Molecular Transport in a Nonequilibrium Droplet Microemulsion System

Langmuir 2001, 17, 6893-6904

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Molecular Transport in a Nonequilibrium Droplet Microemulsion System Alex Evilevitch,* Bengt Jo¨nsson, Ulf Olsson, 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 June 13, 2001. In Final Form: August 13, 2001 In this paper, we consider the problem of oil molecular transport between nonequilibrium microemulsion oil droplets in water. In particular, we have investigated the kinetics of solubilization of big oil drops by smaller microemulsion droplets transforming a bimodal size distribution into an equilibrium microemulsion with a unimodal size distribution. The process involves the diffusion of oil monomers across the aqueous solvent. Solubilization experiments are presented on a well-characterized nonionic microemulsion system, where interfacial area and dispersed volume are conserved and where the excess oil chemical potential is dominated by interfacial curvature energy. An analysis of the experiments shows that the solubilization kinetics depend significantly on the concentrations and sizes of big and small droplets. We formulate a quantitative model for the solubilization kinetics where the effects of size and concentrations are treated within the framework of a cell model. A quantitative agreement between model and experiment is obtained, and the analysis also shows that the majority of oil monomers are captured by small droplets already in the vicinity of the big drop surface when the concentration of small droplets is high.

1. Introduction Microemulsions are thermodynamically stable liquid mixtures of water, oil, and surfactant. While being macroscopically homogeneous, they are locally structured into polar and apolar domains separated by a surfactantrich dividing surface. Because of the many ways of dividing space, microemulsions may show a large variation in microstructure. Under certain conditions, it is possible to stabilize spherical droplets of, for example, oil in water with a low polydispersity and a concentration invariant size. This occurs when the spontaneous curvature of the surfactant film is finite but not too low and the system is saturated with the dispersed oil.1,2 Droplets can exchange material through coalescence and break up or exchange via the small but finite solubility in the continuous solvent.3-8 The former is rare when the droplets are covered by a dense (saturated) surfactant layer. The latter is always occurring and, for example, leads to Ostwald ripening of emulsions. There are numerous technical applications of the solubilization of apolar molecules in surfactant aggregates in aqueous solution.9,10 The equilibrium aspects of the solubilization phenomenon are largely understood, while for the dynamic aspects, which are crucial in many applications, there remains several unsolved issues. The * To whom correspondence should be addressed. E-mail: [email protected]. (1) Olsson, U.; Wennerstro¨m, H. Adv Colloid Interface Sci. 1994, 49, 113. (2) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Addison-Wesley Publishing Company: Reading, MA, 1994; Vol. 90. (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) Falbe, J. Surfactants in Consumer Products: Theory, Technology, and Application; Springer-Verlag: New York, 1986. (10) Schwunger, M. J.; Stickdorn, K.; Schoma¨cker, R. Chem. Rev. 1995, 95, 849.

general rule is that the dynamics of solubilization and its reverse is transport-limited, and there are no barriers on the molecular level. However, the particular transport process that 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.3,4,5,11 All three alternatives have been observed, but no fully consistent picture has so far emerged.6-8,12-15 Solubilization of emulsion droplets by surfactant solutions has been studied by several investigators.3,11,12,13 Kabalnov and Weers11 have analyzed the solubilization of infinitely diluted emulsion droplets by a solution of surfactant micelles, where the authors present a model for the flow to and from a macroscopic interface. Using a well-characterized nonionic surfactant-wateroil system, we have recently investigated the phase separation of excess oil from an oil-in-water droplet microemulsion.16-18 The nonionic surfactants of the ethylene oxide type have a strongly temperature-dependent spontaneous curvature, and phase separation is conveniently generated by lowering the temperature. The process involves the disproportionation, by nucleation and growth, of a few droplets of the population that grow in (11) Kabalnov, A.; Weers, J. Langmuir 1996, 12, 3442. (12) Carroll, J. B. J. Colloid Interface Sci. 1981, 79, 126. (13) McClements, D. J.; Dungan, S. R. Colloids Surf., A 1995, 104, 127. (14) Soma, J.; Papadopoulos, K. D. J. Colloid Interface Sci. 1996, 181, 225. (15) Taisne, L.; Walstra, P.; Cabane, B. J. Colloid Interface Sci. 1996, 184. (16) Morris, J.; Olsson, U.; Wennerstro¨m, H. Langmuir 1997, 13, 606. (17) Wennerstro¨m, H.; Morris, J.; Olsson, U. Langmuir 1997, 13, 6972. (18) Egelhaaf, S.; Olsson, U.; Schurtenberger, P.; Morris, J.; Wennerstro¨m, H. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 60, 5681.

10.1021/la010899d CCC: $20.00 © 2001 American Chemical Society Published on Web 10/02/2001

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size, allowing the majority of droplets to shrink. When the shrinking droplets have reached their new equilibrium size, the big drops continue to evolve by Ostwald ripening. In this work, we consider the reverse process where oil is solubilized by the microemulsion droplets. A droplet microemulsion sample is initially temperature-quenched from the phase boundary into the two-phase area, where the droplets at equilibrium coexist with excess oil and then are left for some time to allow the nucleation and growth of larger droplets. Thus, the original unimodal size distribution is split into a bimodal size distribution of larger and smaller droplets as compared to the original average size. At a certain time after the quench, the temperature is reversed back to the initial temperature. As a result, the system reverts back into the original unimodal size distribution by having the large drops solubilized by the small droplets. In a previous analysis19 of the resolubilization kinetics, we assumed, as a first attempt, the infinite dilution limit (IDL), that is, where both big and small droplets are infinitely dilute. This case is particularly simple because we can utilize the analytical framework outlined in the original analysis of Ostwald ripening by Lifshitz, and Slyozov20 and Wagner.21 The analysis, however, showed significant effects of finite concentrations (total volume fraction φ ≈ 0.1), with a higher solubilization rate as compared to the IDL. In this paper, we therefore extend our analysis to incorporate also the effects of the concentrations of big and small droplets. This is done using a cell model approach. The paper is organized so that in Section 2, we first describe the experimental system and the experiments. Here, we also consider the energetics of the droplets. In Section 3, we review the IDL analysis. In Section 4, we present our model for the molecular transport between droplets at finite droplet concentrations. In Section 5, we use the new transport model to analyze our experimental results. 2. Experimental System Our experimental system is a well-characterized microemulsion system22 composed of water, decane, and the nonionic surfactant pentaethylene glycol dodecyl ether (C12E5). The interface separating the oil and the alkyl chain of C12E5 from the ethylene oxide chain and water is known to have an essentially invariant area, as, per molecule, independent of the curvature.22 It is therefore useful to define the curvature at this particular “neutral interface”, and the corresponding sphere radius, which we denote the hydrocarbon radius, Rhc, becomes

