Analysis of Yield Rate in Single Crystal Casting Process Using an Engineering Simulation Model

Transcription

1 Materials Transactions, Vol. 44, No. 5 (23) pp. 829 to 835 Special Issue on Solidification Science and Processing for Advanced Materials #23 The Japan Institute of Metals Analysis of Yield Rate in Single Crystal Casting Process Using an Engineering Simulation Model Hisao Esaka, Manabu Tamura and Kei Shinozuka National Defense Academy, Department of Materials Science and Engineering, Yokosuka , Japan A 2-D engineering model for grain selection has been developed taking the columnar dendrite growth theory into consideration. After evaluating this model via a unidirectional solidification experiment, the single-crystal casting process was simulated. Since the time required for calculation is rather short, a statistical analysis has been performed for the first time. The yield rate of well-oriented single crystal is increased by increasing the initial number of grains on the chill plate. However, the yield rate does not exceed approximately 9%. A detailed investigation of the formation mechanism of misorientation has revealed two possible processes (Type A and Type B) that may occur during single crystal casting process. (Received November 2, 22; Accepted March 28, 23) Keywords: unidirectional solidification, grain selection, turbine blade, preferred growth direction, yield rate, engineering model 1. Introduction Increasing the operating temperature of turbine blades in generators or jet engines results in energy savings. Structure control performed by process control, as well as material design, is important for the development of heat resistant materials. Turbine blades are made of nickel-base or cobaltbase complex poly-phase alloys. These materials are very difficult or impossible to forge or weld; therefore, turbine blades are usually processed by casting. 1,2) Since the mechanical strength at elevated temperatures of the unidirectionally solidified blade is superior to that of the polycrystalline blade, unidirectionally solidified turbine blades have replaced polycrystalline turbine blades. Recently the mechanical strength of single crystal blades at high temperature has been found to be much higher than polycrystalline and unidirectionally solidified blades. Two processes are used to perform single-crystal casting. 2) The first is the seeding method, which uses well-defined seed crystals. The second is the selector method, which involves a chill plate and a selector. During the growth in the selector, favored grains are automatically selected until only one grain remains. The preferred growth direction of the abovementioned alloys is known to be h1i. The mechanical properties of such alloys, for example creep strength, are not isotropic, and h1i single crystals oriented along the length of the turbine blade have excellent mechanical strength. 1,3) The goal of single-crystal casting is to align the h1i single crystals along the length of the turbine blade. However, this process does not always yield such turbine blades. 2 5) The purpose of the present study is to analyze the grain selection during growth. A simple engineering model based on the columnar dendrite theory has been developed recently, and using this model, the yield rate of good products and the mechanism of misorientation, which occurs by chance, have been analyzed. 2. Engineering Model 2.1 Background Each crystal has a preferred growth direction that is based on the anisotropy of surface tension. Alloys used in the construction of turbine blades have a cubic lattice and a preferred growth direction of h1i. 1 5) A technique used for unidirectional solidification in turbine blades controls heat flow unidirectionally via external heaters and coolers. Here, the grains, the growth direction of which is approximately the direction of heat flow, grows over other grains. This process is referred to as grain selection 6 8) and consists of the following two steps. 6,9) The first step is blocking, which is observed when two neighboring grains converge as solidification proceeds. Figure 1 illustrates the process of blocking, where grain B, which is growing more closely to the heat flow direction, α A Grain Boundary Fig. 1 A schematic drawing of solid-liquid interfacial morphology at a grain boundary, which is formed by blocking. β B

