Journal of Colloid and Interface Science 253, 443–454 (2002)doi:10.1006/jcis.2002.8449

Interfacial Resistance in Solubilization Kinetics

Rajesh Shah and P. Neogi1

Department of Chemical Engineering, University of Missouri—Rolla, Rolla, Missouri 65409-1230

Received January 16, 2002; accepted April 30, 2002; published online August 27, 2002

An experiment has been designed to measure the kinetics of solu-bilization of oils in a micellar solution. The results show a very largerange of behaviors, but in one system the solubilization rate is deter-mined completely by what is called the interfacial resistance, whichquite likely arises out of the structural changes that micelles see. Themodel given here is successful in showing that this resistance is notthe usual resistance due to diffusion and that it is quite prelevantin systems that are chemically similar but other conditions varyingwidely (Appendix). We conclude by showing that some of the exist-ing theories of interfacial resistance are not reasonable, suggest howthe real system may be more complex, and show that the interfacialconductance k∗∝ c−1

m at large surfactant concentrations where cm

is the concentration of micelles. C© 2002 Elsevier Science (USA)

Key Words: solubilization; rate of; interfacial resistance.

INTRODUCTION

There are many instances of mass transfer in the presenceof surfactants where the interfacial resistance is controlling in-stead of the usual convective-diffusive effects. It takes placein detergency where there is no surfactant transfer across theinterface (1, 2) and where there is such a transfer (3) and insystems where a simple solute is being transferred across an in-terface covered by a surfactant (4). There are of course manyexamples where convective-diffusive transfer is controlling (5,6) and many more where it competes with interfacial resistance(7, 8), and only a few references are cited. As shown in Fig. 1,micelles in the aqueous phase diffuse toward the oil–water in-terface, demicellize, and then adsorb. Eventually, the surfactanttogether with some oil desorb and then remicellize, but now themicelle contains a solubilizate. This is followed by diffusioninto the bulk liquid. All these structural changes can be lumpedinto a single resistance. Carroll (1) measured the interfacial con-ductance, ki, for solubilization of oils by nonionic surfactants atlarge surfactant concentrations and found that ki was indepen-dent of surfactant concentration. Williams et al. (8) used sodiumdodecyl sulfate (SDS) at low concentrations to deliver a solu-bilizate from the aqueous phase to the oil phase and found thattheir ki decreased with increasing surfactant concentrations. Theinverse of demicellization time constant τ−1

2 also shows a similar

1 To whom correspondence should be addressed.

44

decrease followed by an increase (9, 10) shown schematicallyin Fig. 2. In fact, if the demicellization step in Fig. 1 is ratecontrolling, then ki is related to τ−1

2 (they have different dimen-sions). The suggestion that demicellization was rate controllingwas made by Oh and Shah (11) who showed that the effectiverate of solubilization exhibited a maximum at the bottom of the“V” in Fig. 2, at about 200 mM for SDS.

Along with data on solubilization kinetics, investigators alsoreport data on Ostwald ripening of emulsions in the presenceof surfactants. In Ostwald ripening, the Laplace pressure placesthe oil in small droplets at a higher chemical potential over theoil in the larger droplets. As a result, the oil in smaller dropletsdissolve in water and move to the larger droplets. That is, thelarge droplets grow and the small droplets disappear. The aver-age volume of a drop is seen to grow linearly with time (12),which is called the coarsening process. Since the solubility of oilin water is important, Kabalnov (13) suggested that the coars-ening process could be hastened by adding a surfactant. Instead,he found (with all surfactant concentrations at least nominallyabove the cmc) that initially the coarsening rates decreased withincreasing surfactant concentrations, then increased, and then re-mained constant. The surfactant used was sodium dodecyl sul-fate (SDS). When the surfactant concentrations remain abovethe cmc, which is a necessary condition for solubilization, theinterfacial tension and hence its effect on Ostwald ripening re-main independent of the total surfactant concentrations. Theconcentration ranges of Williams et al. (8), Oh and Shah (11),and Kabalnov (13) are shown in Fig. 2. Of the other data onOstwald ripening with SDS, the rate of coarsening data obtainedby Taylor (14) shows a maximum as a function of SDS concen-tration at about 80 mM, and those by Soma and Papadopoulos(15) are in the low-concentration range where they see only anincrease in the coarsening rate with SDS concentration.

