1 Introduction

In general, Co-based superalloys have excellent thermal corrosion, hot fatigue performance, and welding property.[1] However, the high-temperature mechanical properties of traditional Co-based superalloys, which are strengthened through carbide and solid-solution strengthening, cannot compete with γ–γ'-reinforced Ni-base superalloys. Therefore, the use of these Co-based superalloys is limited as a high-temperature structural material. Since the discovery of Co3(Al,W) phase by Sato et al.,[2] research on new γ'-phase reinforced Co-based superalloys has developed rapidly.[3,4,5,6,7,8] The γ' solvus temperature of Co–Ni–Al–W-based superalloys was improved to over 1100 °C using alloying methods.[3,4] The Co–Ni–Al–W-based superalloys show stable γ–γ' microstructures at elevated temperatures and significantly improve the high-temperature strength by precipitation strengthening. Therefore, Co–Ni–Al–W-based superalloys have great prospects for engineering applications.

Casting process performance is the key to engineering applications of superalloys. Due to a large range of crystallization temperature, large linear shrinkage of the superalloys, and the development of superalloys castings with light, thin, and complex shapes, higher requirements are put forward for superalloys’ castability especially hot-cracking resistance during casting.[9,15] Alloy composition, solidification process, and casting shape are key factors that affect the casting process performance, hot-cracking resistance, and solidification defects of superalloy castings, which in turn affect the quality of castings.[16,17,18,19,20,21,22] Among these factors, the alloy composition affects mainly the casting process properties by affecting the solidification behavior of the alloy. In our previous study,[23] we found that the γ' phase precipitated in the solidification final stage will be replaced by Ti-rich γ/γ' eutectic when Ni is added to the Co–Al–W-based alloy. The Co–Al–W-based alloy also has Ti-rich β/γ' eutectics at the same time, indicating that the change in Ni content will inevitably affect the fraction of two eutectics formed in the later stage of solidification and then significantly affect the casting process performance of the alloy. Therefore, the effect of Ni content on the solidification behavior and hot-tearing susceptibility of Co–Ni–Al–W-based superalloys was studied in this paper. The results could offer an experimental reference for the Ni content design of Co–Ni–Al–W-based superalloys.

2 Experimental

In this study, four γ'-reinforced Co–Ni–Al–W-based superalloys, Co–xNi–11Al–4W–5Cr–1Ta–4Ti (x = 10, 20, 30, 40), with different Ni content were designed. The alloys were fabricated by a vacuum induction melting method. The alloys are referred as 10Ni, 20Ni, 30Ni, and 40Ni alloy, respectively. Table I shows the actual chemical composition (at. pct) of experimental alloys.

Table I The Chemical Composition of Experimental Alloys in This Study (Atomic Percent)
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Single crystal bars and tube-like castability test samples were prepared by high-rate solidification (HRS) directional solidification, which is performed in the Bridgeman vacuum-directional solidification furnace. The heating temperature of the directional solidification experiment was 1520 °C, the holding time was 5 min, and the withdrawal rate was 3 mm/min. According the previous study,[16,17,18,19,20] the specific parameters of the tube-like castability test are shown in Figure 1. The core and shell for the hot-cracking experiments were prepared using alumina with the purity of 99.7 pct. Every three hot-cracking molds were combined in one group and mounted on water-cooled copper disks. The inner diameter of the hollow tube is 6 mm, and the hollow section length is 120 mm. The wall thicknesses are 1.5, 2, and 2.5 mm. The heating temperature of the hot-cracking experiment was 1520 °C, and the pull down was started after holding for 5 minutes with a pulling speed of 3 mm/min. The tube-like castability test samples were analyzed, and the crack length and width were determined. According to the previous studies,[17,18] the hot-tearing susceptibility of the alloys was evaluated by measuring the crack ratio r. The r was then defined as follows :[18]

Fig. 1
figure 1

Schematic diagram of the tube-like castability test device (units: mm)

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$${r}=\sum_{i=1}^{n}{l}_{i}{w}_{i}/LC,$$
(1)

where n is the number of cracks, wi is the crack width, li is the crack length, C is the outer circumference of the tube, and L is the length of the hollow section of the casting tube.

