Evaluation of the Solidification Control Parameters in the Bridgman Process

Abstract

Single crystals are widely applied in semiconductors and turbine blades and manufactured by directional solidification. The solidification control parameters such as the withdrawal rate, axial temperature gradient and cooling rate are the key parameters in directional solidification. Through a mathematical analysis, we derive the explicit relationships between the solidification control parameters and the physical and geometrical parameters. The accuracy of these relationships is high which is proven by the comparisons between the accurate values and the approximate values. These relationships give deep understanding into the directional solidification process. Moreover, they provide guidance for the manufacturing of directionally solidified crystals. Additionally, these relationships establish a connection between the microstructure and the physical and geometrical parameters.

1. Introduction

Directional solidification has been an important research focus in materials science communities¹⁾ and the pattern formation.^2,3) In the practical application, directional solidification is used to produce columnar crystals and single crystals which are widely applied in the turbine blades⁴⁾ and semiconductors⁵⁾ due to the unique mechanical, optical and electrical properties. The Bridgman process is one of the main directional solidification method and is applied widely in industrial manufacture. The temperature distribution during directional solidification is important for the understanding of the crystal growth and the control of microstructure and properties. The temperature distribution and its evolution are affected by the solidification control parameters such as the withdrawal rate (V), axial temperature gradient at the liquid-solid interface (G) and cooling rate (V_c) etc. Hence, the microstructure and the properties are determined by these solidification control parameters. For example, in the directional solidification of alloys, the primary and secondary dendrite arm spacings are determined by G and V.^6–11) The interface stability is determined by G/V.^12,13) The shape of the γ′ phase related to the endurance property is influenced by V.¹⁴⁾ The MC carbide geometry depends on the coarsening process and the coarsening time is positively related to V_c.¹⁵⁾ The content of shrinkage porosities is negative correlated with V¹⁶⁾ and the segregation of alloy elements is also related to V.¹⁷⁾ However, the explicit relationships between the solidification control parameters and the physical and geometrical parameters are not clear.

In this work, we focus on the temperature distribution in the Bridgman process. The approximate analytic relationships between the solidification control parameters and the physical and geometrical parameters will be derived through the thermal analysis with the aid of mathematical analysis. Precisely, the expressions of the withdrawal rate V, axial temperature gradient G and cooling rate V_c will be obtained when the Biot number is much less than 1. High thermal conductivity materials such as metals,¹⁸⁾ casts with small radii or low heat transfer coefficients at the mold surfaces lead to small Biot numbers. Moreover, expressions such as the relationship between the microstructure of directionally solidified crystals and the physical and geometrical parameters can be derived based on these expressions. For instance, the expression of primary dendrite arm spacings can be obtained according to the research of Hunt⁶⁾ and Kurz and Fisher.⁷⁾ Additionally, discussions and applications of these relationships are given.

2. Mathematical Analysis and Results

In this section, the analytic relationships between V, G and V_c and the physical and geometry parameters will be derived based on the thermal analysis. Regarding the thermal analysis, the Bridgman process can be simplified to the following model.^18,19) An infinitely-long cylinder with constant isotropic properties and negligible latent heat of fusion was employed and the steady state differential equation for temperature T is   

\begin{equation} \frac{\partial^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial^{2}T}{\partial z^{2}}-\frac{\mathrm{P}_{\text{e}}}{R}\frac{\partial T}{\partial z}=0, \end{equation} (1)

where r is the radial position, z is the axial position down from the border between the hot and cool zones, R is the radius of the cast and P_e is the Péclet number. P_e is written as   

\begin{equation} \mathrm{P}_{\text{e}}=\frac{V\rho C_{p}R}{k}, \end{equation} (2)

where k is the thermal conductivity, ρ is the density, C_p is the heat capacity and V is the growth rate (can be considered as the withdrawal rate). Hence, in the analysis, P_e is proportional to V. The thermal diffusivity of the mold is assumed the same as the solid and the melt. The boundary conditions are Refs. 18, 19:

• as z → ∞, T → T_c (cooler temperature),
• as z → −∞, T → T_h (heater temperature),
• at r = 0, ∂T/∂r = 0 (symmetry),
• at r = R for z > 0, ∂T/∂r = −B_i(T − T_c)/R,
• at r = R for z < 0, ∂T/∂r = −B_i(T − T_h)/R,
• at z = 0, T₊ = T₋ and (∂T/∂z)₊ = (∂T/∂z)₋, where   
  \begin{equation} \mathrm{B}_{\text{i}}=\frac{hR}{k} \end{equation} (3)

is the Biot number and h is the heat transfer coefficient at the mold surface.

