J . Phys. Chem. 1991, 95, 2550-2555
2550 c
I
r l
Figure 11. CI2E5/H20/CR (a = 40%. y = 7%. T = 32.4 " C ) microemulsion. Higher magnification of cohesive fracture. Planar and rounded surfaces and broken tubule-like or vestibule-likestructures are visible as in Figure 4.
electrons. Comparison with Figures 6 and 7 reveals the presence of adhesive (Y)and cohesive (Z) fracture regions. Figure 8 demonstrates the ability of this technique to distinguish between adhesive and cohesive fractures, on the basis of macroscopic features of the replica imaged at low magnification. At an intermediate magnification of the FFTEM replica, as shown in Figure 9,morphological differences between adhesive (Y)and cohesive (Z) fracture regions can be studied. These morphological differences are more easily identified at higher magnifications as shown in Figures IO and 1 1. Figure IO shows the adhesive fracture at higher magnification and reveals rounded domains -1000 A wide. Many of the rounded domains have small (-50 A) bumps on the surface. Comparison of the adhesive fracture in Figure 10 to the micrograph of Bodet et al. in Figure 3 reveals striking similarities. From this evidence, we conclude that Bodet et al. imaged adhesive fracture surfaces. Images of adhesive fracture surfaces allow
visualization of microemulsion structure as it exists near solid surfaces. Figure 1 1 is a higher magnification image of the cohesive fracture surface. The cohesive fracture shows planar regions, rounded domains, and fractured vestibular structures. The presence of fractured tubule-like or vestibule-like structures supports previous evidence that these microemulsions are bicontinuous. Comparison of the cohesive fracture in Figure 1 1 to the micrograph of Jahn and Strey shown in Figure 4 reveals that the planar and rounded domains and fractured vestibular structures are common to both micrographs. Therefore, it is concluded that Ja hn and Strey were observing cohesive fractures, Le., fractures through the bulk microemulsion, whereas Bodet et al. were viewing adhesive fractures. Summary
The development of a new method of preparing samples for FFTEM has elucidated the origin of conflicting images from identical microemulsions in the studies by Bodet et aI.l5 and Jahn and Strey.I6 It was determined that the different images produced by Bodet et al. and Jahn and Strey corresponded to adhesive and cohesive fractures, respectively. Adhesive fracture surfaces contain information about microemulsion structure near solid surfaces. Cohesive fractures follow the bulk microstructure morphology and therefore contain information related to the bulk microstructure. The sample preparation technique developed here allows simultaneous observation of adhesive and cohesive fractures and affords a new avenue for studying interfacial phenomena.
Acknowledgment. We are grateful to Dr. J. R. Bellare and Prof. R. Strey for helpful discussions of the issues and for providing the micrographs used in Figures 3 and 4,respectively. Support for this research came from the U.S.Department of Energy (Grant DOE/DEACI9-79BCl0116-A013) and the NSF Center for Interfacial Engineering. Registry No. CllESl3055-95-6; n-octane. 1 1 1-65-9.
Aggregation Kinetics of Oil-in-Water Microemulsion Droplets Stabilized by C,2E5 Paul D. I. Fletcher* and Josef F. Holzwartht School of Chemistry, University of Hull, Hull HU6 7RX. U.K., and Fritz Haber Institut der Max Planck Gesellschaf. Faradayweg 4-6, IO00 Berlin 33. West Germany (Received: May 9, 1990)
We have measured the kinetics of aggregation of oil-in-water microemulsion droplets stabilized by the nonionic surfactant n-dodecyl pentakis(ethy1eneglycol) ether (CI2E5).The rate of droplet aggregation, which is induced by increasing temperature, was determined by using an iodine laser temperature-jump (ILTJ) technique. The second-order rate constant for the droplet aggregation is found to be significantly lower than the diffusion-controlled limiting value, indicating an energy barrier of magnitude several kT must be overcome before the equilibrium aggregated state is reached. This energy barrier is found to decrease with increasing droplet size.