To evaluate the solidification segregation of experimental alloys, the point matrix scanning technique was used to measure the solid–liquid partition coefficients of the experimental alloys.[24,25] In representative dendritic microstructure, the composition of the 14 × 14 point grid with a grid spacing of 60 μm was obtained by electron probe microanalysis. By fitting the sorted EPMA data, the solid–liquid partition coefficients of the individual element were obtained. The solid–liquid partition coefficients were assumed constant and limited back-diffusion was accounted for as follows[26]:

$$C_{{\text{s}}} = C_{0} k(1 - (1 -2\alpha k)f_{{\text{s}}} )^{(k -1)(1 - 2\alpha k)},$$
(2)

where C0 is the nominal composition; Cs is the mole fraction of solute in the solid; fs is the solid fraction; k is the solid–liquid partition coefficients; and α is the Fourier number, which in superalloys is usually 0.01.[21]

The solidification path of experimental alloys was analyzed by the isothermal solidification experiments.[27,28] The specimens of isothermal solidification experiment were cut from the ingots with a size of 10 mm × 8 mm × 8 mm, and the specimens were put into the graphite blocks with holes and sealed with the mixture of silica sol and alumina powder to avoid oxidation during solidification process. The sealed specimens were heated up to 1450 °C for 15 minutes to homogenize the melt and then cooled to different isothermal solidification temperatures at 10 °C/min. The specimens were held at that temperature for about 15 minutes and then rapidly quenched into the cold water. The isothermal solidification experiments were done at 10 °C intervals in the range of 1160 °C to 1400 °C.

The specimens were polished and etched in a solution of 1 vol pct HF + 33 vol pct CH3COOH + 33 vol pct H2O + 33 vol pct HNO3. Nikon LV150 optical microscope and Zeiss Ultra 55 field emission scanning electron microscope were employed to observe the microstructure. The volume fraction of intergranular phases and residual liquid was determined by the intercept method. The residual liquid fraction was measured by the metallographic pictures with 100 multiples. For the measurement of intergranular phases fraction, the images of a 5 × 5 matrix at 800 multiples (the different intergranular phases can be clearly distinguished at this multiple) were obtained using a scanning electron microscope in backscattered electron mode (BSE). Then the 25 pictures were spliced into a large picture, which is large enough to show the overall morphology of the cross section. This large picture was used to measure the volume fractions of different intergranular phases. The electron probe microanalysis (EPMA: JXA-8230) was used to obtain the chemical composition. Differential thermal analysis (DTA) measurement was performed in a NETZSCH STA449C apparatus to obtain the transformations temperatures of main phases. And the heating rate and cooling rates of DTA testing are both 10 °C/min

3 Results

3.1 Effect of Ni Content on the Solidification Characteristics of Co–Ni–Al–W-Based Superalloys

Figure 2 shows the representative as-cast microstructure after directional solidification of the alloys with different Ni contents. In the low magnification microstructure, the as-cast microstructure of different alloys can be seen to show a dendritic structure. For four alloys with different Ni contents, a distinct secondary phase appears in the interdendritic area. The interdendritic area of different alloys was analyzed by high-magnification observation and energy dispersive spectrometry (EDS). According to our previous study,[23] there are two secondary phases of blocky black β/γ' eutectics and network γ/γ' eutectics in alloys with different Ni contents (as shown in Figure 2). Among these eutectics, the β/γ' eutectics is enriched in Ti and Al, and the γ/γ' eutectics also contains high levels of Ti. And with increasing Ni content, the content of Al in γ/γ' eutectics increases, while the content of Ti in γ/γ' eutectics decreases. Table II shows the volume fraction of secondary phases calculated by the intercept method. In the 10Ni alloy, the volume fractions of the β/γ' eutectics and γ/γ' eutectics are 7.96 pct and 0.37 pct, respectively. With increasing Ni content, the β/γ' eutectics’ volume fractions gradually decrease, while the γ/γ' eutectics’ volume fractions gradually increase. In the 40Ni alloy, the volume fraction of the β/γ' eutectics was less than 0.01 pct, while the γ/γ' eutectics’ volume fraction increased to 7.55 pct. This result indicates that the increase in Ni content decreased the precipitation of the β/γ' eutectics and promoted the formation of the γ/γ' eutectic.

