Study on the morphology and distribution of Fe phase in solidification structure of Cu-15Fe alloy ingots

The realization of high strength of copper-iron (Cu-Fe) alloy is related to its solidification structure. The morphology and distribution characteristics of Fe phase in the solidification structure of Cu-15Fe alloy ingot were analyzed, and the deformation strength of the alloy was compared. The results show that the cooling conditions can affect the size, morphology and distribution of Fe phase in the solidified structure. The average distribution uniformity in the water-cooled copper mold ingot (W-ingot) is 0.45% higher than that in the quartz mold ingot (Q-ingot). The distribution quality of Fe phase in solidification structure can be evaluated by fractal dimension and average Fe phase area. The larger the fractal dimension is, the smaller the average Fe phase area is, where Fe phase is smaller and more uniform in the corresponding region. In the experiment, the strength of the strip increased from 510 to 547 Mpa corresponds to the Q-ingot and the W-ingot.


Introduction
Copper-iron (Cu-Fe) alloys have the properties of both copper (Cu) and iron (Fe) components and exhibits the advantages of high strength, electrical conductivity, thermal conductivity, electromagnetic wave shielding, and low cost, and can be used in connectors, springs, electrical wires, transmission wires, and cables, lead frames, and other electrical devices [1][2][3][4]. The structure of alloy after solidification is composed of a mixture of Fe phase and Cu matrix. Through the mechanical deformation, Fe phase in the Cu matrix can be aligned filaments with a ribbon-like cross-section to obtain a microcomposites, which have the high strength and good electrical conductivity [5][6][7][8]. The strength of Cu-Fe alloy during the deformation process is directly affected by the volume fraction, dendrite size and distribution uniformity of Fe phase in the solidification structure [6][7][8][9][10]. Therefore, in order to control the quality of solidification structure to improve the properties of Cu-Fe alloy, it is of great significance to quantitatively describe and characterize the Fe phase in the solidification structure.
The solidification structure of the material includes two aspects: macrostructure and microstructure. Macrostructure refers to the microstructure inside the material observed with the naked eye. Microstructure refers to the internal morphology of the material observed through the microscope. The growth of γ-Fe in Cu-Fe alloy is limited by the solid-liquid two-phase region, and the morphology and size of Fe phase are difficult to be observed directly by naked eyes. Therefore, it is very important to study the characteristics of Fe phase microstructure. The traditional metallographic examination mainly relies on the comparative analysis, so it can only reach the qualitative and semi quantitative state [11]. The element distribution or segregation ratio can be used to quantitatively express the uneven distribution of elements over a long distance. However, because the precipitation and distribution of Fe phase in the solidification process of Cu-Fe alloy are affected by multiple repeated miscellaneous factors, the correlation between the distribution of Fe elements and the distribution of Fe phase structure is low, and the distribution uniformity obtained is significantly different. Dendrite arm spacing and the curvature of dendrite tip are mainly used to describe the local solidification structure of a single dendrite [12]. The parameter lack a quantitative description of the overall morphology of solidification Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. structure, especially for the requirements of the properties of Cu-Fe alloy. Therefore, a new method or parameter is needed to quantitatively describe the Fe phase in the solidification structure.
The distribution uniformity of Fe phase is reflected in the dispersion degree of the cross-sectional distribution of the structure in the metallographic analysis. Due to the complexity of Fe phase distribution in solidification structure, fractal dimension was proposed to analyze the solidification structure of Cu-Fe alloy in this study. Fractal theory takes the irregular and unsmooth geometric bodies in nature and nonlinear systems, as well as the disordered (irregular) and self-similar systems in social activities as the objects, directly recognizes the internal laws from the nonlinear complex systems, and uses a more convenient quantitative method to express the complex objects that could not or were difficult to be quantitatively described before. It is widely used in many fields to reveal the laws behind complex and disordered chaotic phenomena and irregular forms and the essential relationship between the whole and the part [13][14][15][16][17][18][19]. Zhang et al [16] used fractal dimension to describe the fractal dimension of particles in materials and its relationship with heat transfer. Macek [17] studied the bending torsional fatigue fracture process of steel using fractal dimension. Yang et al [18] described dendrite and cellular structure in Ni-based super alloy under various cooling conditions by fractal dimension. CaO et al [19] evaluated carbon segregation morphology of high carbon steel and die steel based on fractal dimension.
In this study, Fe content (w), Fe phase fraction (V), average Fe phase area (S) and secondary dendrite arm spacing (λ 2 ) in solidification structure were measured. The shape factor (F) was used to describe the complexity of Fe phase morphology in solidification structure. The fractal dimension (D) was calculated to quantitatively the distribution characterize of Fe phase in solidification structure and the influence mechanism of morphology characteristics and distribution uniformity were discussed. In addition, the alloy was deformed and the tensile strength was measured.

