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Über dieses Buch

100 years after the first observation of ripening by Ostwald and 40 years after the first publication of a theory describing this process, this monograph presents, in a self-consistent and comprehensive manner, all the bits and pieces of coarsening theories so that the main issues and the underlying mathematics of self-similar coarsening of dispersed systems can be understood. It contains all of the background material necessary to understand growth and coarsening of spherical particles or droplets in a liquid or solid matrix. Some basic knowledge of heat and mass transfer, thermodynamics and differential equations would be helpful, but not necessary, as all the concepts required are introduced. The text is suitable for advanced undergraduate and graduate students as well as for researchers. Rather than giving a complete survey of the field, it presents a careful derivation of the existing results and places them into some perspective.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
Whenever a single phase system is put into a two-phase metastable state, for example by quenching from a high temperature or changing pressure, a second phase nucleates, grows, and coarsens. Nucleation of the second phase occurs since the energy of the single phase system can be reduced by forming regions of the second phase. Nanometer-sized precipitates of the new phase thus appear. Growth of nuclei then proceeds due to heat diffusion away from the nucleus into the matrix or mass diffusion from the matrix to the nuclei. This results in a dispersion of second phase particles in a matrix. Coarsening, also called Ostwald ripening or competitive growth, then occurs where large particles grow at the expense of small particles. The nucleation, growth, and coarsening is very common, occurring in systems ranging from solid alloys to the precipitation of drops from clouds [246].
Lorenz Ratke, Peter W. Voorhees

2. Thermodynamics of alloys

Abstract
In this chapter we will present briefly some basic thermodynamic relations that are useful in understanding the fundamentals of growth and coarsening of particles embedded in a matrix. The reader who is already familiar with the concepts of free energy, chemical potential, equilibrium of heterogeneous alloys, the Gibbs-Thomson effect etc. may skip this chapter.
Lorenz Ratke, Peter W. Voorhees

3. Transport of heat and mass

Abstract
Growth and coarsening of particles or precipitates in a matrix require the transport of heat or mass or both simultaneously. Heat and mass can be transported by diffusion, where heat diffusion in a crystal occurs by the random transport of enthalpy via electrons and phonons, quantized lattice vibrations, with a direction set by the temperature gradient, where heat always flows from high to low temperatures. Mass diffusion occurs in crystals by a random walk of atoms utilizing lattice vacancies. The details on heat and mass transport are discussed in standard textbooks on solid state physics and material sciences. In liquids heat and mass can be transported by diffusion, convection, or both. To describe the diffusion process we derive the conservation equations for heat and mass under diffusive and convective-diffusive conditions. These basic equations are later used for particle growth and coarsening.
Lorenz Ratke, Peter W. Voorhees

4. Growth

Abstract
In this chapter we will examine the growth of particles in a matrix via a number of mechanisms First we consider growth or solidification from a supercooled melt, second growth in a supersaturated matrix, and third growth by interface kinetics of first order where mass or heat is transported. We then discuss the growth by interface kinetics of second order and growth of particles or droplets where convective heat or mass fluxes play an important role. The closing section deals with growth under varying, time dependent supersaturation.
Lorenz Ratke, Peter W. Voorhees

5. Statistics of growth

Abstract
In this chapter we will study the evolution of a number of spherical particles described by a particle size density distribution. We assume that the particles do not interact, meaning for example if they grow by heat or mass diffusion at constant supercooling or supersaturation the diffusion fields do never overlap. The particles never touch each other and grow without any limit. Thus, the description of a particle ensemble does not take into account for example mass conservation applied to the entire sample. This simplification allows us to treat the statistics in a simple and analytical way. However, the situation described is not purely theoretical: E.g. when cooling a metal one can have for a limited period of time following nucleation just growth of these particles — very much as described here. As the particles grow the matrix concentration decreases and competitive growth — coarsening — sets in. This situation will be treated in chapters 7,8. The general statistical concepts presented below are useful in these chapters.
Lorenz Ratke, Peter W. Voorhees

