Kinetics diffusion-controlled model

J.B.Schwartz, A.P.Simonelli and W.I.Higuchi, Drug release from wax matrices I. Analysis of data with first-order kinetics and with the diffusion-controlled model, J. Pharm. Sci., 57,274 (1968). 

Padding, J.T., and Boek, E.S. "Evidence for diffusion-controlled recombination kinetics in model wormlike micelles". Eurcrphys. Lett. 66, 756762 (2004). 

Finally, a brief overview was presented of important experimental approaches, including GITT, EMF-temperature measurement, EIS and PCT, for investigating lithium intercalation/deintercalation. In this way, it is possible to determine - on an experimental basis - thermodynamic properties such as electrode potential, chemical potential, enthalpy and entropy, as well as kinetic parameters such as the diffusion coefficients of lithium ion in the solid electrode. The PCT technique, when aided by computational methods, represents the most powerful tool for determining the kinetics of lithium intercalation/deintercalation when lithium transport cannot be simply explained based on a conventional, diffusion-controlled model. 

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 above diffusion-controlled model assumes transport by difiusion of the surface-active molecules to be the rate-controlled step. The so-called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state, and vice-versa [17]. 

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... 

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... 

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... 

The analysis of the kinetic data was performed on the basis of the diffusion-controlled model, using the Langmuir and the aggregation isotherm, given by Eq. (2.16) and Eqs. (2.107) -(2.111), respectively. As one can see, the agreement with the theory is not satisfactory. The models developed mainly by the Bulgarian school [33] requires extensive numerical calculations so that its application to experimental data will be possible only after the elaboration of effective computer programmes. 

Reaction (D) The kinetics of decay of the polymer radicals is studied by observing the time dependent concentration of the radicals (the intensity of ESR spectra). Equation (7.18) can be modeled by using a diffusion-controlled bi-molecular reaction. Equation (7.19) and (7.20) can be addressed in the same way using a diffusion-controlled model with the reactivity of ROO being higher than R-. 

Modelling the Diffusion-Controlled Overall Kinetics and Cure Rate Law of Epoxy Systems... 

For a free radical polymerisation system, an unsaturated polyester resin, an auto-acceleration was observed close to the onset of vitrification. To model the curing kinetics for these systems, including the mobility-controlled regions, a specific diffusion control model will need to be incorporated in a mechanistic reaction model. The heat capacity and the mobility factor can still give information about how vitrification is occurring, and how it is related to the auto-acceleration effect. 

In diffusion-controlled adsorption models, one assumes that there is no activation energy barrier to the transfer of surfactant molecules between the subsurface and the surface [85]. Thus diffusion is the only mechanism needed in establishing adsorption equilibrium. The time required for the molecules to transfer from the bulk to the subsurface is much longer than the time required for equilibration between the surface and the subsurface. On the contrary, if the adsorption or desorption rate at the interface is slow or comparable to the diffusion rate, the adsorption process is significant. This model is called the mixed-kinetic adsorption model. This condition may depend not only on the properties of the system but also on the diffusion length and possibly on convection conditions. The diffusion-controlled model of Eqs. (3) and (4) have been given by Fainerman et al. [86,87]. 

Most spraying processes work under dynamic conditions and improvement of their efficiency requires the use of surfactants that lower the liquid surface tension yLv under these dynamic conditions. The interfaces involved (e.g. droplets formed in a spray or impacting on a surface) are freshly formed and have only a small effective age of some seconds or even less than a millisecond. The most frequently used parameter to characterize the dynamic properties of liquid adsorption layers is the dynamic surface tension (that is a time dependent quantity). Techniques should be available to measure yLv as a function of time (ranging firom a fraction of a millisecond to minutes and hours or days). To optimize the use of surfactants, polymers and mixtures of them specific knowledge of their dynamic adsorption behavior rather than equilibrium properties is of great interest [28]. It is, therefore, necessary to describe the dynamics of surfeictant adsorption at a fundamental level. The first physically sound model for adsorption kinetics was derived by Ward and Tordai [29]. It is based on the assumption that the time dependence of surface or interfacial tension, which is directly proportional to the surface excess F (moles m ), is caused by diffusion and transport of surfeictant molecules to the interface. This is referred to as the diffusion controlled adsorption kinetics model . This diffusion controlled model assumes transport by diffusion of the surface active molecules to be the rate controlled step. The so called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state and vice versa [28]. 

While a need remains to validate the mathematical model for adsorption in micellar solutions against additional experimental data sets, indications are that the kinetics of adsorption of C jTAB surfactants both above and below the cmc can be explained quantitatively on a diffusion-controlled model if finite micelle breakdown kinetics are allowed for. 

The accumulated evidence from studies on the OFC suggests that the adsorption of C ,TABs is diffusion-controlled below the cmc. Above the cmc, there are deviations from a diffusion-controlled model for CigTAB + NaBr, which can be quantitatively explained by slow micellar breakdown kinetics. The alternative of an adsorption barrier cannot be ruled out, though there is as yet no evidence of structm es at the air-water interface akin to those observed in the SAR at the solid-liquid interface. More limited studies on other femiUes of ionic scu -factants in the OFC and MBP apparatus do not show large deviations from diffusion control. The importance of well-defined hydrodynamics and accru-ate equilibriiun adsorption isotherms cannot be overstressed in quantitative studies of adsorption mechanisms. There is still a need for measurements at higher strain rates, such as occur in tcffbulent foams, jet breakup and impacting drops, and for additional studies with micellar systems to establish quantitatively the connection between micellar breakdown kinetics and rates of adsorption. 

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