A novel prediction model of the freckle defects for single-crystal superalloy blades

The freckle is a typical surface defect formed during the directional solidification of SC (single crystal) components of Ni-based superalloys. It generally appears as a long and narrow trail of equiaxed grains aligned roughly parallel to the direction of gravity, which breaks the integrity. Once appears, the freckle can never be avoided by further treatment. With the turbine blade requirements increasing, the state-of-the-art methods include adding more refractory elements into Ni-based SC superalloys, and the design of blades with a more complex geometric shape make the freckle defects grow easier at the special surface zones. This leads to a significant challenge in controlling grain defect formation in SC blades and vanes for freckles. The current consensus to freckle formation is described as the Rayleigh-Taylor instability (RTI) flow that occurred in the interdendritic zone. During solidification, the compositional segregation (CS) occurred as soon as the solid interface forwarding led to the density changes between the interdendritic melting phases and the residual liquids. For nickel-based superalloys, the enrichment of low-density solutes like Al, and Ti at the interdendritic zone will make the liquid lighter. However, the widely used Ra model can’t correctly match the freckle tendency of the components with complex shapes (called geometrical effects of freckles), especially for blades. Recent reports indicate that the freckles occur at the preferred positions in the casting. In this work, a novel model is designed to quantitatively discuss the geometrical effects on freckle formation and to combine existing Ra number models with solidification models to enable freckle predictions at a smaller scale. The proceeding of this work can make the design of complex blades easier.


Introduction
Directional solidification (DS) [1] technology is an important process for preparing defect-free singlecrystal high-temperature alloy parts.This technology is based on the theory of interface stability [2] and can minimize the number of grain boundaries in the material system, thereby avoiding the weakening effect of grain boundaries under high-temperature conditions.However, in practical manufacturing processes, it is still challenging to obtain theoretically perfect single crystals due to the combined effects of alloy composition, shape, and solidification process.Common single-crystal defects such as impurities, freckles [3][4][5], and recrystallization may occur during the solidification process [6].A freckle is a common defect that occurs during the directional solidification of single-crystal (SC) components made of Ni-based superalloys.It is characterized by a trail of equiaxed grains that are aligned roughly parallel to the force of gravity.Freckles are known to be caused by thermosolutal convection resulting from a density inversion in the mushy zone due to interdendritic segregation.That is, Ni-based SC superalloys with higher concentrations (especially for the next-generation superalloys) of Re, W, Ta, and Al are more susceptible to freckling (higher segregation) during directional solidification [7][8][9].Many studies have explored the simple relationship between freckling and the convection factor known as the Ra number, which is described by the Rayleigh-Taylor instability (RTI).Higher Ra values are generally associated with a greater tendency for freckling to occur [10][11][12][13][14].However, it can be challenging to pinpoint the exact location of freckles using Ra alone (but the calculation is easier), as the actual convection field can be more complex than anticipated.
When using complex methods for CFD simulation (including the direct Ra method), the convective zone is often assumed to be located at the center of the solution domain.However, numerous experiments have shown that freckles tend to appear first at the edges and corners of the casting (introduced as 'wallattachment effect' or 'edge effect') [11,[15][16][17], which is inconsistent with typical simulation results.This inconsistency has been preliminarily resolved by using a combination of computational fluid dynamics (CFD) and phase field (PF) methods [18,19], considering the hindrance effect of dendrites on fluids.However, this method is difficult to apply for solving problems on a larger scale due to limitations in computing power and algorithms.That is, it is not cost-effective and seems to be impossible to manufacture iteration.Therefore, if the foregoing defects of the Ra model can be resolved, a more accurate and computationally efficient model can be obtained, which makes the manufacturing iteration by freckling controlling available.

