Primary dendrite spacing selection during directional solidification of multicomponent nickel-based superalloy: multiphase-field study

Abstract

The primary dendrite spacing selection in a multicomponent Ni-based superalloy during directional solidification was systematically studied using two-dimensional phase-field simulations. The alloy thermodynamic and kinetic data were obtained from Pandat software with PanNickel database and directly coupled into the multiphase-field model. All the simulations were performed on a GPU server, and an optimized computing scheme using GPU shared memory was adopted. First, the morphology of the solidification front was studied, and the segregation pattern was investigated and compared with the experimental results. Then, the dendritic spacing distribution under a wide range of pulling velocities V_p (10–500 μm s⁻¹) and temperature gradients G (2–200 K mm⁻¹) was obtained and analyzed. The simulation results agree well with analytical model that the primary dendrite spacing scales as $\varLambda \propto V_{\text{p}}^{ - b} G^{ - c}$. The coefficient b is near a constant value of 0.38 and varies slightly between 0.34 and 0.42, while coefficient c increases monotonously from 0.27 to 0.56 with the increasing G. The predicted dendritic spacing agrees well with the experimental data, but exhibits a major difference when under very low cooling rate (R < 0.1 K s⁻¹). The effect of grain inclination angle θ on the final primary dendritic spacing was also studied, and an abnormal decrease in dendritic spacing was found under low grain orientation where θ < 10°. When the grain inclination angle exceeds 20°, the dendritic spacing increases with θ as the power law.

Introduction

Nickel-based superalloy single-crystal turbine blades, which are produced by directional solidification technique, have been extensively used in advanced aero-engines. The primary dendrite arm spacing (PDAS), as one of the most important microstructure characteristics in superalloy directional solidification, can influence the material’s mechanical properties. Up to now, a variety of directional solidification techniques have been developed, such as Bridgman high-rate solidification technique (HRS) [1], liquid metal cooling (LMC) [2] and zone melting liquid metal cooling (ZMLMC) [3]. The different solidification techniques have provided a range of solidification conditions and thus led to a different distribution of primary spacing in superalloy products [3,4,5,6,7,8,9,10,11].

The variation of dendritic spacing with pulling velocity V_p and thermal gradient G has long been analyzed in classical theories and numerous experiments. In analytical models [12, 13], the primary dendritic spacing Λ observes the power law $\Delta T_{0}^{a} V_{\text{p}}^{ - b} G^{ - c}$, where $\Delta T_{0}$ is the alloy solidification range, a = b = 0.25 and c = 0.5. The experimental results found that coefficients b and c do not remain constant under different cooling conditions, and they vary from 0.19 to 0.75 and 0.3 to 0.56, respectively. As for the orientation dependence of the primary dendrite spacing, Gandin et al. [14] proposed a model based on experiments with SCN-3.61 wt% acetone alloy as:

$$ \varLambda \propto \Delta T_{0}^{a} V_{\text{p}}^{ - b} G^{ - c} F\left( \theta \right) $$

(1)

by putting forward an orientation correction

$$ F\left( \theta \right) = 1 + d\left[ {\left( {\cos \theta } \right)^{ - e} - 1} \right] $$

(2)

From the experiments and analysis, the coefficients they estimated are d ≈ 0.15 and e ≈ 8 for range of velocity V_p (20–1000 μm s⁻¹) and temperature gradient G (0.1–10 K mm⁻¹).

The primary spacing in directionally solidified nickel-based superalloy has been studied since 1970s. Quested and McLean [4] investigated four nickel-based superalloy and showed that the Λ was indeed scaling as $V_{\text{p}}^{ - 0.25} G^{ - 0.5}$. Schneider et al. [15] used an expression $\varLambda { = }147\left( {V_{\text{p}} G} \right)^{ - 0.3384}$ that has been found to be applicable to a wide range of superalloys. Li [5, 6] presented detailed data on the dependence of primary dendrite arm spacing on a wide range of thermal gradients and solidification velocities, and especially under lower-temperature gradients. Liu [3] used the ZMLMC technique and studied the dendritic spacing under high-temperature gradients up to 1000 K cm⁻¹. Although experiments have been done to investigate the primary spacing of superalloy variation, the parameters of the solidification were mostly not well defined, especially the temperature gradient. In general, there is not a well-accepted expression for the primary dendrite spacing with relationship to temperature gradient and pulling velocity. The coefficients in Eq. (1) seem to vary with solidification conditions according to existing experimental results. Moreover, the systematic investigation of primary spacing distribution under a wide range of solidification conditions has not yet been accomplished.