Rhc )

3(φo + 0.5φs)ls φs

(1)

Here ls ) vs/as ≈ 14.5 Å is the surfactant volume to area ratio resulting from vs ≈ 700 Å3 and as ) 48.3 Å2.23 The factor 0.5φs comes from the alkyl chain volume of C12E5 having approximately one-half of the total molecular volume. For alkyl oligo ethylene oxide surfactants, CnEm, the preferred spherical curvature, c0, can be tuned by changing the temperature,24,25 and in this way one can explore a whole range of aggregate geometries. For C12E5 in water-decane mixtures, we (19) Evilevitch, A.; Olsson, U.; Jo¨nsson, B.; Wennerstro¨m, H. Langmuir 2000, 16, 8755. (20) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35. (21) Wagner, C. Z. Elektrochem. 1961, 65, 581. (22) Olsson, U.; Schurtenberger, P. Prog. Colloid Polym. Sci. 1997, 104, 157. (23) Olsson, U.; Schurtenberger, P. Langmuir 1993, 9, 3389. (24) Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1993, 49, 113. (25) Strey, R. Colloid Polym. Sci. 1994, 272, 1005.

Figure 1. Partial phase diagram of the system C12E5-decanewater. The surfactant/oil volume ratio (φs/φo) is fixed constant at 0.815. The labeled phases are oil-in-water microemulsion (L1), oil (O), lamellar (LR), a bilayer continuous liquid phase (L3), and water (W). The figure also shows a schematic diagram of the structural changes involved in decreasing the temperature below the L1/(L1 + O) phase boundary. have c0 ≈ β(T0 - T), where T0 ) 38.3 °C and β ) 1.0 × 10-3 Å-1 K-1.26 At a given oil to surfactant ratio, the hydrocarbon radius, Rhc, of a spherical droplet is determined by the composition through the constraints of a fixed area to volume ratio. At an elevated temperature, corresponding to c-1 0 > Rhc, large nonspherical aggregates are formed,27,28 while when c-1 0 ≈ Rhc, there is a match between the optimal conditions for the surfactant film and the compositional constraints. At a still lower temperature, c-1 0 < Rhc, and the optimal spheres can no longer form in a homogeneous system. This leads to a separation into a microemulsion phase with a smaller radius (∼c-1 0 ) and an excess pure oil phase.2,27,29 At a surfactant to oil volume fraction ratio of φs/φo ) 0.815, a water-rich (oil-in-water) microemulsion phase is stable in the temperature range from 25 to 32 °C, while below 25 °C, the microemulsion phase separates with pure oil as the second phase (see Figure 1).30 At the L1/(L1 + O) phase boundary at 25 °C, which is remarkably independent of water concentration, the solution consists of spherical microemulsion oil droplets corresponding to the maximum curvature toward oil given the constraint of the area to enclosed volume ratio imposed by the ratio φs/φo.24 The hydrocarbon radius is approximately 75 Å, the polydispersity is low, and structural and dynamical properties follow closely those of a hard sphere fluid.22 When the temperature is decreased below 25 °C, the preferred curvature toward oil increases, and when equilibrium is reached, the microemulsion droplets have decreased in size, and excess oil has diffused from (26) Le, T. D.; Olsson, U.; Wennerstro¨m, H.; Schurtenberger, P. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 60, 4300. (27) Leaver, M. S.; Olsson, U.; Wennersto¨m, H.; Strey, R. J. Phys. II 1994, 4, 515. (28) Leaver, M. S.; Furo´, I.; Olsson, U. Langmuir 1995, 11, 1524. (29) Turkevich, L. A.; Safran, S. A.; Pincus, P. A. Theory of Shape Transitions in Microemulsions. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1986; Vol. 6; p 1177. (30) Leaver, M. L.; et al. J. Chem. Soc., Faraday Trans. 1995, 91, 4269.

Molecular Transport in a Microemulsion System

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small droplets to the newly nucleated larger drops. From there on, the population of large drops continues to evolve through a classical Ostwald ripening process.18 Because of the 2 orders of magnitude higher monomeric solubility of the surfactant as compared to the oil, we can assume that the surfactant redistributes fast and that the rate-limiting step of the relaxation process is the redistribution of oil. Curvature Energy of a Spherical Droplet. The free energy of a microemulsion droplet system can be described as the sum of the curvature energy and the entropy of mixing of the droplets:

G ) Gc + Gmix

(2)

The mixing entropy term can be neglected here since the change in it at the temperature jump will be very small as compared to the change in the curvature energy. The curvature free energy is an integral over the area of the surfactant film of a curvature free energy density, gc, so that

Gc )

∫ dA g

c

(3)

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 form:31

gc ) 2κ(H - H0)2 + κjK

(4)

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. H0 is the spontaneous curvature, and the two modulii, the bending rigidity, κ, and the saddle splay constant, κj, describe the elastic properties of the polar/apolar interface. It is particularly convenient to use eq 4 when comparing surfactant films of constant topology, since the Gaussian curvature term then only contributes as a constant by virtue of the Gauss-Bonnet theorem. In the present paper, we consider spherical aggregates only, where the topology of the system changes as the number of droplets changes. It is then more convenient to, instead of the Gaussian curvature, consider the difference curvature, ∆c ) (c1 - c2)/2, as the second variable. This yields32

gc ) 2κ′(H - c0)2 - κj(∆c)2

(5)

where c0 is the preferred spherical curvature discussed above. Its relation to H0 is given by

