Trans. Nonferrous Met. Soc. China 31(2021) 1518-1528

Free dendritic growth model considering both interfacial nonisothermal nature and effect of convection for binary alloy

Shu-cheng LIU¹, Li-hua LIU¹, Shu LI^2,3, Jin-zhong WANG¹

1. School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, China;

2. School of Science, Harbin University of Science and Technology, Harbin 150080, China;

3. Heilongjiang Provincial Key Laboratory of Quantum Manipulation & Control, Harbin University of Science and Technology, Harbin 150080, China

Received 25 May 2020; accepted 30 December 2020

Abstract: Considering both the effect of the nonisothermal nature of the interface as well as the effect of forced convection, an extended free dendritic growth model for binary alloys was proposed. Comparative analysis indicates that the effect of convection on solute diffusion is more remarkable compared with the ignorable effect of convection on thermal diffusion at low bath undercooling, due to the fact that solute diffusion coefficient is usually three orders of magnitude less than thermal diffusion coefficient. At high bath undercooling, the effect of convection on the dendritic growth is very slight. Furthermore, a satisfying agreement between the model predictions with the available experiment data for the Cu₇₀Ni₃₀ alloy was obtained, especially at low bath undercoolings, profiting from the higher values of interfacial migration velocity predicted by the present model with nonideal fluid case than that predicted by the one ignoring the effect of convection.

Key words: modeling; dendritic solidification; binary alloys; nonisothermal interface; convection

1 Introduction

Theoretical analysis and mathematical modeling implicated in the investigation of crystal growth are as important as experimental observation [1-6]. To describe the free dendritic growth, three aspects should be taken into account, which are the interface kinetics, the thermal or solutal transport in the bulk liquid ahead of the solid-liquid interface and the morphological stability for the interface [7-14]. The interface kinetics, i.e. the interface response function [15], is used to set up a relationship between the interfacial migration velocity and thermodynamic driving force determined by the interfacial temperature and compositions of solid and liquid phases. The thermal or solutal transport in liquid is commonly dealt with by solving the classical Fick diffusion equation or the extended hyperbolic diffusion equation [16,17]. The morphological stability analysis should also be made, which gives a relationship between the interfacial migration velocity and the dentritic tip radius of curvature [18].

Most of the free dendritic growth models have a deficiency, i.e. the isothermal and isosolutal solid-liquid interface assumption. In fact, the interface is nonisothermal and nonisosolutal one [19]. Strictly speaking, the solid-liquid interface is non-planar one, which leads to the variable interfacial curvature and normal velocity along the curved interface. Additionally, the crystalline anisotropy results in the interfacial energy anisotropy and the kinetic growth anisotropy. Recently, for eliminating the isothermal and isosolutal solid-liquid interface assumption, LI et al [20-24] have successfully obtained the exact solution of steady state Fick diffusion equation and made a series of comparative analysis. It was further demonstrated that the effect of the nonisothermal nature of interface is significant and the effect of the nonisosolutal nature of interface is very slight and really negligible.

Those models [20-24], however, ignore the influence of convection on dendrite growth. It is interesting and necessary to consider the convection effect when modeling the free dendrite growth [25-28]. There are two kinds of convection, i.e., the forced convection and the natural convection. The forced convection is caused by an external force. The natural convection occurs naturally and is caused by the density change due to the variation in concentration or temperature. In the present work, the focus of our attention is the forced convection to model the free dentritic growth during solidification. The forced convection affects the thermal or solutal transport in the liquid as well as morphological stability for the interface. According to the Navier-Stokes equation [26,28], the new thermal diffusion equation was re-solved. Based on microscopic solvability theory, the tip radius of dendrite given by ALEXANDROV and GALENKO [28-31] was adopted. Comparisons with the related models and the available experiment data were also made.

2 Modeling

In this section, considering a forced convection in alloy melts, the thermal diffusion equations with the nonisothermal boundary condition were re-solved to discuss the influence of convection on thermal diffusion. The results for solutal transport and morphological stability criterion given by ALEXANDROV and GALENKO [28] were adopted considering convection and isosolutal interface assumption.