The phase-field method [16,17,18,19,20], which adopts the concept of diffuse interface, has proven to be a powerful tool to study microstructure evolution in alloy solidification. For the simulation of complex multicomponent technical alloy like superalloy, the link of CALPHAD (CALculation of PHase Diagram) databases with phase-field model has become the common choice. Warnken [21] and Böttger [22] have used the multiphase-field model coupled with thermodynamic database to investigate the dendrite growth and segregation behavior in nickel-based superalloy. However, due to expensive computational cost of calculating multicomponent diffusion, the computational domain is limited and cannot be applied to large-scale superalloy directional solidification simulation. Recently, the GPU (graphical processing unit)-based parallel computing technique has been extensively used in accelerating phase-field computation [23,24,25]. By applying the GPU parallel computing scheme to phase-field model, it is now possible to study the primary dendrite selection in a multicomponent superalloy.

In this work, the multiphase-field model proposed by Steinbach [18] is adopted, and the software Pandat with PanNi database is used to present realistic superalloy thermodynamic data for multiphase-field model. The computation process is accelerated by a previously developed GPU computing scheme [23], which achieves more than 200 times speed-up ratio compared with that using a single CPU core. The main target of this research is to investigate the primary dendrite spacing in a wide range of pulling velocities (10–500 μm s⁻¹) and temperature gradients (2–200 K mm⁻¹) by large-scale phase-field simulations. The simulation results are analyzed using the analytical model in Eq. (1) and then compared with the experimental results. In addition, the effect of grain inclination angle on the final dendritic spacing is also investigated, and the results are different from the analytical model prediction when under small inclination angles (θ < 10°).

Methods

The directional solidification of the superalloy involves a complex solidification path with coupled heat and multicomponent solute diffusion. The phase-field model for binary alloy with a linearized phase diagram can hardly describe the solidification behavior of the superalloy. The multiphase-field model, which take account the effect of all the individual alloy components, has provided a better choice to simulate the superalloy solidification.

Multiphase-field model

The multiphase-field model used in this work was first proposed by Steinbach and Eiken et al. [18, 19], and the model is able to simulate multicomponent superalloy phase transformations [21]. In this model, all phases are represented by a set of scalar phase fields $\left\{ {\phi_{\alpha } \left( {\vec{x},t} \right)} \right\}$, with the constraint $\sum\limits_{\alpha = 1}^{v} {\phi_{\alpha } } = 1$. The alloy phase compositions $\left\{ {\vec{c}_{\alpha } \left( {\vec{x},t} \right)} \right\}$ have individual value in each phase and vary smoothly at the phase interfaces. The free energy functional is the integration of local density functional over the domain Ω.

$$ F\left( {\left\{ {\phi_{\alpha } } \right\},\left\{ {\vec{c}_{\alpha } } \right\}} \right) = \int_{\varOmega } {f\left( {\left\{ {\phi_{\alpha } } \right\},\left\{ {\vec{c}_{\alpha } } \right\}} \right)} $$

(3)

The density functional is the sum of chemical energy density $f^{\text{chem}} \left( {\left\{ {\phi_{\alpha } } \right\},\left\{ {\vec{c}_{\alpha } } \right\}} \right)$ and interface energy density $f^{\text{intf}} \left( {\left\{ {\phi_{\alpha } } \right\}} \right)$. For an alloy system with v phases, the free energy density can be expressed as

$$ \begin{aligned} f & = f^{\text{intf}} \left( {\left\{ {\phi_{\alpha } } \right\}} \right) + f^{\text{chem}} \left( {\left\{ {\phi_{\alpha } } \right\},\left\{ {c_{\alpha }^{k} } \right\}} \right) \\ & = \sum\limits_{\alpha = 1}^{v} {\sum\limits_{\beta = \alpha + 1}^{v} {\frac{{4\sigma_{\alpha \beta } }}{v\eta }\left( { - \frac{{\eta^{2} }}{{\pi^{2} }}\nabla \phi_{\alpha } \nabla \phi_{\beta } + \phi_{\alpha } \phi_{\beta } } \right)} } + \sum\limits_{\alpha = 1}^{v} {\phi_{\alpha } f_{\alpha } \left( {\vec{c}_{\alpha } } \right)} \\ \end{aligned} $$

(4)

where $\sigma_{\alpha \beta }$ denotes the interface energy between phase α and phase β in a multiphase junction with v phase and η is the width of the interface.

