Kinetics and Dynamics. Local Equilibrium

The aim of this chapter is to clarify the conditions for which chemical kinetics can be correctly applied to the description of solid state processes. Kinetics describes the evolution in time of a non-equilibrium many-particle system towards equilibrium (or steady state) in terms of macroscopic parameters. Dynamics, on the other hand, describes the local motion of the individual particles of this ensemble. This motion can be uncorrelated (single particle vibration, jump) or it can be correlated (e.g., through non-localized phonons). Local motions, as described by dynamics, are necessary prerequisites for the thermally activated jumps responsible for the movements over macroscopic distances which we ultimately categorize as transport and solid state reaction.. 

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

Before continuing with the discussion on the dynamics of SE s in crystals and their kinetic consequences, let us introduce the elementary modes of SE motion. In a periodic lattice, a vacant neighboring site is a necessary condition for transport since it allows the site exchange of individual atomic particles to take place. Rotational motion of molecular groups can also be regarded as an individual motion, but it has no macroscopic transport component. It may, however, promote (translational) diffusion of other SE s [M. Jansen (1991)]. 

In addition to the individual and uncorrelated particle motions, we also have collective ones. In a strict sense, the hopping of an individual vacancy is already coupled to the correlated phonon motions. Harmonic lattice vibrations are the obvious example for a collective particle motion. Fixed phase relations exist between the vibrating particles. The harmonic case can be transformed to become a one-particle problem [A. Weiss, H. Witte (1983)]. The anharmonic collective motion is much more difficult to treat theoretically. Correlated many-particle displacements, such as those which occur during phase transformations, are further non-trivial examples of collective motions. 

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Thermodynamic models have been a mainstay of geochemistry since the early twentieth century. They are especially effective for deep earth conditions where local equilibrium conditions prevail. However, at and near the Earth s surface extensive amounts of mass transport and low temperatures keep many reactions from reaching equilibrium. Kinetics models are needed to properly describe these situations. This makes the models of kinetic and dynamic processes described in this book complementary to thermodynamic... 

The earth s subsurface is not at complete thermodynamic equilibrium, but parts of the system and many species are observed to be at local equilibrium or, at least, at a dynamic steady state. For example, the release of a toxic contaminant into a groundwater reservoir can be viewed as a perturbation of the local equilibrium, and we can ask questions such as. What reactions will occur How long will they take and Over what spatial scale will they occur Addressing these questions leads to a need to identify actual chemical species and reaction processes and consider both the thermodynamics and kinetics of reactions. 

To develop the kinetic equations in condensed phases the master equation must be formulated. In Section 3 the master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The latter set of equations permits consideration of history of formation of the local solid structure as well as its influence on the subsequent elementary stages. The many-body problem and closing procedure for kinetic equations are discussed. The influence of fast and slow stages on a closed system of equations is demonstrated. The multistage character of the kinetic processes in condensed phase creates a problem of self-consistency in describing the dynamics of elementary stages and the equilibrium state of the condensed system. This problem is solved within the framework of a lattice-gas model description of the condensed phases. 

The initial conditions for the nonzero moments in the Riemann shock problem are shown in Figure 8.9. On the left half of the domain the normalized number density is Moo = 1, and on the right half it is 0.1. Initially the RMS velocity is 1 on both sides of the domain and the mean velocity is null. The moments are initialized as Maxwellian, so that the initial conditions are at local equilibrium. As time increases, a shock wave starting at x = 0 moves to the right and a deflation wave moves to the left. In the limit of t = 0, the solution is the same as for the Euler equation of gas dynamics. Sample results for the moments with T = 100 at time t = 0.5 are shown in Figure 8.10. For this case, the collisions are very weak, and thus there is little transfer of kinetic energy from the M-component to the... 

Some heat flows in connection with entropy production are associated with other thermodynamic variables. Typical single fluxes and forces are summarized above. It may be noted that steady fluxes are considered. Kinetic theory provides theoretical justification of some of these flux force relations (J = LX). Here, L is called phenomenological coefficient. But kinetic theory has limitation in the sense that first approximation to distribution function corresponds to local equilibrium hypothesis. It may be noted that non-equilibrium molecular dynamics (model and simulation) provides justification of these laws for a wide range. Nevertheless, justification has to be provided by experiments (Table 2.1). 

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