Effect of substrate orientation on the columnar-to-equiaxed transition in laser surface remelted single crystal superalloys

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ScienceDirect Acta Materialia 88 (2015) 283–292 www.elsevier.com/locate/actamat

Effect of substrate orientation on the columnar-to-equiaxed transition in laser surface remelted single crystal superalloys ⇑

L. Wang, N. Wang, W.J. Yao and Y.P. Zheng The Key Laboratory of Space Applied Physics and Chemistry, Ministry of Education, School of Science, Northwestern Polytechnical University, Xi’an 710072, China Received 15 January 2015; revised 23 January 2015; accepted 25 January 2015

Abstract—For a successful repair of single-crystal (SX) components, an epitaxial growth of columnar dendrites is required, i.e. a columnar to equiaxed transition (CET) has to be avoided. In this study it was found that changing the angles of laser treatment with respect to the standard orientation [1 0 0](0 0 1) can have a strong and positive effect on the quality of the part. By rotating the SX substrate around x-, y-, and z-axis, which coincide with the [1 0 0], [0 1 0], and [0 0 1] crystallographic directions, respectively, the effect of the substrate orientation for different rotation angles on the columnar-to-equiaxed transition (CET) in laser remelting process was determined. It was found that from the [1 0 0] (0 0 1) initial state, the CET varies strongly with the substrate orientation when the rotation is made around the y-axis, whereas there is nearly no change when the rotation is performed around the x- or z-axis. This drastic difference is caused by the fact that for rotation around the y-axis by 45° the melt pool contains no intersection point where dendrite domains with three preferred h1 0 0i growth directions meet. For the rotation about the x or z-axis, at least one intersection point appears. The intersection point is the most vulnerable place for the CET since the thermal gradient tends to reach its minimal value there. The number and location of the intersection points jointly control the overall CET tendency. Reducing those points and/or moving them to the positions with high thermal gradient are beneficial for avoiding CET generally. Our results clearly demonstrate that the laser processing window for the repair of the single crystal superalloy can be enlarged if the substrate orientation is rotated correspondingly. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Single crystal superalloy; CET; Intersection points; Substrate orientation

1. Introduction Nickel-base superalloys are materials of choice for manufacturing the hot-section components in the gas turbine due to their superior mechanical properties at high temperatures [1]. For maintaining and improving the properties, this kind of material is often cast into single crystal (SX) form. Nevertheless, under the severe service conditions of high temperature and high pressure, tip and platform damages often take place. Due to the high cost of the single crystal components, effective repair techniques which extend their life are highly desirable [2–4]. SX superalloy parts have been repaired by the epitaxial laser metal forming process, in which metal powder is injected into a melt pool created by a moving laser beam [5–8]. During this process, if specific solidification conditions are satisfied, the epitaxial growth of cells/dendrites along the original orientation in the substrate occurs and no equiaxed stray grains1 (SGs) form. Otherwise, equiaxed SGs are produced, which

⇑ Corresponding author; e-mail: [email protected] 1

The stray grains (SGs) could be equiaxed or disoriented columnar grains. Since we discuss the substrate orientation-dependent CET in this paper, SGs are referred to equiaxed ones.

results in a rejection of the part [9,10]. The grain boundaries resulting from the SGs lead to the occurrence of solidification cracking [11–14]. The columnar-to-equiaxed transition [15–18], CET, has been regarded as one of the underlying mechanisms for the formation of equiaxed SGs. Some theoretical and experimental works on this topic have been conducted. Hunt [19] developed the first analytical model to describe CET under the steady state conditions, in which the growth direction of the columnar dendrite was anti-parallel to the heat-flux direction. Later, this model was extended to the case of rapid solidification [20] and applied to laser treatments of Ni-base commercial SX superalloys [5]. An analytical relationship among the volume fraction of SGs /, solidification conditions, and material properties was obtained: (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )n Gnsl 4pN 0 1 3  ¼a ð1Þ V sl 3 ln½1  / n þ 1 where Gsl and Vsl are the temperature gradient and the growth velocity normal to the solidification front, N0 is the number of nucleation sites, and a and n are materialdependent constants. Eq. (1) shows that the CET is controlled by the thermal gradient and growth velocity for a

http://dx.doi.org/10.1016/j.actamat.2015.01.063 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

