Multicomponent Diffusion in Solute-Containing Micelle and

The refractive index increments of the solute and surfactant, R1 and R2, ... The time t0 at which the two solutions came into contact was recorded and...
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Multicomponent Diffusion in Solute-Containing Micelle and Microemulsion Solutions Wyatt J. Musnicki,† Stephanie R. Dungan,*,†,‡ and Ronald J. Phillips† †

Department of Chemical Engineering and Materials Science and ‡Department of Food Science and Technology, University of California, Davis, California 95616, United States ABSTRACT: Holographic interferometry was used to obtain new results for the four coefficients that determine rates of multicomponent diffusion of hydrophobic solutes and surfactants in microemulsions. The three solutes pentanol, octanol, and heptane were examined in microemulsions formed from decaethylene glycol monododecyl ether (C12E10) and sodium dodecyl sulfate (SDS). These coefficients are compared with relevant binary and effective binary diffusion coefficients, and also with ternary diffusion coefficients reported in the literature. It is shown that a strong coupling exists between the diffusion of hydrophobic solutes and surfactant in solute-containing microemulsions. In particular, the presence of a gradient in the concentration of the solute can induce a surprisingly large flux of surfactant either up or down the solute gradient. Within the framework of irreversible thermodynamics, these results indicate that hydrophobic solute molecules significantly alter the chemical potential of the surfactant in microemulsions. These effects are present to a comparable degree for both the nonionic C12E10 and ionic SDS microemulsions.

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INTRODUCTION The diffusion of hydrophobic solutes in a microemulsion plays an important role in many applications, including the delivery of poorly soluble compounds such as pharmaceuticals, micronutrients, preservatives, flavors, and antimicrobials. The use of microemulsions to modify the delivery of these compounds has several advantages compared to traditional emulsified systems. One is that the sizes of the droplets within microemulsions are on the order of 4−10 nm, which allows the hydrophobic compounds to access interstitial positions that would be inaccessible to larger aggregates. In pharmaceutical applications, the small size of microemulsion droplets is thought to contribute to high bioavailability for the active ingredients.1,2 Another advantage of microemulsions is that they are thermodynamically stable. Therefore, they do not need energy intensive operations to disperse the solution components, and they do not phase separate, which can cause the solute flux to fluctuate when control of solute delivery is essential. The diffusion of hydrophobic compounds in surfactant solutions has received much attention in recent years, most of it focused on self-diffusion.3,4 However, gradient diffusion of solute in surfactant solutions presents a fundamentally different process,5,6 and it is often gradient diffusion that is of central importance in delivery applications. The differences are particularly significant at higher surfactant and solute concentrations, where effects of solute−surfactant interactions between micelles or microemulsion structures, nonideal solute partitioning behavior, and solute−surfactant interactions during self-assembly can become important. In this study, we fixed the surfactant concentration at 0.2 M and investigated solute concentrations near the maximum concentration attainable. © 2014 American Chemical Society

The rate at which hydrophobic compounds diffuse through a micellar solution or microemulsion can be heavily influenced by the solubilization of the compound into the micelles. Under conditions where only a small number of solute molecules are contained within each micelle, several researchers have attempted to describe the transport process theoretically. Most studies express the total flux of solute through a homogeneous micellar solution as a sum of two contributing terms: the flux of solute moving through the continuous aqueous solvent and the flux of solute from the diffusion of solute-containing micelles.7−11 If the solute and surfactant within micelles are treated as a secondary pseudophase and ideal solution thermodynamics is applied, then the solubilization of the solute can be treated as simple partitioning between the two phases, with K s,c =

Cms Cas

(1)

Here Csm is moles of solute per volume of the micelles, Csa is the moles of solute per volume in the aqueous solution outside the micelles, and Ks,c is the partition coefficient. Because the diffusion time scale of interest in this study is on the order of hours, the solubilization of solute into the micelle can be considered to be at equilibrium along the diffusion pathway. Using Fick’s law to describe the solute fluxes in each phase, and eq 1 to relate the concentration of solute in the two phases, Received: April 29, 2014 Revised: August 4, 2014 Published: August 19, 2014 11019

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surfactant fluxes. If the multicomponent diffusion is described by a generalized form of Fick’s law,17 then the flux of solute (1) and surfactant (2) within a microemulsion can be written, respectively, as

the effective diffusion coefficient Deff of the solute within the micellar solution is found to be Deff =

Da + ϕK s,cDm 1 + ϕK s,c

(2)

where ϕ is the volume fraction of the microemulsion particles, Da is the diffusion coefficient of the solute in the aqueous phase, and Dm is the diffusivity of the solute-containing micelles. Equation 2 predicts that the effective diffusion coefficient of the solute is the weighted average of the solute and micelle diffusivity. This weighting is controlled by the solute’s affinity for the micellar interior through the value of Ks,c. A small value of Ks,c indicates a more polar solute that resides predominantly in the aqueous phase, and in the limit Ks,c → 0 the effective diffusivity becomes exactly equal to Da. A large value of Ks,c indicates a more nonpolar solute that resides mostly in the micellar phase, and in the limit of large Ks,c the effective diffusivity becomes equal to Dm. In deriving eq 2, it has been assumed that the net flux of surfactant is zero, the surfactant concentration is constant, and any coupled transport of surfactant by the solute gradient or vice versa has been neglected. However, these conditions and assumptions may not be generally valid. In a study of medium chain alcohols diffusing in an anionic micellar solution of sodium dodecyl sulfate (SDS), an alcohol concentration gradient was shown to generate fluxes in not only alcohol but also surfactant.12 This coupled transport of surfactant was shown to be non-negligible, and at times much larger in magnitude than the diffusion of solute or surfactant from their own concentration gradients. The presence of micelle charge and counterions was suspected to play a major role in the coupled transport of the solute and to be the cause of the large coupled fluxes. However, coupled fluxes were also measured for hydrocortisone in tyloxapol micelles,13 indicating that even for nonionic micelles a flux of surfactant can be induced by a concentration gradient in a hydrophobic solute. Here we explore this question further, by examining solute diffusion within anionic and nonionic surfactant solutions using the same solutes, and identical surfactant concentrations. Our results will show that eq 2 does not generally yield reliable predictions for coupled transport of solutes and surfactants. Surfactant transport can also take place via monomer and molecular mechanisms. In the absence of solute, Turq et al.14 treated the flux of the monomer and micelle separately, with the two species related by a local equilibrium condition.15 The resulting net flux of surfactant has contributions from both monomer and micelle diffusion mechanisms, with the latter heavily weighted by larger aggregation numbers m and low monomer concentrationsi.e., low values of the critical micelle concentration (CMC).16 This basic mechanism has the potential to be altered by the presence of solute in several aspects. Solute addition can alter the CMC and monomer concentration, and change the size, shape, and interactions of micelles. More fundamentally, the addition of solute can alter the chemical potential of surfactant in micelles and in monomer form and thus alter the energy landscape in which diffusion takes place. These effects are likely responsible for the coupled transport of surfactant discussed above, and for other aspects of multicomponent diffusion. Because a mixture of water, solute, and surfactant forms a ternary solution, it is appropriate to use a multicomponent diffusion expression to capture any coupling of the solute and

