Measurement of the Dynamic Surface Excess in an Overflowing

Bain, C. D.; Burnett-Hall, G. D.; Montgomerie, R. R. Nature 1994, 372, 414. Domingues dos ..... Anna Angus-Smyth , Richard A. Campbell , and Colin D. ...
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Langmuir 1998, 14, 990-996

Articles Measurement of the Dynamic Surface Excess in an Overflowing Cylinder by Neutron Reflection Samantha Manning-Benson, Stephen R. W. Parker, and Colin D. Bain* Physical & Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, U.K.

Jeffrey Penfold ISIS, CCLRC, Chilton, Didcot, Oxon OX11 0QZ, U.K. Received September 30, 1997 Neutron reflection has been used to measure the dynamic surface concentration, Γdyn, of the cationic surfactant CTAB in an overflowing cylinder (OFC). This is the first direct measurement of Γdyn for a surfactant at an expanding liquid surface. Γdyn was measured for bulk concentrations between 1.8 and 0.125 mM, corresponding to areas per molecule of 42 to >300 Å2. Careful account had to be taken of the curvature of the surface in establishing calibration procedures. Γdyn was less than the equilibrium surface concentration Γe at all concentrations below the cmc but approached Γe at bulk concentrations well above the cmc, as expected for an expanding surface. Ellipsometry has been used previously to measure the coefficient of ellipticity of surfactant solutions in an OFC. The neutron reflection data were used to calibrate the ellipticity of the surface in terms of Γdyn, and ellipsometry was then used to map changes in Γdyn across the surface of the OFC with a precision of 2 × 10-8 mol m-2. The ellipticity can also be calibrated against Γe for static solutions, and this calibration curve can be used to calculate Γdyn from the dynamic ellipticities. The values of Γdyn determined by this indirect route were in excellent agreement with the direct measurements by neutron reflection.

Introduction In both natural and industrial processes the surfaces of surfactant solutions are frequently far from equilibrium. Examples from the industrial sphere include foams and emulsions, multiphase flow, detergency, coating, and wetting. The surface of the lung is an important natural example. Whereas the equilibrium properties of surfactant solutions are reasonably well-understood, at least from a phenomenological perspective, the dynamic behavior can be much more complex.1,2 The addition of a surfactant to a solution lowers the equilibrium surface tension σe due to adsorption of the surfactant at the surface of the solution. At thermal and chemical equilibrium the surface tension is uniform over the surface. If the surface area changes, either by expansion or contraction, the surface excess Γ deviates from its equilibrium value. Except in the case of spherical symmetry, the dynamic surface excess Γdyn and hence the dynamic surface tension σdyn are not uniform over the surface. The surface tension gradient generates a shear stress at the surface that is balanced by a viscous stress associated with a velocity gradient in the bulk fluid; in other words, the surface tension gradient accelerates the underlying liquid. This phenomenon is termed the Marangoni effect. Marangoni flow induced by surface tension gradients can dominate (1) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier, Amsterdam, 1995. (2) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann, Boston, 1991.

the hydrodynamic behavior near the surface and can even lead to the self-sustained motion of a liquid.3 Modeling of Marangoni flow is extremely complex: the hydrodynamic flow determines the mass transport of the surfactant to an interface, yet the hydrodynamic flow is itself determined by the distribution of surfactant at the interface. To develop a theoretical model, we need to understand the adsorption dynamics and the relationship between σdyn and Γdyn. To test a theoretical model, we need to be able to measure the dynamic interfacial properties of the surface. Several techniques have been developed for the measurement of σdyn, of which the most widely used is the maximum bubble pressure (MBP) method.4 Γdyn is more difficult to measure. Γdyn can be inferred from σdyn but only if one assumes that Γdyn(σ) ) Γe(σ), where Γe is the surface excess at equilibrium. To our knowledge, there have been no direct measurements of the dynamic surface excess for a soluble surfactant under controlled hydrodynamic conditions. In this paper, we report the measurement of Γdyn by neutron reflection from a cationic surfactant at the expanding surface of an overflowing cylinder. The kinetics of adsorption of surfactants on long time scales (minutes) has been studied by a number of techniques including ellipsometry5 and second-harmonic (3) Bain, C. D.; Burnett-Hall, G. D.; Montgomerie, R. R. Nature 1994, 372, 414. Domingues dos Santos, F.; Ondarc¸ uhu, T. Phys. Rev. Lett. 1995, 75, 2972. (4) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990; p 18. (5) Tiberg, F. J. Chem. Soc., Faraday Trans. 1996, 92, 531.

