Diffusion interface reactions

Figure 4-24 A numerical solution of the diffusion-interface reaction equation. The conditions are = 0.4 and — 4. After Zhang et al. (1989).

Recent applications of e-beam and HF-plasma SNMS have been published in the following areas aerosol particles [3.77], X-ray mirrors [3.78, 3.79], ceramics and hard coatings [3.80-3.84], glasses [3.85], interface reactions [3.86], ion implantations [3.87], molecular beam epitaxy (MBE) layers [3.88], multilayer systems [3.89], ohmic contacts [3.90], organic additives [3.91], perovskite-type and superconducting layers [3.92], steel [3.93, 3.94], surface deposition [3.95], sub-surface diffusion [3.96], sensors [3.97-3.99], soil [3.100], and thermal barrier coatings [3.101]. 

Fig. 3. Reduced time plots, tr = (t/t0.9), for the contracting area and contracting volume equations [eqn. (7), n = 2 and 3], diffusion-controlled reactions proceedings in one [eqn. (10)], two [eqn. (13)] and three [eqn. (14)] dimensions, the Ginstling— Brounshtein equation [eqn. (11)] and first-, second- and third-order reactions [eqns. (15)—(17)]. Diffusion control is shown as a full line, interface advance as a broken line and reaction orders are dotted. Rate processes become more strongly deceleratory as the number of dimensions in which interface advance occurs is increased. The numbers on the curves indicate the equation numbers.

Dehydration reactions are typically both endothermic and reversible. Reported kinetic characteristics for water release show various a—time relationships and rate control has been ascribed to either interface reactions or to diffusion processes. Where water elimination occurs at an interface, this may be characterized by (i) rapid, and perhaps complete, initial nucleation on some or all surfaces [212,213], followed by advance of the coherent interface thus generated, (ii) nucleation at specific surface sites [208], perhaps maintained during reaction [426], followed by growth or (iii) (exceptionally) water elimination at existing crystal surfaces without growth [62]. 

SO sharply defined that they are called surfaces. Well-defined surfaces occur between solids and either gases or liquids and thus are commonly found in catalytic and electrode reactions. More diffuse interfaces may occur between solids, as in microelectronic devices, and between fluids or semifluids, as in many polymeric and colloidal systems. 

We first explain the setting of reactors for all CFD simulations. We used Fluent 6.2 as a CFD code. Each reactant fluid is split into laminated fluid segments at the reactor inlet. The flow in reactors was assumed to be laminar flow. Thus, the reactants mix only by molecular diffusion, and reactions take place fi om the interface between each reactant fluid. The reaction formulas and the rate equations of multiple reactions proceeding in reactors were as follows A + B R, ri = A iCaCb B + R S, t2 = CbCr, where R was the desired product and S was the by-product. The other assumptions were as follows the diffusion coefficient of every component was 10" m /s the reactants reacted isothermally, that is, k was fixed at... 

Strictly speaking, the validity of the shrinking unreacted core model is limited to those fluid-solid reactions where the reactant solid is nonporous and the reaction occurs at a well-defined, sharp reaction interface. Because of the simplicity of the model it is tempting to attempt to apply it to reactions involving porous solids also, but this can lead to incorrect analyses of experimental data. In a porous solid the chemical reaction occurs over a diffuse zone rather than at a sharp interface, and the model can be made use of only in the case of diffusion-controlled reactions. 

Figure 9.7 shows concentration profiles schematically for A and B according to the two-film model. Initially, we ignore the presence of the gas film and consider material balances for A and B across a thin strip of width dx in the liquid film at a distance x from the gas-liquid interface. (Since the gas-film mass transfer is in series with combined diffusion and reaction in the liquid film, its effect can be added as a resistance in series.)... 

The rate of diffusion is proper tional to the difference of concentrations between the bulk of the fluid and at the interface. In the steady state, the rates of diffusion and reaction are equal. 

Description of mixing with diffusion and reaction in terms of the concept of material interfaces. Journal of Fluid Mechanics 114, 83-103. 

The interface where the reaction takes place must be accessible to these reactants. In such cases, the diffusion of reactants to the reaction front becomes an important factor and diffusion of reactions to each other is rate determining step. 

Fig. 3.1 The I—V characteristic of (a) a p-type and (b) an n-type silicon electrode under the assumption that the current is dominated by the properties of the semiconductor and is not limited by interface reactions or by diffusion in the electrolyte, (c) The characteristic I—V curve in an alkaline electrolyte under the...

