 any
structural metals--steels, aluminum alloys, and superalloys--are products of solid-state
diffusional phase transformations. These transformations occur when the temperature
of an alloy is abruptly lowered and a thermodynamically stable single phase separates
into multiple phases at these lower temperatures. This separation, which occurs by
diffusion of matter among the phases, depends on the thermodynamics of the system,
the elastic fields generated by the transformation, and the surface energy of the
interfaces between phases. The end result of the transformation process is the formation
of a multiphase microstructure, which is a key variable in setting the mechanical
properties (stiffness, strength, toughness) of the alloy.
In many alloys (especially those used at high temperatures), there is an in situ
transformation process called coarsening in which a dispersion of very small precipitates
evolve to a system of a few very large precipitates. This coarsening process severely
degrades the properties of the alloy, in some cases leading to failure. While reduction
of surface energy is the overall driving force for coarsening, the process also depends
strongly on the elastic properties and crystal structures of the alloy phases. By
carefully choosing the alloy components, it may be possible to use the elastic properties
to slow or eliminate coarsening and improve material performance over time.
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| Figure 1: Evolution of 10 Ni3Al
precipitates in aNi matrix. |
Over the past several years, Professors Perry Leo of the Aerospace Engineering and
Mechanics Department and John Lowengrub of the School of Mathematics at the University
of Minnesota, together with collaborators Herng-Jeng Jou (Colorado School of Mines)
and Qing Nie (University of Chicago), have been developing mathematical models and
computational methods to predict microstructural features in alloys in two space
dimensions. Professor Lowengrub is a Fellow of the Supercomputing Institute, and
Professor Leo is an Associate Fellow. These investigators work involves understanding
and simulating the interactions among diffusion, elastic stresses, and surface energy
during microstructural evolution. Their research group is the first to develop methods
that incorporate elastic inhomogeneity and anisotropy in dynamic simulations.
Leo and Lowengrub's research group has focused on two types of techniques to study
microstructural evolution–boundary integral methods (BIM) in which the precipitate-matrix
boundaries are assumed to have zero thickness and diffuse interface methods (DIM)
in which the boundaries have finite but narrow thickness. In BIM, field equations
are mapped to sharp boundaries between phases, and boundary conditions are used to
formulate boundary integral equations. In DIM, evolution of the smooth fields is
given by a coupled set of partial differential equations.
The BIM is efficient because dimensionality of the governing equations is reduced.
Furthermore, Leo, Lowengrub, and Nie have developed highly efficient and accurate
algorithms using parallel computations. Efficiencies of 90% are regularly achieved.
The BIM results include studies of growth shapes and coarsening shapes in elastically
inhomogeneous systems using both isotropic and anisotropic elasticity. A calculation
of multiparticle evolution is shown in Figure 1. Parameters appropriate to a Ni-Al
system are chosen, with the matrix phase being essentially pure Ni and the precipitate
phase being Ni3-Al. Both phases have cubic anisotropy. The precipitates are
circular at t = 0. As they evolve, their shapes become squarish, reflecting
the underlying crystal, and they begin to align along the horizontal direction. In
addition, the particles attract each other, which is shown to be a characteristic
of the elastic constants used in the simulation. Such behavior may indicate that
the particles eventually merge.
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| Figure 2: Evolution of 12 isotropic
precipitates in an isotropic matrix. Shear modulus of precipitates is one half that
of the matrix. |
One drawback of BIM is that equations break down when topological changes such as
merging or vanishing of precipitates occur. To compute the system in Figure 1, precipitates
were simply removed when their area dropped to less than 0.1. On the other hand,
while the DIM is more expensive than the BIM, the DIM is useful because it naturally
handles particle merging and vanishing via a smooth transition region between phases.
This is illustrated in Figure 2 where the evolution of a system of twelve precipitates
in an isotropic matrix is shown. Here, a smooth coarsening evolution towards a plate-like
structure is observed.
Leo and Lowengrub's research group is continuing to refine and enhance their methods
to perform successively more realistic simulations of microstructure evolution. The
group intends to study the coarsening statistics of systems of many precipitates
in the future, developing three-dimensional models and including non-equilibrium
effects such as interface kinetics in systems where the precipitate and matrix have
very different crystal structures. By accurately simulating the formation of microstructure
in alloys, Leo and Lowengrub hope to eventually be able to provide metallurgists
with a prescription for generating alloys with desirable material properties. |
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