Diffuse phase boundaries

Simple Diffusion Phase-Boundary Controlled Material Transport... 

Analysis of S-SEBS by SAXS has revealed the presence of cylindrical morphologies for a degree of sulfonation of <34%. Interestingly, different morphologies can also be observed when membranes are cast from different solvents. Membranes (27 mol% degree of sulfonation) cast from THE form lamellar morphologies, as seen in Figure 3.27, while those cast from MeOH/ THE (tetrahydrofuran) (20/80 v/v) exhibit a diffusive phase boundary with disorderly interconnections between domains. This is due to the differences... 

Fig. 7.30. The thickness of the diffuse phase boundary as a function of composition measured by SAXS and SANS. The values remain constant as the composition is varied however, there is a difference ( 15A) between the two. (Reproduced with permission from [66]. Copyright 1993 American Chemical Society.)...

It is now interesting to recognize that application of these relations is not restricted to the ordered periodic system but can be extended, with slight modifications, to real systems which may show variations in the thicknesses of the crystalline and amorphous regions, and may also possess diffuse phase boundaries. The changes in K z) resulting from a successive perturbation of... 

Koberstein, J. T., Rs, S. (1983), Small-angle X-ray scattering measurements of diffuse phase-boundary thickness in segmented polyurethane elastomers, J. Polym. ScL, Polym. Phys.Ed.,2, 2m-lQ0. 

These y(r) curves were analyzed using a pseudo-two-phase structure model, which has a broad distribution of long period and lamellar thickness and a diffuse phase boundary at the interface of the crystal and the amorphous regions as shown by Figure 11.4. 

Because the reaction takes place in the Hquid, the amount of Hquid held in the contacting vessel is important, as are the Hquid physical properties such as viscosity, density, and surface tension. These properties affect gas bubble size and therefore phase boundary area and diffusion properties for rate considerations. Chemically, the oxidation rate is also dependent on the concentration of the anthrahydroquinone, the actual oxygen concentration in the Hquid, and the system temperature (64). The oxidation reaction is also exothermic, releasing the remaining 45% of the heat of formation from the elements. Temperature can be controUed by the various options described under hydrogenation. Added heat release can result from decomposition of hydrogen peroxide or direct reaction of H2O2 and hydroquinone (HQ) at a catalytic site (eq. 19). 

FIG. 16-9 General scheme of adsorbent particles in a packed bed showing the locations of mass transfer and dispersive mechanisms. Numerals correspond to mimhered paragraphs in the text 1, pore diffusion 2, solid diffusion 3, reaction kinetics at phase boundary 4, external mass transfer 5, fluid mixing. 

The factor B = D/RT is the mobility and contains the diffusion coefficient D, the gas constant R, and the absolute temperature T. The equation includes a diffusion and a migration term. Correspondingly Eq. (2-23) gives the first diffusion law for Zj = 0 and Ohm s Law for grad /i, = 0. For transfer across a phase boundary ... 

In this section we discuss the basic mechanisms of pattern formation in growth processes under the influence of a diffusion field. For simphcity we consider the sohdification of a pure material from the undercooled melt, where the latent heat L is emitted from the solidification front. Since heat diffusion is a slow and rate-limiting process, we may assume that the interface kinetics is fast enough to achieve local equihbrium at the phase boundary. Strictly speaking, we assume an infinitely fast kinetic coefficient. 

The general theoretical treatment of ion-selective membranes assumes a homogeneous membrane phase and thermodynamic equilibrium at the phase boundaries. Obvious deviations from a Nemstian behavior are explained by an additional diffusion potential inside the membrane. However, allowing stationary state conditions in which the thermodynamic equilibrium is not established some hitherto difficult to explain facts (e.g., super-Nemstian slope, dependence of the selectivity of ion-transport upon the availability of co-ions, etc.) can be understood more easily. 

The explicit mathematical treatment for such stationary-state situations at certain ion-selective membranes was performed by Iljuschenko and Mirkin 106). As the publication is in Russian and in a not widely distributed journal, their work will be cited in the appendix. The authors obtain an equation (s. (34) on page 28) similar to the one developed by Eisenman et al. 6) for glass membranes using the three-segment potential approach. However, the mobilities used in the stationary-state treatment are those which describe the ion migration in an electric field through a diffusion layer at the phase boundary. A diffusion process through the entire membrane with constant ion mobilities does not have to be assumed. The non-Nernstian behavior of extremely thin layers (i.e., ISFET) can therefore also be described, as well as the role of an electron transfer at solid-state membranes. 

