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Dynamic Surface Excesses of Fluorocarbon Surfactants† Julian Eastoe,* Alex Rankin, and Ray Wat School of Chemistry, University of Bristol, Bristol BS8 1TS, United Kingdom
Colin D. Bain and Dmitrii Styrkas Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom
Jeff Penfold ISIS Facility, Rutherford Appleton Laboratory, Chilton, OXON OX11 0QX, United Kingdom Received October 10, 2002. In Final Form: December 19, 2002 Dynamic adsorption and surface tension behavior of aqueous fluoro-surfactant solutions has been investigated using a range of experimental techniques, including neutron reflection (NR). Equilibrium tensions, γeq, have been measured by drop volume tensiometry (DVT), and dynamic surface tensions, γdyn, have been determined using the nonperturbative method, surface light scattering (SLS). Dynamic conditions for NR and SLS measurements were established using an overflowing cylinder (OFC). The OFC provides a dynamic surface on the 0.1-1 s time scale and offers a large (∼50 cm2), near flat surface for interrogating interfacial properties. To exploit these techniques effectively, a fluorocarbon anionic surfactant, sodium bis(1H,1H-nonafluoropentyl)-2-sulfosuccinate (di-CF4) has been specifically selected. Molecular structure effects have been explored with the C6 analogue sodium bis(1H,1H,7H-dodecafluoro-n-heptyl) sulfosuccinate (di-HCF6). Using OFC-NR, dynamic surface excesses, Γdyn, have been measured directly, and these values have been compared to equilibrium coverages, Γeq, determined by DVT and NR. Close to the critical micelle concentration (cmc) of di-CF4 (1.58 mmol dm-3), Γdyn and Γeq are very similar, and the ratio φ ) Γdyn/Γeq is unity to within the precision of the experiment. For moderate differences in surface tension, up to ∆γ ) γdyn - γeq e 15 mN m-1, φ remains close to 1. At concentrations of 0.2-0.7 mmol dm-3, the dynamic surface excess Γdyn is measurably different from the equilibrium value Γeq (φ < 1). This concentration range coincides with the largest differences in surface tension, ∆γ. For both di-CF4 and di-HCF6, the maximum values of ∆γ and ∆Γ occur around the same bulk concentration, ∼0.7 mmol dm-3, suggesting that dynamic surface behavior is determined mainly by mass transport (which is related to the bulk concentration), rather than surfactant properties such as cmc, for these surfactants.
Introduction When a new interface is formed in a surfactant solution, the equilibrium surface tension (γeq) is not instantly reached. To reduce γ, surfactant molecules first must diffuse to the surface from the bulk, adsorb, and orient themselves at the interface: these processes give rise to a dynamic surface tension (DST, γdyn). This key property is important in many industrial and biological processes [e.g., ref 1 and references therein]. Examples are fast coating processes, such as photographic film manufacture; delivery of agricultural sprays; and gas transport across the expanding-contracting pulmonary membrane. Despite decades of work and the obvious industrial relevance, some important questions concerning dynamic adsorption remain unanswered. Numerous techniques have been developed for measuring γdyn, of which maximum bubble pressure (MBP) is perhaps the most widely used. Others such as Wilhelmy plate (WP) or rod methods are invasive, and an important concern is how much the surface is perturbed. The feasibility of surface light scattering (SLS), which is a noninvasive, optical, laser-based method, for * To whom correspondence should be addressed. E-mail:
[email protected]. Tel: + 44 117 9289180. Fax: + 44 117 9250612. † Part of the Langmuir special issue dedicated to neutron reflectometry. (1) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995.