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L. Wang et al. / Acta Materialia 88 (2015) 283–292

specific alloy. High thermal gradient and low growth rate tend to avoid the CET whereas low thermal gradient and high growth rate will promote the equiaxed SG formation. It should be noticed that Eq. (1) is derived under the condition that the dendrite growth direction is parallel to the thermal gradient. If this is not the case, Eq. (1) should be modified. In the laser remelting or laser cladding process, the dendrites within a SX weld pool grow along several preferred crystallographic directions rather than along the thermal gradient direction. In this case, the thermal gradient component Gd and the dendritic growth velocity Vd along the primary stem should be used. Moreover, if the substrate orientation is changed, the dendrite growth direction also varies, which makes the situation more complex. Therefore, in order to obtain the concise CET distribution in a laser weld pool, the influence of the substrate orientation needs to be included. This situation has been considered by Vitek [21]. For characterizing the effect of the substrate orientation, he proposed an average volume fraction of equiaxed grains,  to describe the overall CET ability, by which the contri/, bution of the local CET of all the points in the laser weld pool has been taken into account. In this case, the author found that although the substrate orientation influences the CET locally, it has no significant impact on the overall CET tendency. It seems that one cannot depress the CET by adjusting the substrate orientation. Later, Anderson et al. [22,23] used a more sophisticated heat-transfer and fluid-flow model to compute the thermal field more precisely for studying the CET mechanism in laser and electron beam welds. Their results showed that SGs are more likely to appear in the zones around the intersections where the dendrite domains with different preferred crystallographic directions meet. Since the SG formation is related to the CET, it indicates that, if one can eliminate and/or reduce those intersection points by proper control of the substrate orientation, the CET tendency could be decreased. This is the starting point of the present paper. Now the question is if the intersections can be eliminated or reduced by the variation of the substrate orientation? Normally, the variation of the substrate orientation is operated by rotating a workpiece around a certain axis clockwise or counterclockwise through an angle. For a SX sample, several axes of rotation can be selected. The simple and direct selection is to use the axes which are

z [001]

x

y ξx

[100]

[010]

r Vb

r n

Fig. 1. Schematic diagram illustrating the initial orientation [1 0 0](0 0 1). The different rotation manners of the sample around the x-, y-, and z-axis are marked by the blank arrows, respectively. One of the rotation angles nx is given.

parallel to the crystallographic directions. To show it clearly, a schematic illustration of the different rotational axes relative to the solidification front in the present work is given in Fig. 1, where ~ n is the unit vector normal to the solidification front. The initial orientation is set to be [1 0 0](0 0 1), which means the laser beam scans along the [1 0 0] crystallographic direction on the (0 0 1) crystal plane. In this case, the orientation variation can be operated by the rotation of the substrate around the x-, y-, or z-axis through an angle ni (i = x, y, and z), and the x-, y-, and z-axis coincide with the [1 0 0], [0 1 0], and [0 0 1] directions, respectively. ni has a positive value if the rotation is operated clockwise as the hollow arrows designate whereas it takes a negative value if the rotation is performed counterclockwise. Vitek’s conclusion [21] that the substrate orientation does not have apparent influence on the overall CET tendency was drawn under the conditions that the rotation was operated about the x- or z-axis. One should note that the change of the orientation by x- or z-axis rotation leads to varied changes in the positions of the intersections on the two sides of the weld pool in the transverse section. It will result in different solidification conditions and, therefore, the contributions of the Vd and Gd on the left and right sides could be counteracted. This may be the reason why orientation changes performed about x- and z-axis have a slight effect on the overall CET tendency. However, for the orientation change by the rotation of the sample around y axis, the dendrite domains distribute symmetrically about the centerline of the weld pool since the orientation is also centerline symmetric. This suggests that the variations of the intersections on the two sides of the weld pool have the same behavior, which will cause the same solidification condition on the left and right sides. Thus, the variation in the substrate orientation in this way can lead to different distributions of the intersections and will result in a stronger or weaker contribution to CET. Such a difference between the effect of the y-axis rotation and that of x or z axis has never been determined before. The aim of the present work is to check this point theoretically. For such a purpose, the variation in the solidification conditions, Gd and Vd, for the rotation of the sample around the x-, y-, and z-axis were calculated, and their effects on the local and the overall CET tendencies were determined and compared. It is shown that the overall CET tendency varies significantly with the orientation when the rotation was operated around the y-axis whereas it is nearly constant if the rotation is performed around the x- or z-axis. The underlying mechanism on this difference was then analyzed. Preliminary experiments showed the substrate orientation-dependent CET for y-axis rotation. More detailed experimental work which will be compared with the present theoretical results is presently in progress and will be presented in a following paper. 2. Modeling methods The modeling approach used here is similar to those of Vitek [21] and Anderson et al. [22] and the simulation work was carried out in three steps. The first was to determine the shape of the weld pool and local solidification conditions by calculating the temperature field numerically; then Gd and Vd were calculated by utilizing the geometry model

