Microstructure Evolution of Binary and Multicomponent Manganese Steels During Selective Laser Melting: Phase-Field Modeling and Experimental Validation

Abstract

In additive manufacturing processes, solidification velocities are extremely high in comparison to ordinary directional solidification. Therefore, the dependencies of the primary dendrite arm spacing (PDAS) on the process parameters deviate from the dependencies predicted by standard analytical methods. In this work, we investigate the microstructure evolution and element distribution in Fe-18.9Mn and Fe-18.5Mn-Al-C alloys solidified during the selective laser melting process. A quantitative multicomponent phase-field model verified by Green-function calculations (Karma, Rappel: Phys. Rev. E, 1998, 57, 4323) and the convergence analysis is used. The resulting non-standard dependencies of the PDAS on the process parameters in a wide range of solidification velocities are compared with analytical calculations. It is shown that the numerical values of the PDAS are similar to the values predicted by the Kurz–Fisher method for the low and intermediate solidification velocities and are smaller for the solidification velocities higher than 0.03 m/s. The PDAS and the Mn distribution in a Fe-18.5Mn-Al-C alloy are compared to the experimental results and a very good agreement is found.

1 Introduction

During the last 10 years, the microstructure formation in additive manufacturing (AM) was intensively investigated by means of cellular automaton, finite element, and phase-field methods.[1,2,3,4,5,6,7,8] The motivation of these studies was triggered by the fact that the growth morphology and the extent of the microsegregation strongly influence the properties of the additively manufactured parts. The materials under modeling investigation were alloys used for high-cost applications, e.g., aerospace and biomedical applications such as Al, Ni, and Ti-base alloys. NiTi and Ni-base superalloys are, for example, produced by selective laser melting (SLM) allowing the production of constructional elements with a wide geometrical freedom. The microstructure of SLM-fabricated materials is usually anisotropic with columnar grains elongated in the direction of the highest heat flow, i.e., parallel to the building direction in most cases. SLM-built parts have a predefined shape, and one tends to avoid post-processing to ensure efficient fabrication. The understanding of the effect of process parameters on the solidification microstructure in AM is, therefore, essential to realize the full potential of this fabrication method.

Most research in the field of AM of austenitic steels has focused on the well-known stainless steels 304L and 316L, e.g., Reference 9 However, AM of austenitic high-manganese steels (HMnS) has recently gained increased attention.[10,11,12] The concept of these alloys is based on stabilization of the austenitic phase by high amounts of manganese (Mn) and in many cases carbon. Due to their low stacking-fault energy (SFE), which is usually in the range of 10 to 50 mJ/m$^2$, multiple deformation mechanisms can be activated in addition to dislocation glide. In HMnS with SFE values below 18 mJ/m$^2$ transformation-induced plasticity (TRIP) is observed, whereas twinning-induced plasticity (TWIP) is activated in the range between 12 and 50 mJ/m$^2$.[13,14,15] The activation of these additional deformation mechanisms combined with the strong planarity of dislocation motion and reduced dynamic recovery promotes a high work-hardening capacity. It has been shown that HMnS take advantage of the unique processing conditions during SLM since pronounced elemental segregation, which is usually a severe issue during conventional processing,[11] can be strongly suppressed. Furthermore, SLM-produced specimens with filigree lattice structures made of HMnS showed a superior energy-absorption capacity as compared to other austenitic steels.[12] Therefore, HMnS are promising alloys for AM applications and are investigated in this study.

The solidification conditions during SLM are similar to directional solidification in the sense that the microstructure is controlled by the local temperature gradient and the cooling rate, which both define the solidification velocity. To investigate the dependency of the Primary Dendrite Arm Spacing (PDAS) on the temperature gradient, a large number of simulations were carried out by means of phase-field modeling,[8,16,17,18,19,20,21,22,23] and the results were compared to the analytical values predicted by the methods of Kurz–Fisher (KF)[24] and Hunt[25] as well as to experimental data.

The main phase-field methods used in the directional solidification of alloys are the model of dilute binary alloys of Echebarria et al.,[26,27,28] the multi-phase model of Steinbach et al.,[29,30] and the similar model of Kim et al.,[31,32] which includes the detailed development of the chemical driving force. All these methods use the thin interface limit to calculate the appropriate interface mobility and anti-trapping current to avoid solute trapping in case of the different diffusion coefficients in solid and liquid phases because both factors (mobility and solute trapping) strongly influence the distribution of elements along the dendrite arms perpendicular to the building direction. The model of dilute binary alloys[26] uses a standard double-well potential with the phase-field variable varying between − 1 and 1, and it is verified by convergence analysis for isothermal and directional solidification. The multi-phase model[29] uses a piecewise bilinear potential with the phase-field variable varying between 0 and 1 and is more efficient in application to multi-phase systems.

The competitive growth of multiple columnar cells in directional solidification was simulated in two and three dimensions (2D and 3D) in the works of Takaki et al.[19,20] based on the model of dilute binary alloys.[26] The dependencies of primary dendrite arm spacing (PDAS) showed an appropriate agreement to theoretical predictions at low gradients and low solidification velocities. The competitive dendritic growth of grains with different orientations during directional solidification for low pulling velocities was studied by Li et al.[16] by means of the model of Steinbach et al.[29,30] The influence of grain orientation and temperature gradient on the selection of primary dendritic spacing was investigated by Tourret et al.[17,18] based on the model of dilute binary alloys[26] and with application of an effective mathematical method for a low pulling velocity. The authors identify a range of grain orientations in which grain selection follows the classical description as well as conditions that favor unusual overgrowth of favorably-oriented dendrites. The authors discussed that the transition to a cellular regime should promote such events as found in Reference 20. The dependence of the grain boundary orientation upon the azimuthal angles was also studied in 3D. Diepers et al.[21] and Viardin et al.[22] carried out simulations for low solidification velocities and low cooling rates with the help of the multi-phase models with different potentials[30,33] and showed that the phase-field model can correctly describe the transient adjustments of primary spacing. Furthermore, it can reproduce the evolution of grain boundaries between columnar grains with a different crystallographic orientation for small and moderate misorientation angles. Recently, Keller et al.[23] and Yang et al.[8] modeled the PDAS in a multicomponent Ni-based superalloy at intermediate pulling rates and temperature gradients. However, the investigation of the PDAS at high gradients and high cooling rates during the SLM process is still needed because, as shown in experiments and simulations,[6,34,35] the effect of the temperature gradient in SLM is smaller, and the effect of the cooling rate is larger in comparison to the ordinary directional solidification.

Recently, a number of experiments and simulations were carried out to investigate the solidification microstructure in AM.[7,36,37] The model based on Kobayashi[38] and Boettinger et al.[39] was used by Acharya et al.[37] to study the non-equilibrium solidification behavior of a binary Ni-Nb alloy in AM with convection effects and heat transfer, and it turned out that the phase-field method allows to predict microsegregations and microstructure evolution in presence of the temperature variation. Furthermore, a strong microsegregation of Nb between the cells during single-line laser resolidification was shown by Ghosh et al.[7] in simulations where the solidification velocity varies between 0.05 and 0.2 m/s, showing that the effect of convection on the PDAS is small due to the extremely fine cellular structures in AM.

The main goal of this study is to understand the deviation between experimental data and the theoretical predictions and the effect of the high solidification velocity in SLM on the PDAS in detail. We will use the multicomponent phase-field model developed in References 4, 40, and 41, which was applied to the investigation of the PDAS and microsegregations in SLM-produced Ni-based superalloys. The model uses the double-well potential with the phase-field variable varying from 0 to 1 and a linear phase diagram. We verify the method by comparing the tip-velocities with the theoretical predictions obtained by the Green-function calculations given in Reference 42 and by the convergence analysis, and we use it to investigate the microstructure formation in an Fe-18.9Mn and an Fe-18.5Mn-Al-C alloys and to study the dependency of the PDAS on realistic SLM process conditions.

