Accumulation of Surface-Active Solutes in the Aerosol Particles

We measured the composition of the droplets of the aerosols generated by ... of the solutes, sonication was performed under the atmosphere of carbon d...
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J. Phys. Chem. B 1998, 102, 4337-4341

4337

Accumulation of Surface-Active Solutes in the Aerosol Particles Generated by Ultrasound Dmitrii N. Rassokhin† Faculty of Chemistry, Moscow LomonosoV State UniVersity, 119899 Moscow, Russia ReceiVed: NoVember 12, 1997

We measured the composition of the droplets of the aerosols generated by ultrasound from aqueous solutions containing various concentrations of Triton X-100 (an octylphenyl ethoxylate nonionic surfactant) and D-glucose, the latter being a reference nonsurfactant solute. To prevent the sonochemical decomposition of the solutes, sonication was performed under the atmosphere of carbon dioxide. It was found that the concentration of the surfactant in the droplets of the ultrasonically generated aerosol was significantly (up to 10 times) higher than that in the solution being atomized, while the concentration of D-glucose in the droplets did not differ from its concentration in the bulk of the solution. The enrichment of the aerosol’s droplets with the surfactant was found to increase with the decrease of the surfactant concentration in the solution being atomized. The observed effect is explained by the specific mechanism of the ultrasonic aerosol formation: under the influence of ultrasound small droplets tear off the solution’s surface enriched with the dissolved surfactants owing to the adsorption of the surfactants at the water/gas interface.

Introduction Ultrasound has long been used for aerosolization of liquids. The effects of the ultrasound frequency and intensity, temperature, pressure, and composition of the liquid being atomized, as well as many other factors, on the efficiency of the aerosol generation and on the size distribution of the particles produced have received extensive study.1 However, no attention has been paid to the chemical composition of the ultrasonically generated aerosols. The formation of the aerosol under the influence of ultrasound takes place at the interface between the liquid and atmosphere: ultrasonic waves fall on the surface from inside the liquid and generate surface microwaves, which, in turn, eject small droplets of the liquid into the atmosphere, where they become the aerosol’s particles. The ultrasonic atomization of liquids is somewhat similar to the phenomenon known as drop entrainment. The latter is observed when a liquid is boiled or a gas is bubbled through it; the vapor or gas bubbles rise to the surface of the liquid and burst there, ejecting small drops into the atmosphere. It has been shown that the concentrations of the ions in the drops generated due to the drop entrainment from the aqueous solution containing ionic solutes differ from the concentrations of these ions in the bulk of the solution. The drop entrainment mechanism has been extensively studied by Khentov and co-workers,2 and their research suggest that the adsorption of the solutes at the liquid-atmosphere interface can dramatically influence the chemical composition of the aerosol particles in the case that they break off the liquid’s surface. This idea led us to the assumption that the concentration of the surface-active solute in the small droplets taking off the surface of the solution under the influence of ultrasound could be greater than the average concentration of the solute in the bulk, because, owing to the adsorption, the concentration of such a solute in a thin layer of the liquid adjoining the surface is greater than its concentration in the bulk. In other words, it can be expected

Figure 1. Experimental setup.

that the aerosols generated with ultrasound from the solutions containing surface-active solutes would be enriched with such solutes. In order to test this assumption, in the present work we measured the composition of the droplets of the aerosols generated with ultrasound from aqueous solutions containing various concentrations of Triton X-100, which is a mixture of an octylphenyl ethoxylates OPE9 and OPE10 with the following formula HO–(CH2–CH2–O)n

where n ) 9 or 10 for OPE9 and OPE10, respectively. Unlike Khentov and co-workers, whose research was mentioned above, we decided on choosing nonionic surface-active solutes, because their adsorption is easier to consider theoretically. Experimental Section

†

Email: [email protected]. Present address: 3-Dimensional Pharmaceuticals, Inc. Eagleview Corporate Center, 665 Stockton Drive, Suite 104, Exton, PA 19341.

