dielectric permittivity. Bedeaux has derived this relation for two-phase flow.16 In the Appendix we present a simplified derivation that is restricted to solid spheres. In order to account for thermodynamic interactions between the spheres one has to add terms proportional to higher powers of cp to the right-hand side (rhs) of eq 2.2.17J8 This can be written as
.. - 1 (2.3) where the function p(T,q) gives the modification of the moment of the friction forces on the surface of a single sphere due to the ensemble-averaged hydrodynamic interactions with the other spheres; it will simply be referred to as the friction moment. In the analysis of the experimental data we use the virial expansion AT,v) = 1 + a2(T)fv+ a3(T)3f2vz+ ...
(2.4)
In the literature usually a virial expansion of qr is used, which is related to the one for the friction moment by 7, = 1 + (5/2)fv
clusters with a Gibbs free energy AF per mole of contact points.25 This gives a contribution to the pair correlation function at contact proportional to e-mfRT. In a similar fashion as performed for the dielectric permittivity measurements by van Dijk et aL4 one may derive an expression for the virial coefficienta2 in terms of an integral over the pair correlation function. One then obtains
+ (51210 + a Z f v 2+ (5/2)(1 + 2a2 + a3)f3v3+ ...
(2.5)
Aside from the theoretical considerations mentioned above, the advantage of using the virial expansion of I.L is that it does not diverge at higher volume fractions, like vr,l8 which already diverges in the physical domain. For low values of the shear rate the best theoretical value for the second virial coefficientof the viscosity of a solid sphere suspension is 6.2, which corresponds to a2 = 1.48.19120The best experimental value is a2 = 3.08 f 0.11 for suspensions of latex18Pz1and silica18*22 spheres. The value of a3 has also been determined experimentally for the same suspensions: a3 = -3.15 f 0.21.18121~22 In the present case of water/AOT/isooctane microemulsions it has been established on the basis of dielectric permittivity measurements4 and on dynamic Kerf effect measurementsz3that the droplets may aggregate (coalescence does not seem to occur in these systemsz4)and form (15) Saiti3, N. J. Phys. SOC.Jpn. 1950, 5, 4. (16) Bedeaux, D. Physica A 1983,121,345. (17) Bedeaux, D.; Kapral, R.; Mazur, P. Physica A 1977, 88, 88. (18) Bedeaux, D. J . Colloid Interface Sci. 1987, 118, 80. (19)Batchelor, G. K.; Green, J. J. Fluid Mech. 1972, 56, 375. (20) Batchelor, G. K. J. Fluid Mech. 1977, 83, 97. (21) Krieger, I. M. Adu. Colloid Interface Sci. 1972, 3, 111. (22) de Kruif, C. G.; van Iersel, E. M. F.; Vrij, A.; Russel, W. B. J . Chem. Phys. 1985,83,4117. (23) van der Linden, E.; Bedeaux, D.; Hilfiker, R.; Eicke, H.-F. Ber. Bunsen-Ges. Phys. Chem. 1991,95, 876. (24) Cazabat, A. M. Adu. Colloid Interface Sci. 1992, 38, 33.
where the first term (az,~)is due to the temperatureindependent contribution to the pair correlation function which originates from hard sphere interactions. The second term (a2,l)is the contribution due to the aggregated droplets. For the third-order coefficient a3 the expression is expected to be
The terms on the rhs of eq 2.7 are due to the monomermonomer-monomer (~3,0),monomer-dimer (a3,1), and trimer ( ~ 3 , zand a3,3) contributions to the three-particle correlation function. The trimer binding Gibbs free energies AFr and AF" are expected to be roughly double and triple the dimer energy, because trimers contain two or three contact points. In the interpretation of the experimental results it is convenient to write AF = AH - TAS, where AH is the binding enthalpy and A S is the binding entropy. It is now important to realize that AH and A S have a negligible temperature dependence in the temperature domain considered. As a consequence one finds from the experimentally determined temperature dependence of az(T) the quantities UZ,O, UZJ exp(AS/R), and AH. It is not possible to obtain UZJ and A S seperately. Using however the fact that AF must be on the order of R T and that AH is considerably larger than this, one may use as a rough estimate A S i= AHIT0 with TOequal to room temperature. The relatively large value of the experimentally determined product a2,1 exp(AS/R) is thus a consequence of the large value of AS. The prefactor U Z , should ~ be on the order of unity.
