A new mechanism for freckle initiation based on microstructural level simulation

Abstract

Freckle formation was directly simulated in three dimensions with a microstructure level model for unidirectional solidified Pb–Sn alloys. The microscale solidification model resolves the complex interaction of solute partition, interdendritic thermosolutal convection, dendrite formation and remelting. These simulations show that it is competition between all off these phenomena at the microstructural level that determines both the initiation and survival of solute channels and, hence, the propensity for freckle formation. The solute enriched interdendritic flow remelts both primary and secondary arms, and can deflect the primaries. This enables solutal channels to become self-sustaining, forming freckles. The microstructural model was then applied to predict the critical Rayleigh number, with excellent agreement with a large range of experimental data, demonstrating its potential for applications to new alloy systems and solidification processes.

Introduction

Freckles, channel-like macrosegregation defects, are commonly observed in directionally solidified or single crystal castings of nickel-based superalloys [1], [2], [3], [4]. Understanding and predicting the mechanisms of freckle formation is desirable for designing Ni-based superalloy components, their processing, and new compositions [5]. It is generally agreed that freckles arise due to a complex interaction of solute segregation, thermal variation and dendrite morphology, all of which contribute to the onset of thermosolutal convection in the mushy zone [6], [7], [8], [9], [10], [11], [12], [13].

The Rayleigh number, a ratio proportional to the buoyancy driving force over the viscous resistance force, has been recommended as a criterion for freckle initiation [14], [15], [16], [17]:

$${\text{Ra}}_h=\frac{\mathit{gKh}}{\alpha \nu }\left(\frac{\mathrm{\Delta }\rho }{{\rho }_0}\right)$$

where h is the characteristic length scale, g is the acceleration due to gravity, K is the mean permeability of the mushy zone, α is the thermal diffusivity, ν is the kinematic viscosity, and Δρ/ρ₀ is the density inversion due to thermal and/or compositional variation.

The Rayleigh number provides an efficient method to evaluate the stability of freckle formation and it has been extensively examined for both complex Ni-based superalloys and Pb–Sn alloys [16]. Determination of the critical value indicating freckle existence, however, requires significant experimentation. Further, evaluating the Rayleigh number from experimental results is also difficult since: (1) density varies with local temperature and chemical composition; (2) permeability depends on the stochastic dendritic microstructure; (3) the thermophysical properties may not be accurately known.

A number of computational models have been previously developed that simulate the heat and mass transport phenomena during alloy solidification using the full set of conservation equations to study freckles [7], [11], [18], [19], [20]. These prior models treat the mushy zone as a porous medium with mean isotropic or anisotropic permeability. The natural convection problem becomes that of analysing the motion of a fluid overlying a porous medium, termed mixture theory, as described by Bennon and Incropera [21]. These prior models have successfully predicted channel-like macrosegregation in both two and three dimensions that qualitatively match experimental measurements [8], [22], [23]. In comparison with mathematical criteria like the Rayleigh number, such models directly account for a range of solidification conditions and alloy compositions. However, the numerical noise (either inherent or applied) can affect the initiation of convective instabilities in the macrosegregation models, providing imposed uncertainty in the quantitative results.

Physically, solute segregation at the microstructural level plays a key role in driving interdendritic convection, while the dendritic morphology limits the flow. Therefore, a microscale solidification model that predicts both microsegregation and a dendritic morphology can provide a better understanding of their interaction so as to directly predict freckle initiation, perhaps giving further qualitative and quantitative insights. Very few microscale solidification models on columnar dendritic growth under thermosolutal convection have been published, and the ones that have are two-dimensional (2-D) [24], [25], [26]. The nature of interdendritic convection, as demonstrated for constrained dendritic growth under natural convection [27], requires simulations to be carried out in three dimensions. In this paper a three-dimensional (3-D) microscale numerical model that predicts microsegregation, dendrite morphology and interdendritic convection is presented and applied to investigate the mechanisms of freckle initiation at the dendritic level.