Adsorption diffusion-controlled kinetics model

Surfactant conservation equation. This interfacial condition is a statement of solute balance on the interface. Several physicochemical effects can be modelled here including diffusion-controlled kinetics, adsorption-controlled kinetics, insoluble surfactants and surface solute production rate per unit area (for example by condensation or evaporation of solute across the interface). 

Formation and stripping of a cobalt adlayer on/from a polycrystalline Au electrode have been studied [469] applying electrochemical methods under underpotential conditions. The kinetics of deposition fitted a model of a simultaneous adsorption and diffusion-controlled two-dimensional instantaneous nucleation of cobalt on the electrode surface. 

Innocenti et al. have studied the kinetics [101] of two-dimensional phase transitions of sulfide and halide ions, as well as electrosorption valency [102] of these ions adsorbed on Ag(lll). The electrode potential was stepped up from the value negative enough to exclude anionic adsorption to the potential range providing stability of either the first or the second, more compressed, ordered overlayer of the anions. The kinetic behavior was interpreted in terms of a model that accounts for diffusion-controlled random adsorption of the anions, followed by the progressive polynucleation and growth. 

Adsorption Kinetics Diffusion and Kinetic Controlled Models... 

Active Oxygen Method (AOM), 535, 544 Adsorption, and interfacial properties diffusion and kinetic controlled models, 617-618 (figs.), 620-622 Gibbs adsorption isotherm, 617-619 kinetics of surface-active substances, 639... 

The rate of contaminant adsorption onto activated carixm particles is controlled by two parallel diffusion mechanisms of pore and surface diffusion, which operate in different manners and extents depending upon adsorption temperature and adsorbate concentration. The present study showed that two mechanisms are separated successfully using a stepwise linearization technique incorporated with adsorption diffusion model. Surface and pore diffiisivities were obtained based on kinetic data in two types of adsorbers and isothermal data attained from batch bottle technique. Furthermore, intraparticle diffiisivities onto activated carbon particles were estimated by traditional breakthrough curve method and final results were compared with those obtained by more rigorous stepwise linearization technique. 

The same steps as discussed above for the case of isotope exchange (diffusion in liquid film, surface reaction, intraparticle diffusion) were considered in a kinetic model [771] of metal ion adsorption from solution. This model was presented in a book with diskettes (FORTRAN program, rate controlled by reaction, by transport or mixed control). 

Barrow [772] derived a kinetic model for sorption of ions on soils. This model considers two steps adsorption on heterogeneous surface and diffusive penetration. Eight parameters were used to model sorption kinetics at constant temperature and another parameter (activation energy of diffusion) was necessary to model kinetics at variable T. Normal distribution of initial surface potential was used with its mean value and standard deviation as adjustable parameters. This surface potential was assumed to decrease linearly with the amount adsorbed and amount transferred to the interior (diffusion), and the proportionality factors were two other adjustable parameters. The other model parameters were sorption capacity, binding constant and one rate constant of reaction representing the adsorption, and diffusion coefficient of the adsorbate in tire solid. The results used to test the model cover a broad range of T (3-80°C) and reaction times (1-75 days with uptake steadily increasing). The pH was not recorded or controlled. 

The first physically sound model for adsorption kinetics, which was derived by Ward and Tordai [18], is based on the assumption that the time dependence of a surface or interfacial tension (which is directly proportional to the surface excess F, in mol m ) is caused by diffusion and transport of surfactant molecules to the interface. This is referred to as diffusion-controlled adsorption kinetics model . The interfacial surfactant concentration at any time t, T(t), is given by the following expression,... 

The adsorption kinetics of interfacial active molecules at liquid interfaces, for example surfactants at the aqueous solution/air or solution/organic solvent interface, can be described by quantitative models. The first physically founded model for interfaces with time invariant area was derived by Ward Tordai (1946). It is based on the assumption that the time dependence of interfacial tension, which is directly correlated to the interfacial concentration T of the adsorbing molecules, is caused by a transport of molecules to the interface. In the absence of any external influences this transport is controlled by diffusion and the result, the so-called diffusion controlled adsorption kinetics model, has the following form... 

