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Dissolution Rates of Pure Nonionic Surfactants Bing-Hung Chen† and Clarence A. Miller* Department of Chemical Engineering, Rice University, Houston, Texas 77251-1892
John M. Walsh, Patrick B. Warren, J. Noel Ruddock, and Peter R. Garrett Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral L63 3JW, U.K.
Francoise Argoul and Christophe Leger Centre Recherche Paul Pascal, CNRS, Avenue Albert Schweitzer, 33600 Pessac, France Received October 13, 1999. In Final Form: March 21, 2000 Small drops of the pure linear alcohol ethoxylates C12E5 and C12E6 were injected into water at temperatures below their cloud points, and the times required for their dissolution were measured using videomicroscopy. Separately the rates of growth of the various liquid crystalline intermediate phases formed during penetration experiments with these surfactants in vertical linear cells were measured using videomicroscopy. It was found from both types of experiments that the dissolution process was controlled by diffusion, not by kinetics of phase transformation at interfaces. Effective diffusivities of the various phases were calculated from the data obtained and were found to be of order 10-10 m2/s. Finally, interferometry was used to measure concentration distributions as a function of time during dissolution of the lamellar phase of C12E5. Diffusivity in the micellar solution was found to increase with increasing surfactant concentration with a further increase occurring at the concentration where the lamellar phase formed. The results were consistent with the effective diffusivities determined from the videomicroscopy experiments and with available values in the literature obtained by other techniques.
Introduction For surfactants used in aqueous washing processes to be effective, they must dissolve completely in the washing bath in a time period much shorter than the washing time. The trends toward decreased water usage, which implies higher surfactant concentrations, and lower temperatures in household laundry processes have drawn attention recently to this rapid dissolution requirement. One issue is whether viscous liquid crystalline phases formed during dissolution of neat nonionic surfactants cause significant decreases in dissolution rates. Complicating factors exist such as the possibility that these mesophases can trap solid particles of builders such as zeolites. In the present work, however, we address only the simplest situation, e.g., dissolution of pure liquid nonionic surfactants. In one set of experiments described below videomicroscopy was used to measure (a) the overall dissolution times of small surfactant drops injected into water and (b) the rates of growth of the various liquid crystalline intermediate phases which formed when comparable volumes of neat surfactant and water were contacted in a long, linear cell. Using the information obtained from these experiments and theoretical work described below, we concluded that the dissolution process was controlled by diffusion, not by kinetics of phase transformation at interfaces, and determined the effective diffusion coefficients in the various phases which formed. These coefficients were found to be of order 10-10 m2/s for dissolution of pure C12E6 and C12E5 in the temperature * To whom correspondence should be addressed. † Present address: Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260.
range 25-30 °C. In the former case three intermediate liquid crystalline phases formed: lamellar, viscous isotropic, and hexagonal. In the latter case only the lamellar phase formed. In both cases the number and type of intermediate phases were consistent with the published phase diagrams.1-3 In another experiment interferometry was employed to measure concentration distributions as a function of time during dissolution of the lamellar phase LR of pure C12E5 at 26 °C. The data were used to calculate diffusivity D as a function of concentration. It was found that D increased with increasing surfactant concentration in the micellar solution L1 and increased further when transformation to LR occurred. Average values of D in both L1 and LR phases were in general agreement with the results of the videomicroscopy experiments. Moreover, results of both experiments were consistent with available information on diffusivities obtained by others in the same systems using different techniques. Theory If two semi-infinite phases in a binary system are brought into contact so that one-dimensional linear diffusion occurs with uniform density and no convection, the governing equation is
( )
∂ωi ∂2ωi ) Di ∂t ∂x2
(1)
(1) Mitchell, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T., McDonald, M. P. J. Chem. Soc., Faraday Trans. 1 1983, 79, 975. (2) Strey, R.; Schoma¨cker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86, 2253. (3) Strey, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 182.
