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01.Evolution Equations for Grain Growth and Coarsening (pp.5-60)
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Evolution Equations for Grain Growth and Coarsening (pp.5-60) $100.00
Authors:  (F.D. Fischer, J. Svoboda, Institute of Mechanics, Montanuniversitat Leoben, Leoben, Austria, and others)
A real classical topic of research for evolving systems of microstructures with grains
like metals and ceramics is grain growth and coarsening (Ostwald Ripening). Both
phenomena are driven by the decrease in total surface energy, while the chemical
composition of the individual objects remains unchanged. Only the morphology of the
system evolves and gives the impression that smaller objects are “eaten” by the larger
ones. Evolution equations for these phenomena were published in the last century mostly
as phenomenological equations. The authors show, however, that the evolution equations
can be derived from the Thermodynamic Extremal Principle (TEP). This principle, based
on the works of L. Onsager (1931, 1945) and later of H. Ziegler (1977), allows
calculation of the evolution equations of the state parameters, e.g., the effective radii of
individual grains, precipitates, etc., by maximizing the dissipation, expressed in a
quadratic form of the kinetic parameters, with the constraint that the dissipation is equal
to the negative rate of Gibbs energy in the isothermal case. Later studies have allowed
working with distribution functions of the effective radii of objects and the evolution of
those to stationary distribution functions. Also, the inverse problem has been solved, by
which the evolution equations of the grain or precipitate effective radii can be derived
from steady state distribution functions. This concept was extended from one-parameter
distribution functions to various two-parameter distribution functions. The concept of
TEP also allows successful simulations of the microstructure evolution in multicomponent
multiple particle systems. Practical applications of this concept of evolution
equations for multi-component materials are published and used in the metals processing
industry—see the program package MatCalc,
The main progress of using the TEP to derive evolution equations can be seen in a
rigorous variational approach compared to the numerous heuristic approaches to grain
growth and coarsening. 

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Evolution Equations for Grain Growth and Coarsening (pp.5-60)