There are two general ideas to describe the dynamics of adsorption at liquid interfaces. The diffusion controlled model assumes the diffusional transport of interfacially active molecules from the bulk to the interface to be the rate-controlling process, while the so-called kinetic controlled model is based on transfer mechanisms of molecules from the solution to the adsorbed state and vice versa. A schematic picture of the interfacial region is shown in Fig. 4.1. showing the different contributions, transport in the bulk and the transfer process. 

Theoretical Models of Diffusion-Controlled Adsorption Kinetics... 

The quantitative description of adsorption kinetics processes is much more complicated than the use of the simplified models mentioned above. An introduction into the variety of theoretical models and appropriate boundary conditions is given in a recent review (Miller et al. 1994a). The diffusion-controlled model assumes that the step of transfer from the subsurface... 

Although this is a very complex equation, it allows to take into consideration any function of R(t), and consequently A(t), resulting from experiments with growing drops or bubbles. In combination with an adsorption isotherm (diffusion-controlled case) or a transfer mechanism (mixed diffusion-kinetic-controlled model) it describes the adsorption process at a growing or even receding drop. Eq. (4.48) can be applied in its present form only via numerical calculations and an algorithm is given by MacLeod Radke (1994). 

The disadvantage of this thermodynamic criterion is that it can be used only in combination with a purification procedure. Thus, if such a procedure is not available, the purity of a surfactant solution with respect to its useful interfacial study caimot be checked. Therefore, a second criterion was developed, which is based on an adsorption kinetics model. If adsorption of both, the main component and the impurity is assumed to be diffusion-controlled, the difference between the surface tension values, measured after a definite time t j after adsorption and desorption, respectively, is a measure of the purity of the solution. 

In the first case, either the theoretical model has to allow for the evaporation process or evaporation has to be avoided by the establishment of special experimental conditions. MacLeod Radke (1994) report on the adsorption kinetics of 1-decanol at the aqueous solution interface using the growing drop method. They distinguish between three cases decanol in the aqueous phase only, decanol in the air phase only, decanol in both phases. The adsorption kinetics shows different behaviour and is fastest for the case of decanol in both phases (Fig. 5.34). The application of a proper theory (for example Miller 1980, MacLeod Radke 1994) in all three cases is a diffusion-controlled mechanism of the decanol adsorption kinetics. 

In experiments on nonionic surfactants, namely Triton X-405 Geeraerts at al. (1993) performed simultaneously dynamic surface tension and potential measurements in order to discuss peculiarities of nonionic surfactants containing oxethylene chains of different lengths as hydrophilic part. Deviations from a diffusion controlled adsorption were explained by dipole relaxations. In recent papers by Fainerman et al. (1994b, c, d) and Fainerman Miller (1994a, b) developed a new model to explain the adsorption kinetics of a series of Triton X molecules with 4 to 40 oxethylene groups. This model assumes two different orientations of the nonionic molecule and explains the observed deviations of the experimental data from a pure diffusion controlled adsorption very well. Measurements in a wide temperature interval and in presence of salts known as structure breaker were performed which supported the new idea of different molecular interfacial orientations. At small concentration and short adsorption times the kinetics can be described by a usual diffusion model. Experiments of Liggieri et al. (1994) on Triton X-100 at the hexane/water interface show the same results. 

As mentioned above, beside the diffusion-controlled models, others exist to describe the adsorption kinetics and exchange of matter. De Feijter et al. (1987) have developed a relation taking into consideration simultaneous adsorption of proteins and surfactants at an interface. As a special case a relation results which describes the equilibrium state of adsorption of polymer molecules at a liquid interface. 

Appendix 4E Application of the Laplace Transform to solve the diffusion-controlled ADSORPTION KINETICS MODEL... 

Models considering diffusion in the bulk as the only rate controlling process are called pure diffusion controlled. When the diffusion is assumed to be fast in comparison to the transfer of molecules between the subsurface and the interface the model is called kinetic-controlled or barrier-controlled. Both steps are taken into account in so-called mixed diffusion kinetic controlled models. Van den Tempel proposed processes within the adsorption layer to be considered instead of hypothetical adsorption barriers [18, 19, 20]. We believe that such models, which account for actual physical processes within adsorption layers, such as reorientation of molecules, their dimerisation and formation of clusters, although explanations for all known cases of anomalous adsorption kinetics do not exist yet, have to be preferred over any formal model. However, reliable experimental evidence for a slower surface tension decrease caused by aggregation within the adsorption layer does not allow the conclusion that this is an exclusive mechanism. 

