Fluid Flow and Defect Formation in the Three-Dimensional Dendritic Structure of Nickel-Based Single Crystals

Directional solidification provides control of the orientation and growth of columnar grain and single-crystal materials.[1,2] This advancement in processing has yielded a significant improvement in the high-temperature creep and fatigue properties of nickel-based superalloys.[1‐4] Unfortunately, depending on alloy and process conditions, directional solidification may result in the formation of grain defects such as misoriented, high-angle grain boundaries and/or freckle chains.[4‐7] The frequency of such defects has been shown to be related directly to slower cooling rates that characteristically produce coarser dendritic structures and larger interdendritic porosity.[7] Advanced alloys with higher levels of refractory alloying elements and large, more geometrically complex cast components are even more prone to defect formation. Thus, accurate solidification models that can predict the conditions under which these defects form as well as provide input to alloy and component design can clearly provide substantial benefits.[5,6,8‐16]

During directional solidification, an inversion in the solute density gradient can lead to localized fluid flow, resulting in flow channels compositionally rich in interdendritic solute.[5‐11] These channels or “chimneys” are the precursor event to the breakdown of single-crystal solidification, as the local transport of molten alloy upward through the melt can fragment, erode, and transport solidified material to various locations in the melt. These transported fragments migrate primarily toward the walls of the casting and may form large misoriented grains or chains of fragments, termed “freckle chains” shown in Figure 1.

These grain defects develop when a critical solute-induced density gradient allows the buoyant forces in the melt to overcome the frictional drag forces of the environment; therefore, the Rayleigh number (Ra _h ) is often used as a quantitative predictor for the conditions associated with defect formation.[8,11,13,14]

The density gradient in the liquid is denoted by the term (Δρ/ρ _o), g is the acceleration because of gravity, α is thermal diffusivity, ν is kinematic viscosity, and \( \bar{K} \) is the average permeability. Among the aforementioned examples, permeability is the only factor derived typically by means other than direct assessment from the chemistry and thermodynamics of the system. Permeability can vary by orders of magnitude within the mushy zone,[17‐21] so the details of the dendritic structure can influence the Rayleigh number strongly. Permeability has typically been defined empirically with consideration of flow through porous media.[17,20‐28] More recently, computational analyses of permeability have been performed with more accurate representations of the dendritic structure, albeit with relatively small volumes considered.[18,19,29‐31] Other difficulties associated with assessment of the Rayleigh number in superalloy solidification include (1) accurate prediction of solute density gradients in the mushy zone,[32,33] (2) selection of an optimal length scale that encompasses the scale of convective potential accurately,[12,13] and (3) accurate knowledge of thermal diffusivity and kinematic viscosity in multicomponent alloy systems.

The current investigation uses experimentally derived, 3D dendritic networks removed from the solidification front for simulation of isothermal, noncompressible fluid flow for direct permeability assessment. The relatively large volume of the reconstructed dendritic structure, with micrometer-scale resolution, permits the analysis of flow properties across multiple primary and secondary dendrites and from dendrite tips to deep within the mushy zone. Two different alloys with substantially different dendritic structure have been studied. The relationship of dendritic structure to interfacial surface area (ISA), flow tortuosity (T _avg.), and resulting permeability (K) are examined, and the implications for the development of convective instabilities are addressed.