The experimental data of the microstructure characterization for the SLM-produced Fe-18.5Mn-Al-C steel are given in Section II. The phase-field model for multicomponent systems is briefly described in Section III–A, and the materials properties and thermodynamic parameters of the investigated alloys are listed in Section III–B. The verification of the model for the isothermal solidification is presented in Section IV–A, and in Sections IV–B and IV–C we describe the simulation results for the directional solidification and analyze observed effects of large cooling rates and temperature gradients on the PDAS discussing the general dependencies in comparison to the analytical predictions. In Section IV–D the microstructure evolution and competition of arms during rapid solidification in the SLM-produced multicomponent Fe-18.5Mn-Al-C alloy is presented and the PDAS and microsegregations are compared with experimental data.

2 Experimental Results for Fe-18.5Mn-Al-C Steel

In order to validate numerically reproduced arm spacing, we carried out the microstructure characterization of the SLM-produced Fe-18.5Mn-Al-C steel by means of optical microscopy, scanning electron microscopy (SEM) analysis, and electron backscatter diffraction (EBSD). Samples were built using a M1 cusing SLM device (Concept Laser GmbH) equipped with an Yb fiber laser. The microstructure at the constant laser power of 180 W and the scan speed of 571 mm/s is shown in Figure 1. Figure 1(a) illustrates the structure resulting from layer-wise build-up of the material by individual laser-melted tracks during the SLM process. The varying morphology of these tracks can be related to the scanning strategy used, i.e., 33 deg rotations between consecutive layers (a detailed description of the experimental procedure is given in Reference 11). The underlying grain structure becomes visible using EBSD analysis and is depicted in Figure 1(b). The grains were preferably elongated along the building direction (BD) and extended over several laser-melted tracks, which agrees with preferred epitaxial growth in the direction of the highest heat flow, i.e., towards the substrate plate. Additionally, a very fine dendritic solidification structure can be recognized (see Figure 1(c)). Although the local conditions of the heat transfer are quite complex and depend on various factors like melt pool size, heat conductivity, and atmosphere, the morphology of the dendrites can be described as cellular dendritic without presence of secondary dendrite arms (see inset of Figure 1(c)). The cross section of the primary dendrite arms, i.e., the PDAS, was determined from SEM images to be about 615 $\,\pm 195$ nm. According to electron probe micro analysis and energy-dispersive X-ray spectroscopy, the variation of the manganese content is about $\,\pm 1.3$ wt pct.

Fig. 1

Microstructure of the Fe-18.5Mn-Al-C steel produced by SLM: (a) optical micrograph, (b) image quality-EBSD map, and (c) SEM image using secondary electrons. The structure of layer-wise build-up is evident in the optical micrograph (a), whereas the microstructure consisting of grains elongated in BD (b) and fine cellular primary dendrites (c) is visible using SEM

3 Model Description

3.1 Model Equations

The phase-field model for multi-phase and multicomponent systems contains two types of evolution equations: (1) the kinetic equation for the evolution of the phase-field variables and (2) multicomponent diffusion equations. Here, we deal with the solidification process and consider only two phases. For the evaluation of the chemical driving force in a multicomponent system, we use an approach which utilizes the entropy change between phases and the equilibrium concentrations.[4] The anisotropy is calculated according to the method described in Reference 42.

For the simulation of the solidification, the phase-field equation for the evolution of the solid phase is used in the form

$$\begin{aligned} \tau a_s( {\mathbf {n}})^2\frac{ \partial \phi }{ \partial t}= & {} W^2\vec {\nabla } a_s( {\mathbf {n}})^2 \vec {\nabla } \phi +W^2\partial _x\left( |\vec {\nabla } \phi |^2 a_s( {\mathbf {n}})\frac{\partial a_s( {\mathbf {n}}) }{\partial (\partial _x \phi )}\right) \nonumber \\&+ W^2\partial _y\left( |\vec {\nabla } \phi |^2 a_s( {\mathbf {n}})\frac{\partial a_s( {\mathbf {n}}) }{\partial (\partial _y \phi )}\right) - \frac{ \partial f }{ \partial \phi } + \frac{a_1 W}{\sigma } \frac{ \partial g }{ \partial \phi }\Delta G , \end{aligned}$$

(1)

where $\phi$ is the phase-field order parameter in [0, 1] which is equal to 0 in the liquid phase and 1 in the solid phase, $\tau$ is the relaxation time of the system, W is the width of the interface, $a_s( {\mathbf {n}})=1+\varepsilon _4\cos (4\theta )$ is a four-fold anisotropic function, $\varepsilon _4$ is the anisotropy strength, $\theta$ is the angle between the outer normal vector ${\mathbf {n}}$ at the interface and the constant building direction, $a_1$ is a numeric constant, and $\sigma$ is the interface energy. The model function f is the double-well potential specified as $f=\phi ^2(1-\phi )^2$ and the model function g is chosen as $g=\phi ^3(10-15\phi +6\phi ^2)$, $\Delta G$ is the chemical driving force of the solidification defined as

$$\begin{aligned} \Delta G=\sum _A^N X_{L}^{A}\Delta A^A_{SL}\frac{(C^A-C^{A,eq})}{h^A(\phi )}, \end{aligned}$$

(2)

where the sum is taken over all N components indicated by index A excluding the base component. Here, $C^A$ is the mean concentration, $C^{A,eq}=C^{A,eq}_S \phi +C^{A,eq}_L(1-\phi )$ is the mean equilibrium concentration, $\Delta A^A_{SL} = C^{A,eq}_S - C^{A,eq}_L$ is the difference of the equilibrium concentrations in the liquid and solid phases, $C^{A,eq}_{S/L}$, $h^A(\phi )=\phi +(1-\phi )k_{L}^A$ is the partition function, $k_{L}^A=X_{S}^A/X_{L}^A$ is the partition coefficient, and $X_L^{A}$ and $X_S^{A}$ are the thermodynamic factors of the liquid and solid phases.

The equilibrium concentrations at a temperature T are calculated as

$$\begin{aligned} C^{A,eq}_{S/L}=C^{A,eq}_{S/L}(T_0) + \frac{ (T-T_0)}{m_{S/L,p}^A}, \end{aligned}$$

(3)

where $m_{L,p}^A$ and $m_{S,p}^A$ are the liquidus and solidus slopes along the solidification path, which are assumed to be constant according to the linear phase diagram, and $T_0$ is the melting temperature. For the sake of simplicity, we assume that the thermodynamic factors of components are similar. In this case, they can be calculated using the entropy change between the phases, $\Delta S$, as $X_{L/S}^{A}=\Delta S\left( \sum _A \Delta A^A_{SL}/m_{L/S,p}^A\right) ^{-1}$.[4]

The diffusion equation for the mean concentration of a component A is given by

$$\begin{aligned} \frac{\partial C^A}{\partial t}={\mathbf {\nabla }} \left[ D^{A}(\phi ) {\mathbf {\nabla }}\frac{(C^A-C^{A,eq})}{h^A(\phi )} + \frac{W}{\sqrt{2}}\Delta A^A_{SL}\frac{\mathbf {\nabla } \phi }{|{\mathbf {\nabla }} \phi |}\frac{ \partial \phi }{ \partial t}\right] , \end{aligned}$$

(4)

where $D^A(\phi )= D_S^{A}\phi + D^{A}_L (1-\phi )$ is the weighted diffusion coefficient, $D_L^{A}$ and $D^{A}_S$ are the diffusion coefficients in the liquid and solid phases. The second term on the right-hand side is the anti-trapping current which is used for the investigated alloy with $D_{S}^A\ll D_L^A$. This is the so-called one-sided diffusion model. Note that for the verification of the model by theoretical calculations we will use the two-sided symmetric model where $D_S^A=D_L^A$.

