Numerical Simulation and the Phase-Field Method

Numerical models of conserved order-parameter evolution and of nonconserved order-parameter evolution produce simulations that capture many aspects of observed microstructural evolution. These equations, as derived from variational principles, constitute the phase-field method [9]. The phase-field method depends on models for the homogeneous free-energy density for one or more order parameters, kinetic assumptions for each order-parameter field (i.e., conserved order parameters leading to a Cahn-Hilliard kinetic equation), model parameters for the gradient-energy coefficients, subsidiary equations for any other fields such as heat flow, and trustworthy numerical implementation. 

The phase-field simulations reproduce a wide range of microstructural phenomena such as dendrite formation in supercooled fixed-stoichiometry systems [10], dendrite formation and segregation patterns in constitutionally supercooled alloy systems [11], elastic interactions between precipitates [12], and polycrystalline solidification, impingement, and grain growth [6]. 

7This ordering transition occurs at constant composition and is accomplished by microscopic re-arrangement of atoms into two sublattices. 

The simple two-dimensional phase-field simulations in Figs. 18.4 and 18.5 were obtained by numerically solving the Cahn-Hilliard (Eq. 18.25) and the Allen-Cahn equations (Eq. 18.26). Each simulation s initial conditions consisted of unstable order-parameter values from the top of the hump in Fig. 18.1 with a small spatial 

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