This simplified model involves the basic heat diffusion mechanism in the directional solidification process and is easy to perform the theoretical analysis. Then we can obtain the explicit expression. Although the expressions are approximate, they show the fundamental mutual dependence of the parameters clearly.

Carslaw and Jaeger²⁰⁾ provided abundant analytical solutions of classical problems in thermal analysis. Chang¹⁹⁾ (see also Ref. 18) applied these results and obtained the analytical solution of the above equation. The solution is Refs. 18, 19, for z > 0 (within the cool zone)   

\begin{align} \Phi(r,z)&\equiv\frac{T-T_{c}}{T_{h}-T_{c}}=\sum_{n=1}^{\infty}C_{n}J_{0}\left(\frac{a_{n}r}{R}\right)\\ &\quad\times\exp\left(\frac{1}{2}(\mathrm{P}_{\text{e}}-(\mathrm{P}_{\text{e}}^{2}+4a_{n}^{2})^{\frac{1}{2}})\frac{z}{R}\right), \end{align} (4)

and for z < 0 (within the hot zone)   

\begin{align} \Phi(r,z)&=1+\sum_{n=1}^{\infty}\bar{C}_{n}J_{0}\left(\frac{a_{n}r}{R}\right)\\ &\quad\times\exp\left(\frac{1}{2}(\mathrm{P}_{\text{e}}+(\mathrm{P}_{\text{e}}^{2}+4a_{n}^{2})^{\frac{1}{2}})\frac{z}{R}\right), \end{align} (5)

where J₀ is the Bessel function of the first kind of zero order,   

\begin{equation*} C_{n}=\frac{\mathrm{B}_{\text{i}}(\mathrm{P}_{\text{e}}+(\mathrm{P}_{\text{e}}^{2}+4a_{n}^{2})^{\frac{1}{2}})}{(\mathrm{B}_{\text{i}}^{2}+a_{n}^{2})J_{0}(a_{n})(\mathrm{P}_{\text{e}}^{2}+4a_{n}^{2})^{\frac{1}{2}}}, \end{equation*}

  

\begin{equation*} \bar{C}_{n}=\frac{\mathrm{B}_{\text{i}}(\mathrm{P}_{\text{e}}-(\mathrm{P}_{\text{e}}^{2}+4a_{n}^{2})^{\frac{1}{2}})}{(\mathrm{B}_{\text{i}}^{2}+a_{n}^{2})J_{0}(a_{n})(\mathrm{P}_{\text{e}}^{2}+4a_{n}^{2})^{\frac{1}{2}}} \end{equation*}

and a_n are positive eigenvalues determined by   

\begin{equation} a_{n}J_{1}(a_{n})-\mathrm{B}_{\text{i}}J_{0}(a_{n})=0. \end{equation} (6)

Next, some preliminary results will be presented. As is well known,²¹⁾ $J_{1} = - J'_{0}$ and J₀ can be written as   

\begin{equation*} J_{0}(x)=1-\left(\frac{x}{2}\right)^{2}{}+\frac{1}{4}\left(\frac{x}{2}\right)^{4}{}-\frac{1}{36}\left(\frac{x}{2}\right)^{6}{}+\cdots. \end{equation*}

Let b_n denote the nonnegative critical points of J₀, i.e., $J'_{0}(b_{n}) = 0$. By a simple observation on (6), we know that b_n < a_n < b_n+1. Indeed, let f(x) = xJ₁(x) − B_iJ₀(x). Then f(b_n) = −B_iJ₀(b_n) and f(b_n+1) = −B_iJ₀(b_n+1). Since b_n and b_n+1 are adjacent critical points of J₀, f(b_n) · f(b_n+1) < 0. Hence, there exists a (unique) point, which is exactly a_n, such that b_n < a_n < b_n+1 and f(a_n) = 0.

