Interface local equilibrium

In most circumstances, it can be assumed diat die gas-solid reaction proceeds more rapidly diaii die gaseous transport, and dierefore diat local equilibrium exists between die solid and gaseous components at die source and sink. This implies diat die extent and direction of die transport reaction at each end of die temperature gradient may be assessed solely from diermodynamic data, and diat die rate of uansport across die interface between die gas and die solid phases, at bodi reactant and product sites, is not rate-determining. Transport of die gaseous species between die source of atoms and die sink where deposition takes place is die rate-determining process. 

The physics underlying Eqs. (74-76) is quite simple. A solidifying front releases latent heat which diffuses away as expressed by Eq. (74) the need for heat conservation at the interface gives Eq. (75) Eq. (76) is the local equilibrium condition at the interface which takes into account the Gibbs-Thomson correction (see Eq. (54)) K is the two-dimensional curvature and d Q) is the so-called anisotropic capillary length with an assumed fourfold symmetry. 

Regarding the electrode/electrolyte interface, it is important to distinguish between two types of electrochemical systems thermodynamically closed (and in equilibrium) and open systems. While the former can be understood by knowing the equilibrium atomic structure of the interface and the electrochemical potentials of all components, open systems require more information, since the electrochemical potentials within the interface are not necessarily constant. Variations could be caused by electrocatalytic reactions locally changing the concentration of the various species. In this chapter, we will focus on the former situation, i.e., interfaces in equilibrium with a bulk electrode and a multicomponent bulk electrolyte, which are both influenced by temperature and pressures/activities, and constrained by a finite voltage between electrode and electrolyte. 

There is zero resistance to mass transfer at the interface, itself, and therefore the concentrations at the interface are in local equilibrium. 

Potentiometric transducers now belong to the most mature transducers with numerous commercial products. For potentiometric transducers, a local equilibrium is established at the transducer interface at near-zero current flow, where the change... 

The symbol [ ], indicates concentrations at the extreme limit of the diffusional film (i.e., volume concentrations in the region in direct contact or very close to the liquid-liquid interface). Because of the fast nature of the distribution reaction, local equilibrium always holds at the interface. 

Equation 9e expresses the assumption of local equilibrium of the partitioning process at the stationary phase - mobile phase Interface. 

Local equilibrium is assumed to exist at the interface of the growth of ferrite which is considered to be momentarily stationary . Under these conditions it is possible to derive a position for a planar interface as... 

A critical part of the calculations is to calculate the tie-line at the interface corresponding to local equilibrium, and Enomoto (1992) used the central atoms model to predict the thermodynamic properties of a and 7. Some assumptions were made concerning the growth mode and the calculation of this tie-line is dependent on whether growth occurred under the following alternative conditions ... 

For liquid-vapor interfaces, the correlation length in the bulk is of t he order of atomic distance unless one is close to the critical point Hence the concept of local equilibrium is well justified in most practical circumstances For. solid surfaces above the roughening temperature, the concept also makes sense. Since the surface is rough adding (or removing) an atom to a particular part of the surface docs not disturb the local equilibrium state very much, and this sampling procedure can be used to determine the local chemical potential. This is the essence of the Gibbs-Thomson relation (1). 

Equality (1.20) is of primary importance because of the following reason. It is customary in most ionic transport theories to use the local electroneutrality (LEN) approximation, that is, to set formally e = 0 in (1.9c). This reduces the order of the system (1.9), (l.lld) and makes overdetermined the boundary value problems (b.v.p.s) which were well posed for (1.9). In particular, in terms of LEN approximation, the continuity of Ci and ip is not preserved at the interfaces of discontinuity of N, such as those at the ion-exchange membrane/solution contact or at the contact of two ion-exchange membranes or ion-exchangers, etc. Physically this amounts to replacing the thin internal (boundary) layers, associated with N discontinuities, by jumps. On the other hand, according to (1-20) at local equilibrium the electrochemical potential of a species remains continuous across the interface. (Discontinuity of Cj, ip follows from continuity of p2 and preservation of the LEN condition (1.13) on both sides of the interface.)... 

