Casting and Solidification

An Application of Fractal Theory to Complex Macrostructure: Quantitatively Characterization of Segregation Morphology

Keywords:
segregation morphology,
fractal dimension,
perimeter-area method,
box-counting method,
local solidification time

2020 Volume 60 Issue 6 Pages 1188-1195

Details

Abstract

Segregation of solute elements is an inherent characteristic of alloy solidification. Macro/semi-macro segregation seriously affects the mechanical properties of the final products. High-carbon steel billets is an important base material for producing high-end rod wire, while macro/semi-macro segregation is more serious due to its high carbon element content and low distribution coefficient. In order to control the segregation defects of high-carbon steel delicately, the morphology characteristics of segregation in 82B cord steel billet (the carbon content is 0.82 wt%) produced by continuous casting were studied based on fractal theory. It is shown that segregation morphology has fractal characteristics. Different calculation methods of fractal dimension describe segregation characteristics from different angles; fractal dimension calculated by perimeter-area method (*D _{PA}*) can quantitatively characterize the complexity of segregation profile, while fractal dimension calculated by the box-counting method (

1. Introduction

Segregation is the result of redistribution of solute elements between liquid and solid phases during solidification of alloys, and it is an inherent characteristic of alloy solidification. The degree of segregation increases with decreasing the equilibrium partition coefficient of solute elements.^{1,2)} Segregation can be classified into microsegregation and macrosegregation. Microsegregation is the difference of composition within the interdendritic in a short distance, which can be alleviated or even eliminated during subsequent heat treatment.^{3)} Macrosegregation in billets is non-uniformity of composition over large areas, and their size can vary from several millimeters, centimeters or even meters.^{4)} Due to large sizes, macrosegregation is considered more harmful to finished steel properties, resulting in serious quality problems (such as crack or failure) of continuous casting billet.^{5,6,7)} According to Brimacombe,^{8)} problem of macrosegregation becomes more acute in case of high carbon steels, in high strength low alloy (HSLA) steels, as well as in high alloy steel. At the same time, Haida *et al.*^{9)} found that the centreline segregation is composed of interconnected “punctate” segregation, which is defined as semi-macro segregation. With the improvement of steel quality, semi-macro segregation has become an important factor restricting the improvement of high quality steel quality. Therefore, in order to control segregation defects delicately and effectively, it is very meaningful to study the segregation characteristics in more detail.

In general, the research of segregation is mainly carried out from two aspects: composition content and morphology. The former is mainly based on the chemical analysis of drill samples,^{6)} electron probe^{10)} and Glow Discharge Optical Emission Spectroscopy (GDOES).^{11)} However, these methods have a small analysis range or can not observe the morphology of segregation. For the latter, the macrostructure of the billet is usually obtained via hot pickling,^{12,13)} and then the segregation degree is qualitatively judged according to the macro rating diagram. These methods are convenient, however, quantitative measurement of components and qualitative description of segregation morphology are difficult to satisfy the goals of delicately control of segregation and improvement of product quality.

Morphology of cast structure in alloys is complex, diverse and irregular due to the influence of solidification conditions and liquid fluid.^{14)} To evaluate the complexity and irregularity of structure, the fractal theory has been applied to characterize the morphology of materials structure.^{15,16,17,18,19,20,21,22,23)} Yang *et al.*^{17)} described dendrite and cellular structure in Ni-based superalloy under various cooling conditions by fractal dimension. Ohsasa *et al.*^{22,23)} evaluated dendrite structure of an Al–Si alloy and predicted dendrite morphology in Fe-base-ternary alloys based on fractal dimension. Sanyal *et al.*^{24)} described the mushy zone of an alloy with continuous fractal structure, and they calculated the permeability of the mushy zone based on fractal dimension of dendrite. In previous work of our group, the fractal dimension has been applied to describe morphology of solidification structure,^{25)} carbon element distribution along casting direction in continuous casting billets,^{26)} as well as morphology of carbon segregation in electro-slag remelting (ESR) billets.^{27)} Compare to ESR technique, the cooling rate is larger, the liquid flow is stronger, and the morphology of segregation is more complex in the process of continuous casting. In additional, to the authors’ knowledge, the study of applying fractal theory to describe segregation morphology in high carbon steel billet had not been reported. In view of these, the present research is mainly based on fractal theory to analyze the segregation morphology from the perspective of morphology, thus providing guidance for delicately controlling segregation defects.

