Diffusional theory

Shalashilin D V and Thompson D L 1996 Intrinsic non-RRK behavior classical trajectory, statistical theory, and diffusional theory studies of a unimolecular reaction J. Chem. Phys. 105 1833—45... 

Inequality (6.67) is the softest criterion of perturbation theory. Its physical meaning is straightforward the reorientation angle (2.30) should be small. Otherwise, a complete circle may be accomplished during the correlation time of angular momentum and the rotation may be considered to be quasi-free. Diffusional theory should not be extended to this situation. When it was nevertheless done [268], the results turned out to be qualitatively incorrect orientational relaxation time 19,2 remained finite for xj —> 00. In reality t0j2 tends to infinity in this limit [27, 269]. 

The reaction in the bulk can be really neglected if the free radicals are unstable as a result of fast decomposition, discharging, or other reasons. But even in this case neither of the abovementioned models is an appropriate tool for the explanation of the phenomenon. The contact diffusional theory provides the alternative interpretation of the effect originating from either geminate recombination alone or together with the reaction in the bulk. 

An unambiguous interpretation of these well-known experimental facts in the framework of the diffusional theory is hardly possible. To overcome considerable difficulties arisen, use is usually made of different additional (not very convincing) assumptions and suggestions. In contrast, from a physicochemical viewpoint, the phenomena and dependences observed in practice seem to be quite natural and easily explainable.134 136 139 141 199 These can therefore readily be expected to hold in binary systems of whatever chemical nature. 

It should be noted that the indifference of the silicon layer and the disappearance of the CoSi layer in the Co-Co2Si-CoSi-Si system and also of the PtSi layer in the Pt-Pt2Si-PtSi-Si system cannot be explained from a diffusional viewpoint. Indeed, according to the diffusional theory, the existing CoSi and PtSi layers should have grown parabolically. Evidently, their growth could not proceed without the participation of silicon atoms from the substrate. 

It is clear that in general the kinetic dependences considered in this chapter gradually transform into each other with passing time. In contrast to the diffusional theory, the physicochemical approach thus gives a more complicated, not simply parabolic, relationship between the thickness of two chemical compound layers and the time, in accordance with the available experimental data in binary systems. 

It should be emphasised that according to the diffusional theory any chemical compound layer once formed cannot then disappear during isothermal annealing of the A-B reaction couple because its growth rate increases with decreasing thickness dx/dt ac ox, tending to infinity at... 

Both systems are suitable to check whether or not there is a directly proportional relationship between the width of the homogeneity range of a compound and the growth rate of its layer, predicted by the diffusional theory.5 It is clear that in view of the presence of the liquid zinc phase during preparation of Ni-Zn and Co-Zn reaction couples, all the inter-metallic phases had equal and favourable conditions to form their nuclei at the interface between nickel or cobalt and zinc, which could then readily grow during subsequent isothermal annealing. 

Hence, the diffusional theory does not allow any existing compound layer to disappear during dissolution. Intuitively, this conclusion appears to be quite evident because any increase in the dissolution rate, resulting in a decrease of the layer thickness, automatically leads to an increase in its growth rate (see equations (5.27) and (5.28) in which x is in the denominator of the term kx x responsible for the layer-growth rate). Due to such a compensation effect, the thickness of the ApBq layer exceeds zero at any real values of the diffusion coefficients of the components across its bulk. 

Again, the diffusional theory rests on the assumption of local equilibrium or quasi-equilibrium. However, it is clear that no local equilibrium can exist in any diffusion couple in which the layers of some part of thermodynamically stable compounds are missing. Also, if successive layers of reactants and products are in equilibrium with each other, then all the system is in local equilibrium. Therefore, applying the Gibbs phase rule, it is easy to come to the logical conclusion that under constant temperature and pressure conditions no compound layer can occur at all between two reactants in a binary system since in this case the largest number of co-... 

The unjustified neglect of a chemical interaction step in analysing the process of compound-layer formation appears to be the main source of discrepancies between the diffusional theory and the experimental data. The primary aim of this book is, on the basis of physicochemical views regarding solid state reaction kinetics, to attempt... 

In Figure 3, a comparison of the interferometrically measured fluxes with the theoretical ones shows that the transfer of aniline into pure water follows Pick s diffusional theory. This is expected since the interface was shown to be stable in the Schheren image (case la) in Figure 2, and therefore mass transfer in this case is by diffusion only. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Olivet, E. D., Diffusional Separation Processes Theory, Design and Evaluation. Wiley, New York, 1966. Reid, R. C., Prausnitz, J. M., and Sherwood, T. K., The Properties of Gases and Liquids. McGraw-Hill, New York, 1977. 

The theory of seaweed formation does not only apply to solidification processes but in fact to the completely different phenomenon of a wettingdewetting transition. To be precise, this applies to the so-called partial wetting scenario, where a thin liquid film may coexist with a dry surface on the same substrate. These equations are equivalent to the one-sided model of diffusional growth with an effective diffusion coefficient which depends on the viscosity and on the thermodynamical properties of the thin film. 