Fig. 2
figure 2

The as-cast microstructure of different alloys: (a), (b) 10Ni alloy, (c), (d) 20Ni alloy, (e), (f) 30Ni alloy, and (g), (h) 40Ni alloy

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Table II The Composition and Volume Fractions of Precipitation in Different Alloys
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The solid–liquid partition coefficients (ki) is usually used to evaluate the segregation degree of alloying elements. As shown in Figure 3, the solid–liquid partition coefficients of Ti, Ta, Ni, and Al are smaller than 1, suggesting that these elements segregate to the interdendritic area. Co, W, and Cr partition to the dendritic cores due to the distribution coefficient are greater than 1. Comparing the effects of the Ni content on the segregation behavior, a change in the composition can be found to have small effect on the degree of Cr, Al, and Ni segregation. However, with increasing Ni content, the degree of W and Ti segregation increased, while the degree of Ta segregation decreased. According to the distribution behavior of alloying elements during solidification, the content of W and Ta in the residual liquid decreases, while the content of Ti increases. Ta and W are heavy metal elements, and Ti is a light metal element. Therefore, the changes in the segregation behavior of W, Ta and Ti caused by the Ni content increase will cause the gradient of liquid density in the mushy zone to increase during the directional solidification of the alloy, showing that the formation tendency of freckle defects in the alloy is increased. In the subsequent alloy design, reducing the formation tendency of freckle defects can start from two aspects:[29,30](1) adding alloying elements that can reduce the segregation of W and Ti elements, and (2) adding heavy metal elements similar to Ta that are segregated to the interdendritic area, such as Mo.[31] Both methods can reduce the gradient of liquid density in the mushy zone during the directional solidification of the Co–Ni–Al–W-based alloy.

Fig. 3
figure 3

The solid–liquid partition coefficients of each element

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Figure 4 is the DTA curves for the alloys with different Ni contents. In the 10Ni alloy, the solidus temperature is 1326 °C, and the liquidus temperatures 1376 °C. With increasing Ni content, the liquidus temperatures of the alloy did not change significantly, while the solidus temperatures gradually decreased, and the solidus temperature of the 40Ni alloy was only 1291 °C. This results in the freezing range of the alloy increasing from 50 °C to 77 °C with increasing Ni content. Moreover, there are endothermic peaks in the heating curves of the 10Ni, 20Ni, and 30Ni alloys. The endothermic peak temperature decreases with increasing Ni content and disappears in the 40Ni alloy. According to the secondary phase volume fraction in the alloy with different Ni contents, the endothermic peak is induced by the dissolution of β/γ' eutectics. For the cooling curve of the 10Ni alloy, an exothermic peak appears at 1260 °C. Because the γ/γ' eutectics volume fraction in the 10Ni alloy is very small, the exothermic peak is induced by the precipitation of the β/γ' eutectics. For the cooling curve of the 20Ni alloy, two exothermic peaks appear at 1206 °C and 1235 °C. According to the results of a previous study on the alloy solidification path,[23] the exothermic peak at 1235 °C is the precipitation peak of β/γ' eutectics, and the exothermic peak at 1206 °C is the precipitation peak of γ/γ' eutectics. There is only one exothermic peak in the cooling curves of the 30Ni alloy and 40Ni alloy. According to the volume fraction of the secondary phase in the two alloys, we can consider that these two exothermic peaks are the precipitation peaks of the γ/γ' eutectics. Therefore, as the Ni content increases, the γ/γ' eutectics’ precipitation temperature gradually increases, and the β/γ' eutectics precipitation temperature gradually decreases.