Materials and methods
2.1. Materials preparation 12 kg Cu-15Fe alloy ingot was casted by vacuum induction furnace. Oxygen free copper (Cu > wt99.95%) and pure iron (Fe > 99.86 wt%) were used as raw materials. The raw materials were heated to 1450°C and then were casted into the water-cooled copper mold and the quartz mold to obtain a round ingot with a diameter of 110 mm and a height of about 140 mm. A 10 mm thick disc was taken from the bottom 70 mm. Figure 1 shows the cutting and marking of the sample. The Fe content at the marking position was measured by using X-ray fluorescence spectrometer (XRF). The sample were mirror polished and clean before the etched in a solution of 10 g of FeCl 3 + 30 ml of HCl +120 ml of H 2 O. The microstructure feature was analyzed by Olympus optical microscopy (OM).

Fractal dimension
It is necessary to preprocess the obtained metallographic structure diagram before calculating the fractal dimension. Preprocessing is just to convert the original image into a gray-scale image, adjust the contrast, and do not change any morphological features. The basic principle of calculating the fractal dimension of Fe phase distribution is to take the structural center as the origin, cover the solidification structure with a box with size z and obtain the number of Fe phase N(z) covered in the box. Then, the box size, z, was changed to obtain different the number of Fe phase numbers N(z). the fractal dimension can be obtained by equation (1): Where D is the fractal dimension of Fe phase distribution in solidified structure; z is the box size; N(z) is the amount of Fe phase covered in the box. The particles are distributed on the plane. If the distribution of particles has self-similarity, there is N(z) ∼ z 2 . The more uniform the particle distribution is, the closer the fractal dimension is to 2 of the ideal distribution. A series of points composed of z and N(z) was obtained firstly. Then, a dot plot with ln (z) as the horizontal coordinate and ln (N(z)) as the longitudinal coordinate were obrained. The slope was calculated by fitting all data points and the absolute value of the slope is the fractal dimension. Figure 2 shows the fitting relationship between ln (N(z)) and ln (z) corresponding to the solidification structure of the quartz mold ingot at 1.0 cm. The slope of the fitting line is 1.5254, and the fitting coefficient R 2 is 0.9795. So, the fractal dimension of Fe phase distribution of the quartz mold ingot at 1.0 cm is 1.5254. When the box size is constant, the larger the number of Fe phase, the more dispersed the Fe phase distribution. Therefore, the larger the fractal dimension, the more dispersed and uniform the Fe phase distribution.

Results and discussion
3.1. Microstructure morphology Figures 3 and 4 are the microstructure diagrams of different positions in the quartz mold ingots (Q-ingot) and water-cooled copper mold ingots (W-ingot), respectively. The black is Fe phase (α-Fe, which is transformed from γ-Fe), the yellow part is Cu matrix. It can be seen that there are different solidification structure in the ingot from the surface to the center, where there are two types of Fe phase: cellular crystal and dendrite. The solidification structure in the quartz mold ingot presents dendrite morphology from the surface to the center, and the dendrite near the center area is thick. In the water-cooled copper mold ingot, the size of Fe phase is significantly smaller than the former, indicating that the cooling rate affects the morphology and size of Fe phase. The solidification structure near the surface of water-cooled copper mold ingot is mainly fine cellular crystal, and the solidification structure mainly presents dendrite morphology away from the surface. The size of Fe dendrite increases with the distance from the surface. In addition, there are a large number of randomly distributed Fe phases cellular in the W-ingot, which is different from the regular dendrite arrangement in the Q-ingot.

