Modeling Grain Growth

As noted above, ostensibly, part of the empirical background concerning grain growth that demands explanation is the power law growth in time of the grain size. One of the justly famous contributions in that regard is the work of Burke 

As yet, the model is purely descriptive and says nothing about the possible subsequent evolution of the microstructure. To endow this geometric model with an underlying dynamics , we first require a statement of the energetics. One conventional choice is a Hamiltonian of the form 

Once these kinematic preliminaries have been settled, the total free energy functional is then written as the integral 

In addition to the definition of the energy functional, it is necessary to institute a dynamics associated with a model of this sort. In the present setting, the dynamics is obtained by asserting that the temporal evolution of a given field variable is linear in fhe driving force, namely. 

Notice that one of the challenges of a model of this type is that there are as many equations as there are components to the order parameter. As a result, for the simulation of a polycrystal with many different grains, there will be as many field variables as fhere are grains resulting in both conceptual and computational 

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An equation of the form provided in Equation (8) is often used to fit experimental coarsening data. The exponent, n, is used to infer the rate-controlling step for the growth process. This has proven useful for modeling grain growth in a number of metallic alloy... 

Modeling grain growth experimentally is illustrated in Figure 24.22. Such studies are an extension of the measurement of GB migration. 

German, R. M. and Olevsky, E. A., Modeling grain growth dependence on the liquid content in liquid phase sintered materials, MetalL Mater. Tram. A, 29A, 3057-66, 1998. 

The merits and demerits of the many computer-simulation approaches to grain growth are critically analysed in a book chapter by Humphreys and Hatherly (1995), and the reader is referred to this to gain an appreciation of how alternative modelling strategies can be compared and evaluated. A still more recent and very clear critical comparison of the various modelling approaches is by Miodownik (2001). 

Grain growth involves no phase transformation, but a number of such transformations have been modelled and simulated in recent years. A recently published overview volume relates some experimental observations of phase... 

M. K. Venkitachalam, L.-Q. Chen, A. G. Khachaturyan, G. L. Messing. A multiple-component order parameter phase field model for anisotropic grain growth. Mater Sci Eng A 238 94, 1997. 

Ultimately, a knowledge of kinetics is valuable because it leads to prediction of the rates of materials processes of practical importance. Analyses of the kinetics of such processes are included here as an alternative to a purely theoretical approach. Some examples of these processes with well-developed kinetic models are the rates of diffusion of a chemical species through a material, conduction of heat during casting, grain growth, vapor deposition, sintering of powders, solidification, and diffusional creep. 

The ejection mechanisms considered by Watson and Salpeter are mainly photodesorption and ejection during molecule formation, the former being the most important process. Other mechanisms involving interaction with cosmic rays or with IR photons are of minor importance. Since ejection by UV photons is highly unlikely in very dense clouds, condensation of the interstellar gas and grain growth in the Watson-Salpeter-model would proceed without limit, which of course is in contradiction to observations. 

In real systems the final stage of sintering almost generally involves also grain growth, which complicates confrontation of the theoretical models with experimental data. 

The grain growth models predict a kinetic equation of the form ... 

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