When B_i ≪ 1, from (6), we have a₁ ≪ 1. Hence, to obtain a₁, the higher order terms can be neglected in the expansion. Taking the first three terms of J₀ and inserting it into (6) gives   

\begin{equation} a_{1}=\sqrt{\frac{8\mathrm{B}_{\text{i}}}{4+\mathrm{B}_{\text{i}}}}. \end{equation} (7)

Hence, for B_i ≪ 1,   

\begin{equation} a_{1}=\sqrt{2\mathrm{B}_{\text{i}}}. \end{equation} (8)

If P_e = 0, the isotherm of Φ(r, z) is symmetric to that of Φ(r, −z) with respect to the plane z = 0 and Φ(r, z) + Φ(r, −z) ≡ 1. Hence, Φ(r, 0) ≡ 1/2 and we can obtain from (4) or (5) that   

\begin{equation*} \sum_{n=1}^{\infty}\cfrac{\mathrm{B}_{\text{i}}J_{0}\biggl(\cfrac{a_{n}r}{R}\biggr)}{(\mathrm{B}_{\text{i}}^{2}+a_{n}^{2})J_{0}(a_{n})}=\frac{1}{2},\quad 0<r<R. \end{equation*}

In particular, let r = R and we have   

\begin{equation} \sum_{n=1}^{\infty}\frac{\mathrm{B}_{\text{i}}}{(\mathrm{B}_{\text{i}}^{2}+a_{n}^{2})}=\frac{1}{2}. \end{equation} (9)

Now, we intend to deduce the expressions for the solidification control parameters. Chang and Wilcox¹⁸⁾ indicated that the interface is the most planar when it lies at the border of the hot and cool zones, i.e., at z = 0. In addition, the axial temperature gradient is the maximum at z = 0. Let T_m denote the melting point temperature and Φ_m is the normalized temperature (i.e., Φ_m = (T_m − T_c)/(T_h − T_c)). A proper withdrawal rate V (equivalent to P_e) should be set such that the isotherm at Φ_m lies at z = 0. Thus, substituting z = 0 and r = R to (4) or (5) and combining with (9), gives   

\begin{equation*} \Phi_{m}=\frac{1}{2}+\sum_{n=1}^{\infty}\frac{\mathrm{B}_{\text{i}}}{\mathrm{B}_{\text{i}}^{2}+a_{n}^{2}}\left(\frac{\mathrm{P}_{\text{e}}}{\sqrt{\mathrm{P}_{\text{e}}^{2}+4a_{n}^{2}}}\right). \end{equation*}

Since B_i ≪ 1, $a_{1} = \sqrt{2\text{B}_{\text{i}}} \ll 1$ and a_n ≥ a₂ > b₂ = 3.8317 (n ≥ 2), the first term is dominant. Thus, the following can be obtained   

\begin{equation} \Phi_{m}=\frac{1}{2}+\frac{\mathrm{B}_{\text{i}}}{\mathrm{B}_{\text{i}}^{2}+a_{1}^{2}}\cdot\frac{\mathrm{P}_{\text{e}}}{\sqrt{\mathrm{P}_{\text{e}}^{2}+4a_{1}^{2}}}. \end{equation} (10)

Then substitute (7) into (10) and by a calculation we obtain   

\begin{equation} \mathrm{P}_{\text{e}}=\cfrac{\biggl(\Phi_{m}-\cfrac{1}{2}\biggr)\sqrt{\cfrac{32\mathrm{B}_{\text{i}}}{4+\mathrm{B}_{\text{i}}}}}{\sqrt{\biggl(\cfrac{4+\mathrm{B}_{\text{i}}}{8+4\mathrm{B}_{\text{i}}}\biggr)^{2}-\biggl(\Phi_{m}-\cfrac{1}{2}\biggr)^{2}}}. \end{equation} (11)

When Φ_m = 0.75 and B_i ≤ 0.2, the relative error is less than 0.60%. Although the accuracy is high, the expression is complicated. Since B_i ≪ 1, above equation can be reduced to   

\begin{equation} \mathrm{P}_{\text{e}}=L_{1}\sqrt{\mathrm{B}_{\text{i}}}, \end{equation} (12)

where $L_{1} = \sqrt{2} (2\Phi _{m} - 1)/\sqrt{\Phi _{m}(1 - \Phi _{m})} $.

The accuracy of (12) is shown in Table 1 where Φ_m = 0.75 and R = 0.1 m (similarly hereinafter). The comparisons between the accurate values calculated by (4) (or (5), similarly hereinafter) and the approximate values calculated by (12) are shown in Fig. 1. It can be seen that (12) is an appropriate simplification.

Table 1 The relative errors δ of P_e, G and V_c.¹

Fig. 1

Comparisons between the accurate values and the approximate values of P_e on B_i.