The kinetic parameter $\tau$ in Eq. [1] is defined using the diffuse interface limit as

$$\begin{aligned} \tau =\frac{a_1 a_2 W^3 }{\sigma } \sum _A \frac{X_L^A\left|\Delta A^{A}_{SL}\right|^2}{ D_L^A}, \end{aligned}$$

(5)

where numerical constants are defined as $a_1=\sqrt{2}/6$ and $a_2=2.35$ (see Appendix A).

The temperature during the SLM process is calculated as in directional solidification, i.e.,

$$\begin{aligned} T(y,t)=T_0 -\Delta T_{\mathrm{in}} +G(y-2N_y/3) -\dot{T} t +GN_{\rm shift}\Delta y, \end{aligned}$$

(6)

where $\Delta T_{\mathrm{in}}$ is an initial undercooling, G is the temperature gradient, $\dot{T}$ is the cooling rate, t is the current simulation time, and y is the growth direction. $N_{\rm shift}$ is a number of shifts of the moving frame and $\Delta y$ is the shift length. A moving frame is used, which is shifted when the dendrite tip reaches 1/3 or 2/3 of the domain height $N_y$ in the convergence or solidification tests, respectively. The solidification velocity is calculated as $V_{\mathrm{tip}} = N_{\rm shift}\Delta y/t$. The temperature in the simulation of isothermal solidification is constant, $T=T_0 -\Delta T$, with a constant undercooling $\Delta T$. The dimensionless undercooling is calculated as $\Omega =\dfrac{\Delta T}{\Delta T_0} =\dfrac{T_0-T}{T_0-T_s'}$, where $\Delta T_0 = T_0-T_s'$ is the total undercooling and $T_s'$ is the solidification temperature.

In order to produce the secondary dendrite arms, the thermal fluctuations (noise) are added in the phase-field equation.[41] The boundary conditions are no flux conditions on the left and right sides and on the bottom of the simulation box. The concentration on the top of the box is fixed to the initial alloy composition $C_0$ for a binary alloy or $C_0^A$ for a multicomponent alloy. The system size is chosen to be much larger than the diffusion length. The ratio of the system size in x-direction to the diffusion length lies in the range between 15 and 60 for small velocities and between 100 and 230 for high velocities.

A dimensionless tip velocity for a multicomponent system is defined as $\tilde{V}_{\rm tip}=V_{\rm tip}\bar{d}_0/\bar{D}_L$, where the mean capillary length and the mean diffusion coefficient are calculated by the following relations

$$\begin{aligned} \bar{d}_0&=\frac{\sigma }{\sum _A X_L^A\left|\Delta A_{SL}^A\right|^2/D_L^A\sum _AD_L^A}, \,\,\, \bar{D}_L=\frac{\sum _A X_L^A\left|\Delta A_{SL}^A\right|^2}{\sum _A X_L^A\left|\Delta A_{SL}^A\right|^2/ D_L^A}. \end{aligned}$$

(7)

An alternative method to get the same ratio $\bar{d}_0/\bar{D}_L$ is to define

$$\begin{aligned} \bar{d}_0=\frac{\sigma }{\sum _A X_L^A\left|\Delta A_{SL}^A\right|^2}, \,\,\, \bar{D}_L=\sum _AD_L^A. \end{aligned}$$

(8)

It could be proven that for a binary alloy these equations simplify to $\bar{d}_0=d_0=\dfrac{\sigma }{ X_L\left|\Delta A_{SL}\right|^2}$ and $\bar{D}_L=D_L$, where $X_L$, $\Delta A_{SL}$, and $D_L$ are parameters for a binary system.

The analytical values of the PDAS are calculated by the Kurz–Fisher (KF) method,[24] i.e.,

$$\begin{aligned} \lambda =4.3\left( \frac{(1-\Omega )}{G}\right) ^{0.5}\left( \frac{\Gamma \Delta T_0 D_Lk_L}{ V_{\rm tip} }\right) ^{0.25}=4.3(1-\Omega )^{0.5}\left( \frac{\Gamma \Delta T_0 D_L k_L}{ \dot{T} G }\right) ^{0.25}, \end{aligned}$$

(9)

where $V_{\rm tip} =\dot{T}/G$, $k_L$ is the partition coefficient of a binary alloy, and $\Gamma =\sigma /\Delta S$ is the Gibbs-Thomson coefficient. Note that in Eq. [9] we replaced the difference between the tip and solidification temperatures $T^*-T_s'$ (see Eq. [4.15] in Reference 24) by $(1-\Omega )\Delta T_0$. It should be also noted that the numerical constant in Eq. [9] is valid for a 3D dendrite geometry and can be different in 2D simulations.

For multicomponent alloys, the Kurz–Fisher model is extended:

$$\begin{aligned} \lambda =4.3\left( \frac{(1-\Omega )\Delta T_0}{G}\right) ^{0.5}\left( \frac{\Gamma }{ V_{\rm tip}\sum _A m_L^A\Delta A_{SL}^A/ D^A_L}\right) ^{0.25}, \end{aligned}$$

(10)

where $m^A_L=X_L^A\Delta A_{SL}^A/\Delta S$ is the liquidus slope and $\Delta T_0= \sum _A m^A_Lk_L^A\Delta A_{SL}^A$. The theoretical value of the undercooling corresponding to a chosen tip velocity is calculated by a fitting equation from the Green-function calculations[42]

$$\begin{aligned} \Omega = \left( \frac{\pi ^4\tilde{V}_{\rm tip}}{820\varepsilon _4}\right) ^\frac{1}{5.4}. \end{aligned}$$

(11)

The derivation of Eqs. [7], [10], and [11] is done in Reference 4.

3.2 Estimation of Model Parameters

The thermodynamic parameters for Fe-18.9Mn and Fe-18.5Mn-Al-C alloys are evaluated using the TCFE9 database.[43,44] The parameters are listed in Table I. For the investigated alloys, the temperature dependencies of the equilibrium concentrations can be approximated by linear functions.

Table I Equilibrium and Material Parameters for the Fe-18.5Mn-Al-C Alloy and for the Fe-18.9Mn Alloy Used in the Simulation

The parameters of the SLM process such as the laser power $Q=180$ W and the scanning speed $v=571$ mm/s were chosen in accordance to Reference 11, where the microstructure of manganese steel was investigated in detail. The thermal conditions during additive manufacturing were calculated in works[5,7,23] using finite element models. The parameters were estimated for different alloys and velocity combinations. In order to utilize these data for our conditions, the mean values of the thermal gradient and the cooling rate were rescaled proportionally to the ratio $\lambda v/Q$ ($\lambda$ is the thermal conductivity) according to the Rosenthal’s analytical solution as described in Reference 4. The estimated values for the temperature gradient and the cooling rate for our simulations are in intervals from 2 to $5\times 10^{6}$ K/m and from 1 to $8\times 10^{5}$ K/s, respectively.