J1 = −D11∇C1 − D12∇C2

(3)

J2 = −D21∇C1 − D22∇C2

(4)

Here Ci and Ji are the concentration and flux of the ith species, respectively. In eqs 3 and 4, Dij is the diffusion coefficient that relates the flux contribution of the ith species to the concentration gradient of the jth species. For example, if the cross-term D21 is zero, then there is no flux of surfactant created from a solute concentration gradient. Two commonly used general methods for measuring diffusion in liquids include interferometry and the Taylor dispersion method. In the Taylor dispersion method, the spread of solute is measured at the exit of a long, cylindrical tube, which typically is more than 10 m in length and ∼0.5 mm in diameter. The analysis and practical considerations for implementation of this method are well established.18,19 Interferometric methods generally rely on differences in refractive index between the diffusion cell and a reference cell. In holographic interferometry, a holographic image is substituted for the traditional reference cell. A holographic interferometer is therefore a “single-path” device. There is no need for a separate, physical reference cell or sample holders of high optical quality, and alignment is easier than in classical interferometry.20,21 Holographic interferometry has been found to be useful for measuring diffusion in complex materials that may exhibit local heterogeneity, such as polymer gels, which are commonly used in controlled release devices and for bioseparations. It is also suitable for highly viscous fluids and solids that cannot be subjected to the flow required in the Taylor dispersion method. Taylor-tube dispersion and interferometry have been adapted previously to measure Dij from a series of complementary experiments.12,13,22−24 In this study, holographic interferometry5,6,16,21,25,26 was used to measure Dij following the suggestions provided by Miller.24 Values of Dij were measured for three different solutes: n-pentanol, n-octanol, and n-heptane, listed in order of increasing extent to which the solute partitions into the microemulsion. Two different surfactants were investigated in forming the microemulsions: sodium dodecyl sulfate (SDS) and decaethylene glycol monododecyl ether (C12E10). SDS (CMC 8 mM) contains a negatively charged headgroup, whereas C12E10 (CMC 0.09 mM27) is nonionic. Both surfactants self-assemble to form spherical aggregates,28,29 with aggregation numbers that vary little with surfactant concentration.30−32 Aggregation numbers at room temperature for SDS and C12E10 in the absence of solute are reported to be 6333 and 122,29 respectively.

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MATERIALS AND METHODS

Materials. The surfactants sodium dodecyl sulfate (99.0%) and decaethylene glycol monododecyl ether (99.0%), and the solutes pentanol, octanol, and heptane, were purchased from Sigma-Aldrich and used without modification. Slavich 4 in. × 5 in. PFG-01 holographic plates were obtained through Integraf L.L.C. and stored at room temperature. Kodak D-19 developer and Kodak Photo-Flo were obtained from Sigma-Aldrich and were prepared according to the manufacturer’s instructions. A plate-bleaching solution was prepared using 35 g of copper sulfate pentahydrate, 100 g of potassium bromide, 11020

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where n0 is the mean refractive index of the sample, C1 is the concentration of solute, C2 is the concentration of surfactant, and C̅ 1 and C̅ 2 are the mean concentrations of the solute or surfactant, respectively. The refractive index increments of the solute and surfactant, R1 and R2, respectively, are the first derivative of the refractive index with respect to the concentration of that component. Because the refractive index varies linearly with concentration, then according to eq 6, the fringes develop in response to concentration gradients of the solute and the surfactant that exist only in the ydirection. To set up this unidirectional transport process, one solution was added to a modified cuvette. Next, a second solution, with slightly different composition and slightly higher density, was injected below the first. A cut-away illustration of the modified cuvette is shown in Figure 2. A 2 in. × 16 gauge stainless steel dispensing needle was

5 g of sodium bisulfate monohydrate (all from Sigma-Aldrich), and 1000 mL of deionized water. The microemulsions were prepared by mixing the surfactant, solute, and deionized water in a glass vial. The contents were then stirred with a magnetic stir-bar at room temperature for at least 24h prior to a measurement. All samples were visually inspected to verify that only a single, clear, isotropic phase was present. Holographic Interferometry. The gradient-diffusion of solute within the microemulsion was monitored by holographic interferometry, using an apparatus that was described previously.16,21 A diagram of the experimental setup is shown in Figure 1. A 20 mW HeNe laser

Figure 1. Diagram of the holographic interferometer. was used to create a coherent monochromatic light source at a wavelength of 633 nm. This source was split and expanded to create two identical beams roughly 6 cm in diameter. One of these beams, the object beam, passes through a diffusion cell and becomes phase shifted. The second beam, the reference beam, bypasses the diffusion cell and retains the same phase as the source. A holographic plate is exposed to both beams at time te for a duration of 20 ms, and after the plate has been developed, the object beam at te can be then be reconstructed at any time later by illuminating the plate with the reference beam. The holographically reconstructed beam and the object beam at a later time ti are superimposed on a projection screen, where any subsequent interference can be recorded digitally. The phase shift of the object beam is proportional to the refractive index of the sample within the diffusion cell. As a consequence, interference with the reference beam depends directly on the change in the refractive index profile of the sample between te and ti. Because of the evolving concentration variations within the sample, this difference also depends on vertical position y. For convenience, in our experiments the object beam at ti is tilted a small angle θ, which introduces a linear, horizontal, x dependence into the intensity pattern. The intensity Ii of the interference pattern at time ti can then be represented mathematically by21

{ 2πλL [n(y,t ) − n(y,t )] + 4πθλx }

Ii = A2 cos

i

e

Figure 2. Diagram of the custom cuvette. Optical axis points into the page. attached to one corner of a standard 40 mm × 10 mm × 5 mm rectangular cuvette by using a drop of UV curing adhesive. The tip or outlet of the needle was positioned so that it was approximately 1 mm from the bottom of the cuvette. The inlet of the needle was attached to a syringe pump. The interface between the upper and lower liquids was visible during the filling process and was monitored to be sure there is no convection. If any convection develops later in the experiment, it would be immediately apparent, because the fringes move much more rapidly than would be the case for diffusion. On top of the cuvette was placed a standard PTFE cap whose bottom was machined to be slightly spherically concave and to which was attached a 1/16 in. i.d. PTFE outlet tube. The cuvette was first filled completely with the less dense solution, so that all the gaseous headspace was removed. The presence of any headspace can cause a significant loss of any volatile solute to the upper portion of the cuvette, creating a density-driven flow within the sample, and ultimately invalidating the experiment. The second, slightly more dense solution, was then pumped into the bottom of the cuvette via the attached needle, displacing the lighter solution through the outlet tube, and creating a flat interface between the two solutions. The time t0 at which the two solutions came into contact was recorded and taken as the starting point for the diffusion of solute or surfactant from higher concentration to lower. The two solutions were prepared so as to have solute or surfactant concentrations that differed by 0.03 M. The more dense solution was continuously pumped into the

(5)

where A is the amplitude of the light wave, λ is the vacuum wavelength of the laser, L is the path length of the cuvette (5 mm), and n(y,t) is the refractive index of the sample at the vertical position y within the diffusion cell at time t. To track the interference pattern, only the minima of the dark bands (i.e., the “fringes”) are of interest, so one can express the horizontal position xj of the jth fringe as a function of y, the position where Ii in eq 5 is minimized:

xj =

(2j − 1)λ L − [n(y ,t i) − n(y ,te)] 4θ 2θ

(6)

The refractive indices used in eq 6 are linearly related to the concentration of surfactant and solute within the diffusion cell as long as the concentration of either component does not deviate far from the mean concentration. In other words, the refractive index n can be expressed by a linear relationship of the form

n = n0 + R1(C1 − C1̅ ) + R 2(C2 − C̅ 2)

j = 1, 2, ...

