Langmuir 2000, 16, 8755-8762
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Kinetics of Oil Solubilization in Microemulsion Droplets. Mechanism of Oil Transport† Alex Evilevitch, Ulf Olsson, Bengt Jo¨nsson, and Håkan Wennerstro¨m* Division of Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, SE 22100 Lund, Sweden Received April 1, 2000. In Final Form: June 22, 2000 We have studied the kinetics of the solubilization of oil through a temperature jump into a droplet microemulsion phase in the system water-pentaethylene oxide dodecyl ether-decane at 25 °C. The initial state is formed by subjecting the equilibrium system at 25 °C to a temperature quench to 22, 20, and 14 °C, respectively. At this lower temperature, which at equilibrium corresponds to a two-phase system, oil droplets form and grow in size with increasing time. By varying the time between the quench and the T-jump, the size of the initial oil drops is varied in a systematic and known way in the relaxation study. The relaxation process is monitored by following the turbidity of the system. We find that for all the systems the relaxation back to equlibrium is much faster than the drop growth process observed after the temperature quench. This general observation is explained by realizing that the redissolution of the oil drops is analogous to the oil transfer phase, which in the quench experiment occurs prior to the Ostwald ripening phase. More significant is that we observed a qualitative transition in the relaxation behavior when the initial aggregate distribution is varied. In all cases we have the same initial temperature and overall composition and one population of many small droplets and fewer larger drops. The size of the larger drops only affects the relaxation in a quantitative way. If the small droplets are only slightly smaller than the equilibrium size, equilibration occurs through the diffusion of oil molecules in the bulk phase. When the initial droplets are sufficiently small, a new kinetic route is available where there is an efficient direct oil transfer between the small droplets and large drops. This allows for a fast relaxation of the oil distribution between the two populations of drops.
Introduction There are numerous technical applications of the solubilization of apolar molecules in surfactant aggregates in aqueous solution.1,2 The equilibrium aspects of the solubilization phenomenon are largely understood, while for the dynamic aspects, which are crucial in many applications, there remain several unsolved issues. The general rule is that the dynamics of solubilization and its reverse are transport limited and there are no barriers on the molecular level. However, which particular transport process dominates can vary from system to system. The three typical possibilities are that the rate of solubilization is determined by (i) the diffusion of single apolar molecules in the aqueous phase, (ii) diffusion of micellar aggregates followed by fusion/fission between aggregates, and (iii) diffusion of aggregates followed by exchange of solubilized molecules leaving the aggregates otherwise intact.3-5 All three alternatives have been observed, but no fully consistent picture has so far emerged.6-9 The balance between the different processes depends on a rather subtle balance between interaggregate interactions, the solubility of the apolar substace in water, and the interaction between the apolar substance and the polar groups of the † Part of the special issue “Colloid Science Matured, Four Colloid Scientists Turn 60 at the Millenium”.
(1) Falbe, J. Surfactants in Consumer Products: Theory, Technology and Application; Springer-Verlag: New York, 1986. (2) Schwunger, M. J.; Stickdorn, K.; Schoma¨cker, R. Chem. Rev. 1995, 95, 849. (3) Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Rippon, S.; Lubetkin, S. D.; Mulqueen, P. J. Langmuir 1999, 15, 4495. (4) Kabalnov, A. S. Langmuir 1994, 10, 680. (5) Brinck, J.; Jo¨nsson, B.; Tiberg, F. Langmuir 1998, 14, 5863. (6) Taisne, L.; Cabane, B. Langmuir 1998, 14, 4744. (7) Vollmer, D.; Strey, R.; Vollmer, J. J. Chem. Phys. 1997, 107, 3619. (8) Vollmer, D.; Strey, R.; Vollmer, J. J. Chem. Phys. 1997, 107, 3627. (9) Carroll, J. B. J. Colloid Interface Sci. 1981, 79, 126.
surfactant. To arrive at a detailed understanding of the dynamic behavior of such a three component system, it is usually a prerequisite that one has a detailed understanding of its equilibrium properties. The structure and phase behavior of nonionic microemulsions can be understood within the flexible surface model, with a curvature free energy ascribed to the surfactant loaded polar-apolar interface. The curvature free energy is an integral over the area of the surfactant film of a curvature free energy density, gc, so that
Gc )
∫gc dA
(1)
where gc in turn is usually developed to second order in the local principal curvatures c1 and c2. Normally one writes gc in the Helfrich form10
gc ) 2κ(H - H0)2 + κjK
(2)
Here the two variables are the mean curvature H ) (c1 + c2)/2 and the Gaussian curvature K ) c1c2. The equation also contains three system specific parameters, where H0 is the spontaneous curvature and the two modulii; the bending rigidity κ and the saddle splay constant κj. It is particularly convenient to use eq 2 when comparing surfactant films of constant topology, since the Gaussian curvature term then only contributes as a constant in eq 1 by virtue of the Gauss-Bonnet theorem. In the present paper we specifically chose to consider spherical aggregates only, where the topology of the system changes as the number of droplets changes. It is then more convenient to rewrite eq 2 and use instead of the Gaussian curvature the difference curvature, ∆c ) (c1 - c2)/2, as the (10) Helfrich, W. Z. Naturforsch. 1973, 28c, 693.