determining γdyn has already been demonstrated.2 SLS measures light scattered by thermally excited capillary waves in the 2 f 4 × 104 s-1 range [e.g., ref 3]. The frequency and damping of this scattered light are sensitive to surface tension. For quantitative analysis of DST, the dynamic adsorbed amount Γdyn must also be determined, and this is much more difficult to measure directly; it can be inferred from γdyn, but only if the assumption that Γdyn(γ) ) Γeq(γ) is correct. Neutron reflection (NR) is an alternative, direct method for determining Γdyn. The main advantage of neutron reflection is that it essentially “counts” interfacially adsorbed molecules. On the other hand, tensiometry is indirect, and interpretation of the curve of γ versus ln(a) (where a is the activity) always involves assumptions in terms of an adsorption equation: this problem is particularly acute for off-equilibrium systems. Hence, the aim of this work was to determine dynamic surface excesses Γdyn directly using neutron reflection with model fluorocarbon surfactants. Dynamic conditions were established by use of an overflowing cylinder (OFC) described in detail elsewhere.2,4,5 The OFC is suitable for measurements on the 0.1-1 s time scale, which is of considerable interest for (2) Manning-Benson, S.; Bain, C. D.; Darton, R. C.; Sharpe, D.; Eastoe, J.; Reynolds, P. Langmuir 1997, 13, 5808. (3) Earnshaw, J. C.; McGivern, R. C.; McLaughlin, A. C.; Winch, P. J. Langmuir 1990, 6, 649. (4) Bain, C. D.; Manning-Benson, S.; Darton, R. C. J. Colloid Interface Sci. 2000, 229, 247.
10.1021/la0266740 CCC: $25.00 © 2003 American Chemical Society Published on Web 02/19/2003
Dynamic Surface Excesses of Surfactants
Figure 1. (a) The overflowing cylinder (OFC). (b) Schematic setup of the OFC with neutron reflection. Slits: S1, S2, S3, S4. M: incident beam monitor. D: final detector. Redrawn from ref 5.
practical applications of surfactants. The cell is shown in Figure 1a; it consists of a stainless steel cylinder, 80 mm in diameter and 140 mm tall. Surface properties, in particular the relationship between surface excess, surface expansion rate, and bulk concentration, depend solely on adsorption kinetics at any given concentration and surface excess. The OFC is particularly attractive as it offers a large (∼50 cm2), almost flat surface for analysis by ellipsometry, spectroscopy, and scattering techniques. Since the surface is at a steady state, experiments lasting up to several hours can be performed to improve signalto-noise, which is essential for NR experiments. Marangoni effects determine the surface expansion rate, which is found to be independent of the bulk flow rate.4 Consequently, adsorption kinetics can only be studied at a single surface concentration for any given bulk concentration. An OFC suitable for NR experiments requires a solution volume of approximately 1.5 L, and thus when working with deuterated materials for neutron studies, the amount and cost of surfactant required becomes a significant consideration. The OFC has been successfully used to investigate cationic hydrocarbon surfactants and has been interfaced to NR, SLS, and ellipsometry.2,4,5 The presence of micelles above the cmc can be a complicating factor, as they are likely to contribute toward bulk-to-interface transport. A micelle can be considered as an active reservoir, both as a source of and as a sink for surfactant monomer. Most of the experiments described (5) Manning-Benson, S.; Parker, S. R. W.; Bain, C. D.; Penfold. J. Langmuir 1998, 14, 990.
