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Back | Show only these items: | Adsorption kinetics of ammonium perfluorononanoate at the air–water interface

1
Adsorption kinetics of ammonium perfluorononanoate at the air–water interface

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

The adsorption of the fluorocarbon surfactant, ammonium perfluorononanoate (APFN) has been studied in an overflowing cylinder (OFC).

Type: Object | Advantage: None | Novelty: None | ConceptID: Obj1

The equilibrium adsorption isotherm, determined by drop shape tensiometry, is well-described by the Volmer isotherm.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

The dynamic surface excess at the expanding surface of the OFC was measured by neutron reflectometry (NR).

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

The dynamic surface excess was in good agreement with the predictions of a diffusion-controlled adsorption model.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

Significant Marangoni effects were observed.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

The relationship between the mass transport equation in the OFC and the short and long-time asymptotes of the Ward–Tordai equation is noted.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

Introduction

The overflowing cylinder is a useful platform for studying adsorption kinetics and Marangoni effects (i.e., surface-tension-driven hydrodynamic flows) in surfactant solutions.^1–15

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

The overflowing cylinder provides an extremely stable, continuously expanding, air–water interface with a surface age in the range of 0.1–1 s.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

The large flat surface of the OFC is well-suited for a range of sensitive surface analytical techniques.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

In the presence of surfactants, surface tension gradients lead to large accelerations of the surface: the surface velocity can increase by more than an order of magnitude compared with pure water.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

More remarkably still, the interfacial properties are determined autonomously by the surfactant and become independent of the bulk flow rate or the dimensions of the cylinder (within certain limits).^1,11

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

Although the range of timescales accessible in the OFC is less than in some other methods, such as the maximum bubble pressure (MBP) technique,^16,17 the well-defined hydrodynamics and time-independence of the OFC allow precise testing of kinetic models for adsorption and provide a valuable complement to more traditional methods.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

We have shown recently⁸ that adsorption of the cationic surfactants C_nTAB (CH₃(CH₂)_n−1N⁺(CH₃)₃Br⁻, n = 12–16) at the surface of the OFC is diffusion-controlled at all concentrations below the critical micelle concentration (c.m.c.).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

For C₁₄TAB + NaBr, a model was developed incorporating micellar diffusion and adsorption was shown to be diffusion-controlled with fast micelle breakdown kinetics above the c.m.c.⁸

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

For C₁₆TAB + NaBr, micelle breakdown kinetics become important and a rate constant for micelle breakdown was deduced.¹⁴

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

In this note, we present data on the adsorption of the fluorocarbon surfactant ammonium perfluorononanoate (APFN) in an OFC.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

This study is the first on the adsorption kinetics of a perfluorocarboxylate surfactant at the air–water interface.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj1

Perfluorocarboxylate surfactants are highly surface-active molecules and can reduce the surface tension of water to <20 mN m⁻¹.^18,19

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

Synergistic effects in surfactant mixtures result in even lower surface tensions.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

Studies of the kinetics of adsorption of mixtures of APFN with hydrocarbon surfactants (where the interactions are primarily antagonistic) will be described elsewhere.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

In this preliminary study, we look at the adsorption kinetics of pure APFN.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

The specific choice of APFN for this study was dictated by the desirability of a Krafft point that is below room temperature and by a critical micelle concentration (c.m.c. = 8.9 mM) that is not too high.^18,19

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

To model adsorption kinetics, one needs reliable data on both the equilibrium and dynamic properties of a surfactant.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

In the Results section, we first present measurements of the equilibrium surface tension and deduce the adsorption isotherm.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

We then present measurements of the dynamic surface excess, Γ_dyn, on the OFC by neutron reflectometry at the spallation neutron source, ISIS.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

To interpret Γ_dyn, we need also to know the surface expansion rate, which is measured by laser Doppler scattering.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met2

The surface expansion rate, θ, was found to be in the range θ = 1–3 s⁻¹, which therefore defines the timescale over which information on the adsorption kinetics is accessible.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

The experimental values of Γ_dyn are then compared with the predictions of a diffusion-controlled adsorption model and found to be in excellent agreement.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

Finally, we draw an analogy between the kinetics of the overflowing cylinder and adsorption to a stagnant interface.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