L. Wang et al. / Acta Materialia 88 (2015) 283–292

derived by Rappaz et al. [24–26]; finally the local and over were computed. all CET tendencies, / and /, DD3, a Chinese first-generation SX superalloy, was chosen for the experiments. The composition of DD3 and the material properties used for the thermal field calculation are listed in Tables 1 and 2, respectively [27–30]. 2.1. Thermal field modeling A 3-D steady-state heat transfer model which contains the heat conduction and heat convection equations [22] was employed to calculate the temperature field in the weld pool. It should be noted that during the laser scanning process, Marangoni convection could be induced which will have an effect on the weld pool shape [31]. However, the extent of such effect depends highly on the specific processing conditions [32]. To check if the present selected laser processing conditions would cause strong convection, the calculations were conducted with and without Marangoni convection [22,33]. When the convection was not considered, only the heat conduction equation was solved, whereas when the convection was involved, the momentum and continuity equations were computed as well. The calculated weld pool shapes were then compared with the experimental ones which were obtained under the same processing conditions. The CW 1000 W Nd: YAG laser used had a beam with a Gaussian power distribution factor of 2 and radius of 0.75 mm. The processing parameters were 230 W and 2.5 mm/s and the heat absorption coefficient was set to be 0.5. As shown later, under the present processing conditions, Marangoni convection is very weak and can be neglected. Therefore, the thermal field without convection was used to further calculate the thermal gradient Gd and the dendrite growth velocity Vd in the weld pool. 2.2. Solidification conditions for different rotation manners The variation in the solidification conditions, Gd and Vd, with orientation for the rotation around the x-, y-, and z-axis are related to Vsl and Gsl by: Gd ¼ Gsl  cos whkl

ð2aÞ

V d ¼ V sl = cos whkl

ð2bÞ

where whkl is the angle between the [hkl] preferred dendrite growth direction and vector normal to the solidification front, as shown in Fig. 2(a). In order to determine cos whkl, the geometrical model proposed by Rappaz et al. [24–26] was applied, in which cos whkl ¼ ~ n ~ uhkl was used and ~ uhkl is the unit vector along the [hkl] growth direction. The general matrix for transforming ~ uhkl from the [1 0 0], [0 1 0], and [0 0 1] crystallographic reference system into the x–y–z reference system was given as: 2 3 cos a1 0 0 cos a0 1 0 cos a0 0 1 6 7 R ¼ 4 cos b1 0 0 cos b0 1 0 cos b0 0 1 5 ð3Þ cos c1 0 0

cos c0 1 0

cos c0 0 1

where ahkl, bhkl, and chkl are the angles between the [hkl] growth direction and the x-, y-, and z-axis, respectively. The angles a001, b001, and c001 are shown in Fig. 2(b). In principle, the dendrite growth directions at any location can be determined if the matrix R is given. To make the

285

calculation simple, we decomposed an arbitrary orientation into a combination of the rotations of the sample around the x-, y-, and z-axis. In this case, only the parameter ni for i axis is required and the corresponding matrix given in Eq. (3) can be decomposed into R = Rx(nx)Ry(ny)Rz(nz). Rx(nx), Ry(ny), and Rz(nz) are given by: 2 3 1 0 0 6 7 Rx ðnx Þ ¼ 4 0 cos nx  sin nx 5 ð4aÞ 0 sin nx cos nx 2 6 Ry ðny Þ ¼ 4 2

