Phases of growth

Although the above is complicated, it does aptly illustrate the various mechanisms involved when atoms (ions) migrate by diffusion and start to form a new structure by formation of incipient embryos, then nuclei and finally the growth of phase boundaries. 

With these relations, we can perform quantitative calculations of the reaction kinetics. We start with Eqns. (6.35) and (6.36), which now contain Vp (= volume increase if one mole - or one equivalent — of species / is transported across the reaction layer p). In contrast to the growth of a single phase layer, however, transport occurs now in the two adjacent phases (p—1) and (p+ 1). This additional transport moves the interfaces p- )/p and p/(p+1) in addition to their shift due to the transport in p itself. Therefore, the kinetic differential equation for the growth of phase p has the following form... 

In this context, it is again advisable to distinguish between rate constants of the first and second kind. kp, as introduced in Eqn. (6.41), obviously is the rate constant k of the first kind. It describes the growth of phase p when all the other phases form simultaneously. The rate constant kf] of the second kind describes the growth of phase p from phases (p- 1) and (p+ 1) only. 

CHAPTER 20 GROWTH OF PHASES IN CONCENTRATION AND THERMAL FIELDS... 

In a UCST system, when the temperature is reduced to a final value 7/ that is below the critical temperature Tc, a mixture with a concentration 0 not too far from the critical composition phase separate into two phases whose compositions lie on the opposite sides of the binodal envelope line of Fig. 9-1. The dynamics of the separation process of a single phase into these two phases is controlled by Tf, the composition , the rate of the quench dT/dt, the viscous (or viscoelastic) properties of the phases formed, and the interfacial tension F between the two phases. Although a variety of different kinds of behavior can occur, there are two generic types of phase separation, namely, spinodal decomposition (SD) and nucleation and growth (NG). SD occurs when the mixture is quenched into a part of the phase diagram where the mixture is unstable to small variations in composition, leading to immediate growth of phase-separated domains. When the quenched... 

Fig. 13. Coexistence, hysteresis and kinetics, (a) Schematic of interface-controlled growth of phase ft into phase a (b) superposition of spectra and thermal hysteresis (c) time evolution of volume fraction of phase ft Avrami model (Eq. (14)) for different exponents n.

The maximum in the first stage reflects a fluctuation in composition that travels rapidly and forms a web-like structure. The second stage arises from the growth in size of the new phase. In a third stage, the growth of phases is controlled by viscous flow. At the end of the phase separation processes, it is difficult to tell whether they occured by nucleation and growth or by spinodal decomposition. One must observe... 

For almost fifty years, attempts have been made to interpret the experimental results in Fig. 8-13 by an ever increasingly sophisticated phenomenological theory [39]. This theory is concerned with the nucleation and growth of phase B under the assumption that the rate of decomposition is equal to the rate of the phase boundary reaction. This phase boundary reaction rate, in turn, is assumed to be either constant, or to depend uniquely upon the area of the interface already formed [8]. The growth of phase B is generally three-dimensional. Thus, if either the reactant or the reaction product are anisotropic, it is expected that the rate of growth will be different in the direction of each different crystal axis. In certain cases, microscopic observations seem to confirm that the growth rate of B is constant. Whether this supports the postulate that the phase boundary reaction is constant on a submicroscopic scale or not remains an open question. 

If it is assumed that nucleation of phase B is equally probable on all inner and outer surfaces of the crystal of A, and that the rate of the phase boundary reaction for the growth of phase B is constant, then the rate law for the dissociation reaction can be calculated, provided that the nucleation probability is known as a function of time. In the simplest case, there will be An possible nucleation sites, each of which has an equal a priori probability of becoming an actual nucleus. If n is the number of nuclei already present, then the rate of nucleation is ... 

Values of the exponent n other than n — A will occur if a nucleation law other than that in eq. (8-41) is obeyed, or if the growth of phase B is one- or two-dimensional. It can easily be shown that the essential features of the rate curve following the incubation period, as shown in Fig. 8-13, are well-fitted by the Erofeev equation. Also, it is not difficult to refine the assumptions and the derivation of this equation without altering the physico-chemical concepts upon which it is based. However, to what extent these refined treatments of the phenomenological theory are meaningful will remain an open question as long as the nucleation processes are not independently observed but - as is the case today with few exceptions - are only inferred from the overall decomposition kinetics. 

The growth of phase separated domains may be analyzed by examining the shift of wavenumber maximum (qm) and the corresponchng peak intensity (Im) as a function of elapsed time. A power law scheme has been customarily employed for characterizing the phase growth as follow ... 

Reactions in the diffusion zone during the formation and growth of phase layers can be divided into two types ... 

As it follows from Equation 3.40, (dA%/dt) becomes positive, and growth of phase 1 is allowed at... 

With some (incubation) time, the nuclei of phase 2 between 1 and B will stop being suppressed and a switch to case b wiU take place. Using the above given approximations and writing balance equations for cases a and h, taking into account the difference between partial diffusion coefficients D, Df in phase 1, one can show that T2, that is, the moment of time at which the value becomes positive and the possibility of diffusion growth of phase 2 layer appears, is defined by the following formula ... 

To express the gradient in the dilute solution in the presence of a growing intermediate phase, it is necessary to solve the following set of algebraic equations for simultaneous parabolic growth of phase 1 layer and of S-solution with parabolic movement of interfaces A/1 and 1 // ... 

At t - 00, phase 2 continues growing while the growth of phase 1 stops. The system of equations (Equation 8.15) is then reduced to the following two equations ... 

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