2020 Volume 60 Issue 6 Pages 1188-1195
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 (DPA) can quantitatively characterize the complexity of segregation profile, while fractal dimension calculated by the box-counting method (DBC) reflects the spatial distribution characteristics of segregation in billets. Secondary dendrite arm spacing (SDAS) mainly affects the complexity of segregation profile. In additional, negative-correlation is shown between DPA and cube root of local solidification time (the fitting coefficient is 0.79). This result demonstrated the potential of DPA as a parameter for estimating local solidification time of the billet in which the measurement of SDAS is difficult.
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 probe10) 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.
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.
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.
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.
Macrostructure of 82B cord steel billet produced via continuous casting. (Online version in color.)
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 method15,29,30) is often used to calculate fractal dimension of fractal islands, whereas box-counting method19,20,21,22,23,31) 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, lnA is taken as horizontal coordinate and lnP 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.
Figure 4 shows the relationship between lnP and lnA of sample 1# for 29°C superheat. The slope of the fitted curve is 0.6755, and the fitting coefficient R2 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, i.e., 1.3510. The same method was used to calculate the fractal dimension and corresponding fitting coefficient of the remaining samples for superheats of 29°C and 35°C. According to Table 1, the fitting coefficients of perimeter-area method are close to 1. This indicates that the use of the fractal dimension calculated by perimeter-area method to quantitatively describe the morphology of segregation is effective. As shown in Fig. 4, fractal dimension increase with increasing perimeter when the area is constant. Meanwhile, the larger the perimeter, the more complex the profile of the graphic when the area is constant. Therefore, fractal dimension calculated by perimeter-area method can quantitatively describe the complexity of the profile morphology of segregation, and the larger fractal dimension, the more complex the profile of segregation.
Relationship between lnP and lnA for fractal dimension of 1# sample under 29°C (P-perimeter, A-area). (Online version in color.)
Fractal dimension calculated by perimeter-area method of the segregation (denoted as DPA) at different locations and average DPA of the left-right symmetrical position for 29°C and 35°C superheats as shown in Figs. 5(a) and 5(b), respectively. It can be seen that the variation trend of DPA with position is basically the same under the different superheats. That is, DPA decreases gradually from the outside to the center of billet. For the profile of segregation point in two-dimensional plane, the fractal dimension is between 1 and 2.19) The closer the fractal dimension is to 1, the smoother the profile of the segregation point. The closer the fractal dimension is to 2, the more rough the profile of the segregation point. The larger fractal dimension, it indicates that the more complex the profile of the segregation point. As can be seen from macrostructure of the billet in Figs. 2 and 3, DPA of equiaxed grain zone is the minimum. Therefore, the morphology complexity of segregation point in equiaxed grains zone is less than columnar grains.
DPA 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, DPA 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.
The relationship between lnN(r) and lnr 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 R2 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, i.e., 1.7930. The same method was used to calculate the fractal dimension and fitting coefficient of the remaining samples for superheats of 29°C and 35°C. According to Table 1, the fitting coefficients of box-counting method are close to 1. This indicates that fractal dimension calculated by box-counting method is applied to quantitatively describe the morphology of segregation is also effective. As indicated in Fig. 6, fractal dimension increase with increasing the number of boxes when the size of the box is constant. Meanwhile, when r is constant, the greater N(r), the more dispersed the segregation point. Therefore, fractal dimension calculated by box-counting method is another measure of segregation characteristics, it reflects spatial distribution characteristics of segregation point in macrostructure of the billet. The larger fractal dimension, the more complex of the segregation point distribution in billets.
Relationship between lnN(r) and lnr for fractal dimension of 1# sample under 29°C (N(r): Box count; r: Box size; R2: fitting coefficients).
Fractal dimension calculated by box-counting method of the segregation (denoted as DBC) at different locations and average DBC of the left-right symmetrical position for 29°C and 35°C superheats is shown in Figs. 7(a) and 7(b), respectively. It can be seen that the variation trend of DBC with position is basically the same under the different superheats. That is, DBC gradually decreases and then increases from the outside to the center of billet. This indicates the complexity of segregation point distribution in macrostructure gradually decreases and then increases.
DBC 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 undercooling32) 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, DBC gradually decreases and then increases from the outside to the center of billet.
It should be noted that the value of DPA and DBC is different as shown in Figs. 5 and 7. This is mainly because the definition of fractal dimension is given on the basis of the corresponding measure, and different definitions often have different measures. As shown in Figs. 2 and 3, the morphology of carbon segregation in billets varies with the change of solidification structure. Therefore, the value of DPA and DBC is closely related to the morphology of solidification structure. The perimeter-area method is based on perimeter and area to describe the complexity of the segregation profile. The box-counting method is mainly based on the ratio of space occupied to describe the distribution characteristics of segregation in billets. Thus, the value of DPA depend on the morphology of dendritic interface, while the value of DBC depend on the spatial distribution of dendrite in billet. In additional, the box-counting method is relatively popular, but it also has shortcoming. When counting the total number of meshes N(r), the meshes including one point and many points have the same weight, lacking weight factor. Therefore, different calculation methods of fractal dimension describe the fractal characteristics of objects from different angles, and the value of fractal dimension may also be different. Fractal dimension obtained by different calculation methods can not be simply compared. The relationship between fractal dimension and secondary dendrite arm spacing will be discussed in next section in order to further study the influence parameters of fractal dimension.
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 DPA decreases with increasing SDAS, and there is no obvious relationship between DBC and SDAS. This indicates that SDAS affects the complexity of segregation profile, while there is no obvious relationship between SDAS and spatial distribution characteristics of segregation in billet.
Relation between fractal dimension and SDAS. (a) DPA, (b) DBC. (Online version in color.)
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 DPA. 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 DBC and SDAS.
The SDAS of billet depends mainly on the local cooling rate as shown in Eq. (4).35,36) Jacobi and Schwerdtfeger37) 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).
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.
Local solidification time at different locations. (Online version in color.)
Figure 10 presents the relationship between DPA 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.38) There is a negative correlation between DPA and SDAS as shown in Fig. 8(a). Thus, there is also a negative correlation between DPA and the cubic root of local solidification time, and the fitting coefficient is 0.79 in Fig. 10. This indicates that the complexity of segregation profile of billet during solidification is mainly controlled by local solidification time. The smaller the local solidification time, the more complex the segregation profile.
Relation between DPA and cube root of local solidification time, θf1/3. (Online version in color.)
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 DPA and local solidification time as shown in Eq. (6). This result demonstrates that local solidification time of billets can be quantitatively evaluated by DPA in which the measurement of SDAS is difficult. At the same time, the functional relationship between DPA and local solidification time is only for high-carbon steel billet and the number of samples is limited in present work. In order to verify the universality of Eq. (6), further research is needed.
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.
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 (DPA) can quantitatively characterize the complexity of segregation profile, while fractal dimension calculated by the box-counting method (DBC) reflects the spatial distribution characteristics of segregation in billets. This is mainly due to the definition of fractal dimension is given on the basis of the corresponding measure, and different definitions often have different measures.
(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 DPA 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.
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.