(2κ2κ+ κj)

c0 ) H0

(6)

The bending modulus κ′ is related to those of eq 4 by

κ′ ) κ + κj/2

(7)

For spherical aggregates ∆c ) 0 and by rewriting the expression for the curvature free energy, we need only consider the first term in eq 5. The curvature free energy of a spherical droplet of radius R is thus

Gc ) 8πκ′(1 - Rc0)2

(8)

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 the curvature toward oil as positive. Phase Separation Experiment. Experiments have been performed on the φ ) 10% (v/v) C12E5 and decane microemulsion oil droplets in water. Turbidity was measured at λ ) 406 nm using a Perkin-Elmer Lambda 14 UV/visible spectrophotometer, (31) Helfrich, W. Z. Naturforsch. 1973, 28C, 693. (32) 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.

Figure 2. Variation of turbidity with time for the C12E5-decane system (φs/φo ) 0.815) at φ ) 10% (v/v) 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 120 min at 22 °C. The dotted lines are the extrapolations to the time of the temperature jump. Temperature variation during the quench followed by the temperature jump is also displayed in the same figure. 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 using two water baths, where one was kept at a temperature of 25 °C corresponding to the L1/(L1 + O) phase boundary, and the second one was kept at a temperature of 22 °C in the two-phase region. The sample was first equilibrated at 25 °C for a couple of hours, and the turbidity was measured to ensure that its value was constant in time. Then, the sample was temperature-quenched in the twophase region by placing it in the 22 °C water bath and keeping it there for 2 h. The entire sample was always immersed into the thermostated water. All the turbidity measurements were performed in 1-mm quartz cells, and it took 20 s to change the sample temperature by 3° (from 25 to 22 °C and vice versa). In this way, temperature gradients within the sample could be minimized. Immediately after the temperature quench, the situation is that the droplets will have a lower curvature than the optimal one at the new temperature. Therefore, the majority of the droplets will shrink in size, expelling the excess oil via molecular diffusion across water to the new, nucleated phase of larger drops. After a short time, as we have shown in our previous work,18 the small droplets will reach the new equilibrium size, while the big drops will continue to evolve in size via an Ostwald ripening process.17,18 After 2 h, the initially monodispersed microemulsion droplet system with Rhc ) 75 Å has transformed into a droplet solution with two narrow size distributions: one population of small droplets, dominating in number and containing ∼70% of the oil, with hydrocarbon radius Rsmall ) 61 Å, and another population of big drops with hydrocarbon radius Rbig ) 150 Å (as we have previously evaluated from turbidity data).19 This solution of two well-defined droplet populations was chosen to be the “starting point” system for the experimental evaluation and the test of our theoretical model describing the oil transport from the big drops to the small droplets (see Figure 2). Resolubilization Experiment. The phase-separating oil was resolubilized by bringing the sample back to 25 °C after 2 h at 22 °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 copperconstantan thermocouple. Then, the sample was placed in the 25 °C thermostated cell holder in the spectrophotometer, and the turbidity was traced continually with time (see Figure 2).

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Figure 4. Cell model illustration of the resolubilization process.

Figure 3. Variation of turbidity with time for the temperature jump from 22 (b) to 25 °C after 120 min. The dotted line is the extrapolation to the time of the temperature jump. The solid line is the relaxation rates obtained from the IDL approximation calculations.

3. Resolubilization Experiment: Infinite Dilution Limit More detailed turbidity data for the resolubilization experiment (taken from ref 19) are presented in Figure 3. As can be seen, the turbidity drops by approximately 20% during the heating time, that is, the first 20 s. The extrapolated value at t ) 0, however, agrees well with the turbidity measured prior to the temperature jump. It took 20 min for turbidity to relax back to the equilibrium value at 25 °C. In ref 19, experimental resolubilization rates were compared with calculated rates based on the IDL approximation to understand the mechanism of the resolubilization process. The calculated change in droplet radius with time was expressed as turbidity (see Figure 3). Equations relating the turbidity to the droplet radius are presented in Appendix 1. In the IDL approximation, one assumes molecular exchange between droplets on the basis of the following assumptions:33 (i) the mass transport is limited by the molecular diffusion in the dispersion medium (in our case, diffusion of single oil monomers across water), (ii) the particles of the dispersed phase are spherical and separated from each other by distances which are much larger than the particle sizes (the volume fraction of the disperse phase φ f 0), and (iii) inhomogeneities of the concentration distribution of the molecular solution of the particles’ substance in space caused by diffusion are neglected. That is, after the temperature jump from 22 to 25 °C, oil molecules in the solvent are quickly solubilized by the small droplets, 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 big drops to small droplets. However, it is evident from Figure 3 that the experimental resolubilization rate is significantly faster than the one predicted by the IDL model. The main problem with the IDL approximation is that the concentration profile of oil monomers from the droplet approaches (33) Kabalnov, A. S.; Shchukin, E. D. Adv. Colloid Interface Sci. 1992, 38, 69.