Experimental section

Materials

Ammonium perfluorononanoate was prepared by neutralisation of heptadecafluorononanoic acid (Fluorochem, 99%) with ammonium hydroxide solution (Aldrich, 28% NH₃ in water, 99.99+%) followed by freeze-drying and recrystallisation (3 times) from dichloromethane/isopropanol.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

The tetra-ammonium salt of ethylenediamine tetra-acetic acid (EDTA) was prepared as either 74 or 130 mM aqueous solutions by neutralisation of EDTA (Aldrich, 99.999%) with 4 equivalents of ammonium hydroxide solution.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

Ultra-high purity (UHP) water was obtained from a variety of water polishing systems.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

Overflowing cylinder

The design and operation of the overflowing cylinder has been described in detail previously.¹

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

Briefly, a gravity-fed solution flows upwards in a stainless steel cylinder (8 cm in diameter) and overflows the rim.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The flow at the surface is radially outwards from a stagnation point at the centre of the cylinder.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The surface expansion rate, θ = d lnA/dt (where A is the area of an element of surface) is approximately uniform over the surface of the cylinder.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

Measurements on the OFC were conducted at a temperature of 298 K and a volume flow rate of 17 cm³ s⁻¹, corresponding to a mean vertical velocity in the cylinder of 0.34 cm s⁻¹.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The total fluid volume in the OFC and pumping system was 1.6 dm³.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

Surface tensiometry

Surface tensions were measured at 298 K by axisymmetric drop shape analysis (ITConcept, Longessaigne, France).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

Bubbles of air with a typical volume of 5 μl were formed in aqueous solutions of APFN containing 1 mol% of the tetra-ammonium salt of EDTA to complex calcium ions (the EDTA will be present in solution with varying degrees of ionisation, with the speciation dependent on the APFN concentration).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

EDTA was not added to the solutions for dynamic measurements by NR and LDV (see below), since trace ions do not have time to exert a measurable effect on the expanding air–water interface.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

Bubbles were allowed to equilibrate for 1–45 min before measurements were taken.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

The drift in the surface tension values was <0.1 mN m⁻¹ min⁻¹; 2–3 readings were acquired at each concentration.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

Tetra-ammonium EDTA was tested to ensure that it was not surface-active at the concentrations employed here.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

Laser Doppler velocimetry (LDV)

A 10 mW HeNe laser beam was split into two equal parts, focused, and recombined on the surface of the OFC.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp3

The APFN solution was seeded with a few mg of 2-μm TiO₂ particles.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp3

When these particles pass through the interference fringes formed by the crossed laser beams they scatter light with an intensity that is modulated at a frequency (in the range of 10⁴ Hz) that is determined by the spacing of the fringes and the velocity of the particles.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp3

Experimental details may be found elsewhere.¹

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp3

The surface velocity, u_s(r) was measured as a function of the radial position, r, and the surface expansion rate deduced from θ = r⁻¹d(ru_s)/dr.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

θ was almost constant across the central 5 cm of the cylinder: the value at r = 0 is reported.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

Neutron reflectometry (NR)

Neutron reflection experiments on the OFC were carried out on the SURF beamline at the spallation neutron source ISIS (Didcot, UK) according to a protocol that has been described in detail previously.^3,15

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

The reflectivity was calibrated with a sample of pure D₂O in the OFC before the sample runs and checked with a second calibration after the runs were complete.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

The calibration factors differed by 2%.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

The sample solutions were prepared in null-reflecting water (nrw = 0.08 D₂O: 0.92 H₂O by mole).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

Reflectivity profiles were acquired for between 20 min for the highest concentration of APFN and 130 min for the lowest concentration.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

The raw time-of-flight data were binned into intervals with a width of 5% of the momentum transfer and normalised by the calibration factor determined from the D₂O runs.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

The profiles were analysed on a simple three-layer model (air/monolayer/water) to yield a thickness, τ, and scattering length density, ρ, of the adsorbed monolayer.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

These two parameters are strongly correlated, but the surface excess can be determined reliably from the equationwhere b_i is the scattering length of atom i and n_i is the number of times that atom i appears in the molecular formula for the surfactant.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

Assuming that the ammonium counterion is fully exchanged with solvent and that all the counterions contribute to the reflectivity, the scattering length of APFN, ∑n_ib_i = 165.3 fm.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