If the transport process is rate-determining, the rate is controlled by the diffusion coefficient of the migrating species. There are several models that describe diffusion-controlled processes. A useful model has been proposed for a reaction occurring at the interface between two solid phases A and B [290]. This model can work for both solids and compressed liquids because it doesn t take into account the crystalline environment but only the diffusion coefficient. This model was initially developed for planar interface reactions, and then it was applied by lander [291] to powdered compacts. The starting point is the so-called parabolic law, describing the bulk-diffusion-controlled growth of a product layer in a unidirectional process, occurring on a planar interface where the reaction surface remains constant ... 

Axe,L. Anderson, P.R. (1995) Sr diffusion and reaction within Fe oxides. Evaluation of the rate limiting mechanism for sorption J. Coll. Interf. Sd. 175 157—168 Axe,L. Anderson, P.R. (1997) Experimental and theoretical diffusivities of Cd and Sr in hydrous ferric oxide. J. Coll. Interface Sci. 

Figure 1-11 Concentration profile for (a) crystal growth controlled by interface reaction (the concentration profile is flat and does not change with time), (b) diffusive crystal growth with t2 = 4fi and = 4t2 (the profile is an error function and propagates according to (c) convective crystal growth (the profile is an exponential function and does not change with time), and (d) crystal growth controlled by both interface reaction and diffusion (both the interface concentration and the length of the profile vary).

If the growth rate is controlled by both interface reaction and diffusion (Figure 1-1 Id), then (i) the concentration profile is not flat, (ii) the interface concentration changes with time toward the saturation concentration, (iii) the diffusion profile propagates into the melt, and (iv) the growth rate is not constant, nor does it obey the parabolic growth law. 

Geochemical kinetics is stiU in its infancy, and much research is necessary. One task is the accumulation of kinetic data, such as experimental determination of reaction rate laws and rate coefficients for homogeneous reactions, diffusion coefficients of various components in various phases under various conditions (temperature, pressure, fluid compositions, and phase compositions), interface reaction rates as a function of supersaturation, crystal growth and dissolution rates, and bubble growth and dissolution rates. These data are critical to geological applications of kinetics. Data collection requires increasingly more sophisticated experimental apparatus and analytical instruments, and often new progresses arise from new instrumentation or methods. 

Even when the principles of interface reaction and diffusion are thought to be understood, the integrated results may still require major new work. For example, the growth rate of an individual crystal in an infinite melt can be predicted if parameters are known, but the growth rates of many crystals (and different minerals), i.e., the kinetics of crystallization of a magma, is not quantitatively understood. 

Nucleation is necessary for the new phase to form, and is often the most difficult step. Because the new phase and old phase have the same composition, mass transport is not necessary. However, for very rapid interface reaction rate, heat transport may play a role. The growth rate may be controlled either by interface reaction or heat transport. Because diffusivity of heat is much greater than chemical diffusivity, crystal growth controlled by heat transport is expected to be much more rapid than crystal growth controlled by mass transport. For vaporization of liquid (e.g., water vapor) in air, because the gas phase is already present (air), nucleation is not necessary except for vaporization (bubbling) beginning in the interior. Similarly, for ice melting (ice water) in nature, nucleation does not seem to be difficult. 

Examination of Figure 1-12 provides some clue to qualitatively gauge the interface reaction rate for reactions in water. Figure 1-12 shows that, for mineral with low solubility and high bond strength (characterized by (z+z )max, where z+ and z are valences of ions to be dissociated), the overall dissolution rate is controlled by interface reaction otherwise, it is controlled by mass transport. Because diffusivities of common cations and anions in water do not differ much (by less than a factor of 10 Table l-3a), when the overall reaction rate is controlled by interface reaction, it means that interface reaction is slow when the overall reaction rate is controlled by mass transport, the interface reaction rate is rapid. Therefore, from Figure 1-12, we may conclude that the interface reaction rate increases with mineral solubility and decreases with bond strength (z+z )max to be dissociated. 

Zhang et al. (1989) treated the interplay between diffusion and interface reaction during the initial and transient stages of crystal dissolution in a silicate melt. Using the interface reaction rate of diopside, they found that the period for... 

The new reference frame is known as the interface-fixed reference frame, and the old reference frame is called the laboratory-fixed reference frame. The melt consumption rate u depends on whether the growth is controlled by interface reaction, or by diffusion, or by externally imposed conditions such as cooling. 

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