Fig. 4
figure 4

DTA curve of alloys with different Ni contents: (a) heating and (b) cooling. The peaks marked with red circles are β/γ′ eutectics dissolution peaks, the peaks marked with blue circles are β/γ′ eutectics precipitation peaks, and the peaks marked with green circles are γ/γ′ eutectics precipitation peaks (Color figure online)

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3.2 Effect of Ni Content on Microstructure Evolution of Alloy During Solidification

To analyze the microstructure evolution during directional solidification, isothermal solidification experiments were carried out on alloys with different Ni contents. Figure 5 shows the solidification microstructures and residual liquid fraction of the 10Ni alloy quenched at different temperatures. As shown in Figure 5, fully fine dendritic structures were observed when the sample was quenched from 1380 °C. Fine dendrites are formed during rapid cooling of the alloy liquid phase, indicating that the 10Ni alloy is a liquid phase at 1380 °C. When cooling to 1370 °C, the solidification microstructure of the 10Ni alloy consists of coarse γ dendrites and fine γ dendrites, indicating that the solid phase of the 10Ni alloy begins to precipitate at this temperature during solidification, and the residual liquid phase is transformed into fine γ dendrites. With a further decrease in the quenching temperature, the residual liquid fraction in the alloy gradually decreases. When the 10Ni alloy is quenched at 1290 °C, the residual liquid fraction is only 11.85 pct. At this time, the residual liquid network is still continuous in the quenched microstructure (as seen in Figure 5(d)). When cooling to 1280 °C, the coarse γ dendrites overlap with each other, and the residual liquid network is disconnected to form isolated regions. The flow of residual liquid is restricted at this temperature. In addition, coarse β/γ' eutectics appear in the solidification microstructure after quenching (as seen in Figure 5(e)), showing that the β/γ' eutectics precipitation temperature during solidification is between 1280 °C and 1290 °C. When the 10Ni alloy is further cooled to 1240 °C, the volume fraction of the residual liquid is already less than 1pct. However, the residual liquid phase in the 10Ni alloy decreases very slowly with a continuous decrease in the quenching temperature. The 10Ni alloy did not fully solidify until the temperature was reduced to 1190 °C, and a small part of the interdendritic area appeared as blocky γ/γ' eutectics, showing that the γ/γ' eutectics precipitation temperature in the 10Ni alloy is between 1190 °C and 1200 °C (as seen in Figure 5(g)).

Fig. 5
figure 5

Microstructure of 10Ni alloy after isothermal quenching at different temperatures: (a) variation of residual liquid fraction with isothermal quenching temperature, (b) 1380 °C, (c) 1370 °C, (d) 1290 °C, (e) 1280 °C, (f) 1240 °C, and (g) 1190 °C

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Figure 6 shows the solidification microstructures and residual liquid fraction of the 20Ni alloy quenched at different temperatures. As shown in Figure 6, the solidification behavior of 20Ni alloy is very similar to the solidification behavior of the 10Ni alloy. The 20Ni alloy begins to precipitate coarse γ dendrites after quenching at 1370 °C. When the 20Ni alloy is cooled to 1290 °C, the residual liquid fraction is only 12.31 pct. At this time, the residual liquid network is still continuous in the quenched microstructure (as seen in Figure 6(d)). When cooling to 1280 °C, the coarse γ dendrites overlap with each other, and the flow of residual liquid is restricted (as seen in Figure 6(e)). As the isothermal quenching temperature decreases to 1260 °C, coarse β/γ' eutectics appeared in the solidification microstructure of the 20Ni alloy (as seen in Figure 6(f)), showing that the β/γ' eutectics precipitation temperature during solidification is between 1260 °C and 1270 °C. When the 20Ni alloy is further cooled to 1220 °C, the residual liquid fraction is already less than 1pct, and blocky γ/γ' eutectics appeared in the solidification microstructure, showing that the γ/γ' eutectics precipitation temperature in the 20Ni alloy is between 1220 °C and 1230 °C (as seen in Figure 6(g)). The complete solidification temperature of 20Ni alloy is 1200 °C.