(7) 11021

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bottom of the cuvette until the interface between the top and bottom solutions reached halfway up the cuvette, and the flow rate was controlled so that the interface remained flat during filling. The micellar systems we studied in this work, SDS and C12E10, have approximately spherical shapes and only weak tendencies toward nonspherical growth with the changes in solution conditions used in these experiments. At concentrations similar to those used here and with no added salt, SDS micelles increase in radius by less than 17% with addition of heptane34 or octanol,35−37 with an even smaller effect with pentanol.35,37 Aggregation numbers vary by less than 10% over an SDS concentration range of 0.15−0.25 M.31 C12E10 micelles have been investigated by only a few researchers,29,32,38 but these results indicate that its properties are similar to those of other micelles such as C12E8 and C10E8, which similarly remain nearly spherical with only small (∼20%) increases in radius with a change in surfactant concentration39,40 or alcohol addition.41,42 Thus, assuming constant properties over the small concentration variation around the average composition of the diffusion cell is well-justified. The difference in the time between the contact of the top and bottom solutions and the time that the holographic plate was exposed to the laser, i.e. te, was varied to allow the refractive index gradient to relax sufficiently to reduce the mirage effect near the interface of the two solutions.43−45 The time te was selected to range from 30 to 80 min and generally depended on the rate that the refractive index gradient near the interface decreased with time, which in turn depends on the magnitude of the solute and surfactant diffusivities. The cuvette remained at room temperature, 23.4 ± 0.7°C, throughout the holographic interferometry experiments. Although in previous studies of gel diffusion a water-cooled copper cuvette holder was used to control temperature,5 it was found in the current studies that this approach created temperature gradients large enough to disrupt the flat interface. Refractive Index. The refractive index of aqueous solutions of pentanol or octanol and SDS was measured with a Extech RF11 portable refractometer at several solute and surfactant concentrations. The refractometer was calibrated with deionized water. All solutions were equilibrated at room temperature (21.1 °C) for at least 24 h prior to measurement, and the refractive indices were then measured at room temperature. The refractive indices were corrected to 23.0 °C, which is the approximate temperature at which the diffusion experiments were conducted in the holographic interferometer.

Figure 3. Typical interference pattern of pentanol diffusing in an SDS microemulsion (a) without and (b) with the extracted interference fringes.

interference pattern at each of k rows of pixels. A 1-D Fourier transform along each vertical row of pixels was used to obtain an approximate νk that was provided as a guess to the search routine. The initial phase shift δ0 for the first row of pixels was set to zero. The phase shift initialization values for the succeeding row were taken from the solution to the previous row. Given the values of νk and the phase shift obtained from the search program, the location of the minima (i.e., the fringes) can be obtained from eq 6. A typical extracted fringe pattern is shown in Figure 3b, which shows the extracted fringes superimposed on the interference pattern of Figure 3a that was used for the analysis. Fringe Analysis for Binary Diffusion. In previous studies, holographic interferometry has been used to monitor binary diffusion, in solutions in which there is only one solute, or in which multicomponent interactions are negligible.5,16,21,25,26,46,47 Based on the solution for one-dimensional, time-dependent diffusion, the refractive index profile produced by the binary diffusion of a solute diffusing between two semiinfinite slabs is

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THEORY: FRINGE ANALYSIS The position of the fringes from a digitally recorded interference pattern can be found through visual inspection. Indeed, as discussed below, for binary diffusion a knowledge of the position of the minimum and maximum is sufficient to permit calculation of a diffusion coefficient. However, to increase the precision of the fringe location, in this work a new numerical method was developed that makes use of the entire interference fringe, rather than only the extrema. This approach is more accurate for the analysis of binary diffusion and is also flexible enough to be applied to the more complicated theory needed to analyze multicomponent diffusion. Note from eq 5 that the first term in the argument of the cosine is constant with respect to x at a fixed value of yk, whereas the second term is linear in x, so the intensity at a fixed vertical position can be expressed as I(x ,yk ) = A2 cos(δk + νkx)

n(y ,t ) = n0 +

⎛y − y ⎞ R1ΔC1 int ⎟⎟ erf ⎜⎜ 2 ⎝ 4D bt ⎠

(9)

Here ΔC1 is the initial concentration difference of the solute between the top and bottom solutions, Db is the “effective” binary diffusivity of the solute, yint is the location of the interface, and erf(z) is the error function. Combining eq 9 with eq 6 yields xj =

(8)

where δk is the phase shift at vertical position yk and νk is proportional to the wavenumber. This dependence can also be observed visually in Figure 3a, where the dark bands remain equally spaced left-to-right, but the locations of the bands change along the vertical axis of the pattern. A search routine that varied A2, δk, and νk was used to fit eq 8 to each

⎡ (2j − 1)λ L R1ΔC1 ⎢ ⎛ y − yint ⎞ ⎟⎟ erf ⎜⎜ − 4θ 2θ 2 ⎢⎣ ⎝ 4D bti ⎠ ⎛ y − y ⎞⎤ int ⎥ ⎟⎟ − erf ⎜⎜ ⎝ 4D bte ⎠⎥⎦

(10)

which was fit to the measured interference fringes to obtain the effective binary diffusivity. A nonlinear least-squares approach 11022

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with Newton’s method was used to obtain Db and yint. One noteworthy trait of the fringes created through the binary diffusion of a solute is that there is only a single minimum and maximum; i.e., the equation ∂xj/∂y = 0 has only two solutions for y. Fringe Analysis for Ternary Diffusion. For ternary systems, the flux of one species can be affected by the concentration gradient of another, as shown in eqs 3 and 4. The analytical solution for the concentration profiles for ternary diffusion within two semi-infinite slabs in contact, using eqs 3 and 4 to describe the fluxes, is provided by Fujita and Gosting.48 Combining this analytical solution and eq 7 allows the refractive index to be written as

elements of the diffusivity matrix were found by solving eqs 13−16 to yield

R1ΔC1 + R 2ΔC2 2

(13)

D+ = D11 + D22 +

(D22 − D11)2 + 4D12D21

(14)

D22 +

b=

R1 D R 2 12

− D11 −

(15) R2 D R1 21

D+ − D−

(16)

and

α1 =

R1ΔC1 R1ΔC1 + R 2ΔC2

(17)