In the present work, the fractal theory was applied to describe the morphology characterizations of segregation in 82B cord steel billet with 0.82 wt% carbon content. Fractal dimension of segregation was calculated by perimeter-area method and box-counting method. Exploring the similarities and differences between perimeter-area method and box-counting method, and the physical meaning of the two methods were identified. At the same time, effects of solidification conditions on the segregation morphology characterizations were discussed.

2. Experimental Procedures

High carbon steel billet samples (an 82B cord steel billet with 0.82 wt% carbon content) have been collected from the continuous casting shop, the size of cross-sectional is 150 mm × 150 mm. The billets were etched with 1:1 warm hydrochloric acid-water solution for revealing the cast structure more clearly. The temperature of hydrochloric acid-water solution is 60–80°C, and macroetching time is 22 min. After macroetching, the cast structure of each billet was obtained. As shown in **Fig. 1**, a total of 13 samples (1#–13#) were symmetrically chosen along the centerline of the billet, and each sample has a size of 10 mm × 10 mm. These macrostructures of all samples are photographed by a Sony a6000 high-definition digital camera under the same lighting conditions. According to the principle of hot pickling, the content of solute element in segregation zone is higher and the free energy of Gibbs is high, the reaction with hydrochloric acid is intense and the color becomes black when the billet is etched. After hot pickling experiment, the segregation morphology can be identified by the black zone in the macrostructure of the billet, and the white zone is solidification structure. In order to make the research results more representative, the billets with superheat of 29°C and 35°C were selected for analysis.

Fig. 1.

Schematic of sample locations in cross section of billet.

After obtaining the macrostructure of the billet, the image was preprocessed with the image processing software Image-Pro Plus 6.0 without changing any morphology features. The preprocessing includes converting the original image into a grayscale image, sharpening and defogging, and adjusting the contrast. The preprocessed image of the macrostructure can distinguish clearly between the solidification structure and segregation zone, thus reducing analysis errors to a great extent.

3. Results and Discussion

3.1. Morphology of Cast Structure
**Figures 2** and **3** show macrostructure images of the overall and local positions of the billet for 29°C and 35°C superheats.^{25)} Dark regions and white dendrite correspond to carbon segregation and solidification structure, respectively. As indicated in the Fig. 2, macrostructure consists of three regions: chill, columnar and equiaxed zones in the billet, *i.e.*, the morphology of solidification structure is different from the outer to middle zone of the billet. Solidification structures in the outer zone (samples 1#, 2#, 12# and 13#) are mainly columnar grains, samples 3#, 4#, 10# and 11# are columnar to equiaxed transition (CET) zone, whereas samples 5#–9# are mainly equiaxed grains. As shown in the Fig. 3, samples 1#, 4# and 6# are typical columnar grains, CET and equiaxed grains, respectively. Meanwhile, the morphology of carbon segregation in billets varies with the change of solidification structure.

Fig. 2.

Macrostructure of 82B cord steel billet produced via continuous casting. (Online version in color.)

Fig. 3.

Typical morphology of columnar grains, CET, and equiaxed grains. (Online version in color.)

Fractal theory, presented by American scientist B.B.Mandelbrot, describes the phenomenon with irregular, unstable and highly complex structures in nature from the perspective of self-similar. The introduction of fractal theory provides guidance for people to measure the irregularity of the real world, and fractal dimension was used to evaluate the complexity of fractal graphics quantitatively.^{28)} Fractal dimensions are important because they can be defined in connection with real-world data, and they can be measured approximately by means of experiments. There are many ways for calculating fractal dimension in fractal geometry. In generally, different calculation methods describe fractal objects from different angles, and the value of fractal dimension is different. The perimeter-area method^{15,29,30)} is often used to calculate fractal dimension of fractal islands, whereas box-counting method^{19,20,21,22,23,31)} is one of the most widely used method to calculate fractal dimension at present. Therefore, in the present work, fractal dimension of segregation is calculated based on perimeter-area method and box-counting method, and the physical meaning of the two methods were identified.

The perimeter-area method is mainly used to calculate the fractal dimension of single or group irregular island patterns (*i.e.*, fractal islands). In generally, the ratio of perimeter to square root of area of the regular pattern is only related to the geometry of the graphic, but not to the size. However, the boundary length is closely related to the measurement scale for irregular fractal islands. When the scale becomes smaller until it approaches zero, the length of the boundary will increase or tend to infinity, while the area surrounded by fractal island will tend to be finite. In view of these, the researchers propose to use fractal dimension calculated by perimeter-area method to quantitatively characterize the complexity of fractal island boundaries. As indicated in the Figs. 2 and 3, carbon segregation (black regions) is separated by solidification structure into separate segregation points (similar to fractal islands), its morphology is complex and irregular. Thus, perimeter-area method can be used to calculate fractal dimension of segregation.