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Dynamic Surface Excess in an Overflowing Cylinder

generation (SHG)6 under conditions of natural convection. For adsorption measurements on the second and millisecond time scales, forced convection is essential. Steadystate techniques such as a jet,1 inclined plate,1 and overflowing cylinder (OFC)7-9 confer significant advantages over time-dependent techniques such as expanding bubbles. High-pressure jets are suitable for studying dynamics on the millisecond time scale, and Eisenthal has reported SHG measurements of p-nitrophenol adsorption in a jet.10 Measurements on conventional surfactants by this technique have not yet been reported, largely due to the weak signals and the difficulty in interpreting the SHG signal in terms of surface coverage. The inclined plate is suitable for measurements on longer time scales (∼1 s), but attempts to measure Γdyn by neutron reflection from a liquid flowing down an inclined plate have been hampered by instabilities in the flow.11 The OFC, which we employ here, is suitable for measurements on the 0.1-1-s time scale. The OFC is particularly attractive because it offers a large, almost flat surface for analysis by spectroscopic or scattering techniques and because the flow is strictly two-dimensional, which simplifies hydrodynamic modeling. In an OFC, water is pumped vertically upward through a cylinder and allowed to flow over the horizontal rim.8 Liquid at the surface flows radially outward from the center toward the rim of the cylinder. In the presence of a surfactant, the surface tension gradient at the surface can increase the surface velocity by an order of magnitude compared to that for pure water. The surface expansion rate12 is typically in the range 1-10 s-1, which for dilute surfactant solutions leads to a surface that is far from equilibrium. In previous papers, we have described the noninvasive measurement of surface properties on an OFC by ellipsometry8 and surface light scattering.9 We have also obtained vibrational spectra from surfactants at the surface of an OFC by external reflection FTIR spectroscopy.13 While both the coefficient of ellipticity Fj obtained by ellipsometry14 and line strengths in FTIR spectra are determined by the structure of the monolayer, neither quantity is simply and unambiguously related to Γdyn.15 In contrast, neutron reflection gives a direct quantitative measurement of the surface concentration. The reflection of neutrons from a surfactant monolayer is formally identical to the reflection of s-polarized light, (6) Rasing, Th.; Stehlin, T.; Shen, Y. R.; Kim, M. W.; Valint, P., Jr J. Chem. Phys. 1988, 89, 3386. (7) Padday, J. F. Proc. Int. Congr. Surf. Act. 1957, 1. Piccardi, G.; Ferroni, E. Ann. Chim. (Rome) 1951, 41, 3. Bergink-Martens, D. J. M. Interface Dilation. The Overflowing Cylinder Technique. Ph.D. thesis, Agricultural University, Wageningen, 1989. Bergink-Martens, D. J. M.; Bos, H. J.; Prins, A.; Schulte, B. C. J. Colloid Interface Sci. 1990, 138, 1. Bergink-Martens, D. J. M.; Bos, H. J.; Prins, A. J. Colloid Interface Sci. 1994, 165, 221. Bergink-Martens, D. J. M.; Bos, H. J.; van Kalsbeek, H. K. A. I.; Prins, A. In Food Colloids and Polymers: Stability and Mechanical Properties; Dickenson, E., Walstra, P., Eds.; Royal Society of Chemistry: Cambridge, U.K., 1993; pp 291, 376. Darton, R. C.; Grunnet-Jepsen, H.; Thomas, P. D.; Whalley, P. B. Paper 132C, AIChE Conference, San Francisco, 1994. (8) Manning-Benson, S.; Bain, C. D.; Darton, R. C. J. Colloid Interface Sci. 1997, 189, 109. (9) Manning-Benson, S.; Bain, C. D.; Darton, R. C.; Sharpe, D.; Eastoe, J.; Reynolds, P. Langmuir 1997, 13, 5808. (10) Castro, A.; Ong, S.; Eisenthal, K. B. Chem. Phys. Lett. 1989, 163, 412. (11) Cook, D.; Thomas, R. K. Private communication. (12) The surface expansion rate is defined as d ln A/dt, where A is the area at the air-water interface of an element of fluid at the surface. (13) Parker, S. R. W. Thesis, Oxford University, 1997. (14) Meunier, J. In Light Scattering by Liquid Interfaces and Complementary Techniques; Langevin, D., Ed.; Marcel Dekker: New York, 1992; Chapter 17. (15) For dilute surfactant solutions, the distinction between the surface concentration and the surface excess is insignificant, and we will use Γdyn to refer to either.

