Surfactants diffusion-limited adsorption

One can even go so far as to derive the isotherm from dynamic measurements. Note that this does not mean that a dynamic Langmuir isotherm is derived the theory is based on diffusion-limited adsorption, so the surface is taken to be fully relaxed with respect to the sub-surface concentration. In other words, the isotherm is taken to be identical to that in the static case. This is probably correct, unless under dynamic conditions the surfactant assumes a different conformation. 

Abstract We review a new theoretical approach to the kinetics of surfactant adsorption at fluid-fluid interfaces. It yields a more complete description of the kinetics both in the aqueous solution and at the interface, deriving all equations from a free-energy functional. It also provides a general method to calculate dynamic surface tensions. For non-ionic surfactants, the results coincide with previous models. Non-ionic surfactants are shown to usually undergo diffusion-limited adsorption, in agreement with the experiments. Strong electrostatic interactions in salt-free ionic surfactant solutions are found to... 

Fig. 1 Diffusion-limited adsorption exhibited by non-ionic surfactants. Four examples for dynamic surface tension measurements are shown decyl alcohol at concentration 9.49 x 10" M (open circles) adapted from ref. [17] Triton X-100 at concentration 2.32 x 10 M (squares) adapted from ref. [8] CiaEOg at concentration 6 x 10 M (triangles) and CioPY at concentration 4.35 x 10 M (solid circles), both adapted from ref. [18]. The asymptotic t dependence shown by the solid fitting lines is a footprint of diffusion-limited adsorption...

Since a for common surfactants is of order lOT, we expect T/t to be much smaller than t. In other words, the adsorption of many non-ionic surfactants, not hindered by any high potential barrier, is expected to be diffusion-limited. The asymptotic time dependence (9) yields a distinct footprint for diffusion-limited adsorption, as demonstrated in Fig. 1. 

We infer that ionic surfactants with added salt behave much like non-ionic surfactants, i.e, undergo diffusion-limited adsorption provided that no additional barriers to adsorption exist. The departure from the non-ionic behavior depends on the salt concentration and is described to first approximation by Eq. (26). The footprint of diffusion-limited adsorption, i.e. the asymptotic time dependence, is observed in experiments, as demonstrated in Fig. 5. Consequently, the scheme described in Section 2 for solving the adsorption problem and calculating the dynamic surface tension in the non-ionic case is applicable also to ionic surfactants in the presence of salt, and good fitting to experimental measurements can be obtained [13]. 

Fig. 5 Diffusion-limited adsorption exhibited by ionic surfactants with added salt Dynamic interfacial tension between an aqueous solution of 4.86 X 10 M SDS with 0.1 M NaCl and dodecane (open circles and left ordinate), adapted from ref. [13] Dynamic surface tension of an aqueous solution of 2.0xl0 M SDS with 0.5 M NaCl (squares and left ordinate), adapted from ref. [30] Surface coverage deduced from second-harmonic-generation measurements on a saturated aqueous solution of SDNS with 2% NaCl (filled circles and right ordinate), adapted from ref. [31]. The asymptotic dependence shown by the solid fitting lines is a footprint of diffusion-limited adsorption...

Fig. 12. The complex capacitance plot and a corresponding equivalent circuit for diffusion-limited adsorption of electro-inactive surfactant represented by an additional capacity C. ...

The first theoretical model of surfactant adsorption from micellar solutions, proposed by Lucassen [142], uses the simplifying assumptions that the micelles are monodisperse and that the micellization happens as a single step, which is described as a reversible reaction of order n (the micelle aggregation number). Later, more realistic models, which account for the multi-step character of the micellar process, were developed [143-145]. The assumption for a complete local dynamic equilibrium between monomers and micelles makes possible to use the equilibrium mass action law for the micellization reaction [142,146,147]. In such a case, the surfactant transfer corresponds to a conventional diffusion-limited adsorption characterized by an effective diffusion coefficient, Deff, which depends on the micelle diffusivity, concentration, and aggregation number. Dgff is independent of the rate constants of the fast and slow demicellization processes and k. Joos et al. [146,147] confirmed experimentally that in some cases the adsorption from micellar solutions could be actually described as a diffusion-limited process characterized by an apparent diffusivity,... 

The role of adsorption kinetics and the diffusion of surfactants is especially important in controlling capillary impregnation. According to studies by N.N. Churaev, the solution impregnating the capillary quickly loses its dissolved surfactant due to adsorption of the latter on capillary walls, so the rate of impregnation may be limited by the diffusional transport of surfactant from the bulk of the solution to the menisci in the pores. 

