Introduction. Equilibrium Phase Boundaries

This section is devoted to the basic kinetics of interfaces in solids. In Chapter 10 we shall work out some ideas in more detail and introduce atomic models for the determination of kinetic parameters. 

Interfaces are necessarily narrow, their smallest width being of atomic dimension. Therefore, thermodynamic potential gradients or potential changes across interfaces are often large compared with corresponding quantities in the bulk crystal. As a consequence, the linear regime of transport rates across interfaces is readily exceeded. 

The experimental determination of a potential change across a solid/solid interface is a most difficult task since it means that potential probes have to be placed very near the interface. Electrochemists face a similar problem when they study electrode kinetics, but the handling of fluids in this respect is much easier. Nevertheless, we will exploit their concepts and methods to some extent in what follows. 

Let us begin with the analysis of dynamic equilibrium. For the interior of phase a to be in dynamic equilibrium, all the particle fluxes must vanish. We can formulate these fluxes explicitly as 

It follows that the gradients Vpjn vanish individually if one deals with independent potentials and therefore extended equilibrium phases must be homogeneous. 

---

The appearance of two stable steady states X, X3 allows the system to exist in two phases with different densities X and X3 of the species X. It may even happen that these two phases coexist in the same system separated by a phase boundary. The whole situation is very similar to the phenomenon of phase transitions in equilibrium systems such as gas-liquid or liquid-solid systems. According to this similarity, the phenomenon of different phases in a nonequilibrium system is called a nonequilibrium phase transition or a "dissipative structure". Clearly, the inclusion of coexistence between X and X3 and of phase boundaries into our theory requires the introduction of additional diffusion terms into the equation of motion (6.5) in order to account for spatial variations of X. The analogies between our autocatalytic system (for v = 2) and equilibrium phase transitions have been worked out by F. SCHLOGL (1972) on a phenomenological and by JANSSEN (1974) on a stochastic level. 

The heptane water and toluene water interfaces were simulated by the use of the DREIDING force field on the software of Cerius2 Dynamics and Minimizer modules (MSI, San Diego) [6]. The two-phase systems were constructed from 62 heptane molecules and 500 water molecules or 100 toluene molecules and 500 water molecules in a quadratic prism cell. Each bulk phase was optimized for 500 ps at 300 K under NET ensemble in advance. The periodic boundary conditions were applied along all three directions. The calculations of the two-phase system were run under NVT ensemble. The dimensions of the cells in the final calculations were 23.5 A x 22.6 Ax 52.4 A for the heptane-water system and 24.5 A x 24.3 A x 55.2 A for the toluene-water system. The timestep was 1 fs in all cases and the simulation almost reached equilibrium after 50 ps. The density vs. distance profile showed a clear interface with a thickness of ca. 10 A in both systems. The result in the heptane-water system is shown in Fig. 3. Interfacial adsorption of an extractant can be simulated by a similar procedure after the introduction of the extractant molecule at the position from where the dynamics will be started. 

---