In the development below, we leave out nonionics becausethe charge effects in ionics may have an important contribution(13) that may make nonionics qualitatively different from theionics. First, we report the results of a set of solubilizationexperiments using SDS where a model is used to obtain theinterfacial conductance ki. Described below are experimentson how fast a small drop of oil lying on a glass slide disap-pears into a detergent solution covering it. Next, the resultsof the previous experiments discussed earlier are analyzed us-ing the model given in the next section to show that they too

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444 SHAH AN

FIG. 1. From left to right are shown in sequence diffusion of micelles tothe interface, demicellization, and adsorption. Desorption (both surfactant andoil), micellization, and diffusion back to the bulk phase follow.

demonstrate the same ki–SDS concentration trend as the presentwork.

The importance of kinetics of solubilization is not confinedto commercial detergency, but applies as well to physiologicalsystems such as dissolution of cholesterol by bile salts (7), incleanup of underground pollutants (16), and in delivery of reac-tants (17) and drugs (18). Very often, interfacial resistances areseen to be controlling.

MODEL

The drop and the coordinate system used are shown in Fig. 3.Then the flux of oil at the interface is

N = ki(Kocm − ci), [1]

where cm is the concentration of micelles assumed to be the same

FIG. 2. Schematic rates of demicellization as a function of concentrationof SDS (9, 10) as well as the ranges of SDS concentrations over which mass-

transfer data are available (8, 11) and over which Ostwald ripening data (13) areavailable are shown.

D NEOGI

FIG. 3. Shape of the drop on the solid surface and the coordinates are shown.The quantities measured in the experiments ro and θ are indicated.

everywhere and Ko is the partition coefficient between the oilin micelles and pure oil. Hence, Kocm is the oil concentration atthe interface if there had been no interfacial resistance, but dueto the interfacial resistance there is no equilibrium between bulkoil and the solubilized oil at the interface, and

N = kLci, [2]

where kL is the liquid-side mass-transfer coefficient and the con-centration far away from the drop is assumed to be zero. Com-bining Eqs. [1] and [2] leads to

N = Kokcm;1

k= 1

ki+ 1

kL. [3]

The conservation equation for oil is

codV

dt= −SN , [4]

where co is the molar density of the bulk oil and S is the interfacialarea, which can be written as

S = λV 2/3, [5]

where λ is given by

λ = 2π (1 − cos θ )

sin θ

[3 sin θ (1 + cos θ )

π (1 − cos θ )(2 + cos θ )

]2/3

, [6]

where θ is the dynamic contact angle. It is being assumed herethat the basic shape of the small drop is that of a spherical cap,which is often observed in dynamic systems (19). Equation [6]has been plotted in Fig. 4. The measured values of dynamiccontact angles usually do not fall below 12◦. As a result, basedon the near constancy of values of λ above this value, λ hasbeen replaced by an average up to θ = 12◦ to obtain λ = 4.69.Turning now to a model for kL, it is noted in Fig. 3 that if the drop

is thin, then the curvature of the upper surface can be ignored.In this case, the conservation equation for the oil in the aqueous

445

1 − V

Vo

INTERFACIAL RESISTANCE IN SOLUBILIZATION KINETICS

FIG. 4. From Eq. [6], λ has been plotted against θ . The average value of λ used has been indicated with a dashed line.

phase can be written as

∂c

∂t= Do

∂2c

∂z2[7]

in the region over the drop following the coordinate systemshown in Fig. 1. Here Do is the effective diffusivity of the solu-bilized oil and c is the concentration of oil in the micellar phase.The solution is

c = ci erfc

(z − h√4Dot

), [8]

where ci is the concentration of solubilized oil at the interfaceand the concentration of the solubilized oil at infinite distanceaway has been taken to be zero. Now the flux of oil from thedrop to the aqueous phase is

N = −Do∂c

∂z

∣∣∣∣z=h

= ci

√Do

π t= kLci. [9]

This gives us the value of kL. Combining with Eqs. [3]–[5], onehas

dV

V 2/3= − Koλcm

co

√Doπ t ki

ki +√

Doπ t

dt. [10]

Integration from time t = 0 where V = Vo leads to

( )1/3 ¯

= 2Koλcm

3V 1/3o co

Do

kiπ

[τ 1/2 − ln

(1 + τ 1/2

)], [11]

where

τ 1/2 = ki

√π t

Do.