The phase-field equation can be derived from the variation of the free energy functional, with the aim of minimizing the total free energy F

$$ \frac{{\partial \phi_{\alpha } }}{\partial t} = - \sum\limits_{\beta \ne \alpha }^{v} {M_{\alpha \beta } \left( {\frac{\delta F}{{\delta \phi_{\alpha } }} - \frac{\delta F}{{\delta \phi_{\beta } }}} \right)} $$

(5)

where $M_{\alpha \beta }$ is the anisotropic interface mobility between α and β phases. The multiphase problem can be decomposed into a sum of dual-phase transitions as follows:

$$ \frac{{\partial \phi_{\alpha } }}{\partial t} = \sum\limits_{\beta \ne \alpha }^{v} {\tilde{M}_{\alpha \beta } \left[ {\frac{{\sigma_{\alpha \beta } }}{v}\left( {\frac{1}{2}\left( {\nabla^{2} \phi_{\alpha } - \nabla^{2} \phi_{\beta } } \right) + \frac{{\pi^{2} }}{{2\eta^{2} }}\left( {\phi_{\alpha } - \phi_{\beta } } \right)} \right) + \sum\limits_{\alpha \ne \beta \ne \gamma } {J_{\alpha \beta \gamma } } + \frac{\pi }{\eta }\sqrt {\phi_{\alpha } \phi_{\beta } } \Delta G_{\alpha \beta } } \right]} $$

(6)

where $J_{\alpha \beta \gamma } = \frac{1}{2}\left( {\sigma_{\beta \gamma } - \sigma_{\alpha \gamma } } \right)\left( {\frac{{\pi^{2} }}{{\eta^{2} }}\phi_{\gamma } + \nabla^{2} \phi_{\gamma } } \right)$ and $\Delta G_{\alpha \beta }$ is the thermodynamic driving force which is related to local temperature T and composition $\vec{c}$.

$$ \Delta G_{\alpha \beta } = \frac{1}{{V_{m} }}\left( {g_{\beta } - g_{\alpha } - \sum\limits_{i = 1}^{n - 1} {\tilde{\mu }^{i} \left( {c_{\beta }^{i} - c_{\alpha }^{i} } \right)} } \right) $$

(7)

The $g_{\alpha }$ and $g_{\beta }$ are the molar Gibbs energy of α and β phases, and the Lagrange multiplier $\tilde{\mu }^{i} = \mu^{i} - \mu^{n}$ is the diffusion potential of the ith component, with the $\mu^{i}$ and $\mu^{n}$ being the chemical potential of the ith component and the solvent. And by multiplying molar volume $V_{\text{m}}$, the molar free energy can be transferred to volume free energy.

In the multiphase-field model, the extra freedom of separate phase compositions $\left\{ {\vec{c}_{\alpha } } \right\}$ is fixed by quasi-equilibrium condition [19]. In this condition, the diffusion process is assumed to be much faster than the phase transformation process, and equal diffusion potential is stressed over the interface [17, 19].

$$ \tilde{\mu }^{i} = \frac{{dg_{\alpha } }}{{dc_{\alpha }^{i} }} = \frac{{dg_{\beta } }}{{dc_{\beta }^{i} }} = \cdots = \frac{{dg_{v} }}{{dc_{v}^{i} }} $$

(8)

Together the with the mass constraint $\vec{c} = \sum\limits_{\alpha = 1}^{v} {\phi_{\alpha } \vec{c}_{\alpha } }$, the Newton’s method can be used to solve these nonlinear equations. However, frequent access to thermodynamic databases and the Newton iteration process for all interface cells is time intensive. The extrapolation method [19, 22], which uses the Taylor’s expansion, has provided a feasible way to solve the quasi-equilibrium condition with acceptable computational efficiency, and it is adopted in this work. The expression for the extrapolated thermodynamic driving force can be written as

$$ \Delta G_{\alpha \beta }^{\text{Ext}} = \sum\limits_{i = 1}^{n - 1} {\left( {\frac{{\partial \Delta G_{\alpha \beta } }}{{\partial c_{\alpha }^{i} }}} \right)\left( {c_{\alpha }^{{i,{\text{Ext}}}} - c_{\alpha }^{{i,{\text{eq}}}} } \right)} + \left( {\frac{{\partial \Delta G_{\alpha \beta } }}{\partial T}} \right)\left( {T - T^{\text{eq}} } \right) $$

(9)

where $c_{\alpha }^{{i,{\text{eq}}}}$ is the equilibrium concentration of ith component in α phase at reference temperature $T^{\text{eq}}$. In quasi-equilibrium condition, the phase compositions are not independent variables, and their relationship can be expressed by multibinary approximation [19, 22] using the coefficient $K_{\alpha \beta }^{ii} = \frac{{\partial c_{\alpha }^{i} }}{{\partial c_{\beta }^{i} }} = \frac{{{{\partial \Delta G_{\alpha \beta } } \mathord{\left/ {\vphantom {{\partial \Delta G_{\alpha \beta } } {\partial c_{\beta }^{i} }}} \right. \kern-0pt} {\partial c_{\beta }^{i} }}}}{{{{\partial \Delta G_{\alpha \beta } } \mathord{\left/ {\vphantom {{\partial \Delta G_{\alpha \beta } } {\partial c_{\alpha }^{i} }}} \right. \kern-0pt} {\partial c_{\alpha }^{i} }}}}$. And the full derivation of the solution of local phase composition $\left\{ {\vec{c}_{\alpha }^{\text{Ext}} } \right\}$ under quasi-equilibrium constraint can be found in Ref. [19].