4 Numerical Results

4.1 Verification of the Steady-State Tip Velocity by Green-Function Calculations

In this section we describe the verification of the model by means of convergence tests with different numerical resolution. The steady-state tip velocity at a constant undercooling was estimated for a single dendrite and compared with the analytical value predicted by the Green-function method for two-sided symmetric diffusion. Note that for the one-sided diffusion in directional solidification of a real alloy, we have verified the model by convergence tests without comparison with analytical predictions and have found the regions where the tip velocity or the tip undercooling does not depend on the discretization size (see next sections). The convergence tests were carried out for interface anisotropy coefficients of 0.05 and 0.06 and cannot be systematically extrapolated to other parameters.

The simulation of the isothermal crystal growth for the binary Fe-18.9Mn alloy was carried out at $\Omega =0.55$ and $\varepsilon _4=0.05$ in a box of size 12 $\times$ 32.4 $\mu$m with different numerical resolutions in the interval of $W/d_0\in [2,9]$ and for different ratios of the discretization size to the interface width $\Delta x/W=0.46$ and 0.73. In Figure 2, the dependencies of the tip velocity on $W/d_0$ are shown. The tip velocity is compared to the value predicted by Green-function calculations $\tilde{V}_{\rm tip}=0.0171$ (dashed line). The tip velocity increases with decreasing discretization size and converges to the theoretical value at $\Delta x/W=0.46$ in the region $W/d_0<7.5$ and at $\Delta x/W=0.73$ in the region $W/d_0<4.5$. The value of $\Delta x/W=0.46$ will be used in all tests below as more numerically stable. Note that for other anisotropies, the dependency of the tip velocity on $W/d_0$ can be non-monotonic.

Fig. 2

Dimensionless tip velocity as a function of $W/d_0$ at $\Omega =0.55$, $\varepsilon _4=0.05$. The dashed line corresponds to the value obtained from the Green-function calculation

Now we introduce an interface stability parameter of dendrite growth, $\sigma _{{\text{W}}}^{*} = \frac{{Dd_{0} }}{{V_{{{\text{tip}}}} W^{2} }}$, which is useful for estimating the accuracy of the method. The larger this parameter is, the larger is the precision of the estimation of the tip velocity. From the convergence test in Figure 2 we can find that the interface stability parameter for isothermal solidification should be in the region $\sigma ^*_W>1.2$ to reproduce the theoretical value of the tip velocity. Note that the interface stability parameter depends on the interface width and is different from the commonly used stability parameter $\sigma ^*$.

In the SLM process the growth velocities are much larger than 0.01 m/s. Therefore, for further verification we have chosen the tip-velocities in the interval $V_{\rm{tip}}\in [0.0076,0.117]$ m/s (see Table BI in Appendix B). The following parameters were chosen: the size of the simulation box $N_x\times N_y=300\times 800$ grid points, $W=0.01,0.02$$\mu$m. The discretization size is smaller for larger velocity to ensure $\sigma ^*_W>1.2$. The anisotropy coefficient was chosen as 0.05 and 0.06 that is higher than in most metallic alloys. This choice is motivated by the convergence at a larger discretization size, that simplifies the numerical calculations. Furthermore, for smaller anisotropy strength the undercooling $\Omega$ becomes close to 1 in 2D case leading to extremely small PDAS (see Eq. [9]), which contradicts to experimental observations.

In Table BI, the numerically estimated tip velocities are presented in comparison to the theoretical values calculated by Eq. [11]. The numerical values are close to the theoretically predicted one with errors $<1.2\%$. Notice that the quantitative agreement is good over the whole range of velocities. The examples of simulated tip shapes in steady-state growth for different undercoolings for the Fe-18.9Mn alloy are visualized in Figure 3 by means of the concentration distribution of Mn. By comparison of the tip shapes in Figures 3(b) and (c), it can be seen that for larger tip velocity the tip radius is smaller in accordance to the theory of dendrite growth.

Fig. 3

Comparison of steady-state tip shapes: (a) $\Omega =0.71$, $\varepsilon _4=0.05$, $V_{\rm{tip}}=0.03$ m/s; (b) $\Omega =0.6$, $\varepsilon _4=0.06$, $V_{\rm{tip}}=0.0148$ m/s; (c) $\Omega =0.55$, $\varepsilon _4=0.05$, $V_{\rm{tip}}=0.0076$ m/s. The concentration field of Mn is shown in mass fraction. The x and y axes have units of $\Delta x$

4.2 Directional Solidification at a Constant Undercooling

Numerical tests for directional solidification were carried out at a constant undercooling on the dendrite tip. In this method, we substitute in Eq. [6] $\dot{T} t=GN_{\mathrm{shift}}\Delta y$ and $\Delta T_{\text{in}}=\Delta T={\text{const}}$. So that the tip velocity is equal to the solidification velocity, $V_{\text{tip}}=N_{\text{shift}}\Delta y/t=\dot{T}/G$.

After a competitive growth between primary arms growing from the nuclei, a stable columnar microstructure is formed. The average PDAS, $\lambda$, is estimated after long-time simulations as the system width dividing by the number of cells on the line perpendicular to the y-axis. The one-sided model with the different diffusion coefficients in liquid and solid phases is used with corresponding anti-trapping current.

In the simulations, a set of randomly distributed nuclei with the density of one nucleus per 15 $\Delta x$ was located on the bottom of the simulation domain. This allows to avoid the influence of initialization at different discretization sizes. In Diepers et al.,[21] it was shown that the simulated PDAS depends on the distance between the initial nuclei. The use of a thermal noise to produce the nucleation of dendrite arms on the planar interface, as it was applied in many simulations, can influence the resulting arm spacing because it depends on the discretization size and on the ratio $\Delta x/W$. The mean distance between nuclei can be changed and, therefore, the PDAS can be different. In order to exclude this effect, we assumed a constant mean distance between the nuclei. Our test simulations showed a difference in the PDAS between the two initialization methods at $\sigma _W^*<0.6$.

In order to verify the model of directional solidification, a convergence test was carried out with different numerical resolutions $\sigma _W^*\in [0.25,1.32]$ and $G=1.8\cdot 10^{6}$ K/m. The box size was chosen as $N_x\times N_y=1000\times 1600$ grid points. The undercoolings were chosen as $\Omega =0.715$ for $\varepsilon _4=0.05$ and $\Omega =0.8$ for $\varepsilon _4=0.06$ that corresponds to the tip velocities $V_{\text{tip}}=0.03$ and 0.067 m/s according to Eq. [11]. The velocity of the dendrite tip closest to the upper border of the box was monitored during the simulation. The numerical tip velocities were compared to the theoretical values, and numerically estimated $\lambda$ were compared to the analytical values calculated by the Kurz–Fisher equation [9].

In Figures 6(a) and (b), the dependencies of the reduced velocity, $V_ {\text{tip}}$(num)/$V_ {\text{tip}}$(theory), and the reduced PDAS, $\lambda$(num)/$\lambda$(KF), on the interface stability parameter are shown. Both quantities tend to constant values at $\sigma _W^*>0.8$. Hence, the value of $\sigma _W^*=0.8$ can be used as a limit value for estimating an appropriate grid resolution. From our results also follows that $\lambda$ decreases with the increasing tip velocity as predicted by Eq. [9].

The next test shows the effect of the temperature gradient on the tip velocity and the PDAS. $\Omega$ and $\varepsilon _4$ were chosen corresponding to the tip velocities $V_{\text{tip}}=0.03$ and 0.067 m/s. The discretization size was chosen as $\Delta x =9.2$ nm that gives $\sigma _W^*\ge 0.62$.