cos ny

0

0  sin ny

1 0

cos nz

6 Rz ðnz Þ ¼ 4 sin nz 0

sinny

3

0 7 5 cos ny

 sin nz cos nz 0

0

ð4bÞ

3

7 05 1

ð4cÞ

ni is defined positive if the rotation is conducted clockwise when one observes along the coordinate axis direction. To show the effect of ni, we only need to calculate the solidification conditions within the regime of [45°, 45°] with the interval of 15° since the other regime of ni will give the same results. 2.3. Local and overall CET After the solidification conditions for the three rotation manners were obtained, the local volume fraction of equiaxed SGs was calculated by: 2 !3=3:4 3 V d 5 / ¼ 1  exp 42:356  1019 ð5Þ G3:4 d Eq. (5) is directly obtained from Eq. (1) by replacing Gsl and Vsl with Gd and Vd with the values a = 1.25  106 Kn m1 s1, n = 3.4, and N0 = 2  1015 m3. These values were taken from the work of Gaumann et al. for CMSX-4 [5] since no such experimental data are available for DD3 now. It should be noted that different values of a and N0 of CMSX-4 and DD3 do not influence the relative effect of orientation on the CET. As the CET tendency varies from point to point in a weld pool, an area-weighted average volume fraction of  was used to assess the overall CET tenequiaxed grains, /, dency, in which the contributions from all points were  taken into account. The detailed method for calculating / has been given previously [21].

3. Results The reason why the substrate orientation affects the CET is that it changes the solidification conditions (Vd and Gd) by altering the dendrite growth direction through interaction with the weld pool shape. In this part, the weld pool shape will be given firstly, then the distributions of Vd and Gd will be shown, and finally, the variation of the local and overall CET tendency will be presented. Only the results of clockwise x- or z-axis rotation are given since counterclockwise rotation leads to the same results. For y-axis, both the results of clockwise and coun-

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Table 1. Nominal composition of DD3 single crystal superalloy [29]. Element

Cr

Al

Ti

W

Co

Mo

Ni

Weight %

9.5

5.9

2.2

5.2

5

3.8

Balance

Table 2. Physical parameters of DD3 used in the calculation of the temperature field [27–30]. Physical parameter

Unit

Value

Density Solidus temperature Liquidus temperature Thermal conductivity Specific heat of solid Specific heat of solid Coefficient of thermal expansion Latent heat of fusion Effective viscosity Temperature coefficient of surface tension

kg m3 K K W m1 K1 J kg1 K1 J kg1 K1 K1 J kg1 Pa s1 N m1 K1

8.2  103 1649 1601 24.62 603 603 1  105 2.43  105 7 1  106

ny = 0° and ny = 45° obtained under the same processing conditions. It can be seen that if Marangoni convection is involved, the weld pool has a flat and wide bottom. In contrast, if Marangoni convection is not considered, a concave pool was obtained which agrees reasonably well with the experimental pool shape. This indicates that for the selected parameters, the effect of Marangoni convection can be neglected. This is necessary for the present work in which the equiaxed grain formation from the nucleation ahead of columnar dendrites is considered, but not from the fragments of the dendrite arms caused by the convection. Note that the weld pool shape does not change with orientation.

(a)

(b) Fig. 2. Schematic diagrams of the weld pool and associated angles used in the analysis. (a) Angle whkl between [hkl] preferred dendrite growth direction and the vector normal to solidification front ~ n, and (b) the angles a, b, and c between [hkl] preferred dendrite growth direction and the x-, y-, and z-axis. Only a001, b001, and c001 were marked.

terclockwise rotations are presented because the solidification conditions of these two cases are different. 3.1. Weld pool shape Fig. 3 presents two calculated weld pool shapes with and without Marangoni convection, marked as dash-dotted and dashed curves respectively, compared with experiment for

3.2. Variation in Vd and Gd distributions with substrate orientation Fig. 4 shows the variation in the distribution of the thermal gradient with orientation for different angles. For the three rotation manners (Fig. 4a–d), the value of Gd ranges from 7.0  105 to 1.8  106 K m1. Most parts within the weld pool have the high values from 1.0  106 to 1.4  106 K m1, as the green regions show. In some small regions, there exist minima. For the initial orientation, two minima of Gd, about 8.0  105 K m1, appear for all cases, which locate at the two sides of the weld pool in the transverse section as the dark blue points show. The variations in the distributions of the minima with orientation angle, however, are different for x-, y-, and zaxis rotations. For the x-axis rotation, Fig. 4(a), with the increases of nx, the position of the right minimum moves to lower left whereas the left one moves to upper right. Finally, when nx is 45°, the right minimum stops at the lower center part of the weld pool and only this minimum value exists, resulting in a symmetrical distribution of Gd. The situation of the z-axis rotation is similar to that of x-axis rotation except that the moving directions of the minima are opposite and the final position of the minimum is higher at the centerline, as shown in Fig. 4(d). For the yaxis rotation, nevertheless, the situation is quite different. Both of the minima either move down for the counterclockwise rotation (Fig. 4(b)) or move up for clockwise rotation (Fig. 4(c)). When ny is around 30° or 30°, the two minima approach the lower or upper boundary, and they finally leave the weld pool when ny reaches 45° or 45°.