2.1 Thermal transport in liquid

Crystal growth in the incoming flow of fluid is described in terms of the nonlinear Stefan type diffusion problem [32,33] with the moving free boundary of phase transition. The flow can be described by the Navier-Stokes equation [26,28]:

(1)

where ω is the fluid flow velocity, is the vector gradient operator, ρ is the density, P is the pressure, and η is the kinematic viscosity of the fluid. Considering convection in the liquid and ignoring diffusion in the solid, the thermal transport phenomena can be described by the extended Fick diffusion equation as

(2)

where T_L is the temperature in bulk liquid, t is time, and D_T is the thermal diffusivity in the liquid.

In a small domain of the tip, the solid-liquid interface can be approximated by a paraboloid of revolution under the steady state dendritic growth condition in undercooled melts [18,34]. It is convenient to use a parabolic coordinate system (α, β, φ) to solve temperature and solute fields in the liquid phase, instead of the Cartesion coordinate system (x, y, z). The coordinate transition is adopted: x=rαβcos φ, y=rαβsin φ, and z′=0.5r(α²-β²), where r is the tip radius of curvature, z′=z-Vt is a conversion to fix the reference frame on the moving interface, and V is the interfacial migration velocity in the z-direction. Then α=1 represents the solid-liquid interface with a tip radius of curvature r [22]. The temperature and solute concentration fields in the bulk liquid have the forms T_L(α, β) and C_L(α, β), respectively. Here, the coordinate φ is not included due to rotational symmetry. With the parabolic coordinate system (α, β, φ), Eq. (2) is rewritten as follows [28]:

(3)

where u_α and u_β are the normal and tangent components of relative velocities between the interface of growing dendritic crystal and the undercooled melt, which are defined as follows:

(4)

(5)

where U is the incoming fluid flow velocity far from the interface, and g(α) is described by

(6)

where Re is the Reynolds number defined by Re=rU/η and the exponential function E₁(z) is defined by

(7)

It is hard to obtain the exact solution of Eq. (3), due to the complexity of the expression of g(α) defined by Eq. (6). In order to discuss the effect of convection in the liquid on the thermal diffusion field, a simplified version of g(α)=1/α is taken. This corresponds to the ideal fluid condition. Then, Eq. (3) can be rewritten as

(8)

where P_t0=P_t+P_tf, P_t is the thermal growth Peclet number defined by P_t=rV/(2D_T), and P_tf is the thermal flow Peclet number defined by P_tf=rU/(2D_T). In this equation, a term -2P_t0βT_L/β is neglected with the approximation 1/β ＞＞2P_t0β, since the solidification behavior at the tip of dendrite (β=0) is only influenced by the regions in the vicinity of β≈0 [23].

By adopting the separation variable method, Eq. (8) is divided into a Bessel equation and a confluent hypergeometric equation. Its solution can be obtained as

(9)

where Φ(a, b, z) is the confluen thypergeometric function of the second kind, J₀(λβ) is the zeroth order Bessel function of the first kind, T_∞ is the temperature of the undercooled melt far from the interface as an integration constant, A_t(λ) is a coefficient to be determined, and λ is a constant. Considering the nonisothermal nature of the interface, caused by the non-planar interface and the crystalline anisotropy, with the isosolutal interface assumption [25], the interface response function can be given as [24]

(10)

where T_L(1, β) represents the interfacial temperature, T_M is the melting point of binary alloy with the nominal composition C₀, Q is the latent heat of fusion, c_p is the specific heat capacity, d₀ is the capillary constant, is the kinetic coefficient, m_L is the slope of equilibrium liquidus, and is the solute concentration on the liquid side at the tip. In addition, the kinetic liquidus slope m(V) and the parameters f₁(β) and f₂(β) are defined as follows with the assumption θ_d=0 and θ_β=π/4 [24]:

(11)

(12)

(13a)

(13b)

(14a)

(14b)

where a_d and a_β are the stiffnesses of anisotropy, V_D is the solute diffusive speed in bulk liquid, V_n is the interfical normal speed, k_e and k are the equilibrium and nonequilibrium partition coefficients, respectively, and V_DI is the interfacial solute diffusive velocity, at VD and ψ=0 at V ≥V_D.