Introduction
n-Tetradecane may be solubilized in aqueous solutions of ndodecyl pentakis(ethy1ene glycol) ether (C12E5)to give single-phase oil-in-water (o/w) microemulsions. The single-phase microemulsion domain is bounded at low temperatures by the solubilization phase boundary at which an excess oil phase separates and at high temperatures by the cloud-point curve where a surfactant-rich phase separates. The temperatures of these phase boundarics depend on the molar ratio of tetradecane to ClzE5( = R ) do not significantly vary with the concentration of dispersed oil
and surfactant when R is kept constant.' At a microscopic level, the structure of these microemulsions is one of spherical, fairly monodisperse oil droplets coated with a monolayer of surfactant. At temperatures close to the solubilization phase boundary, the hydrodynamic radius (rh)of the droplets has been found to be proportional to R and independent of surfactant + oil concentration (at constant R ) , according to eq 1, where V is the molecular volume of the dispersed oil, A, is rh = 3( V / A , ) R + 6 (1) the area occupied by a surfactant molecule at the interface between
To whom correspondence should be addressed at the University of Hull.
'Max Planck Gesellschaft.
( I ) Aveyard,
R.;Binks. B. P.; Fletcher, P. D. 1. Lungmuir 1909.5, 1210.
0022-3654/91 /2095-2550$02.50/0 0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No.6, 1991 2551
Aggregation Kinetics of Microemulsion Droplets separated droplets
0
droplets to form a short-lived transient, coalesced droplet dimer can be broken into steps as shown in Figure l.s The two droplets initially form a noncoalesced encounter pair in a preequilibrium step. The transient coalescence of the droplets then occurs within the encounter pair. The rate of the overall droplet coalescence process is some (as yet unknown) combination of the encounter-pair formation and dissociation rates together with the rate of coalescence within the droplet encounter pair. The present study is an investigation of the kinetics of the initial encounter-pair formation step (droplet aggregation). The present microemulsion system forms a useful model system to investigate the kinetics of the droplet-clustering process. One can easily prepare droplets of a known size (by adjusting R ) and concentration (by adjusting the surfactant concentration at a given R value). Additionally, the clustering of the droplets can be controlled by varying the temperature within the stable microemulsion range. The rates of the droplet clustering have been measured by monitoring the solution turbidity changes following a rapid temperature jump produced by a I-ps pulse from an iodine laser.
0
+
11
co
droplet encounter pair
II
0
shod-liveddroplet dimer
Figure 1. Postulated mechanism of droplet coalescence in microemulsions.
the oil core and the surfactant monolayer, and 6 is the thickness of the surfactant monolayer, which includes entrapped solvent. The microemulsion droplet concentration is given by [droplets] = ([surfactant] - c m c ) ( A , 3 / 3 6 ~ P R ~ ) (2)
For tetradecane-in-water microemulsions stabilized by C,2ESthe parameters of eqs 1 and 2 have the following values:' V = 0.431 nm3; A, = 0.29 f 0.33 nm2, 6 = 3.0 f 0.3 nm, and the aqueous phase cmc is approximately 5 X lo-' M. Hence, the values of rh and the droplet concentration can be easily calculated from the solution composition according to rh/nm = 4.51R
+ 3.0
(3)
[droplets] = ([surfactant] - cmc)/861R2
(4)
It has further been shown that, as the temperature range of the single-phase microemulsion region is crossed (Le. as the system is taken from a low temperature close to the solubilization phase boundary toward the cloud-point boundary), the droplet size remains constant but the droplets cluster together.1.2 Viscosity measurements indicate the clusters formed are rather nonspherical.' At the lowest possible temperatures (i.e. within about 0.2 OC above the solubilization phase boundary) the droplets behave virtually ideally for droplet volume fractions less than approximately IO%.' Although the equilibrium behavior of microemulsions has been well studied, the dynamics of these droplet systems remains relatively poorly understood. It is important to gain such an understanding, since much of the interest in oil water surfactant systems is centered on their behavior in kinefically stabilized structures such as emulsions and foams. The droplet size distribution in microemulsions is maintained in a rapid dynamic equilibrium via a mechanism of continual droplet coalescence and reseparation."' The overall process of the coalescence of two
+
+
(2) Clark, S.;Fletcher, P. D. I.; Ye, X. bngmuir 1990, 6, 1301. (3) Eicke, H. F.;Shepherd, J. C. W.; Steinmann, A. J . Colloid Inreflace Sci. 1976, 56, 168. (4) Atik, S.S.;Thomas, J. K. J . Am. Chem. Soc. 1981, 103, 3543. (5) Fletcher, P. D. 1.; Howe, A. M.; Robinson, 8.H. J. Chem. Soc., Faraday Trans. I , 1987,83.985. (6) Fletcher, P. D. I.; Parrott, D. In Chemical and Biological Reactions in Compartmentalised Liquids; Knoche, W., Schomacker, R., Eds.; Springer-Verlan: .. -_ Berlin. - - --.1989. ( 7 ) Bommarius, A . S . ; Holzwarth, J. F.; Wang, D. 1. C.; Hatton, T. A. J. Phys. Chem. 1990, 94.1232.