The error of the estimation of $\lambda$ can be calculated as $100\ {\rm pct}/N_{\rm{cell}}$, where $N_{\rm{cell}}=N_x\Delta x/\lambda$ is the number of cells in the simulation domain. Thus, in the box sized $N_x\times N_y=1000\times 1600$ grid points, the error for $G=0.5\times 10^{6}$ K/m and $V_{\rm{tip}}=0.03$ m/s is about 10.9 pct, the error for $G=1.8\times 10^{6}$ K/m and $V_{\text{tip}}=0.067$ m/s is about 6.3 pct.

The simulated microstructure in the binary Fe-18.9Mn alloy is shown in Figures 4 and 5 for different undercoolings and temperature gradients. The formation of the liquid droplets along the cell boundaries can be seen at a larger undercooling, i.e., a larger solidification velocity. Moreover, at a larger thermal gradients the cell size distribution becomes less regular.

Fig. 4

Columnar microstructure in the Fe-18.9Mn alloy simulated at the constant undercooling $\Omega =0.715$, $\varepsilon _4=0.05$ ($V_{\text{tip}}({\text {theory}})=0.03$ m/s) and different temperature gradients: (a) $G=5\times 10^5$ K/m; (b) $G=1.8\times 10^6$ K/m; (c) $G=4.6\times 10^6$ K/m. The concentration field of Mn is shown in mass fraction. The x and y axes have units of $\Delta x=9.2$ nm

Fig. 5

Columnar microstructure in the Fe-18.9Mn alloy simulated at the constant undercooling $\Omega =0.8$, $\varepsilon _4=0.06$ ($V_{\text{tip}}({\text {theory}})=0.067$ m/s) and different temperature gradients: (a) $G=5\times 10^5$ K/m; (b) $G=1.8\times 10^6$ K/m; (c) $G=4.6\times 10^6$ K/m. The concentration field of Mn is shown in mass fraction. The x and y axes have units of $\Delta x=9.2$ nm

The examples of numerically estimated values of $\lambda$ and $V_{\text{tip}}$ are listed in Table BII in Appendix B in comparison to the analytical values. By analyzing the results, we want to understand the effect of the temperature gradient on the steady-state growth velocity and on the PDAS. The dependencies of the reduced velocity, $V_ {\text{tip}}$(num)/$V_ {\text{tip}}$(theory), and the reduced PDAS, $\lambda$(num)/$\lambda$(KF), on the temperature gradient are plotted in Figures 6(c) and (d). It can be seen that the numerical tip velocity slightly decreases with an increasing temperature gradient and deviates from the theoretical value by maximum 25 pct for 0.03 m/s and by − 10 pct for 0.067 m/s. The reason for the large difference between the numerical and theoretical values, with the exception that we use theoretical velocities for the two-sided model, is that in directional solidification tests the simulation time is smaller than in tests for isothermal solidification, and the velocity cannot reach a steady-state value. From Figure 6(d), it can be seen that the values of $\lambda$(num)/$\lambda$(KF) noticeably increase with increasing G. This behavior indicates that the dependency of $\lambda$(num) on G is much weaker than the theoretically predicted one.

For the comparison, the numerical values of $V_ {\text{tip}}$ and $\lambda$ estimated for the symmetric diffusion we provide in Figures 6(c) and (d) by triangles and in Table BII in brackets. The numerical tip velocity is smaller than in the one-sided case by 17 to 19 pct. That leads to the larger values of $\lambda$.

It should be noted that our results for the PDAS in 2D should be verified in the future for 3D because the difference between 2D and 3D simulations is non-trivial.

Fig. 6

Directional solidification of the Fe-18.9Mn alloy at a constant undercooling which corresponds to $V_ {\text{tip}}$(theory) $=0.03$ m/s and 0.067 m/s: (a) $V_ {\text{tip}}$(num)/$V_ {\text{tip}}$(theory) against $\sigma _W^*$, $\varepsilon _4=0.05$; (b) $\lambda$(num)/$\lambda$(KF) against $\sigma _W^*$, $\varepsilon _4=0.05$ (comparison with constant solidification velocity); (c) $V_ {\text{tip}}$(num)/$V_ {\text{tip}}$(theory) against G; (d) $\lambda$(num)/$\lambda$(KF) against G

4.3 Directional Solidification at a Constant Solidification Velocity

The directional solidification was simulated at a constant solidification velocity which is estimated as a ratio between the cooling rate and the temperature gradient $\dot{T}/G$. The temperature in the system is calculated by Eq. [6].

A set of randomly distributed nuclei, as in the previous tests, was used. The anisotropy was set to $\varepsilon _4=0.05$ and the box size to $N_x\times N_y=1200\times 1600$ grid points.

The initial undercooling, $\Delta T_{\text{in}}=\Delta T$(theory), was taken in accordance to $\Omega$ calculated by Eq. [11] for a given solidification velocity. The choice of initial undercooling is important because it influences the coarsening in the beginning of the simulation. During the solidification, the temperature on a dendrite tip closest to the upper border of the box and the tip velocity were monitored. The PDAS and the steady-state tip undercooling were measured after long-time simulation when $\lambda$ reaches a constant value.

The results of convergence tests are shown in Figure 7 for small and high solidification velocities (100 $\mu$m/s and 0.03 K/s) and different temperature gradients. The steady-state tip undercooling $\Delta T$ is plotted as a function of the interface stability parameter. The theoretical value of the tip undercooling, $\Delta T$(theory), is shown by a dashed line. $\Delta T$ decreases with the increasing grid resolution and converges to a constant value which is close to the theoretically predicted one in the region $\sigma _W^*>0.5$ for the small velocity (Figure 7(a)) and in the region $\sigma _W^*>0.6$ for the high velocity (Figure 7(b)). In comparison to the convergence test in Figure 6(a) the limit value of $\sigma _W^*$ is smaller allowing the simulation at a coarser resolution.

The dependency of $\lambda$(num)/$\lambda$(KF) against $\sigma _W^*$ for $V_{\text{tip}}=0.03$ K/s and $G=1.8\cdot 10^{6}$ K/m is plotted in Figure 6(b). It can be seen that $\lambda$(num) increases with the increasing resolution in contrary to the test at a constant undercooling. It can be explained by the effect of the tip undercooling, which decreases with $\sigma _W^*$ and causes the increase of $\lambda$ according to Eq. [9]. As we already mentioned before, the interface stability parameter $\sigma ^*_W$ indicates the precision of the method. The larger the stability parameter is, the closer $\lambda$ is to a stable value. Hence, the precision of the estimation of $\lambda$ should be larger. Note that for other anisotropies and tip velocities, the dependency of $\lambda$ on the stability parameter can be non-monotonic.

The examples of the microstructure developed during the directional solidification with $V_{\text{tip}}=0.067$ m/s at different temperature gradients are shown in Figure 8. The columnar structure is similar to the tests at a constant undercooling. The formation of a large amount of thin liquid droplets along the cell boundaries can be observed. The Mn-content in the droplets is extremely high. This can lead to the formation of inclusions and degradation of mechanical properties.

The numerical results including the tip undercooling and the PDAS for different $V_{\text{tip}}$ and G are listed in Table BIII in Appendix B. The analytical values of $\lambda$ calculated by Eq. [9] are presented for comparison. The numerical undercooling on the tip, $\Delta T$(num), is close to the theoretical value for all tests.

In a real SLM process the thermal gradient and the solidification velocity vary together in the solidifying melt pool. Our aim is to investigate the effect of G and cooling rate on the PDAS separately to understand in detail the contribution of each process parameter and at the same time to verify the model results. An additional factor complicating the analysis is the tip undercooling which also depends on the solidification velocity. For this reason, we simplify the process and use some constant solidification conditions. For instance, by keeping the solidification velocity constant (by changing the cooling rate proportionally to G), we can study the pure effect of the thermal gradient.