10.1021/la9913497 CCC: $19.00 © 2000 American Chemical Society Published on Web 05/17/2000
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Here ωi is the mass fraction of one component, for instance surfactant, in phase i, Di is the corresponding diffusivity, which is assumed to be independent of composition, x is the distance from the initial surface of contact, and t is time. This equation has the following similarity solution applicable in both initial phases and in all intermediate phases formed:4
(
ωi ) Ai + Bi erf
)
x (4Dt)1/2
u(r,t) )
[ (
) (
( )
where u ) (ω - ω∞)/(ω0 - ω∞). At r ) 0 the concentration is given by
(2) u(0,t) ) erf
In this equation Ai and Bi are constants to be determined by the boundary conditions. These are that the initial compositions exist at x ) (∞ and that local equilibrium and conservation of mass of the surfactant apply at all interfaces. For consistency with the similarity solution, the boundary conditions should contain x and t only in terms of the single variable (x/t1/2), as in eq 2 above. Consideration of the surfactant conservation equations shows that this condition leads to a requirement that the position of interface j satisfy the following equation:
)]
r + r0 r - r0 1 erf - erf + 1/2 2 (4Dt) (4Dt)1/2 2 Dt 1/2 -(r+r0)2/4Dt (e - e-(r-r0) /4Dt) (5) 2 πr
(
r0
)
(4Dt)1/2
(6)
The dissolution time td in this case is obtained by setting u(0,td) ) u*, where u* is obtained by letting ω ) ω* in the definition for u given above. Experimental Section
The boundary conditions are (∂ωi/∂r) ) 0 at r ) 0, existence of the bulk composition ω∞ at very large r, and, as before, local equilibrium and conservation of mass of surfactant at each interface. The initial condition is ω ) ωo, the concentration in the neat surfactant, for r e ro and ω ) ω∞ for r g ro. The time td for complete dissolution is that for which the surfactant concentration at r ) 0 is ω*, the highest surfactant concentration possible in the aqueous micellar solution, according to the equilibrium phase diagram. A finite difference method was used to obtain the solution and, in particular, to determine td. For the present system with no discontinuity in concentration at the interfaces, according to available phase diagrams, and for the special case that all the Di are equal, i.e., the binary diffusion coefficient D is uniform, an analytical solution for the concentration distribution can be found using Fourier transforms:5
The pure linear alcohol ethoxylates n-dodecyl tri-, tetra-, penta-, and hexaoxyethylene monoethers (C12E3-C12E6) were obtained from Nikkol Chemical Company, Japan. Water was distilled and deionized with a SYBRON Barnstead glass still and Nanopure II system. Two types of dissolution experiments were conducted with videomicroscopy corresponding to the two situations considered in the preceding section. In one type a pure nonionic surfactant was carefully layered on water or a dilute solution of the same surfactant in a vertically oriented glass capillary cell having a thickness of 400 µm. Such experiments were performed at Rice with a microscope having a vertically oriented stage used for similar experiments in previous studies.6 The cell was maintained at a constant temperature using a thermal stage, and the position of the various interfaces was measured as a function of time using videomicroscopy. Similar experiments were conducted at Unilever Research, Port Sunlight Laboratory with some modifications. (a) The cell interior was made hydrophobic by soaking it first in “Sylon CT” for 1 min and then in methanol twice for 45 and 30 s, respectively, before drying with nitrogen. Sylon CT is a 5 vol % solution of dimethyldichlorosilane in toluene, available from Sigma-Aldrich. With this procedure interfacial curvature was reduced so that the surface of initial contact between surfactant and water was more nearly flat. (b) After the cell was half filled with nonionic surfactant, it was immersed in ice to freeze the surfactant. It was then placed in a refrigerator at a temperature below the surfactant freezing point. A glass pipet containing water and having a long tip which would fit inside the cell was placed in the same refrigerator. After the temperatures had equalized, the water was carefully injected from the pipet to fill the remainder of the cell, which was then placed in thermal stage at the desired temperature. Because some interfaces, especially that between the lamellar phase and the liquid surfactant, were somewhat irregular, the rates of growth of the various phases were determined by measuring their areas in the cell as a function of time. The second type of experiment involved injection of small drops of nonionic surfactant into water contained in the same cells but this time placed on the horizontal stage of a conventional microscope. Here too a thermal stage maintained constant temperature. Videomicroscopy was used to monitor changes in drop diameter with time. The injection procedure was different from that used previously7 in that drops were injected through glass micropipets having radii between 30 and 100 µm instead of through a hypodermic needle with an outside diameter of 210 µm as in previous work. The use of micropipets was necessary to obtain drops of the rather viscous surfactants in the desired size range of 50-100 µm. Moreover, the gas injection system used to drive the surfactant through the micropipets into the drops (Picospritzer II, Parker Hannifin Company) assured reproducibility of drop size. Both pressure of the driving gas (nitrogen) and duration of the pulse could be varied. The
(4) Danckwerts, P. V. Trans. Faraday Soc. 1950, 46, 701. (5) Crank, J., The Mathematics of Diffusion; Clarendon Press: Oxford, 1975; p 29.