10.1021/la000511z CCC: $19.00 © 2000 American Chemical Society Published on Web 09/29/2000
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second variable. This yields11
gc ) 2κ′(H - c0)2 - κj(∆c)2
(3)
where we call c0 the preferred curvature
(2κ2κ+ κj)
c0 ) H0
(4)
since it represents the optimal curvature of the symmetrically curved film. (H0 is the optimal curvature of a cylindrically curved film.) The bending modulus κ′ is related to those of eq 2 by
κ′ ) κ + κj/2
(5)
For spherical aggregates ∆c ) 0, and by rewriting the expression for the curvature free energy, we need only consider the first term is eq 3. The curvature free energy of a spherical droplet of radius R is thus
Gc ) 8πκ′(1 - Rc0)2
(6)
which has its lowest value at Rc0 ) 1. This is our expression for the curvature free energy of a spherical microemulsion droplet, where we count curvature toward oil as positive. For the present system c0 shows a linear temperature dependence over a large temperature interval with12
c0(Τ) = 9.3 × 106(311.5 - (T/K)) (m-1)
(7)
On the lower phase boundary of the microemulsion phase (25 °C), the droplets coexist with pure oil and the excess oil’s chemical potential is consequently zero. From the free energy expression, eq 6, it follows that this corresponds to R ) c0-1, i.e., the model predicts spherical droplets having their optimal radius.13 In a series of investigations14-17 we have studied the equilibrium properties of the ternary system C12E5decane-water. Figure 1 shows a plane of the water rich part of the phase diagram for a fixed surfactant to oil volume ratio of 0.815. The characteristic property of these systems is the spontaneous curvature of the surfactant film which decreases with increasing temperature.18 This has the consequence that one observes a typical sequence of aggregate structures of spherical droplets at low T (2527 °C in Figure 1) and these grow in size and transform to a bicontinuous structure (28-32 °C) when the spontaneous curvature approaches zero. In this region there is a transition to a lamellar phase (33-34 °C in Figure 1) followed at higher temperatures by a sponge phase. Below 25 °C there is a two-phase region where one phase contains spherical droplets with a higher surfactant/oil ratio and the second phase is virtually pure oil. The oil swollen droplets occurring just above 25 °C repel one another, and the system behaves very similar to an ideal hard sphere system.17 If one subjects such a solution of swollen (11) Wennerstro¨m, H.; Anderson, D. M. In Statistical Mechanics and Differential Geometry of Micro-Structured Materials; Friedman, A., Nitsche J. C. C., Davis, H. T., Eds.; Springer-Verlag: Berlin, 1991. (12) Le, T. D.; Olsson, U.; Wennerstro¨m, H.; Schurtenberger, P. Phys. Rev. E 1999, 60, 4300. (13) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Addison-Wesley Publishing Co.: 1994; Vol. 90. (14) Olsson, U.; Schurtenberger, P. Langmuir 1993, 9, 3389. (15) Leaver, M. S.; Olsson, U. Langmuir 1994, 10, 3449. (16) Fukuda, K.; Olsson, U.; Wu¨rz, U. Langmuir 1994, 10, 3222. (17) Olsson, U.; Schurtenberger, P. Prog. Colloid Sci. 1997, 104, 157. (18) Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1994, 49, 113.