L. Wang et al. / Acta Materialia 88 (2015) 283–292

287

(a)

(b) Fig. 3. Solidification microstructures and the comparison of experimental and calculated melt pools in the transverse section for (a) ny = 0° and (b) ny = 45°. The dashed curve presents the calculated pool shape without convection and the dashed-dotted curve presents the one with convection.

(a)

(b)

(c)

(d)

Fig. 4. Variations in the distribution of thermal gradient, Gd, with orientation for (a) x-axis clockwise rotation, (b) y-axis counterclockwise rotation, (c) y-axis clockwise rotation, and (d) z-axis clockwise rotation.

It can be seen that the variation in the positions of Gd minima in the case of the y-axis rotation is quite different from those of x- and z-axis rotations. As will be shown later, the positions of the minima correspond to the intersections where the dendrite domains with different preferred crystallographic h0 0 1i directions meet. This indicates that thermal gradients at the dendrite domain intersections have minimum values, which will affect the CET. The variations in distribution of the local dendrite growth velocity, Vd/Vb, with the substrate orientation for the different rotation manners are shown in Fig. 5. Vd is smallest and tends to be zero near the weld pool boundary

regardless of the specific orientation. With the increase of the distance from the pool boundary to the center and upper part, Vd increases. However, the intensities of the increase are different, leading to that the maximum values and their distributions are different for the varied rotation manners. For the x-axis rotation, Fig. 5(a), Vd increases and its value reaches Vb in the other part except the pool boundary region. For the counterclockwise y-axis rotation, Fig. 5(b), the value of Vd can exceed 1 a little bit only near the surface region, as the yellow part presents in the case of ny = 45°. For the clockwise y-axis rotation, Fig. 5(c), the situation is similar as that of the counterclockwise one except that the maximum value of Vd increases to about

288

L. Wang et al. / Acta Materialia 88 (2015) 283–292

(a)

(b)

(c)

(d)

Fig. 5. Variations in the distribution of relative dendrite growth velocity, Vd/Vb, with orientation for (a) x-axis clockwise rotation, (b) y-axis counterclockwise rotation, (c) y-axis clockwise rotation, and (d) z-axis clockwise rotation.

1.4 Vb in the upper part of the weld pool, as the red areas show. If the rotation is operated about the z-axis, Fig. 5(d), the maximum value of Vd also can reach 1.4 Vb, and its location is in a small region in the upper center part of the weld pool. It can be seen that if Vd is greater than Vb, it always appears near the upper center part. Although the Vd distribution varies with the orientation, no special feature in the y-axis rotation can be identified from the other two cases. 3.3. Variation in / distribution with substrate orientation The / distributions at the solidification front for different orientations corresponding to the x-, y-, and z-axis rotations are given in Fig. 6. For all cases, it can be seen that near the fusion line, / is smaller than 0.03. In some specific positions, / has the maximum value about 0.25. The positions of the maxima vary with orientation, which also show different characteristics for different axis rotations. Comparing Fig. 6 with Fig. 4, one can see that the distribution of the maxima of / exhibits some similarities to that of the minima of Gd with orientation. That is, the position with small thermal gradient has a high volume fraction of equiaxed grains, where CET should take place preferentially. The variation in the position with the maximum / is also same as that of the position with the minimum Gd.

(a)

(b)

Therefore, for the x- and z-axis rotations, Fig. 6(a) and (d), with the increase of orientation angle, two maxima of / evolve to one, which locates at the centerline. On the other hand, for y-axis rotation by ±45°, Fig. 6(b) and (c), the two maxima move outside of the weld pool and the value of / is smaller than 0.09 in all positions. This indicates that it could be possible to avoid CET by a rotation of the sample around the y-axis. A preliminary experiment has been conducted for ny = 0° and ny = 45° via y-axis rotation under the same processing conditions (Fig. 3). Clearly, the stray grains for ny = 0° appear at the right intersection and boundary between [1 0 0] and [0 0 1] dendrite domains, as marked by the arrows in Fig. 3(a). However, for ny = 45°, no stray grain can be seen, Fig. 3(b). This proves that the intersections and boundaries of dendrite domains are vulnerable regions for CET. Further CET can be avoided when ny reaches 45°. The detailed experimental results for different orientation angles via x-, y-, and z-axis rotations will be given in a following paper.  with substrate orientation 3.4. Variation in / To assess the overall CET tendency within a melt pool, the area-weighted average volume fraction of equiaxed SGs  was calculated and its variation with the orientation for /

(c)

(d)

Fig. 6. Variations in the distribution of / with orientation for (a) x-axis clockwise rotation, (b) y-axis counterclockwise rotation, (c) y-axis clockwise rotation, and (d) z-axis clockwise rotation.