(6) Mori, F.; Lim, J.-C.; Raney, O. G.; Elsik, C. M.; Miller, C. A. Colloids Surf. 1989, 40, 323. (7) Lim, J.-C.; Miller, C. A. Langmuir 1991, 7, 2021.
xj ) j t1/2
(3)
If the phase diagram, all diffusivities, and initial compositions are known, one can solve for all Ai, Bi, and j. As discussed below, our interest here is in situations where the phase diagram and initial compositions are known, but not the diffusivities. Moreover, j can be determined experimentally. Then it can be shown that one can find the ratios (Di/Dr), where Dr is the diffusivity in one of the phases chosen as a reference. However, there is not enough information to determine individual diffusivities because the number of interfaces is 1 less than the number of phases. Individual diffusivities can be obtained if these results are combined with results of a separate experiment involving measurement of the time required for a drop of the surfactant to dissolve in water or an aqueous solution of the surfactant. For the purpose of analyzing this latter experiment, it is necessary to solve numerically the governing equation describing radial diffusion in spherical geometry:
[ ( )]
∂ωi Di ∂ 2 ∂ωi ) 2 r ∂t ∂r r ∂r
(4)
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Table 1. Results of Vertical Cell Experiments surf- temp actant (°C)
laba
C12E6
30
Rice
C12E6
25
URPS
C12E5
28
Rice
C12E5
25
URPS
a
phase or surfactant wt interface fraction range LR V1 H1 L1-H1 LR V1 H1 LR L1-LR LR
0.64-0.84 0.60-0.64 0.41-0.60 0.41 0.68-0.85 0.62-0.68 0.39-0.62 0.54-0.86 0.54 0.56-0.87
slope of growth plot (m/s1/2) 1.70 × 10-5 3.35 × 10-6 6.84 × 10-6 -5.44 × 10-6 2.00 × 10-5 2.01 × 10-6 7.60 × 10-6 4.65 × 10-5 ∼0 3.94 × 10-5
Note: URPS ) Unilever Research, Port Sunlight Laboratory.
micropipets were pulled by gravity from capillaries with outside diameters of 1 mm. Further details of the experimental procedure are given elsewhere.8 Concentration profiles during one-dimensional diffusion in a rectangular cell were determined by phase shift interferometry using a Mach Zehnder interferometer. The technique is described in detail elsewhere.9 Water and the lamellar phase of C12E5 were contacted in a long horizontal cell. Its upper and lower surfaces were two optical glass flats separated by two 50 µm diameter copper wire spacers. A third spacer separated the lamellar phase and water initially and was withdrawn to initiate the interdiffusion process.
Results. 1. Videomicroscopy Experiments. The published phase diagram for C12E6/water1 shows that at 30 °C three liquid crystalline intermediate phases should develop between the neat surfactant (L2) and the aqueous phase (L1). These are the lamellar liquid crystal (LR), a bicontinuous cubic phase (V1), and the normal hexagonal phase (H1). It also indicates that the composition is approximately continuous at each interface with compositions as indicated in Table 1. Experiments with the vertical stage microscope confirmed that these intermediate phases did, in fact, appear and that the displacements of the various interfaces from the initial surface of contact were proportional to the square root of time; i.e., the dissolution process was controlled by diffusion, not interfacial kinetics of phase transformation. Results for one experiment are shown in Figure 1. Data were not obtainable at earlier times than those indicated in Figure 1 because a small amount of mixing inevitably occurred on contact. As a result, a short time was required before the various interfaces could be clearly resolved. The slopes of plots showing thicknesses of the various intermediate phases as a function of t1/2 are given in Table 1 based on data such as those shown in Figure 1. According to the least-squares analysis for the data of Figure 1, the standard deviations in the reported slopes are approximately 5% for the L1/H1 interface and the thickness of the LR phase and 2% for the thicknesses of the H1 and V1 phases. While diffusivity does vary with composition in nonionic surfactant/water systems, as shown by the interferometry experiments discussed later, the theoretical analysis described above can be used to obtain information on “effective” diffusivities in the various phases. Plots of (Di/ Dh) developed using this approach are shown in Figure 2 for plausible values of Dh, the effective diffusivity of the hexagonal phase. The assumption of uniform density made (8) Chen, B.-H. Ph.D. Thesis, Rice University, 1998. (9) Leger, C.; Elezgaray, J.; Argoul, F. Phys. Rev. Lett. 1997, 78, 5010.