The dependencies of the numerical values of the PDAS on G (see Table BIII) as well as the analytical dependencies are plotted in Figure 9. The tests at a constant solidification velocity and at a constant undercooling are compared. Interestingly, the ratio $\lambda$(num)/$\lambda$(KF) for the tip velocity 0.03 m/s is by 3 to 6 pct larger than in the previous test at a constant undercooling (see Tables BII and BIII). In both methods, this ratio increases with increasing G. By fitting the curves in Figure 9 for 0.03 m/s, we found the power $G^{-0.23}$, which is much smaller than the theoretical power $G^{-0.5}$ for $\lambda$(KF). Presumably, at a high $V_{\text{tip}}$ and a relatively small G, the tip radius $R_{\text{tip}}$ is much smaller than the height of the ellipsoid dendrite $a=(1-\Omega )\Delta T_0/G$, so that the calculation of the PDAS as the geometrical mean of both lengths ($R_{\text{tip}}$ and a) is not precise.

Fig. 7

Tip undercooling in directional solidification at a constant solidification velocity as a function of the interface stability parameter, $\sigma _W^*$: (a) $V_{\text{tip}}=100$$\mu$m/s; (b) $V_{\text{tip}}=0.03$ m/s. The dashed line corresponds to the theoretical value obtained by Eq. [11]

Fig. 8

Columnar microstructure simulated at the solidification velocity $V_{\text{tip}}=0.067$ m/s: (a) $G=5\times 10^5$ K/m, $\dot{T}=3.35\times 10^4$ K/s; (b) $G=1.8\times 10^6$ K/m, $\dot{T}=1.2\times 10^5$ K/s; (c) $G=4.6\times 10^6$ K/m, $\dot{T}=3.07\times 10^5$ K/s. The concentration field of Mn is shown in mass fraction. The x and y axes have units of $\Delta x=9.2$ nm

Fig. 9

PDAS for the Fe-18.9Mn alloy as a function of the temperature gradient at a constant solidification velocity (solid symbols) and at a constant undercooling (empty symbols). The curves without symbols shows the Kurz–Fisher calculations

In Figure 10, the dependencies of the PDAS on the cooling rate at a constant G and on the term $G^{-0.5}V_{\text{tip}}^{-0.25}$ are shown. Here, the dashed lines correspond to the numerical results and the solid lines correspond to the theoretically predicted PDAS calculated by the KF method, Eq. [9]. It can be seen from Figure 10(a) that $\lambda$ decreases with the increasing cooling rate, i.e., with the increasing solidification velocity at a constant G. Fitting the curves at $G=4.6\times 10^{6}$ K/m gives the power of order $\dot{T}^{-0.51}$ for $\lambda$(KF) and $\dot{T}^{-0.6}$ for $\lambda$(num). Fitting the curves at $G=1.8\times 10^{6}$ K/m gives the power of order $\dot{T}^{-0.59}$ for $\lambda$(KF) and $\dot{T}^{-0.6}$ for $\lambda$(num). From our results at high solidification velocities, we can state that the dependency of $\lambda$ on the cooling rate is stronger than the standard theoretical power $\lambda (G={\text{const}})\sim \dot{T}^{-0.25}$. This finding coincides with the experimental data and with previous numerical calculations.[6,34]

The comparison of three samples with different G at the same cooling rate $\dot{T}=1.4\times 10^{5}$ K/s (see Figure 10(a)) gives the power of order $G^{0.09}$ for $\lambda$(KF) and $G^{0.20}$ for $\lambda$(num). This is even inverse to the standard dependency.

Fig. 10

PDAS for the Fe-18.9Mn for high solidification velocities at $G=4.6\times 10^{6}$ K/m and $G=1.8\times 10^{6}$ K/m: (a) PDAS against the cooling rate; (b) PDAS against $G^{-0.5}V_{\text{tip}}^{-0.25}$. The solid line shows the Kurz–Fisher calculations, the dashed line shows the numerical results

It is known from the experimental data[34] and the theoretical model[45] that the thermal gradient in rapid solidification does not play a role in microstructure selection. At the same time, the dependency of the PDAS on the cooling rate becomes stronger and tends to the power $\dot{T}^{-0.5}$. In order to understand this behavior, we turn to the KF model. As described in Reference 24 (page 86), at high solidification velocities the tip temperature ($T^*$) decreases very fast and reaches the solidification temperature ($T_s'$). In this case, the tip undercooling, which is defined as $\Omega = \frac{{(T_{0} - T^{*} )}}{{(T_{0} - T_{{\rm{s}^{\prime}}} )}}$, tends to 1, and $\lambda$ decreases in accordance to Eq. [9]. This means that the dependency $\lambda (\dot{T}={\text{const}})\sim G^{-0.25}$ is not valid for high undercoolings related to high solidification velocities.

In order to get a common power law dependence for the PDAS against G, we suggest to approximate the term $(1-\Omega )^{0.5}$ in Eq. [9] by a power function of $\Omega$, which itself is a power function of G. For example, we can choose the function $0.25(\Omega )^{-1.5}$ as an appropriate approximation in the range of the undercooling between 0.5 and 0.92 as shown in Figure 11. The power can be even larger for the function $0.20(\Omega )^{-1.8}$ in the range between 0.5 and 0.95. Then using $\Omega \sim V_{\text{tip}}^{1/5.4}$ and $V_{\text{tip}}=\dot{T}/G$ we get $\lambda \sim G^{0.03}\dot{T}^{-0.53}$. Hence, the dependency of $\lambda$ on G at high undercoolings should be much weaker than the standard dependency. It must be noted that this is only due to the presence of the term $(1-\Omega )$ in the KF equation, which is neglected in a large number of publications because, for most important range of dendritic growth velocities, the interval $T^*-T_s'$ is assumed to be equal to $\Delta T_0$. Taking into account the dependency of the tip undercooling on the solidification velocity allows us to predict the PDAS by KF theory at high solidification velocities with a large precision.

Fig. 11

Comparison of dependencies $(1-\Omega )^{0.5}$, $0.25\Omega ^{-1.5}$ and $0.20\Omega ^{-1.8}$

By fitting the dependencies of the PDAS on $10^3G^{-0.5}V_{\text{tip}}^{-0.25}$ in Figure 10(b) for the phase-field simulation and the KF model by a linear function, we get the following slopes which are the prefactors of the power law: $k({\text{PF}})=\{1.25, 1.45\}$ and $k({\text{KF}})=\{2.11, 2.11\}$ for $G=\{1.8\times 10^{6},4.6\times 10^{6}\}$ K/m. From these dependencies, we can state that the power law for the PDAS is overall in agreement in terms of exponents with the theoretical predictions by Eq. [9]. Note that in 2D simulations the prefactor of the power law can be different from the predictions for the 3D geometry.

For comparison, the results for slow solidification velocities are shown in Table BIV in Appendix B and in Figure 12. Samples N1 and N2 show that the change of the anisotropy strength from 0.002 to 0.03 does not affect the precision of the estimation of $\lambda$. The comparison of samples with different numerical resolutions (N2 and N3, N5 and N6, N7 and N8) shows that the increase of $\sigma ^*_W$ causes the decrease of the tip undercooling and the change of $\lambda$. This behavior is in agreement with the results of the full convergence test for the tip velocity 100 $\mu$m/s which are shown before in Figure 7(a). Note that the dependencies of the tip undercooling on the numerical resolution are non-monotonic, however, in general view, the tip undercooling decreases with $\sigma ^*_W$ and tends to the theoretically predicted value in the region $\sigma ^*_W>0.5$.