3. Experimental Section AOT (sodium bis(2-ethylhexyl) sulfosuccinate) was obtained from Fluka, 98% purum. It was purified as follows: A 145.5-g sample of AOT was dissolved in 1 L of n-pentane. The solution was cooled to -15 O C in a cold room, where all the following extractions were carried out. The AOT solution was extracted three times with 75% methanol (25% water). The first time 65 mL and the second and third times 50 mL was used. The collected 75% methanol fractions contained most of the polar impurities and were discarded. It has been checked that further extractions contain mostly AOT and do not improve the final result. Subsequently the AOT/n-pentane solution was extracted four times with 100% methanol, the first two times with 150 mL of methanol and the last two times with 100 mL methanol. The collected AOT/methanol fractions were combined and extracted, still at -15 "C, with 50 mL of n-pentane. This n-pentane and the mother liquid contained most of the apolar impurities and were discarded. Most of the solvent from the AOTimethanol solution was then carefully evaporated under reduced pressure while the temperature was kept below 40 O C to prevent the hydrolysis of AOT. The solidified AOT was further dried for 24 h in U ~ C U O(0.05 mmHg) a t room temperature. The yield was 123.6g of AOT (85 % ),which (25) Koper, G. J. M.; Smeets,J. To be published inhog. Colloid Polym.
Sci.
Langmuir, Vol. 10, No. 5, 1994
WaterlAOTlIsooctane Microemulsions Table 1. Efflux Times ( 8 ) for Microemulsions with wo5 [HzO]/[AOT] = 25, Droplet Volume Fraction 'p, and Temperature T
Table 3. Relative Viscosities qr for wo= 25, Calculated from the Efflux Times in Table 1, Relative to the Viscosity of Isooctane!
cp
T ("C) 4.5 8.5 12.5 17.5 21.0 25.0 30.0
isooctane 174.65 166.78 159.58 151.26 145.91 140.18 133.62
0.01
0.05
163.46
180.76 171.53
149.47 159.40 152.67
136.85
cp
0.070 203.60 195.1 184.77 178.63 172.49 166.22
0.10
210.02 199.78 193.7 188.1 182.8
P
T ("C) 4.5 8.5 12.5 17.5 21.0 25.0 30.0
0.19 329.4 314.37 302.6 291 287.7 288.7 297.4
0.24 412.8 394.4 378.54 370.31 371.8 386.2 416.7
0.28 538.2 508.75 494.0 491.6 498.8 529.4 597.3
0.32 678 646.8 629.5 637 662.0 717 823
0.36 852.4 840.6 858.4 899.3 1036 1306
Table 2. Efflux Times ( 8 ) for Microemulsions with wo= 35, Droplet Volume Fraction 'p, and Temperature T P
T 0.29 0.39 ("C) -isooctane 0.03 0.057 0.089 0.15 0.19 3 177.93 352 581 1151 171.70 185.84 201.78 223.47 272.84 336 565 1087 6 165.87 179.50 194.91 215.89 263.68 325 554 1055 9 160.41 173.63 188.55 209.14 255.70 318 543 1029 12 155.34 15 202.74 248.98 308.73 540 1041 244.03 305.68 528 1060 18 150.46 145.91 241.5 307.87 553 1169 21 242.3 319 141.62 607 1353 24 246.9 334 27 137.56 695 1628 133.62 257.3 371.47 836 1937 30 was dissolved to a 500-mL AOT/isooctanesolution and was stored at -15 "C until further use. Isooctane was obtained from Merck (4727) and was used without further purification. Water was purified with a Milli-Q filtering apparatus from Millipore. Microemulsions were prepared by mixing proper amounts of water ( Vw),isooctane (VJ, and stock solution (V.) with concentration