Variational Theory of Crystal Growth in Multicomponent Alloys

Abstract

The provisions for a new variational theory of crystal growth in multicomponent metal melts were formulated. The developed theory is the generalization of the previously conducted studies of crystal growth under conditions of deviation from local equilibrium at the phase boundary. The description of the methods of non-equilibrium thermodynamics of interrelated physico-chemical processes occurring in the initial phase, on the interface of phases and inside the growing crystal, was compared with the variational description of the crystal growth as a macrobody. The developed approach made it possible to find the general expression for the crystal growth rate, considering the influence of thermal and diffusion processes, as well as taking into account the influence of nonstationary effects associated with deviation from the local equilibrium on the surface of the growing nucleus. The justification of the new method showed that when the condition of the local equilibrium on the surface of the growing crystal is satisfied, the resulting equations take the form of expressions that can be obtained by constructing the equation of a mass and internal energy balance for the system under consideration. As an example, the problem of crystal growth from a melt of eutectic composition was considered. The equation of the growth rate of the two-component nucleus of the stoichiometric composition was obtained, taking into account the influence of the local non-equilibrium effects on growth. The expressions obtained were compared with the known equations of the solute trapping theory.

Rapid Multicomponent Alloy Solidification with Allowance for the Local Nonequilibrium and Cross-Diffusion Effects

“…The additive manufacturing is a new and advanced technology where the final 3D products are produced by the layer-upon-layer joining of the material. The high technological potential of the 3D printing and additive manufacturing has attracted much attention to rapid multicomponent alloy solidification and raised challenging questions from a fundamental point of view, which have been discussed by many authors [ 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 ].…”

“…Additive manufacturing of metallic alloys is a rapid solidification process when the solid–liquid interfaces may move with velocities as large as 1–30 m/s (see, for example [ 12 , 18 , 19 ], and references therein), which can be reached during re-solidification after ultrashort pulse laser melting [ 20 , 21 , 22 , 23 , 24 ]. At such velocities, solute diffusion in the liquid phase occurs under far from local equilibrium conditions and classical diffusion equation of parabolic type is not adequate for describing space–time evolution of solute concentrations [ 14 , 17 , 18 , 19 ]. To consider the deviation from local equilibrium of solute diffusion the liquid phase under rapid solidification conditions, the local nonequilibrium diffusion model (LNDM) [ 25 , 26 , 27 ] has been introduced by using diffusion equation of hyperbolic type, which is not based on the local equilibrium assumption.…”

“…Another important feature of multicomponent alloy solidification is the diffusional interaction between the species or cross-diffusion effects when diffusion fluxes of each component depend on the concentration gradient of all the components [ 14 , 17 , 38 ]. Off-diagonal diffusion terms of the diffusion matrix are usually neglected for multicomponent liquids, yet there is little justification, because in some situations the diffusional interaction between the species in multicomponent alloys plays an important role but the effect of the off-diagonal diffusion terms on diffusional transformation kinetics in multicomponent systems is still an open question.…”