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with the neutron scattering length taking the role of the polarizability.16 The refractive index for neutrons η is determined by the neutron scattering length density Fs

η ) 1 - (λ2/2π)Fs

(1)

and the scattering length density is determined by the composition of the film

Fs )

∑nibi

(2)

where ni is the number density and bi the scattering length of the ith nucleus. A characteristic of neutron scattering is that the scattering lengths of deuterium (b ) 6.67 × 10-5 Å) and hydrogen (b ) -3.74 × 10-5 Å) have opposite signs. Since the scattering length of 16O is 5.80 × 10-5 Å, D2O and H2O scatter neutrons with opposite phases. A mixture of ∼8% D2O in H2O has the same scattering length density as air and is consequently referred to as nullreflecting water (nrw). The scattering length of a CH2 group is small, and normal surfactants adsorbed at the surface of nrw are virtually invisible to neutrons. Deuterated surfactants, however, are strong scatterers and are readily detected by neutron reflection (NR). The neutron reflectivity from deuterated surfactants in nrw provides a sensitive, direct, and absolute measure of the surface concentration of the surfactant. In this paper we first describe the OFC and the NR experiment. We then present data acquired for solutions of the cationic surfactant hexadecyltrimethylammonium bromide (CTAB) over a range of concentrations and for three different sampling areas centered on the axis of the overflowing cylinder: 20 mm × 20 mm, 30 mm × 30 mm, and 40 mm × 40 mm. We refer to these neutron footprints as A20, B30, and C40, respectively. We discuss the problems introduced by the curvature of the surface and the conditions required to optimize the signal-to-noise ratio without introducing calibration errors. Finally, we compare the surface concentrations determined directly by NR from the OFC with those calculated indirectly from ellipsometry and show how the two techniques can be used in combination to map the variation in Γ across the surface of the OFC. Experimental Section The OFC was constructed of stainless steel and had an internal diameter of 80 mm and a height of 140 mm but was similar in other respects to the glass OFC that we have described previously.8 The OFC was mounted on an active antivibration platform (Sandercock JRS, Mod-2) on a computer-controlled vertical translation stage. The dimensions of the sample station did not allow for a separate leveling plate, so thin shim stock was used to ensure that the top rim of the OFC was accurately horizontal. Two reservoirs were used to decouple the OFC from pump vibrations, and a water bath was employed to thermostat the solutions at 298 ( 0.5 K. The total volume of liquid in the system was 1.5 dm3. The flow rate was 16 cm3 s-1, corresponding to a vertical velocity of 3.2 mm s-1 in the cylinder. The length of the wetting film on the outside of the cylinder was approximately 120 mm. Both these parameters lay in the regime where the interfacial properties are independent of the values of the parameters.8 The NR measurements were carried out on the reflectometer “SURF” at the neutron spallation source “ISIS” (Rutherford(16) Lee, E. M.; Thomas, R. K.; Penfold, J.; Ward, R. C. J. Phys. Chem. 1989, 93, 381. Lekner, J. Theory of Reflection; Nijhoff: Dordrecht, 1987.