In eqs 12 and 17, ΔC1 and ΔC2 are the initial concentration differences between the top and bottom solutions for the solute and surfactant, respectively. Substituting eq 11 into eq 6 produces the governing equation describing the fringe positions: xj =

⎧ ⎡ ⎛y−y ⎞ ⎛ y − y ⎞⎤ ⎪ (2j − 1)λ L int int ⎥ ⎟⎟ ⎟⎟ − erf ⎜⎜ − Δn⎨(a + bα1)⎢erf ⎜⎜ ⎢⎣ ⎝ 4D−t i ⎠ 4θ 2θ ⎪ 4 D te ⎠⎥⎦ ⎝ − ⎩ ⎡ ⎛y − y ⎞ ⎛ y − y ⎞⎤⎫ ⎪ int ⎟ int ⎟⎥ + (1 − a − bα1)⎢erf ⎜⎜ − erf ⎜⎜ ⎟ ⎬ ⎟ ⎪ ⎢⎣ ⎝ 4D+t i ⎠ ⎝ 4D+te ⎠⎥⎦⎭

(20)

R 2 (D+ − D−)a(1 − a) R1 b

(a + b)(1 − a)D+ − a(1 − a − b)D− b

(21)

(22)

RESULTS The refractive indices of pentanol−SDS solutions were measured at pentanol concentrations ranging from 0 to 0.45 M, at pentanol concentration steps of 0.11 M, and at SDS concentrations ranging from 0 to 0.35 M, at steps of 0.04 M. Similarly, the refractive indices of octanol−SDS solutions were measured at octanol concentrations over the range 0−0.06 M, at steps of 0.015 M, and at SDS concentrations over the range 0−0.35 M, at steps of 0.04M. The refractive indices of SDS− heptane solutions were also measured, but a change in refractive index with heptane concentration was not evident due to the resolution of the refractometer and the relatively small heptane concentrations that could be solubilized within these solutions. Measuring the refractive index of C12E10 solutions at surfactant concentrations of 0.2 M was found to be impossible due to the limitations of the refractometer and the relatively high refractive index of these solutions. The refractive index of the solute−surfactant solutions in all cases showed a linear relationship with concentration in both SDS and the solute. The refractive index increments for solute and surfactant were determined by a multiple linear regression to these data points. The two slopes from this analysis are equal to the refractive index increments, R1 and R2. The coefficient of determination R2 was found to be above 0.999 for pentanol in SDS and octanol in SDS, indicating that the refractive index data are linear at these conditions. Diffusion measurements were performed for binary and ternary aqueous solutions formed from pentanol, octanol, heptane, SDS, and C12E10. The binary solutions all yielded interference fringes with a single maximum and minimum, as expected from the development above. These results were analyzed by using eq 10. In addition, three of the solute− surfactant ternary solutions measured in this study also exhibited interferometric fringes with only a single minimum and maximum. The effective or pseudobinary diffusion coefficient Deff of solutes in the latter solutions were calculated for purposes of comparison. All of the results for ternary systems, containing a hydrophobic solute in a microemulsion, were analyzed with the multicomponent analysis and eqs 18−22. Solute diffusion can then be shown to be truly

R1 D R 2 12

D+ − D−

R1 (D+ − D−)(a + b)(1 − a − b) R2 b

■

(12)

(D22 − D11)2 + 4D12D21

a=

D21 =

The MATLAB code that was used to locate interference fringes via eq 8 and perform the binary and ternary analyses via eqs 10 and 18, is available in the appendices of Musnicki.49 Note that the cross-terms, D12 and D21, are the only terms that depend on the ratio of the refractive index increments. The main terms, on the diagonal of the diffusivity matrix, are unaffected by values of R1 and R2.

(11)

D− = D11 + D22 −

D+ − D22 −

(19)

D22 =

where Δn =

(a + b)(1 − a)D− − a(1 − a − b)D+ b

D12 = −

⎡ ⎛ y − yint ⎞ n(y ,t ) = n0 + Δn⎢(a + bα1) erf ⎜ ⎟ ⎢⎣ ⎝ 4D−t ⎠ ⎛ y − y ⎞⎤ int ⎥ ⎟⎟ + (1 − a − bα1) erf ⎜⎜ ⎝ 4D+t ⎠⎥⎦

D11 =

(18)

To determine the four parameters a, b, D−, and D+ in eq 18, two separate experiments with different values of α1 were used. Some experiments were conducted in which the initial concentration difference was only in the solute (ΔC2 = 0, α1 = 1), and others in which the initial concentration difference was only in the surfactant (ΔC1 = 0, α1 = 0). Both sets of the resulting experimental interference fringes were fit simultaneously to eq 18. The parameters D−, D+, yint, a, b, R1, and R2 were obtained using a nonlinear least-squares approach with Newton’s method. After the parameters were obtained, the 11023

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pseudobinary if D21 is negligible. A nonzero value of D21 indicates that there is a flux of surfactant (SDS or C12E10) coupled to the concentration gradient of solute (pentanol, octanol, or heptane). With this coupled flux, the pseudobinary diffusivity no longer depends only on the solute’s diffusion pathway, invalidating the pseudobinary approximation. For three of the solute−surfactant pairs studied, the interference fringes exhibited two sets of maxima and minima, making it impossible to fit the extracted fringes using a binary analysis. When there is more than one minimum and maximum, the cross-terms, D21, must be nonzero, and therefore there must also be a coupled flux between the surfactant and solute. Thus, holographic interferometry provided useful visual evidence of the importance of multicomponent effects, even prior to performing the full analysis. The results that follow are divided into three sections. The first section presents measured values of the binary diffusion coefficients for the individual solute molecules in water (i.e., values for Da) and the binary diffusion coefficients of SDS and C12E10 micelles (i.e., solute-free values for Dm). In the second section, the pseudobinary analysis is applied to the gradient diffusion of the three solute-surfactant pairs that exhibited fringes with one minimum and maximum. The effective diffusion coefficient for each solute in each type of microemulsion is reported; in the Discussion the validity of the pseudobinary diffusion coefficients are analyzed. The final section of results presents the ternary diffusion coefficients for all sets of solute-surfactant pairs, with the analysis of the selfterm diffusivities and the implications of nonzero values for the cross-term diffusivities given in the Discussion. Binary Diffusion of Solute and Micelles. Binary solute diffusion coefficients Da were measured for solutions of pentanol and octanol in water. The diffusivity of heptane in water could not be measured due to its very low aqueous solubility, and the resulting inability to produce a large enough heptane concentration difference. The concentration difference in these binary experiments was created by contacting pure water with a solution of water saturated with the respective solute. An interference pattern that is representative of all binary diffusion experiments can be seen in Figure 4a. Extracting the fringe positions using the procedure described above, and fitting the result to eq 10, yielded the diffusion coefficients given in Table 1. The coefficient of determination R2 was calculated for the fit to each fringe, and because several