asymptotically the value of the bulk monomer concentration; that is, it describes the case of infinite dilution. Here, we are dealing with a droplet system of a finite concentration. Refinements of the IDL theory have been previously done for the coarsening kinetics of Ostwald ripening processes, where the volume fraction of the precipitate was taken into account with respect to the size distribution of droplets typical for an Ostwald ripening process.34-36 Here, we are elaborating a more general model for the molecular exchange mechanism only, applied to the case of oil resolubilization in droplets for a system with bimodal droplet size distribution. 4. Effect of Droplet Concentrations: Cell Model Approach In our resolubilization model, we are assuming that the droplet microemulsion solution has a discrete bimodal size distribution. To analyze how the solubilization rate depends on the concentrations of big and small droplets, we apply a spherical cell model.37,38 Here, the system is divided into equally sized spherical cells, each containing one big oil drop in the center surrounded by the small oil droplets, as illustrated in Figure 4. Here, we assume a homogeneous distribution of small droplets within each cell. Conservation of Area and Volume. Dimensions of the Cell. At all times, we have drop(let)s of oil covered by a monolayer of the surfactant C12E5, and these drop(let)s undergo Brownian motion in the continuous 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 amount of surfactant and oil. During the experiment, the size distribution of the drops change, 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 rapid equilibration on the time scale of the oil transport.39 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 number of smaller droplets in the size range 61-75 Å in radius and a population of larger drops 100150 Å in radius (containing up to 30% of the oil). The (34) Ardel, A. J. Acta Metall. Sin. 1972, 20, 61. (35) Brailsford, A. D.; Wynblatt, P. Acta Metall. Sin. 1979, 27, 489. (36) Marqusee, J. A.; Ross, J. J. Chem. Phys. 1984, 80, 536. (37) Jo¨nsson, B.; Wennerstro¨m, H. J. Colloid Interface Sci. 1981, 80, 482. (38) Evilevitch, A.; Lobaskin, V.; Olsson, U.; Linse, P.; Schurtenberger, P. Langmuir 2001, 17, 1043. (39) Hedin, N. NMR Studies of Complex Fluids and Solids Formed by Surfactants. Ph.D. Thesis, Royal Institute of Technology, 2000.


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Figure 5. Free energy of the cell as a function of big drop’s radius for the system C12E5-decane-water at φ ) 10% (v/v).

distributions around the two most probable sizes are rather narrow,18 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

{

2 A ) 4πNsmallRsmall + 4πNbigR2big 3 + 4/3πNbigR3big V ) 4/3πNsmallRsmall

(9) (10)

Thus, by fixing one of the parameters Nsmall, Rsmall, or Rbig in eqs 9 and 10 and setting Nbig ) 1 within the cell, one will obtain the other two. The volume of the cell is thus equal to

Vcell )

3 4/3π(R3big + NsmallRsmall ) φhc

(11)

where φhc is the hydrocarbon volume fraction defined as φhc ) φo + 0.5φs. The radii, Rsmall and Rbig, are then also defined as hydrocarbon radii. Free Energy of a Cell. The free energy expression for the entire cell will be the sum of curvature energies for all drop(let)s. Considering one big drop surrounded by small droplets within the cell, we obtain the following: 2

2

Gc ) 8πκ′[(1 - Rbigc0) + Nsmall(1 - Rsmallc0) ]

Figure 6. Oil’s chemical potential for a spherical drop(let), µo, as a function of the drop(let)’s radius, R, for the system C12E5decane-water at φ ) 10% (v/v).

can be expressed as a function of radius of the droplet:

µc )

( ) ∂Gc ∂no

)

ns,nw

∂Gc ∂R ∂Gc R ) ∂R ∂no ∂R 3no

(13)

We obtain the following expression for the droplet’s oil chemical potential:

µoil ) µ0oil + µc ) µ0oil -

4κ′v0(1 - Rc0)c0 R2

(14)

where µ0oil is the standard chemical potential of oil-in-oil, no, ns, and nw are the number of oil, surfactant, and water molecules, respectively, and vo is the molecular volume of the oil. We have plotted the excess chemical potential of oil in a spherical drop(let), µc, as a function of the drop(let) radius, R, in Figure 6. Concentration of Oil Monomers Surrounding the Droplet. Knowing the chemical potential of oil in a droplet, we can calculate the equilibrium concentration of oil monomers in the solution surrounding the droplet. The chemical potential of oil in water is

µoil ) µ0oil + µHE + kT ln Coil,w

(15)

(12)

At equilibrium after the temperature jump, the curvature energy is at its minimum value of zero with all droplets of the same size Rbig ) Rsmall ) Rf ) (c0f )-1, where index f indicates the final state. For a better visualization of the system energetics during the resolubilization process, we calculate the free energy of the cell as a function of the big drop’s radius for the system C12E5-decane-water (Figure 5). The energy is calculated using area and volume conservation, eqs 9 and 10 in combination with eq 12, with Rf ) (c0f )-1 ) 75 Å corresponding to the (L1 + O)/L1 phase boundary equilibrium state at 25 °C. Chemical Potential of Oil in a Droplet. The chemical potential of an oil molecule in a droplet depends on the change in curvature energy as one oil molecule is expelled or adsorbed by the droplet. Therefore, the curvature part of the chemical potential of an oil molecule in a droplet

where µHE is the hydrophobic effect, the difference in chemical potential for an oil molecule to be in an aqueous environment compared with a hydrocarbon environment. By combining eqs 14 and 15, the concentration of oil in the solution surrounding a droplet becomes

(

Coil,w ) exp

)

( )

µc - µHE µc ) C∞oil exp kT kT

(16)

where C∞oil is the solubility of oil in water. The oil concentration in the solution surrounding a droplet is, in Figure 7, plotted as a function of the droplet’s radius. Here, we assume that this oil concentration in water is an equilibrium concentration due to a fast exchange between oil in droplets and oil in water, in comparison with the resolubilization process. The parameters used are for the system C12E5-decane-water:

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Figure 7. Difference between the oil (decane) concentration in water at the surface of the droplet and the solubility of decane is plotted as a function of drop(let)’s radius for the system C12E5decane-water at φ ) 10% (v/v).