2 83 H. Esaka, M. Tamura and K. Shinozuka blocks the growth of grain A. Assuming that the traveling velocity of the isotherm is V and that the growth angles are and ( >), as shown in Fig. 1, the growth velocities of dendrites in grains A and B are V/cos and V/cos, respectively. Since the growth velocity of the dendrite in grain A is higher than that of the dendrite in grain B, the driving force for the growth of the dendrite in grain A should be greater than that of the dendrite in grain B. Therefore, the dendrite in grain A requires significant undercooling, so that the tip temperature of the dendrite in grain A is lower than that of the dendrite in grain B. Due to the positive temperature gradient, the dendrite in grain B leads to that in grain A. Thus, when the dendrite in grain A approach that in grain B, the dendrites in grain B stop the growth of the dendrite in grain A, since the solutal field around the dendrite in grain B blocks the path of the dendrite tip of grain A, and thus the dendrite in grain A loses its driving force for growth. Recently, D Souza et al. 1) reported that the primary trunk of grain B stops the growth of grain A instead of the secondary arms of grain B. In any case, grain B always stops the growth of grain A. The grain boundary between grains A and B forms along the direction of growth of grain B and is a straight line, as shown in Fig. 1 (dotted line). When a grain approaches the sidewall, blocking occurs on the wall, as shown in Fig. 2. Here, the wall acts as a grain boundary. The second step is branching, which is observed when the two neighboring grains are diverging as solidification proceeds. Figure 3 shows schematically the process of branching. At the grain boundary, which is a relatively wide boundary in this case, not only secondary arms, but also tertiary arms grow into this area, which is largely undercooled. By competitive growth, one of the tertiary arms becomes a new primary arm. The probability of branching may be independent of the growth direction. Therefore, the grain boundary generated is the bisector of two growth directions and again is a straight line. When a grain heads away from the sidewall, branching takes place near the wall, as shown in Fig. 4. The wall again acts as a grain boundary for this grain. As explained above, both in the cases of blocking and Grain Boundary A B Fig. 3 A schematic illustration of solid-liquid interface at a grain boundary, which is formed by branching. Fig. 4 A schematic view of dendrites, heading away from the inert wall. Fig. 2 A schematic drawing of dendrites, approaching the inert wall. branching, the grain boundaries formed are straight lines. In order to verify this, a unidirectional solidification experiment was performed using succinonitrile-1.3 mass% acetone alloy. 11) The experimental result is shown in Fig. 5. In this case, the temperature gradient and growth velocity are 1:9 1 3 Km 1 and 8: ms 1, respectively. Blocking between grains A and B occurs, and the grain boundary formed is parallel to the growth direction of grain A. On the other hand, branching takes place between grains B and C. The grain boundary between grains B and C is the bisector of both growth directions and is a straight line. Grain B disappears when the solid/liquid interface advances up to the point X, as shown in Fig. 5. In other words, grain B disappears where its grain boundaries intersect, this is a very important criteria for grain selection.

3 Analysis of Yield Rate in Single Crystal Casting Process Using an Engineering Simulation Model 831 Fig. 5 Grain selection observed in succinonitrile-acetone alloy. Temperature gradient: 1:9 1 3 Km 1, Growth velocity: 8: ms 1. Grain B disappeared at X due to grain selection. 2.2 Simulation model The single-crystal casting process involves a chill plate and a space for grain selection, called the selector. In addition, a cavity exists in which to form the turbine blade. Molten metal is poured into the mold, and many grains nucleate on the chill plate. Grain selection takes place in the selector, and a single crystal turbine blade is produced. For simplicity, we assume that no special devices, such as a pigtail or a zigzag selector, are used in the present study. Figure 6 illustrates the domain of this model. The growth length from the chill plate, L, is normalized by the width of the chill plate, D. As mentioned above, many grains nucleate randomly on the chill plate. In the model, the diameter and growth direction of the grain are given randomly as the initial condition. Since this is a 2-D model, the grain has two-fold symmetry. Therefore, the range of the growth direction lies between 45 to 135 from the isotherm, as shown in Fig. 7. The initial diameter of the grain ranges from :1d av to 1d av, where d av is the average initial diameter of the grain. The initial number of grains is a parameter of this model. A simplified flow of the calculation is shown in Fig. 8. Comparing the growth directions of two neighboring grains, Fig. 6 Growth Direction D Chill Plate The domain of the present model. Fig Heat Flow Direction Definition of the growth direction. 