Table 1. Solute (Saturated) and Surfactant (0.2 M) Binary Diffusivities in Water Da or Dm (×10−6 cm2/s) solute/surfactant

measured

literature

reference

pentanol octanol heptane SDS C12E10

8.1 5.5

8.6 5.9 7.0 5.1 0.9

Leaist22 Leaist22 Oelkers51 Leaist50 Kong et al.16

4.9 1.1

fringe patterns were taken over the course of an experiment, the average R2 for the entire experiment was also calculated. For each solute, the fit to the interference fringes yielded R2 values greater than 0.98, indicating excellent agreement between the results and eq 10. Micelle diffusivities Dm for SDS and C12E10, at a surfactant concentration of 0.20 M, are included in Table 1. A concentration difference of 0.03 M was used to measure these micelle diffusivities. Values obtained in the literature5,16,22,50,51 are presented for comparison, and are in good agreement with the measured values. We note that the diffusivity of heptane was calculated by Oelkers,51 rather than being measured. In addition, the literature result quoted for C12E10 micelles was measured by Kong et al.,16 and corrected for comparison at 0.2 M by using eq 79 in Buck et al.5 Pseudobinary Analysis. Diffusion experiments with pentanol and octanol in SDS, and pentanol in C12E10, yielded interference patterns with one minimum and one maximum. A typical example is shown in Figure 4b. The close similarities between Figure 4b and the binary diffusion interferometric pattern shown in Figure 4a suggest that a pseudobinary approximation might be applied to analyze these experiments. For octanol diffusing in C12E10, and for heptane in either SDS or C12E10, the fringes, as illustrated in Figure 4c, had more than one maximum and minimum. They therefore differed qualitatively from the type of curve predicted by eq 10 and could not be treated as a pseudobinary system. The effective binary diffusivities for pentanol and octanol in SDS, and pentanol in C12E10, are reported in Table 2. The value Table 2. Measured Effective Binary Diffusivities with 0.2 M Surfactant solute

surfactant

C1 (M)

Deff (×10−6 cm2/s)

pentanol octanol pentanol

SDS SDS C12E10

0.20 0.03 0.20

10.2 1.8 4.7

of R2 was again calculated for the fit to each fringe, and the average values were all above 0.90, indicating that the regressed curves fit the extracted fringes very well. The solute concentrations C1 are given in the table; the surfactant concentrations C2 were 0.2 M. Ternary Analysis. For several of the systems under investigation, the holographic interferometric fringes gave immediate visual evidence of significant multicomponent effects in the diffusion process (Table 1). These experiments yielded interference patterns with four peaks (two maxima and two minima) as shown, for example, in Figure 4c. Although the presence of four peaks clearly indicates that multicomponent effects are present, it does not necessarily follow that multicomponent effects are absent if only two peaks are present, as discussed in the next section. The ternary analysis

Figure 4. Interferometric pattern of (a) the binary diffusion of SDS micelles, (b) pentanol diffusion in an SDS microemulsion, and (c) heptane diffusion in an SDS microemulsion. 11024

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Table 3. Ternary Parameters (C2 = 0.2 M) solute

surfactant

C1 (M)

D− (×10−6 cm2/s)

D+ (×10−6 cm2/s)

a

b

R1/R2

pentanol octanol heptane pentanol octanol heptane

SDS SDS SDS C12E10 C12E10 C12E10

0.20 0.03 0.03 0.20 0.13 0.03

4.3 0.8 0.5 1.0 0.2 0.2

6.2 5.1 5.7 4.8 0.9 1.1

0.86 −0.03 −0.14 0.99 0.00 −0.01

−3.84 0.65 14.9 −0.96 14.9 15.2

0.34 0.44 0.08 0.13 0.09 0.02

was therefore used to analyze the patterns for all sets of solutes and surfactants, regardless of the number of peaks displayed. The fringes for all solute−surfactant pairs were successfully fit, even ones with multiple maxima and minima such as the one shown in Figure 4c. The four parameters D−, D+, a, and b, of the sets of experiments for each solute−surfactant pair, are given in Table 3. The average coefficient of determination R2 was also calculated for the ternary fits. The values of R2 for the ternary fits were all greater than 0.90, again showing that the fitting procedure represents the extracted fringes very well. Because the refractive index increments R1 and R2 could not all be measured independently, they were extracted from the fit for each set of experiments using the appropriate values of ΔC1 and ΔC2. The ratio of the two is shown in Table 3. For pentanol and octanol, the refractive index increments, R1 and R2, were also measured independently with a refractometer as described above. The results are presented in Table 4, along

Table 5. Ternary Diffusion Coefficients (C2 = 0.2 M)

solute pentanol octanol

R1 (×10

−1

M )

9.8 ± 0.3 16.6 ± 1.4

−3

R2 (×10

−1

M )

38.2 ± 0.3 38.9 ± 0.2

surfactant

pentanol octanol heptane pentanol octanol heptane

SDS SDS SDS C12E10 C12E10 C12E10

0.20 0.03 0.03 0.20 0.13 0.03

D11 (×10−6 cm2/s)

D21 (×10−6 cm2/s)

D12 (×10−6 cm2/s)

D22 (×10−6 cm2/s)

6.0 0.9 0.0 4.8 0.2 0.2

1.1 0.7 −4.0 0.0 −0.6 −0.3

0.7 0.5 0.7 0.4 0.0 0.0

4.5 5.0 5.7 1.0 0.9 1.1

two-thirds of the main term D22 in the case of heptane in SDS or octanol in C12E10. In three cases the coupling coefficients D21 are negative, whereas in two cases they are positive. The largest values, relative to D22, are negative, indicating a flux of surfactant up the gradient in solute concentration. These trends are discussed further below.

■

Table 4. Measured Refractive Index Increments in SDS −3

solute

C1 (M)

DISCUSSION Binary and Pseudobinary Results. The measured binary diffusivities for pentanol and octanol in Table 1 agree with values reported in the literature to within 10%, providing further validation of the experimental method. As with the pentanol and octanol solutes, there is good agreement between our holographic interferometry values and those obtained by other methods for SDS. We note that the measured diffusion coefficient of the SDS micelles is roughly 2−3 times larger than the value at the CMC, a difference that can be attributed to the charged headgroups of SDS and the resulting electrostatic micelle−micelle interactions at this finite concentration.6 A similar but much weaker effect influences the uncharged C12E10 micelles.5 The infinite-dilution diffusion coefficient of C12E10 micelles measured by Kong et al.,16 0.69 × 10−6 cm2/s, is roughly 30% smaller than the diffusivity measured here at a surfactant concentration of 0.2 M. The higher value measured here results from an alteration of the thermodynamic driving force and hydrodynamic resistance to diffusion and is caused by intermicellar interactions that include a steric repulsion between micelles, as well as a repulsion due to the overlap forces of the surfactant headgroups.5 Upon comparing the results in Tables 1 and 2, one finds that the value of the effective binary diffusion coefficient Deff obtained for pentanol in the SDS solution is higher than that of molecular pentanol in water (Table 1). The value of Deff for octanol in SDS solution, on the other hand, is lower than the micelle diffusion coefficient Dm in Table 1. These results are inconsistent with eq 2, which indicates that Dm < Deff < Da. Measurements of aggregation properties for octanol-containing SDS micelles,36 near the concentrations we use, predict only a moderate increase in the micelle radius, and therefore a micelle diffusion coefficient that is about 10% lower than that of solutefree micelles. This change is much smaller than what is reported in Table 2.