Fig. 6
figure 6

Microstructure of 20Ni alloy after isothermal quenching at different temperatures: (a) variation of residual liquid fraction with isothermal quenching temperature, (b) 1380 °C, (c) 1370 °C, (d) 1290 °C, (e) 1280 °C, (f) 1260 °C, and (g) 1220 °C

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Figure 7 shows the solidification microstructures and residual liquid fraction of 30Ni alloy quenched at different temperatures. As seen in Figure 7, the 30Ni alloy begins to precipitate coarse γ dendrites after quenching at 1370 °C. When the 30Ni alloy is cooled to 1310 °C, the residual liquid network is still continuous, and the residual liquid fraction is 12.45 pct (as seen in Figure 7(d)). When cooling to 1300 °C, the coarse γ dendrites overlap with each other, and the flow of residual liquid is restricted (as seen in Figure 7(e)). As the isothermal quenching temperature decreases to 1260 °C, coarse β/γ' eutectics appear in the 30Ni alloy, indicating that the β/γ' eutectics precipitation temperature is between 1260 °C and 1270 °C (as seen in Figure 7(f)). When the 30Ni alloy is further cooled to 1240 °C, blocky γ/γ' eutectics appear, indicating that the γ/γ' eutectics precipitation temperature is between 1240 °C and 1250 °C (as seen in Figure 7(g)). At this time, the residual liquid fraction is 2.56pct. The complete solidification temperature of the 30Ni alloy is 1220 °C.

Fig. 7
figure 7

Microstructure of 30Ni alloy after isothermal quenching at different temperatures: (a) variation of residual liquid fraction with isothermal quenching temperature, (b) 1380 °C, (c) 1370 °C, (d) 1310 °C, (e) 1300 °C, (f) 1260 °C, and (g) 1240 °C

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Figure 8 shows the solidification microstructures and residual liquid fraction of the 40Ni alloy quenched at different temperatures. As seen in Figure 8, the 40Ni alloy begins to precipitate coarse γ dendrites after quenching at 1370 °C. When the 40Ni alloy is cooled to 1320 °C, the residual liquid network is still continuous, and the residual liquid fraction is 11.94 pct (as seen in Figure 8(d)). When cooling to 1310 °C, the coarse γ dendrites overlap with each other, and the flow of residual liquid is restricted (as seen in Figure 8(e)). As the isothermal quenching temperature decreases to 1250 °C, blocky γ/γ' eutectics appears, indicating that the γ/γ' eutectics precipitation temperature is between 1250 °C and 1260 °C (as seen in Figure 8(f)). When the 40Ni alloy is further cooled to 1230 °C, the residual liquid fraction is 1.01pct. The complete solidification temperature of the 40Ni alloy is 1220 °C. In addition, the precipitation of β/γ' eutectics is not observed in isothermal solidification experiments due to the small volume fraction of β/γ' eutectics.

Fig. 8
figure 8

Microstructure of 40Ni alloy after isothermal quenching at different temperatures: (a) variation of residual liquid fraction with isothermal quenching temperature, (b) 1380 °C, (c) 1370 °C, (d) 1320 °C, (e) 1310 °C, (f) 1250 °C, and (g) 1230 °C

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3.3 Effect of Ni Content on Hot-Tearing Susceptibility of Co–Ni–Al–W-based superalloy

Figure 9 shows the macromorphology and typical hot-tearing fracture surface of tube-like castability test samples. As seen in Figure 9, the cracks propagate along the tube wall, and the length and width of the cracks become larger with increasing Ni content. By observing the fracture surface of the samples, the fracture surface exhibits features such as smooth dendrite tips, rounded ends, and droplets. No cleavage planes and dimples can be found, indicating that cracks are formed in the mushy zone during solidification. The crack ratio r of four alloys with different wall thicknesses is calculated, as shown in Figure 10. For tube-like samples with different wall thicknesses, the crack ratio of the alloy gradually increases as the Ni content in the alloy increases, indicating that the hot-tearing susceptibility of the alloy increases gradually with increasing Ni content.