One particular case where τ is small is of importance. There,Eq. [11] becomes

1 −(

V

Vo

)1/3

= Koλkicm

3co

t

V 1/3o

. [12]

Whereas Eq. [12] holds at short times or for large interfacialresistances, it is clear from Eq. [9] that the mass-transfer coef-ficient due to diffusion decreases with time and this resistancebecomes controlling at large times. The part in the square brack-ets in Eq. [1] has been plotted in Fig. 5 and a transition betweenthe two modes can be seen.

In cases where the dynamic contact angle is large, the sec-ond part of Eq. [9] will not hold. Some curvature needs to beconsidered. For a full drop kLd/D = 2 where d is the drop diam-eter. Using the radius of curvature of the drop as d/2, we obtainkL = D sin θ/ro where can be related to V and θ using Eq. [14].If we assume that θ is constant at large times, we obtain

1 −(

V

Vo

)2/3

= sin θ2λKo Dcm

3coω

t

V 2/3o

. [13]

However, Vo here is not the initial volume but some adjustableparameter which may have a simple relation to the initial volume.

V = ω3r3

o and Eq. [14] can be used. It is important to note thatthe small time asymptote, Eq. [12], has a very different volume

446 SHAH AND NEOGI

FIG. 5. The time dependence of solubilization is shown following Eq. [11]

resistance from the region controlled by diffusion in the bulk.

(and initial volume) dependence than the large time asymptoteEq. [13].

EXPERIMENTAL

As mentioned earlier, the object was to solubilize a drop of oillying on a solid surface into a surfactant solution covering thedrop. A biological slide was cut and cleaned in KOH–normalpropanol solution and then in distilled water and placed insidethe environmental chamber of a Rame-Hart contact angle go-niometer. A drop of oil (reagent grade) of volume 5–15 µl waslaid on the slide and then the environmental chamber was floodedwith the surfactant solution. Upon flooding, the drop streamedupward and disintegrated. Since this did not happen when thechamber was flooded with distilled water alone, it was concludedthat the surface tension decreased the oil–water interfacial ten-sion to a point that it could no long hold it together againstgravity and the drop disintegrated.

An attempt was made to place a drop on the lower side ofthe slide using a syringe with a bent needle, but we could notmake a proper drop because it would not coalesce with the slide;it would do so if water was used instead of surfactant solution.Hence, the slide was lowered to as close to the floor as possibleand the chamber was filled with distilled water just covering thelower surface of the slide. The drop was now placed on the lowersurface of the slide and then the rest of the chamber was floodedwith the surfactant solution. Thus, a minimum amount of wateris used. There is a phase at the start of the experiments wherethe surfactant solution and water have to mix and result in ansolution of uniform concentration. This is the “initial transient”phase and it is hoped that by using the smallest amount of waterthat the duration of this phase is minimized. The volume of water

used was 20 ml and the surfactant solution 52 ml. The reportedsurfactant concentrations are the overall values.

. The small knee in the plot separates the region controlled by the interfacial

The object of using a very small drop is twofold. First, theeffect of gravity is kept very small and the stable drops underdistilled water show segment-of-sphere profiles undistorted bygravity. Second, the small amount of oil when solubilized doesnot change appreciably the properties of the micellar solution.

The systems used were toluene–sodium dodecyl sulfate(SDS) solution, n-heptane–SDS solution, and toluene–dodecyltrimethyl ammonium bromide (DTAB). The surfactants werepurchased from Aldrich and used as supplied. The basal radiusro and the dynamic contact angle θ were measured as functionsof time.

RESULTS

The results of solubilization of drops of toluene in SDS solu-tion are shown in Fig. 6. Only a single concentration of surfactantsolution was used, but the initial drop volumes were varied. Theresults are plotted using Eq. [12], that is, interfacial resistancecontrolled. It is seen that all data fall on the same line irrespec-tive of the initial drop volumes Vo as predicted. The duration ofthe experiments ranged from 10 to 50 h. In Fig. 7, the surfac-tant concentrations were varied. Not shown here are the data forconcentrations 240 and 270 mM, which lie on the largest slopesshown. The volumes are calculated assuming that drop shapesare segments of spheres, hence,

V = π (1 − cos θ )(2 + cos θ )

3 sin θ (1 + cos θ )r3

o . [14]