In studies of anionic surfactants with the tetramethyl ammonium counterion, Thomas and coworkers found that a fraction of the counterions in the electrical double layer (up to 90% in one case) were not detected by NR.^20,21

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

However, since the scattering length density of NH₄⁺ in nrw is close to zero, the counterions make very little contribution to the reflectivity here.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

Neutron reflection can be used to determine the equilibrium adsorption isotherm of surfactants,^15,22 but due to the limited availability of beamtime, surface tensiometry was used instead (see above).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

Results and discussion

Equilibrium adsorption

The equilibrium adsorption isotherm of APFN was determined by surface tensiometry.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

The presence of divalent counterions, especially Ca²⁺, is known to interfere with interpretation of surface tension data for anionic surfactants,²³ so the tetra-ammonium salt of EDTA was added to the APFN solutions at a level of 1% of the surfactant concentration (by mole) to complex any trace Ca²⁺ ions present in the surfactant or leached from the glassware.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

To determine the appropriate level of EDTA, we measured the surface tension of a 0.7-mM solution of APFN containing varying concentrations of EDTA.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

In the presence of trace divalent ions, the surface tension is expected to show a maximum with increasing EDTA concentration: at low concentrations, divalent ions complex to the surfactant at the interface and lower the surface tension; at high concentrations EDTA increases the ionic strength, which screens the electrostatic repulsions between adsorbed APFN molecules, increases the extent of adsorption and lowers the surface tension.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp1

For 0.7-mM APFN, the surface tension was constant at low EDTA concentrations and decreased at concentrations of EDTA above 8 mol%.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

A value of 1 mol% falls within the plateau region.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

The ammonium salt of EDTA was used in preference to the commercially available tetrasodium salt so as not to add a second counterion to the system.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

The surface tension of APFN as a function of bulk concentration, c, is shown in Fig. 1.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

The Debye–Hückel Limiting Law was employed to convert the bulk concentrations into activities.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

Various adsorption isotherms were explored to fit the surface tension data.

Type: Method | Advantage: None | Novelty: None | ConceptID: Met4

The most appropriate isotherm is the van der Waals’ isotherm with counterion binding.^24,25

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

To determine the four unknown parameters in this model with confidence it is usually necessary to make measurements both with and without added electrolyte.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

For APFN, however, we found that the interaction parameter between adsorbed molecules was very small, so the simpler Volmer isotherm could be used eqn. (2).where K₁ = 4.6 × 10³ m³ mol⁻¹, K₂ = 1.2 × 10⁻³ m³ mol⁻¹, Γ_∞ = 5.74 μmol m⁻² and the concentrations of the surfactant ion (c₁) and counterion (c₂) immediately below the surface (z = 0) are determined by Gouy–Chapman theory.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

For the ammonium salt, counterion binding is weak (K₂ is very small); the isotherm is thus insensitive to the exact value of K₂ and can be determined with reasonable confidence in the absence of data with added electrolyte.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

The equivalent surface tension isotherm is shown as a solid line in Fig. 1.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

The intersection between this isotherm and a linear fit to the surface tension above the c.m.c. yields a value for the c.m.c. of 8.9 mM.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

This value is in agreement with a c.m.c. of 9.1 mM²⁶ obtained by electrical conductivity.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

The value of Γ at the c.m.c. is 4.3 μmol m⁻²; the limiting value of Γ < Γ_∞ because micelle formation places a cap on the chemical potential of the surfactant.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

Γ_∞⁻¹ is the excluded area per molecule, A₀, in the van der Waals’ model.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs7

The value we obtain for A₀ = 0.29 nm² is very close to the cross-sectional area of a helical fluorocarbon chain (0.27 nm²)²⁷.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

Dynamic surface excess

Sample reflectivity profiles are shown in Fig. 2 for three concentrations of APFN, together with theoretical fits to a three-layer model (air, monolayer, nrw).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs8

The dynamic surface excess obtained from such fits is plotted against the bulk concentration in Fig. 3.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

The equilibrium isotherm from surface tensiometry is shown as a solid line.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

We would expect agreement between Γ_e and Γ_dyn at high concentrations where the mass transport to the surface is fast on the timescale of the OFC.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp2

However, the limiting value of Γ_dyn = 4.7 μmol m⁻² above the c.m.c. is significantly higher than the tensiometric value of 4.3 μmol m⁻².