The determination of fractal dimension by perimeter-area method is mainly based on the relationship between perimeter and area, the principle is shown in Eqs. (1) and (2). Procedure of perimeter-area method is as follows; the perimeter and area of all segregation points is calculated by image analysis software Image-Pro Plus 6.0. Then, ln*A* is taken as horizontal coordinate and ln*P* as vertical coordinate in the dot diagram. The slope *S* is obtained by using the least-squares method to expand the linear quasi-combination of all data points in the graph. Then the fractal dimension is twice as high as its slope, that is, *D*=*2S*.

(1) |

(2) |

**Figure 4** shows the relationship between ln*P* and ln*A* of sample 1# for 29°C superheat. The slope of the fitted curve is 0.6755, and the fitting coefficient *R ^{2}* is 0.9837. According to the calculation principle of the perimeter-area method, the fractal dimension of sample 1# for 29°C superheat is twice as high as its slope,

Fig. 4.

Relationship between ln*P* and ln*A* for fractal dimension of 1# sample under 29°C (*P*-perimeter, *A*-area). (Online version in color.)

Table 1. Fitting coefficients for perimeter-area method and box-counting method.

R^{2}Sample | R (perimeter-area method)^{2} | R (box-counting method)^{2} | ||
---|---|---|---|---|

29°C | 35°C | 29°C | 35°C | |

1# | 0.9837 | 0.9847 | 0.9997 | 0.9997 |

2# | 0.9791 | 0.9823 | 0.9994 | 0.9997 |

3# | 0.9831 | 0.9861 | 0.9994 | 0.9996 |

4# | 0.9825 | 0.9847 | 0.9994 | 0.9994 |

5# | 0.9851 | 0.9842 | 0.9988 | 0.9995 |

6# | 0.9809 | 0.9799 | 0.9991 | 0.9994 |

7# | 0.9830 | 0.9811 | 0.9992 | 0.9995 |

8# | 0.9826 | 0.9796 | 0.9993 | 0.9994 |

9# | 0.9865 | 0.9879 | 0.9990 | 0.9993 |

10# | 0.9858 | 0.9823 | 0.9991 | 0.9992 |

11# | 0.9881 | 0.9799 | 0.9993 | 0.9994 |

12# | 0.9841 | 0.9845 | 0.9997 | 0.9997 |

13# | 0.9859 | 0.9854 | 0.9997 | 0.9997 |

Fractal dimension calculated by perimeter-area method of the segregation (denoted as *D _{PA}*) at different locations and average

Fig. 5.

*D _{PA}* at different locations for 29°C and 35°C superheats. (a) different locations, (b) average value of left and right sides. (Online version in color.)

Macrostructure of billet is composed of solidification structure and segregation, and the segregation morphology is closely related to solidification structure. A lot of fine equiaxed grains are formed on the surface of billet due to a large number of heterogeneous nucleation, and excess solute is discharged from equiaxed grains into the surrounding liquid phase. These solute-rich liquid phases will form point segregation (semi-macrosegregation) after solidification around the dendrites. Due to the large undercooling, the shape of the equiaxed grains formed on the surface of the billet is different, which makes the morphology of point segregation complex and diverse. Dendrites in the columnar grains region grow in the opposite direction from heat flux. The farther away from the mold wall, the more concentrated the dendritic growth direction, and the larger the average grain size. Thus, the distribution of dendrite is regular, and the complexity of segregation morphology around columnar grains decreases gradually. Dendrite has enough time coarsening in equiaxed regions due to the lower cooling rate, which makes the dendrite morphology simple. Thereby, the segregation morphology in equiaxed regions is also relatively simple. Therefore, *D _{PA}* decreases gradually from the outer to the middle zones of the billet.