A quantitative description of adsoiption kinetics processes is so far usually based on the model derived in 1946 by Ward and Tordai [3], The various models developed on this basis use mainly different boundary and initial conditions [2], as it becomes clear from the schematic in Fig. 4.1. The diffusion-controlled adsorption model of Ward and Tordai assumes that the step of transfer from the subsurface to the interface is fast compared to the transport from the bulk to the subsurface. It is based on the following general equation,... 

As mentioned above many adsorption kinetics models were discussed in the literature. These models consider specific mechanisms of the molecular transfer from the subsurface to the interface, or vice versa in the case of desorption [5, 6, 7, 8, 9, 10, 11, 12, 14, 23], The models, which assume only the transfer mechanism as rate determining step are called kinetic-controlled. More advanced models, the so-called mixed models, consider both the transport by diffusion in the bulk and the transfer mechanism (cf. Fig. 4.1). Such mixed models were first derived by Baret in 1969 [9] who combined Eq. (4.1) with a certain transfer mechanism. 

The diffusion controlled adsorption model is the most useful one as it is at present the main basis for all particular adsorption kinetics theories. It has also been demonstrated that for most of the systems for which an adsorption barrier controlled mechanism was discussed surface active impurities had been detected. These impurities can simulate a kinetic controlled adsorption mechanism [53, 54, 55]. 

Calculations show that the model of a non-equilibrium surface layer is an alternative to kinetic-controlled adsorption models. On the basis of the purely diffusion-controlled adsorption mechanism the proper consideration of a non-equilibrium diffusion layer leads to a satisfactory agreement between theory and experimental data for various studied systems, systematically demonstrated for the short-chain alcohols [132], The non-equilibrium model is applicable in the concentration range from 10 to 10 mol/cm at different values of the Langmuir constant at- For l < 10 mol/cm a consideration of non-equilibrium layer effects is not necessary. For ai > 10 mol/cm and large surfactant concentration the Ay values calculated from the proposed theory do not compensate the discrepancy to the experimental data so that other mechanisms have to be taken into account. An empirical formula also proposed in [132] for the estimation of the non-equilibrium surface layer thickness leads to a better agreement with experimental data, however this expression restricts the validity of the non-equilibrium surface layer model as alternative to non-diffusional adsorption kinetics. 

Most surfactants adsorb diffusion controlled at liquid interfaces. It was discussed above that exceptions observed in the literature and interpreted in terms of adsorption and desorption barriers have been understood later by the pure diffusion model when the respective experimental conditions were considered properly. One of the most important points in this respect was the systematic analysis of impurity effects on the adsorption kinetics of surfactants. This point was for example discussed in detail in the book by Dukhin et al. [2]. Another reason for the observation of an adsorption process slower than expected from diffusion is the... 

For a modelling of adsorption processes the well-known integro-differential equation (4.1) derived by Ward and Tordai [3] is used. It is the most general relationship between the dynamic adsorption r(t) and the subsurface concentration e(0,t) for fresh non-deformed surfaces and is valid for kinetic-controlled, pure diffusion-controlled and mixed adsorption mechanisms. For a diffusion-controlled adsorption mechanism Eq. (4.1) predicts different F dependencies on t for different types of isotherms. For example, the Frumkin adsorption isotherm predicts a slower initial rate of surface tension decrease than the Langmuir isotherm does. In section 4.2.2. it was shown that reorientation processes in the adsorption layer can mimic adsorption processes faster than expected from diffusion. In this paragraph we will give experimental evidence, that changes in the molar area of adsorbed molecules can cause sueh effectively faster adsorption processes. 

The kinetic data for the adsorption of Hg(II) by the copolymer satisfactorily follow both the pseudo-second-order model (where linear plots of t vs t/qt) as well as intra-particle diffusion model (up to 3h after that data does not satisfy intra-particle diffusion model, indicating that adsorption in the beginning is diffusion controlled. The correlation coefficients (R ) and the rate constants for CJ-g-PAA are shown in Table 10.4. While the isotherm studies indicate unilayer... 

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