droplets to a rapid temperature quench into the two-phase area, a separate macroscopic oil phase will eventually form. We have previously also studied the kinetics of the initial phase of this phase separation process,19-21 which in fact is the inverse of a solubilization. Initially there is a nucleation phase where a minority of the initially equal size droplets grow to a larger size, as illustrated in Figure 1. For small temperature quenches the formation of the larger droplets is an activated process with a small nucleation rate, while for deep quenches the formation of the larger droplets is more akin to a spinodal decomposition. In both cases the initial nucleation phase is followed by a transfer phase where oil diffuses from small droplets to the newly nucleated drops. The net transfer of oil from small (shrinking) droplets to the larger (growing) drops essentially terminates when the small droplets reach their (nearly) equilibrium size. From there on, however, the population of large drops continues to evolve through classical Ostwald ripening.21 In the present paper we present studies of the reverse process where a sample that has initially been temperature quenched into the two-phase area and then left for some time to allow the nucleation and growth of the larger droplets. At a controlled time after the quench the temperature is reversed and brought back to the initial conditions. In this state the large drops are unstable and gradually shrink in size. We follow the approach to equilibrium, the resolubilization, by measuring the turbidity of the solution. By studying the dynamics of the solubilization process with an initial state consisting of small droplets of a radius in the range of 12-40 nm, we eliminate the rate-limiting character of bulk transport for a solubilization of a macroscopic domain. Thus the experiments become sensitive to the molecular nature of the transfer process. Experimental Section Materials. The nonionic surfactant pentaethylene glycol dodecyl ether (C12E5) was obtained from Nikkol Ltd. Tokyo, and decane (99%) was from Sigma. These chemicals were used as received. The samples were prepared using Millipore filtered water. Phase Diagram Determination. Samples for the phase diagram were prepared from two stock solutions of φs ) 4.5% C12E5, φo ) 5.5% decane, water, where φ is the volume fraction. Theses stock solutions were mixed in 5 mm NMR tubes (≈400 µL) that were flame sealed. The NMR self-diffusion experiments were made with the same samples. Phase boundary temperatures were determined by visual inspection in transmitted light. For the microemulsionn˜oil equilibrium, the kinetics of the phase separation is slow, while the reverse process is faster (of the order of minutes). Therefore the L1/(L1 + O) phase boundary was determined by solubilization upon increasing the temperature from the microemulsion-excess oil equilibrium. The kinetics involved in the microemulsion-excess water phase boundary is fast. The volume fractions for the samples were calculated using the following densities (g/cm3): 0.967 (C12E5); 0.73 (decane); 0.997 (H2O). Self-Diffusion Measurements. The proton self-diffusion studies were performed on a Bruker 200 MHz NMR spectrometer using the Fourier transform pulsed gradient spin-echo (FTPGSE) technique. The temperature was kept constant at 25 °C and controlled by a thermostated air-flow. The temperature control unit was calibrated with a copper-constantan thermocouple. (19) Morris, J.; Olsson, U.; Wennerstro¨m, H. Langmuir 1997, 13, 606. (20) Wennerstro¨m, H.; Morris, J.; Olsson, U. Langmuir 1997, 13, 6972. (21) Egelhaaf, S.; Olsson, U.; Schurtenberger, P.; Morris, J.; Wennerstro¨m, H. Phys. Rev. E 1999, 60, 5681.
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Figure 1. Partial phase diagram of the system C12E5-decane-water. The surfactant/oil volume ratio (φs/φo) is fixed constant at 0.815. The labeled phases are oil-in-water microemuslion (L1), oil (O), and lamella (LR). The figure also shows a schematic diagram of two cases for the structural changes involved at the T-jump transition from the L1 + O region to the L1 phase boundary. T-jumps from 14 to 25 °C and from 22 to 25 °C are shown with arrows. Turbidity Measurements. L1 f L1 + O, T-Quench. A sample at a concentration of φ ) 10% of C12E5 and decane at the volume ratio 0.815 in water has been studied. Turbidity was measured at λ ) 406 nm using a Perkin-Elmer Lambda 14 UV/visible spectrophotometer, containing a thermostated sample cell holder. The temperature was controlled by a Haake circulating water bath, which kept the temperature constant within (0.1 °C. The temperature quenches were carried out with the help of two water baths, where one was kept at the L1/(L1 + O) phase boundary temperature of 25 ˚C and the other one at a temperature below 25 ˚C, in the two-phase region. The sample was first preequilibrated at 25 ˚C for a couple of hours, and the turbidity was measured. Then at time zero, the sample was placed in the other water bath at a lower temperature and was kept there for the entire time, while the turbidity of the sample was followed with 5-10 min time intervals. The sample was temperature quenched from 25 ˚C down to 14, 20, and 22 ˚C. Since, the entire sample was always immersed into the thermostated water, temperature gradients within the sample could be avoided. All the turbidity measurements were performed in 1 mm quartz cells. L1 + O f L1, T-Jump. The phase separating oil was resolubilized by bringing the sample back to 25 ˚C. The sample was first placed in the 25 ˚C water bath for 20 s (the time it takes to heat up the sample homogeneously). The temperature in the
quartz cell was controlled with a coppar-constantan thermocouple. Then the sample was placed in the 25 ˚C thermostated cell holder in the spectrophotometer, and the turbidity was traced continually with time.