Box-counting dimension is one of the most widely used dimension. Its popularity is largely due to its relative ease of mathematical calculation, and there are no special requirements for image morphology. Procedure of the box-counting method is as follows; the macrostructure is covered by the square meshes with the size of *r*. The number of meshes, *N* (*r*), in which segregation regions is included, is counted. Then, the mesh size, *r*, is changed and the same procedure is repeated. If the following relationship (as shown in Eq. (3)) is made up between the number of the meshes, *N* (*r*), and the mesh size, *r*, the geometry of the segregation is fractal. In the actual calculation, a series of *r* and *N* (*r*) are obtained. Then, the dot diagram is made with ln(*r*) as horizontal coordinate and ln(N (r)) as vertical coordinate. The slope is obtained by linear fitting of all data points by least-square method in the diagram. Then, fractal dimension is the absolute value of slope.

(3) |

The relationship between ln*N*(*r*) and ln*r* of sample 1# for 29°C superheat as shown in **Fig. 6**. The slope of the fitted curve is −1.7930, and the fitting coefficient *R ^{2}* is 0.9997. According to the calculation principle of the box-counting method, the fractal dimension of sample 1# for 29°C superheat is the absolute value of slope,

Fig. 6.

Relationship between ln*N*(*r*) and ln*r* for fractal dimension of 1# sample under 29°C (*N*(*r*): Box count; *r*: Box size; R^{2}: fitting coefficients).

Fractal dimension calculated by box-counting method of the segregation (denoted as *D _{BC}*) at different locations and average

Fig. 7.

*D _{BC}* at different locations for 29°C and 35°C superheats. (a) different locations, (b) average value of left and right sides. (Online version in color.)

Due to the existence of a large number of fine equiaxed grains in chill, the solute-rich liquid phases solidifies around equiaxed grains, which increases the complexity of segregation point distribution. With the formation of columnar grains, the direction of dendritic growth is basically the same (opposite to the direction of heat flow), and there is a certain regularity. The segregation points around columnar grains are also regularly distributed, thus the complexity of their distribution decreases gradually. The billet is solidified with equiaxed grains due to constitutional undercooling^{32)} and heat transfer in the later stage of solidification. The cooling rate is small and dendrite has sufficient time to grow in the center of billets, the dendritic morphology in equiaxed zone is relatively simple. However, the central equiaxed grains float in the residual liquid phase with the flow of liquid phase, and the dendrite is randomly distributed in billet. Thus, the segregation is also randomly distributed in the center of billets. Therefore, *D _{BC}* gradually decreases and then increases from the outside to the center of billet.

It should be noted that the value of *D _{PA}* and

Secondary dendrite arm spacing (SDAS) is a very important parameter in the solidification process of metals. It reflects the solidification conditions of the billet and is closely related to the internal quality of the products. In addition, SDAS can predict the local cooling rate and local solidification time during the solidification of the billet.^{33)} The SDAS of the billet is measured with the linear intercept method, multiple typical dendrites are measured and their average values are used as SDAS values. Relationship between fractal dimension and SDAS as shown in **Fig. 8**. It can be seen that *D _{PA}* decreases with increasing SDAS, and there is no obvious relationship between

Fig. 8.

Relation between fractal dimension and SDAS. (a) *D _{PA}*, (b)

The complexity of macro/semi-macro segregation morphology is closely related to solidification structure. Solidification structure grows in the form of dendrite in billets because of the high cooling rate in the continuous casting process, and the liquid phase of the rich solute is distributed around dendrites. Flow of residual solute enriched liquid in the mushy zone is induced by suction created by the solidification shrinkage, solutal convection, sedimentation of free crystallites, bulging of solidifying strands between the support rolls, deformation of dendrites, *etc.*^{1,11)} Flow of residual solute enriched liquid will lead to macrosegregation in the center of billet. The other part of the liquid phase solidifies gradually around the dendrite with decreasing temperature, and finally forms semi-macro segregation point. According to solidification theory, the solid-liquid interface is more stable when there is no undercooling or low undercooling in the front of the solid-liquid interface. When the degree of undercooling is high, the stability of the solid-liquid interface is destroyed, which makes the morphology of the interface complex. The smaller SDAS, the greater the local cooling rate as shown in Eq. (4). The degree of undercooling increases with increasing cooling rate.^{34)} The increase of undercooling makes the solidification interface (segregation morphology) more complex, resulting in the increase of *D _{PA}*. On the other hand, semi-macro segregation is distributed among dendrites and separated into independent segregation points by dendrite arm. The morphology of the segregation profile depends on dendrite arm. Therefore, SDAS affects the profile of segregation points, whether from the point of view of solidification theory or geometric relationship. However, the distribution of segregation point in billet mainly depends on the liquid phase flow. When the liquid phase flow is strong, the segregation is mostly concentrated in the center of the billet, while the more dispersed the segregation distribution when the liquid phase flow is weak. In the actual solidification process, SDAS only affects the flow of liquid phase to a certain extent, and does not play a decisive role. Thus, there is no obvious relationship between

The SDAS of billet depends mainly on the local cooling rate as shown in Eq. (4).^{35,36)} Jacobi and Schwerdtfeger^{37)} proposed that m, n values can be 109.2 and 0.44 for high carbon steel, respectively. Then, the local solidification time of each location in billets can be calculated according to Eq. (5).