By using the Hankel transform for coordinate β, combined with Eqs. (9) and (10), A_t(λ) can be further determined as follows:

(15)

where δ(λ) is the delta function. F₁(λ) and F₂(λ) are defined by [23]

(16)

(17)

Additionally, the thermal transport balance equation is adopted at the nonisothermal solid- liquid interface:

(18)

where K_L is the thermal conductivity in the liquidin the absence of melt convection, and is the unit vector normal to the interface. Substituting the temperature field described by Eq. (9) into the conservation equation, Eq. (18), the following relationship at the tip (β=0) can be finally given as follows:

(19)

where the new Ivantsov function Iv(P_t, P_tf) and the parameters N₁(P_t, P_tf) and N₂(P_t, P_tf) are defined by

(20)

(21)

(22)

Here, a relationship Φ(a, a, z)/Φ(a, a+1, z)= is used, which can be demonstrated mathematically.

As the value of g(α) is in the range of (0, 1) and is monotonically decreasing with α, there are two limit cases for the effect of convection on the thermal diffusion in liquid phase. g(α)≡1 represents the condition that there is no convection in the liquid. With this assumption, the present result is reduced to the previous one [24]. Under another limit condition, g(α)≡0, one can obtain the same expression as described by Eq. (19), in which related parameters are redefined as follows:

(23)

(24)

(25)

2.2 Solutal transport in liquid

For solutal transport in the liquid, the isosolutal assumption along the solid-liquid interface has been shown to be reasonable approximation [22]. With this assumption the interfacial solute composition at the tip is given by ALEXANDROV and GALENKO [28] as

(26)

where P_c is the solute growth Peclet number defined by P_c=rV/(2D_L) (where D_L is the solute diffusivity in the liquid), and P_cf is the solute flow Peclet number defined by P_cf=rU/(2D_L). At the given parabolic coordinate system (α, β, φ), Iv(P_c, P_cf) is defined as

(27)

where P_c0 is defined by P_c0=P_c+P_cf.

Considering the relaxation effect of local nonequilibrium solute diffusion in bulk liquid, the interfacial solute composition at the tip can be rewritten as [9,11]

(28a)

(28b)

Similar to the thermal diffusion, there are also two limit cases for the solute diffusion. One is with the assumption g(α)≡1, i.e. the condition ignoring the effect of convection on solute diffusion in the liquid. The other limit case is with the assumption of g(α)≡0. Under this condition, Eq. (27) can be re-solved as

(29)

2.3 Morphological stability criterion

The analysis of solid-liquid interface morphological stability should be made to uniquely determine solidification behavior with steady state dendritic growth. Taking into account the interfacial nonisothermal nature, the microscopic solvability theory can give a correlation between the tip radius of curvature r and the interfacial migration velocity V at a given bath undercolling △T. According to Ref. [28], the selection criterion derived from microscopic solvability theory is described as

(30a)

(30b)

where σ* is defined by σ*=2D_Td₀/(r²V), σ₀ is the selection constant, , T_Q is defined by T_Q=Q/c_p, and the other parameters are defined as follows:

(31)

(32)

(33)

where a₁ is a constant defined by σ₀ (a₁=(8σ₀/7)^1/2(3/56)^3/8), , is defined by δ₀=1，the mass and thermal transport functions, d_0CD is the chemical capillary length, and the parameters α(Re) and p are defined by

(34)

(35)

Then being derived from Eq. (31), the tip radius of curvature r can be obtained as

(36a)

(36b)

Up to now, considering forced flow in the fluid, the entire free dendrite growth model has been described. By solving Eqs. (19), (28) and (36) simultaneously, with any given bath undercooling △T, one can determine the solidification behavior, uniquely.

3 Results and discussion

In this section, model comparisons were made to analyze the effect of convection on thermal diffusion and solutal diffusion. Experimental comparison was also made for the Cu₇₀Ni₃₀ alloy. The numerical results are shown in Figs. 1-6. The thermo-physical data, used in the present computations, are listed in Table 1. In order to express more clearly, the distinction between these models discussed in the present work is shown in Table 2.