Figure 1. Intemediate phase thicknesses and interfacial positions as a function of t1/2 for vertical cell experiment with C12E6 at 30 °C: (a) L1-H1 interface; (b) thicknesses of H1 and V1 phases; (c) thickness of LR phase.
in the theoretical section is valid in this system as the density of neat C12E6 was found to be 998 kg/m3 at 25 °C. With uniform density the continuity equation can be invoked to show that no convection is induced by the diffusion process and hence that eq 1 is valid.
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Figure 2. Plots of (Di/Dh) for C12E6 at 30 °C calculated from results of the vertical cell experiment.
The times required for complete dissolution at 30 °C of drops of C12E6 injected into water using the micropipet technique described above are shown in Figure 3. Dissolution time is proportional to the square of the initial drop radius as expected for a process controlled by diffusion. Also shown in Figure 3 are predicted dissolution times obtained with the finite difference solution described previously for two values of Dh and corresponding values of the other diffusivities given by Figure 2. The best agreement between the theory and the experimental data is found for Dh ) 1.08 × 10-10 m2/s. The predicted line for Dh ) 1.01 × 10-10 m2/s is included to show the sensitivity to changes in Dh. The best value of Dh is between the values of 0.85 and 2.3 × 10-10 m2/s recently reported in the same system for diffusivities perpendicular to and parallel to the aligned rodlike micelles in the hexagonal phase.10 Since the hexagonal phase formed in our experiments consists of multiple domains with various alignments, this result is reasonable. The calculated values for the effective diffusivities of the various phases are given in Table 2. It was found that the predicted dissolution time td was almost unaffected by the value chosen for the diffusivity DL2 of the L2 phase (Figure 4). That is, this method does not provide an accurate estimate of DL2. Presumably, the reason is that this phase is rapidly converted to the lamellar liquid crystal soon after drop injection. Accordingly, the low value of DL2 shown in Figure 2 for the optimum value of Dh indicated above is of little concern, and a value of 1 × 10-10 m2/s for DL2 was used in the calculations. In the vertical cell experiments the interface between L2 and LR moved rapidly and, in addition, deviated somewhat from being planar. This behavior introduced some uncertainty (10) Sallen, L.; Oswald, P.; Sotta, P. J. Phys. II 1997, 7, 107.
in the ratio (DL2/Dh) obtained from analysis of the experimental data and presented in Figure 2. Both vertical cell and drop dissolution experiments were conducted for pure C12E5 and water at temperatures between 25 and 28 °C. In this case the only intermediate phase observed was LR, which is consistent with known equilibrium phase behavior.1-3 Results are summarized in Tables 1 and 2. The effective diffusivity for the lamellar phase at 28 °C was 2.0 × 10-10 m2/s, about the same as that found for the lamellar phase in the C12E6 system at 30 °C. Experiments were also carried out for conditions above the cloud points of pure surfactants. For C12E5 at 35 and 40 °C two intermediate phases were observed: the lamellar (LR) phase which appeared immediately upon injection of a surfactant drop, and the surfactant-rich L1 phase, which was visible later at the interface between the LR phase and water although presumably present in small amounts from the time of initial injection. Eventually, the LR phase dissolved, leaving a single drop of the L1 phase in equilibrium with the aqueous phase. For other surfactants such as C12E3 and C12E4 where the phase diagrams1 showed that water and the lamellar phase coexisted at equilibrium in dilute solutions for temperatures near 30 °C, myelinic figures of the LR phase developed at the interface between neat surfactant and water shortly after contact. Some additional swelling of the myelinic figures occurred over time, but even after 1 h they formed a compact and viscous mass, as shown in Figure 5 for C12E3 at 35 °C. Similar behavior was seen at 30 °C for C12E3 and at 30, 35, and 40 °C for C12E4. At 40 °C in the C12E3 system, where water and the L3 phase are present at equilibrium, myelinic figures again formed shortly after contact. However, a few minutes later