C of AOT. The total volume of the dispersion is taken to be the sum Vof the three volumes. After shaking the mixture for about 1 min a homogeneous microemulsion is obtained. The ratio of the water volume and the stock solution volume determines the wo of the dispersion according to (3.1) where n is the average number of water molecules that are bound to one AOT molecule (approximately l ) , and u, is the molar volume of water (55.4 dm3/mol at 23 "C). The droplet volume fraction is given by
vw+ V,C(U, + nu,) cp=
V
1389
(3.2)
with us the AOT molar volume (391 dm3/mol a t 23 "C). The viscosity measurements have been performed with an Ubbelohde capillary viscometer (Schott, Germany), tube type Oa. Prior to the measurements the capillary was flushed with acetone and n-pentane. After each microemulsion sample the capillary was flushedwith isooctane. The capillary was immersed in a water bath, which was kept at the desired temperature. The microemulsion was given 10-15 min to equilibrate with the bath. The temperature of the bath oscillated around the mean value with an amplitude of 0.04 K; the oscillation time was always shorter than the efflux time. The efflux time measurementa
?"("C) 0.01
4.5 8.5 12.5 17.5 21.0 25.0 30.0
0.05
1.020 1.161 1.163 1.030 1.165 1.025 1.167 1.029 1.173
0.070 0.10 0.19 0.24 0.28 0.32 1.27 1.27 1.27 1.27 1.28 1.29
1.38 1.39 1.40 1.41 1.44
2.07 2.07 2.09 2.12 2.17 2.27 2.46
2.66 2.66 2.67 2.76 2.87 3.11 3.52
3.53 3.50 3.56 3.74 3.94 4.35 5.16
0.36
4.53 4.52 6.06 4.61 6.25 4.93 6.75 5.31 7.33 6.0 8.4 7.2 10.4
The uncertainty in the viscosity ranges from 0.005 at cp = 0.01 to 0.05 for cp = 0.25 and higher. The viscosities for T = 25 and 30 "C at cp = 0.34 and 0.38 have a higher uncertainty. Table 4. Relative Viscosities qr for wo= 35 Calculated from the Efflux Times in Table 2, Relative to the Viscosity of I sooctane. cp
T("C) 3 6 9 12
0.030
0.057
0.089
0.15
1.10 1.10 1.10
1.21 1.21
1.36 1.37 1.37 1.37
1.72 1.72 1.72 1.74 1.76 1.79 1.86 1.95 2.09
1.21
15 18 21 24 27 30
0.19 2.19 2.17 2.17 2.20 2.21 2.26 2.34 2.51 2.70 3.10
0.29 3.78 3.81 3.87 3.93 4.04 4.08 4.41 4.99 5.89 7.2
0.39 7.82 7.66 7.70 7.77 8.1 8.5 9.7 11.6 14.4 17.6
The uncertainty in the viscosity ranges from 2 to 16 in the last digit specified. (I
Table 5. Virial Coefficients of j~ as a Function of Temperature and wo(See E s s 2.4,2.6, and 2.7) WO = 20 f = 1.01 f 0.05 wo = 25 f = 1.04 f 0.04 02 = (2.7 f 0.3) + exp((37f 7) (89 f 18 kJ mol-l)/RT) a3 = (-2.5 f 0.4) - exp((43f 9) (104 f 22 kJ mol-l)/RT) w o = 30 f = 1.06 f 0.04 w = 35 f = 1.15 f 0.04 02 = (2.5 f 0.3) + exp((50f 4) (121 f 11kJ mol-1)lRT) a3 = (-2.5 f 0.4) - exp((55& 5) (132 f 12 kJ mol-l)/RT) were repeated 3-10 times in order to obtain an accuracy of typically 5 X lW. The shear rate dependence was checked by performing the measurement with one microemulsion for a few capillary diameters: no significant dependence waa observed. Furthermore, for one microemulsionthe viscosity was measured with a variable-speed rotation viscometer. Within experimental accuracy, which was limited by evaporation, no shear rate dependence was observed.