Results Turbidity Measurements. After a sample had equilibrated at 25 °C, it was directly temperature quenched at time zero down to one of three different temperatures in the unstable two-phase region: 14, 20, and 22 °C. Turbidity was then measured as a function of time. An almost linear increase in turbidity was observed directly after the temperature drop for all the samples due to the instability. The turbidity increases notably faster for the deep quench; compare Figure 2a,b. Parts a and b of Figure 2 show the time evaluation of turbidity as a function of time for the samples that were temporarily kept at 22 and 14 °C, respectively. In Figure 2a the sample was heated to 25 °C after it was kept at 22 °C for 1 or 2 h, respectively. As it was described above, the sample was first immersed into 25 °C water for 20 s (to ensure rapid and homogeneous heating) and then it was placed in a thermostated cell holder in the spectrophotometer and turbidity was reg-
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Figure 3. Variation of turbidity with time for the temperature jumps from 14 °C (b), 20 °C (9), and 22 °C (2) to 25 °C after 2 h. The dotted line is the extrapolation to the time of the temperature jump. The solid lines are the relaxation rates obtained from the model calculations.
f 25 °C jump, while for the 14 °C f 25 °C jump turbidity has dropped from a very high value of 4.9 cm-1 down to an almost noise level at the baseline value of 0.27 cm-1 after only 20 s. Turbidity and Drop Size. Turbidity is the measure of the reduction in intensity of light as it passes through the sample, due to the scattering process. The fraction of the incident light scattered in all directions by a collection of particles is obtained by integrating the angular intensity function I(θ) over the surface of a sphere and dividing by the initial intensity, I0,u:22 Figure 2. (a, top) Variation of turbidity with time for the C12E5decane system (φs/φo ) 0.815) at φ ) 10% droplets. The figure shows a turbidity increase for the temperature quench from 25 to 22 °C at time zero, together with a turbidity decrease for the reverse temperature jump from 22 to 25 °C after 60 and 120 min at 22 °C, respectively. The dotted lines are the extrapolations to the time of the temperature jump. (b, bottom) Variation of turbidity with time for the C12E5-decane system (φs/φo ) 0.815) at φ ) 10% droplets. The figure shows a turbidity increase for the temperature quench from 25 to 14 °C at time zero, together with a turbidity decrease for the reverse temperature jump from 14 to 25 °C after 30, 60, 75, 105, and 135 min at 14 °C, respectively. The dotted lines are the extrapolations to the time of the temperature jump.
istered with time. From the figure one can observe that some of the data points are lost during the first 20 s, yet we see that the turbidity drop is only around 20% within the first 20 s, so that the measured turbidity could be extrapolated back to the time of the temperature jump (see the dotted lines). From the figure it is obvious that the resolubilization process is much faster than the phase separation. It took 15 min for turbidity to relax back to the initial value it had before the sample was left to phase separate for 2 h at 22 °C. An even more dramatic turbidity drop could be observed for the sample which was heated back to 25 °C after being kept at 14 °C for 30, 60, 75, 105, and 135 min; see Figure 2b. Here on the contrary to the sample in Figure 2a, turbidity drops a factor of 10 during the first 20 s after the 14 °C f 25 °C T-jump (see the extrapolated dotted lines). These observations are summarized in Figure 3, where the turbidity is plotted versus time for the temperature jumps to 25 °C after 2 h at 14, 20, and 22 °C. One can see that it takes ≈10 min for resolubilization to complete for the 22 °C f 25 °C jump. It takes ≈15 min for the 20 °C
τ)
π 2π I(θ)r2 sin θ dθ dφ ∫θ)0 ∫φ)0
1 Ι0,u
(8)
When R , λ, where λ is the wavelength of light, one can use the Rayleigh-Gans-Debye (RGD) scattering approximation
I(θ)r2 ) P(θ)(Rθ) I0,u
(9)
where P(θ) is the so-called form factor, which for small radii can be described by a Guinier approximation:
P(θ) ) 1 -
(qRg)2 3
(10)
with scattering angle θ, scattering vector q, and a radius of gyration, Rg, which for a uniform sphere is given by the following: Rg2 ) 3/5R2. (Rθ) is the Rayleigh ratio, given by
Rθ ) Rπ/2(1 + cos2 θ)
(11)
and
Rπ/2 )
2π2nw2 dn 2 VsphereφS(0) λ4 dφ
( )
(12)
Here S(0) is the structure factor which we assume to be q independent in the light scattering q-range. Vsphere is the volume of the droplets, and φ is the total volume fraction of scattering particles and is given by the sum of (22) Hunter, R. J. Introduction to Modern Colloid Science; Oxford University Press Inc.: New York, 1993.
