The problem or microstructural evolution is the following:
Given an initial composition profile, how does the composition profile evolve with time?
Our interest is in solving this problem in the presence of elastic stresses (due to transformational strains). We assume that the system reaches the elastic equilibrium instantaneously for any given composition profile. This assumption helps us split the problem problem of morphological evolution into solving the following two problems iteratively: the problem of elastic equilibrium, and the (free boundary) diffusion problem. In this page, I describe our efforts in using a diffuse interface model to solve for the morphological evolution in elastically stressed solids.
Let us consider a solid which has been quenched from a homogeneous single phase region into a two phase region. The homogeneous phase will now transform into a two phase mixture. During such a (solid state phase) transformation, elastic stresses arise due to compositional heterogeneities, interfacial stresses, and the differences in crystal structures and symmetry between the parent and the product phases. In addition, there can also be externally applied stresses.
The elastic stresses manifest themselves in several ways during the microstructural evolution. For example, the precipitates change their shape with increasing size and align themselves along specific crystallographic directions. Further, there are reports that elastic stresses also lead to dendritic morphologies and/or splitting of precipitates (which are yet to be completely investigated).
However, the most important (and, useful) question regarding microstructural evolution in an elastically stressed system is the following:
What are the factors (especially from the point of view of elastic stresses) that stabilise a given microstructural feature?
The parameters which are of significance from the elasticity point of view are the characteristic length (L) and misfit (which decide the length and energy scales at which the elastic stress effects become important), and the elastic anisotropy and inhomogeneity.
Characteristic length scale (L)
The ratio of the precipitate-matrix interfacial energy to the elastic energy stored in the precipitate decides the characteristic length scale at which the elastic stress effects dominate the microstructural features.
The elastic energy scales as the volume while the interfacial energy scales as the surface area (of the precipitate). So, the equilibrium shape of the precipitate will be dominated by the interfacial energy at small sizes and by the elastic energy at larger sizes.
If the shear modulus of the precipitate is 100 GPa, the interfacial energy is 50mJ/m^2, and the misfit (see below the definition) is 4x10^3, then the characteristic length scale is of the order of 0.1 micro metre.
Misfit
The ratio of the difference in lattice parameters of the precipitate and matrix to the lattice parameter of the matrix is known as the misfit.
The misfit determines the elastic energy of the precipitate. Higher the misfit, higher would be the elastic energy stored in the precipitate.
In a Ni-rich matrix, for Ni-Al precipitates (in Ni-base superalloys), the misfit values range from 0.004 to 0.006. In a Ni-rich matrix, for Ni-Si precipitates (in Ni-base superalloys), the misfit value is of the order of -0.003.
Elastic inhomogeneity
The ratio of the elastic constants of the precipitate to that of the matrix is known as the elastic inhomogeneity. Typically, the ratio of the shear moduli is used to indicate the inhomogeneity.
The inhomogenity indicates the comparative hardness of the precipitate and matrix.
For Ni-rich matrix and Ni3Al precipitates, the Ni3Al precipitates are softer and so the inhomogenity is less than unity.
Elastic anisotropy
Let us consider a system with cubic anisotropy in elastic constants. Then the ratio of twice C44 to the difference between C11 and C12 is known as the anisotropy parameter.
If the anisotropy parameter is unity, the system is elastically isotropic. If it is less than unity, the <111> directions are elastically softer while, if it greater than unity, the <100> directions are elastically softer.
In the case of Nickel, the anisotropy parameter is of the order of 4.
Elastic stress effects on morphological evolution can be studied using either atomistic models, or continuum models. The continuum models can be classified as sharp interface models, and diffuse interface models.
The diffuse interface models have been used (typically with a Fourier spectral implementation) in the past decade for studying the microstructural evolution in elastically stressed alloys. My efforst has been on doing a parametric study by varying the misfit, inhomogeneity and elastic anisotropy and study the elastic stress effects on morphological evolution using diffuse interface models.
There are several recent reviews which discuss elastic stress effects on morphological evolution. Here are some, which I found to be very useful:
(1) Peter Fratzl, Oliver Penrose, and Joel L. Lebowitz
Modeling of Phase Separation in Alloys with Coherent Elastic Misfit
Journal of Statistical Physics, Vol. 95, Nos. 5/6, 1999.
An article, dedicated to Prof. John W. Cahn, on the occasion of his 70th birthday, which reviews the theoretical approaches that are used to model the elastic stress effects on microstructural evolution. The reference list (with 237 references) is pretty exhaustive and useful.
(2) William C. Johnson
Influence of Elastic Stress on Phase transformations
Lectures on the theory of phase transformations, 2nd edition, Edited by Hubert I. Aaronson, Published by TMS, Warrendale, Pennsylvania, 1999. p. 35.
An extremely readable treatment of the thermodynamics and elasticity theory underlying the models used for studying the elastic stress effects
(3) Minoru Doi
Elasticity Effects on the Microstructure of Alloys Containing Coherent Precipitates
Progress in Materials Science, Vol. 40, pp. 79-180, 1996.
A nice place to look for experimental results.
(4) William C Johnson and Peter W Voorhees
The Thermodynamics of Elastically Stressed Crystals
Solid State Physics, Vol. 59, pp.1-201, 2004. Edited byHenry Ehrenreich and Frans Spaepen.
All that you would ever need to know about the thermodynamics of elastically stressed solids and were afraid to ask!