Furthermore, combining (2), (3) with (12), the relationship between the withdrawal rate V and the physical and geometrical parameters can be obtained:   

\begin{equation} V=\frac{L_{1}}{\rho C_{p}}\sqrt{\frac{hk}{R}}. \end{equation} (13)

In the following, the expression of the axial temperature gradient G at the interface will be deduced. First, approximations of a_n for n ≥ 2 are presented. Expanding J₀ at b_n by Taylor’s formula gives (note that $J'_{0}(b_{n}) = 0$)   

\begin{equation} J_{0}(x)=J_{0}(b_{n})+\frac{1}{2}J''_{0}(b_{n})(x-b_{n})^{2}+\cdots. \end{equation} (14)

Taking the first two terms and inserting into (6), we have for n ≥ 2   

\begin{equation*} a_{n}=b_{n}+\frac{-b_{n}J''_{0}(b_{n})\pm\sqrt{b_{n}^{2}J''_{0}(b_{n})^{2}-2\mathrm{B}_{\text{i}}(2+\mathrm{B}_{\text{i}})J_{0}(b_{n})J''_{0}(b_{n})}}{(2+\mathrm{B}_{\text{i}})J''_{0}(b_{n})}, \end{equation*}

where ± is − if n is odd and is + if n is even. Rationalizing the numerator and noting $b_{n}^{2} \gg 2\text{B}_{\text{i}}(2 + \text{B}_{\text{i}})J_{0}(b_{n})$ for n ≥ 2, we have   

\begin{equation*} a_{n}=b_{n}+\frac{\mathrm{B}_{\text{i}}|J_{0}(b_{n})|}{b_{n}|J''_{0}(b_{n})|}. \end{equation*}

Since J₀ satisfies [Ref. 21, (1.11) in p. 3]   

\begin{equation*} J''_{0}(x)+\frac{1}{x}J'_{0}(x)+J_{0}(x)=0 \end{equation*}

and $J'_{0}(b_{n}) = 0$, $|J''_{0}(b_{n})| = |J_{0}(b_{n})|$. Hence, for n ≥ 2   

\begin{equation} a_{n}=b_{n}+\frac{\mathrm{B}_{\text{i}}}{b_{n}}. \end{equation} (15)

Taking the derivative of Φ with respect to −z at r = 0 and z = 0 gives   

\begin{align} G&=-\frac{\partial\Phi}{\partial z}(0,0)=\sum_{n=1}^{\infty}A_{n}\\ &=\sum_{n=1}^{\infty}\frac{2\mathrm{B}_{\text{i}}a_{n}^{2}}{R(\mathrm{B}_{\text{i}}^{2}+a_{n}^{2})J_{0}(a_{n})(\mathrm{P}_{\text{e}}^{2}+4a_{n}^{2})^{\frac{1}{2}}}. \end{align} (16)

Since this series converges slowly, more terms need to be estimated. For the first term A₁, recall (10) and we have   

\begin{equation*} A_{1}=\frac{(2\Phi_{m}-1)a_{1}^{2}}{RJ_{0}(a_{1})\mathrm{P}_{\text{e}}}. \end{equation*}

When B_i ≪ 1, J₀(a₁) $ \simeq $ 1 and we have by recalling (8) and (12)   

\begin{equation} A_{1}=\frac{2(2\Phi_{m}-1)\sqrt{\mathrm{B}_{\text{i}}}}{RL_{1}}=\frac{\sqrt{2\Phi_{m}(1-\Phi_{m})}}{R}\sqrt{\mathrm{B}_{\text{i}}}. \end{equation} (17)

Now we estimate the second term A₂. Recalling (14) and (15), we have   

\begin{equation*} J_{0}(a_{2})\simeq J_{0}(b_{2})+\frac{\mathrm{B}_{\text{i}}^{2}}{2b_{2}^{2}}J''_{0}(b_{2})\simeq J_{0}(b_{2}). \end{equation*}

On the other hand, $a_{2}^{2} \gg \text{B}_{\text{i}}^{2}$ implies $a_{2}^{2}/(\text{B}_{\text{i}}^{2} + a_{2}^{2}) \simeq 1$. Hence,   

\begin{equation} A_{2}=\frac{2\mathrm{B}_{\text{i}}}{RJ_{0}(b_{2})(L_{1}^{2}\mathrm{B}_{\text{i}}+4(b_{2}^{2}+2\mathrm{B}_{\text{i}}))^{\frac{1}{2}}}. \end{equation} (18)