4. Results and Discussion For this study we have restricted ourselves to microemulsions with w o of 25 and 35,corresponding to radii of the water core of 4.2 and 5.7 nm. Furthermore, wo = 20 and 30 were also studied, but less extensively. 4.1. Density. For several of the microemulsions that were prepared the mass density p has been measured at 23 "Cby weighing a known volume. By fitting the observed densities to the calculated cp, one finds a dependence which can be described by a linear relation:
+ 342cp kg/m3 p = 692 + 360cp kg/m3 p
= 692
for w o = 25
for w o = 35
(4.1)
1390 Langmuir, Vol. 10,No.5, 1994
Smeets et al.
w, = 35
w, = 25 12
-_-_____T ---- T T T
= 30 "C
= 25 "C = 21 "C = 4.5 OC
. . . . . . . . . . . . . . . . . . . . . . 0.0 0.1 0.2 0.3 0.4
v
0.0
0.1
0.2
v
0.3
0.4
Figure 1. Relative viscosity qr of a water/AOT/isooctanemicroemulsion as a function of the droplet volume fraction cp, the temperature T,and w o [H*Ol/[AOT].
The intercept (cp = 0) equals the density po of pure isooctane. The slope can be compared with a calculation based on ideal mixing of the three constituents with densities 1140 kg/m3 (AOT), 692 kg/m3 (isooctane), and 998 kg/m3 (water), which for w o = 25 gives a slope of 370 kg/m3. Within experimental accuracy these values are the same. 4.2. Viscosity. The measured efflux times are given in Tables 1 and 2. These times are to be corrected according to the manufacturer's instructions (Schott, Germany)? t, = t, - 25000/tm2
(4.2)
where t , and tc are the measured and corrected efflux times, respectively. With this corrected efflux time the relative viscosity can be calculated through (4.3) with p calculated by eq 4.1 and t and t o the corrected efflux times for the microemulsionand the oil, respectively. These relative viscosities are presented in Tables 3 and 4, and in Figure 1. The uncertainty in the viscosities is predominantly determined by the reproducibility of the microemulsion preparation. The viscosities for w o = 25, cp = 0.38, and T = 25 and 30 "C have been corrected for a gradual increase of viscosity with age, which is caused by evaporation of the solventa27 4.3. Results. For w o = 25 the relative low droplet volume fraction viscosities clearly show the linear dependence on volume fraction which is contained in both SaitB's formula (2.2)and Einstein's formula (2.1). The form factor f equals 1.04 f 0.04, independent of the temperature. For nonvanishing volume fractions particle interactions have to be taken into account as is demonstrated in Figure 1. (26) According to the manufacturer's specifications, the correction coefficientfor our tube equals (32 k 2) X lO*s3. Calibrationmeasurementa with water showed that relation 4.2 does not hold for high viscosities for this value; a correction coefficient of 25 OOO ss gives a better description. (27)Measurementa at 21 "C showed that the viscosity of this microemulsion increased. Through a second determination of the density it was concluded that the increase can be ascribed to evaporation of the oil. The correction consists of interpolating from the new 9 to p = 0.38. The uncertainty involved in this procedure accounts for the large errors in the viscosities derived from it.
