Jong-Kyu Yoon

A phase field model for isothermal solidification of multicomponent alloys

With the ability to model the kinetics and the pattern formation for solidification, a phase field model has been studied by many scientists. Currently available models, however, are restricted not only to binary alloys but also to those with substitutional solute elements. In this work, a new phase field model is developed to study solidification of a multicomponent alloy containing substitutional as well as interstitial solute elements. By employing the number of moles per unit volume as the concentration variable, the evolution equations of both the phase field and the concentration fields are derived from the free energy functional in the thermodynamically consistent way. In the model, the interfacial region is assumed to be a mixture of solid and liquid with the same composition, but with different chemical potentials. Based on this assumption, the phase field parameters are matched to the alloy properties and an interface thickness limitation is also deduced. Using the chemical rate theory, the phase field mobility is determined at a thin-interface limit condition under the assumption of negligible diffusivity in the solid phase. Another advantage of the model is that any thermodynamic database available in the literature can be directly ported to the model such that quantitative results for solidification of the real alloy systems could be made. As an example, a dendritic growth in an Fe-Mn-C ternary alloy is examined with the thermodynamic data from the commercial software Thermo-Calc code.

A phase field model for the solute drag on moving grain boundaries

Abstract We propose a model based on a phase-field approach to study the effect of the solute drag on moving grain boundaries in a binary alloy system. By considering the grain boundary as a distinguishable phase and adopting a “segregation potential” in the grain boundary region, the effect of solute drag is automatically incorporated into the model. It is shown at equilibrium that the model can reproduce the equilibrium solute segregation and Gibbs adsorption. It is also demonstrated at a one-dimensional steady state that the model includes both the solute drag proposed by Cahn and the free energy dissipation by Hillert and Sundman. In the dilute solution limit, the simple expressions for the concentration distribution around the interfacial region and the solute drag are obtained as functions of boundary velocity, diffusivity and segregation potential and they are found to be consistent with the previous theories for solute drag phenomenon. In two-dimensional quasi steady state, the phase field model reduces to the relationship between normal velocity and the curvature of the boundary and the relationship between phase field mobility and the grain boundary mobility is obtained.

The effects of pulling rates on the shape of crystal/melt interface in Si single crystal growth by the Czochralski method

Abstract The temperature profile and the shape of the crystal/melt interface in a Czochralski (CZ) furnace for large Si single crystal growth was simulated taking into consideration the fluid flow and surface radiational heat transfer. The view factors of the surface elements were calculated for radiation heat transfer. Two phases (crystal and melt) were treated as a continuous phase by assigning artificially large viscosity to the solid phase and the latent heat was accounted for by iterative heat evolution method, This simulation model of a CZ system was applied to the growth process of a 6 inch Si single crystal to study the effects of pulling rates on the interface shape. It was found that the interface becomes more concave to the melt as the crystal grows or as the pulling speeds become higher, and that the interface height is linearly dependent upon the pulling rate.

A phase field model for phase transformation in an elastically stressed binary alloy

We introduce a phase field model which permits the study of the morphological evolution of second phase particles coherently precipitated from the mother phase in elastically stressed alloys. The model has three field variables: the concentration of solute atoms, the displacement field, and the phase field. These fields are coupled via interaction terms in the free energy functional and through kinetic factors. In the model, the interfacial region is defined as a mixture of the mother and second phases, which have the same chemical potential but different compositions. A matched asymptotic expansion analysis of the model is carried out. In the sharp interface limit, our model produces the modified Gibbs–Thomson equation at the coherent interface, which includes not only the effect of interface curvature but also that of coherent elastic strain energy. Two-dimensional computations are performed for the microstructural evolution, and precipitate growth in the elastically anisotropic media. Our results are consistent with previous experimental and theoretical studies.