C∞oil ) 2.16797 × 1020 m-3 (for decane in water at 25 °C),40 vo ) 323 × 10-30 m3, c0 ) 1/75 × 10-10 m-1,40 κ′ ) 1.5kT.26 The analysis of the free energy changes during the oil transport provides us with a qualitative understanding of the parameters controlling and driving the oil exchange between the nonequilibrium microemulsion droplets. However, to get a quantitative understanding on the mechanism and dynamics, it is necessary to analyze the diffusion kinetics of the molecular transport process. Diffusion Model for the Oil Transfer through the Continuous Aqueous Phase. After the temperature jump from 22 to 25 °C, the small droplets will have a higher curvature, and the big drops will have a lower curvature than the optimal curvature at 25 °C. As it was described above, this deviation in the interfacial curvature from the preferred curvature results in a higher oil monomer concentration at the surface of the big drops and lower oil monomer concentration at the surface of the small droplets than the average free oil concentration in the bulk. Thus, the temperature jump creates a concentration gradient between oil surface concentrations of small droplets and big drops. As a result, the oil will be desolubilized from the big drops and resolubilized in the small droplets; that is, there is a net oil transport of oil from the big drops to the small ones. Cell Model. The oil droplet solution is thus assumed to consist of Nbig larger spherical oil droplets of radius Rbig and Nsmall small spherical oil droplets of radius Rsmall. In our spherical cell model, the solution is divided into Nbig spherical cells with a big spherical droplet in the center of each cell and a surrounding solution consisting of solvent (water), small droplets, and small amounts of monomeric oil and amphiphiles. The radius of the cell, Rcell, can be calculated from the condition that the volume of the cells equals the total volume of the system, V:

4πR3cell Nbig 3

)V

(17)

Since all the cells are of equal size and contain equal amounts of small droplets and dissolved material, there is no net flow of oil between the cells, and at the cell (40) McAuliffe, C. J. Phys. Chem. 1966, 70, 1267.

Figure 8. In- and outflow of free oil monomers to a spherical shell between radii r and r + dr around the big drop.

boundary we have the condition:

dCoil (Rcell) ) 0 dr

(18)

Oil Transport to or from the Big Drop in the Center of the Cell. Let us now consider the diffusive transport of oil out from a big drop placed in the center of a spherical cell. In the cell, there is only one big oil drop surrounded by an aqueous solution of oil monomers and small oil droplets. We assume that the small droplets diffuse fast on the solubilization time scale so that we can use the approximation that they are all of equal size at each instant. We also assume that they are randomly distributed. By considering the in- and outflow of free oil monomers to a spherical shell between the radii r and r + dr around the big drop (see Figure 8) and also the amount of free oil disappearing from this shell due to the binding to the small droplets, the following equation is obtained:

(

)

dCoil d dCoil(r) ) Doil4π dr r2 4πr2 dr dt dr dr dCoil,to small droplets (19) 4πr2 dr dt We are now assuming that the binding of free oil monomers to the small droplets in a spherical shell between r and r + dr can be approximated by a steady-state flow of oil to the corresponding droplets in the bulk solution with the free oil concentration Coil(r). Now, dCoil,to small droplets/dt can be written as (see Appendix 2)

dCoil,to small droplets ) dt (Coil(r) - C(Rsmall))Doil4πCsmallRsmall (20) where Csmall is the concentration of small droplets and C(Rsmall) is the equilibrium concentration of free oil at the surface of a small droplet. By inserting eq 20 into eq 19, the following flow equation is obtained:

(

)

1 dCoil 1 d 2dCoil(r) r ) 2 Doil dt dr r dr (Coil(r) - C(Rsmall))4πCsmallRsmall (21)


Langmuir, Vol. 17, No. 22, 2001 6899

At steady state, eq 21 can be rewritten as

0)

(

)

d 2dCoil(r) r dr dr r2(Coil(r) - C(Rsmall))4πCsmallRsmall (22)

The solution to this differential equation is

1 Coil(r) ) C(Rsmall) + (A exp(Rr) + B exp(-Rr)) (23) r where

R ) x4πCsmallRsmall

(24)

Constants A and B can be calculated from the boundary conditions:

Coil(r) ) C(Rbig) when r ) Rbig

(25)

dCoil (Rcell) ) 0 dr

(26)

and

Using eqs 23-26, the equation for the free oil concentration profile in the cell is obtained:

Rbig f(r) (27) Coil(r) ) C(Rsmall) + (C(Rbig) - C(Rsmall)) r where

f(r) ) {(RRcell - 1) exp(R(Rcell - r)) + (RRcell + 1) × exp(-R(Rcell - r))}/{(RRcell - 1) exp(R(Rcell - Rbig)) + (RRcell + 1) exp(-R(Rcell - Rbig))} (28) Using eqs 27 and 28, we can now calculate how much oil is diffusing per unit time out from a big oil drop:

dC(Rbig) ) dr Doil4πRbig(C(Rbig) - C(Rsmall))(1 - Rbigf ′(Rbig)) (29)

Qout ) -Doil4πR2big

Here, we have used the fact that f(Rbig) ) 1, and we have

Rbigf ′(Rbig) ) (RRcell - 1) (RRcell + 1)

-RRbig (RRcell - 1) (RRcell + 1)

model analysis. They provide us with an expression for the solubilization rate of the big drops as a function of concentrations and sizes of big and small droplets and the driving force, represented by the differences in monomer solubility between the big and small droplets. In the limit of small φbig, eq 31 reduces to Rbig f ′(Rbig) ) -RRbig, provided that R is not too small, and the flow can be written as

Qout ) (C(Rbig) - C(Rsmall))Doil4πRbig(1 + RRbig) (33) A more formal requirement for eq 33 is that . exp(-2RRbig(φ-1/3 big - 1)). In this limit, we can recover the expression given by Kabalnov and Weers,11 who considered the solubilization of very large droplets, so that C(Rbig) ) C∞oil, by empty micelles, so that C(Rsmall) ) 0. We also obtain the correct infinite dilution limit, where both the big and small droplets are dilute. This situation is described by eq 33 when we set R ) 0. 5. Model Analysis and Discussion

- exp(-2R(Rcell - Rbig)) (30) + exp(-2R(Rcell - Rbig))

By introducing Rbig/Rcell ) φ1/3 big, where φbig is the volume fraction of big drops, eq 30 can be rewritten as

- exp(-2RRbig(φ-1/3 big - 1)) (31) Rbig f ′(Rbig) ) -RRbig -1/3 + exp(-2RRbig(φbig - 1)) where

)

Figure 9. Variation of turbidity with time for the temperature jump from 22 to 25 °C after 120 min, with a volume fraction of small droplets prior to the temperature jump of φhc,small ≈ 5% (v/v) (b) and φhc,small ≈ 14% (v/v) ([). The dotted line is the extrapolation to the time of the temperature jump. The solid lines are the relaxation rates obtained from the cell model calculations.