(4) |

(5) |

**Figure 9** shows local solidification time at different locations. It can be seen that local solidification time increases gradually from the outside to the center of billet. This is mainly because the cooling medium acts on the surface of the billet during the solidification process, the local cooling rate decreases gradually from the surface to the center, hence the local solidification time increases.

Fig. 9.

Local solidification time at different locations. (Online version in color.)

**Figure 10** presents the relationship between *D _{PA}* and cube root of local solidification time. Fractal dimension of segregation point decreases with increasing cube root of local solidification time. The data in the diagram are fitted linearly by least-square method, and the function relation shown in Eq. (6) is obtained. It is well known SDAS is proportional to the cube root of local solidification time.

(6) |

Fig. 10.

Relation between *D _{PA}* and cube root of local solidification time,

Local solidification time is a very important parameter in alloy solidification process. The majority of studies calculate the local solidification time of the alloy by SDAS. It is found that it is relatively easy to measure SDAS in the columnar grains regions. However, for equiaxed grains, the measurement of SDAS is usually difficult and the measurement error is large because of the influence of dendrite coarsening. It is difficult to obtain the local solidification time of the alloy accurately. Compared with SDAS, fractal dimension is easy to obtain and the error is small, and there is a functional relationship between *D _{PA}* and local solidification time as shown in Eq. (6). This result demonstrates that local solidification time of billets can be quantitatively evaluated by

From a computational point of view, fractal dimension is relative simple. Fractal dimension provides us with a new perspective to understand the characteristics of segregation defects in billets, which makes us have a deeper understanding of the formation of segregation, thus providing guidance for controlling segregation defects. In additional, Fractal dimension can also be attempted to describe the morphology characteristics of segregation points in other types steel grade.

4. Conclusions

Present study was undertaken to investigate the segregation morphology characteristics in 82B cord steel billet (carbon content of 0.82 wt%). Macrostructure of the billet was obtained with a 1:1 warm hydrochloric acid-water solution. Then the morphology characteristics of segregation were analyzed based on fractal theory. Some interesting results of the present study are summarized as follows:

(1) Morphology of segregation in billets has fractal characteristics. Fractal dimension calculated by perimeter-area method (*D _{PA}*) can quantitatively characterize the complexity of segregation profile, while fractal dimension calculated by the box-counting method (

(2) SDAS mainly affects the complexity of segregation profile, while there is no obvious relationship between SDAS and spatial distribution characteristics of segregation in billet.

(3) Negative-correlation is shown between *D _{PA}* and cube root of local solidification time (the fitting coefficient is 0.79). This finding indicates that fractal dimension can be used as a parameter for estimating local solidification time of the billet in which the measurement of SDAS is difficult.

Acknowledgement

The authors are very grateful for support from United Funds between National Natural Science Foundation and Baowu Steel Group Corporation Limited from China (No.U1860101) and Chongqing Fundamental Research and Cutting-Edge Technology Funds (No.cstc2017jcyjAX0019). Many thanks to companies for providing Image-Pro Plus ® 6.0 Software Company.