L. Wang et al. / Acta Materialia 88 (2015) 283–292 0.10

x y z

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 -60

-45

-30

-15

0 (deg)

15

30

45

60

Fig. 7. Variation in the area-weighted average volume fraction of SGs  with orientation for the x-, y-, and z-axis rotations. (/)

the different rotation manners is plotted in Fig. 7. When the  stayed rotation was performed around the x- or z-axis, / nearly constant with respect to the rotation angle. This agrees with Vitek’s conclusion that the substrate orientation has no significant impact on the overall CET tendency  is symmetrical with the variation of [21]. Additionally, / the substrate orientation for the rotations around the xand z-axis clockwise and counterclockwise, i.e.  ðnx Þ ¼ /  ðnx Þ and /ðn  z Þ ¼ /ðn  / z Þ. Nevertheless, the case for the y-axis rotation is quite different. Firstly, the value of  varies significantly with n. The general tendency is that /  / decreases with the increase of the absolute value of ny and reaches its minimal value at ny = ± 45°. The exceptional  is slightly higher case is the orientation ny = 15°, in which / than that of the initial orientation. Secondly, the value of  y Þ is not equal to that of /ðn  /ðn y Þ except for ny = 45°. 4. Discussion 4.1. Effect of orientation on Vd and Gd For constant Vsl and Gsl, according to Eq. (2), different values of Vd and Gd originate from different cos w. Therefore, in order to show the distributions of dendrite growth velocity and thermal gradient, the variations of cos w under different rotation manners should be considered. If the dendrite growth direction is aligned strictly with the heat flux, w is 0° and cos w equals to 1. For the dendrite growth in SX alloys, w does not coincide with normal vector of the solidification front in the laser weld pool, hence, cos w is smaller than unit. This will give rise to higher Vd and lower Gd for the same Vsl and Gsl, which tends to promote CET. Since the thermal gradient and local CET have minima and maxima at intersections in the weld pool, the distribution of cos w has its unique characteristics there. The pffiffiffi analysis indicates that cos w has the smallest value, 3=3; at intersection points for different kinds of rotation. As shown in Fig. 2(a), w100, w010, and w001 are the angles between the preferred directions [1 0 0], [0 1 0] and [0 0 1] and the normal vector of the solidification front, ~ n; respectively. Following the minimal velocity criterion in the weld pool [26], the dendrite with the smallest growth rate will be selected. This means that the dendrite having the smallest whkl will be activated and thereby w equals to the smallest item among