RRbigφ-1/3 big - 1 RRbigφ-1/3 big + 1

(32)

Equations 29-32 represent the central result of the cell

Comparison with Experiment. Above, we have derived analytically a theoretical model for molecular exchange kinetics in a nonequilibrium droplet microemulsion system of finite droplet concentration with a bimodal size distribution. The motivation for this effort is that we have an experimentally realizable situation with a nonionic surfactant-water-oil system having unusually well-defined initial conditions, and where both the energetics and the dynamic molecular processes can be given a quantitative description. In Figure 9, we compare the experimental turbidity data with a simulated curve based on the cell model calculations of the previous section. For this experimental case, φbig ) 0.024, and eq 33 applies with RRbig ≈ 1. The other input parameters in the calculations are Rbig (initial) ) 150 Å, Rsmall (initial) ) 61 Å, C∞oil ) 2.16797 × 1020 m-3 (for decane in water at 25 °C),40 Doil ) 9.5 × 10-10 m2/s (for decane in water at 25 °C),19 vo ) 323 × 10-30 m3, c0 ) 1/75 × 10-10 m-1,40 and κ′ ) 1.5kT.26 The tubidity trace was calculated in an iterative manner, as outlined in Appendix 1, conserving area and volume. As can be seen, we find a

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quantitative agreement between the modeled and the experimental data, and, in particular, we reproduce the fact that finite concentration of droplets gives a faster resolubilization compared to the infinite dilution case. Remaining at low φbig, we see from eq 33 that, for RRbig . 1, the solubilization rate increases with the square root of the small droplet concentration. To test the effect of small droplet concentration, we also performed a second experiment. We took the same sample as in our previous experiment, that is, a solution of two droplet populations, one with big drops of radius 150 Å and another one with small droplets with radius 61 Å. Now, we wanted to follow the transport of oil from the big drops to the small droplets after the temperature jump from 22 to 25 °C, but with a higher small droplet concentration than in the previous resolubilization experiment. This could be realized by mixing this prequenched sample with a solution of φ ) 30% (v/v) C12E5 and decane (at the volume ratio 1.11) microemulsion droplets in water at 22 °C, where the hydrocarbon radius of the droplets is Rhc ) 61 Å. By mixing these two microemulsion solutions at the ratio 1:1 at 22 °C, we prepared a solution of big drops (Rhc ) 150 Å) and small droplets with a higher concentration, φhc,small ≈ 14% (v/v), compared to φhc,small ≈ 5% (v/v) in the previous experiment. We have checked via calculations that the effect of dilution of the big drops on the transport kinetics can be neglected due to very low concentration of the big drops. Immediately after mixing, the sample was brought back to the one-phase equilibrium region by a temperature jump from 22 to 25 °C (in the same way as before), and the turbidity was continuously traced with time. The results from these experiments are also shown in Figure 9 together with the theoretically predicted curve. As can be seen, the increase in small droplet concentration leads to a faster resolubilization, and the model calculations are again in quantitative agreement with the data. Because of a different surfactant to oil volume ratio in the final mixture than in the previous experiment, the final hydrocarbon radius of droplets at 25 °C is smaller than 75 Å. However, this does not affect the initial rate of the resolubilization of the big drops, which we are interested in. The good agreement between model and experiments is a strong indication that the cell model describes the important features of the resolubilization experiment. Below, we will further analyze the model and address the question where the net oil leaving the big drops is captured by the small droplets in the solution. Effect of Small Droplet Concentration on Oil Solubilization Rate. Let us now go back to eq 33 and analyze it to understand how the small droplet concentration affects the way in which the oil is solubilized. We keep the cell dimensions constant, Rcell ) constant, and vary the parameter, RRbig, by changing the concentration of small droplets, Csmall. For a better visualization, instead of the oil flow, Qout, from the big drop, we choose to look at what distance from the big drop the solubilization of oil monomers in the small droplets is taking place. Since we have a cell model with the condition dCoil/dr(Rcell) ) 0 at the cell boundary and a steady-state flow assumption, it implies that all the oil, if desolubilized from the big drop, will be solubilized in the small droplets. We define the function P(r) as the probability for the oil flow leaving the big drop to be solubilized in the small droplets at the distance, r, from the center of the cell. P(r) is thus the difference between the oil flow entering the spherical shell within the cell at the distance r and leaving this shell at the distance r + dr, normalized by the total flow of oil out from the big drop:

Evilevitch et al.

P(r) dr )

Qoil(r) - Qoil(r + dr)

) Qoil(Rbig) dCoil(r) dCoil(r + dr) + Doil4π(r + dr)2 -Doil4πr2 dr dr w dC (R ) oil big 2 -Doil4πRbig dr dC (r) d 2 oil r R2rf(r) dr dr P(r) ) ) (34) dCoil(Rbig) 1 - Rbigf ′(Rbig) 2 -Rbig dr

(

)

