The transformation sequences in the cubic → tetragonal decomposition
Introduction
High-temperature homogeneous multicomponent solid solutions are usually supersaturated at lower temperatures and undergo phase transformations involving either dissolution (decomposition) with the separation of concentration of phases, or crystal lattice rearrangement changing the crystal lattice symmetry (displacive transformation or atomic ordering), or both. The equilibration process in such systems is characterized by several relaxing internal thermodynamic parameters. These parameters can be composition, transformation strain and atomic long-range order (lro) parameters. Their coordinate dependence determines the system microstructure, namely, the spatial pattern formed by structural and compositional domains. The phase transformations in multicomponent and multi-parametric systems do not necessarily develop in accordance with the conventional scenario of the nucleation and growth of the equilibrium tetragonal phase within the parent phase matrix. As a rule, the internal parameters have significantly different typical relaxation rates and this provides opportunities to manipulate the transformation pathway by changing the heat treatment schedule. The cooling rate, annealing temperature and an appropriate choice of composition may channel the transformation pathway through a sequence of transient phases that do not appear at the equilibrium phase diagram but can be plotted on its quasi-equilibrium extension. Given the fact that these metastable and transient phases may have advanced physical properties, the study of multi-parametric phase transformation is an important area of research. Analysis of a metastable extension of the phase diagram and the corresponding non-equilibrium free energies of the transient phases already provides some clues for the prediction of a sequence of the structural transformation along its path to equilibrium.
The possibility that precipitation of the equilibrium phase does not necessarily occur straightforwardly, as is usually assumed, was first proposed by Khachaturyan et al. in a theory of decomposition in Al–Li alloys [1]. The thermodynamic theory of cascade transformations for a wider class of systems was developed by Soffa and Laughlin [2]. The energy consideration in these analyses may indicate the direction of the transformation pathway, but does not predict the important kinetic details of the transformation path [3], [4], [5], which depends on the topology of the phase diagram, the concentration dependence of free energies of thermodynamically competing phases and the position of the representative point of the equilibrium system with respect to non-equilibrium free energies. In these analyses, the important effect, the contribution of elastic energy caused by crystal lattice misfit between phases and their structural domains, is neglected, although strain energy is the major factor in the majority of the cases, determining microstructure and transformation path.
In this paper, we focus on a generic case of the decomposition of a supersaturated cubic binary alloy into a mixture of cubic and tetragonal phases. The results obtained may also be applicable to some cases of precipitation of the face-centered cubic (fcc) from the body-centered cubic (bcc) phase and vice versa, because these transformations are also described by tetragonal crystal lattice rearrangement (Bain strain). By definition, the system under consideration is multi-parametric – its internal thermodynamic parameters are the composition and the components of the transformation strain providing the cubic → tetragonal martensitic transformation (MT). Whereas composition change is caused by the diffusional separation of atoms and takes a comparatively long time, crystal lattice rearrangement is caused by the transformation strain and occurs practically instantaneously. Therefore, this kind of system is a primary candidate for showing cascade transformations with unusual transient structures. A study of conditions resulting in such cascade transformations and the microstructures developed along the transformation pathways is a subject of this paper.
It will be shown that the sequence of structural transformations can be vary greatly, depending on the temperature and composition of the alloy – there are three temperature–composition ranges where these sequences are radically different. We especially emphasize the case of cascade transformations that develop through isostructural spinodal decomposition, producing a metastable coherent mixture of two cubic phases, which serves as a template for the following MT developing within the solute-rich cubic phase. The possible transient microstructures formed along the transformation path to equilibrium are investigated. In particular, we discuss the mechanism of formation of the checkerboard (CB)-type structures observed in several systems [6], [7], [8], [9], [10], [11].
We use two- (2D) and three-dimensional (3D) phase field microelasticity (PFM) modeling. This choice is dictated by two factors. First, PFM, as with any phase field method, has predictive power since it does not use ad hoc assumptions about structural transformation pathways. Secondly, PFM method explicitly takes into consideration the transformation-induced elastic strain, which is the major factor responsible for formation of the domain structure. 3D modeling is employed to verify the validity of the results obtained via 2D modeling.
Section snippets
Theoretical model
The PFM approach is very flexible. The PFM kinetic equations, which explicitly include the strain energy contribution, can be straightforwardly extended to take into consideration the magnetostatic and/or electrostatic energy as well [12], [13], [14]. PFM modeling has been successfully used for the study of the MT [15], [16], [17], [18], [19], [20], [21], [22] and decomposition involving precipitation and atomic ordering that produces a tetragonal atomic arrangement [23], [24], [25], [26], [27]
Microstructural evolution in the cascade transformation
As discussed in Section 2.1, the transformation pathway may vary significantly depending on the temperature and composition range of isothermal aging. Different decomposition scenarios are denoted as cases A, B and C in Fig. 1. The information obtained by thermodynamic interpretation of the “chemical” free energies plotted in Fig. 1 gives only approximate classification of the expected transformation sequences. It neglects the contribution of the strain and interfacial energies and thus is not
Discussion and conclusion
We have investigated phase transformation sequences and microstructure evolution during coherent decomposition of a cubic solid solution into a mixture of equilibrium cubic and tetragonal phases in systems with typical generic thermodynamic properties. Our computational modeling has demonstrated that the textbook vision of the decomposition of a supersaturated solid solution into a mixture of phases of different symmetry as a one-stage process is only one of several decomposition mechanisms. It
Acknowledgements
The financial support by Office of Naval Research through MURI N0000114-06-1-0204 and NSF grant DMR-0242619 (YN and AK) are gratefully acknowledged. YJ is supported by the startup fund of Texas A&M University. This research was supported in part by the San Diego Supercomputer Center.
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