Diffuse phase boundaries interface

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. 

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. 

In this case, we have given both the starting conditions and those of the intermediate stage of solid state reaction. It should be clear that A reacts with B, and vice versa. Thus, a phase boundary is formed at the interface of the bulk of each particle, i.e.- between A and AB, and between B and AB. The phase boundary, AB, then grows outward as shown above. Once the phase boundary is established, then each reacting specie must diffuse through the phase AB to reach its opposite phase boundary in order to react. That is- A must difiuse through AB to the phase boundary... 

Figure 4 illustrates the dependence of on Aq for the case when r = 1 at several different values of [Fig. 4(a)] and when = 0.5 and at several different values of r [Fig. 4(b)]. From Fig. 4(a), one can see that takes a maximum around y = 0, i.e., Aq The volume ratio affects strongly the value of as shown in Fig. 4(b), which is ascribed to the dependence of the equilibrium concentration on r through Eq. (25). This simple example illustrates the necessity of taking into account the variation of the phase-boundary potential, and hence the adsorption of i, when one tries to measure the adsorption properties of a certain ionic species in the oil-water two-phase systems by changing the concentration of i in one of the phases. A similar situation exists also in voltammetric measurements of the transfer of surface-active ions across the polarized O/W interface. In this case, the time-varying thickness of the diffusion layers plays the role of the fixed volume in the above partition example. The adsorption of surface-active ions is hence expected to reach a maximum around the half-wave potential of the ion transfer. 

PBP model considers the membrane potential as a sum of the potentials formed at the membrane-solution interfaces (phase boundary potentials), and generally neglects any diffusion potential within the membrane ... 

The only potential that varies significantly is the phase boundary potential at the membrane/sample interface EPB-. This potential arises from an unequal equilibrium distribution of ions between the aqueous sample and organic membrane phases. The phase transfer equilibrium reaction at the interface is very rapid relative to the diffusion of ions across the aqueous sample and organic membrane phases. A separation of charge occurs at the interface where the ions partition between the two phases, which results in a buildup of potential at the sample/mem-brane interface that can be described thermodynamically in terms of the electrochemical potential. At interfacial equilibrium, the electrochemical potentials in the two phases are equal. The phase boundary potential is a result of an equilibrium distribution of ions between phases. The phase boundary potentials can be described by the following equation ... 

In contrast, a fast reaction rate will result in steep concentration gradients for the reactants and a higher reaction rate near the solvent interface. This concept is represented diagrammatically in Figure 2.13b, where the concentration of reactant A is almost as high as that in phase 1 at the solvent interface, but plummets as it is rapidly consumed by the reaction. Thus, for a fast reaction, the majority of reactant is converted to product near the phase boundary layer and the rate of the reaction is limited by the rate of phase transfer and diffusion. 

GITT also provides very comprehensive information about the kinetic parameters of the electrode by analysis of the electrical current. The current 1, which is driven through the galvanic cell by an external current or voltage source, determines the number of electroactive species added to (or taken away from) the electrode and discharged at the electrode/ electrolyte interface. A chemical diffusion process occurs within the electrode and the current corresponds to the motion of mobile ionic species within the electrode just inside the phase boundary with the electrolyte (at x = 0)... 

Although the SOFC community has generally maintained an empirical approach to the three-phase boundary longer than the aqueous and polymer literature, the last 20 years have seen a similar transformation of our understanding of SOFC cathode kinetics. Few examples remain today of solid-state electrochemical reactions that are not known to be at least partially limited by solid-state or surface diffusion processes or chemical catalytic processes remote from the electrochemical—kinetic interface. 

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

The resistance in each phase is made up of two parts the diffusional resistance in the laminar film and the resistance in the bulk fluid. All current theories on mass transfer, i. e. film, penetration, and surface renewal assume that the resistance in the bulk fluid is negligible and the major resistance occurs in the laminar films on either side of the interface (Figure 3-2). Fick s law of diffusion forms the basis for these theories proposed to describe mass transfer through this laminar film to the phase boundary. 

Let us conclude this section with a few general remarks. If we assume phase boundary rate control, the rate of advance is co-determined by the interface mobility, which in turn is related to the mobilities of the atoms in the interface. We note that 1) the directional dependence of mobilities or diffusivities in the interface may be quite pronounced (depending on 5) and 2) the mobilities or diffusivities depend on the component chemical potentials, which change over time at the interface until diffusion control eventually becomes rate determining. 

Thus far we have examined diffusion under infinite conditions, where no phase boundaries exist. Some practical situations may be described by the above treatment. More frequently, the diffusion process will be initiated in the neighborhood of one or more phase boundaries as, for example, in chromatography and electrochemistry. The phase boundaries may be either permeable or impermeable to the diffusing solute. In electrochemical techniques, the boundary (e.g., the working electrode) is usually impermeable however, this is not always so (e.g., some ion-selective electrodes, membranes, liquid-liquid interfaces). In the... 

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