The temperature and droplet volume fraction dependent friction moment, calculated according to eq 2.3, is shown in Figure 2. Fitting eq 2.4 to the data for each temperature and subsequently fitting eq 2.6 to the obtained values for the second-order coefficients a2 yields a binding enthalpy of 89 f 18 kJ mol-'; see Figure 3. Furthermore, the temperature-independent part can be determined at a2,o = 2.7 f 0.3, and for the third-order coefficient a3,o = -2.5 f 0.4; see Table 5. The virial coefficients a2 and a3 are found to have a linear interdependence as is shown in Figure 4. As can be seen from eqs 2.6 and 2.7, this means that in the third-order coefficient contributions due to monomer-dimer correlations dominate over those from trimers. For w o = 35 the data have been analyzed in the same way as described above. From the analysis it follows that the form factor f deviates from 1: f = 1.15 f 0.04. From the temperature dependence of the second-order coefficient a2 follows an enthalpy of binding of AH = 121 f 11 kJ mol-' and a low-temperature limit a2,o = (2.5 f 0.3) and a3,0= -2.5 f 0.4; see Table 5. As is the case with wo = 25, the coefficients a2 and a3 have a linear interdependence (Figure 4), so contributions to the third-order coefficient due to monomer-dimer correlations dominate over those from trimers. For w o = 20 and w o = 30 the viscosity has only been studied for volume fractions up to cp = 0.1. From these data we have deduced values for the form factor only: 1.01 f 0.05 and 1.06 f 0.04, respectively; no enthalpy of binding could be determined. 4.4. Discussion. The form factor f differs no more than 15 % from its ideal value of 1. We believe that solvent molecules entrapped between the surfactant tails account for this discrepancy. A deformation of the droplets in shear flow would lead to a shear rate dependence of the viscosity, which is not observed. It is therefore concluded that hydrodynamically the droplets behave as solid spheres with stick boundary conditions. This conclusion is further supported by the fact that the values of the temperature-independent contributions to the virial coefficients are, again within experimental accuracy, in reasonable agreement with values found for essentially solid hard spheres made of silica or latex, a2,o= 3.08 f 0.11and ~ 3 . = 0 -3.15 i 0.21.1'3921922
Langmuir, Vol. 10, No. 5, 1994 1391
WaterlAOTlIsooctane Microemulsions
w, = 25
w, = 35
2.2
-
2.2
2.0
-
1.0
1.8
-
1.8
~
-
-€- --*
1.8
CL
CL 1.4
___.__._ T = 30 'C
_ _ _ _ T = 25 'C - - - T = 21 'C -T = 4.5 'C 0.8
0.0
1
I
I
,
,
0.1
I
, ,
1
Oh
I
1
,
1
0.5
, , , ,
,
0.4
___.____ T = 30 _ _ _ _ T = 2 7 "C
1.2
OC
___ __
1.0
T
-T=
= 24
T = 18
O C OC
6 O C
0.8
B P Figure 2. Friction moment per droplet p as a function of the droplet volume fraction cp, the temperature T, and wg.
w, = 25
w, = 35 '1
I
i
I I
/
I'
,F 2 ' . . , . 235 240
I
,
,
,
, , , , , , , , I , , , , ,
265
, , , , 360 365
Figure 3. Second-order coefficient a2 of j~ vs temperature T. The insets show van't Hoff graphs of the temperature-dependentpart of a2. The binding enthalpy of 89 kJ mol-l is equal to 37 kBTo One may also compare a2,o with the best theoretical value per droplet contact. Van Dijk et al. argue that this value a2,o = 1.48.18Jg~20There is as yet no explanation for the relatively large difference between the theoretical and corresponds to 1/2 kBT per surfactant molecule in the experimental values. contact region of two coagulating droplets.' As explained The values for the binding enthalpy have also been in section 2 this enthalpy is balanced by an approximately determined with other methods: Van Dijk et al. have equal entropic contribution of 37 kg. performed dielectric permittivity measurements on water/ RiEka et aL31have performed static light scattering and AOT/isooctane microemulsions for w o= 25 and 35 as well, self-diffusivity measurements on water/AOT/n-hexane yielding 74 f 1 and 119 f 1 kJ mol-', re~pectively.~ microemulsions. Their conclusion, obtained in particular Robertus et al. have performed small-anglex-ray scattering from the value of the virial coefficientof the self-diffusivity, (SAXS) experiments on the same systems.12 Their data is that their microemulsions do not behave like hard were further analyzed by Koper and BedeauxZ8who, using spheres. In view of our measurements their conclusion an extension of Baxter's sticky hard sphere mode1,29*30 indicates, that at the temperature and droplet radius they extracted the enthalpies of binding, resulting in 36 and 43 considered, aggregation is important. kJ mol-', respectively. Within experimental uncertainty To recapitulate our conclusions, droplet-phase water1 our results and those from van Dijk et al. are identical, but differ significantly from the values that follow from AOT/isooctane microemulsions behave hydrodynamically the work of Robertus et al. The origin of this discrepancy as a suspension of solid spheres. Furthermore, they behave is unclear to us. Preliminary results of electrooptical thermodynamically as a suspension of sticky hard spheres; birefringence measurements give AH = 80 kJ/mol for w o i.e., there is an attractive interaction between the droplets = 25. that causes aggregation. (28) Koper, G. J. M.; Bedeaux, D. Physica A 1992, 187,489. (29) Baxter, R. J. J. Chem. Phys. 1968,49, 2770. (30)Baxter, R. J. A u t . J. Phys. 1968,21, 563.