S. R. Lee and J. S. Kim, On the sublimit solution branches of the stripe patterns formed in counterflow diffusion flames by diffusional-thermal instability. Combust. Theory Model. 6(2) 263-278,2002. 

Sivashinsky, G.L, Diffusional-thermal theory of cellular flames. Combust. Sci. Technol., 15,137,1977. 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

The treatment of the two-phase SECM problem applicable to immiscible liquid-liquid systems, requires a consideration of mass transfer in both liquid phases, unless conditions are selected so that the phase that does not contain the tip (denoted as phase 2 throughout this chapter) can be assumed to be maintained at a constant composition. Many SECM experiments on liquid-liquid interfaces have therefore employed much higher concentrations of the reactant of interest in phase 2 compared to the phase containing the tip (phase 1), so that depletion and diffusional effects in phase 2 can be eliminated [18,47,48]. This has the advantage that simpler theoretical treatments can be used, but places obvious limitations on the range of conditions under which reactions can be studied. In this section we review SECM theory appropriate to liquid-liquid interfaces at the full level where there are no restrictions on either the concentrations or diffusion coefficients of the reactants in the two phases. Specific attention is given to SECM feedback [49] and SECMIT [9], which represent the most widely used modes of operation. The extension of the models described to other techniques, such as DPSC, is relatively straightforward. 

Various models of SFE have been published, which aim at understanding the kinetics of the processes. For many dynamic extractions of compounds from solid matrices, e.g. for additives in polymers, the analytes are present in small amounts in the matrix and during extraction their concentration in the SCF is well below the solubility limit. The rate of extraction is then not determined principally by solubility, but by the rate of mass transfer out of the matrix. Supercritical gas extraction usually falls very clearly into the class of purely diffusional operations. Gere et al. [285] have reported the physico-chemical principles that are the foundation of theory and practice of SCF analytical techniques. The authors stress in particular the use of intrinsic solubility parameters (such as the Hildebrand solubility parameter 5), in relation to the solubility of analytes in SCFs and optimisation of SFE conditions. 

When (DEB), is much smaller than unity, the polymer relaxation is relatively rapid compared to diffusion. In this case, conformational changes take place instantaneously and equilibrium is attained after each diffusional jump. This is the type of diffusion encountered ordinarily and is called viscous diffusion. Therefore, the transport will obey classical theories of diffusion. When (DEB), is much larger than unity, the molecular relaxation is very slow compared to diffusion and there are no conformational changes of the medium within the diffusion time scale. In this case, Fick s law is generally valid, but no concentration dependence of the diffusion coefficient is expected. This is termed elastic diffusion. When (DEB), is in the neighborhood of unity, molecular rearrangment... 

The original theory of diffusional coagulation of spherical aerosol particles was developed by von Smoluchowski (1916,1917). The underlying hypothesis in this theory is that every aerosol particle acts as a sink for the diffusing species. The concentration of the diffusing species at the surface of the aerosol particle is assumed to be zero. At some distance away, the concentration is the bulk concentration. 

One of earliest approaches of estimating the diffusion coefficient through a polymer carrier is that of Eyring (1936). In this theory, diffusion of a solute through a medium is presented as a series of jumps instead of a continuous process. Therefore, in Eq. (18) in Table I, which comes from the Eyring analysis, X is the diffusional jump of the drug in the polymer and v is the frequency of jumping. 

A convenient concept for introducing the surface boundary condition into the mathematical formulation of migration theory is that of what may be called a diffusional offset length d. Suppose that the external and surface conditions are describable by a set of parameters X, which we do not need to specify in detail we also allow the surface conditions to depend on the internal hydrogen concentration just beneath the surface. If the hydrogen complexes that are continually forming in the crystal are sufficiently immobile, the balance between inflow and outflow across the surface will depend only on X and on the concentration no(0) of H0 just beneath the surface. (If mobile H+ or H are present, the statement just... 

Several investigators have attempted to modify the basic Deutsch equation so that it would more nearly describe precipitator performance. Cooperman ( A New Theory of Precipitator Efficiency, Pap. 69-4, APCA meeting, New York, 1969) introduced correction factors for diffusional forces arising from variations in particle concentration along the precipitator length and also perpendicular to the collecting surface. Robinson [Atmos. Environ. 1(3), 193 (1967)] derived an equation for collection efficiency in which two erosion or reentrainment terms are introduced. 

In reality, the diffusional rate constant is time-dependent, as explained at the end of Section 4.2.1, and should be written as ki(t). Several models have been developed to express the time-dependent rate constant (see Box 4.1). For instance, in Smoluchowski s theory, ki (t) is given by... 

The design of fixed-bed ion exchangers shares a common theory with fixed-bed adsorbers, which are discussed in Chapter 17. In addition, Thomas(14) has developed a theory of fixed-bed ion exchange based on equation 18.21. It assumed that diffusional resistances are negligible. Though this is now known to be unlikely, the general form of the solutions proposed by Thomas may be used for film- and pellet-diffusion control. 

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