Continuum solidification models

The transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

Materials in the macroscopic sense follow laws of continuum models in which the nanoscale phenomenon is accounted for by statistical averages. Continuum models and analysis separate materials into solids (structures) and fluids. Computational solid mechanics and structural mechanics emphasize the analysis of solid materials and its structural design. Computational fluid mechanics treats material behaviors that involve the equilibrium and motion of liquid and gases. A relative new area, called multiphysics, includes materials systems that contain interacting fluids and structures such as phase changes (solidification, melting), or interaction of control, mechanical and electromagnetic (MEMS, sensors, etc.). 

Bennon WD, Incropera FP (1987) A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems-II. Application to solidification in a regular cavity. Int J Heat Transfer 30(10) 2171-2187... 

Fig. 12.19. Results of coupled continuum/ceUular automaton model for solidification (courtesy of M. Rappaz).

The current part of the present chapter has had as its aim the use of the study of microstructural evolution as a case study in the techniques for bridging scales described earlier in the chapter. The examples that were recounted in our discussion of microstructure and its evolution drew from a variety of the resources discussed earlier in the chapter in the context of bridging scales . In particular, we have seen how kinetic Monte Carlo models adopt an information passage philosophy in which calculations of one type are used to inform those of another. Similarly, the description of solidification, including information on the local crystal orientations, using a linkage of cellular automata with continuum descriptions of heat flow illustrates how more than one computational scheme may be brought under the... 

Typically, a phase-change system is treated as a single continuum i.e. there is no explicit differentiation between the solid and liquid phases. Numerical algorithms based on such single-domain approaches are by now well established and have produced useful and insightful models of solidification system. Some authors, however, have recognized that continuum approaches often lack a comprehensive treatment of the general nature of the systems. Recently full multiple-phase models have been proposed by Beckermann and coworkers [37-45]. 

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