For n ≥ 3, (15) indicates that a_n $ \simeq $ b_n and J₀(a_n) $ \simeq $ J₀(b_n). Note also that $\text{P}_{\text{e}}^{2} \ll 4a_{n}^{2}$ and we have   

\begin{equation} A_{n}=\frac{\mathrm{B}_{\text{i}}}{RJ_{0}(b_{n})b_{n}}. \end{equation} (19)

Combining (16), (17), (18) and (19) gives   

\begin{align} G&=\frac{\sqrt{2\Phi_{m}(1-\Phi_{m})}}{R}\sqrt{\mathrm{B}_{\text{i}}}+\frac{2\mathrm{B}_{\text{i}}}{RJ_{0}(b_{2})(4b_{2}^{2}+(L_{1}^{2}+8)\mathrm{B}_{\text{i}})^{\frac{1}{2}}}\\ &\quad+\frac{\mathrm{B}_{\text{i}}}{R}\sum_{n=3}^{\infty}\frac{1}{J_{0}(b_{n})b_{n}}. \end{align} (20)

When Φ_m = 0.75 and B_i ≤ 0.2, the relative error is less than 0.91%. Similar to the expression for P_e, above equation can be further simplified to   

\begin{equation} G=L_{2}\frac{\sqrt{\mathrm{B}_{\text{i}}}}{R}+L_{3}\frac{\mathrm{B}_{\text{i}}}{R}=L_{2}\sqrt{\frac{h}{Rk}}+L_{3}\frac{h}{k}, \end{equation} (21)

where $L_{2} = \sqrt{2\Phi _{m}(1 - \Phi _{m})} $ and $L_{3} = \sum\nolimits_{n = 2}^{\infty } \,1/(J_{0}(b_{n})b_{n})$ are constants. The accuracy of (21) is also shown in Table 1 and the comparisons between the accurate values and the approximate values calculated by (21) are shown in Fig. 2. It can be seen that (21) is an appropriate simplification.

Fig. 2

Comparisons between the accurate values and the approximate values of G on B_i.

Since the expressions for the withdraw rate V and the axial temperature gradient G have been obtained, the expression for the cooling rate V_c follows easily:   

\begin{align} V_{c}&=V\cdot G=\frac{L_{1}k}{\rho C_{p}R^{2}}(L_{2}\mathrm{B}_{\text{i}}+L_{3}\mathrm{B}_{\text{i}}^{\frac{3}{2}})\\ &=\frac{L_{1}L_{2}h}{\rho C_{p}R}+\frac{L_{1}L_{3}h}{\rho C_{p}}\sqrt{\frac{h}{Rk}}. \end{align} (22)

The accuracy of (22) is shown in Table 1 as well and the comparisons between the accurate values and the approximate values calculated by (22) are shown in Fig. 3. It can be seen that (22) is an appropriate simplification.

Fig. 3

Comparisons between the accurate values and the approximate values of $\tilde{V}_{c}$ on B_i, where $\tilde{V}_{c} = G\cdot \text{P}_{\text{e}} = V_{c}\cdot \rho C_{p}R/k$. The relative error of $\tilde{V}_{c}$ is the same as that of V_c.

3. Discussion

In the last section, the explicit relationships between the solidification control parameters and the physical and geometrical parameters have been obtained. In this section, some discussions and applications based on these relationships will be presented.

According to (13), the following relationships can be obtained:   

\begin{equation} V\propto\frac{1}{\sqrt{R}},\ V\propto\sqrt{k},\quad V\propto\sqrt{h},\ V\propto\frac{1}{\rho},\ V\propto\frac{1}{C_{p}}, \end{equation} (23)

where ∝ denotes “is proportional to”. Additionally, the relationships between V and the temperatures of the melting point T_m, the hot zone T_h and the cool zone T_c are illustrated in Table 2 where   

\begin{equation*} C_{1}=\frac{1}{\rho C_{p}}\sqrt{\frac{hk}{2R}}\frac{(T_{h}-T_{c})}{((T_{m}-T_{c})(T_{h}-T_{m}))^{\frac{3}{2}}} \end{equation*}

and   

\begin{equation*} C_{2}=\sqrt{\frac{h}{2Rk}}\frac{(2T_{m}-T_{h}-T_{c})}{(T_{h}-T_{c})^{2}\sqrt{(T_{m}-T_{c})(T_{h}-T_{m})}}. \end{equation*}

These relationships provide deep understanding of the directional solidification process. For instance, if the radius of the cast R increases by 4 times, the withdrawal rate V should be half of the previous setting. The withdrawal rate V will increase by 2 times when h increases by 4 times. Similar assertions with respect to k, ρ and C_p can be made. In addition, V increases in T_m and decreases in T_h and T_c according to Table 2. These results imply that a superalloy cast with small lateral dimension, large thermal conductivity, large heat transfer coefficient (heat dissipation efficiency), small density and small specific heat capacity can be manufactured with a large withdrawal rate (i.e. the high production efficiency).