The dependencies of $\lambda$ on the term $G^{-0.5}V_{\text{tip}}^{-0.25}$ for slow solidification velocities with $\sigma ^*_W>0.66$ are plotted in Figure 12. It can be observed that the numerical values of $\lambda$ are close to the theoretical values. The numerical results from Reference 19 are shown by stars. For the pulling rate 100 $\mu$m/s, they are as follows: $\lambda =107$$\mu$m for $G=20$ K/mm and $\lambda =44$$\mu$m for $G=200$ K/mm. Notice that the pulling rate in the directional solidification corresponds to the tip velocity in our simulations. It can be seen that the values of $\lambda$ in the present work and in Reference 19 are close to the analytical prediction by the KF model.

Fig. 12

PDAS for the Fe-18.9Mn alloy for slow solidification velocities as a function of $G^{-0.5}V_{\text{tip}}^{-0.25}$. The solid line shows the Kurz–Fisher calculations, the dashed line shows the numerical results. The results from Ref. [19] are shown by stars

The examples of the developed microstructure for slow solidification velocities are shown in Figure 13 (samples N6, N7, and N8 in Table BIV). It is interesting that the sample in Figure 13(b) has a dendritic structure with well-developed side-branching. The increase of the numerical resolution from $\sigma _W^* =0.24$ in Figure 13(b) to $\sigma _W^* =0.73$ in Figure 13(c) causes the decrease of $\lambda$ and the reduction in secondary arms. Hence, the developing of side branches predicted in Figure 13(b) is due to the calculation being poorly resolved numerically.

Fig. 13

Microstructure simulated for slow solidification velocities: (a) $V_{\text{tip}}=100$$\mu$m/s, $G=2\times 10^4$ K/m, $W=0.5$$\mu$m; (b) $V_{\text{tip}}=1000$$\mu$m/s, $G=2\times 10^5$ K/m, $W=0.26$$\mu$m, $\sigma _W^*=0.24$; (c) $V_{\text{tip}}=1000$$\mu$m/s, $G=2\times 10^5$ K/m, $W=0.15$$\mu$m, $\sigma _W^*=0.73$. The concentration field of Mn is shown in mass fraction. The x and y axes have units of $\Delta x$

4.4 Simulation Results in the SLM Process for Fe-18.5Mn-Al-C Alloy

The simulations were performed for Fe-18.5Mn-Al-C alloy in a 2D domain of 1200 $\times$ 1600 grid points with $W=0.02$$\mu$m (Figures 14 through 16) and 1000 $\times$ 1600 grid points with $W= 0.015$$\mu$m (Figure 17). We start with a set of randomly distributed nuclei as in previous tests. After some coarsening process, the distance between the dendrite arms turns to a constant value. The full simulation time was 2 ms. The temperature gradient was chosen as $4.6\times 10^{6}$ K/m, the cooling rate was ranging from $1.4\times 10^{5}$ to $4.6\times 10^{5}$ K/s, and the mean growth velocity was ranging from 0.03 to 0.1 m/s. The model parameters and numerical results including $\lambda$ and tip undercoolings are listed in Table BV in Appendix B. $\sigma _W^*$ is calculated using the mean capillary length and the mean diffusion coefficient from Eq. [8].

Fig. 14

Microstructure evolution of the Fe-18.5Mn-Al-C alloy during SLM. The temperature gradient is $G=4.6\times 10^6$ K/m and the cooling rate $\dot{T}=1.4\times 10^5$ K/s ($V_{\text{tip}}=0.03$ m/s). The concentration fields of Mn, Al, C are shown in mass fraction. The liquid phase is shown by black color and the solid phase by yellow color. The x and y axes have units of $\Delta x=9.2$ nm ($\sigma _W^*=1.53$)

Fig. 15

Microstructure evolution of the Fe-18.5Mn-Al-C alloy during SLM. The temperature gradient is $G=4.6\times 10^6$ K/m and the cooling rate $\dot{T}=3\times 10^5$ K/s ($V_{\text{tip}}=0.065$ m/s). The concentration fields of Mn, Al, C are shown in mass fraction. The liquid phase is shown by black color and the solid phase by yellow color. The x and y axes have units of $\Delta x=9.2$ nm ($\sigma _W^*=0.89$)

Fig. 16

Microstructure evolution of the Fe-18.5Mn-Al-C alloy during SLM. The temperature gradient is $G=4.6\times 10^6$ K/m and the cooling rate $\dot{T}=4.6\times 10^5$ K/s ($V_{\text{tip}}=0.1$ m/s). The concentration fields of Mn, Al, C are shown in mass fraction. The liquid phase is shown by black color and the solid phase by yellow color. The x and y axes have units of $\Delta x=9.2$ nm ($\sigma _W^*=0.58$)

Fig. 17

Microstructure evolution of the Fe-18.5Mn-Al-C alloy during SLM. The temperature gradient is $G=4.6\times 10^6$ K/m and the cooling rate $\dot{T}=4.6\times 10^5$ K/s ($V_{\text{tip}}=0.1$ m/s). The concentration fields of Mn, Al, C are shown in mass fraction. The liquid phase is shown by black color and the solid phase by yellow color. The x and y axes have units of $\Delta x=6.9$ nm ($\sigma _W^*=1.03$)

Figures 14 through 17 show the concentration fields of Mn, Al, and C as well as the phase field (see samples N1, 3, 5, 6 in Table BV). A regular columnar structure without side-branching can be observed in all tests. The thickness of the liquid layers between the cells decreases with the increasing tip velocity. At $V_{\text{tip}}=0.067$ and 0.1 K/s liquid droplets can be observed. In Figure 16 with $\sigma _W^*=0.58$, the columnar structure is less regular, and liquid layers between cells are reduced in comparison to Figure 17 with larger numerical resolution $\sigma _W^*=1.03$.

Figure 18 shows the dependency of the PDAS on the cooling rate at a constant G and on the term $G^{-0.5}V_{\text{tip}}^{-0.25}$. The dashed lines correspond to the numerical results. The solid lines correspond to the theoretically predicted PDAS calculated by the KF method for multicomponent alloys, Eq. [10].

Fig. 18

PDAS for the Fe-18.5Mn-Al-C alloy at $G=4.6\times 10^{6}$ K/m and $G=1.8\times 10^{6}$ K/m: (a) PDAS against the cooling rate; (b) PDAS against $G^{-0.5}V_{\text{tip}}^{-0.25}$. The solid line shows the Kurz–Fisher calculations, the dashed line shows the numerical results

It can be seen from Figure 18(a) that $\lambda$ decreases with the increasing cooling rate, i.e., the increasing solidification velocity. The ratio $\lambda$ (num)/$\lambda$ (KF) decreases too. The linear fitting of the dependency on the cooling rate at $G=4.6\times 10^{6}$ K/m gives the power of order $\dot{T}^{-0.468}$ for $\lambda$(KF) and $\dot{T}^{-0.47}$ for $\lambda$(num). This is similar to results for the Fe-18.9Mn alloy.

The comparison of $\lambda$ for three samples in Figure 18(a) at the same cooling rate $\dot{T}=1.4\times 10^{5}$ K/s gives the power for the temperature gradient of order $G^{-0.095}$ for $\lambda$(KF) and $G^{0.21}$ for $\lambda$(num). This is again similar to results for the Fe-18.9Mn alloy.

By fitting the dependencies of the PDAS on $10^3G^{-0.5}V_{\text{tip}}^{-0.25}$ in Figure 18(b) by a linear function for the phase-field simulation and the KF model, the following prefactors are found: $k({\text{PF}})=\{1.53, 1.65\}$ and $k({\text{KF}})=\{1.96, 2.00\}$ for $G=\{1.8\times 10^{6},4.6\times 10^{6}\}$ K/m, respectively. The prefactors $k({\text{PF}})$ are larger than for the binary alloy in Figure 10(b).

