William J. Boettinger

Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method

Abstract A comprehensive model is developed for solving the heat and solute diffusion equations during solidification that avoids tracking the liquid—solid interface. The bulk liquid and solid phases are treated as regular solutions and an order parameter (the phase field) is introduced to describe the interfacial region between them. Two-dimensional computations are performed for ideal solutions and for dendritic growth into an isothermal and highly supersaturated liquid phase. The dependence upon various material and computational parameters, including the approach to conventional sharp interface theories, is investigated. Realistic growth patterns are obtained that include the development, coarsening, and coalescence of secondary and tertiary dendrite arms. Microsegregation patterns are examined and compared for different values of the solid diffusion coefficient.

Solidification Microstructures: Recent Developments, Future Directions

The status of solidification science is critically evaluated and future directions of research in this technologically important area are proposed. The most important advances in solidification science and technology of the last decade are discussed: interface dynamics, phase selection, microstructure selection, peritectic growth, convection effects, multicomponent alloys, and numerical techniques. It is shown how the advent of new mathematical techniques (especially phase-field and cellular automata models) coupled with powerful computers now allows the following: modeling of complicated interface morphologies, taking into account not only steady state but also non-steady state phenomena; considering real alloys consisting of many elements through on-line use of large thermodynamic data banks; and taking into account natural and forced convection effects. A series of open questions and future prospects are also given. It is hoped that the reader is encouraged to explore this important and highly interesting field and to add her/his contributions to an ever better understanding and modeling of microstructure development.

Development of a diffusion mobility database for Ni-base superalloys

Abstract For the fcc phase of the Ni–Al–Co–Cr–Hf–Mo–Re–Ta–Ti–W system, diffusion data in various constituent binary systems were assessed to establish a multicomponent diffusion mobility database. The diffusion assessment relied on an existing thermodynamic database for the calculation of needed thermodynamic factors. The mobilities determined for the self-diffusion of the components in the fcc phase (a metastable state for some components) were consistent with the correlation of the diffusivity with the melting point. The general agreement of calculated and measured diffusion coefficients in the Ni–Co–Cr–Mo and Ni–Al–Cr–Mo quaternary systems demonstrated the ability of the database to extrapolate to higher order systems. Finally, the mobility database, in conjunction with an available thermodynamic database and a finite-difference diffusion code, was used to simulate a multicomponent diffusion couple between two commercial Ni-base superalloys.

Convective and interfacial instabilities during unidirectional solidification of a binary alloy

Abstract The onset of coupled convective and constitutional interfacial instabilities during the directional solidification of a single phase binary alloy at constant velocity vertically upwards (positive z -direction) is treated by a linear stability analysis. We consider a system for which the temperature gradient alone would cause a negative density gradient and the solute gradient alone would cause a positive density gradient. The temperature and concentration fields are coupled through the hydrodynamic equations. The solidification boundary conditions at the solid-liquid interface couple the hydrodynamic and interfacial stability phenomena. Specific calculations were made for physical properties appropriate to the solidification of lead containing tin. Results indicate that the stability-instability criterion differs substantially from the criterion of a net neutral density gradient. For a temperature gradient in the liquid of 200 K/cm and for velocities in the range 1–40 μm/s, a convective-like long wavelength instability occurs at a critical concentration that increases with velocity; whereas for V > 40 μm/s, the concentration at which instability occurs decreases as velocity is increased and the values of concentration and wavelength at the onset of instability correspond to the predictions of previous morphological stability theory in which density changes and convection are neglected. Application of a vertical static magnetic field increases the critical concentration for convective instabilities but a field of a tesla (10 4 gauss) is needed to cause an order of magnitude change.

On the Sn-Bi-Ag ternary phase diagram

The selection and evaluation of Pb-free solders requires information that is best determined through a knowledge of ternary and higher order phase diagrams. As part of an ongoing program on Pb-free solder phase diagrams at the National Institute of Standards and Technology, a thermodynamic model is formulated for the Sn-Bi-Ag phase diagram. Thermodynamic functions for the various phases obtained by fitting measured data for the three constituent binary systems are extrapolated to the ternary system using the method of Muggianu. Modeling results are compared to preliminary experimental data for the ternary system and are applied in the calculation of the solidification path.