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Figure 1. Schematic illustration of the overflowing cylinder (OFC) and the neutron beam line: S ) slit; M ) monitor; D ) detector. Appleton Laboratory, Didcot, U.K.).17 A pulsed beam of neutrons with wavelengths λ in the range 0.5-6.5 Å was reflected from the flowing surface of the OFC at an angle of incidence θ ) 1.5° and detected by a time-of-flight detector. The wavelength of the neutrons can be calculated from the time of arrival of the neutrons at the detector. The sample station is shown schematically in Figure 1. A pair of slits S1 and S2 before the OFC collimate the neutron beam. A monitor, M, measures the incident flux from the source. The surface of the water becomes increasingly curved near the rim of the OFC, so slit S2 was used to restrict the area of the OFC illuminated by the neutron beam. The dimensions of S2 were 0.5 mm × 20 mm, 0.8 mm × 30 mm, and 1 mm × 40 mm for footprints A20, B30, and C40, respectively. Slit S3, immediately after the sample, and slit S4, in front of the detector and 2.43 m from the sample, were used to suppress the background. Slits S1 and S4 were set to 2 mm × 40 mm throughout, and S3 was 2 mm high and the same width as S2. Solutions of CD3(CD2)15N+(CH3)3Br- (dCTAB) were prepared in nrw. Deuterated hydrocarbon chains scatter neutrons strongly, so the specular reflection from the dCTAB solutions is assignable to the adsorbed surfactant, superimposed on a weak background signal from incoherent scattering in the subphase. The data were obtained in the form of reflectivity, R(Qz), where the momentum transfer Qz ) 4π sin θ/λ, and fitted with a three-layer model (air-monolayer-nrw) in which the thickness τ and scattering length density F of the monolayer and the background level are free parameters. The area per molecule A was then calculated from A ) ∑cibi/Fτ, where ci is the number of atoms of type i in the molecule. Any residual hydrogen in the dCTAB will reduce ∑cibi and lead to a systematic overestimation in A. Pure D2O (Fluorochem) was used as a calibration standard in the OFC. To conserve dCTAB, solutions were prepared by successive dilution of stock solutions of initial concentration 1.8 and 0.56 mM. The dCTAB was a gift from Dr. R. K. Thomas and was synthesized by established procedures.18 Contrast-matched water was prepared with 22.4 g of D2O per 250 mL of solvent. Ultra-high-purity H2O (Elga) was used throughout.

Results (i) Calibration. The determination of the surface concentration of surfactants by neutron reflection is wellestablished.19 In the usual procedure, pure D2O is employed as a calibrant to establish a scale factor that is then applied to the reflectivity profiles R(Qz) of the surfactant solutions. This calibration procedure relies on the collection efficiency of the detector being the same for the calibrant and sample. For static samples, the use of a large trough ensures that the analysis area is sufficiently flat that all the specularly reflected neutrons pass through slit S4 and fall onto the detector. In an (17) Penfold, J.; Richardson, R. M.; Zarbaksh, A.; Webster, J. R. P.; Bucknall, D. G.; Rennie, A. R.; Jones, R. A. L.; Cosgrove, T.; Thomas, R. K.; Higgins, J. S.; Fletcher, P. D. I.; Dickinson, E.; Roser, S. J.; McLure, I. A.; Hillman, A. R.; Richards, R. W.; Staples, E. J.; Burgess, A. N.; Simister, E. A.; White, J. W. J. Chem. Soc., Faraday Trans. 1997, 93 (22), 3899-3917. (18) Simister, E. A.; Thomas, R. K.; Penfold, J.; Aveyard, R.; Binks; B. P.; Cooper, P.; Fletcher, P. D. I.; Lu, J. R.; Sokolowski, A. J. Phys. Chem. 1992, 96, 1383. (19) Bradley, J. E.; Lee, E. M.; Thomas, R. K.; Willatt, A. J.; Penfold, J.; Ward, R. C.; Gregory, D. P.; Waschkowski, W. Langmuir 1988, 4, 821.

Manning-Benson et al.