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here were performed in dilute solutions below the cmc, thereby minimizing these effects. Hence, choice of surfactant, in terms of physicochemical properties, is an important aspect of experimental design. The principal surfactant selected for this work is a dichain anionic surfactant, di-CF4 (sodium bis(1H,1H-nonafluoro-n-pentyl) sulfosuccinate), which is especially appropriate for the following reasons: (i) Its synthesis and dilute aqueous phase behavior (critical micelle concentration, cmc) have been reported.6-9 (ii) Its surface purity has been painstakingly characterized.8,9 (iii) Being a fluorocarbon, it has appropriate refractive indices for both light (SLS) and neutrons (NR). (iv) For hydrocarbon surfactants, the equilibrium postcmc surface tensions (γcmc) are typically 35 mN m-1, whereas this is much lower for di-CF4 at ∼18 mN m-1. Hence with di-CF4 and other fluorinated compounds, a wide surface pressure window (γwater - γcmc) of 54 mN m-1 can be achieved, compared to 37 mN m-1 for hydrocarbon surfactants. This gives a broader scope to DST studies. (v) Raw materials costs are a factor of 10 cheaper than the equivalent deuterated compound, an important consideration when the sample volume for this OFC experiment is 1.5 L. Effects of the fluorocarbon chain structure have been investigated with the C6 analogue, sodium bis(1H,1H,7Hdodecafluoro-n-heptyl) sulfosuccinate (di-HCF6) (note the molecular structures given in Figure 7, so these compounds are not strictly homologous). This introduction of H-CF2 groups at the hydrophobic chain termini has previously been shown to decrease the surface excess compared to that for a fully fluorinated chain; effects of these subtle changes in chemical structure on physicochemical properties have been fully documented elsewhere.8 In this paper, new insight into the dynamic adsorption process is obtained by comparing values of Γdyn with the normal equilibrium adsorbed amounts, Γeq, as a function of concentration and surfactant type. Experimental Section Materials. Fluorinated surfactants di-CF4 and di-HCF6 were synthesized based on established procedures first developed by Yoshino et al., among others.6-9 Results from elemental analysis and 1H, 19F, and 13C NMR (JEOL Lambda 300 MHz in CDCl3 for intermediates but d-acetone for surfactants) were consistent with the desired products. Mass spectroscopy (Fisons Autospec) gave no peaks for higher/lower chain homologues, indicating the surfactants were effectively single molecular weight compounds. Particular attention was paid to methods of purification to obtain surface chemically pure surfactants, and protocols detailed elsewhere8-11 were rigorously followed. H2O was taken from a Millipore Milli-Q system, and D2O (Fluorochem, 99.9%) was doubly distilled before use in NR experiments. Measured cmc’s were 1.58 mmol dm-3 for di-CF4 and 0.49 mmol dm-3 for diHCF6. Methods. Equilibrium and Dynamic Surface Tensions. All glassware, syringes, and capillaries (tensiometry) were scrupulously cleaned, as documented elsewhere.8-11 Equilibrium surface tensions γeq were determined using a Lauda TVT1 drop-volume tensiometer (DVT), which was operated as described before.8-11 (6) Yoshino, N.; Komine, N.; Suzuki, J.-I.; Arima, Y.; Hirai, H. Bull. Chem. Soc. Jpn. 1991, 64, 3262. (7) Yoshino, N.; Morita, M.; Ito, A.; Abe, M. J. Fluorine Chem. 1995, 70, 187. (8) Downer, A.; Eastoe, J.; Pitt, A. R.; Simister, E. A.; Penfold, J. Langmuir 1999, 15, 7591. (9) Eastoe, J.; Nave, S.; Downer, A.; Paul, A.; Rankin, A.; Tribe, K.; Penfold, J. Langmuir 2000, 16, 4511. (10) Nave, S.; Eastoe, J.; Penfold, J. Langmuir 2000, 16, 8733. (11) Eastoe, J.; Paul, A.; Rankin, A.; Wat, R.; Penfold, J.; Webster, J. R. P. Langmuir 2001, 17, 7873.
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Figure 2. Scan to optimize height of the D2O surface with respect to the incident beam. The maximum window here (∼1.6 mm) represents a traverse of the beam footprint from the front to the back rim edge. This DVT method allows long time scales (up to 15 min per drop) to be investigated, and so the decay of γ with time may be followed to ensure that a true equilibrium (plateau) value is reached. The temperature was 30 °C. The equilibrium surface excesses, Γeq, and limiting areas per molecule at the cmc, Acmc, were obtained by applying the Gibbs equation to a quadratic fit through the pre-cmc data points.
