Phase-field modeling of eutectic Ti–Fe alloy solidification

Abstract

The eutectic microstructure formed during the solidification of a binary titanium–iron alloy (Ti–29.5 at.% Fe) has been simulated using the phase-field method. The model uses the chemical free energy contributions of phases with different thermodynamic factors. It is demonstrated that the simulated microstructure exhibits phenomena which are also observed during eutectic solidification in experiments. The obtained microstructure consist of a combination of circular and lamellar phase structures. The growth behavior and growth morphology of the phases are found to be a strong function of the front undercooling and the temperature gradient in the system. The simulated dependency of the steady state lamellar width versus the front undercooling corresponds to the theoretical prediction following from the “Jackson–Hunt” model. It is also shown that the relation between the mobilities of the phases strongly affects the eutectic microstructure.

Introduction

Titanium-based alloys are suitable for commercial applications due to their high strength (1000 MPa) and good corrosion resistance. Under stress, a plastic elongation to failure of 10–15% can be achieved [1]. Commonly the mechanical properties of a material are directly related to its internal microstructure. In recent years, lamellar nano-/ultrafine eutectic microstructures in Ti–Fe and Ti–Fe–Sn alloys have been observed [2]. Nano-/ultrafine structures are formed in Ti–Fe eutectic alloys at cooling rates of 10–100 K/s [2], [3]. The presence of such very fine lamellar structures in these alloys is the main reason for their very high strength (2000 MPa) compared to their coarse-grained counterparts [3]. Therefore, the phase selection process on the nano-scale during the solidification has become a highly interesting topic. The morphology of the eutectic colonies and the shape of the phases in the microstructure strongly depend on the actual cooling rate, the adopted casting conditions [4] and ternary alloy additions. In addition, it has been observed that the degree of super-saturation alters the lattice parameters of the formed A2 and B2 structures, thus strongly affecting the resulting mechanical properties [3], [4], [5], [6]. Hence, it requires an in-depth understanding of the solidification behavior of these nano-/ultrafine eutectic alloys at different degrees of undercooling and its dependence on the processing conditions.

In the past couple of decades, the solidification behavior of eutectic and near-eutectic alloys has been widely studied [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. The wide variety of parameters, such as melt undercooling, growth velocity, temperature gradient at the solid–liquid interface and the post solidification cooling rate creates challenges in predicting microstructures in rapid solidification processes [14]. For various eutectic alloys, such as Ag–Cu [15], Ni–Sn [16], Co–Sb [17], Co–Sn [18] and Co–Mo [19], a transition from the lamellar eutectic to an anomalous morphology beyond a critical undercooling in container-less solidification experiments has been reported. Even the mechanism of microstructure formation during undercooling of Nb–Al hypereutectic alloys revealed the formation of metastable phases (σ-Nb₂Al) beyond a critical cooling rate [11]. Such phases nucleate in the undercooled melt in competition with the eutectic structure due to slow diffusion. However, the analysis of the solidification process suggests that the anomalous eutectic morphology results from the decomposition of the metastable phase on post solidification cooling. Systematic studies on the Ni₆₀Nb₄₀ eutectic alloys during drop tube experiments reveal amorphous phase formation up to a thickness of 2 mm near the surface [10]. Therefore, a wide range of phase formation and microstructural features with a wide range of achievable physical properties pushes the interest for more detailed fundamental research on the solidification of eutectic alloys.

The phase-field method is a powerful tool for simulating interface driven mechanisms such as nucleation and microstructure evolution during solidification. The main principle of the method is the construction of thermodynamic potentials depending on the phase-field function, as proposed by the works of Cahn and Hilliard [20], [21]. In recent years, phase-field models have been successfully applied to simulate various solidification phenomena such as dendritic growth [22], peritectic solidification [23], [24] and eutectic growth [25], [26]. This method allows the in-depth numerical simulation of the microstructure evolution during the solidification of alloys on a microscopic scale. Recently, the phase-field method has been re-investigated with respect to its conformity to the underlying sharp-interface model. The result was the so-called thin-interface limit [27]. Such improved models can show fundamentally improved convergence in terms of the interface width of the phase-field model and allow high accuracy in the computations of the microstructure evolution for the considered mesh sizes. A great advantage of this class of models is that they make it possible, i.e. computationally feasible, to calculate under conditions of local equilibrium, which are a good approximation in many experimentally important cases. These are called quantitative phase-field models [22], [26] or second order models by Almgren [28] and recently by Caginalp and Eck [29]. These models were initially developed for pure materials. In a more recent contribution by Folch and Plapp [26], the thin interface limit was established for the first time for the simulation of eutectic growth. This model has been used to study the peritectic growth with fluid flow in the melting phase [24] as well as the directional eutectic solidification of lamellae [30].

A quantitative phase-field model approach describing periodic eutectic growth of lamellae based on [26] for the solidification of Ti–Fe is considered in the present study.