The experimental setup is shown in Figure 1. The diskshaped transducer (1) was made of lead titanate-zirconate with

S1089-5647(98)00183-7 CCC: $15.00 © 1998 American Chemical Society Published on Web 05/07/1998

4338 J. Phys. Chem. B, Vol. 102, No. 22, 1998 a diameter of 32.9 mm, height of 2.8 mm, and main resonance frequency of 724 kHz. The transducer was cable-connected to a high-frequency generator (not shown in the figure) and mounted on the bottom of the ultrasonic bath (2) filled with vacuum-degassed water. A 100-mL glass sonochemical cell (3) with a thin polypropylene film bottom was filled with 50 mL of the solution to be atomized. The aqueous solutions used in the experiment contained 5 × 10-3 M of D-glucose3 (used as a reference nonsurfactant solute) and various concentrations of Triton X-100 (Serva, pure grade, cat. no. 37240, with the average molecular weight of 628 g/mol). The cell was positioned over the ultrasonic bath so that only the cell’s bottom touched the water in the bath. The carrier gas (carbon dioxide or argon) was bubbled vigorously through the solution for about 15 min through a thin Teflon capillary (4). Then, the capillary was closed with a cap, and a flow of the same gas was fed through a gas inlet (6), the flow rate being 375 ( 20 mL/min. The ultrasonic generator was switched on. The acoustic power introduced in the cell was 38 ( 3 W, as it was estimated calorimetrically using the technique described elsewhere.4 The temperature in the cell was maintained at 25 ( 0.5 °C5 by the thermostatically controlled flow of water through the water jacket (5). The beam of ultrasound entering the cell through its bottom formed an ultrasonic fountain and aerosol in the cell. The flow of the gas carried the aerosol out of the cell into the separator (7) to capture large drops of the solution splashed by the fountain and occasionally carried away by the gas flow. Then, the aerosol was fed into the cyclone (8), where the centrifugal force caused the aerosol’s particles to precipitate and collect in the test tube (9). A magnified cross section of the cyclone (8′) is shown in the lower right corner of Figure 1. Its inner diameter was 10 mm, the diameter of the inlet nozzle being 0.9 mm. The effectiveness of the cyclone was observed visually: if it was detached when the ultrasonic generator was working and the gas flow was on, a thick fog coming out of the separator’s (7) outlet could be clearly seen in the side illumination, whereas, when the cyclone was attached, no fog could be seen leaving its outlet. In each run of the experiment 0.6 ( 0.1 mL of the precipitated aerosol sample was collected in 20 min of sonication. After completion of the aerosol sample collection, the ultrasonic generator was switched off, and a sample of the solution of about 1 mL was taken from the sonochemical cell (3). The glucose concentrations in the samples were measured spectrophotometrically, using glucose reaction with K3Fe(CN)6. We utilized a modification of the standard method found in the literature.6 The ferricyanide reagent was prepared by dissolving 1.5 g of anhydrous Na2CO3 and 0.15 g of K3Fe(CN)6 in about 100 mL of distilled water and diluted with water up to 250 mL exactly. A sample solution (0.1 mL) was added to 3 mL of the ferricyanide reagent, and the mixture was held at 90 °C for 10 min. Then, its optical density was measured at 400 nm. D-Glucose reduced [Fe(CN)6]3-, so the optical density at 400 nm decreased linearly with increase of the glucose concentration in the sample solution in the range of 1 × 10-3 to 1 × 10-2 M, as was verified using glucose solutions of known concentrations. It was also checked that admixtures of Triton X-100 up to 1 × 10-2 M in the samples did not interfere with the measurements. The accuracy of glucose concentration measurement by this method was 4%. UV spectra of the samples were recorded on a Shimadzu UV2401PC UV-vis spectrophotometer. A narrow (4-mm wide) quartz cuvette with optical length of 10 mm was used, allowing for small samples of about 1 mL. To prevent the gas dissolved

Rassokhin

Figure 2. UV spectra of aqueous Triton X-100 solutions obtained from the run of the experiment with the initial solution containing 5 × 10-5 M Triton X-100 and 5 × 10-3 M D-glucose. The curves are listed in the order of decreasing absorbance at 223 nm: (1) The condensed aerosol sample. Sonication was performed under CO2. 0.2 mL of the sample was diluted with 0.8 mL of water before measurement. (2) The initial solution containing 5 × 10-5 M Triton X-100 and 5 × 10-3 M D-glucose; 0.5 mL of the sample was diluted with 0.5 mL of water before measurement. (3) A sample of the solution taken from the sonochemical cell after aerosol collection. Sonication was performed under CO2. 0.5 mL of the sample was diluted with 0.5 mL of water before the measurement. (4) Same as 1, but sonication was performed under Ar atmosphere. (5) Same as 3, but sonication was performed under Ar atmosphere.

