Diffusion-controlled model kinetic rate

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]. 

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

For the epoxy resins studied, the mobility factor based on heat capacity coincides very well with the diffusion factor, calculated from the nonreversing heat flow via chemical kinetics modelling, and describing the effects of diffusion control on the rate of conversion of the cure reaction. Although the two resins behave quite differently, this coincidence between the mobility factor and diffusion factor is valid for both systems. Therefore, the mobility factor can be used for a quantitative description of then-altered rate of conversion in the (partially) vitrified state for the decrease in rate during vitrification, the increase in rate during devitrification and the diffusion-controlled rate in the (partially) vitrified region in between both processes. 

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]. 

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. 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

Mechanisms of dissolution kinetics of crystals have been intensively studied in the pharmaceutical domain, because the rate of dissolution affects the bioavailability of drug crystals. Many efforts have been made to describe the crystal dissolution behavior. A variety of empirical or semi-empirical models have been used to describe drug dissolution or release from formulations [1-6]. Noyes and Whitney published the first quantitative study of the dissolution process in 1897 [7]. They found that the dissolution process is diffusion controlled and involves no chemical reaction. The Noyes-Whitney equation simply states that the dissolution rate is directly proportional to the difference between the solubility and the solution concentration ... 

Spiro [27] has derived quantitative expressions for the catalytic effect of electron conducting catalysts on oxidation-reduction reactions in solution in which the catalyst assumes the Emp imposed on it by the interacting redox couples. When both partial reaction polarization curves in the region of Emp exhibit Tafel type kinetics, he determined that the catalytic rate of reaction will be proportional to the concentrations of the two reactants raised to fractional powers in many simple cases, the power is one. On the other hand, if the polarization curve of one of the reactants shows diffusion-controlled kinetics, the catalytic rate of reaction will be proportional to the concentration of that reactant alone. Electroless metal deposition systems, at least those that appear to obey the MPT model, may be considered to be a special case of the general class of heterogeneously catalyzed reactions treated by Spiro. 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

The product cystine is presumably formed in the recombination of two thiyl radicals. This free-radical model is suitable for formal treatment of the kinetic data however, it does not account for all possible reactions of the RS radical (68). The rate constants for the reactions of this species with RS-, 02 and Cu L, (n = 2, 3) are comparable, and on the order of 109-10loM-1s-1 (70-72). Because all of these reaction partners are present in relatively high and competitive concentrations, the recombination of the thiyl radical must be a relatively minor reaction compared to the other reaction paths even though it has a diffusion controlled rate constant. It follows that the RS radical is most likely involved in a series of side reactions producing various intermediates. In order to comply with the noted chemoselectivity, at some point these transient species should produce a common intermediate leading to the formation of cystine. 

Many high-pressure reactions consist of a diffusion-controlled growth where also the nucleation rate must be taken into account. Assuming a diffusion-controlled growth of the product phase from randomly distributed nuclei within reactant phase A, various mathematical models have been developed and the dependence of the nucleation rate / on time formulated. Usually a first-order kinetic law I =fNoe fi is assumed for the nucleation from an active site, where N t) = is the number of active sites at time t. Different shapes of the... 

If a reaction is diffusion-controlled then reaction occurs on encounter of two particles. In a simulation employing finite time steps, a pair of reactants may encounter during a time step and then separate before the end of the time step. This pair should have reacted, but the reaction is not registered. Consequently, the modeled kinetics underestimates both the rate and the amount of reaction. This problem of encounter during a time step can be... 

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