Table 1 Thermo-physical data for Cu₇₀Ni₃₀ alloy used in model computations

3.1 Effect of convection on thermal diffusion

The dendritic tip temperature T_i as function of the bath undercooling △T is shown in Fig. 1, for three models considering different expressions of g(α).

Table 2 Definition of g(α) used in different models

Fig. 1 Evolution of tip temperature as function of bath undercooling △T for Cu₇₀Ni₃₀ alloy

As the value of g(α) is in the range of (0, 1) and is monotonically decreasing with α, there are two limit cases for the effect of convection on the thermal diffusion in liquid phase. As shown in Table 2, limit case 1 represents the model with assumption of g(α)≡1, and this is the condition that there is no convection in the liquid. Another limit case, called limit case 2, denotes the model with assumption of g(α)≡0. It is a limiting condition that the influence of the convection on thermal diffusion is the most remarkable. The ideal fluid case means the model with the assumption of g(α)=1/α. It should be stressed that in order to focus on the discussion about the effect of convection on thermal diffusion, different expressions of g(α) are only adopted in the description of thermal diffusion, while the same equations were adopted in the description of solutal diffusion, Eqs. (27) and (28) with g(α)≡1, as well as morphological stability criterion, Eq. (36), for the three models. This can be seen from Table 2 more clearly. Figure 1 shows that there are slight differences of the dendritic tip temperatures between these models at low bath undercoolings while the differences are more negligible at high bath undercoolings.

As well know, the total undercooling △T can be divided into the following five parts: the thermal undercooling △T_T=T_M-T_∞, the constitutional undercooling the under- cooling due to the kinetic liquidus line which deviates from the equilibrium position △T_N= (m_L-m(V))C₀, the curvature undercooling and the kinetic undercooling So, we have △T=△T_T+△T_C+△T_N+△T_R+△T_K. Based on this relationship and the result given by the present model, Eq. (19), one can further determine the thermal undercooling △T_T as △T_T=IvQ/c_p+△T_R(N₁-1)+△T_K(N₂-1) (where N₁ is the parameter mentioned above as N₁(P_t, P_tf); N₂ is the parameter mentioned above as N₂(P_t, P_tf)). Thus, the difference between the tip temperatures predicted by the three models discussed above can be analyzed through the values of the Ivantsov function (Iv) and the parameters N₁ and N₂, due to the relation The values of the Ivantsov function (Iv) and the parameters N₁ and N₂ as functions of both Peclet number P_t and undercooling △T are shown in Figs. 2 and 3, respectively. It can be seen from Fig. 2 that the values of Iv for the three conditions are almost equal to each other. This means that the effect of convection on thermal diffusion with the isothermal interface assumption is ignorable (N₁=N₂≡1). Under the nonisothermal interface condition, the parameters N₁ and N₂ are not constantly equal to 1. So, the difference between the tip temperatures predicted by the three models is mainly caused by the parameters N₁ and N₂. As shown in Fig. 3, at low undercoolings △T, the value of parameter N₂ given by limit case 1 has an obvious deviation from that predicted by limit case 2, while there is a relatively small difference between the parameters N₁. However, it is just the value of parameter N₁ that plays an important role in determining the difference between the tip temperatures predicted by the three models. This is because the curvature undercooling △T_R is much larger than the kinetic undercooling △T_K, at low bath undercooling △T, in the relationship △T_T=IvQ/c_p+△T_R(N₁-1)+ △T_K(N₂-1). Under this condition, the value of parameter N₂ almost does not influence the difference between . Therefore, considering the small difference between the values of N₁, the effect of convection on thermal diffusion is also very slight under the condition with nonisothermal solid-liquid interface. Additionally, in Fig. 4 it can also be seen that there is a very slight deviation of the interfacial migration velocity predicted by limit case 1 from that calculated by limit case 2, especially at high undercooling.