in the samples from forming bubbles in the cuvette, we held the samples at 65 °C in small open test tubes for about 5 min and then centrifuged before placing them in the cuvette. If it was necessary, the samples were diluted to a certain degree with distilled water before measurement. Using solutions of Triton X-100 of known concentrations, we found the extinction coefficient at 223 nm to be  ) 9.33 × 103 dm3 cm/mol. Triton X-100 concentration measurements relied on this coefficient and were accurate to 6% or better in the range of concentrations from 1 × 10-5 to 1 × 10-3 M. Results and Discussion The goal of the experiment was to test whether surface-active dissolved substances concentrate in the droplets of ultrasonically generated aerosols. To do so, we would compare the concentrations of Triton X-100 in the following three samples: 1. the initial solution (that is, the solution from which the aerosol was generated), 2. the condensed aerosol, and 3. the solution taken from the sonochemical cell after the collection of the aerosol. In each run of the experiment, the concentration of Triton X-100 in the initial solution was varied. Figure 2 shows UV spectra of aqueous Triton X-100 solutions obtained from the run of the experiment with the initial solution containing 5 × 10-5 M Triton X-100.7 It can be seen that, if the solution was saturated with carbon dioxide, the spectra of the initial sample, the sample taken from the cell after the sonication, and the sample of aerosol were similar to each other; therefore, no decomposition of Triton X-100 under the influence of ultrasound took place. At the same time, the spectra of the samples obtained from the solution saturated with argon show almost complete sonochemical decomposition of Triton X-100. This is in accordance with the fact described in the literature8 that saturation with polyatomic gases suppresses the sonolysis of low molecular weight solutes in aqueous solutions. Having

Surface-Active Solutes in Aerosol Particles

J. Phys. Chem. B, Vol. 102, No. 22, 1998 4339 after we have withdrawn a certain significant amount of the aerosol from the sonochemical cell. Suppose we can capture all aerosol particles leaving the sonochemical cell (position 3 in Figure 1). In other words, allow no aerosol to escape the cyclone (position 8 in Figure 1). Then, from the material balance, we can obtain the following differential equation (eq 2)

KC dV ) C dV + V dC

(2)

where C is the concentration of the surfactant in the sonochemical cell and V is the volume of the solution in the cell. Solving eq 2 with the initial condition (eq 2a) Figure 3. Observed enrichment factor (circles) and surface tension isotherms of aqueous solutions of OPE10 (squares) and OPE9 (triangles) at 25 °C. The surface tension data are taken from Crook et al.9 Dotted line, the enrichment factor computed by eq 8 with Γmax ) 2.3 × 10-6 mol/m2 and r ) 10-5 m; dashed line, the enrichment factor computed by eq 10 with n ) 0.5 and A ) 1.862 × 10-2 M-0.5.

checked the assumption that under argon Triton X-100 would undergo sonolysis, whereas under carbon dioxide it would not, we used carbon dioxide as the carrier gas throughout in the subsequent experiment, and computed the concentrations of Triton X-100 in the samples from their absorbances at a fixed wavelength of 223 nm, as it was described above in the Experimental Section. Obviously, the composition of the aerosol could have been affected by evaporation of water, which would have increased the measured concentration of Triton X-100 in the aerosol samples. To take the evaporation effect into account, we added 5 × 10-3 M D-glucose to the initial solutions as a reference nonvolatile, nonsurfactant solute. However, in the experiment we discovered that the concentrations of glucose were always the same in all three samples obtained in each run when carbon dioxide was used as a carrier gas; therefore, the water evaporation side effects could be ruled out. Figure 3 shows how the observed enrichment factor K′, defined as CA/C0, where C0 is the initial concentration of Triton X-100 and CA, the concentration of Triton X-100 in the collected aerosol sample, depends on C0. Surface tension isotherms of aqueous Triton X-100 solutions measured at 25 °C by Crook et al.9 are plotted on the same chart for the theoretical analysis of the effect. It can be seen that the concentrations of the surfactant in the aerosol samples were higher than its concentrations in the solutions from which the aerosols were generated; the less the concentrations, the more the observed enrichment factors. At the concentration of Triton X-100 of 5 × 10-6 M, the enrichment factor was found to be about 10, with a tendency to grow further at even lower concentrations of the surfactant; at the concentration of Triton X-100 of 1 × 10-3 M the enrichment factor was found to be about 1.6. One can suppose that, if the concentration of Triton X-100 increased even further, the enrichment factor would approach 1, i.e., almost no enrichment would take place.10 Strictly speaking, the observed enrichment factor K′ ) CA/ C0 must differ from the value of the “true” enrichment factor defined by eq 1