Experimental Section Materials. The surfactant CI2ESwas a chromatographically pure sample supplied by Nikkol. Water was doubly distilled from a quartz vessel. n-Tetradecane was a Koch Light puriss sample, which was passed over an alumina column prior to use in order to remove polar impurities. Microemulsion samples were prepared by addition of the required quantity of alkane to a solution of CI2Es in water. The microemulsion phases can be very slow (several hours) to reach the final, equilibrium state, since it appears a transient liquid-crystalline phase is formed on initial mixing of the components at room temperature. The quickest way to obtain the equilibrium microemulsions was found to be to heat the mixtures to a temperature above that of the upper temperature phase boundary of the system, to shake the solution, and to cool to the desired temperature. As long as the resultant microemulsions were not cooled too far below the lower temperature phase boundary, cycling the temperature up and down gave reproducible values of the turbidity. The microemulsions formed by this system (particularly those with high R values) are generally bluish, as they scatter light relatively strongly in comparison to water-in-oil microemulsions of a similar droplet size. Methods. The iodine laser temperature-jump (ILTJ) apparatus has been described e l s e ~ h e r e . ~In- ~the mode employed in this study, the light pulse energy was 0.5-0.8 J produced in approximately 1 ps a t a wavelength of 1.3 15 pm. The polarization of the incident beam was approximately 10-2096. The sample path length with respect to the iodine laser was 5 mm, and it is estimated that approximately 70% of the incident pulse was absorbed by the sample. The temperature rise following a laser pulse was 0.5-1 O C . The turbidity of the sample was monitored at 90' to the incident iodine laser heating pulse with a stabilized 0.27-W Coherent laser emitting at 413 nm. It is worth noting here that the ILTJ technique is the optimum method for the study of these noionic systems, as the conventional Joule-heating temperaturejump technique can only be applied to electrically conducting samples (which would necessitate the addition of electrolyte). Additionally, Joule-heating techniques necessarily involve an electric field jump that can introduce spurious relaxation processes.I0 Equilibrium turbidity measurements were made with a Shimadzu UV-2100 spectrophotometer. For both kinetic and equilibrium turbidity measurements, the samples were contained within quartz cuvettes and thermostating was achieved to a (8) Holzwarth, J. F.; Schmidt, A.; Wolff, H.; Volk, R. J . Phys. Chem. 1977, 81, 2300. (9) Holzwarth, J. F. In Techniques and Applications of Fast Reactions in Solution; Gettins, W. J., Wyn-Jones, E., Eds.; Rcidel: Dordrecht, The Netherlands, 1979; p47. (IO) Marcandalli, B.; Stange, G.; Holnvarth, J. F. J . Chem. Soc., Faraday Trans. I , 1988.84, 2807.
