PhD Thesis Colloquium: Ms. Soumya Mishra 28/08/25)

Thesis title:

Core-Shell Precipitates in Ternary Alloys: A Phase Field Study

Faculty advisor(s):

Prof. Abinandanan T A

When?

28^th August, 2025 (Thursday), 11:00 AM (India Standard Time)

Where

KPA Auditorium, Department of Materials Engineering

Abstract:

Since the discovery of age hardening, a large number of alloys have been produced that exploit the presence of one or more precipitate phases in the microstructure for strength enhancement. In recent decades, alloys with two precipitate phases in a core-shell configuration (in such diverse alloys as aluminium alloys [1, 2, 3], steels [4, 5] and complex concentrated alloys [6, 7]) have attracted much attention. In such alloys, researchers have demonstrated the phenomenon of size focusing [2] and speculated about the potential for bimodal size distribution of precipitates [2]. These developments call for a theoretical understanding of microstructure formation and evolution in such systems. Further, while coarsening in systems with a single precipitate phase is well understood (both theoretically and computationally), such understanding is lacking for systems with multiple precipitate phases in general and for core-shell morphology in particular.

This thesis is the first systematic study of the formation of core-shell precipitates and their evolution well into the coarsening regime. This study employs phase-field simulations using a ternary version of the Cahn-Hilliard model in two-dimensional systems (2D); the parameters are chosen in such a way that we study only equiaxed precipitate shapes (with circular core precipitates embedded in a larger annular shell phase, surrounded by the matrix phase). A ternary alloy system has been chosen since it is the simplest system in which three phases (two precipitate phases and a matrix) can coexist; 2D simulations are chosen in order to get reliable statistics without losing the essence of the phenomenon under study.

This study focuses on the following three aspects of core-shell microstructures: (a) Size focusing during the early stages, for which we provide a new explanation involving the classical growth theory due to Zener; (b) Emergence of a bimodal size distribution through a novel mechanism, in which an initially unimodal distribution spontaneously transforms into a bimodal one; and (c) Validity of LSW-type (Lifshitz–Slyozov–Wagner) scaling laws during coarsening of precipitates with a bimodal distribution consisting of core-shell precipitates coexisting with isolated shell precipitates.

Size Focusing

In our computer simulations, core-shell morphologies are prepared using a two-step heat treatment, similar to experiments [2]. The first heat treatment begins with randomly placed nuclei of core precipitates that are allowed to grow and coarsen. In the second treatment, the shell phase spontaneously forms and grows around the core precipitate due to the wetting condition. During the early stages of the second treatment, size focusing is observed, in which an initially broad scaled size distribution (inherited from the first treatment) becomes narrower during the formation of the shell phase around the cores. Zener’s growth theory predicts that the growth rate of a smaller precipitate is higher than that of a larger precipitate, leading, naturally, to the narrowing of the size distribution. Thus, size focusing is primarily a growth phenomenon, and it ends when growth ends (coarsening begins). We examine this phenomenon using two parameters: duration of size focusing t_sf (i.e., how long it lasts) and the strength of size focusing s_sf (i.e., how much narrowing can be achieved).

Our simulations show that the duration of size focusing t_sf depends only on the inter-precipitate distance, and the strength of size focusing s_sf is higher in systems with (a) larger size dispersity of the cores, (b) larger inter-precipitate spacing, (c) higher shell volume fraction, and (d) lower core mobility.

Emergence of bimodal distribution

We demonstrate a novel mechanism by which, in systems with core-shell precipitates, an initially unimodal size distribution (inherited from size focusing) transforms into a bimodal one. When size focusing ends and coarsening sets in, the core may dissolve away, leaving behind smaller isolated shell precipitates; under suitable circumstances (explained below), this leads the scaled size distribution to become bimodal, with small isolated shell precipitates contributing to the first peak, and larger core-shell precipitates contributing to the second peak. Our results reveal that the emergence of bimodality is promoted in systems with a higher shell volume fraction and higher core mobility.

Scaling laws for bimodal size distribution

The bimodal nature of the distribution is preserved well into coarsening, where the LSW-type scaling laws hold: the r^3 and n^(−3/2) increase linearly with time and the scaled size distribution is time-invariant. Thus, our study expands the range of applicability of LSW-type scaling laws to three-phase microstructure (a) with a bimodal size distribution and (b) core-shell precipitates coexisting with isolated precipitates.

Janus and Isolated morphologies

In a ternary three-phase microstructure containing a matrix (major phase) and two different types of precipitates (minor phases), if sources of anisotropy are ignored, two other possibilities exist: isolated [8, 9] and Janus (two-faced) [10] morphologies. In this study, we have considered such morphologies in which the two precipitate phases are symmetric in terms of (a) interfacial energy, (b) mobility, and (c) precipitate volume fraction. We further restrict our study to systems with low precipitate volume fraction, for which theoretical predictions are possible.

In the case of isolated morphology, even if species diffusivity is different across different phases, in the limit of a small volume fraction of each precipitate phase, it is reasonable to expect that the coarsening of one phase will not be affected by the other. This expectation is confirmed in our simulation, in which the volume fraction of each precipitate phase is 0.1.

In the case of Janus morphology, if all the precipitates are in Janus configuration, the coarsening rate would be lower because of the lower curvature of the precipitate-matrix interface. In our simulations, however, not all the precipitates are in Janus configuration. Our study shows that LSW-type scaling laws hold even in such microstructures with Janus precipitates coexisting with isolated ones, and consistent with our prediction, the coarsening rate is lower.

The novel contributions from this thesis may be summarised as follows: (a) size-focusing of core-shell precipitates is established to be a growth phenomenon; (b) a bimodal size distribution could emerge from an initially unimodal one in systems with core-shell precipitates; and, (c) the range of applicability of LSW-type scaling laws is greatly expanded to systems with two precipitate phases in (i) core-shell, (ii) Janus, and (iii) isolated morphologies.

References

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