A numerical analysis of fluid flow, heat transfer and solidification in the bending-type square billet continuous casting process

A numerical modeling system was developed which can simulate the transport phenomena of a bending type square billet continuous casting process. Fluid flow and heat transfer were analyzed with a 3-dimensional finite volume method (FVM) with the aid of an effective heat capacity algorithm for the solidification. For a complex geometry of the bending type billet caster, a body-fitted-coordinate (BFC) system was employed. The bent structure of the caster allows a recirculating flow to develop in the upper and outer-radius region and the main stream to shift toward inner radius. This causes the thinner solid shell in the inner radius region than in the outer one. Besides standard operation conditions, we have analyzed the results when casting speed, caster shape, and tundish superheat changes. Lower casting speed makes the solid shell thicker by reducing heat flux from the mold. In the vertical caster, solid shell thickness are more uniform than that in the bending-type in entire region. When superheat increases by 5°C, solid shell thickness at the mold exit becomes thinner by 1 mm.

The photorefractive effects of Fe and Fe + Ce doped LiTaO3 single crystal

Abstract LiTaO 3 single crystals doped with Fe and Ce transition metals were grown by the Czochralski method. The variations of birefringence change with light irradiation time were measured with the variation of light intensity, the concentration of transition metals, heat treatment and poling condition. The absorption coefficients were also examined in the UV–VIS wavelength range and compared with the results of the birefringence change. The saturated value of the birefringence change in Fe:LiTaO 3 had a proportional relationship with light intensity and the concentration of doped transition metal, and in poled Fe:LiTaO 3 , the variation degree of Δ( n e − n o ) can be estimated by the magnitude of the absorption band which represents the transition state of the doped metal.

Application of an axial magnetic field to vertical gradient freeze GaAs single crystal growth

Abstract We have developed a growth technique for vertical gradient freeze GaAs single crystals, in which an axial magnetic field can be applied up to 1500 G. From calculations, the application of an axial magnetic field of 1000 G was found to be enough for suppressing the Marangoni flow and promote the stability of the thermal environment throughout the whole melt during the crystal growth, enhancing the production yield. Through the growth and characterization of the AM-VGF (axial magnetic field applied-vertical gradient freeze) GaAs single crystal lightly doped with indium of ∼ 10 18 cm −3 , the effects of the application of an axial magnetic field to improve crystal quality were confirmed. Low defect, nearly striation-free single crystals can be obtained with this technique. The maximum deviations of the electrical properties for the growth direction and the optical characteristics for the radial direction in the crystal are 15% and 5%, respectively.

Capillarity and electromigration effects on asperity contact evolution in microelectromechanical systems switches

Morphology evolution of asperity contacts in microelectromechanical systems switches is examined in the framework of the phase field method. Surface mass diffusion, driven by capillarity, rapidly increases the asperity contact radius at early times, decreases it slowly at later times, and, possibly, pinches off the contact due to a Rayleigh instability. Electromigration accelerates this process but retards or prevents the pinch off at late times. The evolution occurs faster when the asperity size and/or density are smaller. Approximate analytical results for the evolution kinetics are provided.

Numerical prediction of operational parameters in Czochralski growth of large-scale Si

Abstract Numerical calculation was performed on the fluid flow and mass transfer in large-scale Czochralski growth of silicon. The κ – ϵ two equation model was used in order to simulate the turbulent characteristics of the fluid motions. The calculated axial and radial profiles of oxygen concentration showed good agreements with experiments in the case of 6 inch silicon growth. The average concentrations and the radial uniformities of oxygen for various crucible and crystal rotation rates for the case of 10 inch Si growth were calculated at the solidification interface. On the basis of the above results, the optimum starting conditions of the operation for the 10 inch size silicon grown by the Czochralski method were numerically calculated.

A phase field model for electromigration-induced surface evolution

In this study, a phase field model is presented to study the effects of electromigration on the surface evolution of single crystal Al metal interconnects. Two dimensional computer simulations are performed for the surface evolution of metal interconnects due to electromigration in various conditions, such as anisotropy in diffusivity, different initial void sizes, and different crystallographic directions compared to the direction of the ambient electric field. From the results of computer simulations, it may be seen that the types of anisotropy and the relative direction of the diffusivity are the decisive factors in motion and shape change. As the symmetry of anisotropy in diffusivity decreases, the void evolves into a more unstable shape. Moreover, the voids of the system with two-fold diffusion symmetry are most likely to evolve into slits when the crystallographic direction is toward a specific orientation compared to the direction of the ambient electric field.