*3.2. Dendritic Growth Kinetics*

The dendritic growth velocity is an important feature under the rapid solidification condition, which has an influence on the microstructure and physical properties of binary Co-4.54%Sn alloy. Figure 4a displays the measured and calculated growth velocity of the primary αCo phase, which presents two characteristics. At first, the dendritic growth velocity of primary αCo phase increased with the increase of undercooling, and a maximum velocity of 0.95 m/s was obtained at 175 K undercooling, which demonstrates a sluggish tendency for this alloy. However, the dendritic growth velocity slowed down once the undercooling exceeded 175 K and the dendrite decreased to 0.85 m/s growth velocity at the largest undercooling of 208 K. A Gaussian function was derived by fitting the actual dendrite growth versus undercooling to

$$V = 1.37 \times 10^3 \exp\left(-\frac{1.71 \times 10^{-21}}{k\_B \cdot \Delta T}\right) \exp\left(-\frac{-1.14 \times 10^{-19}}{k\_B \cdot T}\right) \tag{1}$$

**Figure 4.** Measured and calculated dendritic growth velocity and solute content of rapidly solidified binary. Co-4.54%Sn alloy versus undercoolings, (**a**) dendritic growth velocity, and (**b**) solute content.

The LKT model for dendritic growth developed by Lipton, Kurz and Trivedi [20], which was further extended by Boettinger, Coriell and Trivedi (BCT) [21], has proven to be the most competent model for predicting dendritic growth characteristics with linear liquidus and solidus lines during rapid solidification. Cao et al. [22] modified the LKT/BCT dendritic growth theory to be applicable to predicting the kinetic characteristics of dendritic

growth in the alloy systems which have extremely curved liquidus and solidus lines. This model is comprised by the following seven principal equations:

$$
\Delta T = \Delta T\_l + \Delta T\_c + \Delta T\_r + \Delta T\_k \tag{2}
$$

$$R = \frac{\Gamma}{\sigma^\*} \left[ \frac{\Delta H}{\mathbb{C}\_{PL}} P\_l \tilde{\xi}\_l - \frac{2m\_\perp' C\_0 (1 - k\_V) P\_c}{1 - (1 - k\_v) I\_v (P\_c) \tilde{\xi}\_c} \right]^{-1} \tag{3}$$

Δ*T*: bulk undercooling; Δ*Tt*: thermal undercooling; Δ*Tc*: solutal undercooling; Δ*Tr*: curvature undercooling; Δ*Tk*: kinetic undercooling; *V*: dendritic growth velocity; *R*: the dendrite tip radius; Δ*H*: heat of fusion; *CPL*: specific heat of the liquid phase; *P*<sup>t</sup> and *Pc*: thermal and solutal Peclet numbers; *ξ<sup>t</sup>* and *ξc*: thermal and solutal stability functions; *m L*: actual liquidus slope; *C*0: alloy composition; *kv*: actual solute partition coefficient; *Iv*(*Pc*): solutal Ivantsov function; Γ: Gibbs–Thomson coefficient; and *σ*∗: the stability constant equal to 1/ 4*π*<sup>2</sup> .

From the LKT/BCT dendritic growth model, it can be seen that the dendritic growth velocity increased remarkably when the undercooling was larger than 25 K, and was 1~3 orders of magnitude larger than the actual dendritic growth velocity at the undercooling of 58~208 K. Therefore, the actual dendritic growth velocity was quite sluggish under the rapid solidification. According to the theoretical calculation of the LKT/BCT model, the dendritic growth is mainly controlled by the solutal diffusion if the undercooling is smaller than 91 K, which corresponds with the slower velocity. Subsequently, the effect of thermal diffusion becomes an increasingly important factor and finally replaces solutal diffusion as the dominant controlling factor once the undercooling exceeds 91 K.