Fig. 9
figure 9

The macromorphology and typical hot-tearing fracture surface of tube-like castability test samples. (a), (b) Macromorphology, (c) typical hot-tearing fracture surface

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Fig. 10
figure 10

Crack ratio of the tube-like castability test for alloys with different Ni contents

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The morphology and element distribution of hot cracks in the 10Ni alloy and the 40Ni alloy were analyzed by SEM and EDS, as shown in Figure 11. A large amount of Ti and Al elements can be seen to be enriched near the hot cracks. Since Ti and Al elements are usually concentrated in the residual liquid at the end of solidification, hot cracks are formed in the solidification final stage and propagate along the residual liquid at the end of solidification. Comparing the segregation behavior of Ti and Al elements between the 10Ni alloy and 40Ni alloy, Al in the 10Ni alloy only can be found to segregate in the β/γ' eutectics at interdendritic area. Most of the Ti elements are segregated in the β/γ' eutectics at interdendritic area, and a small part is concentrated in the coarse γ' phase at interdendritic area. In the 40Ni alloy, Ti and Al elements are also enriched in interdendritic area. Among these elements, Al does not have special segregation at the γ/γ' eutectics. Ti has slight segregation at the γ/γ' eutectics, but the segregation is not obvious, showing that the β/γ' eutectics precipitation can significantly reduce the Ti and Al elements in the residual liquid phase, while the precipitation of γ/γ' eutectics can only slightly reduce the content of Ti.

Fig. 11
figure 11

Morphology and element distribution of typical hot cracks in alloys with different Ni contents: (a1)–(h1) 10Ni alloy, (a2)–(h2) 40Ni alloy

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4 Discussion

According to the analysis of the as-cast microstructure and isothermal quench microstructure for alloys with different Ni contents, the change in Ni content is found not to change the type of secondary phase in the alloy but significantly changes the volume fraction and precipitation temperature of the secondary phase in the alloy. Combining the results of the DTA analysis, the microstructure evolution during solidification, and our previous study,[23] Figure 12 illustrates the effect of the Ni content on the solidification process of the alloy. As shown in Figure 12, the solidification paths of the 4 alloys are L → L1 + γ → L2 + γ + β/γ' → L3 + γ + β/γ' + γ/γ' → γ + β/γ' + γ/γ'. With increasing Ni content, the β/γ' eutectics precipitation temperature decreases, while the γ/γ' eutectics precipitation temperature increases. In the 40Ni alloy, although the precipitation of β/γ' eutectics is not observed, the precipitation temperature should be close to γ/γ' eutectics according to the analysis of experimental results. According to previous studies for Co-Ni-Al-W-based superalloys,[6,32,33] the Ni element can increase the dissolution temperature and stability of the γ' phase and expand the γ+γ' two phase area, which will increase the γ/γ' eutectics precipitation temperature during solidification. Ti and Al are the β/γ' eutectics forming elements, while Ti is the γ/γ' eutectics forming element. The β/γ' eutectics will precipitate only when the Ti and Al concentration in the residual liquid achieve the required concentration for eutectic formation. Due to the increase in the γ/γ' eutectics precipitation temperature, the Ti concentration in the residual liquid decreases. Therefore, the precipitation of β/γ' eutectics decreases.

Fig. 12
figure 12

Schematic diagram of the solidification process of alloys with different Ni contents

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Regarding the hot tearing of the alloy during the casting process, a variety of theories with respect to hot-tearing formation have been proposed to describe the formation mechanism, including liquid film theory,[34] solidification shrinkage compensation theory,[35,36] strength theory,[37] and intergranular bridge theory.[38,39] However, various theories are consistent in essence. That is, in the temperature range of the hot-crack-sensitive zone, hot cracks will be formed when the solidification shrinkage stress of the solidifying body exceeds its strength. The hot-tearing criterion proposed by Clyne and Davies[39] is based on the theory that in the last stage of freezing, the liquid has difficulty moving freely so that the strain applied during this stage cannot be accommodated by liquid feeding. They suggest that the crack caused by the solidification shrinkage can be eliminated by liquid feeding at a solid fraction between 0.4 and 0.9. When the solid fraction is between 0.9 and 0.99, due to the obstacles of liquid feeding, the stress generated by solidification shrinkage will gradually accumulate. At the same time, since the strength and plasticity of the alloy are relatively poor at this stage, this stage more easily forms hot cracks (which can be referred to as the “hot-crack-sensitive zone”). When the volume fraction of the residual liquid is lower than 1 pct, the bonding strength between the dendrites is significantly increased, so the formation of hot cracks is hindered.