2552 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 1
Fletcher and Holzwarth 0.25
f
0.2 turtidity/cm-1
turbiditylcm-1 0.15
0.1
\
0.1
’\
0.05
0
.L
0.01
0 35
30
40
0
temp/C
0.2
0.4
0
[C12E5YM
Figure 2. Equilibrium turbidity versus temperature for microemulsion containing 0.013 96 mol dm” CI2E5.The curves (in order of increasing turbidity) refer to R values of 1.0, 2.0,3.0, 3.5, and 4.0.
Figure 4. Equilibrium low-temperature limiting values of the turbidity versus droplet volume fraction for R = 2.0 microemulsions. The solid and dashed theoretical curves were calculated as described in the text.
The turbidity of a dispersion of noninteracting spheres of radius much less than the wavelength of light is given by the following equation:I I
P
0.4
1 I
turbidity/cm-l
0.3
I
T
I
b
0
1
0
3
2
4
R
Figure 3. Equilibrium low-temperature limiting values of the turbidity versus R for microemulsions containing 0.013 96 mol dm-’ CI2E5.The solid curve is calculated as described in the text.
precision of fO.l
O C
by use of Haake water circulators.
Results and Discussion Equilibrium Turbidity Measurements. Figure 2 shows the measured turbidity as a function of temperature for microemulsions of different R values within the ranges corresponding to thermodynamically stable microemulsions. The turbidity increases strongly as the temperature is increased from the solubilization phase boundary toward the cloud point. It has been shown previously that this increase is due to increasing attractive interactions and not to a growth in the droplet radius.’,* The plateau values at low temperatures close to the solubilization phase boundaries correspond to minimum attractive interdroplet interactions. The low-temperature limiting values of the turbidity are plotted versus R in Figure 3.
= (24r3/X4)((n,2- n,2)/(n:
+ 2n,2))2NV,2
(5)
T is the turbidity, X is the wavelength, t+ and n, are the refractive indices for the particle and solvent, respectively, N is number of particle per unit volume, and V, is the particle volume. In order to model the size dependence of the turbidity, we assume here that the surfactant tail groups have the same refractive index as the tetradecane core and the head group refractive index matches that of the water solvent. The theoretical curve in Figure 3 is calculated by using eq 5 together with the known values of the droplet radii (eq 1) and an assumed value of 1.9 nm for the effective thickness of the surfactant tail group region. The refractive indices of the droplets and the solvent were taken as 1.429 and 1.333, respectively.I2 The experimental low-temperature turbidity values are in good agreement with the calculated curve shown in Figure 3. The concentration dependence of the turbidity of R = 2.00 droplets at a temperature close to the solubilization phase boundary is shown in Figure 4. The experimental values are compared with a theoretical curve calculated by assuming the droplets behave as “hard spheres”. In this case, the variation of turbidity with the volume fraction of the droplets 0 is given by eq 6. In the T
= (constant)O(l - 0)4/[(1
+ 20)2- 03(4 - O)]
(6)
calculation of the theoretical curves, the value of the constant in eq 6 was adjusted so that the maximum calculated values matched that of the experimental curve. For the solid line shown in Figure 4, the volume fraction of the microemulsion droplets was assumed to be equal to the sum of the volume fractions of the dispersed oil plus the surfactant ( d 0.924 g/cm3). For the dashed curve, the volume fraction was assumed to include 10 molecules of ‘bound” water of solvation per surfactant molecule. The experimental values of the turbidity fall between the two calculated (1 1) Hunter, R. J. Foundutions of Colloid Science; Clarendon Rcss: Oxford, U.K., 1986; Vol. 1. (12) Handbmk of Chemistry und Physics, 62nd 4.CRC ; Press: Boca Raton, FL, 1981. (13) Cambat, A. M.;Langevin, D.; Pouchelon, A. J. Colloid Interfie Sci. 1980, 73, 1.
The Journal of Physical Chemistry, Vol. 95, No. 6,1991 2553
Aggregation Kinetics of Microemulsion Droplets
TABLE [: Summarv of Derived Kinetic Parameters ~
R
r,/nm
1 2
7.5 12.0 21.0
4
0
3
2
1
timums
Figure 5. Typical kinetic trace showing the best fit to a single-exponential curve (dashed line).
reciprocal *O0 relaxatlcn tlme/8- 1
300 .