The solute content of primary αCo phase was detected using the EDS method, and the results show the increasing tendency of undercooling, as seen in Figure 4b. For example, the solute content *C*∗ *<sup>S</sup>* was 0.96 wt.%Sn when the undercooling Δ*T* was 11 K, which is the smallest level of undercooling acquired by the experiment. If the undercooling Δ*T* attained the largest undercooling 189 K of the experimental sample, the solute content also achieved 4.19 wt.%. According to the equilibrium diagram of Figure 1a,b, the solute content of primary αCo phase was only 0.56 wt.%Sn once the temperature decreased to 1760 K and the primary αCo phase began to grow from the liquid phase. Obviously, the solute trapping takes place during the rapid solidification. From Figure 2, it can be seen that the dendritic structure of the primary αCo phase refined remarkably with the increase of undercooling caused by the strong recalescence, the volume fraction of the βCo3Sn2 phase decreased apparently at the same time and a large number of Sn solutes diffused in the solvent interstitial of the primary αCo phase and led to the higher solute content. The variations of the liquid concentration *C*∗ *<sup>L</sup>* and the solid concentration *C*<sup>∗</sup> *<sup>S</sup>* at the liquid/solid interface with undercooling were calculated using the LKT/BCT model, as illustrated in Figure 4b; the experimental value was close to the calculated *C*∗ *<sup>S</sup>* result at the largest undercooling of 189 K. Apparently, the segregationless solidification may occur once the undercooling exceeds 189 K for binary Co-4.54%Sn alloy at t rapid solidification.

#### *3.3. Microhardness and Electrical Resistivity*

The microhardness of the binary Co-4.54%Sn alloy is illustrated in Figure 5a. As the undercooling was small, Δ*T* = 11 K, the microhardness *Hv* was 228.6 HV, and then increased with undercooling. The maximum microhardness attained was 335.5 HV where the undercooling Δ*T* was 189 K, which is 1.47 times larger than the value of 11 K undercooling. The liner fitting function at different undercoolings is demonstrated as:

$$H\_v = 221.13 + 0.65\Delta T\tag{4}$$

**Figure 5.** Microhardness and resistivity of Co-4.54%Sn alloy at different undercoolings, (**a**) microhardness and (**b**) resistivity.

Apparently, undercooling has a significant influence on microhardness. From the analysis above, it can be seen that large undercooling had more obvious recalescence, which led to the refined microstructure, and also enhanced the dendritic growth velocity and solute content. Therefore, the refined structure had increased grain boundary and larger solute content, which improved the microhardness under rapid solidification [23]. In addition, the microstructure also affected the hardness of Co-4.54%Sn alloy. Similarly, the undercooling also prompted the resistivity of binary Co-4.54%Sn alloy to rise, as displayed in Figure 5b. To describe the relationship between resistivity *ρ*<sup>r</sup> and undercoolings Δ*T*, an expression is proposed:

$$
\rho\_{\tau} = 67.71 + 0.63 \Delta T - 1.15 \times 10^{-3} \Delta T^2 \tag{5}
$$

At the small undercooling of 11 K, the electrical resistivity *ρ*<sup>r</sup> was only 76.4 μΩ·cm. The electrical resistivity grew rapidly with the increase of undercooling, and the largest electrical resistivity *ρ*<sup>r</sup> of 145.2 μΩ·cm was obtained at the undercooling of 189 K, which is 1.9 times larger than that of 11 K. There are two reasons for this phenomenon. According to the classical nucleation energy definition, the nucleation energy of melt varies inversely with the square of the undercooling. Clearly, the nucleation energy decreases rapidly with the increase of undercooling, which enhances the number of crystal nuclei and refines the microstructure obviously. The growing crystal boundary hinders the electronic transmission, resulting in the electrical resistivity increasing. On the other hand, the reciprocal of relaxation time for lattice scattering caused by the impurity defect is proportional to the content of impurity concentration. Since the lattice distortion will cause strong electron scattering and reduce the mobility of free electrons, the more serious the lattice distortion of the total solute matrix in the constituent phase, the higher the resistivity of the alloy [24]. As shown in Figure 4b, the solute content of the primary (αCo) phase enhances with the increase of undercooling, which generates the lattice distortion influence significant on the electronic transmission, thus improving the electrical resistivity.