For dual interface, the mobility of the solid and liquid interface follows the thin interface limit [26, 27], and can be written as

$$ \tilde{M}_{sl} = M_{sl} \frac{8\eta }{{\pi^{2} }} = \frac{16}{{15a_{2} \eta }}\frac{{V_{m} }}{RT}\sum\limits_{i = 1}^{n - 1} {\frac{{D_{\text{l}}^{i} }}{{c_{\text{l}}^{{i,{\text{eq}}}} \left( {1 - k_{sl}^{i} } \right)^{2} }}} $$

(10)

where a₂ = 0.6276, T is the temperature, R is the gas constant, $D_{\text{l}}^{i}$ is the diffusion coefficient of ith component in the liquid and $k_{sl}^{i} = {{c_{\text{s}}^{{i,{\text{eq}}}} } \mathord{\left/ {\vphantom {{c_{\text{s}}^{{i,{\text{eq}}}} } {c_{\text{l}}^{{i,{\text{eq}}}} }}} \right. \kern-0pt} {c_{\text{l}}^{{i,{\text{eq}}}} }}$ is the equilibrium partition coefficient, with $c_{\text{s}}^{{i,{\text{eq}}}}$ and $c_{\text{l}}^{{i,{\text{eq}}}}$ being the equilibrium concentration in solid and liquid phases.

To perform quantitative phase-field simulations, the anti-trapping current term is added into the solute transfer equation to eliminate anomalous interface effects [26,27,28]. In multicomponent superalloy system, if only liquid and solid (FCC) phases are considered, the time evolution equation of the concentration field can be expressed as

$$ \frac{{\partial c^{i} }}{\partial t} = \nabla \left[ {\sum\limits_{\alpha = l,s} {\sum\limits_{j = 1}^{n - 1} {D_{\alpha }^{ij} \phi_{\alpha } \nabla c_{\alpha }^{j} } } + \frac{\eta }{\pi }\left( {c_{\text{l}}^{i} - c_{\text{s}}^{i} } \right)\sqrt {\phi_{\text{s}} \phi_{\text{l}} } \frac{{\partial \phi_{\text{s}} }}{\partial t}\frac{{\nabla \phi_{\text{s}} }}{{\left| {\nabla \phi_{\text{s}} } \right|}}} \right] $$

(11)

where $D_{\alpha }^{ij}$ is the diffusion coefficient of component i and j in α phase.

Implementation

PF Eqs. (6) and (11) are solved in two dimensions on uniform finite difference grids using an explicit Euler time scheme. A high-precision nine-point stencil Laplacian operator is used in solving phase-field equation, and a standard five-point stencil Laplacian operator is used in solving solute diffusion equation. To reduce the amount of calculations, Eq. (6) and the anti-trapping current term in Eq. (11) are only computed near the solid–liquid interface where $\left| {\nabla \phi } \right|^{2} > 1 \times 10^{ - 10}$. In addition, no thermal noise is added into the phase-field equation.

In directional solidification, the primary dendrite array needs quite a long time to reach a steady state [25, 29]. Therefore, to reduce the simulation cost, the moving-frame algorithm [24, 30] is used. At initial time, the dendrites at the bottom of the computational domain grow upwards naturally. When the maximum dendrite height reaches a critical value $H_{\hbox{max} }$, the moving-frame algorithm works, and all the dendrite arms are kept under $H_{\hbox{max} }$. For each lattice advance, the values of lattice points, including phase-field variables $\left\{ {\phi_{\alpha } \left( {\vec{x},t} \right)} \right\}$ and solute concentrations $\left\{ {\vec{c}_{\alpha } \left( {\vec{x},t} \right)} \right\}$, are transfer to the lower lattice points along y-direction. And the lattice points at the top of computational domain are set with the initial values as in the liquid.

GPU-based parallel computation

The large-scale phase-field simulation of multicomponent nickel-based superalloy dendrite growth in directional solidification condition requires large computational cost. To accelerate the computation, we adopt a previously developed GPU-based parallel computing scheme [23], and this algorithm has been optimized by using the GPU shared memory as user managed cache. All the simulations were performed on a GPU server with 8 NVIDIA GTX1080Ti GPUs, and each simulation case was processed by one GPU. The relative large computational domain size and long simulation time are chosen to ensure reliable results on the steady primary dendritic spacing. As for the simulation time, the shortest simulation with the computational domain of 4096 × 758 μm² over 2 × 10⁷ steps lasts about 7 h, and the longest simulation with the domain size of 8192 × 1536 μm² over 1.5 × 10⁷ steps lasts about 80 h. Compared with the performance on an Intel Xeon CPU E5-2680 v4 at 2.4 Ghz, the GPU-based computing scheme generally achieves more than 200 times speed-up ratio.