-40
42
41
43
temp./C
k2/10’
M-’s-’ 0.6 f 0.1 1.9 f 0.1 10.0 f 0.5
k-zls-’ 0 f 50 80 f 50 230 50
*
k’:’ 9.3 10.2 11.0
AG’fRT 1.3 6.3 4.1
at both longer and shorter times showed that there were no other significant relaxation processes occurring. The temperature dependences of the reciprocal relaxation time k and the relaxation amplitude are shown in Figure 6. The value of k is a maximum and is fairly insensitive to temperature at temperatures close to the solubilization phase boundary, where the droplet interactions are weakest, but decreases as the cloudpoint boundary is approached at higher temperatures. The amplitude of the relaxation increases as the rate decreases. This behavior is consistent with the following droplet behavior: At low temperatures, the droplets are present as single isolated species and hence relaxation at these temperatures corresponds to two droplets coming together to form a droplet pair. At higher temperatures, the aggregation of droplet clusters (rather than isolated droplets) is likely to be occurring to give droplet clusters containing more than two droplets. This process of “clustering of clusters” is expected to be slower than that of two droplets aggregating, since the initial concentration of droplet clusters will be lower than the original single-droplet concentration. We are concerned here with the primary process of two individual droplets sticking together, and hence the following data all refer to low temperatures close to the relevant solubilization phase boundary where only single droplets and droplet dimers should be present (Le. there are no droplet clusters containing more than two droplets) provided the droplet concentration is low enough. The dynamic equilibrium between droplets D and droplet encounter pairs D2 is represented by k
D + D & k-2 D,
60 I
(8)
Dz is formed with a second-order rate constant k2 and dissociates with a first-order rate constant k+ The equilibrium constant K (=[Dz]/[Dl2) is equal to k2/k+ The reciprocal relaxation time k is related to the droplet concentration by k = 4kz[droplets] + k-2 (9)
40
42
41
43
temp./C
Figure 6. Temperature dependence of (a) the reciprocal relaxation time k and (b) the relaxation amplitude for an R = 4 microemulsion con-
-~ taining 0.01396 mol d ~ n CIzES. curves, and hence the equilibrium behavior of the droplets at temperatures close to the solublization phase boundary can be reasonably well described in terms of a “hard-sphere” interaction potential. Kinetic Measurements. Figure 5 shows a representative kinetic trace together with the “best-fit” single-exponential curve calculated according to eq 7, where I,, Io, and B are the measured I, = Io exp(-kt)
+B
(7)
transmitted light intensities at times t , zero, and infinity, respectively, and k is the reciprocal relaxation time. As expected, the shape of the experimental curve corresponds to a decrease in light transmittance following the temperature jump (corresponding to a turbidity increase). The measured trace appears to be accurately described by a single-exponential curve, although there is no a priori reason to expect this. Hence, the relaxation of the turbidity can be characterized in terms of a single relaxation time (k-I) and an amplitude that is quoted here as a fraction of the inital signal (amplitude = (Io - B ) / I o ) . Monitoring the signal
Hence, the association and dissociation rate constants may be obtained from the slope and intercept, respectively, of a plot of k versus [droplets]. The dependences of the rates on droplet concentration for three different droplet sizes are shown in Figure 7. In all cases, the temperatures were chosen so as to be close to the solubilization phase boundary, where the rate constants correspond to two droplets aggregating, (Le. the first stage of the droplet clustering). Consistent with the kinetic scheme shown above, the reciprocal relaxation time shows a linear dependence on the droplet concentration for the three droplet sizes. The relaxation amplitude passes through a maximum with increasing droplet concentration as shown in Figure 8. This behavior parallels that of the equilibrium turbidity (Figure 4). The derived values of the rate constants for the different droplet sizes are given in Table I. The values of k2 may be compared with the diffusion-controlled limiting value kdcestimated by using the Smoluchowski equation:14 kdc = 8RT/37 (10) where 7 is the viscosity of the continuous solvent. The rate of droplet aggregation is slower than the diffusion-controlled limiting value for all the droplet sizes. The rate constant ratio kdc/k2varies from 1600 for R = 1 droplets to 110 for R = 4 droplets. This result has implications for the shape of the droplet interaction energy versus distance curve. If the interaction energy decreased monotonically as the droplets approached to form the droplet (14) Smoluchowski, M. Z . Phys. Chem.; Sroechiom. Venvundtschuffsl. 1917, 92, 129.