289

w100, w010, and w100 at any position on the solidification front. Meanwhile, as the [1 0 0], [0 1 0], and [0 0 1] directions are perpendicular to each other, one has cos2w100 + cos2w010 + cos2w001 = 1. In this case, the distribution of cos w in a weld pool can be clarified: (1) at the intersections where different dendrite domains meet, whkls are the pffiffiffi same and, therefore, cos w100 = cos w010 = cos w001 = 3=3 holds. This value is the smallest one of cos w. (2) Within the regime offfiffiffi a certain [hkl] dendrite domain, cos w is larger than p 3=3. The farer the distance from the intersections, the higher the value. (3) At the boundaries of two dendrite domains, pffiffiffi the value of cos w is larger than that at intersections, 3=3; but smaller than those within the domains. Bearing the variation in the values of cos w in mind, one can see that the intersections of the dendrite domains have some connections with the results we presented in Figs. 4 and 5. Therefore, to show the underlying mechanism on the distributions of Vd and Gd under different rotation manners, the change in the number and location of the intersection should be checked. By using the geometric model of Rappaz et al. [24–26], the movements of intersection points of different dendrite domains with n when the sample is rotated around the x-, y-, and z-axis were calculated and shown schematically in Fig. 8, respectively. At the initial orientation angle n = 0°, the boundaries of different dendrite domains are illustrated by the dash-dotted lines. There are two intersections locating at the center positions on the two sides of the weld pool, as the solid triangles present. With the increase of n from 0° to 45° by every 15° increment, the variations in the position of the intersection points for the three kinds of rotation are different. For the x-axis rotation, Fig. 8(a), the left intersection point, at which the ½0  1 0; [1 0 0], and [0 0 1] dendrite domains meet, moves to upper left side and finally leaves the pool, while the right one shifts to the lower left and stops at the centerline. When nx reaches 45°, only this intersection exists. For the z-axis rotation, Fig. 8(c), both of the intersections move nearly horizontally to the right side with the increase of n and only the left one exists finally at the centerline when nz reaches 45°. The situation for the y-axis rotation, however, is quite different. The two intersection points shift upwards or downwards with ny when the rotation is performed clockwise or counterclockwise and disappear when ny reaches 45° or 45°. The most obvious feature of this rotation manner that differs from the other two is that the intersection points can be removed entirely when ny is either 45° or 45°. Moreover, their locations are symmetrical with respect to the centerline for the same ny. From the variations in the number and the location of the intersection with n, one can see that the minima of Gd always appear at the intersection via cos w which locates at or near the centerline whereas the maxima of Vd do not show this tendency, indicating the Gd distribution is more sensitive to the orientation than Vd. As the Gd minima always appear at intersections, the variations in solidification conditions with the orientation for the y-axis rotation are different from those for x- and z-axis due to the absence of the intersections for ny = ± 45°. In those cases, Gd at any point within the melt pool is higher than 1.0  106 K m1 and its average value is larger than that in the orientations by the x- or z-axis rotation. It is this difference that results in weak CET for the yaxis rotation since the high thermal gradient would lower the tendency of CET for similar velocities.

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L. Wang et al. / Acta Materialia 88 (2015) 283–292

centerline has a higher / than the other, as seen in cases ni = 15° and 30° (i = x, z), revealing that an intersection located near the centerline will have higher CET tendency. Due to this reason, it indicates that the CET can be eliminated by a proper control of the number and location of the intersection points by rotating the substrate by ny. Note that in the positions near the fusion line, the value of / is always small regardless of Gd. This results from the extremely low growth velocity there, which tends to be zero. The CET is not likely to occur in those regions due to the absence of the constitutionally undercooled zone. This can be seen for the case ny = 30°, in which the intersections are located near the fusion line. Although the lowest Gd is present, / is still small.

z 45o 30

y x

o

15o

[0 1 0]

[010]

[100]

0o

30

45o

0o

o

15o

[001]

(a) z

y

45o

30

[0 1 0]

45o

o

15o

o

x

30

[100]

15o

[010]

0o

0o

- 15o

- 15o - 30

o

[001]

- 30 o

- 45o

- 45o

(b) z y [0 1 0]

x [100] 30 o

0o

45o

15o

0o

[010] 30 o 15o

45o

[001]

(c) Fig. 8. Movement of intersection points of different dendrite domains with orientation for (a) the x-, (b) y-, and (c) z-axis rotations.

4.2. Effect of orientation on local CET tendency (/) The favored locations for CET and how the local CET tendency changes depend upon the distributions of Vd and Gd. Since Vd does not change much, the distributions of / show similar tendencies to those of Gd with orientation angle for the three rotation manners. As stated before, the positions of / maxima correspond to Gd minima at intersection points in the weld pool. This suggests that Gd has a stronger impact on the CET than Vd. Such a conclusion is expected because / is a function of G3:4 d =V d rather than Gd/Vd, as Eq. (5) shows, which is also the reason why the variation in / with position is more stronger than that in Gd. Recently, Anderson et al. [22] modified the value of n from 3.4 to 5.3, implying that the effect of Gd on the CET could be even more dominant. As pointed out in the above section, due to the lack of intersection points at orientations of ny = ± 45° for y-axis rotation, Gd is higher than 1.0  106 K m1 and the average is larger than that for the x and z-axis rotations. This leads to lower /, less than  . 0.09, and thus results in the smallest / One interesting point, which can be seen in Fig. 6, is that the value of / at the intersection varies with the location. If the two intersections within a melt pool are located symmetrically about the centerline, such as the orientations ny = 0° and ± 15°, the values of / are equal at the two points. If this is not the case, the intersection closer to the

