Transient diffusion modeling

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-inhnite medium. We give here a brief account of surface renewal theory. 

An argument against the defect mediated diffusion model is the same one used earlier that is, there are not enough defects as determined by ESR (Brodsky and Title, 1969, 1976) or Deep Level Transient Spectroscopy measurements (Johnson, 1983) to account for the motion of all of the bonded hydrogen in a-Si H. This objection is removed if the floating bonds are 104-106 times more mobile than the hydrogen atoms. However, such highly mobile defects would rapidly self-annihilate via the process. 

The nearly constant peroxynitrite concentration observed in neutral solution changes dramatically when [Ft] of the solution is increased. Fig. 2 compares the transient absorption of aqueous nitrate at [Ft ] = 10-7 M and [Ft] = 0.140 M. The peroxynitrite concentration drops rapidly as protonation leads to the formation of peronitrous acid (peroxynitrous acids absorbs relatively weakly around 240 nm and is not observable in Fig. 2.). In Fig. 3 the peroxynitrite concentration is represented by the transient absorption at 310 nm as a function of [it]. As expected the formation of peroxynitrous acid increases with the concentration of protons. The protonation of peroxynitrite can be viewed as a prototypical diffusion limited bimolecular reaction and thus constitutes an excellent test bed for diffusion models. 

Fig. 3. The transient absorption pertaining to peroxynitrite measured as a function of [Ft]. The reaction kinetics are compared to those predicted by diffusion models assuming a. steady-state and b. time-dependent proton distribution.

Transient Diffusion During Rapid Thermal Annealing. When the B implant dose is less than 3 X 1014/cm2 and an RTA is performed, transient diffusion is observed (66). Examples are shown in Figure 27 for implants of 1 X 1014-2 X 1014 B atoms per cm2 performed at energies from 1 to 60 keV. The previously published data of Sedgwick (66) are included. This case is modeled similarly as the low-dose B implant-furnace anneal case, except for the magnitude of Denh. Thus, if neutral and donor point defects contribute to B diffusivities D and Df-+, respectively, then the total B diffusion coefficient (Db) is given by (59)... 

The difference between the solution-diffusion and pore-flow mechanisms lies in the relative size and permanence of the pores. For membranes in which transport is best described by the solution-diffusion model and Fick s law, the free-volume elements (pores) in the membrane are tiny spaces between polymer chains caused by thermal motion of the polymer molecules. These volume elements appear and disappear on about the same timescale as the motions of the permeants traversing the membrane. On the other hand, for a membrane in which transport is best described by a pore-flow model and Darcy s law, the free-volume elements (pores) are relatively large and fixed, do not fluctuate in position or volume on the timescale of permeant motion, and are connected to one another. The larger the individual free volume elements (pores), the more likely they are to be present long enough to produce pore-flow characteristics in the membrane. As a rough rule of thumb, the transition between transient (solution-diffusion) and permanent (pore-flow) pores is in the range 5-10 A diameter. 

Reverse osmosis, pervaporation and polymeric gas separation membranes have a dense polymer layer with no visible pores, in which the separation occurs. These membranes show different transport rates for molecules as small as 2-5 A in diameter. The fluxes of permeants through these membranes are also much lower than through the microporous membranes. Transport is best described by the solution-diffusion model. The spaces between the polymer chains in these membranes are less than 5 A in diameter and so are within the normal range of thermal motion of the polymer chains that make up the membrane matrix. Molecules permeate the membrane through free volume elements between the polymer chains that are transient on the timescale of the diffusion processes occurring. 

This electro-optical effect, commonly observed as transient changes in optical birefringence of a solution following application, removal, or reversal of a biasing electric field E(t), has been used extensively as a probe of dynamics of blopolymer solutions, notably by O Konski, and is a valuable tool because it gives information different in form, but related to, results from conventional dielectric relaxation measurements. The state of the subject to 1975 has been comprehensively presented in two review volumes edited by O Konski (25). The discussion here is confined to an outline of a response theory treatment, to be published in more detail elsewhere, of the quadratic effect. The results are more general than earlier ones obtained from rotational diffusion models and should be a useful basis for further theoretical and experimental developments. 

The experimental method used in TEOM for diffusion measurements in zeolites is similar to the uptake and chromatographic methods (i.e., a step change or a pulse injection in the feed is made and the response curve is recorded). It is recommended to operate with dilute systems and low zeolite loadings. For an isothermal system when the uptake rate is influenced by intracrystalline diffusion, with only a small concentration gradient in the adsorbed phase (constant diffusivity), solutions of the transient diffusion equation for various geometries have been given (ii). Adsorption and diffusion of o-xylene, / -xylene, and toluene in HZSM-5 were found to be described well by a one-dimensional model for diffusion in a slab geometry, represented by Eq. (7) (72) ... 