Alloy and simulation conditions

Owing to its extensive research and usage, a second-generation superalloy CMSX-4 [31,32,33] is investigated in this work, and the alloy chemical compositions [33] are listed in Table 1. To reduce the computation time, the alloy components in CMSX-4 with less than 0.5 at.% are excluded, and the model alloy is used in all the simulations.

Table 1 Chemical compositions of the investigated alloys [33]

The equilibrium parameters used in multiphase-field simulations are calculated by Pandat and PanEngine software using PanNi database. All the parameters are updated with a temperature interval of 0.2 K [22] during simulations to ensure accuracy. The equilibrium parameters of the model alloy at T^* = 1621.2 K are shown in Table 2 as an example. And the equilibrium concentrations of Al, Co, Cr, Re, Ta versus temperature are shown in Fig. 1 during solidification interval of 1656–1573 K. The concentration data are calculated using shell mode in Pandat software, and the derivatives of chemical driving force with respect to phase concentrations are calculated using PanEngine software.

Table 2 Equilibrium parameters of the model alloy at T^* = 1621.2 K

Figure 1

The equilibrium concentration of component Al, Co, Cr, Re, Ta in liquid and FCC phases in the solidification interval of 1656–1573 K (the concentration of Ti and W is not shown in the figure)

To investigate the superalloy primary dendrite spacing selection in directional solidification, the effects of temperature gradient G, pulling velocity V_p and the grain inclination angle θ are considered. Initially, the computational domain is set to be rectangle with a temperature gradient along y-direction. And the temperature is assumed to follow the frozen temperature approximation $T = T_{0} + G\left( {y - V_{\text{p}} t} \right)$, where T₀ is the reference temperature at y = 0 and t is the time. Because the initial setting of the seed space has a history effect [29, 30] on the final primary dendrite spacing, a continues seed bed with random noise on the surface was adopted to imitate the natural incubation of dendrite arms, which is shown as case 1 and case 2 in Fig. 2. The seeds in the seed bed have the same crystal orientation. In addition, several types of random noises were tested to make sure the initial condition does not influence the final steady primary dendritic spacing. The misoriented continuous seed bed was set to study the effect grain inclination angle with respect to the heat-flow direction, which is shown as case 2 in Fig. 2. During each simulation, the periodic boundary condition was applied to the y-boundaries, and the adiabatic boundary condition was applied to the x-direction. The moving-frame algorithm was triggered when the maximum dendrite tip height reaches 0.7 × ny. The other parameters used in multiphase-field simulations are listed in Table 3.

Figure 2

Schematic of the setup of the computational domain

Table 3 Simulation parameters in directional solidification of model alloy

In case 1, an orthogonal experimental design was used, the pulling velocities were set to V_p = 10, 20, 50, 100, 200 and 500 μm s⁻¹, and the temperature gradients were set to G = 2, 5, 10, 20, 50, 100 and 200 K mm⁻¹, as listed in Table 4. The simulation conditions have covered the typical processing conditions of high-rate solidification technique (HRS, V_p = 10–100 μm s⁻¹, G = 2–5 K mm⁻¹), liquid metal cooling (LMC, V_p = 50–400 μm s⁻¹, G = 8–20 K mm⁻¹) and high-gradient directional solidification technique by zone melting liquid metal cooling (ZMLMC, G = 10–100 K mm⁻¹) [2, 3, 36]. The computational domain was set to nx × ny = 4096 × 768 μm² for G = 200 K mm⁻¹, nx × ny = 4096 × 1024 μm² for G = 200, 100, 50, 20 and 10 K mm⁻¹, and nx × ny = 8192 × 1536 μm² for G = 5 and 2 K mm⁻¹. And the simulation times were set to 2–15 × 10⁶ steps to ensure a minimum pulling distance of 10 mm under different pulling velocities. In case 2, the pulling velocity V_p was set to be fixed at 100 μm s⁻¹, the temperature gradient G was set to 10, 20 and 50 K mm⁻¹, and the grain inclination angle varied from 0° to 40° with a 5° interval as listed in Table 4.

Table 4 The solidification conditions in the orthogonal design of simulations (full range means all the corresponding values listed in Table 3 are used)

Results and discussion

Interface morphologies and microsegregation

In superalloy directional solidification, the pulling velocity along with the temperature gradient has a great impact on the morphology of the solidification front [3,4,5,6,7,8,9] and the microsegregation pattern [31, 32]. This part gives an overview of the observed microstructure under pulling velocities V_p = 0.5–100 μm s⁻¹ and temperature gradients G = 20–100 K mm⁻¹ as Liu [3] did in experiments.