45 the direction of the grain boundary can be determined. A grain disappears when the grain boundaries intersect each other or the grain boundary intersects the wall of the selector. Every time a grain disappears, the growth directions of neighboring grains are checked again and the direction of the grain boundary is determined. This procedure occurs repeatedly as solidification proceeds. The growth direction of each grain, as determined randomly as an initial condition, is maintained constant as solidification proceeds. Therefore, once the initial condition is fixed, the process of grain selection would be decided deterministically. One advantage of the present engineering model is that this model requires little time to obtain calculated results, less than a few seconds when using a personal computer. The cellular automaton method or modified cellular automaton is a powerful tool to simulate the solidified structure of unidirectional and/or equiaxed solidification ) In addition, these models can predict the dendritic morphology. However, hours or days are required in order to obtain the result of the single crystal casting process, even when calculated using a workstation. If the purposes of the calculations are to estimate the change in the number of grains with growth or the change in the distribution of

4 832 H. Esaka, M. Tamura and K. Shinozuka Input data Initial number of grains, n Final number of grains, n f Grain diameter Growth direction blocking? yes no branching Normalized Number of Grains,n / n Fig. 8 Compute grain boundary Calculate Intersection of grain boundary Growth length if N<=n f yes Output calculation results crystallographic orientation with growth, a detailed physical model, such as the cellular automaton method or phase field method, is not necessary. The newly developed engineering model has the following disadvantages: (1) Solidification conditions, such as growth velocity and temperature gradient, are not taken into account. Thus, the dimensions of the dendrite, such as dendrite tip radius and primary dendrite spacing, cannot be estimated. (2) Length and time are arbitrary. Definite length cannot be predicted, although the geometry remains similar. Thus, the length can be normalized by a specified length, such as the width of the starter. (3) Distribution of solute during solidification cannot be estimated. Although this model has a number of disadvantages, the method enables the events that may occur to be estimated statistically, because this method allows calculation in a short time. 3. Results and Discussion no N=N-1 Program flow sheet of the engineering model for grain selection. 3.1 Validity of grain selection model From an engineering perspective, showing the change in number of grains as a function of growth length may be more important than drawing the grain structure. Unidirectional solidification experiments have been performed 9) using asreceived succinonitrile. The change in the number of grains with growth length was measured when the solid/liquid morphology was dendritic. Here, the number of grains was normalized by the initial number of grains, 16. The growth Normalized Growth Length, L / d av Fig. 9 Relation between number of grains and growth length. Solid circles indicate the experimental results and solid line is an example of calculated result of the present model. length was normalized by the average initial diameter of the grain, d av. Figure 9 shows the experimental results, which indicates that the number of grains decreases with increasing growth length. The calculated result of the present model is also shown in Fig. 9. When the initial number of grains is small, the scatter of the calculated results becomes large. 9) Thus the initial number of grains is fixed to be 16 in this calculation. There is little difference between the experimental and simulation results. The normalized number of grains decreases rapidly in the first stage of solidification and gradually in the latter half of solidification. The present model can also predict this phenomenon. 3.2 Distribution of crystallographic orientation as a function of growth length The change in the distribution of preferred growth direction of each grain as a function of growth length has been analyzed. The initial number of grains was fixed to be 16. The growth length was normalized by the width of the calculated domain, D, which corresponds to the width of the starter in the single crystal casting process. The growth directions of dendrites are measured as shown in Fig. 7, and histograms of the growth angles are analyzed based on growth length. Here, the angles between 45 and 135 are classified into five groups: group a, ; group b, ; group c, ; group d, ; and group e, A typical example of the distribution of crystallographic orientation as a function of growth length is shown in Fig. 1. In the initial stage, at which the normalized number of grains (n=n ) is 1., the growth direction is equally distributed from 45 to 135, since the angles are given randomly. When the normalized number of grains reaches.6 as solidification proceeds, the ratio of largely inclined grains decreases and that of well-aligned grains (81 99 in this study) increases. When n=n reaches.2, approximately 7% of grains fall in the range of 81 to 99. At the final stage of solidification, when, n=n is.1, the growth directions of almost all grains are between 81 and 99. Therefore, one can obtain a good single crystal in this trial. A number of casting trials produced similar results, as shown in Fig. 1.