R1/R2 0.26 ± 0.01 0.43 ± 0.04

with the ratio of R1 to R2. The error bounds provided in Table 4 for the refractive index increments are the standard deviations recovered from the regression analysis. Error estimates for the ratio R1/R2 were determined by propagation of errors, using the standard deviations in R1 and R2. The value of R1/R2 for octanol in SDS agrees quantitatively both with the value in Table 3, and the value from Leaist and Hao.12 The independently measured value of R1/R2 for pentanol in SDS is roughly 30% smaller than the value obtained from fitting the holographic interferometric fringes. Notice that in eqs 19−22, the value of R1/R2 only affects the terms D21 and D12, so the propagation of this difference is limited to those terms. Because the terms D21 and D12 are proportional and inversely proportional to R1/R2, respectively, a 30% decrease in R1/R2 would then decrease |D21| by 30% and increase |D12| by 40%. The ternary diffusion coefficients D11, D12, D21, and D22 for all solutes and surfactants were calculated using the parameters from Table 3 and eqs 19−22. These ternary diffusivities are presented in Table 5. All values of D22 lie within 10% of the binary micelle diffusion coefficients for SDS and C12E10 given in Table 1. The values of D11 for pentanol in SDS and C12E10 microemulsions are consistent with what one would intuitively expect, because they are less than the diffusion coefficient of pentanol alone, and greater than the micelle diffusion coefficient. However, octanol and heptane in solutions of both SDS and C12E10 have values of D11 that are surprisingly low, smaller than even the micelle diffusivity, which one would expect to be the slowest diffusing component. By contrast, the magnitudes of the coupling term D21 are quite large, roughly 11025

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concentration gradient in heptane can produce a flux of SDS that is larger than the flux of heptane in the reverse direction. These relatively large values of D21 for most solute−surfactant pairs indicate that the pseudobinary approximation cannot be used to describe the gradient-diffusion of hydrophobic compounds in microemulsions. The only exception is pentanol diffusing in a C12E10 microemulsion, where the value of D21 is equal to zero, and therefore the pseudobinary assumption is truly valid. It is worth noting that nonzero cross-terms and significant multicomponent effects are present in the C12E10 mixtures, despite the absence of micelle charge and counterion contributions in these nonionic systems. Nonzero values of D12, on the other hand, indicate a counteror co-transport of solute down the concentration gradient of surfactant. However, the measured values of D12 were found to be either slightly positive or zero, which suggests that the flux of solute produced by a concentration gradient in surfactant is generally a minor effect. Because |D12|/|D22| is only 10−12%, the change in the refractive index profile over time when α1 = 0 (or ΔC1 = 0, eq 17) is due primarily to the flux of surfactant; i.e., the experiment is pseudobinary. The micelle diffusion coefficient Dm is therefore very close to D22. That D22 and Dm are approximately equal may be attributed to the fact that the transport of surfactant is mainly due to the diffusion of the microemulsion particles. It is clear that results such as those in Table 5 have important implications for applications involving solute migration or release via micelle carriers. Formulation of poorly water-soluble drugs provides one example with concentrations and solute hydrophobicity similar to those of the systems investigated here. Pharmaceutical agents dissolved in micelles (see, e.g., Savic et al.52), when exposed to tissues or other material low in both solute and surfactant,53 might at first be expected to exhibit transport at rates proportional to the solute concentration and the micelle diffusion coefficient, as predicted in eq 2. Instead, a mixture with properties like those of heptane and SDS in Table 5 would have a solute flux independent of the solute loading; this flux would be given by −D12∇C2 as specified by eq 3. Implications of Irreversible Thermodynamics. The fluxes of solute (component 1) and surfactant (component 2) can be related to their gradients in chemical potential, ∇μ1 and ∇μ2, respectively, by the relation54

These surprisingly high and low values of Deff in the SDS solutions are likely due to a coupled flux of surfactant caused by the concentration gradient in the solute. The co-transport of surfactant caused by a gradient in solute would contribute to the refractive index profile, and because it is unaccounted for in the binary analysis, it would then be recorded as an artificially large value of Deff. For octanol, the surfactant may instead be “counter-transported” by a gradient in the solute, i.e., transported in the direction opposite to its own gradient. Then there would again be a contribution of surfactant to the refractive index profile, which would be seen as an artificially low value of Deff. These explanations are borne out by the ternary analysis below. Both the proposed co-transport (D21 > 0) and the counter-transport (D21 < 0) of surfactant indicate that a multicomponent analysis using eqs 3 and 4 would be more appropriate for these systems. Only for the solution of pentanol in C12E10 does Deff fall between the binary diffusivity values of the solute and micelles. Ternary Results. There is limited ternary diffusivity data in the literature for surfactant and solute solutions, especially at concentrations sufficient to form a microemulsion. However, a comparison can be made between the SDS−octanol ternary diffusivities in Table 5 and the octanol−SDS ternary Taylortube dispersion diffusivities collected by Leaist and Hao.12 The latter were performed at an SDS concentration of 0.1 M, which is half of the concentration used to obtain the diffusivities in this study. Nonetheless, the diagonal terms D11 and D22 that they measured agree quantitatively with the diffusivities in Table 5. Even the value of D11 in Table 5, which for octanol in SDS is much lower than the measured solute-free micelle diffusion coefficient, is in good agreement with the Taylor-tube dispersion result. The agreement between these two very different methods of measuring rates of diffusion provides confirmation of both sets of results. The cross-terms, D12 and D21, also show good agreement with the Taylor-tube measurements if one compares results obtained at similar solute-tosurfactant ratios. However, this agreement does not extend to other ratios or to lower SDS concentrations. In the Taylor-tube experiments,12 the cross-terms were found to be strongly influenced by the concentration of the surfactant. The positive value of D21 for pentanol and octanol in SDS (cf. Table 5) represents a flux of surfactant that is induced by the solute concentration gradient. This coupling, and its effect on the refractive index, is likely the reason for the unusually large value of the effective binary diffusivity of pentanol measured in a pentanol-SDS microemulsion, as given in Table 2. Because the effective binary diffusivity of octanol in an octanol−SDS microemulsion was lower than Dm, the lower bound predicted by eq 2, co-transport of SDS along an octanol concentration gradient does not similarly explain the low effective binary diffusivity of octanol in Table 2. However, in the octanol−SDS system the coefficient D11 is much lower than the micelle diffusion coefficient. In the ternary system, therefore, the flux of octanol down a gradient in octanol concentration is lower than what one would predict if the octanol molecules were divided among weakly interacting SDS micelles, and interactions between the solute and surfactant were otherwise negligible. The negative value of D21 for octanol in solutions of C12E10, and for heptane in solutions of both surfactants, indicates a flux of surfactant up the concentration gradient of the solute. In the case of heptane in SDS, the magnitude of D21 is comparable to the main terms D11 and D22 and indicates that an imposed