Fletcher and Holzwarth
2554 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 8000 1
1
1
8000 ' reclprocal rrlaxrtlon tlmek.1 4ooo
9,
Amplitude/%
4
*Oo0 0 0
I
0.1
0.05
0.15
0.25
0.2
0.3
0
0.05
[droplets]/mM 4000 3000
reciprocal relaxation 2000 time/s.l
loo0
15
A m p 11tu de/% 10
t
0
5
0.01
0.03
0.02
0.04
0
0.05
0.05
[droplets]/mM
I 15001
R=4 60
reciprocal relaxation 1000
-
- 0
0.25
20
.
--
time/s-1
0.2
25 I
1
1
0.1 0.15 [C12E5]/M
L/
0
1
I
2
3
4
dimer, then the rate would be expected to equal the diffusioncontrolled limit. The observation that the rate is significantly slower than the diffusion-controlled rate implies the droplets have to surmount an energy barrier before moving into the energy minimum at a separation distance corresponding to droplet dimer formation. The possible interaction energy versus separation distance curves are illustrated schematically in Figure 9. The free energy barrier to aggregation (AG*) is given by
k2 = kdc exp(-AG*/RT) The magnitude of AG*/RT decreases with increasing droplet size from 7.3 for R = 1 to 4.7 for R = 4, as shown in Table I. The rates of exchange of pyrene molecules between microemulsion droplets for this system have been measured recently.z Although these exchange rates have been assumed to be determined by the overall droplet coalescence rates: it should be noted that the measured exchange rate constants are generally higher than the droplet aggregation rate constants measured here, implying that processes other than droplet coalescence contribute to the rate of exchange of pyrene molecules between the microemulsion droplets. Furthermore, for the case of R = 1 droplets, the lower temperature phase boundary of the system appears to be shifted by the presence of pyrene, indicating that this probe molecule may perturb the droplet structure. Clearly, further work is required to resolve these ambiguities before attempting to separate the contributions to the overall droplet coalescence process.
0
0.2
I
/
30 .
[droplets]/mmM Figure 7. Plots of the reciprocal relaxation time k versus droplet concentration for the different R value microemulsions.
0.15
R=4
Amplitude/%
I
0.1 [Cl2E5]/M
0.01
0.02 0.03 [Cl2E5]/M
0.04
0.05
Figure 8. Plots of relaxation amplitude versus droplet concentration for the different R value microemulsions.
Whereas the equilibrium measurements provide information concerning the interaction energy at the equilibrium separation distance within the droplet encounter pair, the kinetic measurements yield the interaction energy at (presumably) longer separation distances corresponding to the transition state formed during the process of encounter-pair formation. The interaction through water between monolayers of CIZESadsorbed onto mica plates have been measured dire~t1y.l~The force versus separation distance curve for these virtually flat monolayers was found to be repulsive, with no minimum present at low temperatures. At high temperatures, the interaction curves developed a minimum, indicating the presence of an attractive force that increased with increasing temperature. The temperature-dependent force was found to extend to separation distances of 1-3 nm. A temperature-independent repulsive force extending over 100 nm was present and was ascribed to electrostatic repulsion. A combination of the long-range electrostatic force and the short-range attractive force gives an interaction energy versus distance curve of the type shown schematically in Figure 9b, which would account for the kinetic behavior observed here. However, the repulsive energybarrier component of the interaction curve for the microemulsion droplets m a y not be due to a long-range electrostatic force, as the electrostatic potential observed in the mica plate experiments was not associated with the nonionic surfactant but was thought to (15) Claesson, P. M.;Kjellander, R.;Stenius, P.; Christenson, H. K. J . Chem. Soc., Faraday Trans. I 1986, 82, 2135.