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs10

It is not uncommon for there to be a small difference in scaling factors between surface coverages determined by NR and tensiometry, even in measurements on static surfaces.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

There are several possible reasons for this discrepancy²³ including impurities in the surfactant, approximations in the isotherm used to fit the surface tension data, and errors in the determination of the calibration factor in the NR experiments.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

In the case of NR measurements on the OFC, obtaining an accurate calibration is complicated by the fact that the surface of the D₂O calibration sample is significantly more curved than the surface of the surfactant solutions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

This curvature defocuses the reflected neutron beam and reduces the transmission through the final slit before the detector.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

The consequence is that the scale factor is too small and the surface excesses of the surfactant are more or less uniformly overestimated.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

To mitigate this effect, the footprint of the neutron beam on the OFC is limited to 20 × 20 mm.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

There are no equilibrium NR measurements on APFN in large planar troughs for comparison with the OFC data, but we can compare with measurements on the 8-carbon homologue, APFO.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

In the presence of 0.1 M ammonium chloride, Simister et al.²⁸

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

Downes et al.,²⁹ and Lu et al³⁰. all reported a value of Γ_e = 4.1 ± 0.2 μmol m⁻² for APFO at the c.m.c.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

While APFN may have a slightly higher surface excess at the c.m.c. than APFO, it is unlikely to be 15% higher.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

We therefore assign the discrepancy in Fig. 3 to an underestimation of the scale factor in the NR measurements.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

100

To enable comparison of the equilibrium and dynamic surface excesses, the NR data were scaled (by 0.9) to agree with the surface tension isotherm at high surfactant concentration (dotted line in Fig. 4).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

101

The dynamic surface excesses then fall slightly below the equilibrium values at lower surfactant concentrations, as would be expected on physical grounds.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

Marangoni effects

102

The surface expansion rate determined by LDV is shown in Fig. 5 as a function of the bulk concentration of APFN.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met5

103

Surface accelerations are observed at all concentrations of APFN, with a maximum at 1.5 mM.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs11

104

The general shape is characteristic of monomeric surfactants (cf. C₁₄TAB^⁸).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs12

105

At low concentrations, there is very little adsorbed surfactant and hence the gradients in surface excess and consequently in surface tension are small.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs12

106

At high surfactant concentrations, mass transport to the surface is fast and the surface is close to equilibrium: again large surface tension gradients do not arise.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

Adsorption kinetics

107

Analysis of mass transport and convection at a surface under constant dilation yields a relationship between Γ_dyn, θ, the diffusion coefficient, D, the bulk concentration, c, and the subsurface concentration, c_s, at the edge of the electrical double layer.^4,31,32

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

108

In the absence of added electrolyte, the effective diffusion coefficient D = 2D_PFND_NH₄/(D_PFN + D_NH₄), where D_PFN = 5.2 ± 0.2 × 10⁻¹⁰ m² s⁻¹ is the self-diffusion coefficient of perfluorononanoate and D_NH₄ = 1.96 × 10⁻⁹ m² s⁻¹ is the diffusion coefficient of ammonium ions.³³

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

109

(D_PFN was determined by I. Furó by pulsed field gradient NMR of a 3-mM APFN solution in H₂O at 295 K.) If adsorption is diffusion-controlled, then the interface is locally at equilibrium: i.e. the relationship between Γ_dyn and c_s is the same as between Γ_e and c.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

110

For a mixed kinetic-diffusion model, Γ_dyn (c_s) < Γ_e(c).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

111

The diffusion-controlled values of the surface excess are shown with a solid line in Fig. 4.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs13

112

For each pair of experimental parameters, θ and c, there are an infinite number of values of Γ and c_s that satisfy eqn. (3).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs13

113

However, in eqn. (3), Γ_dyn is a monotonically decreasing function of c_s, while in the equilibrium isotherm Γ_e is a monotonically increasing function of c = c_s.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs14

114

Consequently, there is only one pair of values, Γ and c_s, that simultaneously satisfy the mass transport equations and the local equilibrium isotherm.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

115

This value of Γ is plotted in Fig. 4.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

116

As one would anticipate, the calculated values of the dynamic surface excess fall below the equilibrium values, since there has to be a concentration gradient to replenish the surfactant removed from the surface by dilation.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