There are several parameters which can influence the effect of convection on thermal diffusion, such as the velocity of convection U, the coefficient of the fluid viscosity η and the stiffness of anisotropy a_d. The faster the velocity of convection U is or the smaller the coefficient of the fluid viscosity η is, the more remarkable the influence of convection on solute diffusion is. Besides these two factors, the anisotropy strength a_d is also a factor that can determine the degree of the convection effect. As described by Eqs. (16) and (21), the value of N₁ depends on the anisotropy strength a_d. Taking these parameters given in Table 1, the effect of convection on thermal diffusion is slight enough to be ignored. Furthermore, as discussed in the next section, the effect of convection on solute diffusion is remarkable compared with the ignorable effect of convection on thermal diffusion.

Fig. 2 Ivantsov function on thermal diffusion Iv(P_t,P_tf) as function of thermal Peclet number P_t, at given velocity of forced convection U=0.3 m/s

Fig. 3 Values of parameters N₁ and N₂ predicted by three models, as function of bath undercooling △T for Cu₇₀Ni₃₀ alloy

Fig. 4 Interfacial migration velocity V as function of bath undercooling △T for Cu₇₀Ni₃₀ alloy

Based on above discussion, it is also concluded that the effect of nonisothermal nature of the solid-liquid interface is comparable with the effect of convection on thermal diffusion. However, it should be stressed that above discussion is made with the assumption θ_d=0 and θ_β=π/4 as well as a set of parameters given in Table 1. Under other conditions, the effect of non-isothermality may be obvious or even remarkable [23-27]. For example, as discussed in Ref. [27], the effect of non- isothermality is not ignorable and should be taken into account, under the condition that θ_d=π/4 or a_d decreases to 0.1.

3.2 Effect of convection on solute diffusion

In this section, three cases are used for comparative analysis, including limit case 1, limit case 3 and nonideal fluid case. Limit case 1 is the one mentioned above that there is no convection in the liquid. g(α)≡1 is adopted for limit case 1 in the equations describing both thermal diffusion and solute diffusion in the liquid phase. Limit case 3 represents the model with g(α)≡0 in Eqs. (28) and (30), for solute diffusion, which means the maximum limit of the influence of convection on solute diffusion. The nonideal fluid case represents the model, in which g(α) is defined by Eq. (6) for Eqs. (27) and (28) in the solutal transport (Section 2.2). In order to focus on the discussion about the effect of convection on solute diffusion, the same expressions of g(α)≡1 are adopted in the description of thermal diffusion (see also the column “Thermal diffusion” in Table 2). That is to say, in this section it is assumed that the influence of convection on thermal diffusion in liquid is neglected completely in all of the three cases.

As shown in Fig. 4, there is an obvious distinction between the nonideal fluid case and the limit case 1 for the interfacial velocity at low bath undercooling. In contrast, as discussed in Section 3.1, the deviation of the interfacial velocity predicted by limit case 2 from that calculated by limit case 1 is very slight. It is indicated that the effect of convection on solute diffusion is greater than that on thermal diffusion with a set of parameters given in Table 1. This is due to the fact that solute diffusion coefficient is usually three orders of magnitude less than thermal diffusion coefficient. Therefore, the solute diffusion is remarkably slower compared to thermal diffusion. It is just the slower diffusion that leads to more obvious influence of the forced convection on the solutal diffusion compared with the thermal diffusion. This can also be understood by using the solute diffusion scale l_C and the hydrodynamic spatial scale l_H, which are defined as l_C=2D_L/V and l_H=η/U, respectively [35]. The ratio of l_C to l_H (l_C/l_H) is shown in Fig. 5, as a function of the bath undercooling. It is indicated that the value of l_C is comparable with the value of l_H at low undercoolings. Thus, the effect of forced convection on the solutal diffusion is obvious [35].

Fig. 5 Ratio of solutal length scale l_C to hydrodynamic length scale l_H as function of bath undercooling △T for Cu₇₀Ni₃₀ alloy

In order to further demonstrate the effect of convection on solute diffusion, the tip solute concentration of the dendrite as a function of the bath undercooling △T is shown in Fig. 6, for different models including the nonideal fluid cases with η=5×10^-8 m²/s and η=5×10^-10 m²/s. Firstly, it can be seen that the larger the coefficient of viscosity η is, the slighter the effect of the convection on solute diffusion is. Secondly, the forced convection results in the lower tip solutal concentrations on the liquid side compared with the condition that there is not forced convection. Thus, the forced convection can depress solute segregation at the solid-liquid interface. Thirdly, as indicated by Fig. 5, the effect of convection on solute diffusion is very slight and can further be ignored at high bath undercooling, since the solute trapping or even complete solute trapping occurs. Therefore, the effect of convection on solute diffusion should be taken into account in modeling free dendritic growth, under the condition that there exists forced convection, especially at low bath undercoolings.

