Precipitate coarsening

This concentration difference will result in diffusion of component 2 from the small particle to the larger particle. A simple diffusion analysis 

Evaluation of the right-hand side of Equation (6.88) at r = rp yields 

Insertion of Equation (6JS) for the concentrations into Equation (6.89) yields 

Insertion of fp = into Equation (6.90) yields the growth rate of the 

This equation may be separated and integrated to give the time dependence of the most rapidly growing particles  

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We will be looking at kinetics in Chapter 6. But before we can do this we need to know what we mean by driving forces and how we calculate them. In this chapter we show that driving forces can be expressed in terms of simple thermodynamic quantities, and we illustrate this by calculating driving forces for some typical processes like solidification, changes in crystal structure, and precipitate coarsening. 

Fig. 5.7. Schematic of precipitate coarsening. The small precipitate is shrinking, and the large precipitate is growing at its expense. Material travels between the two by solid-state diffusion.

This may not always be the case. For example, if the particles are formed by precipitation, coarsening may begin before complete precipitation has occurred [4]. 

A.J. Ardell. Precipitate coarsening in solids Modern theories, chronic disagreement with experiment. In G.W. Lorimer, editor, Phase Transformations 87, pages 485-490, London, 1988. Institute of Metals. 

Figure C2.11.6. The classic two-particle sintering model illustrating material transport and neck growtli at tire particle contacts resulting in coarsening (left) and densification (right) during sintering. Surface diffusion (a), evaporation-condensation (b), and volume diffusion (c) contribute to coarsening, while volume diffusion (d), grain boundary diffusion (e), solution-precipitation (f), and dislocation motion (g) contribute to densification.

We saw in Chapter 5 that there is a driving force tending to make dispersions of precipitates in alloys coarsen and we would expect a dispersion of droplets in water vapour to do the same. Water droplets in clouds, however, carry electrostatic charges and this gives a different result for the driving force. 

Coarsening of fine precipitates kept in the mother liquor for protracted time was observed in numerous instances the number of very fine particless in the precipitate decreased in time, and the volume of several large particles simultaneously grew. This effect, after 1899, was frequently referred to as Ostwald ripening. 

C. Wagner Theory for the Coarsening of Solid Precipitates Caused hy Ostwald Ripening. Z. Elektrochem. 65, 581 (1961). 

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