The region of the CMC (n (c +o c ) c l) requires special consideration. Substitution of of t2 and ti from Eq. (5.264) into (5.272), and the transition to the limit c -> 0 leads to the dynamic surface elasticity of sub-micellar solutions [165] and thus to a rather obvious conclusion if a solution contains mainly monomers, micelles do not influence the dynamic surface properties. Therefore, even for low frequencies (diffusion controlled adsorption kinetics) there is a concentration range close to the CMC where the surface elasticity is almost constant and begins to increase gradually only at further increasing concentration. Finally the surface elasticity takes values given by relations (5.275) - (5.278). This concentration dependence was observed in experiments with nonionic surfactants [95]. The oscillating barrier... 

Lin, S.-Y., McKeigue, K., and Maldarelli, C., Diffusion-limited interpretation of the induction period in the relaxation in surface tension due to the adsorption of straight chain, small polar group surfactants theory and experiment, Langmuir, 7, 1055, 1991. 

This analysis leads us to the conclusion that ionic surfactants in salt-free solutions undergo kinetically limited adsorption. Indeed, dynamic surface tension curves of such solutions do not exhibit the diffusive asymptotic time dependence of non-ionic surfactants, depicted in Fig. 1. The scheme of Section 2, focusing on the diffusive transport inside the solution, is no longer valid. Instead, the diffusive relaxation in the bulk solution is practically immediate and we should concentrate on the interfacial kinetics, Eq. (21). In this case the subsurface volume fraction, t, obeys the Boltzmann distribution, not the Davies adsorption isotherm (15), and the electric potential is given by the Poisson-Boltzmann theory. By these observations Eq. (21) can be expressed as a function of the surface... 

Some classification of parameters in their connection with physical or mechanical processes is to be done. The main parameter connecting hydrodynamic and diffusion parts of the film flow problem with surfactant is Marangoni number Ma. The both variants of positive (Ma > 0) and negative (Ma < 0) solutal systems are considered. The main hydrodynamic parameters are Re, 7 or equivalently S. 7. This two values determine the mean film thickness i/, mean velocity and flow rate as well as parameter k. The diffusion parameters Pe,co determine the local thickness of diffusion boundary layer h and smallness parameter e. Two values T, Di characterize the masstransfer of surfactant by the adsorption-desorption and the intensity of dissipation by the surface diffusion. Besides the limiting case of fast desorption (T = 0) the more general case (T 1) are considered. Intensity of the surfactant evaporation by parameter Bi is determined. The remaining parameter G gives an indication to the typical value of surface excess concentration A in comparison with c. ... 

Limitations on neutron beam time mean that only selected surfactants can be investigated by OFC-NR. However, parametric and molecular structure studies have been possible with the laboratory-based method maximum bubble pressure tensiometry (MBP). This method has been shown to be reliable for C > 1 mM.2 Details of the data analysis methods and limitations of this approach have been covered in the literature. Briefly, the monomer diffusion coefficient below the cmc, D, can be measured independently by pulsed-field gradient spin-echo NMR measurements. Next, y(t) is determined by MBP and converted to F(0 with the aid of an equilibrium equation of state determined from a combination of equilibrium surface tensiometry and neutron reflection. The values of r(f) are then fitted to a diffusion-controlled adsorption model with an effective diffusion coefficient which is sensitive to the dominant adsorption mechanism 1 for... 

As a result, an electric field must arise within the limits of the diffusion boundary layer which is associated with the dynamic adsorption layer of ionic surfactants. The electric field is caused by the deformation of the DL, i.e. by its slight deviation from electroneutrality. 

To obtain Eqs. 5-10, it was assumed that the concentration of solute within the adsorption boundary layer is related to the solute-surface interaction energy by a Boltzmann distribution. The essence of the thin-layer polarization approach is that a thin diffuse layer can still transport a significant amoimt of solute molecules so as to affect the solute transport outside the diffuse layer. For a strongly adsorbing solute (e.g., a surfactant), the dimensionless relaxation parameter fila (or Kid) can be much greater than imity. If all the adsorbed solute were stuck to the surface of the particle (the diffuseness of the adsorption layer disappears), then L = 0 and there would be no diffusiophoretic migration of the particle. In the limit of [l/a 0 (very weak adsorption), the polarization of the diffuse solute in the interfacial layer vanishes and Eq. 5 reduces to Eq. 1. 

As stated above, the adsorption and desorption of molecules on an interface must be considered to understand the dynamic variations of surface tension. Let us consider a newly formed interface of a liquid which contains a bulk concentration c of surfactant. At the initial moments, the interface is almost free of surfactant molecules and may resemble the situation of Fig. la. In this case, any surfactant molecules which happen to be in the neighborhood of the interface will stick to it, and the rate at which the molecules arrive at the interface will limit their adsorption. In the absence of other driving mechanisms, this rate will depend only on the diffusion of molecules to the surface and is typically very short. 

When surface diffusion is negligible, Pe, the adsorption and desorption rates are slow (the insoluble surfactant limit) and the bubble motion is steady, one obtains stagnant cap behavior for the steady motion. In this regime, Eq. (39) reduces to... 

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