The conclusion is that the solubilization of toluene by SDSover a large range of surfactant concentrations is governedby interfacial resistance. Another important conclusion can be

reached if the data in Fig. 6 are plotted instead as cos θ versustime t for different values of initial drop volumes Vo. As shown

INTERFACIAL RESISTANCE IN SOLUBILIZATION KINETICS 447

FIG. 6. The time dependence of solubilization of toluene is shown following Eq. [12]. The initial volumes Vo are shown in the inset in microliters (µl). The

surfactant concentration (SDS) is the same in all cases at 83.3 mM.

in Fig. 8, the plots do not fall in the order of increasing drop vol-umes. The same is observed in Fig. 9 where the basal radius ro

has been plotted against time. However, if the mean curvature sinθ/ro is plotted against time t, as shown in Fig. 10, then the plotsfollow increasing order of Vo. As noted earlier, above the critical

micellar concentration the value of the surface tension does not mechanism of detergency is the “roll back” (21), where the drop change with surfactant concentration. Consequently, the mean increases its curvature and finally detaches itself from the solid

FIG. 7. The time dependence of solubilization as a function of surfactant conrange indicated in Fig. 6. The bold line is the least-squares fit from Fig. 6.

curvature calculated by assuming that the drop shapes are seg-ments of spheres represent the Laplace pressure. These followan increasing order of Vo. That suggests that it is a key vari-able in the system, and hence that fluid mechanics plays a role.

Yet another feature emerges from the above discussion. One

centration (SDS) is shown. All drop initial volumes are different but lie in the

448 SHAH AND NEOGI

FIG. 8. Data used in Fig. 6 have been replotted in the form of cos θ as function of time. The initial volumes V are shown in the inset in microliters. Note that

against SDS concentrations together with τ data from the

the plots do not follow in the increasing order of Vo.

substrate. It is indeed observed in Fig. 10 that the curvature in-creases with time. However, it is often suggested that the increasein curvature is due to an increase in the contact angle, which is

2 seen not to be true in Fig. 8, at least in this case. smooth curve given by Lessner et al. (10). There seems to be

FIG. 9. Data used in Fig. 6 have been replotted in the form of ro as function oplots do not follow in the increasing order of Vo.

o

From the slopes in Fig. 7, it is possible to calculate Koki = k∗,a new parameter. These values have been plotted in Fig. 11

−1

f time. The initial volumes Vo are shown in the inset in microliters. Note that the

INTERFACIAL RESISTANCE IN SOLUBILIZATION KINETICS 449

FIG. 10. Data used in Fig. 6 have been replotted in the form of the mean curvature sin θ/r as a function of time. The initial volumes V are shown in the inset

in microliters. Note that the plots do follow in the increasing order of Vo.

virtual disagreement between the two. Ko is not expected toshow significant variations. The fact that k∗ shows no extremumwith SDS concentration should be stressed. In addition, as thesolubilization rates reach a saturation value at large surfactantconcentration, asymptotically k∗ ∝ c−1

m there. In the first sectionwe have discussed the other data in this area where the interfacialresistance could be playing a role. In the Appendix we haveanalyzed these data, which show a variety of behaviors, but onusing the present model, we find that in all cases k∗ decreaseswith SDS concentration, reaching a c−1

m limit.

In Fig. 12 are shown the solubilization of n-heptane by SDS fer when plotted using the present variables. The range of times

solution. Two stages can be seen with a cusp separating them. taken is about 5–35 h, about half of that for toluene–SDS. No

FIG. 11. The k∗ values obtained are plotted against SDS concentrations and tvaries inversely with SDS concentration.

o o

The one exception is the response at the largest surfactant con-centration of 240 mM where there is a change in slopes butno cusp. The black diamond has a volume of 17.5 µl and thewhite diamond has a volume of 11.4 µl. It is suggested here theresponse at short times is controlled by interfacial resistance. Al-though the rate is too fast to take much data, what are availableindicate that the two plots are coincident at short times, hencesatisfying the requirements of Eq. [12]. It also suggests that theinitial transients do not play a role. At large times diffusion inthe bulk is rate controlling and the two cases should indeed dif-

he trend is compared to the τ−12 trend from Ref. (10). At large concentrations, k∗

450 SHAH AND NEOGI

FIG. 12. The time dependence of solubilization of n-heptane as a function of surfactant concentration (SDS) is shown. Note that every plot has a cusp indicatedby an arrow. The two plots at largest concentrations (240 mM) seem to show only a constant difference.