Table 2 The partial derivatives of the solidification control parameters V, G and V_c to T_m, T_h and T_c.

As regards the axial temperature gradient G, it can be seen from (21) that G decreases in R and k and increases in h with an explicit expression. The relationships between G and T_m, T_h and T_c are also illustrated in Table 2 which shows that G decreases in T_m and increases in T_h and T_c. Moreover, the value of G at the solidification front can be estimated directly by (21) when the physical and geometrical parameters of casts are given. In addition, note that $L_{2} = \sqrt{2\Phi _{m}(1 - \Phi _{m})} $. It implies that G will be large if Φ_m is closed to 1/2, i.e., T_m is close to the middle of T_h and T_c. In real manufacturing, Φ_m is about 0.8–0.9. It indicates that increasing the heater temperature T_h can enlarge the axial temperature gradient.

Inserting L₁ and L₂ into (22) gives   

\begin{align} V_{c}&=V\cdot G=2(2\Phi_{m}-1)\frac{h}{\rho C_{p}R}\\ &\quad +L_{3}\frac{\sqrt{2}(2\Phi_{m}-1)}{\sqrt{\Phi_{m}(1-\Phi_{m})}}\frac{h}{\rho C_{p}}\sqrt{\frac{h}{Rk}}. \end{align} (24)

Therefore, V_c increases in h and decreases in R and k with an explicit expression. Additionally, from (24) we have the following relationships:   

\begin{equation} V_{c}\propto\frac{1}{\rho},\quad V_{c}\propto\frac{1}{C_{p}}. \end{equation} (25)

The variations of V_c over T_m, T_h and T_c are shown in Table 2. It can be seen that the trend of V_c for T_m, T_h and T_c are similar to that of V. Note that V increases in Φ_m and G decreases in Φ_m, i.e., V and G are contrary to Φ_m. However, the largest V_c can be obtained by a proper Φ_m. That is, given the physical and geometrical parameters, the largest V_c through (24) can be obtained by setting proper T_h and T_c for the manufacturing of a cast.

In directional solidification process, V, G and V_c are key parameters. They are related to many other important quantities. The stable condition of the interface is related with G/V.¹³⁾ The expression of G/V can be derived by (13) and (21) which is   

\begin{equation} \frac{G}{V}=\frac{L_{2}}{L_{1}}\frac{\rho C_{p}}{k}+\frac{L_{3}}{L_{1}}\frac{\rho C_{p}}{k}\sqrt{\frac{hR}{k}}. \end{equation} (26)

Inserting L₁ and L₂ into (26) gives   

\begin{equation} \frac{G}{V}=\frac{\Phi_{m}(1-\Phi_{m})}{2\Phi_{m}-1}\frac{\rho C_{p}}{k}+\frac{L_{3}\sqrt{\Phi_{m}(1-\Phi_{m})}}{\sqrt{2}(2\Phi_{m}-1)}\frac{\rho C_{p}}{k}\sqrt{\frac{hR}{k}}. \end{equation} (27)

Equation (27) shows that G/V decreases in R, k and Φ_m with an explicit expression. In addition, we have   

\begin{equation} \frac{G}{V}\propto\rho,\quad\frac{G}{V}\propto C_{p}. \end{equation} (28)

For a cast, the unstable interface can be adjusted by the control of the heater and cooler temperatures through (27).