K ) limVAf0 CA/C0

(1)

where VA is the volume of the aerosol having been generated after a certain sonication time. Obviously, the less the sonication time, the less the aerosol sample taken in the experiment, and the closer K′ to K. Let us estimate how different they might be

C(V0) ) C0

(2a)

assuming that K does not depend on C, we obtain a formula to compute the concentration of the surfactant in the cell after a certain amount of aerosol VA has been taken out of the cell and the volume of the solution left there is V:

C ) C0(V/V0)(K-1)

(3)

From eq 3 it follows that if the value of K of a given solute is more than 1, then the concentration of the solute will decrease as more aerosol is removed from the cell, which explains the decrease of the Triton X-100 concentration in the cell after a certain amount of aerosol was withdrawn, even if CO2 was used as a carrier gas and no decomposition of Triton X-100 took place (see the UV spectra in Figure 2). Using eq 3, we can draw the formula to predict what the measured concentration of the surfactant in the sample of condensed aerosol of the volume VA will be if the enrichment factor K equals a certain given value. Provided that all the surfactant leaving the cell with the aerosol is completely captured in the cyclone, we can find its concentration in the sample of the condensed aerosol CA from the equation of material balance written either as eq 4 or 4a:

CA )

1 VA

(

(V0 - VA) C0

∫VV-V KC0(V/V0)(K-1) dV 0

0

(4)

A

)

V0 - VA V0

(K-1)

+ VACA ) V0C0

(4a)

From either eq 4 or eq 4a (which needs more laborious transformations), we derive the following solution (eq 5):

[ ( )]

VA V0 CA ) C0 1 - 1 VA V0

K

(5)

Two important conclusions (which are in a full accordance with what we usually call “common sense”) follow from eq 5:

1. CA f KC0 when VA f 0 2. CA f C0 when VA f V0 Equation 5 allows us to estimate the difference between the “true” and experimentally measured enrichment factors: as it was described in the Experimental Section, Va was 0.6 mL and V0 was 50 mL. If K ≈ 10, then the difference between K′ and K would be about 5%, which is within the experimental error and could be neglected. The fact that we neglected the dependence of K on C while deriving eqs 3 and 5 does not contribute much to the error either, because the variation of C

4340 J. Phys. Chem. B, Vol. 102, No. 22, 1998

Rassokhin

Figure 4. Droplet breaking off the surface of the solution containing a surface-active solute.

in the cell during the sonication did not exceed 11%, and such a variation does not significantly affect the value of K, as is clear from Figure 3 (note the logarithmic X-scale). If higher values of K were reached, then the difference between K′ and K should be taken into account, and K should be computed from experimental values of CA by numerically solving eq 5 with respect to K. Again, doing so, one should check whether K does not significantly change when the concentration of the surfactant in the cell changes as more of the solution converted to the aerosol has gone from it; otherwise, one would not have any other choice but numerically solving differential equation 2. Let us try to construct a model of the aerosol formation in the presence of surfactants in order to estimate the enrichment factor and compare it with the one found in the experiment. Assume that a droplet of radius r is tearing off the surface of the solution containing a surface-active solute, as shown in Figure 4. Let C be the concentration of the surface-active solute in the bulk of the liquid and Γ be the quantity of the solute adsorbed per unit area of the surface. Then, the average concentration of the solute in the droplet C h , that is, the total quantity of the solute in the droplet divided by the droplet’s volume, can be calculated by eq 6, which can be reduced to eq 6a:

4 4 C h ) C πr3 + Γ4πr2 / πr3 3 3

(6)

C h ) C + 3 Γ/r

(6a)

(

)( )

Let us make the following two additional assumptions: (1) the adsorption layer marked out with the darker color in Figure 4 is in thermodynamic equilibrium with the solution, and (2) the concentration of the surfactant and its surface activity are high enough so Γ can be expressed by adsorption isotherm 7:

Γ(C) ) Γmax

(7)

With these assumptions, eq 6a can be transformed into eq 8

C h /C ) 1 + 3Γmax/(rC)