Recently, a new model for hot tearing during solidification is proposed by Kou.[40] This model presumes that the crack initiates as the net increase of space between two adjacent columnar dendritic grains is greater than liquid feeding. The change of net space includes the space expansion under tension and the space reduce caused by two grains grow toward each other. Assuming that the hot tearing initiates at the end of solidification(\(\sqrt{{f}_{s}}\)→1), the condition for hot tearing is expressed as follows:[40]

$$\frac{{{\text{d}}\varepsilon _{{{\text{local}}}} }}{{{\text{d}}T}} > \left\{ {\sqrt {1 - \beta } \frac{{{\text{d}}\sqrt {f_{{\text{s}}} } }}{{{\text{d}}T}} + \frac{1}{{{{{\text{d}}T} \mathord{\left/ {\vphantom {{{\text{d}}T} {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t}}}}\frac{{\text{d}}}{{{\text{d}}z}}\left[ {\left( {1 - \sqrt {1 - \beta } \sqrt {f_{{\text{s}}} } } \right)} \right]v_{{\text{z}}} } \right\},$$
(3)

where T is the temperature, β is the solidification shrinkage, εlocal is the strain, fs is the fraction of solid, z is the growth direction of the columnar dendritic grain, vz is the velocity of intergranular liquid flow in the negative z-direction, and dT/dt is the cooling rate.

In this equation, the three terms are closely related to the strain rate, dendrite growth rate, and liquid feeding rate, respectively. To more conveniently indicate the formation tendency of hot tearing, the hot-tearing index (named the hot-tearing sensitivity coefficient) is proposed as follows:[40]

$${\text{HSC}}={|\frac{{\text{d}}T}{\mathrm{{\text{d}}}\sqrt{{f}_{\text{s}}}}|}_{{f}_{\text{s}}\to 1},$$
(4)

where fs is the solid fraction, and T is the temperature.

In this model, the variation of the solid-phase fraction fs with temperature reflects the rate at which two adjacent dendrites approach each other. Therefore, a greater value of HSC means that the lateral growth rate is smaller, resulting in columnar dendritic grains that can grow very long without bridging. As shown in Figure 13, under these conditions, the liquid channel between two adjacent grains can be very long. Furthermore, according to the Hagen–Poiseuille law,[41] the liquid volumetric flow rate through a channel decreases with increasing channel length (and with decreasing channel opening) due to the resistance to flow caused by the viscosity of liquid. Thus, liquid feeding can be expected to be more difficult. Additionally, a long liquid channel may serve as a long sharp notch that promotes cracking. Therefore, the higher value of the HSC is, the greater formation tendency of hot tearing. This model has been successfully used to predict the propensity for hot crack formation in Magnesium alloys,[42] aluminum alloys,[43,44,45] and superalloys.[46,47]

Fig. 13
figure 13

Schematic diagram of the effect of the hot-tearing sensitivity coefficient on hot tearing of the alloy. (a) Lower hot-tearing sensitivity coefficient; (b) Higher hot-tearing sensitivity coefficient