4.3. Effect of the orientation on the overall CET tendency  (/) The variations in the local CET tendency with the orientation discussed in the above section can be applied to  shows a strong dependence on the oriexplain why also / entation for y-axis rotation. Since the overall CET is an area-weighted average volume fraction over all local CET for a given section in weld  also shows dependence on the number and location pool, / of intersection point as / for different kinds of rotations. For this reason, three points can be clarified. (1) For the case of x- or z-axis, at least one intersection point appears. If only one intersection point appears, it always appears in or near the center on the centerline where the value of / is high. In contrast, if a melt pool contains two intersection points, those points are at the two sides with relatively low /. Hence one intersection at the center has a similar  as the two intersections at the two sides. contribution to / Note that the variations in the location of the two intersections are not independent. With one intersection point approaching the center, the other moves away from it. This results in that the higher value of / at one intersection moving to the center is counteracted by the lower / of the other  stays nearly constant. (2) approaching the edge, and thus / For the case of y-axis, however, it is different. Either two or no intersection point exists. At small ny, due to the existence  is high. At large ny such as 30° of the intersection points, / to ±45°, otherwise, since there is no intersection point in  is small. In addition, for y-axis rotation, the weld pool, / the intersection points are always symmetrically distributed about the center plane, which means that / are equal at both points and no counteraction takes place. This is the  depends substantially on substrate orientareason why /  about tion. 3) As to the non-symmetrical distribution of / clockwise and counterclockwise rotations, it is caused by the intersection points for these two cases locate on the positions with different thermal gradients. Compared to ny = 0°, the intersection points for ny = 15° move to the upper part of the weld pool where Gd decreases. It leads  In contrast, for ny = 15° and 30°, the to high / and /.  are situation is opposite and the corresponding / and / lower. As a general rule, the number and location of the  Reducintersection points determine the magnitude of /. ing or even eliminating the intersection points and moving them to the fusion line would lower / and finally  This can be achieved by rotatleads to a low value of /. ing the substrate orientation around the y-axis by different angles.

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Our results may find some applications in laser single crystal repair processes as the y-axis rotation could widen the processing window. If a repair under certain processing conditions for a given orientation fails, it could be feasible by the rotation of the substrate around y-axis. A schematic illustration is given in Fig. 9. If some cracks form in the upper right corner in a turbine blade, the part will usually be cut along the dash-dotted section and then the repair will be performed. This corresponds to the case of n = 0°. If the processing conditions in this case lead to equiaxed grains, one should consider a rotation of the blade by 45° around the [0 1 0] direction and cut the corner along the dashed section. In such a case, a successful repair is possible. Acknowledgements Appreciation is expressed to W. Kurz for going through the manuscript and for his constructive comments. This work was supported by the Aviation Science Foundation of China (Grant No. 2013ZF53080), the National Natural Science Foundation of China (Grant No. 51271149) and the Fund of the Innovation Base of Graduate Students of NPU. Fig. 9. Schematic diagram showing a repair procedure under the orientation rotation around the y-axis by 45°.

5. Conclusions The effects of the substrate orientation on CET in a laser weld pool for single crystal superalloys were investigated for different rotation angles of the substrate around the x-, y-, and z-axis. It is shown that CET depends significantly on the substrate orientation for y-axis rotation whereas there is nearly no change for x- or z-axis rotation. The following points should be emphasized: (1) CET is low for high Gnd =V d . Hence in the weld pool, the location with high Gd and low Vd shows reduced CET. (2) In the laser remelted SX weld pool, two or three preferred dendrite growth orientations form domains which intersect at lines or points. At the intersection point where three dendrite domains meet, Gnd =V d is lowest. This makes such points vulnerable for CET. (3) The substrate orientation alters the dendrite growth velocity Vd and the component of thermal gradient Gd in dendrite growth direction simultaneously. Gd is more sensitive to the orientation change than Vd, hence, the thermal gradient has a stronger effect on controlling CET than growth velocity. (4) As the orientation angle increases and reaches ±45° for y-axis rotation, the laser melt pool can be free of intersection points and Gd has a minimum value. However, at least one intersection point exists by the rotation around the x or z axis. This is the reason why y-axis rotation shows quite different CET tendency when compared with x- or z-axis rotation. (5) The number and the location of the intersections determine the overall CET tendency interactively. Therefore, reducing and even eliminating the intersection points or moving them to the positions with high Gd or near the fusion line where Vd is extremely small would lower / and finally leads to a  low value of /.

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