(8)

where C h /C obviously equals K, the enrichment factor defined by eq 1. Using Gibbs’ eq 9

Γ ) -R-1T-1 dσ/d ln C

(9)

where dσ/d ln C is the derivative of the surface tension of the solution with respect to the natural logarithm of the surfactant

concentration, we can find the value of Γmax ) (2.3 ( 0.2) × 10-6 mol/m2 for Triton X-100 in aqueous solutions at 25 °C from the surface tension curves in Figure 3.11 Now we can substitute the values of Γmax and r in eq 8 to check out whether the enrichment factors predicted by our simple model are any closer to the experimentally obtained values. Even though we did not measure the size distribution of the particles of the aerosols generated in our experiment, the aerosol generated appeared as a fine fog; thus, it would be reasonable to suppose that the particles’ sizes lay within the range between 10-7 and 10-5 m. One can see from Figure 3 that the curve computed by eq 8 with Γmax ) 2.3 × 10-6 mol/m2 and r ) 10-5 m does pass through the experimental point at C ) 1 × 10-3 M, but it predicts much higher values of the enrichment factor than those found in the experiment at the lower concentrations of the surfactant. Thus, the simple model of the droplet formation in the presence of surface-active compounds that we derived above seems to be capable of explaining the enrichment phenomenon only qualitatively. The fact that the enrichment factor was found to rise much slower with the decrease of the surfactant concentration than it would follow from eq 8 is most likely due to a very short time of the aerosol’s droplet formation. It is so short that the adsorption layer cannot approach the thermodynamic equilibrium with the solution, and therefore, the value of Γ reached at the moment the droplet takes off is significantly smaller than Γmax. We can suppose that, at the moment when a droplet of the aerosol tears off, Γ is much lower than Γmax, and thus the adsorption at the droplet’s surface is fully controlled by the diffusion of the surfactant from the bulk of the solution. Although no theory has been elaborated to quantitatively describe the adsorption of surfactants at the liquid-gas interface disturbed by acoustic waves, we can guess that the adsorption rate (and therefore the value of Γ reached at the moment the droplet leaves the surface of the liquid) might be proportional to Cn, where C is the concentration of the surfactant in the bulk of the solution. If this is the case, then the enrichment factor will be expressed by eq 10 rather than eq 8

C h /C ) 1 + A/C1-n

(10)

where A depends, among many other factors, on the radius of the aerosol’s droplets and the surfactant’s diffusion coefficient. It is seen from Figure 3 that eq 10 most satisfactory fits the experimental data if n ) 0.5 and A ) 1.862 × 10-2 M-0.5. Although at this point it is not possible to extract any useful information from the value of A, the value of n suggests that the diffusion-controlled adsorption rate is proportional to the square root of the concentration of the surfactant in the bulk of the solution. Conclusions We found that the concentration of the surfactant in the droplets of the ultrasonically generated aerosol was significantly higher than that in the solution being atomized. The enrichment of the aerosol’s droplets with the surfactant was found to increase with the decrease of the surfactant concentration in the solution being atomized. The enrichment effect has not only theoretical, but also practical, value: First, the side effect of surfactants accumulation in the aerosol should be taken into account in the applications of the ultrasonic aerosol generators: the concentrations of the solutes in the aerosol’s particles might unexpectedly prove to be very different from the concentrations of the solutes in the atomized solution. Second, the effect can be used as a kind of “ultrasonic distillation” to remove active surfactants from solutions, and/

Surface-Active Solutes in Aerosol Particles

J. Phys. Chem. B, Vol. 102, No. 22, 1998 4341 References and Notes

Figure 5. Normalized concentrations of the surfactant as functions of the relative volume of the solution converted to aerosol and removed from the cell, computed by eqs 3 and 5: (1) in the solution remaining in the cell (K ) 5, upper dotted line; K ) 10, lower dotted line); (2) in the condensed aerosol sample (K ) 5, lower solid line; K ) 10, upper solid line).