The principle of treatment of the wall-attachment effect
For freckling, the main deficiency of the classical Ra model and general CFD methods lies in the incorrect estimation of the velocity distribution of the macro convective field.Under normal conditions, due to the action of viscous resistance, the velocity of the area near the boundary in the flow field is usually smaller than that of the area far from the boundary.However, under the directional solidification conditions, the velocity inside the mushy zone biased towards the center of the flow field may be lower than that at the boundary.This anomalous phenomenon is believed to be caused by the obstruction of the dendritic arms in the mushy zone, which is stronger than the wall resistance.This hypothesis has been verified [17,20] in some fast-throwing experiments, as shown in Fig. 1.Obviously, when considering the permeability change caused by the dendritic arm resistance, the wallattachment effect or edge effect of freckles becomes easier to explain.Since the wall flow is less obstructed, the flow near the wall is actually faster, which leads to a higher convective risk near the wall, resulting in stronger freckles.As for the edges, this wall-attachment effect is superimposed, making it easier to form freckles.The next focus is on modeling and quantifying this wall-attachment effect.Considering the permeation and wall-preferring characteristics, similar to the porous structure of the mushy zone of dendrites, the fluid velocity can be approximated by Darcy's law [21], which is given by: [21] where in the above equation, K is the permeability, μ is the liquid viscosity, gL is the liquid fraction of the region, and Δp/L is the pressure term.Regarding the shape characteristics of dendrites, the permeabilities parallel to the primary (Πp) and secondary (Πn) dendritic arms are given by [21,22]: (3) [21,22] where λ1 and λ2 are the spacings between primary and secondary dendritic arms, respectively, and τ is the tortuosity, defined as the ratio of the actual streamline length to the straight-line distance between two points.Some studies have shown that the tortuosity of dendritic flow models is related to their geometric shape characteristics and can be calculated from their geometric fractal dimensions, which is constant about 1.8 [21,23].Therefore, as shown in Fig. 1(b), it can be seen that when low-density liquid in the mushy zone flows upwards due to buoyancy, its available paths can be classified into two types: • Path-w (w means wall): the convective liquid starts from the wall, so its flow is least affected by the tortuosity of dendrites, and its flow can be approximated as the flow in a hypothetical pipe with an inner radius of r and a length of mushy depth L. • Path-d (d means dendrite): The convective liquid starts between two dendrites inside the casting and flows along the dendrite primary arms, being hindered by the flow resistance.Its flow velocity is the seepage velocity with a permeability of Πp.The average flow velocity of path-w can be written based on fluid mechanics knowledge as Eq. ( 4) according to Fig. 2: As shown in Fig. 2, near the wall, if the cross-section that can be left empty at the wall when viewed in the direction of solidification is equivalent to a hypothetical pipe with a radius of r, then it is about 0.4λ1.If we consider the possibility of gaps due to transverse solidification shrinkage between the dendritic arms and the mould wall, to simplify the subsequent calculations, it is approximately assumed here that: Therefore, the average convective velocity u here can reach at least: For the two cases of path-w and path-d, assuming that the pressure terms (Δp/L) are equal, their convective time terms can be written as follows, respectively: Then: Since the tortuosity τ can be regarded as constant as 1.8, the ratio of tw/td could reach 0 to 0.2183 while gL varies (0 to 1).This means that even under extreme conditions, the convective efficiency inside the dendrite only accounts for 21.83% of the wall flow.In the case of lateral permeation, the closer it is to the core of the casting, the lower the convective efficiency will be, which explains the wall-attachment effect in Fig. 1(c).
As shown in Fig. 3, the width δ of the 'overlapping zone' could be approximated as varying linearly as Fig. 1(c).The depth could be regarded as 3-5 λ1 as the observing results [24,25].
Therefore, we can assume that for a point coordinate selected in the XY plane of the solidification section, if we take a certain characteristic radius δ=x• λ1 as the base circle, and assume that the channel formed by scanning the base circle along the solidification direction is the convective channel, then the contribution of the momentum diffusion rate to the convective velocity at each point in the channel is integrated and averaged over the entire base circle.This is the convective average momentum diffusion rate at that point, with units equivalent to m 2 /s.Obviously, only when the contributions of all points in the base circle come from path-w (i.e., wall flow), the momentum diffusion rate can reach its maximum value.Therefore, the dimensionless shape factor Fxy of the XY plane can be defined as: In the above equation, the base circle is divided into n points of equal area ΔS, ti represents the consumption time of the final flow model selected for the i-th point (i=1,2,3...n), δ=x• λ1 represents the radius of the base circle, and tw represents the fastest time for wall flow.
The Fxy shape factor considers the characteristics of the permeation flow, so it can restore the stimulating effect of the geometric factor of the XY section on the formation of shrinkage porosity.When Fxy is relatively large, it corresponds to a point selection closer to the outer wall/corner, which is more likely to freckling.When Fxy is relatively small, it corresponds to a point selection closer to the center region of the casting, which is less likely to freckling.