The Effect of Rapid Solidification Velocity on the Microstructure of Ag-Cu Alloys

Electron beam solidification passes have been performed on a series of Ag-Cu alloys between 1 wt pct Cu and the eutectic composition (28.1 wt pct Cu) at speeds between 1.5 and 400 cm per second. At low growth rates conventional dendritic or eutectic structures are obtained. The maximum growth rate of eutectic structure is 2.5 cm per second. At high growth rates microsegregation-free single phase structures are obtained for all compositions. The velocity required to produce this structure increases with composition for dilute alloys and agrees with the theory of absolute stability of a planar liquid-solid interface with equilibrium partitioning. For alloys between 15 and 28 wt pct Cu, the velocity required to produce the microsegregation-free extended solid solution decreases with composition and is related to nonequilibrium trapping of solute at the liquid solid interface. At intermediate growth rates for alloys with 9 wt pct Cu or greater, a structure consisting of alternating bands of cellular and cell-free material is obtained. The bands form approximately parallel to the local interface.

The formation of ordered ω-related phases in alloys of composition Ti4Al3Nb

During cooling of an alloy of composition Ti4Al3Nb from a B2 phase field above 1100°C, a metastable trigonal (P3m1) ω-related phase, designated ω″, forms along with small amounts of D019 and L10 phases. The ω″ phase exhibits partial collapse of 111 planes and reordering relative to its B2 parent. An apparently equilibrium low temperature phase with the B82 structure was found after 26 days of annealing at 700°C. Both ω″ and B82 structures were verified by means of transmission electron microscopy and by single crystal X-ray diffraction. The latter permitted detailed analysis of the collapse parameters and site occupancies. The observed transformation path, B2(Pm3m)→ω″(P3m1)→B82 (P63/mmc), occurs in two steps. The first involveds a subgroup transition during cooling that is primarily displacive with reordering consistent with the trigonal symmetry imposed by the ω-collapse. The second involves a supergroup transition during prolonged annealing that is primarily replacive and constitutes a chemical disordering. The direct equilibrium transformation, B2→B82, without the formation of an intermediate trigonal phase, can only occur by a reconstructive transformation.

Application of ternary phase diagrams to the development of MoSi2-based materials

Abstract A review of the literature reveals ternary phase diagram data for a number of systems involving MoSi2. Although incomplete, this literature provides the initial basis for a rational approach to alloy design. For example, one can assess the high temperature stability of various artificially introduced reinforcements, such as niobium or SiC in an MoSi2 matrix and the possibilities for the development of stable two-phase microstructures in quasibinary alloys in the MoSi2TiSi2 and MoSi2TaSi2 systems. Revised phase diagrams for these latter systems are presented that indicate the absence of the C11b-to-C40 high temperature polymorphic transformation in pure MoSi2.

The structure of directionally solidified two-phase Sn-Cd peritectic alloys

The structure of Sn-Cd two-phase peritectic alloys directionally solidified at various values ofG/υ (temperature gradient in the liquid divided by growth rate) is reported. The minimum value of G/υ as a function of composition required for the solidification of two-phase peritectic alloys with a planar liquid-solid interface is estimated using a simple constitutional supercooling stability criterion. At a value ofG/υ just below this minimum value, these alloys solidify with a nonplanar interface consisting of cells of α (the high temperature phase) and intercellularβ (the low temperature phase). This produces a coarse rod-like microstructure consisting of rods of α phase imbedded in aβ matrix. At a value ofG/υ above this minimum value, these alloys solidify with a planar interface which alternately deposits bands of α andβ transverse to the growth direction. No coupled growth of α andβ at a planar interface is observed in Sn-Cd two-phase peritectic alloys as was expected. To understand this, an analysis of coupled (eutectic-like) growth of two-phase peritectic alloys is presented and contrasted with the results of the Jackson-Hunt theory of lamellar eutectic growth. This calculation indicates that the coupled growth of two-phase peritectic alloys is unlikely on theoretical grounds.

DTA AND HEAT-FLUX DSC MEASUREMENTS OF ALLOY MELTING AND FREEZING

Publisher Summary This chapter focuses on differential thermal analysis (DTA) and heat-flux differential scanning calorimetry (HF-DSC) of metals and alloys. A thermal analysis guide focused only on metals and alloys is appropriate because metals and alloys behave quite differently from molecular materials such as polymers and organics. Freezing and melting occur rapidly in response to changes in temperature compared to other materials. Melting and freezing transformations, once initiated, take place within, at most, a degree of local thermodynamic equilibrium. Therefore, the chapter also focuses on melting and solidification behavior because special methods can be employed that are not necessarily useful for a broader class of materials and processes. The chapter intends to provide the thermal analysis user with the considerations that are necessary for proper sample preparation and to illustrate how the sample characteristics influence the proper interpretation and analysis of measurements. The chapter describes different types of information usually sought from DTA/HF-DSC during the melting and freezing of alloys. The details of instruments, operation and calibration are described. The goal is to describe the thermal lags between the sample and sample thermocouple that must be understood to enable good analysis of data. The chapter also details the response of the DTA to binary and ternary alloys, respectively.