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In this experiment, it is considered that when the residual liquid-phase network is interconnected, the alloy is easier to make up the shrinkage and no hot cracking occurs. And when the residual liquid phase in the alloy is less than 1 pct, the strength of the interdigitated alloy can resist the formation of hot cracking. Therefore, the temperature range of the thermal crack-sensitive zone of the alloy should be the difference between these two temperatures. According to the morphology and residual liquid-phase fraction obtained from isothermal quenching experiments at different temperatures, the temperature before the residual liquid-phase network breaks and γ dendrites lap each other was used as the upper limit of the hot-cracking-sensitive zone of the alloys (1290 °C, 1290 °C, 1310 °C, 1320 °C for the four alloys) and the lower limit of the hot-cracking sensitive zone of the alloys (1240 °C, 1220 °C, 1230 °C, 1230 °C for the four alloys) with the residual liquid-phase volume fraction close to 1pct. These critical temperatures and the corresponding solid-phase fractions were used to calculate the HSC values of different alloys, and the results are shown in Table III. The table shows that with increasing Ni content, the HSC value gradually increases, indicating that the hot crack formation tendency of the alloy increases with increasing Ni content, basically consistent with the experimental results, indicating that the hot tearing of the Co–Ni–Al–W-based superalloys can be evaluated by this coefficient. Combined with the effect of Ni content on the solidification behavior and the distribution of elements near the hot crack, a large number of β/γ' eutectics can be seen to be precipitated in the 10Ni alloy at higher temperatures, which significantly reduces the content of low melting point elements Ti and Al in the residual liquid. The residual liquid fraction reduces rapidly during the hot-crack-sensitive phase. Therefore, the 10Ni alloy stays a short time in the hot-crack-sensitive zone and has a small hot-tearing formation tendency. With increasing Ni content, the precipitation temperature and volume fraction of the β/γ' eutectics decrease. The segregation of Ti in the alloy also increases gradually. The Ti and Al concentrations in the residual liquid phase of the alloy gradually increase, resulting in a drop in the solidification rate of the alloy in the hot-tearing sensitive area. Therefore, the hot-tearing susceptibility of the alloy gradually increases with increasing Ni content.

Table III Hot-Tearing Sensitivity Coefficient of Alloys With Different Ni Contents
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Although the decrease in Ni content can significantly reduce the hot-tearing susceptibility of Co–Ni–Al–W-based alloys, the Ni element can increase the dissolution temperature and stability of the γ' phase and expand the γ + γ' two phase area.[6,32,33] Therefore, in the composition design of Co–Ni–Al–W-based superalloys, it is necessary to comprehensively consider various factors, such as the solidification behavior, castability, solvus temperature, and stability of the γ' phase, to determine the Ni content. Since the β/γ' eutectics formation can reduce the hot-tearing susceptibility, the precipitation of β/γ' eutectics can be promoted by adding Al or other β/γ' eutectics forming elements, thereby reducing the hot-tearing susceptibility.

5 Conclusions

This research studied the effect of Ni content on the solidification behavior and hot-tearing susceptibility of Co–xNi–11Al–4W–5Cr–1Ta–4Ti (x = 10–40 at. pct) superalloys. The relevant research results are as follows:

(1) The change in Ni content does not change the solidification path of the alloy. With increasing Ni content, the precipitation temperature and volume fraction of β/γ' eutectics decrease, while the precipitation temperature and volume fraction of γ/γ' eutectics increase.

(2) In Co–Ni–Al–W-based superalloys, Al, Ni, Ta, and Ti segregate to the interdendritic area, and Co, W, and Cr partition to the dendritic core. With increasing content of Ni, the degree of W and Ti segregation increases, while the degree of Ta segregation decreases, indicating that the formation tendency of freckle defects in the alloy is increased. In the subsequent alloy design, reducing the segregation of W and Ti elements or adding heavy metal elements that segregate to the interdendritic area can decrease the formation tendency of freckle defects.

(3) With increasing Ni content, the precipitation temperature and volume fraction of the β/γ' eutectics decrease. The Ti and Al concentrations in the residual liquid phase of the alloy gradually increase, resulting in a reduce in the solidification rate of the alloy in the hot-tearing sensitive area, and the hot-tearing susceptibility of the alloy gradually increases. In the subsequent alloy design, hot-tearing susceptibility can be reduced by promoting the precipitation of β/γ' eutectics.