⎡ ∂μ ⎤ ⎢ 1⎥ ⎡ J1,0 ⎤ ⎡ L11 L12 ⎤⎢ ∂y ⎥ ⎢ ⎥ = −⎢ ⎥⎢ ⎥ ⎢⎣ J2,0 ⎥⎦ ⎣ L 21 L 22 ⎦⎢ ∂μ2 ⎥ ⎢⎣ ∂y ⎥⎦

(23)

Here Ji,0 is the molar flux of the ith component, relative to a solvent-fixed reference frame. Although a solution of surfactant and solute in water has three components, the three equations governing the fluxes are not independent and can be simplified to the form shown in eq 23 by using the Gibbs−Duhem equation. For solutions with charged surfactants, in the absence of any added salt, the concentrations and fluxes of the surfactant and counterions are coupled by electroneutrality and the requirement that there be no net current. These requirements provide expressions for the diffusion potential that is caused by the different diffusivities of the charged species, and again the fluxes in the overall system are described by eq 23. 11026

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according to a Poisson distribution, so that the average number of solutes per micelle ⟨i⟩ is

To obtain a generalized form of Fick’s law, as shown in eqs 3 and 4, the gradients in chemical potential in eq 23 must be expressed in terms of gradients in the molar concentration, yielding ⎡ J1,0 ⎤ ⎡L ⎢ ⎥ = −⎢ 11 ⎢⎣ J2,0 ⎥⎦ ⎣ L 21

⎡ ∂μ ∂μ ⎤⎡ ∂C ⎤ 1 ⎢ 1 ⎥⎢ 1 ⎥ L12 ⎤⎢ ∂C1 ∂C2 ⎥⎢ ∂y ⎥ ⎥⎢ ⎥ L 22 ⎦⎢ ∂μ2 ∂μ2 ⎥⎢ ∂C2 ⎥ ⎢ ⎥ ⎢⎣ ∂C1 ∂C2 ⎥⎦⎢⎣ ∂y ⎥⎦

⟨i⟩ =

⎡ D11 D12 ⎤ ⎡ L11 ⎢ ⎥=⎢ ⎣ D21 D22 ⎦ ⎣ L 21

μ2 − μ2⊖

(24)

RT

C1 1 + (K s,c − 1)ϕ

μ2 − μ2⊖ RT

(25)

RT

= ln Cas = ln C1 − ln[1 + (K s,c − 1)ϕ]

(26)

(27)

⊖

In eq 27, μ is a reference chemical potential and ϕ is the volume fraction of the microemulsion droplets. At the surfactant concentrations used in this work, the CMC is negligible relative to C2, and the surfactant is therefore distributed among solute-free micelles or micelles containing varying numbers of solute molecules. If the local average micelle aggregation number is denoted by m, then the surfactant distribution is such that C2 = Cm + m

∞

∑ CMD

i

i=1

⎤ K s,cϕ C ⎡ γC 1 ⎥ ln 2 − 1 ⎢ m m C2 ⎢⎣ 1 + (K s,c − 1)ϕ ⎥⎦

(30)

=

1 ln γCm m

(31)

Equations 27 and 30 provide a basis for evaluation of the derivatives of the chemical potentials in eq 25. Our experimental results indicate that the solute may have a significant impact on the chemical potential of the surfactant. In differentiating the chemical potentials in eqs 27 and 30, we therefore extend the work of Zhang and Annunziata13 by letting the activity coefficient γ and aggregation number m vary with solute concentration C1. Changes in the aggregation number of SDS in the presence of the alcohol and alkane solutes in Table 5 have been measured experimentally.34−37 Although independent measurements of m for C12E10 micelles are not available, data by Nakagawa et al.41 and Kuriyama42 for structurally similar CnEm surfactants and decanol or decane also show a dependence of aggregation number on the concentration of the hydrophobic solutes. These works provide support for a finite value for ∂m/∂C1. A dependence of the activity coefficient γ on the solute concentration could arise from nonideal mixing of the solute-free micelles with surrounding micellar particles; the contributions of intermicellar interactions to the activity coefficient are addressed by Zoeller et al.55 It is likely that such interactions would be affected by the concentration of solute in the surrounding particles, giving rise to a variation of the activity coefficient with C1. We note that the variations in the activity coefficient and aggregation number are coupled by a result derivable from the Gibbs−Duhem equation, in the form of eq 4 in Zhang and Annunziata13,56 or eq 154 (Ch. XI) in the monograph by de Groot and Mazur.54 Differentiating μ1 and μ2 then yields

The solute concentration in the aqueous phase outside the microemulsion particles is typically low enough that it may be treated as an ideal solution. Because the solute in the aqueous phase is in equilibrium with the solute in the microemulsion droplets, their chemical potentials μ1 are equal and given by μ1 − μ1⊖

=

Here μ⊖ 2 is a reference chemical potential for the surfactant, and γ is the activity coefficient that relates the chemical potential μ2 of the surfactant to the micelle concentration Cm, according to

Physically, the coefficients Lij relate the flux of component i to the “thermodynamic force” ∂μj/∂y that acts on component j. The coefficients ∂μi/∂Cj relate those forces to gradients in concentration. Zhang and Annunziata13 propose a model of diffusion in which the solute molecules either diffuse individually, driven by the gradient in the solute chemical potential, or are transported as a result of solute-containing micelles that move in response to their own chemical potential gradient (cf. their eqs 13 and 14). They use a two-phase model to obtain approximate relationships between the chemical potentials and concentrations. In the two-phase model, the microemulsion droplets are treated as a separate phase in equilibrium with the surrounding solvent. Using the partition coefficient Ks,c defined in eq 1, the solute concentration Csa in the aqueous phase is related to the total solute concentration C1 in the two-phase mixture by Cas =

(29)

Zhang and Annunziata13 show that the chemical potential μ2 of the surfactant is given by

The product of the 2 × 2 matrices in eq 24 is the diffusivity matrix, ⎡ ∂μ ∂μ ⎤ 1 ⎢ 1 ⎥ L12 ⎤⎢ ∂C1 ∂C2 ⎥ ⎥⎢ ⎥ L 22 ⎦⎢ ∂μ2 ∂μ2 ⎥ ⎢⎣ ∂C1 ∂C2 ⎥⎦

K s,cϕ C1 (C2/m) 1 + ϕK s,c(ϕ − 1)

(28)

where Cm and CMDi are the concentration of solute-free micelles and micelles containing i solute molecules, respectively. With the assumption that the solute is divided among the micelles

C1 ∂μ1 =1 RT ∂C1

(32)

(K s,c − 1)ϕ C2 ∂μ1 =− RT ∂C2 1 + (K s,c − 1)ϕ

(33)

