Interface equilibrium

Schemes II and III represent the equilibrium interface energetics for ideal n- and p-type semiconductors, respectively, for...

In conclusion, we observe that the crossing of crystal phase boundaries by matter means the transfer of SE s from the sublattices of one phase (a) into the sublattices of another phase (/ ). Since this process disturbs the equilibrium distribution of the SE s, at least near the interface, it therefore triggers local SE relaxation processes. In more elaborated kinetic models of non-equilibrium interfaces, these relaxations have to be analyzed in order to obtain the pertinent kinetic equations and transfer rates. This will be done in Chapter 10. 

The equilibrium interfaces of fluid systems possess one variant chemical potential less than isolated bulk phases with the same number of components. This is due to the additional condition of heterogeneous equilibrium and follows from Gibbs phase rule. As a result, the equilibrium interface of a binary system is invariant at any given P and T, whereas the interface between the phases a and /3 of a ternary system is (mono-) variant. However, we will see later that for multiphase crystals with coherent boundaries, the situation is more complicated. 

Chemical kinetics concerns the evolution in time of a system which deviates from equilibrium. The acting driving forces are the gradients of thermodynamic potential functions. Before establishing the behavior and kinetic laws of interfaces, we need to understand some basic interface thermodynamics. The equilibrium interface is characterized by equal and opposite fluxes of components (or building elements) in the direction normal to the boundary. Ternary systems already reflect the general... 

By changing 6, the surface planes (hm) may become strained or even reconstruct. We conclude that, given P and T, the boundary composition Nb depends on a) the chemical potential of the components in the adjacent phases and b) the orientation o. In equilibrium, 8G = 8Gb = 0 which determines, at any given P, T, and /xh the set 6 (equ). In other words, we can formally express the equilibrium interface state (i.e., structure, strain, composition) as... 

Figure 10-6. Various thermodynamic potentials and the electric charge distribution at and near an equilibrium interface (schematic).

One notes that RA is inversely proportional to the exchange flux (/a) °f the dynamic equilibrium interface. 

Minimizing T = / /( , S7 )dV produces equilibrium interface profiles (r). An equilibrated planar interface is characterized by its excess energy per unit area, 7,... 

The quantity 7/c may be regarded as the local potential due to the interface curvature to add a chemical species per unit volume of the species. On interfaces where the mean curvature is constant everywhere (such as on a sphere where k = 2/Rc, on a cylinder where k = 1/RC. and on a plane and a catenoid where k = 0), this potential is uniform and thus these are equilibrium interfaces. There is an infinite number of equilibrium interfaces a three-parameter family of minimal interfaces has been described [9]. 

The following discussion is excerpted from Mullin (1993) and Elwell and Scheel (1975). Diffusional boundary theory is well-established (see e.g., Bird et al., 1960) and the concept of a boundary unstirred layer was introduced a century ago. Noyes and Whitney (1897) proposed that the change in the rate of crystal growth (dm/dt) was controlled by diffusion from the bulk concentration to the crystal (equilibrium) interface. 

It is the hyperbolic relations (Ag )(Br ) = K and (M )(Y ) = (M+)(Y)/KZ that provides t e basic analogy be Seen the two kinds of systems. In the latter, K is the ionic salt partition coefficient relating membrane and bathing solution activities at an equilibrium interface. The latter form can also be derived for insoluble salt membranes. However the salt activities (super bar quantities) are constant and so are hidden in the value of the solubility product... 

The interfacial potential difference (pd) for the partition equilibrium interface is given by the equality of electrochemical potential in terms of all ions in equilibrium, equation (4). 

Formation of the equilibrium interface characteristic for coexisting phases... 

The observations show that growth in the c-axis direction is not continuous but occurs by step propagation across the basal plane up to vapour density excesses of at least 3 x io g cm within the experimental chamber, such density excesses being reached at — 20 °C when the vapour saturation ratio pjps is about 1.5. This is consistent with the discussion of chapter 4 which predicts, since aLjkT 15 for the crystal-to-vapour transition, that the equilibrium interface should be smooth. From (5.2) and (5.4) we also find that the rate of nucleation of new layers of unit step height should be small for piPs less than about 1.5 at the interface, which is again consistent with observations. 

Up to now we have only discussed the structure of interfaces that have reached equilibrium. However, in many processes only a finite amount of time is available for an interface to be formed. For example, when a pair of polymers is coextruded, the time available for the interface to develop is limited to the time at the process temperature. Alternatively, if one polymer is applied as a solution to coat a second pol)mier, then molecular motion of the polymers near the interface will often only be possible before all the solvent has evaporated. This question of kinetics becomes particularly important when we come to consider interfaces between miscible polymers here there is no equilibrium interface width at aU and the width of the interface that is achieved in practice... 

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