5 Analysis of Yield Rate in Single Crystal Casting Process Using an Engineering Simulation Model 833 Normalized Number of Grains, n/n n/n =1. n/n = n/n =.6 n/n = Normalized Growth Length, L/D Fig. 1 Change in number of grains and distributions of h1i with growth length. A well-oriented single crystal is obtained with this trail. Group a: 45 63, group b: 63 81, group c: 81 99, group d: and group e: Normalized Number of Grains, n/n n/n=1. n/n= n/n= However, some casting trials indicate a different change in growth direction. An example of this is shown in Fig. 11. The change in growth angle distribution during the initial stage of solidification is the same as that shown in Fig. 1. However, even at the final stage of solidification, inclined grains remain. In this case, a grain, the h1i of which is 63.8,is selected. The direction of h1i of this grain is 26.2 different from the heat flow direction. This single crystal may exhibit weak mechanical properties and is regarded as a misorientation. Thus, this casting trial can be concluded to be a failure. 3.3 Yield Rate The number of grains (n) decreases with growth length, as mentioned above. In the latter half of solidification, n decreases slowly, until one grain is selected. In this study, the simulation was continued until a single grain with a constant width was selected in a semi-infinite selector. The crystallographic orientation of the residual grain was then examined. 1 n/n=.1 5 Normalized Growth Length, L/D Fig. 11 Change in number of grains and distributions of h1i with growth length. A misoriented single crystal remains with this trail. Group a: 45 63, group b: 63 81, group c: 81 99, group d: and group e: In practice, the single crystal, h1i of which is within 1 from the heat flow direction, may be applied for operation. 2,4) Therefore, in the present study, the residual grain, h1i of which is within 9 from the heat flow direction, is defined to be a good single crystal. Since the time required for calculation is rather short (within a few seconds), several trials were conducted in order to carry out a statistical analysis. The ratio between the numbers of trials in which good single crystals are obtained to the total number of trials was calculated, and this ratio () is the yield rate Effect of initial number of grains In order to investigate the effect of the initial number of grains (n ) on the yield rate, the calculations were performed over 1 times while varying n. The result is shown in Fig. 12. The yield rate rapidly increases with increasing n. This deduces that enhancement of the nucleation density on the chill plate is important in order to obtain good single crystal. In addition, the yield rate approaches a constant value with increasing n. Within the range of the present study, the maximum yield rate is approximately 9%. In other words, approximately 1% of all castings will fail even if the initial condition of casting can be improved. Few reports indicating the yield rate of good crystallographic orientation in single crystal casting process exist in the literature. 2,4) Okamoto et al. have studied the mechanical properties of single crystal turbine blades 4) and conducted 14 casting trials and obtained 13 casts with good single crystal. They reported that the direction of h1i in the failure was approximately 16 from the heat flow direction. The yield rate was 13/14 in their casting trials. This value is similar to the calculated result obtained using the present engineering model. Higginbotham 2) reported a schematic diagram of the deviation of h1i in the single crystal casting process and showed that, for some single crystals, the h1i deviated greatly from the heat flow direction. The present model may predict the yield rate of the actual casting process for single-crystal products Reasons for the limit of yield rate There are two cases in which the inclined grain is selected. The first case is that the grain, which is near the inert wall and grows inward, remains (Type A). The second case is that the grains, which converge or diverge, and make the grain boundary approximately parallel to the heat flow direction, Yield Rate, η ( % ) Fig Initial Number of Grains, n Relationship between initial number of grains and the yield rate.