References

- 1) M. C. Flemings:
*ISIJ Int.*,**40**(2000), 833. - 2) G. Lesoult:
*Mater. Sci. Eng. A*,**413–414**(2005), 19. - 3) A. Ghosh:
*Sadhana*,**26**(2001), 5. - 4) D. Z. Li, X. Q. Chen, P. X. Fu, X. P. Ma, H. W. Liu, Y. Chen, Y. F. Cao, Y. K. Luan and Y. Y. Li:
*Nat. Commun.*,**5**(2014), 5572. - 5) E. J. Pickering and H. K. D. H. Bhadeshia:
*Metall. Mater. Trans. A*,**45**(2014), 2983. - 6) S. K. Choudhary and S. Ganguly:
*ISIJ Int.*,**47**(2007), 1759. - 7) F. Mayer, M. Wu and A. Ludwig:
*Steel Res. Int.*,**81**(2010), 660. - 8) J. K. Brimacombe:
*Metall. Mater. Trans. A*,**30**(1999), 1899. - 9) O. Haida, H. Kitaoka, Y. Habu, S. Kakihara, H. Bada and S. Shiraishi:
*Trans. Iron Steel Inst. Jpn.*,**24**(1984), 891. - 10) N. Yoshida, O. Umezawa and K. Nagai:
*ISIJ Int.*,**43**(2003), 348. - 11) S. K. Choudhary, S. Ganguly, A. Sengupta and V. Sharma:
*J. Mater. Process. Technol.*,**243**(2017), 312. - 12) K. Miyamura, S. Y. Kitamura, S. Sakaguchi, C. Hamaguchi and M. Hirai:
*Trans. Iron Steel Inst. Jpn.*,**24**(1984), 718. - 13) B. Sang, X. Kang and D. Li:
*J. Mater. Process. Technol.*,**210**(2010), 703. - 14) T. Li, X. Lin and W. D. Huang:
*Acta Mater.*,**54**(2006), 4815. - 15) B. B. Mandelbrot, D. E. Passoja and A. J. Paullay:
*Nature*,**308**(1984), 721. - 16) M. Tanaka:
*J. Mater. Sci.*,**31**(1996), 3513. - 17) A. Yang, Y. Xiong and L. Liu:
*Sci. Technol. Adv. Mater.*,**2**(2001), 101. - 18) S. Kobayashi, T. Maruyama, S. Tsurekawa and T. Watanabe:
*Acta Mater.*,**60**(2012), 6200. - 19) H. Ishida, Y. Natsume and K. Ohsasa:
*ISIJ Int.*,**49**(2009), 37. - 20) M. Yamamoto, I. Narita and H. Miyahara:
*Tetsu-to-Hagané*,**99**(2013), 72 (in Japanese). - 21) F. Satou, H. Esaka and K. Shinozuka:
*Tetsu-to-Hagané*,**99**(2013), 108 (in Japanese). - 22) R. Sugawara, T. Itoh, Y. Natsume and K. Ohsasa:
*Tetsu-to-Hagané*,**99**(2013), 126 (in Japanese). - 23) T. Hatayama, Y. Natsume and K. Ohsasa:
*Tetsu-to-Hagané*,**103**(2017), 695 (in Japanese). - 24) D. Sanyal, P. Ramachandrarao and O. P. Gupta:
*Chem. Eng. Sci.*,**61**(2006), 307. - 25) J. H. Cao, Z. B. Hou, D. W. Guo, Z. A. Guo and P. Tang:
*J. Mater. Sci.*,**54**(2019), 12851. - 26) Z. B. Hou, G. G. Cheng, C. C. Wu and C. Chen:
*Metall. Mater. Trans. B*,**43**(2012), 1517. - 27) Z. B. Hou, J. H. Cao, Y. Chang, W. Wang and H. Chen:
*Acta Metall. Sin.*,**53**(2017), 769 (in Chinese). - 28) B. B. Mandelbrot:
*Science*,**156**(1967), 636. - 29) P. McAnulty, L. V. Meisel and P. J. Cote:
*Phys. Rev. A*,**46**(1992), 3523. - 30) H. P. Xie and Z. D. Chen:
*Eng. Mech.*,**6**(1989), No. 4, 1 (in Chinese). - 31) A. L. Genau, A. C. Freedman and L. Ratke:
*J. Cryst. Growth*,**363**(2013), 49. - 32) W. A. Tiller, K. A. Jackson, J. W. Rutter and B. Chalmers:
*Acta Metall.*,**1**(1953), 428. - 33) Y. M. Won and B. G. Thomas:
*Metall. Mater. Trans. A*,**32**(2001), 1755. - 34) J. F. Xu, M. Xiang, B. Dang and Z. Y. Jian:
*Comput. Mater. Sci.*,**128**(2017), 98. - 35) S. C. Michelic, J. M. Thuswaldner and C. Bernhard:
*Acta Mater.*,**58**(2010), 2738. - 36) C. Cicutti and R. Boeri:
*Scr. Mater.*,**45**(2001), 1455. - 37) H. Jacobi and K. Schwerdtfeger:
*Metall. Trans. A*,**7**(1976), 811. - 38) W. Kurz and D. J. Fisher: Fundamentals of Solidification, Trans Tech Publications, Zurich, Switzerland, (1998), 85.

© 2020 by The Iron and Steel Institute of Japan