From dependencies in Figure 18(b), we can state that the power law for the PDAS is overall in agreement in terms of exponents with the theoretical predictions by Eq. [10] for multicomponent systems. The prefactor in 2D simulations is expected to be different from the 3D case.

Distributions of the manganese content along the samples in x-direction at the height of $400 \Delta x$ of the simulation domain are shown in Figure 19 that correspond to the microstructures in Figures 14 through 17. The main quantities which can be evaluated from these plots are the space averaging deviation of the Mn-content, $\Delta C^{\text{Mn}}$, and the deviation of the Mn-content in the solid phase, $\Delta C^{\text{Mn}}_S$. Here, $\Delta C^{\text{Mn}}$ decreases with an increasing cooling rate due to the decrease of the liquid fraction. In opposite, $\Delta C^{\text{Mn}}_S$ increases with an increasing cooling rate because the temperature interval for solidification increases. Moreover, $\Delta C^{\text{Mn}}_S$ is larger in Figure 19(c) (sample N5) with $\sigma _W^*=0.58$ than in Figure 19(d) (sample N6) with $\sigma _W^*=1.03$ due to a smaller grid resolution. The comparison between the numerical results and experimental data reported in Section II and in Reference 11 shows that sample N6 with $G=4.6\times 10^6$ K/m and $\dot{T} =4.6\times 10^5$ K/s, and with microstructure parameters $\Delta C^{\text{Mn}}= \pm$ 1.8 wt pct, $\lambda ({\text{num}}) = 0.431$$\mu$m, and $\lambda {\text {(KF)}} = 0.921$$\mu$m gives the best fit to the experiment. It can be also seen that the minimum and maximum concentrations of Mn are in agreement with the experimental values.

Fig. 19

The distribution of Mn along the samples with $G=4.6\times 10^6$ K/m shown in Figs. 14 through 17: (a) $\dot{T}=1.4\times 10^5$ K/s; (b) $\dot{T}=3\times 10^5$ K/s; (c) $\dot{T}=4.6\times 10^5$ K/s, $\Delta x=9.2$ nm; (d) $\dot{T}=4.6\times 10^5$ K/s, $\Delta x=6.9$ nm. The composition lines were extracted at the height of $400 \Delta x$ of the simulation domain

5 Conclusion

We present a numerical study carried out by means of phase-field modeling to investigate the microstructure formation during directional solidification of a binary Fe-18.9Mn and a multicomponent Fe-18.5Mn-Al-C alloy produced by SLM. The resulting microstructure is validated by experimental data. The model is verified in a wide range of solidification velocities by convergence tests. An interface stability parameter of dendrite growth is used as a criterion of accuracy of the modeling results. The PDAS, $\lambda$, is measured and compared with the analytical values calculated by the Kurz–Fisher model. The main findings of the study are:

• The numerical dependency of $\lambda$ on the temperature gradient at high solidification velocities (larger 0.01 m/s) corresponding to SLM conditions is much weaker in comparison to the standard dependency in ordinary directional solidification. At a constant solidification velocity, the dependency is of order $G^{-0.23}$, and at a constant cooling rate, the dependency is of order $G^{0.20}$ ($G^{0.21}$ for multicomponent system).

• The numerical dependency of $\lambda$ on the cooling rate at a constant G is of order $\dot{T}^{-0.6}$ ($\dot{T}^{-0.47}$ for multicomponent system). This is much stronger than in the ordinary directional solidification. These results are in agreement with the dependencies observed experimentally in SLM.

• The ratio between the PDAS obtained from numerical results and calculated by the KF method ($\lambda$(num)/$\lambda$(KF)) is close to 1 at small solidification velocities and becomes smaller than 0.6 at solidification velocities larger than 0.03 m/s.

• The analytical dependency of $\lambda$ on the temperature gradient calculated by the KF model (Eqs. [9] and [10]) at a constant cooling rate is of order $G^{0.09}$ ($G^{-0.095}$ for multicomponent system), and the dependency on the cooling rate at a constant G is of order $\dot{T}^{-0.45}$ ($\dot{T}^{-0.47}$ for multicomponent system). These results are in agreement with the numerical dependencies.

• The simulated microstructure in Fe-18.5Mn-Al-C alloy exhibits the best fit to the experiment at $G=4.6\times 10^6$ K/m and $\dot{T} =4.6\times 10^5$ K/s. At these thermal conditions, the regular columnar growth without secondary arms can be observed. The PDAS and the Mn distribution are in a good agreement with experimental data. Therefore, we can state that the phase-field model can predict the PDAS and the elemental distribution for SLM processing conditions. The simulations in three dimensions will be performed in the future for similar SLM parameters to verify the prediction accuracy of both the binary and the multicomponent phase-field models.

Appendices

Appendix A

The constants $a_1$ and $a_2$ are calculated using the integrals I, J, K from analysis of Karma and Rappel.[42] Thus by substitution of the functions

$$\begin{aligned} \phi (x)&= \frac{1}{2} - \frac{1}{2}\tanh \left( \frac{x }{\sqrt{2}}\right) , \nonumber \\ \frac{\partial \phi (x)}{\partial x}&\equiv \partial _x\phi =-\frac{1}{2\sqrt{2}} + \frac{1}{2\sqrt{2}}\tanh \left( \frac{x}{\sqrt{2}}\right) , \nonumber \\ \frac{\partial g(\phi )}{\partial \phi }&\equiv g_\phi =30\phi ^2(1-\phi )^2, \end{aligned}$$

(A1)

we get

$$\begin{aligned} I&= \int \limits _{-\infty }^{\infty }dx(\partial _x\phi )^2=\frac{\sqrt{2}}{6}=0.2357,\,\,\, J = -\int \limits _{-\infty }^{\infty }dx\,\partial _x\phi \, g_\phi =1,\nonumber \\ F&= \int \limits _{0}^{\infty }dx\,\phi =0.4901,\,\,\, K = \int \limits _{-\infty }^{\infty }dx\,\partial _x\phi \,g_\phi \times \int \limits _{0}^{x}dx'\,\phi (x')=0.06377. \end{aligned}$$

(A2)

Here x is taken in units of interface width. Then according to[42] we get

$$\begin{aligned} a_1 = \frac{I}{J}==\frac{\sqrt{2}}{6} \end{aligned}$$

(A3)

and

$$\begin{aligned} a_2 ({\text{KR}}) = \frac{K+JF}{2I}=1.174. \end{aligned}$$

(A4)

Since Eq. [59] in Reference 42 is derived for $\phi \in [-1,1]$, in our model with $\phi \in [0,1]$ we should have

$$\begin{aligned} a_2=2\cdot a_2 ({\text{KR}}) = 2.35. \end{aligned}$$

(A5)

Appendix B

The results of the simulations are listed in Tables BI, BII, BIII, BIV and BV.

Table BI Comparison of the Steady-State Tip Velocities Calculated by the Phase-Field Method (num) and by Green-Function Method (theory) for the Fe-18.9Mn Alloy

Table BII Results of the Simulation of the Directional Solidification at a Constant Undercooling for the Fe-18.9Mn Alloy

Table BIII Results of the Simulation of the Directional Solidification for the Fe-18.5Mn-Al-C Alloy

Table BIV Results of the Simulation of the Directional Solidification for the Fe-18.9Mn Alloy for Slow Solidification Velocities

Table BV Results of the Simulation of the Directional Solidification for the Fe-18.5Mn-Al-C Alloy