⎛ γC2 ⎞⎤ C1 ∂μ2 C ∂m ⎡ C1 ∂ ln γ ⎟⎥ + = − 12 ⎢1 + ln⎝⎜ ⎠ RT ∂C1 m ⎦ m ∂C1 m ∂C1 ⎣ ⎡ ⎤ K s,cϕ C ⎥ − 1⎢ C2 ⎢⎣ 1 + (K s,c − 1)ϕ ⎥⎦

(34)

and 11027

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2 C2 ∂μ2 C K s,c(K s,c − 1)ϕ β = + 1 RT ∂C2 m C2 (1 + ϕ(K s,c − 1))2

D22, V = (1 − ϕ)βDm (35)

Diffusion coefficients relative to a volume-fixed reference frame are thought to be the best approximation to experimental measurements.60 The diffusivity matrix predicted by eqs 45−48 shows some interesting qualitative trends. Recalling that the parameter β, given by eq 36, becomes unity in the limit of low surfactant concentrations, eq 48 indicates that the coefficient D22,V should be similar in magnitude to the binary, micellar diffusion coefficient Dm. In fact, comparison of the right column in Table 5 with the measured diffusivities for SDS and C12E10 micelles in Table 1 shows very good agreement with that prediction. In each case, D22,V is within 10% of the micellar diffusion coefficients. In addition, because the solute concentration C1 in experiments with octanol and heptane was small compared to the surfactant concentration C2, eq 46 implies that the coefficient D12,V should be small compared to D22,V. With the exception of pentanol in the C12E10 solution, the coefficients D12,V in Table 5 are 15% of D22,V or less. The parameter κ, which captures the effect of the solute concentration on the micelle chemical potential and aggregation number, appears in the expressions for D11,V and D21,V. If κ is zero, as in the development of Zhang and Annunziata,13 then these results predict that D21,V = 0. In Table 5, the coefficients D21,V are negative, D21,V < 0, for the heptane and octanol solutes in C12E10 solution, and for heptane in SDS solution. According to eq 47, this negative value of D21,V requires a negative value of κ, which in turn is predicted to reduce the magnitude of D11,V in eq 45. In fact, the results in Table 5 do show that negative values of D21,V are associated with surprisingly low values of D11,V. The most extreme example is the solution of heptane and SDS, for which D11,V is essentially reduced to zero. The same trend is also evident in the results for octanol and heptane in C12E10 solution. Similarly, the largest positive result for D21,V, indicating κ > 0, is for pentanol in SDS solution, which also shows the largest result for D11,V. However, the results for the alcohol solutes are not uniformly in agreement with eqs 45 and 47. The results for pentanol in C12E10 solution and octanol in SDS solution do not conform to the predicted trends. That the differences between eqs 45−48 and the results occur for the alcohol solutes, particularly pentanol, may indicate that the constraint Ks,cϕ ≫ 1 is less appropriate for those solutes.

where ∂ ln γ ∂C2

β=1+

(36)

Application of these results is rendered difficult by the need to know the partition coefficient Ks,c. However, for very hydrophobic solutes it is often the case that Ks,c ≫ 1 and Ks,cϕ ≫ 1. Here the micellar volume fraction ϕ is approximately 0.06. Then, based on partition coefficients reported in the literature,57−59 these criteria are satisfied for all of our solutes with the exception of pentanol, for which we estimate Ks,cϕ ∼ 1−10. In the large Ks,c limit, eq 32 is unchanged and eqs 33−35 simplify to C2 ∂μ1 = −1 RT ∂C2

(37)

⎛ γC2 ⎞⎤ C1 ∂μ2 C ∂ ln m ⎡ C1 ∂ ln γ C ⎟⎥ + =− 1 − 1 ⎢1 + ln⎝⎜ RT ∂C1 m ∂C1 ⎣ m ⎠⎦ m ∂C1 C2 (38)

and C2 ∂μ2 C β = + 1 RT ∂C2 m C2

(39)

Then, using eqs 17a−c in Zhang and Annunziata determine the Lij coefficients of eq 25, one finds D11,0 = Dm(1 + κ ) D12,0 =

to

(40)

C1 (β − 1)Dm C2

D21,0 = κ

13

(41)

C2 Dm C1

(42)

and D22,0 = βDm

(43)

where the parameter κ is defined by κ=

⎛ γC2 ⎞⎤ ∂ ln γ ∂ ln m ⎡ ⎟⎥ − ⎢1 + ln⎝⎜ m ⎠⎦ ∂ ln C1 ∂ ln C1 ⎣

■

SUMMARY Holographic interferometry was used to measure the diffusion of pentanol, octanol, and heptane in microemulsions created with the surfactants SDS or C12E10. A new method of analysis was developed to use the entire length of the interference fringes to obtain one or more diffusion coefficients. For surfactant solutions with the more hydrophilic solutes the interference fringes had a single minimum and maximum, similar to the case for solutes undergoing binary diffusion. However, these pseudobinary results, with the exception of the pentanol−C12E10 solutions, yielded solute diffusivities in the microemulsion that were either anomalously large or small, indicating that the pseudobinary diffusion coefficients were inherently inadequate to describe these systems, because solute−surfactant interactions were too strong. The importance of a multicomponent analysis for systems with the more hydrophobic solutes was shown more directly, by the presence of four peaks instead of two in the interference fringes. Our

(44)

In the analysis of Zhang and Annunziata,13 the dependence of γ and m on the solute concentration C1 was neglected, so that in their development, κ = 0. Converting from a solvent-fixed reference frame, denoted by the subscript 0 in eqs 40−43, to a volume-fixed frame of reference, denoted by a subscript V, gives13,60 D11, V = D11,0 = Dm(1 + κ ) D12, V =

(45)

C1 (β(1 − ϕ) − 1)Dm C2

D21, V = (1 − ϕ)D21,0 = κ(1 − ϕ)

(46)

C2 Dm C1

(48)

(47)

and 11028

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paper provides new evidence that when molecules are solubilized within surfactant solutions, a single effective diffusion coefficient is insufficient to describe the transport of either component. Contributions from cross-terms are substantial and qualitatively modify rates of transport in these mixtures. A ternary, multicomponent method was developed for use with these strongly interacting, ternary systems, thereby allowing interference fringes with multiple minima and maxima to be analyzed. The qualitative transport behavior was most affected by the multicomponent interactions in systems with the more hydrophobic solutes. In microemulsions with heptane and octanol, the values of D11 were found to be significantly lower than the micelle diffusion coefficient, whereas the magnitude of D21 was relatively large compared to D11. For heptane diffusing in a SDS microemulsion, the flux of surfactant caused by a solute concentration gradient was greater than the flux of the solute from its own gradient! For all solutions tested, the values of D22, relating the flux of surfactant to the gradient in surfactant concentration, were found to be similar to the corresponding micelle diffusion coefficients. Much of the qualitative behavior observed in the Dij coefficients can be interpreted by irreversible thermodynamics in conjunction with a two-phase model for the microemulsion.

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AUTHOR INFORMATION

Corresponding Author

*S. R. Dungan. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding for this research was provided by USDA CSREES Project #07-35603-17739.

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REFERENCES

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