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Figure 3. Square of drop radius as a function of dissolution time for C12E6 at 30 °C: O, experimental data; 0, calculation 1; 3, calculation 2. Table 2. Summary of the Best Estimates of Diffusion Coefficients in the Various Phases diffusion coefficients (10-10 m2/s) L1 H1 V1 LR L2
bulk temp surfactant solution (°C) C12E5 C12E6
water water
28 30
0.73 0.86
1.08
2.32
2.0 2.19
1.0 1.0
small droplets of an isotropic liquid appeared at the interface (see Figure 6). These droplets grew and coalesced until the liquid phase completely surrounded the lamellar phase. Myelinic figures could be seen dissolving into the liquid during this time. When all the LR phase had dissolved, a single large liquid drop remained and reached equilibrium with the aqueous phase. In view of the known phase behavior mentioned above, this liquid phase must have been the L3 phase. 2. Interferometry Experiments. Refractive index was measured as a function of composition for the C12E5/water system at 26 °C. As shown in Figure 7, a linear dependence was found in the concentration range of interest with no indication of a discontinuity at the L1/LR phase boundary (approximately 54 wt % surfactant). An experiment was performed in which a lamellar phase containing 67.3 wt % surfactant was contacted with water at 26 °C. The interference pattern observed approximately 2 h after contact is shown in Figure 8. Analysis of this pattern and others obtained at shorter times yielded the concentration profiles shown in Figure 9. The profiles at short times have been shifted slightly to ensure that all are consistent with maintaining zero mass-average velocity throughout the system, as required by conservation of mass in a system of uniform density (the density of pure C12E5 is 966.5 kg/m3 at 20 °C2). Apparently the small (11) Reference 5, pp 230-232.
amount of mixing caused by removal of the wire at initial contact made this correction necessary. Crank11 has shown that the diffusivity at any mass fraction ω1 between those initially contacted may be obtained from the concentration distribution by the following equation:
D(ω1) ) -
|
1 dx 2t dω ω1
∫0ω x dω 1
(7)
Figure 10 shows D(ω) calculated from the concentration profile found at 45 min after initial contact using this equation. D increases severalfold as surfactant mass fraction increases in the L1 region and then jumps by another 50% at the transition to LR. The dashed lines show the effective diffusivities in the two phases calculated from the videomicroscopy results at 28°. Agreement is satisfactory. Probably diffusivity in the lamellar phase is slightly higher in the interferometry experiment because this phase, being present initially, was more uniformly aligned with bilayers parallel to the upper and lower surfaces of the cell (homeotropic). The circles in Figure 10 are values found by Martin et al.12 for C12E5/D2O at 26° using light scattering. Agreement with the interferometry results is good. Discussion The dissolution times for surfactant drops shown in Figure 3 were reproducible within 10%. The slopes shown in Figure 1 for variation of interfacial positions during dissolution of C12E6 at 30 °C in the vertical cell experiments were reproducible within a few percent for the experiments (12) Martin, A.; Leseman, M.; Belkoura, L.; Woermann, D. J. Phys. Chem. 1996, 100, 13760.
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Figure 4. Effect of change in the diffusivity of the L2 phase of the C12E6/water system on calculated dissolution time of C12E6 drop with initial radius of 80 µm at 30 °C.
Figure 5. Video frame showing myelinic figures for a drop of C12E3 injected into water at 35°C.
at Rice. The difference between these results and those obtained at Port Sunlight at 25 °C was somewhat larger, as may be seen from Table 1. For the Port Sunlight experiments the slopes obtained when the thickness of the lamellar phase was plotted as a function of t1/2 for three different experiments can be expressed as (2.0 ( 0.2) × 10-5 m/s1/2. The growth rates measured in individual experiments were found to correlate well with the fraction of the lamellar phase exhibiting homeotropic texture as determined late in the experiment. For this texture no birefringence was observed because the bilayers were all parallel to the cell surfaces. Diffusivity is expected to be relatively high in this case because water can readily move
Figure 6. Video frame for drop of C12E3 injected into water at 40 °C showing myelinic figures and developing L3 phase.
