Abhik Choudhury

Theoretical and numerical study of lamellar eutectic three-phase growth in ternary alloys

We investigate lamellar three-phase patterns that form during the directional solidification of ternary eutectic alloys in thin samples. A distinctive feature of this system is that many different geometric arrangements of the three phases are possible, contrary to the widely studied two-phase patterns in binary eutectics. Here, we first analyze the case of stable lamellar coupled growth of a symmetric model ternary eutectic alloy, using a Jackson-Hunt-type calculation in thin film geometry, for arbitrary configurations, and derive expressions for the front undercooling as a function of velocity and spacing. Next, we carry out phase-field simulations to test our analytic predictions and to study the instabilities of the simplest periodic lamellar arrays. For large spacings, we observe different oscillatory modes that are similar to those found previously for binary eutectics and that can be classified using the symmetry elements of the steady-state pattern. For small spacings, we observe a new instability that leads to a change in the sequence of the phases. Its onset can be well predicted by our analytic calculations. Finally, some preliminary phase-field simulations of three-dimensional growth structures are also presented.

Theoretical and numerical study of lamellar eutectoid growth influenced by volume diffusion.

We investigate the lamellar growth of pearlite at the expense of austenite during the eutectoid transformation in steel. To begin with, we extend the Jackson–Hunt-type calculation (previously used to analyze eutectic transformation) to eutectoid transformation by accounting for diffusion in all the phases. Our principal finding is that the growth rates in the presence of diffusion in all the phases are different compared to the case when diffusion in growing phases is absent. The difference in the dynamics is described by a factor ’ρ’ which comprises the ratio of the diffusivities of the bulk and the growing phases, along with the ratios of the slopes of the phase coexistence lines. Thereafter, we perform phase-field simulations, the results of which are in agreement with analytical predictions. The phase-field simulations also reveal that diffusion in austenite as well as ferrite leads to the formation of tapered cementite along with an overall increase in the transformation kinetics as compared to diffusion in austenite (only). Finally, it is worth noting that the aim of present work is not to consider the pearlitic transformation in totality; rather it is to isolate and thereby investigate the influence of diffusivity in the growing phases on the front velocity.

New Metallographic Method for Estimation of Ordering and Lattice Parameter in Ternary Eutectic Systems

Ternary eutectics develop a rich variety of microstructures depending on the solidification conditions. This article describes a new procedure to analyze the three-phase arrangement in metallographic sections following solidification; this method provides a clear definition of both the interphase spacing and the degree of ordering. Distance and angle between the phase areas are calculated after determination of the centers of mass for each phase area. A polar plot of these data allows an easy determination of the spacing and the order. The new method is discussed with both experimental as well as simulated microstructure images. It is first tested on artificial phase arrangements, which are fully ordered, semi-ordered, or random.

Pattern-Formation During Self-organization in Three-Phase Eutectic Solidification

Eutectic growth is an interesting example for exploring the topic of pattern-formation in multi-phase systems, where the growth of the phases is coupled with the diffusive transport of one or more components in the melt. While in the case of binary alloys, the number of possibilities are limited (lamellae, rods, labyrinth etc.), their number rapidly increases with the number of components and phases. In this paper, we will investigate pattern formation during three-phase eutectic solidification using a state-of-the art phase-field method based on the grand-canonical density formulation. The major aim of the study is to highlight the role of two properties, which are the volume fraction of the solid phases and the solid–liquid interfacial energies, in the self-organization of the solid phases during directional growth. Thereafter, we will show representative phase-field simulations of a micro-structure in a real alloy (Ag–Al–Cu) using an asymmetric phase diagram as well as interfacial properties.

Composition pathway in Fe–Cu–Ni alloy during coarsening

In this work the microstructure evolution for a two phase Fe–Cu–Ni ternary alloy is studied in order to understand the kinetic composition paths during coarsening of precipitates. We have employed a quantitative phase-field model utilizing the CALPHAD database to simulate the temporal evolution of a multi-particle system in a two-dimensional domain. The paths for the far-field matrix and for precipitate average compositions obtained from simulation are found to be rectilinear. The trends are compared with the corresponding sharp interface theory, in the context of an additional degree of freedom for determining the interface compositions due to the Gibbs–Thomson effect in a ternary alloy.