Using an ALD Vacuum Technologies furnace, ingots ranging from 4.5 to 5.5 kg were directionally solidified at the University of Michigan by the conventional Bridgman process to produce single crystals. ALD Vacuum Technologies, Inc. is located in Hanau, Germany and is a wholly owned subsidiary of Advanced Metallurgical Group. Two alloys were investigated: the second-generation commercial superalloy René N4, with a nominal composition of Ni-4.2Al-0.05C-7.5Co-9.8Cr-0.15Hf-1.5Mo-0.5Nb-4.8Ta-3.5Ti-6.0W (wt pct) and a model ternary, Ni-6.5Al-9.5W (wt pct). Withdrawal rates of 2.5 mm/min and 3.3 mm/min were used for René N4 [T _liquidus = 1618 K (1345 °C), T _solidus = 1573 K (1300 °C)] and the Ni-Al-W ternary alloy [T _liquidus = 1729 K (1456 °C), T _solidus = 1706 K (1433 °C)], respectively. During withdrawal, controlled fractures of the mold wall were initiated, allowing the evacuation of molten liquid and producing isolated dendritic structures accessible for investigation. Material containing the transition from fully solidified regions to dendrite tips was extracted from the castings, serial sectioned with the prototype RoboMet.3D^{TM 1} system[34] located within the Materials and Manufacturing Directorate of the Air Force Research Laboratory (AFRL/RX), and imaged parallel to the primary growth direction. Reconstructions of the solidification front in both materials are shown in Figure 2. After image segmentation and reconstruction, both 3D datasets were converted into tetrahedral meshed domains. This mesh was then subdivided into domains that spanned different volume fractions of the solid and were subsequently used as input structures in full Navier-Stokes finite-element, fluid-flow models in two orthogonal flow directions: vertical (i.e., in the primary solidification direction) and cross (i.e., in the secondary dendrite arm growth direction). Additional details of the casting and decanting process,[35] serial sectioning and reconstruction,[36] and fluid-flow simulations[37] have been reported previously for René N4. In this investigation, a substantially different Ni-Al-W ternary alloy has been investigated, and more detailed analyses of the interfacial surface area and fluid flow in subdomains of both mushy zones are presented. The implications of fluid-flow behavior within these experimental domains for the development of convective instabilities are also considered for the first time.

It has been demonstrated that direct assessments of permeability based on actual dendritic structure in the mushy zone can be used for improved assessment of permeability and Rayleigh number (Ra _h ). In this study, 3D datasets of material at the solid–liquid interface in directionally solidified René N4 and a model Ni-6.5Al-9.5W (wt pct) ternary were used to assess vertical and cross-flow permeabilities directly over the mushy zone. The Rayleigh numbers calculated with experimentally determined permeabilities are higher than those determined with conventional permeability approximations.[11,14] Additionally, it has been shown that permeability scales inversely with interfacial surface area independent of the degree of flow path tortuosity. As a result, the dimensionless liquid ratio \( f_{\text{L}}^{3} /\left( {1 - f_{\text{L}} } \right)^{ 2} \) combined with accurate measures of the interfacial surface area can yield more accurate estimates for permeability.

In the systems investigated here, Rayleigh numbers do not exceed a critical value of 1.0 with conventional or 3D dendritic array permeabilities, suggesting instabilities should not occur in either alloy. However, because freckles have been observed in these castings (Figure 1(c)), it is apparent that other physical quantities in the Rayleigh analysis other than permeability, as was focused on here, are not determined adequately. The density gradient (Δρ/ρ _o), kinematic viscosity (α), and the thermal diffusivity (ν) are likely sources of error and require additional investigation. With regard to liquid density gradients, there may be errors in the thermodynamic assessments, and currently, insufficient information is available on the liquid density as a function of temperature. Partial molar volume density calculations have been shown to be accurate to within 5 pct[32,33] for certain alloys; however, a larger error may arise because of uncertainty in the prediction of liquid compositions themselves. The accuracy of phase calculation software is limited by the database used and can vary substantially for unexplored compositions as well as in select commercial nickel-based superalloys with typically three or more elements. Because (Δρ/ρ _o) is directly proportional to Ra _h , a variation in density gradient of an order of a magnitude would affect the local Rayleigh number by the same amount. Alternatively, the kinematic viscosity and thermal diffusivity are maintained as constants and are, as a product, considered equal to the value 5 × 10^–12 m⁴/s².[11,13,14] Because this consistent approximation is used by various investigators, no discrepancies between the results of this and other investigations are introduced by its use. However, it is unlikely that kinematic viscosity, thermal diffusivity, or their product remain constant over the entire liquidus to solidus temperature range in most alloys, including those presented in this study. The absence of variation with temperature or composition is likely to have an uncertainty on the order of ± 5 pct for each quantity. Because (αν) is inversely proportional to Ra _h , a 25 pct increase in the product of the thermal diffusivity and kinematic viscosity would produce a 25 pct decrease in the local Rayleigh number or vice versa. It is, therefore, also reasonable to assert that the greatest amount of error in the current Rayleigh calculations likely reside in either the lack of variation in the quantity (αν) or the assessment of the actual density gradient across the liquidus to solidus temperature range.