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surfactant’s and oil’s volume fractions, φ ) φs + φo and is related to hydrocarbon volume fraction (see Appendix) φhc by φ ) 1.29φhc. nw ) 1.3417 is the refractive index of the solvent (H2O). S(0) for a hard sphere system, most accurately given by Carnahan and Starling,23 is
S(0) )
(1 - φ)4 (1 + 2φ)2 - φ3(4 - φ)
(13)
Finally, the refractive index increment dn/dφ can be calculated from modeling the droplets as core-shell spheres,14,24
(
)
φs dn 3 1/2 φo ) w A + B dφ 2 φ φ
(14)
with Α ) (o - w)/(o + 2w) and B ) (s - w)/(s + 2w). φs and φo can be calculated from Rhc and φhc using equations in the Appendix. The dielectric constants used for calculations were w ) 1.800159, s ) 2.137739, and o ) 2.02265. Combining these equations and performing the integration in eq 8 give the expression for turbidity:
[
τ ) Rπ/2
]
128 R2π3 16 π3 15 λ2
(15)
In the analysis we have assumed that the turbidity is given by the sum of the turbidity from small droplets and from big drops,τtot. ) τsmall + τbig, thus neglecting any crossterms. This simplifying assumption gives an uncertainty in the size determination when size differences are small. However, far from equilibrium the turbidity is dominated by the large drops and we have τtot. ≈ τbig. Discussion We have studied the solubilization of oil into microemulsion droplets by varying the initial condition where the oil is present in drops of varying size. Another aspect is that also the initial microemulsion droplets have a varying size depending on the temperature prior to the jump to 25 °C. Figures 2 and 3 reveal two major qualitative results. In all cases we find that the resolubilization is substantially faster than the phase separation process induced by a temperature quench. In the latter case we have established that the rate is determined by an Ostwald ripening process,21 where individual oil molecules diffuse from smaller to larger drops. The difference in rate is much larger than can be accounted for by the temperature effect of diffusion coefficients and solubilities, and we can thus conclude that the dynamics of the solubilization process is determined by a different process than the phase separation. We are apparently outside the (linear) regime where the forward and backward processes necessarily occur through the same molecular path. A second finding is that the solubilization process occurs faster for the deep quench case where the initial drops are larger and the equlibrium state is reached in a shorter time, although one is much further from the equilibrium state at the start of the process. The most obvious explanation would be that one has two different kinetic paths leading to the equilibrium state. However, to arrive at a mechanistic interpretation of the data, we must first present a more detailed description of the system. (23) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (24) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969.
Figure 4. Time dependence of the mean big drop radius Rbig estimated from the turbidity data for the 25 f 14 °C temperature quench of the C12E5-decane system (φs/φo ) 0.815) at φ ) 10% droplets, compared with the values determined previously17 using the SANS technique.
At all times, we have drop(let)s of oil covered by a monolayer of the surfactant C12E5 and these drop(let)s are diffusing in an aqueous medium containing some dissolved (monomeric) surfactant (5.8 × 10-5 M) and oil (3.6 × 10-7 M) molecules, which is only a very small fraction of the total amounts of the oil and the surfactant. During the experiment the size distribution of the drops changes, triggered by the changes in temperature. There are two important constraints on the size distribution. Both the total area A and the total volume V are given by the composition. Furthermore the surfactant is soluble enough to ensure a more rapid equilibration than the oil. We will also assume that the number of small droplets equilibrates fast relative to the slowest process. During the experiments we have a distinctly bimodal distribution with one population dominating in the number of smaller droplets in the size range 4.5-7.5 nm in radius and a population, containing up to 60% of the oil, of larger drops 10-40 nm in radius. The distributions around the two most probable sizes are rather narrow, and for the purpose of understanding the mechanism of the solubilization process, it is sufficient to consider simply two different classes of droplets of radii Rsmall and Rbig of numbers Nsmall and Nbig, respectively. The conservation of area and volume then leads to the equations
{
A ) 4πNsmallRsmall2 + 4πNbigRbig2 V)
4/3πNsmallRsmall3
+
4/3πNbigRbig3
(16) (17)
With four unknowns we need two additional pieces of information in addition to eqs 16 and 17. The measurement of turbidity can provide one, and the remaining may be obtained from assumptions based on the established model of the equilibrium properties of the system. For example, we may assume that the population of small droplets has reached its equilibrium radius Req(T) at the given temperature. We have previously studied the Ostwald ripening of a quenched system by time-resolved small-angle neutron scattering (SANS).21 For experimental reasons we used heavy water as the solvent, and this affects the equilibrium properties slightly. The quench from 25 to 14 °C in the present study was chosen to mimic the conditions of the SANS experiment. In Figure 4 we compare the time evolution of the mean radius of the large drops determined from the turbidity measurements with the corresponding values determined previously,21 using the SANS tech-
