Elsevier

Journal of Crystal Growth

Volume 334, Issue 1, 1 November 2011, Pages 170-176
Journal of Crystal Growth

Numerical modeling of crystal growth of a nickel-based superalloy with applied direct current

https://doi.org/10.1016/j.jcrysgro.2011.08.011Get rights and content

Abstract

A cellular automaton-finite difference (CA-FD) model is developed to simulate crystal growth with applied direct current (DC) in this study. The simulated dendrite growth is controlled by the CA rules that involved the effects of the supercooling, interfacial curvature and crystal anisotropy. The electric current distribution, the temperature field and the electromagnetic flow caused by the Lorentz force are solved by the FD calculation. The directional solidification of a model nickel-based superalloy with DC field is simulated. The results show that Joule heat brought by electric current delays the dendrite growth and changes the primary dendrite arm spacing (PDAS). Excessive Joule heat will completely stop the crystal growth. The maximum allowable current intensity increases with the cooling efficiency of directional solidification. It is found that the dendrites growing with the DC field have a self-adjustment ability to achieve flat S/L interface in macroscale during directional solidification. The electromagnetic flow induced by the DC field may theoretically accelerate the dendrite growth, but usually the influence is negligible for the directional solidification of the superalloy.

Highlights

► A numerical model is developed to simulate crystal growth with direct current. ► Joule heat delays the dendrite growth and changes the primary dendrite arm spacing. ► Dendrite growing with direct current can self-adjust to achieve flat interface. ► Electromagnetic flow induced by direct current is usually negligible.

Introduction

Controlling solidification process of metals with applied DC fields, as one of the important approaches in electromagnetic processing of materials, has been investigated for decades [1], [2], [3], [4]. It is expected that the crystal growth process can be controlled and thus the demanded microstructure of metals can be achieved by the generated physical effects of electric current, including Joule heat, Peltier heat, electromagnetic flows and electromigration [5], [6], [7], [8], [9], [10]. Many metal materials including tin alloy, plumbum alloy, alumina alloy, superalloy and some pure metals were used for the experimental research of solidification with DC fields [11], [12], [13], [14], [15]. The microstructures of the metals were all modified in different degrees and the composition segregation in some alloys was also influenced by DC fields. Recently we found that not only the microstructure but also the mechanical properties of a single crystal superalloy were markedly improved by introducing DC fields to its directional solidification process [16].

These interesting experimental results show that the application of DC field might be a promising approach to control crystal growth. However, the microscopic mechanisms of DC fields influencing the crystal growth processes are not very clarified yet. The puzzles come from at least two aspects. Firstly, it is hard to discern the microscopic distribution of electric current by experiments, because the current field is always asymmetric and variational with the interface evolution during the growth process due to the different electric conductivity between the solid and the melt. Secondly, it is often difficult to judge the dominative one among the several physical effects of electric field only by experiments because the experimental results are brought about by the integrated action of the physical effects. Rather than the qualitative estimation, the quantitative and dynamic analysis of the effects is required for the accurate mechanism study. In such case, numerical simulation might be more preponderant than the experimental research.

The previous numerical simulations of the crystal growth with the electric current are focused on the variations of macroscopic parameters, including the thermal profile and the melt flow [17], [18], [19], [20], [21]. Recently the simulation is expanding to the microscopic examination. Brush [22], [23] simulated the free growth of a pure metallic single crystal with an electric current by a phase field model. Nikrityuk and Eckert [24] simulated the dendrite growth with electromagnetic flow in steady electromagnetic fields and direct electric current.

This paper presents a CA-FD model for crystal growth with a DC field. The model couples the crystal growth kinetics and the physical effects of the DC field, including the Joule heat, Peltier heat, electromagnetic flow and electricmigration. A binary model alloy approximated to a nickel-based superalloy is used in the numerical research. Exceptionally the influences of Peltier effect and electricmigration cannot be investigated due to the lack in the corresponding parameters currently, the effects of Joule heat and electromagnetic flow induced by the DC field on the dendrite growth are examined by simulation.

Section snippets

Electromagnetic field model

The electric field is generated by introducing an electric potential between the top and the bottom sides of the square calculated domain. The DC field can be written as [21]E=φ,j=0,j=σEwhere E is the intensity of the electric field, φ is the electric potential, j is current density and σ is the electric conductivity given asσ=σL(1fS)+σSfSwhere σL and σS are the electric conductivities of liquid and solid, respectively, fS is the volume fraction of solid. According to Eqs. (1), (2),

Numerical solution

The numerical solution of the model is performed in a square domain in the rectangular coordinates. For simplifying the calculation, the influence of latent heat is ignored though it is included in the governing equation of the model, because the error in thermal profile caused by neglecting latent heat is not signification for the crystal growth controlled by solute diffusion during the directional solidification for most alloys [26], [27]. In addition, the density, partition coefficient,

Effect of Joule heat

The growth of single dendrite with the effect of Joule heat is simulated in a zone of 100 μm×100 μm. The coefficient of heat transfer for the bottom boundary is 800 W/(m2 K). The typical feature of the simulated dendrite growth without applied current (shown in Fig. 1a) agrees qualitatively with both the experimental observation [33] and the predictions by the phase field theories [34]. As shown in Fig. 1b, the primary dendrite grows much slower with Joule heat than that without Joule heat. The

Conclusions

A CA-FD model is developed to simulate crystal growth of a nickel-based superalloy with a DC field. The physical effects of the DC field, including Joule heat, Peltier heat, electromagnetic flow and electricmigration, are involved in the model.

The simulation results show that Joule heat can delay the dendrite growth, but has no remarkable influence on the microsegregation. With the appropriate current density, the PDAS of the directionally solidified alloy decreases and the microstructure of

Acknowledgment

This work was financially supported by the National Natural Science Foundation of China (No.50931004) and the National Basic Research Program of China (No. 2010CB631205).

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