117

The difference between Γ_dyn and Γ_e is greatest at low concentrations, where Γ is most sensitive to changes in c_s, and near the maximum in θ, where the mass transport away from the surface by dilation is greatest.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs15

118

The theoretical difference between Γ_dyn and Γ_e is never greater than 0.2 μmol m⁻², or about 5% of a monolayer of APFN.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res13

119

In the diffusion-controlled model, the difference, Δσ, between the dynamic surface tension, σ_dyn, and equilibrium surface tension, σ_e, is <2 mN m⁻¹ and increases with increasing θ.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

120

Since it is surface tension gradients that drive Marangoni effects, it is not surprising that there is a correlation between θ and Δσ.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

121

We note that much larger values of Δσ (≤30 mN m⁻¹) are observed with more efficient surfactants, such as the cationic surfactant, CTAB [CH₃(CH₂)₁₅N⁺(CH₃)₃Br⁻], or the non-ionic surfactant, C₁₂E₈ [CH₃(CH₂)₁₁(OCH₂CH₂)₈OH].

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs16

122

The maximum surface expansion rates for these surfactants are twice as high as for APFN.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs17

123

While this observation is intuitively reasonable, we do not yet have a sufficiently good understanding of the boundary conditions in the OFC to make quantitative predictions of surface expansion rates from the adsorption isotherm and transport properties of a surfactant alone.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

124

A comparison of the predicted and experimental values of Γ_dyn shows quantitative agreement (Fig. 4).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

125

We have shown in a previous paper that electrostatic barriers to adsorption are not expected to be important at strain rates accessible on an OFC.⁸

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

126

The agreement we observe here with a diffusion-controlled model shows that, on the timescale of the OFC, there are no steric barriers to adsorption for APFN either.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con5

Relationship to kinetic measurements at a static liquid surface

127

Finally, we comment briefly on the relationship between steady-state adsorption with convection on an OFC and adsorption at a clean surface in the absence of convection.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa2

128

In the latter case, the extent of adsorption is given by the classic Ward–Tordai equation³⁴(where t is the time since creation of the clean surface and τ is a dummy variable of integration).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

129

At long times, we can take c_s outside of the integral and eqn. (4) becomes¹⁷

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

130

Comparing eqns. (3) and (5), we see that there is an equivalence between the surface expansion rate, θ, and (2t)⁻¹ in the long-time asymptote of the Ward–Tordai equation.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs18

131

Consequently, measurements made on the OFC should map onto measurements of stagnant adsorption onto a clean surface in the long-time limit if the substitution t = (2θ)⁻¹ is made.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp3

132

For the experiments presented here, c_s > 0.85c at all concentrations (see Fig. 6), so the long-time asymptote should be a good approximation.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp3

133

One would therefore predict agreement with dynamic surface tension measurements made by, for example, the MBP method.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

134

Similarly, to obtain the short-time asymptote of eqn. (4) (c_s → 0), we neglect the back-diffusion term and obtainwhich once again has the same form as the mass-transport equation in the OFC (eqn. (3)), with substitution t = (2θ)⁻¹.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res17

135

While there is no direct parallel between the Ward–Tordai equation and eqn. (3) in the general case, since the mapping between θ and ½t holds in both the long-time and short-time limits, it is unlikely to lead to large errors at intermediate times.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

136

We note, however, that eqn. (3) holds under all conditions (below the c.m.c.) while eqns. (5) and (6) apply only in their respective asymptotic limits.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

Conclusion

137

An analysis of the kinetics of adsorption of the fluorocarbon surfactant, APFN, at the air–water interface shows that adsorption is diffusion-controlled at all concentrations below the c.m.c.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con5

138

The small deviations of the dynamic surface from equilibrium (Δσ ≤ 2 mN m⁻¹, ΔΓ ≤ 0.2 μmol m⁻²) are nevertheless sufficient to yield surface expansion rates in an OFC of up to 3 s⁻¹, six times greater than pure water at the same flow rate.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res19

139

Significant Marangoni effects can occur even when interfaces are very close to equilibrium.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con6

140

Under these conditions (c → c_s) there should be a direct correspondence between data from the OFC and from adsorption measurements at stagnant surfaces.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con7