A terminological remark is due. An equilibrium between two media with different fixed charge density (e.g., an ion-exchanger in contact with an electrolyte solution) is occasionally termed the Donnan equilibrium. The corresponding potential drop between the bulks of the respective media is then termed the Donnan potential. By the same token, we speak of the local Donnan equilibrium and the local Donnan potential, referring, respectively, to the local equilibrium and the interface potential jump at the surface of discontinuity of the fixed charge density, considered in the framework of the LEN approximation. 

Let us begin the discussion of the last example of solid state kinetics in this introductory chapter with the assumption of local equilibrium at the A/AB and AB/B interfaces of the A/AB/B reaction couple (Fig. 1-5). Let us further assume that the reaction geometry is linear and the interfaces between the reactants and the product AB are planar. Later it will be shown that under these assumptions, the (moving) interfaces are morphologically stable during reaction. 

The right hand side is the result of integration. As long as local equilibrium prevails, the average value, LA, of the transport coefficient, taken across the reaction layer, is determined by the thermodynamic parameters at the interfaces A/AB and AB/B, and thus is independent of the reaction layer thickness A . If one inserts Eqn. (1.27) into Eqn. (1.26), a parabolic rate law is found... 

The increase A will occur at interface A/AB if LA/LR< 1, and it will occur at AB/B if La >Lr (Fig. 1-5). We conclude that parabolic rate laws in heterogeneous solid state reactions are the result of two conditions, the prevalence of a linear geometry and of local equilibrium which includes the phase boundaries. 

We deal in this section with quasi-binary systems in which more than one product phase A, B forms between the reactants A(=AX) and B(=BX) (Fig. 6-9). The more interfaces separating the different product phases, the more likely it is that deviations from local equilibrium occur (the interfaces become polarized during transport as indicated in Fig. 6-9, curve b). Polarization of interfaces is the theme of Chapter 10. If, however, we assume that local equilibrium is established during reaction, the driving force of each individual phase (p) in the product is inversely... 

A distinction between solid/fluid and solid/solid boundaries is irrelevant from the point of view of transport theory. Solid/fluid boundaries in reacting systems are, for example, (A,B)/A, B, X (aq) or (A,B)/X2(g). More important is the distinction according to the number of components. In isothermal binary systems, the boundary is invariant if local equilibrium prevails. In higher than binary systems, the state of the a/fi interface is, in principle, variable and will be determined by the reaction kinetics, including the diffusion in the adjacent bulk phases. 

Note that we have assumed the vacancies to be ideally diluted. We can then introduce a perturbation of the planar boundary, z = A +0(x,y,t), and define °(x,y) = Cartesian coordinates perpendicular to z. In this way, the morphological stability becomes a two-dimensional problem. Since we also assume that local equilibrium prevails at both interfaces (surfaces), the boundary conditions are... 

In many non-equilibrium situations, this local equilibrium assumption holds for the crystal bulk. However, its verification at the phase boundaries and interfaces (internal and external surfaces) is often difficult. This urges us to pay particular attention to the appropriate kinetic modeling of interfaces, an endeavour which is still in its infancy. 

The basic mechanisms by which various types of interfaces are able to move non-conservatively are now considered, followed by discussion of whether an interface that is moving nonconservatively is able to operate rapidly enough as a source to maintain all species essentially in local equilibrium at the interface. When local equilibrium is achieved, the kinetics of the interface motion is determined by the rate at which the atoms diffuse to or from the interface and not by the rate at which the flux is accommodated at the interface. The kinetics is then diffusion-limited. When the rate is limited by the rate of interface accommodation, it is source-limited. Note that the same concepts were applied in Section 11.4.1 to the ability of dislocations to act as sources during climb. 

The geometry of a B-rich /9-phase platelet growing edgewise in an a-phase matrix is shown in Fig. 20.7. The growing edge is modeled as a cylindrical interface of radius R where local equilibrium between precipitate and matrix is maintained. Adapting Eq. 15.4 to this cylindrical interface, the concentration in the a phase... 

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