(31) RiEka, J.; Borkovec, M.; Hofmeier, U.; Eicke, H.-F. Europhys. Lett. 1990, 11, 379.
Smeets et al.
1392 Langmuir, Vol. 10, No. 5, 1994
where the sum runs over all spheres in the suspension and Vi is the volume of particle i. For an isolated sphere one has
-1
Jvdr
us =
Javdr userr = s
(A.4)
with S the stresslet of the sphere. Faxen’s theorem for a sphere in a velocity field with strain rate e giveslg
S = (20/3)aqoa3(1+ (1/10)a2V2)e
-10
82
Figure 4. Coefficients a3 and a2. The linear interdependence shows that dimer-monomer effects dominate over contributions from trimers for both w o = 25 (solid lines and markers) and w o = 35 (dashed lines and open markers). See eqs 2.6 and 2.7.
Acknowledgment. The authors thank J. W. A. Smulders for performing a part of the measurements and W. Sager for comments on the paper. Part of this work has been performed under the auspices of the EC network Thermodynamics of Complex Systems (Contract No. CHRX-CT92-0007). Appendix: Derivation of SaitG’s Relation We consider a suspension of spheres of radius a dispersed in a medium of viscosity qo. Given the velocity field v(r) the local strain rate is defined by32
e 5 sym vv = ( I / ~ ) ( V + V (vv)~)
(-4.1)
The average strain rate is calculated by averaging the local strain rate over the volume V as
(e) = ( l / m J d r e
We shall assume that the particles are sufficiently small compared to the gradients of the velocity field so that we can restrict ourselves to the first term for the stresslet of a sphere. If we further assume that the strain rate is not noticeably perturbed by the presence of the suspended particles, as is the case for dilute suspensions, then ( u s )= 500’peo
(A.6) with cp = 4na3N/3V the volume fraction of suspended particles and eo the unperturbed strain rate. This then leads to the well-known Einstein relation q/qo = 1 + (5/2)(c.13 For higher volume fractions, even if we restrict ourselves to noninteracting particles, the local strain rate is no longer equal to the unperturbed strain rate. From Green’s function for the creeping flow equation one derives16
e(r) = eo- (l/5qo)us(r)+ J dr’ H(r-r’) us(r’) (A.7) with H a four-index tensor that accounts for the modification of the strain rate due to the suspended spheres. The local field at the site of a particle consists of the unperturbed strain rate eo (in the absence of the suspended particles) plus the contribution due to the other particles
eloc(r)= e, (A.2)
The stress in the fluid consists of three parts: (i) the hydrostatic pressure -p6 with 6 the unit tensor, (ii) the viscous contribution from the solvent, uf= 2qoe, and (iii) the part due to the surface forces on the spheres, us. The average viscous stress is (32) Russel, W. B.; Saville, D. A,; Schowalter, W. R. Colloidal dispersions;Cambridge UniversityPress: Cambridge,Great Britain, 1989; section 2.6.
04.5)
+
dr’ H(r-r’) us(r’) = e(r) + (1/5qo)us(r)
(A.8) Using this local field with eq A.5 yields (A.9) and this results in Saita’s relation15 7
= 710
1 + (3/2)cp 1-9
(A.10)