Aggregation Kinetics of Microemulsion Droplets
The Journal of Physical Chemistry, Vol. 95, No.6, 1991 2555 concentrations. These could not be obtained for. the smaller droplet sizes owing to the vanishingly small relaxation amplitudes under these conditions. However, it appears that the dissociation rate increases with increasing droplet size (Table I). For the case of the R = 4 droplets, it is possible to estimate the equilibrium constant K for D2 formation from the ratio of the association and dissociation rate constants. The value is (4.3 1.3) X los M-l. The value for hard spheres Kh, can be estimated by using the equation due to Fuoss:”
*
Khs = (4~/3)(2r#L
wpuatkndrtrno
L is Avogardro’s number. The calculated value is 1.9 X lo5 M-l and is the value expected at the solubilization phase boundary. The experimental value estimated from the kinetics is higher, since it refers to a temperature following the temperature jump (Le. approximately 1 OC higher than the temperature of the solubilization phase boundary), where the droplets have become more “sticky”. A value of K at the appropriate temperature can be estimated from the temperature dependence of the turbidity (Figure 2) as follows: From eq 5 , it can be seen that the turbidity at constant sample composition is proportional to the average particle volume. Since the limiting value of the turbidity at low temperatures corresponds to isolated droplets, the ratio of the turbidity to the low-temperature limiting value is equal to the average number of droplets present in a droplet cluster (n). Assuming that no droplet clusters contain more than two droplets, K is given by K = ( 1 - l/n)/Do(2/n - 1)2
separationdistance Figure 9. Schematic plots of interaction energy versus droplet separation
distance for (a) a diffusion-controlled aggregation process and (b) an energy-barrier-controlled process.
be due to a residual charge on the mica ~ 1 a t e s . IFurthermore, ~ the electrostatic component of the interactions, if present in the microemulsion systems, would be expected to be rather sensitive to the addition of small quantities of electrolyte. Contrary to this expectation, the microemulsion phase boundaries are insensitive to the addition of electrolytes until rather high concentrations are reached.I6 Hence, it remains a plausible possibility that the interaction forces between the approaching microemulsion droplets are restricted to short-range interactions possibly associated with an intermingling of the surfactant pentakis(ethy1ene glycol) chains. Such an interaction might be expected to be sensitive to the curvature of the approaching monolayers, as we observe here. Values of the dissociation rate constant k-2 could be estimated only approximately, since their accurate determination requires accurate values of the reciprocal relaxation time at low droplet
(13)
where Do is the droplet concentration in the absence of clustering. For this system, the final temperature following the temperature jump is estimated to be 42.1-42.6 “C, and n is in the range 1.1-1.44 when the droplet concentration is I .01 X 106 M. Thus, K is calculated to be in the range (0.14-2.0) X lo6 M-I. Hence, although the uncertainty is high, there appears to be no inconsistency between the values of the equilibrium constant derived from the kinetic and equilibrium measurements.
Conclusions The aggregation of o/w microemulsion droplets stabilized by CI2ESto give n o n c o a l d encounter pairs occurs with a rate that is significantly slower than the diffusion-controlled limiting value. This observation indicates there is a significant free energy barrier to this process that appears to be larger for smaller droplets. Acknowledgment. We thank Dr. Dieter Bauer for technical assistance and the Deutsche Forschungsgemeinschaft (DFG) for financial support. Registry No. CI2E5,3055-95-6; tetradecane, 629-59-4.
(16) Aveyard, R.; Lawless, T. A. J . Chcm. Sa.,Faraday Trans. 1 1986, 82, 295 I .
(12)
( I 7) Fuoss, R. M.J . Am. Chem. Soc. 1958,80, 5059.