Figure 3 shows the solid–liquid interface morphology under different simulation conditions. The solidification front exhibits a planar interface at V_p = 0.5 μm s⁻¹ and G = 20 K mm⁻¹, as shown in Fig. 3a. As the pulling velocity increases from 0.5 to 100 μm s⁻¹, the interface gradually evolves in the sequence of planar, coarse dendrite (Fig. 3b) and fine dendrite (Fig. 3c, d). At high-temperature gradients G ≥ 50 K mm⁻¹, the growth of side branches is suppressed, and fine dendrite–cellular pattern and superfine cellular structure are found in Fig. 3e, f, respectively.

Figure 3

Interface morphologies at solidification front under different processing conditions: a plane interface under V_p = 0.5 μm s⁻¹ and G = 20 K mm⁻¹, b coarse dendrite under V_p = 5 μm s⁻¹ and G = 20 K mm⁻¹, c fine dendrite under V_p = 50 μm s⁻¹ and G = 20 K mm⁻¹, d fine dendrite under V_p = 100 μm s⁻¹ and G = 20 K mm⁻¹, e fine dendrite-cellular under V_p = 100 μm s⁻¹, G = 50 K mm⁻¹, f superfine cellular under V_p = 100 μm s⁻¹ and G = 100 K mm⁻¹

The multiphase-field model takes account of the multicomponent solute transfer in the solid and liquid, as well as the anti-trapping current. When coupling with thermodynamic database, it is capable of investigating the microsegregation pattern in superalloy directional solidification. The segregation pattern (Fig. 4a) under solidification condition of V_p = 50 μm s⁻¹ and G = 5 K mm⁻¹ is taken as an example, the processing condition of which is close to the conventional Bridgman high-rate solidification technique (HRS). Figure 4b–f shows the solute distribution of components Al, Co, Cr, Re, Ta, Ti and W. The components Co, Cr, Re and W segregate to the solidified dendrite arms, while Al, Ta, Ti segregate to the remaining melt. The segregation tendency is in consistent with experiment results of CMSX-4 [31].

Figure 4

a Dendrite morphology and b–h solute distribution of component Al, Co, Cr, Re, Ta, Ti, W under steady state at puling velocity of 50 μm s⁻¹ and temperature gradient of 5 K mm⁻¹

To quantitatively investigate the segregation behavior of each alloy component, the concentration profile at the root of the dendrite arm in Fig. 4a is extracted and plotted in Fig. 5a. The partition coefficients $\bar{k}^{i} = {{\bar{c}_{\text{s}}^{i} } \mathord{\left/ {\vphantom {{\bar{c}_{\text{s}}^{i} } {\bar{c}_{\text{l}}^{i} }}} \right. \kern-0pt} {\bar{c}_{\text{l}}^{i} }}$ of all components are calculated by using the average concentration in the solid $\bar{c}_{\text{s}}^{i}$ and liquid $\bar{c}_{\text{l}}^{i}$. Then, the calculated average partition coefficients $\bar{k}^{i}$ along with the results based on EPMA analysis of CMSX-4 samples produced under cooling rate of 0.25 K s⁻¹ [31] are presented in Fig. 5b. The simulation results of the partition coefficient agree well with the experiment in terms of Al, Co, Re, Ta, Ti and W, while the result of Cr exhibits a major difference with the experiment. It is true that Cr segregates moderately during γ solidification ($\bar{k}^{\text{Cr}}$ close to unity) [37], but the solubility of Cr is much lower in γ′ phase, which results in a pile up between the γ dendrite and γ/γ′ eutectic. Since our multiphase-field simulation did not take account the growth of γ′ phase, the segregation pattern of Cr cannot be correctly predicted.

Figure 5

a The concentration profile at the root of the dendrite arm at puling velocity of 50 μm s⁻¹ and temperature gradient of 5 K mm⁻¹, which presents a cooling rate of 0.25 K s⁻¹. b The interface partition coefficient of all the components calculated from multiphase-field simulation and obtained from experimental results in Ref. [31] using F–G sort method. The CMSX-4 samples are produced under cooling rate of 0.25 K s⁻¹

Effect of temperature gradient and pulling velocity

In this part, the effects of temperature gradient G and pulling velocity V_p on the final primary arm spacing are systematically investigated. In classical theories [12, 13], the primary dendrite spacing Λ can be estimated with analytical function as shown in Eq. (1). As mentioned above, the initial condition of the artificially settled seeds has a history effect on the final dendrite arm spacing. Therefore, we employ a continuous seed bed with random noise on the surface to imitate the natural development of dendrite arm. Figure 6 shows a typical growth process of the directional solidified model superalloy. This process includes initial planar instability, adjustment of primary dendrite arm spacing through elimination and steady-state dendrite growth. In the orthogonal test, the solidification conditions in all the simulations are listed as case 1 in Table 4.