Secondary dendrite arm spacing (SDAS)
In the solidification structure of ingot, there are Fe phases growing in the form of dendrites. The secondary dendrite arm spacing is an important solidification process parameter and it reflects the solidification conditions of the alloy. Figure 7 illustrates the change of SDAS at different positions in the ingot. It can be seen that SDAS increases gradually from the surface to the center. The variation range of SDAS of Fe phase in the Q-ingot and the W-ingot is 11.44 ∼ 15.67 and 3.19 ∼ 9.02 μm, respectively. The average values are 12.50 and 6.69 μm, respectively.
In general, SDAS is largely determined by the cooling process during solidification. In the process of dendrite growth, the different curvatures between the secondary dendrite arms result into the different concentration of solute in the liquid phase around each dendrite arm. The smaller the dendrite curvature, the smaller the  concentration of solute in the nearby liquid phase. The existence of concentration gradient makes the solute diffuse from the coarse dendrite to the fine dendrite, resulting in the melting of the fine dendrite and the coarsening of the coarse dendrite [12]. The solid-liquid region stays longer during the solidification, the more sufficient the above process is, and the greater SDAS is. Because the heat transfer during solidification depends on the ingot surface, the local cooling rate gradually decreases from the surface to the center, and the local solidification time gradually increases. Therefore, the SDAS increases in the central area of the ingot.

Shape factor (F)
The shape factor (F) is a parameter to describe the overall morphological characteristics, which can describe the complexity of Fe morphology. The calculation formula of average shape factor is F = 4 πA/P 2 , where A is the average area of Fe phase, P is the average circumference of Fe phase. The closer the value of shape factor is to 1, the rounder the morphology of the structure is, and the lower the complexity of the morphology is [20]. Figure 8 shows the change of Fe phase shape factor in ingot. The shape factor in the solidification structure from the surface to the center of the ingot decreases gradually. The average shape factor in the symmetrical position of the Q-ingot and the W-ingot is 0.60 ∼ 0.49 and 0.62 ∼ 0.47, respectively. Therefore, the above results means that the complexity of Fe phase morphology increases with the decrease of cooling rate from outside to inside.
The solidification morphology of alloy is affected by heat transfer and mass transfer. When Fe phase precipitates in liquid phase during solidification, in order to make the reaction continue, Fe atoms in the distant liquid phase also need to diffuse to the solid-liquid interface, while the excess Cu atoms at the solid-liquid interface need to diffuse to the liquid phase. The diffusion of Fe atoms will form a Fe rich solute layer at the front of the solidification interface, causing undercooling. The criterion of composition undercooling was proposed  by Rutter and Tiller et al [21]. The degree of component undercooling is mainly affected by the temperature gradient (D L ) at the front of the solid-liquid interface, the dendritic growth rate (V ) and the thickness of the diffusion layer. The cooling rate is the product of temperature gradient and growth rate, C = D L ·V.
In the experiment, the higher cooling rate of the alloy may lead to a larger temperature gradient, which will reduce the undercooling area of the composition and the complexity of the crystal morphology. Figure 9 shows the schematic diagram of the relationship between the cooling rate and the composition undercooling at the front of the solidification interface. T L (C 1 ) and T L (C 2 ) represent the temperature gradient at low cooling rate and high cooling rate, respectively. It can be seen that a greater component undercooling zone is generated (shadow area under T L (C 1 ) straight line) at low cooling rate. Therefore, the undercooling increased from the outside to inside and the solidification morphology became complex.