Considering the case RRbig . 1, the probability function P(r) shows that all of the oil will be solubilized in the small droplets close to the big drop (see Figure 10a). Further out from the surface of the big drop, at some critical distance r ) δ, there will be no oil monomers left. Because of the high, direct oil flow from the big drop to the small droplets at short distances (steeper concentration gradient with decreasing distance between particles, see illustration in Figure 11a), all the oil will be quickly resolubilized in the small droplets within the layer of thickness δ (from hereon we will call this layer of thickness δ the solubilization layer). Assuming a realistic situation where the small droplets are moving randomly within the cell and exchange the material between each other with relaxation rates much faster than the resolubilization process (we have estimated the droplet exchange to occur on the time scale of microseconds for our system, while the resolubilization time scale is in minutes), we conclude that at the distances r > δ, the oil is equally distributed within the cell solely by the small droplet transport. Thus, small droplets in our model are all of the same size due to the equal “appetite” for the free oil. Since the exchange of small droplets is fast as compared to the resolubilization process time scale, the solubilization within the solubilization layer, δ, will be rate-determining. The shorter the solubilization layer is, the faster the oil will be resolubilized due to the shorter diffusion distance for the oil monomers. Conclusively, taking into account the finite small droplet concentration explains the experimentally observed higher resolubilization rates. Now, let us consider the case of infinite dilution of droplets in the surrounding aqueous medium. We will then have the case with RRbig , 1. The probability to solubilize oil within the cell will be equally low anywhere in the cell due to the very low small droplet concentration (see Figure 10b). The thickness of the solubilization layer, δ, is equal to the cell radius, that is, the oil is transported by diffusion of single oil monomers across the water within the entire cell and not by the droplets such as in the previous case, which results in much slower solubilization rates. Setting RRbig , 1, we obtain the same equation for the flow as the one given by the IDL approximation model. Just like in the IDL theory, our model assumes that there is no direct flow of oil between the big drop and the small droplets. The oil is first solubilized in the aqueous medium surrounding the big drop, which contributes to an increased bulk oil concentration. The excess oil in the bulk is then resolubilized by the small droplets (see the illustration in Figure 11b). The observed increase in probability for oil solubilization within the cell with increasing distance r is due to the increasing volume where the solubilization can take place. Thus, in the limit of infinite dilution, our solubilization model yields the mechanism and the formalization for the molecular flow given by the IDL theory, where molecules are entirely


Langmuir, Vol. 17, No. 22, 2001 6901

Figure 11. (a) Illustration showing that there is a direct oil flow from the big drop to the small droplets at short distances (steeper concentration gradient with decreasing distance between particles. (b) In the IDL theory (like in the cell model at low droplet concentrations), there is no direct flow of oil between the big drop and the small droplets. The oil is first solubilized in the aqueous medium surrounding the big drop which contributes to an increased bulk oil concentration. The oil excess in the bulk is then resolubilized by the small droplets.

Figure 10. (a) Probability function P(r) for the case of finite, high, small droplet concentration RRbig . 1, showing that all of the oil will be solubilized in the small droplets close to the big drop within the solubilization layer δ. (b) Probability function P(r) for the case with RRbig , 1, showing that the probability to solubilize oil within the cell will be equally low anywhere in the cell due to the very low (almost nonexistent) small droplet concentration. (c) Probability function P(r) for the temperature jump experiment from 22 to 25 °C after 120 min at 22 °C. The system is C12E5-decane-water at φ ) 10% (v/v).

transported by molecular diffusion in the surrounding medium. This demonstrates that our cell model is of

general character, where the IDL approximation is a special case of our model in the limit of φ f 0. In Figure 10c, we have also calculated the probability function P(r) for our temperature jump experiment described in Section 2 (from ref 19). For this system of microemulsion droplets in water with φ ) 10% (v/v) C12E5 and decane at the volume ratio 0.815, where initially Rbig ) 150 Å and Rsmall ) 61 Å, we obtain RRbig ≈ 1. The P(r) function versus r, in the figure, shows also here a maximum in the probability of solubilizing oil closest to the surface of the big drop. Thus, it is also obvious that, in our resolubilization experiment, oil solubilization rate is affected by small oil droplets’ volume fraction. Since the presence of the small droplets makes the solubilization layer thinner, the experimentally observed resolubilization rate is faster than what is predicted by the IDL approximation model. Effect of Cell Size (Big Drop Number Density) on Oil Solubilization Rate. The realization of the abovementioned experiment demonstrates that it is fully possible to study the effects on kinetics and the mechanism of molecular exchange between droplets with bimodal size distribution by varying the number and size of big and small drop(let)s. Thus, we have experimentally and theoretically studied the case where we are varying the number density of small droplets and found that our cell model adequately describes the experimental observations. Now we will analyze with this model the case where we vary the number density of big drops, while retaining the number density of small droplets constant. Changing the concentration of big drops is the same as changing the size of the cell in our cell model. Again, we

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Evilevitch et al.

are employing the probability function P(r) to see how big drop concentration can affect the resolubilization mechanism. In Figure 12, we have plotted function P(r) versus r for the same experimental system with Rbig ) 150 Å and Rsmall ) 61 Å. The cell radius in this experimental system with φ ) 10% (v/v) is 520 Å. In Figure 12a, we have calculated P(r) for this system with all the same parameters but for a 2 times smaller cell radius (that is, 8 times higher concentration of big drops). P(r), in this case, shows that the probability to solubilize oil monomers expelled from the big drop is increasing with increasing distance from the surface of the big drop, due to an increasing volume and, therefore, increasing number of small droplets that can solubilize oil within this volume. The figure shows us that the solubilization of oil monomers occurs everywhere in the cell. Thus, there is no transport of oil in the cell mediated by the small droplets. Now, going to the case with low concentration of big drops and the same high concentration of small droplets, P(r) predicts that all of the oil will be solubilized by the small droplets not across the entire cell like in the previous case but within the solubilization layer δ. Decreasing the number density of big drops further on does not change the thickness of the solubilization layer (see Figure 12b and c) and thus should not affect the solubilization rate. 6. 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. To understand and explain the mechanism and kinetics of oil solubilization observed in our experiments, we have developed a cell model for molecular transport in nonequilibrium microemulsion droplet systems with finite droplet concentrations including the infinite dilution limit. From this we conclude that in finite concentration systems, molecular solubilization of oil expelled from one population of drops and solubilized by another population of droplets occurs via a direct diffusive transport of oil between droplets only within a so-called solubilization layer, δ, of a finite thickness. On the contrary, the commonly used IDL approximation predicts the solubilization to occur with diffusive molecular transport across the entire system (δ f ∞), where the molecular diffusion between the droplets is occurring indirectly via the surrounding molecular concentration in the bulk medium, thus independently of the sink-droplet concentration. We have shown that the finite sink-droplet concentration affects the thickness of the solubilization layer δ, where δ is decreasing with increasing droplet concentration. This results in faster resolubilization kinetics, as verified by experiments. However, in the limit of φ f 0, our model reduces to the formalization given by the IDL approximation. Because of its flexibility, we believe that our cell model can be very useful in description of other nonequilibrium droplet systems with finite concentrations undergoing molecular resolubilization or particle coarsening. We have also presented an experimental system where a specific bimodal size distribution can be prepared. This distribution evolves into a unimodal distribution by a simple and mild temperature jump. Appendix 1 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

Figure 12. (a) Probability function P(r) for the temperaturejump experiment from 22 to 25 °C after 120 min at 22 °C. The system is C12E5-decane-water at φ ) 10% (v/v). All parameters used in the calculation correspond to the real experimental system, but the cell radius is set 2 times smaller (i.e., an 8 times higher concentration of big drops). (b) Probability function P(r) predicts that all of the oil will be solubilized by the small droplets not across the entire cell like in panel a but within the solubilization layer δ. (c) Decreasing the number density of big drops further on does not change the thickness of the solubilization layer and thus does not effect the solubilization rate.