FIG. 13. The dynamic contact angles for the data shown in Fig. 12 are shown as a function of time. There are significant swings and in some instances the

data do not settle down.

INTERFACIAL RESISTANCE IN SOLUBILIZATION KINETICS 451

FIG. 14. The time dependence of solubilization of toluene as a function of surfactant concentration (DTAB) is shown. Note that every plot has a positive

intercept on the y axis, indicating two steps in the history of solubilization, a very quick but not insignificant first step (interfacial resistance) followed by a slower second step (diffusional resistance).

waves nor ejected droplets were seen in the silhouettes as ob-served in the goniometer, nor were any liquid crystals detectedusing the polarizer–analyzer pair. Contrary to Fig. 8 and to the

observations in all toluene–SDS systems, the dynamic contact function of time for the toluene–DTAB system. All cases seem angles did not change monotonically, but instead showed large to have positive intercepts on the y axis, implying a very short but

FIG. 15. Solubilization from a mixture of squalane (8 wt%) and n-heptane insolubilized. The initial volume Vo = 7.88 µl.

increases and decreases in a very unorganized manner. The val-ues have been shown in Fig. 13.

In Fig. 14 has been plotted the progress of solubilization as a

180 mM SDS. In principle, the curve should reach about 0.6 when all heptane is

452 SHAH AN

very effective initial period of solubilization that is controlled byinterfacial resistance. The time scales of the experiments variedfrom 15 s (in 260 mM solution) to 11 h (in 30 mM solution).

When the oil to be solubilized was made more polar by addingnitro or hydroxy groups, the solubilization times were reducedto seconds.

Some attempts were also made to study mixtures when onecomponent could not be solubilized. However, the drop silhou-ettes showed distortions very likely due to nonuniform masstransfer and Marangoni effect on the surface. The only systemthat was not affected was n-heptane–squalane (8%), shown inFig. 15. Here, squalane is not solubilized. The process is slowbecause there is a diffusional resistance inside the oil phase aswell. When all heptane is solubilized, the plot would reach avalue of 0.6.

DISCUSSION

The interfacial resistance in the presence of surfactants hasbeen known since the time it was first reported by England andBerg (25). We have presented here an experimental system thatis capable of measuring such resistance when the mass transfertakes place from the oil to the water phase (solubilization). Pre-viously, one of us had conducted a similar analysis for the casewhen the mass transfer was from the water to the oil phase (drugdelivery) (8).

One mechanism that has been suggested before was that thedemicellization step controls interfacial resistance (11). Basedon Fig. 11, we find that the concentration dependence of thesetwo rates are so different that it is unlikely that demicellizationis rate controlling. As shown in the Appendix, the very basis ofOh and Shah’s (11) conclusion that demicellization is rate con-trolling becomes shaky when their data are replotted using thepresent quantitative model. Both data show that the interfacialconductance decreases with inverse dependence on surfactantconcentration.

The suggestion by Kabalnov (13) that the interfacial resistanceis due solely to the hindrance caused by the charged oil–waterinterface to the approach of charged micelles is also not tenable.The reason is that the magnitudes of times taken for solubiliza-tion change so dramatically (by at least an order of magnitude)when the oil is changed from toluene to heptane or the surfactantis changed from SDS to DTAB: all for ionic surfactants.

It is possible that all steps shown in Fig. 1 contribute to theoverall resistance. Because there are as many as four steps, it ispossible that from system to system small differences in everystep add up to a significant change in the overall rates.

Although we are quite successful in quantifying the mass-transfer rates when the interfacial resistance is rate controllingor at short times, there are a few features that we have not quan-tified. The first of these is that we do not have a good modelfor the case where the diffusion in the aqueous phase is rate

controlling, although some approximate models have been pro-vided. For a flat surface no quasi-static solution is possible and

D NEOGI

the solution given is an unsteady-state solution that does notshow a dependence on the length scale of the problem. How-ever, for a spherical drop, a quasi-static solution is feasible, butone that is strongly dependent on the length scale of the prob-lem. Since we known that the shape of the drop is a segmentof shpere profile, one should solve the unsteady-state problemin bispherical coordinates, which is very difficult. In toluene-SDS system, the contact angles remain small, but large valuesare reached in others. The second omission is that we have nofluid mechanical treatment to offer, without which we are oneequation short in predicting θ and ro separately as functions oftime. Tanner’s rule of wetting kinetics can be derived by con-sidering the wedge where the viscous dissipation is equated tothe rate of surface work. One may add to it the change in freeenergy due to solubilization, but it does not bring in the Laplacepressure, which is an important variable in our system. Insteadof local considerations, when global ones are made, a very dif-ferent kinetics results, as shown in reactive spreading (20). Suchmodeling has not been attempted here.

