Adsorption diffusion-limited

In any case, exceptions to the FIAM have been pointed out [2,11,38,44,74,76,78]. For example, the uptake has been shown to depend on the Cj M or rMI (e.g. in the case of siderophores [11] or hydrophobic complexes [43,50]), rather than on the free c M. Several authors [11,12,15] showed that a scheme taking into account the kinetics of parallel transfer of M from several solution complexes to the internalisation transporter ( ligand exchange ) can lead to exceptions to the FIAM, even if there is no diffusion limitation. Adsorption equilibrium has been assumed in all the models discussed so far in this chapter, and the consideration of adsorption kinetics is kept for Section 4. Within the framework of the usual hypotheses in this Section 3, we would expect that the FIAM is less likely to apply for larger radii and smaller diffusion coefficients (perhaps arising from D due to the labile complexation of M with a large macromolecule or a colloid particle, see Section 3.3). 

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. 

Regarding the chemical processes, sediments are heterogeneous at various sample, aggregate, and particle scales. Adherent or entrapped nonaqueous-phase liquids and combustion residue particulate carbon (e.g., chars, soot, and ashes) can also function as sorbents. Complex assemblages of these constituents can cause complex mass transfer phenomena, and the term sequestration refers to some combination of diffusion limitation, adsorption, and partitioning (Luthy et al. 1997). Some geosorbents exhibit typical nonlinear sorption behavior (Farrell and Reinhard 1994 Huang and Weber 1998). 

Differential capacitance measurements by Niki et for cytochrome C3 from D. vulgaris, strain Miyazaki, were consistent with irreversible, diffusion-limited adsorption for 4-s drop times above a concentration of 10 fiM. The surface excess of cytochrome C3 was calculated to be 0.92 x 10 " mole/cm. Niki etal also investigated the a.c. polarographic behavior of cytochrome C3 at the reversible half-wave potential. The capacitive peak height was frequency independent while the resistive peak height decreased with increasing frequency to a value of zero above 2000 Hz. These results were fit to a Laitinen-Randles equivalent circuit yielding an n value of... 

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. 

Fig. 2A Dependence between surface tension and surface coverage in diffusion-limited adsorption [Eq. (4)]. The values taken for the parameters match the example in (b). B Typical dynamic surface tension curve in diffusion-limited adsorption (reproduced from ref. [20]). The solution contains 1.586 x 10 M decanol. The solid line is a theoretical fit using the following parameters a = 4.86 A, a = 11.6T, = 3.90T (all three fitted from independent equilibrium measurements), and D = 6.75 x 10 cm /s...

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 adsorption of asphaltenes is practically irreversible. Significantly larger masses and molecule sizes of asphaltenes appear to be the reason. Diffusion of such molecules to solid surface is embarrassing. The mechanism of diffusion limited adsorption is realized (Syunyaev et al., 2009 Diamant Andelman, 1996). Gibbs energy values are more or less the same for surfaces of all investigated materials quartz, dolomite, and mica. It is known that quartz and dolomite are the main components of oil reservoir framework rocks. The porosity has no influence on kinetic parameters of adsorption. Asphaltenes adsorption at the surfaces of quartz and dolomite is the most active. 

The amount adsorbed increases with time but the rate of adsorption decreases. To obtain a characteristic time of diffusion-limited adsorption, we calculate the time required for a full monolayer to adsorb. Let us for simplicity further... 

Monolayer coverage is reached at F = 1 /nr. This leads to a characteristic time T for diffusion limited adsorption of... 

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