Γ)-
dγ 1 2RT d ln a
(1)
For such 1:1 anionic F-surfactants, the solute activities, a, are calculated using a Debye-Hu¨ckel formula, and the prefactor of 2 has been confirmed.9 Dynamic surface tensions, γdyn, were measured on the flowing surface of the OFC using a custombuilt SLS rig as described in the literature.2 Neutron Reflection. Experiments were performed using the OFC on the SURF reflectometer at ISIS at the Rutherford Appleton Laboratories, Didcot, U.K., as described elsewhere.5 The cell was positioned on kinematic mounts and an active antivibration table (adjustable in the z-direction) to assist in the removal of surface vibrations. The specular reflectivity R(Q) was measured normal to the flowing interface as a function of momentum transfer,
4π sin θ Q) λ
Figure 3. (a) D2O calibration of the SURF reflectometer using a free-flowing D2O surface on the OFC. The line represents the fit to yield the scale factor (s ) 0.0599) and background (6.0 × 10-6). T ) 30 °C. (b) Reflectivity profile for di-CF4 in NRW at the cmc (1.58 mmol dm-3) using a free-flowing surface on the OFC. T ) 30 °C. isotherm work.12 Hence, the area per molecule, As, and adsorbed amount, Γ, at each concentration were determined from eq 3.12
∑b
i
As )
i
Fτ
)
1 ΓNa
(3)
(2)
A grazing angle θ of 1.5° and an incident wavelength range of 0.5-6.5 Å gave an accessible Q range of 0.05-0.65 Å-1. To determine the optimal height of the flowing interface, the surface of the OFC was scanned through the ribbon-shaped incident beam, and an example “height-scan” is given in Figure 2. Measurements were made at 30 °C. The reflectometer was calibrated using D2O in the OFC, and Figure 3a shows a representative R(Q) profile for this standard. Surfactant solutions were prepared in null reflecting water (NRW, 8.0 mol % D2O in H2O); therefore the reflectivity was from the adsorbed monolayer only. Typically, it took 1 day of experimental time to set up the cell, minimize surface vibrations, optimize the surface position, and perform D2O calibrations. For surfactant solutions, counting times depended on concentration, but on average, this was 1 h per sample. An important limiting factor was the sample changeover time, which typically added an extra hour to each sample run. The dynamic data reported here were taken on two separate three-day runs on SURF. Repeat measurements were made, with fresh solutions, and separated by about 1 year. Equilibrium NR experiments, using standard liquid troughs, were carried out as before [e.g., refs 8-12]. NR Data Analysis. Reflectivity profiles were fitted using a least-squares routine to determine layer thickness, τ, assuming a uniform scattering length density, F, as usually employed in
∑b represents the sum of isotopic scattering lengths in the molecule. To determine reliable adsorption parameters, R(Q) data were analyzed in different ways: (i) both τ and F were allowed to float to find a minimum χ2 value; (ii) to ensure a global minimum in χ2, τ was fixed at values from 10 to 30 Å and then F was fitted. A minimum in χ2 was obtained at the same thickness as obtained in (i), and a constant value for As was obtained (which is related to the product τF).
Results and Discussion Alignment and Calibration of the OFC on the Reflectometer. Figure 1a shows an image of the OFC, and Figure 1b is a schematic of the experimental setup: specular neutron reflectivity R(Q) is measured normal to the flowing a-w interface. Two slits (S1, S2) collimate the beam before monitor M measures the incident flux. For pure D2O in the OFC, there is an increasing surface curvature near the rim owing to the high surface tension and absence of Marangoni stresses to drive the surface flow. (This curvature effect is also evident at the lower end of surfactant concentration but disappears for tensions (12) Lu, J. R.; Thomas, R. K.; Penfold, J. Adv. Colloid Interface Sci. 2000, 84, 143.
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Figure 4. Dynamic (SLS-OFC) and equilibrium (DVT) surface tension of di-CF4 solutions at 30 °C. For the equilibrium data, the pre-cmc data have been fitted by a quadratic function. Table 1. Slit Dimensions of SURF Reflectometer, as Shown in Figure 1b, Used in Dynamic Neutron Reflection Experiments on the OFC S1
S2
S3
S4
width/ height/ width/ height/ width/ height/ width/ height/ mm mm mm mm mm mm mm mm 40
2.0
20
0.5
20
1.0
30
2.0