Figure 6

A typical growth process of the directional solidified model superalloy: a small bubs from a continuous seed bed, solidification time = 28 s, b elimination of primary dendrite arms and adjustment of dendrite arm spacing, solidification time = 56 s, c primary dendrite arm array under steady state, solidification time = 420 s

Figure 7 shows the final dendrite morphology under all solidification conditions, and the primary dendrite arms exhibit a prominent morphology change. In general, monotonously increasing temperature gradient and pulling velocity can both decrease the primary dendrite spacing. The number of primary arms under temperature gradients of 5 and 2 K mm⁻¹ is more than that in the case of 10 K mm⁻¹, because the computational domain is larger in the former. When under large temperature gradients, the secondary dendrite arm is suppressed even if the pulling velocity is small. This phenomenon is similar to Tien’s findings [38].

Figure 7

The final dendrite morphologies under different solidification conditions of pulling velocity V_p = 500, 200, 100, 50, 20 and 10 μm s⁻¹ and temperature gradient G = 200, 100, 50, 20 and 10 K mm⁻¹

Because the sizes of the computational domain are different, the number of primary dendrite arms in x-direction is normalized by a width of 4 mm during simulations, and the results are shown in Fig. 8. Here, the tips of primary dendrite arms which grow higher than 0.5 × ny were taken into account. The number of dendrite arms in all the simulations has undergone three typical stages: an explosive rise and sharply fall process, a gradual decline process and an unchanged steady state. A phony increase in the number of dendrite arms under temperature gradients G = 2 and 5 K can be observed during the second stage. This is because the dendrite arms under 0.5 × ny were not taken into account, and the primary dendrite arms under low-temperature gradient need a quite long time to adjust their positions through competitive growth. In fact, the number of dendrite arms is gradually decreasing. The number of primary dendrite arms generally decreases with temperature gradient in all the cases. In order to quantitatively investigate the effect of temperature gradient G and the pulling velocity V_p, the primary dendrite arm spacing under steady state is fitted using the analytical model in Eq. (1).

Figure 8

Number of dendrite arms per 4 mm in x-direction under pulling velocity range of: a–f V_p = 500, 200, 100, 50, 20 and 10 μm s⁻¹, with the temperature gradient G being 200–2 K mm⁻¹

The fitting results are shown in Figs. 9 and 10. The graphs are plotted with two sets of axis: one for the temperature gradient G or pulling velocity V_p on a logarithmic scale and one for primary dendrite arm spacing (PDAS) on a linear scale. The points at each solidification condition represent the maximum, average and minimum primary dendrite spacing under final steady state obtained from three repeated simulations. The fitting results of the average value of PDAS (Λ_ave) are shown as curves, along with the formulas. The range [Λ_min, Λ_max] of PDAS is also shown in the figures. In contrast to classical theory where b = 0.5 and c = 0.25, the value of b and c is generally changing under different cooling conditions. The value of b varies slightly from 0.34 to 0.42 in our measurements, and b is near 0.38 in most cases, while the value of c increases monotonously from 0.27 to 0.56 with decreasing temperature gradient G. The details of each b and c are shown in Table 5, and the goodness of the fitting results is evaluated by adjusted coefficient of determination R², which is also provided in Table 5.

Figure 9

The maximum, average and minimum primary dendrite arm spacing (PDAS) as a function of temperature gradient G (2–200 K mm⁻¹) under different pulling velocities a–f V_p = 500, 200, 100, 50, 20, 10 μm s⁻¹. The fitting curves are presented along with the formula of Λ_ave

Figure 10

The maximum, average and minimum primary dendrite arm spacing (PDAS) as a function of pulling velocity V_p (500–10 μm s⁻¹) under different temperature gradients a–f G = 200, 100, 50, 20, 10, 5 K mm⁻¹. The fitting curves are presented along with the formula of Λ_ave