6 834 H. Esaka, M. Tamura and K. Shinozuka 6 13 Normalized Growth Length, L/D A 63.8 B C 93.3 Normalized Growth Length, L/D A B Fig. 13 Calculated grain structure in the final stage of solidification. In this case, the grain, the angle between h1i and isotherm is 63.8, is finally selected. This is an example of type A defect. The digits in this figure indicate the growth angle of dendrites. Fig. 14 Calculated grain structure in the final stage of solidification. In this case, the grain, the angle between h1i and isotherm is 111.6, is finally selected. This is an example of type B defect. The digits in this figure indicate the growth angle of dendrites. remain (Type B). An example of Type A is shown in Fig. 13 in the final stage of grain selection. In order to show clearly the grain selection, the scale of normalized growth length is modified. One of outstanding points in this case is that the grain, the preferred growth direction of which is 63.8, is in the left side of the domain. This inclined grain would remain after grain selection. Another important point is that the preferred growth directions of almost all other grains are approximately, but less than, 9. Furthermore, the growth angle of grain B, which is just to the right-hand side of grain A, is 89.3, which is quite close to 9. Grain B blocks the growth of grains A and C. However, the width of grain B gradually decreases as solidification proceeds. Since the growth direction of grains B and C is inclined to the right, grain B finally disappears at the inert wall of the growth domain. No grains would stop the growth of grain A, even though the growth direction is quite far from the heat flow direction. As a result, grain A, h1i of which deviates 26.2 from the heat flow direction, is selected. This is a failure for the singlecrystal casting process. An example of Type B is shown in Fig. 14 in the final stage of grain selection. In this figure, the scale of normalized growth length is again modified. The growth directions of grains A and B are diverging, and the grain boundary is formed by branching. In this case, the direction of the grain boundary is 89.7, which is rather close to the heat flow direction. Other grains disappear at the inert wall of growth domain. Finally, grain A, the growth direction of which is closer to the heat flow direction compared to grain B, is selected. Here, in Fig. 14, the diverging case is shown as an example. In the converging case, when the direction of the grain boundary, which is formed by blocking, is parallel to the heat flow direction, the same phenomenon occurs. The numbers of cases in which the inclined grain is selected are compared between Type A and Type B in the present study. The probabilities of Type A and Type B failures are not the same, and approximately 9% or more failures are classified as Type A. Therefore, eliminating the Type A failures is important in order to increase the yield rate in the single-crystal casting process. 4. Conclusions An engineering model for grain selection has been developed and applied to the single crystal casting process. Since the time required for calculation is short, a statistical analysis has been performed. (1) Increasing the yield rate requires the grain density on the chill surface to be increased. (2) The yield rate does not reach 1%. Approximately 1% of cast trials fail in the single crystal casting process. (3) There are two cases in which the inclined grain is selected during grain selection. Type A: A grain exists near the inert wall and grows inward the growth

7 Analysis of Yield Rate in Single Crystal Casting Process Using an Engineering Simulation Model 835 domain. Type B: Grains exist which converge or diverge, making the grain boundary parallel to the direction of heat flow. (4) Type A is more probable than Type B as the mechanism of failure in the single crystal casting process. Acknowledgments The authors would like to thank to Tanikawa Fund Promotion of Thermal Technology for financial support. REFERENCES 1) M. Durand-Charre: The Microstructure of Superalloys, (Gordon and Breach Science Publishers, Netherlands, 1997) pp ) G. J. S. Higginbotham: Mat. Sci. Technol. 2 (1986) ) Y. Ohta, Y. G. Nakagawa, A. Ohtomo and Y. Saiga: Bulletin of the Jpn. Inst. of Metals 24 (1985) ) K. Okamoto, Y. Kondou, J. Kaneda, A. Yoshinari and Y. Aono: J. Japan Inst. Metals 63 (1999) ) K. Okamoto, A. Yoshinari, Y. Aono and T. Kato: J. Japan Inst. Metals 63 (1999) ) D. Walton and B. Chalmers: Trans. Metall. Soc. AIME 215 (1959) ) W. Kurz and D. J. Fisher: Fundamentals of Solidification, 3rd ed. (Trans. Tech. Publication, Aedermannsdorf, Switzerland, 1992) pp ) H. Esaka, W. Kurz and R. Trivedi: Solidification Processing 1987, (The Institute of Metals, London, 1988) pp ) H. Esaka, K. Fujita, H. Daimon, M. Tamura and K. Shinozuka: J. Japan Inst. Metals 64 (2) ) N. D Souza, M. G. Ardakani, A. Wagner, B. A. Shollock and M. McLean: J. Mater. Sci. 37 (22) ) H. Esaka: Ph. D. Thesis, No. 615, Ecole Polytechnique Federale de Lausanne, Switzerland (1986). 12) M. Rappaz: Int. Mater. Rev. 34 (1989) ) Ch.-A. Gandin, M. Rappaz and R. Tintillier: Metall. Trans. 24A (1993) ) Ch.-A. Gandin and M. Rappaz: Acta Metall. Mater. 42 (1994) ) M. Rappaz, Ch.-A. Gandin, J.-L. Desbiolles and Ph. Thevoz: Metall. Mater. Trans. A 27A (1996) ) M. F. Zhu and C. P. Hong: ISIJ Int. 41 (21)