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nique. In both systems the radius increases as the cube root of time but the two curves are slightly displaced. The discrepancy might be explained by the fact that we have not properly compensated for the isotope effect of the equilibrium properties when choosing the conditions for the experiment in ordinary water, but it might also be due to differences in the details of the experimental procedure. In the SANS experiments the radius of the small droplets was found to be close to their equilibrium size of 4.5 nm at the temperature of the sample (here and below equilibrium size refers to the hydrocarbon radius of the droplets, Rhc; see Appendix). This is also consistent with the observed Ostwald ripening kinetics, which presupposes a constant volume of the rearranging drop population. For the samples that have been exposed to a less deep temperature quench, it is more problematic to independently determine the mean sizes of small and large droplets. We make the assumption that also in these cases the small droplets reach a size consistent with their equilibrium size before we subject the sample to a temperature jump back to 25 °C. At 22 °C the equilibrium size (hydrocarbon radius) is 6.1 nm and at 20 °C 5.4 nm. The growth process prior to the temperature jump is governed by diffusion of single oil molecules in the aqueous phase.21 It is then the obvious alternative that the equilibration of the reverse process after the jump involves the same elementary process. The situation just after the T-jump is that the small droplets have a higher curvature than what is optimal at the new temperature. Oil molecules in the solvent will quickly be solubilized by these drops, establishing, within a short time, a lower value of the monomer concentration in the bulk. Concomitantly the larger drops are destabilized and refurnish the bulk solution with oil molecules. A (quasi) steady state is established with a flow of oil from large drops to small droplets. On the basis of this general picture, we have formulated a quantitative model for the equilibration based on a cell model with one large drop and many small droplets per cell. The details of the model and the calculations are explained in the Appendix. In Figure 3 we show the simulated turbidity for the three different temperature jumps. With this simple model we find a reasonable agreement with the experimental results for the jumps from 20 to 22 °C. The model predicts slower resolubilization kinetics for the T-jumps with increasing ∆T, and this trend is also experimentally observed when going from 22 to 20 °C. We thus conclude that we can understand the resolubilzation from the 20 and 22 °C experiments in terms of simple oil monomer diffusion, similar to Ostwald ripening. The fact that this process is faster than the turbidity increase with time prior to the temperature jump can be ascribed to larger differences in aggregate size, resulting in a larger concentration gradient. At the lower temperature, the turbidity increase with time is related to the Ostwald ripening occurring within the population of large drops. The resolubilzation, on the other hand occurs with oil transfer from the large to the small drops. However, turning to the 14 °C experiment, we find a completely different behavior. Here, the relaxation of the turbidity is very fast. The diffusion model predicts that the turbidity should reach the equilibrium value at 25 °C after 45 min. Experimentally we find that the relaxation is complete already after a minute or so, so that the relaxation of the 14 °C size distribution is significantly faster compared to the 20 and 22 °C ones, which is opposite from the prediction of the model. When the temperature jump starts from the lower value of 14 °C, most of the relaxation to equilibrium occurs on
Evilevitch et al.
Figure 5. Two-dimensional phase diagram cut of the system C12E5-deacane-water, at φs ) 4.49% and φo varied between 0 and 5.5%. The phase diagram forms a one phase microemulsion channel (L) over the whole range of decane concentrations. At high temperatures water-in-oil microemulsion is formed with excess water (L + W). At low temperatures oil excess is expelled out of oil-in-water microemulsion droplets (L + O). The insert shows a linear dependence of the inverted radius of microemulsion droplets on the L/(L + O) phase boundary temperature, where filled circles are the radius values calculated from the chemical composition and the solid line is calculated with eq 7 in the text.
a time scale shorter than we can resolve in the present form of the experiment. We concluded above that the process is clearly faster than can be accounted for by considering monomer diffusion in the bulk as determining the rate. The alternative to this possibility is that there is a direct transfer of oil between drops, either by drop coalescence or by drop encounters and subsequent oil transfer before the drops separate again. To see why such a possibility might vary with the size of the small drops, it is useful to consider the equilibrium properties of the system. The succession of phases in the system is best understood on the basis of the temperature dependence of the spontaneous curvature of the film, which decreases with increasing temperature. In Figure 5 we show the phase plane at constant surfactant concentration of φs ) 4.49%, which is the concentration we have chosen for the kinetic experiments. The figure insert gives the inverted radius of the microemulsion droplets at the lower phase boundary where oil separates out, calculated as Rhc ) 3ls(φo/φs + 1/2); see Appendix. The insert also displays a line calculated with eq 7, showing a linear relationship between the inverted radius, R-1(T), and the lower phase boundary temperature. At 14 °C the figure shows that the surfactant-to-oil ratio in the small droplets has increased to φs/φo ) 2.25, corresponding to a droplet radius of approximately 4 nm at equilibrium, and prior to the temperature jump one is close to this state of the small drops in the kinetic experiments. If we neglect the influence of the large oil drops, the surfactant and oil of the small droplets will after the temperature jump still be in the one phase region. However in previous studies25,26 we have established that near the upper end of the isotropic one phase region the equilibrium structure is no longer spherical droplets but the aggregates have grown to ultimately form a bicontinuous structure. This transition from discrete spheres to a bicontinuous aggregate is most (25) Olsson, U.; Shinoda, K.; Lindman, B. J. Phys. Chem. 1986, 90, 4083. (26) Leaver, M. S.; Olsson, U.; Wennersto¨m, H.; Strey, R. J. Phys. II 1994, 4, 515.