OFC, the solution near the surface accelerates radially outward. In the presence of surfactant, the necessary force at the interface arises from a surface tension gradient and the surface is extremely flat until one approaches the rim of the cylinder. A pure, isothermal liquid cannot generate a surface tension gradient. The radial force that accelerates the D2O in the calibration experiment must arise from a hydrostatic pressure gradient, which results in a slight doming of the liquid surface. As a consequence, one cannot assume a priori that the surface curvature and hence the collection efficiency of the detector are the same for the D2O and the CTAB solutions in the OFC. We therefore made measurements for three different analysis areas of size 20, 30, and 40 mm; the smallest analysis area sampled only the flattest part of the surface but at the cost of a poorer signal-to-noise ratio. Table 1 shows the calculated areas per molecule, A, for a 1.8 mM solution of dCTAB in nrw for the three slit combinations A20, B30, and C40. Separate calibration profiles from D2O were used for each footprint. 1.8 mM is twice the critical micelle concentration (cmc) of CTAB. Ellipsometry shows that at this concentration the surface coverage in the OFC is very close to the limiting coverage in static solutions above the cmc. NR measurement by Thomas and co-workers20 on static solutions yields an area per molecule of 43 ( 2 Å2 for CTAB at the cmc. For the smallest footprint, A20, the measured value A ) 42 Å2 was within the experimental error of this value. For the two larger footprints, B30 and C40, the values of A were too low. These results suggest that the collection efficiency for the reflected neutrons was higher for the surfactant solution than for the pure D2O for B30 and C40. We tested the effect of curvature on the collection efficiency by reflecting a He-Ne laser beam from the surface and measuring the deflection of the beam at a distance of 2.4 m from the sample, comparable to the distance from the sample to the final slit. Slit S4 in front of the detector is 2 mm high and therefore limits the collection angle of the detector to 1 mrad. For a 1-mm high beam (C40) deviation of the beam through more than 0.2 mrad (0.5 mm) will cause some of the beam to fall outside the slit, assuming a parallel beam and perfect alignment of the sample. For the smallest footprint (A20) the beam is only 0.5 mm high and therefore a larger deflection can be tolerated. Figure 2 shows the deflection of the He-Ne beam as a function of radial distance, r, from the center of the OFC for pure H2O and for three solutions of CTAB at the concentrations 0.125, 0.68, and 1.8 mM (all under flowing conditions). The curvature of the surface is greatest for pure water and decreases progressively as the bulk concentration of the CTAB increases. For r < 10 mm, the maximum deflection in the reflected beam is 0.5 mm, even for the pure water. For the A20 configuration a deflection of 0.75 mm in the vertical direction will still pass through slit S4. Provided that the sample is centrally aligned, the collection efficiency for the reflected neutrons will be close to 100% for all the samples (a small loss from the extreme corners of the irradiated square may occur for pure water). The calibration is therefore valid. We note that an error of 0.1 mm in the height of the surface of the OFC changes the location of the neutron beam on the surface by 5 mm, at which point the edge of the reflected beam from the D2O would fall outside slit S4. Accurate alignment is therefore essential. (20) Lu, J. R.; Hromadova, M.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys. Chem. 1994, 98, 11519.

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Figure 2. Magnitude of the deflection h of a He-Ne beam reflected from the flowing surface of the OFC as a function of the radial distance r from the center of the OFC (h ) 0). Negative values of r correspond to the side of the OFC nearest the detector. The displacement was measured 2.4 m from the OFC. The precision of the measurements is (0.3 mm. Concentration of CTAB in H2O: 0 mM (O); 0.125 mM (2); 0.68 mM (0); 1.8 mM (1).

Figure 3. Reflectivity of the surface of the OFC as a function of momentum transfer Qz for solutions of dCTAB in nrw. Concentration of dCTAB: 1.8 mM (2); 0.46 mM (3); 0.125 mM (b). The footprint was A20 for the 1.8 and 0.46 mM solutions and C40 for the 0.125 mM solution. Acquisition times were approximately 1 h for the two higher concentrations and 5 h for the lowest concentration. Single-layer fits to the reflectivity curves are shown.

Table 1. Calculated Area per Molecule A for a 1.8 mM Solution of dCTAB with Three Different Footprints A20 B30 C40

footprint/mm2

S2/mm2

S4/mm2

A/Å2

20 × 20 30 × 30 40 × 40

40 × 1 30 × 0.8 20 × 0.5

40 × 2 30 × 2 20 × 2

38 37 42

For the middle footprint, B30, the collection efficiency remains close to 100% for the two most concentrated dCTAB solutions, but for the 0.125 mM dCTAB and pure water more than 10% of the reflected beam will be lost. Since S4 has a width of 40 mm and S2 has a width of only 30 mm, no signal will be lost due to horizontal deflection of the neutron beam. For the largest footprint, C40, neutrons fall on a 40 mm × 40 mm area and even the most concentrated dCTAB solution will start to lose its signal from neutrons reflected from the corners of the square. Since S2 and S4 have the same width for C40, the signal will be lost due to horizontal deflection as well as vertical deflection. Pure D2O is not an acceptable calibrant for the more concentrated CTAB solutions for the two larger footprints, B30 and C40. For the 0.125 mM CTAB solution, the surface profile is very similar to that of pure water and therefore the calibration is reasonably reliable even for the larger footprints where an appreciable fraction of the reflected signal is lost. We stress that these calculations on slit sizes are specific to our OFC and to the SURF beamline. Nevertheless, the same considerations will apply in the study of any flowing liquid on any beamline, and similar calculations will be required to establish the most appropriate calibration procedure. (ii) Dependence of Γdyn on CTAB Concentration. Figure 3 shows the neutron reflectivity R(Qz) as a function of the momentum transfer Qz for three representative concentrations of dCTAB: 2 × cmc (1.8 mM), 0.5 × cmc (0.46 mM), and the lowest concentration studied (0.125 mM). For the two higher concentrations, footprint A20 was employed. For the lowest concentration, the reflectivity was extremely weak and C40 was used to increase the signal. The NR profiles were analyzed on a single-slab model of the monolayer to obtain the areas per molecule Adyn