below ∼50 mN m-1.) Therefore, the slit S2 (Figure 1b) is used to reduce the beam footprint, eliminating edge effects, and slits 3 and 4 limit background scattering at the final detector. The slit dimensions used in all the reflectivity measurements presented here are given in Table 1. To align the flowing surface, an approximate height for the surface was first ascertained by eye using reflection of a HeNe laser, which is coincident with the incident neutron ribbon. Next, a more sensitive height scan was performed, by detecting the intensity of reflected neutrons, to ensure the beam illuminates only the OFC center. An example of the sensitivity of reflected neutron intensity to height z is demonstrated in Figure 2. Owing to the grazing incidence angle of 1.5°, a change in height of 1 mm is equivalent to moving 38 mm across the surface of the OFC. Hence, it is possible to optimize scattered intensity from the flowing surface for D2O normalization runs and to account for slight variations in the height of surfactant solutions with different concentrations. For surfactant solutions, the reduction in surface curvature caused by lower surface tensions would potentially result in a larger “flat” surface region than for D2O. Hence, in principle, it would be possible to relax the slit geometry and increase the neutron footprint size to gain some signalto-noise. The downside would be to invalidate the previously run D2O calibration and determined scale factor. Therefore, there is a trade-off between optimizing incident flux (maximizing S2) and maintaining an accurate scale factor. Figure 3a shows a representative D2O calibration run. In this case, the scale factor was found to be 0.0599 and the flat background was ∼6 × 10-6. Figure 3a shows example dynamic reflectivity data with di-CF4, which are discussed below. Dynamic Surface Tensions and Surface Excesses for di-CF4. Equilibrium tensions measured using DVT are given in Figure 4. The cmc break point at (1.58 ( 0.3) mmol dm-3 is clearly defined with no minimum or shoulder, confirming the purity of the surfactant. As expected for a fluorocarbon, the surface tension at the
Figure 5. Variation of surface excess Γ for di-CF4 with bulk concentration at 30 °C. Example error bars are shown at the extremes of concentration. Table 2. Calculated and Derived Parameters for di-CF4 Solutions from OFC-NR and Drop Volume Tensiometry (*) NR Repeat Measurements from a Separate Experiment dynamic-NR c/(10-3 mol dm-3)
τ/Å
4.0 3.5 3.0 2.0 1.6 1.2 0.80 0.65 0.50 0.25 0.80(*) 0.30(*)
19.09 18.65 16.60 20.56 17.63 16.47 18.08 15.62 14.81 11.96 20.04 25.96
equilibrium-DVT
F/(10-6 Γdyn/(10-6 Å-2) mol m-2) As/Å2 2.03 2.11 2.22 2.03 2.17 2.29 1.98 1.61 1.58 1.50 1.63 0.88
3.03 3.08 2.89 3.04 2.99 2.96 2.80 1.97 1.83 1.40 2.55 1.79
54.8 53.9 57.6 50.7 55.5 56.2 59.4 84.1 90.8 118.4 65.1 92.6
Γeq/(10-6 mol m-2)
As/Å2
2.93 2.81 2.64 2.56 2.45 2.16 2.64 2.24
56.7 59.1 62.8 64.9 67.8 76.8 62.8 74.2
cmc is low, at (17.7 ( 0.1) mN m-1. Adsorption parameters derived from the fitted curve at pre-cmc values are given in Table 2. Also shown is the dynamic tension curve determined using SLS on the OFC; at the cmc, the dynamic surface tension γdyn ) (20.6 ( 0.5) mN m-1. At this concentration, mass transport to the surface is sufficiently fast that the surface is nearly at equilibrium with the bulk. However, below the cmc, γdyn quickly deviates from γeq, as diffusion of surfactant to the surface can no longer keep up with loss of surfactant by surface expansion, that is, Γdyn < Γeq. To confirm this interpretation, neutron reflectivity measurements have been used to determine ∆Γ ) Γeq - Γdyn directly. An example of a reflectivity curve for a di-CF4 solution at the cmc in the OFC is given in Figure 3a, together with the fit to the single-layer model. The concentrations studied and respective adsorption parameters derived from single-layer fits are given in Table 2. Figure 5 shows a comparison of dynamic and equilibrium surface excesses, Γdyn and Γeq, as a function of concentration. The data labeled “equilibrium NR” have been taken from ref 8. The agreement between the equilibrium isotherms obtained by NR and tensiometry is reasonably good, as has been discussed previously.8-12 For concentrations above cmc/ 2, the dynamic surface excess is similar to the equilibrium surface excess determined by NR or tensiometrically. Below this concentration, the dynamic adsorbed amount deviates from the equilibrium value with Γdyn < Γeq. Repeat NR measurements demonstrated a typical reproducibility
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Figure 6. Comparison of fractional coverage φ against dynamic surface pressure ∆γ for di-CF4.