Oil Solubilization in Microemulsion Droplets
Figure 6. Self-diffusion coefficients of decane (circles) and C12E5 (triangles) of the system C12E5-decane-water with φs ) 4.49% and 0 e φο e 5.5% measured at temperature 25 °C.
easily demonstrated by self-diffusion measurements,27 and Figure 6 shows the measured self-diffusion coefficients of the oil and the surfactant at 25 °C for the present system. Below φo ) 2%, the decane and C12E5 have very different diffusion coefficients which strongly indicates the presence of oil continuous paths for the decane translation. Above φo ) 2% the two species have the same translational motion, indicating that they diffuse in a common aggregate. The low value of the diffusion constant in the range 2% < φo < 3.5% shows that this aggregate is larger than the minimal sphere. At the higher concentrations φo > 3.5% the observed value of Ds is the one expected for spherical droplets of the size determined by the composition.28 We now return to the fast resolubilization observed when jumping in temperature from 14 °C. The sizes of the initial big drops are not significantly different in the different temperature jump experiments. The difference in kinetic behavior therefore has to be related to a difference in the behavior of the small droplets. Just before the temperature jump these are spherical with a size near the equilibrium one at the given temperature. As illustrated in Figure 5 the equilibrium radius of the droplets is determined by the preferred curvature of the surfactant monolayer, and their composition, i.e., their surfactantto-oil ratio, depends on the radius according to Rhc ) 3ls(φo/ φs + 1/2). As seen in Figure 6 the microemulsion structure at 25 °C depends strongly on the surfactant-to-oil ratio. At 14 °C, the equilibrium droplets have a surfactant-tooil ratio of about 2.2. When these droplets are brought to 25 °C, they would by themselves coalesce to form larger structures of a type found near the droplet-to-bicontinuous transition. On the other hand when the equilibrium droplets at 20 and 22 °C are brought to 25 °C, there is essentially no structural change in a constrained equilibrium. The droplets remain spherical with the same size as initially. Fast structural relaxations of oil droplets have indeed been measured in temperature jumps. Iodine laser temperature jump (ILTJ) experiments performed on oil-inwater microemulsions with different nonionic surfactants29,30 have shown that structural relaxation times, after a rapid temperature increase, can occur on the (27) Wennerstro¨m, W.; So¨derman, O.; Olsson, U.; Lindman, B. Colloids Surf. A 1997, 123-124, 13. (28) Olsson, U.; Nagai, K.; Wennerstro¨m, H. J. Phys. Chem. 1988, 92, 6675. (29) Fletcher, P. D. I.; Holzwarth, J. F. J. Phys. Chem. 1991, 95, 2550.
Langmuir, Vol. 16, No. 23, 2000 8761
millisecond time scale. We have verified that equilibration is fast also for the larger temperature jumps involved in the present experiments. Thus we conclude that the rapid resolubilization when jumping from 14 to 25 °C correlates with the fast aggregation of small droplets into larger nonspherical assemblies. The big drops interfere with this aggregation process, and we can identify two different possibilities for the path to the final equilibrium. One possibility is that some of the small droplets interact directly with the large drops prior to aggregation. Such encounter could result in the transfer of oil and generating two drops of approximately equal size. The other major alternative is that there is first an equilibration within the population of the initial small droplets and the further relaxation involves the transfer of oil from the drops rich in oil to the nonspherical aggregates that are poor in oil. This process should also involve a fragmentation of the aggregates. At present we have no basis for discriminating between these two possibilities since the essential kinetic information is lost in the dead time of our experimental procedure, and further work using more rapid T-jump methods is needed to resolve this issue. Conclusions We have studied the dynamics of the solubilization of oil in a droplet microemulsion by following the turbidity of the system after a temperature jump into an equilibrium microemulsion phase creating an initial state that can be seen as an emulsion. By controlling the time of evolution at a lower temperature where the system phase separates, we have systematically varied the initial aggregate distribution function. Additionally all aggregates are small, less than 50 nm radius, which eliminates transport between bulk systems as the rate determining factor. The main finding is that we observe a transition from a kinetic behavior determined by bulk diffusion of monomers to one where a direct transfer of oil between small droplets and large drops dominate. The latter is much more efficient and leads to a more rapid relaxation, and samples that are further displaced from equilibrium actually show the most rapid relaxation. The switch in the relaxation pathway can be interpreted in terms of the curvature energy of the surfactant film. For droplets with a radius substantially smaller than the preferred curvature there appears an attractive component to the droplet-droplet interaction.31 The kinetic experiments show that also the larger drops rich in oil interact sufficiently favorably with the droplets or aggregates poor in oil to allow for direct transfer of oil between the two populations. The study thus reveals that in one and the same system at given temperature and total composition the relaxation rate can vary by at least an order of magnitude and that the route to equilibrium depends on the initial aggregate distribution function. Appendix Calculation of Resolubilization Rates. Resolubilization rates were calculated within a cell model involving area and volume conservation assumptions described above (eqs 16 and 17) plus an assumption that the concentration of oil monomers in the cell is the same everywhere in the cell, due to fast concentration equili(30) Morris, J. S. The structure and kinetics of oil-in-water microemulsions stabilized by nonionic surfactants, Ph.D. Thesis; University of Hull: Hull, U.K., 1995. Fletcher, P. D. I.; Morris, J. S. Colloids Surf. A 1995, 98, 147. (31) Aurvay, L. J. Phys. 1985, L 163.