Finally, we point out that in most cases in Figs. 12 and 14,although over almost all of the time of observation we see onlythe diffusional resistance, a significant part of solubilization hastaken place in the brief initial period where the interfacial re-sistance is controlling. Thus, the analyses that ignore interfacialrates can be good in predicting the rates of solubilization sincethey are current quantities but may not be good in predicting totaloil solubilized as a function of time since that also covers theperiod where the interfacial resistance is rate controlling (5, 6).

APPENDIX

(a) Consider first the rates of solubilization data of Oh andShah (11). They solubilize a column of benzene of constantcross section. Equations [3], [4], and [9] remain unchanged (andEq. [9] is exact here), but dV = Sdh, where h is the height of thecolumn. Integration to calculate the total time taken to solubilizea column of height ho leads to

ho = 2Kocm Do

cokiπ

[τ 1/2

o − ln(1 + τ 1/2

o

)], [A-1]

where dimensionless τ has been defined in the text and is eval-uated here at time to. If interfacial resistance is controlling,

ho ∼ k∗cmtoco

. [A-2]

Oh and Shah hold ho constant and report to as a function ofcm. Instead of trying to obtain the absolute value of k∗, we re-port in Fig. 16 below a relative k∗

r as a function of total SDSconcentration:

k∗r = 1

cmto[A-3]

The values both decrease with SDS concentration and reach the

INTERFACIAL RESISTANCE IN SOLUBILIZATION KINETICS 453

FIG. 16. Relative rate constants as a function of SDS concentration from the solubilization kinetics data by Oh and Shah (11) using the present model. If we

join the point 10000 on the y axis to the point 1000 on the x axis, a line of slope −1

predicted asymptotic limit of inverse dependence. Note that aquite different functional form would result if the bulk diffusionwere rate controlling.

(b) Consider now the results of Ostwald ripening. The ap-proximate solution for the case where the mass transfer in thebulk phase is rate limiting is called Lifshitz–Slezov–Wagner(LSW) solution (12) and is available in the form of the thirdmoment of the radius

r3 ∼ ωt. [A-4]

There are many assumptions and approximations made in the to some extent its value depends on the initial conditions. Nev-

derivation, and although the final result compares well with the

FIG. 17. Relative rate constants as functions of SDS concentrations from Ostwald ripening data of Kabalnov (13) (triangles), Soma and Papadopoulos (15)

ertheless, it is proportional to kicm. This leads to the expression

(two circles), and Taylor (14) (three squares). The line joining 10−12 on the y axisasymptotic limit is being attained.

results. This line can be used to verify that the asymptotic limit is being attained.

experiments, Kahlweit (22) shows how many errors cancel outone another. If the LSW method is used to analyze the casewhere the interfacial resistance is controlling, one has the sameexpression as Eq. [A-3], but with the second moment replacingthe third. The data of Weiss et al. (23) on Ostwald ripening usingnonionic surfactant cannot tell us clearly which one is applicable.As a result Neogi and Narsimhan (24), abandoned the approx-imate method and obtained a numerical solution for the casewhere interfacial resistance is controlling and obtained resultsidentical to Eq. [A-4]. The constant ω cannot be uniquely deter-mined both because of the numerical nature of the solution and

to 1000 on the x axis has a slope of −1. This line can be used to verify that the

D

454 SHAH AN

for the reduced k∗ of

k∗r = ω

cm. [A-5]

One set of data from Kabalnov (3), two sets (with two differentkinds of initial conditions) from Soma and Papadopoulos (15),and three under the same conditions from Taylor (14) have beenplotted in Fig. 17. They all show the same behavior and thesame asymptotic limit. If the diffusional resistance had beenrate controlling, then ω would have been proportional to theequilibrium solubility of oil in a micellar solution and hence tothe surfactant concentration. Consequently, the left-hand-sideof Eq. [A-5] would then have worked out to be independent ofsurfactant concentration.

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