into or out of the space between bilayers without having to diffuse across bilayers. In the Rice experiments very little homeotropic alignment was observed, probably because the experiments were conducted for shorter times and the glass surfaces were not made hydrophobic. Accordingly, the average slope for the corresponding plot for the lamellar phase was smaller, 1.7 × 10-5 m/s1/2. As shown in Figure 10 diffusivity D(ω) in the L1 phase of C12E5 increases with increasing surfactant mass fraction ω in the composition range of interest here, which is greater than that at the lower consolute point (ωc ≈ 0.01). According to the Stokes-Einstein equation, D should be inversely related to the correlation length. Martin et al.,12
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Figure 9. Concentration distributions at various times obtained from interference patterns for the same experiment as in Figure 8.
Figure 7. Refractive index as a function of surfactant concentration at 26 °C in C12E5/water system. The dashed line indicates the phase boundary between L1 and LR phases.
Figure 10. Diffusivity as function of surfactant mass fraction ω calculated from concentration distribution 45 min after initial contact for the same experiment as in Figure 8.
Figure 8. Interference pattern 2 h after contact of lamellar phase of C12E5 with water at 26 °C.
who studied mass fractions ω up to 0.10 C12E5 in D2O, attributed the increase in D they observed as ω increased above ωc mainly to a decrease in correlation length as composition shifted away from the consolute point. In the liquid crystalline phases a concentration gradient is necessarily associated with variation in spacing of large surfactant aggregates such as long rods or large sheets in a regular array. Accordingly, binary diffusion involves relative motion of the aggregates and the associated transport of water into or out of the spaces separating them. For instance, in the homeotropic portion of the lamellar phase in the vertical cell experiments, the number of bilayers between the glass surfaces must vary with position when a concentration gradient exists, as shown schematically in Figure 11. Moreover, at a fixed position, water content increases and the number of bilayers decreases with time, so that bilayers, such as those designated a, which terminate within the phase must have
Figure 11. Schematic of homeotropic lamellar phase with a concentration gradient in a vertical cell. The arrow shows the direction of viewing in the microscope. See text for discussion of items marked a and b.
a net upward motion. However, since there is a net flux of surfactant in the downward direction, other bilayers, such as those designated b, must move downward. The resistance to this relative motion is presumably a significant factor influencing diffusivity. Figure 10 shows fluctuations in diffusivity of the lamellar phase as a function of concentration but no clear increasing or decreasing trend. Probably the fluctuations are caused by variations in average alignment of the lamellar phase, e.g., variations in the fraction with homeotropic alignment. Since the lamellar phase was present initially in the interferometry experiments, the homeotropic fraction is likely larger than that in the Rice vertical cell experiments, where this phase formed only
Dissolution Rates of Pure Nonionic Surfactants
after contact of two isotropic phases. This difference could account for the lower effective diffusivity found using the videomicroscopy results (see Figure 10). With regard to the fluctuations in lamellar phase diffusivity, it is noteworthy that Laughlin13 concluded from his experiments that the usefulness of interferometric methods to determine concentration profiles in anisotropic phases was limited because refractive index was a function of orientation. Summary
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diffusivities for the various liquid crystalline intermediate phases formed during the dissolution process and for the L1 phase. These effective diffusivities were of order 10-10 m2/s for C12E5 and C12E6 at 28 and 30 °C, respectively. Measurement of concentration profiles as a function of time using interferometry during dissolution of the lamellar phase of C12E5 at 26 °C showed that diffusivity in the L1 phase increased severalfold with increasing surfactant concentration and that an additional increase accompanied formation of the lamellar phase.
Videomicroscopy experiments showed that dissolution of pure nonionic surfactants below their cloud points is controlled by diffusion, not by kinetics of phase transformation at interfaces, and yielded values of effective
Acknowledgment. The research at Rice was supported by Unilever Research. Thanks are due to Professor K. Papadopoulos and his group for discussion of their use of micropipets in studying emulsions. The interferometry experiments were conducted during a four-week visit by P. Garrett to the Centre de Recherche Paul Pascal.
(13) Laughlin, R. G. The Aqueous Phase Behavior of Surfactants; Academic Press: New York, 1994; pp 537-538.
LA9913497