The calculation method for the modified Rayleigh number (Ra)
Based on the aforementioned model, a novel Ra model considering the casting shape factor can be established.Here, the freckle generation factor F is defined as follows: 1 where Ra is the classical Beckermann [27] Rayleigh number for freckle prediction, Fi is the correction factor for each considered shape effect (i.e. the watt-attachment effect of Fxy).Since each correction factor (Fi, i=1,2,3…n) considers the shape effect, after introducing them, it is easy to calculate the differences in the Ra number at various positions in the casting, thereby obtaining the distribution of the Ra number for the entire casting is available.

The settings of the control group
In order to verify the accuracy of considering the shape factor based on the Ra number dominated by the wall-attachment effect, a control experiment was conducted using CMSX-4 alloy to compare the effectiveness of the simulation.The alloy parameters used in the calculation of the Ra number distribution are shown in Table 1, and the process parameters set in the experiment are shown in Table 2 To obtain verification results for complex shapes, directional solidification experiments were conducted on three different complex-shaped samples, as shown in Fig. 4.These samples were arranged in a circular pattern in the furnace, and their placement positions are shown in Figs. 5 (a) and (b).In the calculation, the shadow effect caused by changes in the temperature field will be considered in the form of Figs.

5(b) and (c).
To enable numerical solving, the freckle depth in Fig. 1(c) was set to 4λ1 by assuming a linear variation of Fxy.

The comparisons of freckle predictions and experiments of different shapes
For the cylindrical variable cross-section sample, as shown in Fig. 6, the shape in the XY direction should be the same everywhere.Therefore, under Bridgeman solidification, the solidification interface scans from bottom to top, and the macroscopic freckle factor F starts at 0 and gradually increases to a constant value as the solidification interface advances.This is consistent with the "incubation" characteristics required for freckle formation.It can also be seen that in the freckle model that considers the shape factor in this paper, the freckle factor in all the core parts of the casting is close to 0, indicating that freckles will not appear here, which is consistent with the actual situation.When the cross-section shrinks, the freckle factor suddenly enlarges, which may promote the generation of freckles, consistent with experimental observations.For the rectangular variable cross-section sample, as shown in Fig. 14, the model includes both edge effects and shrinkage effects, making its characteristics more representative.Compared with experimental results, it can be found that freckles will preferentially grow at the corners and invade to a depth of about 4λ1, or about 2 mm.In addition, the rectangular variable cross-section sample will also produce freckles without delay after it shrinks along the solidification direction, which is consistent with the model's prediction.Therefore, the freckle model proposed in this paper can not only describe the classical freckle formation based on the Ra number model but also evaluate the risk of generation in different shapes and regions of castings.Therefore, the proposed model has better adaptability to actual casting production.
For the cross-shaped variable cross-section sample, as shown in Fig. 8, in the actual sample, freckles tend to appear preferentially at the outer convex part of the cross-shaped sample rather than the inner concave part.Comparing with the simulation results, it can be seen that Fxy is higher at the outer convex part and lower at the inner concave part, which is close to the core, and therefore is not conducive to the appearance of freckles.
In addition, the above basic samples all exhibit the effect of shadowing, which causes more severe freckle zones in areas with smaller temperature gradients, and weaker in areas with larger temperature gradients, consistent with observations.It should be noted that the above calculations did not involve CFD, but only made simple modifications to the Ra number distribution that can be easily calculated.However, it can be seen that this simple modification that considers the quantification of shape effects has achieved a high degree of accuracy in locating freckle zones.In other words, for analyzing the occurrence and location of freckles, the weight of accurate solving of physical fields, as classical knowledge believes, is not significant, while the weight of shape effects on freckle generation is much higher than general perception.Therefore, it can be considered that shape effects are the core element for analyzing the occurrence and location of freckles.Figure 6.Comparison between the calculation results of the sample with a cylindrical variable cross-section (λ1=500 μm, R=1.5 mm/min).The yellow arrow indicates the direction of the temperature gradient decrease.