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bration, and is given by the mean oil concentration dictated in most cases by the small droplets. Calculations were performed for a O ) 10% sample with droplet radius Rhc ) 75 Å at 25 °C (at the L1/(L1 + O) phase boundary) as it was previously estimated by SANS measurements17 and is defined as the hydrocarbon radius:
3(φo + 1/2φs)ls Rhc ) φs
4/3π(Rbig3 + NsmallRsmall3) φhc
dR Dνm(Cm - C(R)) ) dt R
(A1)
where φ are the volume fractions of surfactant (φs) and oil (φo) and ls is the length of the surfactant molecule (14.5 Å).14 The hydrocarbon volume fraction is defined as φhc ) φo + 1/2φs. Knowing the equilibrium radius, Rhc, the number of droplets, N, and thus the total area and volume of the droplets can be calculated and are kept constant. Now, fixing one of the parameters Nsmall, Rsmall, or Rbig in eqs 16 and 17, we will obtain the other two. The volume of one cell is equal to
Vcell )
in water),33 and C(R) is concentration of monomers at the droplet’s surface.34 For the case of classical Ostwald ripening,35 R/δ , 1 when the distance between particles is much larger than their radius, we obtain a well-known expression:
Here C(R), the concentration of oil monomers at the surface of the sphere, can be calculated using an equation, initially proposed by Kelvin:36,37
{ }
C(R) ) C(∞) exp
γ)
(A3)
The model describes the resolubilization process by a diffusion-controlled transport of the oil monomers from the big drop in the center of the cell to the small droplets via the continuous water phase. The derivation of equations is not treated in detail here. Solving Fick’s law of diffusion in the steady-state case, we obtain an expression for the rate of oil monomer transport from the surface of a sphere of radius R, expressed in terms of the radius variation of the sphere:
R Dνm(Cm - C(R)) dR ) 1+ dt δ R
(
)
{
(A7)
}
4κ′νm(1 - Rc0)c0 kT R2
C(R) ) C(∞) exp -
(A8)
Since, the net flow of oil monomers should be equal to zero, the mean oil concentration in the bulk phase can be calculated be setting the equality A9 and combining it with the result in eq A5:
Jbigout - NsmallJsmallin ) 0
(A9)
Finally, R(t) can be calculated by an iteration of eq A10 with some set time interval, ∆t,
R(t + ∆t) ) R(t) +
(A4)
where δ is the thickness of a stagnant boundary layer along which diffusion takes place. D is the diffusion coefficient of the oil monomers (9.5 × 10-10 m2/s for decane in water at 25 °C),32 vm is the molar volume of a monomer (323 × 10-30 m3 for decane), Cm is the mean concentration of monomers in the bulk (2.16797 × 1020 m-3 for decane (32) Hayduk, W.; Laudie, H. AIChE J. 1974, 20, 611.
Gc 8πκ′(1 - Rc0)2 ) A 4πR2
where c0 is the so-called preferred curvature and κ′ is the bending modulus described by κ′ ) κ + κj/2.20 Inserting equation for γ into eq A6 gives equation for calculation of the oil’s surface concentration that we have been using in our calculations,
4/3πRbig3 Vcell
4/3πNsmallRsmall3 ) Vcell
(A6)
(A2)
and
φhcsmall
2γνm RkT
where γ is the interfacial tension, C(∞) is the bulk phase solubility of oil (at R ) ∞). In the present case the curvature free energy, Gc, expresses a size dependent surface tension and γ can be expressed as
Using these equations, hydrocarbon volume fractions of big and respectively small droplets can be calculated:
φhcbig )
(A5)
dR ∆t dt
(A10)
LA000511Z (33) McAuliffe, C. J. Phys. Chem. 1966, 70, 1267. (34) Sugimoto, T. AIChE J. 1978, 24, 1125. (35) Wagner, C. Z. Elektrochem. 1961, 65, 581. (36) Thomson, W. (Lord Kelvin). Proceedings of the Royal Society; Edinburgh, 1871. (37) Kabalnov, A. S.; Shchukin, E. D. Adv. Colloid Interface Sci. 1992, 38, 69.