Langmuir, Vol. 17, No. 22, 2001 6903

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:41

τ)

π 2π I(θ)r2 sin(θ) dθ dφ ∫θ)0 ∫φ)0

1 I0,u

(A1.1)

When R , λ, where λ is the wavelength of light, one can use the Rayleigh-Gans-Debye (RGD) scattering approximation:

given by the sum of turbidity from small droplets and from big drops, τtot ) τsmall + τbig, thus neglecting any cross terms. This simplifying assumption gives an uncertainty in the size determination when size differences are small. However, far from equilibrium, the big drops dominate the turbidity (because of the Vsphere ∼ R3 term in eq A 1.5), and we have τtot ≈ τbig. The radius of the droplet, R(t), was calculated as a function of time by an iteration of eq A1.9 with some set time interval, ∆t,

R(t + ∆t) ) R(t) +

I(θ)r2 ) P(θ)(Rθ) I0,u

(A1.2)

dR(t) ∆t dt

(A1.9)

where dR(t)/dt is obtained from the flow equation: where P(θ) is the so-called form factor, which for small radii can be described by a Guinier approximation:

(qRg)2 P(θ) ) 1 3

Rθ ) Rπ/2(1 + cos2 θ)

(A1.4)

and

2π2n2w dn 2 VsphereφS(0) λ4 dφ

( )

(A1.5)

Here S(0) is the structure factor that 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 the surfactant’s and oil’s volume fractions, φ ) φs + φo. The refractive index, nw, of the solvent (H2O) is 1.3417. S(0) for a hard sphere system is given by Carnahan and Starling:42

S(0) )

(1 - φ)4 2

3

(1 + 2φ) - φ (4 - φ)

(A1.6)

Finally, the refractive index increment dn/dφ can be calculated from modeling the droplets as core-shell spheres,23,43

(

)

φs dn 3 1/2 φo ) w A + B dφ 2 φ φ

[

where vo is the molecular volume of an oil monomer (323 × 10-30 m3 for decane). Appendix 2 Oil Transport to or from the Small Droplets. Assume that we have a spherical droplet with the radius Rsmall so that the oil’s concentration at the surface of the droplet is always in local equilibrium with the oil inside the droplet. The concentration of oil at the surface of a small droplet is designated with C(Rsmall). If the droplet is now placed in an environment with an oil concentration, Coil,bulk, the oil will be transported to or from the droplet, depending on whether Coil,bulk is larger or smaller than C(Rsmall), with a diffusive transport. Let us now look at how this transport of oil varies in time after the droplet has been placed in its new environment. The parameters describing our system are the radius of the droplet, the diffusion constant for oil monomers in water, Doil, oil concentrations, Coil,bulk and C(Rsmall), and the time, t. Fick’s second law gives

(

(A2.1)

where r is the radial distance from the center of the droplet. With boundary conditions Coil(r) ) C(Rsmall) when r ) Rsmall, and Coil(r) ) Coil,bulk when r f ∞, and the initial condition Coil(r, 0) ) Coil,bulk for r > Rsmall, the solution to this differential equation is

]

Coil(r) ) Coil,bulk + (C(Rsmall) - Coil,bulk) × (r - Rsmall) Rsmall 1 - erf r x4D t

( (

(A1.8)

In the analysis, we have assumed that the turbidity is (41) Hunter, R. J. Introduction to Modern Colloid Science; Oxford University Press Inc.: New York, 1993. (42) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (43) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969.

oil

))

(A2.2)

With the help of eq A2.2, the inflow of oil per unit time to the small droplet can be written as

2 3

128R(t) π 16 π3 15 λ2

)

dCoil 1 d 2dCoil (r) ) Doil 2 (r) r dt dr r dr

(A1.7)

with A ) o - w/o + 2w and B ) s - w/s + 2w. φs and φo can be calculated from Rhc and φhc. 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 A1.1 gives the expression for turbidity:

τ ) Rπ/2

(A1.10)

(A1.3)

with scattering angle θ, scattering vector q, and a radius of gyration Rg, which for a uniform sphere is given by R2g ) 3/5R2. Rθ is the Rayleigh ratio, given by

Rπ/2 )

Qout dR(t) ) vo dt 4πR(t)2

2 Qin ) Doil4πRsmall

dCoil(Rsmall) ) dr

(

(Coil,bulk - C(Rsmall))Doil4πRsmall 1 +

)

Rsmall 1 xDoilt xπ (A2.3)

We see from eq A2.3 that the steady-state condition

6904


2 appears when t . Rsmall /Doil. At the steady state, the inflow of oil per second to the small droplet is

Qin ) (Coil,bulk - C(Rsmall))Doil4πRsmall (A2.4) This implies that the reduction of free oil in a volume, V, as a result of the inflow of oil to the small droplets, can be written as

Evilevitch et al.

dCoil,to small droplets ) -CsmallQin ) dt - (Coil,bulk - C(Rsmall))Doil4πRsmallCsmall (A2.5) where Csmall is the concentration of small droplets. Note Added After ASAP Posting Equation 34 was formatted incorrectly in the version released ASAP on 10/02/01. The equation has been corrected and re-released ASAP on 10/04/01. LA010899D

Molecular Transport in a Nonequilibrium Droplet Microemulsion System

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