in Γdyn of (5%. The corresponding uncertainty in surface excesses determined by DVT is 3% (at the cmc).8-11 Fractional Coverage. In comparing Figures 4 and 5, it is noteworthy that the dynamic tensions are measurably higher than equilibrium values even when the adsorbed amounts Γdyn and Γeq are very similar. Hence, only small changes in adsorption Γ, within the (5% error of NR, translate to significant differences in tension, ∆γ ) γdyn - γeq. This observation reflects a general feature of surfactant adsorption isotherms that near the cmc the surface tension changes rapidly with bulk concentration (since Γ is high) but Γ only changes slowly since it is near its limiting value and therefore on the plateau of the adsorption isotherm. From NR, DVT, and SLS measurements under equilibrium and dynamic conditions, a relationship can be found between the dynamic surface pressure, πdyn, defined as πdyn ) ∆γ ) γdyn - γeq, and the fractional coverage, φ, defined as the ratio Γdyn/Γeq. Close to the cmc, the surface coverages Γdyn and Γeq are essentially indistinguishable with φ ∼ 1. Since, for an expanding surface, the dynamic coverage Γdyn cannot be greater than that under equilibrium conditions, Γeq, the ratio φ cannot be greater than unity. Figure 6 shows how this coverage ratio φ depends on πdyn. Interestingly, for moderate differences in surface tension (below 15 mN m-1) the dynamic and equilibrium coverages appear to be the same to within the experimental precision. Only above this surface pressure, where the dynamic tension curve pulls away from the equilibrium isotherm (Figure 4), does the measured dynamic adsorption become significantly perturbed from that at equilibrium (φ < 1). Effect of Surfactant Chain Structure. NR-OFC, equilibrium neutron, and DVT studies were also made with a longer-chain surfactant, di-HCF6. The cmc of diHCF6 is 0.49 mmol dm-3, and the equilibrium molecular area is Acmc ) 82.6 Å2.8,13 The larger hydrophobic tails of di-HCF6 result in a higher Acmc compared with di-CF4. At first glance this is surprising, but these effects have been well-documented elsewhere and the reader is directed to refs 8 and 9. The molecular structures of these two surfactants are shown inset in Figure 7. The lower cmc for di-HCF6 means that low surface tensions are achieved at lower concentrations as compared to di-CF4, and it is of interest to explore this change (surface tension data not shown). The equilibrium and dynamic behavior of these two compounds are best compared in terms of a normalized
Neutron reflectivity has been used to study adsorption of two fluoro-surfactants at the dynamic interface of the overflowing cylinder cell. Although these experiments are not simple, experimental procedures have been established to obtain reliable adsorption isotherms for fluorinated ionic surfactants under dynamic conditions. For the neutron reflection experiments, a conservative selection of slit geometries, appropriate for D2O calibration, is essential for consistent measurements between samples. The maximum adsorbed amount on the OFC measured at high concentrations, Γmax, has been shown to be in broad agreement with equilibrium values of Γmax as obtained by tensiometry and equilibrium NR experiments. Although OFC-NR data have been obtained before, for the cationic surfactants CnTAB,5,14 the data presented here represent the first study of chain length effects for anionic surfactants by this method. Most importantly, the ability to measure Γdyn directly permits this value rather than Γeq to be used
(13) Rankin A. J. Ph.D. Thesis, University of Bristol, Bristol, U.K., 2002.
(14) Battal, T.; Shearman, G.; Valkovska, D.; Bain, C. D.; Darton, R. C.; Eastoe, J. Langmuir, in press.
Figure 7. Comparison of equilibrium surface excesses (lines) and corresponding dynamic surface excesses for di-CF4 (b) and di-HCF6 (9).
surface excess Γ/Γcmc, as shown in Figure 7. Note that the higher scattering length of di-HCF6 allows measurements to lower surface concentrations than for di-CF4. Figure 7 illustrates a similarity in behavior for the two surfactants with respect to concentration. The decrease in adsorption on the OFC surface occurs at around the same bulk surfactant concentration for both surfactants, irrespective of the cmc. This decrease occurs in the submillimolar concentration range (∼0.7 mol dm-3), and thus this surface deficit is clearly present at the cmc of di-HCF6, but not at the cmc of di-CF4. The cmc itself is of no intrinsic significance in the determination of φ; the total bulk concentration of surfactant is more important. Conclusions
Dynamic Surface Excesses of Surfactants
in evaluating dynamic surface tension data. For the fluorinated surfactants studied here, two points can be made: (i) Dynamic surface tensions may be quite different from equilibrium values, although this does not mean that the dynamic surface excess is measurably different from that at equilibrium value (Figure 6). Dynamic surface tension is very sensitive to small perturbations in the surface coverage when that coverage is near its limiting value. In this respect, γdyn behaves in a manner similar to γeq. (ii) Studies with two related surfactants, but with different fluorocarbon chain structures, indicate that the deficits in the surface adsorbed amount occur in the same bulk concentration range (