Figure 4. Area per molecule, A, determined by neutron reflection as a function of the concentration of dCTAB in the OFC for three footprints: A20 (9); B30 (b); C40 (2). The monolayer was modeled as a uniform slab.

shown in Figure 4. As expected, Adyn decreases with increasing concentration even above the cmc because the subsurface concentration is lower than the bulk concentration. Values of Adyn are shown for all three footprints. As the bulk concentration of dCTAB decreases, the surface curvature more closely resembles that of the D2O calibration sample, and consequently the percentage difference between the values of Adyn determined for the different footprints also decreases. We note that Γdyn is not constant across the surface (vide infra) but decreases toward the rim. We would therefore not expect different footprints to give identical areas per molecule. The calculated difference in the mean value of Γdyn between A20 and C40 is, however, less than 1 × 10-7 mol m-2. Discussion (i) Acquisition of NR Data from an OFC. The reflection of neutrons from solutions of dCTAB in the OFC is readily detectable even at areas per molecule as low as 300 Å2. The key issue is one of calibration. The smallest

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footprint (20 mm × 20 mm) suffers least from the curvature of the surface and gives a satisfactory signal-to-noise ratio for all except the lowest concentrations. Increasing the height of the final slit S4 from 2 mm to 3 mm would reduce the sensitivity to alignment at the cost of a small increase in background scatter. The width of slit S4 could, however, be reduced to 24 mm with no loss in signal. For the most dilute solutions, a larger footprint is desirable to maximize the signal level. The curves shown in Figure 2 suggest that there is little benefit in increasing the footprint beyond 30 mm × 30 mm (B30), since most of the extra signal will fall outside the slit S4. The height of S4 can, of course, be increased, but the consequent rise in the background level will rapidly outweigh any benefit from the enhanced signal. A final slit dimension of 3 mm × 40 mm (compared to a beam dimension of 0.8 mm × 30 mm) would admit most of the reflected neutrons, though further work would be required to establish definitively the dimensions that maximized the signal-to-noise ratio. The data shown with squares in Figure 4 (A20) are the most reliable for the reasons explained above. The areas per molecule obtained with different footprints converge as the concentration of dCTAB is reduced because the surface curvature at low concentrations of CTAB approaches that of the calibration. There are two principal sources of error in this experiment. First, errors in the alignment of the calibration sample lead to errors in the scale factor used to normalize the reflectivity profiles from the surfactant solutions. It is difficult to ascertain precisely the magnitude of the potential errors, but an upper limit of (10% in the scale factor is a plausible estimate. The resulting error limit for A is (5%, since the reflectivity scales as A-2. An increase in the vertical height of slit S4, as discussed above, would reduce the sensitivity to alignment errors. Second, the preparation of solutions by serial dilution leads to cumulative errors in the concentration of several percent. These errors could be reduced by flushing the system with each concentration before use, but at the cost of an increase in the consumption of expensive deuterated chemicals. Random errors in the data and in the fitting routine used to derive areas per molecule are negligible compared to the two principal sources of error. Two systematic errors in Adyn arising from incomplete deuteration of the dCTAB and as a result of neutrons from the D2O calibrant falling outside slit S4 act in opposite directions: the good agreement between the measured and expected areas for the 1.8 mM solution suggests that the net effect of these two errors is small. (ii) Comparison with Other Experiments. Figure 5 compares our values of Γdyn with static measurements of Γe obtained by Lu et al. by neutron reflection.20 The values of Γdyn lie below Γe at all concentrations below the cmc. For sufficiently high bulk concentrations we would expect Γdyn to approach the value of Γe at the cmc, which we indeed observe. The trends in Γdyn and Γe are in excellent agreement with previous comparisons of dynamic and equilibrium properties by surface tensiometry and ellipsometry.8,9 Given the analogy between the reflection of neutrons and light, one might hope to obtain the surface excess from ellipsometry in a manner analogous to neutron reflection. In comparison with neutron reflection, ellipsometry has several advantages: it is rapid (∼1 s), precise (errors , 1% for a monolayer), local (footprint ∼ 1 mm), and laboratory-based. The analysis of the reflection of neutrons from surfactant monolayers is, however, much more straightforward than the reflection of light for three reasons. First, the neutron scattering length is a scalar

Manning-Benson et al.