Microstructures in a ternary eutectic alloy: devising metrics based on neighbourhood relationships

Ternary eutectics, where three phases form simultaneously from the melt, present an opportunity to study the fundamental science of microstructural pattern formation during the process of solidification. In this paper we investigate these phenomena, both experimentally and by phase-field simulations. The aim is to develop necessary characterisation tools which can be applied to both experimentally determined and simulated microstructures for a quantitative comparison between simulations and experiments. In SEM images of experimental cross sections of directionally solidified Ag-Al-Cu ternary eutectic alloy at least six different types of microstructures are observed. Corresponding 3D phase-field simulations for different solidification conditions and compositions allow us to span and isolate the material parameters which influence the formation of three-phase patterns. Both experimental and simulated microstructures were analysed regarding interface lengths, triple points and number of neighbours. As a result of this integrated experimental and computational effort we conclude that neighbourhood relationships as described herein, turn out to be an appropriate basis to characterise order in patterns.

Spinodal decomposition and droplets entrapment in monotectic solidification

In this article, we present two models to simulate solidification morphologies in monotectic alloys. With the first model, we investigate the morphological evolution under the influence of spinodal decomposition. The model requires that a gradient energy contribution for the concentration field should be incorporated, in order to stabilize phase separation when the liquid concentration is inside the region of miscibility gap. The free energy of the system in this model is derived from direct interpolation of the bulk energy densities. This, however, results in simulation regions in nanometer scale due to contributions from the chemical free energy of the system to the total surface excess. With the second model, our purpose is to develop a phase-field model to simulate scales that are larger than nanometer, where the departures from equilibrium are very small resulting in phase concentrations outside the spinodal region. In view of this, we exclude the concentration gradient contribution to the grand chemical potential functional, and develop a model based on [M. Plapp, Phys. Rev. E 84, 031601 (2011); A. Choudhury and B. Nestler, Phys. Rev. E 85, 021602 (2012)]. The advantage is that the free energy excess across the interface at equilibrium disappears, and hence it is easier to derive the required surface energies with higher interface widths. Due to this benefit, we employ the method to simulate the dynamic entrapment process in the monotectic reaction and study the influence of liquid(1) - liquid(2) surface energy and undercooling on the entrapment process.

Effect of Surface Energy Anisotropy on the Stability of Growth Fronts in Multiphase Alloys

Eutectic growth offers a variety of examples for pattern formation which are interesting both for theoreticians as well as experimentalists. One such example of patterns is ternary eutectic colonies which arise as a result of instabilities during growth of two solid phases. Here, in addition to the two major components being exchanged between the solid phases during eutectic growth, there is an impurity component which is rejected by both solid phases. During progress of solidification, there develops a boundary layer of the third impurity component ahead of the solidification front of the two solid phases. Similar to Mullins–Sekerka type instabilities, such a boundary layer tends to make the global solidification envelope unstable to morphological perturbations giving rise to two-phase cells. This phenomenon has been studied numerically in two dimensions for the conditions of directional solidification, by Plapp and Karma (Phys Rev E 66:061608, 2002) using phase-field simulations. While, in the work by Plapp and Karma (Phys Rev E 66:061608, 2002) all interfaces are isotropic, in our presentation, we extend the phase-field model by considering interfacial anisotropy in the solid–solid and solid–liquid interfaces and characterize the role of interfacial anisotropy on the stability of the growth front through phase-field simulations in two dimensions.

Modelling of transient heat conduction with diffuse interface methods

We present a survey on different numerical interpolation schemes used for two-phase transient heat conduction problems in the context of interface capturing phase-field methods. Examples are general transport problems in the context of diffuse interface methods with a non-equal heat conductivity in normal and tangential directions to the interface. We extend the tonsorial approach recently published by Nicoli M et al (2011 Phys. Rev. E 84 1-6) to the general three-dimensional (3D) transient evolution equations. Validations for one-dimensional, two-dimensional and 3D transient test cases are provided, and the results are in good agreement with analytical and numerical reference solutions.

Influence of solid-solid interface anisotropy on three-phase eutectic growth during directional solidification

We utilize a quantitative phase-field model to simulate three-phase eutectic growth and the oscillatory instabilities at large spacings. We analyze the effect of symmetry in the selection of the oscillatory mode for the case of directional solidification. Going further from our previous article (Choudhury A. et al., Phys. Rev. E, 83 (2011) 051608), we manipulate the symmetry elements in a given configuration through the use of solid-solid anisotropy. Characteristic modes are compared between the case of isotropic surface energies and those retrieved in the presence of solid-solid anisotropy. We end with certain general arguments with regards to the growth rate of oscillatory modes and symmetry elements of the obtained microstructure depending on the symmetry elements of the starting configuration.