The rate of transmembrane diffusion of ions and molecules across a membrane is usually described in terms of a permeability constant (P), defined so that the unitary flux of molecules per unit time [J) across the membrane is 7 = P(co - f,), where co and Ci are the concentrations of the permeant species on opposite sides of membrane correspondingly, P has units of cm s. Two theoretical models have been proposed to account for solute permeation of bilayer membranes. The most generally accepted description for polar nonelectrolytes is the solubility-diffusion model [24]. This model treats the membrane as a thin slab of hydrophobic matter embedded in an aqueous environment. To cross the membrane, the permeating particle dissolves in the hydrophobic region of the membrane, diffuses to the opposite interface, and leaves the membrane by redissolving in the second aqueous phase. If the membrane thickness and the diffusion and partition coefficients of the permeating species are known, the permeability coefficient can be calculated. In some cases, the permeabilities of small molecules (water, urea) and ions (proton, potassium ion) calculated from the solubility-diffusion model are much smaller than experimentally observed values. This has led to an alternative model wherein permeation occurs through transient hydrophilic defects, or pores , formed by thermal fluctuations of surfactant monomers in the membrane [25]. 

This bulk state of secular equilibrium applies to the total amount of the U-series nuclides, but does not necessarily say where the different elements reside within the system. If the bulk system has a single phase (such as a melt or a monomineralic rock) then that phase will be in secular equilibrium. If the material has multiple phases with different partitioning properties, however, the individual phases can maintain radioactive dis-equilibria even when the total system is in secular equilibrium. There are two basic sets of models that exploit this fact, the first assumes complete chemical equilibrium between all phases and the second assumes transient diffusion controlled sohd exchange. 

This classification has been discussed extensively within the context of a one-dimensional advection-diffusion model, along with simple solutions to the relevant equations (Craig, 1969). It should be noted, however, that specific tracers may fall into different categories depending on the nature of the specific application. For example, radiocarbon is a transient tracer in the surface waters of the ocean because its natural inventory (due to cosmic ray production) has been affected... 

The sorption of ethane from dilute mixtures with helium by 4A sieve crystal powder and pellets made without binder has been studied with a microbalance in a flow system at temperatures between 25° and 117°C. Results show clearly that intracrystalline diffusion is the rate-controlling process and that it is represented well by a Pick s law diffusion model. Transient adsorption and desorption are characterized by the same effective diffusivity with an activation energy of 5660 cal/gram mole. 

Menon and Landau [52] developed a model to describe transient diffusion and migration in stagnant binary electrolytes. Nonuniformity at a partially masked cathode was found to increase during electrolysis as the diffusion resistance develops. The calculations were done using an alternating-direction implicit (ADI) finite difference method. 

Of particular interest in Eq. (2.3-65). as well as In the earlier results of fltm theory and transient diffusion into a Buid, is the predicted dependence of the flux on geometry, hydrodynamics (0, y, or Re), physical properties and Sc), and composition driving force. In all cases (he flux varies linearly with the composition driving force but the dependence of flux on difiusivify ranges from (he first to the one-half power. These simple models can be used to guide or interpret mass transfer rate observations In more complex situations. 

Fig. 3 Transient temperature response for CH30H-NaX, pressure step 48-80 Pa (step A, run 13) showing conformity between experimentally observed temperature and theoretical curve, calculated from diffusion model with Do = 2.6 x 10 m s h = 2.3 Wm" From Grenier et al. [27]...

Quintard M. and Whitaker S. 1993b. One- and two-equation models for transient diffusion processes in two-phase systems. In, Advances in Heat Transfer, Harnett J.P., Irvine T.F. Jr and Cho Y.I. (Eds.), Vol. 23. Academic Press, New York, pp. 369-465. 

The models and material property data for predicting fission metal release from fuel particles and fuel elements are described in Ref. 4. The transport of fission metals through the kernel, coatings, fuel rod matrix, and fuel element graphite is modeled as a transient diffusion process in the TRAFIC code (Section 4.2.5,2.2.1.2). The sorption isotherms which are used in the calculation of the rate of evaporation of volatile metals from graphite surfaces account for an increase in graphite sorptivity with increasing neutron fluence. 

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