Figure 5. Dynamic surface excess (b) and equilibrium surface excess ([) for solutions of dCTAB measured by neutron reflection. The dynamic surface excess (O) calculated from ellipsometric data (see text) is also shown.

Figure 6. Generation of calibration curve Fje(Γ) for solutions of CTAB: main figure, curve of Γe(C) obtained from the neutron reflection data of Lu et al. (ref 20 and unpublished data); inset, plot of Fje(Γ) obtained from equilibrium measurements of Fje(C) and the curve fitted to Γe(C) in the main figure.

while the polarizability is a tensor: neutron measurements are independent of the orientation of the molecules; optical measurements are not. Second, there are no local field corrections in NR: the neutron refractive index is linearly related to the density of the monolayer, the optical refractive index is not. Third, there is no optical analogue of nrw: only the difference between the optical properties of the monolayer and water is measured.21 Consequently, while NR provides absolute surface coverages, the analysis of ellipsometry is highly model-dependent. Nevertheless, the benefits of ellipsometry make it desirable to use neutron reflection to calibrate the ellipticity in terms of surface excess. We have reported previously the coefficient of ellipticity, Fj, for CTAB solutions both at equilibrium and under dynamic conditions in the OFC.8 Figure 6 shows how we can use ellipsometry and neutron reflection on static solutions under equilibrium conditions to create a calibration plot Fje(Γ). First, we fitted a smooth curve through (21) NR has the more general advantage that the wavelengths of neutrons are better matched than visible light to the thickness of a monolayer, which allows much greater access to structural detail. For simple determinations of surface excess, the wavelength is less important.

Dynamic Surface Excess in an Overflowing Cylinder

the neutron data of Lu et al.20 to generate a function Γe(C) and then used this function to convert the ellipsometric data Fje(C) into Fje(Γ). We would expect Fjdyn(Γ) ≈ Fje(Γ) for the following reason. Both ellipsometry and neutron reflection probe only the interfacial region. It is therefore immaterial whether or not the interface is in equilibrium with the bulk. The surface age in the overflowing cylinder (which is conveniently defined as the reciprocal of the surface expansion rate) is in the range 0.1-1 s. The typical time scale for equilibration of the structural degrees of freedom in a monolayer at constant surface density is of the order of 10-9 s,22 which is very much less than the surface age. We would therefore expect the structure of the monolayer, and hence the ellipticity, to be determined only by the surface excess.23 In Figure 5 we have plotted Γdyn(C) obtained from the experimental values of Fjdyn and the calibration plot Fje(Γ). The agreement between the dynamic surface excess calculated this way and the direct measurements by neutron reflection is extremely good, from which we can draw two conclusions. First, the values of Γdyn obtained by neutron reflection from the OFC are reliable. Second, the calibration curve Fje(Γ) also applies to Fjdyn(Γ). We have shown that it is possible to determine Γdyn(C) from static and dynamic measurements of the ellipticity if equilibrium measurements of Γe(C) already exist.24 It is still, of course, beneficial to measure Γdyn(C) directly by NR to eliminate the errors inherent in the calibration procedure. In multicomponent systems, neither the surface tension nor the ellipticity uniquely determines the surface composition and NR is required to obtain the dynamic surface excess of individual components. Since we have measured Γdyn(C) directly for CTAB in the OFC, we can use these data to generate a function Fjdyn(Γ) without recourse to the equilibrium data (Figure 7). There is no a priori reason why Fjdyn should be proportional to Γ, but the experimental curve deviates only slightly from a straight line. In fact, if one were to approximate Fjdyn(Γ) by the best fit straight line to the experimental data, the surface excess deduced from measurements of the ellipticity would be in error by less than 1 × 10-7 mol m-2 over most of the concentration range. The simplest physical interpretation of a linear relationship between Fj and Γ is that the density of the monolayer remains constant with coverage and only its thickness changes.25 Further studies on other surfactant systems will be required to establish whether the linearity (22) Tarek, M.; Tobias, D. J.; Klein, M. L. J. Phys. Chem. 1995, 99, 1393. (23) There is also a secondary contribution, FjR, to the ellipticity from the roughness of the interface. FjR depends on the surface tension.14 We have shown previously that Fje(σ) ) Fjdyn(σ), where σdyn was measured by surface light scattering, and hence FjR(Γ) will also be the same under static and dynamic conditions. (24) The Gibbs equation can be integrated to give πcmc ) 2RT∫Γ d(ln C), where πcmc is the surface pressure at the cmc. According to this equation, the static neutron data Γe(C) in Figure 5 overestimate πcmc by 50%. The two most probable explanations for this disagreement are that trace amounts of other surfactants increase the surface excess at low concentrations or that trace divalent anions reduce the prefactor in the Gibb’s equation to a value less than 2. Neither of these effects is likely to play a role in the OFC, where trace impurities do not have time to diffuse to the surface. Since ellipsometry and neutron reflection are to a large extent measuring the same propertysa density profile normal to the interfaceswe would expect trace impurities or divalent counterions to have only a limited effect on the calibration curve F(Γ). The excellent agreement between the two measurements of Γdyn in Figure 5 supports this belief. Equilibrium surface tension measurements can also yield Γe(C), through the Gibbs equation, but calibration curves Fje(Γ), obtained in this way are much more sensitive to impurities. (25) After correcting for the effect of capillary wave roughness, Fjdyn(Γ) is linear for Γ > 1.5 × 10-6 mol m-2 and slightly convex at lower coverages. This curvature could be interpreted in terms of a decrease in density for the sparsest monolayers.