The several prediction cases
Although there are still some problems, the current model can well characterize the shape effects of freckle generation, which enables the consideration of freckle risk zones.Therefore, after the aforementioned model verification, it is possible to detect the possible effects of other variables on the results.For the rectangular variable cross-section sample with good display effect, when the primary dendrite arm spacing is 400, 500, and 600 μm, respectively, while other conditions remain unchanged, the results calculated by the model in this paper are shown in Fig. 9.According to actual production experience, the larger primary dendrite arm spacing should make the flow easier to occur and result in stronger segregation, thus aggravating the occurrence of freckles.From the simulation results, it can be seen that when the primary arm spacing is increased, the freckle factors obtained at the corners of the rectangular variable cross-section sample and at the smaller cross-sections are significantly increased, indicating that the variation is sensitive to the size of the primary arm spacing and consistent with experimental knowledge.Therefore, controlling the size of the primary arm spacing may have a certain effect on the suppression of freckles.For the rectangular variable cross-section sample, when other conditions remain unchanged, the freckle factor simulation results established in this paper are shown in Fig. 10 when the solidification rate increases from 1.5 mm/min to 4.5 mm/min.According to general knowledge, if the solidification system can achieve a higher solidification rate in an equilibrium state, the temperature gradient in the equilibrium state will also be higher, resulting in stronger interface stability and a shorter mushy zone, which should lead to a rapid decrease in Ra number.From the simulation results, it can be seen that when the solidification rate is increased, the freckle factors of the entire rectangular variable cross-section sample are significantly reduced, especially at the corners, which is consistent with the high-speed withdrawing experiment that is not prone to freckles.In addition, since this model considers the effect of shape factors, even at high-speed withdrawing, some corners where the cross-section suddenly shrinks still have the risk of freckle generation due to containing strong shape factors, which is consistent with the actual situation.Therefore, the results of this study also prove the fact that increasing the withdrawing speed can suppress freckle defects.However, it should be pointed out that shape effects are still strong for freckle generation under certain conditions, and there is still a risk of freckle generation at some corners where the cross-λ1=400 μm R=1.5 mm/min λ1=500 μm R=1.5 mm/min λ1=600 μm R=1.5 mm/min λ1=500 μm R=1.5 mm/min λ1=500 μm R=3.0 mm/min λ1=500 μm R=4.5 mm/min section suddenly shrinks, even at a high solidification rate.This means that to eliminate freckles, it is necessary to destroy the RTI conditions that can occur for the Ra number from the bottom up in the solidification process.

Conclusions
Based on the previous understanding and summary of the freckle phenomenon, this paper constructs a freckle model based on the widely used Beckerman Ra number model.The following conclusions are obtained: (1) The freckle mechanism considering shape factor effects is a breakthrough.In the past, the research on freckles has mostly started from the perspective of solidification, and it was usually difficult to accurately locate the actual position of freckle generation in practical results.This means that for a specific part, although the occurrence of freckles can be preliminarily judged through criteria, it is difficult to track the specific position, which brings difficulties to iterative improvement.However, the freckle mechanism considering shape factors solves this problem.In this paper, the introduction of Fxy corrects the velocity field distribution in the casting that was easily misunderstood in the past, thereby eliminating the conflict between the results obtained by Ra number criteria and experiments.This makes it possible to significantly improve the prediction accuracy based on a simple model.
(2) The model is an open model, and its composition does not conflict with the existing research system.In practice, the calculation results of the model can well reflect the influence of shape, arm spacing, solidification rate, and composition change on the freckle factor, and achieve the expected results.