Fig. 6 Evolution of tip solute concentration (mole fraction) on liquid side as function of bath undercooling, for Cu₇₀Ni₃₀ alloy

3.3 Experimental comparison

An experimental comparison for the interfacial migration velocity V versus the bath undercooling △T for the Cu₇₀Ni₃₀ alloy is shown in Fig. 7 [36]. As discussed above, the effect of convection on solute diffusion is remarkable compared with the ignorable effect of convection on thermal diffusion. Thus, in Fig. 7, the interfacial velocity V is predicted by the present model with the nonideal fluid case shown in Table 2. The related thermophysical parameters are shown in Table 1. In addition, as one of the series of models [20-24], the present model also takes into account the nonisothermal nature of the solid-liquid interface caused by curved interface and crystalline anisotropy and takes the isosolutal interface assumption. It is indicated that the present model can give a satisfying description to the experiment data, especially at low bath undercoolings. One of the main reasons lies in the fact that the present model introduces the effect of the forced convection on solutal diffusion. It can be further supported by Fig. 4, in which the interfacial migration velocity V predicted by the present model with nonideal fluid case is obviously faster than that predicted by the one ignoring the effect of convection.

Fig. 7 Comparison of experimental and predicted results of interfacial migration velocity V as function of bath undercooling △T for Cu₇₀Ni₃₀ alloy

4 Conclusions

(1) The larger the coefficient of viscosity, the slighter the effect of the convection on solute diffusion. The forced convection can depress solute segregation at the solid-liquid interface.

(2) Model comparison indicates that the effect of convection on solute diffusion is remarkable compared with the ignorable effect of convection on thermal diffusion, at low bath undercooling. At high bath undercooling, the effect of convection is very slight.

(3) Experimental comparison shows that the present model can give a satisfying description to the experimental data for the Cu₇₀Ni₃₀ alloy, especially at low bath undercoolings, benefiting by introducing the effect of forced convection on solutal diffusion.

Acknowledgments

The authors are grateful for the financial supports from the National Natural Science Foundation of China (No. 51671075), the Heilong-jiang Postdoctoral Fund for Scientific Research Initiation (No. LBH-Q16118), and the Fundamental Research Foundation for Universities of Heilong- jiang Province, China (No. LGYC2018-JC004).

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同时考虑界面非等温特性和对流影响的二元合金自由枝晶生长模型

刘书诚¹，刘礼华¹，李 述^2,3，王金忠¹

1. 哈尔滨工业大学 材料科学与工程学院，哈尔滨 150001；

2. 哈尔滨理工大学 理学院，哈尔滨 150080；

3. 哈尔滨理工大学 黑龙江省量子调控重点实验室，哈尔滨 150080

摘 要：在同时考虑界面非等温性质以及强制对流影响的情况下，提出二元合金的扩展自由枝晶生长模型。模型对比表明：由于溶质扩散系数通常比热扩散系数小3个数量级，当低过冷时，对流对溶质扩散的影响比其对热扩散的影响更强烈。当高过冷时，对流对枝晶生长的影响很小。此外，本模型能够对现有Cu₇₀Ni₃₀合金实验数据给出一致的描述，特别是在低过冷时。这得益于当前模型获得了比忽略对流影响的模型更快的界面迁移速率。

关键词：建模；枝晶凝固；二元合金；非等温界面；对流

(Edited by Wei-ping CHEN)

Corresponding author: Li-hua LIU, Tel: +86-510-86907777, E-mail: liulh@chinafasten.com;

Shu LI, Tel: +86-451-86390978, E-mail: lishu@hrbust.edu.cn

DOI: 10.1016/S1003-6326(21)65595-4

1003-6326/ 2021 The Nonferrous Metals Society of China. Published by Elsevier Ltd & Science Press