Interfaces, solid-state reactions

Norris et al. [1254] discuss the application of several numerical methods to the determination of rate coefficients and of orders of solid state reactions of the contracting interface type. 

As an example, consider the following. Suppose we have a crucible half-filled with a powder. We now fill the crucible with another powder of different composition and then heat the filled crucible. Any solid state reaction which does occur can only do so at the boundary of the two layers of powders. If the reaction is A -t- B = AB, then we find that the reaction product, which is also a solid, forms as a phase boundary between the two layers. The same condition exists in a solid state reaction between two crystalline particles having differing compositions. That is, they can only react at the interface of each particle. This is illustrated in the following diagram, which is a model of how the components react through a phase boundary ... 

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... 

We find that this solid state reaction is very slow, even at 800 °C., and occurs at the interface of the two types of particles. The reaction is slow because it is diffusion-limited. What is happening is that since the silica-network is three-dimensionally bound, the only reaction that occurs is caused by the difiusion of Ba2+ atoms within the network, as shown in the following ... 

Another solid state reaction problem to be mentioned here is the stability of boundaries and boundary conditions. Except for the case of homogeneous reactions in infinite systems, the course of a reaction will also be determined by the state of the boundaries (surfaces, solid-solid interfaces, and other phase boundaries). In reacting systems, these boundaries are normally moving in space and their geometrical form is often morphologically unstable. This instability (which determines the boundary conditions of the kinetic differential equations) adds appreciably to the complexity of many solid state processes and will be discussed later in a chapter of its own. 

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. 

In heterogeneous solid state reactions, the phase boundaries move under the action of chemical (electrochemical) potential gradients. If the Gibbs energy of reaction is dissipated mainly at the interface, the reaction is named an interface controlled chemical reaction. Sometimes a thermodynamic pressure (AG/AK) is invoked to formalize the movement of the phase boundaries during heterogeneous reactions. This force, however, is a virtual thermodynamic force and must not be confused with mechanical (electrical) forces. 

In Chapter 11, growth morphologies are dealt with and the question is raised as to which conditions make the moving phase boundaries morphologically stable or unstable during solid state reactions. One criterion for instability is met if the interface moves against the flux direction of the rate determining (slow) reaction partner. 

Boundaries between solids transmit shear stress, particularly if they are coherent or semicoherent. Therefore, the strain energy density near boundaries changes over the course of solid state reactions. Misfit dislocation networks connected with moving boundaries also change with time. They alter the transport properties at and near the interface. Even if we neglect all this, boundaries between heterogeneous phases are sites of a discontinuous structural change, which may occur cooperatively or by individual thermally activated steps. 

Figure 10-10. Representation of the chemical potential of A during the heterogeneous solid state reaction A+B = AB. a) Diffusion control, b) interface control at b2, c) rate control by rearrangement (relaxation) of A in B in zone A (B), d) simultaneous diffusion and interface control (bj).

The definition of a solid state reaction implies that the reaction product is a solid. If, for example, one of the reactants is a fluid, no deviatoric stresses are transmitted across the common interface. This situation simplifies the mechanical boundary condition significantly and explains why studies on boundary morphology are often performed with solid/fluid systems. 

So far, we have tacitly assumed that the stresses were applied externally. However, stresses which are induced by local changes in component concentrations and the corresponding changes in the lattice parameters during transport and reaction are equally important. These self-stresses can strongly influence the course of a solid state reaction. Similarly, coherent, semicoherent, and even incoherent interfaces during heterogeneous solid state reactions are sources of (local and nonlocal) stress. The... 

Contamination of the sample being formed in a solid-state reaction may occur from the walls of the containment vessel, but is reduced under microwave conditions because of the lower temperatures at the interfaces between the sample and the crucible. 

The growth of layer i — 1, on the other hand, will require an analogous decomposition of layer i by solid-state reaction at the interface xt = 0. This will lead to a negative contribution to the growth kinetics of layer i. Thus, from the time-dependence of the cation interstitial balance at the interfaces we can write the net growth rate of any inner layer i as... 

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