Langmuir, Vol. 14, No. 5, 1998 995

Figure 7. Relationship between dynamic ellipticity Fjdyn and dynamic surface excess Γdyn obtained by NR for CTAB solutions in the OFC. A linear fit to the data is shown.

Figure 8. Variation in the dynamic surface excess, Γdyn(r), as a function of the radial distance from the center of the OFC. Negative values of r correspond to the side of the OFC nearer the neutron detector. Values of Fjdyn(r) from ref 8 were converted into surface excesses from a fit to the data in Figure 7. Concentration of CTAB: 1.73 mM (2); 0.81 mM (O); 0.58 mM ([); 0.31 mM (3); 0.10 mM (9).

of Fjdyn(Γ) is a general feature of soluble surfactants or specific to CTAB. NR by itself does not have the spatial resolution or the precision to measure how Γ varies with radial distance, r, in the OFC. The plot of Fjdyn(Γ) can be used to convert the surface profiles Fjdyn(r) that we have published previously8 into plots of Γdyn(r). Examples of Γdyn(r) are shown in Figure 8 for solutions of CTAB at five concentrations. These curves closely follow the shape of the curves Fjdyn(r). The surface excess is approximately a quadratic function of r, decreasing toward the rim of the cylinder. Even in the concentration regime where large Marangoni effects are observed (0.3-1 mM), the variation in surface excess over the central 40 mm of the OFC is only 2 × 10-7 mol

996 Langmuir, Vol. 14, No. 5, 1998

m-2, or about 5% of a monolayer of CTAB. The precision arising from the ellipsometric measurements is better than 2 × 10-8 mol m-2, so even these small changes are readily measurable. Conclusions We have reported the first successful application of neutron reflection in the determination of the surface excess of a surfactant at a flowing liquid surface. Reliable data can be obtained from an area of size 20 mm × 20 mm at the center of an overflowing cylinder over a wide range of bulk concentrations. We were able to measure the dynamic surface excess of CTAB at areas per molecule ranging from 42 Å2 to greater than 300 Å2. An appropriate choice of slit sizes is essential to ensure that sample curvature is properly taken into account in the calibration procedure.

Manning-Benson et al.

The values of Γdyn(C) obtained by neutron reflection are lower than the equilibrium values Γe(C) at all concentrations below the cmc, as one would expect for an expanding surface. The agreement between the direct measurements of Γdyn reported here and the indirect measurements by ellipsometry is excellent. We have shown how a combination of neutron reflection and ellipsometry can be used to measure variations in the surface excess across the surface of the overflowing cylinder with a precision of 2 × 10-8 mol m-2. Acknowledgment. We thank Drs. R. C. Darton and R. K. Thomas for valuable discussions and the EPSRC (Grant GR/K 79611) and Unilever Research Port Sunlight Laboratory for financial support. We are particularly grateful to Dr. R. K. Thomas for the gift of dCTAB. LA9710785