An application will be given based on the relationships obtained above. According to the study of Pollock and Murphy,²²⁾ defects can be predicted by critical G, G · V or G/V criteria. Thus, combing these criteria with the equations we derived, some guidance for the manufacturing of directionally solidified column crystal and single crystal can be obtained. For example, based on (21) and G > 15°C/cm (cf. Fig. 22 in Ref. 22), the following relationship can be obtained   

\begin{equation} R<\cfrac{2h\Phi_{m}(1-\Phi_{m})}{k\biggl(15-L_{3}\cfrac{h}{k}\biggr)^{2}}. \end{equation} (29)

It means that the size of a cast without defects should be set no more than a value solved by (29). The inequality related to Φ_m is   

\begin{align} &\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{2Rk}{h}\left(15-L_{3}\frac{h}{k}\right)^{2}}<\Phi_{m}\\ &\quad <\frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{2Rk}{h}\left(15-L_{3}\frac{h}{k}\right)^{2}}. \end{align} (30)

That is, when the physical and geometrical parameters are given, the heater and cooler temperatures should be set to satisfy (30) for avoiding defects.

Also, combing G · V > 0.1°C/s which is shown in Fig. 22 (Ref. 22) with (24), it implies that the withdrawal rate value should be set in order to guarantee no defects when the size and the alloy of casts is fixed. Or, the maximum size of cast (R) can be determined according to a given V which is connected into the thermal conditions (T_h, T_c) and the alloy (T_m) (cf. (13)). Similarly, we can use criterion G/V > 2700°C·s/cm² to estimate the maximum size of the cast or set the proper T_h and T_c. Likewise, the critical size (R) for the presence of nucleated grains and for lateral growth can be obtained based on the study of Miller and Pollock²³⁾ (cf. Fig. 16).

Another application is that, from Refs. 6, 7, 10, the primary dendrite arm spacing has the following relationship λ₁ ∝ V^−1/4G^−1/2. Based on the relationships (13) and (21), the explicit relationships between the dendrite arm spacing and the physical and geometrical parameters can be obtained as follows   

\begin{align} \lambda_{1}&=CV^{-\frac{1}{4}}G^{-\frac{1}{2}}\\ &=C\left(\frac{h}{\rho C_{p}}\sqrt{\frac{2h}{Rk}}(2\Phi_{m}-1)\left(\sqrt{\frac{2}{R}}(\Phi_{m}(1-\Phi_{m}))^{\frac{1}{4}}+L_{3}\sqrt{\frac{h}{k}}(\Phi_{m}(1-\Phi_{m}))^{-\frac{1}{4}}\right)^{2}\right)^{-\tfrac{1}{4}}, \end{align} (31)

where C is the alloy parameter related to diffusion coefficient, distribution coefficient, Gibbs Thompson parameter and the solidus and liquidus temperature (cf. eq. (23) in Ref. 10). From (31), the primary dendrite spacing λ₁ decreases in h and increases in R, ρ and C_p with an explicit expression. A smallest λ₁ can be obtained by choosing a proper Φ_m, i.e., adjusting T_h and T_c. Moreover, the primary dendrite spacing can be predicted by the physical and geometrical parameters of the alloy and the solidification conditions through (31).

4. Conclusions

Through the thermal analysis and mathematical derivation, the explicit relationships ((13), (21) and (22)) between the solidification control parameters (V, G and V_c) and the physical and geometrical parameters are presented when the Biot number is much less than 1 (Bi ≤ 0.2). It is proven by the comparisons between the accurate values and the approximate values that the simplified expressions achieve high accuracy. These relationships give deep understanding into the Bridgman process. These relationships provide a connection between the microstructure and the physical and geometrical parameters. Additionally, these relationships are helpful for the manufacturing of directionally solidified columnar crystals and single crystals. The main conclusions are the following.

1. (1)    The explicit relationships of V, G and V_c are obtained. V is proportional to $1/\sqrt{R} $, $\sqrt{k} $, $\sqrt{h} $, 1/ρ and 1/C_p. For instance, the withdrawal rate should be half when the radius increases by 4 times. G increases in h and decreases in R and k. V_c increases in h and decreases in R, k ρ and C_p. The largest V_c can be obtained through (24) by choosing proper T_h and T_c when given a cast.
2. (2)    Combining the explicit relationships with the critical criteria of the appearance of defects, a maximum size of casts can be obtained when given the physical parameters. Or proper heater temperature (T_h) cooler temperature (T_c) can be chosen for no defects when given a cast.
3. (3)    The explicit relationship of the primary dendrite arm spacing (λ₁) is derived which shows that λ₁ decreases in h and increases in R, ρ and C_p. Additionally, a smallest λ₁ can be obtained by adjusting the heater temperature (T_h) cooler temperature (T_c) when given a cast.

Acknowledgments

This work was supported by NSFC 11701454.

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