6. Coarsening — Basics and Growth Laws

Abstract
Imagine a system consisting of a huge number of particles of a certain phase with different radii embedded in a matrix. Some examples are shown in the introduction in figs.1.3 and 1.6. Assume further that the alloy is held at a constant annealing temperature. The concentration of solute in the matrix shall be at a level given approximately by the phase diagram and the annealing temperature. What will happen to the dispersion of particles? Generally one will observe in experiments that the particle radii change with annealing time, the average radius will grow, the number of particles decreases, but the overall volume fraction remains nearly a constant. An example of such a behavior is shown in Fig. 6.1. Why do these changes in particle size occur? Generally speaking, a dispersion of solid or liquid particle in a matrix increases the overall energy due to the existence of interfacial energy associated with the interfacial area. Since the system tries to minimize the energy, the amount of interfacial area decreases with time. A large number of small particles has, however, more interfacial area than one large particle with the same volume. Thus, the energy contained in the interfacial area of a dispersion increases with the number of particles in the dispersion. Reducing the number of particles decreases the excess energy associated with the interfaces. Therefore, a dispersion coarsens or ripens.
Lorenz Ratke, Peter W. Voorhees

7. Ostwald ripening — Wagner analysis

Abstract
In this chapter we treat coarsening under different conditions of heat and mass transport using mainly the Wagner technique to analyse the problem [295], which is somewhat different from the Lifshitz-Slyozov analysis [194]. We will not present the LS analysis in detail since the Marqusee and Ross analysis presented in the next chapter covers the LS analysis and also is much more complete and easier to understand.
Lorenz Ratke, Peter W. Voorhees

8. Ostwald ripening — Marqusee and Ross type analysis

Abstract
The Wagner-analysis rests on the introduction of a critical radius and the assumption that the volume fraction is a conserved quantity. This is inconsistent since the small supersaturation decays during coarsening and thus the volume also varies with time. It is possible to remove this inconsistency using instead the more general mass or enthalpy conservation laws1. The asymptotic analysis employs a power series representation for the particle size distribution in time and shows that asymptotically a time independent state is reached under the appropriate scaling, which is unique and independent of the initial conditions. This analysis was performed by Marqusee and Ross and is especially suitable to incorporate effects beyond simple mass diffusion. Moreover, the analysis is self-consistent unlike the Wagner-analysis. We therefore present their approach in some detail. We first treat coarsening due to interface kinetics in a supersaturated matrix and then treat coarsening due to diffusional heat transfer.
Lorenz Ratke, Peter W. Voorhees

9. Multiparticle diffusion analysis

Abstract
The LSW theory for coarsening has a serious drawback: it does not account for diffusional interactions between particles. As a result the coarsening rate of a particle is a function only of the radius of a particle relative to the average particle size. This is not the case in nonzero volume fraction systems, for when a particle of a given size is surrounded by particles larger than itself it must coarsen in a different manner than when this particle is surrounded by particles that are smaller than itself. Diffusional interactions between particles also must lead to a coarsening rate of the system that is a function of the volume fraction. As the volume fraction increases the interparticle separation decreases but the concentration in the matrix at the interfaces of the particles is not a function of the interparticle separation, it set by the Gibbs-Thomson equation. Thus the concentration gradients in the matrix at the interfaces increases with increasing volume fraction and the coarsening rate of the system must therefore increase. Both the physics of local diffusional interactions and the dependence of the coarsening rate on volume fraction are missing from the LSW theory.
Lorenz Ratke, Peter W. Voorhees

10. Nucleation, Growth and Coarsening

Abstract
In the preceding chapters we treated growth and coarsening as separate processes. As sketched in the introduction (see chapter 1) there are several ways to prepare a two- or multi-phase system:
1.
Nucleation from a supersaturated or supercooled liquid
 
2.
Spinodal decomposition
 
3.
Artificial mixing of powders
 
In the simplest picture transformations that proceed by nucleation can be divided into several steps: the nucleation process itself, followed by growth of the nuclei and finished by coarsening of the precipitates or particles or drops (in solid state transformations it is common to name the new second phase appearing in the matrix as precipitates, in transformations from a liquid to a solid they are called particles or grains and in liquid-liquid decomposition they are termed droplets). A prerequisite for the nucleation type of phase transformation is the existence of a limited miscibility in the phase diagram, see for example the eutectic phase diagram, fig.2.11, where a limited solubility exists in the solid state below the eutectic temperature. The miscibility gap in a eutectic can be described by, for example, the regular solution model. This gives the miscibility gap shown in Fig. 2.6 where the critical point does not exist since it is terminated by the eutectic reaction.
Lorenz Ratke, Peter W. Voorhees

Backmatter

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