The characteristic feature of solid—solid reactions which controls, to some extent, the methods which can be applied to the investigation of their kinetics, is that the continuation of product formation requires the transportation of one or both reactants to a zone of interaction, perhaps through a coherent barrier layer of the product phase or as a monomolec-ular layer across surfaces. Since diffusion at phase boundaries may occur at temperatures appreciably below those required for bulk diffusion, the initial step in product formation may be rapidly completed on the attainment of reaction temperature. In such systems, there is no initial delay during nucleation and the initial processes, perhaps involving monomolec-ular films, are not readily identified. The subsequent growth of the product phase, the main reaction, is thereafter controlled by the diffusion of one or more species through the barrier layer. Microscopic observation is of little value where the phases present cannot be unambiguously identified and X-ray diffraction techniques are more fruitful. More recently, the considerable potential of electron microprobe analyses has been developed and exploited. 

Model Phase boundary control (n) Diffusion control (171)... 

Analyses of rate measurements for the decomposition of a large number of basic halides of Cd, Cu and Zn did not always identify obedience to a single kinetic expression [623—625], though in many instances a satisfactory fit to the first-order equation was found. Observations for the pyrolysis of lead salts were interpreted as indications of diffusion control. More recent work [625] has been concerned with the double salts jcM(OH)2 yMeCl2 where M is Cd or Cu and Me is Ca, Cd, Co, Cu, Mg, Mn, Ni or Zn. In the M = Cd series, with the single exception of the zinc salt, reaction was dehydroxylation with concomitant metathesis and the first-order equation was obeyed. Copper (=M) salts underwent a similar change but kinetic characteristics were more diverse and examples of obedience to the first order, the phase boundary and the Avrami—Erofe ev equations [eqns. (7) and (6)] were found for salts containing the various cations (=Me). 

An unusual variation in kinetics and mechanisms of decomposition with temperature of the compound dioxygencarbonyl chloro-bis(triphenyl-phosphine) iridium(I) has been reported by Ball [1287]. In the lowest temperature range, 379—397 K, a nucleation and growth process was described by the Avrami—Erofe ev equation [eqn. (6), n = 2]. Between 405 and 425 K, data fitted the contracting area expression [eqn. (7), n = 2], indicative of phase boundary control. At higher temperatures, 426— 443 K, diffusion control was indicated by obedience to eqn. (13). The... 

As a result of the diffusional process, there is no net overall molecular flux arising from diffusion in a binary mixture, the two components being transferred at equal and opposite rates. In the process of equimolecular counterdiffusion which occurs, for example, in a distillation column when the two components have equal molar latent heats, the diffusional velocities are the same as the velocities of the molecular species relative to the walls of the equipment or the phase boundary. 

The theoretical treatment which has been developed in Sections 10.2-10.4 relates to mass transfer within a single phase in which no discontinuities exist. In many important applications of mass transfer, however, material is transferred across a phase boundary. Thus, in distillation a vapour and liquid are brought into contact in the fractionating column and the more volatile material is transferred from the liquid to the vapour while the less volatile constituent is transferred in the opposite direction this is an example of equimolecular counterdiffusion. In gas absorption, the soluble gas diffuses to the surface, dissolves in the liquid, and then passes into the bulk of the liquid, and the carrier gas is not transferred. In both of these examples, one phase is a liquid and the other a gas. In liquid -liquid extraction however, a solute is transferred from one liquid solvent to another across a phase boundary, and in the dissolution of a crystal the solute is transferred from a solid to a liquid. 

Figure 3.6. Spatial variation of the electrochemical potential, jl02-, of O2 in YSZ and on a metal electrode surface under conditions of spillover (broken lines A and B) and when equilibrium has been established. In case (A) surface diffusion on the metal surface is rate limiting while in case (B) the backspillover process is controlled by the rate, I/nF, of generation of the backspillover species at the three-phase-boundaries. This is the case most frequently encountered in electrochemical promotion (NEMCA) experiments as shown in Chapter 4.

Equations 4.31 and 4.32 also suggest another important fact regarding NEMCA on noble metal surfaces The rate limiting step for the backspillover of ions from the solid electrolyte over the entire gas exposed catalyst surface is not their surface diffusion, in which case the surfacediffusivity Ds would appear in Eq. 4.32, but rather their creation at the three-phase-boundaries (tpb). Since the surface diffusion length, L, in typical NEMCA catalyst-electrode film is of the order of 2 pm and the observed NEMCA time constants x are typically of the order of 1000 s, this suggests surface diffusivity values, Ds, of at least L2/t, i.e. of at least 4 10 11 cm2/s. Such values are reasonable, in view of the surface science literature for O on Pt(l 11).1314 For example this is exactly the value computed for the surface diffusivity of O on Pt(lll) and Pt(100) at 400°C from the experimental results of Lewis and Gomer14 which they described by the equation ... 

Formation of phase boundaries Rate process changes in solids Nucleation Diffusion processes... 

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