A comparative study of dendritic growth by using the extended Cahn–Hilliard model and the conventional phase-field model

Numerical research of solidification dynamics with anisotropy and thermal fluctuations

“…The first approach to describe the instability phenomena during solidification was presented by Mullins and Sekerka [10,18]. Interfacial energy anisotropy in the model was introduced based on the assumption that scalar gradient coefficient depends on the interface orientation [19]. Function of that coefficient in 2-D Cartesian space has the following form [19,20]:…”

“…Interfacial energy anisotropy in the model was introduced based on the assumption that scalar gradient coefficient depends on the interface orientation [19]. Function of that coefficient in 2-D Cartesian space has the following form [19,20]:…”

“…1. Secondly, the mathemathical model for instability of the freezing pattern was compared with a pure component by the directional freezing of pure metal (nickel) cast [19]. The obtained PFM results are presented in Fig.…”

Simulation of Intermetallic Solidification Using Phase-Field Techniques

“…An alternative approach which has shown considerable promise in the simulation of faceted morphologies is the so called Extended Cahn-Hilliard Model (ECHM), in which, rather than introducing the crystallographic anisotropy via an orientation depended coefficient, , to the gradient energy, higher order tensorial gradient energy terms are introduced [7]. The approach has a number of advantages, including that the stationary crystal shape does not exhibit sharp corners [8,9] even when the interfacial energy is high enough for the corresponding Wulff shape to do so.…”

Interfacial free energy anisotropy driven faceting of precipitates

“…Phase field models, which are best suited for the study of the formation and evolution of microstructures (see [1,2,3,4,5] for some recent reviews), have been used quite successfully to study the effect of interfacial energy anisotropy on microstructures and their evolution: see ] for some representative examples.The phase field models that incorporate interfacial energy anisotropy do so in one of two ways: (A) replace the gradient energy coefficient by an anisotropic function or polynomial, and (B) include higher order terms in the Taylor series expansion of the free energy functional. In this paper, we take the second approach -which is an extension of the original Cahn-Hilliard equation along the lines shown by Abinandanan and Haider [6] (hereafter referred to as ECHAH) and Torabi and Lowengrub [7] (hereafter referred to as ECHTL); this approach is argued to be advantageous in terms of the levels of anisotropy one can incorporate [6] and the kinetics remaining diffusionlimited [13]; in addition, the first method requires regularisation [22] for large anisotropies while ours does not.During solid-solid phase transformations interfacial energy anisotropy is known to lead to faceted precipitates: see for example, PbS precipitates in Na-doped PbTe system [31], Al 3 Sc precipitates in Al(Sc) alloys [32], Pt precipitates in sapphire [33], several metallic precipitates in internally reduced oxides [34], and Al 3 Ti precipitates in Al [35]. Our objective in this paper is to obtain, using phase field modelling, faceted precipitates in cubic systems that distinguish between 100 and 111 (for which one has to necessarily include fourth rank tensor terms [6]) and those that prefer 110 over both 100 and 111 (for which one has to necessarily include sixth rank tensor terms [8]); additionally, the sixth rank tensor terms can also be used to study…”