Figure 1 .
Figure 1.Decanted three-dimensional dendrite morphology of the directional solidified Al-Si alloy (a).Schematic of the decanted directional solidified SC mushy zone (b) and the distribution of the permeability in the mushy zone (c) (Πin inner permeability, Πw outer permeability, δ depth of the freckle defect) [17].

Figure 2 .
Figure 2. Conservative relationship between the minimum equivalent radius R of the wall flow pipe in path-w and the spacing between primary dendrite arms

Figure 3 .
Figure 3. Referring to the existing wall effect observations [26], the region with a distance from the wall <δ can undergo convective flow (i.e., first flowing vertically to the wall under Πn flow and then flowing along the wall), which makes strong convective flow occur in the vicinity of the wall.Meanwhile, the convective flow in the core of the casting is weaker.

Figure 4 .
Figure 4. Schematic diagrams of the three shapes used in the calculation: (a) cylindrical variable cross-section; (b) rectangular with sharp corners variable cross-section; (c) cross-shaped with constant cross-section.

Figure 5 .
Figure 5. (a) The actual placement position of the casting and the orientation of the shadow effect; (b) explanation of the shadow effect; (c) approximation of the temperature field of the shadow effect.

Figure 7 .
Figure 7.Comparison between the calculation results of the rectangular sample with sharp corners variable cross-section (λ1=500 μm, R=1.5 mm/min).The yellow arrow indicates the direction of the temperature gradient decrease.

Figure 8 .
Figure 8.Comparison between the calculation results of the sample with cross-shaped with a constant crosssection (λ1=500 μm, R=1.5 mm/min).The yellow arrow indicates the direction of the temperature gradient decrease.

Figure 9 .
Figure 9. Prediction of the freckle factor effect of the sample with rectangular variable cross-section under different primary dendrite arm spacing based on the parameter calculation results (λ1=400-600 μm, R=1.5 mm/min).The yellow arrow indicates the direction of the temperature gradient decrease.

Figure 10 .
Figure 10.Prediction of the freckle factor effect of the sample with rectangular variable cross-section under different solidification rates based on the parameter calculation results (λ1=500 μm, R=1.5-4.5 mm/min).The yellow arrow indicates the direction of the temperature gradient decrease.For the rectangular variable cross-section sample, when other conditions remain unchanged, the freckle factor simulation results established in this paper are shown in Fig.10when the solidification rate increases from 1.5 mm/min to 4.5 mm/min.According to general knowledge, if the solidification system can achieve a higher solidification rate in an equilibrium state, the temperature gradient in the equilibrium state will also be higher, resulting in stronger interface stability and a shorter mushy zone, which should lead to a rapid decrease in Ra number.From the simulation results, it can be seen that when the solidification rate is increased, the freckle factors of the entire rectangular variable cross-section sample are significantly reduced, especially at the corners, which is consistent with the high-speed withdrawing experiment that is not prone to freckles.In addition, since this model considers the effect of shape factors, even at high-speed withdrawing, some corners where the cross-section suddenly shrinks still have the risk of freckle generation due to containing strong shape factors, which is consistent with the actual situation.Therefore, the results of this study also prove the fact that increasing the withdrawing speed can suppress freckle defects.However, it should be pointed out that shape effects are still strong for freckle generation under certain conditions, and there is still a risk of freckle generation at some corners where the cross-

Table 1 .
. Physical properties for the cases of Ni-based superalloys.

Table 2 .
The process parameters used in the calculation example