Fractal dimension (D)
The fractal dimension and corresponding fitting coefficient at different positions of ingots were listed in table 1. The average fractal dimensions of Fe phase distribution in the W-ingot is 1.5531 which is 0.45% higher than those (1.5462) in the Q-ingot. The average fractal dimension of the left-right symmetrical position in the ingots was shown in figure 10. The fractal dimension increases gradually from the surface to center of ingot and the distribution of Fe phase in the solidification structure tends to be uniform. The fractal dimensions of figures 3(g) Figure 9. Schematic diagram of the relationship between the cooling rate and the composition undercooling. According to the meaning of fractal dimension, the distribution of Fe phase of the former is more uniform. It is also not difficult to see that the distribution uniformity of Fe in the former is better than that in the latter based on subjective judgment. In the surface area of W-ingot, there are a large number of cellular crystals with single structure because of rapid cooling. At the same time, there are also areas where Fe phase is not precipitated, resulting in poor uniformity of Fe phase distribution. In the central area of the ingot, the growth directionality of Fe precipitates decreases, and dendrites are short and uniformly distributed because the liquid temperature is roughly uniform (figures 2 and 3). So, the fractal dimension of Fe phase distribution increases. Figure 11 illustrates the relationship between the SDAS and the fractal dimension. It can be seen that the fractal dimension increase with the increase of SDAS, which indicates SDAS is related to the complexity distribution of Fe phase. The relationship between SDAS of Fe phase and the cooling rate can be expressed as: λ 2 = k × C −n , where λ 2 is secondary dendrite spacing, μm. C is the cooling rate,°C min −1 ; K is 90.28; n is taken as −1/3 [22]. It is well known that the secondary dendrite arm spacing is proportional to the cube root of local solidification time. Figure 12 shows the relationship between fractal dimension and the cube root of local solidification time. The fractal dimension is also proportional to the cube root of local solidification time (equation (2)), and the fitting coefficient is 0.6555 and 0.1896, indicating that the complexity distribution of Fe phase during solidification is related to local solidification time.  Where, k and C are constants. In the actual solidification process, Fe phase will move irregularly because the density difference between Fe phase and liquid phase cause Stokes floating and thermal convection, solidification contraction, resulting in its arbitrary distribution in the central region and the complexity of Fe phase distribution increasing and the fractal dimension from the surface to the center increasing nonlinearly. Figure 13 presents the relationship between the fractal dimension and Fe content, Fe phase fraction and the average Fe phase area at different positions in the ingot. The Fe content generally shows an increasing trend with the increase of fractal dimension under two cooling conditions, but the relationship is not as obvious as the linear relationship between Fe phase fraction, average Fe phase area and fractal dimension. Because there is very different cooling under the Q-ingot and W-ingot, so the difference of Fe phase dendrite size is obvious and there are two independent linear relationships between the average Fe phase area and fractal dimension under 2 cooling conditions, where the average Fe phase area increase with the fractal dimension and have a good fitting relationship. The fitting coefficients are 0.5415 and 0.8134, respectively. It should be noted that different average Fe phase areas in the solidified structure can have the same fractal dimension (figure 13(c)). Therefore, when evaluating the quality of Fe phase distribution in solidification structure, in addition to the fractal dimension, if the size of Fe phase differs greatly, the average Fe phase area needs to be investigated. The larger the fractal dimension is, the smaller the average Fe phase area is, the smaller the Fe phase in the corresponding region is and the more uniform the distribution is.

Strength of alloy after deformation
In order to further evaluate the influence of the solidification structure and distribution characteristics of Fe phase on the strength, the specimen (length × width × height: 50 × 30 × 20 mm) was cut at the same position of the ingot along the cross section direction. The homogenization process was conducted before processing to 990°C × 3 h, hot rolled to 10 mm at 900°C along the height direction, and finally cold rolled to 0.3 mm. Figure 14 illustrates the tensile strength results of Cu-15Fe alloy. It can be seen that the strength of the alloy is different under two solidification conditions. The strength of ingot alloy for Q-ingot and W-ingot is 510 ± 24 and 547 ± 19 Mpa, respectively. The strength of the latter is higher than that of the former, and the deviation of the latter is also smaller than that of the former. As the improvement of deformation strength of Cu-Fe alloy depends on the degree of fibrosis after Fe phase deformation, under the same deformation degree, the Fe phase with smaller size deformation fibrosis will be better in the W-ingot, thus improving the strength and its better distribution uniformity reduces the strength fluctuation range.

Concluding remarks
The average Fe phase area (S), secondary dendrite arm spacing (λ 2 ), shape factor (F) and fractal dimension (D) were used to analyze the morphology and distribution characteristics of Fe phase in the solidification structure of Cu-15Fe alloy ingot, and the strength deformation of the alloy under two cooling conditions was compared. We found that there was a deviation according to the traditional element segregation ratio to evaluate the distribution of Fe phase in the solidification structure. Because the influence of temperature gradient during solidification, the components undercooling increases and the solidification morphology becomes more complex as the cooling rate decreases from outside to inside. The distribution uniformity of Fe phase in the water-cooled copper mold ingot is 0.45% higher than that of in the quartz mold ingot. The distribution quality of Fe phase in solidification structure can be measured by fractal dimension and average Fe phase area. The larger the fractal dimension is, the smaller the average Fe phase area is, the smaller the Fe phase in the corresponding region is and the more uniform the distribution is, and the better the strength of the alloy after