Table 5 Fitting results of variables b and c under different cooling conditions

In 2D simulations, the solute at the dendrite tip cannot diffuse so efficient as in 3D; therefore, the predicted average primary dendrite spacing in 3D should also be smaller than in 2D. To compare the Λ_ave calculated from 2D multiphase-field simulations with the experimental data, the Λ_ave are modified by multiplying a factor of ${{\sqrt 2 } \mathord{\left/ {\vphantom {{\sqrt 2 } 2}} \right. \kern-0pt} 2}$ [29]. The final results are shown in Fig. 11. The experiments in Ref. [5] use second-generation superalloys PWA1484 and RenéN5 with conventional Bridgman HRS technique, and the experiments in Ref. [3] use a first-generation CMSX-2 superalloy with ZMLMC technique, which can provide a very high temperature gradient of 100 K mm⁻¹. The simulation results have a good approximation for Λ_ave when under high pulling velocities, while the predicted Λ_ave under lower pulling velocities exhibit higher values than that in experiments. Moreover, the coarse cellular structure [3, 38] under low pulling velocity of 10 μm s⁻¹ is not reproduced by multiphase-field simulation. The deviation of PDAS the under low pulling velocities could be ascribed to the convection in the liquid [39,40,41], which is not considered in the current multiphase-field model. The thermal and solutal gradients, combined with the effect of gravity, can lead to significant natural convection in the liquid alloy and thus change the morphology of the solid front.

Figure 11

a Average primary dendrite arm spacing obtained from experiments in Refs. [3, 5, 6], b the modified average primary dendrite arm spacing $\tilde{\varLambda }_{ave}$ calculated from multiphase-field simulations

Effect of grain inclination angle

Despite a series of controlling techniques [42, 43], a fraction of single-crystal turbine blades are produced with their <001> crystal orientation deviate from the withdraw (heat-flow) direction up to 15° [44,45,46]. The understanding of primary dendrite arm spacing under misalign withdraw direction can be vital to industrial production. Therefore, in this part, the effect of grain inclination angle on the final PDAS was investigated. All the solidification conditions are listed as case 2 in Table 4.

Figure 12 shows the final dendrite morphology with grain inclination angle of 0°, 10°, 20°, 30° and 40° under temperature gradient of G = 10 K mm⁻¹. It can be seen that, with the increase in inclination angle, the dendrites have more developed secondary arms in the side toward which it drifts. The fully developed secondary arms can block the undeveloped primary dendrite arms and thus increase the PDAS. The calculated average PDAS with different crystal orientations is shown in Fig. 13, and the discrete points are fitted using the power function in Eq. (2). An abnormal decrease in PDAS with grain inclination angle θ was observed when θ < 10°, and this is not consistent with classical theory that PDAS always increases with inclination angle. Moreover, the PDAS almost remains constant within inclination angle of 20° in the three temperature gradients. As long as the inclination angle exceeds 20°, the PDAS shows a significant rise and increases as a power function. The coefficients in Eq. (2) we estimated with our fitting results are d = 0.07 and e = 2.8.

Figure 12

Final dendrite morphology under with different grain orientations a–e θ = 0°, 10°, 20°, 30° and 40°. The cooling condition is V_p = 100 μm s⁻¹ and G = 10 K mm⁻¹

Figure 13

Variation of primary dendrite arm spacing with inclination angle θ under temperature gradients of 10, 20, 50 K mm⁻¹ and pulling velocity of 100 μm s⁻¹

Summary and conclusions

The primary dendrite arm spacing selection in a directional solidified superalloy CMSX-4 was studied using two-dimensional phase-field simulations. The alloy thermodynamic and kinetic data are obtained from Pandat software with PanNickel database, and extrapolation method was used to couple the thermodynamic data with multiphase-field model. With the help of a previously developed GPU parallel computing scheme, we are able to perform hundreds of large-scale phase-field simulations on a multi-GPU server. In the orthogonal simulations, a wide range of processing conditions including pulling velocities (V_p = 10, 20, 50, 100, 200, 500 μm s⁻¹) and temperature gradients (G = 2, 5, 10, 20, 50, 100, 200 K mm⁻¹) are applied to systematically study the primary dendrite spacing variation. The simulation results are analyzed using a classical analytical model $\varLambda \propto \Delta T_{0}^{a} V_{\text{p}}^{ - b} G^{ - c} \left\{ {1 + d\left[ {\left( {\cos \theta } \right)^{ - e} - 1} \right]} \right\}$, and the coefficients b, c, d, e are determined by the fitting results. The main findings are listed as follows:

• 1. The multiphase-field model coupled with CALPHAD can be used to study the multicomponent superalloy microstructure evolution and segregation behavior, and the simulation results are quantitatively in consistent with experimental results.

• 2. In the phase-field simulations, the fitting results of primary dendritic spacing exhibit excellent goodness with the power law, the coefficient b is near 0.38 under different pulling velocities, while the coefficient c increases from 0.27 to 0.56 with increasing temperature gradient.

• 3. The primary dendrite spacing experienced a subtle decrease with increasing grain inclination angle θ when θ < 10°. As θ exceeds 20°, the primary dendrite spacing increases significantly with increasing θ. The coefficients estimated from our fitting results are d = 0.07 and e = 2.8.