or to separate these surfactants out. The performance of such a method is illustrated by Figure 5, which shows normalized concentrations of the surfactant in the solution remaining in the cell and in the condensed aerosol sample as functions of the relative volume of the solution converted to aerosol and removed from the cell, computed by eqs 3 and 5. To make the process more efficient, ways to increase the enrichment factor K should be found. It is likely that the efficiency should grow as we produce finer aerosol with smaller particles. In order to do so, higher ultrasonic frequencies should be used. On the other hand, as we already discussed above, owing to short time of the aerosol’s droplet formation, the adsorption layer at the droplet’s surface is likely not to approach the thermodynamic equilibrium with the solution, and therefore, the value of Γ reached at the moment the droplet breaks away is smaller than that computed from eqs 7 and 9. The time of the droplet formation decreases as the frequency increases, so we can expect a certain optimal frequency to exist at which maximum enrichment factor can be reached. Acknowledgment. The author expresses his gratitude to Professor K. S. Suslick, Professor V. N. Izmailova, Dr. A. M. Parfenova, and M. V. Poteshnova for their interest in this work and many helpful discussions.

(1) Eknadiosyants, O. K. Polucheniye aerozolei (Generation of aerosols). In Fizika i tekhnika moschnogo ul’trazVaka. Fizicheskiye osnoVy ul’trazVukoVoi tekhnologii (Physics and technology of power ultrasound. Physical principles of the ultrasonic technology); Rosenberg, L. D., Ed.; Nauka: Moscow, 1970; pp 337-392 (in Russian). (2) (a) Khentov, V. Ya. Fizikokhimiya kapel’nogo unosa (Physical Chemistry of Drop Entrainment); Rostov State University: Rostov, 1979, 128 pp (in Russian). (b) Vlasov, Yu. V.; Khentov, V. Ya.; Gaponova, T. V.; Gasanov, V. M. Russ. J. Phys. Chem. 1984, 58 (10), 2550-2554 (Russ. J. Phys. Chem., Engl. Transl. 1984, 58 (10), p 1545-1548). (3) Concentrations are expressed in the units of M throughout. 1 M ) 1 mol/dm3 at 25 °C. (4) Rassokhin, D. N.; Kovalev, G. V.; Bugaenko, L. T. J. Am. Chem. Soc. 1995, 117, 344-347. (5) In the preparatory experiments, we examined how the temperature in the cell depended on the temperature of the cooling water in the water jacket, the acoustic power of the ultrasonic waves entering the cell, the volume of the solution in the cell, and the sonication time; the temperature in the cell was monitored using a thermocouple. We found what under the conditions of our experiment the temperature of cooling water should be 2.8 °C lower than the desired temperature of the solution in the cell; the latter would reach the equilibrium state in about 1.5 min after the ultrasonic generator was switched on and remain constant ((0.5 °C) while the ultrasonic generator was working. (6) Ghose, T.; Montenecourt, B. S.; Eveleigh, D. E. Measure of cellulase actiVity (substrates, assays, actiVities and recommendations); Preprint of IUPAC Commission on Biotechnology, 1981. (7) With other concentrations of Triton X-100 studied in our experiment, similar spectra and the same effects of the dissolved gases were observed. (8) Henglein, A.; Gutie´rrez, M. J. Phys. Chem. 1988, 92, 3705-3707 and references therein. (9) Crook, E. H.; Fordyce, D. B.; Trebbi, G. F. J. Phys. Chem. 1963, 67, 1987-1994. (10) At the concentrations of Triton X-100 lower than 5 × 10-6 M, the accuracy and reproducibility of the experiment were too poor, so we could not find the maximum value of the enrichment factor. Rich foam formation in the sonochemical cell during sonication prevented us from examining the solutions with the concentration of Triton X-100 higher then 1 × 10-3 M. (11) There are two surface tension curves in Figure 3: one for OPE9 and another for OPE10. As it can be seen, both OPE9 and OPE10 have the values of the critical micelle concentration (cmc) of about 3 × 10-4 M. The curves’ parts corresponding to the concentrations from 1 × 10-5 to 3 × 10-4 M are almost linear in the logarithmic X-scale, which means that at the concentration of 1 × 10-5 M the maximum adsorption has already been reached, and therefore the adsorption isotherm expressed by eq 7 is observed at the higher concentrations for both OPE9 and OPE10. This allows us to compute the values of dσ/d ln C and thus, using eq 9, find Γmax for both species from the slopes of the curves’ parts corresponding to the concentrations less than the cmc. As we noted in the Experimental Section, Triton X-100, the surfactant used in our experiment, is a mixture of OPE9 and OPE10. The values of Γmax for OPE9 and OPE10 differ by less than 10%, so for Triton X-100 we can use the average value of Γmax ) (2.3 ( 0.2) × 10-6 mo/m2.